Driven responses of periodically patterned superconducting films

We simulate the motion of a commensurate vortex lattice in a periodic lattice of artificial circular pinning sites having different diameters, pinning strengths, and spacings using the time-dependent Ginzburg-Landau formalism. Above some critical DC current density $J_{\mathrm{c}}$, the vortices depin, and the resulting steady-state motion then induces an oscillatory electric field $E\left(t\right)$ with a defect “hopping” frequency $f_0$, which depends on the applied current density and the pinning landscape characteristics. The frequency generated can be locked to an applied AC current density over some range of frequencies, which depends on the amplitude of the DC as well as the AC current densities. Both synchronous and asynchronous collective hopping behaviors are studied as a function of the supercell size of the simulated system and the (asymptotic) synchronization threshold current densities determined.

Figure 1

(a) a simple square lattice drawn as centered square lattice with two pinning sites per square. (b) Moving the site centers along the square diagonal removes those at the corners.

Figure 2

Surface plots of ${\left|\psi \right|}^2$ in the simulated system of size $\sqrt{2}L\times \sqrt{2}L$ with two circular defects of diameter $D$ (indicated by the circles) separated by $L$ representing a unit cell of a large pinning array for different vortex positions in the system corresponding to different phases of a cycle (see also Fig. 3): (a) vortex is moving inside the defect, (b) vortex is the farthest from defects, and (c) vortex is about to enter the defect. The projection at the bottom of (b) indicates regions with different $T_{\mathrm{c}}/\epsilon$ values in dark gray ($\epsilon =1$) and light gray ($\epsilon =0.75$). At the center position of the vortex, the direction of applied current, magnetic field, and resulting Lorentz force ($F_L$) are indicated.

Figure 3

Three periods of the time-dependent electric field $E\left(t\right)$ for $L=10, \epsilon =0.75$, and $D=5$ with applied current $J_{\mathrm{dc}}=0.1$. Snapshots of the order parameter and vortex configurations at three distinct times (a)–(c) of a period are shown in Fig. 2.

Figure 4

$J-E$ characteristics for five different sets of system parameters with $L=10$ given in the legends. The electric field is averaged over $\gtrsim 10$ periods of the vortex motion in the steady state for each current density and parameter set.

Figure 5

(Top) Time-dependent electric field curves for the $L=10, D=5$, and $\epsilon =0.75$ system with $J_{\mathrm{dc}}=0.051,0.06,0.12,0.15$. (Bottom) The Fourier transforms of the $E\left(t\right)$ curves reveal the hopping frequencies $f_0=0.00217$, 0.00547, 0.01577, and 0.02080 for each of the applied currents respectively. The Fourier analysis also shows higher harmonics.

Figure 6

Hopping frequencies as a function of applied DC current density, $f_0\left(J_{\mathrm{dc}}\right)$, for various system parameters with $L=10$. The underlying oscillations of the electric field are caused by repeated periodic pinning and depinning events. The response frequencies are obtained by using the dominant peak of the Fourier transform of the electric field (calculated for $\gtrsim 65$ periods of the vortex motion in the steady state), see Fig. 5.

Figure 7

The averaged voltage, $\langle V\rangle =\sqrt{2}L\langle E\rangle$, for eight different sets of system parameters with $L=10$ given in the legends, the slope of the line is $2·2\pi$. The electric field is averaged over $\gtrsim 10$ periods of the vortex motion in the steady state for each current density and parameter set.

Figure 8

Frequency locking for $f_{\mathrm{ext}}=f_0$ for the system of $J_{\mathrm{dc}}=0.12, L=10, D=5$, and $\epsilon =0.75$; the applied amplitude is $J_{\mathrm{ac}}=0.02$. The plot shows a single-mode oscillation.

Figure 9

Time-dependent electric field behavior of a $L=10, D=5$, and $\epsilon =0.75$ system with applied $J_{\mathrm{dc}}=0.12, J_{\mathrm{ac}}=0.02$, and $f_{ext}=1.046f_0$. the envelope function is $f_{\mathrm{env}}=5.632\times {10}^{-4}$, i.e., about 28 times lower than $f_0$.

Figure 10

Frequency locking region of the benchmark system with $L=10, D=5$, and $\epsilon =0.75$ for different $J_{\mathrm{ac}}$ and fixed $J_{\mathrm{dc}}=0.12$. Here, the electric field is averaged over $\sim {10}^3$ periods of the vortex motion in the steady state.

Figure 11

The current difference, $J_{\mathrm{th}}-J_{\mathrm{c}}$, for the onset of asynchronous hopping with increasing super cell size, $N_{\mathrm{sc}}$ for two systems of $L=10$ and $D=5$ using different defect strength. The value of the critical currents $J_{\mathrm{c}}=0.05$ for the blue line, and $J_{\mathrm{c}}=0.022$ for the red line.