Surface supercurrent diode effect

The free energy of a 2D superconductor per area is

up to the quartic order of the order parameter $\Delta_{\bm{q}}$ with Cooper pair momentum $\bm{q}$, with quadratic and quaric coefficients $\sigma,\gamma$ respectively. In order to describe SDE, the quartic expansion of $\sigma$ is carried out

where $\bm{H}$ is the in-plane external magnetic field, $\hat{\bm{n}}$ is the normal vector of the 2D superconductor, $t=(T-T_{c})/T_{c}$ is the reduced temperature with critical temperature $T_{c}$, and $a_{0,1},b_{0,1},\gamma$ are coefficients to be given later. The expansion of $\gamma$ in terms of $q$ may also be carried out Ilic , which leads to higher order corrections.

By minimizing $f$, equilibrium Cooper pairs are found to have the finite momentum boosted by $\bm{H}$

In the presence of a gauge potential $\bm{A}$, the free energy is changed upon minimal coupling $\bm{q}\to\bm{q}-2e\bm{A}$, and the supercurrent line density can be computed $\bm{J}=-\partial f_{2{\rm D}}/\partial\bm{A}$. In the 2D limit under in-plane fields we take the limit $\bm{A}\to\bm{0}$, the supercurrent thus reads $\bm{J}=\int\frac{d^{2}\bm{q}}{(2\pi)^{2}}\bm{J}_{\bm{q}}$, where $\bm{J}_{\bm{q}}=e|\sigma|\partial_{\bm{q}}\sigma/\gamma$ is a function of $\bm{q}$, defined in the range such that $\sigma\leq 0$.

Although $q$-linear term is sufficient to describe finite-momentum superconductivity at zero current, higher order $q$-terms are needed for the critical current to be nonreciprocal. When $\sigma=t+a_{0}q^{2}+b_{0}(\hat{\bm{n}}\times\bm{H})\cdot\bm{q}$ is truncated at $q^{2}$-term, although $\sigma(\bm{q})\neq\sigma(-\bm{q})$ is not an even function of $\bm{q}$, it is found that $\sigma(\bm{q}_{0}+{\bm{q}})=\sigma(\bm{q}_{0}-{\bm{q}})$ with respect to the equilibrium momentum $\bm{q}_{0}$. As a result, $\bm{J}_{\bm{q}_{0}+{\bm{q}}}=-\bm{J}_{\bm{q}_{0}-{\bm{q}}}$. Such symmetry guarantees the critical current reciprocity $J_{c}^{+}=J_{c}^{-}$ and $\eta=0$.

To describe SDE, we consider the following expansion near the equilibrium momentum

and find that the finite-momentum Cooper pairs boosted by external field would change the odd order coefficients $b_{0}\to 0,b_{1}\to b^{\prime}_{1}$. Due to this modification, up to the linear order in $\bm{H}$, the supercurrent diode coefficient reads

where $\hat{\bm{i}}$ is the current direction. Details of the calculations can be found in the Appendix.

Next we compute the Ginzburg-Landau coefficients $a_{0,1},b_{0,1},\gamma$ in Eq. (3) from a microscopic model of the Rashba superconductor Yuan . Under an in-plane magnetic field $\bm{H}$, the orbital effect is screened, and the Zeeman effect dominates. Denote $c_{s}$ as the electron with spin $s=\uparrow,\downarrow$, then on the Nambu basis $(c_{\uparrow},c_{\downarrow},-c^{\dagger}_{\downarrow},c^{\dagger}_{\uparrow})$, the Bogouliubov-de Gennes (BdG) Hamiltonian reads

where $H_{\bm{k}}$ describes the normal state electrons with momentum $\bm{k}$, spin $\bm{\sigma}$, mass $m$, Fermi energy $\varepsilon_{\rm F}$, Rashba spin-orbit coupling (SOC) strength $\alpha_{\rm R}$ and the magnetic moment $\mu_{\rm m}$. Electrons form Cooper pairs with momentum $\bm{q}$ under the pairing potential $\Delta_{\bm{q}}$, and $\mathcal{T}=i\sigma_{y}K$ is the antiunitary time-reversal operator with complex conjugation $K$.

To the linear order in $\bm{q}$, we have

The zero-field normal Hamiltonian $H_{0}=H_{\bm{k}}|_{\bm{H}=\bm{0}}$ gives rise to two energy bands $\xi_{\pm}={|\bm{k}|^{2}}/({2m})-\varepsilon_{\rm F}\pm\alpha_{\rm R}|\bm{k}|$ split by SOC. Correspondingly the two Fermi surfaces $\xi_{\pm}=0$ will have the same Fermi velocity $v_{\rm F}=\sqrt{2\varepsilon_{\rm F}/m+\alpha_{\rm R}^{2}}$ but different density of states (DOS) $N_{\pm}=N_{0}(1\mp{\alpha_{\rm R}}/{v_{\rm F}})$ with $N_{0}=m/(2\pi)$ EdelME ; Samokhin ; Dimi1 ; Dimi2 .

There are three depairing terms, from the Zeeman effect $\mu_{\rm m}\bm{H}\cdot\bm{\sigma}$, the Doppler effect $\bm{k}\cdot\bm{q}/(2m)$ of the kinetic energy, and most importantly the Edelstein effect $\frac{1}{2}{\alpha_{\rm R}}(\bm{q}\times\bm{\sigma})\cdot\hat{\bm{n}}$ of the SOC EdelEE ; EdelEE1 , respectively.

The even order coefficients $a_{0,1}$ and the coefficient $\gamma$ can be derived as in the conventional BCS theory

where $C_{0}=7\zeta(3)/8$, $C_{1}=93\zeta(5)/2^{8}$ are numerical constants, $v_{\rm F}=\sqrt{2\varepsilon_{\rm F}/m+\alpha_{\rm R}^{2}}$ is the Fermi velocity. The odd order coefficients $b_{0,1}$ are determined by SOC, through both band splitting effect and Edelstein effect:

From the coefficients above, we can work out the conventional supercurrent diode coefficient

where $H_{P}=1.25T_{c}/\mu_{\rm m}$ is the Pauli limit. This formula has been numerically verified in Ref. Yuan with the correct numerical factor.

There have been a series of works in deriving the correct formula for the SDE. As far as we know, Victor M. Edelstein himself was one of the pioneers to theoretically study the possible SDE in a superconductor without inversion symmetry under an external magnetic field EdelSDE . However, the expansion of $\sigma$ was truncated at $q^{2}$ term, and hence was not able to describe the CSDE as elaborated above. Nevertheless, in Ref. EdelSDE , the Ginzburg-Landau coefficient $b_{0}$ is the same as that in Eq. (11).

Liang Fu and the author studied the SDE of a Rashba superconductor (which we dub as CSDE) in Ref. Yuan by both numerical and analytical methods. The numerical model includes the full effect of Zeeman field, the Doppler effect and the Edelstein effect, hence numerical calculations lead to the supercurrent diode coefficient consistent with Eq. (12). However, in the analytical calculations, the Edelstein effect was absent and the the expansion of $\sigma$ was truncated at $q^{3}$ term only. Interestingly, the analytically derived supercurrent diode coefficient was still the same as Eq. (12), in spite of the absence of two ingredients in the expansion of free energy ($q^{4}$ term and Edelstein term).

Recently, S. Ilić and F. S. Bergeret further studied the SDE Ilic by expand $\sigma$ up to $q^{4}$ term. It was then found that the supercurrent diode coefficient was zero to the linear order of the field without the Edelstein effect.

The Edelstein effect is in fact the driving force of the CSDE. Due to the Edelstein effect, Cooper pairs with momentum $\bm{q}$ will also carry spin polarization $\bm{S}\propto\hat{\bm{n}}\times\bm{q}$ EdelME . The external magnetic field $\bm{H}$ then couples to Cooper pairs via the Zeeman effect $\bm{S}\cdot\bm{H}\propto(\hat{\bm{n}}\times\bm{H})\cdot\bm{q}$ and hence favors a particular direction for $\bm{q}$ and for the supercurrent, leading to the CSDE. To explicitly demonstrate this, one can calculate the odd order coefficients without the Edelstein effect, which leads to smaller values $b_{0}=C_{0}{\alpha_{\rm R}}\mu_{\rm m}/{(\pi T_{c})^{2}},\;b_{1}=C_{1}{v_{\rm F}^{2}\alpha_{\rm R}}\mu_{\rm m}/{(\pi T_{c})^{4}}$ compared with those in Eq. (11). Since $b_{1}/b_{0}=2a_{1}/a_{0}$ without the Edelstein effect, the supercurrent diode coefficient vanishes $\eta\equiv 0$ according to Eq. (6). Moreover, the critical field diverges at zero temperature when the Edelstein effect is not included Samokhin ; Dimi1 ; Dimi2 . To have nonzero supercurrent diode effect at finite temperatures and finite critical field at zero temperature, it would be necessary to include the Edelstein effect in a consistent microscopic theory of the two-dimensional Rashba superconductors.

To improve the supercurrent diode coefficient, we can consider strong-SOC materials. In particular, we can consider the strongest limit of SOC $\alpha_{\rm R}\to v_{\rm F}$, which corresponds to the surface states described by a single Dirac cone Hamiltonian $H_{\bm{k}}=v_{\rm F}(\bm{k}\times\bm{\sigma})_{z}$, so that $\alpha_{\rm R}=v_{\rm F}$.

However, a single Dirac cone cannot exist by itself, but can live with the whole bunch of bulk states. As a result, to describe the supercurrent diode effect of superconducting surface states, one may include the bulk superconductor as well.

We consider a semi-infinite superconductor in the lower half space $z\leq 0$, which its normal conducting phase possesses surface states. In the Ginzburg-Landau (GL) regime, the temperature and the field are near the superconducting phase transitions and the length scales are greater than the zero-temperature bulk coherence length. Since surface states usually extend to the bulk by a few atomic layers TI , in the GL regime the surface states only live on the surface $z=0$ of the superconductor. After washing out short-range details, the GL free energy is composed of the bulk term and the surface term

where the bulk energy density is

and the surface energy density is

Here, $\psi(\bm{r})$ is the superconducting order parameter, $\bm{D}=\nabla-2ie\bm{A}$ is the covariant derivative, and $\bm{A}(\bm{r})$ is the vector potential. Due to the Meissner effect in 3D superconductors, besides the internal field $\bm{B}=\nabla\times\bm{A}$, we also introduce the external field $\bm{H}$.

The bulk free energy density $f_{3{\rm D}}$ is characterized by three parameters $m,\alpha$ and $\beta$, where $m$ is the electron mass, $\alpha=\alpha_{0}(T-T_{c0})$ is a function of temperature $T$, and $T_{c0}$ is the bulk critical temperature. The stability of the bulk free energy requires $m,\alpha_{0},\beta$ to be all positive.

The surface free energy density $f_{2{\rm D}}$ is characterized by the quadratic surface energy $\sigma$ Noah ; Simonin ; Andryushin ; deGennes ; saint and quartic term $\gamma$ as described in Section I. As the bulk free energy describes the second-order phase transition between two phases (i.e. superconducting versus normal), in the boundary free energy the quartic term is irrelevant Diehl and hence can be neglected in the rest of this manuscript.

When surface states are present, the surface energy is negative $\sigma<0$, suggesting the preferential accumulation of Cooper pairs on the surface. As a result, the superconducting phase transition consists of two steps. The first step is at temperatures above the bulk critical temperature, known as the superconducting nucleation, where the electrons condense into Cooper pairs in the surface layer, while the bulk electrons stay mostly normal. In the second step, the temperature is lower than the bulk critical temperature, the bulk is superconducting as well as the surface layer.

By minimizing the GL free energy with respect to $\psi$, we derive the GL equation

and the de Gennes boundary condition Noah ; Simonin ; Andryushin ; deGennes ; saint

where $l_{s}=1/(2m|\sigma|)>0$ is the extrapolation length denoting the thickness of surface superconductivity, and $\hat{\bm{n}}$ is the normal vector of the surface pointing outward.

By minimizing the GL free energy with respect to $\bm{A}$, the Ampere’s law is derived with supercurrent density $\bm{j}$

According to the de Gennes boundary condition in Eq. (17), no bulk supercurrent penetrates through the surface $\hat{\bm{n}}\cdot\bm{j}|_{\partial V}=0$, since $\psi^{*}\hat{\bm{n}}\cdot\bm{D}\psi$ is real on the surface.

Near the superconducting phase transition, $|\psi|$ will be small and the GL equation Eq. (16) can be linearized. We then find the solutions to the linearized GL equation to obtain properties on the superconducting phase transitions. Details on the critical temperature and critical field at zero current can be found in Ref. Noah . We review the main results below.

At zero field and zero current, the surface superconductivity is confined within the surface layer of thickness $\sim l_{s}$, and the onset critical temperature for the surface superconductivity is $T_{c}=T_{c0}+({2m\alpha_{0}l_{s}^{2}})^{-1}$, higher than the bulk critical temperature. Near $T_{c}$, we find

where $\phi=\sqrt{{2\alpha_{0}(T_{c}-T)}/{\beta}}$ is determined by the nonlinear GL equation as shown in the Appendix.

Under finite fields and zero current, superconductivity is in general suppressed by the fields. It is well known that the upper critical field for the bulk is $H_{c2}=\Phi_{0}/(2\pi\xi_{0}^{2})\propto(T_{c0}-T)^{1}$, where $\xi_{0}(T)\equiv\sqrt{1/(2m|\alpha|)}$ is the bulk coherence length and $\Phi_{0}=h/(2e)$ is the flux quantum Abrikosov1 . When $H>H_{c2}$, bulk superconductivity is killed, while surface superconductivity can still survive until a new critical field, which is the critical field $H_{c3}$ of the surface superconductivity. For out-of-plane fields $H_{c3\perp}=\Phi_{0}/(2\pi\xi^{2})\propto(T_{c}-T)^{1}$, where the coherence length is $\xi\equiv 1/\sqrt{2m\alpha_{0}(T_{c}-T)}$, while for in-plane fields $H_{c3\parallel}=0.525{\Phi_{0}}/{\left(\xi^{2}l_{s}\right)^{\frac{2}{3}}}\propto(T_{c}-T)^{2/3}$ Noah .

In this work, we consider weak in-plane magnetic fields. In conventional superconductors we would expect Meissner effect, where magnetic fields are repelled from the bulk. As a result, the magnetic field can only penetrate into the surface layer, and supercurrent can only flow on the surface layer as well. The thickness of this surface layer is another fundamental length scale of the superconductor known as the London penetration depth $\lambda_{L}\equiv(2e)^{-1}\sqrt{m\beta/|\alpha|}\propto|T_{c0}-T|^{-1/2}$.

However, in our case of superconductors with surface states, when $T_{c0}<T<T_{c}$ superconductivity only survives in the surface layer of thickness $\sim l_{s}$ already, and the Meissner effect is different. To see this, we plug in the order parameter Eq. (19) of the surface superconductivity into the Ampere’s law Eq. (18), then near $T_{c}$

It can be seen that the London penetration depth $\lambda_{L}$ does not appear in the spatial distribution of the internal magnetic field $\bm{B}$ at least to the leading order. Instead, the magnetic field fully penetrates into the surface layer of thickness $\sim l_{s}$ where surface superconductivity lives. Deep in the bulk, both magnetic field and superconductivity vanish. This is the Meissner effect in the case of superconductors with surface states.

To be concrete, one can calculate that at zero external current but finite field $\bm{H}$, surface Cooper pairs have zero momentum, while a persistent current $\bm{J}_{0}=\int_{-\infty}^{0}\bm{j}dz=\hat{\bm{n}}\times\bm{H}$ will be induced to screen magnetic field in the bulk. Notice that the supercurrent line density $\bm{J}_{0}$ has the same dimension as the magnetic field $\bm{H}$. At finite external current and field, surface Cooper pairs can have finite momentum $\bm{q}$, and the supercurrent line density is

where $\bm{q}$ is confined in the range $q<1/\xi$ such that the system remains superconducting, and

Notice that $H_{c3\perp}=\Phi_{0}/(2\pi\xi^{2})$ is the out-of-plane critical field of the surface superconductivity with the coherence length $\xi\equiv 1/\sqrt{2m\alpha_{0}(T_{c}-T)}$ and the bulk GL parameter $\kappa=\lambda_{L}/\xi_{0}$, and $\bm{J}_{0}$ is the field-induced current at $\bm{q}=\bm{0}$.

We can then calculate the critical currents parallel and anti-parallel to a given direction $\hat{\bm{i}}$ by maximizing and minimizing $\bm{J}\cdot\hat{\bm{i}}$ over $\bm{q}$ in the range $q<1/\xi$ respectively. Namely, $J_{c}^{+}={\rm max}(\bm{J}\cdot\hat{\bm{i}})$ and $J_{c}^{-}=-{\rm min}(\bm{J}\cdot\hat{\bm{i}})$. At weak field, we find $J_{c}^{\pm}=J_{c}(1\pm\eta)$, the average is $J_{c}=\frac{2\sqrt{3}}{9}I_{0}/\xi$, and the surface supercurrent diode coefficient reads

This surface supercurrent diode coefficient $\eta$ is plotted in the phase plane of magnetic field $H$ and temperature $T$, with supercurrent perpendicular to the field, as shown in Fig. 1. At weak fields, $\eta\propto H(T_{c}-T)^{-3/2}$ and $J_{c}\propto H(T_{c}-T)^{3/2}$, hence the critical current difference $J_{c}^{+}-J_{c}^{-}\propto H$ is temperature independent. At higher fields when $H\geq H_{\rm M}\equiv(l_{s}/\xi)H_{c3\perp}/\kappa^{2}$ we find $\eta=1$ until the critical field $H_{c3\parallel}$ where surface superconductivity vanishes. A perfect supercurrent diode hence could be realized in the field regime $H_{\rm M}<H<H_{c3\parallel}$ on the surface of a superconductor with surface states, as shown in Fig. 1 where the Meissner field $H_{\rm M}\propto(T_{c}-T)^{3/2}$ is denoted by the dashed orange line and the critical field of surface superconductivity $H_{c3\parallel}\propto(T_{c}-T)^{2/3}$ is denoted by the solid blue line.

For further lower temperatures $T<T_{c0}$, the surface together with the bulk are all superconducting. Since the surface states contribution now becomes negligible compared with the bulk, the supercurrent becomes reciprocal deep in the bulk superconducting state. Without surface states, the surface energy is zero $\sigma\to 0$, $l_{s}\to\infty$, and the critical current also becomes reciprocal according to Eq. (23) Abrikosov2 . Thus to realize SSDE, surface states are needed on the surface of a superconductor and the temperature range is close to the onset critical temperature so that only surface superconductivity exists.

We have considered an ideal model of the semi-infinite superconductor with surface states on its only one surface (the top surface). However, realistic materials will at least have two surfaces, the top and the bottom surfaces. We suppose surface states exist on both top and bottom surfaces, with extrapolation lengths ${l_{s}^{\rm top}}$ and ${l_{s}^{\rm bot}}$ respectively, and states on the top surface are decoupled from the bottom surface. Without loss of generality, we assume $l_{s}^{\rm top}<l_{s}^{\rm bot}$, then the onset critical temperature for the surface superconductivity is $T_{c}=T_{c0}+({2m\alpha_{0}|l_{s}^{\rm top}|^{2}})^{-1}$. In the temperature regime $T_{c1}<T<T_{c}$ with $T_{c1}=T_{c0}+({2m\alpha_{0}|l_{s}^{\rm bot}|^{2}})^{-1}$, only the top surface is superconducting while the bulk and the bottom surface are not. Eq. (23) then applies in this temperature regime. For lower temperatures $T_{c0}<T<T_{c1}$, both the top and the bottom surfaces are superconducting while the bulk is not, and the total surface supercurrent diode coefficient from these two surfaces is $\eta=\eta(l_{s}^{\rm top})-\eta(l_{s}^{\rm bot})$.

The iron-based superconductors FeSe_1-xTe_x ($x\sim 0.5$) Gang ; Zhang ; Wang ; Kong provide candidates of superconductors with surface states and hence of SSDE. It has been theoretically calculated that Gang , the $3d$ electrons from Fe atoms will exhibit band inversion in the bulk spectrum, and topological surface states would emerge on the (001) surfaces (top and bottom) as observed experimentally Zhang ; Wang ; Kong . However, due to the topological protection, both the top and bottom surfaces will host surface states, and one cannot eliminate states on one surface while remaining another by local operations. Therefore, in usual samples of FeSe_1-xTe_x, we expect $l_{s}^{\rm top}\approx l_{s}^{\rm bot}$, and hence the SSDE would be weak, unless the top and bottom surfaces can be made sharply different.

Notice that our theory of SSDE in fact does not specify the type and origin of the surface states. Surface states in SSDE may have various origins, such as topological surface states in FeSe_1-xTe_x Gang ; Zhang ; Wang ; Kong as discussed above, surface reconstructions in cleaved or covered superconductors Gao ; Heumen ; Par as will be discussed below, and other crystal defect states Matsu ; Khl .

We thus turn to superconductors with surface reconstruction, which fall into another family of iron-based superconductors, the doped iron arsenides BaFe_2-xCo_xAs₂ Gao ; Heumen ; Par . The surface Ba atoms predominantly favor a $\sqrt{2}\times\sqrt{2}$ order Gao , which leads to the lattice reconstruction of the surface layer. As a result, surface related Fe 3$d$ states are present in the electronic structure in the form of surface bands as both theoretically calculated and experimentally observed Heumen . By changing the surface condition upon cleavage or coverage, one can change the reconstruction and hence the onset critical temperature as observed experimentally Par . Compared with topological surface states in FeSe_1-xTe_x, the surface reconstruction in BaFe_2-xCo_xAs₂ can be well adjusted on just one surface. Consequently, the SSDE in BaFe_2-xCo_xAs₂ is expected to be significant upon proper surface conditions.

Finally we address the dependence of supercurrent diode coefficient $\eta$ on field $\bm{H}$ and current $\hat{\bm{i}}$ in the two types of SDEs in this work, which turns out to be dictated by the underlying point group of the superconducting system. Notice that $\eta$ is time-reversal even while $\bm{H}$ and $\hat{\bm{i}}$ are time-reversal odd, thus we expect $\eta$ to be expressed as a quadratic form of $\bm{H}$ and $\hat{\bm{i}}$, which is invariant under the point group. On the surface of a three-dimensional superconductor or in the plane of a two-dimensional Rashba superconductor, the point groups are of the same type, namely $C_{nv}$ or its subgroup, where $n=2,3,4,6,\infty$. When $n>2$, $\bm{H}$ and $\hat{\bm{i}}$ furnish the same 2D irreducible representation, while the normal vector $\hat{\bm{n}}$ furnishes the 1D irreducible representation $A_{2}$. In order to furnish the trivial representation we find $\eta\propto({\bm{H}}\times\hat{\bm{n}})\cdot\hat{\bm{i}}$ as shown in Eqs. (12) and (23). In a two-dimensional superconductor without the inversion center, however, other quadratic invariants will become possible, since the point group is not limited to $C_{nv}$ Yuan ; Yuan1 . For example $\eta\propto\bm{H}\cdot\hat{\bm{i}}$ when the point group is $D_{n}$ ($n=2,3,4,6$), and $\eta\propto\bm{H}\cdot\hat{\bm{i}}-2(\bm{H}\cdot\hat{\bm{m}})(\hat{\bm{m}}\cdot\hat{\bm{i}})$ when the point group is $D_{2d}$, where $\hat{\bm{m}}$ is the normal vector of the mirror plane.