научная статья по теме LONG-RANGE SPIN IMBALANCE IN MESOSCOPIC SUPERCONDUCTORS UNDER A ZEEMAN SPLITTING Физика

Текст научной статьи на тему «LONG-RANGE SPIN IMBALANCE IN MESOSCOPIC SUPERCONDUCTORS UNDER A ZEEMAN SPLITTING»

Pis'ma v ZhETF, vol. 101, iss. 2, pp. 124-130 © 2015 January 25

Long-range spin imbalance in mesoscopic superconductors under a

Zeeman splitting

I. V. Bobkova+*1\ A. M. Bobkov+

+Institute of Solid State Physics, 142432 Chernogolovka, Russia * Moscow Institute of Physics and Technology 141700 Dolgoprudny, Russia Submitted 2 December 2014

We develop a theory ol spin relaxation in Zeeman-splitted superconducting films at low temperatures. A new mechanism of spin relaxation, specific only for Zeeman-splitted superconductors is proposed. It can explain the extremely high spin relaxation lengths, experimentally observed in Zeeman-splitted superconductors, and their strong growth with the magnetic field. In the framework of this mechanism the observed spin signal is formed by the spin-independent nonequilibrium quasiparticle distribution weighted by the spin-split DOS. We demonstrate that the relaxation length of such a spin signal is determined by the energy relaxation length at energies of the order of the superconducting gap.

DOI: 10.7868/S0370274X15020101

Effective control of spin-polarized transport forms a basis of spintronic applications. In particular, it is very important to transmit spin signals over mesoscopic length scales. Usually at low temperatures the spin relaxation length is limited by elastic spin-flip at magnetic impurities or spin-orbit scattering processes. For example, it was shown in transport experiments [1-3] that for A1 thin films in the normal state the spin relaxation length An is of the order of 400-500 nm. However, recently is has been reported for superconducting A1 films that in the presence of a significant Zeeman splitting of the quasiparticle density of states (DOS) the spin signal can spread over distances of several /im [2-4]. In these experiments the superconducting spin relaxation length exceeds considerably the superconducting coherence length, the normal-state spin relaxation length and the charge-imbalance length. It is also important that the relaxation length grows with the applied magnetic field. But a mechanism for such a long-distance spin relaxation is not understand yet.

In the present paper we develop a theory of spin relaxation in Zeeman-splitted superconducting films at low temperatures. It is known that in the absence of the magnetic field (Zeeman splitting of the DOS) and at low temperatures the main mechanisms of the spin relaxation in superconductors are elastic spin flips by magnetic impurities and by spin-orbit interaction [5-9]. Here, we show that it is unlikely that the experimentally observed long-distance spin relaxation is provided by

-^e-mail: bobkova@issp.ac.ru

such elastic spin-flip processes. Instead we suggest a new mechanism, which controls spin relaxation in Zeeman-splitted superconductors.

It is generally believed that the length of a spin signal spread is controlled by the characteristic length of any spin relaxation processes. We show that in the case of Zeeman-splitted superconductor this is not necessary so. The spin relaxation length can be much larger than the length determined by fast elastic spin flip processes. The role of these elastic processes is only to rapidly relax the distribution function to the spin-independent value. The observed spin signal is formed by the spin-independent nonequilibrium quasiparticle distribution weighted by the spin-split DOS. We demonstrate that the relaxation length of such a spin signal is the energy relaxation length. This energy relaxation is provided by inelastic processes such as electron-electron and electron-phonon scattering. At low temperatures these inelastic processes are rather weak, so the corresponding spin relaxation lengths are large. Our theoretical estimates of the expected length scales for A1 agree well with the experimental data [2-4].

We show that the relaxation length for such a mechanism should grow with the magnetic field, as it is observed. The main qualitative reason is the following. It is well-known that the scattering rates of inelastic processes are energy dependent at low temperatures: the electron-electron scattering rate r~_}e ~ e2 and the electron-phonon scattering rate h ~ e3. The most essential energies for the considered here spin imbalance are of the order of the superconducting energy gap A.

124

IhicbMa b >K3TO tom 101 bhh. 1-2 2015

The superconducting gap is suppressed by the magnetic field. This leads to the suppression of the characteristic energy scale important for the relaxation. Consequently, the characteristic scattering rate decreases and the relaxation length grows.

Now we proceed to the calculation. Following the experiments [2-4] we consider the system depicted in Fig. la. It consists of a thin superconducting film (S) overlapped by the injector (I) and detector (D) electrodes. A current is injected into the superconducting film via I. This electrode can be normal or ferromagnetic. The detector electrode is ferromagnetic. The magnetic field is applied in plane of the film and is parallel to the ferromagnetic wires. It is interesting that the spin transport for misaligned magnetic field and injected spins was also studied recently [10]. In our study the quantization axis is chosen along the magnetic field. Both the injector and the detector are coupled to the film by tunnel contacts.

In this case it can be shown that the non-local electric current, measured at the detector is proportional to PdS. Here Pd is the detector polarization and S is the local nonequilibrium spin accumulation in the film at the detector point. Further, we focus on this nonequilibrium spin accumulation S. This quantity can be written in terms of the Keldysh quasiclassical Green function as

oo

S = J deTr [T303 (gK — g^q)] /16, where r« and o^ are

—00

Pauli matrices in the particle-hole and spin spaces, Respectively, gK is the Keldysh component (4x4 matrix)

(gR gK\

of the quasiclassical Green's function g = ,

V 0 9 J

gR(A) are retarded and advanced Green's functions. g^q means the value of the Keldysh component in equilibrium. We assume the superconductor to be in the diffusive limit, so the matrix g obeys the Usadel equation [11, 12]

Ddy(gdyg) + i [A - Sso - Smi - tin, g] = 0. (1)

Here A = er3 — /103T3 — Air2, e is the quasiparticle energy, h = ¡jl-qH is the Zeeman field, A is the order parameter in the film, D is the diffusion constant. dy is a matrix in particle-hole space, accounting for the orbital suppression of superconductivity by the magnetic field. For a general matrix G in particle-hole space dyG = dyG - (2ie/c)(Hx + A0) [PnGP22 - P22GPn], where Pu(22) = (1± r3)/2 and x is the coordinate normal to the film. Eq. (1) should be supplemented by the normalization condition g1 = 1.

The terms Sso = T^J-(aga) and Smj = = Tmi((7T39(TT3) in Eq. (1) describe elastic spin

relaxation processes of spin-orbit scattering and exchange interaction with magnetic impurities, respectively. The last term Sin describes inelastic processes of energy relaxation.

We assume that the transparencies of the injector and detector interfaces are small, so that up to the leading (zero) order in transparency the retarded, advanced Green's functions and the order parameter take their bulk values in the presence of the magnetic field. The Green's functions can be represented in the form gR = = 9oT3 + 9^o-3t3 + fRir2 + ftRa3ir2. It is convenient to use the following 0-parametrization, which satisfies the normalization condition: gRt = (cosh6>+ ± cosh6>_)/2 and fRt = (sinh6>+ ±sinh6>_)/2. The advanced Green's functions can be found as gA = —gR*. Then one can obtain from Eq. (1) that 0± obey the following equation:

(e =F h) sinh 9± + A cosh 9± +

e2

+Di—TjH2d? cosh0± sinh 0±±2iT~1 sinh(6>+ - <9_)+ 6 cl

+ 2rrs}1[cosh0± sinh 9± + sinh(6>+ + #_)]= 0. (2)

Here the third term describes the orbital depairing of superconductivity. Usually this orbital deparing can be disregarded for thin films in parallel magnetic field. However, it can be estimated that for magnetic fields of the order of 1-2 T, which are applied in experiment, the orbital depairing can even exceed the other depairing factors (spin-orbit and magnetic impurity scattering). So, it cannot be neglected in Eq. (2). In order to obtain Eq. (2) we integrate the retarded part of Eq. (1) over the width d of the film along the x-direction. A is calculated self-consistently.

The term Ein, describing inelastic energy relaxation, in principle, also enters Eq. (2) as another depairing factor, but it is neglected because at low temperature it is small as compared to other depairing factors. It is important only for the calculation of the distribution function.

The normalization condition allows to write the Keldysh component as gK = gR(p — <fgA, where <f is the distribution function with the following general structure in particle-hole and spin spaces: <f = (l/2)(^(j_ + + f^Vz + <f°-Tz + (p^_Tzcjz). Physically the distribution function is responsible for the charge imbalance and ip+ for the spin imbalance in the system. The components ^ describe the spin-independent part of the quasiparticle distribution, while accounts for its spin polarization. In the equilibrium = 2tanh(e/2T) and the other components of <f are zero. Via the distribution

1.0 0.5

= 0.05 0.20 0.30 0.418 1.5 1.0 0.5 cl 0

-0.5

-1.0

, , , 1.5

0.5

L = 6.251 s 12.5 18.75 25.0

0.02A0 0.08 0.14 0.20

T=

Fig. 1. (Color online) (a) - Sketch of the system under consideration, (b), (c) - Nonlocal conductance as a function of V for different magnetic fields. Panel (b) corresponds to elastic mechanisms of spin relaxation, L = 1.0. (c) - gnj for the energy relaxation mechanism (see text), L = 12.5. (d) - gnj as a function of V for different L for the energy relaxation mechanism. h = 0.20Ao. For panels (b)-(d) Pi = 0.2. (e) - gnj as a function of V for the normal injector (Pj = 0) and different L; h = 0.20Ao. For panels (b)-(e) T = 0.02Ao. (f) - g„i as a function of

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