Nonequilibrium spin-dependent phenomena in mesoscopic superconductor–normal metal tunnel structures

Nonequilibrium spin-dependent phenomena in mesoscopic superconductor–normal metal tunnel structures

Francesco Giazotto,^1,

*

Fabio Taddei,¹Pino D’Amico,^1,2Rosario Fazio,^1,3 and Fabio Beltram¹

1NEST CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy

2Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

3International School for Advanced Studies (SISSA), I-34014 Trieste, Italy

共Received 1 June 2007; revised manuscript received 27 August 2007; published 20 November 2007兲 We analyze the broad range of spin-dependent nonequilibrium transport properties of hybrid systems com- posed of a normal region tunnel coupled to two superconductors with exchange fields induced by the proximity to thin ferromagnetic layers and highlight its functionalities. By calculating the quasiparticle distribution functions in the normal region, we find that they are spin dependent and strongly sensitive to the relative angle between exchange fields in the two superconductors. The impact of inelastic collisions on their properties is addressed. As a result, the electric current flowing through the system is found to be strongly dependent on the relative angle between exchange fields, giving rise to a huge value of magnetoresistance. Moreover, the current presents a complete spin polarization in a wide range of bias voltages, even in the quasiequilibrium case. In the nonequilibrium limit, we parametrize the distributions with an “effective” temperature, which turns out to be strongly spin dependent, though quite sensitive to inelastic collisions. By tunnel coupling the normal region to an additional superconducting electrode, we show that it is possible to implement a spin-polarized current source of both spin species, depending on the bias voltages applied.

DOI:10.1103/PhysRevB.76.184518 PACS number共s兲: 72.25.⫺b, 85.75.⫺d, 74.50.⫹r, 05.70.Ln

I. INTRODUCTION

Although the interest in nonequilibrium superconductivity dates back to the 1970s,¹ nonequilibrium transport phenom- ena inhybridsuperconducting structures are currently under the spotlight. One of the key experiments that renewed this interest was probably the control of the supercurrent flowing through a Josephson junction, and even the reversal of its sign, accessible by altering the quasiparticle population in the weak link 共see Ref. 2 and references therein兲. Out-of- equilibrium electron population can be realized in mesos- copic conductors subject to a bias voltage in which electrons cannot exchange energy either with one another or with lat- tice phonons, so that their energy distribution is not Fermi-like.³ Quasiequilibrium is reached if electrons can thermalize, while still decoupled from the phonons, so that they can reach a temperature which is different from the one relative to the phonon bath. In ballistic Josephson junctions, supercurrent control occurs by inducing a nonequilibrium population of Andreev levels either by injecting a current through an additional normal terminal connected to the weak link^4,5 or by applying an electromagnetic radiation on the weak link.^6,7 The diffusive long-junction limit was considered^8–11and experimentally realized too.^12,13The con- trol of supercurrent by cooling electrons in the weak link was proposed in Refs.14–17and experimentally realized.¹⁸ It is worthwhile stressing that electron temperature can be low- ered below the phonon temperature, thus realizing electron microrefrigeration,^19,20by exploiting the superconducting en- ergy gap共see Refs. 21and22and references therein兲.

Spin-dependent properties in out-of-equilibrium hybrid systems were investigated in a limited number of articles. In Refs. 23–26, ferromagnet-superconductor-ferromagnet double tunnel junctions were considered in order to study the spin imbalance induced in S by nonequilibrium. In the anti-

ferromagnetic alignment of the magnetizations of the F lay- ers, a strong suppression of superconductivity was found, leading to a large magnetoresistive effect. In Josephson junc- tions, the effect of spin injection²⁷and the presence of weak ferromagnets²⁸were considered, while the effect of the An- dreev reflection on spin accumulation in a ferromagnetic wire was reported in Ref.29. In Ref. 30, the possibility of manipulating magnetism through the interplay of supercon- ductivity and nonequilibrium transport was investigated.

Recently, we have proposed³¹ a hybrid ferromagnet- superconductor共FS兲spin valve whose operation is based on the interplay between out-of-equilibrium quasiparticle dy- namics and proximity-induced exchange coupling in super- conductors. Huge tunnel magnetoresistance values as high as several 10⁶% has been predicted, leading to a fully tunable structure which shows high potential for application in spin- tronics. In this paper, we comprehensively investigate the physics and functionality of the setup analyzed in Ref.31, extending our study to the presence of finite electron- electron interaction and to the quasiequilibrium limit, as well as to the presence of nonidealities in the superconductors. In this setup, a spin-dependent “effective” temperature for the electrons in the N region emerges, thus leading to possible spin-dependent thermoelectric effects.

The paper is organized as follows. In Sec. II, we describe the system under investigation, and in Sec. III, we derive the quasiparticle distribution functions in different regimes. In particular, we consider the nonequilibrium limit in Sec. III A, we include the effect of inelastic collisions in Sec. III B, and we describe the quasiequilibrium regime in Sec. III C. In Sec. IV, we discuss the behavior of the electric current, fo- cusing on the magnetoresistive effects and on the spin- filtering properties of the system in Sec. V. Section VI is devoted to the characterization of the nonequilibrium distri- bution through an “effective” temperature and to the exploi-

1098-0121/2007/76共18兲/184518共12兲 184518-1 ©2007 The American Physical Society

tation of the system as a source of spin-polarized current through the introduction of an additional superconducting electrode. Finally, we draw our conclusions in Sec. VII.

II. SETUP

We consider a device consisting of two identical FS bi- layers共FS1,2兲symmetrically connected to a mesoscopic nor- mal metal region共N兲of lengtht_Nthrough tunnel contacts共I兲 of resistanceR_t. The concentration of impurities is such that quasiparticle transport is diffusive. The resulting system, a FS-I-N-I-SF heterostructure, is shown in Fig.1 in two dis- tinct experimental implementations. Figure 1共a兲 shows a spin-valve-like structure, which consists of a sequence of stacked metallic layers, while Fig.1共b兲displays a planar sys- tem. Although the two implementations are equivalent on theoretical footing, the planar configuration allows the mea- surement of local properties共e.g., the quasiparticle distribu- tion functions as well as the local temperature兲by connecting the N region to additional metallic probes. This will be ad- dressed in Sec. VI. For the sake of simplicity, we assume a symmetric system共a resistance asymmetry would not change the overall physical picture兲, t_F 共tS兲 labels the F 共S兲 layer thickness and a bias voltageVis applied across the structure.

The exchange field in the left ferromagnet 共h1兲 is aligned along thezaxis for the setup in Fig.1共a兲or along theyaxis for the setup in Fig.1共b兲, while that in the right F layer共h2兲 is misaligned by an angle␾关see Figs.1共a

⬘

^兲and1共b

⬘

^{兲兴. For} simplicity, we set兩h1兩=兩h2兩=h. In real structures,h₂ can be rotated by applying an in-plane magnetic field as low as some millitesla. Moreover, we assume that共i兲 the FS inter- face is transparent and 共ii兲 R_t is much larger than both the resistance of the N layer共RN兲and the FS contact resistance.

The first condition ensures that the superconductor is

strongly affected by the proximity of the F layer,³²while the second ensures that all the voltage drop occurs at the tunnel barriers共so that any spatial variation of the chemical poten- tial within the N region can be neglected兲and that each FS bilayer is in local equilibrium.

The electronic properties of a FS bilayer can be analyzed within the quasiclassical Green’s function formalism.³² We are interested in the situation in which the influence of the F layer on the superconductor becomes nonlocal. This occurs in the limitt_S⬍␰S=

冑

_បD/2␲^kBT_candt_F⬍␰F=

冑

_បD/h, where

␰Sand␰Fare the superconducting coherence length and the length of condensate penetration into the ferromagnet, re- spectively. Ddenotes the diffusion coefficient, Tc is the su- perconducting critical temperature, andk_Bis the Boltzmann constant. In this situation, the ferromagnet induces in S a homogeneouseffectiveexchange field共analogous to the one present in magnetic superconductors³²兲through proximity ef- fect and modifies the superconducting gap共⌬兲. The effective values of the exchange field共h^*兲and gap共⌬^*兲are given by³³

⌬^*/⌬=␯St_S共␯St_S+␯Ft_F兲⁻¹,

h^*/h=␯Ft_F共␯St_S+␯Ft_F兲⁻¹, 共1兲 where␯S共␯F兲is the normal-state density of states共DOS兲in S 共F兲. In particular, if␯F=␯Sand fort_FⰆt_S, it follows that

⌬^*/⌬⯝1,

h^*/h⯝t_F/tSⰆ1, 共2兲 i.e., h^*turns out to be much smaller than in an isolated F layer. As a matter of fact,h^*can take values of the order of magnitude of ⌬^*. These conditions can be achieved quite easily in a realistic structure. We assume that the only effect of h^* on the quasiparticles is to lead to a spin-dependent superconducting DOS, i.e., we neglect any influence of the induced magnetic moment on the orbital motion of electrons.

Furthermore, we assume a negligible spin-orbit interaction.³⁴ The superconductor DOS 共N_␴^S兲 thus will be BCS-like but shifted by the effective exchange energy 共equivalent to that of a Zeeman-split superconductor in a magnetic field³⁵兲. By choosing the spin quantization axis along the direction of the exchange field, we have

N_␴^S共␧,h^*兲=1

2

冏

^Re

冋 ^冑

^{共␧+␧␴+h␴*h+*i⌫兲+i⌫2−⌬*2}

册 ^冏

^{, 共3兲}

where␧is the energy measured from the condensate chemi- cal potential,␴= ± 1 refers to the spin parallel 共antiparallel兲 to the direction ofh₁, and⌫ is a smearing parameter.³⁶ The latter allows quasiparticle states within the gap due to inelas- tic scattering in the superconductor³⁷ or inverse proximity effect from the nearby metallic layers. Typical values for⌫ lie in the range⌫⬃1⫻10⁻⁵⌬, . . . , 1⫻10⁻³⌬for Al as a thin- film superconducting electrode.³⁶ In the following calcula- tions, we set⌫= 10⁻⁴⌬^*, unless differently stated.

In order to describe our system, we make use of the tun- neling Hamiltonian approach and neglect proximity effects at NIS interfaces.

F S1 N S2 F

Rt Rt

-V/2 V/2

tN

x z y (a)

tS tF

(a’)z

y h1

h2 φ

S1 N S2

F F

(b)

tN

Rt Rt

(b’)y

x h1

h2 V/2 φ -V/2

FIG. 1. 共Color online兲 Sketch of two possible implementations of the FS-I-N-I-SF structure analyzed in this work.共a兲 Spin-valve setup consisting of a sequence of stacked metallic layers. 共b兲 A planar structure. Ferromagnetic layers共F兲induce in each supercon- ductor, through the proximity effect, an exchange field共h_1,2兲whose relative orientation can be controlled by an externally applied mag- netic field. The F exchange fields are confined共a⬘兲to they-zplane for the setup shown in 共a兲 and 共b⬘^{兲 to the x-y} plane for the setup shown in共b兲and are misaligned by an angle␾. A voltage bias V, applied across the structure, allows to control the energy distributions in the N region. The structure is assumed quasi-one-dimensional.

III. QUASIPARTICLE DISTRIBUTIONS A. Negligible inelastic scattering: Full nonequilibrium limit

At finite bias V and in the limit of negligible inelastic scattering, quasiparticles in the N layer will be out of equi- librium and thus, in general, not distributed according to the Fermi function. The steady-state nonequilibrium distribution functions can be calculated by equating, at each energy value, the tunneling rate of quasiparticle entering the N re- gion from the insulating layer on the left-hand side to the tunneling rate of those exiting through the right-hand-side barrier.³⁸In the general case of noncollinear exchange fields, the spin eigenstates relative to S₂ 共兩↑典 and 兩↓典兲 can be ob- tained by rotating the spin eigenstates relative to S₁共兩⫹典and 兩⫺典兲by the angle␾共representing the misalignment between h₁ and h₂兲. As a consequence, spin up 共␴= + 1 with eigen- state兩⫹典兲quasiparticles exiting the N layer through the right- hand-side barrier will now consist of two contributions. One describes tunneling into spin up 共with eigenstate兩↑典兲quasi- particles, proportional to cos²关␾/2兴, and the other describing tunneling into spin down共with eigenstate兩↓典兲quasiparticles, proportional to sin²关␾/2兴. As a result, the nonequilibrium distribution function in the N layer is spin dependent and can be written as

f_␴共␧,V,h^*,␾兲=N_␴^S1F^S1+关a共␾兲N_␴^S2+b共␾兲N_−␴^S2兴F^S2 N_␴^S1+a共␾兲N_␴^tS2+b共␾兲N_−␴^S2 , 共4兲 where a共␾兲= cos²关␾/2兴, b共␾兲= sin²关␾/2兴, F^S1共S2兲

=f₀共␧±eV/2兲,N_␴^S1=N_␴^S共␧+eV/2兲,N_␴^S2=N_␴^S共␧−eV/2兲, f₀共␧兲 is the Fermi function at bath temperatureT_bath, andeis the electron charge.

Figures2共a兲and 2共b兲 show the nonequilibrium distribu- tion functions关calculated from Eq.共4兲兴for spin up and spin down quasiparticles, respectively, vs energy␧for the parallel configuration共i.e.,␾= 0兲atT_bath= 0.1Tc,h^*= 0.2⌬^*, and dif- ferent values of V 共we assume the superconducting gap to follow the BCS relation⌬^*= 1.764kBTc兲. Figure2shows that by increasing the bias voltage V, spin up and spin down

distributions are shifted in opposite directions on the energy axis, similarly to what is expected in the presence of an effective spin-dependent chemical potential共␮_␴^eff兲. In particu- lar, f₊共␧兲 is shifted toward negative energies, while f₋共␧兲 toward positive energies. Moreover, for eVⲏ⌬^*, the spin- dependent chemical potential saturates at ␮_␴^{eff= −}␴^{h*. As} shown in Ref.30, this effect can be used to electrostatically manipulate the magnetic properties of the N region. The role of a finite⌫共i.e., the presence of quasiparticle states within the gap兲can be appreciated in Fig.2. By increasingeVfrom 0 to⌬^*, the distributions broaden and reflect theheatingof the N region, as discussed in Refs.22and36. This effect is absent for ⌫= 0. By further increasing the bias voltage, the distribution functions sharpen due tocoolingprovided by the superconducting energy gap.²²

Analogously, in Figs.3共a兲and3共b兲, we plot the nonequi- librium distribution functions for spin up and spin down qua- siparticles, respectively, for the antiparallel configuration 共i.e.,␾⁼␲兲. Distribution functions are shown vs energy␧for different values of V and were calculated for the same pa- rameters as in Fig.2. In this case, up and down distributions remain centered around ␧= 0 upon biasing 共equivalently, their effective chemical potential is always␮_␴^eff= 0兲, but at a given bias voltage, the features of the distributions are more pronounced for the spin down case. As we shall see in Sec.

VI, up and down distributions are characterized by different effective electronic temperatures 共T_␴^eff兲. In general, for any angle ␾ differing from 0 or␲, the spin-dependent distribu- tion functions f_␴共␧兲 will be characterized by both an effec- tive chemical potential and an effective electronic tempera- ture.

B. Intermediate inelastic scattering

In the presence of scattering, the approach of Sec. III A cannot be used and one has to resort to the kinetic equation theory. Electrons in metals experience both elastic and in- elastic collisions. The latter drive the system to equilibrium and can be expected to hinder the manifestation of the phe-

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.5 1.0 0.0 0.5

1.0 eV= 0

0.5∆^∗ 1.0∆^∗ 2.0∆^∗ 2.5∆^∗

f+(ε) ^{Tbath= 0.1Tc}_h^∗_{= 0.2∆}^∗

φ= 0 (a)

ε/∆^∗ f−(ε)

(b)

FIG. 2.共Color online兲Spin-dependent quasiparticle distribution functions f_␴共␧兲 in the full nonequilibrium limit vs energy ␧ for several bias voltages at␾= 0,T_bath= 0.1T_c, andh^*= 0.2⌬^*.共a兲f₊共␧兲 and共b兲 f₋共␧兲.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.5 1.0 0.0 0.5

1.0 eV= 0

0.5∆^∗ 1.0∆^∗ 2.0∆^∗ 2.5∆^∗

f+(ε) ^{Tbath= 0.1Tc}_h^∗_{= 0.2∆}^∗

φ=π (a)

ε/∆^∗ f−(ε)

(b)

FIG. 3. 共Color online兲Spin-dependent quasiparticle distribution functions f_␴共␧兲 in the full nonequilibrium limit vs energy ␧ for several bias voltages at␾=␲,T_bath= 0.1T_c, andh^*= 0.2⌬^*.共a兲f₊共␧兲 and共b兲 f₋共␧兲.

nomena discussed in the previous section. At low tempera- tures 共typically below 1 K兲, electron-electron scattering³⁹ and scattering with magnetic impurities^40,41are the dominant sources of inelastic collisions.^3,41,42 Since R_t is, in general, large compared to the wire resistance 关RN=t_N/共N_F^Ne²DA兲兴, whereN_F^N is the N-region DOS at the Fermi energy andA the wire cross section兲, we can assume that f_␴ does not de- pend on the position in the wire.¹⁵

In the following, we shall analyze the role of inelastic electron-electron relaxation on the quasiparticle distribution.

The effect of electron-electron scattering due to the Coulomb interaction on the spin-dependent distributions can be ac- counted for by solving a pair of coupled stationary kinetic equations,

D⳵^2f+共␧兲

⳵^{x2 =Icoll+ 共␧兲,}

D⳵^2f−共␧兲

⳵^{x2 =Icoll− 共␧兲, 共5兲} together with the Kuprianov-Lukichev boundary conditions at the NIS interfaces.⁴³In Eq.共5兲,I_coll^␴ 共␧兲is the net collision rate at energy␧, functional of the distribution functions f_␴, defined by

I_coll^␴ 共␧兲=I_coll^in␴共␧兲−I_coll^out␴共␧兲, 共6兲 where

Icollin␴共␧兲=关1 −f_␴共␧兲兴

冕

^{d␻k共2␻兲f␴共␧−␻兲}

冕

^{dE兵f+共E+␻兲}

⫻关1 −f₊共E兲兴+f₋共E+␻兲关1 −f₋共E兲兴其 共7兲 and

I_coll^out␴共␧兲=f_␴共␧兲

冕

^{d␻k共2␻兲关1 −f␴共␧−␻兲兴}

冕

^{dE兵f+共E兲}

⫻关1 −f₊共E+␻兲兴+f₋共E兲关1 −f₋共E+␻兲兴其. 共8兲 In Eqs.共7兲and共8兲,k共␻兲=␬ee␻^−3/2according to the theory of the screened Coulomb interaction⁴⁴ for a quasi-one- dimensional wire, where␬ee=共␲

冑

2Dប^3/2N_F^NA兲⁻¹.^45,46 By re- writing Eq.共5兲in dimensionless units,¹⁵ the strength of the electron-electron interaction can be expressed as Kcoll

=共Rt/R_N兲共t_N²␬ee/D兲

冑

⌬=共tN/

冑

2兲共Rt/RK兲

冑

⌬/បD, where RK

=h/2e². We note that the strength of the electron-electron interaction turns out to be proportional to the length of the wire as well as to the tunnel barrier resistance.

We solved Eq. 共5兲 with h^*= 0.2⌬^*, eV=⌬^*, and T

= 0.1Tc for several Kcoll values.^44,45 The effect of electron- electron scattering on the quasiparticle distribution functions is displayed in Figs.4and5for␾^{= 0 and}␾⁼␲, respectively.

For the␾= 0 case, by increasing Coulomb interactions, the quasiparticle distributions are forced toward thermal ones still characterized by different chemical potentials for both spin species.³⁰ In the antiparallel configuration 共see Fig. 5兲, the effect of inelastic relaxation is similar, but now the spin- dependent distribution function will coincide for sufficiently largeKcollvalues. It is easy to recognize that, in both cases,

a thermal Fermi-like distribution is reached for Kcoll

of the order of 10. Assuming parameters for a realistic Al/Al₂O₃/Ag SINIS microstructure^21,36共with⌬⯝200␮^eV, D= 0.02 m²s⁻¹, and R_t= 1 k⍀兲, Kcoll= 10 corresponds to a rather long N region,t_N⬇47␮^m.

C. Strong inelastic scattering: Quasiequilibrium limit This is the regime characterized by the fact that the electron-electron interaction is so strong that quasiparticles can reach an equilibrium共Fermi-like兲distribution, while the electron-phonon coupling is negligible.²² Such distributions are characterized by the quasiequilibrium chemical potential and temperature. Since electron-electron interaction occurs between quasiparticles irrespective of their spin 共in the ab- sence of spin-mixing mechanisms兲, the quasiequilibrium temperature 共T^qe兲 will be independent of spin and different from the temperature of the phonon bathT_bath. On the con- trary, since electron-electron interaction redistributes the en- ergy among electrons of a given spin species, in the absence of spin-mixing mechanisms, the quasiequilibrium chemical potential 共␮_␴^qe兲 will depend on the spin. This is a conse-

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.0 0.0 0.0 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 1.0 1.0 1.0

1.0 KKKK_{collcollcollcoll}= 0= 0= 0= 0 0.01 0.01 0.01 0.01 0.1 0.1 0.1 0.1 1.0 1.0 1.0 1.010101010 ffff++++((((εεεε))))

TTT T_{bathbathbathbath}= 0.1= 0.1= 0.1= 0.1TTTT_cccc hhh h^∗∗∗∗= 0.2= 0.2= 0.2= 0.2∆∆∆∆^∗∗∗∗ eV eV eV eV= 1.0= 1.0= 1.0= 1.0∆∆∆∆^∗∗∗∗ φφφ φ= 0= 0= 0= 0 (a) (a) (a) (a)

εεεε////∆∆∆∆^∗∗∗∗ ffff−−−−((((εεεε))))

(b) (b) (b) (b)

FIG. 4. 共Color online兲Spin-dependent quasiparticle distribution functions f_␴共␧兲 vs energy ␧ calculated for severalKcollvalues at

␾= 0,eV=⌬^*,T_bath= 0.1T_c, andh^*= 0.2⌬^*.共a兲 f₊共␧兲and共b兲 f₋共␧兲.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.5 1.0 0.0 0.5

1.0 K_coll= 0

0.01 0.1 1.0 f(ε)+ 10

T_bath= 0.1T_c h^∗= 0.2∆^∗ eV= 1.0∆^∗ φ=π (a)

ε/∆^∗ f−(ε)

(b)

FIG. 5. 共Color online兲Spin-dependent quasiparticle distribution functions f_␴共␧兲 vs energy ␧ calculated for severalKcollvalues at

␾=␲,eV=⌬^*,T_bath= 0.1T_c, andh^*= 0.2⌬^*.共a兲f₊共␧兲and共b兲 f₋共␧兲.

quence of the fact that the number of electrons of a given spin must be conserved. Furthermore, both quasiequilibrium chemical potential and quasiequilibrium temperature will de- pend on␾; therefore, they will be different for parallel and antiparallel configurations.

In the absence of spin-flip mechanisms, the quasiequilib- rium distribution functions can be calculated by imposing the conservation ofparticlecurrents, independently for the two spin species, together with a balance equation for the heat currents. In particular, in the former case, we require that

I_␴^L共V,h^*,␾兲=I_␴^R共V,h^*,␾兲, 共9兲 where

I_␴^L共V,h^*,␾兲= 1

eR_t

冕

^{d␧N␴S1共␧兲关FS1共␧兲−f␴共␧兲兴 共10兲}

and

I_␴^R共V,h^*,␾兲= 1

eR_t

冕

^{d␧关N␴S2共␧兲a共␾兲+N−␴S2共␧兲b共␾兲兴关f␴共␧兲}

−F^S2共␧兲兴 共11兲

are the electric currents flowing through the left共L兲and right 共R兲NIS interface. Note that in contrast to the full nonequi- librium regime where the tunneling rates are set to be equal at each energy, here the conservation involves the total cur- rents since the electron-electron interaction mixes the energy of the electrons. In the absence of electron-phonon coupling, the only contribution to the heat flux is the heat current flow- ingoffthe N region through each NIS interface. The latter is given by

J_␴^L共V,h^*,␾兲= 1

e²R_t

冕

^{d␧␧N␴S1共␧兲关f␴共␧兲−FS1共␧兲兴, 共12兲}

for the left NIS contact, and by J_␴^R共V,h^*,␾兲= 1

e²R_t

冕

^{d␧␧关N␴S2共␧兲a共␾兲+N−␴S2共␧兲b共␾兲兴}

⫻关f_␴共␧兲−F^S2共␧兲兴, 共13兲 for the right contact. The balance equation for the heat flux thus simply reads

兺

_␴ ^{关J␴L共V,h*,␾兲+J␴R共V,h*,␾兲兴= 0. 共14兲}

By assuming that f_␴=f₀共␧−␮_␴^qe,T^qe兲 and solving Eqs. 共9兲 and 共14兲, the temperature and chemical potentials can be easily determined. It turns out that while in the antiparallel alignment spin up and down distributions are equal, in the parallel one, the two spin components have equal effective electronic temperature共though different from the antiparallel alignment兲 but opposite effective chemical potential 共see Figs.4and5for largeKcollvalues兲. Although the quasiequi- librium regime might seem an unrealistic limit, it actually describes the case of a strong electron-electron interaction quite well. Indeed, according to our calculations共Sec. III B兲, quasiequilibrium distributions are already reached for an electron-electron collision strengthKcoll⯝10. In the follow-

ing sections, we shall investigate the impact of quasiequilib- rium on spin-dependent transport properties.

IV. ELECTRIC CURRENT

The transport properties of the FS-I-N-SF structure are determined by the spin-dependent distribution functions f_␴. We note that although a Josephson current can flow through the system, its theoretical description is beyond the scope of the present paper. As a matter of fact, we shall be only con- cerned with the quasiparticle transport. Furthermore, al- though similar results for tunnel magnetoresistance and cur- rent polarization could be obtained in a FS-I-SF structure 共i.e., without the N interlayer兲 and not relying on nonequi- librium, the present system possesses a crucial advantage. In fact, a FS-I-SF structure implies an additional undesired Jo- sephson current, which can be fairly large as compared to the quasiparticle current 共around 1 order of magnitude larger than the quasiparticle current relevant for high tunnel mag- netoresistance and current polarization, see, for example, Ref. 33兲. Such supercurrent could be suppressed, for in- stance, by the application of an additional in-plane magnetic field. This field, however, would largely exceed that required to control the orientation of h^*. By contrast, in the FS-I-N- I-SF system, the supercurrent can be kept extremely small up to a large extent, depending mainly on t_N, on the tunnel barrier transmissivity, and on the N-interlayer material pa- rameters. A simple estimate for the Josephson coupling in our structure reveals that the supercurrent can be from 1 to several orders of magnitude smaller than the quasiparticle current共see, for example, Ref.47兲.

The quasiparticle currentI 共e.g., evaluated at the left in- terface兲is given by

I共V,h^*,␾兲=

兺

␴

I_␴^L共V,h^*,␾兲. 共15兲

Figure 6共a兲 displays the electric current in full nonequilib- rium vs bias voltage V calculated for several angles ␾ ^at h^*= 0.2⌬^*andT= 0.1Tc. A sizable current starts to flow only when the voltageV is such that the DOS is finite for both superconductors in some range of energies. For ␾^{= 0, the} current rises sharply at兩eV兩= 2⌬^*, similarly to the quasipar- ticle current of a SIS junction 共also in the presence of an in-plane magnetic field³⁵兲. In this case, in fact, the DOS of a given spin is shifted by the Zeeman energy in the same di- rection for both superconductors. In contrast, for␾=␲^{, cur-} rent sets off at兩eV兩= 2共⌬^*−h^*兲.

Figure6共b兲shows the nonequilibrium differential conduc- tance

G共V,h^*,␾兲=dI共V,h^*,␾兲

dV , 共16兲

calculated for the same values as in Fig. 6共a兲. Additional features are present at兩eV兩= 2h^*, which are strongly tempera- ture dependent and vanish in the limit T→0 共the zero-bias conductance peak for␾⫽␲ resembles that typical of a SIS junction composed of identical superconductors⁴⁸兲. These are a consequence of the overlapping of the superconducting

DOSs where only thermally activated quasiparticles exist at finite temperature.

All these simply reflects how the spin-dependent DOS in each superconductor contributes to the total quasiparticle current at different V. This can be easily visualized by in- specting Fig.7, which shows idealized finite-temperature ex- change field-split superconducting DOS for parallel spin spe- cies, at different bias voltages V and for the case ␾⁼␲^{. In} this case, the DOS of S₁ is shifted in the opposite direction with respect to that of S₂ 关see Fig.7共a兲for V= 0兴. Then, by biasing the structure, the required voltage for a current to flow is smaller with respect to the ␾= 0 case, i.e., eV

= 2共⌬^*−h^*兲 关see Fig. 7共b兲兴. In the same way, for negative voltages, the current sets off ateV= −2共⌬^*+h^*兲, as shown in Fig. 7共d兲. It is also clear that antiparallel spin species will give rise to features at the opposite bias voltage, therefore explaining the origin of additional feature appearing at eV

= 2共⌬^*+h^*兲. For intermediate values of ␾, features are present at兩eV兩= 2共⌬^*±h^*兲 and at 兩eV兩= 2⌬^*since contribu- tions from both␾^{= 0 and}␾⁼␲configurations are present. Of particular relevance is the voltage interval 2共⌬^*−h^*兲艋兩eV兩 艋2⌬^*. By increasing␾from 0 to␲, the current is enhanced from a vanishingly small value up to a finite value leading to aspin-valveeffect.

It is noteworthy to mention that the nonequilibrium con- dition is essential for the observation of the spin-valve effect.

At equilibrium, the distribution functions in the N layer would be thermal and spin independent.

V. MAGNETORESISTANCE

The spin-valve properties of the FS-I-N-I-SF setup can be evaluated quantitatively by analyzing the tunnel magnetore- sistance ratio共TMR兲, defined as

TMR共V,h^*,␾兲=G共V,h^*,␾兲−G共V,h^*,0兲

G共V,h^*,0兲 . 共17兲 Figure8共a兲displays the absolute value of the nonequilibrium

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00 0.25 0.50 0.75

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 5.5

6 6.5 7 7.5 8

2(∆^∗-h^∗) eRtI/∆∗

eV/∆^∗ φ= 0

π/4 π/2 3π/4 π

T_bath= 0.1T_c h^∗= 0.2∆^∗

(a)

2(∆^∗+h^∗)

-2h^∗ RtG

eV/∆^∗ φ=0

π/4 π/2 3π/4 π T_bath= 0.1T_c h^∗= 0.2∆^∗

(b)

2h^∗ x10^-5

FIG. 6. 共Color online兲 共a兲Nonequilibrium current vs bias volt- ageVfor several angles␾atT_bath= 0.1T_candh^*= 0.2⌬^*.共b兲Non- equilibrium differential conductanceGvsVcalculated for the same values as in共a兲.

0 ∆^∗

-∆^∗ ε

^S1(ε)

eV= -2h^∗∗∗∗

h^∗

^S2(ε)

-∆^∗ 0 ∆^∗ ε h^∗

0 ∆^∗

-∆^∗ ε

^S1(ε)

eV= -2(∆∆∆∆^∗∗∗∗+h^∗∗∗∗) h^∗

^S2(ε)

ε h^∗

0 ∆^∗ -∆^∗ h^∗

0 ∆^∗

-∆^∗ ε

^S1(ε)

eV= 0 ^S2(ε)

-∆^∗ 0 -∆^∗ ε h^∗

h^∗

0 ∆^∗

-∆^∗ ε

^S1(ε)

eV= 2(∆∆∆∆^∗∗∗∗-h^∗∗∗∗) ^S2(ε)

ε h^∗

-∆^∗ 0 ∆^∗

φ = π

(a)

(c) (d)

(b)

FIG. 7. 共Color online兲 Idealized finite-temperature exchange field-split density of statesN^S1,2of S_1,2for parallel spin species, at

␾=␲ and different bias voltagesV. In particular, 共b兲,共c兲, and共d兲 show how features in the tunneling current originate at eV= 2共⌬^*

−h^*兲, eV= −2h^*, and eV= −2共⌬^*+h^*兲, respectively. Antiparallel spin species gives rise to features at opposite voltages. Green- dashed lines represent the superconducting DOS in the absence of the exchange field.

10^-1 10¹ 10³ 10⁵ 10⁷

-3 -2 -1 0 1 2 3

10^-1 10¹ 10³ 10⁵ 10⁷

2(∆^∗+h^∗) φ=π/16

π/8 π/4 π/2 π

|TMR|(%)

T_bath= 0.1T_c h* = 0.2∆*

2h^∗

2(∆^∗-h^∗) (a)

h^∗= 0.4∆^∗ 0.3∆^∗ 0.2∆^∗ 0.1∆^∗

eV/∆^∗

|TMR|(%)

(b) T_bath= 0.1T_c φ=π

FIG. 8. 共Color online兲 共a兲Nonequilibrium tunnel magnetoresis- tance ratio 兩TMR兩 vs V calculated for several angles ␾ at T_bath

= 0.1T_c and h^*= 0.2⌬^*. 共b兲 兩TMR兩vs V for differenth^* values at T= 0.1T_cand␾=␲.

TMR vs bias voltageVcalculated for several angles ␾atT

= 0.1Tcandh^*= 0.2⌬^*. For 2共⌬^*−h^*兲艋兩eV兩艋2⌬^*, the TMR increases monotonically by increasing␾and is maximized at

␾⁼␲where it reaches huge values exceeding 10⁶%. We note that in the limitT= 0 and⌫= 0,兩TMR兩diverges, realizing an ideal full spin-valve effect. The nonequilibrium TMR behav- ior for several exchange field values is shown in Fig.8共b兲, at T= 0.1Tc and␾⁼␲. By decreasing h^*, the maximum TMR value reduces and so does the voltage interval of larger mag- netoresistance. Largerh^*values are thus preferable in order to extend the voltage window for optimized operation and to maximize the TMR.

The spin-filtering properties of this system can be quanti- fied by inspecting the current polarization共P_I兲, defined as

P_I共V,h^*,␾兲=I₊^L共V,h^*,␾兲−I₋^L共V,h^*,␾兲

I₊^L共V,h^*,␾兲+I₋^L共V,h^*,␾兲. 共18兲 The calculated nonequilibrium PI vs V is displayed in Fig.

9共a兲for several ␾^{values, at T= 0.1T}c andh^*= 0.2⌬^*. Upon increasing␾, two intervals of 100% spin-polarized current develop for 2共⌬^*−h^*兲艋兩eV兩艋2⌬^*, extending to wider re- gions关2共⌬^*−h^*兲艋兩eV兩艋2共⌬^*+h^*兲兴as␾^approaches␲^{. For}

␾= 0,PIvanishes like in SIS junctions with an in-plane mag- netic field.³⁵Depending on bias, fully spin-polarized currents of both parallel and antiparallel spin species can be obtained.

The structure can thus also be operated as acontrollablespin filter by changing the orientation ofh₂as well as by varying V. Figure9共b兲shows PI vsVfor severalh^*atT= 0.1Tcand

␾⁼␲. The net effect of increasingh^*is to widen the regions of 100% spin-polarized current.

It is important to discuss the effect of the smearing pa- rameter⌫共which controls the presence of quasiparticle states within the superconducting gap兲 on the magnetoresistance

and current polarizations. As shown in Fig.10共a兲, by increas- ing⌫, the TMR value decreases mostly in the region 2共⌬^*

−h^*兲艋兩eV兩艋2⌬^*, while for other values of V, almost no changes are found apart from some smoothing of sharp fea- tures. In particular, the normal character of transport is strengthened by increasing⌫, which causes a suppression of the large TMR value. The latter, indeed, is a consequence of the presence of the superconducting gap. On the contrary, the impact of⌫onPI, plotted in Fig.10共b兲as a function of the voltage V, is much weaker: the polarization in the range 2共⌬^*−h^*兲艋兩eV兩艋2共⌬^*+h^*兲is almost insensitive to ⌫, be- ing slightly reduced only for⌫ values as large as 10⁻²⌬^*.

TMR values are expected to be marginally affected by the presence of electron-electron relaxation in the N layer. In- deed, as discussed in Secs. III B and III C, Coulomb interac- tion allows quasiparticles to exchange energy 共through in- elastic collisions兲 without coupling the two spin species. As shown in Figs.4 and 5, inelastic scattering leaves the two distributions f₊共␧兲and f₋共␧兲 strongly spin dependent in the parallel configuration, while making them to coincide in the antiparallel configuration, so that both magnetoresistance and current polarizations are expected to be only slightly af- fected. Indeed, TMR is only marginally affected even in the quasiequilibrium regime, as shown in Fig. 11共a兲, where we compare TMR at␾⁼␲, as a function ofV, for the full non- equilibrium and the quasiequilibrium regimes. The effect of energy redistribution characteristic of quasiequilibrium con- sists merely in a smoothing of some of the sharp features present in the nonequilibrium limit. In Fig. 11共b兲, we com- pare the plots of PI at ␾⁼␲ as functions of V for both re- gimes. In particular, quasiequilibrium displays a reduction of polarization for兩V兩⬎2共⌬^*−h^*兲and an increase of polariza- tion for兩V兩⬍2共⌬^*−h^*兲. Nevertheless,PI values as large as 100% can be obtained in the quasiequilibrium limit as well.

-100 -50 0 50 100

-4 -3 -2 -1 0 1 2 3 4

-100 -50 0 50 100

T_bath= 0.1T_c h^∗= 0.2∆^∗

PI(%)

φ = π/16 π/4 π/2 3π/4 φ= 0 π

(a)

h^∗= 0.4∆^∗ 0.3∆^∗ 0.2∆^∗ 0.1∆^∗ 0.01∆^∗

eV/∆^∗ PI(%)

(b)

T_bath= 0.1T_c φ=π

FIG. 9. 共Color online兲 共a兲 Nonequilibrium current polarization P_I vs V calculated for several angles ␾ at T_bath= 0.1T_c and h^*

= 0.2⌬^*. 共b兲 P_I vsV for differenth^* values atT_bath= 0.1T_cand ␾

=␲.

-3 -2 -1 0 1 2 3

-100 -50 0 50 100 10^-1 10¹ 10³ 10⁵ 10⁷ 10⁹

Γ= 10^-6∆^∗ 10^-5∆^∗ 10^-4∆^∗ 10^-3∆^∗ 10^-2∆^∗

|TMR|(%)

T_bath= 0.1T_c h^∗= 0.2∆^∗ φ=π

(a)

Γ= 10^-6∆^∗ 10^-5∆^∗ 10^-4∆^∗ 10^-3∆^∗ 10^-2∆^∗

eV/∆^∗ PI(%)

(b) T_bath= 0.1T_c h^∗= 0.2∆^∗ φ=π

FIG. 10. 共Color online兲 共a兲 Nonequilibrium tunnel magnetore- sistance ratio兩TMR兩vs V calculated for several⌫ values atT_bath

= 0.1T_c, h^*= 0.2⌬^*, and␾=␲. 共b兲 Nonequilibrium P_I vsV calcu- lated for the same values as in共a兲.