Interfacial control of vortex-limited critical current in type-II superconductor films

In a small subset of type-II superconductor films, the critical current is determined by a weakened Bean-Livingston barrier posed by the film surfaces to vortex penetration into the sample. A film property thus depends sensitively on the surface or interface to an adjacent material. We theoretically investigate the dependence of vortex barrier and critical current in such films on the Rashba spin-orbit coupling at their interfaces with adjacent materials. Considering an interface with a magnetic insulator, we find the spontaneous supercurrent resulting from the exchange field and interfacial spin-orbit coupling to substantially modify the vortex surface barrier, consistent with a previous prediction. Thus, we show that the critical currents in superconductor-magnet heterostructures can be controlled, and even enhanced, via the interfacial spin-orbit coupling. Since the latter can be controlled via a gate voltage, our analysis predicts a class of heterostructures amenable to gate-voltage modulation of superconducting critical currents. It also sheds light on the recently observed gate-voltage enhancement of critical current in NbN superconducting films.

Figure 1

Schematic depiction of the system under consideration. A type-II superconductor film S (pink) is sandwiched between two insulators (yellow). An in-plane magnetic field ${\mathit{H}}_e$ is applied and a current $I_{\mathrm{ext}}$ is driven through the film. Interfacial Rashba SOC is assumed to exist in thin layers (blue) adjoining the interfaces. Its strength can be tuned via an applied gate voltage $V_G$. When the adjacent insulators are magnetic, exchange field $\mathit{h}$ is induced in the interfacial layer (blue) of S. The critical current in such films depends on Rashba SOC and the induced exchange field. These can be controlled via the gate voltage $V_G$ and the applied magnetic field ${\mathit{H}}_e$ providing multiple handles on critical current in the superconductor. The coordinate system on the right depicts the orthogonal relation between the symmetry-breaking vector $\widehat{\mathit{n}}$ on the upper S surface, the exchange field $\mathit{h}$, and the effective field $\mathit{\alpha }$ experienced by the S [Eq. (1)].

Figure 2

(a) Schematic depiction of the system under consideration. An external magnetic field $H_e\widehat{\mathit{z}}$ is applied and a current $I_{\mathrm{ext}}\widehat{\mathit{y}}$ is injected. We consider the superconductor film to host a chain of vortices with flux ${\varphi }_0\widehat{\mathit{z}}$, with ${\varphi }_0$ the flux quantum. (b) Normalized Gibbs free energy density $\tilde{G}\equiv G_{1}4\pi ad{\mu }_0{\lambda }^2/\left(V_S{\varphi }_0^2\right)$ [Eq. (13)] vs. the vortex position ${\tilde{x}}_v\equiv 2x_v/d$. In this work, we focus on low fields $H_e<H^{\prime }$ for which the energy profile does not harbor stable or metastable vortex states in the superconductor. (c) Schematic depiction of the forces experienced by the upper vortex with flux along $\widehat{\mathit{z}}$ situated close to the left surface and the lower vortex with flux along $-\widehat{\mathit{z}}$ located near the right surface. The force on each vortex is comprised of attraction towards an image antivortex (purple), interaction with the Meissner currents resulting from the applied field (blue), and the Lorentz force due to externally injected current (green). The applied field along $\widehat{\mathit{z}}$ lowers the surface barrier experienced by the upper vortex close to the left, while it increases the surface barrier experienced by the lower vortex on the right.

Figure 3

(a) Schematic depiction of spontaneous supercurrent density ${\mathit{j}}_L$ (blue arrows) and the resulting forces (green arrows) on a vortex located at different places in the superconductor. The combined effect of SOC and exchange field results in a finite $\mathit{\alpha }$ and consequently, a spontaneous supercurrent directed parallel to $\mathit{\alpha }$ [Eq. (3)] in the interfacial layer, shaded light blue. A much smaller oppositely directed supercurrent density throughout the film (depicted via small blue arrows pointing upwards) ensures zero net current in equilibrium. (b) The normalized finite-$\mathit{\alpha }$ Gibbs energy density contributions ${\tilde{G}}_{\alpha }=G_{\alpha }4e\lambda da{\mu }_0/(V_S{\varphi }_0{\alpha }_{L,R}cos{\theta }_{L,R})$ vs. the normalized vortex position ${\tilde{x}}_v=2x_v/d$. The blue and red curves respectively depict the contributions from left and right interfaces.

Figure 4

(a) Schematic of the MI-superconductor (S) bilayer that enables a control of the critical supercurrent in S layer. The external current is injected along $\widehat{y}$ and the MI magnetic order determines the exchange field $\mathit{h}$, which can be controlled by, for example, an external magnetic field. [(b) and (c)] Change in the superconducting critical current $\mathrm{\Delta }{I}_c$ due to the combined action of exchange field and interfacial SOC resulting in a finite $\mathit{\alpha }$ vs. orientation of the exchange field $\mathit{h}$. $I_c^0$ is the critical current without interfacial SOC. We assume ${\alpha }_L/e{\mu }_{0}d{H}_{sL}^0=0.1$. In (c), we consider $H_e/H_{sL}^0=0.02$.

Figure 5

[(a) and (b)] Schematic depiction of the trilayer device that allows a control of critical current in S layer via magnetic orders in the two adjacent MIs. (c) Fractional change in the critical current vs. orientations of the exchange fields induced by the two adjacent MIs. We assume $H_e=0$, $H_{sL}^0=H_{sR}^0$, $|{\alpha }_L|=|{\alpha }_R|$, and ${\alpha }_L/e{\mu }_{0}d{H}_{sL}^0=0.1$.