Inverse Spin Hall Effect in SNS Josephson Junctions
We consider DC supercurrents in SNS junctions. Spin-orbit coupling in combination with Zeeman fields can induce an effective vector potential in the normal conductor. As a consequence, an out-of-plane spin-density varying along the transverse direction causes a longitudinal phase difference between the superconducting terminals. The resulting equilibrium phase coherent supercurrent is analogue to the non-equilibrium inverse spin Hall effect in normal conductors. We explicitly compute the effect for the Rashba spin orbit coupling in a disordered two-dimensional electron gas with an inhomogeneous perpendicular Zeeman field.
The spin-Hall effect (SHE) and inverse SHE (ISHE) are remarkable demonstrations of the influence of the spin-orbit coupling on electron transport. Via this coupling, a longitudinal electric current can induce a perpendicular spin current and vice versa. These effects take place in metals and semiconductors, where the spin-orbit interaction (SOI) arises from impurity scattering Dyakonov () or band structure effects Sinova (). Utilizing spin injection, SHE, and ISHE, electron spins can be controlled, as recently demonstrated experimentally Valenzuela ().
We discuss the intrinsic SHE and ISHE, where the dominant spin-orbit coupling is from the electron band structure. The study of SHE has been focused on normal conductors, e.g. normal metals and semiconductors. New, interesting, and rich physics occurs in superconductors where electron transport is dissipationless and the ground state exhibits macroscopic coherence. Some superconductivity induced features of the intrinsic SHE has recently been analyzed in bulk superconductors Contani () and SNS Josephson junctions MalshukovSNS (). The latter work revealed an equilibrium spin accumulation at lateral sample edges, similar to non-equilibrium spin accumulation in normal conductors, but the spin Hall current vanished due to time-reversal symmetry in the DC Josephson effect.
We focus on ISHE in Josephson junctions. There are two scenarios depending on how the spin current (density) is created in the normal metal. In a dissipative setup, additional normal/ferromagnetic terminals in the transverse direction inject a non-equilibrium spin current. Subsequently, the ISHE induces an electric potential difference between superconducting terminals, causing Josephson oscillations at frequency . Transport is dissipative due to the spin flow between transverse normal/ferromagnetic terminals. This phenomenon is interesting from an experimental point of view and we will study it quantitatively elsewhere, but we consider here a dissipationless effect.
We predict a novel inverse dissipationless SHE: An out-of-plane equilibrium spin density spatially varying in the transverse direction induces a longitudinal electric supercurrent. Equivalently, it induces a phase shift between two superconducting terminals. In general, since the equilibrium spin-density controls ISHE, Zeeman interaction from magnetic or exchange fields manipulates the resulting Josephson supercurrent. As an explicit illustration, we consider the interplay of spin-orbit coupling and Zeeman fields in a disordered two-dimensional electron gas (2DEG), and compute the magnitude of the equilibrium Josephson ISHE.
The interplay of Zeeman field and SOI leading to an effective phase difference between superconducting terminals has recently also been studied in two quite different systems, but neither exhibits the ISHE we discuss: A supercurrent in response to a spatially homogenous magnetic field has been predicted for Josephson tunneling through a 1D wire Krive () and appears in numerical simulations of the superconducting transport through a ballistic point contact Reynoso () in a spatially homogenous parallel magnetic field. In additon to our main finding of an inverse SHE, we provide an improved understanding of these phenomena by showing how the interplay of Zeeman field and SOI can result in the appearance of an effective electromagnetic vector potential. Such a vector potential, in direct analogy with the Meissner effect, gives rise to a supercurrent.
Let us outline our model. The spin-orbit interaction arises from the band structure, , where ) is a vector of Pauli matrices. We ussume that the spin-orbit field is given by Rashba SOI where and . Two examples of spin density manipulations in 2DEG will be considered: i) a perpendicular to 2DEG Zeeman field spatially varying in the transverse direction , as shown in Fig. 1 and ii) homogeneous Zeeman field directed along the -axis. We will show that setup i) exhibits an equilibrium inverse SHE. Setup ii) also changes the current-phase relation in SNS contacts.
All relevant length scales are assumed larger than the mean free path , and we are in the metallic regime , where and are the Fermi wave-vector and velocity, respectively. These conditions allow a diffusion approximation in the description of electron transport. In this regime, the transport properties are described by a generalized Usadel equation, which we will now derive. The resulting Usadel equation is similar to the one in Ref. MalshukovSNS (), but important non-trivial new terms essential for the effects we discuss are added due to the Zeeman interaction where () are the perpendicular (in-plane) components of the Zeeman field. We start from the anomalous retarded thermal equilibrium Green function , which is the Fourier transform of
with respect to the relative coordinate and relative time . It is convenient to use a singlet-triplet basis representing the Green function,
where denotes a spin projection opposite to , . denotes the singlet component. and are triplet components corresponding to and projections of the Cooper’s pair total spin on the -axis. Using a standard method starting from Gor’kov equations Efetov (), we derive the semiclassical equation
where subscripts and attain the values , or , and
Here , is the 33 matrix spin operator in the triplet subspace, and operators and provide mixing of triplet and singlet components:
where . In the right-hand side of Eq. (3) and the unperturbed retarded Green functions are
The diffusion equation can be derived from Eq. (3) by expanding the operator for small and averaging over . The resulting Usadel equation is
where the angular brackets denote averaging over the Fermi surface. The matrix originates from the SOI and the Zeeman interaction expressed via the operators and . Its off-diagonal terms describe singlet-triplet transitions. The relevant matrix elements for our further analysis are :
where and so that e.g. the singlet-singlet diagonal element is proportional to .
In order to understand some of the underlying physics described by Eq. (7), we will demonstrate that SOI in combination with the Zeeman field gives rise to an effective Meissner effect. Let us first discuss this in the most transparent ”local” approximation when the SOI is strong enough/the system long enough, so that the spin diffusion length , where is the length of the junction, is the spin relaxation rate and is the diffusion constant. In this approximation derivatives in triplet parts of Eq. (7) can be disregarded, except in a narrow range near the boundaries. is assumed to vary slowly on the scale. Expressing the triplet components of via the singlet and substituting them into the singlet projection of Eq.(7), the latter takes the form
where is a real coefficient obtained from the equation
Here we have only included dominant terms proportional to and , where is the SOI coupling constant. Higher order contributions to (9) proportional to and have been disregarded.
The diffusion equation (9) demonstrates that is an effective weak electromagnetic vector-potential. Therefore, similar to the Meissner effect it will induce a supercurrent. To order the solution of Eq. (9) is , where satisfies Eq. (9) with . The exponential factor gives rise to an additional phase difference between the superconducting terminals, the Josephson current is , where is the initial phase difference between the terminals and is the critical current determined by the function . The coefficient is simple for Rashba SOI. For a parallel Zeeman field . For a perpendicular field it vanishes, which is expected since it is similar to the behavior of the spin Hall conductance. Continuing such an analogy, one can expect that for the cubic Dresselhaus Dresselhaus () SOI MalshDress ().
In order to find a finite ISHE even for the Rashba SOI, we must extend our consideration beyond the local approximation. In this case the diffusion equation (7) cannot be reduced to the simple form (9). We consider superconducting leads with equal real order parameters connected via two SN interfaces wit a low transparency . The barriers are assumed to extend into the 2DEG under the superconducting leads, so that the range of a free electron motion is between and at the left and right leads, respectively. Depending on contact fabrication, other models can be similarly studied. For example, the electrons in the 2DEG could move freely under contacts, with the barriers present only in -direction, as shown in Fig.1. The choice of the model is not important for the main qualitative results obtained below.
To the lowest order in the tunneling transparency , the superconducting current can be expressed Aslamazov () as a sum over Matsubara frequencies :
where is the boundary resistance Kupriyanov (), and , with , is the Green function of Eq. (7), i.e. a solution of Eq. (7) with a delta source in its right-hand side. The equations for retarded and advanced functions must be properly continued to the upper and lower complex semiplanes of , respectively. Treating in (7) perturbatively one can express the correction to as
where the unperturbed diffusion propagators and , with or , are obtained from Eq. (7) with hard-wall boundary conditions, at and , while the triplet components in the case of Rashba SOI satisfy the boundary condition MalshAccumulation ().
To illustrate the ISHE, we consider the case of Rashba SOI with finite , and . The parameter of interest is the effective phase difference
From Eq. (7), , where and denotes the average value of the the magnetic field gradient in the contact range. is shown in Fig. 2 as a function of ratio of the spin relaxation rate versus the Thouless energy , a convenient measure of the SOI strength. For large SOI the ”local” approximation is obtained by using in the second term of Eq. (Inverse Spin Hall Effect in SNS Josephson Junctions) the approximate form of . In this case both terms in (Inverse Spin Hall Effect in SNS Josephson Junctions) are proportional to and precisely cancel each other, as in the factor in (9). Beyond this leading ”local” approximation there are terms increasing slower than . They contribute to Fig. 2.
Larger Zeeman fields cannot be treated perturbatively. A strong depairing effect takes place when the characteristic length is small, min. Then, for , both and decay exponentially near contacts with superconducting terminals and the latter become effectively disconnected. On the other hand, as it follows from Eq. (7), components are not subject to the depairing effect and can propagate at the relatively large distance . Such a long-range triplet effect has been studied in SFS junctions, where a link between triplet and singlet Cooper pairs has been induced by an inhomogeneous (rotating) magnetization (see Efetov () and references therein). In our case, a coupling of to and can be provided by SOI through the matrix elements and the spin precession operator originating from the first term in the right-hand side of Eq. (7). Indeed, assuming that and , it is easy to show that the modified Eq. (Inverse Spin Hall Effect in SNS Josephson Junctions) is represented by its second term, where the integrand has the form
The unperturbed functions and are obtained from and projections of Eq. (7), where the precession term and all , except and are ignored. The physics of the process described by (14) is clear: the magnetic field mixes 0-triplet and singlet components of the pairing function within the short range near the left boundary. Further, due to the spin precession in the SOI field the 0-triplet transforms to triplet components. The latter propagate to the right contact where they convert to the singlet through . Integrating (14) over and gives a power law dependence of Im on the magnetic field:
where and are -coordinates of the junction edges.
In contrast to a perpendicular Zeeman field, in a parallel field triplets exponentially decay near boundaries, as can be seen from Eqs. (7,Inverse Spin Hall Effect in SNS Josephson Junctions). So they cannot provide a long-range link between superconducting terminals.
In conclusion, an analogue to the ISHE exists in DC Josephson SNS junctions. Unlike the normal ISHE, the supercurrent through the SNS contact can be induced by a static Zeeman interaction by magnetic or exchange fields oriented normal to the 2DEG and varying in the direction transverse to the electric current. A destructive depairing effect of the strong Zeeman field is diminished by Rashba SOI leading to a power-low dependence on this field. We show that a supercurrent through the junction can also be induced by a uniform parallel Zeeman field, corroborating thus the numerical analysis of Ref. Reynoso (). On the other hand, the depairing effect of such a field was found to be strong (exponential). In both cases an appearance of the supercurrent can be explained in terms of the Meissner effect produced by an effective vector potential, which is a combined effect of the Zeeman field and Rashba spin-orbit interaction.
We considered the diffusive transport regime which is relevant in low mobility metals and (magnetic) semiconductors. Furthermore, the diffusive regime, allows an elucidation of the main physics and parameters governing this phenomena. We expect a strong Josephson ISHE in ballistic junctions containing a metallic normal layer with a strong Rashba interaction, for example in Bi films on some substrates Ast (). Ballistic quantum wells of narrow gap semiconductor are also expected to exhibit an increased Josephson ISHE.
A.G.M. gratefully acknowledges hospitality of NTNU.
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