Impact of Kinetic Inductance on the Critical-Current Oscillations of Nanobridge SQUIDs

In this work, we study the current-phase relation ($\mathrm{C}\mathrm{\Phi }\mathrm{R}$) of lithographically fabricated molybdenum germanium (${\mathrm{Mo}}_{79}{\mathrm{Ge}}_{21}$) nanobridges that is intimately linked with the nanobridge’s kinetic inductance. We do this by imbedding the nanobridges in a superconducting quantum interference device (SQUID). We observe that, for temperatures far below $T_c$, the $\mathrm{C}\mathrm{\Phi }\mathrm{R}$ is linear, as long as the condensate is not weakened by the presence of a supercurrent. We demonstrate lithographic control over the nanobridge kinetic inductance, which scales with the nanobridge aspect ratio. This allows the $I_c(B)$ characteristic of the SQUID to be tuned. The SQUID properties that can be controlled in this way include the SQUID’s sensitivity and the positions of the critical-current maxima. These observations can be of use for the design and operation of future superconducting devices, such as magnetic memories or flux qubits.

Figure 1

Scanning-electron-microscopy (SEM) image of a prototypical SQUID device. Area indicated in red corresponds to the Dayem bridge, while the yellow area indicates the nanobridge. Width $W$ and length $L$ of the latter are indicated. White scale bar represents 200 nm. Four-point current and voltage contacts are indicated as $I\pm$ and $V\pm$, respectively. White circuit diagram presents an equivalent electronic circuit of the SQUID. $L_{K1}$ and $L_{K2}$ represent inductances of each branch, while $I_{c1}$ and $I_{c2}$ represent two critical currents of each branch. $I_{c,\mathrm{tot}}$ is the total critical current of the SQUID. Applied magnetic field, $B$, is oriented as shown.

Figure 2

(a) Critical current versus magnetic field for device $g$, which contains two Dayem bridges. Measured critical currents for positive (negative) bias are shown in blue (red). Solid lines represent vorticity diamonds generated by the model (see text). Fitting parameters of the vorticity diamonds are $I_{c1}=(27.5\pm 0.8)\mu \mathrm{A}$, ${\phi }_{c1}=(10.2\pm 0.1)\mathrm{rad}$, $I_{c2}=(24.7\pm 0.9)\mu \mathrm{A}$, ${\phi }_{c2}=(11.2\pm 0.1)\mathrm{rad}$, $L_{K1}=(122\pm 4)\mathrm{pH}$, and $L_{K2}=(149\pm 6)\mathrm{pH}$. $n_v=0$ vorticity diamond is indicated in gray. (b) Scanning-electron-microscopy image of the investigated device, in which the scale bar corresponds to 200 nm. Applied magnetic field, $B$, is oriented as shown. Bottom SQUID arm corresponds to $j$ = 1 in Eq. (2), top SQUID arm to $j$ = 2. (c) Magnification of the top (bottom) vertices of the diamonds; positive currents are again shown in blue, absolute values of the negative critical currents in red. Dotted lines show how taking a nonlinear $L_K$ into account can capture the shape of the diamond top.

Figure 3

(a) The $n_v=0$ vorticity diamond for current-asymmetry $\alpha =0.3$ and phase-asymmetry $\gamma =0.1$. Vertices of the diamonds are determined by asymmetries in the critical currents and critical phase differences. The diamond extends over a range $B=\mathrm{\Delta }B{\phi }_{c,\mathrm{tot}}/\pi$ in magnetic field. The $n_v\mathrm{th}$ vorticity diamond is identical to the $n_v=0$ diamond, but shifted along the magnetic field axis by $B=n_v\mathrm{\Delta }B$. (b) Evolution of the diamond shape for $\alpha =0$ and increasing $\gamma$ from 0 to 0.8 in steps of 0.2, as indicated by the arrows. (c) Evolution of the diamond shape for $\gamma =0$ and for increasing $\alpha$ from 0 to 0.8 in steps of 0.2, as indicated by the arrows.

Figure 4

(a) Critical current versus magnetic field for device $u$, which contains a nanobridge of length $L\sim 319\mathrm{nm}$ and width $W\sim 34\mathrm{nm}$. Measured critical currents for positive (negative) bias are shown in blue (red). Solid lines represent vorticity diamonds generated by the model. Fitting parameters of the vorticity diamonds are $I_{c1}=(17.1\pm 0.8)\mu \mathrm{A}$, ${\phi }_{c1}=(13.5\pm 0.6)\mathrm{rad}$, $I_{c2}=(16.7\pm 0.6)\mu \mathrm{A}$, ${\phi }_{c2}=(7.9\pm 0.6)\mathrm{rad}$, $L_{K1}=(261\pm 17)\mathrm{pH}$, and $L_{K2}=(156\pm 15)\mathrm{pH}$. $n_v=0$ vorticity diamond is indicated in gray. (b) Scanning-electron-microscopy image of the investigated device; scale bar corresponds to 200 nm. Applied magnetic field, $B$, is oriented as shown. Bottom SQUID arm corresponds to $j=1$ in Eq. (2), top SQUID arm to $j=2$. Due to incomplete lift-off, the inner part of the SQUID is not completely removed. (c) Magnification of the top (bottom) vertices of the diamonds; positive currents are again shown in blue, the absolute values of negative critical currents in red. Dotted lines show how taking a nonlinear $L_K$ into account can capture the shape of the diamond top.

Figure 5

(a) Field dependence of the current-normalized $n_v=0$ vorticity diamond for devices $g$ (blue), $c$ (red), $m$ (purple), $u$ (light blue), $v$ (burgundy), and $w$ (green). Arrow at the top shows the increase in the nanobridge length $L$. (b) Kinetic inductance of the nanobridges, as obtained from the fitting procedure, versus aspect ratio of the bridge for devices $c$, $m$, $u$, $v$, and $w$. Solid line shows linear fitting, using $L_K=\beta (L/W)({\hslash R}_s/k_B{T}_c)$, where $T_c=6\mathrm{K}$ and $R_s=88\mathrm{\Omega }$. Fitting yields $\beta =0.25\pm 0.04$.

Figure 6

(a) $I_c^{+}(B)$ characteristics of device $v$, shown for different sweep rates (0.55, 37.5, and 75 mA/s). Gray dotted line shows that the current-maximum position is only impacted by high sweep rates. (b) Dependence of the average nanobridge positive critical current on sweep rate (S_R) for device $v$ at $B=0\mathrm{mT}$, with sweep rates ranging from 0.275 to 150 mA/s. Inset shows the stochastic distribution for selected sweep rate values. (c) Dependence of the standard deviation of the nanobridge positive critical current on sweep rate for device $v$ at $B=0\mathrm{mT}$, with sweep rates ranging from 0.275 to 150 mA/s. At least 2500 critical currents per field value are collected to calculate the average critical current and its standard deviation. Above this value, both the average and standard deviation converge towards an asymptotic value for all sweep rates.

Figure 7

Positive flux oscillations, $I_c^{+}(B)$, of a ${\mathrm{Mo}}_{79}{\mathrm{Ge}}_{21}$ nanobridge SQUID. Unique vorticity diamond of $n_v=3$ is indicated by the gray area. System is prepared in the $n_v=3$ vorticity state by repeatedly applying a bias current in the unique vorticity diamond, as indicated by the red square. After that, the field is changed to the readout value under zero bias current. Once there, the critical current is measured by taking a regular $VI$, as shown by the green triangle. By repeating this process, both branches of the diamond can be probed. Same procedure is applicable to the other vorticity diamonds.

Figure 8

Schematic overview of the ac setup.