commensuration and incommensuration in the van …...commensuration and incommensuration in the van...
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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
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