Anomalous Fraunhofer interference in epitaxial superconductor-semiconductor Josephson junctions

We investigate patterns of critical current as a function of perpendicular and in-plane magnetic fields in superconductor-semiconductor-superconductor (SNS) junctions based on InAs/InGaAs heterostructures with an epitaxial Al layer. This material system is of interest due to its exceptionally good superconductor-semiconductor coupling, as well as large spin-orbit interaction and $g$ factor in the semiconductor. Thin epitaxial Al allows the application of large in-plane field without destroying superconductivity. For fields perpendicular to the junction, flux focusing results in aperiodic node spacings in the pattern of critical currents known as Fraunhofer patterns by analogy to the related interference effect in optics. Adding an in-plane field yields two further anomalies in the pattern. First, higher-order nodes are systematically strengthened, indicating current flow along the edges of the device, as a result of confinement of Andreev states driven by an induced flux dipole; second, asymmetries in the interference appear that depend on the field direction and magnitude. A model is presented, showing good agreement with experiment, elucidating the roles of flux focusing, Zeeman and spin-orbit coupling, and disorder in producing these effects.

Figure 1

(a) Device and measurement schematic illustrating the extended superconducting Al banks (gray), InAs quantum well (yellow), and InGaAs barrier (green). The top gate (orange) is shown suspended above the junction, for clarity we have omitted the intervening ALD layer. $L$ and $W$ denote the junction length and width respectively. $L_c$ indicates the physical aluminum contact length. The coordinate system is illustrated in the inset. (b) Local magnetic-field focusing parameter $\gamma$ as a function of position $x$ for three different ratios $\beta =B_z/B_f$. On the upper horizontal axis we highlight $2L_{\mathrm{Al}}$, the contact length entering the model. (c) Differential resistance $R$, as a function of bias current $I$ and perpendicular magnetic field $B_z$. (d) Total magnetic field enhancement in the junction $\mathrm{\Gamma }$ as a function of $B_z$, calculated by extraction of the nodes visible in (c,e) (markers), and a fit using Eqs. (3) to (6) (solid line). (e) Critical current $I_c$, plotted logarithmically to highlight periodicity, extracted from (c) (markers). Overlaid are the expectation of Eq. (1) (green) and the modified form taking into account field enhancement due to flux focusing (red).

Figure 2

(a) Differential resistance $R$ as a function of bias current $I$ and in-plane magnetic field $B_x$, applied in the $x$ direction (along the direction of current flow). (b) As in (a) but with the in-plane field $B_y$ along the $y$ direction. (c) Schematic indicating how an in-plane field along $\widehat{x}$ can result in an effective flux dipole in the normal region. (d) Normalized critical current $I_c$ as a function of the angle $\theta$ between the in-plane field and $\widehat{x}$; the field has a fixed magnitude of $B_r=150\mathrm{mT}$. The dots represent the experimental data, the solid line is a theory curve based on a one-parameter fit of $\alpha$ at $\theta =\pi$, using the model based on Eq. (7) (see below). The red and yellow markers highlight the correspondence with (a) and (b), respectively.

Figure 3

(a) Differential resistance $R$ as a function of bias current $I$ and $B_z$, measured for different values of fixed $B_y$: $B_y=\pm 150\mathrm{mT}$ (top) and $B_y=\pm 400\mathrm{mT}$ (bottom). The white numbers in the upper left panel indicate the lobe indices. (b) As (a), for an in-plane magnetic field applied along $\widehat{x}$, using $B_x=\pm 150\mathrm{mT}$ (top) and $B_x=\pm 200\mathrm{mT}$ (bottom).

Figure 4

(a) Numerically calculated critical current as a function of $\mathrm{\Phi }$, normalized by $I_c^{\left(0\right)}$. The in-plane field is assumed along $\widehat{x}$ and the different curves correspond to $\alpha =0$, 0.21, 0.32, 0.43, and 0.64, from bottom to top (each offset by 1). (b) Symmetrized side-lobe maxima extracted from experimental data, for different in-plane fields. The field magnitudes indicated in the plot refer to $\mathbf{B}\parallel \widehat{x}$; all data points for $\mathbf{B}\parallel \widehat{y}$ (black dots) fall on top of the set marked 0 mT. (c) Side-lobe maxima obtained from the numerical data shown in (a).

Figure 5

(a) Normalized asymmetry $\mathcal{A}$ in the lobe maxima as a function of $B_y$, for the first two side-lobes (shown in blue and red respectively). (b) As (a), for magnetic fields oriented along $\widehat{x}$. (c) Magnitude of the asymmetry parameter $\left|\mathcal{A}\right|$ as a function of in-plane field angle $\theta$. Solid(dashed) lines connect points indicating where ${\mathcal{A}}_n$ are positive(negative). The in-plane field is fixed to $B_r=150\mathrm{mT}$. To emphasize the deviation of this angular dependence from the anisotropy observed for $B_z=0$ (see Sec. 4), we include in gray the height of the central lobe $I_c^{\left(0\right)}$ as a function of $\theta$ (arbitrary units). The panels along the edges show the differential-resistance data from which the asymmetries are extracted.

Figure 6

(a) Differential resistance $R$, as a function of gate voltage $V_{\mathrm{g}}$ and bias current $I$. (b) Normalized critical current as a function of side-lobe index $n$, for varying gate voltages, denoted by the colored markers in (a). (c)–(f) Differential resistance $R$, as a function of bias current $I$ and out-of-plane magnetic field $B_z$, for the different values of gate voltage marked in (a). Insets show the extracted supercurrent density $J_c\left(y\right)$. (c) is based on the same data set as shown in Fig. 1

Figure 7

(a) Measured magnetic field asymmetry ${\mathcal{A}}_1$ for the first side-lobe as a function of in-plane field angle $\theta$. (b) as (a) for the second side-lobe ${\mathcal{A}}_2$. Curves are colored according to gate voltage. Note that the zero gate voltage data is the same as that displayed in Fig. 5.