Intrinsic Spin Hall Effect in s-wave Superconducting State: Analysis of Rashba Model

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by  H. Kontani

Intrinsic Spin Hall Effect in s-wave Superconducting State:
Analysis of Rashba Model
H. Kontani, J. Goryo and D.S. Hirashima
Department of Physics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan.
(Dated: January 27, 2009)
A general expression for the spin Hall conductivity (SHC) in the s-wave superconducting state
at finitetemperatures is derived. Based on the expression, we studytheSHCin a two-dimensional
9 electron gas model in the presence of Rashba spin-orbit interaction (SOI). The SHC is zero in the
0 normalstate,whereasittakesalargenegativevalueassoonasthesuperconductivityoccurs,dueto
0 thechange in thequasiparticle contributions. Since this remarkable behavioris independentof the
2 strengthoftheSOI,itwillbewidelyobservedinthinfilmsofsuperconductorswithsurface-induced
RashbaSOI,or in various non-centrosymmetricsuperconductors.
n
a
PACSnumbers: 72.25.Ba,74.70.-b,72.10.-d
J

7
2
The spin Hall effect (SHE), which is a phenomenon the superconducting state. Based on the expression, we

that an electric field E induces a transverse spin cur- analyze the two-dimensionalelectrongas (2DEG) model
]
l rent JS, has attracted considerable attention. The in- with Rashba SOI as a typical spin Hall system, and re-
l
a trinsic SHE, which is independent of impurity scatter- veal that a giant SHC emerges in the superconducting
h
ing, was first predicted in semiconductors [1–4]. Later, state due to the CVC, independently of the strength of
-
s it was revealed that the large intrinsic SHE emerges in the SOI. This phenomenon will be observed in thin film
e
transition metals and their compounds, since the com- superconductors with the aid of surface-induced Rashba
m
plex d-orbital wavefunction induced by spin-orbit inter- SOI [18], or in various non-centrosymmetric supercon-
.
at action (SOI) gives rise to the “orbital Berry phase” that ductors such as CePt3Si, Li2Pt3B, MgSi, and Y2C3.
works as a spin-dependent effective magnetic field [5–8]. The Rashba 2DEG model describes the electrons in a
m
In severalmetals, the spinHall conductivity (SHC, σ ) semiconductorinversionlayer. Ithadbeenstudiedinten-
SH
-
d hadbeenexperimentallydeterminedbyobservingthein- sivelynot onlyas the issue ofsemiconductorspintronics,
n verseSHEsignal[9–11]. TheobservedSHCinPtexceeds but also as a model for non-centrosymmetric supercon-
o 200~e 1 Ω 1cm 1 [10],andtheSHCsinNbandMoare ductor [19]. The SHC in Rashba 2DEG model was first
− − −
c ·
negative [12]. These results can be explained in terms of studied by Sinova et al [1]. They found that the SHC
[
the intrinsic SHE [8, 13], suggesting the importance of takesauniversalvalue e/8πintheballisticregime. ( e
3 the intrinsic mechanism in transition metals. is the electron charge.)−Later, Inoue et al [3] had show−n
v
Inspintronics,superconductivityiswidelyusedtocon- that the SHC vanishes identically in the diffusive regime
7
3 trol the spin state. In this sense, a natural question is due to the CVC induced by short-rangeimpurities. This
2 whether and how the SHE emerges in the superconduct- fact is also explained by the relation Sˆ˙ i[Hˆ,Sˆ ] JˆS
4 ing state. Although DC electric field E cannot exists in in the Rashba model [4, 20], where JˆSx ≡is the spxin∝curx-
. x
6 bulksuperconductors,theSHCinsuperconductorsisob- rent. However, this relation does not hold in the super-
0 servable as follows: For example, the Hallvoltage due to conducting state. For this reason, the SHC in Rashba
8
inverse SHE should appear in the superconducting tun- 2DEG model shows a considerably large value just below
0
neling junction, thanks to the charge imbalance effect T unless the SOI is zero, as we will show below.
: c
v [14]. Moreover,the gradient of temperature will induces The s-waveBCS-Rashba 2DEGmodel is expressedby
i
X the transverse spin current in a bulk superconductors thefollowing4 4formintheNamburepresentation[19]:
×
when the SHC is finite: The observed spin Nernst con-
ar SdHucCtivbiytythαeSHM≡otJtySre/l(a−ti∇onxTσ)SH[15∝]wToσuS′Hld(ǫbFe),realsatdeidsctuossthede Hˆ = 12Xk φˆ†kHˆkφˆk, Hˆk =(cid:18)i∆hˆ0kσˆy −−ihˆ∆0−σ∗ˆky (cid:19), (1)
in the intrinsic anomalous Hall effect [16]. In addition,
theSHCintype-IIsuperconductorswillbemeasurablein where σˆi (i = x,y,z) is the Pauli matrix, and φˆ†k =
the mixedstate under the magnetic field, since the resis- (c†k ,c†k ,c k ,c k ); ckσ is an annihilation operator of
tivity is nonzero [17]. Therein, SHC will not be affected elec↑tron↓. −hˆ0k↑=−ǫk↓ˆ1 + λ(σˆxky σˆykx) is the 2 2 k-
by moving vortices since they do not convey spin. linear Rashba 2DEG Hamilton−ian in the normal×state,
In this letter, we present a general expression for the where λ is the SOI parameter and ǫk = k2/2m µ; µ
−
intrinsicSHC inthe superconductingstate. This expres- is the chemical potential. ∆ is the superconducting gap;
sion indicates that the current vertex correction (CVC), we assume that 0 ∆ µ hereafter. The quasiparticle
≤ ≪
which is a consequence of the conservation laws in the spectrumisgivenbyEk, = (ǫk λk)2+∆2. Wealso
± ±
fieldtheory,cancauseasignificantchangeintheSHC in we assume that the spin-orbit splitting, ∆ = 2λk , is
p SO F
2
much smaller than µ. In this case, the triplet pairing Thesecondterm,∆ΛˆC n I2 Tˆ GˆRΛˆCGˆATˆ ,rep-
y ≡ imp k 0 y 0
induced by the Rashba SOI is negligible [19]. resentstheCVCforI-term. AsshowninRef. [3],thefac-
Below,westudytheSHCinthepresenceofshort-range tor n in eq. (7) cancels with P GˆRGˆA O(γ 1)
imp k −
nonmagneticimpurities,bywhichTcisunchangeddueto O(n−im1p). Therefore, the CVC fPor I-term i∼s importan∼t
Anderson’s theorem [21]. In the Nambu representation even in the clean limit (n 1). We will show this
imp
≪
[21], a single impurity potential term is given by fact explicitly in later calculation.
On the other hand, CVC for II-term is negligible in
Hˆimp = I φˆ†kTˆ0φˆk′, (2) the clean limit: The charge CVC for II-term, ∆ΛˆCyII,
2
k,k′ is given by eq. (7) by replacing GˆA with GˆR. Since
X
GˆRGˆR is regular for γ +0, ∆ΛˆCII O(n ),
where I is the impurity potential. Here and hereafter, k → y ∼ imp
whichisnegligibleinthecleanlimit. Moreover,∆ΛˆCII is
we define the following 4 4 unitary matrices: P y
× related to the self-energy Σˆk through the Ward identity.
Tˆ = ˆ1 0 , Tˆ = σˆx 0 , Since Σˆk is k-independent in the SCBA for short-range
0 (cid:18)0 −ˆ1(cid:19) 1 (cid:18) 0 −σˆx (cid:19) impurities, ∆ΛˆCyII vanishes in the present study. Thus,
bare currents enter into eq. (5).
σˆ 0 0 σˆ
Tˆ2 = 0z σˆ , Tˆ3 = σˆ 0z . (3) By following Ref. [8], we rewrite eqs. (4) and (5) in
z z
(cid:18) (cid:19) (cid:18) (cid:19) the band-diagonal basis. In this basis, (Uˆk†GˆRUˆk)l,m
≡
TGˆhke(ωG) r=een(ωˆ1funcHtˆiokn)−i1n[2t1h]e, aNnadmtbhue rreetparredseedntaatnidonadis- [iGsˆtRh]el,mun=itaδrly,mm/(aωtr−ixEfoklr),thwehcehraenEgekio=f b±asEisk.,±Thaennd, Uwˆke
vanced Green−function are GˆRk(ω) = Gˆk(ω + iγ) and canperformω-integrationsusingtherelationshipsIm(z
dGˆaAkm(ωpi)ng=rGˆatke(ωis−giiγv)e:nInbythγe B=ornnimappπprIo2xNim(ωa)t,ionw,htehree iσγII)−1is=giπveδn(za)sandIm(z−iγ)−2 =−πδ′(z). Forexampl−e,
SH
N(ω) = (m/2π)Re 1/ 1 (∆/ω)2 is the density of
{ − }
states (DOS) per spin. 1 [JJ] ∂f 1
p m,l +2f(El) ,
it SwiinllcebethiensSeHnsCitiisveintdoeptheendeelencttroifctfiheeldMferiesqsnueernccyurωrefnotr, 2kX,l6=mEkm−Ekl "(cid:18)∂ω(cid:19)Ekl k Ekm−Ekl #
ω γ . In the linear response theory [8, 22], the DC
| | ≪ 0 where [JJ] Im [JˆS] [JˆC] . The first (second)
SHC is expressed in the 4 4 Nambu representationas m,l ≡ { x m,l y l,m}
× term corresponds to σIIa (σIIb) in Refs. [8]. As a result,
SH SH
σI = dω ∂f 1Tr JˆSGˆRΛˆCGˆA , (4) σSH is given by the summation of σSI+HIIa and σSIIHb:
SH 2π −∂ω 2 x y
σSIIH = X−kXkZ Z d2ω(cid:18)πf(ω)R(cid:19)e21Trh"JˆxS∂∂GˆωRJˆyCGˆiR σSI+HσIIIIab == kX,l6=m2([EJ[∆kJmJΛ−]]mmE,,llkl)(cid:18)−f(∂∂Eωfl(cid:19))Eklf,(Em) ,((89))
∂GˆR SH 2(Em El)2 k − k
JˆSGˆRJˆC , (5) k,l=m k − k
− x y ∂ω # X6 (cid:2) (cid:3)
where [J∆Λ] Im [JS] [∆ΛC] . Now, we can
where f(ω) = (eω/T + 1)−1. JˆyC is the bare charge calculatethemS,Hl C≡ing{enexraml,sluperycoln,mdu}ctorsusing Eqs.
current operator, which is given by δhˆk+eA/δAy A=0 (8) and (9), which had not been derived previously.
− |
(JˆδShˆ∗−k+eJˆAC/,δsAy/|A(=20)e)fiosrthpearbtaicrelesp(ihnolceu)rrcehnatnonpeelra[2to2r].. k k’
Inxt≡he{prxesenzt}m−odel, they are given by JxS k λk+ I −λk’ JyC
k k
JˆyC =−e myˆ1+λTˆ1 , JˆxS = 2mx Tˆ2, (6) k I −k’
(cid:18) (cid:19) −k
where eλTˆ inJˆCiscalledtheanomalousvelocity[1,3],
which−is esse1ntialyfor the SHE. FIG. 1: An example of terms for σSIH that is finite under Tc.
ΛˆC in eq. (4) is the total charge current dressed by This term is proportional to ∆∝p1−T/Tc.
y
the CVC. When the elastic scattering dominates the in-
σI+IIa ineq. (8)representsthequasiparticlecontribu-
elastic scattering, the CVC is derived from the following SH
tionatthe Fermilevel,andσIIb ineq. (9)representsthe
Bethe-Salpeter equation in the self-consistent Born ap- SH
Berrycurvaturetermthatisrewrittenasthesummation
proximation (SCBA):
oftheBerrycurvatureofBlochelectronsinsidetheFermi
ΛˆC = JˆC+∆ΛˆC, (7) sea. In the normal state in bulk transition metals, CVC
y y y
3
due to localimpurities almostvanishes inthe cleanlimit for x = 0; ΛC = JC+∆ΛC = e(k /m)ˆ1 [3]. However,
y y y − y
since the main matrix elements of current operators are this cancellation does not occur in the superconducting
odd functions of k. Therefore, σI+IIa 0 and the total state, and therefore σI =0 below T .
SHC is approximately given bySσHIIb f≈or γ 0 [8]. In Now,wederiveσI+ISIHa t6 hatrepresenctsthequasiparticle
the Rashba 2DEG model, in contrSaHst, σI+II→a = e/8π in contribution. In a gSeHneral basis, eq. (8) is rewritten as
SH
the normal state because of the CVC for I-term. Since
dω ∂f
σIIb = e/8π, the total SHC vanishes identically in the σI+IIa = B(ω) (14)
SH − SH 2π −∂ω
normal state [3]. k Z (cid:18) (cid:19)
X
Interestingly, such a balance between the quasiparti-
cle term and the Berry curvature term in the normal where B(ω) ≡ 21Tr JˆxSGˆR∆ΛˆCyGˆA . Since B(ω) is sim-
s(∆tat=e i0s)dsreatsstiinca:lσlyI+chIIaanvgaendiswhhesenatthTe=su0pebreccoanudsuecotfivtihtye pfolrifiγed as1,(ew/e32ombt(a1i+nh txh2e))follαow=±in1g(ǫiekx+prαesλskio)2nδ:(|ω|−Eα)
fact6or ∂f/∂ω. TShHus, σI+IIa should show a prominent ≪ P
change−below T in spin HSHall systems where the CVC is σI+IIa = e X(∆,T), (15)
c SH 8π
significant: such examples are metals with Rashba-type
or Dresselhaus-type SOI [3, 4] and doped graphene [23]. X(∆,T) = ∞ dǫ ∂f ǫ2 .
Ontheotherhand,σSIIHb isfiniteevenatT =0unless∆is Z−∞ (cid:18)−∂ω(cid:19)√ǫ2+∆2 ǫ2+2∆2
much larger than the SOI splitting. Therefore, the SHC
It should be stressed that σI+IIa is independent of λ(=
can show a nontrivial temperature dependence below T SH 6
c 0). The asymptotic behavior of X(∆,T) for ∆ 0 is
due to the imbalance between σI+IIa and σIIb. ≥
SH SH
Hereafter, we calculate both σI+IIa and σIIb in the π ∆
SH SH X(∆,T) 1 for ∆ T (T T ),(16)
BCS-Rashba2DEGmodel,andshowthattheSHCtakes ≈ − 2√2Tc ≪ ∼ c
a large value just below T . According to eq. (7), the
c πT
lowest order CVC for ∆ΛCy is given as ≈ 2∆e−∆/T for ∆≫T (T ≪Tc).(17)
r
∆ΛˆC(1)(ω) = n I2 Aˆ(ω), (10) In the same way, we derive that σIIa = (e/8π)Y(∆,T),
y imp SH
Xk where Y = ∞ dǫ(−∂f/∂ω)√ǫ2+∆2 is the Yosida func-
tion. Since Y−∞= 1 O((∆/T )2) for T T , σI
where Aˆ(ω) = Tˆ GˆRJˆCGˆATˆ . Hereafter, we assume the R − c ∼ c SH ≈
0 y 0 ( e/16√2)(∆/T). Figure 1 shows an example of terms
damping rate in the normal state, γ0 = nimpI2m/2, is fo−r σI that is of order O(∆) after k,ω-integrations. It
the smallest parameter. Considering that the pole of repreSsHents the spin current due to the triplet (k , k )
GˆR is ω = E iγ and the relation (z2 + γ2) 1 = ↑ − ↑
(π/γ)δ(z), th±e e±xp−ression of Aˆ(ω) is greatly simp−lified particle-particle excitation induced by the Rashba SOI
and the impurity scattering.
2π
forγ0 ≪1. After performingthe integration 0 dφk/2π Next, we discuss σSIIHb, which is called the “Berry cur-
(φk =tan−1(ky/kx)), Aˆ(ω) is simply given asR vature term” [8]. It is caused by electrons in the Fermi
sea that satisfy El Em < 0. In the normal state, the
Aˆ(ω)= −8eγ·π αmk +λ δ(|ω|−Eα) Tˆ1−xTˆ3 k-summationineqk.·(9)kis restrictedto kF− <|k|<kF+,
αX=±1(cid:18) (cid:19) (cid:16) (cid:17) where kF± are two Fermi momenta [8]. This restric-
tion approximately holds in the superconducting state
where x=∆/ω. As a result, eq. (10) becomes in the present model. Moreover, [JˆS] = [JˆC] = 0
x m,l x m,l
∆ΛˆC(1)(ω) = e(λ/2)( Tˆ +xTˆ ), (11) for (Ekm,Ekl)=(Ek,α,−Ek,α), since there is no number-
y − − 1 3 nonconservingelementsineq. (6)[22]. Forthesereasons,
where the relation γ =πnimpI2N(ω) is considered. in contrast to σSI+HIIa, σSIIHb is insensitive to T and ∆ if
(T,∆) ∆ ,whichwillbeverifiedlaterbyperforming
The second lowest order CVC is given by eq. (10) SO
≪
by replacing JˆC with ∆ΛˆC(1). After taking the k- numerical calculation of eq. (9).
y y Now, we show numerical results for m=µ=1, under
summation, it is simply obtained as
the condition that ∆,∆ γ . Figure 2 (a) shows the
SO 0
≫
∆ΛˆC(2)(ω) = (1 x2)/2 ∆ΛˆC(1)(ω). (12) ∆-dependenceoftheSHCforλ=0.1andT =0.01. The
y − · y quasiparticle term σI+IIa and the Berry curvature term
SH
σIIbaregivenbyeqs. (15)and(9),respectively. Toverify
Thus, the total CVC, ∆ΛCy = ∞l=1∆ΛˆCy(l), is given as thSeHcorrectness of eq. (15), we have also calculated the
CVC by solving the Bethe-Salpeter eq. (7) numerically
λ P
∆ΛCy = −e1+x2 −Tˆ1+xTˆ3 . (13) for γ0 = 0.001, and derived σSI+HIIa using eq. (14). The
(cid:16) (cid:17) obtained result is shown by square dots. Since σSIH =
σI vanishes in the normal state since the off-diagonal (e/16√2)(∆/T )+O(∆2)andσII =O(∆2), σI gives
SH − c SH SH
anomalousvelocitycancelsoutinthetotalchargecurrent the increment in σ just below T .
SH c
| |
4
(a) 1 λ=0.1 SOI shows a prominent increment in the superconduct-
σIIa T=0.01 ing state. Similar drastic change in the SHC will be ob-
E=1
0.5 SH I+IIa F servedincasesofthek3-typeRashbaorDresselhausSOI,
I+IIa (numerical) sincetheCVCfortheanomalousvelocityislargeinthese
π8] cases. The present study opens the way to distinguish
C [1/ 0 between the quasiparticle contribution (eq. (8)) and the
H Berry curvature contribution (eq. (9)), which had been
S σI
desired to understand the mechanism of intrinsic SHE.
SH
−0.5
total SHC Insummary,wehavepresentedthefirststudyofthein-
σIIb trinsic SHE in the superconducting state. Inthe Rashba
SH
−1 ∆ 2DEGmodel,theSHCchangesfromzerotoalargenega-
0 0.05 0.1
(b) 0 tivevaluejustbelowTcunlessλ=0,duetothechangein
Tc=0.2 I−term the quasiparticle contribution σI+IIa. This phenomenon
T=0.1 (indep. of T, λ) SH
c c willbeobservedinthinfilmsuperconductorswiththeaid
π] λ=0.05 of surface-induced Rashba SOI, in non-centrosymmetric
8
1/ Tc=0.05 superconductors,andinsemiconductorsusingthesuper-
C [ conducting proximity effect. Moreover, the SHC in su-
−0.5
H
S perconductors will also be observed using the AC mea-
al surement, since σ (ω) is finite for ω = 0 whereas the
ot xx 6
t T=0.02 intrinsic SHC is independent of ω for 0 ω .γ 1 [3].
c ≤ −
T=0.005 X(∆,T)−1 We are grateful to Y.Otani, T. Kimura,and E. Saitoh
c (indep. of T, λ)
−1 c for valuable discussions.
0 0.5 1 t
FIG. 2: (color online) (a) ∆-dependence of the SHC; com-
parison of I-, IIa-, and IIb-terms. (b) t ≡T/Tc-dependence
of the total SHC, σSH, for Tc=0.2∼0.005.
[1] J. Sinovaet al., Phys.Rev.Lett. 92 (2004) 126603.
[2] S. Murakami et al., Science 301 (2003) 1348.
[3] J. Inoueet al., Phys.Rev.B70 (2004) 041303(R).
Figure 2 (b) shows the T-dependence of the SHC
[4] K.Nomuraetal., Phys.Rev.B 72, 165316 (2005); O.V.
for λ = 0.05, assuming that ∆ = ∆ 1 T/T and
0 − c Dimitrova, Phys.Rev.B 71, 245327 (2005).
∆ = 1.8T in accord with the weak-coupling BCS the-
0 c p [5] H. Kontaniet al., Phys.Rev.Lett. 100, 096601 (2008) .
ory. ItisnoteworthythatbothσI andσI+IIaareunique [6] H. Kontaniet al., J. Phys. Soc. Jpn. 76 (2007) 103702.
SH SH
functions of t T/T , and independent of T and ∆ . [7] G.Y. Guoet al., Phys. Rev.Lett. 100, 096401 (2008).
c c SO
≡
σI takes a minimum value σImax 0.48 at t 0.78. [8] T. Tanaka et al., Phys. Rev.B 77, 165117 (2008).
InSHthecaseof∆ ∆ (i.e.,STH ≈∆− ),σIIb ≈ e/8π [9] E. Saitoh et al., Appl.Phys. Lett. 88 (2006) 182509.
0 ≪ SO c ≪ SO SH ≈− [10] T. Kimuraet al., Phys. Rev.Lett. 98 (2007) 156601.
for0<T <T sincethe modificationofelectronicstates
c [11] S. O. Valenzuela and M. Tinkham, Nature 442 (2006)
for k < k < k is limited, as discussed above. For
F− | | F+ 176.
this reason,σSH e/8π (X(∆,T) 1)for∆0 ∆SO, [12] The observed SHC in Nb (Mo) is −20 (−14) [~e−1 ·
≈− · − ≪
as shown in Fig. 2 (b). σSH(T = 0) = σSIIHb(T = 0) de- Ω−1cm−1]; Y. Otani et al, (unpublished).
creaseswith∆ sincetheformationofsingletparingpre- [13] H. Kontaniet al., Phys. Rev.Lett. 102 016601 (2009).
0
vents the spin current. In the opposite case, ∆ ∆ , [14] S. Takahashi and S. Maekawa, J. Phys. Soc. Jpn 77
0 SO
the relation σ σI is recognized, which mea≫ns that (2008) 031009.
SH ∼ SH [15] S.G. Cheng et al., Phys. Rev.B 78, 045302 (2008).
σII = σIIa + σIIb is very small. Thus, SHC shows a
SH SH SH [16] D. Xiao et al, Phys.Rev.Lett. 97, 026603 (2006).
striking enhancement just below T for any value of λ.
c [17] E. Saitoh et al., privatecommunication.
In the present study, we consider that the conduction
[18] F. Meier et al., Phys.Rev.B 77, 165431 (2008).
electrons are delocalized for kF < k < kF+, and they [19] P.A. Frigeri et al., Phys. Rev. Lett. 92, 097001 (2004);
contribute to σIIb. In convent−ional| i|nsulators, in con- S. Fujimoto, J. Phys. Soc. Jpn. 76 (2007) 051008.
SH
trast, σIIb is expected to vanish since the electrons near [20] O.V. Dimitrova, Phys. Rev.B 71, 245327 (2005).
SH
[21] R.J. Schrieffer, Theory of Superconductivity, Benjamin,
the band edge are localized [24, 25]. (Note that local-
New York (1964).
ization cannot be described in the SCBA.) Interestingly,
[22] A.C. Durst and P.A. Lee, Phys. Rev.B 62 (2000) 1270.
recenttheoreticalandexperimentalefforts have revealed
[23] N.A.Sinitsynetal., Phys.Rev.Lett. 97,106804 (2006);
that σIIb takes a finite value in “topological insulators”
SH S. Onari et al., Phys. Rev.B 78, 121403(R) (2008).
such as graphene [24] and HgTe [25], owing to the delo- [24] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95 (2005)
calized nature of massless Dirac fermions. 146802.
WehaveshowsthattheSHEdrivenbyk-linearRashba [25] B.A. Bernevig et al., Science 314, 1757 (2006).