Spin Transport in Diffusive Superconductors

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by  Jan Petter Morten

Spin transport in diffusive superconductors
Jan Petter Morten,1,∗ Arne Brataas,1 and Wolfgang Belzig2
1Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway
2Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
(Dated: Received 28 January 2004)
5
0
We employ the Keldysh formalism in the quasiclassical approximation to study transport in a
0 diffusive superconductor. The resulting 4×4 transport equations describe the flow of charge and
2
energyaswellasthecorrespondingflowofspinandspinenergy. Spin-flipscatteringduetomagnetic

n impuritiesisincluded. Wefindthatthespin-fliplengthisrenormalizedinthesuperconductingcase
a and propose an experimental system tomeasure thespin-accumulation in a superconductor.
J

5
2 Manipulation of spin-polarized currents can be used as well as electron-hole correlations. Spin-flip scattering
tostudyfundamentaltransportprocessesandmightalso from magnetic impurities is included as the dominant

]
n provide new functionality in electronic devices. In fer- spin relaxation process inside the superconductor. We
o romagnets (F), the current is spin-polarized due to the find that the spin-flip length is renormalized in the BCS
c spin-dependent density of states and the spin-dependent case, and propose an experimental system to measure
-
r scattering potentials. In contrast, in s-wave supercon- the properties resulting from the superconducting corre-
p ductors (S), electrons with spin up and spin down and lations. Many, but not all, experimental systems involv-
u
oppositemomentumformCooperpairswithnonetspin. ing spin-transport in superconductors are in the elastic
s
. Nano-scale superconductors therefore display strikingly transport regime [6], which is considered here. Comple-
t
a different properties when driven out of equilibrium by mentary studies based on the Boltzmann equation for
m spin transport than by charge transport. spin-transport by quasiparticles in the inelastic trans-
- port regime have recently been published [7]. Note that
d Most of the recent activities on the transport proper-
spin-injection is qualitatively different in these opposite
n ties of F/S junctions have studied effects caused by the
transport regimes due to the strong energy-dependence
o physical properties on the F side of the junction. The
c of quasiparticle flow in superconductors [7].
zerospinCooperpairspreventspin-polarizedelectronsto
[ Let us now outline the derivation of our main results.
flowintoS.Consequently,aspinpolarizedcurrentfromF
2 injectedintoScanresultinnonequilibriumspinaccumu- Weusenaturalunitssothat~=kB =1,andtheelectron
v charge is e = e. To describe the out-of-equilibrium
lation near the F/S interface. The competition between −| |
0 electron-hole correlations as well as spin accumulation,
electron-holecorrelationsandspinaccumulationontheF
8
we define the Keldysh Green’s function as
5 side has recently attracted considerable interest[1]. Pos-
1 sible influence of the ferromagnetic order parameter on GˆK(1,2)= ( i)(ρˆ ) [(ψ(1)) , ψ†(2) ] , (1)
40 the superconductor has received less attention. Singlet ij Xk − 3 ikD k (cid:0) (cid:1)j −E
pairing does not allow a spin accumulation in the super-
/0 conductor. Consequently, spin accumulation can reduce where ψ = [ψ↑,ψ↓,ψ↑†,ψ↓†]T is a four-vector and ψ†
at the superconducting gapandchangethe transportprop- the corresponding adjoint vector. The matrix ρˆ3 is
m erties both for transport via quasiparticles and for the the third Pauli matrix generalized to 4 4 space, ρˆ3 =
×
- supercurrent. Experimentally, spin transport in diffu- diag(1,1,−1,−1). The coordinates are 1 = (r1,t1) and
d sive S has recently been studied [2]. Here, the reduced 2 = (r2,t2). Similarly, we define 4 4 retarded and ad-
n quasiparticle penetration due to spin accumulation re- vancedGreen’sfunctions(GˆR, GˆA)×inspin-andparticle-
o sults in loss of spin memory which can be measuredas a hole space. 4 4 matrices are denotedby a “hat” super-
c ×
: decreased magnetoresistance. script. Acompactnotationcanbeobtainedbyconstruc-
v tion of an 8 8 matrix in the Keldysh space (denoted by
i Although the theory of nonequilibrium superconduc- ×
X a “check” superscript) [5].
tivity is widely used and developed, it has to the best of
The quasiclassical Green’s function is defined by
r
a our knowledge not been completely generalized to study gˇ(R,T,p ,E) = i dξ Gˇ(R,T,p,E). This function
spintransport. InthisworkwethususetheKeldyshfor- F π p
is determined by theREilenberger equation which in the
malismandthequasiclassicalapproximation[3–5]torig-
mixed representation for a stationary state can be writ-
orouslyobtainasetofequationsdescribingthetransport
ten
of charge and energy in a diffusive weak coupling S, as
p
well as the transportof spin. This will describe the pen- Eρˆ3+i ∂ˆ eφˆ1 ∆ˆ σˇ, gˇ =0. (2)
etration of spins into S and the associated suppression h m · − − − i−
of the superconducting order parameter. Our descrip- Here∂ˆ= ˆ1 ieAρˆ isthe gaugeinvariantderivative,ˆ1
3
∇ −
tion of the transport properties will be based on a 4 4 isthe4 4unitmatrix,φistheelectromagneticscalarpo-
matrix equation formalism to include spin accumulati×on tential,×∆ˆ containsthe superconductinggapandσˇ is the
2
self-energy due to elastic impurity scattering and spin- We derive kinetic equations and find,
flip scattering by magnetic impurities in quasiclassical
approximation. In the case of strongimpurity scattering jL =0, (4a)
∇·
(dirtylimit)transportisdiffusive. Expansionofthequa- j = 2∆α h , (4b)
T TT T
∇· − | |
siclassical Green’s function in spherical harmonics then
1
givestheUsadelequations. Thesymmetriesandnormal- jLS = 2∆αTT+ αLSLS hLS, (4c)
∇· −(cid:18) | | τ (cid:19)
sf
ization of the Green’s function allows for a parameteri-
1
zation of the quasiclassical,retarded component [3] j = α h . (4d)
TS TSTS TS
∇· −τ
sf
The right-hand side terms represent renormalized scat-
tering because of superconductivity:
¯1cosh(θ) iτ¯ sinh(θ)eiχ
gˆsR =(cid:18)iτ¯2sinh(θ)e−iχ 2¯1cosh(θ) (cid:19), (3) αTT =Im{sinh(θ)}, (5a)
−
α =(Re cosh(θ) )2 (Im sinh(θ) )2, (5b)
LSLS
{ } − { }
α =(Re cosh(θ) )2+(Re sinh(θ) )2. (5c)
TSTS
{ } { }
where ¯1 is the 2 2 unit matrix, τ¯ is the second Pauli The ∆α terms on the right hand side in (4) are
2 TT
× | |
matrix and θ and χ are position and energy dependent due to conversion of quasiparticle current into super-
functions. We assume colinear magnetizations along the current, and the α /τ , α /τ terms are due
LSLS sf TSTS sf
z-axis and s-wave singlet superconducting state. We to spin-flips. The spin-flip time in the normal state is
choose a gauge where the superconducting order param- τ−1 =8πn N S(S+1)v 2/3,wheren isthemagnetic
sf sf 0 | sf| sf
eter ∆ is real and positive, and then the supercurrent is impurity density, N the density of states at the Fermi
0
contained in the electromagnetic vector potential A and level, S the impurity spin quantum number and v is
sf
the chemical potential of the Cooper pairs is included the Fouriertransformedspin-flipimpurity potential. We
in φ. Inspection of the self-consistency relation for ∆ assume isotropic scattering. Our definition of τ differs
sf
reveals that χ = 0,π depending on the boundary con- fromtheusualspin-fliplifetimebyarenormalizationfac-
ditions. This ansatz simplifies the calculations consider- tor4/3. Thisdefinitionreproducesthediffusionequation
ably. The advanced Green’s function is related to the with a spin-flip length l(N) =√Dτ in the normal state.
retarded through gˆA = ρˆ gˆRρˆ †. Because of normal- Thus there is a differenscfe betweensfthe spin-flip lifetime
3 3
−
ization, the Keldysh Gree(cid:2)n’s func(cid:3)tion can be expressed measured in e.g. electron spin resonance and spin-flip
as gˆK = gˆRhˆ hˆgˆA where hˆ is a diagonal distribution transport time.
−
function matrix. We introduce generalized energy-dependent diffusion
coefficients
We will now consider a stationary state. A kinetic
D =D (Re cosh(θ) )2 (Re sinh(θ) )2 , (6a)
equation can be derived from the Usadel equations if L h { } − { } i
we include Keldysh components. The important quan- D =D (Re cosh(θ) )2+(Im sinh(θ) )2 , (6b)
T
tities are the physical particle and energy currents (in- h { } { } i
cluding particles and holes), which we will denote by
whereD =τv2/3is the diffusionconstant. The currents
j and j respectively, with the corresponding distribu- F
T L can then be expressed as
tion functions carrying the same indices, h and h [3].
T L
The physical spin current is denoted jTS and the spin jL = DL hL+Im jE hT, (7a)
energy current j , with distribution functions h and − ∇ { }
LS TS j = D h +Im j h , (7b)
T T T E L
h . The spin-resolved distribution functions can be ex- − ∇ { }
LS
j = D h +Im j h , (7c)
pressedbytheparticledistributionfunctionash = LS T LS E TS
TS(LS) − ∇ { }
−(f↑(E)−f↓(E))/2−(+)(f↑(−E)−f↓(−E))/2. Thecur- jTS =−DL∇hTS+Im{jE}hLS. (7d)
rent components j etc. are spectral quantities, and the
T
Here we have defined the spectral supercurrent as j =
total charge current is given as an integral j (r,t)= E
∞ charge D( χ 2eA)sinh2(θ). The self-consistency relation is
eN dEj (r,t,E), andthe spincurrentis obtained
| | 0 −∞ T ∇ −
by a sRimilar integral of jTS. Energy current is given by 1 ∞
jenergy(r,t) = |e|N0 −∞∞dEEjL(r,t,E), and the differ- ∆(r)=− 2sgn(∆0)N0λZ dEsinh(θ)hL, (8)
ence in energy currRent carried by opposite spins by a −∞
similar integral of jLS. where the factor sgn(∆0) is determined from the bound-
ary condition to give the correct sign and λ is the inter-
Theequilibriumsolutionsforthedistributionfunctions action parameter. The complex part of this equation is
are h = tanh(βE/2) and h = h = h = 0. neglected as a consequence of charge conservation [8].
L,0 T,0 LS,0 TS,0
3
The functions θ andχ are determined by the retarded perconductor in this experimental system has also been
components of the Usadel equation. We obtain calculated theoretically [7]. We will consider the simpli-
fiedgeometryshowninFigure1,wherethereisnocharge
jE =0, (9) transport in the superconductor, and calculate the spin
∇·
D 2θ 1( χ 2eA)2sinh(2θ) = 2iEsinh(θ) accumulation signal in the elastic regime. The F1/N/F2
(cid:20)∇ − 2 ∇ − (cid:21) −
(cid:0)(cid:1)(cid:0)(cid:1)µ(cid:0)(cid:1)(cid:0)(cid:1)=e(cid:0)(cid:1)V/(cid:0)(cid:1)2(cid:0)(cid:1)(cid:0)(cid:1)
3 1 1 (cid:0)(cid:1)
2icosh(θ)∆ + sinh(2θ), (10)
− | | 4τsf (cid:0)(cid:1) L(S)
F
1 (cid:0)(cid:1)(cid:0)(cid:1)
where Eq. (9) implies that the spectral supercurrent is (cid:0)(cid:1) (cid:0)(cid:1) S (cid:0)(cid:1)F(cid:0)(cid:1)
conserved. In addition we have the following symmetry (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
conditions, θ∗(−E) = −θ(E), χ∗(−E) = χ(E). Equa- N (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
tions (4)-(10) determine all transport properties of S. (cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1)
In general, in a hybrid F/S system, the superconduc- S
F2 (cid:0)(cid:1) L
tor cannot be described as in terms of BCS-formulas
(cid:0)(cid:1)
close to the F/S interface due to the proximity effect.
µ2 = eV/2 (cid:0)(cid:1)(cid:0)(cid:1)
Nevertheless, to gain insight into the physics implied by −
the above-mentioned formulas let us now consider the
FIG. 1: Spin battery connected to a superconductor. The
limit of a homogeneous BCS superconductor, and select
thick solid line indicates a tunnelbarrier.
χ = 0. This is relevant for the proposed experiment
below. For energies E < ∆ α = ∆/√∆2 E2
TT
| | | | − systems act as a spin-battery which is connected via a
and the spin-flip renormalization factors are α =
TSTS
tunnel junction to the superconductor. A voltage bias
0, α = ∆2/(∆2 E2). The generalized diffusion
LSLS
− − betweenF andF inducesaspinaccumulationthatcan
constantD =0while D =D∆2/(∆2 E2). FromEq. 1 2
L T
− flowintoS.Thesuperconductingwireis connectedtoan
(7a) this means that there is no energy current carried
S reservoir in equilibrium BCS state by a good metal-
by quasiparticles with energy E < ∆. Gap scatter-
| | | | lic contact at distance L(S) from the N/S interface. On
ing for quasiparticle energies below the superconducting
top of the S wire there is a ferromagnet connected by
gap corresponds to a transformation of the charge cur-
tunnel barrier which upon switching of the magnetiza-
rent (j ) into supercurrent. Such scattering is not possi-
T
tiondirectionactsasadetectorforthespinsignal. Mea-
ble for the physical spin current (j ). Consequently, in
TS
surementoftherelativevoltageofthiselectrodebetween
the absence of spin-flip scattering the quasiparticle spin-
parallelandantiparallel(withrespecttothe topFreser-
current into the superconductor vanishes for E < ∆
| | | | voir) magnetization gives ∆µ = µ(P) µ(AP) which de-
since DL = 0 in Eq. (7d) and αTSTS = 0 in the kinetic −
scribes the difference between electrochemical potential
equation (4d). Note that this result relies on the fact
of spin-up and spin-down quasiparticles located a dis-
that there are different effective diffusion coefficients for
tance L from the N/S interface. This quantity can be
charge current (D ) and for spin current (D ). We also
T L calculated ∆µ = ∞ dEP(D)h (L,E), where P(D)
observe that the term αLSLS is negative below the gap, − −∞ TS
is the spin polarizatiRon of the tunnel barrier between S
acting as a source of spin energy.
andtheFdetector. Weassumeahomogeneousorderpa-
Above the gap (E > ∆) the factor α vanishes
TT
| | | | rameter and BCS spectral properties in the S wire since
whileα =E2/(E2 ∆2), α =(E2+∆2)/(E2
LSLS TSTS
− − therearetunnelbarriersbetweentheN,FandSelements
∆2). For the generalized diffusion coefficients we find
and perturbation from current and spin-flip is weak.
that D = D and D = DE2/(E2 ∆2). Now con-
L T
− We can express the difference between the spin-up
sider the kinetic equations in the BCS case. A charge
and spin-down distribution functions in N close to S as
current carried by quasiparticles with energy E > ∆
can propagate into S. For quasiparticles at |E||>||∆||we| ∆f(N) ≡ f↑(N) − f↓(N) = P(FN)(f(E − eV/2) − f(E +
see that there is no renormalization for the spin-energy eV/2)), where P(FN) = (G G )/(G +G )
maj min maj min
−
diffusion length in Eq. (4c), whereas the spin diffusion is the spin polarization between the F reservoirs and N,
length in Equation (4d) has an energy dependent renor- f(E eV/2) is the Fermi-Dirac distributions in the F
±
malization factor which diverges for energies E = ∆ reservoirs and G is the conductance of major-
| | | | maj(min)
causing massive spin-flip scattering. ity (minority) spin electrons from ferromagnetic reser-
We will now apply this formalism to study spin diffu- voir to the middle of N. There is thus no charge current
sion, and demonstrate the significance of the renormal- or supercurrent anywhere in S, however there may be a
izationofthespindiffusionlength. Experimentalstudies spin-current. Equation (7d) states that there is no spin-
ofspinaccumulationandspininjectionhasrecentlybeen current for energies below the gap, thus for these ener-
performed [9] in metallic spin valves. The spin accumu- gies the N/S interface is effectively insulating. Since the
lationinthephysicallydifferentinelasticregimeforasu- S wire is connected to a reservoir in the other end for
4
E < ∆ the spin distribution function equals the equi- 1
| |
librium value h = 0. We solve the TS kinetic equa-
TS
tion (4d) for energies E > ∆. This equation reduces
| | 0.8
toadiffusionequationwithrenormalizedspin-fliplength
l(S)(E) = l (E2 ∆2)/(E2+∆2), where l = √Dτ
issfthe normsfapl state−spin-flip length. The bousnfdary cons-f V0.6
e
dition at the S reservoiris that the distribution function /
µ
attainstheequilibriumvalue,andattheS/Ninterfacewe ∆
0.4
matchat eachenergy the tunnel spincurrentto the spin
currentinsideS, eN j . WeassumethatL(S)/l(S) 1
| | 0 TS sf ≫
which is a relevant physical situation. 0.2
Thepositionandenergydependentsolutionh issub-
TS
stituted into the expression for the measured difference
0
in electrochemical potential for parallel and antiparallel 0 0.2 0.4 0.6 0.8 1
T/T
configuration, and we obtain c
FIG. 2: Temperature dependence of ∆µ/eV. We use R(S) =
∆µ=2P(D) ∞dE∆f(N)e−L/l(sSf) Rs(fS) , (11) R(I) in the normal state. For the dotted lines L/lsf s=f 6,
Z∆ Rs(fS)+R(I) and for the solid lines L/lsf = 7. The bias eV is 0.1∆(T =
0), 3∆(T =0), 10∆(T =0)forthelowercurvetothehigher
where R(I)(E)=1/(T 2N (E)N ) is the resistanceof curve,respectively.
BCS 0
| |
the N/S tunnel barrier, T is the tunneling matrix ele-
| |
(S)
ment, N (E) is the BCS density of states, R (E) =
BCS sf thatthespinsignaldecreasesduetosuperconductingcor-
l(S)(E)ρ/A is the resistance of the S wire within a spin-
sf relations. For a large energy gap the spin accumulation
flip length and ρ is the resistivity of the material in S
vanishes completely at low temperatures. These effects
when in the normal state (T > T ). This result can be
c can be explained by suppressed subgap spin-currentand
understood as follows. The spin accumulation close to
massive spin-flip at energies close to the gap because of
the tunnel interface is exponentially attenuated by spin-
the superconducting correlations.
flip scattering in S. The spin signal is also decreased by
In conclusion, we have presented a formalism to de-
the tunnel resistance, and since spin current is strongly
scribe elastic spin transport in superconductors with
suppressed for energies E <∆ only quasiparticles with
spin-flip scattering. We find different effective diffusion
| |
energieshigherthanthegapcontribute. Theeffectiveto-
coefficients for charge- and spin-current. The spin-flip
talresistanceis a seriesof the tunnel interface resistance
length is renormalized in the superconducting case, and
with the resistance of S within one spin-flip length.
atenergiesclose to the gapthere is massivespin-flip. As
We will now consider some simplified limits for
anillustrationwecomputethedifferenceinelectrochem-
the quantity ∆µ defined above. In the normal
icalpotential due to spinaccumulationin anexperiment
state where ∆ 0 we find that ∆µ/eV =
sensitive to the renormalization of spin-flip length.
→
2P(D)P(FN)Rs(fS)exp(−L/lsf)/(Rs(fS) + R(I)) where Rs(fS) This work was supported in part by The Research
and R(I) assume their normal state (energy indepen- Council of Norway,NANOMAT Grants No. 158518/431
dent) values. At kBT ∆ the signal measured and 158547/431, RTN Spintronics, the Swiss NSF and
≪
by ∆µ vanishes when the bias is lower than the en- the NCCR Nanocience.
ergy gap eV < ∆ since spin current is suppressed
for quasiparticles below the gap. For higher bias,
eV > ∆, and at zero temperature when the bulk resis-
tance dominates, R(S) R(I), an approximate solution
is ∆µ = 2P(D)P(FNsf)∆≫e−L/lsf e−Lr2/2lsf/r e−L/2lsf + ∗ Electronic address: [email protected]
{ − [1] V. T. Petrashov, I. A. Sosnin, I. Cox, A. Parsons, and C.
πL/2lsf(erf[r L/2lsf] erf[ L/2lsf]) , where r = Troadec, Phys. Rev. Lett. 83, 3281 (1999); M. Giroud,
− }
2p∆/eV. In thpis case the relaption between the energy H. Courtois, K. Hasselbach, D. Mailly, and B. Pannetier,
gap and the bias determines the magnitude of the spin Phys.Rev.B58,11872(1998);V.I.Fal’ko,C.J.Lambert,
signal, and the exponential decrease of the signal. and A. F. Volkov, Pis’ma Zh. Eksp. Teor. Fiz. 69, 497
The temperature dependence of ∆µ in the general (1999), JETP Lett. 69 532 (1999); F. J. Jedema, B. J.
van Wees, B. H. Hoving, A. Filip, and T. M. Klapwijk,
case is given by decrease from a constant value above
Phys.Rev.B60,16549(1999);D.Huertas-Hernando,Yu.
T as the temperature approaches zero. An example
c V. Nazarov, and W. Belzig, Phys. Rev. Lett. 88, 047003
of this behavior is shown in Figure 2. Here we have
(2002).
used the approximate temperature dependence ∆ = [2] J. Y. Gu, J. A. Caballero, R. D. Slater, R. Loloee, and
1.76Tctanh(1.74 Tc/T 1). Our calculations show W. P. Pratt, Jr., Phys.Rev.B 66, 140507 (2002).
−
p
5
[3] W.Belzig,F.K.Wilhelm,C.Bruder,G.Sch¨on,andA.D. Maekawa, Phys. Rev. B 65, 172509 (2002); S. Takahashi
Zaikin, Superlatt. Microstruc. 25, 1251 (1999). and S. Maekawa, Phys. Rev.B 67, 052409 (2003).
[4] N. Kopnin, Theory of Nonequilibrium Superconductivity [8] A.Schmid(1981),vol.65ofNATOAdvanced Study Insti-
(Oxford SciencePublications, 2001). tute Series B, pp.423–480.
[5] J.RammerandH.Smith,Rev.Mod.Phys.58,323(1986). [9] M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790
[6] Y.TserkovnyakandA.Brataas, Phys.Rev.B65,094517 (1985);F.J.Jedema,A.Filip,andB.J.vanWees,Nature
(2002). (London) 410, 345 (2001).
[7] T. Yamashita, S. Takahashi, H. Imamura, and S.