Pinning effects on self-heating and flux-flow instability in superconducting films near $T_c$

The effect of pinning on self-heating triggering the Larkin-Ovchinnikov (LO) flux-flow instability (FFI) in superconducting thin films is theoretically investigated. The problem is considered relying upon the Bezuglyj-Shklovskij (BS) generalization of the LO theory, accounting for a finite heat removal from the quasiparticles at temperature $T^{*}$ to the bath at temperature $T_0$. The FFI critical parameters, namely the current density $j^{*}$, the electric field $E^{*}$, the power density $P^{*}$, and the vortex velocity $v^{*}$ are calculated and graphically analyzed as functions of the magnetic field and the pinning strength. With increasing pinning strength at a fixed magnetic field value $E^{*}$ decreases, $j^{*}$ increases, while $P^{*}$ and $T^{*}$ remain practically constant. Vortex pinning may hence be the cause for eventual discrepancies between experiments on superconductors with strong pinning and the generalized LO and BS results.

Figure 1

Left axis: The nonlinear current-voltage curve (CVC) $E\left(j\right)$ calculated for a cosine washboard pinning potential (WPP) of Ref. [30] in the limit of low temperatures. The dashed line corresponds to the free flux-flow regime $E/{\rho }_f=j$, where ${\rho }_f$ is the flux-flow resistivity. Right axis: The respective nonlinear function $\nu \left(j\right)$ calculated by Eq. (27) of Ref. [30]. Upper inset: Experimental geometry. The transport current $I$ is applied along the WPP channels (grooves). The vortex dynamics across the grooves leads to the appearance of the voltage $V$. Lower inset: Atomic force microscope image of a Nb film surface with a nanogroove array milled by focused ion beam [38] and inducing a pinning potential of the washboard type.

Figure 2

Normalized electric field $E^{*}/E_0$ versus normalized current density $j^{*}/j_0$ at the instability point for a series of values of the pinning strength parameter $\alpha \equiv j_c/j_0$. For all curves, the blue spots outline values obtained at $b\equiv B/B_T=10$, while the red ones mark those at $b=0$. The curve for $\alpha =0$ coincides with the BS curve in Fig. 1 in [16]. Inset: Normalized power density $P^{*}/P_0$ versus normalized current density $j^{*}/j_0$ at the instability point for the same series of values of the pinning strength parameter $\alpha$.

Figure 3

Dependence of the normalized critical electric field $E^{*}\left(t\right)/E_0$ (a), critical current density $j^{*}\left(t\right)/j_0$ (b), critical power density $P^{*}\left(t\right)/P_0$ (c), and critical temperature $T^{*}\left(t\right)/T_0$ (d) on the dimensionless pinning strength parameter $\alpha \equiv j_c/j_0$ and the dimensionless magnetic field $b=B/B_T$ calculated by Eqs. (25), (29), (31), and (23), respectively. The parameters $B_T$ and $E_0$ are given by Eqs. (16) and (24). The thick black curves at $\alpha =0$ reproduce the BS results [16] without pinning.

Figure 4

Dependence of the normalized instability velocity $v^{*}/v_0$ (a) and the instability current density $j^{*}/j_0$ (b) on the dimensionless pinning strength parameter $\alpha \equiv j_c/j_0$ and the dimensionless magnetic field $b=B/B_T$ calculated by Eqs. (25) and (29), respectively, in the low-field range. The parameters $B_T$ and $E_0$ are given by Eqs. (16) and (24) while $v_0\equiv c{E}_0/B_T$. The thick black lines correspond to $b=0.1$ for a comparison with the most closely related experiment of Silhanek et al. [21], as detailed in the text.