liang fu et al- the quantum spin hall effect and topological band theory

8/3/2019 Liang Fu et al- The Quantum Spin Hall Effect and Topological Band Theory

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EE

The Quantum Spin Hall Effectand Topological Band Theory

k=a k=bk=a k=b


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The Quantum Spin Hall Effect

and Topological Band TheoryI. Introduction- Topological band theory

II. Two Dimensions : Quantum Spin Hall Insulator

- Time reversal symmetry & Edge States- Experiment: Transport in HgCdTe quantum wells

III. Three Dimensions : Topological Insulator

-

- Experiment: Photoemission on BixSb1-x and Bi2Se3

IV. Superconducting proximity effect- Majorana fermion bound states- A platform for topological quantum computing?

Thanks to Gene Mele, Liang Fu, Jeffrey Teo,Zahid Hasan + group (expt)


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The Insulating State

Covalent Insulator

Characterized by energy gap: absence of low energy electronic excitations

The vacuumAtomic Insulator

e.g. solid Are.g. intrinsic semiconductor

DiracVacuum

Egap ~ 10 eV

Egap ~ 1 eV

3p

4s

Silicon

Egap = 2 mec2

~ 106 eV

electron

positron ~ hole


4/21

The Integer Quantum Hall State

2D Cyclotron Motion, Landau Levels

gap cE = hE

Quantized Hall conductivity :

Ex

B

Jy

y xy xJ E=2

xyh

ne

=

Integer accurate to 10-9

Energy gap, but NOT an insulator


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IQHE in a crystal with zero net magnetic field

Graphene in a periodic magnetic field B(r)

B(r) = 0Zero gap,Dirac point

+ + + +

+ + +

Haldane Model

(Haldane PRL 1988)

B(r) 0Energy gap

Egap

k

+ + + +

Band structureindistinguishable froman ordinary insulator

xyeh

=


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Topological Band Theory

21 ( ) ( ) Integer2 BZ

n d u ui

= = k kk k k

The distinction between a conventional insulator and the quantum Hall stateis a topological property of the manifold of occupied states

| ( ) : ( )kr

aBrillouin zone a torus Hilbert space

Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982)

g=0 g=1

Analogy: Genus of a surface : g = # holes

Insulator : n = 0IQHE state : xy = n e

2/h

e nvar ant can on y c angeat a quantum phase transition where theenergy gap goes to zero


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Edge States

Gapless states must exist at the interface between topologically distinct phases

IQHE staten=1

Vacuumn=0

Edge states ~ skipping orbits

n=1 n=0

2 Band Model : Dirac Eq.

x

y

Smooth Interpolation

E 0

v( ) ( ) x x y y z H i M x = + +

0 | | /v

0 ( , )yik y M x

yk x e e

0 ( ) vy y E k k =

M0

Gapless Chiral Fermions

E

ky

Eg

M(x) =M0: Gap Eg = 2M0

2 2 2

0( , ) v | |x y E k k k M = +

M(x) ~ M0 sgn(x) : Domain Wall bound state

1

i

xEg


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Quantum Spin Hall Effect in Graphene

The intrinsic spin orbit interaction leads to a small (~10mK-1K) energy gap

Simplest model:|Haldane|2

(conserves Sz)

Haldane

*

Haldane

0 0

0 0

H HH

H H

= =

Kane and Mele PRL 2005

J J

E

Bulk energy gap, but gapless edge statesEdge band structureSpin Filtered edge states

Edge states form a unique 1D electronic conductor HALF an ordinary 1D electron gas Protected by Time Reversal Symmetry (conservation of Sz is NOT essential) Elastic Backscattering is forbidden. No 1D Anderson localization

0 /a k

QSH Insulator

vacuum


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Topological Insulator : A New B=0 Phase

2D Time reversal invariant band structures have a Z2 topological invariant, = 0,1

=0 : Conventional Insulator =1 : Topological Insulator

Kramers degenerate attime reversal

invariant momenta

k* = k* + G

E EEdge States

k*=0 k*=/a k*=0 k*=/a

is a property of bulk bandstructure. Easiest to compute if there is extra symmetry:

1. Spin rotation symmetry : Sz conserved

,mod 2n

=

2. Inversion (P) Symmetry :

determined by Parity of occupied 2D Bloch statesat time reversal invariant points.in bulk Brillouin zone

n n

= Quantum spin Hall effectIndependent spin Chern integers


10/21

Quantum Spin Hall Effect in HgTe quantum wells

Bernevig, Hughes and Zhang, Science 06

HgTe

HgxCd1-xTe

HgxCd1-xTed

d < 6.3 nm : Normal band order d > 6.3 nm : Inverted band order

E E

Conventional Insulator

Quantum spin Hall Insulator

with topological edge states

6 ~ s

8 ~ p

k

6 ~ s

8 ~ p k

Egap~10 meV


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Observation of QSH insulator in HgTe quantum Wells

Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007

V 0

Landauer Conductance

d< 6.3 nm

2

2

e

I Vh=I

Conductance 2e2/h independent of W for short samples (L 6.3nminverted band orderQSH insulator


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Three Dimensional Topological Insulators

In 3D there are 4 Z2 invariants: (0 ;123) characterizingthe bulk. These determine how surface states connect.

Fu, Kane & Mele PRL 07Moore & Balents PRB 07Roy, cond-mat 06

Surface Brillouin Zone

4

1 2

3

2D DiracPoint

E

k=a k=b

E

k=a k=b

OR

0 = 1 : Strong Topological Insulator

Fermi surface encloses odd number of Dirac pointsTopological Metal

Can only exist at a surface

Robust to disorder (antilocalization)

0 = 0 : Weak Topological Insulator

Fermi surface encloses even number of Dirac points

Normal Metal less robust.

EF


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Bi1-xSbx

EF

Pure Bismuthsemimetal

Alloy : .09


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Experiments on Bi1-x Sbx

Map E(kx,ky) for (111) surface states below EF usingAngle Resolved Photoemission Spectroscopy

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nature (08)

5 surface state bands cross EF between and M

Proves that Bi1-x Sbx is a Strong Topological Insulator


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Experiments on Bi2 Se3

Y. Xia, L. Wray, D. Qian, D. Hsieh, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J.Cava, M.Z. Hasan, arXiv:0812.2078

Bi

2Se

3is a strong topological insulator with a simple

surface Fermi surface.

Similar to graphene, except only a

single Dirac point

EF


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Superconducting Proximity Effect

s wave superconductor

Topological insulator

Fu, Kane PRL 08

BCS Superconductor :

i

-k

Surface states acquiresuperconducting gap due to Cooper pair tunneling

k k

kSuperconducting surface states

surface

ik kc c e

Half an ordinary superconductor

Highly nontrivial ground state

-k

k

Dirac point

(s-wave, singlet pairing)

(s-wave, singlet pairing)


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Majorana Fermion at a vortex

Ordinary Superconductor :Andreev bound states in vortex core:

0

E

E ,

-E ,

Bogoliubov Quasi Particle-Holeredundancy :

, ,E E

=

0 =2 =

/ 2h e =

Majorana fermion :Half a state

Two separated vortices define

one zero energy fermion state

(occupied or empty)

Surface Superconductor :

Topological zero mode in core of h/2e vortex:

0

E 0 0 =

E=0


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Topological Quantum Computation

2 separated Majoranas = 2 degenerate states (1 qubit) 2N separated Majoranas = N qubits

Quantum information immune to local decoherence Adiabatic braiding of vortices performs unitary operations on N qubits

Kitaev 2003

Mani ulate Ma oranas b controllin hases on su erconductin unctions

S S

T I

0

S - TI - S line junction Network of line junctions

A wire for Majoranas

2/3

2/30

phase

Extra quasiparticle in eachjunction affects current


19/21

Conclusion

A new electronic phase of matter has been predicted and observed

- 2D : Quantum spin Hall insulator in HgCdTe QWs- 3D : Strong topological insulator in Bi1-xSbx and Bi2Se3

Experimental Challenges

- Spin dependent Transport Measurements- Transport and magneto-transport expts on Bi1-xSbx and Bi2Se3

- Superconducting proximity effect :- Characterize S-TI-S junctions- Create the Majorana bound states- Detect the Majorana bound states

Theoretical Challenges

- Effects of disorder on surface states and critical phenomena- Effects of electron-electron interactions- Other Materials?


20/21

The challenges : Find suitable topological insulator (Bi1-x Sbx ? Eg ~ 30 meV)

Find suitable superconductor which makes good interface ( Nb ? )

Optimize proximity induced gap and discrete Andreev bound states Control the superconducting phases with Josephson junctions Measure current difference when Majoranas are fused

Evidence for good contact between BiSb and Nb : minimal Shottky barrier

Observed super current may be dominated by bulk electrons


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A first step:

Control phases on a single tri-junction with currents I1, I2

12Zero energy boundstate predicted at+

0

r - unc on

Detect zero energy state by tunneling

Predict a zero bias tunneling anomaly when a bound

state is present.

I1I2

liang fu et al- the quantum spin hall effect and topological band theory

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