Nonlinear Nanodevices Using Magnetic Flux Quanta

All devices realized so far that control the motion of magnetic flux quanta employ either samples with nanofabricated spatially-asymmetric potentials (which strongly limit controllability), or pristine superconductors rectifying with low-efficiency time-asymmetric oscillations of an external magnetic field. Using layered ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ materials, here we fabricate and simulate two efficient nonlinear superconducting devices with no spatial asymmetry. These devices can rectify with high-efficiency a two-harmonic external current dragging vortices in target directions by changing either the relative phase or the frequency ratio of the two harmonics.

Figure 1

Out-of-plane flux quanta rectifier.—Panel (a): Rectified dc voltage (which is proportional to the drift vortex velocity) as a function of the out-of-plane magnetic field $H_{\perp }$ at different temperatures. Inset in Panel (a): Nonlinear voltage-current characteristics of the device shown in panel (a) at the second matching field $H_2^{\mathrm{match}}$ (red line), and at two other fields shifted $\pm 5\mathrm{Oe}$ with respect to $H_2^{\mathrm{match}}$. Micrograph in (b): Scanning ion microscopy (SIM) image of a nanofabricated ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ single-crystal film which has a triangular lattice of holes, which were milled by FIB. The holes have a diameter of about 300 nm and are separated by $a_0=1\mu \mathrm{m}$. Holes trap out-of-plane flux quanta resulting in the very nonlinear vortex dynamics needed for nonlinear rectifiers. Panel (b): Measured rectified dc voltage $V_{\mathrm{dc}}$ (red symbols for $H_1^{\mathrm{match}}$ and blue symbols for $H_2^{\mathrm{match}}$, both pointed by arrows labeled “experiment”) versus relative phase $\phi$ between the two harmonics of the driving current $I_{\mathrm{drive}}({\nu }_1,{\nu }_2=2{\nu }_1,\phi )$. The output dc voltage $V_{\mathrm{dc}}$ is normalized by its value $V^{*}=V_{\mathrm{dc}}(\phi =0,H_1^{\mathrm{math}})$ at zero phase shift $\phi =0$ between the harmonics and for the first matching field, $H_1^{\mathrm{match}}$ (i.e., when the number of vortices coincides with the number of holes). The frequency of the second harmonic is chosen here to be twice the frequency of the first harmonic (${\nu }_2=2{\nu }_1=20\mathrm{Hz}$); the amplitude of each harmonic is $I_0=10\mathrm{mA}$, and the temperature 86.8 K. Simulated dc voltages are shown by black symbols for $H_1^{\mathrm{match}}$ and by the green line for $H_2^{\mathrm{match}}$, both pointed by arrows labeled “theory.” Normalized simulation parameters are ${\nu }_1=0.01$, $r^{(p)}=0.2$, $\max (|f_L|)=0.2$, $\max (|f_{pv}|)=0.3$. Note that by changing $\phi$ we can: (i) easily tune the output dc voltage from negative $V_{\mathrm{dc}}(\phi =\pi )$ to positive $V_{\mathrm{dc}}(\phi =0)$ values and (ii) obtain a very smooth and predictable dependence for $V_{\mathrm{dc}}(\phi )$. Note that the $V_{\mathrm{dc}}(\phi )$ dependence is similar to the $I(\phi )$ behavior of a Josephson current between two superconductors as a function of the phase difference of the order parameter. Bottom inset in (b): geometry of the problem.

Figure 2

In-plane flux quanta rectifier.—Panel (a): The blue curve (also pointed by arrow labeled “voltage at a fixed current”) shows the flux-flow voltage, measured at a fixed time-independent current of $100\mu \mathrm{A}$, versus an externally applied magnetic field $H$, which is almost parallel to the ${\mathrm{CuO}}_2$ layers ($H_{\perp }\ll H_{\parallel }$). When increasing the field $H$, two regimes are clearly distinguished: (1) the “JV only” regime corresponds to the absence of out-of-plane magnetic flux quanta—only JVs are in the system; (2) the “$\mathrm{JV}+\mathrm{PV}$” regime, when $H_{\perp }$ ($\propto H$) is strong enough to create PVs, which act as pinning centers (traps) for JVs—resulting in strongly nonlinear dynamics. The rectified dc voltage (red curve, also pointed by arrow labeled “rectified voltage”), measured when a two-harmonic ac current, $I_{\mathrm{drive}}({\nu }_1=100\mathrm{Hz},{\nu }_2=2{\nu }_1,\phi =0)$, with amplitude $I_0=1\mathrm{mA}$ was applied, is low for the weakly nonlinear “JV only” regime, and strongly enhanced in the “$\mathrm{JV}+\mathrm{PV}$” regime. Micrograph in (b): SIM image of the fabricated ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ single-crystal bar for the measurements of Josephson-vortex flow. The device is shaped as a narrow bridge joining large parts of the superconducting bar. The $14.8\times 24.5\times 0.6\mu {\mathrm{m}}^3$ bridge was fabricated by FIB. Panel (b): Red squares are the measured rectified dc voltage $V_{\mathrm{dc}}({\nu }_2,{\nu }_1)$, normalized by $V^{+}=V_{\mathrm{dc}}({\nu }_1,{\nu }_2=2{\nu }_1)$, versus ${\nu }_2$, for a fixed ${\nu }_1=100\mathrm{Hz}$ when the two-harmonic driving current $I_{\mathrm{drive}}({\nu }_1,{\nu }_2,\phi =0)$, with amplitude $I_0=1\mathrm{mA}$, is applied. The green line with sharp peaks corresponds to a simulation of $V_{\mathrm{dc}}({\nu }_2,{\nu }_1)/V^{+}$ versus ${\nu }_2$ using the following normalized parameters ${\lambda }_c=2$, ${\lambda }_{PJ}=0.5$, $f_{JJ}^0=0.5$, $f_{PJ}=0.5$, $\max (|f_L|)=3$, ${\eta }_P/{\eta }_J=5$, ${\nu }_1=0.01$. All the measured peaks of the rectified voltage are perfectly reproduced in our simulations. Right inset in panel (b): Experimental $V_{\mathrm{dc}}({\nu }_2=2{\nu }_1,{\nu }_1=100\mathrm{Hz})$, normalized by $V^{*}=V_{\mathrm{dc}}(\phi =0)$, versus relative phase $\phi$ between the two harmonics. The $V_{\mathrm{dc}}(\phi )$ Josephson-type dependence shown is very similar to the one [Fig. 1b] obtained for our out-of-plane quanta rectifier. The temperature was 10 K for all measurements. Left inset in (b): geometry of the problem.