Gate-tunable Josephson diode in proximitized InAs supercurrent interferometers

The Josephson diode (JD) is a nonreciprocal circuit element that supports a larger critical current in one direction compared to the other. This effect has gained growing interest because of promising applications in superconducting electronic circuits with low power consumption. Some implementations of a JD rely on breaking the inversion symmetry in the material used to realize Josephson junctions (JJs), but recent theoretical proposals have suggested that the effect can also be engineered by combining two JJs hosting highly transmitting Andreev bound states in a Superconducting Quantum Interference Device (SQUID) at a small, but finite flux bias. We have realized a SQUID with two JJs fabricated in a proximitized InAs two-dimensional electron gas (2DEG). We demonstrate gate control of the diode efficiency from zero up to around $30\%$ at specific flux bias values which comes close to the maximum of $\sim 40\%$ predicated in Souto et al. [Phys. Rev. Lett. 129, 267702 (2022)]. The key ingredients to the JD effect in the SQUID arrangement is the presence of highly transmitting channels in the JJs, a flux bias, and an asymmetry between the two SQUID arms.

Figure 1

(a) Circuit schematic of a dc SQUID threaded by the external flux ${\mathrm{\Phi }}_{\mathrm{ext}}$, formed by two gate tunable JJs with nonsinusoidal CPRs with critical currents $I_{\text{c1}}$, $I_{\text{c2}}$ and transparencies ${\tau }_1$, ${\tau }_2$. (b) False-color electron micrograph of the device. The loop consists of a 10 nm Al film (blue) grown on top of an InAs 2DEG (green). The JJs are defined by selectively removing the Al over 150-nm-long stripes on each branch of the loop. Electrostatic gates (yellow and orange) tune the charge carrier density in the junction. We use 15 nm of ${\mathrm{HfO}}_2$ (light blue) as a gate dielectric. On the right, a zoom-in of ${\mathrm{JJ}}_2$ is shown before adding the $\mathrm{FG}$. On top, we show a cross-sectional schematic of the gate configuration of ${\mathrm{JJ}}_2$ along the dashed black line. The scale bar in the main figure is $1\mu \mathrm{m}$, and in the zoom-in it is 300 nm. Dc and ac current bias are defined through the voltage drop over a large series resistor with value $R_b=1\mathrm{M}\mathrm{\Omega }$. The SQUID is shunted to ground with a parallel resistor of value $R_s=10\mathrm{\Omega }$. (c) Differential resistance of ${\mathrm{JJ}}_1$ (left) and ${\mathrm{JJ}}_2$ (right) as a function of gate voltage and current bias. While one junction is being measured, the other is pinched off. The top junction has a slightly higher critical current due to the different channel widths of ${\mathrm{W}}_1=3\mu \mathrm{m}$ and ${\mathrm{W}}_2=2.5\mu \mathrm{m}$.

Figure 2

(a) SQUID oscillations with $V_{\mathrm{G}1}=V_{\mathrm{G}2}=0$. The critical current $I_{\text{c}}^{+}$ and the retrapping current $I_{\text{r}}^{-}$ over one flux period are highlighted in orange and red, respectively. At fixed magnetic field, the absolute value of the critical current in the two sweep directions is not the same. This is best seen in the region $-5<B_{\perp }<0\mu \mathrm{T}$ with a visible example taken at the red and orange arrows, where the SDE has a magnitude of $\sim 23$%. (b) Measurement for a strongly asymmetric SQUID setting with $V_{\mathrm{G}1}=0$ V and $V_{\mathrm{G}2}=-1.1$ V. The junction with the large critical current ${\mathrm{JJ}}_1$ serves as the reference junction. As a consequence, the critical current as a function of flux now reflects the CPR of the weaker junction ${\mathrm{JJ}}_2$. The CPR is strongly nonsinusoidal, and a fit (black dashed line) yields ${\tau }_2^{*}=0.8$. (c) Plot of the extracted $I_{\text{c}}^{+}$ (orange) and $I_{\text{c}}^{-}$ (red) taken from the measurement in (a) and from a measurement where we sweep the current bias from positive to negative values (see Appendixes). The dashed two curves (green and blue) show simplified model fits with ${\tau }_{1,2}^{*}=0.86$ and the critical currents of the junctions taken from Fig. 1.

Figure 3

Magnitude of the diode efficiency $\left|\eta \right|$ as a function of external flux ${\mathrm{\Phi }}_{\mathrm{ext}}$ for different gate configurations as obtained from the measurements (left) and as calculated from the model (right). The sign of $\eta$ is indicated on the visible lobes with $+$ and $-$. The model takes into account the numerically simulated loop inductances, their asymmetry, and the values $I_{c(1,2)}$ of the two junctions obtained from the measurements in Fig. 1. The JJ transparencies were fixed to ${\tau }_1^{*}={\tau }_2^{*}=0.86$. (a) $\left|\eta \right|$ as a function of $V_{\mathrm{G}2}$ at fixed $V_{\mathrm{G}1}$, and (b) $\left|\eta \right|$ as a function of $V_{\mathrm{G}1}$ at fixed $V_{\mathrm{G}2}$. Note that for ${\mathrm{\Phi }}_{\text{ext}}/{\mathrm{\Phi }}_0=0.5$, which equals ${\phi }_{\mathrm{ext}}=\pi$, $\eta =0$ independent on any other parameters.

Figure 4

(a) False color optical image of the full device together with a sketch of the measurement setup. The scale bar is $100\mu \mathrm{m}$. (b) Zoom-in over the SQUID showing the loop area threaded by the external flux ${\mathrm{\Phi }}_{\mathrm{ext}}$. The electron density in the junction region is tuned via a set of gates coloured in yellow and brown. The scale bar is $3\mu \mathrm{m}$.

Figure 5

Sonnet simulations of the loop inductances. The superconducting loop is segmented into an upper (lower) branch 1 (2) indicated by the white dashed boxes. The respective width are $W_1=3\mu \mathrm{m}$, $W_2=2.5\mu \mathrm{m}$, and $W_L=2.75\mu \mathrm{m}$. The two inductances $L_1$, $L_2$ and the mutual inductance $M$ are deduced from the slope of the frequency dependent two-port impedances. It is seen that $M\ll L_{1,2}$ and that there is a small asymmetry of $\sim 6\%$ in the loop inductances.

Figure 6

(a) Two differential resistance plots of the SQUID device for the same gate settings as a function of external flux ${\mathrm{\Phi }}_{\text{ext}}$ and current bias $I$. In the upper plot the current was swept downwards from positive to negative values, while in the lower it was swept upwards. Panels (b) and (c) compare the critical current values $I_{\text{c}}$ with the retrapping ones $I_{\text{r}}$, obtained from (a) and (b) at the position of the peaks in $dV/dI$. The arrows $\uparrow ,\downarrow$ indicate the sweep direction. Panel (d) compares $I_{\text{c}}^{+\uparrow }$ with $I_{\text{c}}^{-\downarrow }$, and in (e) the diode efficiency is shown for three ways using the data in (b)–(d).

Figure 7

SQUID oscillation at different gate voltage configurations. $V_{\mathrm{G}2}$ is fixed at $-0.5$ V, while $V_{\mathrm{G}1}$ is swept from $-0.57$ V to $-0.8$ V. The asymmetry in the SQUID oscillations follows the asymmetry in critical current between the two junctions. We have $I_{\text{c1}}(V_{\mathrm{G}1}=-0.57V)>I_{\text{c2}}(V_{\mathrm{G}2}=-0.5V)$ and $I_{\text{c1}}(V_{\mathrm{G}1}=-0.8V)<I_{\text{c2}}(V_{\mathrm{G}2}=-0.5V)$.

Figure 8

Sequence of simulations for $I_{\text{c}}^{+}$ (green dashed curves) and $I_{\text{c}}^{-}$ (blue dashed curves) to measured (up-sweep) data $I_{\text{c}}^{+}$ (orange dots) and $I_{\text{r}}^{-}$ (red dots). In (a) a sinusoidal CPR with the estimated loop inductance asymmetry is considered, while in graphs (b)–(e) the effective transparencies ${\tau }^{*}$ of the junctions ar increased. Further details are given in the text.

Figure 9

Magnitude of the diode efficiency $\left|\eta \right|$ as a function of the applied external flux ${\mathrm{\Phi }}_{\text{ext}}$ expressed in number of magnetic flux quanta ${\mathrm{\Phi }}_0$, numerically calculated for a SQUID with two sinusoidal CPRs with an asymmetry (a) in loop inductance $\gamma$ and (b) in critical current $\alpha$. The inductances were chosen such that ${\phi }_{\text{L}}=\pi$. In (c) $\left|\eta \right|$ is plotted for a SQUID without loop inductances and two JJs, each with a nonsinusoidal single-channel CPR, as a function of ${\tau }_1$ and normalized external flux for ${\tau }_2=0.7$ and for $\alpha =\gamma ={\phi }_{\text{L}}=0$.