Open issues in the quantitative quasiparticle-dynamics theory of the anomalous ac Josephson effect in ballistic superconducting weak links

Superconducting weak links based on high-mobility InAs quantum wells exhibit an anomalous ac Josephson effect: microwave irradiation produces frequency-doubled Josephson currents and half-integer Shapiro steps, with visibly broadened peaks in the current spectrum. Kroemer (1999) captured the mechanism with an elementary one-dimensional ballistic quasiparticle-dynamics model — phase-dependent Andreev bound states swept by the bias, with stochastic interband Landau–Zener transitions generating a current at twice the Josephson frequency — and posed the open issues separating that model from quantitative comparison with experiment: two-dimensional multi-channel geometry; a phase-preserving statistical treatment of the competition between interband transitions, no transitions, and downward relaxation that predicts the full current frequency spectrum and its observed broadening; incorporation of inelastic scattering and of diffusive Andreev reflection from interface-preparation defects; explanation of the intersubband-scattering puzzle (higher sheet densities improve weak-link performance although intersubband scattering should be phase-breaking); and an experimental test of the predicted weaker length dependence of half-integer compared with integer Shapiro steps. The lineage of the problem is as follows: rigorous microscopic theory exists for adjacent regimes (single channels, point contacts, diffusive junctions), a quasi-two-dimensional extension exists (Jacobs and Kümmel 2005), and the 2019–2024 experimental literature is actively debating whether the strong second harmonic of the current-phase relation in ballistic semiconductor junctions has an equilibrium or non-equilibrium origin, with explicit recent statements that the required theory is still missing and no published systematic length-dependence test. The problem is open.

Background

A Josephson weak link is a superconductor–normal–superconductor (SNS) junction in which two superconducting electrodes (Nb in the experiments discussed here) are separated by a normal region so short that electrons cross it without scattering — the ballistic limit. In the devices that motivated this problem, the normal region is a high-mobility two-dimensional electron gas (2DEG), an InAs quantum well (a thin semiconductor layer that confines electrons in the growth direction while leaving them free to move in the plane) with very long elastic mean free paths.

When an electron in the normal region reaches a superconducting electrode, it can be Andreev reflected: it is converted into a hole retracing its path, while a Cooper pair (the superconducting charge carrier, a bound pair of two electrons) enters the condensate. Repeated Andreev reflection between the two superconducting interfaces of a weak link quantizes the excitation spectrum into Andreev bound states (ABS), whose energies depend on the phase difference $\varphi$ between the two superconducting order parameters. Under a constant voltage $V$, this phase advances at the Josephson frequency $\omega_J = 2eV/\hbar$ (where $2e$ is the Cooper-pair charge and $\hbar$ is the reduced Planck constant) — the ac Josephson effect. In a conventional tunnel junction the pair current then oscillates at $\omega_J$, and under microwave irradiation of frequency $\omega$ the time-averaged voltage locks to plateaus (Shapiro steps) at $V_n = n\hbar\omega/2e$ with integer $n$. Steps at half-integer multiples of $\hbar\omega/2e$ are anomalous: within the standard resistively shunted junction model (the conventional equivalent-circuit description of a Josephson junction) they cannot be produced by a purely sinusoidal current-phase relation (CPR) $I = I_c\sin\varphi$, and they signal either a strong second harmonic $\sin 2\varphi$ in the equilibrium CPR or a genuinely non-equilibrium dynamical response at twice the Josephson frequency. The elementary excitations involved are quasiparticles (QPs), the electron-like and hole-like excitations of the junction.

Lehnert et al. (1999) observed exactly this anomaly in mesoscopic Nb–InAs–Nb weak links: microwave irradiation revealed frequency-doubled Josephson currents producing half-integer Shapiro steps, persisting to high temperatures even in the absence of a dc critical current, with visibly broadened spectral peaks. Kroemer (1999) then developed an elementary, deliberately minimal ('bare-bones') one-dimensional ballistic model: at each instant the QP state is expanded in the instantaneous, phase-dependent ABS set belonging to the phase difference present at that instant; as the bias sweeps the QP band structure through avoided crossings, QPs either follow the adiabatic band or undergo interband transitions, describable as Landau–Zener (LZ) transitions — non-adiabatic tunneling across the avoided-crossing gap — and may subsequently relax downward. The stochastic sequence of these events generates a current component at $2\omega_J$: the frequency doubling. This physics is the ballistic limit of multiple Andreev reflection (MAR), the chain of repeated electron–hole conversions responsible for the subharmonic gap structure in the current–voltage characteristics of voltage-biased superconducting contacts, first modeled with rate equations by Octavio et al. (1983) — equations that track occupation probabilities but destroy phase coherence.

The microscopic theory of MAR and ac Josephson dynamics is well developed in other regimes: the exactly solved single quantum channel (Averin and Bardas 1995), subharmonic Shapiro steps of superconducting point contacts (Cuevas et al. 2002), the microwave-assisted supercurrent of diffusive SNS junctions (Virtanen et al. 2010), ABS dynamics of superconducting quantum point contacts — short, few-mode contacts — under microwave irradiation, treated with a two-level LZ model and the non-equilibrium Keldysh–Green function technique (the standard framework of non-equilibrium Green functions in quantum transport; Bergeret et al. 2011), and coherent MAR transport in wide ballistic junctions in the dc limit (Samuelsson et al. 2004).

Two features specific to 2DEG weak links matter. First, real devices carry many transverse modes ('channels') whose Fermi velocities meet the superconducting interfaces at a distribution of angles; at high sheet electron density, several size-quantized transverse subbands of the quantum well are occupied. Second, intersubband scattering (ISBS) — scattering between these transverse subbands — is expected to randomize the phase of the pair amplitude and hence to degrade supercurrent transport. The open issues posed by Kroemer (1999) ask precisely for the generalizations of his one-dimensional model needed for quantitative comparison with the experiments.

Problem Statement

Kroemer (1999) ends his elementary treatment of quasiparticle dynamics in ballistic superconducting weak links under weak voltage bias by asking what is needed for quantitative comparison with experiment. One scientific objective unifies all items below: a predictive, quantitative theory of the anomalous (frequency-doubled) ac Josephson effect in ballistic SNS weak links — their current-voltage characteristics, Shapiro response, and current frequency spectrum — checked against experiment. Within that objective, the source poses the following linked open issues, each directed toward quantitative comparison with the measured data:

(i) Multi-channel geometry. Generalize the one-dimensional model to the two-dimensional, many-channel structure of real 2DEG weak links, including the angular distribution of Fermi velocities at the superconducting interfaces.

(ii) Phase-preserving transition statistics. Replace the ad hoc treatment of the probabilistic competition between interband (Landau-Zener) transitions, no transitions, and downward transitions by a phase-preserving dynamical analysis — going beyond the phase-incoherent rate equations of Octavio et al. (1983) — and compute from it the current frequency spectrum, in particular the experimentally observed broadening of the peaks at the Josephson frequency $\omega_J$ and at its double $2\omega_J$.

(iii) Inelastic scattering. Incorporate inelastic scattering, estimated to occur on a scale of order 15 micrometers in InAs quantum wells at low temperature, which mimics quasiparticle ejection from the Andreev dynamics and grows in importance in longer devices.

(iv) Diffusive Andreev reflection. Incorporate the possibility that the Andreev reflection process itself is diffusive rather than specular, due to defects produced by the interface-cleaning procedures used in device fabrication, especially for comparisons between devices from different groups.

(v) The intersubband-scattering puzzle. Intersubband scattering in devices with multiple occupied subbands should be phase-breaking and detrimental, yet experimentally higher sheet concentrations improve weak-link performance. Explain this puzzle; the source notes, as a speculative possibility inviting closer investigation, that superconductivity-induced phase correlations might renormalize (hybridize) the subband structure.

(vi) Length dependence of half-integer Shapiro steps. The model predicts that half-integer steps are less sensitive to weak-link length than integer steps; this as-yet-untested prediction calls for sharpening into a quantitative dependence of the integer and half-integer step amplitudes (and of the associated spectral peak widths) on the weak-link length, together with the specification of the discriminating measurement — a controlled series of devices with varying electrode separation, including very long weak links — whose outcome would confirm or falsify the prediction.

Kroemer frames these items as aspects of the model whose elaboration is needed for a more quantitative comparison with actual observation and that had not yet been undertaken; all of them serve the single objective of bringing the quasiparticle-dynamics picture of ballistic weak links into quantitative agreement with experiment.

The phase-preserving, multi-channel quasi-two-dimensional dynamical treatment and the current frequency spectrum it predicts, including the broadening of the peaks at $\omega_J$ and $2\omega_J$ (items (i)-(ii)), form the central objective of this program, with items (iii)-(vi) as linked components of that same objective — further generalizations of the model that quantitative comparison with experiment requires.

References

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