Disordered quantum spin Hall systems

X. C. Xie

International Center for Quantum Materials, Peking University

and

Department of Physics, Oklahoma State University

Collaborators:

H. Jiang, S.G. Cheng, L. Wang and Q.F. Sun

Institute of Physics, Chinese Academy of Science, Beijing, China

Supported by NSF-China, China-MOST, US-DOE, US-NSF

Outline

Introduction

QSHE and its recent developments

Dephasing effect in QSHE (PRL103, 036803, 2009)

The dephasing model and formalismNumerical results and discussion

Disorder effect in QSHE (PRB80, 165316, 2009)

Conclusion and outlook

Hall effect and quantum Hall effect

Edwin Hall (1879)

K.Von klitzing(1980)

y yH

x

E LR B

j= ∝

2

1H

hReν

=

The Hall resistance can be quantized under certain conditions.

The spin Hall effect

Y.K.Kato et al., Science, 306, 1910 (2004）

Carriers with opposite spins are separatedby spin-orbit coupling

ij spin ijk kJ Eσ ε=

Murakmi et al, Science 31, 1248 (2003)

Can spin Hall effect be quantized?

Physical understanding of QSHE: edge states

Since the flow is dissipationless in QHE, the QSHE system might provide a spin circuit without dissipation.

N.Nagosa, Scienece 318,758(2007)

The counter-propagating edge channels for opposite spins in each edge lead to QSHE.

Absence of the magnetic fieldMaintain the time reversal symmetry

(different from the QHE)

Theoretically predicated systems

Graphene (C.L Kane et al., PRL 95, 226801, PRL 95, 146802 (2005) ) (x)demanding extremely low temperature(<10mk)(Yugui Yao et al., PRB 75,041401 (2007) H.Min et al., PRB 74, 165301(2006))

Bismuth layer (S. Murakami PRL 97,236805 (2006) ) (?)spin Hall effect and spin splitting are directly observed

(T. Hirahara et al., PRB, 76,153305(2007)

Jiyu Fan et al., APL 92,142101 (2008))

HgTe/CdTe quantum well (B.A Berneig et al., Science 314,1757 (2006))

3D material BixSb1-x (F. Liang et al., PRL 98,106803 (2007))

Confirmed by experiment using APRESD.Hsieh et al., Nature 452,970 (2008)

InAs/GaAs/AlSb Quantum wells (C.Liu et al., PRL 100,236301(2008) ) (?)2010/10/28

2010/10/28

Transport measurement: theory

S.Datta , Electronic transport in mesoscopic systems M König et al., J. Phys. Soc. Jpn. 77, 031007 (2008)

The longitudinal resistance R14,23 is also quantized in QSHE.

Transport measurement: experiment

B.A. Bernevig, et al., Science 314, 1757 (2006)The experiment set-up inCdTe/HgTe/CdTe samples

c

cddd

d<>

normal insulatorQSH state

Transport measurement: experiment

M. Konig, et al., Science 318, 766 (2007).

d<dc normal insulator

d>dcWith small L and W, R14,23

is quantized, insensitive toW variations.

With large L, R14,2 3 is nolonger quantized.

QSH signal is robust againsttemperature change.

+ +i i i i y i i

i, i, ,[ ] [ ( )]im

x k k Ik k k k kk

H te c c tc c a a t c a t a cσφσ δ σ σ δ σ σ σ σ σ σ σ

σ σ

ε + + ++ += − + + + +∑ ∑

2 2x yi i i t / 2 * , /ef fm a qB aφ= = 2（，），=

The dephasing mode for a QSHE system

L

W1

2 3

4

56

M

L

W1

2 3

4

56 L

W1

2 3

4

56

M

• The first term describes QSHE. This term can also describe QHE if the flux is spin independent.• We use virtual leads to model dephasing processes.

M. Buttiker, Phys. Rev. B 33 3020 (1986)

Two types of dephasing processes

The electron-electron and electron-phonon interactions only reduce the electron phase memory. These dephasing strengths increase with rising temperature.

Magnetic impurities, nuclear spin fluctuations and electro-magnetic noise may reduce spin memory. These dephasing strengths are insensitive to temperature.

Theoretical formalism

The particle current in the lead-p (real or virtual lead) with spin can be expressedσ

Model dephasing processes

First type dephasing processes:

For each virtual lead i i iV V↑ ↓≠

Second type dephasing processes:

The spin flip processes are forbidden.

The spin flip processes are allowed.

For each virtual lead i

i i 0I I↑ ↓= ≡

i i 0I I↑ ↓+ ≡

Theoretical formalism

L

W1

2 3

4

56

M

L

W1

2 3

4

56 L

W1

2 3

4

56

MHall resistance (use QHE Hamiltonian)

Longitudinal leads

Traverse leads 0,p pp p p pI I I V V V↑ ↓ ↑ ↓= + ≡ = =

Hall resistance 2 6 3 5

1 1H

V V V VRI I− −

= =

Longitudinal resistance (use QSHE Hamiltonian)

Longitudinal leads

Traverse leads 0,p pp p p pI I I V V V↑ ↓ ↑ ↓= + ≡ = =

longitudinal resistance 23 2 314,23

14 1

V V VRI I

−= =

1 1 4 4V V V V V↑ ↓ ↑ ↓= = − = − =

1 1 4 4V V V V V↑ ↓ ↑ ↓= = − = − =

The relationship between and Lφ Γ

1 2

L

1 dir depI I I= +

Change L when , dir depI I= L Lφ =

10 20 30 400.0

0.1

0.2

(c)

Γ d

Lφ (a)

B=0.5B=0.2B=0.1

Use wide band limit for virtual leads, the dephasing strength is characterized by Γ

The phase coherence lengthdecreases rapidly with increasing dephasing strength.

Yanxia Xing et al., PRB 77,115346 (2008)

The dephasing in QHE system

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

Beff/n

Γ=0.0tΓ=0.01tΓ=0.02tΓ=0.05t Γ=0.10t

R 14,2

6 (h

/e2 )

L

W1

2 3

4

56

M

L

W1

2 3

4

56 L

W1

2 3

4

56

M

3 , 32 , 32 , 24FE t L a W a M a= − = = =

• The Fermi energy is fixed near the band bottom.• The QHE is robust against dephasing.

Dephasing in QSHE system: I

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

L

W1

2 3

4

56

M

L

W1

2 3

4

56 L

W1

2 3

4

56

M

Beff/n

(a)Γ=0.0tΓ=0.01tΓ=0.02tΓ=0.05t Γ=0.10t

R 14,2

3 (h

/e2 )

(b)

3 , 32 , 32 , 24FE t L a W a M a= − = = =

• Fig.(a) illustrates the first type dephasing while Fig.(b) is for the second type dephasing.

•In the low field, the result is consistent to that of the semi-classical Drude model.

0.00 0.05 0.10 0.15 0.200.2

0.4

0.6

0.8

1.0

0.03 0.05 0.07 0.09 51525354555

0.03 0.05 0.07 0.09 5

15

25

35

45

0.5

1.0

1.5

2.0

(b)

L Φ (a)

Γ (t)

Γ(t)

first category second category

Γ (t)

(c)

(d)

R 14,2

3 (h

/e2 )

L Φ (a

)

(a)

Dephasing in QSHE system: II

eff eff

3 , 32 , 32 , 240.5....( ) 0.3....( )

FE t L a W a M aB a B b

= − = = == =

• The QSH signal is insensitive to the first type dephasing butsensitive to the second type.

• QSH signal is robust against temperature change.

Dephasing in QSHE system: III

25 35 45 55 65 75

0.6

0.8

1.0

1.2

1.4

1.6

10 20 30 40

0.6

0.8

1.0

1.2

1.4

1.6

R14

,23 (h

/e2 )

L (a)

Γ=0.02tΓ=0.05tΓ=0.02tΓ=0.05t

(a)

W (a)

(b)

eff eff

3 , 240.5, 32 ....( ) 0.3, 32 ....( )

FE t M aB W a a B L a b

= − == = = =

• The QSH signal is insensitive to width variation but sensitive to thelength change.

Dephasing in QSHE system: IV

0.0 0.1 0.2 0.3 0.4 0.5

0.1

1

R 14,2

3 (h/

e2 )

Beff/n

Γ=0.02t L=32a Γ=0.05t L=32aΓ=0.02t L=80aΓ=0.05t L=80a

• R exhibits the plateau feature.

eff3 , 24 , 0.5, 32 ....FE t M a B W a= − = = =

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Dephasing in QSHE system: V

0.00.20.40.60.81.0

0.0 0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.0

Γ=0.0tΓ=0.02tΓ=0.05tΓ=0.10t

Beff/n

(a)

I 2,s/I

14 (

h/4π

e)

(b)

3 , 24 , 32 , 32 ....FE t M a L a W a= − = = =

• Spin current is equal to charge current except in band-jumping region.

2, 3, 5, 6,s s s sI I I I= = − = −

spin accumulations: formula and reasoning

L

W1

2 3

4

56

M

L

W1

2 3

4

56 L

W1

2 3

4

56

M Longitudinal leads

Traverse leads 0, 0,p p p pI I V V↑ ↓ ↑ ↓≡ ≡ ≠

resistance p,s1

p pV VR

I↑ ↓−

=

VV

--vv

(a)VV

--vv

(a)VVVVVVVVVVVVVVVVVVVVV

---------------------vvvvvvvvvvvvvvvvvvvvv

(a)VV

--vv

(b)VV

--vv

(b)VVVVVVVVVVVVVVVVVVVVV

---------------------vvvvvvvvvvvvvvvvvvvvv

(b)

for a given edge

in the absence of dephasingin the present of dephasing

spin flip

1

p pV V

I↑ ↓−

1 1 4 4V V V V V↑ ↓ ↑ ↓= = − = − =

2010/10/28

Dephasing in QSHE system: VI

3 , 24 , 32 , 32FE t M a W a L a= − = = =

0.00.10.20.30.40.5

0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5R s (

h/e2 )

(a)

(b)

Beff(h/ea2)

Γ=0.0tΓ=0.02tΓ=0.05tΓ=0.10t

Rs is robust against any dephasing processes

2, 3, 5, 6,s s s sR R R R= = − = −

Conclusion on dephasing effect

The QSHE is currently one of the most exciting research area in condense matter physics. It covers the topics such as: topological insulators, Dirac fermions, and dissipationless spintronics devices, etc.

The spin dephasing plays important roles in QSHE, it makes the quantized longitudinal resistance only observable in mesoscopic systems. However, the spin accumulation measurement that is robust against any dephasing may provide a new playground.

Disorder effect in topological insulator

Topological Anderson insulator

Jian Li , Rui Lin Chen, J. K. Jain and Shunqing Shun arxiv: 0811.3045

2010/10/28

Energy spectrum of HgTe/CdTe QWs

1 2

0.6 Lx

Ly=Na 1 2

0.6 Lx

Ly=Na

1.(a) 1.(b)

1 2

0.6 Lx

Ly=Na 1 2

0.6 Lx

Ly=Na

1.(a) 1.(b)

One dimensional energy bands for device a and b. Ly=80a, a=5nm

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1 2

0.6 Lx

Ly=Na 1 2

0.6 Lx

Ly=Na

1.(a) 1.(b)

1 2

0.6 Lx

Ly=Na 1 2

0.6 Lx

Ly=Na

1.(a) 1.(b)

Anderson impurity influence of HgTe/CdTe QWs： I

Disorder can induce a phase transition in HgTe system.Anomalous plateau vanished in cylinder sample.

The Anderson disorder only act on the red filled region.

2010/10/28

Local current configurations in traditional QSHE region

We only consider spin up parts

W=0meV

W=50meV

W=110meV

W=220meV

EF=7meV, M=-10meV

Ly=80a

2010/10/28

Local current configurations in TAI region: M>0

EF=20meV, M=2meV

Ly=80a

W=0meV

W=100meV

W=150meV

W=250meV

2010/10/28

Local current configurations in TAI region: M<0

EF=20meV, M=-10meV

Ly=80a

W=0meV

W=60meV

W=100meV

W=250meV

2010/10/28

Local current configurations in TAI region: dip point

Bulk states assisted backscattering

The dip feature is clearerfor a large Lx

Conclusion on disorder effect

a)At certain disorder level, bulk states become localized, edge states appear and those edge states give rise to QSHE.

b)At very strong disorder, back scatterings between opposite edges emerge that destroy the topological Anderson insulator.