extreme straintronics of graphene · 1/29/2019 · graphene sustains strains >20%, while...

Extreme Straintronics of Graphene

2DCC-MIP Webinar29 January, 2019

Riju BanerjeePenn State

January 29, 2019 2DCC-MIP Webinar

Berry, Mat. Today (2016)

New questions in FlatlandOne of the most fundamental properties of matter:How it deforms under stress

2004: Discovery of graphene -How are deformations of atom thick materials different from bulk?


• Graphene sustains strains >20%, while silicon breaks at 1.5%. How do materials deform under such extreme strains?

me-mechanicalengineering.com





• Large Deformations also affect the electronic properties endowing certain versatility. Engineer novel electronic properties under extreme strain?

Castro Neto et. al., RMP (2009)

Modify??






• How are ripples in a lattice different from those in a continuous fabric?







• How are ripples in a lattice different from those in a continuous fabric?




Take graphene, apply extreme strain and vary it at the

nanoscale and probe effects

Traditional method of straining

January 29, 2019 2DCC-MIP WebinarPamela C. Burnley

Traditional method of straining

January 29, 2019 2DCC-MIP WebinarPamela C. Burnley

Difficult for atom thick materials

Cannot measure/probe resulting deformations and other physical properties down to atomic scale

Engineering nanoscale ripples in suspended graphene


Grow graphene on Copper Substrate with Step Edges

Step Edges

Preferred tool – Scanning tunneling microscopy


Try to maintain constant current while scanning

𝐼𝐼 ∝ 𝑒𝑒−𝜅𝜅𝑧𝑧

Å change in z gives order of magnitude change in 𝐼𝐼

Scanning tunneling microscope


What does a STM ‘see’?


Measures current 𝐼𝐼 ∝ local electron density

Sharp tip sensitive to electron density at sub-Angstrom length scale





Maps out topographic features by running a feedback loop on current

Record how z needs to change to keep 𝐼𝐼 constant

Z-he

ight







Z-he

ight

Pristine Graphene lattice







Z-he

ight

Dirac cone

Pristine Graphene lattice


Mechanical properties

Studying nanoscale deformations by STM


Levy, et. al Science (2010) Xu, et. al Nano. Lett (2010)Yeh, et. al Surf.Sci. (2011)

Bao, et. al Nat.Nano. (2009) Tapaszto, et. al Nat. Phys. (2012)Zhu, et. al PRB (2014)

• Experimental Nanoscale Strain Manipulation still in its Infancy

• Engineering at nm scale is hard and Wrinkling is inherently somewhat messy

• Ideally : On a suspended sheet, vary, measure and relate wavelength λ, Amplitude, Curvature – This Talk!

Sculpting atomic step edges


Graphene pushes Cu to form higher step edges

Our samples grown by T.G. Nakajima, (Terrones group)

Extreme strain measurement


Measured strain >10%, consistent with first principle calculations

Ripples in a classical fabric


Ripples form due to compressive stress

Wavelength λ ~ Amplitude AWavelength λ ~ 1/Amplitude A

Direct Application Indirectly by Poisson Compression

Cerda et. al, PRL, (2003)

Nanoscale graphene ripples – local curvature


Nanoscale graphene ripples – Draping angle


Δz

Δx

Conserved draping angle

35°

Nanoscale graphene ripples – Rippling angle


~168°

Nanoscale graphene ripples – Beyond classical theory


λ

𝐿𝐿

Classically,𝐿𝐿 ∝ 𝜆𝜆2

A1B1

B2A2

Classical theory of rippling assumes thick, continuous material possible breakdown for 2D material at atomic scale?

𝐿𝐿𝐵𝐵 ≠ 4𝐿𝐿𝐴𝐴


Electronic properties

Graphene electronics


High electron mobility in graphene Attractive for making electronics

Functionalizing by dopants Van der Waals heterostructures Lateral heterostructures

Zhao et. al., Science (2011)

N dopants in graphene

Atomic Lego blocks

X-sec TEM of graphene-hBN

Haigh et. al., N. Mat. (2012)

Zhao et. al., N. Nano (2016)

MonolayerGraphene -MoS2assembly

Graphene -hBN

Levendorf et. al., Nature (2012)

Electronics


B ~ +10 T

B ~ -10 T

Impossible to realize with present technology !

Large fields are uniformStanford University

Non-uniform fields are weak

Kamalakar et. al., Nat. Comm. (2015)

Nanoscale variation of fields ispossible by strain engineering

Manipulate electrons by precise application of Electric and Magnetic fields – ideally down to atomic scale

Present day electronics predominantly use E fields. B fields are often non-trivial to engineer in devices

Graphene – pseudospin and valley


Linear dispersion near Fermi energy

2 inequivalent K points

Can be thought of as a spin

Ψ = 𝜓𝜓𝐴𝐴𝜓𝜓𝐵𝐵

= 𝑒𝑒𝑖𝑖𝑖𝑖.𝑟𝑟/ℏ

21

±𝜉𝜉 𝑒𝑒𝑖𝑖𝑖𝑖

𝜃𝜃 = 𝑡𝑡𝑡𝑡𝑡𝑡−1𝑝𝑝𝑦𝑦𝑝𝑝𝑥𝑥

𝜉𝜉 = ±1 for K,K’

Pseudospin - A new degree of freedom

Castro Neto et. al., RMP (2009)

B

A

What does strain do? Pseudovector potentials


KK’

Reciprocal space

What does strain do? Pseudovector potentials


Reciprocal space

𝐾𝐾 → 𝐾𝐾 + 𝛿𝛿𝛿𝛿𝐾𝐾′ = −𝐾𝐾 → −𝐾𝐾 − 𝛿𝛿𝛿𝛿

For K valley

𝐾𝐾 → 𝐾𝐾 + 𝛿𝛿𝛿𝛿 = 𝐾𝐾 +𝑝𝑝𝑝𝑝𝑐𝑐

For K’ valley

𝐾𝐾′ → 𝐾𝐾′ − 𝛿𝛿𝛿𝛿 = 𝐾𝐾′ −𝑝𝑝𝑝𝑝𝑐𝑐

Like generating magnetic fields of opposite directions

in K and K’ valleys!

𝑝𝑝𝑥𝑥𝑝𝑝𝑦𝑦

∝𝑢𝑢𝑥𝑥𝑥𝑥 − 𝑢𝑢𝑦𝑦𝑦𝑦2𝑢𝑢𝑥𝑥𝑦𝑦

Ando et. al., PRB (2002), Geim et. al., N. Phys (2010)

Vector potential

Pseudo vector potential from strain

What does strain do? Pseudoscalar potentials


Real space

𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖 changes ⇒ 𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 changes to maintain charge neutrality

As 𝑉𝑉 ∝ 𝑒𝑒 𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒, 𝑉𝑉 changes

Ando et. al., PRB (2002)

Scalar potential

Like generating electric fields!𝑉𝑉 ∝ 𝑢𝑢𝑥𝑥𝑥𝑥 + 𝑢𝑢𝑦𝑦𝑦𝑦

Pseudo scalar potential from strain

What does strain do? Pseudo gauge fields


Pseudo gauge fields are created by strain gradients

𝐵𝐵 = ∇ × 𝑝𝑝 ∝ 𝜕𝜕𝑢𝑢𝐸𝐸 = −∇𝑉𝑉 ∝ 𝜕𝜕𝑢𝑢

𝑝𝑝𝑥𝑥𝑝𝑝𝑦𝑦

∝𝑢𝑢𝑥𝑥𝑥𝑥 − 𝑢𝑢𝑦𝑦𝑦𝑦2𝑢𝑢𝑥𝑥𝑦𝑦

𝑉𝑉 ∝ 𝑢𝑢𝑥𝑥𝑥𝑥 + 𝑢𝑢𝑦𝑦𝑦𝑦

Pseudo scalar potential from strainPseudo vector potential from strain

Landau Levels from Pseudomagnetic fields


Electrons in 2D form Landau levels under external B fields

𝐸𝐸𝑖𝑖 ∝ |𝑡𝑡|

Strain induced pseudo magnetic fields

Levy, et. al Science (2010)https://www.nist.gov/programs-projects/measuring-magneto-electronic-properties-graphene-nanometer-scale

Graphene Nanobubbles on Pt

Manipulating electronic states


A

B

𝐸𝐸𝑖𝑖 ∝ 𝑡𝑡

B-A

Electronic states


1

1223

34

45566778

89

910

1011

11

12

1213

1314

1415

1516

1617

17

18

1819

1920

2021

2122

2223

2324

2425

2526

2627

27

Drap

ed R

egio

n

Electronic states


LDOSAmplitudeIncreases at kinks ⇒ Strain

2 models of graphene bending - Calculations


Graphene bending: out of plane and curvature induced in-plane distortions

High LDOS at peaks/troughs

Low LDOS at peaks/troughs

Pseudo electromagnetic superlattice


Large C-C bonds

Small C-C bonds

+/- pseudo B for K/K’

-/+ pseudo B for K/K’

Pseudo E field in-plane

x



Large C-C bonds

Small C-C bonds

+/- pseudo B for K/K’

-/+ pseudo B for K/K’

Pseudo E field in-plane

LDOS of graphene lattice

+periodic strain

Effective 𝐵𝐵𝑚𝑚𝑚𝑚𝑥𝑥 ≈ 200 𝑇𝑇

LDOS of Unstrained graphene

lattice +

periodic E, B fields

First principle calculations

X (n

m)

X (n

m)

x



STRained Electromagnetic Modulated Superlattices (STREMS)

Usually superlattices require interfacing different materials

Stacked superlattice

Lateral Heterostructures

Same material modulated by strain!

Snake states – theoretical prediction/implication


Oroszlány et. al., PRB (2008)

Non uniform B fields create snake states

K and K’ electrons see opposite pseudomagnetic fields

Counterpropagating snake states akin to topological insulators

K

K’

Settnes et. al., PRL (2016)

Summary


1. Developed a technique to study mechanical and electrical properties of nanoscale ripples in suspended Graphene

2. Draping and Rippling angles seems to be Conserved for a wide range of parameters.

3. Possible breakdown of continuum elasticity theory

4. Spectral signatures of strong nonuniform pseudo gauge fields

5. Demonstrated STRained Electromagnetic Modulated Superlattices (STREMS)

Thank You


Questions?

&All of You

Eric HudsonLavish PabbiAnna BinionBill DuschVincent Crespi

Tomotaroh Granzier-NakajimaMauricio TerronesViet-Hung NguyenJean-Christophe CharlierYuanxi Wang

extreme straintronics of graphene · 1/29/2019 · graphene sustains strains >20%, while...

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