quantum spin hall effect and topological insulators · pdf filequantum spin hall effect and...
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![Page 1: Quantum Spin Hall Effect and Topological Insulators · PDF fileQuantum Spin Hall Effect and Topological Insulators ... gap = 2 m ec2 ~ 106 eV electron ... (Thouless et al,](https://reader031.vdocuments.net/reader031/viewer/2022022004/5aabc5067f8b9a693f8c5cce/html5/thumbnails/1.jpg)
E
k=a k=b
Quantum Spin Hall Effectand Topological Insulators
Laurens W. MolenkampPhysikalisches Institut (EP3), Universität Würzburg
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Quantum Spin Hall Effect and Topological Insulators
I. Introduction- Topological Classification of Insulators - Edge States with and w/o Time Reversal Symmetry
II. Two Dimensional TI : Quantum Spin Hall Effect
- Transport in HgTe quantum wellsIII. Three Dimensional TI :
- Topological Insulator & Surface States- Photoemission on BixSb1-x and Bi2Se3- Transport in strained bulk HgTe
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C.L.Kane and E.J. Mele, Science 314, 1692 (2006)
C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
The Insulating State – Topologically Generalized
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The Usual Boring Insulating State
Covalent Insulator
Characterized by energy gap: absence of low energy electronic excitations
The vacuumAtomic Insulatore.g. solid Ar
Dirac Vacuum
Egap ~ 10 eV
e.g. intrinsic semiconductor
Egap ~ 1 eV3p
4s
Silicon
Egap = 2 mec2
~ 106 eV
electron
positron ~ hole
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C.L.Kane and E.J. Mele, Science 314, 1692 (2006)
C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
The Insulating State – Topologically Generalized
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The Integer Quantum Hall State
2D Cyclotron Motion, Landau Levels
gap cE E
Energy gap, but NOT an insulator
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bulkinsulating
Edge States (as experimentalists see them)
Quantized Hall conductivity :
y xy xJ E2
xy hn e
Integer accurate to 10-9
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Topological Band Theory
g=0 g=1
21 ( ) ( )2 BZ
n d u ui
k kk k k
The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states
Analogy: Genus of a surface : g = # holes
Insulator : n = 0IQHE state : xy = n e2/h
The TKNN invariant can only change at a phase transition where the energy gap goes to zero
Classified by Chern (or TKNN) integer topological invariant (Thouless et al, 1982)
( ) :H k Bloch Hamiltonians Brilloui wn zone (torus) ith energy gap
u(k) = Bloch wavefunction
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Edge StatesGapless states must exist at the interface between different topological phases
IQHE staten=1
Vacuumn=0
Edge states ~ skipping orbits
Bulk – Boundary Correspondence : n = # Chiral Edge Modes
This approach can actually be generalized to a spinfull QHE at zero magnetic field:the Quantum Spin Hall Effect
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C.L.Kane and E.J. Mele, Science 314, 1692 (2006)
C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
The Insulating State – Topologically Generalized
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Topological Insulator : A New B=0 Phase There are 2 classes of 2D time reversal invariant band structures
Z2 topological invariant: = 0,1 is a property of bulk bandstructure, but can be understood by
from the bulk - boundary correspondence
=0 : Conventional Insulator =1 : Topological Insulator
Kramers degenerate attime reversal
invariant momenta k* = k* + G
k*=0 k*=/a k*=0 k*=/a
Edge States for 0<k</a
Even number of bandscrossing Fermi energy
Odd number of bandscrossing Fermi energy
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5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.76.6
1.0
1.5
0.5
0.0
-0.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
5.5
Bandgap vs. lattice constant(at room temperature in zinc blende structure)
Ban
dgap
ene
rgy
(eV
)
lattice constant a [Å]0 © CT-CREW 1999
(Hg,Cd)Te compound semiconductors
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band structure
D.J. Chadi et al. PRB, 3058 (1972)
fundamental energy gap
meV 30086 EE meV 30086 EE
semi-metal or semiconductor
HgTe bulk band structure
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
Eg
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Layer Structure
gate
insulator
cap layer
doping layer
barrier
barrierquantum well
doping layer
buffer
substrate
Au
100 nm Si N /SiO
3 4 2
25 nm CdTe
CdZnTe(001)
25 nm CdTe10 nm HgCdTe x = 0.79 nm HgCdTe with I10 nm HgCdTe x = 0.74 - 12 nm HgTe10 nm HgCdTe x = 0.7 9 nm HgCdTe with I10 nm HgCdTe x = 0.7
symmetric or asymmetricdoping
Carrier densities: ns = 1x1011 ... 2x1012 cm-2
Carrier mobilities: = 1x105 ... 1.5x106 cm2/Vs
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 80
100
200
300
400
500
µ=1.06*106cm2(Vs)-1
nHall=4.01*1011cm-2
Q2134a_Gate
B[T]
Rxx
[]
-15000
-10000
-5000
0
5000
10000
15000
Graph2
Rxy
[]
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Type-III QW
VBO = 570 meV
HgCdTeHgCdTeHgTe
HgCdTe
HH1E1
QW < 63 Å
HgTe
inverted normal
band structure
conduction band
valence band
HgTe-Quantum Wells
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123456
k (0.01 -1)
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Ener
gyE(
k)(e
V)
k || (1,1)k || (1,0)k = (kx,ky)
k || (1,1)k || (1,0)k = (kx,ky)
4 nm QW 15 nm QW
normal
semiconductor
inverted
semiconductor
1 2 3 4 5 6
k (0.01 -1)
-0.20
-0.15
-0.10
-0.05
0.00
0.50
0.10
0.15
0.20
E2
H1H2
E1L1
0.6 0.8 1.0 1.2 1.4
dHgTe (100 )
E2E2
E1E1H1H1
H2H2H3H3
H4H4 H5H5
H6H6L1L1
QW Band Structure from k.p Model
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Dirac Bandstructure near dc
E
k
E1
H1
invertedgap
4.0nm 6.2 nm 7.0 nm
normalgap
H1
E1
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
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Edge StatesGapless states must exist at the interface between different topological phases
Egap
Domain wall bound state 0
n=1 n=0
Band inversion – Dirac Equation
x
y
M<0
M>0
Smooth transition : gap must pass through zero
Jackiw, Rebbi (1976)Su, Schrieffer, Heeger (1980)
This is the zero B-field generalization of the Quantum Hall effect:the Quantum Spin Hall Effect
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QSHE, Simplified Picture
normalinsulator
bulk
bulkinsulating
entire sampleinsulating
m > 0 m < 0
QSHE
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Experimental Signature
normal insulator state
QSHI
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Need small Samples
L
W
(L x W) m
2.0 x 1.0 m1.0 x 1.0 m1.0 x 0.5 m
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First Observation of QSHI state
M. König et al., Science 318, 766 (2007).
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Multi-Terminal Probe
210001121000012100001210000121100012
T
heIG
heIG
t
t
2
23
144
2
14
142
232
generally
22 2)1(
ehnR t
3exp4
2 t
t
RR
heG t
2
exp,4 2
Landauer-Büttiker Formalism normal conducting contacts no QSHE
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0.0 0.5 1.0 1.5 2.00
5
10
15
20
25
R (k
)
V* (V)
I: 1-4V: 2-3
1
3
2
4
R14,23=1/4 h/e2
R14,14=3/4 h/e2
Non-Local data on H-bar
A. Roth et al., Science 325, 294 (2009).
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Configurations would be equivalent in quantum adiabatic regime
-1 0 1 2 30
5
10
15
20
25
30
35
40
R (k
)
V* (V)
I: 1-4V: 2-3
R14,23=1/2 h/e2
R14,14=3/2 h/e2
I: 1-3V: 5-6
R13,13=4/3 h/e2
R13,54=1/3 h/e2
-1 0 1 2 3 4
V* (V)
Multi-Terminal Measurements
A. Roth et al., Science 325, 294 (2009).
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Metallic Spin-Hall Effect
Intrinsic SHE
Rashba effect
J.Sinova et al.,Phys. Rev. Lett. 92, 126603 (2004)
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H-bar for detection of Spin-Hall-Effect
(electrical detection through inverse SHE)
E.M. Hankiewicz et al ., PRB 70, R241301 (2004)
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200 nm 200 nm
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00
100
200
300
400
500
0
2
4
6
8
10
12
R12
,36
()
Vg (V)
I (n
A)
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
T = 2 K
Rno
nloc
al /
k
VGate / V
I / n
A
n-conductingp-conducting
insu
latin
g
– Suppress non-local QSHE using long leads or narrow wires
– Intrinsic metallic SHE only shows up for holes: larger spin-orbit
– Amplitude in agreement with modeling (E. Hankiewicz, J. Sinova)
H-bar experiments
C. Brüne et al., Nature Physics 6, 448 (2010).
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QSHE and iSHE as spin injector and detector
Split-gated H-bar
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Detect iSHE through QSHI edge channels
I
U
Gate in 3-8 leg is scanned, 2-9 leg is n-type metallic,
current passed between contacts 2 and 9.
C. Brüne et al., Nature Physics 8, 486–491 (2012)
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Detect QSHI through inverse iSHE
I
U
Gate in 3-8 leg is scanned, 2-9 leg is n-type metallic,
current passed between contacts 3 and 8 C. Brüne et al., Nature Physics 8, 486–491 (2012)
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From traffic jam to info-superhighwayon chip
Traffic jam inside chips today Info highways for the chips in the future
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3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracy
Surface Brillouin Zone
2D Dirac Point
E
k=a k=b
E
k=a k=b
0 = 1 : Strong Topological Insulator
Fermi circle encloses odd number of Dirac pointsTopological Metal :
1/4 grapheneBerry’s phase Robust to disorder: impossible to localize
0 = 0 : Weak Topological Insulator
Related to layered 2D QSHI ; (123) ~ Miller indicesFermi surface encloses even number of Dirac points
OR4
1 2
3
EF
How do the Dirac points connect? Determined by 4 bulk Z2 topological invariants 0 ; (123)
kx
ky
kx
ky
kx
ky
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Bi1-xSbxTheory: Predict Bi1-xSbx is a topological insulator by exploiting
inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07)
Experiment: ARPES (Hsieh et al. Nature ’08)
• Bi1-x Sbx is a Strong Topological Insulator 0;(1,2,3) = 1;(111)
• 5 surface state bands cross EF between and M
ARPES Experiment : Y. Xia et al., Nature Phys. (2009).Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3
• 0;(1,2,3) = 1;(000) : Band inversion at
• Energy gap: ~ .3 eV : A room temperaturetopological insulator
• Simple surface state structure :Similar to graphene, except only a single Dirac point
EF
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Bulk HgTe as a 3-D Topological ‚Insulator‘
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
Bulk HgTe is semimetal,
topological surface state overlaps w/ valenceband.
k(1/a)
E-E
F(eV
)
ARPES: Yulin Chen, ZX Shen, Stanford
C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).
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70 nm layer on CdTe substrate:coherent strain opens gap
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0 2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
12000
14000
16000
0
2000
4000
6000
8000
10000
12000
14000
Rxx (SdH)
R
xx in
Ohm
B in Tesla
Rxy (Hall)
Rxy
in O
hm
Bulk HgTe as a 3-D Topological ‚Insulator‘
@ 20 mK: bulk conductivity almost frozen out - Surface state mobility ca. 35000 cm2/Vs
C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).
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2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
xy [e
2 /h]
B [T]
Bulk HgTe as a 3-D Topological ‚Insulator‘
@ 20 mK: same data, plotted as conductivity
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3D HgTe-calculations
2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
2.73.54.45.67.69.711 33.94.96.78.510.112
experiment
Rxx
in O
hm
B in Tesla
n=3.7*1011 cm-2
n=4.85*1011 cm-2
n=(4.85+3.7)*1011 cm-2
DO
S
Red and blue lines : DOS for each of the Dirac-cones with the corresponding fixed 2D-density,Green line: the sum of the blue and red lines
C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).
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Experiments on a gated Hallbar
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0 2 4 6 8 10 12 14 16
-10
12
34
5 0
5
10
15
20
25
Vgate [V]
B [T]
Rxy
[k
]
Rxy from -1.5V to 5V
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FM FMchiral interconnect
3D topological insulator
Applications of TI in IT
Topological chiral interconnects
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Majorana Fermions
• Quasiparticles in fractional Quantum Hall effect at =5/2Moore, Read ’91
• s-wave superconductor / Topological Insulator structureFu, Kane ‘08
• semiconductor - magnet - superconductor structuresSau, Lutchyn, Tewari, Das Sarma ‘09
• .... among others
• 2 Majorana bound states = 1 fermion- 2 degenerate states (full/empty) = 1 qubit
• 2N separated Majoranas = N qubits• Quantum Information is stored non locally
- Immune from local decoherence• Adiabatic Braiding performs unitary operations
- Non Abelian Statistics
1 2i
Potential Condensed Matter Hosts :
Topological Quantum Computing Kitaev, 2003
Create
Braid
Measure 12 34 12 340 0 1 1 / 2
t
12 340 0
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Superconducting contacts on strained HgTe
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dV/dI (Vbias, B) at 20 mK
dV/dI (Ibias, B) at 20 mK
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Outlook
•A new electronic phase of matter has been predicted and observed- 2D : Quantum spin Hall insulator in (Hg,Cd)Te QW’s- 3D : Strong topological insulator in Bi2Se3, Bi2Te3 and strained HgTe
•Dissipationless transport in spin-polarized 1D channels•Strong Magnetoelectric Effect; Possibilities for domain wall transport?•Superconductor/Topological Insulator structures host Majorana Fermions
- A Platform for Topological Quantum Computation•Some Challenges in the near future:
- Transport Measurements on topological insulators- Superconducting structures :
- Create, Detect Majorana bound states- Magnetic structures :
- Create chiral edge states, chiral Majorana edge states
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Würzburg:Bastian Büttner, Christoph Brüne, Markus König, Andreas Roth, VolkmarHock, Alina Novik, Chaoxing Liu, Ewelina Hankiewicz , Grigory Tkachov, Björn TrauzettelStanford:Xiaoliang Qi, Shoucheng Zhang
Funding: DFG (Schwerpunkt Spintronik, DFG-JST), Humboldt Stiftung, GIF, EU, ERC,DARPA
Acknowledgements