probing excitonic mechanics in suspended and …
TRANSCRIPT
The Pennsylvania State University
The Graduate School
PROBING EXCITONIC MECHANICS IN SUSPENDED AND STRAINED TRANSITION
METAL DICHALCOGENIDES MONOLAYERS
A Dissertation in
Physics
by
Hongchao Xie
© 2019 Hongchao Xie
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2019
The dissertation of Hongchao Xie was reviewed and approved* by the following:
Chaoxing Liu
Associate Professor of Physics
Dissertation Advisor, Chair of Committee
Kin Fai Mak
Associate Professor of Physics, Cornell University
Adjunct Associate Professor of Physics, Penn State University
Mikael C. Rechtsman
Downsbrough Early Career Development Professor of Physics
Venkatraman Gopalan
Professor of Materials Science & Engineering, Professor of Physics
Richard W. Robinett
Professor of Physics
Associate Head for Undergraduate and Graduate Students
*Signatures are on file in the Graduate School
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ABSTRACT
Over the past decade, the interest in two-dimensional (2D) materials, especially for
atomically thin transition metal dichalcogenide (TMD) semiconductors, had dramatically thrived
for both fundamental science and practical applications. The reduced dielectric screening in 2D
mainly attributes to the strong excitonic effect in atomically thin TMD semiconductors. This
pronounced exciton feature can maintain at room temperature, which indicates strong light-matter
interaction and possible optoelectronic application using monolayer semiconductors. Meanwhile,
the absence of inversion symmetry and out-of-plane mirror symmetry jointly endows carriers in
monolayer TMDs with a new valley degree of freedom (DOF). Namely, in hexagonally-arranged
lattice of 2D materials, electrons that residing at band edges of K and K’ valleys can carry opposite
valley magnetic moments and Berry curvatures, which allows the further control of valley-indexed
carriers with polarized light, electrical and magnetic fields. Besides, the large strain sustainability
of monolayer TMDs gives rise to mechanically tunable band gap with 70 meV redshift per 1%
strain up to recorded 10% applied strain. Thus, the interaction of macroscopic mechanical means
with valley electrons makes monolayer TMD semiconductor a promising platform to implement
novel valley-controlled mechanical devices. This motivates the experimental studies demonstrated
in this dissertation.
In this dissertation, we investigate the valley contrasting coupling between optoelectronic
carriers (exciton & flowing electrons) and mechanics in a monolayer TMD semiconductor. In the
first parts (Chapter 1&2), I will present emerging properties of TMD monolayers and discuss
interesting physics that can study after suspending or straining these atomically thin materials. The
fabrication and measurement of typical TMD suspended devices will also be demonstrated in
details.
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In the secondary part (Chapter 3), we demonstrate robust exciton bistability by continuous-
wave optical excitation in a suspended monolayer WSe2 at a much lower intensity level of 103
W/cm2. The observed bistability is originated from a photothermal mechanism, which can provide
both optical nonlinearity and internal passive feedback in a simple cavity-less structure. This is
supported by detailed excitation wavelength and power dependence studies of the sample
reflectance, as well as by numerical simulation including the temperature-dependent optical
response of monolayer WSe2. Furthermore, under a finite magnetic field, the bistability becomes
valley dependent and controllable not only by light intensity but also by light helicity due to the
exciton valley Zeeman effect, which open up an exciting opportunity in controlling light with light
using monolayer materials.
In the following part (Chapter 4), we report the observation of exciton-optomechanical
coupling in a suspended monolayer MoSe2 mechanical resonator. In particular, we have observed
light-induced damping and anti-damping of mechanical vibrations and modulation of the
mechanical spring constant by moderate optical pumping near the exciton resonance with variable
detuning. The observed exciton-optomechanical coupling strength is also highly gate-tunable. Our
observations can be fully explained by a model based on photothermal backaction and gate-induced
mirror symmetry breaking in the device structure. The observation of gate-tunable exciton-
optomechanical coupling in a monolayer semiconductor may find novel applications in
nanoelectromechanical systems (NEMS) and in exciton-optomechanics.
In the last part of this dissertation (Chapter 5), we present the study of magnetization purely
originated from the valley DOF in strained MoS2 monolayers. By breaking the three-fold rotational
symmetry in single-layer MoS2 via a uniaxial stress, we have demonstrated the pure electrical
generation of valley magnetization in this material, and its direct imaging by Kerr rotation
microscopy. The observed out-of-plane magnetization is independent of in-plane magnetic field,
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linearly proportional to the in-plane current density, and optimized when the current is orthogonal
to the strain-induced piezoelectric field. These results are fully consistent with a theoretical model
of valley magnetoelectricity driven by Berry curvature effects. Furthermore, the effect persists at
room temperature, opening possibilities for practical valleytronic devices.
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TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................................. ix
LIST OF TABLES ................................................................................................................. xvii
ACKNOWLEDGEMENTS ................................................................................................... xviii
Chapter 1 Introduction ............................................................................................................ 1
1.1 Monolayer transition metal dichalcogenides (TMDs) ............................................... 3 1.1.1 Crossover from indirect to direct band gap ..................................................... 5 1.1.2 Electronic band structure ................................................................................. 7
1.1.3 Valley-dependent optical selection rules ....................................................... 10 1.1.4 Valley magnetic response .............................................................................. 11 1.1.5 Exciton .......................................................................................................... 12 1.1.6 Mechanical properties ................................................................................... 15
1.2 Nanoelectromechanical Systems (NEMS) .............................................................. 16 1.2.1 2D NEMS ...................................................................................................... 16 1.2.2 Mechanical model of membrane ................................................................... 18
1.3 Optomechanics ........................................................................................................ 23 1.3.1 Modeling optomechanical coupling .............................................................. 24 1.3.2 Optomechanical backaction .......................................................................... 27
Chapter 2 Experimental reviews ........................................................................................... 31
2.1 Device fabrication .................................................................................................... 31 2.1.1 Sample preparation ........................................................................................ 31
2.1.2 Van der Waals assembly ............................................................................... 33
2.1.3 Suspend atomically thin membrane .............................................................. 36 2.2 Optical spectroscopy of monolayer semiconductors ................................................ 37
2.2.1 Photoluminescence spectroscopy .................................................................. 38
2.2.2 Optical reflection spectroscopy ..................................................................... 39
2.2.3 Magnetic circular dichroism spectroscopy .................................................... 41 2.2.4 Characterization of mechanics in 2D NEMS ................................................ 42
Chapter 3 Valley selective exciton bistability in suspended monolayer WSe2 ..................... 45
3.1 Optical bistability ..................................................................................................... 46 3.2 Experimental setup and basic characterization of suspended WSe2 monolayer ...... 47
3.3 Observation of optical bistability in suspended monolayer WSe2 ........................... 50 3.3.1 Power dependence of monochromatic reflectance spectra across exciton
resonance ........................................................................................................ 50
3.3.2 Wavelength dependence of broadband reflectance spectra across exciton
resonance ........................................................................................................ 52
3.4 Modelling mechanism of the observed optical bistability ....................................... 54 3.4.1 Simulation of photothermal model ................................................................ 55
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3.4.2 Comparison to the observed bistable exciton resonance ............................... 58
3.5 Valley contrasting bi-exciton resonance and control of light by helicity ................. 59 3.6 Conclusion ............................................................................................................... 62
Chapter 4 Tunable exciton-optomechanical coupling in suspended monolayer MoSe2 ....... 64
4.1 Exciton-optomechanical backaction through strain ................................................. 65 4.2 Experimental setup ................................................................................................... 66
4.3 Observation of exciton-optomechanical coupling in monolayer MoSe2 resonator .. 68 4.3.1 Optical response of strained MoSe2 resonator ............................................... 69
4.3.2 Spectral dependence of mechanical response................................................ 70
4.3.3 Photothermal backaction of exciton-optomechanical coupling .................... 72
4.4 Modelling exciton-optomechanical coupling in MoSe2 resonator ........................... 77 4.4.1 Gate dependence of exciton spectral shift ..................................................... 78
4.4.2 Gate dependence of mechanical resonance ................................................... 79
4.4.3 Modelling photothermal backaction in strained MoSe2 resonator ................ 80
4.5 Tunable exciton-optomechanical effects .................................................................. 83 4.5.1 Gate dependence of exciton-optomechanical coupling ................................. 83
4.5.2 Power dependence of exciton-optomechanical coupling .............................. 84
4.6 Detection of thermal vibrations of monolayer MoSe2 at cryogenic temperatures .... 86
4.7 Conclusion ............................................................................................................... 87
Chapter 5 Valley magnetoelectricity in single-layer MoS2 ................................................... 89
5.1 Magnetization from valley degree of freedom ......................................................... 90 5.2 Experimental setup ................................................................................................... 93
5.2.1 Device fabrication ......................................................................................... 94
5.2.2 Second-harmonic generation (SHG) ............................................................. 95
5.2.3 Photoluminescence (PL) spectroscopy .......................................................... 96
5.2.4 Magneto-optic Kerr rotation (KR) spectroscopy ........................................... 96
5.3 Observation of valley magnetoelectric effect in single-layer MoS2 ......................... 97 5.3.1 Valley magnetization in single-layer MoS2 ................................................... 97
5.3.2 Dependences of bias current and in-plane magnetic field ............................. 99
5.3.3 Hamiltonian model for valley magnetoelectric effect in strained
monolayer MoS2 ............................................................................................. 100
5.3.4 Dependences of the current and strain directions with respect to the
crystalline axis ................................................................................................ 101
5.3.5 Temperature dependence of valley magnetoelectric effect ........................... 104
5.4 Extended support experiments for the observed valley magnetoelectricity ............. 106 5.4.1 Dependence of Kerr rotation on probe polarization ...................................... 107
5.4.2 Dependence of Kerr rotation angle on probe photon energy ........................ 108
5.4.3 Kerr rotation under pure gate modulation ..................................................... 109
5.4.4 Simulation of valley magnetoelectric effect in Corbino disk geometry ........ 109
5.4.5 Effects of conductivity anisotropy ................................................................ 111
5.4.6 Comparison to the effect of trigonal warping of the band structure .............. 113
5.4.7 Valley Hall effect in single-layer MoS2 ........................................................ 114
5.5 Microscopic model of valley magnetization in strained and biased MoS2 .............. 116
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5.5.1 Derivation of the piezoelectric field in single-layer MoS2 under a uniaxial
stress ............................................................................................................... 116
5.5.2 Electronic band structure of strained single-layer MoS2 ............................... 118
5.5.3 Valley magnetization ..................................................................................... 120
5.5.4 Circular dichroism ......................................................................................... 124
5.5.5 Comparison of microscopic theory with experiment .................................... 125
5.6 Conclusion ............................................................................................................... 127
Chapter 6 Summary and future perspectives ........................................................................ 128
Bibliography .......................................................................................................................... 132
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LIST OF FIGURES
Figure 1-1: Trigonal prismatic structure of monolayer TMDs in 2H phase. The honeycomb
lattice structure with broken inversion symmetry and the first Brillouin zone with the
high symmetry points are shown. Figure is obtained from Ref. [23]. .............................. 4
Figure 1-2: Calculated electronic band structure of atomically thin MoS2. (a) Atomic
orbital projection of band structures for monolayer MoS2 from first-principle
predictions, without spin-orbit coupling. Fermi energy is set to zero. Symbol size is
proportional to its contribution to corresponding state. (1) Contributions from Mo-d
orbital. (2) Total contributions from p orbitals, dominated by S atoms. (3) Total s
orbitals. (b) The simulated band structure of bulk, quadri-layer, bilayer and single-
layer MoS2 (from left to right). A transition from indirect band gap to direct band gap
semiconductor occurs at monolayer limit. Figures are obtained from Ref. [30], [31]. .... 6
Figure 1-3: Optical selection rules and valley magnetic response. (a)Electronic bands (here
for W compounds) around K and K’ points, which are spin-split by the spin-orbit
interactions. The spin (up and down arrows) and valley (K and K’) degree of freedom
are locked together. Azimuthal quantum number (m) for each band is shown, and the
valley-dependent optical selection rules are depicted. (b) Zeeman shift of the K and
K’ valleys under an out-of-plane magnetic field. The dashed and solid lines represent
bands of opposite spin (split by spin-orbit coupling). 𝜇𝑠, 𝜇𝑉 and 𝜇𝑂 are the spin, valley
and intra-atomic orbital magnetic moments, respectively. Figures are obtained from
Ref. [23], [33]. .................................................................................................................. 10
Figure 1-4: Strong excitonic effect in 2D TMDs. (a) Schematics of electrons and holes
bound into excitons for the three-dimensional bulk and a quasi-two-dimensional
monolayer. The changes in the dielectric environment are indicated schematically by
different dielectric constants 휀3𝐷 and 휀2𝐷 and by the vacuum permittivity 휀0 .(b)
Absorption spectrum of monolayer MoS2 at 10K (solid green line). A and B are
exciton resonances corresponding to transitions from the two spin-split valence bands
to the conduction bands. The blue dashed line shows the absorbance (arbitrary units)
if excitonic effects were absent. The inset illustrates the Coulomb bounding between
an optically generated electron-hole pair, forming a bound exciton. Figures are
obtained from Ref. [23], [44]. .......................................................................................... 14
Figure 1-5: NEMS devices made of 2D materials. By fabricating suspended samples over
a trench structure, intrinsic properties include (a) electrical transport (e.g. carrier
mobility), (b) optical response (PL& linear absorbance) (c) mechanical properties (e.g.
Young’s modulus) and (d) thermal transport (thermal conductivity) of 2D materials
were systematically studied, which would be challenged for samples on substrate.
Also, 2D NEMS is a good platform to study interplay between nanomechanics and
other interesting DOF (e.g. charge, spin, valley etc.) in real 2D limit. Figures are
obtained from Ref. [10], [12], [29], [58]. ......................................................................... 18
x
Figure 1-6: Schematic of the cavity optomechanical system. A mechanical oscillator
comprises one end of a Fabry-Perot cavity, coupling mechanical motion to the optical
cavity length. Figure is obtained from Ref. [108]. ........................................................... 24
Figure 1-7: Schematics of back-action effects in cavity optomechanics. (a) Schematic
diagram of light-induced force experienced by the end mirror as a function of
displacement in Fabry-Perot system. The finite decay rate of confined photons induces
a delayed response to mirror motion. (b) “P-V” style schematic depicting the work
done by the radiation force during one cycle of oscillation. The work is given by the
enclosed area swept in the force-displacement diagram, which is due to the retardation
of the force (finite cavity decay rate). The work is negative or positive, depending on
whether one is on the red-detuned or blue-detuned side of resonance. This thus gives
rise to damping or amplification, respectively. Figure is obtained and modified from
Ref. [109]. ........................................................................................................................ 28
Figure 2-1: Optical images of monolayer TMDs exfoliated on (a) silicon substrate with
285nm oxide layer atop and (b) PDMS substrate doubly-capsulated with thin hBN
flakes. ............................................................................................................................... 33
Figure 2-2: Process flow for heterostructure fabrication using PC as adhesive polymer.
This process flow can be used to fabricate heterostructure FET or NEMS device. The
final heterostructure maintains the stacking order. Step 1: Attach a PC/PPC/PDMS
stamp on the glass slide and place it on micro-manipulator of transfer stage; Step 2:
First layer pick-up; Step 3: Unload the bottom substrate and load the other exfoliated
substrate; Step 4: Second layer pick-up, Step 5: Place a new prepatterned substrate
with electrodes for final transfer; Step 6: The new prepatterned substrate makes a
contact with heterostructure and PC stamp at 150 C; Step 7: PC and heterostructure
detached from handle substrate and transferred to the new prepatterned substrate.
Figure is obtained from Ref. [137]. .................................................................................. 35
Figure 2-3: PDMS deterministic dry transfer method for 2D NEMS ..................................... 37
Figure 2-4: Schematics of microscopic confocal system in reflection geometry................... 39
Figure 2-5: Interferometry technique to characterize mechanics of 2D NEMS. (a)
Schematic of reflected beams from both suspended membrane and silicon surface. (b)
The normalized reflectance as a function of displacement of membrane. The blue
arrows indicate the transduction between displacement of membrane and reflectance.
The oscillation of membrane can periodically modulate the sample reflectance (brown
dash lines). Inset: gate dependence of normalized reflectance of suspended sample
with 450 nm probe wavelength. The red solid line is prediction with Eqns. (2-3) to (2-
6).. .................................................................................................................................... 43
Figure 3-1: Schematic of optical bistability. 𝐼𝑓 and 𝐼𝑏 are the critical intensities for
transmittance bistability under forward and backward intensity sweeps, respectively.
Figure is obtained from Ref. [150]. .................................................................................. 46
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Figure 3-2: Experimental setup and suspended WSe2 device. (a) Sketch of experimental
setup. The suspended samples are placed in an optical cryostat. Optical excitation,
either broadband radiation or monochromatic beam, is illuminated onto sample. For
single-colored source, both the incident and reflected laser light are monitored by a
pair of photodiodes. The reflected broadband spectrum from the sample is recorded
by a dispersive spectrometer. (b) Schematic side view of a suspended monolayer WSe2
sample over a drumhead trench of 8 μm in diameter and 600 nm in depth on a Si
substrate. The substrate has a 600-nm oxide layer and pre-patterned Ti/Au electrodes.
The inset is an optical image of a sample illuminated by white light. Gold bars are
electrodes and the dark ring is the edge of the trench. ..................................................... 48
Figure 3-3: Temperature dependence of exciton resonance in suspended WSe2 monolayer.
(a) Reflectance spectrum measured at 4 - 330 K. A sharp exciton resonance is
observed at low temperatures. The resonance redshifts and broadens with increasing
temperature. (b) Simulated reflectance spectrum at 4 - 330 K. (c) Exciton peak
wavelength and linewidth (symbols) as a function of temperature used in the
simulation of (b). Solid lines are fits to Eqns. (3-2) and (3-3). ........................................ 49
Figure 3-4: Optical bistability in suspended monolayer WSe2. (a) Reflectance spectrum
of sample #1 at 4 K. (b) Absorbance spectrum described by a single Lorentzian that
best fits the reflectance spectrum of (a). (c, d) Reflectance at representative
wavelengths (708, 712, 720, 725 and 728 nm) under forward and backward power
sweeps from experiment (c) and simulation (d). The wavelengths are marked by
vertical dashed lines in (a, b)............................................................................................ 51
Figure 3-5: Bistable excitonic reflection spectra under monochromatic pump. (a, b)
Contour plots of reflectance of sample #2 at 4 K as a function of probe wavelength
(bottom axis) and pump wavelength (left axis). The pump wavelength is swept
forward (i.e. increases) in (a) and backward (i.e. decreases) in (b). The pump power is
fixed at 100 μW. The diagonal dashed lines are the residual pumps. (c, d) The
extracted exciton peak wavelength (c) and linewidth (d) from the experimental
reflectance spectra. (e, f) Simulation of the parameters shown in (c, d) based on the
photo-thermal effect as described in the main text. ......................................................... 53
Figure 3-6: (a) Schematic of the photo-thermal effect. The exciton resonance redshifts
under optical excitation for both blue- and red-detuned optical excitation from the
exciton resonance (shown by blue and red vertical arrows, respectively). For the red-
detuned optical excitation, the exciton resonance runs towards the pump and the effect
is cumulative. This internal passive feedback together with nonlinearity can lead to
optical bistability. (b) Simulated average sample temperature as a function of
excitation power at 708 nm (blue-detuned from the exciton resonance) and 728 nm
(red-detuned from the exciton resonance). 𝑃𝑓 and 𝑃𝑏 are the critical powers for
temperature instability under forward and backward power sweeps, respectively. (c)
Simulated average sample temperature as a function of wavelength at 1, 10 and 100
μW. 𝜆𝑓 and 𝜆𝑏 are the critical wavelengths for temperature instability under forward
and backward wavelength sweeps, respectively. The region between the critical
powers (wavelengths) is bistable. .................................................................................... 57
xii
Figure 3-7: Wavelength dependence of the sample reflectance for forward (blue) and
backward (red) wavelength sweeps. (a, b, c) are experiment and (d, e, f) are simulation.
The excitation power is fixed at 1 μW (a, d), 10 μW (b, e) and 100 μW (c, f). ............... 59
Figure 3-8: Valley-dependent optical bistability and control of light by its helicity. (a)
Helicity-resolved reflectance of sample #3 at 4 K under an out-of-plane magnetic field
of 8 T. The K and K’ valley excitons are split by ~ 2 meV. The vertical dashed line
corresponds to 710 nm. (b) Helicity-resolved reflectance under power sweeps at 710
nm. There exists a finite power range between two bistability regions (between the
vertical dotted lines). (c) Magnetic circular dichroism (MCD) under forward and
backward power sweeps at 710 nm. (d) Helicity-controlled optical ‘switching’ in real
time at 710 nm and 39 μW. The helicity of light is modulated at 1 Hz by a liquid
crystal modulator (upper panel) and the sample reflectance follows the helicity
modulation with a change of reflectance by ~ 30% (lower panel). .................................. 61
Figure 4-1: Schematic illustration of the device geometry and experimental setup. The
mechanics of suspended membrane is characterized with optical interferometry
technique. And the incident and reflected beams are recorded by a normal and fast
photodetector, respectively. The static displacement 𝑧 and dynamical oscillation 𝛿𝑧 of membrane can transduce to reflection signals, which are measured with a
multimeter and a network analyzer correspondingly. Meanwhile, the vibration of
monolayer MoSe2 membrane generates a dynamical strain 𝛿𝜖 that give rise to an
oscillating spectral shift of exciton resonance δ𝜔𝑋. In addition, the illumination of the
sample with light near the exciton resonance can generate a photothermal force 𝐹𝑝ℎ.
As a result, the oscillation of monolayer semiconductor (oscillating exciton spectral
shift) can periodically modulate the photothermal force as δ𝐹𝑝ℎ. ................................... 68
Figure 4-2: Gate tuning of excitonic and mechanical resonances in suspended MoSe2
resonator. (a) Optical reflectance spectra of sample #1 are shown at selective gate
voltages (upper panel) and in contour plot (lower panel). The orange dashed line in
lower panel of (a) is prediction of Eqn. (4-6). (b) Mechanical resonance shift as a
function of gate voltages. The grey dashed line is the prediction of Eqn. (4-8). Inset:
A typical mechanical resonance of our resonator at 3.3K, and the corresponding
Lorentzian fit with Q ~ 20,000. ........................................................................................ 70
Figure 4-3: Interaction between excitonic and mechanical resonances in monolayer MoSe2
resonator. (a) Normalized reflectivity change pumped with 1 μW as a function of
driven frequency at selective wavelengths throughout the exciton resonance. The
spectra are displaced with equal spacing for clarity. Data was taken with 𝑉𝑔 = 40𝑉 at
3.3K. The illumination power is 1 μW. (b) The peak response of mechanical
resonance across exciton resonance (symbols with line) as a function of incident
wavelength (bottom axis) and exciton detuning Δ𝜔 (top axis). Data was extracted
from (a). The dashed red line is the illustration of spectral shift due to dynamical strain.
Inset: A sketch of a dynamical shift in the exciton resonance (δ𝜔𝑋). (c) The extracted
dynamic shift of exciton resonance δ𝜔𝑋 at selective gate voltages (symbols). The blue
solid line is obtained by multiplying quantity 𝑑𝜔𝑋
𝑑𝑉𝑔 extracted from the lower panel of
Fig.4-2 (a) with a dynamic voltage δ𝑉𝑔~ 70 mV. ............................................................ 72
xiii
Figure 4-4: Light induced damping and anti-damping of mechanical modes across exciton
resonance in monolayer MoSe2 resonator. (a) Normalized reflectivity change pumped
with 10 μW as a function of driven frequency at selective wavelengths throughout the
exciton resonance. The spectra are displaced with equal spacing for clarity. Data was
taken with 𝑉𝑔 = 40𝑉 at 3.3K. The extracted (b) line width and (c) frequency of
mechanical resonance shown in (a) as a function of exciton detuning (bottom axis)
and incident wavelength (top axis). The green and blue solid lines in (b) and (c) are
extracted contributions from photothermal softening(symmetric) and backaction
(anti-symmetric). The red solid lines in (b) and (c) are fitting lines by summing up
green and blue lines. The data without symmetric contribution are shown in (d) and
(e), where the red solid lines now are fitting from Eqns. (4-1) and (4-13). ..................... 74
Figure 4-5: Gate dependence of optical damping across exciton resonance. (a) The shift of
mechanical linewidth induced by antisymmetric contribution as a function of exciton
detuning at selective gate voltages. Data is displaced with equal spacing for clarity.
The solid lines are fitting based on Eqns. (4-1) and (4-13). (b) The extracted amplitude
of the antisymmetric contribution of mechanical linewidth as a function of gate
voltages. The red dashed line is prediction based on Eqns (4-1) and (4-13). .................. 84
Figure 4-6: Power dependence of optical damping across exciton resonance. (a) The shift
of mechanical linewidth induced by antisymmetric contribution as a function of
exciton detuning at selective incident power. Data is displaced with equal spacing for
clarity. The solid lines are fitting based on Eqns. (4-1) and (4-13). (b) The extracted
amplitude of the antisymmetric contribution of mechanical linewidth as a function of
incident power. The red dashed line is prediction based on Eqns (4-1) and (4-13). ........ 85
Figure 4-7: Thermal motions of monolayer MoSe2 resonator. (a) Optical reflection
spectrum of sample #2 at 𝑉𝑔 = −60𝑉. The blue and green dashed lines are aligned
with 752 nm and 759 nm, which used for measurements of thermal noise in (b) and
(c). The red dash lines in (b) and (c) are Lorentzian fittings. ........................................... 87
Figure 5-1: Valley magnetoelectric effect in strained single-layer MoS2. (a) Piezoelectric
field 𝓔𝒑𝒛 is produced (right) as a result of stress-induced distortion of the hexagonal
lattice (left) and the associated ion charge polarization. Blue and yellow balls
represent Mo and S atoms, respectively. (b) Out-of-plane valley magnetization 𝑴𝑽 is
induced by an in-plane current density 𝑱 in the presence of the piezoelectric field 𝓔𝒑𝒛.
(c) Electronic band structure of n-doped single-layer MoS2 around the K and K’
valleys of the Brillouin zone. The band extrema are shifted from the K/K’ point (black
dashed lines) to opposite directions by a uniaxial strain, which also do not coincide
with the extrema of the Berry curvature distribution (blue dashed lines). The Fermi
level is tilted under an in-plane bias electric field. ........................................................... 92
Figure 5-2: Basic optical characterization in single-layer MoS2. (a) Intensity of the second-
harmonic component parallel to the excitation polarization (𝐼) as a function of the
excitation polarization angle measured from the armchair direction (𝜃) (symbols).
Solid line is a fit to 𝐼 = 𝐼0cos2(3𝜃) with 𝐼0 denoting the maximum intensity. (b)
Photoluminescence (PL) spectrum of single-layer MoS2 on a flexible substrate with
0% (blue), 0.1% (cyan) and 0.2% (green) strain along the armchair direction. Red line
xiv
is the PL spectrum of the flake after being transferred onto a silicon substrate. The
spectra are normalized to the peak intensity and vertically shifted for clarity. ................ 94
Figure 5-3: Magneto-optic KR microscopy setup for strained single-layer MoS2. (a)
Optical image of strained MoS2 device. Black arrow indicates the direction of
electrical current. (b) Schematics of a MoS2 FET with bias voltage (Vsd) applied on
the source–drain electrodes and gate voltage (Vg) applied through the
Si/SiO2 substrate. For a strained single-layer MoS2 FET, a longitudinal electrical
current density J (black arrow) is optimally flowed in the transverse direction of
the strained-induced piezoelectric field Epz (orange arrow). The valley
magnetoelectric effect is detected by focusing a linearly polarized probe beam onto
the device under normal incidence and measuring the Kerr rotation angle θKR of the
reflected beam. ................................................................................................................ 97
Figure 5-4: Valley Hall effect and valley magnetoelectric effect in single-layer MoS2. Kerr
rotation image of an unstrained (a) and strained (b) single-layer MoS2 device under
two opposite bias directions (black arrows). Boundaries of the electrodes and the
device channel are marked in dashed black and green lines, respectively. The
unstrained device was measured with 𝑉𝑑𝑠= 2.5 V and Vg = 0 V (𝐽 = 22 A/m). KR is
observed at channel edges only. The strained device was measured with 𝑉𝑑𝑠 = 2.5 V
and Vg = 20 V (𝐽 = 13 A/m). KR is observed over the entire channel. ............................ 98
Figure 5-5: Dependences of KR angle on magnitude of bias current and in-plane magnetic
field. (a) Gate dependence of the KR at a fixed location on the strained device
(symbols) and gate dependence of channel current density 𝐽 measured with 𝑉𝑑𝑠 = 2.5
V (red solid line). Inset shows the corresponding bias dependences with Vg = 20 V. (d)
KR at a fixed location on the strained device as a function of in-plane magnetic field
B. Inset illustrates the Hanle effect for spin magnetization 𝑀𝑆 and the absence of it for
valley magnetization 𝑀𝑉. ................................................................................................. 99
Figure 5-6: Dependence on current direction of the valley magnetoelectric effect. a,
Piezoelectric field 𝓔𝒑𝒛 is produced along the armchair direction when single-layer
MoS2 is stressed along the armchair direction; b, KR image of sample (a) measured
in the Corbino disk geometry with 𝑉𝑑𝑠 = 1 V and Vg = 25 V (𝐽 = 3.9 A/m); c, Predicted
spatial distribution of valley magnetization by Eqn. (5-1). The direction of 𝓔𝒑𝒛 is
shown in orange lines. Boundaries of the electrodes and the device channel are marked
in dashed black and green lines, respectively. d to f are the same as a to c with single-
layer MoS2 stressed along the zigzag direction. The KR image was measured under
𝑉𝑑𝑠 = 6.9 V and Vg = 20 V (𝐽 = 2.2 A/m). ........................................................................ 103
Figure 5-7: Temperature dependence of the valley magnetoelectric effect. Gate
dependence of the two-point conductance (a) and KR at a fixed location on a strained
device (b) measured with 𝑉𝑑𝑠 = 2 V at four selected temperatures 10, 50, 90, and 110
K. Inset of b displays the gate dependence of the KR (symbols) and conductance (red
solid lines) at room temperature. ..................................................................................... 105
Figure 5-8: Valley ME effect at room temperature. a, Kerr rotation image of the device
shown in Fig. 5-6(b) at room temperature measured with Vds = 2.1 V and Vg = 20 V.
xv
b, The background noise level measured with Vds = 0 and Vg = 0. c, Bias dependence
of the KR at a fixed location of the device channel (symbols) and the corresponding
current density (solid red line) measured with Vg = 20 V. .............................................. 106
Figure 5-9: Kerr rotation as a function of probe light polarization. KR was measured on
the strained device in Corbino disk geometry shown in Fig. 5-6(b) at two locations of
the device channel corresponding to two different bias fields (red and blue lines). The
linear polarization of the probe beam was varied by a half-wave plate and the sample
was fixed. ........................................................................................................................ 107
Figure 5-10: Dependence of KR on probe photon energy. Photoluminescence spectrum
shows the A exciton resonance in single-layer MoS2. The peak energy is marked by
the dashed line. KR was measured as a function of probe photon energy, the range of
which is limited by the tuning range of our laser. ........................................................... 108
Figure 5-11: Kerr rotation image under gate modulation. KR image of the unstrained
device shown in Fig. 5-4(a) measured with bias voltage Vds = 0 and gate voltage Vg
modulated at 4.11 kHz with a peak-to-peak amplitude of 2.5 V. Black and green
dashed lines indicate the electrode and sample boundaries, respectively. No
identifiable signal can be observed anywhere on the device channel. ............................. 109
Figure 5-12: Electrostatic simulation of Corbino disk devices. a, Microscope image of a
monolayer MoS2 device with Corbino disk electrodes: electrodes in gold, Si substrate
in grey and monolayer MoS2 in light blue. b, c, Contour plot of the simulated electric
potential (b) and electric-field strength (c) of the device. Inner and outer electrode are
fixed at 5 and 0 V, respectively. Black arrows in c indicate the local bias field direction
(arrow direction) and amplitude (arrow length). .............................................................. 111
Figure 5-13: Simulation on Corbino disk devices. a, Piezoelectric field ℇ𝑝𝑧 is produced
along the armchair direction when single-layer MoS2 is stretched along the armchair
direction. b, KR image of sample (a) measured in the Corbino disk geometry with 𝑉𝑑𝑐 = 1 V and Vg = 25 V. c, Predicted spatial distribution of valley magnetization based
on the valley ME effect. The direction of ℇ𝑝𝑧 is shown in orange lines. Boundaries of
the electrodes and the device channel are marked in dashed black and green lines,
respectively. d, Simulation of current density J including the conductivity anisotropy
induced by strain (0.5%, left column; 50%, right column). e to h are the same as a to
d with single-layer MoS2 stretched along the zigzag direction. The KR image was
measured under 𝑉𝑑𝑐 = 6.9 V and Vg = 20 V. .................................................................... 112
Figure 5-14: KR mapping of Corbino disk device under reversed bias. KR image with
current flowing radially outward (a) and inward (b). The KR signal flips sign under
reversed bias. .................................................................................................................... 113
Figure 5-15: Valley Hall effect in single-layer MoS2 at 30 K. a, Kerr rotation microscope
image of an unstrained monolayer MoS2 device under Vsd = 2.5 V and Vg = 0 V (J =
22 A/m). The boundaries of the electrodes and the device channel are shown in black
and green dashed lines, respectively. b, Conductivity as a function of gate voltage (Vg).
Inset: current as a function of bias voltage (Vds) with Vg = 20 V and a microscope
xvi
image of the device. c, Gate dependence of the KR at three different locations on the
device channel indicated by circles in (a) measured with Vds = 2.5 V. Red, black and
blue line correspond to KR at the lower edge, center, and top edge of the channel,
respectively. d, Bias dependence of the KR at selected gate voltages near the lower
edge of the channel. ......................................................................................................... 115
Figure 5-16: Image of normalized KR signal by the channel current density for unstrained
(left) and strained (right) monolayer MoS2. ..................................................................... 116
Figure 5-17: Band structure of monolayer MoS2 at the K and K’ point. The shaded regions
illustrate the occupied states in the conduction band. a, Unstrained monolayer MoS2
under a finite bias field, giving rise to a tilted Fermi surface. b, Strained monolayer
MoS2 under bias. Net valley magnetization is generated in this configuration. The
dashed black lines indicate the K/K’ point and dashed blue lines indicate the extrema
of the Berry curvature distribution. .................................................................................. 123
Figure 5-18: Optical absorbance of monolayer MoS2 near the A exciton resonance at 10
K. ...................................................................................................................................... 126
xvii
LIST OF TABLES
Table 4-1: Comparison of backaction effects in cavity optomechanics and this work. .......... 77
Table 5-1: Comparison of the KR measurements and the model based on conductivity
anisotropy in the Corbino disk devices. ........................................................................... 112
Table 5-2: Comparison of the effect reported in Ref. [185] and our work ............................ 114
xviii
ACKNOWLEDGEMENTS
When looking back to my life in graduate school, it is great honor for me to work with lots
of wonderful people. Here I want to thank all the help and inspiration I got in past five years to
bring this dissertation to light.
First and foremost, I would like to acknowledge my academic advisor, Prof. Kin Fai Mak,
for giving me a chance to work on the exciting frontiers of two-dimensional physics. Fai has
incredible insight of physics and unceasing enthusiasm to explore the unknowns. He is always
very creative to come up with research topics and immediately find a smart way to solve the
problem. Along with his knowledgeable background of physics, Fai’s hands-on experimental skills
also impressed me much. When discussing with Fai on my research, I always do my utmost to
digest some of his wisdom and passion for science. I come to realize how generous and patient he
is before my experiments finally work. I would never forget the basic principles of research Fai
used to emphasize, “First, you know what to do. Then you know what you are doing. At last, you
understand what you have seen.” He had shown me how to be a great physicist with constant pursuit
of new ideas, solid understanding of physical picture, and critical attention to experimental details
that I am trying with greatest efforts to approach in the coming years. I am very fortunate to work
with Fai and indebted to him with training academically and personally.
I would like to thank Prof. Jie Shan for mentoring me as a co-advisor. Her uncompromising
questions in group meeting and in private discussion illuminate me to better understand my research.
Jie always explained those tough concepts to me with the greatest patience I can never find
elsewhere. Also, she kindly provided me with numerous suggestions on career development and
professionalism, which I particularly appreciate with. And her persistence in research and great
passion for science are invaluable personalities that I admired much. Without the support and
cultivation from Fai and Jie, I cannot imagine how far I can go on these dissertation works.
xix
I would also like to thank Prof. Chaoxing Liu for serving as chair of my dissertation
committee and helping me finish lots of paperwork. As a student enrolled in Prof. Liu’s two courses:
statistical physics and semiconductor physics, I have benefited a lot from his lectures and achieved
a better understanding of my research problems. I am very thankful to Prof. Mikael C. Rechtsman
and Prof. Venkatraman Gopalan for their careful reading and valuable suggestions on my
dissertation.
I am extremely fortunate to be able to work in two amazing universities within my graduate
life – Penn State and Cornell. As one of the ‘oldest’ members in Mak and Shan group, it would be
a unique experience to witness the ‘never-off’ light shifted from labs on 3rd floor of Davey Lab at
Penn State to those on 2nd basement of Physical Science Building at Cornell. My thanks go to all
the former and current members of Mak and Shan group: Prof. Xiaoxiang Xi, Prof. Jieun Lee, Dr.
Liang Zhao, Prof. Xiuhong Zhu, Dr. Yanyuan Zhao, Dr. Zefang Wang, Egon Sohn, Dr. Yi-Hsin
Chiu, Dr. Shengwei Jiang, , Kaifei Kang, Lizhong Li, Dr. Yanhao Tang, Dr. Xiao-Xiao Zhang, Dr.
Tingxin Li, Tong Zhou, Dr. Yang Xu, Dr. Greg Stiehl, Dr. Chenhao Jin, Zui Tao, Avi Shragai and
all the young members.
Several people deserve my special thanks. I am extremely indebted to Jieun for her selfless
and patient mentoring during my first two years in Mak and Shan lab. Her knowledge background,
work ethic and first-rated skills set a high career model for me. Special thanks to Xiaoxiang, for
constantly helping me with both academical and personal stuff. I really appreciate all the knowledge
about superconductors from him. Another special thanks to Shengwei as my senior collaborator,
for giving me opportunities to work with him on several projects, and for teaching me how to work
smartly and efficiently. Thanks to my ‘long-term’ officemates: Egon and Zefang at Penn State and
Yanhao at Cornell for all the discussion of physics, music, films we had, just to name a few. I also
thank Yang for his experience on sample fabrication and discussion on magnetic materials. I also
xx
thank Lizhong and Tingxin to be my ‘dinnermates’ occasionally.
I would also like to express my thanks to my classmates for discussion and company in
courses: Jiabin Yu, Yang Ge, Jiho Noh, Tsung-yao Wu, Ya-wen Chuang, Yu Jiang, Boyang Zheng,
Tao Wang, Dr. Jianyun Zhao, Yixuan Chen, Tomy Granzier-Nakajima, Sean Thomas, Justin
Rodriguez, Pavlo Bulanchuk, Krishnanand Mallayya, Tim Phillsbury, Umar Mauroof, Eugene
Tupikov, Daniel Schussheim, Leonardo E. Schendes Tavares, Ssohrab Borhanian, Benjamin Katz,
Neel Malvania, Patrick Godwin, Monica Tatiana Ricon Ramirez.
I also want to express my special thanks for friends I met throughout my PhD: Dr. Rui-
Xing Zhang, Prof. Bing Yao, Prof. Cui-Zu Chang, Dr. Zhong Lin, Dr. Yuhe Zhang, Dr. James
Kally, Dr. Gang Yang, Dr. Yufei Shen, Dr. Yixin Yan, Dr. Yu Pan, Dr. Qingze Wang, Dr. Di Xiao,
Dr. Jue Jiang, Dr. Jianxiao Zhang, Dr. Xiao Gan, Yifan Zhao, Songyang Pu, Yicheng Zhang, Run
Xiao, Jiali Lu, Tongzhou Zhao, Jiawei Wen, Xiaoche Wang (at Penn State) and Yanxin Ji, Ruofan
Li, Vishakha Gupta (at Cornell).
It would be a valuable experience to work in two of the best academic cleanrooms with
amazing staffs that include: Michael Labella, Kathy Gehoski, Guy Lavallee (at Penn State Nanofab)
and Garry Bordonaro, Aaron Windsor, Christopher Alpha (at Cornell CNF).
I am also grateful to Prof. Yuanbo Zhang at Fudan University for giving me a chance to do
undergraduate summer research intern in his lab and triggered my interest of two-dimensional
physics. I would also to thank Prof. Kostya Novoselov for his advising and encouragement on my
MPhys degree research at University of Manchester.
Finally, I would like to show my particular thanks to my beloved parents for their
unconditional love, understanding and support throughout my life. And my warmest thanks go to
Rui. With her tacit understanding and witty ideas, my life has been filled with joy and excitement.
Thank you and love you!
xxi
To my parents
Chapter 1
Introduction
Size and restricted geometry play decisive roles in physical properties of materials. In
reduced dimensionality, quantum physics teaches us that the energy of the spatially confined
electrons becomes discrete. In particular for two-dimensional (2D) space, in a strong magnetic field
and at ultra-low temperatures, the electron orbit can be completely quantized. In this case, the
resistance along the direction of the current vanishes completely, indicating a non-dissipating flow
of electrons, just like in superconductors. Additionally, the Hall effect — characterized by the
resistance measured perpendicular to the current demonstrates discrete steps with defined plateaus
over large intervals of magnetic fields. This quantization is called the quantum Hall effect [1].
Discovery of integer and fractional multiples of 𝑒2 ℎ⁄ (where 𝑒 denotes elementary charge of
electron, ℎ denotes Planck constant) in quasi-two-dimensional quantum wells lead to two Nobel
prizes for the study of these magic quantum mechanical effect [1]–[3]. Since then, people never
cease to hunt for realistic systems that can completely confine electrons in 2D and expect to exhibit
even more bizarre quantum effects.
Graphene, with only single layer of carbon atoms, is the first truly 2D system which had
been experimentally demonstrated by Andre Geim and Kostya Novoselov in 2004 [4]. This later
leads to their winning of the 2010 Nobel prize in physics. Date back to 1946, Canadian physicist
Philip Wallace pioneeringly hypothesized the existence of graphene and predicted the linear energy
dispersion without a band gap around the Fermi level [5]. This single hexagonal-structured layer
of carbon atoms has demonstrated exceptional properties more than prediction in versatile fields,
including ballistic transport [4], [6]–[8], quantum hall effect at room temperature [9], excellent
thermal conductivity [10], [11], high mechanical tolerance and flexibility [12], [13], high gate
2
tunability [4], [7], [8]. Thus, graphene was initially expected to revolutionize the high-speed
electronic application [14], [15]. However, as a double-edged sword, the gapless electronic nature
of graphene make it challenged to fulfill applicable on/off ratio [14], [16] for modern electronic
device like transistors. Numerous efforts were applied to induce a band gap in graphene that include
breaking graphene superlattice’s symmetry by substrate [17], chemical doping [18] and electric
field [19]–[21]. However, the ‘gapped’ graphene cannot gain sufficient gap without sacrificing its
high mobility as a trade-off. In addition, the weak spin orbit coupling in graphene [22] hinders
novel quantum phenomena from emerging, which can be useful for further optoelectronic
application. Consequentially, much interest has been stimulated to other 2D materials beyond
graphene for both fundamental science and practical application [23], [24]. And transition metal
dichalcogenides (TMDs) is such a new class of 2D materials that attracts numerous interests
following the ‘gold rush’ of graphene.
Transition metal dichalcogenides (TMDs) is a class of II-VI semiconductors with the type
MX2, where M labels a transition metal atom (e.g. Mo, W, etc.) and X represents a chalcogen atom
(e.g. S, Se, Te). Most of known TMDs are formed with a layered structure, from which we can
isolate monolayer TMD from bulk crystal by either mechanical exfoliation or chemical growth.
Distinctly different from graphene, intrinsic semiconductor gaps with magnitude of 1~2 eV [23],
[24] naturally reside within monolayer TMD crystals. Ranging from visible to near-infrared, such
band gaps offer highly interesting optoelectronic application based on 2D materials. In addition,
restricted geometry in low dimensionality triggers a series of novel electronic and optical properties
in 2D TMDs. Most particular is the robust formation of a quasiparticle, namely as exciton, and
resulted occurrence of strong light-matter interaction in monolayer TMDs [23], [24]. This will
come as the core topic of our discussion in following chapters.
Nanoelectromechanical systems (NEMS) is a family of devices coupling mechanical
motion with electrical or optical functionalities in nanoscale. In NEMS devices, the movable
3
ingredients can be actuated in static displacement or driven into dynamical vibration. Thus, the
mechanical motion in NEMS can be transduced into electrical or optical signals with high
sensitivity [25], [26]. Moreover, external perturbation can easily influence mechanical properties
of NEMS due to the miniscule nature built in such systems. Therefore, it would be highly interesting
to explore the probe-induced back-action effect on the mechanical performance of NEMS. When
integrated with 2D materials, this NEMS structure provides an ideal platform to study intrinsic
properties of atomically thin TMDs with mechanical knob. More importantly, by suspending
single-layered TMDs, it opens up tempting opportunities to study the interplay between mechanical
properties and rich electronic and optical features within this class of 2D materials. Our prior
interest will focus on coupling between mechanical mode and optical pumping through strong light-
matter interaction in suspended monolayer TMD semiconductor.
The main focus of this dissertation is about the NEMS based on TMD monolayers. By
suspension, both novel optical phenomena and interplay between mechanics and strong light-matter
interaction in 2D semiconductors were systematically studied and discussed in the following
sequence. First, we will discuss the emerging optical nonlinearity and its optical control in
monolayer WSe2 in Chapter 3. Second, we will demonstrate dynamical control of suspended
monolayer MoSe2 by light through the material’s strong excitonic resonance in Chapter 4. In
Chapter 5, along with flexible substrate, we will present the study of magneto-electricity effect in
strained single-layered MoS2. Here, we will begin our discussion with the introduction of
monolayer TMDs.
1.1 Monolayer transition metal dichalcogenides (TMDs)
The study of TMDs has a long history which can date back to 1963 for obtaining ultra-thin
MoS2 samples with use of adhesive tape [27]. Over the past decade, the ‘gold rush’ of graphene
4
stimulated the resurgence of interest for this class of van der Waals materials. In general, there are
several prototypes of lattice structure for TMDs including 1T, 1T’, 2H and 3R [28]. Throughout
this dissertation, we only focus on TMDs stacked in 2H phase since the rich physics that we will
discuss only exist in this structure phase. In 2H phase, single-layered TMDs form honeycomb
lattice from top view, from which a layer of transition metal atom M is sandwiched by two layers
of chalcogen atoms X to form trigonal prismatic structure [23]. Similar to graphene, monolayer
layered TMDs also possess K and K’ valleys in momentum space. When thinned down to
monolayer limit, though its atomic structure remains unchanged, the reduced dimensionality
induces renormalization of electronic structure and new crystalline symmetry in 2D TMDs [23].
Thus, emerging electronic and optical features distinguish monolayer TMDs from their bulk
counterparts as we discussed below.
Figure 1-1: Trigonal prismatic structure of monolayer TMDs in 2H phase. The honeycomb lattice
structure with broken inversion symmetry and the first Brillouin zone with the high symmetry
points are shown. Figure is obtained from Ref. [23].
5
1.1.1 Crossover from indirect to direct band gap
Here, we take MoS2 as a sample and discuss layer-dependent evolution of semiconductor
gaps in atomically thin TMDs. The bulk MoS2 is well-known as an indirect band gap semiconductor
with the gap size of 1.29 eV [29]. And this indirect band transition occurred between the valence
band maximum at Γ point and the conduction minimum along the Γ and Κ points of Brillouin zone.
Meanwhile, the smallest direct band gap resides at the K point with a gap size of 1.8 eV [29]. In
addition, the electronic bands associated with band gaps above have different orbital origins. At Κ
point, the direct band gap is mainly contributed from the 𝑑 orbitals of Mo atoms (𝑑𝑧2 for the
conduction band minimum and 𝑑𝑥2−𝑦2 for the valence band maximum, respectively) [29]–[31]. In
contrast, both 𝑑𝑧2 orbital of Mo atoms and 𝑝𝑧 orbital of S atoms contributed mostly to the indirect
band gap along the Γ and Κ points [29]–[31]. Note that orbitals from S atoms not only interact with
atoms within the same layer but also ‘talk’ to orbitals originated from atoms at neighboring layer.
Since Mo atoms sandwiched between two layer of S atoms, the direct bandgap states from Mo-𝑑
orbitals are more localized and less influenced by interlayer coupling in out-of-plane direction
compared to the indirect bandgap states which include S- 𝑝 orbitals. As the thickness reduces, the
indirect band gap gradually opens up while the direct bandgap remains almost unchanged due to
increasing quantum confinement effect. When thinned down to monolayer limit, the indirect band
gap along the Γ and Κ points exceed the direct band gap at Κ point. Thus, MoS2 undergoes a
crossover from an indirect band gap to direct band gap semiconductor in monolayer as depicted in
Fig. 1-2. Mak et al. [29] and Splendiani et al. [30] pioneeringly investigated this layer-dependent
evolution of electronic structure in MoS2 through optical measurements. Through systematical
characterizations, particularly for the work of Ref. [29], optical feature from the indirect band gap
consistently blue shifted down to bilayer and completely disappeared in monolayer limit.
Meanwhile, the feature from direct band gap transition changed little down to monolayer with at
6
least two order of magnitude stronger in scale than the counterpart from indirect band gap. All the
experimental observation demonstrated that there was a crossover between indirect to direct band
gap occurring in monolayer MoS2 [29]–[31]. The indirect to direct band gap transition is essentially
the same for other group VI semiconductor monolayers described above using MoS2.
Figure 1-2: Calculated electronic band structure of atomically thin MoS2. (a) Atomic orbital
projection of band structures for monolayer MoS2 from first-principle predictions, without spin-
orbit coupling. Fermi energy is set to zero. Symbol size is proportional to its contribution to
corresponding state. (1) Contributions from Mo-d orbital. (2) Total contributions from p orbitals,
dominated by S atoms. (3) Total s orbitals. (b) The simulated band structure of bulk, quadri-layer,
bilayer and single-layer MoS2 (from left to right). A transition from indirect band gap to direct band
gap semiconductor occurs at monolayer limit. Figures are obtained from Ref [30], [31].
7
1.1.2 Electronic band structure
In this section, we illustrate the basic properties of electronic band structure near the direct
band gap in monolayer MoS2 and the general description can also be applied to other group VI
TMDs monolayers. The trigonal prismatic structure of one MoS2 unit cell splits the d orbitals of
Mo atom into three subgroups: 𝐴1(𝑑𝑧2), 𝐸(𝑑𝑥𝑦, 𝑑𝑥2−𝑦2) and 𝐸′(𝑑𝑥𝑧, 𝑑𝑦𝑧) [31], [32]. As discussed
in section 1.1.1, the in-plane mirror symmetry allows the hybridization only between 𝐴1 and 𝐸
from the conduction band and valence band [31], [32], respectively, which opens the direct band
gap at K or K’ (namely, K’) point in Brillouin zone. Thus, the wavefunctions that describe states
in the adjacency of K/K’ points are given as the following symmetry-adapted basis, [32]
|𝜙𝑐⟩ = |𝑑𝑧2⟩ |𝜙𝑣𝜏⟩ =
1
√2(|𝑑𝑥2−𝑦2⟩ + 𝑖𝜏|𝑑𝑥𝑦⟩) (1-1)
where |𝜙𝑐⟩ is the state basis for conduction band, |𝜙𝑣⟩ is for valence band, and τ denotes ± with
upper and lower signs for the K (K’) valley index related by time reversal operation. To lowest
energy approximation, two-band k ∙ p theory gives out an effective Hamiltonian as [32]
��0 = 𝑎𝑡(𝜏𝑘𝑥��𝑥 + 𝑘𝑦��𝑦) +Δ
2��𝑧 (1-2)
where �� denotes the Pauli matrices for the tow basis functions above, 𝑎 is the lattice constant, 𝑡 is
the effective hopping integral, and Δ is the energy gap. This effective Hamiltonian of monolayer
MoS2 is akin to the ‘gapped’ graphene since they share the similar symmetry properties. And the
nature of the heavy metal d orbitals provides strong spin orbit coupling (SOC) to lift the spin
degeneracy within electronic bands. In addition, the reflection symmetry within a layer only allows
SOC to have out-of-plane component [31], [32]. Namely, under first order approximation, orbitals
of 𝐸(𝑑𝑥𝑦, 𝑑𝑥2−𝑦2) have non-zero SOC component while SOC from 𝐴1(𝑑𝑧2) vanishes.
Consequently, the top valence bands at K and K’ points are spin-polarized and separated by
hundreds of meV while the conduction bands remain spin degenerate [32]. This spin-polarized top
8
valence bands give rise to two optical transitions from upper and lower valence bands to the
conduction bands at K and K’ points, which denoted as resonance A and B, respectively [29], [30].
The resonance A with lowest transition energy is our prior interest in this dissertation since major
population of carrier involves within this transition process. This induces prominently strong light-
matter interaction and related novel phenomena which we will present more related details in
Chapter 3 and 4. In reality, finite spin splitting of conduction bands at K and K’ points [23] can be
generated by SOC in TMDs with heavier atoms include MoSe2, WS2 and WSe2 monolayers. It
plays an important role on formation of richer quasiparticle complexes [23] and long lifetime of
valley carriers [23], [33]. But all these interesting physics go beyond the interest of this dissertation
and we will not discuss the phenomena induced by spin-polarized conduction bands. The 𝑘 ∙ 𝑝
model-based Hamiltonian above neglects SOC and now we modify ��0 with SOC approximated by
intra-atomic contribution 𝑳 ∙ 𝑺. The total Hamiltonian is then transformed as [32]
�� = 𝑎𝑡(𝜏𝑘𝑥��𝑥 + 𝑘𝑦��𝑦) +Δ
2��𝑧 − 𝜆𝜏
��𝑧−1
2��𝑧 (1-3)
where 2𝜆 is the spin splitting generated by SOC at the valence band edge and ��𝑧 is the Pauli matrix
for spin. It is noteworthy to emphasis that the spin orientation of upper valence band is opposite at
K and K’ points due to time reversal symmetry. This ‘spin-valley coupling’ effect gives rise to
unique valley-contrasting optical selection rules [23], [33] and further magnetic control [23], [33]
of valley index as a new degree of freedom. We will cover these two novel phenomena in next two
sections.
This spin splitting is not limited to specific materials and directly resulted from inversion
symmetry breaking. In monolayer MoS2, the inversion symmetry of lattice is intrinsically broken
due to the trigonal prismatic arrangement of S atoms [31], [32]. Namely, if taking the Mo atom as
the inversion center, an S atom will map to a vacuum position without identical counterparts.
Except for bandgap opening at K and K’ points, another explicit consequence of inversion
9
symmetry breaking is valley-contrasting orbital magnetic moment 𝝁𝑉 [33], [34] in TMD
monolayers, which originated from self-rotation of electron wavepackets. This valley magnetic
moment can impose an anomalous velocity into a bias-driven electron in the vicinity of K valley.
The direction of anomalous velocity [33], [34] is switched at K’ valley to satisfy time reversal
symmetry. Intuitively, this effect can be viewed as the drifted electron experienced a Lorentz force
induced by a magnetic field in momentum space, namely the Berry curvature Ω𝐵 [33], [34]. Note
that the valley magnetic moment 𝝁𝑉 is proportional to the Berry curvature Ω𝐵 since these two
physical quantities share the same geometric origin [33], [34]. We will focus more on the valley
orbital magnetic moment due to its direct response to external fields. For 2D massive Dirac system
as monolayer TMDs, the valley magnetic moment has the simple form 𝝁𝑉 = 𝜏𝑒ℏ
2𝑚∗ �� at K and K’
points [33], [34], where 𝑚∗ is the effective mass of the Bloch band and �� is the unit vector pointing
out-of-plane. It mimics a spin magnetic moment with an effective Bohr magneton 𝜇𝐵∗ =
𝑒ℏ
2𝑚∗. In
addition, the conduction and valence bands carry valley magnetic moments with the same sign [33].
At the presence of an external magnetic field, the valley magnetic moment can generate a Zeeman-
type response as −𝝁𝑉 ∙ 𝐵. This coupling between valley magnetic moment and external magnetic
field, together with the strong SOC introduced above, allows the control of the valley degree of
freedom in TMD monolayers by circularly polarized light [35]–[37] and out-of-plane magnetic
field [38]–[43] as we discuss in section 1.13 and 1.14 for more details below.
10
1.1.3 Valley-dependent optical selection rules
Th opposite valley magnetic moment explicitly gives rise to valley-dependent optical
selection rules in single-layer TMDs. Originated mostly from 𝑑𝑧2 orbitals, the conduction band
edges denote a zero magnetic quantum number, whereas the valence band edges with
|𝑑𝑥2−𝑦2⟩ + 𝑖𝜏|𝑑𝑥𝑦⟩ orbitals carry a magnetic quantum number 2𝝉 [33]. Due to the canceled spin
contribution, inter-band transition is thus forbidden at K and K’ points if only considering atomic
orbital angular moment under electric dipole approximation ∆𝑚 = ±1 [33]. However, as we
discussed in previous section, the valley magnetic moment at K and K’ points carries a magnetic
Figure 1-3: Optical selection rules and valley magnetic response. (a)Electronic bands (here for W
compounds) around K and K’ points, which are spin-split by the spin-orbit interactions. The spin
(up and down arrows) and valley (K and K’) degree of freedom are locked together. Azimuthal
quantum number (m) for each band is shown, and the valley-dependent optical selection rules are
depicted. (b) Zeeman shift of the K and K’ valleys under an out-of-plane magnetic field. The dashed
and solid lines represent bands of opposite spin (split by spin-orbit coupling). 𝜇𝑠, 𝜇𝑉 and 𝜇𝑂 are the
spin, valley and intra-atomic orbital magnetic moments, respectively. Figures are obtained from
Ref.[23], [33].
11
quantum number −𝝉, which naturally lead to the total angular momentum change ±𝝉 at K and K’
valleys, respectively [33]. This allows the occurrence of direct optical transition at K and K’ valleys
exclusively coupled to the incident light with opposite helicity, namely the left- and the right-
circularly polarized light [33], [34]. The valley-contrasting optical selection rules have been
experimentally observed in single-layered TMDs by several groups independently [38]–[42]. Due
to SOC-induced spin valley locking, together with large momentum separation of K and K’ valleys,
the emitted photoluminescent (PL) preserves a large PL handedness ρ that suggests well-
maintained valley polarization upon excitation near the transition energy [33], [34].
1.1.4 Valley magnetic response
The presence of valley magnetic moment opposite at K and K’ valleys also lead to
selectively addressing two valleys by a magnetic field perpendicular to TMD monolayers. The total
magnetic moment 𝝁𝑡𝑜𝑡 imposed on an electron in the vicinity of K and K’ valleys is largely
attributed by summing up three independent contributions: spin magnetic moment 𝝁𝑠, intra-atomic
orbital magnetic moment 𝝁𝑂 and valley (inter-atomic) magnetic moment 𝝁𝑉 [33], [34]. Through
Zeeman interaction in finite magnetic field, both conduction and valence bands experience energy
shifts under the influence of these three magnetic moments [33], [34]. Within those allowed inter-
band transition, the spin-induced Zeeman shifts cancel each other for conduction and valence bands
at K and K’ valleys. The intra-atomic orbital magnetic moments contribute energy shift of 2𝜏𝝁𝐵𝐵
owing to different nature of d orbitals for conduction and valence band edges [33]. Similarly, the
Zeeman contribution from valley magnetic moments is given as 𝜏𝝁𝐵𝐵(𝑚0
𝑚𝑐∗ −
𝑚0
𝑚𝑣∗), which stems
from different effective mass of conduction and valence bands [33], [34]. In the vicinity of K and
K’ points, the effective masses of conduction and valence band edges are approximately equal [31],
12
[32]. Thus, the total Zeeman splitting between K and K’ valleys can further reduce to −4𝝁𝐵𝐵
purely due to the contribution of intra-atomic orbital moment. This valley Zeeman splitting matches
the experimental results with −(3.8 ± 0.2)𝝁𝐵𝐵 surprisingly well [38], [39]. In addition, it lifts the
degeneracy of left-and right circularly polarized emission in energy scale and allows the further
control of K and K’ valleys individually, together with valley-contrasting optical selection rules.
1.1.5 Exciton
Exciton is a quasi-particle made of an electron and hole pair bound by Coulomb interaction.
With optical illumination in gapped materials, an electron can be excited to the conduction band
and left a hole in the valence band by absorbing an incident photon. As the explicit consequence,
such electron-hole pair is correlated by Coulomb interaction and forming a hydrogen-like
compound, namely exciton. Though their different physical origins, it is still helpful to make an
analogy between exciton and well-studied hydrogen atom. First, the binding energy of exciton is
similar to the ionization energy of hydrogen atom, which lowers the total energy from the energy
sum of individual electron and hole. In addition, exciton’s binding energy is easily influenced by
dielectric environment due to its origin from Coulomb interaction. Second, as in hydrogen atom,
Bohr radius is an important physical quantity to characterize the wavefunction of exciton. It
classifies different types of exciton in length scale stemming from competition between Coulomb
interaction and Bloch lattice potential. Like hydrogen atom, excitons demonstrated various of
excited states with different energy and symmetries.
Typically, excitons can be categorized as Frenkel type and Wannier-Mott (Wannier for
simplicity) type. Frenkel excitons generally exist in weak dielectric systems with strong Coulomb
interaction and large binding energy (~ 0.1eV). As the explicit consequence, these tightly-bound
excitons localize within Bohr radius smaller than lattice spacing, of which the Coulomb coupling
13
dominates over lattice potential. In contrast, Wannier-typed excitons reside in material systems
with large dielectric constant. Since the wavefunction of Wannier exciton spread over several unit
cell, the lattice potential can be ‘felt’ by Wannier excitons and be incorporated into the effective
masses of the electron and hole as Bloch particles. As result, the binding energy of Wannier
excitons is around ~ 0.01eV much smaller than localized Frenkel counterparts. All the excitons
discussed in this dissertation are Wannier-typed excitons.
Dimensionality, in addition to intrinsic dielectric capability of materials, is another influential
element to engineer properties of excitons. With the development of nanotechnology, material
systems in reduced dimensionality was fabricated and intensively studied, some of which show
promising potential as platform to study excitons. From 0D quantum dot, 1D carbon nanotube to
quasi-2D quantum wells, geometric restriction enhances the quantum confinement effect and also
further reduces dielectric screening by allowing a large portion of electric fields from electron-hole
pair to extend into the vacuum [44]. Thus, excitonic features are more pronounced due to the
enhanced Coulomb interaction in low dimensionality. And excitons had been intensively studied
in quantum wells system [45] and manifested lots of exciting physics include the strong signature
of exciton Bose-Einstein condensation [46], [47]. Monolayer TMD semiconductor is true 2D
system which basically is surface everywhere. Consequentially, both theoretical calculation and
experimental measurements have showed huge exciton binding energy ranging from 0.3 to 1 eV
for different TMDs [33], which is roughly one orders of magnitude larger than those in quasi-2D
quantum wells [23]. The significantly high binding energy of excitons in TMD monolayers gives
rise to the robust exciton resonance up to room temperature [23]. More interestingly, together with
strong SOC and valley dependent optical selection rules, excitons with contrasting valley index can
be selectively excited and controlled by different handedness of circularly polarized light [35]–
[37]. The valley-dependent optical selection rules also become spin-dependent due to spin-valley
coupling [32] and thus allow excitons being further distinguished by external magnetic field [38]–
14
[42]. In addition, rich exciton species in TMD monolayers had been pronouncedly identified and
characterized at higher temperature [23], [33], which are usually elusive in other systems. These
excitons beyond fundamental excitation include charged exciton (trion), exciton pair (bi-exciton)
and even higher order excitonic complexes [23], [33].
Throughout this dissertation, we only focus on the 1s typed fundamental exciton state in
single-layered TMD semiconductors. Near this fundamental exciton resonance, high reflectivity
(>80%) with narrow linewidth (~ 2 meV) from TMD monolayers had been experimentally
demonstrated [48]–[50], which indicated the occurrence of extremely strong light-matter
interaction. This ‘cavity-like’ optical response originated purely from monolayer semiconductors.
Without substrate screening, the binding energy of excitons in suspended TMD monolayers can be
further enhanced [51]. In addition to the intrinsic properties of excitons, we are more interested in
Figure 1-4: Strong excitonic effect in 2D TMDs. (a) Schematics of electrons and holes bound into
excitons for the three-dimensional bulk and a quasi-two-dimensional monolayer. The changes in
the dielectric environment are indicated schematically by different dielectric constants 휀3𝐷 and 휀2𝐷
and by the vacuum permittivity 휀0.(b) Absorption spectrum of monolayer MoS2 at 10K (solid green
line). A and B are exciton resonances corresponding to transitions from the two spin-split valence
bands to the conduction bands. The blue dashed line shows the absorbance (arbitrary units) if
excitonic effects were absent. The inset illustrates the Coulomb bounding between an optically
generated electron-hole pair, forming a bound exciton. Figures are obtained from Ref. [23], [44]
15
optically tuning physical properties of suspended TMD monolayers by using exciton resonance as
a cavity-less ‘cavity resonance’. Thus, with large on-resonance excitonic reflectivity and narrow
linewidth, suspended TMD monolayer stands out as an ideal platform to study numerous optical
phenomena include optical bistability that conventionally required delicate cavity design and
operation [52]. Meanwhile, it is particularly compelling to study the exciton-optomechanical
coupling with such cavity-less structure. In Chapter 3, we will show that the suspended WSe2
monolayers exhibit power- and wavelength-dependent nonlinearities. Together with unique valley-
contrasting optical rules and magnetic response, this exciton-induced bistability can help to control
light not only with light intensity but also with its polarization using monolayer materials. In
Chapter 4, we will demonstrate the light-induced damping and anti-damping of mechanical
vibrations and modulation of the mechanical spring constant at the μW level near the exciton
resonance with variable detuning in a suspended MoSe2 monolayer. Owing to its strain tunability
of exciton resonance energy, the monolayer MoSe2 resonator generates a dynamical strain as
vibrating and manifested different mechanical responses that stems from contrast photothermal
back-action at opposite detunings of exciton resonance. These two works above will open up
exciting opportunities to study and engineer optical nonlinearity and cavity-less optomechanics
using a monolayer semiconductor.
1.1.6 Mechanical properties
Intrinsically, the stability of the sp2 bounds gives rise to graphene’s exceptional mechanical
properties. Measurements from AFM nano-indentation show that the in-plane elastic stiffness of
graphene yields of 340 N/m [12], which corresponds to a bulk Young’s modulus of 1TPa. The
measurements also reveal the ultra-strength of graphene with a breaking strain of 25% [12]. Such
16
large mechanical strength and strain sustainability greatly exceed those for any of conventional
materials used for NEMS application.
Atomically thin TMDs include MoS2, WS2, and WSe2 also share similar mechanical
properties as their graphene cousin. Through the similar indentation measurements, either
exfoliated or CVD grown MoS2 monolayers were found to have a Young’s modulus around 270
GPa [53], [54] with applied strain of 10% before breaking [53]. The mechanical properties of
tungsten-based TMDs have also been identified. For few-layered (4-15 layers) WSe2 by exfoliation
is found to have a Young’s modulus of 167 GPa [55] with a breaking strain of 7% [55]. Also, the
CVD-grown WS2 monolayers demonstrate a Young’s modulus with 272 GPa [54], similar to that
of single-layered MoS2. With external strain applied, both theoretical calculation and experimental
results show a redshift of band gap energy in both direct and indirect transition of TMDs that is
measured to be linear with strain. And the correspond redshift at a rate of ~70 meV per percent
applied strain for direction gap transitions [56] has been determined by optical and electronical
means for monolayer TMD semiconductors [57]. Such large strain sustainability introduced above,
together with pronounced strain-tunability in band gap, allows the mechanical tuning of electronic
properties of TMD monolayers [57]. Simultaneously, it opens up the possibility of coupling novel
optical properties with mechanical modes in suspended TMD monolayers as we will discussed
below.
1.2 Nanoelectromechanical Systems (NEMS)
1.2.1 2D NEMS
Due to the nature of two-dimensional materials, the emerging properties of TMDs monolayers
illustrated in the above sections can be greatly influenced by the substrate they are residing on. Any
17
external perturbation particularly includes substrate effects (roughness, dielectric screening,
contamination etc.) can limit the study of intrinsic properties in TMD monolayers. The natural
thought to solve this problem is to suspended these atomically thin flakes to get rid of the substrate
effect. Through this way, the ultra-high carrier mobility [58], [59], thermal conductivity[10], [11]
and mechanical properties [12], [13] of graphene, as well as fundamental properties [53]–[55],
[60]–[64]of 2D TMDs, had been studied, which is challenged to investigate in the substrate-
supported samples.
Meanwhile, we can fabricate nanoelectromechanical system (NEMS) within atomic scale
using these suspended 2D membranes. And atomically thin monolayer materials have attracted
considerable interest in the studies of NEMS devices [65]–[86] because of their extremely
lightweight, mechanical flexibility, gate-tuneability and strong inter-mode couplings[75], [81],
[84], [86]. They also hold appealing potential in versatile sensing applications [80], [87]–[98]. In
turns, the high sensitivity of 2D NEMS is a benefit to not only study these intrinsic properties of
materials, but also to study interplay between thin film mechanics and other properties of 2D
materials[60], [72], [74], [99]–[105], especially for using mechanical resonance to probe the charge
density wave[102]–[104], superconductivity[72], [103] in 2D electron-correlated materials.
For 2D semiconductors, although the mechanical aspects of NEMS devices based on
monolayer materials have been extensively studied in the literature [60], [65], [66], [80], [85], the
dynamical coupling between mechanical vibrations and light (i.e. optomechanical coupling)
remains as an open subject. In particular, the presence of strong excitonic resonances in monolayer
TMD semiconductors[23], [33], which strongly interact with light (can reflect ~ 90 % of the
incident light) [48]–[50], presents a very interesting avenue to explore optomechanical coupling
effects in atomically thin materials without the need of an optical cavity.
18
1.2.2 Mechanical model of membrane
In this section, we will review the static and dynamical mechanics of a suspended membrane.
The mechanical model developed below is aimed to practically extract physical quantities (e.g.
displacement of membrane, Young’s modulus, initial strain etc.) from our optomechanical
experiments using gate voltages.
The differential equation of motion for a circular membrane with clamped edges under
uniform tension is given as [106]
𝜌𝜕2𝑧
𝜕𝑡2= −𝐷∇4𝑧 + 𝜎∇2𝑧 + 𝑃 (1-4)
Figure 1-5: NEMS devices made of 2D materials. By fabricating suspended samples over a trench
structure, intrinsic properties include (a) electrical transport (e.g. carrier mobility), (b) optical
response (PL& linear absorbance) (c) mechanical properties (e.g. Young’s modulus) and (d)
thermal transport (thermal conductivity) of 2D materials were systematically studied, which would
be challenged for samples on substrate. Also, 2D NEMS is a good platform to study interplay
between nanomechanics and other interesting DOF (e.g. charge, spin, valley etc.) in real 2D limit.
Figures are obtained from Ref. [10], [12], [29], [58].
19
where z is the displacement of the membrane, 𝜌 is the area density, 𝐷 is the 2D bending stiffness,
𝜎 is the 2D tension, and 𝑃 is the pressure acting on the membrane. The above equation is adapted
from Foppl-von Karman model which is used for describing the deflection of thin flat plates[106],
[107]. Before further derivation, we can just intuitively analysis the contributions from each term
that related to mechanical properties. The 2D bending stiffness is defined as 𝐷 =𝐸𝑌𝑡
2
12(1−𝜈2) [106],
where 𝐸𝑌 is the Young’s modulus of the membrane material, 𝑡 is the thickness of the plate, and 𝜈
is the Poisson ratio. If we plug in the typical numbers for TMD monolayers include 𝐸𝑌 ~ 200 𝐺𝑃𝑎,
𝑡 ~ 0.7 𝑛𝑚 and 𝜈 ~ 0.18 [60], then the area bending stiffness is 𝐷 ~ 10−8 . For the tension
contribution, by taking a typical tension 𝜎 ~ 0.1𝑁/𝑚 and the diameter 𝐿 ~10 𝜇𝑚 of our
monolayer NEMS, the tension term is of typical value around 𝜎𝐿 ~ 10−6 . From the simple
estimation above, we can reasonably conclude that the tension is dominant within our monolayer
sample and the bending term can safely drop out for further discussion. For equilibrium condition,
after setting the motion term to zero at the left-hand side of equation, the equation of motion for
the clamped membrane is rewritten as
𝜎∇2𝑧 + 𝑃 = 0 (1-5)
For simplicity, we neglect the angle contribution and the Laplace notation only has radial
dependence as ∇2=1
𝑟
𝜕
𝜕𝑟+
𝜕2
𝜕𝑟2 in cylindrical coordinate. With the boundary conditions both at the
center and edge of membrane, then the displacement profile 𝑧(𝑟) can be described in a parabola
shape [107]
𝑧(𝑟) = 𝑧0(1 −𝑟2
𝑅2) (1-6)
where 𝑧0 = 𝑃𝑅2/4𝜎 is the displacement at the center of membrane, thus the force acting at the
center is given as 𝐹 = 4𝜋𝜎𝑧.
20
In our experiments, through doping extra charges into the membrane, the suspended flake can
be pulled down and the deflected displacement is well controlled by an electrostatic force that stems
from a DC gate voltage 𝑉𝑔. Thus, the whole suspended device forms a parallel-plate capacitor
structure and the corresponding electrostatic force is described as
𝑃(𝑧) =1
2
𝜕𝐶
𝜕𝑧𝑉𝑔2 =
𝜖0𝑉𝑔2
2(𝑑−𝑧(𝑟))2 (1-7)
where 𝐶 is the unit capacitance of the device, 𝜖0 is the vacuum dielectric constant, 𝑑 is the vacuum
gap between the suspended membrane and the bottom surface of trench. Typically for our
experiments, the maximal vertical displacement of membrane is of ~ 50 nm, which is much
smaller than the vacuum gap of ~ 600 nm. Thus, we can further simplify the electrostatic pressure
above to 1st order approximation as
𝑃 = 𝑃0 (1 −𝑧
𝑑)−2≅ 𝑃0 (1 + 2
𝑧
𝑑) (1-8)
where 𝑃0 =𝜖0𝑉𝑔
2
2𝑑2 is the displacement at the center of membrane. After plugging the electrostatic
pressure back to the equilibrium equation of net force as
𝜎∇2𝑧 + 𝑃0 (1 + 2𝑧
𝑑) = 0 (1-9)
We can reformulate the 2nd order differential equation in differential order of radius
variable as
𝑟2𝑧′′ + 𝑟𝑧′ + Λ2𝑟2(𝑧 +𝑑
2) = 0 (1-10)
where Λ = √2𝑃0
𝜎𝑑 and one can solve this zeroth order Bessel’s equation and obtain the general
solution by considering the boundary condition 𝑧(𝑅) = 0
𝑧(𝑟) =𝑑
2(𝐽0(Λ𝑟)
𝐽0(Λ𝑅)− 1) (1-11)
where 𝐽0 is the Bessel function with first kind. We note that the calculated profile of displacement
based on this electrostatic approximation is similar to that derived from parabola model but ‘bended’
21
deeper at the center of membrane since the force will increase as the membrane pulled close to the
back-gate surface. Since Λ is determined by applied gate voltage 𝑉𝑔, this gate dependence provides
us a very convenient knob to tune and calibrate the static deflection of membrane together with the
interferometry technique introduced in section 2.2.4.
Similar analysis applies to dynamical description for the motion of a clamped membrane. We
start with replacing 𝑧 ⟶ 𝑧 + �� and 𝑃 ⟶ 𝑃 + �� , where the tilde notation represents the time-
dependent oscillation component. Assuming mechanical properties of the membrane are time-
independent, we can split the equation of motion Eqn. (1-5) into the static and dynamic parts
𝜎∇2𝑧 + 𝑃 = 0
𝜎∇2�� + �� = 𝜌𝜕2��
𝜕𝑡2 (1-12)
After setting �� = 0 for the convenience of discussion and using the conventional
separation of variables method, we can obtain the solution for dynamic displacement �� as
��(𝑟, 𝑡) = ∑ 𝐴𝑚𝐽0 (𝜁𝑚
𝑅𝑟) 𝑒𝑖Ω𝑚𝑡∞
𝑚=0 (1-13)
where the 𝐴𝑚 is the complex amplitude of m𝑡ℎ vibration mode and Ω𝑚 is the angular resonant
frequency of the m𝑡ℎ vibration mode as written blow
Ω𝑚 =𝜁𝑚
𝑅√𝜎
𝜌 (1-14)
where 휁𝑚 is the m𝑡ℎ positive root of Bessel function 𝐽0, and 휁0 ≅ 2.405 for the fundamental mode
(of our interest). To obtain the m𝑡ℎ complex amplitude 𝐴𝑚 of mechanical mode, we need to plug
the specific expression of external force into the equation of oscillation above. To determine the
resonance frequency of resonators, an electrostatic force with both DC and AC components is
applied to drive the suspended membrane into vibration. From previous discussion for electrostatic
pressure 𝑃 ≅𝜖0𝑉𝑔
2
2𝑑2(1 + 2
𝑧
𝑑), we can explicitly write down the AC component of the driving force
22
by letting 𝑉𝑔⟶ 𝑉𝑔 + ��𝑔. Only keeping 1st order approximation term, we have the expression for
dynamical electrostatic pressure �� as
�� ≅𝜖0𝑉𝑔
2𝑑2��𝑔 (1-15)
To solve the dynamics part of motion, we can also rewrite the above �� in terms of Fourier
Bessel series to match the format of dynamical displacement as �� = ∑ 𝑃𝑚𝐽0 (𝜁𝑚
𝑅𝑟) 𝑒𝑖Ω𝑚𝑡∞
𝑚=0 .
Plugging the dynamical displacement �� to the dynamics equation of motion, now we are able to
solve for the complex amplitude as 𝐴𝑚 =𝑃𝑚
𝜌(Ω𝑚2 −Ω2)
, where the series coefficient for electrostatic
pressure is defined as
𝑃𝑚 = ∫ 𝑟��𝑅
0𝐽0(
𝜁𝑚
𝑅𝑟)𝑑𝑟
𝑅2
2(𝐽1(휁𝑚))
2⁄ =2��
𝜁𝑚𝐽1(𝜁𝑚) (1-16)
where the electrostatic pressure effectively projects itself onto different modes of motion scaled by
a factor of 2
𝜁𝑚𝐽1(𝜁𝑚). Thus, by adding a damping term ργ
∂z
∂t, the complex amplitude 𝐴𝑚 can be
modified as
𝐴𝑚 =𝐼𝑚
(Ω𝑚2 −Ω2+𝑖Ωγ)
(1-17)
where 𝐼𝑚 =2��
𝜁𝑚𝐽1(𝜁𝑚)𝜌=
𝜖0𝑉𝑔��𝑔
𝜌𝑑2𝜁𝑚𝐽1(𝜁𝑚) is directly related to both DC and AC voltages applied onto
the sample. Namely, the magnitude of membrane motion is a Lorentzian function in the frequency
domain and the maximal amplitude is determined by applied gate voltages (both DC and AC gates).
We emphasis here that the gate-tunability of complex amplitude 𝐴𝑚 allows us to estimate the
dynamical strain induced by vibration purely from mechanical resonance measurement. And the
resultant dynamical shift of exciton resonance opens up possibilities to dynamically control the
motion of suspended TMD monolayers through its exciton resonance as we will introduce next
section.
23
1.3 Optomechanics
Optomechanics is an interdisciplinary field with utilizing laser light to control mechanical
objects, usually for resonators in meso- or nano-scale [108], [109]. In addition to trapping and
manipulating small particles, light-induced forces can also transduce and modulate
micromechanical motion [108], [109]. Specially, those NEMS devices with lightweight exhibit
noticeable advantages for optomechanical study and applications, and the resultant low stiffness in
turn endows this NEMS structure with higher sensitivity to be probed or controlled by feeble forces
induced by light. As consequence, although optomechanical effects have been utilized extensively
in macroscales include the gravitational detection in kilometer length and a gram-scale cavity
mirror [110], the expanding interests of optomechanics growingly stem from the progress of NEMS
technology.
Basically, dynamical control of mechanical motion by optomechanics depends on the use
of a ‘back-action’ force [108], [109]. Back-action can be intuitively understood as the perturbation
induced by a probe (usually laser beam) is no longer negligible for the sample and in turn it can
influence properties of the sample. And in our case, when optically measuring the motion of a
resonator, an extra damping was added by feedback from optical probe and thus modify the
oscillating states of the resonator. Numerous studies and application for these types of effect have
been reported extensively, which includes force sensing [80], non-volatile mechanical storage [111]
and photonic signal conversion [112], [113]. Nevertheless, an overwhelming majority of
optomechanics researches arise from the possible cooling down to the quantum ground state by
optomechanical coupling, where the average phonon occupation is less than one [109]. Meanwhile,
a system that resides around the ground state is a fundamental requirement for testing of quantum
theory and implementation of quantum computing [108], [109]. We will first demonstrate the
backaction dynamics by modeling a movable mirror driven by light-induced force with retardation,
24
then discuss optomechanical backaction effects (cooling and self-oscillation) with derivation, and
our motivation for excitonic optomechanics using TMD monolayers.
1.3.1 Modeling optomechanical coupling
In this section, we will review the classical picture of optomechanical coupling by formulism.
Here, we start with a driven damped oscillator model and demonstrate how the light-induced force
that acts on oscillator with retardation can generates a backaction to modify the mechanical
oscillation. The dynamical motion of a damped harmonic oscillator driven by a light-induced force
is described as the following equation:
𝑚�� +𝑚𝛤𝑀�� + 𝑚Ω𝑀2 𝑧 = 𝐹𝑝ℎ(𝑧(𝑡)) (1-18)
where 𝑚 is the effective mass of the oscillator, Ω𝑀 is the mechanical resonance frequency, 𝛤𝑀 is
the intrinsic damping rate, and 𝐹𝑝ℎ is the driven force induced by light. And this driven force can
Figure 1-6: Schematic of the cavity optomechanical system. A mechanical oscillator comprises
one end of a Fabry-Perot cavity, coupling mechanical motion to the optical cavity length. Figure is
obtained from Ref.[108]
25
be modeled by a delayed force that tends to reach its proper value in terms of position coordinate
𝑧 with some retardation 𝜏
𝐹𝑝ℎ(𝑧(𝑡)) = 𝐹(𝑧0) + ∫𝑑𝐹(𝑧(𝑡′))
𝑑𝑡′𝑇(𝑡 − 𝑡′)𝑑𝑡′
𝑡
0 (1-19)
where 𝑧0 is the mechanical equilibrium position, and the function 𝑇(𝑡) describes the time delay.
Thus, the equation of motion we need to solve is reformulated as:
𝑚�� +𝑚𝛤𝑀�� + 𝑚Ω𝑀2 𝑧 = 𝐹(𝑧0) + ∫
𝑑𝐹(𝑧(𝑡′))
𝑑𝑡′𝑇(𝑡 − 𝑡′)𝑑𝑡′
𝑡
0 (1-20)
We notice that the above equation can be solved easily in Fourier space by Laplace
transformation by adapting the Fourier transformation between frequency and time domains:
z(Ω) = ∫ 𝑧(𝑡)𝑒−𝑖Ω𝑡𝑑𝑡∞
0 (1-21)
The constant term 𝐹(𝑧0) has no time-dependence and just changes the equilibrium position
statically, which can be dropped out by shifting the displacement coordinate to this new equilibrium
position. Then, the equation of motion yields the following form
−𝑚Ω2𝑧(Ω) + 𝑖Ω𝑚𝛤𝑀𝑧(Ω) + 𝑚Ω𝑀2 𝑧(Ω) = ∫ 𝑒−𝑖Ω𝑡𝑑𝑡 ∫
𝑑𝐹(𝑧(𝑡′))
𝑑𝑡′𝑇(𝑡 − 𝑡′)𝑑𝑡′
𝑡
0
∞
0 (1-22)
As 𝐹(𝑧(𝑡′)) implicitly depends on time and its derivative in terms of time can be rewritten as
𝑑𝐹(𝑧(𝑡′))
𝑑𝑡′=𝜕𝐹(𝑧(𝑡′))
𝜕𝑧
𝜕𝑧(𝑡′)
𝜕𝑡′ (1-23)
Under small amplitude of oscillation, 𝐹(𝑧(𝑡′)) is expanded around 𝑧(𝑡0) as: 𝐹(𝑧(𝑡′)) ≈
𝐹(𝑧(𝑡0)) + [𝑧(𝑡′) − 𝑧(𝑡0)]∇𝐹, where we adapt the abbreviation 𝜕𝐹(𝑧(𝑡′)) 𝜕𝑧⁄ |
𝑧=𝑧(𝑡0)= ∇𝐹. And
we can safely drop the subscript note with the small amplitude approximation as 𝜕𝐹(𝑧(𝑡′)) 𝜕𝑧⁄ =
∇𝐹. As consequence, the equation of motion is rewritten as
−𝑚Ω2𝑧(Ω) + 𝑖Ω𝑚𝛤𝑀𝑧(Ω) + 𝑚Ω𝑀2 𝑧(Ω) = ∫ 𝑒−𝑖Ω𝑡𝑑𝑡 ∫ ∇𝐹
𝜕𝑧(𝑡′)
𝜕𝑡′𝑇(𝑡 − 𝑡′)𝑑𝑡′
𝑡
0
∞
0 (1-24)
With the helpful properties of the Laplace transform for convolutions as
𝐹1(Ω)𝐹2(Ω) = ∫ 𝑒−𝑖Ω𝑡𝑑𝑡 ∫ 𝐹1(𝑡′) 𝐹2(𝑡 − 𝑡
′)𝑑𝑡′𝑡
0
∞
0 (1-25)
26
where 𝐹1 and 𝐹2 are general functions used for discussion here. The equation of motion is
reformulated as
−𝑚Ω2𝑧(Ω) + 𝑚Ω𝑀2 𝑧(Ω) + 𝑖Ω𝑚𝛤𝑀𝑧(Ω) = 𝑖Ω∇𝐹𝑇(Ω)𝑧(Ω) (1-26)
And the light-induced force mostly stems from two components as radiation pressure and
photothermal effect. Both these two photo-induced processes react with an exponential behavior.
Thus, we can reasonably assume the profile of retardation function 𝑇 is of exponential type,
𝑇(𝑡) = 1 − 𝑒−𝑡/𝜏 (1-27)
Consequentially, the Fourier counterpart of the retardation function 𝑇(𝑡) is given by
𝑇(Ω) =1
𝑖Ω(1+𝑖Ω𝜏) (1-28)
After plugging the frequency-dependent retardation function back to the equation of
motion, we reformulated this oscillation equation in the power of Ω as
−𝑚Ω2𝑧(Ω) + 𝑖Ω𝑚𝛤eff𝑧(Ω) +𝑚Ωeff2 𝑧(Ω) = 0 (1-29)
with an effective resonance frequency (namely, spring constant) and an effective damping as [114]
Ωeff2 = Ω𝑀
2 (1 −1
(1+Ω2𝜏2)
∇𝐹
𝐾) ,
𝛤𝑒ff = 𝛤𝑀(1 + 𝑄𝑀Ω𝑀𝜏
(1+Ω2𝜏2)
∇𝐹
𝐾) (1-30)
where 𝐾 = 𝑚Ω𝑀2 is spring constant of the oscillator, and 𝑄𝑀 =
Ω𝑀
𝛤𝑀 is the mechanical quality factor.
From the expressions of effective resonance frequency and damping above, several interesting
phenomena can be explicitly concluded. The frequency dependence of the effective mechanical
quantities above indicates the propriate frequency window of oscillation required to observe the
pronounced changes induced by light. At very high mechanical frequency, the light-induced
contribution becomes negligible as the oscillator vibrates in the dark. In the contrast, as the
oscillator vibrates with small frequency, the light-induced terms are constants and no sufficient
damping change generates, especially for mechanical damping. With all other parameters fixed, the
27
maximal optomechanical damping change occurs when Ω𝑀τ ~ 1. This highlights the importance
of τ to estimate the effectiveness of optomechanical modification. For a typical mechanical
resonance of our samples with Ω𝑀
2𝜋 ~ 40 MHz, the optimal time constant 𝜏 would then be 4 ns.
Second, the oscillator systems with low mass (namely, small rigidity) are more preferable to
observe such light-mechanics interplay. Most importantly, sitting at different slopes of light-
induced force (namely, gradient of force ∇𝐹 ) can give rise to completely opposite responses
corresponding to damping (red-detuned) or anti-damping (blue-detuned) of mechanical motion as
we will discuss in details below.
1.3.2 Optomechanical backaction
In optomechanical systems, the damping of mechanical resonator can be optically modified
by introducing a backaction-induced damping [108], [109], [115]. The physics behind this extra
damping can be understood by a simple Fabry-Perot cavity as depicted in Fig. 1-6 with
displacement-dependent resonance frequency 𝜔𝑐𝑎𝑣(𝑧): the displacement z increases as the cavity
elongates, and thus the cavity resonance frequency (of interest) decreases. And the light-induced
force is proportional to the profile of cavity resonance as a function of displacement. Both radiation
pressure and photothermal forces give rise to the same physics as we will illustrate below. Here,
let us start with the movable mirror residing at the rising slope of the resonance in displacement
coordinate, which sustains the optical mode frequency still higher than the laser frequency. Namely,
the incident beam is red-detuned with respect to the cavity resonance.
28
Now we vibrate the movable mirror forwards and backwards in a small cycle. If the mirror
moves around with infinitely slow rate (adiabatically), the light-induced force can be ‘experienced’
by the moving mirror without doing additional work. Thus, no mechanical quantities will be
modified by the light-induced force under such condition. However, as the mirror moves with finite
sweep rate, the mirror ‘feels’ the light-induced force with time delay. Such retardation originates
from the finite decay time that incident photons or photothermal deformations take to build up
equilibrium. Thus, if the time delay is much shorter than one cycle of oscillation (namely, the
system is at equilibrium all the time), the light-induced force acts instantaneously on the mirror and
follows the profile of cavity resonance without any retardation. However, if the time delay is
comparable to one cycle of oscillation, the light-induced force that actually acts on the moving
Figure 1-7: Schematics of back-action effects in cavity optomechanics. (a) Schematic diagram of
light-induced force experienced by the end mirror as a function of displacement in Fabry-Perot
system. The finite decay rate of confined photons induces a delayed response to mirror motion. (b)
“P-V” style schematic depicting the work done by the radiation force during one cycle of
oscillation. The work is given by the enclosed area swept in the force-displacement diagram, which
is due to the retardation of the force (finite cavity decay rate). The work is negative or positive,
depending on whether one is on the red-detuned or blue-detuned side of resonance. This thus gives
rise to damping or amplification, respectively. Figure is obtained and modified from Ref.[109]
29
mirror will be different from the adiabatic counterpart at the same location (see Fig 1-7(a)). In the
first half circle, as the mirror moves towards the resonance with non-adiabatic speed, the light-
induced force remains smaller than it would be in the adiabatic motion. Conversely, as the mirror
moves back towards the starting point, the light-induced force exceeds that in infinitely slow
motion. Overall, the work done by light-induced force (the net enclosed area in force-displacement
plot in Fig. 1-7(b)) is negative at red-detuned pump. Namely, the light-induced force extracts
mechanical energy from the mirror.
Similarly, if we put the movable mirror into the falling slope of the resonance in displacement
coordinate and play the same ‘game’, the net work done by light-induced force is positive and the
mechanical motion is amplified. Intuitively, the optomechanical backaction is like a ‘guide’ that
drives the energy flowing between mechanical motion and light. As the explicit consequence, the
linewidth of mechanical motion is broadened (narrowed) when the work done by light is positive
(negative). Namely, a broadened linewidth of mechanical modes means that the mechanical energy
now decays faster than before, and thus the effective temperature of this mechanical mode is
lowered. This process above is optomechanical ‘cooling’ of mechanical motion. Conversely,
optomechanical ‘amplification’ can be realized by introducing a feedback that effectively reduce
damping in a resonator. This can ultimately lead to self-oscillation of a resonator, in which the
optomechanical backaction reduces the resonator’s damping to zero and the resonator can sustain
its vibration only by thermal fluctuation. Owing to its rich and compelling potential for novel
phenomena in both fundamental science and application, numerous races had been setting off to
either cool a NEMS resonator to its quantum state [116]–[124] or pump a NEMS resonator into
self-oscillation [116], [118], [125]. Both photothermal force [116], [118] and radiation pressure
force[117], [119]–[125] had been exploited to demonstrate sufficient optomechanical cooling or
gain in NEMS systems.
30
All these optomechanical systems above that require delicate cavity design and operation
[108], [109], [126], [127], including taper fiber access, coupling adjustment, and cavity stability,
impedes their potential application to cutting-edge opto-electronic devices, such as photonic
transistors and sensors. It would be particularly compelling to realize such optomechanical effects
in a mechanical resonator with a cavity-less structure. Hence, optical control of micro-resonators
with cavity-less structure is a promising alternative to achieve applicable integration for on-chip
micromechanical systems. Also, higher sensitivity of optical sensing can be reached in a micro-
resonator with low stiffness Thus, to realize the cavity-less optomechanics in mesoscopic
mechanics systems, an optical resonator with materials that possess both good mechanical
properties and strong light-matter coupling is highly demanded. Together with the excellent
mechanical strength and flexibility, the strong excitonic effects in monolayer TMD semiconductors
makes this type of materials as a favorable candidate to study and develop cavity-less
optomechanics using atomically thin materials. In Chapter 4, we will demonstrate our observation
of light-induced damping and anti-damping of mechanical modes and modification of mechanical
stiffness by optical excitation near exciton resonance in a monolayer MoSe2 resonator.
Chapter 2
Experimental reviews
In this chapter, we will introduce experimental methods to fabricate 2D
monolayer/heterostructure devices, especially for suspended MEMS structure. We will also review
basic optical techniques that applied to characterize the optical/mechanical properties of atomically
thin crystals. Experimental detection of static displacement and dynamic resonance for monolayer
resonators will be discussed in detail individually.
2.1 Device fabrication
2.1.1 Sample preparation
Due to the nature of layered materials, external perturbation can easily break down the weakly-
bound van der Waals interactions among layers [128], [129]. Starting from this simple idea,
intensive efforts were employed to obtain atomically thin flakes, or ultimately monolayers till the
first successful isolation and identification of graphene by Andre Geim and Kostya Novoselov [4].
In this groundbreaking work, single-layer graphene was exfoliated using a surprisingly easy method:
repeatedly peeling flakes off bulk crystal by scotch tape [4]. Soon the magic of this ‘scotch tape’
method was applied to a wider range of layered materials, including the family of transition metal
dichalcogenides [6], [128], [129]. Nowadays, mechanical exfoliation still demonstrates its power
on efficient access to atomically thin crystals with high sample quality. And all the TMD
monolayers we studied in this dissertation are mechanically exfoliated from bulk crystals.
32
Here we will briefly illustrate how to prepare atomically thin samples from scratch. First, a
bulk crystal was placed onto the adhesive face of scotch tape served as ‘parent’ tape. And we took
a second scotch tape with similar length and width and peeled a piece of thin flake from the bulk
crystal. This scotch tape with thin flake is ‘daughter’ tape and will be used for further exfoliation.
Then we repeated this peel-exfoliate process between two ‘daughter’ tapes for several times till
exfoliated thin flakes showed desired optical contrast. During this tape preparation, we avoided the
overlapping of thin flakes and tried to cover most of tape with thin-down flakes. Then tapes with
considerable thin flakes were gradually placed onto targeted substrates with some titled angle and
gently massaged with a cotton tip for several minutes. This step is aimed to avoid bubbles forming
between tape and substrate. One should also be careful about the mechanical strength for massage
process. Some external force is needed to form good contact between flakes and the substrate. This
will strengthen the van der Waals force between substrate to pull flakes down from bulk on the
scotch tape. Meanwhile, thin flakes can easily be broken into messy clusters under unnecessarily
strong shear force. After tape massage, we gentle lifted the tape with small tiled angle and left
noticeable thin flakes onto the targeted substrate for further search and identification. Note that one
may want to adjust the type of scotch tapes for different layered materials onto different targeted
substrates. To increase the yield when exfoliating thin flakes, we usually chose white ‘magic tape’
for silicon substrate with oxide layer coating atop whereas blue Nitto tape was used for sample
exfoliation onto flexible substrate include PDMS films. In practice, we noticed that the yield of
obtaining thin flakes significantly enhanced using silicon substrates with oxygen plasma treatment
beforehand. This is a key step to harvest high sample quality monolayers/few-layers with relatively
larger size, up to ~ 100 𝜇m.
To identify exfoliated monolayers, optical contrast varies for materials on different
substrates. Both graphene and TMDs show noticeable optical contrast on 280 nm oxide-coated
silicon substrate [130]–[132], whereas monolayer hBN is almost unidentifiable under such
33
condition [133]. This is caused by interference effect between monolayer material and silicon oxide
layer. Silicon substrate with different oxide thickness can possibly be applied to search of extra thin
hBN flakes. Another commonly-used substrate for exfoliation is polymer PDMS film. The
‘transparency’ of PDMS challenged the search of monolayer graphene on it. Unlike graphene,
monolayer TMDs especially for TMD semiconductors show enough ‘visibility’ on PDMS film due
to strong light-matter interaction. Also, adhesion of PDMS film is controlled by temperature and
beneficial for exfoliating larger monolayer TMD crystals. Thus, one can directly identify and
transfer desired monolayer TMDs onto targeted chips all by one PDMS film [134]. This efficient
transfer method avoids extra lithography or solvent exposure and was mainly used for fabricating
purely monolayer and suspended monolayer devices, which will be discussed in section 2.1.3. In
the section 2.2, I will discuss in details about how to reliably identify monolayers with not only
optical contrast but also other optical spectroscopy methods.
2.1.2 Van der Waals assembly
In this section, we will illustrate a facile and efficient method to fabricate 2D material-based
FET with high sample quality. The basic idea of this fabrication method is as following: after
Figure 2-1: Optical images of monolayer TMDs exfoliated on (a) silicon substrate with 285 nm
oxide layer atop and (b) PDMS substrate doubly-capsulated with thin hBN flakes.
34
identifying monolayers ideal for further study, we utilized a polymer stamp on a glass slide to pick
up and stack these monolayers in a desirable sequence. Once finished stacking layer by layer, the
sandwich-like van der Waals heterostructure was released onto the preprinted substrates and thus
the optoelectronic or optomechanical devices based on van der Waals materials were fabricated.
Thus, as well as probing the materials’ intrinsic properties, the heterostructure stacked by this ‘dry-
transfer’ method also allows us to design and engineer artificial material structure with favorable
sequence or stacking order for novel phenomena and application. This polymer-assisted technique
was first exploited in fabricating ultra-clean graphene field-effect devices [135]. Within the
invention of this intelligent fabrication method, a big band gap (namely insulator) layered materials
– hexagonal boron nitride (hBN) was introduced [135]. This graphene cousin later becomes an
important building block to fabricate 2D heterostructure devices with high quality.
Now we briefly review this ‘dry-transfer’ method firstly introduced in Ref.[136]. First, a piece
of sticky polymer film made from polypropylene carbonate (PPC) was put onto another soft
polymer polydimethylsiloxane (PDMS) with greater thickness to form a stamp. And this polymer
stamp can generate greater adhesion with exfoliated hBN flakes and peel them up from oxide-
coated silicon substrates by controlling the temperature of substrates. Due to the nature of van der
Waals materials, the picked hBN flake can form stronger bound with the subsequent atomically
thin flakes and thus easily pick up these flakes from silicon substrates, which is usually challenged
by directly using PPC film include graphene. Moreover, the hBN-encapsulated graphene was
protected from ‘dirty’ environment and demonstrated high-quality performance in transport
measurement due to atomically flatness and clean interface provided from hBN. In some sense, this
‘dry-transfer’ technique, or more professional terminology as ‘van der Waals assembly’, is a
cornerstone that paves paths towards numerous possibilities of observing interesting physics only
emerged with high sample quality.
35
Inspired by Ref [137], we developed this van der Waals assembly with several modification
for our research interests. First, we replace the PPC film with another polymer with greater
stickiness and higher melting point, named as polycarbonate (PC). The explicit consequence is that
new stamp structure can directly pick up almost all the layered materials studied in our research
group, which greatly enrich the diversities of devices we can fabricate. Second, we add one more
PPC layer between PC film and PDMS substrate as a soft buffer layer, which prevents the falling
or stretching of PC film from PDMS. This PPC film can naturally form a curvature and thus allow
the smooth occurrence of layer stacking with finite tilted angles instead of putting two layers
together instantaneously that caused the formation of bubbles.
Figure 2-2: Process flow for heterostructure fabrication using PC as adhesive polymer. This
process flow can be used to fabricate heterostructure FET or NEMS device. The final
heterostructure maintains the stacking order. Step 1: Attach a PC/PPC/PDMS stamp on the glass
slide and place it on micro-manipulator of transfer stage; Step 2: First layer pick-up; Step 3: Unload
the bottom substrate and load the other exfoliated substrate; Step 4: Second layer pick-up, Step 5:
Place a new prepatterned substrate with electrodes for final transfer; Step 6: The new prepatterned
substrate makes a contact with heterostructure and PC stamp at 150 C; Step 7: PC and
heterostructure detached from handle substrate and transferred to the new prepatterned substrate.
Figure is obtained from Ref.[137].
36
2.1.3 Suspend atomically thin membrane
In this dissertation, most of interesting phenomena originated from suspended TMD
monolayers. Conventional fabrication of a suspended 2D device inevitably exposes the material to
lithography solvents and aggressive acid liquids followed by a delicate critical point drying. Such
low-yielded process usually brings additional contamination and seriously degrades the sample
quality especially for TMD materials. Thus, a lithography-free method is undoubtedly critical to
fabricate a suspended TMD device that sustains high sample quality. Ideally, it would be highly
favorable to transfer the monolayers onto the prepattern trench directly from tapes or substrates that
used for exfoliation. In principle, the high flexibility and large breaking strength of TMD
monolayers make this one-step fabrication possible in practice, which can sufficiently limit extra
physical damage or chemical contamination. Consequentially, two different routines to acquire
suspended TMD devices were conducted in our research. Together with pre-deposited metal
electrodes in substrates, both two fabrication methods required the prepattern trenches etched by
reactive ion etching (RIE). The first strategy we tried to suspend TMD monolayers is to directly
exfoliate TMD monolayers onto silicon substrates prepatterned with dense trench arrays. As normal
exfoliation on silicon substrates, there would be some chance to suspend monolayers over a trench
during this exfoliation process. Obviously, this method can produce high-quality suspended TMD
samples, but its low yield greatly jeopardizes the efficiency of device fabrication. In practice, we
found that the polymer PDMS was a high-yielded exfoliation ‘tape’ to obtain TMD monolayers in
large size (usually > 30 μ𝑚). In addition, we come up with the second fabrication method that
monolayers were exfoliated and transferred onto the target trench all by one piece of PDMS. And
this PDMS deterministic transfer method [134] had been exploited to fabricate free-standing
devices. The working mechanism behind this technique depends on the viscoelasticity of PDMS.
Namely, PDMS behaves like an elastic solid within a short period while it moves like a liquid at
37
much longer timescales [134]. The monolayer flakes were easily ‘peeled’ from its bulk due to the
intimate contact can be formed between PDMS and the flakes. By controlling the temperature
during this transfer, the contact between PDMS and flakes can be weakened. This can explicitly
lead to the detachment of flakes from PDMS, which adhere favorably to the silicon surface. Due to
the absence of capillary force within this PDMS-based transfer, the transfer of monolayer flakes
over a trench is straightforward with surprisingly high yield (> 90%). Note that partial cover of the
trench would likely give rise to tear or damage of flakes. By peeling off very slowly, the suspended
region of monolayers can completely sustain without noticeable wrinkles or mechanical cracks. All
the suspended monolayer samples were fabricated through this PDMS deterministic transfer
technique.
2.2 Optical spectroscopy of monolayer semiconductors
Due to the intrinsically strong excitonic effects, optical spectroscopy turns out to be an
efficient and powerful experimental technique to unveil the properties of monolayer TMD
Figure 2-3: PDMS deterministic dry transfer method for 2D NEMS.
38
semiconductors. Meanwhile, the interesting light-valley interactions in such atomically thin
materials endows versatile knobs (e.g. polarization of light) to resolve and control the internal DOF
(e.g. spin and valley) of these monolayers with optical excitation. In this section, we will mainly
review the photoluminescence spectroscopy and optical reflection spectroscopy as two routine
spectroscopy methods employed in the studies within this dissertation. In addition, the basic
principle of magnetic circular dichroism spectroscopy will be briefly introduced. And we also will
discuss how to transduce the mechanical performance of suspended membranes into optical signals
using interferometry techniques.
2.2.1 Photoluminescence spectroscopy
Photoluminescence (PL) is a light emission process that occurs after absorbing photons in
materials. In general, electrons within a material can be excited to permissible excited states by
gaining excess energy from incident photons and the vacuums (or holes) are correspondingly left
by the excited electrons in valence band. Then the excited electrons will rapidly relax to the
conduction band edge and stay there for a certain time duration (~ femtosecond to picosecond). It
will be energetically favored for the excited electrons relaxing back to the valence bands. As
consequence, the recombined electron-hole pairs can emit photons (PL) that are of lower energy
than those are absorbed. This is a radiative recombination process, which, at the same time,
competes with other possible non-radiative recombination process include Auger process and
defect trapping. Note that electronic transition that occurs at indirect band gaps needs the assistance
of phonons due to the conservation of momentum, which can significantly reduce the quantum
efficiency of PL. Thus, the PL features in monolayer TMD semiconductors are of order of
magnitudes higher than those in bilayer or few-layered samples, which provides a fast and reliable
way to determine monolayer TMD flakes.
39
The experimental setup of PL measurements is shown in Figure 2-4. A monochrome
excitation beam was guided through a beam splitter and focally projected onto the monolayer
samples with ~ 1𝜇m2 spot size using an optical objective. The emitted PL signals was collected
with the same objective and reflected by the beam splitter for spectrum detection. The
corresponding PLD spectrum was measured by a dispersive spectrometer with liquid nitrogen-
chilled charge-coupled device (CCD). Usually, to optimize the signal-to-noise ratio (SNR) in PL
measurements, a long pass filter was employed to filter out the excitation photons for delicate
spectrum features and a spatial filter (pin hole) was used to eliminate the scattered light from
substrates and only allow the signals from sample pass through.
2.2.2 Optical reflection spectroscopy
Optical absorption spectroscopy can unveil rich physical properties in materials include band
gap, excitonic resonance and spin texture of band structure etc. The ideal way of obtaining
Figure 2-4: Schematics of microscopic confocal system in reflection geometry.
40
absorption of a material is to perform optical reflection and transmission measurements to make
use of the following relation as
A(𝜔) = 1 − R(𝜔) − T(𝜔) (2-1)
As a complimentary technique to PL spectroscopy, optical absorption spectroscopy is
employed to directly probe the band-to-band transition of materials from photon absorption in
contrast to detection of light emission in PL spectroscopy. Meanwhile, the oscillation strength of
electronic band-to-band transition is much stronger than the counterpart of band-to-defect transition.
As consequence, optical absorption exhibits clear features of transition between band edges with
least influence from defect states. In thin film limit, the sample absorbance can be directly
calculated from the reflection/transmission contrast [56], [138]:
A(𝜔) =𝑛𝑠2−1
4
𝑅−𝑅0
𝑅0=𝑛𝑠+1
2
𝑇−𝑇0
𝑇0 (2-2)
which indicates the more evident contrast from reflection measurements than that in transmission.
However, one cannot directly extract the sample absorbance from its reflection contrast spectrum
on SiO2/Si substrate due to the distortion from multilayer reflection. Instead, we practically obtain
the sample absorbance from Krames-Kronig constrained variable fitting as the case shown below.
For suspended atomically thin TMD materials studied in this dissertation, we can obtain
the corresponding optical reflectance 𝑅/𝑅0 which is characterizing by the ratio of the reflected
power from the suspended sample 𝑅 to that from the bare trench 𝑅0 under normal incidence. These
two quantities can be expressed through the parameters of the TMD monolayers/vacuum/Si
multilayer structure and the optical properties of its constitutes as [50], [139]
𝑅(𝜔) = |−𝐶−𝐷[1−��(𝜔)]
𝐶+𝐷[1+��(𝜔)]|2
, 𝑅0(𝜔) = |−𝐶−𝐷
𝐶+𝐷|2. (2-3)
Here we have assumed the incident power to be unit. The coefficients C and D are related to the
refractive index 𝑛𝜁 , thickness 𝑑𝜁 , and phase shift 𝛿𝜁 = 𝑛𝜁𝜔𝑑𝜁/𝑐 in the vacuum (휁 = 𝑣) and Si
(휁 = 𝑆𝑖) layers [50]:
41
(𝐶𝐷) = (
𝑠𝑖𝑛𝛿𝑣 𝑖𝑛𝑆𝑖𝑐𝑜𝑠𝛿𝑣𝑖𝑐𝑜𝑠𝛿𝑣 𝑛𝑆𝑖𝑠𝑖𝑛𝛿𝑣
) (𝑠𝑖𝑛𝛿𝑆𝑖 + 𝑖𝑛𝑆𝑖𝑐𝑜𝑠𝛿𝑆𝑖𝑖𝑐𝑜𝑠𝛿𝑆𝑖 + 𝑛𝑆𝑖𝑠𝑖𝑛𝛿𝑆𝑖
). (2-4)
And the complex absorbance ��(𝜔) of monolayer TMD semiconductor is modeled by a series of
Lorentzian arisen from different exciton resonances [50](A, B exciton resonance or charged exciton)
��(𝜔) = ∑𝐴𝑗
1−𝑖ℏ(𝜔−𝜔𝑗)/𝛾𝑗𝑗 . (2-5)
Here 𝐴𝑗 is the area of the jth Lorentzian, 𝜔𝑗 is the jth resonance frequency, and 𝛾𝑗 is the jth total
exciton linewidth.
The setup of optical reflection spectroscopy is similar to the one introduced in the PL
measurement. Here we use a broadband beam (supercontinuum laser or warm color lamp) as the
excitation source instead of a monochromic laser. And the dispersive CCD spectrometer we used
is reliable to measure the reflection spectrum from 400 to 850 nm with 0.1nm resolution.
2.2.3 Magnetic circular dichroism (MCD) spectroscopy
Magnetic circular dichroism (MCD) stems from the different absorption of materials between
left- and right-handed circularly polarized light at the presence of a magnetic field perpendicular to
the sample surface. The intensity of MCD signals is defined as ρ𝑀𝐶𝐷 =𝐴𝐿−𝐴𝑅
𝐴𝐿+𝐴𝑅, where 𝐴𝐿(𝑅) denotes
the sample’s absorbance of left (right) handed circularly polarized light with L(R).
To enhance the signal to noise ratio (SNR), the optical excitation was modulated
between left and right circular polarization by a photo-elastic modulator at certain modulating
frequency. The reflected beam was collected by the same objective and detected by a photodiode.
The MCD was determined as the ratio of the AC component at modulating frequency (measured
by a lock-in amplifier) and the DC component (measured by a multimeter) of the reflected light
intensity.
42
2.2.4 Characterization of mechanics in 2D NEMS
The optical spectroscopy methods illustrated above are employed to characterize the
physical properties of a material by measuring the corresponding spectrum. For the suspended
TMD monolayers, we not only care about the intrinsic properties of the atomically thin materials
but also aim to investigate the mechanical motions of these free-standing membranes. Hence, the
interferometry technique was introduced to calibrate and measure both static displacement and
dynamical vibration of suspended monolayers [65]. In 2D monolayer-based NEMS device, both
suspended membrane and back trench surface can reflect the incident laser beam and the two
reflected beams can form interference pattern with each other. As the membrane position change,
the total intensity from two reflected beams also change due to the modulated interference condition.
Hence, the mechanical motion of suspended membranes can be transduced to optical response
through this interferometry technique.
The suspended TMD monolayers can be charged and actuated vertically by applying a gate
voltage due to capacitive nature of the resonator. Meanwhile, the reflectivity R of suspended
monolayers is dependent on the vertical displacement Δz towards the silicon back surface. Thus,
one can extract the vertical displacement of membrane by measuring the gate-dependent reflectance.
Modeling the gate dependence of 𝑅/𝑅0 at a given wavelength can unveil the gate dependence of
Δz, the separation of the center of the suspended membrane from the underneath silicon substrate.
By balancing the electrostatic force 𝐹𝑔 ≈𝜀0𝜋𝑅
2𝑉𝑔2
2𝑑𝑣2 due to gating with the force due to tension, i.e.
4𝜋𝜎Δ𝑧 − 𝐹𝑔 = 0, (2-6)
the vertical displacement Δz of the center of the membrane as a function of gate voltage 𝑉𝑔 can be
calculated. Here 휀0 is the vacuum permittivity, 𝑅 is the radius of the suspended membrane, and
𝜎 = 𝜎0 + 𝜎′is the total tension in the membrane with 𝜎0 and 𝜎′ being the initial built-in tension and
43
the gate-induced tension, respectively. Furthermore, 𝜎′ can be written in terms of the material’s 2D
Young’s modulus 𝐸𝑌, the Poisson ratio 𝜈, membrane thickness 𝑡 and the strain 𝜖 as 𝜎′ =𝐸𝑌𝑡
1−𝜈2𝜖.
The strain is related to the vertical displacement Δz as 𝜖 =2
3(Δ𝑧
𝑅)2
. The corresponding gate
dependence of the vertical displacement Δz and the strain 𝜖 is numerically shown in Chapter 4.
Furthermore, the exciton peak position as a function of the gate voltage 𝑉𝑔 due to strain can also be
fitted by the model by using a peak shift rate of 0.1 eV/% strain (see Figure 4-2(a)), which agrees
very well with reported values [60].
Thus, we monitor mechanical resonance of our 2D NEMS devices by periodically driving the
resonator capacitively and measure the corresponding reflectance response in terms of driving
frequency. The driven vibration of monolayer membrane reaches the maximum as the AC driving
frequency resides at the natural frequency of the resonator. This on-resonance driven vibration
Figure 2-5: Interferometry technique to characterize mechanics of 2D NEMS. (a) Schematic of
reflected beams from both suspended membrane and silicon surface. (b) The normalized reflectance
as a function of displacement of membrane. The blue arrows indicate the transduction between
displacement of membrane and reflectance. The oscillation of membrane can periodically modulate
the sample reflectance (brown dash lines). Inset: gate dependence of normalized reflectance of
suspended sample with 450 nm probe wavelength. The red solid line is prediction with Eqns. (2-3)
to (2-6).
44
leads to a Lorentzian-shaped reflectivity change of the suspended MoSe2 at natural frequency. To
make the optical response of reflectivity change more evident, a DC gate voltage is needed to break
the out-of-plane mirror symmetry of mechanical motion. This is also consistent with the derivation
of the complex amplitude in mechanical vibration shown in Eqn. (1-17).
Chapter 3
Valley selective exciton bistability in suspended monolayer WSe2
During the past few decades people have witnessed at least two major innovations in
science which have had substantial impact on technology as well as science itself, even our daily
lives. We refer, of course, these two to be transistor and laser. Transistor is a semiconductor device
that used to amplify or switch electrical signals by employing electrons in an effective two-
dimensional region that residing at the interface between the silicon surface and its oxide layer.
Some charges, or current flow, can inevitably convert to heat dissipation during the operation. As
transistors are getting smaller and denser in logic circuit nowadays, the accumulated heat within
nanoscale can cause lots of troubles and degrade the device performance.
Alternatively, a transistor based on photons [140], [141] is proposed to solve this problem.
In principle, it can realize the switching times at the speed of light, and lower the power
consumption during the signal transmission and has promising potential to greatly enhance the
efficiency in current optical communication systems [142]. The most essential building block could
be implemented optical transistor is the concept of optical bistability [52]. In this chapter, we will
demonstrate the realization of optical bistability with strong excitonic effect in monolayer materials,
as well as the repeatable switching of light not only by light intensity also purely by its helicity
using monolayer TMDs semiconductors.
This chapter is based on published work in which I am the first author [50]. This chapter
is reprinted with permission from H. Xie, S. Jiang, J. Shan, K. F. Mak, Nano. Lett. 2018 (5), 3213-
3220 (https://pubs.acs.org/doi/abs/10.1021/acs.nanolett.8b00987).
46
3.1 Optical bistability
Optical bistability, the phenomenon of two well-discriminated stable states depending
upon the history of the optical input, is the key concept for all-optical transistors, switches, logical
gates, and memory [52]. Namely, we can control the light with the intensity of light through the
optical transistor device. So far, optical bistability had been realized in photonic crystals or optical
cavity systems with nonlinear medium, which requires the delicate device configuration and well-
selective nonlinear materials [52]. Monolayer TMD semiconductors, with extremely strong
excitonic effect [44], [139], [143]–[147], provide an ideal playground to study the optical
nonlinearities stem from light-matter interaction. The emerging valley degree of freedom (DOF)
[32] and its coupling to light helicity [35]–[37], [148], [149] and magnetic field [38]–[43], gives us
extra knobs to play with such nonlinearity.
Figure 3-1: Schematic of optical bistability. 𝐼𝑓 and 𝐼𝑏 are the critical intensities for transmittance
bistability under forward and backward intensity sweeps, respectively. Figure is obtained from
Ref.[150].
47
3.2 Experimental setup and basic characterization of suspended WSe2 monolayer
Suspended monolayer WSe2 was fabricated by transferring monolayer WSe2 flakes onto
trenched SiO2/Si substrates. Figure 3-2(b) shows the optical image and schematic side view of a
representative monolayer sample suspended over a circular trench of 8 μm in diameter and 600 nm
in depth. Suspended WSe2 samples were measured in an optical cryostat as shown in Figure 3-2(a).
Light was coupled into and out of it using a high numerical aperture microscope objective under
normal incidence. The beam size on the sample is typically about 1 μm. For reflectance spectrum
measurements, broadband radiation from a super-continuum light source and a spectrometer
equipped with a liquid nitrogen cooled charge coupled device (CCD) were used. The power on the
sample was limited to < 1 μW to avoid any nonlinear effects. For the cw excitation measurement,
a tunable cw Ti-sapphire laser and a Si avalanche photodiode were employed. The laser power on
the sample was controlled by rotating a half waveplate sandwiched between two linear polarizers.
It was limited to 250 μW to avoid sample damage. In the pump-probe measurement, the reflected
pump beam was filtered by a crossed linear polarizer. In the helicity-resolved measurement, a liquid
crystal phase plate was used to change the incident light polarization from linear to circular. For
the MCD measurement and the reflectance switching experiment, the light helicity was modulated
by electrically modulating the liquid crystal phase plate at several Hz’s and the reflected light was
detected by an avalanche photodiode. The MCD signal was recorded with a lock-in amplifier and
the time trace of the reflected light was recorded with an oscilloscope.
48
Figure 3-3(a) is the reflectance spectrum of a suspended WSe2 monolayer at temperatures
ranging from 4 K to 330 K. The reflectance 𝑅/𝑅0 at any given wavelength was determined as the
ratio of the reflected power from the suspended sample 𝑅 to the reflected power from the bare
trench 𝑅0 under normal incidence. It exhibits a sharp exciton resonance with over 80% reflectance
contrast at low temperatures (Figure 3-3(a)). The spectra can be simulated by considering light
propagation in the WSe2/vacuum/Si multilayer structure with the WSe2 optical response described
by a single Lorenzian that arises from the exciton resonance (Figure 3-3(b)). The small discrepancy
around 720 nm at 4 K is due to the contribution of the charged exciton to the optical response of
monolayer WSe2 [149], [151], [152], which has been neglected in the simulation for simplicity.
The exciton peak wavelength and linewidth employed in the simulation are shown in Figure 3-3(c)
as a function of temperature. In the low temperature limit, the exciton is peaked at 709 nm (1.75
eV) with a linewidth of ~ 7 meV. With increasing temperature, the exciton resonance redshifts and
broadens. This behavior is well described by the exciton-phonon interaction (solid lines) with an
Figure 3-2: Experimental setup and suspended WSe2 device. (a) Sketch of experimental setup. The
suspended samples are placed in an optical cryostat. Optical excitation, either broadband radiation
or monochromatic beam, is illuminated onto sample. For single-colored source, both the incident
and reflected laser light are monitored by a pair of photodiodes. The reflected broadband spectrum
from the sample is recorded by a dispersive spectrometer. (b) Schematic side view of a suspended
monolayer WSe2 sample over a drumhead trench of 8 μm in diameter and 600 nm in depth on a Si
substrate. The substrate has a 600-nm oxide layer and pre-patterned Ti/Au electrodes. The inset is
an optical image of a sample illuminated by white light. Gold bars are electrodes and the dark ring
is the edge of the trench.
49
average phonon energy of ~ 16 meV and LO-phonon energy of ~ 30 meV [153], [154]. These
values are in good agreement with the values reported for WSe2 [153].
Figure 3-3: Temperature dependence of exciton resonance in suspended WSe2 monolayer. (a)
Reflectance spectrum measured at 4 - 330 K. A sharp exciton resonance is observed at low
temperatures. The resonance redshifts and broadens with increasing temperature. (b) Simulated
reflectance spectrum at 4 - 330 K. (c) Exciton peak wavelength and linewidth (symbols) as a
function of temperature used in the simulation of (b). Solid lines are fits to Eqns. (3-2) and (3-3).
50
3.3 Observation of optical bistability in suspended monolayer WSe2
In this section, we present the systematical study of the optical bistability through both
monochromatic and broadband probe under optical pump near exciton resonance of suspended
WSe2 monolayers.
3.3.1 Power dependence of monochromatic reflectance spectra across exciton resonance
We investigate the optical nonlinearity in monolayer WSe2 near the exciton resonance at 4
K. Figure 3-4 shows the power dependence of the reflectance for cw excitation tuned through the
exciton resonance in sample #1. The beam size on the sample is typically about 1 μm. The results
are similar up to ~ 100 K, for which thermal broadening is insignificant. They are also reproducible
in all samples studied here except small variations in the exciton resonance wavelengths and
linewidths. Results at selected wavelengths, marked as vertical dashed lines on the experimental
reflectance spectrum (Fig. 3-4(a)) and on the corresponding simulated absorbance spectrum (Fig.
3-4(b)), are shown in Fig. 3-4(c). At 708 nm (blue-detuned from the exciton resonance), the
reflectance increases with increasing power. Onset of nonlinearity occurs at a power level as low
as 1 μW. At all other wavelengths (red-detuned from the exciton resonance), the reflectance
exhibits jumps at critical powers and a hysteretic behavior with power sweeps. These are signatures
of optical bistability. Moreover, both the critical powers and the size of the bistability region
increase for excitation at longer wavelengths. We note that several μW is sufficient to generate
optical bistability in suspended monolayers here, in contrast to 10 mW in monolayer TMDs
encapsulated in hBN under similar experimental conditions [48].
51
Figure 3-4: Optical bistability in suspended monolayer WSe2. (a) Reflectance spectrum of sample
#1 at 4 K. (b) Absorbance spectrum described by a single Lorentzian that best fits the reflectance
spectrum of (a). (c, d) Reflectance at representative wavelengths (708, 712, 720, 725 and 728 nm)
under forward and backward power sweeps from experiment (c) and simulation (d). The
wavelengths are marked by vertical dashed lines in (a, b).
70 0 71 0 72 0 73 00.0
0.6
1.2
0.0
0.6
1.2
0.0
0.6
1.2
0.0
0.6
1.2
0.0
0.6
1.2
0 100 200 300 4000.0
0.6
1.2
Re
flec
tance
Wavelength (nm)
(a)
(c)
708 nm
Reflecta
nce
712 nm
720 nm
725 nm
Power (mW)
728 nm
exp
70 0 71 0 72 0 73 00.0
0.5
1.0
0.0
0.6
1.2
0.0
0.6
1.2
0.0
0.6
1.2
0.0
0.6
1.2
0 100 200 300 4000.0
0.6
1.2
Wavelength (nm)
Abs
orb
ance
(b)
(d)
708 nm
712 nm
Reflecta
nce
720 nm
725 nm
Power (mW)
728 nm
sim
52
3.3.2 Wavelength dependence of broadband reflectance spectra across exciton resonance
To understand the origin of the observed optical bistability, we monitor the reflectance
spectrum with a low-power super-continuum probe of suspended samples that are pumped by cw
excitation at a fixed power of 100 μW. Figure 3-5(a) and (b) are the contour plots of the reflectance
as a function of the pump (left axis) and probe (bottom axis) wavelengths of sample #2. Figure 3-
5(a) corresponds to a forward sweep of the pump wavelength (i.e. increasing wavelength), and Fig.
3-5(b), a backward sweep (i.e. decreasing wavelength). The bright features along the diagonal
dashed lines are the residual of the pump intensity. For the forward sweep, a clear exciton resonance
is initially observed at 715 nm with a width of 10 meV. As the pump wavelength approaches the
exciton resonance, a significant redshift of the exciton by over 20 nm accompanied with a
broadening by ~ 100 meV is observed. A sudden “recovery” of the exciton resonance follows when
the pump wavelength increases beyond a critical value of ~ 745 nm. In contrast, for the backward
sweep of the pump wavelength, the exciton resonance suddenly redshifts to ~ 738 nm at a lower
critical pump wavelength of ~ 735 nm. The exciton gradually returns to the original position as the
pump wavelength further decreases. We summarize the pump wavelength dependence of the
exciton peak and linewidth in Fig. 3-5(c) and (d), respectively. These values were extracted from
the analysis of the reflectance spectra (Methods). Clear hysteresis in both the peak wavelength and
the linewidth can be observed. Figure 3-5(a) and (b) also show that reflectance at probe wavelengths
off the exciton resonance (or far above the resonance at 450 nm, not shown) remains independent
of pump. Since the reflectance from the multilayer structure is very sensitive to the vacuum cavity
length due to optical interference, our result indicates a negligible pump-induced vertical movement
of the suspended membrane.
53
The significant pump-induced exciton redshift and broadening near the bistability region
suggests the importance of a photo-thermal mechanism. We estimate that a temperature rise of ~
300 K is required to explain the observed exciton resonance shift under near-resonance (740 nm)
excitation at 100 μW. Such a large temperature rise, as we discuss below, is plausible due to the
low thermal conductance of suspended samples. As the sample temperature rises under optical
pumping in the photo-thermal effect, the exciton resonance “runs” away from the pump wavelength
for a blue-detuned pump, which in turn decreases the sample absorbance at the pump wavelength
and reduces the temperature rise. No cumulative effect can build up. In contrast, the exciton
resonance “runs” towards the pump wavelength for a red-detuned pump, which increases the
sample absorbance at the pump wavelength and, in turn, causes a further increase in the sample
temperature and more redshift towards the pump wavelength. This process is cumulative, i.e. a
positive feedback, and can cause the frequency of exciton resonance overshooting that of optical
Figure 3-5: Bistable excitonic reflection spectra under monochromatic pump. (a, b) Contour plots
of reflectance of sample #2 at 4 K as a function of probe wavelength (bottom axis) and pump
wavelength (left axis). The pump wavelength is swept forward (i.e. increases) in (a) and backward
(i.e. decreases) in (b). The pump power is fixed at 100 μW. The diagonal dashed lines are the
residual pumps. (c, d) The extracted exciton peak wavelength (c) and linewidth (d) from the
experimental reflectance spectra. (e, f) Simulation of the parameters shown in (c, d) based on the
photo-thermal effect as described in the main text.
720
740
760
0.2150
1.600
(a)
700 720 740 760
720
740
760
(b)
Pu
mp
wa
vele
ng
th (
nm
)
Probe wavelength (nm)
720
740
760
700 720 740 7600
50
100
Re
so
nan
ce
wa
ve
len
gth
(nm
)
exp
(c)
(d)
Lin
ew
idth
(m
eV
)
Pump wavelength (nm)
720
740
760
700 720 740 7600
50
100
Re
so
nan
ce
wa
ve
len
gth
(nm
) (e)
(f)
Lin
ew
idth
(m
eV
)
Pump wavelength (nm)
sim
54
pump. This internal passive feedback, together with optical nonlinearity, leads to optical bistability
in monolayer WSe2 (Fig. 3-6(a)). For a red-detuned pump under a forward power sweep, the system
becomes unstable and switches suddenly from a low-temperature state to a high-temperature state
due to the rapid shift of the exciton resonance, especially when the pump power 𝑃 exceeds a critical
value 𝑃𝑓. Under a backward power sweep, the system starts from the high-temperature state, and
can remain there even below 𝑃𝑓. Only when the pump power decreases below a much lower critical
value 𝑃𝑏 , optical pumping can no longer keep the system in the high-temperature state, and it
switches suddenly to the low-temperature state. This is mainly due to the accumulated heat in
suspended WSe2 monolayers that own low thermal conductivity. As consequence, a hysteretic
bistable behavior occurs.
3.4 Modelling mechanism of the observed optical bistability
We would like to emphasis again that the occurrence of optical bistability mainly
attributing to the nonlinear optical response and internal positive feedback from external excitation.
The sharp exciton resonance and its passive feedback (redshift under illumination) endow WSe2
monolayers with feasibility to realize optical bistability near exciton resonance. Optical pump near
exciton resonance can redshift exciton resonance through either relaxing the existed strain or
heating up the sample. In Section 3.3.2, significant broadening of exciton line width under pump
near exciton resonance implies the overwhelming role of photothermal effect from laser light. This
is also verified by the comparison between temperature dependence of exciton resonance in Figure
3-3(c) and the pump-induced exciton redshift in Figure 3-5(c). In this section, we build up a
photothermal model to simulate the temperature profile of WSe2 membrane that optically pumped
near exciton resonance and predict the corresponding optical bistability.
55
3.4.1 Simulation of photothermal model
Now we turn to a more quantitative discussion of optical bistability from the photo-thermal
effect. The exciton reflection contrast of monolayer WSe2 can be modeled by Eqn. (S1) and (S2)
by a Lorentzian for the exciton resonance
��(𝜔) =𝐴0
1−2𝑖ℏ(𝜔−𝜔0)/𝛾. (3-1)
Here 𝐴0 is the area of the Lorentzian and 𝛾 is the total exciton linewidth. As shown in Fig. 3-3(b),
the reflectance spectrum at 4 K can be fitted very well by choosing 𝐴0 ≈ 0.85, ℏ𝜔0 ≈ 1.748 eV
and 𝛾 ≈ 7 meV.The reflection contrast at higher temperatures can also be simulated by taking into
account the temperature dependent exciton peak position 𝜔0(𝑇) and linewidth 𝛾(𝑇) [153], [154],
𝜔0(𝑇) = 𝜔0(0) − 𝑆⟨𝜔𝐷⟩[coth (⟨ℏ𝜔𝐷⟩
2𝑘𝐵𝑇) − 1], (3-2)
𝛾(𝑇) ≈ 𝛾(0) +𝛾0
𝑒𝑥𝑝(ℏ𝜔𝐷/𝑘𝐵𝑇)−1. (3-3)
Here ℏ𝜔0(0) ≈ 1.75 eV is the exciton resonance at zero temperature, ⟨ℏ𝜔𝐷⟩ ≈ 16 meV and
ℏ𝜔𝐷 ≈ 30 meV are the average phonon energy and LO-phonon energy of WSe2, respectively, 𝑆 ≈
2.25 is the coupling parameter, 𝑘𝐵 is the Boltzmann constant,𝛾(0) ≈ 7 meV is the linewidth at
zero temperature, and 𝛾0 ≈ 76 meV is the characteristic linewidth corresponding to exciton-
phonon scattering [153]. Eqns. (3-2) and (3-3) describe the experimental results in Fig. 3-3(c) very
well. In Fig. 3-3(b) we show the simulated exciton reflection contrast spectra at varying
temperatures. The results agree well with the experimental data. Now we demonstrate a thermal
mechanism to show how the incident laser induces change in optical response of suspended 2D
semiconductor. The temperature profile 𝑇(𝑟) of the membrane induced by central laser spot when
using the heat diffusion equation is as
𝜕𝑇
𝜕𝑡=
𝜅
𝜌𝑐𝑝∇2𝑇 (3-4)
56
where 𝜅 is the sheet thermal conductance of the sample, 𝜌 is the area mass density of the sample,
and 𝑐𝑝 is the specific heat capacity. In steady state, the above heat diffusion equation just
transforms as Laplace’s equation as
0 = ∇2𝑇 =1
𝑟
𝜕𝑇
𝜕𝑟+𝜕2𝑇
𝜕𝑟2 (3-5)
Note that the boundary condition for the above temperature profile is the temperature at the edge
of membrane 𝑇(𝑅) = 𝑇0 = 300 K. Another boundary condition comes from the heat flow 𝑞 across
the edge of laser spot 𝑟𝑠 as 𝜕𝑇
𝜕𝑟|𝑟=𝑟𝑠
= −𝑞
𝜅, where 𝑞 is in unit of W/m. And the equilibrium condition
of membrane can only be satisfied when the heat flow into the membrane is equal to the absorbed
power of incident laser as
𝐴𝐿𝑃 = 2π𝑟𝑠𝑞 (3-6)
where 𝐴 is the real part of complex absorbance �� in the suspended WSe2, 𝐿 is the local field factor
and with the laser power 𝑃. And solving the steady thermal equation with the boundary conditions
gives rise to the temperature profiled 𝑇′(𝑟) as
𝑇′(𝑟) = 𝑇0 +𝐴𝐿𝑃
2𝜋𝜅ln(
𝑅
𝑟) (3-7)
Thus, we can express the average sample temperature 𝑇 under optical excitation at wavelength 𝜆
as
𝑇(𝑃, 𝜆) =2𝜋
𝜋(𝑅2−𝑟𝑠2)∫ 𝑇′(𝑟)𝑟𝑑𝑟𝑅
𝑟𝑠= 𝑇0 +
𝐴𝐿𝑃
4𝜋𝜅(1 −
2𝑟𝑠2 ln
𝑅
𝑟𝑠
𝑅2−𝑟𝑠2 ) ≅ 𝑇0 +
𝐴(𝑇,𝜆)𝐿(𝑇,𝜆)𝑃
4𝜋𝜅(𝑇) (3-8)
where the average temperature of membrane is a good approximation with small spot size applied
to our case (𝑟𝑠 ~0.5 𝜇𝑚). Optical bistability requires the existence of more than one solution to 𝑇
for a given 𝑃 (Fig. 3-6(b)). Therefore, in addition to normal regions of monotonically increasing 𝑇
with 𝑃, there must exist a region of decreasing 𝑇 with 𝑃, i.e. 𝑑𝑇
𝑑𝑃≤ 0. Carrying out the derivatives
gives us the condition
57
𝑑𝐴
𝑑𝑇
𝑇−𝑇0
𝐴+𝑑𝐿
𝑑𝑇
𝑇−𝑇0
𝐿≥ 1 . (3-9)
Meanwhile, the first term in Eqn. (3-9) is significantly bigger than the second term because of the
weak temperature dependence of the local field factor. In fact, the local field factor can be well
approximated by 𝐿 = |1 −𝐶−𝐷[1−𝐴(𝑇,𝜆)]
𝐶+𝐷[1+𝐴(𝑇,𝜆)]|2 ≈ |1 −
𝐶−𝐷
𝐶+𝐷|2
. Ignoring the weak temperature
dependence of the local field factor (which is mainly determined by the substrate properties) and
the sample thermal conductance for simplicity, we obtain from Eqn. (3-9) a criterion for optical
bistability
𝑑𝐴
𝑑𝑇≳
𝐴
𝑇−𝑇0 . (3-10)
Since 𝑑𝐴
𝑑𝑇 is negative (positive) on the blue (red) side of the exciton resonance (Fig.3-3(a), (b)) and
𝐴
𝑇−𝑇0 is always positive, optical bistability can only occur for sufficiently red-detuned optical
pumping. This is fully consistent with our experimental observation and the above intuitive picture
for the phenomenon.
Figure 3-6: (a) Schematic of the photo-thermal effect. The exciton resonance redshifts under
optical excitation for both blue- and red-detuned optical excitation from the exciton resonance
(shown by blue and red vertical arrows, respectively). For the red-detuned optical excitation, the
exciton resonance runs towards the pump and the effect is cumulative. This internal passive
feedback together with nonlinearity can lead to optical bistability. (b) Simulated average sample
temperature as a function of excitation power at 708 nm (blue-detuned) and 728 nm (red-detuned).
𝑃𝑓 and 𝑃𝑏 are the critical powers for temperature instability under forward and backward power
sweeps, respectively. (c) Simulated average sample temperature as a function of wavelength at 1,
10 and 100 μW. 𝜆𝑓 and 𝜆𝑏 are the critical wavelengths for temperature instability under forward
and backward wavelength sweeps, respectively. The region between the critical powers
(wavelengths) is bistable.
700 735 7700
150
300
lb
Te
mp
era
ture
(K
)
Wavelength (nm)
1mW
10mW
100mW
lf
(c)
0 250 5000
150
300
Te
mp
era
ture
(K
)
Power (mW)
708 nm
728 nm
Pf
Pb
(b)
700 720 7400.0
0.5
1.0
Abso
rban
ce
Wavelength (nm)
(a)
58
3.4.2 Comparison to the observed bistable exciton resonance
In this section, we compare in details the predictions of the photo-thermal effect and the
experimental observations. To this end, we solve Eqn. (3-8) numerically for 𝑇(𝑃, 𝜆). For simplicity
we have used a constant thermal conductance (𝜅 ≈ 5 × 10−9 W/K). Figure 3-6(b) shows 𝑇(𝑃) for
two representative wavelengths. At 708 nm (blue-detuned from the exciton resonance), 𝑇 increases
monotonically with 𝑃. In contrast, at 728 nm (red-detuned from the exciton resonance), temperature
instability occurs at critical power 𝑃𝑓 and 𝑃𝑏 for forward and backward power sweeps, respectively.
The system is bistable between these two powers. Figure 3-6(c) is 𝑇(𝜆) for three representative
powers (1, 10 and 100 μW). Temperature instability is observed only at 10 and 100 μW. Again, the
system is bistable between the critical wavelength 𝜆𝑓 and 𝜆𝑏 , at which temperature instability
occurs for forward and backward wavelength sweeps, respectively. Moreover, the bistability region
shifts towards more red-detuned wavelengths for larger powers.
Now equipped with the sample temperature, we can simulate its reflectance using the
measured temperature dependent absorbance. Figure 3-4(d) is the simulated power dependence of
the sample reflectance. The simulation reproduces the overall trend of the experiment (Fig. 3-4(c))
for all wavelengths with critical powers within a factor of 2 – 3 of the measured values. Figure 3-
7(d) – (f) are the simulated wavelength dependence of the sample reflectance at a fixed power of 1,
10 and 100 μW, respectively. The agreement between simulation and experiment (Fig. 3-7(a) – (c))
is also very good including the critical wavelengths. We have also included in Fig. 3-5(e-f)
simulation for the pump-probe experiment (Fig. 3-5(c), (d)). Again, good agreement is seen for the
exciton characteristics as a function of pump wavelength. We note that there is only one free
parameter (the sample thermal conductance) in the simulation. The value of 5 × 10−9 W/K was
chosen to fit the entire set of the experimental observations. It is consistent with the measured [155],
[156] and calculated [157] thermal conductivity of WSe2. The low thermal conductance of
59
suspended WSe2 is primarily responsible for the low critical powers required for optical bistability
here. We conclude that the photo-thermal effect captures the main features of optical bistability in
suspended monolayer WSe2 despite the crude assumptions made in the simulation, including the
temperature-independent thermal conductance and spatially uniform heating.
3.5 Valley contrasting bi-exciton resonance and control of light by helicity
Finally, we demonstrate the unique valley-selective optical bistability and control of light by
its helicity in suspended monolayer WSe2. The idea relies on the valley-dependent optical response
of monolayer TMDs [23], [32]–[43], [148], [151], in which the two degenerate direct gaps at the K
and K’ valleys of the Brillouin zone couple exclusively to the left (𝜎+) and right (𝜎−) circularly
polarized light, respectively. Since the K and K’ valleys are time-reversal copies of each other, the
Figure 3-7: Wavelength dependence of the sample reflectance for forward (blue) and backward
(red) wavelength sweeps. (a, b, c) are experiment and (d, e, f) are simulation. The excitation power
is fixed at 1 μW (a, d), 10 μW (b, e) and 100 μW (c, f).
700 710 720 730
0.0
0.5
1.0
1.5
Reflecta
nce
Wavelength (nm)
1 mW
(d)
700 710 720 730
0.0
0.5
1.0
1.5
Reflecta
nce
Wavelength (nm)
10 mW
(e)
700 720 740 760
0.0
0.5
1.0
1.5
Reflecta
nce
Wavelength (nm)
100 mW
(f)
700 710 720 7300.0
0.5
1.0
1.5
Refle
cta
nce
Wavelength (nm)
1 mW
(a)
700 720 740 7600.0
0.5
1.0
1.5
Re
flecta
nce
Wavelength (nm)
100 mW
(c)
700 710 720 7300.0
0.5
1.0
1.5
Re
fle
cta
nce
Wavelength (nm)
10 mW
(b)
60
valley degeneracy can be lifted by applying an out-of-plane magnetic field (i.e. the exciton valley
Zeeman effect [38]–[43]). This is shown in Fig. 3-8(a) for the helicity-resolved reflectance
spectrum measured under a magnetic field of 8 T. The exciton resonance is split by ~ 1.8 meV, in
agreement with the reported magnitude of the exciton valley Zeeman effect [38]–[43]. We study in
Fig. 3-8(b) the optical nonlinearity for the 𝜎+ and 𝜎− excitation at a red-detuned wavelength (710
nm, marked by the dashed vertical line in Fig. 3-8(a)). Optical bistability is observed for both
circular polarizations with larger critical powers and a wider hysteresis loop for the 𝜎+ excitation.
This is consistent with the fact that 710 nm corresponds to a larger redshift from the exciton
resonance for the 𝜎+ excitation.
Moreover, there exists a finite range of power between the two bistability regions (marked by
two dotted vertical lines). This can be utilized for repeatable switching of the sample reflectance
by light helicity, as illustrated in the power dependence of the magnetic circular dichroism (MCD)
(Fig. 3-8(c)). This observation can be understood in terms of an energy barrier for optical switching.
For the hysteresis loop corresponding to left-handed optical pumping under 8 T, the lowest energy
state is the high-temperature (low-reflectance) state. During the forward scan, switching the
incident light handedness (e.g. by the modulator) from left to right within the hysteretic power
range changes the energy barrier for optical switching, and can cause a one-time only switch to the
high-temperature (low-reflectance) state. Subsequent switching of the light handedness back to left
merely changes the energy barrier, and cannot overcome the energy barrier to bring the state back
to the higher energy low-temperature (high-reflectance) state. As a result, the state is trapped to the
high-temperature (low-reflectance) state and no MCD signal (averaged over many modulation
cycles) can be observed. The same mechanism works for the hysteresis loop corresponding to right-
handed optical pumping, in which the lowest energy state is now the low-temperature (high-
reflectance) state. Rather than merely changing the energy barrier for optical switching, switching
the incident light handedness for the narrow power range in between the left- and right-handed
61
hysteresis loops (shaded) actually changes the relative energy minima of the system between the
low-temperature (high-reflectance) state and the high-temperature (low-reflectance) state. Similar
argument can be made for the backward power scan. This is consistent with the result of Fig. 3-8(c)
if we take into account the broadened hysteresis loops. Repeatable switching of the sample
reflectance by pure helicity modulation is therefore possible in this narrow power range.
Figure 3-8: Valley-dependent optical bistability and control of light by its helicity. (a) Helicity-
resolved reflectance of sample #3 at 4 K under an out-of-plane magnetic field of 8 T. The K and
K’ valley excitons are split by ~ 2 meV. The vertical dashed line corresponds to 710 nm. (b)
Helicity-resolved reflectance under power sweeps at 710 nm. There exists a finite power range
between two bistability regions (between the vertical dotted lines). (c) Magnetic circular dichroism
(MCD) under forward and backward power sweeps at 710 nm. (d) Helicity-controlled optical
‘switching’ in real time at 710 nm and 39 μW. The helicity of light is modulated at 1 Hz by a liquid
crystal modulator (upper panel) and the sample reflectance follows the helicity modulation with a
change of reflectance by ~ 30% (lower panel).
62
We thus choose a power between the bistability regions (39 μW) at 710 nm to demonstrate
repeatable light switching by pure helicity in Fig. 3-8(d). In the upper panel, the light helicity is
modulated by a liquid crystal modulator at 1 Hz. Repeatable switching of the sample reflectance
by ~ 30% is achieved (lower panel). (Note the delayed response in the reflectance when switching
from 𝜎+ to 𝜎− is due to the slow response of our liquid crystal modulator rather than limited by
the intrinsic response of the material. Using an electro-optic modulator, we have observed a
switching frequency up to 50 kHz, which again is limited by the speed of our light modulator.)
Although the unique valley-dependent optical bistability and helicity-controlled optical switching
can only be achieved under a finite magnetic field here, it is, in principle, also feasible to achieve
such a phenomenon under zero magnetic field utilizing the magnetic proximity effect. Recent
experimental studies have shown strong magnetic proximity coupling of monolayer TMDs to
magnetic insulators with a proximity exchange field exceeding 10 T [158], [159]! Our study has
thus opened up exciting opportunities in controlling light with light, including wavelength, power,
and helicity, using monolayer materials.
3.6 Conclusion
In this chapter, we have demonstrated robust optical bistability, the phenomenon of two
well-discriminated stable states depending upon the history of the optical output, in fully suspended
monolayers of WSe2 at low temperatures near the exciton resonance. Optical bistability has been
achieved under continuous-wave optical illumination that is red-detuned from the exciton
resonance at an intensity level of 103 W/cm2. The observed bistability is originated from a
photothermal mechanism, which provides both optical nonlinearity and passive positive feedback,
two essential elements for optical bistability. The low thermal conductance of suspended samples
is primarily responsible for the low excitation intensities required for optical bistability. At the
63
presence of external magnetic field, the exciton bistability becomes helicity dependent due to
exciton valley Zeeman effect, which enables repeatable switching of the sample reflectance by light
polarization (left- or right-handedness). Our study has opened up exciting opportunities in
controlling light with light, including its wavelength, power, and polarization, using monolayer
semiconductors.
Chapter 4
Tunable exciton-optomechanical coupling in suspended monolayer MoSe2
The extremely strong excitonic interaction in monolayer transition metal dichalcogenide
(TMD) semiconductors has enabled many fascinating light-matter interaction phenomena [23],
[33], [34], such as strongly coupled exciton-polaritons [160]–[162] and nearly perfect monolayer
mirrors [48], [49]. The strong light-matter interaction also opens up the possibility to dynamically
control the mechanical motion of a monolayer semiconductor through its exciton resonance [163].
Such exciton-optomechanical coupling in monolayer TMDs has not been studied so far. In principle,
the strong excitonic resonance in atomically thin semiconductor is reminiscent to the cavity
resonance of an optical cavity. Thus, we expect to observe the light-induced damping and anti-
damping of mechanical vibrations and modulation of the mechanical spring constant by moderate
optical pumping near the exciton resonance with variable detuning.
In this chapter, we discuss the first demonstration of exciton-optomechanical coupling in
monolayer MoSe2 resonators, as well as the its tunability using atomically thin semiconductor for
potential cavity-less optomechanics. The observation of gate-tunable exciton-optomechanical
coupling in a monolayer semiconductor may find novel applications in nanoelectromechanical
systems (NEMS) and in exciton-optomechanics.
This chapter is based on unpublished work in which I am the first author [164]. This chapter
is reprinted with permission from a manuscript in preparation to be published with the same title.
65
4.1 Exciton-optomechanical backaction through strain
Atomically thin monolayer materials have attracted tremendous interests in the studies of
NEMS devices [65], [66], [75]–[84], [67], [85], [86], [68]–[74] because of their extremely
lightweight, mechanical flexibility, gate-tuneability and strong inter-mode couplings [75], [81],
[84], [86]. They also hold great promise in many sensing applications [80], [87], [96]–[98], [88]–
[95]. Although the mechanical aspects of NEMS devices based on monolayer materials have been
extensively studied in the literature [60], [65], [66], [68]–[71], [75], [76], [78]–[80], [83], [85], the
dynamical coupling between mechanical vibrations and light (i.e. optomechanical coupling)
remains as an open subject. In particular, the presence of strong excitonic resonances in monolayer
TMD semiconductors [23], [33], [34], which strongly interact with light (can reflect ~ 90 % of the
incident light) [48]–[50], presents a very interesting avenue to explore optomechanical coupling
effects in atomically thin materials without the need of an optical cavity. Although such cavity-less
optomechanical coupling has been demonstrated in quantum well heterostructures [163], the unique
mechanical properties and the extremely strong excitonic effects in monolayer TMDs can enable
fundamental studies and applications into new regimes.
We study in this work gate-tunable exciton-optomechanical coupling in suspended
monolayer MoSe2 (Fig. 4-1). The silicon back gate can induce charge in the MoSe2 membrane and
pull it down, which breaks the mirror symmetry of the mechanical device. Mechanical vibration of
the membrane can then create an oscillating strain in the sample, which in turn produces a
dynamical spectral shift in the exciton resonance due to its linear dependence on strain [56], [165].
As a result, illumination of the sample with light near the exciton resonance can generate a
photothermal force with dynamical backaction, producing interesting exciton-optomechanical
coupling effects we will explore below. Note that exciton-phonon parametric coupling cannot
generate efficient optomechanical backaction acting onto our suspended 2D systems. Under such
66
mechanism, sufficient optomechanical feedback can only be achieved as the frequency magnitude
of mechanical modes is comparable to line width of exciton resonance. In our case, full line width
of exciton resonance (~ 3 meV ≈ 0.73 THz, see Figure 4-2(a)) is five order of magnitudes larger
than the vibration frequency of the suspended membrane (~ 40 MHz, see Figure 4-2(b)). Instead,
such an effect can be straightforwardly understood as exciton analogue of the standard
optomechanics. Similar to cavity resonance with position dependence, strain-tunable band gaps of
TMD semiconductors can periodically modulate exciton resonance by simply vibrating the
suspended membrane. Tensile strain causes redshift of exciton resonance [56], [60], [165] and thus
leads to asymmetrical photothermal deformation for optical excitation at opposite detuning of
exciton resonance. Additional strain is induced onto the sample by this photothermal deformation
potential. And the photon-induced strain takes time to accumulate (~ 10 ns for single-layer MoSe2
in cryogenic temperature) [105], which is determined by thermal properties of the suspended TMD
flakes. Thus, in principle, a backaction effect can build up via strain that mediates exciton resonance
with external optical excitation.
4.2 Experimental setup
Polydimethylsiloxane (PDMS) was employed to exfoliate monolayer MoSe2 flakes from
bulk crystals. After being identified and confirmed by optical contrast and photoluminescence, the
desired monolayers of MoSe2 were transferred onto the prepatterned trench substrates with metal
electrodes nearby. Circular drumhead trenches with diameter ~ 5 μm were structured adjacent to
electrodes and etched to 600-nm depth by photolithography and subsequent reactive ion etching
(RIE).
The suspended samples were measured under normal incidence with a microscope coupled
to a cryostat at 3.3K, of which filled with helium exchange gas ~19 Torr. For the reflectance
67
measurements, either a supercontinuum light source with broadband radiation or a tunable-
wavelength Ti-Sapphire laser were employed to illuminate the samples. Illumination was focused
onto samples with a 40× objective with long working distance and the reflected radiation was
collected with the same objective. With a supercontinuum light source, the reflection was detected
by a spectrometer equipped with a charge coupled device (CCD). For a tunable-wavelength Ti-
Sapphire laser, both incident and reflected beams were recorded by a pair of silicon Avalanche
photodetectors. The reflectance spectra were obtained by comparing the reflectance from the
suspended MoSe2 and bare trench regions.
The suspended MoSe2 can be charged and actuated vertically by applying a gate voltage
due to capacitive nature of the resonator [65], [66], [68], [85]. In addition, the reflectivity 𝑅 of the
suspended MoSe2 is dependent on the vertical displacement 𝑧 towards the silicon backplane [81].
Thus, we characterize the mechanical resonance of our MoSe2 device by driving the resonator
capacitively and monitoring the corresponding reflectivity response as a function of the driving
frequency with a fast photodetector. The driven vibration of MoSe2 membrane reaches the
maximum as the ac driving frequency resides at the natural frequency of the resonator. This on-
resonance driven vibration leads to a Lorentzian-shaped reflectivity change of the suspended
MoSe2 at natural frequency. To convincingly visualize optical response of reflectivity change for
measurement of mechanical resonance, a dc gate voltage is needed to break the out-of-plane mirror
symmetry of periodic oscillation.
To observe the optomechanical effect acting on the MoSe2 resonator, a tunable-wavelength
Ti-Sapphire laser was employed. The illumination beam near exciton resonance was served both
as a pump and probe for measuring mechanical response under different pumping conditions. The
pump power was continuously tuned by rotating the polarization angle of a half-waveplate
sandwiched between two linear polarizers in the incident beam path.
68
4.3 Observation of exciton-optomechanical coupling in monolayer MoSe2 resonator
In this section, we present the experimental findings that exhibit mutual interplay between
exciton resonance and mechanical resonance of monolayer MoSe2 resonators via strain. And such
exciton-optomechanical coupling in turn can modify mechanical modes of monolayer MoSe2
resonator without assistance of a cavity structure.
Figure 4-1: Schematic illustration of the device geometry and experimental setup. The mechanics
of suspended membrane is characterized with optical interferometry technique. And the incident
and reflected beams are recorded by a normal and fast photodetector, respectively. The static
displacement 𝑧 and dynamical oscillation 𝛿𝑧 of membrane can transduce to reflection signals,
which are measured with a multimeter and a network analyzer correspondingly. Meanwhile, the
vibration of monolayer MoSe2 membrane generates a dynamical strain 𝛿𝜖 that give rise to an
oscillating spectral shift of exciton resonance δ𝜔𝑋. In addition, the illumination of the sample with
light near the exciton resonance can generate a photothermal force 𝐹𝑝ℎ. As a result, the oscillation
of monolayer semiconductor (oscillating exciton spectral shift) can periodically modulate the
photothermal force as δ𝐹𝑝ℎ.
69
4.3.1 Optical response of strained MoSe2 resonator
Figure 4-2(a) shows the sample reflection contrast spectra at varying gate voltages (𝑉𝑔 = 0 −
40 V) in a contour plot from device #1. (We only focus on positive 𝑉𝑔 in this study. Results from
negative 𝑉𝑔 are similar.) Spectra at selected gate voltages are also shown. A strong dip in the
reflection contrast due to the neutral exciton resonance in monolayer MoSe2 can be clearly seen.
The sharp exciton linewidth (𝛾 ~ 3 meV) and the large on-resonance reflection contrast (~ 90 %)
are consistent with results reported for nearly perfect monolayer mirrors [48], [49], which illustrate
the high sample quality here. The exciton resonance 𝜔𝑋 redshifts with increasing gate voltage due
to build up of in-plane strain in the sample as the gate pulls it down. It provides the necessary
mechanism for exciton-optomechanical coupling [81], as we will investigate below. Here the
nonlinear gate-induced spectral shift ∆𝜔𝑋 is due to the nonlinear dependence of the sample strain
on 𝑉𝑔 [81], [166], [167]. The theoretical gate-dependence of the exciton resonance 𝜔𝑋 is also
plotted in the lower panel of Fig. 4-2(a), which shows good agreement with the experimental result.
Detailed simulation can be found in section 4.4.1.
To characterize the mechanical resonance of our suspended sample, we drive the MoSe2
into mechanical vibrations by a small ac voltage (δ𝑉𝑔~ 100 mV in amplitude) superimposed on top
of the dc gate voltage 𝑉𝑔. We illuminate the sample by a laser beam with center wavelength 765
nm near its exciton resonance, and detect the reflectivity change as a function of the driving
frequency (details see section 4.4.2). Figure 4-2(b) shows the gate dependence of the fundamental
mechanical mode in a contour plot. With increasing |𝑉𝑔|, the resonance frequency first decreases
slightly and then increases afterwards. The behavior is consistent with results reported in the
literature [66], [71], [76], [78], [166]–[168], which can be explained by gate modulation of the
Young’s modulus in the sample as shown by the solid line fit (Fitting model see section 4.4.2). At
small 𝑉𝑔, our mechanical resonator displays high quality factor Q ~ 20,000 as shown in inset of Fig.
70
4-2(b). The quality factor decreases quickly at high 𝑉𝑔 due to increased mechanical losses,
consistent with results from the literature [68], [70].
4.3.2 Exciton-modified mechanical response of MoSe2 resonator
Next we investigate the spectral dependence of the mechanical response in order to
understand the origin of the detected vibration-induced reflectivity change. To do this we vary the
incident laser wavelength near the exciton resonance and keep a small incident power fixed at 𝑃 =
1 μW. Figure 4-3(a) shows the detected reflectivity change due to the fundamental mechanical
mode at 𝑉𝑔 = 40 V and at varying incident wavelength. The peak response as a function of the
incident wavelength and frequency detuning Δ𝜔 ≡ 𝜔 − 𝜔𝑋 from the exciton resonance 𝜔𝑋 is
Figure 4-2: Gate tuning of excitonic and mechanical resonances in suspended MoSe2 resonator.
(a) Optical reflectance spectra of sample #1 are shown at selective gate voltages (upper panel) and
in contour plot (lower panel). The orange dashed line in lower panel of (a) is prediction of Eqn. (4-
6). (b) Mechanical resonance shift as a function of gate voltages. The grey dashed line is the
prediction of Eqn. (4-8). Inset: A typical mechanical resonance of our resonator at 3.3K, and the
corresponding Lorentzian fit with Q ~ 20,000.
71
shown in Fig. 4-3(b). The response changes sign at the exciton resonance, maximizes in amplitude
at a small detuning (Δ𝜔 ~ 3 THz), and diminishes with further increase in detuning. The vanishing
response exactly at the exciton resonance also provides a very reliable way to calibrate the
frequency detuning Δ𝜔 . The spectral response in Fig. 4-3(b) can be reproduced by taking a
frequency derivative of the measured reflection contrast 1
𝑅
𝑑𝑅
𝑑𝜔 (Fig. 4-2(a)), and multiplied by a
frequency shift δ𝜔𝑋 ≈ 0.1 THz. The spectral dependence of the mechanical response can therefore
be understood in terms of a dynamical shift in the exciton resonance (δ𝜔𝑋) caused by an oscillating
strain (δ𝜖) in the suspended MoSe2 sample undergoing mechanical vibrations (inset of Fig. 4-3(b)).
The dynamical shift δ𝜔𝑋 produces the reflectivity change as observed, which also provides an
effective method of detecting thermal vibrations of a monolayer semiconductor at cryogenic
temperatures (see section 4.6). Such picture can be further tested in Fig. 4-3(c) by comparing the
𝑉𝑔-dependence of the extracted dynamical shift δ𝜔𝑋 with the quantity 𝑑𝜔𝑋
𝑑𝑉𝑔δ𝑉𝑔 computed from the
exciton spectral shift in Fig. 4-2(a) (under ac driving voltage δ𝑉𝑔~ 70 mV). The good agreement
supports our interpretation. The result in Fig. 4-3(c) also suggests stronger exciton-mechanical
coupling at higher gate biases. We will come back to this point later.
72
4.3.3 Photothermal backaction of exciton-optomechanical coupling
Equipped with an understanding of the excitonic and mechanical characters of our device, we
now proceed to examine effects due to exciton-optomechanical backaction. Figure 4-4(a), similar
to Fig. 4-3(a), shows the optically detected fundamental mechanical mode at 𝑉𝑔 = 40 V and at
Figure 4-3: Interaction between excitonic and mechanical resonances in monolayer MoSe2
resonator. (a) Normalized reflectivity change pumped with 1 μW as a function of driven frequency
at selective wavelengths throughout the exciton resonance. The spectra are displaced with equal
spacing for clarity. Data was taken with 𝑉𝑔 = 40𝑉 at 3.3K. The illumination power is 1 μW. (b)
The peak response of mechanical resonance across exciton resonance (symbols with line) as a
function of incident wavelength (bottom axis) and exciton detuning Δ𝜔 (top axis). Data was
extracted from (a). The dashed red line is the illustration of spectral shift due to dynamical strain.
Inset: A sketch of a dynamical shift in the exciton resonance (δ𝜔𝑋). (c) The extracted dynamic shift
of exciton resonance δ𝜔𝑋 at selective gate voltages (symbols). The blue solid line is obtained by
multiplying quantity 𝑑𝜔𝑋
𝑑𝑉𝑔 extracted from the lower panel of Fig.4-2 (a) with a dynamic voltage
δ𝑉𝑔~ 70 mV.
73
varying incident wavelength. Compared to Fig. 4-3(a), the incident light power is increased from
𝑃 = 1 μW to 𝑃 = 10 μW in order to better show effects from dynamical backaction. Very
asymmetric dependence on optical detuning is seen. In particular, the blue-detuned mechanical
resonance is significantly broader than the red-detuned counterpart. The shift in the mechanical
resonance is also very asymmetric with detuning. These behaviors are summarized by the detuning
dependences of the mechanical linewidth and resonance frequency shown in Fig. 4-4(b) and (c),
respectively. For large detuning from the exciton resonance, the fundamental mechanical mode is
largely unperturbed by the incident light. Near the exciton resonance, however, the mechanical
resonance frequency redshifts and the mechanical linewidth broadens with very asymmetric
detuning dependences.
74
Figure 4-4: Light induced damping and anti-damping of mechanical modes across exciton
resonance in monolayer MoSe2 resonator. (a) Normalized reflectivity change pumped with 10 μW
as a function of driven frequency at selective wavelengths throughout the exciton resonance. The
spectra are displaced with equal spacing for clarity. Data was taken with 𝑉𝑔 = 40𝑉 at 3.3K. The
extracted (b) line width and (c) frequency of mechanical resonance shown in (a) as a function of
exciton detuning (bottom axis) and incident wavelength (top axis). The green and blue solid lines
in (b) and (c) are extracted contributions from photothermal softening(symmetric) and backaction
(anti-symmetric). The red solid lines in (b) and (c) are fitting lines by summing up green and blue
lines. The data without symmetric contribution are shown in (d) and (e), where the red solid lines
now are fitting from Eqns. (4-1) and (4-13).
75
The observations can be understood in a model of photothermal backaction and gate-induced
mirror symmetry breaking in the mechanical device. At 𝑉𝑔 = 40 V, the monolayer MoSe2
membrane is pulled down by the applied electrostatic force from the back gate, which lowers the
out-of-plane mirror symmetry of the mechanical device (Fig. 4-1). With the lowered symmetry, a
small driven mechanical vibration in the out-of-plane direction creates an oscillating strain δ𝜖 in
the membrane that produces a dynamical spectral shift in the exciton resonance δ𝜔𝑋. Meanwhile,
illumination of the sample by laser light near the exciton resonance produces a photothermal force
𝐹𝑝ℎ, whose magnitude depends on the detuning from the exciton resonance Δ𝜔 and the overall
vertical displacement 𝑧 of the membrane (Fig. 4-1). As a result, the oscillating membrane (and the
oscillating exciton resonance) periodically modulates the photothermal force 𝐹𝑝ℎ and creates a
dynamical backaction on its mechanical vibration, which changes its mechanical linewidth Γ and
resonance frequency Ω according to [105], [114]
Γ = Γ0 (1 + 𝑄Ω0𝜏
1+Ω02𝜏2
∇𝐹𝑝ℎ
𝛫),
Ω ≈ Ω0 (1 −1
1+Ω02𝜏2
∇𝐹𝑝ℎ
2𝛫). ` (4-1)
Here Γ0 and Ω0 are the unperturbed mechanical linewidth and resonance frequency, respectively,
𝜏 is the time delay for the photothermal force in response to the mechanical movement of the
membrane, ∇𝐹𝑝ℎ is the gradient of the photothermal force in the out-of-plane direction, and 𝛫 is
the unperturbed spring constant of the mechanical resonator, 𝑄 =Ω0
2πΓ0 is the mechanical quality
factor.
To quantitatively model our experiment, we need to calculate the gradient of the
photothermal force ∇𝐹𝑝ℎ.Thus, the contribution that gives rise to dynamical backaction is
𝛻𝐹𝑝ℎ
𝛫= 𝛼𝑃
4(𝛥𝜔)
[(𝛥𝜔)2+𝛾2]2∙ ∆𝜔𝑋 (4-2)
76
where 𝛼 represents the overall amplitude of the photon-induced rigidity compared to the
unperturbed spring constant of the resonator. Here the overall amplitude 𝛼 depends on parameters
of the device such as the Young’s modulus, the thermal expansion coefficient and the thermal
conductivity of MoSe2 etc. Detailed expression of 𝛻𝐹𝑝ℎ
𝛫 will be derived and presented in the
following section 4.4.3. As expected, the result is linearly proportional to the incident power 𝑃 (see
Fig. 4-6(a),(b)) and the gate-induced shift in the exciton resonance ∆𝜔𝑋 (Fig. 4-2(a)), and exhibits
antisymmetric dependence on detuning Δ𝜔. It reflects the fact that the oscillating exciton resonance
dynamically modulates the photothermal force. In addition to the dynamical backaction
contribution, there is also a contribution from photothermal softening [68], [81], [105] of the
mechanical spring constant ∝𝑃
(Δ𝜔)2+𝛾2. Since it is originated from optical absorption near the
exciton resonance, it shows symmetric dependence on detuning Δ𝜔. It is also independent of ∆𝜔𝑋.
The results in Fig. 4-4(b) and (c) can be described by including both the symmetric and the
antisymmetric contributions with the corresponding amplitudes and the exciton linewidth 𝛾 as
fitting parameters. In order to isolate the antisymmetric contribution linear to ∆𝜔𝑋, we subtract the
symmetric contribution from Fig. 4-4(b) and (c), and show the results in Fig. 4-4(d) and (e). The
dependences on detuning Δ𝜔 are reminiscent to the optical spring effect and optical damping
reported in the literature on cavity optomechanics [108], [109], [115], [126]. The underlying
mechanism is in fact very similar. In cavity optomechanics, the mechanical vibrations of the end
mirror of an optical cavity dynamically change the cavity resonance frequency, which in turn
modulates the intracavity optical field and the radiation force on the mirror. The periodically
modulating radiation force creates a dynamical backaction on the mirror’s oscillations, and changes
its mechanical resonance frequency and linewidth. Here the mirror is replaced by the MoSe2
membrane and the cavity resonance is replaced by the exciton resonance (no optical cavity is
required). Because the exciton resonance redshifts (∆𝜔𝑋 < 0) with increasing strain (i.e. increasing
77
downward displacement 𝑧 of the membrane), the sign of the detuning dependences in Fig. 4-4(d)
and (e) is opposite to that in cavity optomechanics. In particular, red-detuning (blue-detuning)
causes anti-damping (damping) and blueshift (redshift) on the mechanical resonance.
Cavity optomechanics
Measurement: optical spring effect and
optical damping of mechanical motion
through the cavity resonance
This work
Measurement: light induced damping and anti-
damping of mechanical motion through the
fundamental exciton resonance
Resonantly reflect or transmit the incident
light at cavity resonance frequency due to the
intracavity interference of trapped light
Resonantly reflect or transmit the incident light
at exciton resonance frequency due to the
radiative damping of oscillating exciton dipole
Cavity resonance with position dependence Exciton resonance with strain dependence
Mirror at movable end Suspended MoSe2 monolayer
Radiation force due to the momentum
transfer of confined photons
Photothermal force due to light induced
tension change
Time delay: decay lifetime of trapped photon
inside cavity.
Time delay: thermal equilibrium time of
photothermal deformation
4.4 Modeling strain induced exciton-optomechanical coupling in suspended MoSe2
We utilize mechanical model of thin film developed in Section 1.2.2 to model the gate-
induced strain effect of excitonic and mechanical resonance in our MoSe2 resonators. Particularly,
integrating with required mechanical and optical theories, we derive in details to show a dynamical
backaction can build up through a photothermal force, which stems from the light absorption near
exciton resonance and the resultant tension change. And such photothermal force acts on the
resonator with some time delay related to thermal diffusion and subsequent deformation.
Table 4-1: Comparison of backaction effects in cavity optomechanics and this work.
78
4.4.1 Gate induced exciton spectral shift
Next, we find the exciton spectral shift as a function of gate voltage. The suspended MoSe2
monolayer was pulled down by electrostatic force and the strained flake redshifted its excitonic
resonance correspondingly. For a circular drumhead trench, the deflection of the membrane under
uniform-loaded force can be described with a parabola distribution 𝜉(𝑟) = 𝑧(1 −𝑟2
𝑅2), where z is
the vertical displacement at the center of the suspended membrane, and 𝑅 = 2.5 𝜇𝑚 is the radius
of the circular drumhead trench. The stored elastic energy within the strained membrane 𝑈𝑒𝑙 is
[166], [167]
𝑈𝑒𝑙 =𝜋𝐸𝑌𝑡
1−𝜈2∫ [(1 − 𝜈2)𝜖0 +
1
2(𝜕𝜉
𝜕𝑟)2]2
𝑅
0𝑟𝑑𝑟 =
𝜋𝐸𝑌𝑡
1−𝜈2[2𝑧4
3𝑅2+ (1 − 𝜈2) (𝜖0𝑧
2 +1
2𝜖02𝑅2)] (4-3)
where 𝐸𝑌 is the 2D Young’s modulus of MoSe2, 𝜈 is the Poisson ratio, 𝑡 = 0.67 𝑛𝑚 is the
thickness of monolayer MoSe2 flake and 𝜖0 is the initial strain that applied during device
fabrication. And the electrostatic energy of the gated resonator 𝑈𝑒𝑠 is
𝑈𝑒𝑠 = −1
2𝐶𝑔𝑉𝑔
2 (4-4)
where 𝐶𝑔 is the electrostatic capacitance of the resonator and 𝑉𝑔 is the gate voltage. Therefore, to
find the equilibrium position of the membrane, we set the total energy to be minimum, namely:
𝐹𝑡𝑜𝑡 = −𝜕(𝑈𝑒𝑙+𝑈𝑒𝑠)
𝜕𝑧= 0, (4-5)
Consequently, we obtain the vertical displacement of suspended flake 𝑧 ≈𝜀0
8𝐸𝑌𝑡𝜖0
𝑅2
𝑑2𝑉𝑔2. The
strain 𝜖 is related to the vertical displacement 𝑧 as 𝜖 =1
2𝑅∫ (
𝜕𝜉
𝜕𝑟)2𝑅
0𝑟𝑑𝑟 =
2
3(z
𝑅)2. Therefore, the
gate induced exciton shift is
∆𝜔𝑋 = 𝛽𝑋 𝜖 ≈𝛽𝑋𝜀0
2𝑅2
96𝐸𝑌2𝑡2𝜖0
2𝑑4𝑉𝑔4 (4-6)
79
where 𝛽𝑋 represents the strain-induced excitonic spectral shift rate. And the extracted exciton peak
position as a function of the gate voltage 𝑉𝑔 can also be fitted by the model using 𝛽𝑋~ 0.1 eV/%
strain (lower panel of Fig. 4-2(a)), which agrees reasonably well with reported values [56], [60],
[165].
4.4.2 Gate induced shift in mechanical resonance
The above membrane mechanics can also illustrate the observed shift in mechanical resonance
of our resonator in terms of gate voltage. The spring constant of the resonator 𝑘𝑒𝑓𝑓 is given by
taking the second derivative of the total energy
𝑘𝑒𝑓𝑓 =𝜕2(𝑈𝑒𝑙+𝑈𝑒𝑠)
𝜕𝑧2=
8𝜋𝐸𝑌𝑡
(1−𝜈2)𝑧2 + 4.924𝐸𝑌𝑡𝜖0 −
1
2
𝜕2𝐶𝑔
𝜕𝑧2𝑉𝑔2 (4-7)
Thus, the corresponding frequency of mechanical vibration Ω0 is as following
Ω0 = √𝑘𝑒𝑓𝑓
𝑀𝑒𝑓𝑓= √
𝜋𝜀02
8(1−𝜈2)𝐸𝑌𝑡𝜖02𝑅2
𝑑2𝑉𝑔4−
𝜀0𝜋𝑅2
3𝑑3𝑉𝑔2+4.924𝐸𝑌𝑡𝜖0
𝑀𝑒𝑓𝑓 (4-8)
Here, 휀0 is vacuum dielectric constant, 𝑑 = 600 𝑛𝑚 is the trench depth between the top SiO2
surface and the back-gate silicon surface. By using young’s modulus 𝐸𝑌 = 310 GPa , effective
mass 𝑀𝑒𝑓𝑓 = 9 × 10−17 𝑘𝑔 and built-in strain 𝜖0 = 0.36% as free fitting parameters at 3.3K, the
measured gate-induced frequency can be well described by the above frequency equation in terms
of gate voltage 𝑉𝑔. The extracted effective mass 𝑀𝑒𝑓𝑓 and young’s modulus 𝐸𝑌, respectively, are
slightly larger than the reported values in room temperature [169].This discrepancy suggests
additional mass and initial strain imparted into the resonator by the polymer residues from
fabrication process. In addition, a stiffened modulus at low temperature was reported in tensioned
graphene resonator [170] likely due to influence of adsorbent molecules, which is similar to our
observation here. Symmetrical frequency shift by gate excludes the negligible doping effect on
80
mechanical frequency (results at negative gates not shown). To minimize the optomechanical effect,
a probe wavelength away from exciton resonance (765 nm) was employed in gate-dependent
measurements with low power intensity (~ 5 𝜇W).
4.4.3 Modelling photothermal backaction in strained MoSe2 resonator
To better understand our experiment, we need to make quantitative predictions for the
photothermal coupling strength (namely, the gradient of photothermal force) ∇𝐹𝑝ℎ for our system.
The strain-mediated feedback mechanism is the following (see lower left panel of Fig. 4-1): the
monolayer MoSe2 membrane is initially deformed by a gate voltage, and it absorbs light
proportional to both the excitonic absorbance �� and the incident laser power P. The membrane
heats up locally near the laser spot, it takes some timescale 𝜏 for the temperature to equilibrate
across the membrane, the tension σ changes, which equivalently apply a photothermal force 𝐹𝑝ℎ
with time delay 𝜏 onto the membrane. This retarded force can exert nonzero work within one
oscillation cycle and cause the resonator to experience a change in both frequency and line width
of mechanical motion.
The quantitative formula for this light-induced change in frequency and line width of
mechanical modes have been given as Eqn. (4-1), which are all proportional to the gradient of
photothermal force 𝛻𝐹𝑝ℎ. Here, we derive the gradient of photothermal force 𝛻𝐹𝑝ℎ ≡ 𝑑𝐹𝑝ℎ/𝑑𝑧 for
our resonator system, where 𝐹𝑝ℎ = 4𝜋𝜎𝑝ℎ𝑧 is the photothermal force felt by the suspended
membrane and 𝜎𝑝ℎ is the photothermal-induced tension. And we have for photothermal force
gradient related to dynamical backaction as
𝛻𝐹𝑝ℎ = 4𝜋𝑧𝑑𝜎𝑝ℎ
𝑑𝑧 (4-9)
81
If we write down the photothermal tension from laser heating onto our monolayer MoSe2 membrane,
then as [81]
𝜎𝑝ℎ = −𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐴𝐿𝑃
4𝜋𝜅 (4-10)
where 𝛼𝑡ℎ is the thermal expansion coefficient, 𝜅 is the sheet thermal conductivity, �� is the
complex absorbance of the monolayer MoSe2 flake, 𝐿 is local field factor, 𝑃 is the power of the
incident laser. Here the complex absorbance is modeled by a single Lorentzian that arises from the
exciton resonance ��(𝜔) =𝐴0
1−𝑖(𝜔−𝜔𝑋(𝑧))
(𝛾)
. Since the exciton resonance of monolayer MoSe2
membrane is linearly proportional to the applied strain, which is also a function of vertical
displacement of the suspended sample. Then the gradient of photothermal tension can be written
by substituting the complex absorbance.
𝑑𝜎𝑝ℎ
𝑑𝑧≈ −
𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐿
4𝜋𝜅
𝜕��
𝜕𝑧𝑃 = −
𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐿
4𝜋𝜅𝑃
𝐴0
(1+(Δ𝜔)2
𝛾2)2
2(Δ𝜔)
𝛾2∂𝜔𝑋
∂ϵ
∂ϵ
∂z (4-11)
Here we use β𝑋 =∂𝜔𝑋
∂ϵ and
∂ϵ
∂z=4
3
𝑧
𝑅2 from section 4.4.2. Then we can substitute the gradient of
absorbance due to exciton resonance into gradient of the photothermal force,
𝛻𝐹𝑝𝑡ℎ = −𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐿𝑃
𝜅(𝑧
𝜕��
𝜕𝑧) = −
𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐿
𝜅𝑃��(
4(Δ𝜔)
𝛾2+ℏ2(Δ𝜔)2Δ𝜔𝑋) (4-12)
Thus, the contribution that gives rise to dynamical backaction is
𝛻𝐹𝑝ℎ
𝛫= −
1
𝐾
𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐿𝐴0
𝜅
𝛾2
4𝑃
4(𝛥𝜔)
[(𝛥𝜔)2+𝛾2]2∙ ∆𝜔𝑋 = 𝛼𝑃
4(𝛥𝜔)
[(𝛥𝜔)2+𝛾2]2∙ ∆𝜔𝑋 (4-13)
where 𝛼 = −1
𝐾
𝐸𝑌𝑡
1−𝜈2𝛼𝑡ℎ𝐿𝐴0
𝜅
𝛾2
4 is the detailed expression for the overall magnitude of the light
induced stiffness compared to the natural spring constant of the resonator as shown in Eqn.(4-2).
Two significant implications can be concluded here. First, when ∆𝜔𝑋 = 0, namely no deflection
of membrane, 𝛻𝐹𝑝ℎ will vanish. In other word, there is no dynamical backaction from photothermal
force without breaking out-of-plane mirror symmetry by gate [81]. Second, the sign of 𝛻𝐹𝑝ℎ is
determined jointly by exciton detuning 𝛥𝜔 and excitonic spectral shift ∆𝜔𝑋 induced by strain.
82
So far, we still need the explicit expression for the time delay 𝜏 that photothermal force in
response to the mechanical movement of the membrane to fulfill the quantitative model of
photothermal backaction. Since a photothermal model within membrane have been built up in
Chapter 3, we also start from the steady heat diffusion shown as Eqn. (3-4) and add a time-
dependent term ��(𝑟, 𝑡) with the following substitution 𝑇 → 𝑇(𝑟) + ��(𝑟, 𝑡) . After applying
separation of variables and boundary condition ��(𝑅, 𝑡) = 0, we acquire the solution for ��(𝑟, 𝑡)
similar to the dynamical displacement ��(𝑟, 𝑡) in section 1.2.2 as
��(𝑟, 𝑡) = ∑ 𝐴𝑚𝐽0 (𝑟√𝜌𝑐𝑝
𝜅𝜏𝑚)∞
𝑚=0 𝑒−𝑡/𝜏𝑚 (4-14)
where the 𝐴𝑚 is the amplitude and 𝜏𝑚 is the time constant of the mth phonon diffusion mode. Apart
from the oscillating counterpart in the dynamical displacement ��(𝑟, 𝑡), the time dependent part in
��(𝑟, 𝑡) is an exponential decay. Thus, the time delay 𝜏 is equivalent to the longest time constant of
the possible phonon diffusion process as
𝜏 =𝑅2𝜌𝑐𝑝
2.4052𝜅 (4-15)
In principle, we now have each piece of modelling photothermal backaction and can
quantitatively predict the light-induced damping and anti-damping acting on our resonators. We
can also estimate the overall amplitude of the photo-induced change in line width ∆Γ =Ω02𝜏
1+Ω02𝜏2
∇𝐹𝑝ℎ
2𝜋𝛫
by combining Eqn. (4-1) with Eqn. (4-13). The estimated ∆Γ = 0.5𝑀𝐻𝑧 (V𝑔 = 40 𝑉) is in good
agreement with the extracted value from the experiment (Fig. 4-4(d)) by using the measured
mechanical resonance Ω0 = 2𝜋 × 41 𝑀𝐻𝑧 (V𝑔 = 40 𝑉 ), spring constant 𝐾 = 4.8 𝑁𝑚−1 , the
extracted young’s modulus 𝐸𝑌 = 208 𝑁𝑚−1, thermal expansion coefficient 𝛼𝑡ℎ = 4 × 10
−6𝐾−1
[171], local field factor 𝐿 = 2.27, sheet thermal conductivity 𝜅 = 2 × 10−8𝑊𝐾−1 [62], [105],
[172], excitonic absorbance 𝐴0 = 0.85 , and exciton line width 𝛾 = 3.8 𝑇𝐻𝑧 , strain-induced
exciton shift ∆𝜔𝑋 = 3.8 𝑇𝐻𝑧, the time for heat travel out of drum 𝜏 =𝑅2𝜌𝐶𝑝
2.4052𝜅= 5 × 10−9s [105]
83
(with specific heat capacity 𝐶𝑝 = 20 𝐽/(𝑘𝑔 ∙ 𝐾) agrees with values in the literature for monolayer
[105] and bulk [173] MoSe2). And we will demonstrate the numerical prediction of photothermal
backaction as developed above and compare the predictions to the extracted experimental findings
within next section.
4.5 Tunable exciton-optomechanical effects
We have performed multiple checks to further examine the observed damping and anti-
damping in mechanical modes through exciton resonance for our system. In this section, we present
gate- and power-dependence of light induced shift of mechanical line width ΔΓ. By comparing
experimental results to the model of photothermal backaction, we thus verify its exciton-
optomechanical origin of the observed modification of mechanical modes across the exciton
resonance in a monolayer MoSe2 resonator. Below we will focus on the mechanical linewidth
because optical damping is generally much more pronounced than the optical spring effect due to
the enhancement factor 𝑄 in Eqn. (4-1).
4.5.1 Gate dependence of exciton-optomechanical coupling
Finally, we note that Eqn. (4-13) also predicts the antisymmetric contribution is important
only when there is a significant gate-induced redshift in the exciton resonance (∆𝜔𝑋 ≳ 𝛾). To test
this prediction, we perform optical detuning studies at different applied gate voltages, which vary
the in-plane sample strain and thus the shift in the exciton resonance ∆𝜔𝑋. Figure 4-5(a) shows the
detuning dependences of the antisymmetric contribution of the mechanical linewidth at different
gate voltages 𝑉𝑔 (the incident power is fixed at 𝑃 = 3.3 μW). We have subtracted the symmetric
contributions from the total (symmetric plus antisymmetric) mechanical linewidth to obtain Fig. 4-
84
5(a). We can see that the amplitude of the antisymmetric contribution increases significantly with
increasing 𝑉𝑔, as summarized in Fig. 4-5(b). In Fig. 4-5(b), we also show the prediction from Eqn.
(4-1) and (4-13) taking into account the 𝑉𝑔-dependence of the exciton resonance shift ∆𝜔𝑋 from
Fig. 4-2(a). The good agreement strongly supports our physical interpretations.
4.5.2 Power dependence of exciton-optomechanical coupling
To further understand the observation on the sample’s mechanical response as a function of
detunings, we perform the measurement on the power dependence of the mechanical resonance
under the same condition (3.3K, Vg ~ 40V). The magnitudes of light induced change of line width
are background-subtracted and summarized in Fig. 4-6(a) at several selected powers. For line width
Figure 4-5: Gate dependence of optical damping across exciton resonance. (a) The shift of
mechanical linewidth induced by antisymmetric contribution as a function of exciton detuning at
selective gate voltages. Data is displaced with equal spacing for clarity. The solid lines are fitting
based on Eqns. (4-1) and (4-13). (b) The extracted amplitude of the antisymmetric contribution of
mechanical linewidth as a function of gate voltages. The red dashed line is prediction based on
Eqns (4-1) and (4-13).
85
of mechanical motion, a clear enhancement of optomechanical damping and anti-damping is
observed with increasing pump power. These amplitude of optical-induced mechanical response
clearly shows a linear dependence as a function of power. By using the young’s modulus 𝐸𝑌, initial
strain 𝜖0 and extracted parameters from fitting Eqn. (4-9), experimental results and theoretical
calculation reach a reasonably good agreement for photo-induced line width shown in Fig. 4-6(b).
Note that there is no sudden jump, namely optical bistability, occurring for the amplitudes of
mechanical resonance as a function of detuning for all our measurement. This indicates the incident
powers still remain at low level without generating any nonlinearity behavior. In addition, the
increased power can lead to a redshift of exciton resonance, which further enlargers the optically
induced change of line width. Thus, the power dependence of line width shifted by optical pump is
in agreement with the prediction from photothermal backaction model.
Figure 4-6: Power dependence of optical damping across exciton resonance. (a) The shift of
mechanical linewidth induced by antisymmetric contribution as a function of exciton detuning at
selective incident power. Data is displaced with equal spacing for clarity. The solid lines are fitting
based on Eqns. (4-1) and (4-13). (b) The extracted amplitude of the antisymmetric contribution of
mechanical linewidth as a function of incident power. The red dashed line is prediction based on
Eqns (4-1) and (4-13).
86
4.6 Detection of thermal vibrations of monolayer MoSe2 at cryogenic temperatures
In driven vibration, the vibrated MoSe2 monolayer generates an oscillating strain (δ𝜖) and
corresponding dynamical shift of exciton resonance (δ𝜔𝑋) that can greatly modulate the reflectivity
change near exciton resonance. Namely, the mechanical response of the suspended MoSe2 sample
can be detected more ‘easily’ with probe wavelength near exciton resonance than those far from
exciton resonance. To verify this exciton-enhanced sensitivity of mechanical resonance, we
measure the thermal vibrations of another MoSe2 mechanical resonator (D2) with the same device
structure at 3.3K. After testing the spectral range of fundamental mechanical mode, the modulated
force is turned off and the resulting thermal vibration noise is recorded with a spectrum analyzer.
To clearly resolve the thermal vibration process at such cryogenic temperature, we used the probe
wavelength near exciton resonance with incident power P = 100 𝜇W.
Figure 4-7(a) shows the reflection contrast measurement of device #2 with gate voltage (~-
60V) at 3.3K, with exciton resonance around 745nm. The above-mentioned 100𝜇W probe power
for resolving thermal motion can significantly heat up the suspended membrane, which further
redshifts the exciton resonance of MoSe2 monolayer to around 750nm. In addition, we can see that
the reflectivity of sample only changes rapidly around the exciton resonance compared to the
regions outside the exciton resonance. In Fig. 4-7 (d) and (e), we demonstrated thermal noise
spectra at the same gate voltage for probe wavelength (b) near exciton resonance (752nm) and (c)
away from exciton resonance (759 nm). Mechanical resonance around 55.2MHz is clearly observed
by 752nm probe as the resonance peak with 759 nm probe remains elusive. This indicates that
dynamical shift of exciton resonance due to thermal motion can amplify the thermal noise signal
by modulating the reflectivity change near exciton resonance. Namely, one can utilize the exciton
resonance with appropriate probe wavelength to enhance the measurement sensitivity of
mechanical motion of suspended monolayers.
87
4.7 Conclusion
In this chapter, we have demonstrated efficient dynamical control of the mechanical motion
of suspended monolayer MoSe2 by light through the material’s strong excitonic resonance. Gate-
tunable optomechanical damping and anti-damping of the mechanical vibrations due to
photothermal backaction have been observed. The damping and anti-damping effects are effective
Figure 4-7: Thermal motions of monolayer MoSe2 resonator. (a) Optical reflection spectrum of
sample #2 at 𝑉𝑔 = −60𝑉. The blue and green dashed lines are aligned with 752 nm and 759 nm,
which used for measurements of thermal noise in (b) and (c). The red dash lines in (b) and (c) are
Lorentzian fittings.
88
(Fig. 4-4 b and d) in the sense that the photo-induced change in the mechanical linewidth at
moderate optical pumping is significantly larger than the unperturbed linewidth Γ0 . Further
optimization of the device that allows larger gate-induced shift in the exciton resonance ∆𝜔𝑋 ,
which favors dynamical backaction, could open up interesting applications in optomechanical
cooling and mechanical lasing in monolayer TMD semiconductors.
Chapter 5
Valley magnetoelectricity in single-layer MoS2
Magnetoelectric (ME) effect, the phenomenon of inducing magnetization by application of
an electric field or vice versa, holds great promise for magnetic sensing and switching applications
[174]. Studies of the ME effect have so far focused on the control of the electron spin degree of
freedom (DOF) in materials such as multiferroics [175] and conventional semiconductors [176].
Meanwhile, the crystalline structure of solids can attribute electrons a valley degree of freedom
(DOF) [23], [32]–[34], which represents the degenerate energy extrema in momentum space. And
the appealing feasibility for dynamic control of valleys has led to the concept of valleytronics
[23], [33], [34], [177], which aims at harvesting valley, together with charge and spin, to store
and transduce information.
In this chapter, we will demonstrate a new form of the ME effect based on the valley DOF
in two-dimensional (2D) Dirac materials [32]–[34], [178]. By manipulating carriers’ momentum
distribution, the valley magnetization can arise from the joint effect of current and strain. And such
magnetoelectric effect from valley DOF that sustains up to room temperature offers a practical
mechanism for converting charge to magnetization in valleytronic devices.
This chapter is based on published work in which I am a co-author [179]. This chapter is
reprinted with permission from J. Lee, Z. Wang, H. Xie, K. F. Mak, J. Shan, Nat. Mater. 2017 16,
887–891. ( https://www.nature.com/articles/nmat4931).
90
5.1 Magnetization from valley degree of freedom
Electrons in two-dimensional (2D) Dirac materials including gapped graphene and single-
layer TMDs possess a new two-fold valley degree of freedom (DOF) corresponding to the K and
K’ valleys of the Brillouin zone. The valley DOF carries orbital magnetic moment [32]–[34], [178].
We would like to emphasis that the magnitude of valley magnetic moment depends on carrier’s
momentum. Due to time reversal symmetry and the limit of 2D nature, valley magnetic moment
points out of 2D plane with opposite direction at K and K’ valleys. Thus, a nonzero valley
magnetization can arise from either a finite population imbalance between the valleys (i.e. a net
valley polarization) or a distribution difference between them without a population imbalance [178].
Such valley magnetization forms the basis for valley-based applications. Whereas the former
relaxes by intervalley scattering, the latter is largely limited by intravalley scattering. The presence
of valley contrasting Berry curvatures in 2D Dirac materials, which can couple to external
electromagnetic excitations, enables the control of valley magnetization [33], [34], [38]–[43],
[180]–[186]. Although the control of valley magnetization by circularly polarized light and by a
vertical magnetic field has now become routine [33], [34], [38]–[43], [181]–[183], [185], [186],
the development of practical valleytronic devices requires the pure electrical control of valley
magnetization. The valley magnetoelectric (ME) effect is an attractive approach for this purpose.
To realize the linear ME effect, a material has to possess broken time-reversal and spatial-
inversion symmetries [174], [187]. For an electrical conductor, time-reversal symmetry can be
broken naturally by application of a bias voltage, under which dissipation through carrier scattering
is caused by a charge current. The magnetoelectricity produced in this manner is known as the
kinematic ME effect [188], [189]. Single-layer TMDs such as MoS2, which are intrinsically non-
centrosymmetric, satisfy these symmetry requirements with finite doping and biasing. The
characteristic three-fold rotational symmetry of these materials, however, renders the relevant
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valley ME susceptibilities zero [187], [190]. In this work, we apply a uniaxial stress to lower the
crystal symmetry and demonstrate the generation of steady-state out-of-plane valley magnetization
by an in-plane charge current in single-layer MoS2. Such an effect can be understood intuitively as
the valley analog of the Rashba-Edelstein effect for spins [189], [191], [192](Fig. 5-1(b)). In the
presence of an in-plane electric field 𝓔𝒑𝒛 , an in-plane current density 𝑱 generates an effective
magnetic field ∝ 𝓔𝒑𝒛 × 𝑱, resulting in finite out-of-plane valley magnetization 𝑴𝑽. In place of
the Rashba field (an out-of-plane built-in electric field at the semiconductor heterostructure
interfaces) in the Rashba-Edelstein effect [189], [191], [192], 𝓔𝒑𝒛 here is the in-plane piezoelectric
field that arises from the strain-induced lattice distortion and the associated ion charge polarization
(Fig. 5-1(a)), a phenomenon that has been recently demonstrated in single-layer MoS2. The effect
provides a possible valley magnetization-current conversion mechanism for magnetization
switching and detection applications [191], [192].
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Figure 5-1: Valley magnetoelectric effect in strained single-layer MoS2. (a) Piezoelectric field 𝓔𝒑𝒛
is produced (right) as a result of stress-induced distortion of the hexagonal lattice (left) and the
associated ion charge polarization. Blue and yellow balls represent Mo and S atoms, respectively.
(b) Out-of-plane valley magnetization 𝑴𝑽 is induced by an in-plane current density 𝑱 in the
presence of the piezoelectric field 𝓔𝒑𝒛. (c) Electronic band structure of n-doped single-layer MoS2
around the K and K’ valleys of the Brillouin zone. The band extrema are shifted from the K/K’
point (black dashed lines) to opposite directions by a uniaxial strain, which also do not coincide
with the extrema of the Berry curvature distribution (blue dashed lines). The Fermi level is tilted
under an in-plane bias electric field.
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5.2 Experimental setup
In our experiment, single-layer MoS2 was mechanically exfoliated from bulk crystals onto
flexible polydimethylsiloxane (PDMS) substrates. The flakes were stressed along a particular high-
symmetry axis by mechanically stretching the substrate. The strained flakes were then transferred
onto SiO2/Si substrates with pre-patterned electrodes to form field-effect transistors with Si as the
back gate. Strain was fixed for all subsequent measurements. During the process, the strain level in
the MoS2 flakes was monitored by the photoluminescence (PL) energy shift based on the strain-
induced decrease of the band gap (~ 20 nm/% strain independent of direction [56], [165]) (Fig. 5-
2(b)). The crystallographic orientation of the flakes was determined by the second-harmonic
generation (SHG) under normal incidence [56]. Figure 5-2(a) shows the intensity of the second-
harmonic component parallel to the excitation as a function of the polarization angle, the maximum
of which corresponds to the armchair direction. Polar magneto-optic Kerr rotation (KR) microscopy
at variable temperatures was employed to image the spatial distribution of the magnetization in the
device channels [184]. To this end, linearly polarized light was focused onto the device under
normal incidence and its polarization rotation (i.e. KR angle) upon reflection was detected. The
polar configuration is sensitive to out-of-plane magnetization only. To enhance the detection
sensitivity, we have used a probe wavelength close to the MoS2 A exciton resonance and a lock-in
detection method with the bias voltage being modulated at 4.11 kHz. The KR sensitivity is about
0.5 μrad/Hz1/2 (details see section 5.2.2).
94
5.2.1 Device fabrication
Atomically thin flakes of MoS2 were mechanically exfoliated from synthetic bulk crystals
(2D Semiconductors) onto flexible polydimethylsiloxane (PDMS) substrates. Single-layer flakes
were identified by the combination of their optical contrast and photoluminescence (PL) spectrum.
The crystallographic orientation of each single-layer flake was determined by performing optical
second-harmonic generation (SHG) on multiple locations of the flake to ensure that it contains a
single domain. The flakes were then stressed uniaxially along a high-symmetry axis by stretching
the PDMS substrates using a mechanical translation stage. The PL spectrum was simultaneously
measured to monitor the strain level. Strain up to about 0.2 - 0.3% (corresponding to a shift in the
PL peak wavelength by 4 - 6 nm) was typically achieved in monolayer MoS2 before the flake started
to slip or break. A glass slide was then employed to support the stretched PDMS for sample transfer
Figure 5-2: Basic optical characterization in single-layer MoS2. (a) Intensity of the second-
harmonic component parallel to the excitation polarization (𝐼) as a function of the excitation
polarization angle measured from the armchair direction (𝜃) (symbols). Solid line is a fit to 𝐼 =𝐼0cos
2(3𝜃) with 𝐼0 denoting the maximum intensity. (b) Photoluminescence (PL) spectrum of
single-layer MoS2 on a flexible substrate with 0% (blue), 0.1% (cyan) and 0.2% (green) strain along
the armchair direction. Red line is the PL spectrum of the flake after being transferred onto a silicon
substrate. The spectra are normalized to the peak intensity and vertically shifted for clarity.
95
onto a Si substrate with a 100-nm SiO2 layer. During the transfer, the plane of the MoS2 flake and
the Si substrate were kept parallel so that strain was best maintained. The Si substrates were pre-
patterned with electrodes (Ti 3nm/Au 30nm) by either e-beam or photo-lithography followed by
metal evaporation so that the MoS2 devices are free of resist contamination.
5.2.2 Second-harmonic generation (SHG)
To determine the crystallographic orientation of single-layer MoS2 flakes, we employed
SHG measurements based on a femtosecond Ti:Sapphire oscillator centered at 820 nm with a
repetition rate of ~ 80 MHz and a pulse duration of ~ 100 fs. The laser beam was focused onto the
sample under normal incidence using a 40x microscope objective. The reflected second-harmonic
(SH) radiation peaked at 410 nm was filtered and detected by a spectrometer equipped with a
nitrogen-cooled charge-coupled device (CCD). The excitation power was kept below 1 mW on the
sample. The crystallographic orientation of the crystal was determined from the SH intensity as a
function of the excitation polarization with respect to the sample’s high-symmetry axes. We chose
to fix the sample while varying the linear polarization of the excitation beam using a broadband
half-wave plate. The reflected SH beam was passed through the same broadband half-wave plate
and an analyzer for the measurement of the SH component parallel to the excitation polarization.
The integration time for each SH spectrum was 5 seconds with binning applied to the CCD readout.
In Fig. 5-2(a), we show the integrated SH intensity as a function of excitation polarization. A six-
fold pattern was observed as expected from D3h symmetry of the crystal. The SH maxima
correspond to the armchair direction.
96
5.2.3 Photoluminescence (PL) spectroscopy
PL spectroscopy was performed on MoS2 flakes to first identify their monolayer thickness
and then to characterize the strain level. A continuous-wave solid-state laser at 532 nm (up to 50
µW on the sample) was employed as the excitation source. PL spectrum was collected with a
confocal microscope setup, filtered, and recorded by a spectrometer equipped with a CCD.
5.2.4 Magneto-optic Kerr rotation (KR) microscopy
KR microscopy was performed on devices in an optical cryostat. The schematics of KR
microscopy setup is shown in Figure 5-3 (b). A linearly polarized probe laser was tuned to a
wavelength slightly red shifted from the PL peak wavelength and focused perpendicularly onto the
devices by a 40x microscope objective. The power on the sample was 200 - 350 µW. The reflected
probe was collected by the same objective, passed through a half-wave plate, split by a Wollaston
prism, and detected by a pair of balanced photodetectors. In order to increase the signal-to-noise
ratio, an oscillating bias voltage at 4.11 kHz was applied to the devices and the KR was detected
using a lock-in amplifier with a time constant of 30 msec. The noise level was about 0.5 µrad/Hz1/2
away from the boundaries of the gold electrodes. The noise level is typically higher near the
boundaries due to distortions of the light polarization upon reflection. For two-dimensional (2D)
spatial mapping, the devices were scanned by an XY piezo-stage while the probe beam was fixed.
The KR measurement in the presence of an in-plane magnetic field was performed in a cryostat
equipped with a superconducting magnet in the Voigt geometry.
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5.3 Observation of valley magnetoelectric effect in single-layer MoS2
In the following section 5.3 and 5.4, we investigate the valley associated magnetization
without external magnetic field in single-layer MoS2 FET devices by employing Kerr rotation
microscopy. Particularly, we focus more on the magnetization induced by a distribution difference
of valley magnetic moments between two valleys instead of a population imbalance.
5.3.1 Valley magnetization in single-layer MoS2
Here we present the KR image of two single-layer MoS2 devices at 30 K, one with
intentional strain and one without in Fig. 5-4 (a) and (b). Drastic differences are observed. In the
Figure 5-3: Magneto-optic KR microscopy setup for strained single-layer MoS2. (a) Optical image
of strained MoS2 device. Black arrow indicates the direction of electrical current. (b) Schematics
of a MoS2 FET with bias voltage (Vsd) applied on the source–drain electrodes and gate voltage
(Vg) applied through the Si/SiO2 substrate. For a strained single-layer MoS2 FET, a longitudinal
electrical current density J (black arrow) is optimally flowed in the transverse direction of the
strained-induced piezoelectric field Epz (orange arrow). The valley magnetoelectric effect is
detected by focusing a linearly polarized probe beam onto the device under normal incidence
and measuring the Kerr rotation angle θKR of the reflected beam.
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unstrained device (Fig. 5-4(a)), the KR is present only near the channel edges and is of opposite
sign on the two edges. When the current direction is reversed, the KR switches sign on the two
edges. (Detailed studies on the gate and bias dependences are shown in Section 5.4.7) Such a KR
signal is originated from valley polarization accumulated on the channel edges that is driven by the
valley Hall effect (VHE), as reported by recent experiments [180], [184], [186]. On the other hand,
in the strained device (Fig. 5-4(b)), a much stronger KR signal per unit current (in Fig. 5-16) is
observed nearly uniformly over the entire device channel. The signal also changes sign under a
reversed current direction. For this particular device, the sample was strained by ~ 0.2% along the
zigzag direction and biased along the same direction.
Figure 5-4: Valley Hall effect and valley magnetoelectric effect in single-layer MoS2. Kerr
rotation image of an unstrained (a) and strained (b) single-layer MoS2 device under two opposite
bias directions (black arrows). Boundaries of the electrodes and the device channel are marked in
dashed black and green lines, respectively. The unstrained device was measured with 𝑉𝑑𝑠= 2.5 V
and 𝑉𝑔 = 0 V (𝐽 = 22 A/m). KR is observed at channel edges only. The strained device was measured
with 𝑉𝑑𝑠 = 2.5 V and 𝑉𝑔 = 20 V (𝐽 = 13 A/m). KR is observed over the entire channel.
99
5.3.2 Dependences of bias current and in-plane magnetic field
We performed multiple control measurements to further examine the observed KR signal
in the bulk of strained single-layer MoS2. First, we studied the dependence of the KR signal on the
channel current density (𝐽). Figure 5-5(a) shows the KR (symbols) and the current density (solid
lines) as a function of gate voltage (𝑉𝑔), as well as bias voltage (𝑉𝑑𝑠) (inset of Fig. 5-5(a)). A typical
n-type transport behavior is seen. The observed KR is linearly proportional to 𝐽. Second, we studied
the effect in the presence of an in-plane magnetic field in the Voigt geometry. No dependence of
the KR signal on the magnetic field is observed up to ± 0.5 T (Fig. 5-5(b)). The absence of the
Hanle effect (that is, Larmor precession of spin magnetization under an in-plane magnetic field
[193]) excludes the spin nature of the observed effect in case of weak spin-orbit coupling (SOC).
It is, however, fully compatible with valley magnetization, which does not couple with an in-plane
magnetic field [148]. Third, KR was measured as a function of the probe beam’s polarization
direction. The observed negligible dependence excludes strain-induced optical birefringence as a
possible origin of the observed KR (more details see Section 5.4).
Figure 5-5: Dependences of KR angle on magnitude of bias current and in-plane magnetic field.
(a) Gate dependence of the KR at a fixed location on the strained device (symbols) and gate
dependence of channel current density 𝐽 measured with 𝑉𝑑𝑠 = 2.5 V (red solid line). Inset shows
the corresponding bias dependences with 𝑉𝑔 = 20 V. (d) KR at a fixed location on the strained
device as a function of in-plane magnetic field B. Inset illustrates the Hanle effect for spin
magnetization 𝑀𝑆 and the absence of it for valley magnetization 𝑀𝑉.
100
5.3.3 Hamiltonian model for valley magnetoelectric effect in strained monolayer MoS2
To understand the microscopic mechanism of the observation, we consider the effect of
strain on the Dirac Hamiltonian of single-layer MoS2 in the k·p approximation [32], [194], [195],
𝐻0 = ℏ𝑣𝐹(𝑘𝑥𝜎𝑥 + 𝑠𝑘𝑦𝜎𝑦) +∆
2𝜎𝑧. Here ℏ and 𝑣𝐹 denote, respectively, the Planck’s constant and
the Fermi velocity, ℏ𝒌 = ℏ(𝑘𝑥�� + 𝑘𝑦��) is the crystal momentum measured from the K (s = +1)
and K’ (s = -1) point of the Brillouin zone, 𝝈 is the Pauli matrix for the sublattice index, and ∆ is
the band gap. We choose unit vector 𝑥 and �� along the armchair and zigzag direction, respectively
(�� corresponds to the out-of-plane direction). The SOC has been ignored since we only consider n-
type MoS2 and the SOC is weak in the lowest-energy conduction bands of MoS2 [31]. Under
uniform uniaxial strain along high-symmetry axes, the Hamiltonian is modified as 𝐻 ≈
ℏ(𝑣𝐹𝑥𝑘𝑥′ 𝜎𝑥 + 𝑠𝑣𝐹𝑦𝑘𝑦
′ 𝜎𝑦) +∆′
2𝜎𝑧 + 𝑠𝛼(𝑢xx − 𝑢𝑦𝑦)ℏ𝑣𝐹𝑘𝑦
′ [194], [195]. Here the crystal
momentum ℏ𝒌 is replaced by the canonical momentum ℏ𝒌′ due to a strain-induced fictitious vector
potential, and a new (third) term that does not couple to the Pauli matrix appears with 𝛼 ~ 1 and
𝑢ij denoting, respectively, a dimensionless parameter and a strain tensor element [194], [195]. The
Hamiltonian also takes into account the anisotropy in the Fermi velocity due to broken three-fold
rotational symmetry (𝑣𝐹𝑥 ≠ 𝑣𝐹𝑦) and the shift of the energy gap from ∆ to ∆′. The major effect of
these changes is the shift of the band extrema and the extrema of the Berry curvature distribution
(blue dashed lines) illustrated schematically in Fig. 5-1(c). The band extrema are shifted from the
original K/K’ point (black dashed lines) to opposite directions, which also do not coincide with the
extrema of the Berry curvature distribution. Under a bias field, a current is formed and the Fermi
surface tilts to the direction of the bias field. Although the two valleys maintain an equal population,
the modified band structure and Berry curvature distribution, in combination with the non-
equilibrium carrier distribution, lead to a net valley magnetization:
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𝑴𝑽 ≈3𝛼𝜀ℏ𝑣𝐹
𝑒𝑥𝑥Δ′ ℇ𝒑𝒛 × 𝑱. (5-1)
Here 휀 and 𝑒𝑥𝑥 denote, respectively, the dielectric constant and the piezoelectric tensor element of
single-layer MoS2, and only the leading-order term in both the piezoelectric field ℇ𝒑𝒛 and the
current density 𝑱 is considered. Similarly, the maximum absorbance difference for left- and right-
handed light (which determines the KR angle) is calculated by ignoring the excitonic effect as 𝐴+ −
𝐴− ≈ 𝑀𝑉𝐴Φ0
3Γ with Φ0 , A and Γ denoting, respectively, the magnetic flux quantum, the peak
absorbance and the width of the optical transition (See Sections 5.5 for detailed derivations).
Equation (5-1) reproduces our intuitive result for the valley ME effect in single-layer
TMDs as illustrated in Fig. 5-1(b). It is also consistent with the symmetry requirements [187], [190].
We consider the ME susceptibility 𝛼𝑖𝑗 that relates the bias electric field ℰ𝑗 to magnetization 𝑀𝑖 =
𝛼𝑖𝑗ℰ𝑗. Since single-layer MoS2 does not spontaneously break time-reversal symmetry, a charge
current under bias that causes dissipation is required for finite 𝛼𝑖𝑗 [188], [189]. For unstrained
single-layer MoS2 with D3h symmetry, the ME susceptibilities 𝛼𝑧𝑥 and 𝛼𝑧𝑦 that are relevant for the
generation of out-of-plane magnetization by an in-plane bias field all vanish [187], [190]. However,
under a uniaxial stress along the armchair or the zigzag direction, the symmetry point group is
reduced to C2v with the mirror line along the 𝑥 (armchair) axis. The 𝛼𝑧𝑦 component becomes
nonzero, and finite magnetization 𝑀𝑧 = 𝛼𝑧𝑦ℰ𝑦 emerges for a bias field with nonzero component
perpendicular to the mirror line (that is, ℰ𝑦 ≠ 0) [187], [190].
5.3.4 Dependences of the current and strain directions with respect to the crystalline axis
Our major experimental findings shown in Fig. 5-4 can also be explained by Equation (5-
1), namely, the need of finite strain to generate valley magnetization and the linear dependence of
102
the KR on the amplitude and sign of the current density. To further verify the vector relation of
𝑴𝑽 ∝ 𝓔𝒑𝒛 × 𝑱, we fabricated devices in the Corbino disk geometry, which allows current along
the radial direction over a wide range of angles relative to the applied stress. Two devices are shown
in Fig. 5-6 with a uniaxial stress applied along the armchair (upper row) and zigzag axes (lower
row). The piezoelectric field 𝓔𝒑𝒛 is along the 𝑥-axis with opposite direction in the two cases (solid
orange lines). Note the effect of tensile strain along the zigzag direction is similar to compressive
strain in the armchair direction. Both devices show positive KR in one half and negative KR in the
second half of the channel (Figs. 5-6(b) and (e)). In particular, the KR signal vanishes along the
orange line and reaches maximum approximately perpendicular to it. The observed patterns agree
reasonably well with the simulation result based on 𝑀𝑉 ∝ �� ∙ (𝓔𝒑𝒛 × 𝑱) for both devices (Figs. 5-
6(c) and (f)). (Details of simulation are provided in Section 5.4.4.) Deviations of the experimental
results from the predictions may originate from the presence of inhomogeneous strain in the
samples. Inhomogeneous doping and optical responses associated with it, as well as strain-induced
conductivity anisotropy may also play a role. However, the observed images with one-fold
symmetry cannot be explained by either the strain-induced conductivity anisotropy or the trigonal
warping effect of the band structure [185]. (See Sections 5.4.5 and 5.4.6 for more details.)
103
Moreover, Eqn. (5-1) predicts a KR signal level that is comparable to the value observed
in experiment as discussed below. The detailed analysis that relates the KR angle 𝜃𝐾𝑅 to the optical
properties of the sample has been discussed in Supplementary Section 6 of Ref.[184]. In short, the
KR angle is proportional to the difference in absorbance of left- and right-handed light at the probe
energy [184]: 𝜃𝐾𝑅 = 𝐼𝑚[𝛽𝑙𝑜𝑐(𝐴+ − 𝐴−)] . Here Im[…] denotes the imaginary part and 𝛽𝑙𝑜𝑐
describes the combined properties of the substrate, oxide layer and the local field factors that
influence wave propagation in the multilayer structure and is on the order of unity (𝛽𝑙𝑜𝑐 ~ 1). For
a typical measurement, a KR angle of ~ 100 𝜇𝑟𝑎𝑑 was observed at 10 K in MoS2 strained by ~ 0.5%
and in the presence of a current density of 𝐽 ~ 10 A/m. Using Eqn. (5-1) and the experimental
parameters we estimate the circular dichroism to be 𝐴+ − 𝐴−~ (2 − 3) × 10−5 (KR angle 𝜃 ~ 20
Figure 5-6: Dependence on current direction of the valley magnetoelectric effect. a, Piezoelectric
field 𝓔𝒑𝒛 is produced along the armchair direction when single-layer MoS2 is stressed along the
armchair direction; b, KR image of sample (a) measured in the Corbino disk geometry with 𝑉𝑑𝑠 =
1 V and 𝑉𝑔 = 25 V (𝐽 = 3.9 A/m); c, Predicted spatial distribution of valley magnetization by Eqn.
(5-1). The direction of 𝓔𝒑𝒛 is shown in orange lines. Boundaries of the electrodes and the device
channel are marked in dashed black and green lines, respectively. d to f are the same as a to c with
single-layer MoS2 stressed along the zigzag direction. The KR image was measured under 𝑉𝑑𝑠 =
6.9 V and 𝑉𝑔 = 20 V (𝐽 = 2.2 A/m).
104
– 30 𝜇𝑟𝑎𝑑) if we take Γ ~ 3.5 meV (thermal broadening), Δ′ ~ 2 eV, 𝑣𝐹 ≈ 0.6 × 106 ms-1 (ref.[32]),
𝛼 ~ 1 (ref.[194], [195]) and peak absorbance 𝐴 ~ 0.3 – 0.5 (in Fig. 5-18). We also estimate the
valley magnetization to be 𝑀𝑉 ~ 4 × 10−11 Amp. It corresponds to a volumetric converse
magnetoelectric coefficient 𝛼𝑧𝑦 ~ 𝜇0𝑀𝑉/ℇ𝑑𝑐𝑡 ~0.5 ps/m. Here 𝜇0 and 𝑡 (≈ 0.67 nm) denote the
vacuum permeability and the approximate thickness of monolayer MoS2, respectively. We thus
conclude that out-of-plane valley magnetization has been generated and observed by KR in strained
single-layer MoS2 through the valley ME effect.
5.3.5 Temperature dependence of valley magnetoelectric effect
Finally, we examine the temperature dependence of the effect. The gate dependence of the
two-point square conductance 𝜎 (Fig. 5-7(a)) and the KR (Fig. 5-7(b)) from the same device is
shown side by side at several selected temperatures. A clear metal-insulator crossover is observed
from the transport data around 𝑉𝑔 ≈ 12 V. The device channel shows a metallic behavior above
this gate voltage (that is, 𝜎 decreases with increasing temperature), and an insulating behavior
below it (that is, 𝜎 increases with temperature). The observed square conductance at the crossover
(𝜎 ≈ 20 μS) is smaller than the Ioffe-Regel criterion (𝜎 ≈ 40 μS) [196], which is likely due to an
underestimated experimental value by the two-point method with Schottky contacts. The KR of the
device under fixed strain shows a similar temperature dependence, as expected from 𝑴𝑽 ∝
𝓔𝒑𝒛 × 𝑱. The detailed temperature dependences of the two quantities, however, are not fully
identical since the parameters involved in these measurements (such as the optical conductivity at
the probe wavelength, the optical resonance width and the Schottky barrier height) are also
temperature dependent. Detailed studies of the temperature dependences are beyond the scope of
this work.
105
However, we draw attention to an important finding here: valley ME effect persists at room
temperature. The gate dependence of the valley magnetization at room temperature is shown in the
inset of Fig. 5-7(b), and the spatial image and bias dependence in Fig. 5-8. The observation of
electric-field induced valley magnetization at room temperature opens up the possibility of valley-
based applications, such as magnetic switching [191], [192] and flexible devices [99], [197], by
engineering the Berry curvature effects.
Figure 5-7: Temperature dependence of the valley magnetoelectric effect. Gate dependence of the
two-point conductance (a) and KR at a fixed location on a strained device (b) measured with 𝑉𝑑𝑠 = 2 V at four selected temperatures 10, 50, 90, and 110 K. Inset of b displays the gate dependence
of the KR (symbols) and conductance (red solid lines) at room temperature.
106
5.4 Extended support experiments for the observed valley magnetoelectricity
The nonzero Kerr rotation in the polar geometry observed in this experiment has been
interpreted as a result of out-of-plane magnetization (either near the channel edges in the VHE or
throughout the entire device channel in the ME effect). As we have argued in the main text and
supported by the in-plane magnetic field dependence (Fig. 5-5(b)), the magnetization is the valley
magnetization. We have also studied the dependence of the observed effect on strain, bias current,
and the direction of the current density with respect to the piezoelectric field through the Corbino
disk geometry. All of these results are consistent with the microscopic model of valley
magnetoelectricity. On the other hand, other effects could also give rise to the observed KR signal.
Figure 5-8: Valley ME effect at room temperature. a, Kerr rotation image of the device shown in
Fig. 5-6(b) at room temperature measured with Vds = 2.1 V and Vg = 20 V. b, The background noise
level measured with Vds = 0 and Vg = 0. c, Bias dependence of the KR at a fixed location of the
device channel (symbols) and the corresponding current density (solid red line) measured with Vg
= 20 V.
107
We performed several control experiments on both strained and unstrained samples to rule out these
potential artifacts. These studies include the dependence of the effect on the probe wavelength,
probe polarization, sample temperature, bias and gate voltage, and gate modulation. The bias and
gate dependences, as well as the temperature dependence, have been discussed in the section 5.3
(Figs. 5-4, 5-5, 5-6 and 5-7). In this section, we present studies on the potential effects from sample
birefringence, sample conductivity anisotropy, and source-gate coupling. We also present results
on the photon energy dependence of the observed KR signal.
5.4.1 Dependence of Kerr rotation on probe polarization
Anisotropy in the equilibrium optical response due to strain can give rise to KR although
such an effect is not expected to be bias dependent and to be detected using the lock-in method
with the bias voltage being modulated. We examined the effect of strain-induced birefringence on
the KR by measuring the KR as a function of the probe beam’s incident polarization. Strain-induced
Figure 5-9: Kerr rotation as a function of probe light polarization. KR was measured on the
strained device in Corbino disk geometry shown in Fig. 5-6(b) at two locations of the device
channel corresponding to two different bias fields (red and blue lines). The linear polarization of
the probe beam was varied by a half-wave plate and the sample was fixed.
KR
(µ
rad
)
108
birefringence would strongly depend on the relative angle between the light polarization and the
applied stress direction. In Fig. 5-9, we show the KR measured on a strained monolayer MoS2
device under two bias directions as a function of the probe beam polarization. No dependence is
found, confirming that strain-induced birefringence has a negligible effect on the measured KR.
5.4.2 Dependence of Kerr rotation angle on probe photon energy
The KR angle measures the left- and right-handed absorbance difference at the probe
photon energy (see sections 5.5.4 and 5.5.5). As a result the KR signal is expected to exhibit
significant photon energy dependence near the A exciton resonance. To verify this we have
measured the dependence of the KR signal on the probe photon energy, tuned by varying the base
temperature of our diode laser (range is thus limited). Figure 5-10 shows the dependence, which
exhibits a significant change near the A exciton resonance. The response is similar to Ref. [193].
Under this condition our probe is sensitive to the presence of a small valley magnetization.
Figure 5-10: Dependence of KR on probe photon energy. Photoluminescence spectrum shows the
A exciton resonance in single-layer MoS2. The peak energy is marked by the dashed line. KR was
measured as a function of probe photon energy, the range of which is limited by the tuning range
of our laser.
0
5000
10000
15000
PL r
eson
ance
(a.u
.)
1.85 1.90 1.95 2.00-40
-20
0
20
40
60
80
KR
(m
rad
)
KR
(m
rad
)
Photon energy (eV)
109
5.4.3 Kerr rotation under pure gate modulation
We evaluated the potential contribution of the source-gate coupling to the KR by measuring
the KR under a pure gate modulation. Instead of applying a modulating bias source-drain voltage
as in the original measurement, we applied a modulating gate voltage at 2.5 V peak-to-peak under
zero bias. The observed KR image (Fig. 5-11) shows negligible signal over the entire device
channel, suggesting that the source-gate coupling effect is negligible in our measurements. Fig. 5-
11 shows the result for the same device as in Fig. 5-4(a) without intentional strain. Similar results
were observed for strained devices.
5.4.4 Simulation of valley ME effect in Corbino disk geometry
To study the valley ME effect as a function of current relative to the stress direction, we
employed devices in the shape of a Corbino disk. The KR image from experiment and simulation
for two devices stressed along two different high-symmetry axes is shown in Fig. 5-6. Here we
Figure 5-11: Kerr rotation image under gate modulation. KR image of the unstrained device shown
in Fig. 5-4(a) measured with bias voltage Vds = 0 and gate voltage Vg modulated at 4.11 kHz with
a peak-to-peak amplitude of 2.5 V. Black and green dashed lines indicate the electrode and sample
boundaries, respectively. No identifiable signal can be observed anywhere on the device channel.
2 µm
KR (µrad)
-120 1200
110
show the details of the simulation. Figure 5-12(a) is the optical microscope image of one of the
devices with electrodes in gold, Si substrate in grey, and monolayer MoS2 in light blue. To find the
local bias electric field (and current density assuming an isotropic conductivity), we numerically
solved the Laplace equation with the boundary conditions given by the device geometry and known
electric potential on the electrodes using the Comsol package. Figures 5-12(b) and 3c show,
respectively, the electric potential and electric field amplitude (√ℰ𝑥′2 + ℰ𝑦′
2) distribution around
the Corbino disk when the inner and outer electrode potential are set at 5 V and 0 V, respectively.
The axes 𝑥′ and 𝑦′ are defined as shown in Fig. 5-12. The black arrows in Fig. 5-12(c) show the
direction and amplitude of the local bias electric field, which is mostly radial except near the edges
of the outer electrode. The result confirms the intuitive expectation that away from the edges, the
field distribution is radially uniform. Valley magnetization on the device channel was then
simulated using 𝑀𝑧 = 𝛼𝑧𝑦ℰ𝑦 by setting 𝛼𝑧𝑦 to be a constant and converting the calculated {ℰ𝑥′,
ℰ𝑦′} to {ℰ𝑥 , ℰ𝑦} in the basis of the high-symmetry axes of the crystal. The crystallographic
orientation of the sample was inferred from the second-harmonic generation (SHG) measurement.
The simulated spatial distribution of the valley magnetization is shown in Figs. 5-6 (c) and (f).
111
5.4.5 Effects of conductivity anisotropy
In Fig. 5-13 we show the simulation result for the spatial distribution of current density in
the Corbino disk devices by including strain-induced anisotropy in conductivity. We have assumed
that the conductivity is larger perpendicular to the strain direction. In the first set of simulation we
have assumed a small conductivity anisotropy (0.5%) and in the second set we have greatly
exaggerated the anisotropy to 50% in order to make the effect clearly seen. For comparison, we
have also included in the figure results of our experiment and simulation based on our model of
Figure 5-12: Electrostatic simulation of Corbino disk devices. a, Microscope image of a
monolayer MoS2 device with Corbino disk electrodes: electrodes in gold, Si substrate in grey and
monolayer MoS2 in light blue. b, c, Contour plot of the simulated electric potential (b) and electric-
field strength (c) of the device. Inner and outer electrode are fixed at 5 and 0 V, respectively. Black
arrows in c indicate the local bias field direction (arrow direction) and amplitude (arrow length).
b c
x' (µm)
y'(µ
m)
Electric potential (V)
2
1.5
1
0.5
0
Field strength (V/µm)
x' (µm)
y'(µ
m)
a
5 µm
112
valley ME effect (Fig. 5-6). The conductivity anisotropy cannot explain the experimental
observations. In Table S1 we compare their key differences.
Furthermore, we have performed two additional control experiments to support the above
conclusion. First, we note that any model that only concerns the magnitude of the conductivity will
not be able to explain the experimental observations since the effect also depends on the direction
Table 5-1: Comparison of the KR measurements and the model based on conductivity anisotropy
in the Corbino disk devices.
Experiment: KR near A
exciton resonance
Model: conductivity
anisotropy (larger
conductivity ⊥ strain)
Spatial distribution 1-fold symmetry 2-fold symmetry
Strain direction
- Armchair
- Zigzag
Maximum KR direction
- Zigzag
- Zigzag
Maximum current direction
- Zigzag
- Armchair
Figure 5-13: Simulation on Corbino disk devices. a, Piezoelectric field ℇ𝑝𝑧 is produced along the
armchair direction when single-layer MoS2 is stretched along the armchair direction. b, KR image
of sample (a) measured in the Corbino disk geometry with 𝑉𝑑𝑐 = 1 V and 𝑉𝑔 = 25 V. c, Predicted
spatial distribution of valley magnetization based on the valley ME effect. The direction of ℇ𝑝𝑧 is
shown in orange lines. Boundaries of the electrodes and the device channel are marked in dashed
black and green lines, respectively. d, Simulation of current density J including the conductivity
anisotropy induced by strain (0.5%, left column; 50%, right column). e to h are the same as a to d
with single-layer MoS2 stretched along the zigzag direction. The KR image was measured under
𝑉𝑑𝑐 = 6.9 V and 𝑉𝑔 = 20 V.
113
of the current. This has been seen above in Table 5-1 in the 1-fold symmetry of the KR image. This
can also be seen in Fig. 5-14, where the KR changes sign when the current direction is reversed.
Second, we performed KR mapping of the Corbino disk devices with different probe polarization
orientations with respect to the strain direction and obtained identical results (also see Fig. 5-9). If
the dominant mechanism were strain-induced conductivity anisotropy, the result would be
dependent on the polarization orientation with respect to the strain direction.
Based on these evidences we summarize that strain-induced conductivity anisotropy cannot
explain our experimental results using the Corbino disk geometry. However, it is conceivable that
the strain-induced conductivity anisotropy contributes to the experimental observation. It may, for
example, distort the ideal KR image (Fig. 5-6 or Fig. 5-13) in addition to doping and strain
inhomogeneities that are present in our devices.
5.4.6 Comparison to the effect of trigonal warping of the band structure
In Table 5-2 we compare the major differences between the effect observed in this work
and the effect reported in Ref. [185]. Ref. [185] reported an observation of polarized
Figure 5-14: KR mapping of Corbino disk device under reversed bias. KR image with current
flowing radially outward (a) and inward (b). The KR signal flips sign under reversed bias.
a b
200 -200 0
KR (µrad)
114
electroluminescence in forward biased p-n junctions in few-layer WSe2. The effect was modeled
as different electron-hole overlap at the K and K’ valleys under a bias field due to trigonal warping.
In our experiment, we observed KR in biased and strained monolayer MoS2 and modeled the effect
as strain- and current-induced valley magnetization, in analogue to the Rashba-Edelstein effect in
semiconductor quantum wells. The effect observed in this work arises from a completely different
mechanism and cannot be explained by the trigonal warping effect of the band structure.
Science, 344, 725 (2014)
Measurement: degree of circular polarization of
electroluminescence
Model: trigonal warping
Our work
Measurement: Kerr rotation near the A
exciton resonance
Model: strain- and current-induced valley
magnetization
(p) 3-fold symmetry as a function of current
direction: ∝ 𝑠𝑖𝑛3𝜑 (𝜑 is the azimuthal angle)
(o & p) 1-fold symmetry as a function of
current direction: ∝ 𝑠𝑖𝑛𝜑
(p) Peaked for current along the armchair
direction
(o & p) Peaked for current along the zigzag
direction
(o) Signal shows no to weak dependence on the
amplitude of the current.
(o & p) Signal depends linearly on the
amplitude of the current.
5.4.7 Valley Hall effect in single-layer MoS2
We show extra measurement results of the valley Hall effect (VHE) at 30 K in the
unstrained single-layer MoS2 device (Fig. 5-4(a)). Figure 5-15(a) (same as Fig. 5-4(a)) shows the
two-dimensional (2D) Kerr rotation (KR) image of the device. A clear KR signal is present only
near the two channel edges, as expected for the VHE [184]. An optical microscope image of the
device is shown in the lower inset of Fig. 5-15(b). The gate dependence of the two-point
conductivity (Fig. 5-15(b)) shows a typical n-type behavior. The upper inset shows the bias (Vds)
Table 5-2: Comparison of the effect reported in Ref. [185] and our work
(p-predicted property; o-observed property)
115
dependence of the current (I) with Vg = 20 V. The two-point mobility of the device was estimated
to be about 120 cm2/Vs. The gate dependence of the KR near the two edges (red and blue lines in
Fig. 5-15(c)) resembles that of the channel conductivity or the carrier density. Negligible KR signal
was measured at the center of the device channel (black line) for all gate voltages. In Fig. 5-15(d),
KR near one of the edges was shown as a function of the bias voltage Vds at several selected gate
voltages.
Figure 5-15: Valley Hall effect in single-layer MoS2 at 30 K. a, Kerr rotation microscope image
of an unstrained monolayer MoS2 device under Vsd = 2.5 V and Vg = 0 V (J = 22 A/m). The
boundaries of the electrodes and the device channel are shown in black and green dashed lines,
respectively. b, Conductivity as a function of gate voltage (Vg). Inset: current as a function of bias
voltage (Vds) with Vg = 20 V and a microscope image of the device. c, Gate dependence of the KR
at three different locations on the device channel indicated by circles in (a) measured with Vds = 2.5
V. Red, black and blue line correspond to KR at the lower edge, center, and top edge of the channel,
respectively. d, Bias dependence of the KR at selected gate voltages near the lower edge of the
channel.
b
c
a
2 µm
KR (µrad)
-120 1200
d
-20 -10 0 10 20
0
50
100
0 1 2 30
20
40
(S
)
Vg (V)
I (
A)
Vds
(V)
0 1 2 3
0
50
100
150
0 V
-20 V
KR
(ra
d)
Vds
(V)
Vg = 20 V
-20 -10 0 10 20-100
-50
0
50
100
KR
(ra
d)
Vg (V)
116
For comparison we also show in Fig. 5-16 the images of normalized KR signals by the
channel current density of unstrained (left) and strained (right) monolayer MoS2. The KR signal
from the VHE is much smaller than that from the magnetoelectric (ME) effect under the same
current density.
5.5 Microscopic model of valley magnetization in strained and biased MoS2
5.5.1 Derivation of the piezoelectric field in single-layer MoS2 under a uniaxial stress
In monolayer MoS2, which lacks inversion symmetry, uniaxial stress results in a nonzero
piezoelectric field 𝓔𝒑𝒛 as a result of strain-induced lattice distortion and the associated ion charge
polarization [99], [197]. In this section, we show the relation between strain and 𝓔𝒑𝒛 by considering
crystal’s symmetry.
The linear piezoelectric effect can be described as the first-order coupling between surface
polarization vector 𝑷 and mechanical strain tensor 𝑢𝑗𝑘: 𝑃𝑖 = 𝑒𝑖𝑗𝑘𝑢𝑗𝑘 with 𝑒𝑖𝑗𝑘 denoting the third-
Figure 5-16: Image of normalized KR signal by the channel current density for unstrained (left)
and strained (right) monolayer MoS2.
2 µm
KR per current density (mrad·m/A)
-250
250 0
117
rank piezoelectric tensor 34. For monolayer MoS2, which belongs to D3h point group, 𝑒𝑖𝑗𝑘 has only
one nonzero independent coefficient. By employing the Voigt notation to reduce the number of
indices, the piezoelectric tensor can be written as [99], [197]–[199]
[𝑒] = (𝑒𝑥𝑥 −𝑒𝑥𝑥 00 0 00 0 0
0 0 00 0 −𝑒𝑥𝑥0 0 0
), (5-2)
where x, y and z are along the armchair, zigzag, and out-of-plane direction. By employing [𝑢𝑇] =
(𝑢𝑥𝑥, 𝑢𝑦𝑦, 𝑢𝑧𝑧, 2𝑢𝑦𝑧, 2𝑢𝑥𝑧, 2𝑢𝑥𝑦), the resulting surface polarization becomes
[𝑃] = [𝑒𝑢] = 𝑒𝑥𝑥 (
𝑢𝑥𝑥 − 𝑢𝑦𝑦−2𝑢𝑥𝑦0
). (5-3)
In our devices, stress 𝑡𝑘𝑙 was applied to monolayer flakes to induce strain 𝑢𝑖𝑗 = 𝑆𝑖𝑗𝑘𝑙𝑡𝑘𝑙,
where 𝑆𝑖𝑗𝑘𝑙 is the symmetry-constrained compliance. For D3h point group, it has the following form
[198]:
[𝑆] =
(
𝑆11 𝑆12 𝑆13𝑆12 𝑆11 𝑆13𝑆13 𝑆13 𝑆33
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
𝑆44 0 0 0 𝑆44 0 0 0 2(𝑆11 − 𝑆12))
. (5-4)
When uniaxial stress is applied along the x-axis (armchair direction), i.e. only 𝑡𝑥𝑥 ≠ 0, the
resulting strain tensor becomes [𝑢𝑇] = 𝑡𝑥𝑥(𝑆11, 𝑆12, 𝑆13, 0,0,0), giving rise to a piezoelectric
polarization 𝑷 = 𝑒𝑥𝑥𝑡𝑥𝑥(𝑆11 − 𝑆12, 0,0) pointing in the armchair direction. When stress is
applied along the y-axis (zigzag direction), however, only tyy ≠ 0 , which induces a piezoelectric
field 𝑷 = 𝑒𝑥𝑥𝑡𝑦𝑦(𝑆12 − 𝑆11, 0,0) of opposite sign.
118
5.5.2 Electronic band structure of strained single-layer MoS2
The 𝑘 ∙ 𝑝 Hamiltonian of single-layer MoS2 ignoring the spin-orbit coupling (SOC) can be
written as [32], [178]
𝐻0 = ℏ𝑣𝐹(𝑘𝑥𝜎𝑥 + 𝑠𝑘𝑦𝜎𝑦) +∆
2𝜎𝑧. (5-5)
Here ℏ = ℎ/(2𝜋) is the Planck’s constant, 𝑣𝐹 is the Fermi velocity, ℏ𝒌 = ℏ(𝑘𝑥�� + 𝑘𝑦��) is the
crystal momentum counted from the K (s = +1) and K’ (s = -1) point of the Brillouin zone, 𝝈 is the
Pauli matrix accounting for the sublattice index, and ∆ is the band gap. The corresponding basis
functions for the conduction band |𝑐⟩ and the valence band |𝑣⟩ are
|𝑐⟩ = |𝑑𝑧2⟩, |𝑣⟩ =1
√2(|𝑑𝑥2−𝑦2⟩ + 𝑖𝑠|𝑑𝑥𝑦⟩), (5-6)
where |𝑑𝑧2⟩ , |𝑑𝑥2−𝑦2⟩ and |𝑑𝑥𝑦⟩ are the d-orbitals of the Mo atoms. The valence band
wavefunction at the two valleys is related by the time reversal operation. The SOC is weak in the
conduction bands of MoS2 and is ignored for n-doped samples that are studied in this work. The
above simple two-band model with electron and hole symmetry is thus a good description of n-
doped single-layer MoS2.
In general, the effects of uniform uniaxial strain on the Hamiltonian include 30,31: 1)
modification of the Fermi velocity because of the broken 3-fold rotational symmetry (the Fermi
velocity along the armchair and the zigzag directions 𝑣𝐹𝑥 ≠ 𝑣𝐹𝑦 ); 2) change of the crystal
momentum ℏ𝒌 to the canonical momentum ℏ𝒌′ = ℏ𝒌 −𝑒
𝑐𝑨, where 𝑨 is a strain-induced fictitious
vector potential, e and c denote the elementary charge and speed of light in vacuum, respectively;
3) modification of the energy gap ∆→ ∆′; and 4) addition of a new valley-dependent term that does
not couple to the sublattice pseudospin Pauli matrices 𝑠𝛼(𝑢𝑥𝑥 − 𝑢𝑦𝑦)ℏ𝑣𝐹𝑘𝑦′ . Here 𝛼 is a
119
dimensionless parameter on the order of unity and 𝑢𝑖𝑗 is a strain tensor element [194], [195]. The
Hamiltonian for single-layer MoS2 strained along high-symmetry axes can be written as
𝐻 ≈ ℏ𝑣𝐹𝑥𝑘𝑥′ 𝜎𝑥 + 𝑠ℏ𝑣𝐹𝑦𝑘𝑦
′ 𝜎𝑦 +∆′
2𝜎𝑧 + 𝑠𝛼(𝑢𝑥𝑥 − 𝑢𝑦𝑦)ℏ𝑣𝐹𝑘𝑦
′ , (5-7)
where higher-order terms in strain 𝑢𝑖𝑗 have been ignored. The eigen-energies of the Hamiltonian
can be found as
𝜖𝑐,𝑣(𝒌′) = ±𝜖(𝒌′) + 𝑠𝛼(𝑢𝑥𝑥 − 𝑢𝑦𝑦)ℏ𝑣𝐹𝑘𝑦′ ,
(5-8)
𝜖(𝒌′) =1
2√∆′2 + 4ℏ2(𝑣𝐹𝑥
2 𝑘𝑥′ 2 + 𝑣𝐹𝑦
2 𝑘𝑦′ 2).
Here the “+” and “-“ sign are for the energy dispersion of the conduction band (𝜖𝑐) and the valence
band (𝜖𝑣 ), respectively. Similarly we can find the eigenstates of the Hamiltonian. The Berry
curvature of the bands, defined through the periodic part of the Bloch wavefunction 𝑢(𝒌) as
𝛀(𝒌) = 𝛁𝒌 × ⟨𝑢(𝒌)|𝑖𝛁𝒌|𝑢(𝒌)⟩, can be shown for the conduction band as [31], [32]
𝛀(𝒌′) = sΩ(𝒌′)�� = −sℏ2𝑣𝐹
2∆′
4[𝜖(𝒌′)]3��. (5-9)
For relatively small strain levels, we can approximate 𝑣𝐹𝑥 ≈ 𝑣𝐹𝑦 ≈ 𝑣𝐹 and the second line
of Eq. (5-8) becomes 𝜖(𝒌′) ≈∆′
2+ℏ2𝑘′
2
2𝑚, where 𝑚 =
∆′
2𝑣𝐹2 is the band mass. With this approximation
the energy dispersion can be expressed as
𝜖𝑐(𝒌′) ≈∆′
2+ℏ2(𝑘′+𝑠𝑘0)
2
2𝑚, 𝜖𝑣(𝒌′) ≈ −
∆′
2−ℏ2(𝑘′−𝑠𝑘0)
2
2𝑚 , (5-10)
where the shift of the band extrema with respect to 𝒌′ = 0 is given by 𝒌𝟎 = 𝑘0�� ≡𝛼(𝑢𝑥𝑥−𝑢𝑦𝑦)Δ
′
2ℏ𝑣𝐹��.
On the other hand, the extrema of the Berry curvature distribution is not shifted from 𝒌′ = 0 since
it is a property of the wavefunctions and the wavefunctions are not affected by the last term in the
120
Hamiltonian that does not couple to the sublattice pseudospin Pauli matrices [190]. As we show
below that this leads to nonzero valley magnetization in the presence of a bias current. We note that
the strained Hamiltonian 𝐻 of Eq. (5-7) is very similar to that for a strained topological crystalline
insulator analyzed in Ref. [190]. The derived results of Eq. 5-8 and 5-9 are also consistent with the
results for the strained topological crystalline insulator [190].
5.5.3 Valley magnetization
It has been shown by Xiao et al. that the valley magnetization 𝑴 of an electron gas in a 2D
Dirac material (n-doped MoS2 in this case) can be calculated as [178]
𝑴 ≈2𝑒
ℏ∫
𝑑𝒌
(2𝜋)2𝑓(𝒌)(
Δ
2+ 𝜇)𝛀(𝒌). (5-11)
Here the integration is over the entire Brillouin zone, 𝑓(𝒌) is the electron distribution function, and
𝜇 is the chemical potential counted from the bottom of the conduction band. Note that 𝑴 is out-of-
plane since the Berry curvature is out-of-plane. We consider strained monolayer MoS2 under a
constant bias field ℇ𝒅𝒄 with both the sample and bias field being spatially uniform. In the absence
of the bias field, the electrons at temperature T are described by the Fermi-Dirac distribution 𝑓𝑐0(𝐸,
Δ
2+ 𝜇, 𝑇) with 𝐸 = 𝜖𝑐(𝒌). It can be shown that the valley magnetization vanishes since the Berry
curvature has opposite values in the two valleys (Eq. 5-9). Under the bias field, a non-equilibrium
correction to the distribution function emerges and can be approximated as
𝑓𝑐1 =𝑒𝜏
ℏℇ𝒅𝒄 ⋅ 𝛁𝒌𝑓𝑐0, (5-12)
where 𝜏 is the electron scattering time. This correction can be obtained under the relaxation time
approximation from the Boltzmann equation for the distribution of electrons.
121
Below we calculate Eq. 5-11 under the non-equilibrium carrier distribution. The integration
can be broken into the contribution from the two valleys explicitly. We note that the energy
dispersion is centered at momentum 𝒌′ = −𝑠𝒌𝟎, whereas the Berry curvature is centered at 𝒌′ =
0. We change variables so that the energy is centered at 0 momentum and the Berry curvature at
𝑠𝒌𝟎 for simplicity of the calculation. Equation 5-11 now can be rewritten for strained MoS2 as
𝑀 ≈𝑒
ℏΔ′ ∫
𝑑𝒌
(2𝜋)2𝑓𝑐1(𝒌)[Ω(𝒌 + 𝒌𝟎) − Ω(𝒌 − 𝒌𝟎)], (5-13)
where the integration is for one valley only and 𝑓𝑐1(𝒌) = 𝑓𝑐1(𝐸,Δ′
2+ 𝜇, 𝑇) with 𝐸 ≈
∆′
2+ℏ2𝑘2
2𝑚. In
Eq. 5-13 we have also used the fact that the band gap energy (~ 2 eV) far exceeds the chemical
potential (~ 10’s meV) so that Δ′
2+ 𝜇 ≈
Δ′
2. Next, we plug Eq. 5-12 into Eq. 5-13 and do the
integration by parts
𝑀 = −Δ′𝑒2𝜏
ℏ2ℇ𝒅𝒄 ⋅ ∫
𝑑𝒌
(2𝜋)2𝑓𝑐0(𝒌)[𝛁𝒌Ω(𝒌 + 𝒌𝟎) − 𝛁𝒌Ω(𝒌 − 𝒌𝟎)]
(5-14)
≈12𝑒2𝜏
𝑚Δ′ℇ𝒅𝒄 ⋅ 𝒌𝟎 ∫
𝑑𝒌
(2𝜋)2𝑓𝑐0(𝒌).
We have used 𝜕Ω
𝜕𝜖≈ −6
ℏ2
𝑚Δ′2. Expressing the remaining integration through the chemical
potential, ∫𝑑𝒌
(2𝜋)2𝑓𝑐0(𝒌) =
𝑚𝜇
2𝜋ℏ2, we finally obtain
𝑀 = 12𝑒
ℎ(𝜇
Δ′)(𝑒𝜏ℇ𝒅𝒄)⋅(ℏ𝒌𝟎)
𝑚=
6
Φ0(𝜇
Δ′)(𝑒𝜏ℇ𝒅𝒄)⋅(ℏ𝒌𝟎)
𝑚. (5-15)
The second half of the equation expresses the valley magnetization in terms of the magnetic
flux quantum Φ0 =ℎ
2𝑒 . The 2D valley magnetization has the unit of [energy]/[magnetic flux
quantum] = Ampere. It is out-of-plane. Its magnitude is determined by ℇ𝒅𝒄 ⋅ 𝒌𝟎. Since 𝒌𝟎 = 𝑘0��,
122
it’s most efficient to generate valley magnetization by biasing the sample in the same direction as
𝒌𝟎, i.e. along the zigzag direction.
We further note that the shift 𝒌𝟎 can also be expressed in terms of the piezoelectric field
ℇ𝒑𝒛 [194], [195], which is related to strain as εℇ𝒑𝒛 = [𝑒𝑢] with 휀 and [𝑒] denoting the dielectric
constant and the piezoelectric tensor. For uniform uniaxial in-plane strain, 𝒌𝟎 and ℇ𝒑𝒛 are related
as
𝒌𝟎 =𝛼𝜀Δ′
2𝑒𝑥𝑥ℏ𝑣𝐹�� × ℇ𝒑𝒛. (5-16)
The piezoelectric field is along the armchair direction, i.e. ℇ𝒑𝒛 = ℇ𝑝𝑧𝑥 for strain along the high
symmetry axes. We rewrite Eq. 5-15 as
𝑴 =3𝛼𝜀ℏ𝑣𝐹
𝑒𝑥𝑥Δ′ ℇ𝒑𝒛 × 𝑱, (5-17)
where 𝑱 is the current density and defined by the bias field, 𝑱 =2𝜇
𝜋ℏ2 𝑒2𝜏ℇ𝒅𝒄. We have thus obtained
the intuitive result of the main text that 𝑴 ∝ ℇ𝒑𝒛 × 𝑱. Although there is no population imbalance
between the two valleys, the net valley magnetization is nonzero in strained and biased monolayer
MoS2 due to the distinct carrier configuration of the two valleys. From the symmetry point of view,
unstrained single-layer MoS2 has D3h symmetry. The uniaxial in-plane strain along high symmetry
axes breaks the rotational symmetry and leaves only one mirror symmetry line along the armchair
direction. The resultant piezoelectric field is along the symmetry line. A net valley magnetization
emerges as long as there is a nonzero current component perpendicular to the symmetry line.
Our results can be intuitively understood based on Fig. 5-17. Figure 5-17(a) shows the
conduction bands (orange) and valence bands (blue) of unstrained monolayer MoS2 at the K and
K’ valley. The Fermi surface tilts at each valley under a bias current, but the extrema of both the
123
bands and the Berry curvature distribution are centered at the K and K’ point (black dashed lines).
The contribution to the valley magnetization from the two valleys cancels exactly since the Berry
curvature is of opposite sign at the two valleys. When monolayer MoS2 is strained, both the bands
and the Berry curvature distribution shift either closer to or further away from each other in
momentum (depending on whether it’s tensile or compressive strain). The dashed blue lines
illustrate the extrema of the Berry curvature distribution, which do not coincide with the band
extrema. The contribution to the valley magnetization from the two valleys no longer cancels
although the carrier population of the two valleys remains equal.
Figure 5-17: Band structure of monolayer MoS2 at the K and K’ point. The shaded regions
illustrate the occupied states in the conduction band. a, Unstrained monolayer MoS2 under a finite
bias field, giving rise to a tilted Fermi surface. b, Strained monolayer MoS2 under bias. Net valley
magnetization is generated in this configuration. The dashed black lines indicate the K/K’ point
and dashed blue lines indicate the extrema of the Berry curvature distribution.
124
5.5.4 Circular dichroism
A material’s response to an oscillating electric field at angular frequency 𝜔, ℇ𝒐𝒑(𝑡) =
𝑅𝑒{ℇ𝒐𝒑𝑒𝑖𝜔𝑡}, can be divided into interband and intraband transitions. For photon energies ℏ𝜔 ~ Δ,
the latter contribution is insignificant and can be ignored. In monolayer MoS2 the interband
transitions are modified by the strong Coulomb interactions to form excitonic states. For simplicity,
below we analyze circular dichroism in strained and biased monolayer MoS2 by ignoring the
excitonic interactions. The optical conductivity tensor element (𝜎𝛼𝛽) for interband transitions under
a two-band approximation can be evaluated as
𝜎𝛼𝛽(𝜔) =𝑖ℏ𝑒2
𝑚2Δ∫
𝑑𝒌
(2𝜋)2(𝑓𝑣 − 𝑓𝑐)𝑝𝑐𝑣,𝛼𝑝𝑣𝑐,𝛽𝛿(𝜖𝑐 − 𝜖𝑣 − ℏ𝜔). (5-18)
Here the integration is over the entire Brillouin zone, 𝑓𝑐 and 𝑓𝑣 (≈ 1) are the conduction band and
valence band carrier distribution functions, respectively, 𝒑𝒄𝒗 = ⟨𝑐|𝒑|𝑣⟩ is the transition matrix, and
𝛿(𝑥) is the Dirac delta function. It has been shown by Xiao et al. [32] that the transition matrix
element near the K (s = +1) and K’ (s = -1) point for left-handed (+) and right-handed (-) light can
be expressed as
|𝒑±(𝒌)|𝟐 ∝ (1 ± 𝑠
Δ
𝟐𝜖(𝒌))𝟐. (5-19)
For strained samples, the transition matrix keeps the same form with Δ → Δ′ and 𝒌 → 𝒌′
since only the wavefunctions are involved. We compute the normalized circular dichroism 𝜌(𝜔) =
𝜎+(𝜔)−𝜎−(𝜔)
𝜎+(𝜔)+𝜎−(𝜔) [or equivalently, 𝜌(𝜔) =
𝐴+(𝜔)−𝐴−(𝜔)
𝐴+(𝜔)+𝐴−(𝜔), defined through the absorbance for the left- and
right-handed light 𝐴+(𝜔) and 𝐴−(𝜔), respectively) following the same approach as for valley
magnetization:
𝜎+(𝜔) − 𝜎−(𝜔) ≈ −𝐶4𝑒𝜏ℇ𝒅𝒄⋅𝒌𝟎
𝜋Δ′ℏ(𝜖
𝜕𝑓𝑐0
𝜕𝜖)|𝜖=(ℏ𝜔−Δ′) 2⁄
(5-20)
125
𝜎+(𝜔) + 𝜎−(𝜔) ≈ 𝐶2𝑚
𝜋ℏ2Θ(ℏ𝜔 − Δ′ − 2μ),
where constant 𝐶 is a combination of the materials parameters. We note that in the above
derivations we have assumed the thermal/lifetime broadening of the energy levels is larger than the
shift of the transition energies due to strain and bias. This is justified since the momentum shift due
to strain 𝑘0 or bias 𝑒𝜏
ℏℇ𝑑𝑐 is a small fraction of the Fermi wavevector (10-2 – 10-1)√2𝑚𝜇 ℏ⁄ under
typical experimental conditions and the corresponding energy shift is <1 meV.
The circular dichroism is dependent on the photon energy. We estimate its peak value by
using ℏ𝜔 ≈ Δ′ + 2μ and (𝜖𝜕𝑓𝑐0
𝜕𝜖)|𝜖=(ℏ𝜔−Δ′) 2⁄
≃ −𝜇
Γ, where the width Γ is given by 4𝑘𝐵𝑇 if the
transition is primarily thermally broadened, or by the experimental transition linewidth if the
transition is primarily disorder broadened. The normalized circular dichroism is thus estimated as
𝜌 =2
Γ(𝜇
Δ′)(𝑒𝜏ℇ𝒅𝒄)⋅(ℏ𝒌𝟎)
𝑚= 𝑀
Φ0
3Γ∝ 𝑀. (5-21)
Our result shows that the circular dichroism is linearly proportional to the net out-of-plane valley
magnetization, which is non-vanishing in strained and biased monolayer MoS2.
5.5.5 Comparison of microscopic theory with experiment
We compare the result of the above microscopic theory (Eq. 5-21) with typical
experimental results: a KR angle 𝜃𝐾𝑅 of ~ 100 𝜇𝑟𝑎𝑑 was observed at 10 K in monolayer MoS2
strained by ~ 0.5% along one of the high symmetry axes and in the presence of a current density of
𝐽 ~ 10 A/m. We estimate the normalized circular dichroism using Eq. 5-21 to be 𝜌 ~ 5 × 10−5 if
we take Γ ~ 3.5 meV (thermal broadening at 10 K), Δ′ ~ 2 eV, 𝑣𝐹 = 0.6 × 106 ms-1 (Ref. [32]) and
𝛼 ~ 1 (Ref. [194], [195]) (note that 𝛼 for single-layer MoS2 has not been well calibrated although
it is expected to be on the order of unity). Furthermore, by combining with the low-temperature
126
peak absorbance of ~ 0.3 – 0.5 (depending on the transition width of the sample) from experiment
(Fig. 5-18), we obtain 𝐴+ − 𝐴−~2− 3 × 10−5.
On the other hand, in the polar Kerr rotation microscopy as employed in this study, linearly
polarized light impinges on a sample under normal incidence and the polarization of the reflected
light is analyzed. The KR angle 𝜃𝐾𝑅 is defined as the rotation of the polarization upon reflection.
We have shown previously that the KR angle 𝜃𝐾𝑅 can be related to the absorbance difference for
left- and right-handed light through a local field factor 𝛽𝑙𝑜𝑐 due to interference of the probe in the
multiple thin film structure of our devices (MoS2 on SiO2/Si substrates) [184]:
𝜃𝐾𝑅(𝜔) = 𝐼𝑚{𝛽𝑙𝑜𝑐[𝐴+(𝜔) − 𝐴−(𝜔)]}. (5-22)
The local field factor is on the order of unity (𝛽𝑙𝑜𝑐 ~ 1). The microscopic theory thus
predicts that the KR angle 𝜃𝐾𝑅 ~ 20 – 30 𝜇𝑟𝑎𝑑. Despite of the approximations employed in the
model (for instance, ignoring the excitonic effects), it predicts a KR level that is comparable to the
measured one.
The theory also predicts valley magnetization 𝑀 ~ 4 × 10−11 Amp. It corresponds to a
volumetric converse magnetoelectric coefficient of 𝜇0𝑀/ℇ𝑑𝑐𝑡 ~ 0.5 ps/m. Here 𝜇0 and 𝑡 (≈
Figure 5-18: Optical absorbance of monolayer MoS2 near the A exciton resonance at 10 K.
127
0.67 nm) denote the vacuum permeability and the approximate thickness of monolayer MoS2,
respectively. Note that the value is comparable to that of Cr2O3, one of the known single-phase
materials with large converse ME coefficients [200].
5.6 Conclusion
In this chapter, by breaking the three-fold rotational symmetry in single-layer MoS2 via a
uniaxial stress, we have demonstrated the pure electrical generation of valley magnetization in this
material, and its direct imaging by Kerr rotation microscopy. The observed out-of-plane
magnetization is independent of in-plane magnetic field, linearly proportional to the in-plane
current density, and optimized when the current is orthogonal to the strain-induced piezoelectric
field. These results are fully consistent with a theoretical model of valley magnetoelectricity driven
by Berry curvature effects. Furthermore, the effect persists at room temperature, opening
possibilities for practical valleytronic devices.
Chapter 6
Summary and future perspectives
In summary, this dissertation has demonstrated the interaction of nanomechanics with
photonics and valley-based optoelectronics using a monolayer TMD semiconductor. In particular,
we have been focusing on the role of exciton resonance with valley DOF in optomechanical
coupling. In Chapter 3, we have shown the realization of a robust optical bistability when optically
pumped with moderate power near exciton resonance of a suspended WSe2 monolayer. A
photothermal mechanism provides both optical nonlinearity and internal positive feedback that are
essential prerequisites of the observed bistable exciton resonance. The presence of an external
magnetic field further allows control of the sample reflectivity by light helicity due to its internal
valley DOF.
Moreover, in Chapter 4, we have presented the first dynamical control of mechanical
motion of a suspended MoSe2 monolayer through its exciton resonance without an optical cavity
structure. The oscillating membrane can periodically modulate a photothermal force that originated
from excitonic absorbance and build up a dynamical backaction on its mechanical motion. In
addition, gate tunable optomechanical damping and anti-damping of the mechanical vibrations due
to photothermal backaction have been observed in real two-dimensional limit.
Another important aspect of this dissertation is the generation of magnetization through the
control of electrons’ valley magnetic moment, which originated from Berry phase effect. Unlike
the magnetization induced from unbalanced population between valleys, valley magnetization can
also arise from unpolarized valleys by manipulating carriers’ momentum distribution. In Chapter
5, we have reported the realization of such magnetization without valley polarization through the
joint effect of current and strain in a monolayer MoS2. By lowering the symmetry of the MoS2
129
lattice with uniaxial strain, the distributions of valley magnetic moment at valleys are displaced
with respect to the corresponding valley centers. In addition, the tilting Fermi pockets at K and K’
valleys, when flowing a charge current, can enclose inequivalent magnitude of magnetic moment.
This valley-unpolarized non-equilibrium momentum distribution thus gives rise to a net
magnetization, namely, valley magnetoelectric effect. Our finding that can persist up to room
temperature implies a practical possibility for integrating the valleytronics with magnetic
information processing.
For future perspectives, it would be still interesting to investigate some unexplored aspects
from the demonstrated studies in this dissertation. In principle, together with extreme lightweight,
the strong excitonic effect in monolayer TMD semiconductor make the material as a promising
platform to study cavity-less optomechanics. In our device structure, the photothermal-based
optomechanical effect overwhelmingly dominates over the radiation pressure-based counterpart.
However, there are several limitations that prevents us from better resolving the effects from
dynamical backactions and obtaining more convincing data. The first issue is that high laser power
could not be applied due to the low thermal conductance of monolayer TMD, which would greatly
limit the induced optomechanical effect. Otherwise, the significant photothermal heating can easily
lead to the bistability shown in Chapter 3 instead of enhanced optomechanical effect. For a larger
photothermal coupling, a high symmetry-breaking force (by gate voltage) would be desired but
simultaneously a large gate voltage could lower the quality factor of mechanical mode due to the
Joule dissipation of displacement current through the oscillation membrane. In low temperature,
exciton resonance in monolayer TMD semiconductor would become sharp with great contrast. But
it also requires much higher gate voltage to pull down the suspended monolayer since the
membrane becomes very stiff at cryogenic condition. Namely, there is a tradeoff between
symmetry-breaking force and quality factor to improve the photothermal coupling with applying
gate voltage. Also, shrinking the area of suspended membrane could help to avoid the generation
130
of bistability but it also requires much higher gate voltage to pull the membrane down. To observe
much cleaner data for optical spring effect and optical damping, one need to do more thoughtful
simulation and optimization in the geometry of device to obtain pronounce effects from
optomechanical backaction.
Although, the optomechanical contribution from radiation pressure was almost negligible,
it would be still interesting to investigate the cavity optomechanics using excitonic effect in
monolayer TMD semiconductor. Instead of directly forming resonator with monolayers, it would
be desirable to put the monolayer semiconductor into a high-finesse cavity or place it very close to
an optical microcavity to enhance the radiation pressure felt by the membrane. Thus, lots of
interesting physics could arise from interfacing such exciton-polariton schematics with
nanomechanics. Compared to photothermal optomechanics, the interaction between exciton and
confined photons paves attractive routes towards cooling to the quantum ground state. In addition,
it would be highly interesting to investigate the generation of frequency comb by an optical micro-
resonator integrated with monolayer TMD semiconductors.
Within the past two years, the discovery of atomically thin magnetic materials opens a
window to a richer and largely unexplored regime in low dimensional physics, as well as studies
done in this dissertation. As indicating in Chapter 3, it would be encouraging to realize the helicity
control of optical switching without an external magnetic field in monolayer TMD semiconductor,
which can be magnetically proximitized by a neighboring 2D magnet (e.g. CrBr3). Meanwhile,
strain can play a very important role in the intrinsic magnetization of 2D magnetic materials. The
phase transition between anti-ferromagnetism (AFM) and ferromagnetism (FM) via strain have
been predicted in bilayer 2D magnetic materials without solid experimental evidence to date. In
addition, it would also be interesting to explore the strain-induced change in internal magnetization
of 2D magnet, where strain bridges the internal spin macroscopic mechanical DOF. The strong
magneto-elastic coupling within magnetic 2D atomic crystals (e.g. CrI3) would be another highly
131
interesting topic that deserves thorough investigation. Meanwhile, as analogy to exciton-
optomechanics in Chapter 4, it would be highly interesting to study the coupling between
ferromagnetic resonance (FMR) and mechanics in a 2D magnet resonator, which can transduce
spin information into mechanical state. Such transduction between quantum mechanics and macro-
mechanics can also serve as a frequency transducer between GHz and MHz, which are normal
frequency range for FMR and mechanical modes in our device, respectively. And these novel spin-
mechanical devices, like above-mentioned exciton-optomechanical counterparts, may enable new
classes of measurements and find more interesting applications in the near future.
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VITA
Hongchao Xie
EDUCATION
Visiting Ph.D. student 01/2018-07/2019
Cornell University, Ithaca, NY Advisor: Prof. Kin Fai Mak
Ph.D. in Physics 08/2014-08/2019
The Pennsylvania State University, University Park, PA Advisor: Prof. Kin Fai Mak
MPhys (Hons) Physics (1st class) 09/2012-06/2014
The University of Manchester, Manchester, UK Advisor: Prof. Kostya Novoselov
B.S. in Physics 09/2009-06/2013
Shandong University, Jinan, China
AWARDS
Homer F. Braddock Graduate Fellow Scholarship, The Pennsylvania State University, 2014
College Scholarship, Shandong University, 2010 & 2011
PUBLICATIONS
• H. Xie, J. Shan, K. F. Mak, “Tunable exciton-optomechanical coupling in suspended
monolayer MoSe2”, in preparation.
• H. Xie, S. Jiang, J. Shan, K. F. Mak, “Valley-selective exciton bistability in a suspended
monolayer semiconductor”, Nano Lett. 2018 (5), 3213-3220.
• J. Lee, Z. Wang, H. Xie, J. Shan, K. F. Mak, “Valley magnetoelectricity in single-layer MoS2”,
Nat. Mater. 2017, 16, 887-891
• G. M. Stiehl, R. Li, V. Gupta, I. El Baggari, S. Jiang, H. Xie, L. F. Kourkoutis, K. F. Mak, J.
Shan, R. A. Buhrman, D. C. Ralph, “Layer-dependent spin-orbit torques generated by the
centrosymmetric transition metal dichalcogenide 𝛽-MoTe2”, arXiv:1906.01068.