Non-equilibrium effects in a Josephson junction coupled to a precessing spin
We present a theoretical study of a Josephson junction consisting of two s-wave 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 time-dependent boundary condition with different tunneling amplitudes for spin-up and spin-down quasiparticles and where the precession produces spin-flip scattering processes. We show that as a result, the Andreev states develop sidebands and a non-equilibrium 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 steady-state Josephson current whose current-phase 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 time-dependent and its polarization axis rotates with the same precession frequency as the classical spin.
Recently, superconducting-ferromagnetic (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 charge-based 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 nano-scaled 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. Spin-polarized 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 spin-angular momentum. The angular momentum lost by spin-polarized 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 spin-transfer torque mediated by electrical currents was theoretically investigated by Slonczewski (14) and Berger (15) who worked out a description for ferromagnet-normal metal (FN) multilayer structures and showed that spin-transfer 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 spin-transfer 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 nano-scaled 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 spin-transfer 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 so-called 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 read-out of quantum-spin 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 spin-triplet correlations induced by the spin precession. The appearance of superconducting spin-triplet correlations in SFS junctions has been used to explain the observation of a long-range proximity effect in a number of experiments (34); (35); (36); (37). Keizer et al. observed a supercurrent through a junction consisting of conventional s-wave superconductors coupled over a layer of the half-metallic ferromagnet CrO much thicker than the decay length of the superconducting spin-singlet correlations (34). Various mechanisms for converting the spin-singlet correlations of the superconducting leads into spin-triplet 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 spin-triplet conversion (38). In reference [(39)], it was suggested that the spin-singlet to spin-triplet conversion is due to interface regions with misaligned averaged magnetic moments breaking the spin-rotation symmetry of the junction producing spin mixing as well as spin-flip processes. A similar trilayer structure with noncollinear magnetizations resulting in a long-range 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 long-range 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 spin-triplet correlations. Other experimental verifications of long-range 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 spin-triplet correlations in such junctions and their effect on the long-range 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 time-dependent Andreev level dynamics and a dynamical inverse proximity effect in the form of induced time-dependent spin-triplet 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 long-range 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 spin-independent (51); (52); (53); (54); (55); (56) as well as spin-dependent, or spin-active, interfaces (57); (58); (59); (60); (61); (62). In many problems, in particular when an explicit time dependence appears, the T-matrix 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 time-dependent spin-active boundary condition for the two superconductors which we solve using the T-matrix formulation. First, the transport equations are solved separately to find the classical trajectories for each lead. Then the T-matrix describing the scattering between the leads is used to connect the trajectories across the time-dependent spin-active interface.
We start by outlining the T-matrix 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 spin-up and spin-down 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 time-dependent spin-triplet correlations and Andreev-level dynamics giving rise to the spin currents. In Sec. III D, we discuss the back-action 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.
We consider two superconductors forming a Josephson junction over a nanomagnet. The nanomagnet may either be magnetic nanoparticle or a single-molecule 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 time-dependent spin-active interface (see figure 1). The nanomagnet together with the two superconducting leads are described by the total Hamiltonian (22); (23)
The left (L) and right (R) leads are s-wave superconductors described by the BCS Hamiltonian
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,
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
where the right-hand 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
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
where and with being the Pauli matrices. The spin-dependent hopping amplitude is
where the first term is a spin-conserving part with different hopping amplitudes for spin-up and spin-down quasiparticles. The second term is time-dependent 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.
We formulate the problem using nonequilibrium Green’s functions in the quasiclassical approximation following references [(58); (64)]. The tunneling Hamiltonian (6) provides a time-dependent and spin-active boundary condition for the quasiclassical Green’s function and is solved by a T-matrix 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
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 Keldysh-Nambu-spin 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 particle-hole, space which is further parameterized using the Pauli spin matrices () (see reference [(50)] for details). The matrix components in the combined Nambu-spin space are conveniently divided into spin scalar () and spin vector () parts,
where are Pauli matrices in Nambu space and includes self-energies 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,
and a set of self-consistency equations such as the one for the superconducting order parameter,
where is an average over the Fermi surface, is the pairing-interaction strength and is the cut-off 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 semi-infinite lead, . The propagators are then connected across the interface by the tunneling Hamiltonian, , whose effects can be incorporated via a quasiclassical t-matrix equation as
The hopping elements of the tunneling Hamiltonian enter the t-matrix equation via a matrix defined as
for the left side of the interface and the right-side matrix is obtained by interchanging and . The time dependence in the current problem enters through the hopping element , which in particle-hole space has the form
and in Keldysh space. Here, the hopping elements in equation (6) have been replaced with their Fermi-surface 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
where and the upper and lower signs, and , refer to the incoming and outgoing propagators, respectively.
The self-energy fields, such as the order parameter, depend on the full propagators and should in principle be calculated self-consistenly 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 self-energies 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 Fermi-surface 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
where . The Green’s functions in the above expressions are the lesser propagators defined as .
ii.2 Solving the time-dependent boundary condition
The time-dependent 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 dc-voltage 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 spin-up and spin-down 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.
To solve the boundary condition (16) dependent on the matrix (14) in the laboratory frame, it is more convenient to Fourier transform the t-matrix equation from the time domain to energy space where it becomes an algebraic equation,
The propagators have the following Nambu-spin structure,
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
which is equivalent to a time dependence of
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 self-energy 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 t-matrix 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 Keldysh-Nambu-spin matrices can be factorized in spin space into generalized diagonal matrices , spin-raising matrices , and spin-lowering matrices . These matrices are still Keldysh-Nambu 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
Using the spin algebra, the t-matrix equation in energy space (equation (19)) may be written as
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 t-matrix 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
The unitary transformation matrix for removing the time dependence of is
resulting in a transformation to a rotating frame of reference with respect to the precessing spin in which the hopping element is
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 spin-up and spin-down quasiparticles, . The hopping element also contains a spin-flip term scattering between the two spin bands.
Next, we apply the unitary transformation to the propagator and obtain
The transformation of the propagators into the rotating frame introduces spin-dependent energy shifts, , displayed in the transformed propagators, e.g. as
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 spin-up and spin-down quasiparticles. The resulting transport equation is
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 self-energy fields in do not contain terms which are off-diagonal in spin space corresponding to spin-flip scattering or equal-spin paring order parameters. For ballistic, or clean, s-wave superconducting leads, and equation (32) is time independent. The propagators obey equation (32) for a specular-scattering 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 spin-up(-down) quasiparticles are scattered or injected into the spin-down(-up) band. This non-equilibrium spin injection, however, leaves the charge current, equation (17), time independent. The charge current’s steady-state solution follows directly from the spin-independent 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 off-diagonal elements of the propagators . The off-diagonal 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.
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 s-wave superconducting leads and their the spin-singlet pairing amplitudes, . When electron- and hole-like 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
which lets quasiparticles tunnel across the junction with their spin directions and energies unaffected but with different hopping amplitude for spin-up and spin-down. This scattering behavior is captured by the matrix (see below). The spin-flip part of the hopping element,
flips a spin-down(-up) quasiparticle into a spin-up(-down) quasiparticle while changing its energy by (). Here, we have defined . This kind of tunneling creates spin-flip 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 spin-preserving 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 spin-flip for quasiparticles tunneling across the nanomagnet.
The angle between the spin and the magnetic field determines the amount of spin-flip 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 spin-flip 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 t-matrix formulation and are included for completeness.
For a static spin with precession frequency , the t-matrix equation (24) reduces to
which can easily be solved analytically since the spin-up and spin-down bands separate into two sets of equations. Straight-forward calculations of the density of states show that Andreev levels form within the superconducting gap and are located at energies given as
which, given the energy dispersion in equation (36), is evaluated to
showing that he critical current is reduced due to the spin-flip scattering generated by the embedded nanomagnet (76).
If only spin-independent tunneling is present and the spin-dependent hopping strength , the spin-transmission coefficient while which is the usual transparency of a Josephson junction. The Andreev levels are now and carry a charge current
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 spin-dependent tunneling until the spin-dependent 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 spin-dependent 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 spin-active 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 spin-independent tunneling is decreased to 0, leaving only spin-dependent tunneling, , the Andreev levels are shifted by to give . The corresponding charge current is then
The cross-over 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 t-matrix equation is simply
with . Since the hopping elements are time independent, the t-matrix equation can be transformed into energy space where the solution can be found as