Destroying superconductivity in thin films with an electric field

In this paper, we use a Ginzburg-Landau approach to show that a suitably strong electric field can drive a phase transition from a superconductor to a normal metal. The transition is induced by taking into account corrections to the permittivity due to the superconductive gap and persists even when screening effects are considered. We test the model against recent experimental observations in which a strong electric field has been observed to control the supercurrent in superconducting thin films. We find excellent agreement with the experimental data and are able to explain several observed features. We additionally suggest a way to test our theoretical proposal via superconductor-superconductor electron tunneling.

Figure 1

Section of the setup. The superconductive film is extended in the $x$ and $y$ directions and has thickness $L$ along the $z$ direction. The electric field $E_z$ pointing along the direction of the small black arrows is applied to both sides of the film, by means of two charge distributions $A$ and $B$, taken to be equal. The penetration length is ${\lambda }_E\simeq 1$ nm.

Figure 2

(a) Numerical computation of the averaged critical current ${\mathcal{I}}_{\mathrm{c}}$ normalized against its values at $T=5$ mK, $\overline{{\mathcal{I}}_{\mathrm{c}}}$, as a function of the applied electric field $E_0$ ]see Eq. (1)] normalized against the 5 mK critical electric field ${\overline{E}}_{\mathrm{c}}$. The curves are the numerical simulations and the dots are the experimental data from Ref. [2]. The values of $\overline{{\mathcal{I}}_{\mathrm{c}}}$ at different temperatures are taken from the experiments. The parameters ${\beta }_1$ and ${\beta }_2$ are fixed as discussed in the text. (b) The suppression of the electric-field effect as a function of the thickness of the film $L$ for $T=5$ mK and $E={\overline{E}}_{\mathrm{c}}$. The red dots are the experimental data from Ref. [2].

Figure 3

Tunneling currents as a function of the applied voltage eV through the junction for $E_0/{\overline{E}}_{\mathrm{c}}=0$ (dashed green curve), $E_0/{\overline{E}}_{\mathrm{c}}=0.98$ (blue curve) and $E_0/{\overline{E}}_{\mathrm{c}}=1.1$ (red dots line). The temperature is $T=250$ mK $\simeq 3T_{\mathrm{c}}/5$.

Figure 4

A plot of the superconductor energy at zero temperature (minus a constant permittivity contribution) against the applied electric field. The blue dots represent our model while the yellow dots are the standard Ginzburg-Landau model ($\epsilon \left[\mathrm{\Delta }\right]={\epsilon }_0$) with a chemical potential. The normal phase has zero energy (modulo the constant permittivity term) and lies along the horizontal axis. We can see as the electric field is increased the energy of our system eventually becomes equal to that of the energy of the normal phase indicating a phase transition.

Figure 5

(a) The profile of the gap in the $z$ direction at the critical electric field ${\overline{E}}_{\mathrm{c}}$ and zero temperature with various values of the coupling ${\beta }_1$. The red line corresponds to the constant permittivity case ($\epsilon \left[\mathrm{\Delta }\right]={\epsilon }_0$) and depends only weakly on the position. For each blue line of smaller $\overline{\mathrm{\Delta }}\left(z\right)$, we are increasing ${\beta }_1$ (from $1.42\times {10}^{-29}{\mathrm{m}}^3$ to $1.13\times {10}^{-28}{\mathrm{m}}^3$ in steps of $1.42\times {10}^{-29}{\mathrm{m}}^3$). (b) The dependence of ${\beta }_1$ on $T/T_{\mathrm{c}}$ normalized by the value of ${\beta }_1$ at zero temperature [given in Eq. (A6)]. The blue line represents how ${\beta }_1$ must change if the critical electric field is unaffected by temperature. The red points are the critical electric field extracted from experimental data [2]. The grey dashed line is a linear fit to the first three experimental points and it has a nonzero, positive intercept with the $y$ axis.

Figure 6

(a) The dependence of the average gap at zero temperature on the electric field applied at the edge of the superconductor for ${\beta }_1$ having the critical value Eq. (A6). The red line corresponds to ${\beta }_2=0$. The purple dashed lines indicate values of ${\beta }_2$ greater than zero (below the red line) from $1.97\times {10}^{-54}{\mathrm{m}}^6$ to $9.83\times {10}^{-54}{\mathrm{m}}^6$ in steps of $1.97\times {10}^{-54}{\mathrm{m}}^6$. The blue dashed lines are for values of ${\beta }_2$ less than zero (above the red line) from $-3.93\times {10}^{-55}{\mathrm{m}}^6$ to $-1.97\times {10}^{-54}{\mathrm{m}}^6$ in steps of $-3.93\times {10}^{-55}{\mathrm{m}}^6$. (b) A plot of the critical current, normalized by its zero temperature value, against the applied electric field. The red dots are data while the solid blue line is a best fit from our model.

Figure 7

The dependence of the ${\beta }_2$ coupling, normalized by the value at 5 mK ($-7.27\times {10}^{-55}{\mathrm{m}}^6$) against temperature extracted by fitting our model to experimental data.

Figure 8

Plots of the energy densities. Each line is a different choice of $E_0/E_{\mathrm{c},T=0}$ with red corresponding to positive values and blue to negative values of $E_0$. (a) is a plot of the potential energy density against position for $E_0>0$ with $E_0/E_{\mathrm{c},T=0}=0,1/10,...,9/10$ (bottom to top). (b) is a plot of the potential energy density against position for $E_0<0$ with $E_0/E_{\mathrm{c},T=0}=0,-1/10,...,-9/10$ (bottom to top). (c) and (d) represent the values of the kinetic and boundary energy densities against electric field and position with the electric field ranging from $E_0/E_{\mathrm{c},T=0}=-9/10,...,9/10$.