Granular aluminum induced superconductivity in germanium for hole spin-based hybrid devices

Abstract

In superconductor-semiconductor hybrid structures, superconductivity and spin polarization are competing effects because magnetic fields break Cooper pairs. They can be combined using thin films and in-plane magnetic fields, an approach that enabled the pursuit of Majorana zero modes, Kitaev chains, and Andreev spin qubits (ASQs), but remains challenging for materials with small in-plane g-factors. Here we show that granular aluminum (grAl), composed of nanometer-scale aluminum grains embedded in an amorphous oxide matrix, can overcome this limitation. By depositing grAl on Ge/SiGe heterostructures, we induce a hard superconducting gap with BCS peaks at 305 $μ$eV and magnetic-field resilience for both the in-plane and out-of-plane directions, allowing Zeeman splitting of Yu-Shiba-Rusinov (YSR) states beyond 50 $μ$eV (12 GHz). Leveraging this robustness, we reveal signatures of hole physics and demonstrate g-tensor tunability.

Figures (16)

• Figure 1: GrAl induced superconductivity in planar Ge.a. High-angle annular dark-field scanning transmission electron microscopy image of a Ge/SiGe heterostructure with a grAl layer on top (left), and the corresponding electron energy-loss spectroscopy (right). The granular structure of the grAl layer is clearly resolved. A thin oxide layer is observed at the interface between the grAl layer and the QW. In addition, a localized Al region is detected within the QW (see SI Fig. S1). b. False-colour scanning electron microscopy image of a copy of the device, with the superconducting grAl contact in red, the Pd contact in light-blue and gates in yellow. B and S are used to electrostatically form a tunable constriction to probe the proximitized region. Together with N and P they are used to form a QD, where N controls the coupling to the normal lead and P the QD electrochemical potential. Gate O tunes the Ge density of states close to the dot. For all the results presented in the main text, it was grounded. c. Differential conductance $G$ = $dI/dV$ in units of $G_0 = 2e^2/h$ as a function of the voltage applied to gate S ($V_{\mathrm{S}}$) and bias voltage ($V_{\mathrm{SD}}$) at $V_\mathrm{B}$ = 4.35 V, with gates $V_{\mathrm{P}}$ and $V_{\mathrm{N}}$ set at 0V so that no QD is formed. At high $V_{\mathrm{S}}$ values (tunneling regime) coherence peaks at $\pm$ 305 $\mu$V are observed. Around $V_{\mathrm{S}}$ = 2.4 V (single channel regime), we measure enhanced in-gap conductance approaching 2$\mathrm{G_0}$ (see SI Fig. S3). d. Normalized $G$ vs $V_{\mathrm{SD}}$ for proximitized Ge (black line) and for grAl (grey line). The black trace was obtained at $V_\mathrm{S}$ = 3.26 V, deep in the tunneling regime, where the peak conductance is $6\times10^{-3}G_0$. The grey trace shows the numerical derivative of the current measured in tunneling spectroscopy of a grAl film with the same resistivity as that used in this study (see methods). For proximitized Ge the BCS peak is at $\Delta^*$ = 305 $\mu$V, to be compared with $\Delta_{\mathrm{grAl}}$ = 360 $\mu$V. e-g.$G$ vs $V_{\mathrm{SD}}$ and magnetic field $B_{z,x,y}$ at $V_\mathrm{B}$ = 4.35V and $V_\mathrm{S}$ = 3.30V. A region of reduced subgap conductance is visible up to 300 mT in the out-of-plane ($z$) direction and above 950 mT in-plane.
• Figure 2: Spectroscopy measurements of a hybrid QD for different couplings to the supercondcutor.a.$G$ through the QD vs $V_\mathrm{P}$ and $V_{\mathrm{SD}}$ calculated by numerical differentiation of the measured current. Coulomb diamonds with a region of suppressed conductance for $|V_{\mathrm{SD}}|$ < 200 $\mu$V and a charging energy of $U$ = 3.2 mV are observed. b.$G$, measured with a lock-in amplifier, at increased coupling to the superconductor compared to the configuration shown in (a). Subgap states emerge displaying the characteristic singlet-doublet ground state sequence associated with YSR physics. At higher bias voltage, additional conductance features that replicate the YSR state appear. Their origin is discussed in the SI Fig. S4. c-e.$G$ vs $V_\mathrm{P}$ and $V_{\mathrm{SD}}$ with increasing $\Gamma_{\mathrm{S}}$ for the YSR enclosed by the white dashed line in (b). For a fixed $\Gamma_\mathrm{S}$ sweeping $V_\mathrm{P}$ corresponds to sweeping the effective dot occupancy $n_g$ in the singlet-doublet phase diagram (see inset), where $n_g=1$ corresponds to the center of the doublet region. At moderate coupling strengths the system exhibits well-defined singlet and doublet regions, while beyond a critical $\Gamma_\mathrm{S}$ (d) a singlet-like ground states persists across the full $n_g$ range (e). For the ZBW model we extract $U =$ 685 $\mu$eV and $\Gamma_\mathrm{S}$ = 156 $\mu$eV (c), 183 $\mu$eV (d), 204 $\mu$eV (e) (see SI section S6).
• Figure 3: Spin splitting of YSRs revealing signatures of hole physics.a-c.$G$ vs $V_\mathrm{P}$ and $V_{\mathrm{SD}}$ in the same coupling regimes as \ref{['fig:fig2']}, respectively, with an applied out-of-plane magnetic field of 80 mT. Pronounced splitting is observed in regions with a singlet ground state. In (a-b) the Zeeman splitting is asymmetric between the left and right sides of the doublet ground-state region. The white dashed lines indicate the plunger voltages for measurements in \ref{['fig:fig4']}. d-f Lowest-energy even-odd parity transitions calculated with the ZBW model. Solid lines include the magneto-tunneling effect due to heavy-hole-light-hole mixing. Dashed lines show the result without magneto-tunneling for comparison. We use $\Gamma_\mathrm{S}=156, \,183 \; {\rm and} \; 204 \; \mu$eV as extracted from \ref{['fig:fig2']}, respectively. g Lowest energy levels of the system as a function of $n_g$, where only a section of the energy spectrum is shown, corresponding to the right-half of (a). The cartoon is obtained using a ZBW model without the magneto-tunneling term and without the Zeeman term in the superconductor Hamiltonian. The gray lines correspond to the Zeeman-split lowest-energy doublet state, whereas the black line correspond to the lowest-energy singlet state. The colored arrows indicate the transport-allowed transitions at different values of $n_g$ corresponding to the colored lines in panel a. h. Cartoon illustrating the lowest-order tunneling processes between the superconductor and the QD in regions I and II. The different tunneling processes lead to a different spin-splitting due to the magneto-tunnel effect. See text for discussion.
• Figure 4: $g$-tensor measurements for different coupling strengths.a.$G$ vs $V_{\mathrm{SD}}$ and $B_z$ (out-of-plane direction) at $\Gamma_\mathrm{S}$ = 204 $\mu$eV and with $V_\mathrm{P}$ fixed at the positions marked in \ref{['fig:fig3']} by the white dashed line. The splitting initially evolves linearly according to the Zeeman effect. Above roughly 100 mT the splitting deviates from a linear trend, likely due to the interaction with the quasiparticles continuum, as in lee2014spin. b. Plot showing the Zeeman splitting as a function of $B_z$ for configurations $\square$ ($\Gamma_\mathrm{S}$ = 156 $\mu$eV) and $\bigcirc$ ($\Gamma_\mathrm{S}$ = 204 $\mu$eV). The data points are extracted from the separation of the conductance peaks at positive bias. Solid lines are the results of the linear fits to the data, yielding a $g$-factor of 7.44$\pm$0.20 for $\square$ and of 6.65$\pm$0.23 for configuration $\bigcirc$. c,d. Polar projections on the laboratory frame of the $g$-tensors for configurations \ref{['fig:fig3']} respectively. Dark data points denote linear fits of the Zeeman splitting vs magnetic field magnitude at a fixed direction. Light data points provide a consistency check (see SI section S10). e. Zeeman splitting $\Delta E$ as a function of in-plane magnetic field angle at $B_\parallel$ = 420 mT, where $\phi$ = 0 corresponds to $y$-direction, for the three configurations reported in \ref{['fig:fig3']}. The shaded regions correspond to the standard deviation on the measured splittings, which we consider to be equal to twice the bias voltage resolution (1 $\mu$V).
• Figure S1: STEM and EELS analysis of Al-rich defects in a Ge/SiGe heterostructure.a-c. Bright-field scanning transmission electron microscopy (BF-STEM) images of a Ge/SiGe heterostructure with a grAl layer on top of a Ge quantum well (QW), showing a defect-free region (a) and regions containing defects in the QW (b,c). The defects exhibit sharp interfaces with the QW and are bounded by facets inclined at roughly 60° with respect to the QW plane. Notably, this angle coincides with the typical dislocations propagation direction in SiGe systems. d. High-angle annular dark-field STEM and the corresponding electron energy-loss spectroscopy (EELS) analysis of a region with a defect in the QW, revealing that the defect is an aluminium-rich crystal embedded inside the germanium and silicon-germanium layers. We speculate that aluminium diffuses from the grAl layer into the QW along structural defects. Possibly, this diffusion happens during the atomic layer deposition of the oxide layer.