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Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions

David J. van Woerkom, Alex Proutski, Bernard van Heck, Daniël Bouman, Jukka I. Väyrynen, Leonid I. Glazman, Peter Krogstrup, Jesper Nygård, Leo P. Kouwenhoven, Attila Geresdi

arXiv:1609.00333v1cond-mat.mes-hallcond-mat.supr-con

TL;DR

Microwave spectroscopy can probe Andreev bound states in semiconductor Josephson junctions, where gate control tunes the nanowire channels. The paper measures their microwave response and reports spectra consistent with short-junction ABS behavior, alongside evidence of symmetry breaking and spin-related effects.

  • Problem

    Microwave spectroscopy is a charge-parity-conserving way to excite Andreev bound states, enabling coherent control and motivating its use to study ABS in proximitized semiconductor systems.

    Source: p. 1

  • Method

    The authors use an InAs nanowire with epitaxial Al shells in a flux-controlled hybrid SQUID and detect its ABS through microwave spectroscopy of the spectrometer junction.

    Source: p. 1

  • Results

    The measured flux-dependent spectra qualitatively agree with the short-junction ABS relation for one or several transport channels, and the ABS-transition visibility agrees with the predicted phase dependence using fitted T = 0.57.

    Source: pp. 2, 4

  • Takeaways & Limitations

    The measurements provide direct evidence for symmetry breaking of ABS in a multichannel ballistic system and demonstrate Zeeman splitting and spin-orbit coupling effects.

    Source: p. 5

    The calculations omit magnetic-field orbital effects on the nanowire, which could influence proximity-induced superconductivity.

    Source: p. 5

Abstract

from arXiv · show

The superconducting proximity effect in semiconductor nanowires has recently enabled the study of novel superconducting architectures, such as gate-tunable superconducting qubits and multiterminal Josephson junctions. As opposed to their metallic counterparts, the electron density in semiconductor nanosystems is tunable by external electrostatic gates providing a highly scalable and in-situ variation of the device properties. In addition, semiconductors with large g-factor and spin-orbit coupling have been shown to give rise to exotic phenomena in superconductivity, such as $\varphi_0$ Josephson junctions and the emergence of Majorana bound states. Here, we report microwave spectroscopy measurements that directly reveal the presence of Andreev bound states (ABS) in ballistic semiconductor channels. We show that the measured ABS spectra is a result of transport channels with gate-tunable, high transmission probabilities up to 0.9, which is required for gate-tunable Andreev qubits and beneficial for braiding schemes of Majorana states. For the first time, we detect excitations of a spin-split pair of ABS and observe symmetry-broken ABS, a direct consequence of the spin-orbit coupling in the semiconductor.

Supplementary online material Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor

The supplementary material describes fabrication of the nanowire devices and the cryogenic measurement setup. The devices combine electrostatic gates, a hybrid SQUID, and a spectrometer junction.

DEVICE FABRICATION

  • The devices were fabricated on Si/SiOx with patterned gates, capacitors, resistors, Al tunnel junctions, and an InAs nanowire contacted into a NbTiN hybrid-SQUID loop.The nanowire channel was defined by removing the Al shell for device 1 and by in-situ patterning for device 2.Source: DEVICE FABRICATION · p. 7

MEASUREMENT SETUP

  • The measurement setup used a 12 mK dilution refrigerator with room-temperature π-LC filtering and base-temperature copper-powder and RC filters.The setup schematic shows the hybrid SQUID and spectrometer junction with on-chip lumped circuit elements.Source: Figure S1 · p. 8, MEASUREMENT SETUP · p. 9

DEVICE CIRCUIT PARAMETERS

The circuit parameters are inferred from the spectrometer’s plasma resonance and junction properties. A single-mode LCR fit agrees near resonance but deviates at higher frequencies.

DEVICE CIRCUIT PARAMETERS

  • The parallel-LCR model places the circuit resonance at ω0 = (LC)^−1/2, while the fit deviates at higher frequencies.The authors attribute the high-frequency discrepancy to additional losses or resonant modes not included in Eq. (S1).Source: DEVICE CIRCUIT PARAMETERS · p. 9
  • The superconducting gap and junction linear resistance were used to infer Josephson energy, inductance, and the circuit parameters listed in Table S2.The supplied table passage identifies the table as containing parameters for the featured devices but provides no numerical entries.Source: DEVICE CIRCUIT PARAMETERS · p. 9, Table S2 · p. 10
  • The spectrometer’s measured I(V) traces for devices 1 and 2 were fitted to Eq. (S1) to characterize the plasma resonance.The fit is reported to agree quantitatively near resonance, with deviations at higher voltages attributed to unaccounted losses or resonant modes.Source: Figure S2 · p. 9, DEVICE CIRCUIT PARAMETERS · p. 9
  • The large-scale spectrometer trace shows back-bending near eVspec ≈ 2∆spec = 482 µeV, attributed to local junction overheating above the gap edge.The proposed mechanism is a large quasiparticle current density above the gap edge.Source: Figure S3 · p. 10

Spectrum analysis

The spectrum analysis uses the spectrometer current trace and its second derivative to identify transitions of the circuit environment. Positive second-derivative peaks correspond to actual transitions.

Spectrum analysis

  • Peaks with −d2I/dV2(V) > 0 correspond to peaks in I(V) and identify transitions detected by the spectrometer junction.The second derivative is used to remove the smooth plasma-mode background and locate peaks in Re[Z(ω)].Source: Spectrum analysis · p. 11, Figure S4 · p. 11
  • The analysis compares the I(V) trace with its −d2I/dV2(V) trace, marking corresponding peaks and treating only peaks above zero as transitions.The illustrated dataset was taken on device 1 at Vg = −1410 mV and phase ϕ = π.Source: Figure S4 · p. 11

I(V) trace of the hybrid SQUID

The hybrid-SQUID I(V) trace changes with gate voltage, with increasing subgap conductance qualitatively consistent with multiple Andreev reflection. The soft nanowire gap limits identification of MAR features in the analyzed bias range.

I(V) trace of the hybrid SQUID

  • Subgap conductance increases with gate voltage, qualitatively agreeing with a multiple-Andreev-reflection contribution to the hybrid-SQUID current.The zero-voltage data corresponds to the supercurrent branch.Source: I(V) trace of the hybrid SQUID · p. 12
  • The conductance G was evaluated over VSQUID = 350–430 µV, above 2∆, but MAR features were not identified after subtracting the tunnel-junction current background.The authors attribute this to the soft superconducting gap in the nanowire junction.Source: I(V) trace of the hybrid SQUID · p. 12
  • At Vg = −1.75 V the nanowire is fully depleted, so the measured I(V) trace represents the Al/AlOx/Al tunnel junction in the hybrid SQUID.The bias voltage was swept from left to right.Source: Figure S5 · p. 12

Fit of ABS with high transmission

The model accurately fits the avoided crossing between the ABS and plasma mode in a high-transmission dataset, with a fitted transmission of 0.90 ± 0.01. The extracted gap is lower than in Fig. 3a, possibly because of fit bias or gate dependence.

Fit of ABS with high transmission

  • T = 0.90 ± 0.01 and ∆ = 97.5 ± 1.7 µeV fit the high-transmission ABS spectrum at Vg = −1.525 V.The model accurately predicts the avoided crossing between the ABS and plasma mode.Source: Figure S6 · p. 13
  • The fitted ∆ is lower than in Fig. 3a, potentially because the fit underestimates the gap near ϕ = π or because ∆ depends on Vg.Source: Figure S6 · p. 13
  • At ϕ = π, the probability densities are shown for the ground state and two observed excited states.Source: Figure S6 · p. 13

Symmetry-broken ABS in bipolar magnetic field

The spectrum is symmetric around ϕ = π at zero magnetic field, while the two lowest ABS transitions become asymmetric at B = ±40 mT. The antisymmetric contribution shifts oppositely for positive and negative fields.

Symmetry-broken ABS in bipolar magnetic field

  • At B = ±40 mT, the two lowermost ABS transitions are asymmetric, with opposite shifts for positive and negative magnetic fields.The antisymmetric contribution is most visible at Vspec ≈ 100 µV.Source: Figure S7 · p. 14
  • At zero magnetic field, the full spectrum is symmetric around ϕ = π.Source: Figure S7 · p. 14
  • The data were taken on device 1 at Vg = −770 mV; grey regions indicate missing data due to circuit bias instability.Source: Figure S7 · p. 14

Estimate of the ABS-plasma resonance avoided crossing

The avoided crossing is estimated by modeling the plasma oscillations and weak link as coupled degrees of freedom. The resulting splitting estimate is lowest order in √z, which is relatively large for device 1 and may contribute to the discrepancy with the observed splitting.

Estimate of the ABS-plasma resonance avoided crossing

Hamiltonian description of the hybrid SQUID

The hybrid SQUID model combines the junction phase and charge with a weak-link Hamiltonian, then numerically diagonalizes the resulting system. Its eigenstates and transition frequencies are used to assess ABS character, visibility, and fit to measured data.

Hamiltonian description of the hybrid SQUID

  • The hybrid SQUID Hamiltonian combines the junction charging and Josephson terms with the weak-link Hamiltonian H_ABS(ϕ − δ).The phase and charge operators obey [δ, N] = i.Source: Hamiltonian description of the hybrid… · p. 15
  • The weak link is represented using Pauli operators on the ground state and an excited state with a pair of quasiparticles.Its Hamiltonian is expressed using two phase-dependent functions V2 and V3.Source: Hamiltonian description of the hybrid… · pp. 15–16
  • The model is numerically diagonalized in a truncated 2M-dimensional phase-and-weak-link basis, with M chosen to converge the eigenvalues.Source: Hamiltonian description of the hybrid… · p. 16
  • Ground-state transition frequencies are fitted to the experimental data, while probability densities and current matrix elements assess ABS character and transition visibility.Probability densities indicate overlap with the weak-link excited state; visibility is set by the squared ground-to-excited-state current matrix element.Source: Hamiltonian description of the hybrid… · p. 16

Equilibrium phase drop

The equilibrium phase drop across the tunnel junction is small, so the weak-link phase remains close to the applied phase. An approximate solution closely tracks the numerical result for the parameters shown in Fig. S8.

Equilibrium phase drop

  • The calculation tests the assumption φ ≈ ϕ by finding the equilibrium δ that minimizes the ground-state Josephson energy at EC = 0.Since φ = ϕ − δ, a small δ establishes the approximation.Source: Equilibrium phase drop · p. 16
  • δ reaches only about 0.12 near ϕ = π/2, leaving the weak-link phase φ = ϕ − δ close to the applied phase.At ϕ = nπ, φ equals ϕ exactly.Source: Equilibrium phase drop · p. 17
  • For the Fig. 3a circuit parameters, the approximate δ solution is very close to the numerical solution in Fig. S8.The caption identifies the black curve as Eq. (S17) and the red curve as the numerical solution of Eq. (S16).Source: Equilibrium phase drop · p. 17, Figure S8 · p. 17

Andreev bound states in a proximitized Rashba nanowire in a parallel magnetic field

A transfer-matrix model of a proximitized Rashba wire is used to calculate Andreev bound states in two Fermi-level regimes. The calculated spectra agree better with experiment in the high-Fermi-level regime, while the model’s linearization and short-junction assumptions delimit its scope.

Andreev bound states in a proximitized Rashba nanowire in a parallel magnetic field

  • The model starts from a Rashba-wire BdG Hamiltonian with proximitized s-wave superconductivity, Zeeman field, and a short-range scatterer representing finite channel transmission.The model defines ESO = mα^2/2 and EZ = 1/2 gµB B.Source: Andreev bound states in a proximitized… · p. 17
  • The analysis linearizes the BdG equations near EF for regimes EF ≫ ESO, EZ, Δ and EF = 0; the latter requires strong spin-orbit coupling, ESO ≫ Δ, EZ.The bound states considered have |E| < Δ.Source: Andreev bound states in a proximitized… · p. 18
  • A normal-state transfer matrix supplies the weak-link boundary condition for the linearized BdG equations, yielding an ABS bound-state equation solved numerically.Neglecting transfer-matrix energy dependence is appropriate in the short-junction limit; regime (b) additionally requires junction length below α/EZ.Source: Andreev bound states in a proximitized… · p. 18
  • The calculated ABS spectrum agrees better with experiment in regime (a); regime (b) misses a preceding parity switch and places the 300 mT final point in the continuum.The authors favor the high-Fermi-level regime but cannot exclude an intermediate 0 ≲ EF ≲ ESO fit; Fig. S9 compares the regimes.Source: Andreev bound states in a proximitized… · pp. 18–19, Figure S9 · p. 19
  • The current linearized model is not extended to arbitrary EF, and a more detailed finite-field analysis of the bound-state equation remains future work.The authors identify extending beyond linearization as a way to calculate spectra at arbitrary EF.Source: Andreev bound states in a proximitized… · p. 18
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