Quasiclassical theory of magnetoelectric effects in superconducting heterostructures in the presence of the spin-orbit coupling
The quasiclassical theory in terms of equations for the Green’s functions (Eilenberger equations) is generalized in order to allow for quantitative description of the magneto-electric effects and proximity-induced triplet correlations in the presence of spin-orbit coupling in hybrid superconducting systems. The formalism is valid under the condition that the spin-orbit coupling is weak with respect to the Fermi energy, but exceeds the superconducting energy scale considerably. On the basis of the derived formalism it is shown that the triplet correlations in the spin-orbit coupled normal metal can be induced by proximity to a singlet superconductor without any exchange or external magnetic field. They contain an odd-frequency even-momentum component, which is stable against disorder. The value of the proximity-induced triplet correlations is of the order of , that is absent in the framework of the standard quasiclassical approximation, but can be described by our theory. The spin polarization, induced by the Josephson current flowing through the superconductor/Rashba metal/superconductor junction, is also calculated.
By now it is already known that spin-orbit coupling (SOC) is a source of many interesting phenomena. Some of them originate from coupling of charge and spin degrees of freedom and often called by magnetoelectric effects. In addition to their fundamental importance these phenomena can be of interest for the spintronics and, in particular, for the superconducting spintronicslinder15 (); eschrig15 (). It is worth to mention here some of them, which are related to the subject of this work. For nonsuperconducting systems these are the spin Hall effect (SHE)dyakonov71 (); dyakonov71_2 (); chazalviel75 (); hirsch99 (); dyakonov06 (); mishchenko04 (); kato04 (); kato04_2 (); wunderlich05 (); raimondi06 (), the inverse SHEvalenzuela06 (); morota11 (); isasa14 (), the direct magneto-electric aronov89 (); edelstein90 (); kato04 (); silov04 () and inverse magneto-electric (spin-galvanic) effectsshen14 (); ganichev02 (); rojas_sanchez13 ().
The direct magneto-electric effect was also predicted for superconducting systems edelstein95 (); edelstein05 (); malshukov08 (); bergeret16 (), where it consists in generation of an equilibrium spin polarization in response to a supercurrent. The analogue of the inverse magneto-electric effect has also been reported for superconducting systems. For homogeneous superconducting systems its physics is that the SO-coupled superconductor turns into the inhomogeneous phase-modulated state (it is also called by the helical phase) in response to an applied exchange field edelstein89 (); samokhin04 (); kaur05 (); dimitrova07 (); houzet15 (). For Josephson junction the inverse magneto-electric effect is a cause of the anomalous phase shift , which modifies the current-phase relation according to . This is the so called -junction, and its interpretation in terms of the inverse magneto-electric effect was reported in konschelle15 (). It was actively studied recently in half-metal junctions, noncoplanar ferromagnetic junctions, ferromagnetic Josephson junctions with spin orbit interaction or TI surface states krive04 (); braude07 (); asano07 (); reinoso08 (); buzdin08 (); tanaka09 (); zazunov09 (); liu10 (); alidoust13 (); brunetti13 (); yokoyama14 (); bergeret15 (); dolcini15 (); campagnano15 (); mironov15 (); konschelle15 (); kuzmanovski16 (); zyuzin16 ().
The other group of effects, which are of entirely superconducting nature, is connected to the generation of triplet superconducting correlations via proximity to a conventional superconductor (S). This singlet-triplet conversion can also be viewed as a coupling between charge and spin degrees of freedom, that is a kind of magnetoelectric effect. The triplet Cooper pairs with non-zero average spin can play the same role in superconducting spintronics as electron spins in conventional spintronics. However, the corresponding spin currents are dissipationless. Most of the research in this area, both experimental and theoretical, has focused on proximity junctions involving ferromagnets (F) (see Ref. eschrig15, and references therein). The ferromagnets induce triplet correlations by lifting the spin degeneracy buzdin05 (); bergeret05 (). But, the proximity between a singlet superconductor and a homogeneous ferromagnet only leads to creation of zero average spin pairs. In order to have non-zero average spin one needs to include a magnetic inhomogeneity or SOC into consideration because they act as spin mixers bergeret05 (); eschrig15 (); bergeret13 (); bergeret14 ().
The SOC also lifts spin degeneracy. It has been demonstrated that for homogeneous superconductors in the presence of SOC the pair wave function is the mixture of singlet s-wave and triplet p-wave components gorkov01 (); alicea10 (). So, the question is if it by itself can induce triplet pair correlations in proximity to a singlet superconductor? The most convenient and commonly used method to treat superconducting hybrid systems is a quasiclassical theory of superconductivity. The SOC can be treated in the quasiclassical approximation when its characteristic energy is much less than the Fermi energy . This situation is typical. Within the framework of the quasiclassical theory it was found that SOC by itself does not induce any triplet pairing bergeret13 (); bergeret14 ().
On the other hand, working in the framework of Gor’kov equations beyond the quasiclassical approximation, Edelstein showed that interfacial spin-orbit scattering generates triplet pairing in 3D ballistic superconductor/normal-metal junctionsedelstein03 (). More recently, it was reported in several works on the basis of the lattice numerical calculation yang09 (), a gauge-covariant analytical approach konschelle15 (), and, at last, on the basis of exact Gor’kov technique, that triplet superconductivity can be generated in Rashba metals by proximity to a singlet superconductorreeg15 ().
However, the Gor’kov equations are of very limited use for inhomogeneous problems. So, it is desirable to generalize the quasiclassical technique in order to be able to describe the SOC-induced triplet correlations and the magneto-electric and spin-galvanic effect, which are also beyond the framework of quasiclassical approximation. A way to such a generalization has already been proposed in the framework of gauge-covariant Green’s functions approach konschelle15 (). But the results for the proximity-induced triplet correlations in the ballistic limit seem to be not fully coinciding with the results of exact Gor’kov’s approach reeg15 (). Also the appropriate normalization condition and boundary conditions for the quasiclassical Green’s functions were not considered in Ref. konschelle15, .
It is also worth to note here that the quasiclassical formalism in the framework of the gauge-covariant approach has been developed also for nonsuperconducting systems gorini10 (); raimondi12 (). On the basis of this formalism a generalized Boltzmann equation for the charge and spin distribution functions was formulated. Then it was applied, in particular, to the investigation of the spin Hall and the inverse spin galvanic (Edelstein) effects.
In this work we generalize the quasiclassical equations, the normalization condition and the corresponding boundary conditions for the absolutely transparent interface in order to be able to calculate the Green’s functions up to the first order with respect to the parameter . This allows us to describe the magneto-electric effects and proximity-induced triplet correlation in the presence of SOC, while the quasiclassical approximation only provides Green’s functions up to zero order with respect to this parameter and is not able to catch them. In order to check our formalism we consider the proximity effect at the interface between the the singlet superconductor and the Rashba metal in the ballistic limit. The result of Ref. reeg15, is recovered, if the exact expression of this work is properly expanded up to the first order in . The statement of this work that the triplet correlations are absent in the first order with respect to this parameter is incorrect.
We also consider the direct magneto-electric effect in superconductor/Rashba metal/superconductor ballistic junction. Its essence is a creation of a stationary spin polarization in response to a Josephson electric current flowing through the junction. As far as we know, this effect has not been quantitatively calculated so far, while the direct magneto-electric effect in homogeneous Rashba superconductors in ballisticedelstein95 () and diffusiveedelstein05 () systems were considered, and the direct magneto-electric effect in superconductor/Rashba metal/superconductor diffusive junction was calculated as wellmalshukov08 ().
The paper is organized as follows. In sections II and III we derive the quasiclassical formalism, which accounts for the corrections up to the first order with respect to the parameter . In sec. II the corresponding equations for the quasiclassical Green’s function and the normalization condition are derived, while sec. III is devoted to the derivation of the boundary conditions. The proximity-induced triplet correlations at Rashba metal/superconductor interface are considered in sec. IV, and the direct magneto-electric effect in superconductor/Rashba metal/superconductor ballistic junction is calculated in sec. V. In sec. VI we summarize our results. The Appendix A is devoted to details of the direct magneto-electric effect calculations.
Ii generalized quasiclassical equations
In this section we derive the equations of motion for the quasiclassical Green’s functions keeping the terms up to the first order with respect to the parameter , while the standard quasiclassical approximation neglects them. It is these terms that provide singlet-triplet conversion in superconducting proximity systems in the absence of an applied magnetic field and/or any ferromagnetic elements. They are also responsible for the magneto-electric effects in SO-coupled systems.
We do not restrict ourselves by equilibrium situations and work with the Green’s functions in the Keldysh technique keldysh (); mahan (). We start with the Hamiltonian of a singlet superconductor in the presence of a generic linear in momentum spin-orbit (SO) coupling bergeret13 (); bergeret14 ():
where is the superconducting parameter and is the Hamiltonian of the normal metal in the presence of the spin-orbit coupling (NSO). The general linear in momentum SO is expressed by the term , where are Pauli matrices in spin space. , is the chemical potential, and is an exchange field. We assume that the system involves nonmagnetic impurities that can be described by a Gaussian scattering potential: .
The advanced (), retarded (), and Keldysh () blocks of Gor’kov Green function in the Keldysh technique are defined in a standard way (see, for example, Ref. bbza16, ).
By averaging the Green function over the impurity scattering potential in the Born approximation we find the following Gor’kov equation:
Here, we introduce the Pauli matrices in particle-hole space , with . is the matrix structure of the superconducting order parameter in the particle-hole space. is the self-energy describing the elastic scattering at nonmagnetic impurities, where is the quasiparticle mean free time and is the density of states at the Fermi level of the normal state. The two time dependent products of operators is equivalent to .
In this work we concentrate on the ballistic systems, therefore below we assume . The including of the impurity self-energy into the resulting equations is straightforward.
The main goal of the present work is to develop the theory for plane interfaces between the SO materials and superconductors. We focus on the two-dimensional case here. For this reason it is convenient to perform the Fourier transformation with respect to the -coordinate, parallel to the considered 2D interface:
where and for generality we allow for the slow dependence of the Green’s function on the center of mass coordinate along the interface. Substituting Eq. (4) into Eq. (3) one can obtain the Gor’kov equation for :
where the envelope functions are slow functions of , varying at quasiclassical length scales, except at , where they are discontinuous.
The equivalent equation in the variable is
Further we reduce the amount of information by defining envelope functions of one variable,
where . Below we also use the analogous definition for . In Eq. (10) the second line contains terms, which have additional small factor with respect to the terms in the first line. Here is the characteristic spin-orbit energy, which is of the order of . The terms in the first line represent the well-known quasiclassical Eilenberger equation eilenberger68 (); larkin69 (); bergeret14 (), and the terms in the second line are corrections to the quasiclassical approximation and usually are neglected. However, as was already mentioned in the introduction, part of them are responsible for the magnetoelectric effects in superconductors and superconducting heterostructures and singlet-triplet conversion in the absence of the exchange field, therefore we should keep them in order to get possibility to treat these effects.
Further we will only keep the terms of the order of , but will neglect the terms, which do not contain (of the order of ), because here we are interested in the limit , which is appropriate for many real spin-orbit materials, such as metal surfaceslashell96 (); hoesch04 (); koroteev04 () and metallic surface alloysnakagawa07 (); ast07 (); eremeev12 (). Working in the framework of the perturbation theory up to the first order in it is enough to change the full Green’s function by its quasiclassical approximation . This allows us to simplify Eq. (10) further. One can find from the quasiclassical version of Eqs. (7) and (8), respectively. For example, for we get
where means anticommutator, and we have neglected all the terms, which does not contain . For example, all the terms, proportional to spatial derivatives of and are disregarded. It is also assumed that the SO coupling does not depend on coordinates.
where the trajectory marked by the subscript ”+”(”-”) is defined by and . Expressing the envelope functions via in Eq. (13) it is easy to obtain the final equation for the quasiclassical Green’s function. The second and third lines in Eq. (13) represent the corrections of the first order in to the well-known quasiclassical equation, expressed by the first line of Eq. (13).
Further we only consider stationary problems, therefore a Fourier transformation with respect to can be performed. We are interested in situations, when zeeman field is absent: . We also assume that the Green’s function does not depend on the -coordinate along the interfaces. Under these conditions Eq. (13) can be simplified considerably and takes the form (it is already rewritten in terms of the quasiclassical Green’s function):
Here we use the fact that at . This equation is one of the central results of our paper and contain all necessary terms to catch the proximity induced triplet correlations a NSO/S interface and the direct magnetoelectric effect in homogeneous superconductors edelstein95 () and in ballistic superconducting heterostructures.
Eq. (15) should be supplied by the normalization condition. In usual quasiclassical theory the normalization condition is . However, we obtain that it should be modified if one would like to take into account the terms of the order of . Below we derive the appropriate normalization condition. Multiplying Eq. (7) by from the left and Eq. (8) by from the right, we obtain the following expression at :
where only terms of zero and first order in are kept, but all the terms of the first order with respect to are neglected. Please note that at Eq. (16) is not valid. If the Green’s functions do not depend on the -coordinate, then the normalization condition for the envelope functions takes the well-known zaitsev84 (); millis88 () form:
for an arbitrary value of . This can be easily found if the Eilenberger equation is solved for a half-space. For example, if we consider the left half-space, then one should take the case . These inequalities cannot be changed because the envelope functions have discontinuities at coinciding arguments. Then taking the limit and we get that . Analogously for the right half-space one should take the case . Then taking the limit and we also get that .
Further, the normalization condition for the quasiclassical Green’s function can be obtained from Eq. (17) at and the sign corresponds to the left (right) half-space. The envelope function is directly connected to the quasiclassical Green’s function according to Eq. (14). In order to connect the envelope function to the quasiclassical Green’s function, we need to calculate the discontinuity of at . It can be obtained by integrating the Gor’kov equation (5) about and taking into account the continuity condition for the full Gor’kov Green’s function at zaitsev84 (); millis88 (). Up to the first order terms with respect to we get the following expression:
The same normalization condition is also valid for the regions, which are not semi-infinite, if is fulfilled there. It can be proven directly by multiplying Eq. (15) by from the left, then from the right and adding the resulting equations.
Iii generalized boundary conditions
Quasiclassical equations are not valid in the vicinity of interfaces, where the normal state hamiltonian of the system changes over the atomic length scales. Therefore they should be supplied by the boundary conditions. In order to derive the appropriate boundary condition we generally follow Refs. zaitsev84, ; millis88, . The main strategy is to solve the interface scattering problem disregarding all the low-energy terms in the hamiltonians of the left and right materials: the superconducting order parameter, the quasiparticle energy and the exchange field should be neglected because we work only up to the zero order with respect to the parameter . In the standard quasiclassical approach the spin-orbit coupling term also should be neglected upon considering the scattering problem. However, our goal is to correctly account for the terms of the first order with respect to . Therefore, we must keep the spin-orbit coupling terms in the normal state hamiltonian of the SO-material.
Further we restrict ourselves by the case of ”absolutely transparent interfaces” only. It means that where is no interface scattering barrier and there is no mismatch of the Fermi surfaces at the interface (without taking into account the spin-orbit coupling term), that is . It is well-known that in this case the boundary conditions take the most simple linear form, while they are highly nonlinear for an arbitrary transparency of the interface and special further efforts are necessary to make them ready for practical use eschrig00 (); eschrig09 (); zhao04 (). We postpone this problem for future consideration and demonstrate that even for the case of ”absolutely transparent interfaces” the boundary conditions should be modified with respect to the standard form if we need to account for the terms of the first order with respect to .
The Schrodinger equation in the interface region takes the form
where and we assume , that is SO coupling is nonzero only on the left side of the interface. Its full solution can be written as follows:
where at are constant Nambu vectors, corresponding to left-moving () and right-moving () solutions. The general solution of the scattering problem at the interface between the SO-material and a material without SO-coupling can be easily found making use of Eq. (20) and the appropriate boundary conditions at the interface ():
The connection between the left and right-moving solutions can be formulated in terms of the so-called interface transfer matrix as follows
where for the considered here problem of the absolutely transparent interface
where upper and lower signs correspond to the NSO/S and S/NSO interfaces, respectively. .
From Eq. (23) and the conjugated equation one obtains that the envelope functions and are connected by
We are interested only in the boundary conditions for the envelope functions for the coinciding subscripts because only these envelope functions are connected to the quasiclassical Green’s functions and are necessary for calculating observables. From Eq. (25) it follows that the boundary condition for at NSO/S interface takes the form
where we have taken into account that is of the first with respect to , consequently all the terms, quadratic with respect to should be disregarded. For the same reason only the quasiclassical approximation for at enters the above equation. The boundary condition at S/NSO interface is obtained from Eq. (26) by substituting .
It can be shown that in the ballistic limit and for the fully transparent interface we consider. In this case, taking into account the definition of the quasiclassical Green’s function Eq. (14), one can obtain from Eq. (25) the following simple form of the boundary conditions
where the signs correspond to the NSO/S and S/NSO interfaces, respectively. It is seen that neglecting the right hand side of the above equation, which is of the first order in , we obtain the well-known quasiclassical boundary condition at a fully transparent interface: , that is just the continuity of the Green’s function. This value of the quasiclassical value of the Green’s function at the interface enters the right hand side of the boundary condition. It is worth to note here that if there is an equal SO coupling in the both materials, the boundary condition reduces to the standard continuity condition .
Iv proximity-induced triplet correlations at NSO/S interface
Here on the basis of the derived formalism we consider the proximity effect at a NSO/S interface, where the spin-orbit coupling in NSO is assumed to be of the Rashba-type for concreteness. It is found that taking into account the corrections of the first order with respect to to the quasiclassical approximation leads to the appearance of the proximity-induced triplet correlations in the NSO region without any exchange or Zeeman term. These correlations are long-ranged, that is they decay on the length scale of the normal state coherence length in the NSO region. They also contain an odd-frequency even-momentum component, which does not disappear after averaging over trajectories. These results are in sharp contrast with the results of the pure quasiclassical approximation, where the spin-orbit interaction by itself cannot be a source of any triplet correlations (induced by the proximity effect with a singlet superconductor), and can only modify the proximity induced triplet correlations in the presence of an exchange field bergeret13 (); bergeret14 ().
The sketch of the system is shown in Fig. 1(a). The interface between the superconductor and the NSO is at [see Fig. 1(a)]. The SO coupling is nonzero only in the normal metal part and is absent in the superconductor. The NSO/S interface is assumed to be fully transparent. However, we have also considered another system, where the superconducting and normal regions have absolutely the same normal state hamiltonians with non-zero spin-orbit interaction term. The corresponding experimental setup could be realized on the basis of a proximity induced superconductivity [see Fig. 1(b)]. We have found that the results for the proximity induced triplet correlations in NSO region are the same for the both setups.
Here we present the detailed calculations only for the case shown in Fig. 1(a). Our calculations are based on Eq. (15). For simplicity we have considered only the linearized case here, when the Eilenberger equations can be linearized with respect to the anomalous Green’s function. Under our conditions it can be realized at , where is a critical temperature of the superconductor. The Green’s function in the Nambu space can be represented as
where it is enough to calculate the normal components for and the anomalous components of the retarded Green’s function can be found from the following linear equations
where we introduce the following expansion of the anomalous Green’s function over the spin basis: . While is the singlet component of the anomalous Green’s function, for are the corresponding triplet components. The last terms in Eqs. (30) and (32) are the corrections of the order of to the quasiclassical approximation, therefore one can use the quasiclassical approximation for the anomalous Green’s function in these terms. As it was mentioned above, has no triplet components in the absence of the exchange field, that is .
can be easily found making use of the quasiclassical version of Eqs. (29)-(32), boundary conditions, which are reduced to continuity of the anomalous Green’s function in the quasiclassical limit, and the asymptotic conditions, which require the anomalous Green’s function to be non-growing functions at . The resulting expressions take the form: in the NSO:
and in the superconductor:
where subscripts and correspond to right-moving () and left-moving () trajectories, respectively. The exponential factors in the above expressions decay at the appropriate infinity due to the fact that for the retarded Green’s functions has an infinitesimal imaginary value with .
In order to find the corrections of the order of to this quasiclassical solution, we need the normal components of the Green’s function up to the same order of magnitude. It is easy to check that the following solution in the NSO region
Substituting Eq. (35) into Eqs. (29)-(32) and making use of boundary conditions (27) one obtains the following expression for the proximity induced anomalous retarded Green’s function in the NSO region:
while . In the superconductor the solution has no corrections to the quasiclassical answer, if the spin-orbit coupling is zero there.
It is seen from Eqs. (37)-(38) that the proximity-induced superconducting condensate in the Rashba metal has triplet components of the first order with respect to in the absence of a Zeeman term. Our answer fully coincides with the proper expansion to the first order with respect to of the general result for the Gor’kov Green’s function, obtained in Ref. reeg15, , what is a good check of the validity of our approach. It is worth to mention here that expressions (37)-(38) are only valid at the distances from the interface (where is a superconducting coherence length), because physically our approximation can be viewed as a projection of two different quasiparticle trajectories, corresponding to two different spin-orbit split Fermi surfaces, onto the same direction, determined by the Fermi surface in the absence of the spin-orbit splitting. However, this restriction is of no practical importance for the problems under consideration because all the proximity-induced superconducting correlations, which are of interest for us, decay much faster, at the characteristic length scale of .
Now we discuss the symmetry classification of the obtained proximity-induced correlation. Pair amplitudes are classified into four types according to their behavior with respect to Matsubara frequency, momentum (parity), and spin eschrig15 (). Type A: spin singlet, even frequency, even parity; type B: spin singlet, odd frequency, odd parity; type C: spin triplet, even frequency, odd parity and type D: spin triplet, odd frequency, even parity.
In order to analyze which types of correlations are present in Eqs. (36)-(38), we should turn to the Marsubara frequency representation and divide the correlations into symmetric and antisymmetric parts with respect to . As for singlet correlations, here we have the both types of them. The type A correlations are the most typical and survive for a dirty case as well. The singlet, odd frequency and odd parity correlations also arise here due to the broken translational invariance, as it was reported for other physical systems with broken translational symmetry tanaka07 (); eschrig07 (). But this type of correlations would disappear in the dirty system after averaging over trajectories due to its odd-parity nature.
As for the triplet correlations, the both possible types are also present here. It is worth to underline that the singlet-triplet mixing, reported for the homogeneous superconductor with SOC gorkov01 (); alicea10 (), is only p-wave, that is of type C. In the homogeneous case the type D of correlations was reported in the presence of a Zeeman term or the applied supercurrent malshukov08 (); konschelle15 (). In spatially inhomogeneous systems the odd-frequency even-parity triplet correlations also arise due to the broken translational symmetry.
It is seen from Eqs. (37)-(38) that both and components of the triplet correlations contain as type C so as type D correlations. But after averaging over trajectories is zero, and does not disappear. It is stable against disorder and it is this triplet component that gives rise to the direct magneto-electric effect, discussed in the next section.
V direct magnetoelectric effect in a S/NSO/S ballistic junction
In this section we predict that the ballistic S/NSO/S Josephson junction responses to a dc supercurrent flowing across the junction by developing a stationary spin density oriented along the junction interfaces. This phenomenon can be viewed as a direct magnetoelectric effect, i.e. the Edelstein effect. The analogous effect also takes place in normal intrinsic spin orbit coupled metals, where it was first theoretically predicted in Refs. aronov89, ; edelstein90, and later observed experimentally in Refs. kato04, ; silov04, . In normal spin-orbit coupled metals the spin polarization is produced by externally applied electric field. The magnetoelectric polarizability was also discussed in the normal phase of topological insulators rev1 (); rev2 (); essin09 (). Further the magnetoelectric effect was also predicted for bulk superconductors edelstein95 (); edelstein05 () and diffusive superconducting heterostructures malshukov08 (), where its essence is that the supercurrent gives rise to a spin polarization along the direction, determined by the particular type of the spin-orbit coupling. It is also predicted in Josephson junctions on the basis of 3D topological insulators surface states bbza16 (). Therefore, it is natural that the same effect should take place in ballistic spin-orbit coupled Josephson junctions. Below we calculate it on the basis of the formalism, developed in the present work.
To uncover this phenomenon, we first evaluate the average spin polarization:
In terms of the Green function, the components of spin polarization take the following form:
where is the Keldysh component of the quasiclassical Green’s function, which can be expressed via the retarded, advanced components and the distribution function. For the equilibrium problem we consider the above expression can be rewritten as follows: