Commensuration and Incommensuration in the van der Waals Heterojunctions. Philip Kim Physics Department Harvard University

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1 Commensuration and Incommensuration in the van der Waals Heterojunctions Philip Kim Physics Department Harvard University

2 Rise of 2D van der Waals Systems graphene Metal-Chalcogenide Bi 2 Sr 2 CaCu 2 O 8-x Charge Transfer Bechgaard Salt C (TMTSF) 2 PF 6 hexa-bn X Lead Halide Layered Organic B N M X M = Ta, Nb, Mo, Eu X = S, Se, Te, Semiconducting materials: WSe 2, MoSe 2, MoS 2, Complex-metallic compounds : TaSe 2, TaS 2, Magnetic materials: Fe-TaS 2, CrSiTe 3, Superconducting: NbSe 2, Bi 2 Sr 2 CaCu 2 O 8-x, ZrNCl, A A C B A

3 Graphen/hBN Moire Pattern 5 o 10 o 30 o 20 o 15 o Not commensurated except for special angles

4 Moire pattern in Graphene on hbn: Scanning Probe Microscopy Graphene on BN exhibits clear Moiré pattern q=5.7 o q=2.0 o q=0.56 o 9.0 nm 13.4 nm Xue et al, Nature Mater (2011); Decker et al Nano Lett (2011) Minigap formation near the Dirac point due to Moire superlattice momentum

A 68 879 (1955) Tight binding on 2D Square lattice with

5 Harper s Equation: Competition of Two Length Scales Proc. Phys. Soc. Lond. A (1955) Tight binding on 2D Square lattice with magnetic field a Harper s Equation Two competing length scales: a : lattice periodicity l B : magnetic periodicity

6 Commensuration / Incommensuration of Two Length Scales Spirograph a / l B = p/q a l B

Hofstadter s Butterfly Quantum Hall Conductance

When b=p/q, where p, q are coprimes, each LL

$degenerate Energy bands develop fractal$ periodic unit cell εnergy (in unit of band

7 Hofstadter s Butterfly Quantum Hall Conductance Harper s Equation 1 TKNN (1982); Stredar (1982) 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 εnergy (in unit of band width) a Diophantine equation for gaps Wannier (1978) 0 1 φ (in unit of φ 0 )

8 Quantum Hall Effect with Two Integer Numbers n/n 0 : density per unit cell; φ : flux per unit cell (t, s) R xx (kω) 0 15 (-4, 1) (-4, 2) (-3, 2) (-2, 1) (-1, 1) (-2, 2) (-1, 2) (3, 0) (4, 0) Quantization of σ xx and σ xy (-2, -2) (-4, 0) (-8, 0) (-3, 1) Fractal Quantum Hall Effect (8, -1 0) σ xy (ms) 0 1 (-12, 0) (12, 0) (-16, 0) B (T) (16, 0) V g (V) Dean et al. Nature (2103)

Charge Density Waves in 1T-TaS 2 T H 10 0

Resistivity (Ω-cm) 10-1 10-2 10-3 Nearly

9 Charge Density Waves in 1T-TaS 2 T H 10 0 Commensurate Ma, arxiv Series of first order phase transition indicated by resistance changes Resistivity (Ω-cm) Nearly Commensurate T (K) Incommensurate

10 Charge Density Waves in 1T-TaS 2 : Icommensurate CDW Incommensurate CDW T = 375 K QQ BBBBBBBBBB ~ 1 aa QQ CCCCCC ~ 1 λλ CCCCCC Tsen et. al., PNAS (2016) λλ CCCCCC mmmm xx + nnnn yy Ta Q CDW a a λ CDW y x

11 Charge Density Waves in 1T-TaS 2 : Commensurate CDW Commensurate CDW T = 100 K QQ BBBBBBBBBB ~ 1 aa QQ CCCCCC ~ 1 λλ CCCCCC Tsen et. al., PNAS (2016) λλ CCCCCC = mmmm xx + nnnn yy Q CDW

12 Charge Density Waves in 1T-TaS 2 : Nearly Commensurate CDW Nearly Commensurate CDW T = ~300 K QQ BBBBBBBBBB ~ 1 aa QQ CCCCCC ~ 1 λλ CCCCCC Tsen et. al., PNAS (2016) Discommensuration lines Wu, Lieber, Science 1989

13 C-IC Charge Density Waves Phase Transition in Atomically Thin 1T-TaS nm Homogeneous Incommesuration nm Resistivity (Ω-cm) nm 8 nm ρ DC C C C C C C C C C C C C D Temperature nm ρ T Tsen et. al., PNAS (2016) T (K) Homogeneous Commesuration

14 Imaging AB/AC domain in Bernal stacked bilayer graphene Image from Spiecker (FAU)

15 Propagation 1-D Transport Channel Along the Domain Boundaries

16 Controlled Stacking of Graphene-Graphene Layers Controlling Twisting Angle: Tear and Stack Integrate with SiN membrane for TEM study Y. Cao et al., (PRL in press)

$(Diffraction) Image plane$ (Real space image) J. M.

17 Dark Field Imaging Graphene Moire Transmission Electron Microscopy Instrument and ray diagram Incommensurate Moire structures Graphene/hBN Graphene/graphene Dark field imaging Graphene/graphene/hBN aperture Specimen Objective lens Back focal plane (Diffraction) Image plane (Real space image) J. M. Yuk et al., Carbon 80, 755 (2014) Dark field imaging Commensurate Moire structures??

$electron diffraction$ pattern Rotational

18 Graphene on Graphene With a Small Twisting Angle TEM dark field image (g = ) and selected area electron diffraction pattern Rotational displacement Uniaxial displacement + =

19 Twisting Engineering C/IC Domain Boundaries

$Twisted Bilayer Graphene: Satellite Diffraction Peaks TEM diffraction pattern Two graphene sheets with twist Angle: 0.$

20 Twisted Bilayer Graphene: Satellite Diffraction Peaks TEM diffraction pattern Two graphene sheets with twist Angle: 0.41 deg on hbn substrate 2110 First Order Bragg Peak BN graphene Second Order Bragg Peak Third Order Bragg Peak 12 10

$Diffraction patterns$ zoomed up (x50) on each

21 Real Space Versus Fourier Space: Satellite Peaks deg 0.41deg 0.79deg 1.6deg Selected Area Diffraction patterns zoomed up (x50) on each graphene peak deg 0.41deg 0.79deg 1.6deg

22 Twisting Angle Dependent Satellite Peak Intensities First Order: Second Order: Third Order: I satellite / I Bragg Satellites 1 Satellites 2 Satellites 1 Satellites 2 Satellite 1 Main Main Satellites 2 Main

23 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

$Simulated Diffractions A/B B/A Qualitative$

24 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 A/B B/A Qualitative agreement with experiments!

$diffraction?$

25 Mathematical Modeling: Inversion Problem How we can model general incommensurate twisted lattices? Can we recover the microscopic relaxation pattern from satellite diffraction? AB/BA Symmetric AB/BA Asymmetric Paul Cazeaux and Mitch Luskin

26 Transport Along Domain Boundaries: Topological Mode AC AB AB stack R Q = h/4e 2 AA AC stack Bernal stacking L. Ju et al., Nature 520, 650 (2015) Vertical electric field Bulk energy band Network resistivity ~ h/4e 2 = 6.4 kω

27 Transport at the Neutrality of Twisted Bilayers 2 µm 0 o 3 o Decoupled stacking Bernal stacking V top =+5 V V top =-5 V V top =+10 V V top =-10 V IC/C domains 0.8 o V top =+5 V V top =-5 V

28 Twisting Engineering C/IC Domain Boundaries in MoS 2 S Mo Bulk Stacking Method Controlling Twisting Angle: Tear and Stack Y. Cao et al., (PRL 2016)

29 MoS 2 on MoS 2 With a Small Twisting Angle MoS MoS AB stack with partial dislocations

30 MoS 2 on MoS 2 With 180 o Twisting Angle MoS MoS Pairs of partial dislocations

31 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 Hofstadter Butterfly C. R.

Ghahari, Y. Gao, J. Katoch, M. Ishigami, P.

Shepard, J. Hone, Charge Density Waves in TaS 2 A.

Okamoto, K. A. Spoth, Y. Liu, W. J. Lu, Y. P.

32 Acknowledgements Hofstadter Butterfly 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, Charge Density Waves in TaS 2 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 Twisted vdw Layers H. Yoo, R. Engelke, M. Kim, G. C, Yi K. Zhang, E. Tadmor P. Cazeaux, M. Luskin R. Hovden Funding:

Bloch, Landau, and Dirac: Hofstadter s Butterfly in Graphene. Philip Kim. Physics Department, Columbia University

Bloch, Landau, and Dirac: Hofstadter s Butterfly in Graphene Philip Kim Physics Department, Columbia University Acknowledgment Prof. Cory Dean (now at CUNY) Lei Wang Patrick Maher Fereshte Ghahari Carlos