Nonequilibrium effects in a Josephson junction coupled to a precessing spin
Abstract
We present a theoretical study of a Josephson junction consisting of two swave superconducting leads coupled over a classical spin. When an external magnetic field is applied, the classical spin will precess with the Larmor frequency. This magnetically active interface results in a timedependent boundary condition with different tunneling amplitudes for spinup and spindown quasiparticles and where the precession produces spinflip scattering processes. We show that as a result, the Andreev states develop sidebands and a nonequilibrium population which depend on the precession frequency and the angle between the classical spin and the external magnetic field. The Andreev states lead to a steadystate Josephson current whose currentphase relation could be used for characterizing the precessing spin. In addition to the charge transport, a magnetization current is also generated. This spin current is timedependent and its polarization axis rotates with the same precession frequency as the classical spin.
pacs:
I Introduction
Recently, superconductingferromagnetic (SF) hybrid devices have received increased attention due to their potential as spintronics devices. In spintronics, the spin degree of freedom is employed to create new phenomena which could be used to create entirely new devices or be used in combination with conventional chargebased electronics (1); (2). Information, e.g., can be stored in the magnetization direction of a small ferromagnet and its state can be read out by measuring a current through a nanoscaled contact determined by the magnetization direction. Nanomagnets such as single molecular magnets or magnetic nanoparticles may be suitable building blocks for such information storage (3); (4).
The interest in single molecular magnets been sparked by their appealingly long relaxation times at low temperatures (5) and experimental breakthroughs in contacting molecules to both superconducting and normal leads has made molecular spintronics a growing field of research. Transport measurements of molecular magnets in normal junctions have been made as a means to characterize the magnetic states (6); (7). Contacting of C molecules (8), metallofullerenes (9) and carbon nanotubes (10); (11) to superconducting leads has also been demonstrated. In addition, single molecular magnets have been suggested for quantum computing applications (12); (13) due to their long relaxation times.
Currents are not only used to read out the state of a magnet but are also used to control the magnetization direction. Spinpolarized currents carry angular momentum. However, a spin current is not a conserved quantity in a ferromagnet and a spin current oriented in such a way that its direction is perpendicular to the interface plane between the ferromagnetic layer and the leads may lose some of its spinangular momentum. The angular momentum lost by spinpolarized electrons transported through a ferromagnet is transferred to the ferromagnet. This transfer of angular momentum generates a torque acting on the feromagnet’s magnetization direction. This spintransfer torque mediated by electrical currents was theoretically investigated by Slonczewski (14) and Berger (15) who worked out a description for ferromagnetnormal metal (FN) multilayer structures and showed that spintransfer torques can lead to precession as well as reversal of the magnetization direction. These theoretical predictions were experimentally verified by Tsoi (16) and Myers (17). Nonequilbrium magnetization dynamics (18); (19) and spintransfer torques (20) in FNF trilayers coupled to superconducting leads have also been studied.
In this paper, we study the coupling between the magnetization dynamics of a nanomagnet or single molecular magnet and Josephson currents through a nanoscaled junction. We consider two superconducting leads coupled over a nanomagnet, consisting of e.g. a molecular magnet or a magnetic nanoparticle as shown in figure 1. The spins of the magnetic molecule or nanoparticle are assumed to be held parallel to each other resulting in a uniform magnetization which can be represented by a macrospin (21). If an external magnetic field is applied, the spin of the nanomagnet starts to precess with the Larmor frequency. This dynamics changes however when it is coupled to conduction electrons in the leads (22); (23). As our starting point, we take the model by Zhu et al. (22); (23) and extend it to include arbitrary tunneling strengths leading to a modified quasiparticle spectrum displaying Andreev levels for energies within the superconducting gap, (24); (25). In reference [(26)], we focused on the dc Josephson charge current, while here, we focus on the coupling between the dynamics of the Andreev levels and the transport properties. The coupling of the two superconducting leads over the precessing spin produces an ac spin Josephson current. The difference between the spin currents on the left and right sides of the interfaces produces a spintransfer torque, , shifting the precession frequency of the rotating spin. At finite temperatures, there is also a spin current carried by quasiparticles generating a damping of the magnetization dynamics, the socalled Gilbert damping (27); (28); (29). A transition of the leads from the normal state into the superconducting state reduces the Gilbert damping (30) since the number of quasiparticles is suppressed for temperatures (31). This interplay between the Josephson effect and a single spin may be used for readout of quantumspin states (32) or for manipulation of Andreev levels in the junction (33).
Furthermore, we find that the ac Josephson spin current is a result of superconducting spintriplet correlations induced by the spin precession. The appearance of superconducting spintriplet correlations in SFS junctions has been used to explain the observation of a longrange proximity effect in a number of experiments (34); (35); (36); (37). Keizer et al. observed a supercurrent through a junction consisting of conventional swave superconductors coupled over a layer of the halfmetallic ferromagnet CrO much thicker than the decay length of the superconducting spinsinglet correlations (34). Various mechanisms for converting the spinsinglet correlations of the superconducting leads into spintriplet correlations that may survive within a ferromagnetic layer have been suggested (38); (39); (40). Bergeret et al. showed that a local inhomogeneous magnetization direction at the SF interface is sufficient to generate spintriplet conversion (38). In reference [(39)], it was suggested that the spinsinglet to spintriplet conversion is due to interface regions with misaligned averaged magnetic moments breaking the spinrotation symmetry of the junction producing spin mixing as well as spinflip processes. A similar trilayer structure with noncollinear magnetizations resulting in a longrange triplet proximity effect was proposed by Houzet and Buzdin (40). Taking into account the importance of interface composition, Khaire et al. (36) devised SFS junctions consisting of conventional superconductors and CrO in which they had inserted weakly ferromagnetic layers between the superconductors and the half metal to produce interface layers with misaligned magnetization directions. A longrange proximity effect was observed in junctions containing the interface layers, but not in junctions without. Confirmation of Keizer’s results were made by Wang et al. (37) who measured a supercurrent through a crystalline Co nanowire. The Co nanowire was a single crystal, but the contacting procedure was likely to cause defects at the SF interfaces and the inhomogeneous magnetic moments needed to create the spintriplet correlations. Other experimental verifications of longrange proximity effects includes Holmium (Ho) wires contacted to conventional superconductors (35). Ho has a conical ferromagnetic structure whose magnetization rotates like a helix along the axis. The appearance of spintriplet correlations in such junctions and their effect on the longrange proximity effect (41); (42) and spin currents (43) have also been studied theoretically. In the present problem, the magnetization direction varies in time rather than in space giving rise to timedependent Andreev level dynamics and a dynamical inverse proximity effect in the form of induced timedependent spintriplet correlations. Houzet (44) studied a related problem in which a Josephson junction consisting a ferromagnetic layer with a precessing magnetization placed between two diffusive superconductors was predicted to display a longrange triplet proximity effect.
We formulate the problem of two superconducting leads coupled over a nanomagnet in terms of nonequilibrium Green’s functions. The quasiclassical theory of superconductivity is based on Landau’s Fermi liquid theory (45); (46) and is applicable to both superconducting (47); (48); (49) and superfluid (50) phenomena as well as inhomogeneous superconductors and nonequilibrium situations. The quasiclassical theory gives a macroscopic description where microscopic details are entered as phenomenological parameters (50). Basically, it is an expansion in a small parameter , where is the Fermi energy, and is suitable for weakly perturbed superconductors. The perturbations should be weak compared to the Fermi energy, , and of low frequency, . Interfaces and surfaces in superconducting heterostructures or point contacts, on the other hand, are strong, localized perturbations with strengths comparable to the Fermi surface energy (50). Within quasiclassical theory, interfaces are handled by formulation of boundary conditions which usually have been expressed as scattering problems, being able to treat spinindependent (51); (52); (53); (54); (55); (56) as well as spindependent, or spinactive, interfaces (57); (58); (59); (60); (61); (62). In many problems, in particular when an explicit time dependence appears, the Tmatrix formulation is more convenient (63); (58); (59). This formulation is also well suited for studying interfaces with different numbers of trajectories on either side as is the case for normal metal/half metal interfaces (61); (64). The two methods have proved to be equivalent and may be applied both in the limit of clean and in the limit of diffuse superconductors (64). In the latter case, the boundary conditions coincides with those of Kuprianov and Lukichev (65) and of Nazarov (66). For a recent review of quasiclassical theory we refer the reader to reference [(67)]. In the present problem, the dynamics of the nanomagnet constitutes a timedependent spinactive boundary condition for the two superconductors which we solve using the Tmatrix formulation. First, the transport equations are solved separately to find the classical trajectories for each lead. Then the Tmatrix describing the scattering between the leads is used to connect the trajectories across the timedependent spinactive interface.
We start by outlining the Tmatrix formulation applicable to scattering via the precessing magnetic moment in Sec. II. We show that the boundary condition can be solved both in the laboratory frame and a rotating frame. In the latter solution, the explicit time dependence is removed by a transformation to a rotating frame rendering this approach suitable for efficient numerical implementations for studies of transport properties. However, the solution comes at the cost of introducing an energy shift of the chemical potentials for the spinup and spindown bands. The laboratory frame approach is, on the other hand, suitable for studying modifications to the superconducting state although the explicit time dependence increases the complexity of the solution. In Sec. III A, we review the results for the Josephson charge current in reference [(26)] in terms of the laboratory frame description. The spin currents are described in Sec. III B, which is followed in Sec. III C by the induced timedependent spintriplet correlations and Andreevlevel dynamics giving rise to the spin currents. In Sec. III D, we discuss the backaction of the scattering processes on the magnetization dynamics while the magnetization induced in the leads is discussed in section III E. In Sec. IV, we conclude with a summary of our results.
Ii Model
We consider two superconductors forming a Josephson junction over a nanomagnet. The nanomagnet may either be magnetic nanoparticle or a singlemolecule magnet and we will assume that contact between the leads and the nanomagnet is made up of a few single quantum channels. The magnetization of the nanomagnet is put in precession and the resulting contact will constitute a timedependent spinactive interface (see figure 1). The nanomagnet together with the two superconducting leads are described by the total Hamiltonian (22); (23)
(1) 
The left (L) and right (R) leads are swave superconductors described by the BCS Hamiltonian
(2) 
where the dispersion, , and the chemical potential, , are assumed to be the same for both leads. The order parameter of the leads is assumed temperature dependent, . Here is the relative superconducting phase difference over the junction which we treat as a static variable that is tunable. The nanomagnet is subjected to an external magnetic field modeled as an effective field acting on the nanomagnet’s magnetic moment, . Included in this effective field are also any r.f. fields to maintain precession, crystal anisotropy fields and demagnetization effects. The magnetic moment of the nanomagnet is viewed as a single spin, or macrospin, which we will treat as a classical entity. This macrospin is related to the magnetic moment by where is the gyromagnetic ratio. The spin and the effective magnetic field couple via a Zeeman term,
(3) 
If the effective field is applied at an angle, , relative to the spin, a torque is produced that brings the classical spin into precession around the direction of the effective field. This precession generated by the tilt angle occurs with the Larmor frequency , where the magnitude of the external field is . Here, we take the direction of the effective field to be along the axis and the angle is defined as . The Larmor precession is captured by the equation of motion
(4) 
where the righthand side is the torque produced by the effective field. The magnitude of the spin is constant, . Consequently, the path traced out by the spin, , has a time dependence which lies totally in the direction of the spin, . In the absence of any other torques than the effective field, the direction of the precessing spin may be written as
(5) 
Tunneling quasiparticles have the possibility to tunnel directly between the leads with hopping amplitude or interact with the precessing spin via an exchange coupling of strength . These processes are described by the tunneling Hamiltonian
(6) 
where and with being the Pauli matrices. The spindependent hopping amplitude is
(7) 
where the first term is a spinconserving part with different hopping amplitudes for spinup and spindown quasiparticles. The second term is timedependent and describes processes where quasiparticles flip their spins while exchanging energy with the rotating spin. The junction’s transport properties as well as the modifications to the superconducting states of the leads depend on the hopping amplitudes as well as the the superconducting phase difference, , the precession of the spin, , and the tilt angle, , between the effective field and the precessing spin.
ii.1 Approach
We formulate the problem using nonequilibrium Green’s functions in the quasiclassical approximation following references [(58); (64)]. The tunneling Hamiltonian (6) provides a timedependent and spinactive boundary condition for the quasiclassical Green’s function and is solved by a Tmatrix equation (68); (69); (70). A quasiclassical Green’s function is a propagator describing quasiparticles moving along classical trajectories defined by the Fermi velocity at a given quasiparticle momentum on the Fermi surface, . The information of a quasiclassical Green’s function is contained in the the object
(8) 
where is the spatial coordinate, is the quasiparticle energy relative to the chemical potential and is time. The propagator is a matrix in the combined KeldyshNambuspin space. The ”check” denotes a matrix in Keldysh space where the components are retarded (), advanced () and Keldysh () Green’s functions while the ”hat” indicates a matrix in Nambu, or particlehole, space which is further parameterized using the Pauli spin matrices () (see reference [(50)] for details). The matrix components in the combined Nambuspin space are conveniently divided into spin scalar () and spin vector () parts,
(9) 
where and are the spinsinglet and spintriplet components of the anomalous Green’s functions (similarly for and ). The Green’s function obeys a Boltzmanlike transport equation (47); (48); (49)
(10) 
where are Pauli matrices in Nambu space and includes selfenergies like the superconducting order parameter, , and impurity contributions, , as well as any external fields, . The ”” product represents a convolution over common time arguments combined with a matrix muliplication (50); (67). The transport equation is complemented by the Eilenberger normalization condition,
(11) 
and a set of selfconsistency equations such as the one for the superconducting order parameter,
(12) 
where is an average over the Fermi surface, is the pairinginteraction strength and is the cutoff energy which may be eliminated by making use of the critical temperature, .
The transport equation (10) is solved separately for each lead, treating the interface as an impenetrable surface where quasiparticles are perfectly reflected. This hard wall boundary condition leads to a solution, , for each semiinfinite lead, . The propagators are then connected across the interface by the tunneling Hamiltonian, , whose effects can be incorporated via a quasiclassical tmatrix equation as
(13) 
The hopping elements of the tunneling Hamiltonian enter the tmatrix equation via a matrix defined as
(14) 
for the left side of the interface and the rightside matrix is obtained by interchanging and . The time dependence in the current problem enters through the hopping element , which in particlehole space has the form
(15) 
and in Keldysh space. Here, the hopping elements in equation (6) have been replaced with their Fermisurface limits, and with being the normal density of states at . Note that for the junction studied, the hopping elements have the symmetries . The t matrices (13) are used to calculate the full quasiclassical propagators which depending on if their trajectories lead up to or away from the interface are divided into ”incoming” () and ”outgoing” () propagators, given by
(16) 
where and the upper and lower signs, and , refer to the incoming and outgoing propagators, respectively.
The selfenergy fields, such as the order parameter, depend on the full propagators and should in principle be calculated selfconsistenly taking into account the interface scattering. However, we assume in this study that the area of the point contact, , is small compared to where is the superconducting coherence length. As a result, the superconducting state does not change considerably and the order parameter, , and other possible selfenergies in the leads do not have to be recalculated (50). We will also assume that the superconducting phase changes abruptly over the contact.
The use of equations (10,11) together with the boundary condition (16) allows for calculation of the transport properties the junction. The charge and spin currents are given by an average over the Fermisurface momentum directions of the full propagators. Here, this average amounts to a difference between incoming and outgoing propagators and the charge, , and spin, , currents evaluated in lead for a single conduction channel are
(17) 
and
(18) 
where . The Green’s functions in the above expressions are the lesser propagators defined as .
ii.2 Solving the timedependent boundary condition
The timedependent boundary conditions can be solved in two different ways. In the first procedure, the boundary conditions are solved in the laboratory frame in which the time dependence is preserved and is manifested as frequency shifts in a difference equation. The treatment is similar to that of dcvoltage biased SIS junctions (71); (72); (73); (63); (58); (59); (20). The second approach involves removing the explicit time dependence of by a transformation to a rotating frame (see reference [(26)] for details). This procedure is numerically more efficient but the transformation, however, introduces an exchange field shifting the chemical potentials of the spinup and spindown bands in the leads making the first approach more suitable for studying changes in the superconducting state in vicinity of the junction due to quasiparticle tunneling via the precessing spin. Below we describe both within a quasiclassical framework.
Laboratory frame
To solve the boundary condition (16) dependent on the matrix (14) in the laboratory frame, it is more convenient to Fourier transform the tmatrix equation from the time domain to energy space where it becomes an algebraic equation,
(19) 
The propagators have the following Nambuspin structure,
(20)  
The gap, , is both temperature and phase dependent and the ””(””) sign of the phase dependence refers to lead (). Furthermore, is the quasiparticle occupation function setting the two superconducting leads in thermal equilibrium with each other. The t matrices in energy space are a sum of t matrices whose energies differ by and satisfy the relation
(21) 
which is equivalent to a time dependence of
(22) 
The Ansatz above and the assumption that the leads are in equilibrium so that their respective Green’s functions may be written as lets one evaluate the coefficient matrices and in equation (19) and subsequently solve the resulting difference equation in terms of (63); (58); (59). This is a quite general procedure capable of handling diverse forms of selfenergy fields . However, the matrix coefficients in equation (19) must be evaluated for each particular kind of junction and lead state. The properties of these matrices then determine the specific tmatrix difference equation and the solution strategy.
For the present calculation certain simplifications can be made due to the spin independence of given by equation (20) and the form of the hopping element in equation (15): the KeldyshNambuspin matrices can be factorized in spin space into generalized diagonal matrices , spinraising matrices , and spinlowering matrices . These matrices are still KeldyshNambu matrices but they have the algebraic properties of spin matrices such as , , and . A matrix factorized in this form may be shown to have the time dependence
(23)  
Using the spin algebra, the tmatrix equation in energy space (equation (19)) may be written as
(24) 
The coefficient matrices in equation (24) are functions of energy and precession frequency and are straight forward to evaluate. Contrary to e.g. the case of a finite dc voltage, where the tmatrix equation is a difference equation solved by recursive methods, we have a matrix equation for which can be solved by simple (numerical) inversion. Factorizing the propagators in equation (16) according to their spin structure results in
(25)  
(26)  
(27) 
with .
Rotating frame
The unitary transformation matrix for removing the time dependence of is
(28) 
resulting in a transformation to a rotating frame of reference with respect to the precessing spin in which the hopping element is
(29) 
The direction of the precessing spin is now static in this rotating frame, , but the hopping element is still spin active with different hopping amplitudes for spinup and spindown quasiparticles, . The hopping element also contains a spinflip term scattering between the two spin bands.
Next, we apply the unitary transformation to the propagator and obtain
(30)  
The transformation of the propagators into the rotating frame introduces spindependent energy shifts, , displayed in the transformed propagators, e.g. as
(31)  
for the retarded Green’s function. The advanced and Keldysh propagators are similarly transformed.


Applying to the transport equation, equation (10), similarly leads to a shift of the energies of the spinup and spindown quasiparticles. The resulting transport equation is
(32) 
where the energy shifts are captured by the appearance of an effective magnetic field in the leads, . Equation (32) is time independent as long as the selfenergy fields in do not contain terms which are offdiagonal in spin space corresponding to spinflip scattering or equalspin paring order parameters. For ballistic, or clean, swave superconducting leads, and equation (32) is time independent. The propagators obey equation (32) for a specularscattering boundary condition (64); (58). Introduction of the tunneling processes lead to a boundary condition problem which is still time independent in this rotating frame and the ”” product in equation (16) reduces to a matrix multiplication in energy space.
The precessing spin introduces spin mixing of the two spin bands in such a way that spinup(down) quasiparticles are scattered or injected into the spindown(up) band. This nonequilibrium spin injection, however, leaves the charge current, equation (17), time independent. The charge current’s steadystate solution follows directly from the spinindependent trace over particles and holes in equation (17) and the fact that the transformation matrix leaves the diagonal terms of the propagators time independent. The spin current, on the other hand, has components which include the offdiagonal elements of the propagators . The offdiagonal elements, which are proportional to the Pauli spin matrices and , are time dependent since these are affected by the time dependence of the transformation matrix . Thus, for a finite tilt angle, , the spin current may be time dependent.
Iii Results
The precessing spin introduces a new energy scale and the superconducting correlation functions can be expected to be modified due to the scattering processes the precessing spin gives rise to. The starting point is the swave superconducting leads and their the spinsinglet pairing amplitudes, . When electron and holelike quasiparticles interfere constructively, sharp states inside the superconducting gap called Andreev states are formed. In regular Josephson junctions without magnetically active interfaces, these Andreev states come in degenerate pairs , which can be described by the spinors . Each member of the spinor, , is subjected to transmission and Andreev retroreflection and the Andreev bound states are formed when the processes lead to constructive interference along closed loops schematically illustrated as . A schematic picture of the scattering processes is shown in the upper panel of figure 2. The tunneling processes described above are captured by the hopping element
(33) 
which lets quasiparticles tunnel across the junction with their spin directions and energies unaffected but with different hopping amplitude for spinup and spindown. This scattering behavior is captured by the matrix (see below). The spinflip part of the hopping element,
(34) 
flips a spindown(up) quasiparticle into a spinup(down) quasiparticle while changing its energy by (). Here, we have defined . This kind of tunneling creates spinflip processes, e.g. , which is a process captured by the elements .
Focusing on the left side of the interface and parameterizing the matrix according to equation (23), leads one to conclude that the spinpreserving component of is a combination of the two processes described above, . There are also mixed tunneling processes where tunneling with and without spin flip are combined into the terms and . These two terms generate a net spinflip for quasiparticles tunneling across the nanomagnet.
The angle between the spin and the magnetic field determines the amount of spinflip scattering ranging from zero for parallel alignment to maximum in the case of . The frequency sets the amount of energy exchanged between a quasiparticle and the rotating spin during a spinflip event as indicated in figure 2 (b). First, we will look at the consequences for the density of states and the charge current due to the scattering caused by the precessing spin. After that, we will take a closer look at the effects on the superconducting pair correlations before we turn to the spin scattering states and the spin current as well as their implications for the leads.
iii.1 Charge currents
This section reviews some of the results presented in reference [(26)]. Here, however, the charge current results are described in the tmatrix formulation and are included for completeness.




Static spin
For a static spin with precession frequency , the tmatrix equation (24) reduces to
(35) 
which can easily be solved analytically since the spinup and spindown bands separate into two sets of equations. Straightforward calculations of the density of states show that Andreev levels form within the superconducting gap and are located at energies given as
(36) 
where we have defined and a spintransmission coefficient . The charge current density is related to the energy dispersion and the occupation function as (74); (75)
(37) 
which, given the energy dispersion in equation (36), is evaluated to
(38) 
showing that he critical current is reduced due to the spinflip scattering generated by the embedded nanomagnet (76).
If only spinindependent tunneling is present and the spindependent hopping strength , the spintransmission coefficient while which is the usual transparency of a Josephson junction. The Andreev levels are now and carry a charge current
(39) 
Both of these relations are shown as solid black lines in figures 3 (a) and (b). As can be seen in these figures, the junction is in a state corresponding to the junction’s energy being minimized for the phase difference . Increasing the hopping strength of the spindependent tunneling until the spindependent tunneling dominates, , causes the junction to shift from being in the state to being a state, as can be seen in figure 3(a). When spindependent tunneling dominates, the junction’s ground state is such that the coupled superconductors have an internal phase shift of , as predicted in reference [(77)]. Such states can also be observed in junctions where the spinactive barrier has been extended to a ferromagnetic region (78). In such junctions, the width of the ferromagnetic layer as well as the strength of the exchange field determine the transport properties.
If, on the other hand, the spinindependent tunneling is decreased to 0, leaving only spindependent tunneling, , the Andreev levels are shifted by to give . The corresponding charge current is then
(40) 
The crossover from the to state occurs at where the Andreev levels are . These Andreev levels are independent of the phase difference between the two superconductors leading to a zero Andreev current consistent with equation (37).
In figures 3 (c) and (d), the current kernel for the left side of the interface is plotted for and , respectively. The current kernel is integrated over energy to give the total current for a specified phase difference, . The kernel indicates which states, i.e. Andreev levels and continuum states, are participating in transporting current through the junction. The direction of the current is given by the sign of . As can be seen in panel (c), the lower of the two Andreev levels is occupied which is consistent with quasiparticle states below the Fermi surface being occupied. The same is true for in panel (d), although the current kernel has been shifted by .
junction with a small tilt angle
For a junction with zero tilt angle, the solution of the boundary condition problem reduces to the static spin case. The tmatrix equation is simply
(41) 
with . Since the hopping elements are time independent, the tmatrix equation can be transformed into energy space where the solution can be found as