Inverse Edelstein effect of the surface states of a topological insulator

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by  Hao Geng

Inverse Edelstein effect of the surface states of a topological insulator
Hao Geng,1 Wei Luo,1 W.Y. Deng,1 L. Sheng,1,2,a) R. Shen,1,2 and D. Y. Xing1,2
1)National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University,
Nanjing 210093, China
2)Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093,
China
The surface states of three-dimensional topological insulators posses the unique property of spin-momentum
interlocking. This property gives rise to the interesting inverse Edelstein effect (IEE), in which an applied
7
spin bias µ is converted to a measurable charge voltage difference V. We develop a semiclassical theory for
1
the IEE of the surface states of Bi Se thin films, which is applicable from the ballistic regime to diffusive
0 2 3
regime. We find that the IEE efficiency ratio γ = V/µ exhibits universal dependence on sample size, and
2
approaches π/4 in the ballistic limit and 1 in the diffusive limit.
n
a
J PACS numbers: 72.25.Dc, 85.75.-d, 73.50.Bk

2
1
Spintronicshasbeenarapidlygrowingfieldofresearch

] in the past two decades because of its potential applica-
l
l tionsinmemory,logic,andsensingdevices,whichutilize
a both spin and charge degrees of freedom of electrons1–7.
h
Among the major tasks in spintronics, electrical detec-
-
s tionofspincurrentandspinbiasremainstobechalleng-
e
ing. One method is to use the inverse spin Hall effect
m
(ISHE),inwhichapurespincurrentgeneratesameasur-
t. able transverse charge current8–11. While the ISHE has
ma beenwidelyemployedinspintronicexperiments12–16,the FIG. 1. Schematic view of the setup for observing the IEE.
electrical signal generated is usually small, e.g., the spin A TI thin film is covered partly by a ferromagnetic metal.
d- Hall angle θsh =0.08 in Pt17. Another method that has When the magnetization of the ferromagnet is stimulated to
n been attracting increasing interest is the inverse Edel- precess around the y axis, by using a radio frequency signal,
o stein effect (IEE)18,19, in which spin injection induces a spin bias polarized along the y direction will be created in
c nonequilibrium spin polarization and in turn generates thecoveredregionoftheTIfilm,andelectrical currentalong
[ x-axiswill be generated dueto theIEE.
a charge current in the longitudinal direction. The IEE

1 has been observed in Bi20, which was attributed to the
v Rashba spin-orbit coupling on the interface.
5 Topological insulators (TIs)21,22 and topological
scattering and sample size dependence in the IEE are
6 Kondo insulators (TKIs)23 are a new quantum state of not addressed in the simplified theory29. In this Letter,
2
3 matter. A three-dimensional (3D) TI has a bulk insu- we follow the model of Luo et al.29 and employ a semi-
0 lating gap with gapless surface states, which are pro- classical approach30 to study the IEE of the topological
tected from impurity backscattering by nontrivial bulk
. surface states. Our analytical theory is applicable from
1
band topology and time-reversal symmetry. The topo-
0 ballistictodiffusiveregime,andmayprovideusefulguid-
logicalsurface states posses the unique property of spin-
7 ance for experimental study of the IEE in 3D TIs.
momentum locking22–24, which are promising for appli-
1 Let us start from the effective Hamiltonian of surface
: cationsinspintronicdevices25,26. In2014,largeIEEwas states of a thin film of 3D TI Bi Se 31,32
v 2 3
realized in bulk insulating TIs Bi Sb Te Se and
i 1.5 0.5 1.7 1.3
X Sn-doped Bi Te Se27, which was interpreted as a result ∆
2 2 H = τˆ σˆ +v (p σˆ −p σˆ ) . (1)
r of the spin-momentum locking of the topologicalsurface 2 z z f y x x y
a states. Recently, in another experimental work28, the
Here, ~p = (p ,p ) is the electron momentum, σˆ with
IEEwasobservedonthesurfaceofTKISmB . Byusing x y α
6
α=x,y,zarethePaulimatricesforelectronspin,andτˆ
a Landauer-Bu¨ttiker like formula, Luo et al. theoreti- z
describes the bonding and antibonding of surface states
callystudiedthe IEEofthesurfacestatesinthe ballistic
on the two surfaces, with ∆ as the hybridization energy.
regime, and predicted that a spin bias polarizedin the y
The eigenenergies for τ =±1 are degenerate, given by
direction can generate a charge current flowing in the x z
direction29, which is in good agreement with the exper-
∆2
imental observation28. However, the effect of impurity E (~p)=± v2p2+ . (2)
τz r f 4
Here, p2 = p2 + p2, and signs + and − are for the
x y
conduction and valence bands, respectively. The cor-
a)Electronicmail: [email protected] responding eigenstates will be denoted as |τ i. The
z
2
Fermi energy E is set to be in the conduction band. Integrating the first-order linear differential equation
F
We now calculate the average of σˆ in the eigenstates (4) and taking the boundary conditions Eqs. (5) and (6)
y
|τ i by using the Feynman-Hellman Theorem, yieding into consideration, it is easy to obtain a self-consistent
z
hτ |σˆ |τ i = −v /v with v = v2p /E , which will be equation for g (x), which can be solved numerically30.
z y z x f x f x F τz
usedlater. TheFermivelocity,beingrenormalizedbythe InRef.30,itisfoundthatalinearapproximationg¯ (x)=
τz
nonzero hybridization energy, becomes vF =vf2pF/EF. a+bx to gτz(x) generally works very well. In particu-
Fig. 1 illustrates the setup for observing the IEE. A lar, the linear approximation becomes exact in the bal-
ferromagnet covers a part of a TI film. When the mag- listic limit, i.e., L ≪ l , and diffusive limit, L ≫ l 30.
y f y f
netization is stimulated to precess around a certain di- By following a similar procedure as that detailed in
rection nˆ, a spin bias polarized along nˆ is generated in Ref.30, we obtain for the coefficients a and b as a =
the covered region of the TI film. In other words, for U (L + κl )(L + 2κl ) and b = −U /(L + 2κl ),
L x f x f L x f
an electron with spin parallel or antiparallel to nˆ, its where l = v τ is the electron mean free path, U =
f F 0 L
chemical potential increases or decreases by an amount (eµ)η 1−∆2/4E2, and
F
(−eµ). The spin bias can be conveniently described by
p
tbheeenopdeermatoonrst(r−ateeµd)σˆth·anˆt.foIrnththeegbeoamlliesttricyrsehgoiwmne,initFhiags. −ππ/2/2cosφe−2lfLcxosφdφ
η = , (7)
t1h,eonIElyEtheffeeyctc29o.mpTohneerneftoroef,tfhoersspiminplbicaistyc,ownterifbouctuessotno R −ππ/2/2e−2lfLcxosφdφ
R
thefavorablesituation,wherethespinbiasispolarizedin
tisheusyedditroecdtieosnc.riTbehethseemelie-cctlarossniiccaltrbaonltszpmoratnnequation30 −ππ/2/2e−2lfLcxosφdφ
κ= . (8)
v ∂fτz =−fτz −f¯τz . (3) −πRπ/2/2 co1sφe−2lfLcxosφdφ
x R
∂x τ
0 In the appendix, we will show that this linear approx-
where fτz(x,vx) is the nonequilibrium distribution func- imation to g¯τz is very accurate in comparison with the
tion of the electrons in the τ band, and τ is the re- exact solution.
z 0
laxation time due to impurity scattering. In the linear- The electrical current is given by
responseregime,thedistributionfunctiontakestheform
eL ∂f
f = f + − ∂f0 g (x,v ), where f is the equilib- I = y v g (x,v ) − 0 dp dp . (9)
riτuzm dis0trib(cid:16)utio∂nEτfzu(cid:17)nctτzion. Ixt follows fro0m Eq. (3) that h2 Xτz Z x τz x (cid:18) ∂Eτz(cid:19) x y
g (x,v ) satisfies the following equation
τz x Following Shen, Vignale, and Raimondi19, we define an
IEEconductanceG =I/µ. Byusingthe abovelinear
∂g g −g¯ IEE
vx ∂xτz =− τz τ τz , (4) approximationto g¯τz, analyticalexpressionforGIEE can
be obtained as
where g¯ (x)= 1 2πdφg (x,v cosφ).
τz 2π 0 τz F G =G0 χbal +χdif (10)
The region coverRed by the ferromagnet is treated as a IEE IEE IEE IEE
(cid:16) (cid:17)
reservoir, and the uncovered region is considered as the
sample region. Since there is a spin bias in the reser- where G0 = G0 1−∆2/4E2 with G0 = N (e2/h)
IEE F ch
voir, so the electron distribution function in the reser- and Nch = 4kFLy/ph as the number of conducting chan-
voir deviates from the equilibrium distribution function, nels, and
tfhτze(sxp,ivnx b>ia0s)i=s pfr0oj+ec(cid:16)te−d∂∂iEnfτ0tzo(cid:17)t(h−eeµsu)hbτszp|aσˆcye|τzoif,twheheτrze χbIEalE = 12Z−π2π (cid:18)cosφ− Lxη+L2xκlf(cid:19)e−2lfLcxosφ cosφdφ ,
band. For right-moving electrons, when they just cross 2
π
the x = 0 boundary between the reservoir and sample χdif = ηlf 2 1−e−2lfLcxosφ cos2φdφ .
region,their distribution function remains to be same as IEE Lx+2κlf Z−π (cid:18) (cid:19)
in the reservoir. As a consequence, 2
We have divided G into two parts, labeled by super-
v IEE
g (x=0,v >0)=eµ x . (5) scripts “bal” and “dif”, corresponding to contributions
τz x vf from electron ballistic and diffusive transport processes.
In the ballistic limit, where L ≪ l , it is easy to ob-
The rightendofthe sampleregionatx=L is assumed x f
x tain G = πG0 . This result is consistent with that
to connect to another equilibrium reservoir. When left- IEE 4 IEE
obtainedbyLuoetal.29usingtheLandauer-Bu¨ttikerfor-
moving electrons cross the boundary x = L , their dis-
x
mulainthe ballisticregimeinthe absenceofthe contact
tributionfunctionremainstobeintheequilibriumstate,
potential barrier. In the opposite diffusive limit, where
such that
L ≫l , we have G = π lf G0 , which is essentially
x f IEE 2Lx IEE
g (x=L ,v <0)=0 . (6) a Drude like formula.
τz x x
3
When the electric current I flows through the system,
it causes a voltage difference V = I/G between the two 0.001 0.01 0.1 1 10 100 1000
endsofthesystem,whereGistheelectricalconductance 1.00 1.00
of the system. We introduce the ratio γ = V/µ to mea-
0.96 0.96
sure the efficiency of the spin-charge conversion. In gen-
eral, γ ≤ 1, and γ = 1 would mean perfect spin-charge 0.92 0.92
conversion, in which a spin bias µ is fully converted to
0.88 0.88
an equal amount of charge bias. Because I = µG by
IEE
definition, the efficiency ratio can also be expressed as 0.84 0.84
γ =GIEE/G. The expression for G is given by30 0.80 /4 0.80
G=G0 χbal+χdif (11) 0.001 0.01 0.1 1 10 100 1000
lf/Lx
(cid:0) (cid:1)
where
FIG. 2. The universal function of γ/γ0 versus lf/Lx, where
χbal = κlf π2 e−2lfLcxosφ cosφdφ , γ0 = q1−∆2/4Ef2. γ/γ0 approaches π/4 in the ballistic
Lx+2κlf Z−π limit, and 1 in the diffusivelimit.
2
π
χdif = lf 2 1−e−2lfLcxosφ cos2φdφ .
Lx+2κlf Z−π (cid:18) (cid:19)
2

(a) (b)
UsingEqs.(10)and(11),onecancalculatethe efficiency 0.9 0.9
0.2 0.8 2 8
ratio. It is easy to find that the efficiency ratio γ nor- 0.8 0.5 1 5 0.8
malizedby γ = 1−∆2/4E2 is a universalfunctionof 0.7 0.7
llaft/eLdx,cuinrdveepoe0fntdhepentunofivaenrysaml foudFnecltpioanraims edtiesprsl.ayTehdeincaFlciug-. 0G00IEE ..56 Numerical 00..56 0/GEIEE
2. We see that in the ballistic and diffusive limits, γ/γ0 I/0.4 Approximation 0.4 GIE
convergestotwodifferentconstants. Infact,usingtheex- 0.3 Diffusive 0.3
Ballistic
pressions for G in the two limits30, G=G for L ≪l , 0.2 0.2
0 x f
and G = π lf G0 for L ≫ l , one can readily obtain 0.1 0.1
γ/γ0 = π4 i2nLtxhe ballisticxlimitf, and γ/γ0 = 1 in the dif- 0.0 0.2 0.4x/L0x.6 0.8 1.0 0 2 4lf/Lx6 8 10
fusive limit. We mentionthatthese asymptotic formulas
for γ/γ0 are exact, because the linear approximation to FIG. 3. (a) Normalized electrical current due to the IEE as
g¯τz becomes exact in the ballistic and diffusive limits30. afunctionofnormalizedcoordinatex/Lx forseveraldifferent
Theresultthatγ approachesγ0 inthe diffusivelimitcan values of lf/Lx. (b) IEE conductances as functions of lf/Lx
be understoodasfollows. Inthe diffusivelimit, L ≫l , calculatedfromexactnumericalsolutionandapproximatefor-
x f
theelectronspropagatingatsmallangleswiththexaxis, mula Eq. (10). The black solid line stands for the result of
the Landauer-Bu¨ttiker like formula in the ballistic regime,
i.e., φ ≃ 0, make dominant contributions to the elec-
G /G0 = π,andtheblackdashlinestandsfortheDrude
tric current. For φ ≃ 0, the boundary condition Eq. IEE IEE 4
(5) reduces to gτz(x=0,vx >0) = eµ(vF/vf)cosφ ≃ like formula in thediffusiveregime, GIEE/G0IEE = 2πLlfx.
eµ(v /v ) = eµγ . Therefore, the spin bias µ is just
F f 0
equivalent to a charge bias γ µ, and as a result, the effi-
0
ciencyratiobecomesγ =γ µ/µ=γ . Whentheelectron
0 0
Fermi energy E is much larger than the hybridization funded by the PAPD of Jiangsu Higher Education Insti-
F
gap ∆, we have γ = 1, so that γ = π in the ballistic tutions (L.S. and D.Y.X.).
0 4
limit and γ = 1 in the diffusive limit. The spin-charge
conversionis perfect in the diffusive limit.
WehaveshownthathighlyefficientIEEorspin-charge Appendix A: Verification of Linear Approximation with
conversioncanbeachievedonaTIsurfacebecauseofthe Exact Solution
spin-momentum interlocking of the surface states. An
analyticaltheoryforthe IEEisdeveloped,whichisvalid In this appendix, we show that our linear approxima-
from the ballistic to diffusive regime. The IEE will be tiong¯ =a+bxisaverygoodapproximationcompared
τz
very useful for electrical detection of spin current and with the exact numerical result. In Fig. 3(a), we show
spin accumulation in spintronics. the exactly calculated electrical current I(x) due to the
This work was supported by the State Key Program IEE as a function of position x, for several different val-
for Basic Researches of China under grants numbers ues of l /L . For a given value of l /L , I(x) is a con-
f x f x
2015CB921202 and 2014CB921103 (L.S.), the National stant independent of x, meaning that the continuity of
Natural Science Foundation of China under grant num- the electrical current is satisfied. This serves as an evi-
bers 11674160(L.S.) and 11474149(R.S.), and a project dencethatournumericalresultis accurate. InFig. 3(b),
4
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