Philip KimPhysics DepartmentHarvard University

Commensuration and Incommensurationin the van der Waals Heterojunctions

graphene

hexa-BN

X

X

M

C

BN

Metal-Chalcogenide

M = Ta, Nb, Mo, Eu …

X = S, Se, Te, …

Bi2Sr2CaCu2O8-x

Charge Transfer Bechgaard Salt

(TMTSF)2PF6

Lead Halide Layered Organic

• Semiconducting materials: WSe2, MoSe2, MoS2, …

• Complex-metallic compounds : TaSe2, TaS2, …

• Magnetic materials: Fe-TaS2, CrSiTe3 ,…

• Superconducting: NbSe2, Bi2Sr2CaCu2O8-x, ZrNCl,…

Rise of 2D van der Waals Systems

AB

AC

A

5o 10o

15o20o30o

Graphen/hBN Moire Pattern

Not commensurated except for special angles

Moire pattern in Graphene on hBN: Scanning Probe Microscopy

Xue et al, Nature Mater (2011);Decker et al Nano Lett (2011)

Graphene on BN exhibits clear Moiré pattern

q=5.7o q=2.0o q=0.56o

9.0 nm13.4 nm

Minigap formation near the Dirac point due to Moire superlattice

momentum

Harper’s Equation: Competition of Two Length Scales

Proc. Phys. Soc. Lond. A 68 879 (1955)

a

Tight binding on 2D Square lattice with magnetic field

Harper’s Equation

Two competing length scales:a : lattice periodicitylB : magnetic periodicity

Commensuration / Incommensuration of Two Length Scales

Spirograph

lBa

a / lB = p/q

a

Hofstadter’s Butterfly

When b=p/q, where p, q are coprimes, each LL splits into q sub-bands that are p-fold degenerate

Energy bands develop fractal structure when magnetic length is of order the periodic unit cell

Harper’s Equation

εner

gy(in

uni

t of b

and

wid

th)

φ (in unit of φ0)0 1

1

Diophantine equation for gaps

Wannier (1978)

Quantum Hall Conductance

TKNN (1982); Stredar (1982)

Quantum Hall Effect with Two Integer Numbers

(t, s)

(-4, 0)

(-8, 0)

(-12, 0)

(-16, 0)

(4, 0)

(8, 0)

(12, 0)

(16, 0)

(3, 0)

(-2, -2)

(-4, 1)

(-2, 1)(-1, 1)

(-3, 2)(-4, 2)

(-2, 2)

(-1, 2)

n/n0: density per unit cell; φ : flux per unit cell

(-3, 1)

0 15Rxx (kΩ)Quantization of σxx and σxy

Vg (V)

B(T

)

σxy (mS) 0 1-1Fractal Quantum Hall Effect

Dean et al. Nature (2103)

Charge Density Waves in 1T-TaS2

Commensurate

Nearly Commensurate

Incommensurate

0 50 100 150 200 250 300 350 40010-4

10-3

10-2

10-1

100

Resis

tivity

(Ω-c

m)

T (K)

T H

Series of first order phase transitionindicated by resistance changes

Ma, arXiv1507.01312

Charge Density Waves in 1T-TaS2 : Icommensurate CDW

Incommensurate CDW𝑄𝑄𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 ~

1𝐵𝐵

𝑄𝑄𝐶𝐶𝐶𝐶𝐶𝐶 ~1

𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶

𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶 ≠ 𝑚𝑚𝐵𝐵𝑥𝑥 + 𝑛𝑛𝐵𝐵 𝑦𝑦

QCDW

Ta

aa

λCDW

xy

T = 375 K

Tsen et. al., PNAS (2016)

Commensurate CDW𝑄𝑄𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 ~

1𝐵𝐵

𝑄𝑄𝐶𝐶𝐶𝐶𝐶𝐶 ~1

𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶

QCDW

T = 100 K

𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑚𝑚𝐵𝐵𝑥𝑥 + 𝑛𝑛𝐵𝐵 𝑦𝑦

QCDW

Charge Density Waves in 1T-TaS2 : Commensurate CDWTsen et. al., PNAS (2016)

Nearly Commensurate CDW𝑄𝑄𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 ~

1𝐵𝐵

𝑄𝑄𝐶𝐶𝐶𝐶𝐶𝐶 ~1

𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶

T = ~300 K

Wu, Lieber, Science 1989

Charge Density Waves in 1T-TaS2 : Nearly Commensurate CDW

Discommensuration lines

Tsen et. al., PNAS (2016)

C-IC Charge Density Waves Phase Transition in Atomically Thin 1T-TaS2

C

C

C

C

C

C

C

C

C C

C C

D

ρDC

Homogeneous Incommesuration

Homogeneous Commesuration

Temperature

10-3

10-2

10-1

50 100 150 200 250 300

10-3

10-2

10-1

10-3

10-2

10-1

10-3

10-2

10-1

50 100 150 200 250 30010-3

10-2

10-1

Resis

tivity

(Ω-c

m)

T (K)

20 nm

8 nm

6 nm

4 nm

2 nm

∆ρ

∆T

Tsen et. al., PNAS (2016)

Imaging AB/AC domain in Bernal stacked bilayer graphene

Image from Spiecker (FAU)

Propagation 1-D Transport Channel Along the Domain Boundaries

Controlled Stacking of Graphene-Graphene Layers

Controlling Twisting Angle: Tear and Stack

Y. Cao et al., (PRL in press)

Integrate with SiN membrane for TEM study

Transmission Electron Microscopy

Specimen

Back focal plane(Diffraction)

Objective lens

Image plane(Real space image)

Instrument and ray diagram

Dark Field Imaging Graphene Moire Incommensurate Moirestructures

J. M. Yuk et al., Carbon 80, 755 (2014)

Graphene/hBN

Graphene/graphene

Graphene/graphene/hBN

Commensurate Moirestructures??

Dark field imaging

aperture

Dark field imaging

TEM dark field image (g = 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏) and selected area electron diffraction pattern

=+Rotational displacement Uniaxial displacement

Graphene on Graphene With a Small Twisting Angle

Twisting Engineering C/IC Domain Boundaries

Twisted Bilayer Graphene: Satellite Diffraction Peaks

TEM diffraction pattern Two graphene sheets with twist Angle: 0.41 deg on hBN substrate

graphene

BN

1010

0110

1100

2110

1120

1210

2020

0220

First Order Bragg Peak

Second Order Bragg Peak

Third Order Bragg Peak

Real Space Versus Fourier Space: Satellite Peaks

0.1-0.2deg 0.41deg 0.79deg 1.6deg

Selected Area Diffraction patterns zoomed up (x50) on each graphene peak0.1-0.2deg 0.41deg 0.79deg 1.6deg

Twisting Angle Dependent Satellite Peak Intensities

Satellites 1Satellites 2

First Order:

Main

Satellites 1

Second Order:

Main Satellites 2 Main

Satellite 1

Satellites 2

Third Order:Isatellite / IBragg

Relaxation of Atomic Lattice and Buckling

• Atomic scale relaxation

• Buckling out of plane

• Electronic properties

…..

Wijk et al., Nano Letters (2016)

Atomic Scale Relaxation Modeling: Commensurate Lattice

vdW interactions modeled using discrete–continuum effective potentials for Lennard–Jones (LJ) and Kolmogorov–Crespi (KC) for full atomistic resolution.

Simulated Diffractions

Qualitative agreement with experiments!

A/B

B/A

Mathematical Modeling: Inversion Problem

Paul Cazeaux and Mitch Luskin

• How we can model general incommensurate twisted lattices?• Can we recover the microscopic relaxation pattern from satellite diffraction?

AB/BA Symmetric

AB/BA Asymmetric

Transport Along Domain Boundaries: Topological Mode

Bernal stacking

AB stack

AC stack

ACAB

AA

L. Ju et al., Nature 520, 650 (2015)

Vertical electric field

Bulk energy band

Network resistivity ~ h/4e2 = 6.4 kΩ

RQ = h/4e2

Transport at the Neutrality of Twisted Bilayers

0 o3 o

0.8 o

2 µm

Bernal stacking

Decoupled stacking

IC/C domains

Vtop=-5 VVtop=+5 V Vtop=-10 VVtop=+10 V

Vtop=-5 VVtop=+5 V

Twisting Engineering C/IC Domain Boundaries in MoS2

Mo

S

Bulk Stacking Method Controlling Twisting Angle: Tear and Stack

Y. Cao et al., (PRL 2016)

MoS2 on MoS2 With a Small Twisting Angle

-6.6 6.6

MoS2 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

MoS2 𝟏𝟏𝟏𝟏𝟐𝟐𝟏𝟏1

2

3

1 2 3

AB stack with partial dislocations

MoS2 on MoS2 With 180o Twisting Angle

1 2 3

4 5 6

1

2

3

4

5

6

MoS2 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏MoS2 𝟏𝟏𝟏𝟏𝟐𝟐𝟏𝟏

Pairs of partial dislocations

Summary and Outlook• Commensuration and incommensuration between electric potential

and magnetic length creates fractalized quantum Hall effect

• C/IC phase transition in CDW lattice in TaS2 can be tuned by dimensionality changes in the samples.

• In graphene bilayer, tunable domain structures of C/IC boundary can be controlled with twist angle control.

• Observation of different types of domain structures in bilayer MoS2 formed as a result of commensurate transition.

Going Forward:Symmetry elements in the materials can be engineered by twist angle control.

Can we generate topologically non-trivial electronic structures from twisted vdW stacks?

Acknowledgements

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone,

Hofstadter Butterfly

Charge Density Waves in TaS2

A. W. Tsen, R. Hovden, D. Z. Wang, Y. D. Kim, J. Okamoto, K. A. Spoth, Y. Liu, W. J. Lu, Y. P. Sun, J. Hone, L. F. Kourkoutis, and A. N. Pasupathy

Funding:

Twisted vdW Layers

H. Yoo, R. Engelke, M. Kim, G. –C, YiK. Zhang, E. TadmorP. Cazeaux, M. LuskinR. Hovden