Spin-Orbit Interaction and Induced Superconductivity in a One-Dimensional Hole Gas.

Low dimensional semiconducting structures with strong spin-orbit interaction (SOI) and induced superconductivity attracted great interest in the search for topological superconductors. Both the strong SOI and hard superconducting gap are directly related to the topological protection of the predicted Majorana bound states. Here we explore the one-dimensional hole gas in germanium silicon (Ge-Si) core-shell nanowires (NWs) as a new material candidate for creating a topological superconductor. Fitting multiple Andreev reflection measurements shows that the NW has two transport channels only, underlining its one-dimensionality. Furthermore, we find anisotropy of the Landé g-factor that, combined with band structure calculations, provides us qualitative evidence for the direct Rashba SOI and a strong orbital effect of the magnetic field. Finally, a hard superconducting gap is found in the tunneling regime and the open regime, where we use the Kondo peak as a new tool to gauge the quality of the superconducting gap.

The large band offset and small dimensions of the Ge–Si core–shell nanowire (NW) lead to the formation of a high-quality one-dimensional hole gas.[1,2] Moreover, the direct coupling of the two lowest-energy hole bands mediated by the electric field is predicted to lead to a strong direct Rashba spin–orbit interaction (SOI).[3,4] The bands are coupled through the electric dipole moments that stem from their wave function consisting of a mixture of angular momentum (L) states. On top of that, the spin states of that wave function are mixed due to heavy and light hole mixing. Therefore, an electric field couples via the dipole moment to the spin states of the system and causes the SOI. This is different from the Rashba SOI, which originates from the coupling of valence and conduction bands. The predicted strong SOI is interesting for controlling the spin in a quantum dot electrically.[5,6] Combining this strong SOI with superconductivity is a promising route toward a topological superconductor.[7,8] Signatures of Majorana bound states (MBSs) have been found in multiple NW experiments.[9,10] An important intermediate result is the measurement of a hard superconducting gap,[11,12] which ensures the semiconductor is well proximitized as is needed for obtaining MBSs.

Here we study a superconducting quantum dot in a Ge–Si NW. The scanning and transmission electron microscopy images of the device (Figure a,b) show a Josephson junction of ∼170 nm in length. The quantum dot is formed in between the contacts. The Ge–Si core–shell nanowires were grown by the vapor–liquid–solid (VLS) method as discussed in detail in the Supporting Information of ref (2). The NW has a Ge core with a radius of 3 nm. The Ge crystal direction is found to be [110], in which hole mobilities up to 4600 cm2/ Vs are reported.[2] The elemental analysis in Figure c reveals a pure Ge core with a 1 nm Si shell and a 3 nm amorphous silicon oxide shell around the wire. Superconductivity is induced in the Ge core by aluminum (Al) leads,[13] and crucially, the device is annealed for a short time at a moderate temperature.[14,15] We believe that the high temperature causes the Al to diffuse in the wire, therefore enhancing the coupling to the hole gas. Note that we do not diffuse the Al all the way through, since we pinch off the wire (Figure S1) and there is no Al found in the elemental analysis (Figure c). Two terminal voltage bias measurements are performed on this device in a dilution refrigerator with an electron temperature of ∼50 mK.

Figure 1

(a) False colored scanning electron microscope image of the device with the NW (yellow) with aluminum contacts (gray) on a Si/SiN wafer (blue). The magnetic field axes, voltage bias measurement setup, and global bottom gate are indicated. (b) Transmission electron microscope (TEM) image of the cross section of the NW. (c) Energy dispersive X-ray spectroscopy of the area displayed in panel b. The colors represent different elements: Ge is green, Si is blue, and oxygen (O) is red, respectively. The Ge–Si core–shell wire is capped by a SiO shell. (d) Voltage bias tunneling spectroscopy measurement of the superconducting quantum dot as the bottom gate voltage Vbg is altered. The superconducting gap, an Andreev level (AL), and multiple Andreev reflections appear as peaks in differential conductance (dI/dV). The AL, Δ, and 2Δ are marked by dashed green, yellow, and white lines, respectively. The even or odd occupation is indicated, and the kink in the observed Andreev level is highlighted by the arrows. (e, f) Same measurement as panel d with a magnetic field, B, applied perpendicular to the substrate (x-direction) of 60 mT and 1 T, respectively. A zero bias Kondo peak is observed as the quantum dot is occupied by an odd number of electrons. At B = 1 T, the resonance is split due to the Zeeman effect. (g) Linear splitting of the Kondo peak at Vbg = −0.098 V as a function of B. The Zeeman effect splits the spinful Kondo peak, which is indicated by the dashed green line.

To perform tunneling spectroscopy measurements, the bottom gate voltage Vbg is used to vary the barriers of the quantum dot and alter the density of the holes as well. From a large source-drain voltage, V, measurement (Figure S1), we estimate a charging energy, U, of 12 meV, barriers’ asymmetry of Γ1/Γ2 = 0.2–0.5, where Γ1(2) is the coupling to the left (right) lead, and a lever arm of 0.3 eV/V. In Figure d, the differential conductance dI/dV as a function of V versus Vbg reveals a superconducting gap (2Δ = 380 μeV) and several Andreev processes within this window. Additionally, an even–odd structure shows up in both the superconducting state at low V and normal state at high V, which is related to the even or odd parity of the holes in the quantum dot. The even–odd structure persists as we suppress the superconductivity in the device by applying a small magnetic field (60 mT) perpendicular to the substrate (Figure e). A zero bias peak appears when the quantum dot has odd parity. This is a signature of the Kondo effect.[16,17] When increasing the magnetic field to 1 T, the Kondo peak splits due to the Zeeman effect by 2gμB. The energy splitting of the two levels is linear as shown in Figure g and thus can be used to extract a Landé g-factor, g, of 1.9. In the remainder of the Letter, we will discuss the three magnetic field regimes of Figure d–f (0 T, 60 mT, and 1T, respectively) in more detail.

The resonance that disperses with Vbg in Figure d is an Andreev Level (AL), which is the energy transition from the ground to the excited state in the dot.[18,19] The ground state of the dot switches between singlet and doublet if the occupation in the dot changes, as sketched in the phase diagram in the top panel of Figure a. Since our charging energy is large, we trace the dashed line in the phase diagram. The AL undergoes Andreev reflection at the side of the quantum dot with a large coupling (Γ2) and normal reflection at the opposite side that has lower coupling (Γ1), as schematically drawn in the bottom panel of Figure a. The superconducting lead with the low coupling serves as a tunneling spectroscopy probe of the density of states. To be more precise, the coherence peak of the superconducting gap probes the Andreev level energy, EAL. For example, if EAL = 0, we measure it at eV = Δ; the resonance thus has an offset of ±Δ in the measurement in Figure d. The ground state transition is visible as a kink of the resonance at V = Δ at Vbg = −0.09 and −0.11 mV. At a more negative Vbg, the coupling of the hole gas to the superconducting reservoirs is strongly enhanced. This eventually leads to the observation of both the DC and AC Josephson effects (Figure S2).

Figure 2

(a, top) A phase diagram of the ground state in the superconducting quantum dot sketched as a function of the quantum dot energy ϵ0 versus the coupling to the superconducting reservoir Γs, both normalized to the charging energy, U. Because of the large U compared to Γs, we expect to trace the dashed line. The bottom panel shows the Andreev level (dashed gray line) with energy EAL that is formed by the Andreev reflection (AR) at one side and normal reflection (NR) at the other side of the dot. The reflection processes are different due to asymmetric barriers Γ1 and Γ2, indicated as the barrier width. The density of states in the NW is probed by the superconductor on the left side by doing voltage bias tunneling spectroscopy. (b) Tunneling spectroscopy measurement at Vbg = −0.85 V. The first- and second-order multiple Andreev reflections are observed. A two-mode model fits the data well with Δ = 190 μeV. (c) Measured current of panel b. The data is fitted with a single- and two-mode model. The latter resembles the data better and is therefore used to extract transmission values. (d) Transmission of the first and second mode, T1 and T2, extracted from the fit of multiple Andreev reflections at a different Vbg. The transmission increases significantly below Vbg = −0.8 V.

In the upper part of Figure d, we measure the multiple Andreev reflection (MAR): resonances at integer fractions of the superconducting gap. Figure b presents a line trace at Vbg = −0.85 V that shows the gap edge and first- and second-order Andreev reflection. Fitting the differential conductance[20,21] (see Supporting Information) allows us to extract Δ = 190 μeV, close to the bulk gap of Al. We also fit the measured current to extract the transmission of the spin degenerate longitudinal modes in the NW (Figure c).[22,23] The two-mode fit resembles the data better than the single-mode fit. Also, we checked that fitting with more than two-modes results in T = 0 outcomes for the extra modes. Therefore, the first provides us with an estimate for the transmission in the two modes, T1 and T2. We interpret the two modes as two semiconducting bands in the NW. The MAR fitting analysis is repeated at a different Vbg, and the resulting T1 and T2 are plotted in Figure d. The strong increase of the transmission below Vbg = −0.8 V is attributed to the increase of the Fermi level and Γ1 and Γ2.

The Landé g-factor g is investigated further by measuring the Kondo peak splitting as a 0.9 T magnetic field is rotated from y- to z-, x- to z-, and x- to y-direction as presented in the second row of Figure a–c. Interestingly, we find a strong anisotropy of the Kondo peak splitting and accordingly of g at Vbg = −0.79 V and Vbg = −0.82 V; see the bottom row of Figure a–c and Figure S4, respectively. Both directions perpendicular to the NW show a strongly enhanced g. Similar anisotropy has been reported before in a closed quantum dot, where g is even quenched in the z-direction.[24−26] In our experiment, the highest g of 3.5 is found when the magnetic field is pointed perpendicular to the NW and almost perpendicular to the substrate.

Figure 3

(a–c) Rotations of the magnetic field with a 0.9 T magnitude in the yz-, xz-, and xy-plane, respectively, at Vbg = −0.79 V. The upper panel shows the schematic of the device and the magnetic field rotation performed. The differential conductance data is plotted in the center panel, and the splitting of the Kondo peak changes as the angles are swept. The sudden changes in conductance are due to small switches in Vbg. The lower panel shows the extracted g of the center panel in cyan and g at Vbg = −0.5 V in magenta. For the xy-plane. the anisotropy is highlighted and calculated. (d) Summary of the measured anisotropies of g at a different Vbg. (e) Simulation result of the quantum dot. The anisotropy of g∥ and g⊥ changes as the Fermi energy is altered. The colors represent the band from where the quantum dot level predominantly stems. The highlighted part shows a similar behavior in the anisotropy values as the data in part d. The inset depicts a schematic representation of the energy ordering of the quantum dot levels originating from two bands along the NW. (f) Simulation as in part e, now with an applied electric field of 10 V/μm. The SOI causes anisotropy with respect to the electric field direction as g is pointed perpendicular and g parallel to the electric field. The anisotropy increases as the Fermi level is raised. The same range as in part e is highlighted. (g) Simulated spin–orbit energies for the first band (k = 0) of the infinite wire model as a function of the electric field along the x-direction. The direct Rashba term is the leading contribution.

On the contrary, at a Vbg = −0.5 V, we find an isotropic g (bottom row of Figure a–c), all of which have a value of around 2. The anisotropies at a different Vbg are summarized in Figure d. The strong anisotropy seems to set in around Vbg = −0.7 V. This sudden transition from isotropic to anisotropic g, which has not been observed before in a quantum dot system, is correlated with the increase in transmission in Figure d. We speculate that the change from isotropic to anisotropic behavior is related to the occupation of two bands in the NW. To test this hypothesis and get an understanding of the origin of the anisotropy, we theoretically model the band structure of our NW and focus on the two lowest bands.

We use the model described in ref (4) and apply it to our experimental geometry (see Supporting Information for details). Simulating the device as an infinite wire, we first consider the anistropy of g between the directions parallel and perpendicular to the NW. We find that there are two contributions to the anisotropy: the Zeeman and the orbital effect of the magnetic field.[27,28] The anisotropy of the Zeeman component is similar for the two lowest bands, where for the orbital part the anisotropy differs strongly. The anisotropy of the total g, therefore, shows a strong difference for the two lowest bands (Figures S6 and S7). This agrees qualitatively with earlier predictions,[3] but we find additionally that strain lifts the quenching of g along the NW such that g∥/g⊥ ∼ 2, in agreement with our measurements. From these observations, we conclude that the observed isotropic and anisotropic g with respect to the NW-axis is due to the orbital effect.

In addition, we include the confinement along the NW, such that a quantum dot is formed and the energy levels are quantized in the z-direction. Besides the lowest-energy states studied before,[6,24] we also consider a large range of higher quantum dot levels. In the regime where two bands are occupied, we observe that the quantum dot levels originating from the first and second band have a unique ordering as a function of Fermi energy, this situation is sketched in the inset of Figure e. We also find that some of the quantum dot levels are a mixture of the two bands (Figure S9), resulting in a different anisotropy for each quantum dot level. In the simulation results (Figure e and Figure S10), the anisotropy values are colored according to the band they predominantly originate from. To compare the simulation with the measured data, we note that a more negative Vbg in the experiment increases the Fermi level for holes E. In the simulation, we observe a regime in E (highlighted in Figure e), where the anisotropy g⊥/ g∥ is around 1 and goes up toward 2 as E increases. This behavior qualitatively resembles the measurement of g/g and g/g in Figure d.

Now we turn to the magnetic field rotation in the xy-plane, the two directions perpendicular to the NW that are parallel and perpendicular to the electric field induced by the bottom gate. The measured anisotropy is gmin/gmax = 0.8 (Figure c). The maximum g of 3.5 is just offset of the y-direction, which is almost parallel to the electric field. This anisotropy with respect to the electric field direction is a signature of the SOI.[24,25] As discussed before, the Ge–Si NWs are predicted to have both the Rashba SOI and the direct Rashba SOI.[3,6] The electric field could also cause anisotropy via the orbital effect or geometry, due to an anisotropic wave function. However, we can rule that out since our simulations show that the wave function does not significantly change as electric field is applied (Figure S8). In the simulation (Figure f) with a constant electric field of 10 mV/μm, we observe anisotropy of g parallel (g) and perpendicular (g) to the electric field. Similar to our data, the anisotropy starts below 1 and goes to 1 as the Fermi level is increased. The spread in the anisotropy values is due to the mixing of the bands for each quantum dot level. Furthermore, we calculated the magnitude of the Rashba and direct contribution to the SOI using the infinite wire model and found that the direct Rashba SOI is dominating in the small diameter nanowires of our study (Figure g). This agrees with the effective Hamiltonian derived in ref (3), which predicts that the direct Rashba SOI dominates in NWs with a Ge core of 3 nm radius. To summarize, we observe anisotropy with respect to the electric field direction that is caused by the SOI, which is likely for the largest part due to the direct Rashba SOI.

Finally, in Figure , we take a detailed look at the superconducting gap as a function of magnetic field. We find the critical magnetic field Bc for different directions: Bc, = 220 mT (Figure a), Bc, = 220 mT (Figure b), and Bc, = 45 mT (Figure g and Figure S3), consistent with an Al thin film. Future devices could be improved by using a thinner Al film to increase the critical magnetic field.[29] In this case, the topological phase could be reachable, with the measured g of 3.5[8]. In the tunneling regime at Vbg = −0.12 V, we observe a clean gap closing (Figure a). The conductance inside the gap is suppressed by 2 orders of magnitude, signaling a low quasiparticle density of states in the superconducting gap. This large conductance suppression remains as the gap size decreases toward Bc (bottom panel in Figure a). In the low conductance regime, we thus measure a hard superconducting gap persisting up to Bc in Ge–Si NWs.

Figure 4

(a) Closing of the superconducting gap, as B is ramped up in the z-direction. The line traces below are taken at 50 mT intervals and show the induced superconducting gap. The vertical line trace shows the conductance at V = 0 V normalized to the conductance extracted at V = 0.5 mV. A 2 orders of magnitude conductance suppression is observed. (b) The superconducting gap closes, and a Kondo peak appears as the magnetic field is increased in the y-direction. The resonances within the gap stem from Andreev processes. The line traces depict the transition from the superconducting gap to the Kondo peak, which takes place from 170 to 190 mT (5 mT step). From the pink trace, a Kondo energy kBTK of 50 μeV is extracted with a Lorentzian fit.

The closing of the superconducting gap in a higher conductance regime is presented in Figure b. Since the transmission is increased, Andreev reflection processes cause a significant conductance within the superconducting gap.[30] Therefore, the conductance suppression in the gap becomes an ill-defined measure of the quasiparticle density of states and with that the quality of the induced superconductivity. However, here we can use the Kondo peak to examine the quasiparticle density of states in the superconducting gap. The Kondo peak is formed by coupling through quasiparticle states within the window of the Kondo energy (kBTK). In the regime where kBTK  ≤  Δ, the existence and size of the Kondo peak are then an indication of the quasiparticle density of states inside the superconducting gap.[31,32] In our measurement, Δ is indeed largerthan kBTK up to a magnetic field B = 170 mT (see the blue and magenta line traces in the bottom panel of Figure b). Since in the measurement the Kondo peak only arises once the gap is fully closed, we have a low quasiparticle density of states within the superconducting gap. This supports our observation of a hard superconducting gap up to Bc. It also illustrates a new way of gauging whether the superconducting gap is hard in a high conductance regime.

Combining all three magnetic field regimes of Figures –4, we observed Andreev levels showing a ground state transition, SOI from the coexistence of two bands in Ge–Si core–shell NWs, and a hard superconducting gap. The combination and correlation of these observations is a crucial step for exploring this material system as a candidate for creating a one-dimensional topological superconductor.