Finite Length Effects and Coulomb Interaction in Ge Quantum Well-Based Josephson Junctions Probed with Microwave Spectroscopy

Abstract

Proximitized Ge quantum wells have emerged as a novel platform for studying Andreev bound states (ABSs), due to their expected strong spin-orbit interaction and high mobility. Here, we used microwave spectroscopy techniques to investigate ABSs in Josephson junctions (JJs) realized in proximitized Ge quantum wells. Spectroscopic signatures observed in a 350 nm junction indicated the presence of multiple ABSs, and were reproduced with a model including finite-length effects. The ABS spectra measured for a $1.2~μ$m junction were explained by a model including three ABSs in two conduction channels and finite Coulomb interaction. Our work highlights the importance of interactions in JJs and serves as a basis for understanding and manipulating ABSs in Ge-based hybrid devices.

Figures (9)

• Figure 1: (a) Schematic of device 1, consisting of a PtSiGe superconducting loop (purple), Ti/Al/Ti/Au gates (yellow) and a Ti/Al/Ti/Au flux line (blue). The dc voltage $V_\mathrm{g}$ controlled the hole density in the Ge quantum well via the accumulation gate. The rf drive tone for two-tone spectroscopy measurements was applied via the side gate indicated with $V_\mathrm{s}$ using a bias-tee. For device 1, the magnetic flux $\Phi$ was controlled by passing a current $I$ through the flux line, while for device 2, $\Phi$ was controlled via a superconducting coil, mounted in the vicinity of the sample. (b,c) False-colored scanning electron micrographs (top view) of exemplary Josephson junctions, identical to device 1 (b) and 2 (c). The dimensions of each junction are defined by the separation $L$ between the superconducting PtSiGe electrodes (purple) and the width $W$ of the metallic accumulation gate (yellow). The side gate (yellow) was used to apply the rf drive tone. The gates were separated from the Ge heterostructure by a SiO${}_\mathrm{x}$ dielectric layer (not visible). (d) Optical microscope image after flip-chip bonding and wire bonding. The device chip ($3\times 3~\mathrm{mm}^2$) hosting the Ge quantum well was connected to the resonator chip ($10\times 10~\mathrm{mm}^2$) via In bumps. The grounded ends of the $\lambda/4$ Nb resonators were aligned to the superconducting loops of the devices (not visible). (e,f) Amplitude $R=\sqrt{I^2+Q^2}$ of the resonator transmission $S_{21}$ (normalized) as a function of readout frequency $f_\mathrm{r}$ for device 1 (e) and device 2 (f). The dashed black lines are fits to the data obtained using the circle-fit method Probst2015.
• Figure 2: (a) Device 1: amplitude $R$ of the resonator transmission $S_{21}$ as a function of offset readout frequency $f_\mathrm{r}-f_0$ and phase $\varphi$ at gate voltage $V_\mathrm{g}=-0.7650$ V, with $f_\mathrm{0}=6.90115$ GHz. (b) Amplitude $R$ of the resonator transmission $S_{21}$ as a function of drive frequency $f_\mathrm{d}$ ($P_\mathrm{d}=-26$ dBm) and phase $\varphi$, measured together with (a). A single Andreev bound state (ABS) pair transition (PT) was observed. The black dashed line is a fit to the PT frequency $f_1=(2E_1)/h$ using Eq. \ref{['eq1']} for $E_1$. (c) Same as (a), measured at $V_\mathrm{g}=-0.7443$ V. Anticrossings (red arrows) indicate ABS interaction with the resonator. (d) Same as (b), measured together with (c). Besides the PT, a single-quasiparticle transition (SQPT) was observed, indicating the presence of two ABSs. The black dashed line is a fit to the PT $f_\mathrm{1}=(2E_1)/h$ and the green dashed line corresponds to the SQPT with frequency $f=(E_2-E_1)/h$, where $E_1$ and $E_2$ are given by Eq. \ref{['eq1']}. (e) Amplitude $R$ of the resonator transmission $S_{21}$ as a function of drive frequency $f_\mathrm{d}$ ($P_\mathrm{d}=-25$ dBm) and gate voltage $V_\mathrm{g}$, measured at $\varphi=\pi$. The PT (black arrow) was observed over nearly the full range of $V_\mathrm{g}$. The dark blue horizontal arrow indicates a replica of the PT. The SQPT is visible around $V_\mathrm{g}=-0.7550$ V (green arrow). (f) Same as (e), measured at $\varphi=0$. The SQPT appeared in a limited range of $V_\mathrm{g}$, indicated with green arrows.
• Figure 3: (a) Energy spectrum of a long junction hosting three spin-degenerate Andreev bound states (ABSs) in two conduction channels (purple, orange) with energies $E_1$ (purple, solid line), $E_2$ (orange, solid line) and $E_3$ (purple, dashed line) as a function of the phase difference across the junction $\varphi$, shown in the excitation picture. The continuum above the induced superconducting gap $\Delta$ is indicated with gray shading. Coulomb interaction manifests as an energy penalty $U_i$ given by Eq. \ref{['eq3']} for states with even occupation. (b) Zoom-in of the energy spectrum in (a). The pair transition (PT) for the ABS with energy $E_1$ is indicated with arrows. The transition frequency is $f=(2E_1)/h$. (c) PT for the ABS with energy $E_2$, resulting in the transition frequency $f=(2E_2)/h$. (d) Single-quasiparticle transition (SQPT) from $E_1$ to $E_2$. The energy levels are shifted as a result of Coulomb interaction, yielding the transition frequency $f=(E_2 -E_1 +U_1 -U_2)/h$. (e) Same as (d), but with the first ABS initially in an excited state, resulting in $f=(E_2 -E_1 -U_1 -U_2)/h$. (f) Mixed PT with frequency $f=(E_1 + E_2 -U_1 -U_2)/h$. (g) SQPT from $E_2$ to $E_3$. Since Coulomb interaction is neglected for $E_3$ due to its close proximity to the continuum, the SQPT frequency is given by $f=(E_3-E_2+U_2)/h$.
• Figure 4: (a) Device 2: amplitude $R$ of the resonator transmission $S_{21}$ as a function of drive frequency $f_\mathrm{d}$ ($P_\mathrm{d}=-28$ dBm) and phase $\varphi$, measured at $V_\mathrm{g}=-0.7990$ V. Multiple pair transitions (PTs) and single-quasiparticle transitions (SQPTs) are visible around $\varphi=\pi$. (b) Same as (a), but overlaid with ABS transitions predicted by the long junction model with three ABSs and including Coulomb interaction. The black and gray dashed lines are PTs, schematically shown in Figs. \ref{['fig3']}(b) and (c). The gray dotted line is a replica of the gray PT, shifted downward by the resonator frequency $f_\mathrm{res}\approx6.86731$ GHz. The green, red and purple dashed lines are SQPTs corresponding to Figs. \ref{['fig3']}(d), (e) and (g), respectively. The cyan line is a mixed PT [Fig. \ref{['fig3']}(f)] and the orange line is a two-photon transition (2-PhT) involving the lowest-energy PT and a resonator photon with energy $E=hf_\mathrm{res}$. (c) Same as (a), but measured over a wider range of $\varphi$ and at $V_\mathrm{g}=-0.6994$ V and $V_\mathrm{s}=-0.0090$ V ($P_\mathrm{d}=-35$ dBm). At this gate voltage configuration, several PTs and SQPTs were observed. (d) Same as (c), overlaid with ABS transitions predicted by the model. The transitions are colored according to the legend in (b). Here, the yellow dashed line is a SQPT from the lowest-energy ABS $E_1$ to the highest-energy ABS $E_3$.
• Figure 5: Schematic representation of the flip-chip bonded device in the QCage.24 sample holder, the dilution refrigerator used to perform the measurements, and the electronic setup.