Magnetoresistance oscillations in superconducting strips: A Ginzburg-Landau study

Within the time-dependent Ginzburg-Landau theory we study the dynamic properties of current-carrying superconducting strips in the presence of a perpendicular magnetic field. We found pronounced voltage peaks as a function of the magnetic field, the amplitude of which depends both on sample dimensions and external parameters. These voltage oscillations are a consequence of moving vortices, which undergo alternating static and dynamic phases. At higher fields or for high currents, the continuous motion of vortices is responsible for the monotonic background on which the resistance oscillations due to the entry of additional vortices are superimposed. Mechanisms for such vortex-assisted resistance oscillations are discussed. Qualitative changes in the magnetoresistance curves are observed in the presence of random defects, which affect the dynamics of vortices in the system.

Figure 1

The model system: a superconducting strip (of width $w$, thickness $d\ll \lambda ,\xi$, and periodically extended in the $y$ direction) in the presence of a dc current $I$ and a perpendicular magnetic field $H$.

Figure 2

(a) Time-averaged voltage $V$ (in the units of normal state voltage $V_n$) as a function of magnetic field $H$ for different values of the applied current $I$. Panels 1–6 show snapshots of the Cooper-pair density at the field and current values indicated in the $V\left(H\right)$ curves. Arrows in panels 2, 4, and 6 indicate the direction of vortex motion. The width of the sample is $w=4\xi$ and the size of the unit cell is $L=4\xi$.

Figure 3

Voltage (in units of normal state voltage $V_n$) vs time characteristics of the sample in Fig. 2 at $I/\left(j_{0}w\right)=0.01$ and $H=0.53H_{c2}$ (a) and $I/\left(j_{0}w\right)=0.012$ and $H=0.83H_{c2}$ (b). Insets show snapshots of ${\left|\psi \right|}^2$ at time intervals indicated on the $V\left(t\right)$ curve.

Figure 4

(a), (b) Time evolution of the free energy (in units of $F_0=H_c^2{V}_0/8\pi$ with $V_0$ being the unit cell volume) for the same sample as in Fig. 2 for $H=0.53H_{c2}$ (a) and $H=0.63H_{c2}$ (b). Vertical arrows indicate the penetration $\Delta F_p$ and expulsion $\Delta F_e$ barriers and horizontal arrow shows the time interval $\Delta t$ over which the vortex is present inside the sample. Insets show snapshots of ${\left|\psi \right|}^2$ at the times indicated on the curves. (c) Time-averaged voltage (solid curve, referred to the right axis, voltage is given in units of normal state voltage $V_n$), $\Delta F_p$ (filled circles) and $\Delta F_e$ (open circles) as a function of the magnetic field. The applied current is $I/\left(j_{0}w\right)=0.01$.

Figure 5

Time-averaged voltage (in units of normal state voltage $V_n$) as a function of flux $\Phi$ (calculated over the unit cell area $S=w\times L$) for different width of the sample. The applied current density is $j=0.04j_0$.

Figure 6

Time-averaged voltage $V$ (in units of normal state voltage $V_n$) as a function of magnetic field $H$ for the sample with $w=5\xi$ and $L=15\xi$ without (a) and with eight randomly generated defects of radius $r=0.5\xi$ (b) inside of which the superconducting transition temperature is suppressed. Results are shown for three different values of the applied current. Panels 1–4 show snapshots of ${\left|\psi \right|}^2$ for current and magnetic field values indicated on the $V\left(t\right)$ curves. White circles in panels 2–4 indicate the positions of the defects.

Figure 7

Voltage vs time characteristics of the sample of Fig. 6 without (dashed red curve) and with (solid black curve) defects at $I/\left(j_{0}w\right)=0.013$ and $H=0.365H_{c2}$. Panels 1–4 show snapshots of ${\left|\psi \right|}^2$ at time intervals indicated on the $V\left(t\right)$ curve.

Figure 8

The average ${\left|\psi \right|}_t$ vs applied magnetic field for the sample with $L=32\xi$ and $W=4\xi$ for different current densities $j$. The inset shows the system setup used for the simulations. Panels 1–9 show snapshots of $\left|\psi \right|$ for the current and field values indicated in the main panel. Panels 3${}^{\prime }$ and 9${}^{\prime }$ show contour plots of the time average of ${\left|\psi \right|}_t$ from which the trajectories of the moving vortices can be clearly identified. The GL parameter is $\kappa =2.2$.

Figure 9

Phase diagrams: ${\left|\psi \right|}_t$ versus the magnetic field $H$ and (a) the current density $j$ for $L=32\xi$ and $W=4\xi$; (b) the width $W$ for fixed length $L=32\xi$ at $j=0.65$; (c) the length $L$ for fixed width $W=4\xi$ at $j=0.65$. Regions I–IV indicate four different phases (see text for the description of different phases) separated by the white dashed lines and the Arabic numbers in (c) indicate the number of vortices pinned in the sample. The vertical red/dotted lines highlight the upper critical magnetic field $H_{c2}$. $H_{c1}^{*}$ and $H_{c2}^{*}$ shows the transitions from the Meissner state to the vortex state and from the vortex state to the normal state, respectively.