Non-equilibrium effects in a Josephson junction coupled to a precessing spin

C. Holmqvist1, S. Teber2 and M. Fogelstrom1

1Department of Microtechnology and Nanoscience - MC2,Chalmers University of Technology, S-412 96 Goteborg, Sweden.

2Laboratoire de Physique Theorique et Hautes Energies,Universite Pierre et Marie Curie, 4 place Jussieu, 75005, Paris, France

(Dated: November 23, 2010)

We present a theoretical study of a Josephson junction consisting of two s-wave superconductingleads coupled over a classical spin. When an external magnetic field is applied, the classical spin willprecess with the Larmor frequency. This magnetically active interface results in a time-dependentboundary condition with different tunneling amplitudes for spin-up and spin-down quasiparticles andwhere the precession produces spin-flip scattering processes. We show that as a result, the Andreevstates develop sidebands and a non-equilibrium population which depend on the precession frequencyand the angle between the classical spin and the external magnetic field. The Andreev states lead toa steady-state Josephson current whose current-phase relation could be used for characterizing theprecessing spin. In addition to the charge transport, a magnetization current is also generated. Thisspin current is time-dependent and its polarization axis rotates with the same precession frequencyas the classical spin.

PACS numbers:

I. INTRODUCTION

Recently, superconducting-ferromagnetic (SF) hybriddevices have received increased attention due to theirpotential as spintronics devices. In spintronics, the spindegree of freedom is employed to create new phenomenawhich could be used to create entirely new devices orbe used in combination with conventional charge-basedelectronics1,2. Information, e.g., can be stored in themagnetization direction of a small ferromagnet and itsstate can be read out by measuring a current througha nano-scaled contact determined by the magnetizationdirection. Nanomagnets such as single molecular mag-nets or magnetic nanoparticles may be suitable buildingblocks for such information storage3,4.

The interest in single molecular magnets been sparkedby their appealingly long relaxation times at lowtemperatures5 and experimental breakthroughs in con-tacting molecules to both superconducting and normalleads has made molecular spintronics a growing field ofresearch. Transport measurements of molecular mag-nets in normal junctions have been made as a meansto characterize the magnetic states6,7. Contacting of C60

molecules8, metallofullerenes9 and carbon nanotubes10,11

to superconducting leads has also been demonstrated.In addition, single molecular magnets have been sug-gested for quantum computing applications12,13 due totheir long relaxation times.

Currents are not only used to read out the state of amagnet but are also used to control the magnetizationdirection. Spin-polarized currents carry angular momen-tum. However, a spin current is not a conserved quantityin a ferromagnet and a spin current oriented in such away that its direction is perpendicular to the interfaceplane between the ferromagnetic layer and the leads maylose some of its spin-angular momentum. The angular

momentum lost by spin-polarized electrons transportedthrough a ferromagnet is transferred to the ferromagnet.This transfer of angular momentum generates a torqueacting on the feromagnet’s magnetization direction. Thisspin-transfer torque mediated by electrical currents wastheoretically investigated by Slonczewski14 and Berger15

who worked out a description for ferromagnet-normalmetal (FN) multilayer structures and showed that spin-transfer torques can lead to precession as well as reversalof the magnetization direction. These theoretical predic-tions were experimentally verified by Tsoi16 and Myers17.Nonequilbrium magnetization dynamics18,19 and spin-transfer torques20 in FNF trilayers coupled to supercon-ducting leads have also been studied.

In this paper, we study the coupling between the mag-netization dynamics of a nanomagnet or single molecular

FIG. 1: (Color online) Two superconducting leads are coupledover a magnetic spin, S(t). The spin precesses with the Lar-mor frequency ωL due to an applied external magnetic fieldH applied at an angle ϑ. Quasiparticles tunneling betweensuperconductors interact with the precessing spin via an ex-change coupling resulting in a dynamical inverse proximityeffect existing within the superconducting coherence lengthξ0 of the junction interface.

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magnet and Josephson currents through a nano-scaledjunction. We consider two superconducting leads cou-pled over a nanomagnet, consisting of e.g. a molecularmagnet or a magnetic nanoparticle as shown in figure 1.The spins of the magnetic molecule or nanoparticle areassumed to be held parallel to each other resulting ina uniform magnetization which can be represented by amacrospin21. If an external magnetic field is applied, thespin of the nanomagnet starts to precess with the Lar-mor frequency. This dynamics changes however when itis coupled to conduction electrons in the leads22,23. Asour starting point, we take the model by Zhu et al.22,23

and extend it to include arbitrary tunneling strengthsleading to a modified quasiparticle spectrum displayingAndreev levels for energies within the superconductinggap, |ε| < ∆(T )24,25. In reference [26], we focused onthe dc Josephson charge current, while here, we focus onthe coupling between the dynamics of the Andreev levelsand the transport properties. The coupling of the twosuperconducting leads over the precessing spin producesan ac spin Josephson current. The difference betweenthe spin currents on the left and right sides of the in-terfaces produces a spin-transfer torque, τ = jsL − j

sR,

shifting the precession frequency of the rotating spin. Atfinite temperatures, there is also a spin current carried byquasiparticles generating a damping of the magnetizationdynamics, the so-called Gilbert damping27–29. A transi-tion of the leads from the normal state into the super-conducting state reduces the Gilbert damping30 since thenumber of quasiparticles is suppressed for temperaturesT < Tc

31. This interplay between the Josephson effectand a single spin may be used for read-out of quantum-spin states32 or for manipulation of Andreev levels in thejunction33.

Furthermore, we find that the ac Josephson spin cur-rent is a result of superconducting spin-triplet correla-tions induced by the spin precession. The appearanceof superconducting spin-triplet correlations in SFS junc-tions has been used to explain the observation of a long-range proximity effect in a number of experiments34–37.Keizer et al. observed a supercurrent through a junc-tion consisting of conventional s-wave superconductorscoupled over a layer of the half-metallic ferromagnetCrO2 much thicker than the decay length of the super-conducting spin-singlet correlations34. Various mecha-nisms for converting the spin-singlet correlations of thesuperconducting leads into spin-triplet correlations thatmay survive within a ferromagnetic layer have beensuggested38–40. Bergeret et al. showed that a local inho-mogeneous magnetization direction at the SF interface issufficient to generate spin-triplet conversion38. In refer-ence [39], it was suggested that the spin-singlet to spin-triplet conversion is due to interface regions with mis-aligned averaged magnetic moments breaking the spin-rotation symmetry of the junction producing spin mix-ing as well as spin-flip processes. A similar trilayerstructure with noncollinear magnetizations resulting ina long-range triplet proximity effect was proposed by

Houzet and Buzdin40. Taking into account the impor-tance of interface composition, Khaire et al.36 devisedSFS junctions consisting of conventional superconductorsand CrO2 in which they had inserted weakly ferromag-netic layers between the superconductors and the halfmetal to produce interface layers with misaligned mag-netization directions. A long-range proximity effect wasobserved in junctions containing the interface layers, butnot in junctions without. Confirmation of Keizer’s re-sults were made by Wang et al.37 who measured a su-percurrent through a crystalline Co nanowire. The Conanowire was a single crystal, but the contacting proce-dure was likely to cause defects at the SF interfaces andthe inhomogeneous magnetic moments needed to createthe spin-triplet correlations. Other experimental verifi-cations of long-range proximity effects includes Holmium(Ho) wires contacted to conventional superconductors35.Ho has a conical ferromagnetic structure whose magne-tization rotates like a helix along the c axis. The ap-pearance of spin-triplet correlations in such junctions andtheir effect on the long-range proximity effect41,42 andspin currents43 have also been studied theoretically. Inthe present problem, the magnetization direction variesin time rather than in space giving rise to time-dependentAndreev level dynamics and a dynamical inverse prox-imity effect in the form of induced time-dependent spin-triplet correlations. Houzet44 studied a related problemin which a Josephson junction consisting a ferromagneticlayer with a precessing magnetization placed between twodiffusive superconductors was predicted to display a long-range triplet proximity effect.

We formulate the problem of two superconduct-ing leads coupled over a nanomagnet in terms ofnonequilibrium Green’s functions. The quasiclassi-cal theory of superconductivity is based on Landau’sFermi liquid theory45,46 and is applicable to bothsuperconducting47–49 and superfluid50 phenomena aswell as inhomogeneous superconductors and nonequilib-rium situations. The quasiclassical theory gives a macro-scopic description where microscopic details are enteredas phenomenological parameters50. Basically, it is an ex-pansion in a small parameter kBT/EF , where EF is theFermi energy, and is suitable for weakly perturbed super-conductors. The perturbations should be weak comparedto the Fermi energy, V EF , and of low frequency,~ω EF . Interfaces and surfaces in superconductingheterostructures or point contacts, on the other hand, arestrong, localized perturbations with strengths compara-ble to the Fermi surface energy50. Within quasiclassicaltheory, interfaces are handled by formulation of boundaryconditions which usually have been expressed as scatter-ing problems, being able to treat spin-independent51–56

as well as spin-dependent, or spin-active, interfaces57–62.In many problems, in particular when an explicit timedependence appears, the T-matrix formulation is moreconvenient58,59,63. This formulation is also well suitedfor studying interfaces with different numbers of trajec-tories on either side as is the case for normal metal/half

3

metal interfaces61,64. The two methods have proved to beequivalent and may be applied both in the limit of cleanand in the limit of diffuse superconductors64. In the lat-ter case, the boundary conditions coincides with thoseof Kuprianov and Lukichev65 and of Nazarov66. For arecent review of quasiclassical theory we refer the readerto reference [67]. In the present problem, the dynamicsof the nanomagnet constitutes a time-dependent spin-active boundary condition for the two superconductorswhich we solve using the T-matrix formulation. First,the transport equations are solved separately to find theclassical trajectories for each lead. Then the T-matrixdescribing the scattering between the leads is used toconnect the trajectories across the time-dependent spin-active interface.

We start by outlining the T-matrix formulation appli-cable to scattering via the precessing magnetic momentin Sec. II. We show that the boundary condition can besolved both in the laboratory frame and a rotating frame.In the latter solution, the explicit time dependence is re-moved by a transformation to a rotating frame renderingthis approach suitable for efficient numerical implemen-tations for studies of transport properties. However, thesolution comes at the cost of introducing an energy shiftof the chemical potentials for the spin-up and spin-downbands. The laboratory frame approach is, on the otherhand, suitable for studying modifications to the super-conducting state although the explicit time dependenceincreases the complexity of the solution. In Sec. III A,we review the results for the Josephson charge current inreference [26] in terms of the laboratory frame descrip-tion. The spin currents are described in Sec. III B, whichis followed in Sec. III C by the induced time-dependentspin-triplet correlations and Andreev-level dynamics giv-ing rise to the spin currents. In Sec. III D, we discussthe back-action of the scattering processes on the mag-netization dynamics while the magnetization induced inthe leads is discussed in section III E. In Sec. IV, weconclude with a summary of our results.

II. MODEL

We consider two superconductors forming a Josephsonjunction over a nanomagnet. The nanomagnet may ei-ther be magnetic nanoparticle or a single-molecule mag-net and we will assume that contact between the leadsand the nanomagnet is made up of a few single quantumchannels. The magnetization of the nanomagnet is putin precession and the resulting contact will constitute atime-dependent spin-active interface (see figure 1). Thenanomagnet together with the two superconducting leadsare described by the total Hamiltonian22,23

H = Hleads +HB +HT . (1)

The left (L) and right (R) leads are s-wave superconduc-tors described by the BCS Hamiltonian

Hleads =∑k,σ

α=L,R

ξkc†α,k,σcα,k,σ+

∑k

α=L,R

[∆αc†α,k,↑c

†α,−k,↓+h.c.]

(2)where the dispersion, ξk = ~2k2/2m− µ, and the chem-ical potential, µ, are assumed to be the same for bothleads. The order parameter of the leads is assumed tem-perature dependent, ∆α = ∆(T )e±iϕ/2. Here ϕ is therelative superconducting phase difference over the junc-tion which we treat as a static variable that is tunable.The nanomagnet is subjected to an external magneticfield modeled as an effective field H acting on the nano-magnet’s magnetic moment, µ. Included in this effectivefield are also any r.f. fields to maintain precession, crystalanisotropy fields and demagnetization effects. The mag-netic moment of the nanomagnet is viewed as a singlespin, or macrospin, which we will treat as a classical en-tity. This macrospin is related to the magnetic momentby µ = −γS where γ is the gyromagnetic ratio. Thespin and the effective magnetic field couple via a Zeemanterm,

HB = −γS ·H. (3)

If the effective field is applied at an angle, ϑ, relative tothe spin, a torque is produced that brings the classicalspin into precession around the direction of the effectivefield. This precession generated by the tilt angle occurswith the Larmor frequency ωL = γH, where the mag-nitude of the external field is H = |H|. Here, we takethe direction of the effective field to be along the ez axisand the angle ϑ is defined as H · S(t) = HS cosϑ. TheLarmor precession is captured by the equation of motion

dS

dt= −γS ×H (4)

where the right-hand side is the torque produced by theeffective field. The magnitude of the spin is constant,S = |S(t)| = const. Consequently, the path traced outby the spin, S(t) = SeS(t), has a time dependence whichlies totally in the direction of the spin, eS(t). In theabsence of any other torques than the effective field, thedirection of the precessing spin may be written as

eS(t) =(

cos(ωLt) sinϑ ex + sin(ωLt) sinϑ ey + cosϑ ez).

(5)Tunneling quasiparticles have the possibility to tunneldirectly between the leads with hopping amplitude V0 orinteract with the precessing spin via an exchange cou-pling of strength VS . These processes are described bythe tunneling Hamiltonian

HT =∑

kσ;k′σ′

c†L,kσVkσ;k′σ′cR,k′σ′ + c†R,k′σ′V†kσ;k′σ′cL,kσ,

(6)where Vkσ;k′σ′ = (V0δσσ′ + VS(S(t) · σ)σσ′)δ(k − k′) andσ = (σx, σy, σz) with σi being the Pauli matrices. The

4

spin-dependent hopping amplitude is

VSS(t) · σ = VSS(

cosϑσz + sinϑ e−iωLtσzσx)

(7)

where the first term is a spin-conserving part with dif-ferent hopping amplitudes for spin-up and spin-downquasiparticles. The second term is time-dependent anddescribes processes where quasiparticles flip their spinswhile exchanging energy ωL with the rotating spin. Thejunction’s transport properties as well as the modifica-tions to the superconducting states of the leads dependon the hopping amplitudes V0, VS as well as the the su-perconducting phase difference, ϕ, the precession of thespin, S(t), and the tilt angle, ϑ, between the effectivefield and the precessing spin.

A. Approach

We formulate the problem using nonequilibriumGreen’s functions in the quasiclassical approximation fol-lowing references [58,64]. The tunneling Hamiltonian (6)provides a time-dependent and spin-active boundary con-dition for the quasiclassical Green’s function and is solvedby a T-matrix equation68–70. A quasiclassical Green’sfunction is a propagator describing quasiparticles movingalong classical trajectories defined by the Fermi velocityvF = vF (pF ) at a given quasiparticle momentum on theFermi surface, pF . The information of a quasiclassicalGreen’s function is contained in the the object

g(pF ,R; ε, t) =

(gR(pF ,R; ε, t) gK(pF ,R; ε, t)

0 gA(pF ,R; ε, t)

)(8)

where R is the spatial coordinate, ε is the quasiparti-cle energy relative to the chemical potential µ and t istime. The g(pF ,R; ε, t) propagator is a 8 × 8 matrix inthe combined Keldysh-Nambu-spin space. The ”check”denotes a 2× 2 matrix in Keldysh space where the com-ponents are retarded (R), advanced (A) and Keldysh (K)Green’s functions while the ”hat” indicates a 2 × 2 ma-trix in Nambu, or particle-hole, space which is furtherparameterized using the Pauli spin matrices (σx, σy, σz)(see reference [50] for details). The matrix components inthe combined Nambu-spin space are conveniently dividedinto spin scalar (s) and spin vector (t) parts,

gX =

(gXs + gXt · σ (fXs + fXt · σ)iσy

iσy(fXs + fX

t · σ) gXs − σy(gXt · σ)σy

)(9)

(X = R,A,K) where fXs and fXt are the spin-singletand spin-triplet components of the anomalous Green’sfunctions (similarly for gXs and gXt ). The Green’sfunction g(pF ,R; ε, t) obeys a Boltzman-like transportequation47–49

i~vF ·∇g + [ετ31− H, g] = 0, (10)

where τi are Pauli matrices in Nambu space and H in-cludes self-energies like the superconducting order pa-rameter, ∆ = ∆1, and impurity contributions, Σimp, as

well as any external fields, hext. The ”” product repre-sents a convolution over common time arguments com-bined with a matrix muliplication50,67. The transportequation is complemented by the Eilenberger normaliza-tion condition,

g g = −π21, (11)

and a set of self-consistency equations such as the one forthe superconducting order parameter,

∆(R, t) = λ

∫ εc

−εc

dε

4πi〈fK(pF ,R; ε, t)〉pF , (12)

where 〈·〉pF is an average over the Fermi surface, λ is thepairing-interaction strength and εc is the cut-off energywhich may be eliminated by making use of the criticaltemperature, Tc.

The transport equation (10) is solved separately foreach lead, treating the interface as an impenetrable sur-face where quasiparticles are perfectly reflected. Thishard wall boundary condition leads to a solution, g0

α,for each semi-infinite lead, α = L,R. The propagatorsare then connected across the interface by the tunnelingHamiltonian, HT , whose effects can be incorporated viaa quasiclassical t-matrix equation as

tα(t, t′) = Γα(t, t′) + [Γαg0α tα](t, t′). (13)

The hopping elements of the tunneling Hamiltonian enterthe t-matrix equation via a matrix ΓL(t, t′) defined as

ΓL(t, t′) = [vg0Rv](t, t′) (14)

for the left side of the interface and the right-side matrixΓR is obtained by interchanging L and R. The timedependence in the current problem enters through thehopping element v(t), which in particle-hole space hasthe form

v(t) =

(v0 + vSeS(t)·σ 0

0 v0 − vSσy(eS(t)·σ)σy

)(15)

and v(t) = v(t)1 in Keldysh space. Here, the hop-ping elements V0, VS in equation (6) have been replacedwith their Fermi-surface limits, v0 = πNFV0 and vS =πNFSVS with NF being the normal density of states atEF . Note that for the junction studied, the hopping ele-

ments have the symmetries vLR(t) = v†LR(t) = vRL(t) ≡v(t). The t matrices (13) are used to calculate the fullquasiclassical propagators which depending on if theirtrajectories lead up to or away from the interface aredivided into ”incoming” (gi) and ”outgoing” (go) propa-gators, given by

gi,oα (t, t′) = g0α(t, t′)+[(g0

α±iπ1)tα(g0α∓iπ1)](t, t′) (16)

where α = L,R and the upper and lower signs, ± and ∓,refer to the incoming and outgoing propagators, respec-tively.

5

The self-energy fields, such as the order parameter, de-pend on the full propagators and should in principle becalculated self-consistenly taking into account the inter-face scattering. However, we assume in this study thatthe area of the point contact, A, is small compared to πξ2

0

where ξ0 = ~|vF |/2πTc is the superconducting coherencelength. As a result, the superconducting state does notchange considerably and the order parameter, ∆, andother possible self-energies in the leads do not have to berecalculated50. We will also assume that the supercon-ducting phase changes abruptly over the contact.

The use of equations (10,11) together with the bound-ary condition (16) allows for calculation of the transportproperties the junction. The charge and spin currentsare given by an average over the Fermi-surface momen-tum directions of the full propagators. Here, this averageamounts to a difference between incoming and outgoingpropagators and the charge, jc, and spin, js, currentsevaluated in lead α for a single conduction channel are

jcα(t) =e

2~

∫dε

8πiTr[τ3(gi,<α (ε, t)− go,<α (ε, t))] (17)

and

jsα(t) =1

4

∫dε

8πiTr[τ3σ(gi,<α (ε, t)− go,<α (ε, t))], (18)

where σ = diag(σ,−σyσσy). The Green’s functions inthe above expressions are the lesser propagators definedas g< = 1

2 (gK − gR + gA).

B. Solving the time-dependent boundary condition

The time-dependent boundary conditions can besolved in two different ways. In the first procedure,the boundary conditions are solved in the laboratoryframe in which the time dependence is preserved andis manifested as frequency shifts in a difference equa-tion. The treatment is similar to that of dc-voltage bi-ased SIS junctions20,58,59,63,71–73. The second approachinvolves removing the explicit time dependence of v(t) bya transformation to a rotating frame (see reference [26]for details). This procedure is numerically more efficientbut the transformation, however, introduces an exchangefield shifting the chemical potentials of the spin-up andspin-down bands in the leads making the first approachmore suitable for studying changes in the superconduct-ing state in vicinity of the junction due to quasiparticletunneling via the precessing spin. Below we describe bothwithin a quasiclassical framework.

1. Laboratory frame

To solve the boundary condition (16) dependent on thematrix (14) in the laboratory frame, it is more convenientto Fourier transform the t-matrix equation from the time

domain to energy space where it becomes an algebraicequation,

tα(ε, ε′) = Γα(ε, ε′)+∑ε′′

Γα(ε, ε′′)g0α(ε

′′)tα(ε

′′, ε′). (19)

The propagators g0α(ε, ε′) = g0

α(ε)δ(ε − ε′) have the fol-lowing Nambu-spin structure,

g0,Rα (ε) =

(gR(ε) fR(ε)iσy

iσy fR(ε) gR(ε)

)= − π

ΩR

(εR ∆(T,±ϕ)iσy

iσy∆∗(T,±ϕ) −εR)

where ΩR =√|∆(T )|2 − (εR)2, εR = ε+ i0+,

g0,Aα (ε) = τ3[g0,R

α (ε)]†τ3, (20)

and

g0,Kα (ε) = (g0,R

α (ε)− g0,Aα (ε)) tanh(ε/2T ).

The gap, ∆(T,±ϕ) = ∆(T )e±iϕ/2, is both tempera-ture and phase dependent and the ”+”(”−”) sign ofthe phase dependence refers to lead R(L). Furthermore,tanh(ε/2T ) is the quasiparticle occupation function set-ting the two superconducting leads in thermal equilib-rium with each other. The t matrices in energy spaceare a sum of t matrices whose energies differ by ωL andsatisfy the relation

tα(ε, ε′) =∑n

tα(ε, ε+ nωL)δ(ε− ε′ + nωL), (21)

which is equivalent to a time dependence of

tα(t, t′) =∑n

e−inωLt∫

dε

2πe−iε(t−t

′)tα(ε+ nωL, ε).

(22)The Ansatz above and the assumption that the leads arein equilibrium so that their respective Green’s functionsmay be written as g0

α(t, t′) = g0α(t− t′) lets one evaluate

the coefficient matrices Γα(ε, ε′) and Γα(ε, ε′)g0α(ε

′) in

equation (19) and subsequently solve the resulting differ-ence equation in terms of tα(ε+nωL, ε)

58,59,63. This is aquite general procedure capable of handling diverse formsof self-energy fields H. However, the matrix coefficientsin equation (19) must be evaluated for each particularkind of junction and lead state. The properties of thesematrices then determine the specific t-matrix differenceequation and the solution strategy.

For the present calculation certain simplifications canbe made due to the spin independence of g0

α(ε) givenby equation (20) and the form of the hopping elementin equation (15): the Keldysh-Nambu-spin matrices canbe factorized in spin space into generalized diagonal ma-trices Xd, spin-raising matrices X↑, and spin-loweringmatrices X↓. These matrices are still Keldysh-Nambumatrices but they have the algebraic properties of spinmatrices such as X↑Y ↑ = X↓Y ↓ = 0, X↓,↑Y ↑,↓ ∝ Zd,

6

and Xd Y ↑,↓ ∝ Z↑,↓. A matrix factorized in this formmay be shown to have the time dependence

X(t, t′) =

∫dε

2πe−iε(t−t

′)

[Xd(ε, ωL) + (23)

+ e−iωLtX↑(ε, ωL) + eiωLtX↓(ε, ωL)

].

Using the spin algebra, the t-matrix equation in energyspace (equation (19)) may be written as 1− Ad1 −B↑1 0

−B↓0 1− Ad0 −B↑00 −B↓−1 1− Ad−1

t↑

td

t↓

=

Γ↑

Γd

Γ↓

. (24)

The coefficient matrices in equation (24) are functionsof energy ε and precession frequency ωL and are straightforward to evaluate. Contrary to e.g. the case of a finitedc voltage, where the t-matrix equation is a differenceequation solved by recursive methods, we have a matrixequation for t which can be solved by simple (numerical)inversion. Factorizing the propagators in equation (16)according to their spin structure results in

gd,i,oα (ε) = g0α(ε) + Md

α,±(ε)tdα(ε, ωL)Mdα,∓(ε) (25)

g↑,i,oα (ε, t) =e−iωLtMdα,±(ε+ωL)t↑α(ε, ωL)Md

α,∓(ε)(26)

g↓,i,oα (ε, t) =e+iωLtMdα,±(ε−ωL)t↓α(ε, ωL)Md

α,∓(ε)(27)

with Mdα,±(ε) = g0

α(ε)± iπ1.

2. Rotating frame

The unitary transformation matrix for removing thetime dependence of v(t) is

U(t) =

(e−i

ωL2 tσz 0

0 eiωL2 tσz

)(28)

resulting in a transformation to a rotating frame of ref-erence with respect to the precessing spin S(t) in whichthe hopping element is

v = U†(t)v(t)U(t) =

(v0 + vSeS ·~σ 0

0 v0 − vSσyeS ·~σσy

).

(29)The direction eS of the precessing spin is now staticin this rotating frame, eS = cosϑez + sinϑex, but thehopping element v is still spin active with different hop-ping amplitudes for spin-up and spin-down quasiparti-cles, v0 ± vS cosϑ. The hopping element also containsa spin-flip term vS sinϑ scattering between the two spinbands.

Next, we apply the unitary transformation to the prop-agator gα(t, t′) and obtain

ˇgα(t, t′) = U†(t) gα(t, t′) U(t′) (30)

= U†(t)∫

dε

2π

dε′

2πe−i(εt−ε

′t′)gα(ε, ε′) U(t′).

(a)

(b)

FIG. 2: (Color online) (a) Schematic illustration of Andreevscattering in a Josephson junction. Electron-like quasipar-ticles (blue) impinging on the right superconductor are An-dreev retroreflected (thick arrows) into hole-like quasiparticles(white) at the superconductor surface. The hole-like quasi-particles are transmitted (thin arrows) to the opposite (left)superconductor, where they are Andreev retroreflected intoelectron-like quasiparticles. This Andreev scattering processis phase coherent and results in the formation of quasipar-ticle states within the superconducting gap, Andreev levels,when the quasiparticles interfere constructively along closeloops. (b) Scattering processes in a Josephson junction cou-pled coupled over a classical spin precessing with frequencyωL. A tunneling quasiparticle has the possibility to tunnelacross the junction with preserved spin and energy, or havingits spin flipped by interaction with the classical spin whilegaining or losing energy ωL. Andreev reflection occurs at thejunction interfaces.

The transformation of the propagators into the rotatingframe introduces spin-dependent energy shifts, ε → ε ±ωL/2, displayed in the transformed propagators, e.g. as

ˆg0,Rα (ε) (31)

=

gRα(ε+ ωL

2 ) 0 0 fRα(ε+ ωL2 )

0 gRα(ε− ωL2 ) −fRα(ε− ωL

2 ) 0

0 fRα(ε− ωL2 ) gRα(ε− ωL

2 ) 0

−fRα(ε+ ωL2 ) 0 0 gRα(ε+ ωL

2 )

for the retarded Green’s function. The advanced andKeldysh propagators are similarly transformed.

Applying U(t) to the transport equation, equation(10), similarly leads to a shift of the energies of the spin-up and spin-down quasiparticles. The resulting transport

7

equation is

i~vF ·∇ˇgα + [(ετ3 + heff ·σz)1− ˇHα, ˇgα] = 0, (32)

where the energy shifts are captured by the appearanceof an effective magnetic field in the leads, heff = ωL

2 ez.Equation (32) is time independent as long as the self-energy fields in Hα do not contain terms which are off-diagonal in spin space corresponding to spin-flip scatter-ing or equal-spin paring order parameters. For ballistic,or clean, s-wave superconducting leads, Hα = ˇ∆(T )αand equation (32) is time independent. The propaga-tors ˇgα(t, t′) obey equation (32) for a specular-scatteringboundary condition58,64. Introduction of the tunnelingprocesses lead to a boundary condition problem which isstill time independent in this rotating frame and the ””product in equation (16) reduces to a matrix multiplica-tion in energy space.

The precessing spin introduces spin mixing of the twospin bands in such a way that spin-up(-down) quasi-particles are scattered or injected into the spin-down(-up) band. This non-equilibrium spin injection, however,leaves the charge current, equation (17), time indepen-dent. The charge current’s steady-state solution followsdirectly from the spin-independent trace over particlesand holes in equation (17) and the fact that the trans-

formation matrix U(t) leaves the diagonal terms of the

propagators gi/o,<α time independent. The spin current,

on the other hand, has components which include the

off-diagonal elements of the propagators gi/o,<α . The off-

diagonal elements, which are proportional to the Paulispin matrices σx and σy, are time dependent since theseare affected by the time dependence of the transforma-tion matrix U(t). Thus, for a finite tilt angle, ϑ 6= 0, thespin current may be time dependent.

III. RESULTS

The precessing spin introduces a new energy scaleand the superconducting correlation functions can be ex-pected to be modified due to the scattering processes theprecessing spin gives rise to. The starting point is thes-wave superconducting leads and their the spin-singletpairing amplitudes, fs ∼ 〈(ψk↑ψ−k↓ − ψ−k↓ψk↑)〉. Whenelectron- and hole-like quasiparticles interfere construc-tively, sharp states inside the superconducting gap calledAndreev states are formed. In regular Josephson junc-tions without magnetically active interfaces, these An-

dreev states come in degenerate pairs ψ↑(↓), which can

be described by the spinors ψ↑(↓) = (ψ↑(↓), ψ†↓(↑))

T . Each

member of the spinor, ψ↑(↓), is subjected to transmissionVLR(RL) and Andreev retroreflection AR(L) and the An-dreev bound states are formed when the processes leadto constructive interference along closed loops schemat-

ically illustrated as ψL↑(↓)(k, ε)VLR−→ ψL↑(↓)(k, ε)

AR−→ψ†R↓(↑)(−k,−ε)

VRL−→ ψ†R↓(↑)(−k,−ε)AL−→ ψL↑(↓)(k, ε). A

schematic picture of the scattering processes is shown inthe upper panel of figure 2. The tunneling processes de-scribed above are captured by the hopping element

vd =

(v0 + vS cosϑσz 0

0 v0 + vS cosϑσz

)(33)

which lets quasiparticles tunnel across the junction withtheir spin directions and energies unaffected but withdifferent hopping amplitude for spin-up and spin-down.This scattering behavior is captured by the matrix vd gR vd (see below). The spin-flip part of the hoppingelement,

v↑(↓) =

(vS sinϑσ+(−) 0

0 vS sinϑσ−(+)

)(34)

flips a spin-down(-up) quasiparticle into a spin-up(-down) quasiparticle while changing its energy byωL(−ωL). Here, we have defined σ+(−) = 1

2 (σx ±iσy). This kind of tunneling creates spin-flip pro-

cesses, e.g. ψL↑(↓)(k, ε)v↓(↑)LR−→ ψL↓(↑)(k−(k+), ε−(ε+))

AR−→

ψ†R↑(↓)(−k−(−k+),−ε−(−ε+))v↑(↓)RL−→ ψ†R↓(↑)(−k,−ε)

AL−→ψL↑(↓)(k, ε), which is a process captured by the elements

v↑/↓g0Rv↓/↑.

Focusing on the left side of the interface and pa-rameterizing the matrix ΓL according to equation (23),leads one to conclude that the spin-preserving compo-nent of ΓL is a combination of the two processes de-scribed above, ΓdL = vdg0

Rvd + v↑g0Rv↓ + v↓g0

Rv↑.There are also mixed tunneling processes where tunnelingwith and without spin flip are combined into the terms

Γ↑L = v↑g0Rvd+ vdg0

Rv↑ and Γ↓L = v↓g0Rvd+ vdg0

Rv↓.These two terms generate a net spin-flip for quasiparti-cles tunneling across the nanomagnet.

The angle ϑ between the spin S and the magnetic fieldH determines the amount of spin-flip scattering rangingfrom zero for parallel alignment to maximum in the caseof S ⊥H. The frequency ωL sets the amount of energyexchanged between a quasiparticle and the rotating spinduring a spin-flip event as indicated in figure 2 (b). First,we will look at the consequences for the density of statesand the charge current due to the scattering caused bythe precessing spin. After that, we will take a closer lookat the effects on the superconducting pair correlationsbefore we turn to the spin scattering states and the spincurrent as well as their implications for the leads.

A. Charge currents

This section reviews some of the results presented inreference [26]. Here, however, the charge current resultsare described in the t-matrix formulation and are in-cluded for completeness.

8

(a) (c)

(b) (d)

FIG. 3: (Color online) Current-phase relation (a) and An-dreev levels (b) for a junction whose hopping elements areparameterized as v0 = cosα and vS = sinα and α is var-ied from 0 (solid black line) to π/2 (solid red line) in stepsof 0.1π/2. The charge current is plotted in units of e∆/~for temperature T → 0. The charge-current kernel jc,<(ε, ϕ)shows the population of the Andreev levels as well as the di-rection in which the populated Andreev levels carry current.In (c), jc,< is plotted for α = 0. The lower Andreev level,which has energies below the Fermi surface, is populated andcarries current in the positive direction (red) for phase dif-ferences 0 < ϕ < π and in the negative direction (blue) forπ < ϕ < 2π. In (d), jc,< is similarly plotted for α = π/2.In this case, the populated Andreev level carries a negativecurrent for 0 < ϕ < π and a positive one for π < ϕ < 2π.The lower gap edge is indicated by the dashed black line.

1. Static spin

For a static spin with precession frequency ωL = 0, thet-matrix equation (24) reduces to

[1− Ad0] td = Γd (35)

which can easily be solved analytically since the spin-up and spin-down bands separate into two sets of equa-tions. Straight-forward calculations of the density ofstates show that Andreev levels form within the super-conducting gap and are located at energies given as

εJ = ±∆

√1−D0 sin2 ϕ

2−DS cos2

ϕ

2(36)

where we have defined D0 = 4v20/(1 + 2(v2

0 + v2S) + (v2

0 −v2S)2) and a spin-transmission coefficient DS = 4v2

S/(1 +2(v2

0 + v2S) + (v2

0 − v2S)2). The charge current density

is related to the energy dispersion and the occupation

function as74,75

jcα =2e

~∂εJ∂ϕ

tanh (εJ2T

), (37)

which, given the energy dispersion in equation (36), isevaluated to

jcα =e

~(D0 −DS)∆ sinϕ√

1−D0 sin2 (ϕ/2)−DS cos2 (ϕ/2)tanh (

εJ2T

).

(38)showing that he critical current is reduced due tothe spin-flip scattering generated by the embeddednanomagnet76.

If only spin-independent tunneling is present and thespin-dependent hopping strength vS → 0, the spin-transmission coefficient DS → 0 while D0 → D =4v2

0/(1 + v20)2 which is the usual transparency of a

Josephson junction. The Andreev levels are now εJ =

±∆√

1−D sin2 ϕ2 and carry a charge current

jcα =e

~D∆ sinϕ√

1−D sin2 (ϕ/2)tanh (

εJ2T

). (39)

Both of these relations are shown as solid black lines infigures 3 (a) and (b). As can be seen in these figures,the junction is in a 0 state corresponding to the junc-tion’s energy being minimized for the phase differenceϕ = 0. Increasing the hopping strength of the spin-dependent tunneling until the spin-dependent tunnelingdominates, vS > v0, causes the junction to shift from be-ing in the 0 state to being a π state, as can be seen in fig-ure 3(a). When spin-dependent tunneling dominates, thejunction’s ground state is such that the coupled super-conductors have an internal phase shift of π, as predictedin reference [77]. Such π states can also be observed injunctions where the spin-active barrier has been extendedto a ferromagnetic region78. In such junctions, the widthof the ferromagnetic layer as well as the strength of theexchange field determine the transport properties.

If, on the other hand, the spin-independent tunnelingis decreased to 0, leaving only spin-dependent tunneling,(v0 = 0, vS = 1), the Andreev levels are shifted by πto give εJ = ±∆

√1−DS cos2 ϕ

2 . The correspondingcharge current is then

jcα = − e~

DS∆ sinϕ√1−DS cos2 (ϕ/2)

tanh (εJ2T

). (40)

The cross-over from the 0 to π state occurs at v0 = vSwhere the Andreev levels are εJ = ±∆/

√3. These An-

dreev levels are independent of the phase difference be-tween the two superconductors leading to a zero Andreevcurrent consistent with equation (37).

In figures 3 (c) and (d), the current kernel for the leftside of the interface jc,<L (ε, ϕ) is plotted for v0 = 1, vS = 0and v0 = 0, vS = 1, respectively. The current ker-nel jc,<L (ε, ϕ) is integrated over energy ε to give the

9

total current for a specified phase difference, jcα(t) =∫dε2π j

c,<α (ε, ϕ). The kernel jc,<(ε, ϕ) indicates which

states, i.e. Andreev levels and continuum states, are par-ticipating in transporting current through the junction.The direction of the current is given by the sign of jc,<.As can be seen in panel (c), the lower of the two Andreevlevels is occupied which is consistent with quasiparticlestates below the Fermi surface being occupied. The sameis true for jc,< in panel (d), although the current kernelhas been shifted by π.

2. π junction with a small tilt angle

For a junction with zero tilt angle, the solution of theboundary condition problem reduces to the static spincase. The t-matrix equation is simply

t0α(t, t′) = Γ0α(t, t′) + [Γ0

αg0α t0α](t, t′) (41)

with Γ0α = Γdα = [vd g0

α vd](t, t′). Since the hoppingelements are time independent, the t-matrix equation canbe transformed into energy space where the solution canbe found as

t0α(ε) = [[1− Γ0αg0

α]−1Γ0α](ε). (42)

The t matrices are then used to calculate the incom-ing and outgoing propagators according to equation (16).The poles of these propagators subsequently give the An-dreev levels, ε0

J , which turn out to be the same as for astatic spin (ωL = 0) with arbitrary tilt angle.

If the classical spin acquires a small tilt angle ϑ = δsuch that sin δ ≈ δ and precesses around the ez axis withfrequency ωL, the spin-dependent hopping element maybe approximated with vSeS(t)·~σ = vS

(σz+δ e−iωLtσzσx

)assuming that v0 = 0. This means that a quasiparti-cle may either tunnel across the junction with its spinand energy conserved, or it may gain or lose energyωL while having its spin reversed as illustrated in fig-ure 2 (b). Then, to first order in δ, the t matricesare tα = t0α + δtα while Γα = Γ0

α + δΓα. Note thatδΓα = e−iωLtδΓ↑α + eiωLtδΓ↓α and has no diagonal time-independent term. Inserting this into the t-matrix equa-tion and disregarding terms of order δ2 and higher givesan equation for δtα as

δtα(t, t′) = [δΓα (1 + g0α t0α)](t, t′) (43)

+[Γ0αg0

αδtα](t, t′).

Since there are no energy shifts in the last term, the t-matrix equation can be written according to equation(24) as 1− Ad1 0 0

0 1− Ad0 00 0 1− Ad−1

δt↑αδtdαδt↓α

=

δΓ′↑α0δΓ′↓α

(44)

where δΓ′↑/↓α = δΓ

↑/↓α (1+gαt0α). The time-independent

change in the t matrix is consequently zero and the total

t matrix is given by tα = t0α + e−iωLtδt↑α + eiωLtδt↓α. Inother words, a small tilt angle adds two time-dependentcomponents to the t matrix. These two components havepoles at ε0

J ± ωL since quasiparticles gain or lose energyωL while being spin flipped. However, these sidebands donot show up in the Andreev levels for the charge currentin this first-order approximation since tdα = t0α. Whenthe tilt angle becomes large enough for higher-order pro-cesses to become important, the sidebands do appear inthe Andreev level spectrum which is shown explicitly infigure 4. Spin-down (spin-up) quasiparticles in the stateε0J are scattered into the upper (lower) sideband, which

has an opposite spin direction and the quasiparticles ex-change the energy ωL in this spin-flip process with theprecessing spin. A cross-section of the spin-resolved den-sity of states for ϕ = 0 is shown in figure 4(a).

In addition to the sidebands, the scattering processesalso lead to a nonequilibrium population of states. Thequasiparticles occupying the lower Andreev level ε0

J arescattered into states with energies larger than the Fermienergy, leading to an occupation of states above theFermi surface. Quasiparticles are also scattered into thecontinuum states below the gap edge. The continuumscattering leads to a reduced life-time for the quasipar-ticles in the state ε0

J and can be seen as a broadeningof the state for energies ε0

J ≤ −∆ + ωL∆. Despite thetime-dependent dynamics of the rotating spin and thenonequilibrium population of states, the charge currentis still time independent consistent with reference [22].

3. π junction with arbitrary tilt angle

As the tilt angle increases, scattering into the side-bands increases. The spin-degenerate state ε0

J now splitsup into a spin-up and a spin-down state and the spin-flipscattering still occurs between states separated by energyωL. The population of the sharp states, the Andreevlevels, can be understood on the basis of the spin-flipscattering processes. As before, the spin-flip scatteringconnects two states in opposite spin-bands separated byenergy ωL. This connection results in a population ofthe upper state under the condition that the lower stateis occupied. In other words, if the spin-down propaga-tor is occupied, the spin-up propagator is occupied aswell. Vice versa, if the spin-down propagator is not occu-pied, the spin-up propagator is not occupied either. Thisnon-equilibrium occupation is an effect of the precessingclassical spin which scatters between the two spin states.The fine details of this are shown in figure 4 where weplot jc,<(ε;ϕ) for some values of the tilt angle ϑ.

This nonequilibrium occupation of states generates acurrent-phase relation quite different from the non-spin-flip current-phase relation as shown in figure 5. In thelaboratory frame, the tilt angle ϑ between the precess-ing spin and the external magnetic field determines thesplitting of the Andreev levels. The splitting in turn de-termines the current-phase relation and the locations of

10

(a)

(b)

(c)

FIG. 4: (Color online) The charge current-phase relation,jc(ϕ), charge current kernel, jc,<(ε, ϕ), and density of statesare plotted for a π junction (v0 = 0, vS = 1) with tilt angle (a)ϑ = 0.1π/2, (b) ϑ = π/4 and (c) ϑ = π/2. The temperatureof the leads is put to 10−5∆. The spectral charge current,jc,<(ε, ϕ), shows the Andreev levels and their population. Atsome phase differences ϕc < ϕ < 2π− ϕc, scattering betweenthe Andreev levels and continuum states, which are indicatedwith the black dashed line, cause a broadening of the Andreevlevels. The charge current, jc(ϕ) (plotted in units of e∆/~),is the energy-integrated spectral current and displays abruptjumps at phase differences where Andreev levels become pop-ulated/unpopulated. The density of states (DoS) on the rightsides are plotted for the phase difference ϕ = 0 and shows thesplitting of the spin-up and spin-down Andreev levels as wellas the scattering of the continuum levels into the gap. Theprecessing spin has a frequency of ωL = 0.5∆ in all plots.

(a) (b)

(c) (d)

FIG. 5: (Color online) (a) Current-phase relation for junctionwith (v0 = 0, vS = 1), ω = 0.5∆ and ϑ = π/4. (b) Current-phase relations for (v0 = 0, vS = 1), ωL = 0.5∆, and ϑ variesbetween 0 and π/4. (c) ω = 0.5∆, ϑ = π/4, v0 = 0, and vSvaries between 1.0 and 0.3 (d) (v0 = 0, vS = 1), ϑ = π/4 andωL varies between 0.00∆ and 2.50∆. As seen sharp step-likefeatures in the current-phase relation may be washed out byeither having a limited transparency vS 1 or by having alarge frequency ωL & ∆. In all four panels the temperatureis set to zero and the currents are plotted in units of e∆/~.

the abrupt jumps as can be seen in panel 5 (b). If theclassical spin is aligned with the magnetic field, thereare no spin-flip scattering processes and the only tunnel-ing processes taking place are the usual ones where thequasiparticles’ energies are conserved (as in figure 2 (a)).Hence there is no splitting of Andreev levels and there areno abrupt jumps in the current-phase relation. If, on theother hand, the classical spin precesses in the plane, thereare only spin-flip scattering processes present. Hence, theAndreev levels split up into a single spin-up and a spin-down levels. Since each spin band only has one Andreevlevel, there is only one jump in the current-phase relation.

When the spin-dependent hopping strength, vs, is de-creased, the abrupt jumps in the current-phase relationdisappear as can be seen in panel 5 (c). The reason is thatthe Andreev levels are closer to the gap edges for lowertransparencies. If they are close enough they even mergewith the continuum states that have been scattered intothe gap. Similar effects are created when the precessionfrequency ωL is increased. The continuum states are scat-tered further into the gap, removing the sharp Andreevlevels. This modification of the Andreev levels causes thejumps in the current-phase relation to be smoothed out(panel 5 (d)).

11

FIG. 6: (Color online) Scattering processes in a Josephsonjunction coupled to a spin precessing with tilt angle ϑ = π/2and v0 = 0. The tunneling processes are such that all quasi-particles traversing the junction reserve their spins while gain-ing or losing energy ωL.

4. π junction with tilt angle ϑ = π/2

In the limit ϑ → π/2 in which the classical spin pre-cesses in the plane, the only spin-dependent tunnelingpresent generates spin-flip scattering processes causingquasiparticles gain or lose energy ωL as shown in figure6 (see also references79,80). Further assuming that thespin-independent tunneling is negligible, i.e. v0 = 0, thehopping element is

v = e−iωLtv↑ + eiωLtv↓ (45)

where

v↑ = vS sinϑ

(σ+ 00 σ−

)(46)

and

v↓ = vS sinϑ

(σ− 00 σ+

)which leads to a Γα where only the diagonal term isnonzero and has the form

Γdα = [v↑g0R/Lv

↓](t, t′) + [v↓g0R/Lv

↑](t, t′). (47)

Carrying out the convolutions, we find that [v↑ g0R

v↓](t, t′) is associated with the spin-up quasiparticleswhile [v↓g0

Rv↑](t, t′) is related to the spin-down quasi-

particles. These Γα matrices result in a t-matrix equationgiven as 1− Ad1 0 0

0 1− Ad0 00 0 1− Ad−1

t↑αtdαt↓α

=

0Γdα0

(48)

which immediately results in the only nonzero t-matrixterm being tdα. These two properties ensure that we candivide the t-matrix equation into two sets of equations,one for each spin band, and Fourier transform the equa-tions into energy space. For the left side, we find

tL↑(ε) =

[1

|vs|2g−1R↓,∞(ε− ωL)− gL↑,∞(ε)

]−1

. (49)

The corresponding t matrix for the right side is foundby interchanging L and R. The spin-down componentis given by substituting ↑↔↓ and reversing the sign ofthe energy shift, −ωL → +ωL. The form of the t matrixindicates that a Green’s function at a given energy ε onone side of the interface is connected to Green’s functionsat energies ε± ωL on the other side of the interface.

As before, the current is carried by sharp Andreev lev-els inside the gap as well as continuum states in regions±ωL around the gap edges ±∆ as shown in figure 4 (c).The Andreev levels are given by

εJ,↑ =ω

2±∆

√1 + f(ω, ϕ, vs) (50)

εJ,↓ = −ω2±∆

√1 + f(ω, ϕ, vs)

where

f(ω, ϕ, vs) =8v4s

(1− v4s)2

cos2 ϕ

2+( ω

2∆

)2

(51)

− (1 + v4s)2

(1− v4s)2

√4v4s

(1 + v4s)2

(1 + cosϕ)2 + 4

(1− v4

s

1 + v4s

)2 ( ω

2∆

)2

.

In the laboratory frame, the number of Andreev levelspresent when ϑ = π/2 is thus half the number of statesin junctions with 0 < ϑ < π/2. The states lie sym-metrically around ±ωL/2 and disperse with the phasedifference ϕ until they touch the gap edges and mergewith the continuum at ϕc(ωL). The sharp states reap-pear again at 2π − ϕc(ωL). The population of the statesis indicated in figure 4 (c) showing the current kerneljc,<(ε, ϕ). Since the number of states is reduced, two forthe incoming propagator and two for the outgoing prop-agator, there is only one abrupt jump, instead of two, inthe current-phase relation for ϑ = π/2 shown in figure 5(b).

5. Tunnel limit

For a low transparency junction with v0, vS 1, wemay use the first-order approximation tα(t, t′) = Γα(t, t′)and obtain an analytic expression for the charge current.The charge current between the two spin-singlet super-conductors is time-independent and is given by the diag-onal hopping element tdL = ΓdL. In the absence of an ap-plied voltage bias, the current is given by the anomalousGreen’s functions and has the form (at zero temperature)

jc =

e∆2~((D0 −DS cos2 ϑ)− 2

πDS sin2 ϑK(ωL2∆

))sinϕ,

forωL < 2∆,e∆2~

((D0 −DS cos2 ϑ)− 4∆

πωLDS sin2 ϑK

(ωL2∆

))sinϕ,

forωL > 2∆,(52)

where K is the complete elliptic integral of the first kind.For an alternative derivation, see reference [26]. The firstterm is due to the processes which do not flip the spin

12

of the tunneling electrons, vd g0R vd. The part of the

current depending on the precession frequency is entirelydue to the spin-flip processes described by v↓g0

Rv↑ andv↓ g0

R v↑. Equation (52) shows that also in the limitof low transparency but arbitrary precession frequency,the critical current is reduced by spin-flip scattering76

and even reversed in junctions dominated by spin-flipscattering77.

B. Spin currents

The charge and spin currents in equations (17) and(18) are evaluated as the difference between the incom-ing and outgoing propagators in each lead α. Defininga current matrix as j<α (ε, t) = gi,<α (ε, t) − go,<α (ε, t) andusing the expressions for the incoming and outgoing prop-agators, equation (16), it is easily found that the currentmatrix is given by the commutator

j<α (ε, t) = 2πi[tα(ε, t), g0α(ε)]< . (53)

The t-matrices have the spin properties of equation (23)and can, without any loss of generality, be divided intoa scalar singlet part tsα = tdα − tzα and a spin-vectortriplet part ttα(ε, ωL) = tzα(ε, ωL) + e−iωLtt↑α(ε, ωL) +eiωLtt↓α(ε, ωL) such that

tα(t, t′) =

∫dε

2πe−iε(t−t

′)

[tsα(ε, ωL) + ttα(ε, ωL)

].

The scalar and spin-vector parts of the t matrix inNambu-spin space are, e.g. for the retarded (R) matrixcomponent,

ts,Rα =

(γs,Rα φs,Rα iσy

iσyφs,Rα γs,Rα

)(54)

and

tt,Rα =

(γt,Rα (φt,Rα )iσy

iσy(φt,Rα ) −σy(γt,Rα )σy

)(55)

where γt,Rα /φt,Rα and γt,Rα /φt,Rα are 2× 2 matrices in spinspace. The spin-triplet t matrix can be parameterizedsimilarly to the Green’s function in equation (9),

tt,Rα =

(γt,Rα · σ (φt,Rα · σ)iσy

iσy(φt,R

α · σ) −σy(γt,Rα · σ)σy

), (56)

although the parameterization using the basis (z, ↑, ↓),e.g. γt,Rα = γt,Rα,zσz + e−iωtγt,Rα,↑σ+ + eiωtγt,Rα,↓σ−, has amore straight-forward time dependence.

From the current matrix, the division of the t matrixinto scalar and triplet components, and the fact that g0

α

is a scalar, it is clear that the spin current is due to thetriplet components of the t matrix

jsα(t) =1

4

∫dε

8πi2πiTr[τ3σ[ttα(ε, t), g0

α(ε)]< ] (57)

since ∫d

2πεTr[τ3σ[tsα(ε, t), g0

α(ε)]< ] = 0. (58)

As a passing note, the charge current is due to the scalarcomponent of the t matrix,

jcα(t) =e

2~

∫dε

8πi2πiTr[τ3[tsα(ε, t), g0

α(ε)]< ], (59)

since ∫d

2πεTr[τ3[ttα(ε, t), g0

α(ε)]< ] = 0. (60)

1. Small tilt angle, ϑ π/2

Contrary to the charge current case, there are first-order contributions to the spin current if the tilt angle isassumed to be small, i.e. sinϑ ≈ ϑ. In section III A 2,it was found that for a small angle ϑ, the change in thet matrix δt compared to the t matrix for zero tilt angle,t0α, i.e. tα = t0α + δtα, was given by δtα = e−iωLtδt↑α +eiωLtδt↓α. Hence, the triplet t matrix is ttα = δtα giving aspin current matrix of

jt,<α = 2πi[δttα, g0α]< . (61)

The z component of the spin current is zero since δtdα = 0(as was found in section III A 2). Instead, the spin currentconsists of the two components

jsα(t) = e−iωLtδj↑α + eiωLtδj↓α (62)

scaling linearly with ϑ π/2 and where

δj↑/↓α =1

4

∫dε

8πi2πiTr[τ3σ[δt↑/↓α (ε, t), g0

α(ε)]< ]. (63)

The upper two panels in figure 7 show the current kernelsj↑,<(ε, ϕ) and j↓,<(ε, ϕ) and the resulting spin-current-phase relation for a small tilt angle, ϑ = 0.1π/2. Thetwo scattering states carry spin currents in opposite di-rections as opposed to the charge current Andreev levelswhich carry a charge in the same direction, see figure 4(a).

2. Arbitrary tilt angle

Increasing the tilt angle modifies only the magnitudeof the spin current. The scattering into the sidebandsincreases as seen in figure 7 (c) and (d) while the zcomponent of the spin current remains zero. Conse-quently, the spin polarization still rotates with the pre-cession frequency ωL and can be written as jα(t) =

j↑αe−iωLt + j↓αeiωLt where j↑/↓α takes the form

j↑/↓α (t) =1

4

∫dε

8πi2πiTr[τ3σ[t↑/↓α (ε, t), g0

α(ε)]< ]. (64)

13

(a) (b)

(c) (d)

FIG. 7: (Color online) The spin current kernel, (a) j↑,< and(b) j↓,<, for a π junction (v0 = 0,DS = 1) with tilt angle ϑ =0.1π/2. Panels (c) and (d) show j↑,< and j↓,<, respectively,for tilt angle ϑ = π/4. The precessing spin has a frequencyof ωL = 0.5∆ and the temperature T = 0. The resultingspin-current-phase relation is plotted in units of ∆ on top ofeach panel. As seen the phase-dependent population of thescattering states makes the spin-current an almost steplikefunction of phase as was the case for the charge-current-phaserelation shown in figure 4.

Similarly to the charge carrying density of states (fig-ure 4), the scattering into the continuum states increaseswith increasing tilt angle. Figure 8 shows the integratedspin currents, i.e. the current-phase relations for the ↑and ↓ components of the spin currents for a transparentjunction with v0 = 0 and DS = 1.0. The abrupt jumpsin the spin-current-phase relations result from the loss orgain of population of spin scattering states as the phasedifference ϕ changes shown in figure 7.

Taking a closer look at the triplet commutator and us-ing equation (55), the spin-current matrix is (suppressingthe index α for the leads)

jt,< = 2πi[tt, g0]<. (65)

Dividing the retarded t matrix into an anomalous t ma-trix,

φt,RA =

(0 (φt,RA )iσy

iσy(φt,RA ) 0

), (66)

and ”normal” t matrix,

γt,RN =

(γt,RN 0

0 −σy(γt,RN )σy

), (67)

(similarly for the Keldysh and advanced components),

(a) (b)

FIG. 8: (Color online) Current-phase relations for the spin-current components (j↑, j↓) for v0 = 0 and DS = 1.0. Thetime-independent jz-component of the spin current is zero forall cases studied (full black line). The precession frequency isω = 0.2∆, the tilt angle ϑ = π/4 and the temperature T = 0.The spin currents are plotted in units of ∆.

and doing the same for the lead Green’s function g0

f0,RA =

(0 (f0

A)iσyiσy(f0

A) 0

)(68)

g0N =

(g0N 00 −σy(g0

N )σy

),

the spin-current matrix is given by

jt,< = 2πi[γtN , g0N ]< + 2πi[φtA, f

0A]< (69)

+ 2πi[γtN , f0A]< + 2πi[φtA, g

0N ]<

where the first term is a contribution from the normalGreen’s functions and the second term is given by theanomalous Green’s functions. The last two terms do notcontribute to the current since their nonzero elements areoff-diagonal in Nambu-spin space. We will separate thespin-current into two contributions, one from the normalparts of Green’s function and t matrix (jsN ) and one fromthe anomalous parts (jsA). Thus we write

jsN =1

4

∫dε

8πi2πiTr[τ3σ[γtN (ε, t), g0

N (ε)]< ] (70)

and

jsA =1

4

∫dε

8πi2πiTr[τ3σ[φtA(ε, t), f0

A(ε)]< ].

The spin current contains contributions from both as canbe seen in figure 9 where jsN and jsA have been plottedas functions of the phase difference in panels (a) and (b).In panel (c), the normal and anomalous contributions tothe charge current have been plotted for comparison. Asthe figure shows, the charge current is completely givenby the corresponding anomalous contributions.

14

(a) (b) (c)

FIG. 9: (Color online) Spin-current-phase relations for (a)the normal spin current, jN , (b) the anomalous spin current,jA, and (c) the normal and anomalous charge currents, jcNand jcA, for v0 = 0, precession frequency ω = 0.2∆, tilt angleϑ = π/4 and temperature T = 0. The spin currents areplotted in units of ∆ and the charge currents in units of e∆/~.All currents are scaled by DS .

The spin current can be expressed in terms of the di-rection of the rotating spin, S, as

js(t) =1

SDSβH cosϑ (γH)× S(t) (71)

− 1

S

√D0DSβ⊥S⊥(t),

where D0 is defined as in section III A 1. The compo-nents βH and β⊥ are plotted in figure 10. β⊥ is finiteonly when we have a mixed scattering, i.e. for v0 > 0.Moreover, β⊥ depends quadratically on the precessionfrequency and has a sinϕ-like dependence on phase andis zero for both 0 and π junctions when in their zero-current or ground state. For high hopping-dependentstrengths, DS = 1 and v0 = 0, the βH component hasa current-phase relation with abrupt jumps which aresmoothed out for DS < 1 and v0 > 0. In contrast to β⊥,the component βH depends linearly on ωL for small tointermediate precession frequencies, ωL . ∆/2.

3. Tunnel limit

In the tunnel limit, the triplet t matrix is given byttα = e−iωLtΓ↑α + eiωLtΓ↓α + Γzα. The normal part of thespin current is given by the normal parts of the t matrix

as in equation (70), such that jsN = e−iωLtj↑N+e−iωLtj↓N .Performing the integral in equation (64) assuming thatωL ∆ results in

j↑N ≈1

4πv2S sinϑ cosϑ

4πi∆− 8i∆E

(√3ωL2∆

)(72)

(a) (b) (c)

FIG. 10: (Color online) The phase relation of the functionsβH and β⊥ at zero temperature for various realizations of thetunneling strengths. The precession frequency is ω = 0.2∆and the tilt angle is ϑ = π/4. The part of spin current that ro-tates with the spin precession ∼ βH is always non-zero with aphase-dependent modulation around a mean amplitude. Notethe different frequency scalings of βH(∼ ωL) and β⊥(∼ ω2

L).

where E(x) is a complete elliptic integral of the second

kind. A Taylor expansion to second order gives j↑N ≈v2S sinϑ cosϑ

3iω2L

16∆ . Similar integrations yield j↓N = −j↑Nand jz = 0. The normal spin current at zero temperatureis then

jsN (t) =1

Sv2S

3ωL16∆

cosϑ (γH)× S(t). (73)

Similarly, the anomalous integrals produce

j↑/↓A =

1

4π

(−iv0vS sinϑ sinϕ∓ v2

S sinϑ cosϑ cosϕ)

×

8i∆K(ωL

2∆

)− 4πi∆

(74)

where K(x) is a complete elliptic integral of the firstkind. In the limit ωL ∆, the ωL dependence reducesto 8i∆K

(ωL2∆

)− 4πi∆ ≈ πi∆(ωL/2∆)2. Also for the

anomalous spin current, jz = 0 and the total anomalousspin current is

jsA(t) = − 1

Sv2S

ωL cosϕ

16∆cosϑ (γH)× S(t) (75)

− 1

Sv0vS

ω2L sinϕ

16∆S⊥(t). (76)

To summarize, the total spin current is

js(t) =1

SDSβH cosϑ (γH)× S(t)− 1

S

√D0DSβ⊥S⊥(t)

(77)

where βH = 164∆ [3 − cosϕ]ωL, β⊥ = sinϕ

64∆ ω2L and the

transmission coefficients reduce to DS ≈ 4v2S and D0 ≈

4v20 for v0, vS 1.

15

(a) (b) (c)

FIG. 11: (Color online) Temperature dependence of the func-tions α, βH and β⊥ for precession frequency ωL = 0.2∆T=0

and tilt angle ϑ = π/4. The functions are plotted withphase difference ϕ = π or ϕ = 0 depending on if the junc-tion is in a π state or a 0 state. Note the strong responseat low temperatures captured in βH . At these tempera-tures the quasiparticle-caused Gilbert damping is frozen out,α(T → 0) = 0, and the Andreev-level dynamics alone willcouple back to the precessing spin. The abrupt jumps atT . Tc are when ωL = ∆(T )/2 and the spin scatteringconnects the square-root singularities in the superconductingdensity of states at ε = ∆(T ).

4. Temperature dependence

The Josephson spin current described above is an effectof quasiparticles interfering constructively along closedloops when the quasiparticles are subjected to Andreevreflection and transmission across the junction as spec-ified by the rotating classical spin. The singlet-pairingnature of the leads does not support spin currents. Thismeans that the spin-dynamics of the Andreev levels willbe restricted to a small volume (or area) with radius givenby the coherence length in vicinity of the nanomagnetjunction.

However, there is another source of spin current notdependent on the Andreev scattering proceses. At tem-peratures T > 0, the normal contribution to the spincurrent, jsN given by equation (70), results in an extraterm

jsqp =1

S2DSαS(t)× S(t) (78)

which is a spin current carried by quasiparticles and istherefore independent of the superconducting phase dif-ference ϕ. The parameter α depends in general on theprecession frequency ωL and the temperature T . Thiscontribution to the spin current, jsqp, may have a finitespin-polarized dc component along the z axis. This situ-ation is similar to a ferromagnetic quantum dot or layercoupled to normally conducting leads where a precessionof the magnetization leads to the junction behaving as a

spin pump28,81. In the tunnel limit, it can be shown thatthe parameter α = 1/2π.

In figure 11, the temperature dependence of the func-tion α(T, ωL) related to quasiparticles is plotted as well asthe function βH(T, ωL) which is related to the Andreev-level dynamics. At temperatures above the critical tem-perature, T > Tc, the spin current is completely given bythe quasiparticle spin current jsqp and α = 1/2π. As thetemperature decreases, the normal quasiparticles freezeout as the superconducting gap opens and α → 0 asT/Tc → 0. The functions α(T, ωL) and βH(T, ωL) de-pend on the hopping amplitudes vS , v0 as well as on thejunction state, i.e. the superconducting phase difference.For a π junction, whose tunneling is dominated by vS ,α(T, ωL) and βH(T, ωL) are only weakly dependent on vSand v0. The function β⊥(T, ωL) has a sinusoidal ϕ de-pendence and since it is zero for both π and 0 junctionsit is not shown in figure 11. The parameter βH(T, ωL)is zero above the critical temperature and increases asT/Tc → 0, saturating at ∼ π

8ωLαT>Tc . For 0 junc-tions, where v0 < vS , the reduction of α(T, ωL) is slower.At the same time, βH(T, ωL) saturates at a lower value,∼ π

16ωLαT>Tc . The parameter α(T, ωL) does not dependon the precession frequency for ωL . ∆/2.

C. Triplet correlations

In section III B, it was shown that a rotating classi-cal spin inside a phase-biased Josephson junction pro-duces a time-dependent spin current. This is somewhatsurprising since the Josephson junction was assumed toconsist of two superconducting leads with s-wave symme-try. The Andreev processes depicted in figure 2 (b) pro-duced by the rotating spin lead to new spin-pairing corre-lations which are formed when these scattering processesresult in positive interference along closed loops. Theadditional pairing correlations to the usual spin-singletones ∼ 1

2 〈ψ↑ψ↓ − ψ↓ψ↑〉 are the spin-triplet components12 〈ψ↑ψ↓+ψ↓ψ↑〉, 〈ψ↑ψ↑〉 and 〈ψ↓ψ↓〉. The induced tripletcorrelations can be expected to form near the junctiondue to the spin mixing and locally broken spin-rotationsymmetry provided by S(t)39,44. A similar situation ex-ists in SFS junctions with conical ferromagnets; spin cur-rents arise due to spin-triplet correlations induced by ahelical rotation of the magnetization direction in the fer-romagnetic layer43. The spin-triplet correlations inducedby the rotating spin are localized near the junction andare evanescent on length scales on the order of the super-conducting coherence length, ξ0 = ~vf/2πTc. The for-mation of triplet Cooper pairs depends on the details ofthe scattering off and the tunneling over the precessingspin, such as the precession frequency, ωL, the tilt an-gle, ϑ, and the relative amplitude of hopping strenghts,v0, vS . It also depends on the leads through the super-conducting phase difference ϕ, and the temperature, T .

We now want to quantify the pairing correlations gen-

16

FIG. 12: The components of the d vector, (d↑,d↓,dz), as functions of temperature. The angle ϑ between H and S is π4

and theprecession frequency is 0.2∆T=0. In each panel, the spin-independent hopping amplitude vo is fixed while the spin-transmissioncoefficient Ds is varied. The components are scaled by DsωL. In the left panel all junctions are π-junctions while in the centerand right panel junctions with Ds = 0.1 and Ds = 0.1, 0.2 respectively are 0-junctions. For the 0-junctions the correspondingd-vector measures even-in k odd-in ε spin-triplet correlations

erated in vicinity of the precessing spin. Turning firstto the spin-singlet components we extract the anoma-lous Green’s functions are f<s (±k, ε) from the matrix[gin,s + gout,s]< = 2[g0

L (1 + [tsL, g

0L])]< and write

ψ(k) =

∫ εc

−εc

dε

8πi[f<s (k, ε) + f<s (−k, ε)] (79)

with ”+(−)” referring to the anomalous scalar matrixcomponent of the incoming (outgoing) propagator onthe left side of the junction and vice versa for the right

side. ψ(k) is a measure of the (singlet) paring corre-lations available to form a singlet order parameter as

∆s(k) = λsη(k)〈η(k′)ψ(k′)〉k′·n>0, where η(k) = η(−k)are basis functions of even parity on which the pairinginteraction may be expanded. The energy εc is the usualcut-off that appears in the BCS gap equation (12).

The anomalous component of the surface propagatoralso has spin-vector parts, f int and foutt , resulting fromthe spin-scattering processes in the junction. These in-duced anomalous components can be quantified in termsof a d vector, which is customary to define in relationto triplet correlations. In general, a 2 × 2 triplet order

parameter is given by ∆k = d(k) ·σiσy and the d vectorpoints along the direction of zero spin projection of theCooper pairs82. For π junctions, we define a vector dowhich is odd in momentum and even in energy as

do(k) = n·k∫ εc

−εc

dε

8πi[f<t (k, ε)− f<t (−k, ε)] (80)

where the direction of the surface normal is n and, for the

left side of the interface, f<t (±k, ε) refers to the incoming

(outgoing) propagator. For 0 junctions, we instead definea d vector which is even in momentum and odd in energy,de, as

de(k) =

∫ εc

−εc

dε

8πisε[f

Kt (k, ε) + fKt (−k, ε)] (81)

where sε is the sign of the energy ε. This definition isbased on the anomalous correlations of [gin,t + gout,t]<

which is related to the matrix in equation (53) by[gin,t + gout,t]< = 2[g0

L (1 + [ttL, g

0L])]<. Spin-triplet

pairing that is even-in k and odd-in ε was first consideredas a candidate pairing state for 3He and is in principlenot forbidden by symmetry83 although not realized forsuperfluid 3He.

The triplet correlations span the spin space in such away that fKz ∼ 1

2 〈ψ↑ψ↓+ψ↓ψ↑〉, fK↑ ∼ 〈ψ↑ψ↑〉 and fK↓ ∼

〈ψ↓ψ↓〉. Moreover, the instantaneous spin direction ofthe triplet correlations depends on the rotating spin S,leading to a time dependence for the d vector of the form

d(t) = dz + d↑e−iωLt + d↓e

iωLt. (82)

The components of the d vector are plotted in figure 12.As expected, the components are finite when the leads aresuperconducting (T/Tc < 1). Setting vo = 0 leads to themagnitudes of the components being equal except for ascaling of DsωL. These properties are modified for finitevo; as the spin-independent tunneling is increased, anasymmetry between d↑ and d↓ emerges and the universalscaling disappears. For low temperatures, T/Tc . 0.1,one can express the d vector in terms of the direction ofthe rotating spin S as

d(t) = δLS(t)×S(t) + δH(γH)×S(t) + δzSz. (83)

17

For π junctions, the term δz is zero, while δH = 0for 0 junctions. In the tunnel limit, vo, vs 1, itis possible to find analytical expressions for these S-dependent components. For the odd d vector in thetunnel limit at low temperatures, δL,o = πDs sin(ϕ/2)and δH,o = 4πivovs sin(ϕ/2). The even-in momentum dvector, on the other hand, is cut-off dependent and di-verges logarithmically with εc. For the plots in figure 12,we used εc = 20∆. Furthermore, the relation between dvectors on either side of the interface is dR(t) = −dL(t).

D. Back-action on the precessing spin

The spin currents on either side of the interface are re-lated by jsR(t, ϕ) = −jsL(t,−ϕ), a difference leading to atorque, τ (t) = jsL(t)−jsR(t), exerted on the rotating spinS. The Josephson spin current consists of two spin-vectorcomponents, jsL/R = jH,L/R(γH)× S + j⊥,L/RS⊥. Theperpendicular spin currents on either side of the interface,j⊥,LS⊥ and j⊥,RS⊥, are equal and therefore cancel. Theother spin currents, jH,L(γH)×S and jH,R(γH)×S, areequal in magnitude but carry spin angular momentumin opposite directions, leading to a torque, here calledAndreev torque, which is given for a single conductionchannel by

τA(t) =2~SDsβH cosϑ (γH)×S(t). (84)

The torque is parallel to the one generated by the exter-nal magnetic field H and hence leads to a shift in theprecession frequency, ωL → ωL[1 + 2~

S DsβH cosϑ]. Aswas seen in section III C, the rotating spin induces localspin-triplet correlations near the junction interface andthe spin-triplet correlations allows the superconductingleads to support a spin current even at low temperatureswhen the quasiparticles are frozen out. However, the spincurrent is nothing but transport of spin-angular momen-tum and the non-conservation of the spin current resultsin a torque acting on the rotating spin. The shift in pre-cession frequency of the rotating spin is therefore a directconsequence of the induced spin-triplet correlations.

The spin current carried by normal quasiparticlestransport angular momentum from the rotating spininto the leads resulting in a damping of the spinprecession14,84. This process is the main contributionto the Gilbert damping, which has been studied quiteextensively (see 28 and references therein). As there aremany possible contributions to the Gilbert damping, it isoften entered as a phenomenological parameter27. Here,the quasiparticle torque, τ qp, is given by the spin currentin equation (78) as

τ qp(t) =2~S2αDsS(t)× S(t) (85)

for one conduction channel. Since the torque is perpen-dicular to S(t) as well as S(t), it leads to an alignment ofS with the effective magnetic field H. At temperatures

above the critical temperature, α = (2π)−1, but as thetemperature decreases, the quasiparticles freeze out andthe quasiparticle spin current as well as the quasiparticletorque vanish as T/Tc → 0. This reduction of Gilbertdamping due to superconducting phase transitions wasinvestigated in references [30,31] where the Gilbert damp-ing in domains with a precessing magnetization was mea-sured at temperatures around Tc.

We propose that the Andreev torque can be mea-sured in the following way: Assume that a nanoparticledoped with a few magnetic atoms giving it large spin,say S ∼ 50~, is placed between two superconductingaluminum (Al) leads. The superconducting coherencelength, ξ0 (∼ 2µm for Al), limits the area within whichthe time-dependent spin dynamics affects the Andreevreflection to ∼ πξ2

0 . A contact width ∼ 100 nm results inn ∼ 600 conduction channels. The superconducting gap∆ ∼ 200µeV in bulk Al, but can be made considerablysmaller in the point contact. The changes in the powerabsorption spectrum of an FMR experiment can nowbe calculated. The width of the resonance peak consti-tutes of inhomogeneous broadening, which is due to sam-ple imperfections such as anisotropy fields, and homoge-nous broadening ∆Hhom, which is due to Gilbert damp-ing. The homogenous broadening may be expressed as85

∆Hhom = 2√3HαG where H = |H| and αG = 2~

S nαDsis the Gilbert constant27. For a typical magnetic field,H = 60 mT, the macrospin precession is ∼ 1µeV corre-sponding to a frequency of 10 GHz. For a junction withDs ∼ 0.1, the freezing out of normal quasiparticles de-creases the width of the resonance peak and results ina difference in homogeneous broadening on the order of∆Hhom(T/Tc > 1) − ∆Hhom(T/Tc → 0) ∼ 26 mT. Asthe temperature is lowered, a shift in the resonance peakH0, corresponding to the shift in precession frequency,appears due to the Andreev torque. The shift can be re-lated to the Gilbert constant as ∆ωL/ωL = αGβH cosϑ.Choosing ϑ = π/4, βH ∼ 1

16ωL∆ in the low transparency

limit for temperatures T/Tc < 1/4 which leads to achange in the resonance peak of ∆H0/H0 ∼ 2%. Thisratio could be made higher by increasing the ratio ~n/Sor by increasing the junction transparency.

E. Magnetization

The proximity effect is the penetration of supercon-ducting order into a normal metal86 or a ferromagnet87.The reverse case, the so-called inverse proximity effect,occurs when ferromagnetic order penetrates into thesuperconductor88,89. In the present case, where a pre-cessing classical spin sits between two superconductingleads, there is an induced magnetization in the leads be-sides the induced triplet correlations. In the case of astatic spin, only the spin-singlet correlations and pos-sibly the spin-triplet correlations with spin projectionalong the direction of the classical spin contribute to themagnetization88. For a finite precession frequency, also

18

(a) (b) (c)

FIG. 13: (Color online) The phase relation of the functionsµL ans µH at zero temperature. The precession frequency isω = 0.2∆ and the tilt angle is ϑ = π/4. The function µH isscaled by DS .

the time-dependent spin-triplet correlations contribute tothe magnetization, which can be written on the form

Mα = Mdα +M↑

αe−iωLt +M↓αeiωLt. (86)

The magnetization is given by50

Mα(t) =µB2

∫dε

8πiTr[τ3σ(gi,<α (ε, t) + go,<α (ε, t))], (87)

where µB is the Bohr magneton. From numerical inves-tigations it can be found that

ML(t) =µBS2DsµLS × S +

µBSµH(γH)× S (88)

where µL, µH and µZ are functions of temperature, phasedifference and the hopping amplitudes vS and v0. Thelow-temperature phase relation of the parameters µL andµH are plotted in figure 13. As low temperatures µZ iszero. The parameter µH has a sinusoidal phase relationwhile µL shows a cosine behavior. In the tunnel limit,µL = 1

6π (2 − cosϕ), µH = 23πv0vS sinϕ while µZ = 0.

The temperature dependence of the parameters µL andµZ for a fixed phase difference is shown in figure 14. Thephase difference is ϕ = π if the junction is in a π state(vS > v0) or a 0 state (vS < v0). For these phase dif-ferences, the parameter µH (not shown) is zero. Theparameter µZ , on the other hand, is finite for tempera-tures 1/2 < T/Tc < 1. The parameter µL ∼ (2π)−1 asT → 0 for π junctions while µL ∼ (6π)−1 for 0 junctions.Note that unlike reference [88], the magnetization is pro-portional to the precession frequency and that M = 0for a static spin.

(a) (b) (c)

FIG. 14: (Color online) The temperature dependence of thefunctions µL and µz at phase differences ϕ = π or ϕ = 0depending on the state of the junction (vS > v0 or vS < v0).The precession frequency and the tilt angle are the same asin figure 13.

IV. CONCLUSIONS

In this work we have presented a detailed analysisof the coupling between the Josephson effect and ananomagnet. The magnetization of the nanomagnet isbrought into precession by an applied external magneticfield. The magnetization of the nanomagnet is modeledas a classical spin and the precession provides a time-dependent as well as spin-active interface which we han-dled by formulating a boundary condition within the qua-siclassical theory of superconductivity. The boundarycondition was formulated in terms of a t-matrix equa-tion describing the scattering between the two supercon-ducting leads58,64. We showed that the t-matrix equa-tion could be solved in the laboratory frame by preserv-ing the explicit time dependence or in a rotating framewhere the time dependence has been removed by a uni-tary transformation. The rotating frame solution is morenumerically efficient although it introduces a shift of thechemical energies of the spin-up and spin-down bandsof the leads making the laboratory frame solution moresuitable for investigations of modifications to the super-conducting state of the leads due to the time-dependentboundary condition.

As was already described in reference [26], the Joseph-son charge current is time independent despite the mag-netization dynamics. Here, we focus on the changes tothe superconducting state and we show that this sta-tionary solution nevertheless exhibits a nonequilibriumpopulation of the density of states due to spin scatteringbetween the Andreev levels. The nonequilibrium popu-lation of the Andreev levels results in a modified current-phase relation and causes abrupt jumps as Andreev lev-els become populated or unpopulated. The spin current,on the other hand, lacks time-independent components.Instead, the spin polarization of the current precessesaround the direction of the external magnetic field.

We have shown that the spin current is due to an

19

Andreev-level dynamics with induced spin-triplet compo-nents. The spin-triplet correlations are due to the break-ing of spin-rotation symmetry caused by the precessingclassical spin. In more detail, the spin-triplet correlationsare a result of the combination of Andreev retroreflectionof electron- and hole-like quasiparticles at the supercon-ductor surfaces and the spin scattering processes causedby the precessing spin in which the quasiparticles undergospin flip while exchanging energy with the classical spin.The triplet correlations are evidence of a dynamical in-verse proximity effect produced by the time-dependentspin-active barrier.

The spin currents generate a torque on the classicalspin changing its dynamics. This back-action is dueto Andreev scattering processes and shifts the preces-sion frequency of the classical spin and can in princi-ple be tuned using the superconducting phase differenceacross the junction. There is also a source of Gilbertdamping at finite temperatures due to the presence of

quasiparticles27. The amount of Gilbert damping can becontrolled by adjusting the temperature since the quasi-particles freeze out as T → 0.

We suggest that the effects of the Andreev torque canbe measured in an FMR experiment similar to that inreference [30]. A measured shift in precession frequencywould be a probe of the Andreev-level dynamics. Con-versely, by controlling the Andreev-level dynamics andthe inverse proximity effect, one may control the mag-netization dynamics of a nanomagnet90 which would beuseful in spintronics applications.

Acknowledgments

The authors are grateful to D. Feinberg, M. Houzet, T.Lofwander, J. Michelsen and V. Shumeiko for stimulatingdiscussions.

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