Comparing tunneling spectroscopy and charge sensing of
Andreev bound states in a semiconductor-superconductor hybrid nanowire structure

Experiments with superconductor-semiconductor hybrid structures have demonstrated the presence of Andreev bound states (ABSs) in proximitized semiconductors by a variety of methods, including tunneling spectroscopy in NIS junctions (N is normal, S is superconductor, I is an insulating tunnel barrier), Coulomb blockade spectroscopy in superconducting islands with normal leads (NISIN) Higginbotham et al. (2015); Albrecht et al. (2016); Shen et al. (2018); Deng et al. (2016); Vaitiekėnas et al. (2020), and SNS spectroscopy of Josephson junctions Laroche et al. (2019); van Zanten et al. (2020); Sabonis et al. (2020); Kringhøj et al. (2021); Tosi et al. (2019); Hays et al. (2020, 2018); van Woerkom et al. (2017). These methods reveal distinct spectroscopic features and raise different technical challenges. With growing interest in detecting and controlling Majorana zero modes (MZMs) across Josephson junctions Ginossar and Grosfeld (2014); Hassler et al. (2011); Hyart et al. (2013) and distinguishing them from ABSs, it is important to compare methods in systems that can be probed several ways.

In this Article, we report measurements of an electrostatically gated multi-segment hybrid InAs/Al nanowire (NW) device [see Fig. 1(a)] that is reconfigurable in situ, allowing a comparison of spectroscopic signatures of the same ABS by three methods: a Josephson junction (SIS), a Coulomb island (SISIN), and a radio-frequency (RF) charge sensor. We observed consistency between methods, noting that RF charge detection was especially fast, and, unlike transport, does not alter charge occupancy during measurement.

The device was fabricated from an InAs NW grown by molecular beam epitaxy, with Al grown on three facets of the hexagonal core. Using conventional lithographic processing, Al was removed by wet etching in $\sim$100 nm segments adjacent to cutter gates C1 and C2, providing gate-controllable barriers. Blanket atomic layer deposited ${\rm HfO}_{2}$ insulated all gates from the NW. Depleting the InAs with C1 and C2 yielded an SISIN device with an S island separated by tunnel barriers (I) from an S lead on the left and N lead on the right. Voltages applied to plunger gates P1-P3 tuned carrier density in adjacent proximitized semiconductor regions as well as the charge offset on the S island. At the far right, a bare InAs segment (again, created by etching of Al), served as RF single-electron transistor (RF-SET) that was capacitively coupled to the main S island by a floating gate [see Fig. 1(a)]. Data acquisition followed Ref. van Zanten et al. (2020) and charge sensing followed Ref. Razmadze et al. (2019). Measurements were performed in a dilution refrigerator with a 6-1-1 T vector magnet.

To generate ABSs in the nanowire, we applied a moderate magnetic field, $B$ = 600 mT, along the nanowire axis and set gate C1 to the tunneling regime (conductance $g\ll e^{2}/h$), junction C2 to the open regime ($g\sim e^{2}/h$), and gates P1-P3 to $-4$ V to reduce density in the NW. In this configuration, SIS spectroscopy [Figs. 1b)] was used to locate a single subgap state by tuning P3. Figure 1(d) shows differential conductance $g$ as a function of bias $V_{B}$ and $B$ in this SIS configuration. Near $B=0$ a precursor of the finite-field ABS is seen at $V_{B}~{}\sim~{}\pm 330~{}\mu$V. Increasing $B$ leads to a single ABS moving down with an effective $g$ factor of $\sim 6$. Starting at $B\sim 0.6$ T [red marker in Fig. 1(d)] the feature at $V_{B}$ = 220 $\mu$V becomes independent of the further increase $B$. This presumably results from competition between a reduction of the induced superconducting gap and a zero-energy-crossing ABS Shen et al. (2018). Unless otherwise noted, measurements described below are at $B$ = 0.6 T.

The location of the ABS within the NW could be inferred from its dependence on P1 and P2. Specifically, spectroscopy showed little response to $V_{P1}$ and a strong response to $V_{P2}$, as shown in Figs. 2(a,b), indicating that the ABS was located on the right side of cutter gate C1, as represented in Fig. 1(b). The energy of the ABS approached the gap edge for $V_{P2}\sim-1.6$ V and $\sim-1.35$ V with a minimum at around $V_{P2}\sim-1.5$ V. Differential conductance as a function of bias, $V_{B}$, and gate P1 [Fig. 2(c)] were carried out at $V_{P2}=-1.49$ V.

Following SIS spectroscopy, gates C1 and C2 were set to the tunneling regime to create a superconducting island, providing SISIN spectroscopy. Figure 3(a) shows the resulting two-dimensional conductance map as a function of bias and island plunger voltage $V_{P3}$. In the SISIN configuration, transport showed 1$e$-periodic resonances at high bias, $V_{B}~{}\gtrsim~{}400~{}\mu$V, and 2$e$-periodic features at low bias. The black dashed diagonal line in Fig. 3(a) indicates the alignment of the chemical potential of the left lead and the island. Four features along this diagonal are highlighted: Near $V_{B}=0$, faint 2$e$-periodic features that become stronger along the diagonal at $V_{B}\sim 65~{}\mu$V ($\bullet$), accompanied by negative differential conductance (NDC); Further increasing bias yielded stronger 1$e$-periodic features starting at $V_{B}\sim 160~{}\mu$V ($\blacksquare$) along with stronger NDC; Around $V_{B}\sim 240~{}\mu$V ($\blacklozenge$) the conductance increase was largely independent of $V_{P3}$; Finally, at $V_{B}~{}\sim~{}300~{}\mu$V the strong 1$e$-periodic conductance resonances ($\bigstar$) were observed.

These marked features in Fig. 3(a) are consistent with a model of an SISIN device with a single ABS at energy $\delta$ on the S island with electrostatic energy $E_{C}(n-n_{g})^{2}$, where $E_{C}$ is the charging energy, $n$ is the number of electrons on the island, and $n_{g}$ is the dimensionless gate voltage. In particular, the 2$e$-periodic features at zero bias arise from resonant Cooper pair tunneling between S lead and S island at odd integer values of $n_{g}$. At increased bias, the resonance occurs at $2V_{B}=4E_{C}n_{g}$ [dashed line in Fig. 3(a)], with the S island on resonance with the left S lead while the energy difference from the Fermi level in the N lead increases. For the case $\delta<E_{C}$, the 2$e$-periodic features are expected to be suppressed around zero bias, as Cooper pair resonances are between excited states above the odd parity ground state of the occupied ABS close to odd integer values of $n_{g}$. The 2$e$-periodic features around zero bias in Fig. 3(a) suggest that indeed $\delta<E_{C}$. At $V_{B}=E_{C}-\delta$ [$\bullet$ in Fig. 3(a)], the ABS can empty into the right normal lead, enabling resonant Cooper pair tunneling, brightening the 2$e$-periodic feature. At yet larger bias with fixed $n_{g}$, the resonance condition for Cooper pair tunneling from left is exceeded, resulting in NDC. At $V_{B}~{}=~{}\Delta_{L}+\delta-E_{C}$ [$\blacksquare$ in Fig. 3(a)] above-gap quasiparticles in the left S lead can excite the ABS thus allowing single-electron transport that no longer requires resonant Cooper pair processes, resulting in 1$e$-periodic resonances in $n_{g}$. Here, NDC presumably results when the coherence peak in the density of states the S lead surpasses the bound state energy. At $V_{B}~{}=~{}\Delta_{L}$ [$\blacklozenge$ in Fig. 3(a)] quasiparticles from the S lead can transfer to the N lead via elastic cotunneling through the ABS or other subgap states, which are only weakly dependent on $n_{g}$. Finally, at $V_{B}=\Delta_{L}+\Delta_{I}-E_{C}$ [$\bigstar$ in Fig. 3(a)] single-electron transport between continuum states of the left S lead and the S island become energetically allowed. Figure  3(b) shows line cuts from Fig. 3(a), indicating both 2$e$ (pink) and 1$e$ (green) features along with the associated NDC. Further details of the four features are discussed in Appendix .2.

We extract a charging energy $E_{C}$ $\sim$ 85 $\mu$eV from independent transport data of Coulomb diamonds (not shown). Using this value, the above model is consistent with an ABS energy $\delta=20~{}\mu$eV and superconducting gaps $\Delta_{L}\sim 220~{}\mu$eV and $\Delta_{I}\sim 160~{}\mu$eV, in the left S lead and S island, both consistent with values seen in similar hybrid nanowires Chang et al. (2015). The condition $E_{C}<\Delta_{I}$ is consistent with the 2$e$ features at low bias in Fig. 3(a).

Transport features observed in SISIN spectroscopy are rather complicated, even with the relatively simple situation of a single ABS on the island. Previous work has showed that RF charge readout offers some advantages compared to transport, particularly speed and simplicity of the data Razmadze et al. (2019); van Veen et al. (2019); de Jong et al. (2019). To perform charge readout on the present device, we use the integrated charge sensor and examine the same ABS investigated above via transport.

Figure 4(a) shows transport-based Coulomb blockade spectroscopy in the SISIN configuration performed over a larger range of P2 gate voltages. The effect of the ABS can be seen as an envelope modulation of Coulomb blockade that follows the shape of the ABS that was previously shown in the Fig. 2(a). To emphasize that the modulation is connected to the ABS, red markers in Fig. 4(a) indicate the tips of Coulomb diamonds. Charge sensing data from the same gate regime as Fig. 4(a) is shown in Fig. 6. Charge sensing shows explicitly the transition in periodicity of superconducting island occupancy whenever the bound state energy $\delta$ is lowered with respect to charging energy $E_{C}$ of the island. This change in occupancy was implied by the model, but not explicitly visible in transport data. Figure 4(b) shows the effect of the 2$e$ to 1$e$ transition as a function of plunger voltage $V_{P2}$ measured with the the integrated RF-SET. The small overshoot of the even level (blue) at $V_{P2}$ = $-$1.51 V is associated to ABS that is oscillating around zero energy as a function of $V_{P2}$. Gate P3 in the Fig. 6 is also responsible for tuning the ABS however with a smaller lever arm compared to P2 compatible with the ABS being localized closer to the barrier C1 side. We note that the suppression of the 2$e$-periodic signal around zero bias in Fig. 4(a) is correlated with the onset of 1$e$-periodicity in Fig. 4(b). This indicates that the exchange of Cooper pairs between the island and superconducting left lead, despite being aligned in chemical potential, is indeed suppressed when $\delta<E_{C}$. The latter requirement is supported by the observation of re-entrant resonant Cooper pair processes at $V_{B}~{}=~{}E_{C}-\delta$.

The three measurements methods of the same ABS are compared in Fig. 4(c). Black data points show the ABS profile extracted using SIS spectroscopy [Fig. 2(a)], red - Coulomb blockade measurements where the ABS energy $\delta$ is determined using the onset of Coulomb blockade features [Fig. 4(a)] and green - RF-SET measurements [Fig. 6]. For SIS and SISIN transport measurements both positive and negative values of $V_{B}$ were used. An offset in $V_{C2}$ of $\sim$ 30 mV between SIS and SISIN measurements, presumably due to cross-capacitance to $V_{C2}$ has been removed. Correspondence between the ABS energy profiles measured using transport (SIS and SISIN) and charge sensing shows that the charge periodicity of the island is indeed affected by the ABS energy. It also shows that RF measurements with increased acquisition rate Razmadze et al. (2019); de Jong et al. (2021) is a useful alternative to transport.