Spin Chern Pumping from the Bulk of Two-Dimensional Topological Insulators

0.64 MB·

by  M. N. Chen

Spin Chern Pumping from the Bulk of Two-Dimensional Topological Insulators
M. N. Chen1, L. Sheng1,∗ R. Shen1, D. N. Sheng2, and D. Y. Xing1†
1National Laboratory of Solid State Microstructures, Department of Physics,
and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
2 Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
Topological insulators (TIs) are a new quantum state of matter discovered recently, which are
characterized by unconventionalbulk topological invariants. Proposals for practical applications of
6 theTIsaremostlybased upontheirmetallicsurface oredgestates. Here,wereportthetheoretical
1
discovery of a bulk quantum pumping effect in a two-dimensional TI electrically modulated in
0
adiabaticcycles. Ineachcycle,anamountofspinproportionaltothesamplewidthcanbepumped
2
n uinstiongaanohnamlf-amgenteatlilciceleelcetcrtordode,e,wuhnicivheirssaalttqruibauntteizdedtochnaorngzeerpousmpipninCghceornndnuucmtivbietrisesC−±C.±Meo2r/ehovcearn,bbye
a measured. Thisdiscoverypavesthewayfordirectinvestigation oftherobusttopological properties
J of theTIs.

3
1 PACSnumbers: 72.25.-b,73.43.-f,73.23.-b,75.76.+j

]
l I. INTRODUCTION (TR)symmetryis consideredtobe aprerequisiteforthe
l
a QSH effect, which protects both the Z index and gap-
2
h
less nature of the edge states. However, based upon the
- Topological transport phenomena have been attract-
s ing a great deal of interest, because they exhibit univer- spin Chern numbers, it was shown that the bulk topo-
e logical properties remain intact even when the TR sym-
m sal properties that are insensitive to perturbations and
metry is broken.[21] This finding evokesinterest to pur-
independent of material details. A classical example of
. sue direct investigation and possibly utilization of the
t suchatransportphenomenonistheintegerquantumHall
a robust topological properties of the TIs, besides using
(IQH) effect in two-dimensional (2D) electron systems,
m
their symmetry-protected gapless edge states which are
first discovered in 1980, [1] which is characterized by an
- more fragile in realistic environments.
d integer quantization of the Hall conductivity in unit of
n e2/h. The IQH effect has been observed in a large va- UnlikethefirstChernnumberunderlyingtheIQHsys-
o riety of materials, ranging from traditional semiconduc- tems, which is embedded into the Hall conductivity, up
c tors, to oxides, [2] graphene, [3] and topological insula- to now the topological invariants in the TIs have not
[
tors (TIs). [4] Laughlin [5] interpreted the IQH effect in been directly observable. Several experimental methods

1 terms ofanadiabatic chargepump. Thouless,Kohmoto, were proposed, but have not been realized. One was to
v Nightingale, and Nijs [6] established a relation between measure the topological magnetoelectric effect, [22, 23]
9 thequantizedHallconductivityoftheIQHsystemanda for which experimental complexities exist. [23] Fu and
2
topological invariant, the first Chern number. Thouless Kane [24] put forward an abstract 1D model, in which
4
3 andNiu [7,8]alsorelatedthe amountofchargepumped thespinpumpingwasrelatedtotheZ2 indexinthelimit
0 in a 1D charge pump to the Chern number. of weak coupling. However, how this fictitious model
. A variant of the IQH effect, the quantum spin Hall could be implemented is still unknown. Furthermore,
1
(QSH) effect, was proposed recently, [9, 10] which has from the viewpoint of application, generalization of the
0
6 beenexperimentallyrealizedinHgTequantumwells[11] idea of the Z2 pump to higher dimension is meaningless,
1 andInAs/GaSbbilayers.[12]Extensionoftheideaofthe because according to the Z theory, [24] only the states
2
: QSH effect has led to the discovery of 3D TIs. [13–16] A at the TR-invariant point of the Brillouin zone can con-
v
QSHsystem,whichisalsocalleda2DTI,hasaninsulat- tribute to the spin pumping, and so the pumping rate
i
X ingbandgapinthebulkandapairofgaplesshelicaledge cannot be enhanced by increment of dimension. In a re-
r statesatthesampleboundary. Whentheelectronspinis cent work, [25] the more general case of finite coupling
a conserved,a QSH system can be viewed as two indepen- between the pump and electrode is investigated by us-
dent IQH systems without Landau levels. [17] Different ing the scattering matrix method. It was found that the
from the charge, the spin does not obey a fundamental spin pumping in the model of Fu and Kane can survive
conservationlaw. Ingeneral,whenthe spinconservation finite scattering of magnetic impurities, and so may be
is absent, unconventional topological invariants, either attributedtothespinChernnumbersratherthantheZ
2
the Z2 index [18] orthe spinChernnumbers,[19–21]are index. Some other authors [26, 27] proposed to pump
needed to describe the QSH systems. The time-reversal quantized charge through the helical edge states by pre-
cessingamagnetcoveringtheedgeofa2DTI,sothatthe
numberofgaplessedgechannelscanbe countedthrough
electrical measurement. This method is indirect, in the
∗ [email protected] sense that the topological invariants are intrinsic prop-
† [email protected] erties of the bulk electron wavefunctions, which do not
2
icmhamnendeilas.tely determine the charge pumping in the edge 6(cid:4)(cid:4)(cid:5)(cid:5)(cid:6)(cid:6)(cid:7)(cid:7)(cid:8)(cid:8)(cid:9)(cid:10)(cid:11)(cid:6)(cid:12)(cid:8) (cid:18)(cid:19)(cid:20)(cid:21) (cid:15)(cid:2)(cid:16)(cid:16)(cid:17)(cid:0)(cid:16) (cid:1)(cid:0)(cid:2)(cid:3)
Herewepredictanintriguingbulktopologicalpumping
effect,directlydrivenbynonzerospinChernnumbers,in
aQSHsystemelectricallymodulatedinadiabaticcycles.
As a consequence of the topological spectral flows of the 4
spin-polarized Wannier functions (SPWFs) in the bulk
t
of the system, spin can be pumped into a nonmagnetic (cid:13)
electrode continuously without net charge transfer. The w 2
totalamountofspinpumpedpercycleisproportionalto
the(cross-section)widthofthesample,andinsensitiveto
thematerialparametersandspin-mixingeffectduetothe 0
Rashbaspin-orbitcoupling. Thiselectricalspinpumpes- -6 -3 0 3 6
tablishesabasis,onwhichspintronicapplicationstaking Wannier centers <x>/a
advantageofthe robusttopologicalpropertiesofthe TIs
(cid:14)
can be developed. Especially, if a half-metallic electrode
FIG.1. PlotofthecentersofmassoftheSPWFs(horizontal
with spin polarization parallel (or antiparallel) to the z-
axis) as functions of ω0t (vertical axis). The parameters are
a−xCis+ies2/uhse(do,ra−qCu−aen2t/izhe)d, ccahnarbgee pmuemaspuirnegdcboyndeuleccttirviictay,l tVa0ke=n0t.o3t0b,eaknyd=d =0.4ak0yc,,wMith0 a=0 avsFetAhe0 l=att~icRe0c/oan0st=ant0.a1nt0d,
means, demonstrating a way to observe the spin Chern t0 = ~vF/a0 (vF′ = vF) as the hopping integral of the tight-
numbers C directly. bindingHamiltonian.
±
II. SPIN CHERN NUMBERS AND SPWFS and t [0,T) with T = 2π/ω0 as the period. The spin
∈
Chern numbers are obtained as
Letusconsidera2DmodelHamiltonianHP =H0+H1 C± = sgn(E0M0) , (3)
with ±
for k < kc, and vanish elsewhere. Not surprisingly,
| y| y
H =v [k sˆ σˆ (k +eA(t))σˆ ] M(t)σˆ . (1) the band touching points k = kc serve as the critical
0 F x z x− y y − z y ± y
points.
Here( e)istheelectroncharge,kisthe2Dmomentum, We now consider a system consisting of a pump for
−
A(t) = A0sin(ω0t) is the vector potential of an ac elec- x< 0, an electrode for x> d, and a potential barrier in
tric field E0cos(ω0t)appliedalongthey directionwith between. The total Hamiltonian of the system reads
−
A = E /ω and frequency ω > 0 being designated,
0 0 0 0
and M(t) = M0cos(ω0t). This model can describe both HP (x<0)
theQSHmaterials,theHgTequantumwells,[28–30]and H = HE +V0σˆz (0<x<d) , (4)

InAs/GaSb bilayers, [31] in the linear order in momen-  HE (x>d)
tum. For definity of discussion, we confine ourselves to
the HgTe quantum wells, for which sˆα with α = x,y,z iwshtehree HHaPmhilatosnbiaenenofgitvheenealebcotvroe,dea.ndAHpEoss=iblveF′kexxpsˆzeσˆrix-
are the Pauli matrices for spin, and σˆ for the electron
α mental setup for realizing this Hamiltonian is explained
and hole bands. As will be discussed below, the time-
in Appendix A in more details. In the barrier region,
dependent mass term M(t) can be induced by varying
the termV σˆ opens aninsulating gapofsize 2V ,which
the voltagesofthe dualgates. H representsthe Rashba 0 z 0
1 accounts for contact deficiencies between the pump and
spin-orbit coupling [32]
electrode. The crucial role of the nonzero spin Chern
R numbersinthespin/chargepumpingprocesscanbevisu-
H1 = 0(ˆ1+σˆz)[sˆykx sˆx(ky +eA(t))] . (2) alizedbyusingtheSPWFs,whichwerefirstintroducedin
2 −
Ref. [25]. We construct a tight-binding Hamiltonian for
To the linear orderin momentum, the Rashbaspin-orbit the effective 1D system at any given k according to Eq.
y
coupling is nonvanishing only in the electron band. [32] (4), and diagonalize the total Hamiltonian of the pump
Within the adiabatic approximation, for a bulk sam- andelectrodenumerically. Followingthesameprocedure
ple there exists a finite energy gap between the con- ascalculatingthespinChernnumbers,[21]thespaceoc-
duction and valence bands for ω t = π/2 or 3π/2. At cupied by electrons is partitioned into two spin sectors
0
6
ω t = π/2 and 3π/2, the conduction and valence bands after diagonalizing the spin operator sˆ in the occupied
0 z
touch at k = 0 and k = kc or kc with kc = eA = space. By definition, the states in the two spin sectors
x y y − y y | 0|
eE /ω . Toclarifythetopologicalpropertiesunderlying areessentiallythemaximallyspin-polarizedstates. Then
0 0
| |
thespin/chargepumping,weconsiderk asaparameter, we construct the Wannier functions [33, 34] for the spin-
y
andcalculatethespinChernnumbersC inthestandard up and spin-down sectors, respectively, which are called
±
way,[21]onthe torusofthe twovariablesk ( , ) the SPWFs.
x
∈ −∞ ∞
3
The evolution of the centers of mass of the SPWFs intotheelectrodeoccurs. Thisresultwillbefurthercon-
for k = 0.4kc and R = 0.1v is shown in Fig. 1. We firmed by direct calculation based upon the scattering
y y 0 F
see that the Wannier centersfor the spin-upsector move matrix theory in the next section.
right and those for the spin-down sector move left, each
TheSPWFsarejustanotherequivalentrepresentation
center shifting on average a lattice constant per cycle.
of the occupied space, and so the counter spectral
Withintheadiabaticapproximation,timet [0,T)plays
∈ flows of the Wannier centers in the two spin sectors
the same role as the momentum of an additional dimen-
represent the true movements of the electrons. If the
sion, [24] namely, k [0,T). Therefore, when k is
t ∈ y Rashba spin-orbit coupling were neglected, the Wannier
considered as a parameter, the evolution of the Wannier
functions would be the eigenstates of sˆ . The nontrivial
functions of the effectively 1D system related to vari- z
spectral flows indicate that at the given k , in each
ous k with time t can be understood from the static y
x cycle a spin-up electron goes from the pump into the
propertiesofa2Dsystemassociatedwithvariousk and
x electrode, and a spin-down electron moves oppositely.
k . In the general theory, [34] the relationship between
t Therefore,no net chargetransfer occurs but a quantized
the Chern number and the spectralflows of the Wannier
spin of 2(~/2) is pumped into the electrode. When the
functions in a 2D system has been established. Accord-
small Rashba spin-orbit coupling is turned on, while the
ingtothistheory,theaveragedisplacementofeachofthe
topological spectral flows remain intact, as seen from
centers of the SPWFs in the spin-up (spin-down) sector
Fig. 1, the spin polarizations of the Wannier functions
withchangingk (ort)from0toT,inunitsofthelattice
t are no longer fully parallel to the z-axis, and may also
constant, must equal to the spin Chern number C = 1
+ vary with time. As a consequence, the amount of spin
(C = 1). Therefore,the nontrivialtransfer of the SP-
−
− pumped per cycle will deviate from the quantized value.
WFs observed in Fig. 1 is a direct manifestation of the
nonzero spin Chern numbers C = 1 in the pump (for
±
±
E M > 0). More interestingly, we see that such spec-
0 0
III. THE PROCESS OF SPIN CHERN
tral flows can go across the finite barrier (V d> 0), and
0 PUMPING
extend into the electrode, even though the barrier and
electrode are topologically trivial. Physically, because
the system needs to recoverits originaleigenstates when A. Spin pumping for a nonmetallic electrode
each cycle ends, the nontrivial spectral flows of the SP-
WFs in the TI need to constitute closed loops through
formation of edge states at the boundary, [35] or extend Ingeneral,theamountofthespinpumpedcanbecon-
into the electrode. However, localized edge states can veniently calculated by using the scattering matrix for-
not exist at the finite barrier due to quantum tunneling mula. [36, 37] The z-component of the spin pumped per
effect, so the transfer of the spectral flows of the SPWFs cycle is given by [36, 37]
~ dr dr dr dr
∆s (k )= dt r∗ ↑↑ r∗ ↓↓ r∗ ↓↑ +r∗ ↑↓ , (5)
z y 4πi ↑↑ dt − ↓↓ dt − ↓↑ dt ↑↓ dt
IT (cid:16) (cid:17)
wherer (α,β = , )isthe reflectionamplitudeforanelectronatthe Fermienergyincidentfromthe spin-β channel
αβ
↑ ↓
of the electrode and reflecting back into the spin-α channel. In the following calculations, the Fermi energy is set to
be zero (E =0), and the Rashba spin-orbit coupling is treated as a perturbation. As shown in Appendix B, to the
F
linear order in R , we obtain
0
cos(2θ)+i[sh(2γ d) sin(2θ)ch(2γ d)]
r = 0 − 0 + (ǫ2) , (6)
↑↑
− ch(2γ d) sin(2θ)sh(2γ d) O
0 0
−
r = ǫ sin(2θ)[1−cos(2θ)] + (ǫ2) , (7) fourth terms in Eq. (5) vanish. Consequently,
↓↑
2ch(2γ d) sin(2θ)sh(2γ d) O
0 − 0 ~
∆s (k )= r∗ dr r∗ dr . (8)
z y 4πi ↑↑ ↑↑− ↓↓ ↓↓
IT
(cid:0) (cid:1)
Thisexpressionhasageometricexplanation: theamount
and r = r and r = r , of spin pumped per cycle equals to the difference be-
↓↓ ↑↑ 2θ→(π−2θ) ↑↓ ↓↑ 2θ→(π−2θ)
whereγ =V /~|v′ and2θ =arg[v (k +e−A(t)|)+iM(t)]. tween the areas enclosed by the directional trajecto-
0 0 F F y
We note that the dimensionless quantity ǫ = R /v ap- ries of r and r on the complex plane, multiplied by
0 F ↑↑ ↓↓
pears as the small expansion parameter. Since r and ~/2π. Due to the relation r = r , yielding
↓↑ ↓↓ ↑↑ 2θ→(π−2θ)
|
r are alwaysreal,the contributions from the third and r = Re(r )+iIm(r ),thetwotermsinEq.(8)make
↑↓ ↓↓ ↑↑ ↑↑
−
4
6 Im 1 R =0.0
2.0
$
Re Re 1 0.05v
jj(t)-(0) 24 -1 (k-0, g1(cid:29) d(cid:30))=(10.4k(cid:31), (cid:29)0)(0.8k(cid:26), 0)(cid:27) 4k,(cid:24) 1.0)(cid:25) sD !1.9 -1 0 1Im0.1v#
0.
(
0 (1.2k(cid:22), 0) -1
(cid:23)
0 2 4 6
0.0 0.5 1.0
w t
g d
(cid:28)
"
FIG.2. Argumentofthecomplexreflectionamplitude,ϕ(t)=
oatrgh(err↑↑p)a,raams eatefursncatrieontaokfenω0ttofobrefRou0r=set0saonfd(kvyF,eγA0d0).=TMh0e F0a.rI4eGkty.ca3ka.enn∆dsttzoh(rkbeyee)dt(hiiffneeusraennmitteovfaasl~u/ien2s)oFafisgR.a02f.u.TnIchntesioeotnt:hotefhrγep0dtarrfaaojmrecekttyoerr=ys
wcoimthplkeycx=plaen|Ae.0|. Inset: trajectories of r↑↑ in a cycle on the of r↑↑ in a cycle for R0 =0.1vF and γ0d=1.0, with the unit
circle indicated by thedotted line.
∆s (k ) as
an equal contribution, so that we can focus on the first z y
term. While the expression (6) for r is independent of
↑↑
the Rashba spin-orbit coupling, as will be shown soon, a ~ 5 R0 2
∆s (k ) 1 (C C ) , (9)
combinationofEqs.(6)and(7)allowsustoevaluatethe z y ≃ 2" − 32(cid:18)vF(cid:19) # +− −
amountofspinpumpeduptothesecondorderinR /v .
0 F
We first consider the case of R0 = 0. From Eq. (6), for |ky| < kyc, and ∆sz(ky) = 0 elsewhere. As expected,
it is easy to show r↑↑(ky) = 1. In Fig. 2, we plot the nonzero Rashba spin-orbit coupling causes ∆sz(ky) to
argument ϕ(t) of r↑|↑(ky) a|s a function of ω0t for several deviate from its quantized value, i.e., (C+−C−)~/2. In
parameter sets. For either γ0d=0 (ideal contact) or 1.0 real materials, R0 is usually much smaller (by an order
(strongpotentialbarrier),ϕ(t)alwaysincrements2πina ofmagnitudeormore)thanvF,[32]sothatthedeviation
cycleaslongas k <kc. Inthiscase,thetrajectoriesof islessthanapercentforanidealconnectionbetweenthe
r (k ) always f|orym| a uynit circle on the complex plane, pump and electrode.
↑↑ y
orientedcounterclockwise,asshownintheinsetofFig.2, For γ0d = 0, ∆sz(ky) can be evaluated numerically
6
suggesting ∆s (k ) = ~ (for E M > 0). For k > kc, andits calculatedresultisplottedinFig.3asa function
however, the szituyation is quite0diff0erent. ϕ(t)|dyo|es noyt of γ0d for three different strengths of the Rashba spin-
changeafter goingthrougha cycle, andthe trajectoryof orbit coupling. For R0 = 0, ∆sz(ky) is quantized to ~,
r↑↑(ky) does not enclose a finite area, so that ∆sz(ky)= independent of γ0d. For R0 = 0.05vF and 0.1vF, weak
0. Apparently, the present result conforms to the spin potential barrier (γ0d 1) has little effect on ∆sz(ky).
≪
Chern numbers given by Eq. (3) and the spectral flows Thisisreasonableastheleading-ordercorrectionofsmall
of the SPWFs. γ0d must be (ǫ2γ0d). Appreciable deviations from the
O
quantized value occur for strong potential barrier (e.g.,
Next we study the correction to ∆s (k ) due to
z y γ d 1). We note that δϕ does not affect the orbit of
0
nonzeroRashbaspin-orbitcoupling. Byexpressingr = ≃
↑↑ r , as it represents a variation in the tangent direction
↑↑
ρeiϕ = 1 δρ2ei(ϕ(0)+δϕ) in the polar coordinate sys- of the orbit, and the orbit can be determined by r =
− ↑↑
tem, whpere ϕ(0) is the argument at R0 = 0, and δϕ and 1 r↓↑ 2eiϕ(0). Inset shows the trajectory of r↑↑ on
δρ2standforthesecond-ordercorrectionstoϕandρ2,re- −| |
the complex plane for R = 0.1v and γ d = 1.0. For
spectively,duetotheRashbaspin-orbitcoupling,Eq.(8) p 0 F 0
such a strong potential barrier, the orbit of r deviates
becomes ∆s (k )=(~/2π)[ (1 δρ2)dϕ(0)+ dδϕ]+ ↑↑
z y T − T from the unit circle visibly. The above result suggests
(ǫ3). We notice that δϕ is a small quantity fluctuating that improving the contact quality between the pump
O H H
around 0 and periodic in time, δϕt=0 =δϕt=T, so that and electrode is helpful for obtaining a nearly integer-
T dδϕ=0. Using the identity δρ2|=|r↓↑|2,|we then ob- quantized value of the pumped spin.
Htain ∆sz(ky) = (~/2π) T(1−|r↓↑|2)dϕ(0), where dϕ(0) By summing over ky between −kyc and kyc, we obtain
can be calculated from Eq. (6) and r↓↑ has been given for the total spin pumped per cycle
H
byEq.(B26). Thisisanexpressionfor∆s (k )accurate
z y
to the second order in R /v . At γ d = 0, we obtain ∆S =σ (2E L /ω ) , (10)
0 F 0 z s | 0| y 0
5
with which in effect modifies the potential barrier between
the pump body and the electrode. As has been shown
2
e 5 R0 above, the charge pumping effect is independent of the
σ 1 (C C ) , (11)
s ≃ 4π " − 32(cid:18)vF(cid:19) # +− − existence and details of the potential barrier, and so we
believe that the pumping effect will survive the charge
foragoodcontact(γ d 1). ∆S is inscalewithwidth and spin accumulations.
0 z
≪
L ofthepump. Bynotingthatω isproportionaltothe
y 0
number of cycles per unit time, σ can be considered as
s
the spin pumping conductivity.
IV. DISCUSSION
B. Charge pumping for a half-metallic electrode
Up to now, all the results obtained from the scatter-
Now we discuss a possible way to experimentally ob- ingmatrixformulaareapparentlyincompleteagreement
serve the spin Chern numbers, by using a half-metallic withthespinChernnumbersgivenbyEq.(3). Thesere-
electrode,in whichconducting channelsfor electronspin sults cannot be explained within the framework of the
antiparallel to the spin polarization are absent. We first Z theory. [24] While one can define a Z index at the
2 2
considerthecase,wherethespinpolarizationoftheelec- TR-invariant point k = 0, the effective 1D Hamilto-
y
trode is parallel to the z axis. The Hamiltonian of the nian given by Eqs. (1) and (2) for any given nonzero k
y
electrodeistakenasHE =vF′(ky)kxsˆzσˆx+V1(ˆ1−sˆz)σˆz/2. does not preserve the TR symmetry, as its TR partner
Inthiscase,asshowninAppendixC,r↑↑ isstillgivenby is at ky, making the Z2 index invalid. The Z2 theory
−
Eq. (6) but r↓↑ 0. It follows that for any ky between predicted that the TR symmetry is crucial for the topo-
≡
kc and kc, the charge pumped per cycle is integer- logicalspinpumping,[24]suggestingthatonlythestates
− y y
quantized and equal to ∆q(ky)=(−e)C+. Similarly, for at the TR-invariant point ky = 0 can contribute to the
the spin polarization of the electrode antiparallel to the spin pumping. This clearly contradicts the present re-
z axis,the chargepumped isequalto∆q(ky)=(−e)C−. sult that all the states with |ky|<kyc contribute equally,
Therefore,the totalchargepumped per cycle is givenby whichisobtaineddirectlyfromthescatteringmatrixfor-
mula. This point is also evidenced by the fact that the
∆Q=σc(2E0 Ly/ω0) , (12) total amount of charge or spin pumped per cycle is in
| |
proportiontothesamplewidthL . Forthesamereason,
y
with
the pumping effect found in this work is also essentially
e2 differentfromthatviaedgestatesinRefs.[26,27],where
σc = C± , (13) the amount of spin or charge pumped per cycle is pro-
− h
portional to the number of the gapless edge channels.
where the spin Chern number C (C ) is taken for the In conclusion, our work uncovers a bulk topological
+ −
spin polarization of the electrode parallel (antiparallel) pumping effect due to direct transfer of the SPWFs be-
to the z axis. We emphasize that Eqs. (12) and (13) ob- tweenthepumpandelectrode,withouttheparticipation
tained above are valid for finite Rashba spin-orbit cou- of edge states. This measurable effect reveals the bulk
pling and finite potential barrier between the pump and topologicalpropertiesofthesystemthatareneithercap-
electrode, indicating that the quantized charge pumping tured by the Z index nor reflected by the number of
2
is robust against small perturbations. Experimentally, gapless edge channels. It can be accurately described by
∆Q can be obtained by measuring the electrical current thespinChernnumbers. ThisspinChernpumpmaylay
in the electrode, and from Eqs. (12) and (13), C can the foundation for direct experimental study and possi-
±
be evaluated, yielding an experimental method to mea- bly utilization ofthe robusttopologicalproperties of the
surethespinChernnumbersdirectly. Thesigninversion TIs.
of ∆Q with reversing the spin polarization of the half- Thepreviousexperimentalwork[38]evidencedthedif-
metallic electrode, as indicated by Eqs. (12) and (13), is ficulty of modulating in time the properties of an open
ahallmarkofthepresentspinChernchargepump,which quantum dot without generating undesired bias voltages
canbeusedtodistinguishitfromtheconventionalThou- due to stray capacitances. This problem might not be
less charge pump [7, 8]. significant in our pumping setup, where a much larger
We have used the single-electron approximation, bulk sample of the TI can be used and the stray ca-
where the electron interaction is not taken into account. pacitances can be greatly reduced. Moreover, a possible
In particular, in the half-metallic electrode case, one way around the obstacle is to use the ac Josephson ef-
of the spin channel is blocked at the boundary, which fect to induce periodically time-dependent Andreev re-
naturally induces some charge and spin accumulations, flection amplitudes in a hybrid normal-superconducting
andconsequentlychangesthepotentialprofile. However, system.[39]ConcretedesignofaspinChernpumpbased
owingto the screeningeffect, the changeinthe potential upon the Josephson effect will await future work. While
profile is expected to be localized at the boundary, the proposedspin pumping scheme may havethe advan-
6
(cid:12)
tage of low noises, its practical application in spintronic (cid:1)(cid:1)(cid:2)(cid:3)(cid:4)
(cid:11)
devices still relies on the discovery of new TIs with bulk
band gaps much greater than room temperature, which (cid:9)(cid:10)(cid:7)(cid:8)
determine the temperature range where the spin Chern (cid:5)(cid:6)(cid:7)(cid:8)
pumping effect can survive. Currently, precessing mag-
(cid:9)(cid:10)(cid:7)(cid:8)
netization is a feasible method to generate robust spin
currents in spintronic devices at room temperature. [40] (cid:1)(cid:2)(cid:2)(cid:3)(cid:4)
(cid:1)
(cid:13)(cid:2)(cid:3)(cid:4)
ACKNOWLEDGMENTS
FIG.4. Aschematicviewofaexperimentalsetupforrealiza-
tionofa2DspinChernpump. ACdTe/HgTe/CdTequantum
This work was supported by the State Key Pro- well heterostructure, with dual gates on its top and bottom,
gram for Basic Researches of China under grants num- is placed between two conductive plates. When the voltages
bers2015CB921202,2014CB921103(LS),2011CB922103 ofthegatesandplatesareadiabatically modulatedinproper
and 2010CB923400 (DYX), the National Natural Sci- cycles, spinorcharge can bepumpedintoelectrodes coupled
enceFoundationofChinaundergrantnumbers11225420 to thequantumwell along thex direction.
(LS),11174125,91021003(DYX)andaprojectfundedby
the PAPDofJiangsuHigher EducationInstitutions. We
also thank the US NSF grants numbers DMR-0906816 M cos(ω t), while keeping the electron Fermi energy
0 0
and DMR-1205734 (DNS). s−till in the band gap.
Theeffectoftheconductiveplatesiseasilyunderstood.
When a voltage drop U(t) is applied across the plates,
Appendix A: A POSSIBLE EXPERIMENTAL a uniform electric field E(t) = E(t)yˆ will be generated
REALIZATION OF THE SPIN CHERN PUMP
in the space between the two plates. The electrons in
the quantum well experience a vector potential A(t) =
In what follows we expand on the model setup and A(t)yˆ with A(t) defined as E(t) = ∂A(t)/∂t. If the
−
possibleexperimentalrealizationofthespinChernpump electric field is chosen to be E(t) = E cos(ω t), one
0 0
−
in more details. gets A(t) = A sin(ω t) with A = E /ω , as desired.
0 0 0 0 0
We point out that the exact time dependencies of M(t)
and A(t) are not essential for realizing the spin Chern
pump, providedthat they havethe same periodicity and
1. The pump
a constant relative phase shift.
A possible experimental realization of the Hamilto-
nian Eq. (1) for the pump is illustrated in Fig. 4.
A HgTe/CdTe quantum-well heterostructure with dual
gates (top and bottom) is placed between two conduc-
tive plates. It is known that when the width of the 2. The nonmagnetic electrode
quantum well (thickness of the HgTe film) is above a
critical size d = 6.3nm, [28, 29] the band structure is
c The pumping effect is insensitive to material details
inverted, characterized a negative mass term M σˆ in
− 0 z of the electrode. The electrode is taken to be a nor-
the Hamiltonian, correspondingto the QSH state. If the
mal metal with a 2D parabolic Hamiltonian H =
E
widthofthequantumwellfallsbelowdc,thebandstruc- E +p2/2m. When E is sufficiently large, for a given
0 0
ture will be aligned in a “normal” way with a positive −
p , we can linearize the effective 1D Hamiltonian H
y E
mass term M0σˆz, corresponding to a normal insulator. at the right and left Fermi points p = mv′(k ) with
As hasbeendiscussedinRefs.[30]and[31], the topolog- x ± F y
v′(k ) = 2m(E +E ) k2/m. A Pauli matrix σˆ is
icalphasetransitionbetweentheQSHphaseandnormal F y F 0 − y x
insulator can also be tuned by applying a gate voltage, introducedqto describe the two branches. To be consis-
which effectively reduces the width of the quantum well. tent with the form of the Hamiltonian in the pump, we
It is assumed that the quantum well under considera- use σ = 1 and 1, respectively, to represent the right-
x
−
tion has a width somewhat greater than d , and so has moving and left-moving branches for s = 1 and oppo-
c z
a negative mass term M σˆ initially. With increasing sitely for s = 1. As a result, the Hamiltonian of the
0 z z
the gate voltage, the −electron mass increases, and can electrode becom−es H =v′(k )k sˆ σˆ at E =0, where
E F y x z x F
invert its sign. Usually, increasing the gate voltage may k =p andk =p mv′(k ). Thespinpumpingeffect
y y x x∓ F y
also adjust the carrier density. Nevertheless, it has been is usually dominated by small k , so that we can further
y
shown[30,31]that,byusingdualgatesandproperlytun- approximate v′(k ) v′(k = 0) v′, with purpose
F y ≃ F y ≡ F
ing their voltagesV (t) and V (t), it is generallypossible to minimize the number of adjustable parameters in the
1 2
to change the electron mass in the desired manner to be model.
7
3. The barrier subband. As a result, the electron density of states is
fully spin-polarized at the Fermi energy. V is set to be
1
infinity in the final result.
For the present Dirac-like Hamiltonian, an ordinary
potential barrier has a very weak effect on the elec-
tron transmission due to the Klein paradox. There-
Appendix B: CALCULATION OF THE
fore, we take the Hamiltonian for the barrier to be
REFLECTION AMPLITUDES FOR A
H =H +V σˆ . The inclusion ofpotential V σˆ opens
B E 0 z 0 z NONMAGNETIC ELECTRODE
upaninsulatingenergygapofsize2V aroundtheFermi
0
level, which presumably is more efficient for describing
1. Electron wavefunctions in the pump and
thecontactdeficienciesandstructuralmismatchbetween
potential barrier
the pump and electrode.
Wenowsolvethescatteringproblemforanelectronat
the Fermienergyincidentfromthe electrode. TheFermi
4. The half-metallic electrode
energy will be taken to be E = 0, which is in the band
F
gap of the pump. Therefore, the incident electron will
The half metal, e.g., CrO , La Sr MnO , etc., is be fully reflected back into the electrode. The Rashba
2 2/3 1/3 3
a substance that acts as a conductor to electrons of one spin-orbit coupling is treated as a perturbation, and the
spinorientation,butasaninsulatortothoseoftheother result will be calculated to the linear order in the small
spin orientation. From the viewpoint of the electronic quantity ǫ = R /v . The wavefunctions of the pump
0 F
structure, one of the spin subbands is metallic, whereas (x < 0), barrier (0 < x < d) and electrode (x > d) are
the Fermi level falls into an energy gap of the other spin denoted by Ψ (x), Ψ (x), and Ψ (x), respectively. We
P B E
subband. To simulate the half-metallic electrode, H is have two boundary conditions: Ψ (0−) = Ψ (0+) and
E P B
taken to be H =v′(k )k sˆ σˆ +V (ˆ1 sˆ )σˆ /2, where Ψ (d 0+)=Ψ (d+0+).
E F y x z x 1 ∓ z z B − E
standsforthespinpolarizationoftheelectrodeparallel We use and to represent the eigenstates of sˆ , and
z
∓ ↑ ↓
and antiparallel to the z-axis, respectively. The second +1and 1torepresentthoseofσˆ . Onthebasis ,+1 ,
z
− |↑ i
term opens an energy gap of size 2V around the Fermi , 1 , ,+1 , and , 1 , the Hamiltonian of pump
1
|↑ − i | ↓ i |↓ − i
level for electron spin antiparallel to the spin polariza- (theEqs.(1)and(2)inthemanuscript)canbeexpanded
tion of the electrode, without affecting the other spin as a 4 4 matrix
×
M(t) v (k +ik˜ ) R ( ik k˜ ) 0
v (−k ik˜ ) F Mx (t) y 0 − 0x− y 0
HP =RF(ikx− k˜y) 0 M(t) v ( k +ik˜ ) (B1)
 0 x0− y 0 v (−k ik˜ ) F −Mx(t) y 
 F − x− y 
 
where k˜ = k +eA(t). For energy E = E = 0, the k =a+ib,a ib, a+ib,and a ib. By substitution
y y F x − − − −
eigen-equation is obtained from Eq. (B1) ofthefourrootsintoEq.(B3),wecaninprincipleobtain
four different eigenfunctions. For the present scattering
[M2(t)+v2k˜2]2 M2(t)R2k˜2 =0 , (B2) problem, we only need the two eigenfunctions that are
F − 0
decaying into the pump, which correspond to the two
with k˜2 =k2+k˜2, and up to a normalizationfactor, the
x y roots with negative imaginary parts.
eigenfunctions are
ForR =0,itiseasytoobtainfortherootsforthetwo
0
A1M(t)/vF decaying modes: kx = −i~η with ~η = M2(t)+k˜y2,
 −A1(kx−ik˜y) eikxx/~ , (B3) which are two-fold degenerate. The correqsponding two
A M(t)/v decaying eigenfunctions are given by
2 F
 A2(kx+ik˜y)  sinθ
  ϕ (x)= eηx , (B4)
where A1 = (ikx + k˜y)M(t) and A2 = [M2(t) + + |↑i⊗(cid:18)icosθ (cid:19)
−
v2k˜2]/R . We need to solve k from the eigen-equation
F 0 x
Eq. (B2). We notice that the equation is a 4th-degree cosθ
ϕ (x)= eηx , (B5)
polynomial of kx with real coefficients, so complex con- − |↓i⊗ isinθ
(cid:18)− (cid:19)
jugate rootsmustappearin pairs. Moreover,Eq.(B2) is
evenink ,sopositiveandnegativerootsappearinpairs. where 2θ = Arg[v k˜ +iM(t)]. For R = 0, we write
Incombinxation,Eq.(B2)musthavefourrootsoftheform the roots of k asFky = i~η+δk , an0d6 also write k2
x x − x x
8
as k2 = (~η)2+δk2. To the second order in ǫ, we can be
solvexfor−δk2 from thxe eigen-equation Eq. (B2)
x
cos2θsinθ 1 iR0 sin2θ
− ∓ vF
 icos3θ 1(cid:16) 2iR0 sin2θ (cid:17) 
δkx2 =(cid:18)±iRvF0 + 2Rv02F2(cid:19)(~η)2sin2(2θ)+O(ǫ3) . (B6) ϕ1,2(x)= −±isin2θ(cid:16)cos∓θ 1v∓F i2Rv0F (cid:17) eη∓x ,
 ±sin3θ 1∓i 12(cid:16)+cos2θ (cid:17)RvF0 
NoticingthattheexpressionforA givenbelowEq.(B3)  h (cid:0) (cid:1) i (B13)
has a factor R0 in the denominato2r, we keep δkx2 to the whSeorme ηe∓re=maηrk1s∓ar2Rev0Finsionr2d(e2rθ.)W. ith respect to the wave-
soercdoenr.dBoryduers,infogrtthheerpeluartpioonseδtko2c=alcu2lai~teηδAk2t+oth(eǫ2li)n,ewaer functionsatR(cid:2)0 =0,namely,Eq(cid:3)s.(B4)and(B5),nonzero
x − x O R leadstononperturbativechangeinthe wavefunctions
derive from Eq. (B6) 0
Eq. (B13), in the sense that Eq. (B13) will not recover
Eqs. (B4) and (B5) in the limit R 0. This is rea-
0
→
sonable, just as what always happens in the degenerate
R
δk = 0~ηsin2(2θ)+ (ǫ2) . (B7) perturbation theory of the quantum mechanics. For the
x
∓2v O
F presentproblem,thewavefunctionΨP(x) inthe pumpis
always expressed as an arbitrary linear superposition of
the two decaying modes: Ψ (x) = B ϕ (x)+B ϕ (x).
With these relations, we obtain for A and A P 1 1 2 2
1 2 By defining ϕ (x) = [ϕ (x) + ϕ (x)] and ϕ (x) =
+ 1 2 −
−
i[ϕ (x) ϕ (x)], we can rewrite Ψ (x) as Ψ (x) =
1 2 P P
− −
D ϕ (x)+D ϕ (x). The final result for the reflection
1 + 2 −
A1 =−vF(~η)2sin(2θ)[1+cos(2θ)] (B8) awmriptelitouudtetshdeepexepnrdessosinolnyfoonrΨΨP((xx==00−−)),asoswfoelleoxwpslicitly
R P
i 0(~η)2sin3(2θ)+ (ǫ2) , (B9)
± 2 O Ψ (x=0−)=D ϕ (0−)+D ϕ (0−) , (B14)
P 1 + 2 −
where
and
sinθ
icosθ
R ϕ+(0−)= sin2θ cosθ R0  , (B15)
A2 = ±i+ 2v0 vF(~η)2sin2(2θ)+O(ǫ2) . (B10) i 1−+cocso2sθ2θ sin2v3Fθ R0 
(cid:18) F(cid:19)  2 cos2θ vF 
 
(cid:0) (cid:1)
Wecanalwayseliminateanycommonfactorthatappears cos2θsinθR0
iBnyalelltimheinfoautirncgomapcoonmenmtosnoffEacqt.o(rB32)v,w(~hηe)n2esviner(2pθo)scsoibslθe,. ϕ−(0−)= 2icos3θRvFv0F  (B16)
F cosθ
we rewrite A1 and A2 as  isinθ 
 − 
 
Now we see that in the limit R 0, the total wave-
0
A = cosθ iR0 sinθsin(2θ)+ (ǫ2) , (B11) functionΨP(0−)willgobacktothe→formofasuperposi-
1
− ± 2v O tionofthetwodecayingwavefunctionsgiveninEqs.(B4)
F
and (B5). In conclusion, while small Rashba spin-orbit
coupling may cause a nonperturbative change of the in-
and dividual decaying wavefunctions, it modifies the “space”
spanned by the two decaying modes in a perturbative
manner. It is this “space” which determines the final
A = i+ R0 sinθ+ (ǫ2) . (B12) result of the reflection amplitudes. This is the physical
2
± 2v O reasonwhyinthefinalresult,theRashbaspin-orbitcou-
(cid:18) F(cid:19)
plingmodifiesthereflectionamplitudesinaperturbative
manner. The wavefunction in the potential barrier can
Then the two decaying wavefunctions can be derived to be written as
Ψ (x)= C1 1 eγ0x+ C2 1 e−γ0x+ C3 1 eγ0x+ C4 1 e−γ0x ,(B17)
B √2|↑i⊗ i √2|↑i⊗ i √2|↓i⊗ i √2|↓i⊗ i
(cid:18)− (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18)− (cid:19)
9
where γ =V /~v′. The forms of the wavefunctions in the pump and barrier
0 0 F
remain to be the same. By some algebra, we arrive at
2. An electron incident from the spin-up channel r↑↓Γ−(2γ0d)
1 r Γ (2γ d)
Ψ (0+)= ↑↓ + 0 . (B28)
For an electronincident from the spin-up channel, the B √2 Γ+(2γ0d)+r↓↓Γ−(2γ0d)
Γ (2γ d) r Γ (2γ d)
wavefunction in the electrode is given by  − 0 − ↓↓ + 0 
 
1 1 r↑↑ 1 Equating Eq. (B14) with Eq. (B28), we obtain
Ψ (x)= +
E √2|↑i⊗ 1 √2|↑i⊗ 1
(cid:18)− (cid:19) (cid:18) (cid:19) icos2θ sinθcosθ Γ− r↑↓
+ r↓↑ 1 . (B18) −sinθcosθ isin2θ Γ+ iR0 cos3θsinθ
√2|↓i⊗ 1 (cid:18)− − (cid:19)(cid:18)− (cid:19) vF
(cid:18)− (cid:19)
Γ (2γ d)+r Γ (2γ d)
First, matching the wavefunctions Eqs. (B17) and (B18) = + 0 ↓↓ − 0 + (ǫ2) (B29)
Γ−(2γ0d) r↓↓Γ+(2γ0d) O
at x=d, one obtain (cid:18) − (cid:19)
1 It follows from Eq. (B29)
C1 = [(1 i)+r↑↑(1+i)]e−γ0d , (B19)
2 −
cos(2θ) i[sh(2γ d) sin(2θ)ch(2γ d)]
1 r = − 0 − 0 + (ǫ2) ,
C2 = 2[(1+i)+r↑↑(1−i)]eγ0d , (B20) ↓↓ ch(2γ0d)−sin(2θ)sh(2γ0d) O
(B30)
r
C3 = ↓↑(1+i)e−γ0d , (B21) and
2
r
C4 = 2↓↑(1−i)eγ0d . (B22) r↑↓ = 1 sin(2θ)[1+cos(2θ)] R0 + (ǫ2) .
−2ch(2γ d) sin(2θ)sh(2γ d) v O
In the next step, we will match wavefunctions at x = 0. 0 − 0 (cid:18) F(cid:19) (B31)
SubstitutingEqs.(B19-B22)intoEq.(B17),wecanwrite
Eq. (B17) at x=0+ as
4. A verification of the result
Γ (2γ d)+r Γ (2γ d)
+ 0 ↑↑ − 0
1 Γ (2γ d)+r Γ (2γ d)
ΨB(0+)= √2− − r↓0↑Γ−(2↑γ↑0d+) 0  (B23) theThtreatnostfaolrmHaatmioinltonianofthesystemisinvariantunder
r Γ (2γ d)
 − ↓↑ + 0 
  ( isˆ σˆ )H( k˜ )(isˆ σˆ )=H(k˜ ) , (B32)
where Γ(ξ) = ch(ξ) ish(ξ). Now equating Eq. (B14) y z y y z y
− −
±
with Eq. (B23), we obtain
so the corresponding transformation of Eq. (B18)
isin2θ sinθcosθ Γ− r↓↑ 1 1
(cid:18)−sinθcosθ icos2θ (cid:19)(cid:18)−Γ+ (cid:19)iRvF0 cosθsin3θ (−isˆyσˆz)ΨE(x)|k˜y→−k˜y = √2|↓i⊗ 1
(cid:18) (cid:19)
Γ (2γ d)+r Γ (2γ d)
=(cid:18)−Γ+−(2γ00d)+r↑↑↑↑Γ−+(2γ00d) (cid:19)+O(ǫ2) (B24) +r↑↑|k˜√y→2−k˜y|↓i⊗ 11
(cid:18)− (cid:19)
It follows from Eq. (B24)
r↓↑|k˜y→−k˜y 1 , (B33)
r = cos(2θ)+i[sh(2γ0d)−sin(2θ)ch(2γ0d)]+ (ǫ2), − √2 |↑i⊗(cid:18)1 (cid:19)
↑↑
− ch(2γ d) sin(2θ)sh(2γ d) O
0 − 0 must also be an eigenstate of H(k˜ ). This result tells
(B25) y
thatanelectronincidentfromthespin-downchannelwill
and
be reflected into the spin-down channel with amplitude
r↓↑ = ǫ sin(2θ)[1−cos(2θ)] + (ǫ2) . (B26) r↑↑|k˜y→−k˜y, and also into the spin-up channel with am-
2ch(2γ0d)−sin(2θ)sh(2γ0d) O plitude −r↓↑|k˜y→−k˜y. Comparing it with Eq. (B27) and
noticing that k˜ k˜ is equivalent to 2θ (π 2θ),
y y
→ − → −
we find immediately the following relations
3. An electron incident from the spin-down channel
r =r , (B34)
↓↓ ↑↑ 2θ→(π−2θ)
Thereflectionamplitudesforanelectronincidentfrom |
the spin-down channel can be solved similarly. Now the and
wavefunction in the electrode is given by
r = r . (B35)
↑↓ ↓↑ 2θ→(π−2θ)
1 1 r↓↓ 1 r↑↓ 1 − |
Ψ (x)= + + .
E √2|↓i⊗ 1 √2|↓i⊗ 1 √2|↑i⊗ 1 The reflection amplitudes given in Eqs. (B25), (B26),
(cid:18) (cid:19) (cid:18)− (cid:19) (cid:18) (cid:19)
(B27) (B30), and (B31) apparently satisfy these relations.
10
Appendix C: CALCULATION OF THE up channel has some probability to be reflected into the
REFLECTION AMPLITUDES FOR A spin-downchannelfromthepump,butthereflectedwave
HALF-METALLIC ELECTRODE will decay quickly to 0 within the barrier, and has no
chance to reach the electrode. Therefore, r 0 and
↓↑
≡
To simulate the half-metallic electrode, the Hamilto- the wavefunction in the electrode becomes
nianoftheelectrodeistakentobeH =v′(k )k sˆ σˆ +
E F y x z x
V (ˆ1 sˆ )σˆ /2. Consider first the case, where the spin
1 z z
∓
polarizationoftheelectrodeisparalleltothez-axis. Now
the Hamiltonian of the potential barrier becomes Ψ (x)= 1 1 + r↑↑ 1 . (C2)
E √2|↑i⊗ 1 √2|↑i⊗ 1
(cid:18)− (cid:19) (cid:18) (cid:19)
0 0
H =H + . (C1)
B E 0 V σˆ
1 z
(cid:18) (cid:19)
For simplicity, we will take V limit in the final The wavefunctioninthe potentialbarriercanbe written
1
→ ∞
result. In this case, an electron incident from the spin- as
Ψ (x)= C1 1 eγ0x+ C2 1 e−γ0x+ C3 1 e−γ1x , (C3)
B √2|↑i⊗ i √2|↑i⊗ i √2|↓i⊗ i
(cid:18)− (cid:19) (cid:18) (cid:19) (cid:18)− (cid:19)
where γ = V /~v′ and γ = V /~v′ . The wave- r 2 (1 r 2) 1uptoanyorderinǫ. Thismeans
function0in th0e puFmp rem1ains 1to bFe →sam∞e. Repeating t|h↑a↑t| ≡(ǫ2)−in|E↓↑q|. (C≡4) must be a correction only to the
O
the same calculationas insec.II, it is straightforwardto argument of r , which as discussed in the manuscript,
↑↑
obtain will not modify the orbit of r on the complex plane.
↑↑
The orbit is always a unit circle for k < kc, and the
cos(2θ)+i[sh(2γ d) sin(2θ)ch(2γ d)] | y| y
r↑↑ = 0 − 0 + (ǫ2). amount of charge pumped per cycle by the ky state is
− ch(2γ0d)−sin(2θ)sh(2γ0d) O(C4) qlauraiznattizioendotof t∆hqe(kelye)ct=ro−deeCa+nt.ipSaimrailllaerllyt,ofothrethze-asxpiisn, pthoe-
This expression is identical in form to Eq. (B25) ob- charge pumped per cycle is ∆q(k )= eC .
y −
tained in Sec. II for a nonmagnetic electrode. How- −
ever, an important difference is r 0. As a result,
↓↑
≡
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