Electric control of superconducting transition through a spin-orbit coupled interface

We demonstrate theoretically all-electric control of the superconducting transition temperature using a device comprised of a conventional superconductor, a ferromagnetic insulator, and semiconducting layers with intrinsic spin-orbit coupling. By using analytical calculations and numerical simulations, we show that the transition temperature of such a device can be controlled by electric gating which alters the ratio of Rashba to Dresselhaus spin-orbit coupling. The results offer a new pathway to control superconductivity in spintronic devices.

Recently, it was demonstrated that the superconductor proximity effect depends strongly on the amount of Rashba and Dresselhaus coupling that is present in the system 52 . Because of this, we set out to determine if T c could be controlled purely electrically by tuning the ratio of Rashba to Dresselhaus interactions with an electric field when a 2DEG is in electrical contact to a superconductor via a FI, the latter one serving as a source of triplet pairs. In this work, we confirm this conjecture and predict that all-electric control of T c is possible in a S/FI/2DEG device. In addition to a gate voltage control, T c also responds to a change in the FI magnetization orientation, causing our proposed device to function as a combined superconducting transistor and magnetic spin-valve. By superconducting transistor, we mean a device where a gate voltage is used to switch on and off superconductivity in the structure, thus controlling to what extent a supercurrent can flow through the superconductor. This type of functionality is of interest since it corresponds to an electrically controlled transition from finite to zero resistance.
Here we investigate a setup where magnetism and spin-orbit coupling are split into two distinct layers rather than coexisting in the same material 52 , the former being experimentally more feasible to achieve. Furthermore, whereas previous works have modelled the superconductor/ferromagnet interface using spin-independent tunneling boundary conditions, we here use the recently derived boundary conditions for strongly spin-polarized interfaces 53 . This means that spin-dependent tunneling, phase-shifts, and depairing effects for arbitrarily strong polarization are included in our new model whereas this has not been possible previously in the literature.

Results
Proposed experimental setup. Our proposed experimental setup is sketched in Fig. 1. The electrically controlled superconducting switch is based on an S/FI bilayer grown on an epitaxial GaAs-based (e.g. AlGaAs/ GaAs) semiconductor thin-film multilayer. To enable electrical control over the Rasha spin-orbit interaction in the 2DEG, Au gate electrodes are fabricated by electron-beam lithography on a few-nanometer-thick insulating SiO 2 layer deposited after the growth and lithographic patterning of the S/FI stack. A four-point probe setup is used to measure changes in the superconducting critical temperature as a function of applied gate voltage V g . Although the insulating SiO 2 layer should minimize possible modulations in the Curie temperature T c of the FI driven by the applied gate voltage V g , which can alone have an effect on the superconducting proximity effect, control samples without the FI layer should also be fabricated to exclude this possibility. We also note that the Rasha spin-orbit coupling is independent on the polarity of V g 50 . In contrast, V g usually has an opposite effect on T c , meaning that a positive V g normally enhances T c , while a negative V g decreases T c 54 . Therefore, modulations in T c due to V g can also be excluded by investigating variations in the spin-orbit-driven superconducting proximity effect as a function of the V g polarity.
Analytical results. To explain the mechanism of the electric control of T c , we first approximate the multilayer structure as an effective monolayer structure where spin-orbit coupling and magnetic exchange fields coexist. This analogy is relevant because the spin-dependent phase-shifts induced by proximity to a FI are known to act as an effective exchange field in thin superconducting structures 55 . Afterwards, we will confirm the analytical treatment by full numerical simulations performed without these approximations.
To the linear order in the superconducting pair amplitudes, the diffusion equations of the system are 52 The symbols ⊥ f f f , , s refer to the electron pair amplitudes with spin-singlet, short-range spin-triplet, and long-range spin-triplet projections, respectively. We have also defined the the triplet mixing factor The ferromagnetism is described by an in-plane exchange splitting h = h(cos θ e x + sin θ e y ), which is parametrized in terms of a magnitude h and direction θ. We also assume an in-plane spin-orbit coupling, which is described in polar coordinates by a magnitude α β ≡ + A 2 2 and type χ ≡ atan(α/β), where α and β are the Rashba and Dresselhaus coefficients. The spin-orbit coefficients are defined by the single-particle Hamiltonian where m* is the effective mass, p the momentum, and σ the spin. Finally, D is the diffusion coefficient of the material, and  is the quasiparticle energy.
The singlet component f s is produced in all conventional superconductors. When these pairs leak into the adjoining ferromagnet, eqs (1)(2)(3) show that a magnetic exchange splitting h induces a nonzero short-ranged triplet component f , as is well-known 13 . When spin-orbit coupling is present (A ≠ 0), with both Rashba and Dresselhaus contributions (sin 2χ ≠ 0), one also generates the long-range 15 triplet component f ⊥ so long as the magnetization direction satisfies cos 2θ ≠ 0. It is the latter observation which offers several ways to control the long-ranged triplet generation. Firstly, since the triplet mixing term is proportional to cos 2θ, we may enable this mechanism by letting θ → 0, or disable it by letting θ → ± π/4. Secondly, since the same term is also proportional to sin 2χ, where we defined χ = atan(α/β), the mechanism is enhanced for α ≅ β, but suppressed when α ≪ β or α ≫ β. Since the magnetization direction θ can be changed using an external magnetic field, and the Rashba coefficient α can be changed using an external electric field, this means that the triplet mixing can be in principle be controlled using either a magnetic field by itself [56][57][58][59][60] , an electric field by itself, or a combination thereof.
It is important to note that the spin-orbit coupling not only introduces a coupling between the different types of spin-polarized Cooper pairs, but that it also has a depairing effect. This is seen by how A modifies the diagonal terms in the equations above, resulting in an alteration of the effective energies in eqs (5) and (6) associated with the superconducting correlation functions f. Imaginary terms in the effective energy can be interpreted as a destabilization and suppression of the given correlations, so the spin-orbit coupling can suppress either f , f ⊥ , or both, depending on the parameters χ and θ. It follows from eq. (5) that increasing the magnitude of A and sin 2χ increases this pair-breaking effect, meaning that the same spin-orbit coupling that maximizes the triplet mixing also maximizes the depairing. However, while the mixing term is proportional to cos 2θ, the depairing terms are proportional to sin 2θ. A key observation which enables the purely electric control over T c is that for a fixed magnetization orientation θ, the depairing energy is controlled by the ratio of Rashba and Dresselhaus spin-orbit coupling χ = atan(α/β). This argument is of importance since we from the numerical simulations find that the dominant effect of the spin-orbit coupling on the critical temperature is not the long-range triplet generation, but rather the short-range triplet suppression. In fact, the most extreme results were obtained for θ = ± π/4, which are precisely the configurations where the linearized diffusion equations disallow triplet mixing.
Numerical results. We have calculated T c numerically and the results are shown in Fig. 2. S is taken as conventional (e.g. Nb), the FI (e.g. GdN, EuO) is treated as a polarized spin-active interface, and the semiconducting layer (e.g. GaAs, InAs) is treated as a normal metal with a Rashba-Dresselhaus spin-orbit coupling. We used the Ricatti-parametrization 61 including the case of spin-orbit coupling 52 together with general magnetic boundary conditions 53 valid for arbitrary polarization of the interface region. We provide a detailed exposition of the computation of the critical temperature in the Methods section.
For all structures, we assumed a thickness of 0.65ξ for the superconductor and 0.15ξ for the 2DEG, where ξ is the zero-temperature coherence length of a bulk superconductor. Assuming ξ = 30 nm, this would imply a superconductor thickness of ~20 nm and thickness of ~4 nm for the spin-orbit coupled layer. As for the magnitude of the spin-orbit coupling, we normalized both α and β to ξ / 2  . If the effective quasiparticle mass m* is assumed equal to the bare electron mass, and we again set ξ = 30 nm, we find that α, β = 1 in dimensionless units corresponds to a coupling α/m * , β/m * = 2.2 × 10 −12 eV m. The spin-active interface was taken to have an experimentally realistic spin-polarization of 50%, a tunneling conductance G T /G ∈ {0.2, 0.3}, and a spin-mixing conductance G ϕ /G T = 1.25, where G is the bulk normal-state conductance of both materials (taken as equal for simplicity). We have run extensive T c calculations for other parameter values as well (not shown here), where we find qualitatively the same behavior as in Fig. 2, but quantitatively less variation if either the tunneling conductance G T is reduced, the spin-mixing conductance G ϕ is reduced, or the spin-polarization is increased. In particular, depending on the quality of the contact between the 2DEG and the FI layer, the tunneling conductance could be very small compared to the normal-state conductance, G T ≪ G.
The results in Fig. 2 display the same basic dependence on the magnetic field direction: the critical temperature is maximal when θ → − π/4, and minimal when θ → + π/4. It is interesting to note how spin-valve functionality is obtained in the present structure with just one magnetic layer, tuning T c from a maximum to minimum upon 90 degrees rotation of the magnetization. The magnitude of this variation depends strongly on the parameters. For strong spin-orbit coupling and moderate interface conductance, we see a variation of nearly 0.6T cs in Fig. 2, where T cs is the critical temperature of a bulk superconductor. This corresponds to 5.5 K for niobium; for comparison, the current experimental record for spin-valve effects is around 1 K 62 . Furthermore, in the region where θ > 0, the critical temperature drops to zero, which means that such a device could in principle function as a spin-valve even at absolute zero. Increasing the interface polarization or weakening the spin-orbit coupling diminishes this effect.
Scientific RepoRts | 6:29312 | DOI: 10.1038/srep29312 The most interesting observation is nevertheless that we can achieve all-electric control over T c for a fixed orientation θ of the FI magnetic moment. One particularly striking example is seen in Fig. 2: for a range of magnetization orientations θ, T c increases from absolute zero at α/β = 1 to a substantial fraction of the bulk critical temperature T cs as α/β is either increased or decreased. For instance, when θ/π = − 0.125, T c = 0 at α/β = 1 while T c = 0.42T cs at α/β = 1.5. For e.g. niobium, this yields a variation of 3.9 K by increasing the Rashba coefficient α by 50%. We highlight this behavior in Fig. 3, where T c is plotted against the spin-orbit coupling ratio α/β for a fixed magnetization orientation. Moreover, we show in Fig. 3 the large change in T c that occurs when altering the in-plane magnetization orientation θ for a fixed α/β.
Let us now interpret the numerical findings in terms of the previous analytical treatment. Although the 2DEG by itself has no intrinsic exchange field, rendering the distinction between short-ranged and long-ranged pairs more accurately described by the terminology "opposite and equal spin-pairing states relative the FI orientation", we will continue to refer to f as short-ranged pairs for brevity and easy comparison with the analytical treatment. When α → β and θ → − π/4, the short-ranged triplet energy → + E i DA 4 2


, resulting in a strong suppression of these triplet pairs. By closing the triplet proximity channel, this reduces the leakage of Cooper pairs from the superconductor, thus increasing the critical temperature of the structure. On the other hand, when α → β and θ → + π/4, the energy  → E , resulting in a minimal suppression of short-ranged triplets. This causes a larger  leakage from the superconductor, and decreases the critical temperature. This leading-order analysis of the physics is in accordance with the numerical results in Fig. 2, as is reasonable since the weak proximity effect described by the linearized equations is expected to be a good approximation for T ≅ T c .

Discussion
In the quasiclassical theory used to compute the critical temperature, one assumes that the thickness of the layer exceeds the Fermi wavelength. This criterion is not satisfied in a 2DEG, which means that phenomena such as weak localization/antilocalization cannot be described by quasiclassical theory. However, the coupling mechanism governing the appearance of a superconducting triplet proximity channel in the system is not expected to change because of this and hence our results should remain qualitatively valid even in this scenario. Moreover, we have considered the diffusive limit of transport which is of relevance for the in-plane motion, whereas the 2DEG thickness is much smaller than the mean free path. There is nevertheless scattering at the multiple interfaces of our structure which is expected to enhance the effective diffusive character of quasiparticle motion considered in our model. 2DEGs can also feature a rather strong spin-orbit interaction, in which case corrections to the Usadel equation have been examined 63 . It could also be of interest to go beyond quasiclassical theory to study T c and other proximity effects in this kind of system 64 , although this is beyond the scope of the present work. Semiconductors such as GaAs and InAs are known to provide both an intrinsic Dresselhaus coupling and an electrically tunable Rashba coupling [48][49][50] . By combining such 2DEG materials with a superconductor and a ferromagnetic insulator, we have shown both analytically and numerically that T c responds to changes in both electric and magnetic fields, either individually or combined. It should therefore be possible to create a device that can function as a superconducting transistor, superconducting spin-valve, or both, depending on whether electric or magnetic stimuli are used as the input signal.

Methods
Diffusion equation. In the diffusive and quasiclassical limit, we can describe the structures discussed herein with the Usadel diffusion equation 15,16,52 τ z where ĝ is the retarded quasiclassical propagator in Nambu ⊗ Spin space, τ z is the third Pauli matrix in Nambu space,  is the quasiparticle energy of the electrons and holes, ∆ = , h is the ferromagnetic exchange field, σ is the Pauli vector in spin space, and D is the diffusion coefficient.
is a background field that accounts for spin-orbit coupling. In this paper, we assume that we have a thin-film structure oriented along the z-axis so ∇ → ∂ z . We assume the exchange field and Rashba-Dresselhaus coupling are both confined to the xy-plane, so they can be parametrized as x y x y x x y y Note that it is the orientation of the spin-orbit field A that defines the x-and y-axes of our coordinate system, since the Dresselhaus spin-orbit coupling is determined by the crystal structure. Thus, the magnetic orientation is measured relative to the crystal structure.
Numerically, solving directly for the propagator ĝ is impractical for two reasons. Firstly, the elements of ĝ are unbounded, and can be arbitrarily large complex numbers. Secondly, the propagator satisfies a normalization condition and particle-hole symmetry which reduces the number of degrees of freedoms, such that solving for each individual matrix element in ĝ would be redundant. Because of this, we have used the so-called Riccati parametrization of the propagators in the numerical simulations 61 : where the Riccati parameters γ and γ  are 2 × 2 matrices in spin space which are related by tilde-conjugation γ ε γ ε + = −  ⁎ ( ) ( ), and the normalization matrices are defined by For an in-plane spin-orbit interaction, eq. (7) parametrizes as Gap equation. Before we can calculate the critical temperature of a material, we require not only a way to calculate the propagator ĝ, but also a way to dynamically update the superconducting gap Δ based on the calculated propagators. This gap equation can be written 52 Scientific RepoRts | 6:29312 | DOI: 10.1038/srep29312 Boundary conditions. Since our purpose is to model a system where the superconductivity, ferromagnetism, and spin-orbit coupling originate from different thin-film layers, we need boundary conditions that connect the propagators of these materials at the interfaces. Numerically, we focused on S/FI/N structures, in which case the ferromagnetic insulator itself is modelled as a strongly polarized spin-active interface. We have used the low-transparency limit of the general spin-active boundary conditions derived in ref. 53, The matrix current is related to the propagators at the interface by = ∇ˆ˜Î GL g g ( ) where G is the normal-state conductance and L the length of the material. It can then be shown that the Riccati parameters must satisfy the boundary condition z 1 12 11 where I 12 and I 11 refers to the top-right and top-left 2 × 2 blocks of the 4 × 4 matrix current Î . In this equation, the matrix current Î should be interpreted as either Î L or Î R , depending on which side of the interface the boundary conditions should describe.
Critical temperature. The critical temperature can be defined as the temperature T c such that the superconducting gap Δ = 0 if and only if T ≥ T c . However, in practice, we cannot expect to obtain the exact result Δ = 0 in simulations due to inexact numerical methods and random floating-point errors. For numerical simulations, we therefore use a more relaxed criterion |Δ | < δ to define the critical temperature T c , where we have set δ = 10 −5 Δ 0s , and Δ 0s is the zero-temperature gap of a bulk superconductor. See the solid black curve in Fig. 4 for a sketch of how Δ (T) typically behaves, and how this is related to the critical temperature T c .
Conceptually, the simplest way to find this critical temperature is to explicitly calculate the superconducting gap Δ as a function of temperature T, and check directly at which temperature we first find |Δ | < δ. However, such a linear search can be very costly when a high accuracy is desired. For instance, to determine the critical temperature to a precision of 0.0001T cs , where T cs is the critical temperature of a bulk superconductor, this would require that Δ (T) be calculated for 10,000 different values of T. For each of these temperatures, we need to solve a set of nonlinear diffusion equations for 150 positions and 800 energies, and repeat this procedure in one material Figure 4. Sketch of the superconducting gap Δ as a function of temperature T for a superconducting hybrid structure. When performing a binary search for the critical temperature, we check whether Δ > δ at a certain number of temperatures-in other words, whether the black solid line is above the red dashed line. The numbered markers show which points on the curve would be evaluated during the first five bisections of such a binary search, and in what order. Note that since the algorithm is actually looking for the intersection between the black solid curve and red dashed curve, we need δ ≪ Δ 0s to obtain accurate results.
Scientific RepoRts | 6:29312 | DOI: 10.1038/srep29312 of the hybrid structure at a time until a selfconsistent solution is found. Thus, the calculation at each of these 10,000 temperatures can in some cases take hours, making this method quite inefficient.
We have instead used a much more efficient binary search algorithm to determine the critical temperature numerically. The main benefit of this algorithm is that after calculating Δ (T) for N particular values of T, we can determine the critical temperature to a precision T cs /2 N+1 . So in contrast to the linear search algorithm, an accuracy of around 0.0001T cs would require calculations at 12 temperatures instead of 10,000. Furthermore, we do not actually need to calculate Δ (T) exactly at these temperatures-it is sufficient to check whether |Δ | < δ or |Δ | > δ to determine whether T > T c or T < T c . Thus, at each of these 12 temperatures, we only have to initialize the entire system to a BCS superconducting state with Δ = δ, then solve the Usadel equation and gap equation a fixed number of times in each material, and finally check whether |Δ | < δ or |Δ | > δ to determine whether T is an upper or lower bound on T c . How the binary search algorithm converges is illustrated in Figs 4 and 5.