Layer Materials in the Flatland: Twisted Geometry and the Strain Effects

Shiang Fang, Stephen Carr, Miguel A. Cazalilla, Efthimios Kaxiras Harvard University

May 19, 2017, Mathematical Modeling of 2D Materials Institute for Mathematics and its Applications, Minneapolis

Supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, and by ARO MURI Award No. W911NF-14-0247. Two-Dimensional Physics

Van der Waals heterostructures

• Superconductivity • CDW • Topological phases • Magnetism • Spintronics • Valleytronics • Optoelectronics • Straintronics

A. K. Geim et al., Nature 499, 419 (2013) 2D Layered Materials

Graphene hBN TMDC

a=2.46A a=2.50A a=3.18A (MoS2) Semimetal Insulator Semiconductor

• Relavisc linear Dirac • Broken inversion • MX2, M=Mo/W, X=S/Se dispersion at K valleys symmetry • Broken inversion • Inversion symmetry • Used to encapsulate symmetry • Mechanical strength graphene • Direct band gap 1-2 eV • Stability at K valleys • Large band gap • Spin-orbit coupling Wannier Transformation

EF EF

Bloch wave/energy and hamiltonian

DFT DFT / TBH Wannier90: A Tool for Obtaining Maximally-Localised Wannier Funcons, A. A. Mostoﬁ, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt and N. Marzari, Comput. Phys. Commun. 178, 685 (2008) TMDC Interlayer Coupling

Empirical interlayer potenal: perspecve view of TMDC 5 interface: 3 1 2 4 1 6

2 layer 2 3 4 5 6 layer 1

Empirical funconal form: V (r)=⌫ exp( (r/R )b ) pp,b b b Ref: Shiang Fang et al., Phys. Rev. B 92, 205108 (2015). Graphene Interlayer coupling

Graphene Wannier pz

Ref: Shiang Fang, Ehimios Kaxiras, Phys. Rev. B 93, 235153 (2016) Twistronics with Bilayer Graphene

Density of states versus twist angle

Van Hove singularities Renormalization of in the density of states Fermi velocity

Stephen Carr et al, Phys. Rev. B 95, 075420 (2017) Experiment at Jarillo-Herrero group

“tear-and-stack” enables sub degree control of sample preparaon the twist angle! Polymer

h-BN ! Graphene

Si Si Fully hBN encapsulated dual-gate twisted bilayer graphene device (cross-secon)

…… Pick up graphene . and bottom h-BN Si

Bottom gate

Si Si

5µm 6 A

5 1 Vxx "↓$%&' 2 ~ =50~200 µV 3

Vxy

4 Ref: Y Cao et al, Phys. Rev. Le. 117, 116804 (2016). TG BG Experiment at Jarillo-Herrero group

B=0

temperature dependence Transport in Twisted Bilayer Graphene

Super-lace induced insulang states with small twist angle for bilayer graphene

Insulang gap from single parcle band structure

Ref: Y Cao et al, Phys. Rev. Lett. 117, 116804 (2016). Electronic Structure for (19,18) TwBLG

Constant energy contour Hall Plateaus and Landau levels

K Energy contour

jump by 8!

• 2 inequivalent K points • 2 sets of orbitals each • 2 spin states • 8 degeneracies Hall Plateaus and Landau levels

Energy contour

jump by 4!

• 1 Gamma point • 2 sets of orbitals • 2 spin states • 4 degeneracies Strained and Rippled Layers

Ubiquitous ripples in graphene! Eﬀects: • Fermi velocity • Work funcon • Pseudo gauge ﬁeld • Scaering and mobility • Topological defects

Graphene kirigami stretchable graphene transistors

Melina K. Blees et al., Nature 524, 204 (2015) Curved Space and Emergent Geometry

Ubiquitous ripples in graphene!

Curved space-me in general Eﬀecve Low-Energy theory: relavity (NASA) Dirac equaon in curved space µ i (@µ µ) =0 with Emergent geometry, metric and connecons µ, ⌫ =2gµ⌫

Ref: Alberto Corjo, Maria A. H. Vozmediano, Europhys.Le.77:47002 (2007 ) Bo Yang, Phys. Rev. B 91, 241403(R) (2015) Strain-induced Pseudo Magnetic Field in Graphene

The magnec ﬁeld is eﬀecvely greater than 300T.

N. Levy et al., Science 329, 544 (2010) Reductionism: Microscopic Models

Deformed graphene unit cell

compress

Tight-binding model for strained graphene/hBN t t0 + ↵ (u + u )+ [!~r(u u )+2!~r u ] ~r ⇡ ~r ~r xx yy ~r y xx yy x xy beyond central force two-center approximaon Manuscript in preparaon. TMDC Hamiltonian with Uniform Strain

Transion Metal Dichalcogenides (TMDC): MX2 (M=Mo, W, X=S, Se).

Orbitals are grouped and classiﬁed (x,y, or z-like) TMDC crystal structure

Hamiltonian for 1st / 3rd neighbor coupling

X is allowed to relax: generalized Cauchy- Born rule

Manuscript in preparaon. DFT/TBH Comparison

Strain-dependent gap

shi ~ -100 meV / %

Manuscript in preparaon. Low-Energy Hamiltonians

Symmetry invariant in kp Hamiltonian Symmetry irreducible representaon

1 dim. uxx + uyy

2 dim. (u u , 2u ) xx yy xy Other objects

(kx,ky) (ˆx, ˆy)

C3v irreducible representaon

Ref: Juan L. Manes et al., Phys. Rev. B 88, 155405 (2013) Strain Physics / Applications

• Band structure engineering

• Interplay between spin / valley / orbital

• Pseudo magnec ﬁeld (300T in graphene)

• Dynamical perturbaon (phonons); Floquet physics

• Probe / control knob for many-body correlated states and quantum phase transion (anisotropy with composite fermions)

Ref: Rodrick Kuate Defo et al, Phys. Rev. B 94, 155310 (2016) Summary

• Wannier transformaon is used to derive the ab inio ght binding Hamiltonians for layers (PRB 92, 205108 (2015)). • For bilayer graphene, the electronic properes depend sensively on the twist angle (PRB 93, 235153 (2016); PRB 95, 075420 (2017)). • Insulang states can be induced at small twist angle(PRL 117, 116804 (2016)). • Strain can be used to engineer desire properes (PRB 94, 155310 (2016)) • The code will be available online: hps://sites.google.com/view/shiangfang

Supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, and by ARO MURI Award No. W911NF-14-0247. Acknowledgements

Ehimios Kaxiras Bertrand Halperin Philip Kim Pablo Jarillo-Herrero Applied Math • Mitchell Luskin • Paul Cazeaux • Daniel Massa

Stephen Carr Jason Luo Yuan Cao Valla Fatemi Grants: • STC CIQM, NSF Grant No. DMR-1231319 • ARO MURI Award No. W911NF-14-0247