A phenomenological theory of superconductor diodes

We study theoretically the superconductor diodes, where the magnitude of the critical current changes as the direction is reversed, in terms of a generalized Ginzburg-Landau model with the higher-order terms in the momentum of the order parameter. This theory is applied to Rashba spin-orbit coupled systems, where analytical relations between the nonreciprocal critical currents and the system parameters are achieved. Numerical calculations with mean-field theory are also obtained to study broader parameter regions. These results offer a rather general description and design principles of superconductor diodes.


INTRODUCTION
Nonreciprocity in materials [1] refers to the phenomenon where physical quantities change as the system is reversed spatially.It has been well studied in semiconductors and plays a key role in modern technologies such as electrical diodes and solar cells.A new developing subject related to this topic is the contribution by the Berry phase of the electronic states [2] such as shift currents [3].
Nonreciprocity in superconductors (SCs) has recently emerged as an active research topic [4][5][6][7].When both inversion and time-reversal symmetries are broken, magnetochiral anisotropy [1,8] is induced and the conductance near the superconducting transition temperature T ≳ T c , i.e. the paraconductivity, becomes different if the current is reversed.The nonreciprocal part is greatly enhanced as the superconducting order parameter ∆ sc develops, i.e. when T → T c .
The research on the nonreciprocity in SCs has been further promoted by the recent discovery of the superconductor diode effect [9], where the critical currents along opposite directions differ, i.e.I c+ ̸ = I c− .As a result, a superconductor diode has zero resistance along one direction but nonzero along the other if the current is set between I c+ and I c− .This discovery is followed by the observation of its Josephson-junction version [10], which shows a stronger nonreciprocal signal.These experiments make great steps towards coherent superconducting devices.However, a theoretical description of the superconductor diode effect is not well developed.Such a theory is needed not only for fundamental understanding but also for further experimental developments.
Here, we show that the SC diode effect can emerge from magnetochiral anisotropy caused by a combination of spin-orbit coupling (SOC) and external Zeeman fields.A description of SC diodes is given with a generalized Ginzburg-Landau (GL) theory, in which the higher-order terms of the order parameter ψ(r) or of its spatial gradient ∇ r ψ(r) must be present to induce nonzero SC diode effect.This is similar to the importance of the third order term ∇ 3 r ψ(r) to the nonreciprocal paraconductivity [4,5].Physically, these terms correspond to the asymmetry in the energy of the Cooper pairs when they propagate in opposite directions.We apply our theory to two-dimensional Rashba SCs [11,12] and obtain the analytical relations between the strengths of the SC diode effects and the corresponding system parameters.Numerical calculations are further done with Bogoliubovde Gennes mean-field Hamiltonians, which can cover a wider parameter range including lower temperatures and stronger Zeeman fields.

Generalized Ginzburg-Landau theory
In presence of SOC and a Zeeman field, a generalized GL free energy of a superconductor can be written as where ψ q is the order parameter in the reciprocal space.The parameters α, β and γ are conventional GL coefficients.The terms η q = h lmn κ lmn q l x q n y q m z (l + m + n being an odd integer) and hβ 1 • q, originating from spinorbit coupling and external Zeeman field h, break both inversion (P) and time-reversal (T ) symmetries and lead to magnetochiral anisotropy.It is assumed that h ≪ T c (we omit the Bohr magneton µ B , the Boltzmann constant k B and the reduced Planck constant ℏ throughout this paper).The higher order terms corresponding to γ ′ and β 2 are also included for reasons that will be clear later.
The structure of the coupling constants κ lmn and β 1 is directly related to the symmetries of the system and thus depends on the form of the SOC.For example, in a system with continuous rotational symmetry C ∞ (rotation axis along ẑ), the group representation leads to η q = (a 0 + a 2 |q| 2 )(h • q) + (b 0 + b 2 |q| 2 )(h × q) • ẑ, up to the linear order in h and the third order in q.The h • q term breaks all mirror symmetries, while (h × q) • ẑ breaks M z , the mirror symmetry in the z-direction.
In general, even if both P and T are broken, nonreciprocal effects are not necessarily expected.To see that, let us define the inversion operators of each dimension, P x , P y and P z , so that the corresponding symmetry invariance requiresP x/y/z H(k x/y/z )P −1 x/y/z = H(−k x/y/z ) respectively, where H(k) is the Hamiltonian of the system.Obviously P = P x P y P z is broken.However, since breaking P x is necessary for any possible difference between the currents along the ±x directions, a term such as hq 2 x q y , although breaking P and T , would not cause such a nonreciprocity.This means that the direction of the magnetic field to induce nonreciprocal effects (in ±xdirections) needs to be determined by symmetries -it should break all possible P x of the Hamiltonian.This will be illustrated with the example discussed in the later part of this paper.
Consider the case where the magnitude of the SC order parameter is uniform and it only varies in its phase along the x-direction , i.e. ψ(r) = |ψ|e iϕ(x) .This assumption is valid as long as we are dealing with superconductors of thicknesses much less than the coherence length.In this simplified case, the free energy becomes with q = ∂ x ϕ(x).The order parameter may be multicomponent in general.In that case, ψ denotes a certain linear combination of these components which minimizes the energy, and the internal structure of the order parameter does not affect our discussion.Also note that the magnetic field appears as a scalar since it is assumed in the proper direction to be determined by the specific form of the spin-orbit coupling in a concrete model.The terms in Eq. ( 2) do not affect the coherence length since the spatial variation is assumed in the phase ϕ only.As for the κ 1 term, there should be a linear derivative term with respect to the magnitude |ψ| correspondingly.However, it vanishes after integrated over space.Higherorder derivative terms of |ψ| could modify the coherence length, but which is a small effect when we consider a weak magnetic field.
The supercurrent along the x-direction is We have introduced the dimensionless variables q ≡ q γ/|α|, γ′ 3) yields The current I as a function of q has a maximum I c+ and a minimum −I c− , I c± being the critical currents along the positive and negative directions respectively.It should be noted that the supercurrent I is nonzero when q vanishes, as can be seen in Eq.( 5).This indicates that the ground state is not with zero q.Instead, the value of ground-state q is determined by I = 0 which leads to q = q 0 ̸ = 0.This kind of finite-momentum pairing usually accompanies the superconductor diode effect.However, while a q-linear term in the free energy, i.e. κ 1 ̸ = 0 in Eq. ( 2), is enough to induce finite-momentum pairing, the superconductor diode effect requires more, as will be clear soon.When h = 0, the maximum and minimum are readily found at q± → ±1/ √ 3, and thus q ± → |α|/3γ ∼ √ ϵ, where ϵ ≡ 1 − T /T c .With nonzero but small h, the variables γ′ , κ3 and β1,2 are much smaller than unity and the solutions q± are only slightly shifted.The extrema can be obtained by expansion, which leads to the critical currents (up to the first order in h √ ϵ) where we defined the diode quality parameter To see whether a given term in Eq. ( 2) is important to the SC diode effect up to the lowest orders in h and ϵ, one may count the exponent of ϵ in it.Since Symmetry operations on the Zeeman fields along the three directions in Rashba systems.A plus (minus) sign means that the Zeeman term is even (odd) under the symmetry operation.
all the terms in Eq. ( 7) are linear in h √ ϵ.On the other hand, one can show that all terms contributing to Q up to the order ∼ h √ ϵ have been included in Eqs.(1-2).Thus, all the terms in Eqs.(1-2) are important while other higher order terms can be neglected.From Eq. ( 7), it is clear that the q-linear term in the kinetic energy of Cooper pairs, i.e., the κ1 term in η q , would not change the critical current alone, because it only shifts the positions of the maximum and the minimum of Eq. ( 5) while keeping their values unchanged.(For this reason, the divergence of κ1 at ϵ → 0 does not cause problems.) Application to Rashba SCs For example of SC diode effects, let us consider twodimensional SCs with Rashba SOC.The normal Hamiltonian can be written as where λ R is the Rashba spin-orbit coupling strength, k = (k x , k y ) is the electron wave vector, h is the magnetic field, µ is the chemical potential, and σ x,y are Pauli matrices.Two x-inverting symmetries, P x1 = σ x and P x2 = σ z , are preserved when h = 0. Their effects on magnetic fields in different directions are shown in TA-BLE I. To break a symmetry, the Zeeman term must be odd under the symmetry operation.TABLE.I shows that only a Zeeman field along the y-direction breaks both P x1 and P x2 .According to our previous symmetry analysis, a nonreciprocity in the ±x-directions is expected only if h y ̸ = 0.The Rashba SOC and the Zeeman field result in the following term of the GL free energy (up to the linear order in h), Thus, if a magnetic field along the y-direction, h = (0, h y , 0), is applied, the critical currents along the ±xdirection will be different, as previously obtained in Eqs.(6-7) and consistent with the symmetry analysis.
Assuming |h| ≪ T c ≪ E R = 1 2 mλ 2 R and treating the problem in the band basis, one may neglect the interband terms and consider only the intra-band pairing ∆.

FIG. 1.
The Rashba superconductor (SC) diode quality parameter Q predicted by the generalized GL theory.The inset shows schematic band structures and the spin momentum locking.
With this simplification, the GL coefficients in Eq. ( 2) can be obtained and the resulting Q-parameter is where μ 2 where αk F is the Rashba splitting energy at the Fermi surface.)Q R as a function of the chemical potential µ is shown in Fig. 1.The parameter Q R has its maximum at the band crossing point µ = 0 and decreases as the Fermi level moves away either towards the band edge µ = −E R or towards the limit µ ≫ E R .At µ = 0, there exists a kink due to the flip of the helicity of the spin-momentum locking.Note that the kink appears also because we took the limit T c /E R → 0 and neglect the inter-band pairing.The calculation is done in the band basis assuming a constant pairing breaking energy near the Fermi surface, which is true when both µ + E R ≫ ∆ and |µ| ≫ ∆ are satisfied.Near µ = 0, moreover, the smallness of the Fermi wave vector k F , compared to the Cooper pair wave vector |q|, invalidates the series expansion over q/k F for the GL theory.Thus, the kink shall become smooth when T c /E R is not infinitesimal.And our GL theory calculations do not apply near µ = 0 or µ = −E R .
The quality parameter Q R may also be obtained using a self-consistent Bogoliubov-de Gennes mean-field Hamiltonian where i, j =↑↓ are matrix indexes in the spin space.This method applies to wider parameter regions although it is feasible only numerically.Note that the pairing gap ∆ de- pends on the wave vector q since it is determined by minimizing the free energy F (q) = −T n,k ln(1 + e −ϵn/T ), where ϵ n are the eigenvalues of ĤBdG .For a given q, the corresponding supercurrent is I x (q) = 2e∂F/∂q x .The critical currents I c+ and I c− are obtained by finding the maximums of I x (q) and −I x (q) respectively.The diode quality parameter Q R , defined in Eq. ( 7), as a function of µ is shown in Fig. 2, which has qualitatively the same features as those of Fig. 1 obtained with the generalized GL method.The kink at µ = 0 becomes smooth since T c /E R is not so small.In the large µ limit, Q R ∼ µ 1/2 , as shown by the log-scale plot in the inset of Fig. 2.
The temperature dependence of Q R is shown in Fig. 3.It gradually increases as T is lowered from T c , consistent with the prediction, Q R ∼ √ T c − T , by the generalized GL theory.However, as T further decreases, Q R starts to increase dramatically.(Results with temperatures near zero cannot be obtained here due to a numerical convergence problem.) Both analytical and numerical calculations show that the SC diode effect in two-dimensional Rashba systems reaches its maximum at the band crossing point.This suggests that stronger experimental signals may be achieved by tuning the chemical potential closer to zero by, for example, gating, as well as by increasing the magnetic field or decreasing the temperature.

DISCUSSION
We have shown that the superconductor diode effects in single superconductors can be understood with a generalized Ginzburg-Landau theory.They originate from the magnetochiral anisotropy induced by the spin-orbit coupling and the Zeeman field, which breaks the inversion and time-reversal symmetries respectively.Applying our theory to two-dimensional Rashba superconductors, we found that this effect is the strongest at the band crossing point, which may be approached by gating.
The experiments [9] were done in multilayer superconductor thin films which break inversion symmetry strongly due to the heterostructure.This shall induce a strong out-of-plane charge polarization which is compatible with the Rashba model.Although the twodimensional treatment is a simplification, we believe such a model captures the essence of the experimental systems in Reference [9].
On the other hand, the SC diode effect may be experimentally realized in (quasi-) two-dimensional Rashba SOC systems such as a LaAlO 3 /SrTiO 3 interface or a InAs quantum well.
While the former is intrinsically superconducting, the later may be put in proximity to a (quasi-) two-dimensional superconductor (a threedimensional superconductor may totally bury the nonreciprocal signal) or it may form a Josephson junction between two superconductors.
While the phenomenological theory provides a rather general illustration of the origin of superconductor diode effects, further studies on concrete models are to follow in order to reveal different features of this effect in var-ious spin-orbit coupled systems, such as Ising superconductors [13][14][15][16][17]. Superconductor diode effect was also obtained on ferromagnet-superconductor interfaces [18], and in topological superconductors where it may be used to manipulate Majorana fermions [19].
Another way of generating superconductor diode effect may be parity mixing of the order parameters, which has been shown to induce nonreciprocal paraconductivity [5].
When we were finalizing our manuscript, we noticed a recent work [42] on a related topic and were informed that another group [43] had been working on a similar problem.

Derivation of GL coefficients
The GL coefficients are obtained in a standard way by applying perturbation method to the Bogoliubov-de Gennes mean-field Hamiltonian, where H R (k) is the normal Hamiltonian defined in Eq. ( 8) and ∆q = ∆ q iσ y is the SC pairing term.The free energy (q-integrand) up to ∆ 2 q is calculated by and the 4-th order term is g > 0 is the on-site attractive interaction strength, which is to be determined self-consistently for a given T c .In a Rashba SC, we neglect the inter-band pairing and get where ξ ± (k) = 1 2 (ξ e ± − ξ h ± ) and δ ± (k) = 1 2 (ξ e ± + ξ h ± ), with ξ e ± (k) and ξ h ± (k) being the eigen-values of H R (q/2 + k) and −H R (q/2 − k) * respectively.The pair-breaking energies δ ± (k) contains contributions from both the Cooper pair wave vector q and the Zeeman field h = h y ŷ.In the second line, we changed the summation over k into the integral over the energy ξ ± by introducing the densities of states ν ± .Assuming ∆ q is small, δ ± and ν ± may be treated as a constants, and thus The calculation of f (4) (q) follows a similar procedure.Keeping the terms in f (2) (q) and f (4) (q) up to the fourth order in q and to the first order in h y , we obtain the GL coefficients as follows.
When µ > 0, we get When µ < 0, it turns out that the results are related to those for positive µ by a factor of 1/ 1 + µ/E R or 1 + µ/E R depending on whether the corresponding terms are even or odd functions of the Zeeman field h y , i.e.
Note that ζ(x) is the Riemann zeta function and E R = mλ 2 R /2.The expression of Q R in Eq. ( 10) is obtained by substituting the above results into Eq.(7).The contributions of the four terms are of the same order of magnitude (see Supplementary) and thus none of them can be neglected.

9 FIG. 2 .FIG. 3 .
FIG.2.The superconductor (SC) diode quality parameter Q R of Rashba spin-orbit coupled systems as a function of the chemical potential µ calculated numerically with the microscopic self-consistent mean-field theory.The dots in the inset show the same numerical data in the large-µ region, but in log scale.The solid line denotes the relation Q R ∼ µ 1/2 .The parameters: The mass m = 0.5, the Rashba strength λR = 1 (or ER = mλ 2 R /2 = 0.25), the zero-field SC transition temperature Tc = 0.02, the temperature T = 0.01, and the Zeeman energy hy = 0.004.