Quantized Spin Transport in Magnetically-Disordered Quantum Spin Hall Systems

Kiminori HATTORI�

Department of Systems Innovation, Graduate School of Engineering Science,

Osaka University, Toyonaka, Osaka 560-8531, Japan

(Received July 1, 2011; accepted October 13, 2011; published online December 2, 2011)

Helical edge states that transport electrons with opposing spins in opposite directions are formed in a quantumspin Hall (QSH) system. These states are essentially robust against time-reversal-invariant perturbations such asnonmagnetic disorder, but are vulnerable to magnetic disorder that breaks time reversal symmetry. In this study, weexplore how magnetic disorder affects charge and spin transport in the QSH system on the basis of the nonequilibriumGreen’s function formalism. We show that quantized charge transport is destroyed by any weak magnetic disorder inan infinitely large system, whereas quantized and conserved spin transport survives magnetic disorder under a specificbias condition as long as the bulk gap remains intact. These two contrasting features are explained in terms of the spin-flip backward scattering allowed in helical edge channels and its effects on charge and spin currents.

KEYWORDS: quantum spin Hall effect, quantum Hall effect, magnetic disorder, edge states, spin transport, quantization

1. Introduction

The quantum spin Hall (QSH) state emerging in thepresence of a spin–orbit interaction is a topologicallynontrivial state of matter, characterized by a bulk insulatinggap and helical edge states providing gapless excitations.1,2)

The simplest picture of the QSH system is two copies ofquantum Hall (QH) systems with different spins arranged sothat time reversal symmetry is recovered. The helical statesformed on each edge constitute a Kramers doublet thattransports electrons with opposing spins in opposite direc-tions, and are essentially insensitive to a single particleperturbation that preserves time reversal symmetry, suchas nonmagnetic impurity scattering.3) The QSH effect wastheoretically predicted for a class of spin–orbit coupledsystems, e.g., a monolayer graphene film,4) an invertedHgTe/CdTe quantum well,5) a zinc-blende semiconductorwith a shear strain gradient,6) and a quantum wire with aharmonic confining potential.7) The first two systems containa single bulk gap within which edge dispersions cross, whilethe latter two entail multiple gaps separating bulk Landaulevels (LLs). Experimentally, the QSH effect was observedin HgTe/CdTe quantum well structures.8) In particular, thelongitudinal charge conductance in a Hall bar geometry isreasonably well quantized without the need for magneticfields for samples of micron length scale. This observation isconsistent with the theoretical prediction. However, largersamples exhibit a clear deviation from the predicted exactquantization.

The stability of helical edge states is an important issuefor both fundamental understanding and potential applica-tions of the QSH effect. On the theoretical side, thus faralmost all studies have addressed the effects of nonmagneticdisorder9–12) or a single Kondo impurity13,14) on chargetransport. In this study, we pursue a quantitative investiga-tion of the effects of magnetic disorder on charge and spintransport in the QSH system. The magnetic disorder violatestime reversal symmetry so that helical edge states are nolonger protected for symmetry reasons. The present studyemploys a generalized LL model to encompass in a unified

manner the little-studied QH system suffering from magneticdisorder. As might be expected, edge conductions in thesetwo systems exhibit marked differences in robustness againstmagnetic disorder. A sharp contrast is also found betweencharge transport and spin transport in the QSH system.In particular, two important features, i.e., the breakdown ofquantized charge transport by any weak magnetic disorder inan infinitely large system and the preservation of quantizedand conserved spin transport under a specific bias condition,are illustrated.

2. Model and Formulation

The LL model for a clean QSH system in the xy-plane isdescribed by introducing a spin-dependent vector potential,which is explicitly expressed as as ¼ bs�zð y; 0; 0Þ in theLandau gauge, where bs is the relevant internal magneticfield and �z is the Pauli spin matrix.6,7) The correspondingHamiltonian is simply written as H0 ¼ ð1=2mÞðp� easÞ2,where p is the canonical momentum and m is the electronmass. We extend this model by incorporating an ordinaryspin-independent vector potential ac ¼ bcð�y; 0; 0Þ for anexternal magnetic field bc. Then, H0 becomes

H0 ¼ 1

2mðp� eAÞ2; ð1Þ

with A ¼ as þ ac, accounting for both QSH and QH effects.The generalized H0 still commutes with �z and is easilydiagonalized. The eigenenergy is "n� ¼ h� j!�jðnþ 1=2Þ,where n ¼ 0; 1; 2; . . . is the LL index, !� ¼ !c � !s� isthe effective cyclotron frequency composed of !s,c ¼jejbs,c=m, and � ¼ �1 for spin-" and # states, respectively.The eigenfunction is represented as ¼nk�ðx; yÞ ¼ eikx�n�ð y�yk�Þ, where k is the propagation wavevector along x, and�n�ð yÞ is the nth eigenfunction of a harmonic oscillator witha natural length scale ‘� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h�=mj!�jp

. The center coordinateyk� ¼ sgnð!�Þk‘2� accounts for the shift of the eigenfunctionin the y-direction by an amount depending on spin andmomentum. This displacement correlates with the dispersionrelation of gapless edge states emerging in a finite-sizesystem. It follows from the factor sgnð!�Þ involved in yk�that these states are helical in the QSH system with !s 6¼ 0

and !c ¼ 0, while they are chiral in the QH system with�E-mail: hattori@ee.es.osaka-u.ac.jp

Journal of the Physical Society of Japan 80 (2011) 124712

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FULL PAPERS

#2011 The Physical Society of Japan

DOI: 10.1143/JPSJ.80.124712

!s ¼ 0 and !c 6¼ 0. Hereafter, we will examine these twosystems comparatively in the presence of magnetic disorder.

Magnetic disorder is caused by random magnetic impu-rities or hyperfine interactions. Generally, an electronundergoes frequent elastic scattering with a spin flip in thepresence of a disorder of this kind. As a result, both spin andmomentum tend to dephase simultaneously. The correspond-ing Hamiltonian is modeled by15,16)

Hm ¼Xj

Bj � ��ðr� RjÞ; ð2Þ

where Bj represents a local magnetic field at position Rj.Thus, the total Hamiltonian we address consists of H ¼H0 þHm.

In the numerical calculation, we assume a strip geometryof magnetically disordered QSH or QH conductor with alength Lx and a width Ly, which is connected at both ends totwo semi-infinite ideal leads without disorder (labeled L andR). In the tight-binding representation on a square latticewith a lattice spacing a, the Hamiltonian H ¼ H0 þHm

describing the disordered region is written as follows:

H0 ¼Xr;r0 ;�

t�rr0cyr�cr0�;

where cr� is the annihilation operator of an electron atposition r with spin �,

t�rr0 ¼�t expði��

rr0 Þ for r ¼ r0 � a

4t for r ¼ r0

0 otherwise

8<: ;

t ¼ h�2=2ma2 is the hopping energy, and ��

rr0 ¼ ðe=h� ÞA� �ðr� r0Þ is the Peierls phase; Hm ¼ ð1=2ÞWP

r;�;�0 mr �cyr����0cr�0 , where W ¼ 2B=a2 and mr is a randomlydirected unit vector.

The spin-resolved particle current in lead � isdescribed by the Landauer–Buttiker formula

J�� ¼ h�1X�;�

T ���� ð�� � ��Þ;

where �� is the spin-dependent chemical potential.17,18)

The transmission coefficient is given by T���� ¼

Tr�Gr�����G

a��Þ, where Gr ¼ ðGaÞy is the retarded

Green’s function and �� is the linewidth function thatdescribes the coupling to lead �. We use the notationsQ� ¼ ð1=2ÞðQ" �Q#Þ to symmetrically express charge(Qþ) and spin (Q�) components of a physical quantity Q.In this terminology, the charge conductance Gcharge

LR ¼IþL =ðVþ

L � VþR Þ is defined with a charge current IþL ¼ eJþ

L

driven by a charge bias þL � þ

R ¼ eðVþL � Vþ

R Þ. The spinconductance Gspin

LR ¼ I�L =ðV�L � V�

R Þ is prescribed similarly.The spin bias �

L � �R ¼ eðV�

L � V�R Þ can be created by

spin pumping at the longitudinal ends.19–23) The magneticdisorder violates spin conservation so that Gcharge

LR and GspinLR

are no longer equivalent.7) Moreover, GspinLR 6¼ Gspin

RL ingeneral because of spin nonconservation, I�L þ I�R 6¼ 0,

although GchargeLR ¼ Gcharge

RL owing to charge conservation,IþL þ IþR ¼ 0. The single-particle density matrix is formu-lated as

� ¼ 1

2i

Z 1

�1G<ð"Þ d"

in terms of the lesser Green’s function G<��0 ¼P

� Gr���

<� G

a��0 , where �<

� ¼ iP

� f����� is the lesserself-energy and f��ð"Þ ¼ f ð"� ��Þ is the Fermi func-tion.17,18) The density matrix is useful in evaluating localquantities in the interior of the conductor. For instance, theparticle current density is represented as

j�ðrÞ ¼ ImðJ����Þrr;where

ðJ�Þrr0 ¼ �at�rr0

h�a2for r ¼ r0 � a

0 otherwise

(:

3. Numerical Calculation and Discussion

In the calculation, the bulk LL gap h�!s,c is taken as theenergy unit. The hopping energy is chosen to be as large ast ¼ 10h�!s,c (which gives ‘" ¼ ‘# ¼� 5a in both QSH andQH states) to reasonably simulate the continuum limit. Thesystem dimension is normally Lx ¼ 1000a and Ly ¼ 50a,unless otherwise stated. The ensemble average is performedover 1000 random disorder configurations to obtain themeans and standard deviations shown below.

Figure 1 explains how magnetic disorder affects thecharge conductance Gcharge

LR . In the absence of disorder,Gcharge

LR varying with the Fermi energy exhibits quantumplateaus at �e2=h with � ¼ 1; 2; 3; . . . for both QSH andQH systems. In the presence of weak magnetic disorder,quantization still persists in the QH system, particularlywhen is situated in between adjacent bulk LLs. Withinthe bulk gap, the W dependence is rigorously flat untilquantization is eventually destroyed by a strong disorder.In the flat region, conductance fluctuations (shown by errorbars) are negligible, signaling the occurrence of perfectquantization due to edge conduction. This behavior is

0

1

2

3

4

5

60 1 2 3 4 0 1 2 3 4 5

-0.20

0.20.40.60.8

1

0.1 1 10 0.1 1 10

(a) (b)

(c) (d)

GL

Rchar

ge(e

2/ h

)

μ ( s ) μ ( c )

G

GC

C

GL

Rchar

ge,

CL

R

W ( s ) W ( c )

Fig. 1. (Color online) (a) and (b) show charge conductance GchargeLR

calculated as a function of Fermi energy at disorder strength W ¼2h�!s,c for QSH and QH systems, respectively. (c) and (d) display Gcharge

LR ¼Gcharge

LR =ðe2=hÞ vs W at ¼ h�!s,c in comparison with variation in spin

coherence CLR in QSH and QH systems, respectively. Solid lines in (a) and

(b) represent GchargeLR in the clean limit for reference. Error bars denote �1

standard deviations for 1000 samples. The system sizes assumed in the

calculation are Lx ¼ 1000a and Ly ¼ 50a.

K. HATTORIJ. Phys. Soc. Jpn. 80 (2011) 124712 FULL PAPERS

124712-2 #2011 The Physical Society of Japan

reminiscent of that for nonmagnetic disorder.24,25) Incontrast, Gcharge

LR decreases markedly and shows a sizablefluctuation for the QSH system even for a weak magneticdisorder. The tendency shown by the means differs from theprevious result derived for a homogeneous phase and spindecoherence, which shows a minimum deviation from aquantum staircase when is in the vicinity of the bulkLLs.26)

To draw further implications of charge transport for weakmagnetic disorder, it is appropriate here to introduce thespin-conserved conductance gsc�� ¼ ðe2=2hÞðT""

�� þ T##�� Þ and

the spin-flip conductance gsf�� ¼ ðe2=2hÞðT"#�� þ T#"

�� Þ. Interms of these elemental conductances, the charge con-ductance is decomposed into Gcharge

LR ¼ gscLR þ gsfLR. Accord-ingly, CLR ¼ ðgscLR � gsfLRÞ=ðgscLR þ gsfLRÞ represents the spincoherence during charge transport,20,27) which is useful incharacterizing a hidden conservation law for edge transportin the disordered QSH system, as shown below. Thespin coherence CLR is plotted in Figs. 1(c) and 1(d), incomparison with Gcharge

LR . The behavior of CLR is in sharpcontrast to that of Gcharge

LR . In the QSH system, CLR remainsunity in the region where Gcharge

LR deviates from the quantizedvalue, while in the QH system CLR is lower than unity in theregion where Gcharge

LR remains quantized. These observationsare easily interpreted as follows. The edge states are helical(chiral) in the QSH (QH) system. The direct consequence ofthis property is that spin-flip forward scattering is forbidden(allowed) in the QSH (QH) system. Therefore, gsfLR ¼ 0

(gsfLR > 0) so that CLR ¼ 1 (CLR < 1) for the QSH (QH)system. On the other hand, spin-flip backward scattering isallowed (forbidden) in the QSH (QH) system. As a result,quantized charge transport is destroyed (protected) in theQSH (QH) system. These interpretations are validated fromthe numerical results for gsfLR and gsfLL shown in Fig. 2, whichreflect spin-flip forward and backward scatterings, respec-tively.

The edge conduction is destroyed by bulk gap closing or,equivalently, by edge-bulk hybridization.24,25,28,29) Theenergy broadening of bulk LLs due to magnetic disorder isgiven by � ¼ W

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih�!s,c=t

pin the self-consistent Born

approximation, providing the critical disorder Wgap ¼�5:6h�!s,c beyond which the bulk gap closes. One the otherhand, the scattering lifetime � in edge states is represented ash�=� ¼ ð=4ÞW2N, where N denotes the density of states persite. The density of edge states numerically calculated at thecenter of the bulk gap leads to Wgap ¼� 3:6h�!s,c beyondwhich the edge and bulk states can mix. These roughestimates rely on the lowest-order perturbation theory.Nevertheless, the derived Wgap is in reasonable agreementwith the critical disorder Wc � 4h�!s,c observed for the CLR

(GchargeLR ) of the QSH (QH) system. Additionally, the spin-flip

length ‘sf ¼ vF�sf in edge states is evaluated from theFermi velocity vF ¼ a=h�N and the spin-flip time �sf ¼ð3=2Þ�,15,16) yielding the critical disorder Wsf ¼� 0:25h�!s,c atwhich ‘sf ¼ Lx ¼ 1000a. This value does not largely differfrom the fall-off point observed for the Gcharge

LR (CLR) of theQSH (QH) system. The first two criteria for Wgap do notdepend on the system size, whereas the last one predicts thatWsf ! 0 as Lx ! 1. The expected behaviors are observedin Fig. 3, which shows the numerical results obtained forvarious Lx values. Thus, we reach an important conclusion

that quantized charge transport in the QSH system isobservable only when Lx < ‘sf , and is always quenched byan arbitrarily weak magnetic disorder in the thermodynamiclimit. This reasonably accounts for the unquantized chargeconductance of a macroscopically large QSH systemmentioned earlier.

It may be worth noting the probable effect on helical edgestates with dispersion crossing in the clean limit. In such acase, mixing due to intraedge scattering may produce ananticrossing gap. That is, the energy dispersion of edgestates is gapped so that the edge conductance vanisheslocally when is situated in the edge gap. In the presentLL model, dispersion crossing is originally avoided [seeFig. 1(b) in ref. 7], and hence edge states remain gapless.This feature is confirmed in Fig. 4, which shows the localdensity of states Nð y; "Þ calculated for a weak magneticdisorder.

Next, we focus on the effects of magnetic disorder onspin transport. Generally, the terminal spin current I�Ldepends on V�

L and V�R separately in a system where spin

conservation is violated. This draws a distinction from theterminal charge current IþL , which depends only on thevoltage difference Vþ

L � VþR . In this study, we particularly

consider the following asymmetric and antisymmetricconfigurations: (R) V�

L ¼ 0 and V�R 6¼ 0, (L) V�

L 6¼ 0 andV�R ¼ 0, and (A) V�

L ¼ �V�R 6¼ 0. In terms of gsc�� and gsf��,

the spin conductances in these three situations are expressedas

-0.20

0.20.40.60.8

11.2

0.1 1 10 0.1 1 10

g(e

2/ h

)

(a) (b)

gLR

sc

gLR

sf

gLL

sfgLR

scgLL

sf

W ( s ) W ( c )

Fig. 2. (Color online) (a) and (b) show elemental conductances gsc�� and

gsf�� as functions of disorder strength W at ¼ h�!s,c for QSH and QH

systems, respectively. Error bars denote �1 standard deviations for 1000

samples. The system sizes assumed in the calculation are Lx ¼ 1000a and

Ly ¼ 50a.

0

0.2

0.4

0.6

0.8

1

1.2

0.1 1 10 0.1 1 10

GL

Rchar

ge(e

2/ h

)(a) (b)

W ( s ) W ( c )

Lx

Lx

Fig. 3. (Color online) Charge conductance GchargeLR as a function of

disorder strength W for various system lengths Lx ¼ 10, 20, 50, 100, 200,

500, and 1000a. (a) and (b) show the results computed for QSH and QH

systems, respectively. The parameters assumed in the calculation are

¼ h�!s,c and Ly ¼ 50a. In the calculation, the disorder average is

performed over 1000 random configurations.

K. HATTORIJ. Phys. Soc. Jpn. 80 (2011) 124712 FULL PAPERS

124712-3 #2011 The Physical Society of Japan

GspinLR jR ¼ gscLR � gsfLR;

GspinLR jL ¼ gscLR þ gsfLR þ 2gsfLL;

GspinLR jA ¼ gscLR þ gsfLL;

which obey the sum rule that

GspinLR jA ¼ 1

2ðGspin

LR jR þGspinLR jLÞ:

Note that if spin conservation is preserved, gsf�� ¼ 0, so that

GchargeLR ¼ Gspin

LR jR,L,A ¼ gscLR.

Figures 5(a) and 5(b) show the spin conductance GspinLR

calculated as a function of the disorder strength W for theQSH and QH systems, respectively. Recall that spin-flipforward (backward) scattering is forbidden in the QSH (QH)system until the bulk gap collapses. For a weak disorder,gsfLR (gsfLL) is therefore vanishingly small for the QSH (QH)system so that Gspin

LR jR (GspinLR jL) becomes formally equivalent

to GchargeLR . This is the reason why Gspin

LR jL is apparentlyquantized for the QH system. The allowed spin-flipscattering significantly affects spin transport, e.g.,Gspin

LR jA ¼� GchargeLR � gsfLR and Gspin

LR jR ¼� GchargeLR � 2gsfLR for

the QH system. The most surprising and importantobservation is that Gspin

LR jA ¼ gscLR þ gsfLL is completelyquantized and unaffected by weak disorder for the QSHsystem. Physically, quantized spin transport in the dis-ordered QSH system is ascribed to the helical property ofedge channels because the intraedge scattering allowed theredoes not reverse spin flux. This is analogous to quantizedcharge transport in a disordered QH system with chiral edgechannels. The intraedge scattering allowed there does notreverse charge flux.

More explicitly, these quantizations are shown from theunitarity of relevant scattering matrices. The scatteringmatrix relating outgoing and incoming mode amplitudes isgenerally represented as

S ¼ t"" r"#r#" t##

� �; ð3Þ

for a pair of helical edge modes at each boundary. Theunitarity of the S matrix requires that Tsc ¼ T"" ¼ T##,Rsf ¼ R"# ¼ R#", and Tsc þ Rsf ¼ 1, where T��0 ¼ jt��0 j2and R��0 ¼ jr��0 j2 are the transmission and reflection

probabilities, respectively. T��0 and R��0 are directly linkedto T��0

�� for the two-terminal QSH conductor withoutinteredge interactions, explaining that Gspin

LR jA ¼ e2=h irre-spective of the details of scattering processes. Analogously,one finds Gcharge

LR ¼ e2=h for the QH conductor.The quantized spin transport in the QSH system does not

immediately imply the conservation of local spin current.Spin conservation is locally violated by a random spintorque

Pj Bj � ��ðr� RjÞ due to magnetic disorder. To

investigate the microscopic details of spin current flow, weconsider a modified structure where a normal region isinserted between each lead and the disordered region. Thenormal region is an extension of the ideal lead where W ¼ 0

so that spin current is conserved. Figures 5(c) and 5(d)show the local spin conductance GspinðxÞ ¼ I�ðxÞ=ðV�

L �V�R Þ computed from the longitudinal spin current

I�ðxÞ ¼ e

Zj�x ðx; yÞ dy

in the QSH and QH systems, respectively. In this calcula-tion, a normal region of length 10a and a disordered regionof length 50a are assumed. It has been verified in thenumerical calculation that the local spin current j�ðx; yÞflows through edge channels over the entire region as longas W < Wc (not shown). As demonstrated in these figures,the longitudinal spin current is generally nonuniform andexhibits a large statistical fluctuation, except in the normalregion connecting to the lead where the quantized terminalspin current flows. Condition (A) is a special case when themean GspinðxÞ stays at the quantized value in the entire QSHconductor, and its fluctuation disappears in normal regions.This feature shows that a global continuity relation isestablished for quantized terminal spin currents, I�L þ

0.1 1 10

0

0.4

0.8

1.2

1.6

2

0.1 1 10

00.20.40.60.8

11.21.4

-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30

GL

Rspin

(e2/ h

)sp

in(e

2/

h)

x (a) x (a)

(a) (b)

(c) (d)

L L

AAR R

L

L

A

A

W ( s ) W ( c )

Fig. 5. (Color online) (a) and (b) show spin conductance GspinLR calculated

as a function of disorder strength W at ¼ h�!s,c in three different spin bias

configurations for QSH and QH systems, respectively. The system sizes

assumed in this calculation are Lx ¼ 1000a and Ly ¼ 50a. (c) and (d)

display local spin conductance GspinðxÞ computed for ¼ h�!s,c and

W ¼ 2h�!s,c in QSH and QH systems, respectively. In this calculation,

a modified structure consisting of a disordered region of length 50a

sandwiched between two normal regions of length 10a is assumed. Error

bars denote �1 standard deviations for 1000 samples.

-20 -10 0 10 20

0.6

0.8

1.0

1.2

1.4

0

0.01

0.02

0.03

0.04

0.05

0.06

y (a)

ε (

) s

N (1 / a )s2

Fig. 4. (Color online) Localdensity of states Nð y; "Þ ¼ L�1x

RNðx; y; "Þ dx

calculated for QSH system at disorder strength W ¼ 2h�!s. In the

calculation, the disorder average is performed over 1000 random

configurations.

K. HATTORIJ. Phys. Soc. Jpn. 80 (2011) 124712 FULL PAPERS

124712-4 #2011 The Physical Society of Japan

I�R ¼ 0, under condition (A). Thus, we obtain anotherimportant conclusion that quantized and conserved spintransport is preserved in the QSH system under theantisymmetric spin bias condition even in the presence ofa weak magnetic disorder.

Finally, we briefly comment on the QSH system in anexternal magnetic field. In this case, the QSH and QH effectsoccur cooperatively, and hence Landau quantization be-comes spin-asymmetric, i.e., "n" < "n#. In the region where"0" < < "0#, current-carrying edge states are fully spin-polarized so that intraedge scattering is completely for-bidden. As a result, charge and spin transport via the lowestedge channel is half-quantized and insensitive to a weakmagnetic disorder. These expected features are confirmed inFig. 6, which illustrates the numerical result obtained for!c ¼ 2!s.

4. Conclusions

In summary, we have numerically investigated the effectsof magnetic disorder on charge and spin transport in QSHand QH systems in a unified manner on the basis of the LLmodel. The nonequilibrium Green’s function formalism isemployed to elucidate charge and spin currents through atwo-terminal conductor, the microscopic profiles of thesecurrents in the interior of the conductor, and their statisticalfluctuations. The main conclusions derived for the QSHsystem are as follows: (i) quantized charge transport issuppressed by magnetic disorder in a macroscopic systemlarger than the spin-flip length, and (ii) quantized andconserved spin transport survives magnetic disorder in theantisymmetric spin bias configuration as long as the bulk gapremains uncollapsed. These two features essentially origi-

nate from the helical property of transport channels inherentto the QSH effect. Therefore, the present conclusions drawnfrom the LL model are widely applicable to various two-dimensional topological insulators exhibiting the QSHeffect. In addition, the LL mechanism suggests that spin-polarized and robust channels can be created by spin-asymmetric Landau quantization in the presence of anexternal magnetic field.

Acknowledgement

This work was supported by a Grant-in-Aid for ScientificResearch (No. 21540320) from the Japan Society for thePromotion of Science.

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0

1

2

3

4

0 1 2 3 4 5μ ( s )

GL

Rchar

ge,

CL

R

G

C

c = 2 s

Fig. 6. (Color online) Normalized charge conductance GchargeLR ¼ Gcharge

LR =ðe2=hÞ and spin coherence CLR calculated as functions of Fermi energy at

disorder strength W ¼ 2h�!s for QSH system subjected to external magnetic

field. The parameters assumed in the calculation are !c ¼ 2!s, Lx ¼ 1000a,

and Ly ¼ 50a. The solid line represents GchargeLR in the clean limit for

reference. Error bars denote �1 standard deviations for 1000 samples.

K. HATTORIJ. Phys. Soc. Jpn. 80 (2011) 124712 FULL PAPERS

124712-5 #2011 The Physical Society of Japan