Current-phase relations of InAs nanowire Josephson junctions: From interacting to multimode regimes

Gate-tunable semiconductor-superconductor nanowires with superconducting leads form exotic Josephson junctions that are a highly desirable platform for two types of qubits: Those with topological superconductivity (Majorana qubits) and those based on tunable anharmonicity (gatemon qubits). Controlling their behavior, however, requires understanding their electrostatic environment and electronic structure. Here we study gated InAs nanowires with epitaxial aluminum shells. By measuring current-phase relations and comparing them with analytical and numerical calculations, we show that we can tune the number of modes, determine the transparency of each mode, and tune into regimes in which electron-electron interactions are apparent, indicating the presence of a quantum dot. To take into account electrostatic and geometrical effects, we perform microscopic self-consistent Schrodinger-Poisson numerical simulations, revealing the energy spectrum of Andreev states in the junction as well as their spatial distribution. Our work systematically demonstrates the effect of device geometry, gate voltage, and phase bias on mode behavior, providing insights into ongoing experimental efforts and predictive device design.

Figure 1

Experimental setup to measure the current-phase relation of an InAs nanowire-based Josephson junction. (a) The sample ring consists of an evaporated aluminum ring (blue), which is interrupted by an InAs nanowire (green with yellow gates) to form a Josephson junction. To measure the current-phase relation of the junction, the sensing head of a SQUID microscope is positioned above the sample ring. A current bias applied to the field coil (light gray) tunes the magnetic flux through the sample ring, providing control of the phase difference across the junction. This phase difference determines the supercurrent in the junction via its current-phase relation. The supercurrent circulates in the sample ring, leading to a signal which is measured by the pickup loop (dark gray). Drawing is to scale. (b) A scanning electron micrograph depicts a top view of the portion of the sample ring spanned by the InAs nanowire. As depicted in the bottom panels, the nanowire has a hexagonal cross section and is coated epitaxially on four facets with a 7-nm layer of aluminum (blue). This epitaxial aluminum layer is chemically removed along a 100-nm span beneath the cutter gate, while the wire is isolated from the gate with a 5-nm layer of ${\mathrm{HfO}}_2$ (pink), shown in bottom left cross section. The center and right cross sections indicate the structure of the wire in ungated sections and in sections with a side gate. The respective locations of the cross sections are labeled by red, blue, and yellow arrows on the SEM image. (c) A cross-sectional side view of the wire shows the layer structure of the device. The sample ring rests on a doped silicon substrate, providing the ability to gate the device globally from below.

Figure 2

CPR measurements probe the evolution of supercurrent-carrying modes with gate voltage. (a) The amplitude and shape of the CPR evolve as the voltage applied to the cutter gate decreases from 0 to $-2$ V, corresponding to changes in the structure of the Andreev bound states as a function of applied electric field. Each curve is a measured current-phase relation at a particular cutter gate voltage. The waterfall plot represents an evenly spaced subset of the gate sweep and is offset vertically for clarity. (b) The dependence of CPR amplitude and shape on cutter gate voltage. The black curve shows the critical current amplitude (left $y$ axis) vs gate voltage. The light blue curve shows CPR skew vs gate voltage (right $y$ axis). The skew is calculated as the ratio between the second and first harmonics (i.e., Fourier component) of the CPR, $S=-\frac{A_2}{A_1}$. Recurring peaks at $V_C<-1.16V$ (red dashed line) signified resonant tunneling in the single-channel regime. (c) Fitting CPR data at $V_C=-1.22$ V (black dots) to Eq. (2) (red line) results in a single channel with $\tau =1$, $T=98$ mK, and $\mathrm{\Delta }=128\mu \mathrm{eV}$. (d) Scatterplot of CPR amplitude vs skew for all measured cutter gate voltages between 0 and $-2$ V reveals distinct regimes. Red dots are taken from $V_C<-1.16$ V, showing highly correlated amplitude and skew expected from a single channel. The blue curve is the expected single-channel behavior calculated from Eq. (2) (with $T=98$ mK and $\mathrm{\Delta }=128\mu \mathrm{eV}$). (e) Extracted transmission coefficients $\{{\tau }_1\cdots {\tau }_6\}$ from best fits to Eq. (2), showing the evolution of the number and transparency of supercurrent-carrying channels with cutter gate voltage. Different colors are only for visual clarity and are not meant to differentiate physical states. The vertical width of shaded regions represents the $68\%$ confidence interval of the fitted parameters.

Figure 3

Numerical simulations of the InAs junction. (a) Top: Schematics of the simulated model, consisting of two wire segments and the junction, each augmented with an electric gate. Empty spaces at the interface of the wire and vacuum in the junction indicate the zero boundary value of potential. Color indicates the calculated electrostatic potential for the left/right gate voltage of $-0.29$ V and cutter voltage of 0.06 V. Bottom: Two-dimensional cuts of the electrostatic potential along the $x$ direction (left) and along the $z$ direction (right). (b) The density of the lowest positive Andreev state at phase $\varphi =\pi$ in the junction at a cutter voltage of (b1) 0.07 V and (b2) 0.06 V, corresponding to the regimes of low and high transparency in the junction, respectively. The color scale shows the probability density calculated as the square of the wave function amplitude. (c) Top: Calculated dependence of the critical current (red and black) and skew (blue) on the cutter gate voltage. Arrows indicate the gate voltages of (b1) and (b2). Bottom: The dependence of the calculated critical current on the skew. In both panels red indicates the single-channel regime. (d) The dependence of the position of the highest occupied level in the junction on the cutter gate voltage.

Figure 4

Using the CPR to elucidate the role of interactions in the single-channel regime. (a) In the single-channel regime, the CPR amplitude develops recurring peaks, indicated by arrows. (b) At two of these peaks, a shoulder appears in the CPR near phases equal to $\pi$ modulo $2\pi$. The colored borders surrounding the plots correspond to colored arrows in (a). The appearance of the shoulders in the CPR is evidence of finite charging energy in the junction region of the device. (c) Modeling our device with finite charging energy allows calculation of CPRs in this regime. The two leads, with gap $\mathrm{\Delta }$ and chemical potential ${\mu }_s$, are coupled to the junction symmetrically with coupling $\mathrm{\Gamma }$. The charging energy is modeled as an exchange energy $J$ in the device, which splits a state at energy $\xi$ in the junction into odd- and even-parity states with respective energies ${\epsilon }_{\downarrow }$ and ${\epsilon }_{\uparrow }$. (d) and (e) The model allows calculation of the spectrum of Andreev states and the CPR, shown here without charging energy (red dashed lines) and with finite charging energy (black solid lines). The appearance of a shoulder is characteristic of finite charging energy when $\xi /\mathrm{\Gamma }\approx 0$ and can be used in fits to the data [red curves in (b)]. (f) Fitting the model to the data throughout the single-channel regime reveals the evolution of energy levels in the device. (g) The model can also be fit to the data at different values of side gate voltage, revealing that more negative side gate voltage corresponds to increased $J/\mathrm{\Gamma }$.