Superconducting size effect in thin films under electric field: mean-field self-consistent model

TL;DR

This paper presents a self-consistent mean-field model analyzing how an external electric field influences superconductivity in thin films, revealing energy scaling behaviors near subband edges and discussing nonlinear effects beyond simple gating.

Contribution

It introduces a mean-field self-consistent model for superconducting thin films under electric fields, highlighting energy scaling near subband edges and nonlinear effects, contrasting with experimental observations.

Findings

Electrostatic effects scale as μ^{-1} near subband edges

Energy scaling follows μ^{-2} away from subband edges

Discussion of nonlinearities beyond gate effects

Abstract

We consider effects of an externally applied electrostatic field on superconductivity, self-consistently within a BCS mean field model, for a clean 3D metal thin film. The electrostatic change in superconducting condensation energy scales as $\mu^{-1}$ close to subband edges as a function of the Fermi energy $\mu$, and follows 3D scaling $\mu^{-2}$ away from them. We discuss nonlinearities beyond gate effect, and contrast results to recent experiments.

Equations

$$F[\Delta ,\varphi ]$$

$$\displaystyle+\int\mathrm{d}^{3}r\,\Bigl{(}\rho\phi-\frac{\epsilon_{0}}{2}(\nabla\phi)^{2}+\int_{0}^{\frac{1}{T}}\mathrm{d}\tau\,\frac{|\Delta(\tau)|^{2}}{\lambda(\tau)}\Bigl{)}\,,$$

$${\mathcal{G}}^{-1}$$

$$\begin{array}{rl}-{ϵ}_0{\nabla }^2\varphi (\mathbf{r})& =\rho (\mathbf{r})-e{n}_e(\mathbf{r})\\ & =\rho (\mathbf{r})+eT\sum _{{\omega }_n}\mathrm{tr}{\tau }_3\mathcal{G}(\mathbf{r},\mathbf{r},{\omega }_n),\end{array}$$

$$\Delta (\mathbf{r})=-\frac{1}{2}T\sum _{|{\omega }_n|<{\omega }_c}\lambda (\mathbf{r})\mathrm{tr}{\tau }_1\mathcal{G}(\mathbf{r},\mathbf{r},{\omega }_n),$$

$$\mathcal{G}$$

$$n_e[\varphi ]=\sum _{n=1}^{\infty }2m|{\psi }_n{|}^2\gamma ({\xi }_n),$$

$$[-\frac{1}{2m}{\partial }_x^2-\mu -e\varphi (x)]{\psi }_n={\xi }_n{\psi }_n,{\psi }_n(\pm \frac{L}{2})=0,$$

$$n(\xi )$$

$$\gamma (\xi )$$

$$f_{ns}(\Delta )$$

$$\equiv \frac{{\Delta }^2}{\lambda }-\frac{m}{2\pi L}{\int }_0^{\Delta }\mathrm{d}\Delta \Delta \sum _{n=1}^{\infty }g(\frac{{\xi }_n}{\Delta }),$$

$$g(y)$$

$$f_{ns}(\Delta )$$

$$\delta f_{ns}$$

$$\displaystyle=-\frac{m}{2\pi L}\int_{0}^{\Delta}\mathrm{d}\Delta\,\Delta\sum_{n=1}^{\infty}\Bigl{(}g(\frac{\xi_{n}}{\Delta})-g(\frac{\xi_{n}^{(0)}}{\Delta})\Bigr{)}$$

$$\simeq \frac{2m}{L}\sum _{n=1}^{\infty }\delta \gamma ({\xi }_n)\delta {\xi }_n,T=0,{\omega }_c\to \infty ,$$

$$\begin{array}{r}-T\mathrm{Tr}\ln {\mathcal{G}}^{-1}+T\mathrm{Tr}\ln {\mathcal{G}}^{-1}{|}_{\varphi ={\varphi }_0}\\ =\int {\mathrm{d}}^{3}r(-e)n_e[\Delta ,{\varphi }_0](\mathbf{r}){\varphi }_1(\mathbf{r})\\ +\frac{1}{2}\int {\mathrm{d}}^{3}r{\mathrm{d}}^3{r}^{\prime }{e}^2\Pi [\Delta ,{\varphi }_0](\mathbf{r},{\mathbf{r}}^{\prime }){\varphi }_1(\mathbf{r}){\varphi }_1({\mathbf{r}}^{\prime })+\dots ,\end{array}$$

$$f[\Delta ]$$

$$=f[\Delta ,{\varphi }_0]+\sum _{\pm }\frac{\mp {ϵ}_0{E}_{\pm }}{L}[{\varphi }_0+\frac{1}{2}{\varphi }_{1,\ast }{]}_{x=\pm \frac{L}{2}}+C,$$

$${\rho }_1$$

$$\delta {\xi }_n\simeq ⟨n|(-e)\varphi |n⟩=\frac{{\lambda }_{TF}^2}{L}e{E}_{+}q(2{\lambda }_{TF}{k}_n),$$

$$\delta f_{ns}\simeq |f_{ns,3D}|\frac{2eE{a}_0}{\sqrt{{\xi }_n^2+{\Delta }^2}+|{\xi }_n|}\frac{{\pi }^2}{4(k_{F}L{)}^2}q(2{\lambda }_{TF}{k}_n),$$

$$\delta f_{ns}^{(2)}$$

$$F[{\Delta }_{\ast },{\varphi }_{\ast },A]=F[{\Delta }_{\ast },{\varphi }_{\ast },0]+\frac{{\hslash }^2\mathcal{V}}{2}{D}_{ij}^s{A}_i{A}_j+\dots .$$

$$n_s(\xi )$$

$$\Pi (x,x^{\prime })$$

$$=2{\int }_{-\infty }^{\infty }\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2{A}_N(\mathbf{r},{\mathbf{r}}^{\prime };{\xi }_1)A_N(\mathbf{r},{\mathbf{r}}^{\prime };{\xi }_2{)}^{\ast }$$

$$\times \frac{n({\xi }_1,\Delta )-n({\xi }_2,\Delta )}{{\xi }_1-{\xi }_2},$$

$$A_N(x,x^{\prime };\xi )=\sum _{p=1}^{\infty }\frac{2}{L}\sin [k_p(x+\frac{L}{2})]\sin [k_p(x^{\prime }+\frac{L}{2})]\delta (\xi -E_p),$$

$$\Pi (x,x^{\prime })$$

$$\times \sin [k_q(x+\frac{L}{2})]\sin [k_q(x^{\prime }+\frac{L}{2})]$$

$$\times \frac{\gamma (E_p-\mu ,\Delta )-\gamma (E_q-\mu ,\Delta )}{E_p-E_q},$$

$${\rho }_e(\mu ,T=0,\Delta )$$

$$\simeq \frac{(2m{)}^{3/2}}{16\pi }\frac{{\Delta }^2}{\sqrt{-\mu }}.$$

$${\partial }_x^2\varphi (x)$$

$$\Rightarrow \varphi (x)$$

Full text

I Introduction

Quantum oscillations in superconducting properties due to size quantization in thin films were predicted early Blatt and Thompson (1963); Falk (1963); Shanenko et al. (2007, 2015), and they were later observed in metallic films Orr et al. (1984); Guo et al. (2004); Zhang et al. (2005); Bao et al. (2005). Modification of superconducting properties by changing the electron density by electrostatic fields was also observed, Mannhart (1992); Mannhart et al. (1993); Ahn et al. (2003); Matthey et al. (2007); Ahn et al. (2006); Shvarts et al. (2005) and is best studied in high-Tc superconductors where the charge density can be low enough to enable efficient gating. Generally, modifications of critical temperature $T_{c}$ and critical current $I_{c}$ have been reported. Modification of $I_{c}$ only was also recently reported in Refs. Shvarts et al., 2005 and De Simoni et al., 2018; *paolucci2018-ues; *paolucci2018-mef in metallic thin-film samples, but the proper interpretation in the latter is still unclear.

Electrostatics of superconductors is an old problem (see e.g. Ref. Lipavský et al., 2002 for a historical review), and the effect of electric fields on superconducting surfaces are theoretically discussed in several works. Shapiro (1984, 1985); Lee et al. (1996); Burlachkov et al. (1993); Lipavský et al. (2006); Morawetz et al. (2008, 2009); Ummarino et al. (2017) In these, effects on the amplitude of superconductivity ($T_{c}$) are usually related to modulation of electronic density of states (DOS), which is also what contributes to the quantum size effects. A common approach is to consider “surface doping” and assume the DOS is modified within a Thomas–Fermi screening length from the surface. Self-consistently screened calculations in superconductors have been previously discussed in Refs. Koyama, 2001; Machida and Koyama, 2003; Woods et al., 2018, in a different context. For the normal state, there is a large literature on microscopic calculations with surface screening, which are routine today e.g. using density functional theory. Lang and Kohn (1970); Ummarino et al. (2017) Modification of $I_{c}$ on the other hand is often assumed to come from changes in the vortex surface pinning potential. Mannhart et al. (1991); Shvarts et al. (2005)

In a simple picture, a static electric field appears as a perturbation of the potential that confines electrons within the thin film. Static fields generally extend up to a screening length from the surface, and so their effect decreases towards high charge density. Although the effects increase with the applied electric field, achievable field magnitude is limited by electric breakdown (e.g. via field emission Gadzuk and Plummer (1973)).

Electrostatic gating of superconductivity in the BCS mean field picture relies on electron-hole asymmetry within an energy window determined by the order parameter and Debye frequency centered at the Fermi level. Adkins and Waldram (1968); Khomskii and Kusmartsev (1992); Khomskii and Freimuth (1995); van Otterlo et al. (1995) In a simple clean thin-film model, strong asymmetry naturally exists in the form of the steplike multiband 2D DOS, which also gives rise to the quantum size effect, and the picture also extends to weakly disordered samples. The only question is to what degree the DOS asymmetry is retained, even though sharp features in the DOS are smeared by disorder, Belitz and Kirkpatrick (1994) and when samples cannot be significantly gated (metallic regime De Simoni et al., 2018; *paolucci2018-ues; *paolucci2018-mef), since the Fermi level is not necessarily fixed at a sensitive point. Regardless, sharp DOS features can increase the charge density range in which electrostatic effects are large enough to be observed. Motivated by the recent experimental results De Simoni et al., 2018; *paolucci2018-ues; *paolucci2018-mef where large effects were seen, we revisit the problem.

In this work, we write down and solve a simple mean-field model for superconductivity in thin films under electric fields, including self-consistent screening. We point out connections between the dependence of electrostatic energy on superconductivity and modulation of superconductivity by fields, and discuss applicability of “surface doping” models in this picture. We also discuss to what degree nonlinear effects beyond linear electrostatic gating could appear in strong fields. We conclude that effects such as observed in Ref. De Simoni et al., 2018; *paolucci2018-ues; *paolucci2018-mef likely are not present in the model considered.

The manuscript is structured as follows. In Sec. II we introduce the mean-field model considered and discuss results obtained for the electric fields and modulation of superconducting properties. Sec. III concludes with discussion.

II Mean-field model

Self-consistent electrostatic screening and the size effect on superconductivity in a clean superconducting metal in a simple mean-field approximation is convenient to consider starting from a Hartree–Bogoliubov free energy. It can be obtained Rice (1967); Ambegaokar et al. (1982); van Otterlo et al. (1995, 1999) by decoupling a long-ranged Coulomb and a (retarded) superconducting contact interaction via Hubbard–Stratonovich transformations, and considering only the classical saddle point in the static limit:

[TABLE]

Here, $\mathcal{G}$ is the electron equilibrium Green function, $U$ is a background potential, $\mu$ chemical potential, $\phi$ is equivalent to the static electric potential, $\Delta$ the superconducting order parameter, and $\rho$ ion and external charge density. The electron charge is $-e$ and we use units with $\hbar=k_{B}=1$. The first term in the free energy is the electronic contribution, and the second part contains the electrostatic and superconducting mean-field contributions. In the absence of currents and magnetic field, at saddle point with suitable gauge $\Delta$ can be chosen real-valued and the values of vector potential and phase are zero. Above, $\phi$ has to be taken as the saddle-point solution, which as typical for variational Poisson does not minimize $F$.

Variations vs. $\phi$ and $\Delta$ give the Poisson and BCS self-consistency equations:

[TABLE]

where $\mathcal{G}$ satisfies the Gor’kov equations $\mathcal{G}^{-1}\mathcal{G}=1$ under the self-consistent potentials. We also here consider a BCS weak-coupling model, with $\Delta(\omega)=\Delta\theta(\omega_{c}-|\omega|)$, with the coupling $\lambda$ taken as constant and the cutoff $\omega_{c}$ similar to the Debye frequency. In bulk, the BCS gap equation is then directly $\Delta=2\omega_{c}e^{-1/(N_{0}{}\lambda)}$ with $N_{0}$ the DOS per spin at Fermi level.

For uniform system, expanding $\mathcal{G}$ in Eq. (3) to lowest order in $\phi$ results to $\epsilon^{\rm RPA}(\bm{q})q^{2}\phi(\bm{q})=\delta\rho(\bm{q})$, where $\epsilon^{\rm RPA}(\bm{q})=\epsilon_{0}-\frac{e^{2}}{q^{2}}\Pi(\bm{q};\Delta)$ is the self-consistent static dielectric function of a clean superconductor. Prange (1963); Seiden (1966) In this model, the static fields are screened, and external charge density affects the electronic DOS, but not the Coulomb effect Morel and Anderson (1962) to $\lambda$. The latter is due to considering mean-field with the decoupling assumed; corrections appear from fluctuations of $\phi$ (see e.g. Ref. Fischer et al., 2018 for explicit calculations), or on mean-field level with different decoupling Schulz (1990).

Aiming to describe effects on a qualitative level, we now consider a simplified model, similar to those used in several previous studies of the quantum size effect in superconducting thin films. Blatt and Thompson (1963); Shanenko et al. (2007) A confining potential $U(\bm{r})$ is taken to be an infinite quantum well at $|x|<L/2$, which supports some number of populated 2D electronic subbands (see Fig. 1). In a static problem without currents, the electric field is perpendicular to the metal surface, and the problem is inhomogeneous only in $x$-direction. Moreover, we take as a variational Ansatz $\Delta$ spatially constant Falk (1963) inside the well; the resulting energies will then be upper bounds to the exact solutions.

With these assumptions, the problem is elementary and mostly given by known results, Anderson (1959); Falk (1963) and can be solved without further approximations. First,

[TABLE]

where $A_{N}$ is the normal-state spectral function (per spin), $u,v=[\frac{1}{2}(1\pm\frac{\xi}{\epsilon})]^{1/2}$ and $\epsilon=\sqrt{\xi^{2}+\Delta^{2}}$. Due to the spatial symmetry, the problem reduces to one dimension. The normal-state DOS per volume is $\nu(\xi)=\frac{2}{\mathcal{V}}\int\mathrm{d}^{3}r\,A_{N}(\bm{r},\bm{r},\xi)=\sum_{n=1}^{\infty}\frac{m}{\pi L}\theta(\xi-\xi_{n})$ where $\xi_{n}$ are the 2D subband edges and $\mathcal{V}$ the film volume. The subbands and the potential $\phi$ are obtained from the Schrödinger–Poisson problem, Eq. (3) with

[TABLE]

where $\psi_{n}(x)$ are the transverse wave functions of the 2D subbands. Here, $\gamma$ describes the contribution to the charge from each subband:

[TABLE]

where $n(\xi)\to u^{2}f_{0}(\epsilon)+v^{2}(1-f_{0}(\epsilon))$ for $\omega_{c}\to\infty$ and $f_{0}$ is the Fermi function. Tinkham (1996) The occupation factor $n$ is broadened by the interactions in a window $\Delta$ around the Fermi level, with the deviation from the Fermi function starting to decay more rapidly beyond the interaction range at $|\xi|\gtrsim\omega_{c}$. Variations in the DOS within this window contribute to the charge response of the amplitude of superconductivity (see Fig. 1). Adkins and Waldram (1968); Hong (1975); Khomskii and Kusmartsev (1992)

To be specific, we assume an external charge density outside the sample (e.g. capacitor plates with constant charge density) such that the amplitudes of the electric fields at the surfaces are fixed, $-\partial_{x}\phi(\pm\frac{L}{2})=E_{\pm}$. Numerically, the nonlinear Poisson problem can be solved iteratively, Eyert (1996) for a fixed value of $\Delta$.

The condensation energy $f_{ns}(\Delta)=(F[\Delta,\phi_{*}[\Delta]]-F[0,\phi_{*}[0]])/\mathcal{V}$ per volume for fixed $\Delta$ now depends only on the density of states. Anderson (1959); Falk (1963) Via direct calculation,

[TABLE]

where we note that $\frac{d}{d\Delta}F[\Delta,\phi_{*}]=\partial_{\Delta}{}F[\Delta,\phi_{*}]$ at the saddle point $\phi_{*}$. Further, Falk (1963)

[TABLE]

Here, $g(y)\to\int_{y}^{\infty}\mathrm{d}x\,\frac{1}{\sqrt{1+x^{2}}}\frac{2}{\pi}\arctan\frac{\omega_{c}/\Delta}{\sqrt{1+x^{2}}}$ for $T=0$ and further $g(y)\to\operatorname{arsinh}(\omega_{c}/\Delta)-\operatorname{arsinh}{}(y)$ for $\omega_{c}\gg\Delta$, $T=0$. For $T=0$ and $\omega_{c}\to\infty$,

[TABLE]

where $\eta(y)=\operatorname{arsinh}y+(\sqrt{y^{2}+1}-|y|)y$. The self-consistent value $\Delta_{*}$ is attained at a solution of $f_{ns}^{\prime}(\Delta_{*})=0$.

Separating out an electrostatic contribution by subtracting the result for some reference potential $\phi_{0}$:

[TABLE]

where $\delta\gamma(\xi)\equiv\gamma(\xi,\Delta)-\gamma(\xi,\Delta=0)$ from Eq. (9), and $\delta\xi_{n}\equiv\xi_{n}-\xi_{n}^{(0)}$. The result (16) includes both gating Blatt and Thompson (1963); Shapiro (1984, 1985) and any nonlinear effects (e.g. energy associated with quantum capacitance) in strong electric fields. Note that the above electrostatic energy contribution depends on the electric fields only via $\xi_{n}=\xi_{n}[\phi]$, an exact statement in the model here.

It is also possible to express the electrostatic energy directly in terms of the self-consistent electric field, at small field strengths. Consider expansion of the electronic energy around a reference electric potential, considering small $\phi_{1}=\phi-\phi_{0}$ and $\rho_{1}=\rho-\rho_{0}$:

[TABLE]

where $n_{e}$ is the electron density and $\Pi$ the density response function. Solving the resulting saddle-point equation for $\phi_{1}$ and substituting the solution into $F$ gives, after integration by parts:

[TABLE]

where $C$ is independent of $\Delta$. In this order of expansion in small $\phi_{1}$, the additional electrostatic field energy in (19) coincides with the standard expression. The linear term $\sim\rho_{1}\phi_{0}$ describes a gate effect on superconductivity, which in this approach we see to be related to the $\Delta$-dependence of the equilibrium potential $\phi_{0}$. Using Eq. (21) the quadratic part can be expressed as $\sim\phi_{1}\epsilon^{\rm RPA}\phi_{1}$. It corresponds to a (quantum) capacitance modulation Morawetz et al. (2008, 2009) by superconductivity. The result (19) can be directly used for computing $\delta{}f_{ns}(\Delta)$ (if $\delta\phi\equiv{}\phi[\Delta]-\phi[0]$ is known) and is equivalent with (16) in the small-field limit. However, due the $\Delta$-dependence of $\phi_{0}$ it is not necessarily very practical to compute, as solving the nonlinear Poisson problem is still required. However, the above expressions can be used as a consistency check.

As noted above, we consider charge density $\rho=\rho_{1}+\rho_{0}$ where $\rho_{1}$ outside the sample fixes the electric field at the surface. Finally, we need to specify the background (“ion”) charge density $\rho_{0}$. The electric potential due to $\rho_{0}$ together with $U$ gives the pseudopotential for the electron system. 111 The confining potential $U$ is not necessary for the formulation, except for enabling the use of the constant-$\Delta$ approximation, see Appendix B. For simplicity, unless otherwise mentioned, below we assume $\rho_{0}=en_{e}[\Delta=0,\phi=0,\mu]$, which results to $\phi_{0}=0$ as the solution in the normal state, and $\mu$ becoming the parameter that fixes the charge density in the normal state. This is of course a crude toy model of the surface electron behavior, even within Hartree-type models Lang and Kohn (1970), but likely modifies mainly the precise positions of the subbands but not the main qualitative features of the effect of the screening of external charges on superconductivity.

II.1 Size effect in electric field

In the same way as the variation in thickness, Blatt and Thompson (1963); Falk (1963) gating by a surface electric field can in principle make a single subband edge $\xi_{n}$ to cross the Fermi level, which results to a jump in superconducting properties. Such response can be larger than in bulk material, and is not captured by “surface doping” models often used for the electric field effect, Shapiro (1984, 1985) where the LDOS $\nu(x,\xi)$ is assumed to be modified in a surface layer of thickness of a screening length $\lambda_{TF}$ according to bulk relations. In addition, the field screening is not exactly Thomas-Fermi type, but this causes less relevant changes than the difference in the DOS.

The order of magnitude of $\delta{}f_{ns}$ can be estimated in a Thomas–Fermi approximation. Taking $\phi(L/2+x^{\prime})\simeq{}-E\lambda_{TF}e^{x^{\prime}/\lambda_{TF}}$ for $x^{\prime}<0$, $\lambda_{TF}=\sqrt{\epsilon_{0}/(e^{2}\nu_{F})}$,

[TABLE]

where $k_{n}=\pi{}n/L$ and $q(z)=z^{2}/(1+z^{2})$. From Eq. (17), and keeping only the smallest $|\xi_{n}|<\omega_{c}$,

[TABLE]

where $f_{ns,3D}=-\frac{1}{4}\frac{mk_{F}}{\pi^{2}}\Delta^{2}$ is the bulk 3D condensation energy and $a_{0}=4\pi\epsilon_{0}/(me^{2})$ the Bohr radius. The above result is valid in the leading order in $\Delta$, as $\phi$ is assumed to be independent of it. The factor $q(2\lambda_{TF}k_{n})$ in reality depends on details of the screening, and below we consider it as a constant of order of magitude 1.

Including the next-order eigenvalue perturbation $\delta\xi_{n}^{(2)}$ in (16) and considering terms of order $E^{2}$ gives the second-order correction,

[TABLE]

where $n=k$ means the limit $\xi_{k}\to\xi_{n}$. This energy contribution is associated with the change $\Pi[\Delta,0]-\Pi[0,0]$ (c.f. Eq. (18), (30)) in the static Lindhard function Prange (1963). However, it is of the same order in $\Delta$ as the change $\Pi[0,\phi_{0}[\Delta]]-\Pi[0,0]$ due to the $\Delta$-dependent shift in the self-consistent equilibrium potential, which we have neglected here. As a consequence, Eq. (24) is not the only contribution to the $E^{2}$ term, and solving the self-consistent electrostatic problem is in general required. 222Alternatively, one could base the expansion (18) around the normal-state potential $\phi_{0}=0$ and include a third-order term $\int\Gamma_{3}(x,x^{\prime},x^{\prime\prime})\phi(x)\phi(x^{\prime})\phi(x^{\prime\prime})$. Conversely, calculation of the effect of superconductivity on the dielectric function requires taking the self-consistency of $\Delta=\Delta_{*}[\phi]$ into account. Seiden (1966)

The overlap factor $q$ above depends on how accurate the Thomas-Fermi screening assumption is close to the surface. For the simple problem here, we can solve the Poisson equation numerically. Such a solution is illustrated in Fig. 2a for $\lambda_{TF}\ll{}L$. Since $\lambda_{TF}\sim{}k_{F}$, screening is not fully exponential, but the electric potential exhibits $1/k_{F}$ oscillations. The correction $\delta\phi=\phi(\Delta)-\phi(\Delta=0)$ to the equilibrium electrostatic potential from superconductivity is small in the high charge density regime considered here. The chemical potential is chosen to be close to a subband edge in the figure.

The corresponding dependence of $\delta{}f_{ns}$ on the electric field magnitude is shown in Fig. 2b, together with the corresponding result from Eq. (23). The electrostatic energy expression (20) is also shown, and coincides with the exact result in the small-field regime. Generally, the electric field effect is appreciable only for $E_{+}L\gtrsim{}\mu$. In the estimate from Eq. (23), we here set $q(z)=1$, to account for the expectation that likely for the true screening length $\lambda_{TF}k_{F}\gtrsim 1$. The second-order correction (24) is neglibile for these parameters, being higher order in $\lambda_{TF}^{2}eE/(L\omega_{c})$, and the nonlinearity visible in the result originates from $g(\xi)$.

The linear gating effect can be suppressed with a charge-neutral field configuration $E_{+}=E_{-}=E$, which corresponds to an experiment using floating gate electrodes (i.e. placing the system inside a plate capacitor). The result from Poisson equation for this case is shown in Fig. 3, together with the result for $\delta{}f_{ns}$. Imposing the field on both sides produces a larger $\delta\phi$. However, as the linear contribution to the free energy cancels, the modulation $\delta{}f_{ns}$ arises from the next-order effect and is an order of magnitude smaller than with the gate effect. Although the energy can still be expressed also via Eq. (20), the eigenvalue perturbation result (24) does not agree as well, as expected.

Whether electrostatic effects are significant depends on how large the modulation $\delta{}f_{ns}$ is compared to $f_{ns}$. The dependence of their ratio on the chemical potential, and hence charge density, is shown in Fig. 4, at a relatively large external field. The magnitude of $\delta{}f_{ns}$ depends strongly on whether the chemical potential is located near a band bottom, where the effect is amplified (see Fig. 1), which produces the oscillations visible in Fig. 4. When $\mu$ is close to a subband bottom, the magnitude appears to be captured well by Eq. 23 (dashed line). When the chemical potential is not close to a band bottom, depending on the ratio between the subband spacing and the cutoff $\omega_{c}$, the electric field effect can vary by order of magnitude. Note that as long as $|\xi_{n}|\ll\omega_{c}$ for some $n$, the result is dominated by the smallest $\xi$ and the cutoff $\omega_{c}<\infty$ is of limited importance. The sum (16) is convergent also for $\omega_{c}\to\infty$. However, these results are based on the simple weak-coupling model for superconductivity, and the precise shape of the modulation may be sensitive to details of the interaction. Regardless, from the above results one can see that the relative magnitude at resonance scales as $\propto{}\Delta/\mu$, and not as $(\Delta/\mu)^{2}$ as one would expect for the amplitude response in 3D bulk Prange (1963); Seiden (1966). Away from the subband edge resonances, $\delta{}f_{ns}\propto(\Delta/\mu)^{2}$.

The self-consistent value of $\Delta_{*}$, $f_{ns}^{\prime}(\Delta_{*})=0$, is shown in Fig. 5a as a function of the film thickness, showing the well-known quantum size effect. Blatt and Thompson (1963); Falk (1963) The corresponding dependence on the chemical potential is shown in Fig. 5b, for several values of the external electric field. In this figure, it is obvious that the electrostatic field simply gates the system: the size effect physics is dominated by the $\xi_{n}$ closest to the chemical potential, so that the gate-induced shift $\delta\xi_{n}$ is identical to a chemical potential shift $-\delta\mu$.

The above discussion can be compared to a surface doping model, where the DOS is assumed to change in a surface layer of thickness $\lambda_{TF}$, and the system is considered a superconducting bilayer in the Cooper limit. In such a model, $\delta{}F_{ns}/F_{ns}\sim{}-\frac{\lambda_{TF}}{L}\delta\frac{1}{\nu_{F}\lambda}\sim{}\frac{\delta\nu_{F}\lambda_{TF}}{\lambda\nu_{F}^{2}L}$. The general form of Eq. (23) can then be recovered by including (ad-hoc) the main features of the multiband DOS in $\delta\nu_{F}$. This can be done by writing $\frac{\delta\nu_{F}}{\nu_{F}}=\frac{\pi}{k_{F}L}\partial_{\xi_{n}}\theta_{\Delta}(\xi_{n})\lambda_{TF}eE$, where $\theta_{\Delta}(\xi)$ is a broadened unit step function with width $\Delta$. For the problem here, although the actual form of $\delta\nu(x,E)$ is different, this simpler model captures the main effects. Surface doping models have indeed been successful in understanding previous experimental results. Mannhart et al. (1991)

II.2 Superfluid weight

The effect of the electrostatic field on the phase fluctuations can be studied via the superfluid weight $D_{ij}^{s}$, Scalapino et al. (1993) which describes the free energy cost of superflow $\Delta(\bm{r})\propto{}e^{2i\bm{A}\cdot\bm{r}}$:

[TABLE]

The “vector potential” $\bm{A}$ describing the superflow can be introduced in Eq. (2) by replacement $\hat{\bm{k}}\mapsto\hat{\bm{k}}+\bm{A}$. The calculation of $D^{s}$ is standard for multiband BCS superconductor. Since the current operators in $y/z$ directions do not here couple different bands, the result for $i,j\in\{y,z\}$ is $D^{s}_{ij}=\delta_{ij}\sum_{n}{}n_{s}(\xi_{n})/(mL)$ where $n_{s}(\xi_{n})$ is the BCS superfluid density: Scalapino et al. (1993)

[TABLE]

where $n(\xi)$ is given by Eq. (8) and $\epsilon^{\prime}=\sqrt{(\xi^{\prime})^{2}+\Delta^{2}}$. As well known, $n_{s}(\xi)\to{}n_{e}(\xi)=2m\gamma(\xi)$ at $T=0$. The electrostatic modulation of the superfluid stiffness is then similar to that of the charge density, i.e., small in the metallic regime. Similar conclusion then applies to the phase stiffness, and quite likely also to the phase-slip energy barrier Langer and Ambegaokar (1967). These results however apply in the clean limit.

III Discussion and conclusions

We discussed an elementary BCS/Hartree–Bogoliubov mean field model for the size effect under self-consistent electrostatic fields in superconducting thin films, and studied it based on numerically exact solutions. As the size modulation in superconducting properties decays relatively slowly with increasing charge density, it increases the response to applied electric fields, effectively changing the small parameter from $(\Delta/\mu)^{2}$ to $\Delta/\mu$ for fine-tuned values of $\mu$, also in films thick compared to the screening length.

The mean-field approach likely is not useful in describing atomically thin, or strongly disordered and resistive samples, where fluctuation effects matter. Phase–plasmon fluctuation effects in principle can be included in the approach above in a standard way by expanding in $\mathop{\mathrm{Re}}\Delta$, $\mathop{\mathrm{Im}}\Delta$ and $V=-i\phi$ around the mean-field solution. A priori, in view of some existing results, Fischer et al. (2018); van Otterlo et al. (1999) however, it’s not clear why such corrections would depend strongly on the external electric field.

Large electrostatic size effects in thin-film systems are expected to be visible mainly in relatively low charge densities, e.g. semiconducting materials. As noted in previous works, Mannhart et al. (1991) it appears likely this is a main effect in high-Tc superconductors. The modulation of screening by superconductivity will also appear in proximity systems, e.g. in semiconductor/superconductor hybrids Vuik et al. (2016); Woods et al. (2018) recently considered as Majorana fermion platforms.

With regard to the large modification of superconducting critical current by electric fields reported in Refs. De Simoni et al., 2018; *paolucci2018-ues; *paolucci2018-mef, it then appears somewhat less likely these results can be understood with electrostatic effects of the type discussed above. At metallic densities $\Delta/\mu\sim{}10^{-4}$, electrostatic effects in the model here, even at a sharp DOS feature, likely can only give $|\delta{}f_{ns}/f_{ns,3D}|\lesssim 10^{-2}$, which is too small to cause large measurable effects. It appears unlikely this is easily rectified by lifting some of the approximations we made. This is simply a manifestation of the “Anderson theorem”: Anderson (1959) the amplitude of conventional superconductivity is insensitive to time-reversal symmetric perturbations, and suppressing it requires perturbations large compared to $\mu$, which are usually not achievable in the metallic regime below the electrical breakdown field. Also, as the linear gate effect generally should dominate nonlinearities, whether superconductivity is suppressed or enhanced depends on the sign of the electric field, quite unlike in Refs. De Simoni et al., 2018; *paolucci2018-ues; *paolucci2018-mef. Previously, reduction in the critical current by an applied field was attributed to modification of vortex pinning. Mannhart et al. (1991); Shvarts et al. (2005) In Ref. De Simoni et al., 2018 effects appear also in aluminum strips with lateral size $\lesssim\xi$, making this explanation less favorable. In the clean-limit model here, it also appears unlikely the phase slip rates would be significantly affected.

In summary, we considered effects of electrostatic fields on superconductivity self-consistently within a BCS model, connected them to questions about electrostatic energy, and commented on their relation to recent experiments. We obtain results for the size and external electric field modulation of superconductivity, and contrast results to a surface doping model. Expanding about this mean field solution, considering electric field effects on phase slips and phase fluctuations is possible. Experimentally, the effects are best visible in low charge density systems, e.g. semiconductor hybrid structures.

Appendix A Density response function in thin film

The static density response in a superconducting infinite potential well can be found, in a situation translationally invariant vs. $y$ and $z$ (i.e. response to a charge sheet). First, we have

[TABLE]

where the trace and the Matsubara sum has been evaluated, and $n(\xi)=u_{\xi}^{2}f_{0}(\epsilon_{\xi})+v_{\xi}^{2}(1-f_{0}(\epsilon_{\xi}))$. The normal-state spectral function for a potential well is

[TABLE]

where $k_{p}=\frac{\pi p}{L}$, $E_{p}=\frac{k_{p}^{2}}{2m}$. Then we have,

[TABLE]

which can be evaluated. Here, the terms $p=q$ imply the limit $E_{p}\to{}E_{q}$.

Appendix B Confining potential

In a more realistic model than in the main text, we would set $U=0$ and the electrons would be confined in the metal film due to the attractive potential from the ionic charge density $\rho_{0}>0$. However, in such calculations the simplifying assumption of a spatially constant $\Delta$ is not permissible, as discussed below.

The charge density in uniform 3D metal for $\mu\to-\infty$ (i.e. deep in the vacuum), with constant $\Delta$, is

[TABLE]

The corresponding Poisson equation in a Thomas-Fermi approximation becomes

[TABLE]

From the solution, we find the electrostatic field fails to confine the “superconducting” electrons, and an infinite amount of total charge $\int_{x_{0}}^{\infty}\mathrm{d}x\,\rho_{e}(x)$ leaks to the vacuum, which is unphysical. In the exact solution, the mean field $|\Delta(\bm{r})|$ would decrease simultaneously with the density, providing a stronger electrostatic confinement. Although the details of this surface effect appear sensitive to the external electric field, it appears unlikely it is important for the stability of superconductivity in the bulk.