Multiple Andreev Reflection in Superconducting Junctions

• Multiple Andreev Reflection (MAR) is the process by which quasiparticles undergo sequential electron-hole conversions in superconducting junctions, enabling charge transport below the superconducting gap.
• MAR generates distinct subharmonic gap structures in I–V characteristics with resonance steps at voltages Vₙ = 2Δ/(n·e) corresponding to the number of reflections.
• MAR analysis aids in extracting device parameters and diagnosing superconducting junction properties, including transparency, coherence, and potential topological signatures.

Multiple Andreev Reflection (MAR) refers to the process by which charge transport occurs in voltage-biased superconducting weak links via sequential Andreev reflections, enabling subgap current flow. In contrast to single Andreev reflection at a normal-superconductor (NS) interface, MAR describes how quasiparticles undergo multiple conversions between electron and hole excitations at the superconducting contacts before attaining sufficient energy to escape above the gap. MAR is the microscopic mechanism underpinning the subharmonic gap structure (SGS) in the current-voltage ($I$–$V$) characteristics of Josephson and SNS (superconductor–normal–superconductor) junctions, especially when the applied bias $eV$ satisfies $eV < 2\Delta$. Its manifestations and theory are central to the spectroscopy of superconducting weak links, the identification of topological excitations, and the understanding of transport in diverse hybrid circuits.

1. Theoretical Framework of Multiple Andreev Reflection

MAR is an inherently nonequilibrium, energy-selective process that emerges when a finite voltage bias, $V$, is applied across a superconducting junction whose interfaces mediate Andreev reflection. For $|eV| < 2\Delta$, direct quasiparticle tunneling is blocked as all states within the superconducting energy gap are unavailable. Instead, charge transport proceeds via a ladder of processes, where quasiparticles gain quanta of $eV$ upon each traversal of the normal region, enabling escape after multiple traversals—each corresponding to an Andreev reflection event (Liao et al., 14 Jan 2026, Zhu et al., 2021, San-Jose et al., 2013).

The MAR order, $n$, denotes the number of such reflections required for a quasiparticle to traverse from subgap energies to the continuum. The dominant resonance condition for the opening of the $n$th MAR channel is

$V_n = \frac{2\Delta}{n e}$

where $\Delta$ is the superconducting gap and $e$ is the electron charge. Subgap current thereby exhibits a staircase in the $I$–$V$ (or features in $dI/dV$), with steps at these subharmonic voltages (“subharmonic gap structure” or SGS) (Zhu et al., 2021, Jauregui et al., 2017, Carrad et al., 2022).

Microscopically, the MAR process—especially in short, highly transparent junctions—is described via recurrence relations for quantum amplitudes or by directly solving the time-dependent Bogoliubov–de Gennes (BdG) equations in Keldysh–Nambu space, incorporating the voltage-induced time evolution of superconducting phases, and constructing a full ladder of multiple sidebands (Liao et al., 14 Jan 2026, San-Jose et al., 2013, Zazunov et al., 2023).

2. Key Signatures and Experimental Observations

The SGS induced by MAR has been observed in a wide array of devices, including:

• STM-based Josephson junctions with asymmetric gaps, showing MAR step energies at $V_n = (\Delta_L+\Delta_R)/(n e)$ (Liao et al., 14 Jan 2026).
• Epitaxial and semiballistic SNS junctions, e.g., Al–MoTe$_2$–Al and Nb–topological insulator nanoribbons, where MAR features are visible up to high harmonics, $n > 10$, demonstrating high interface transparency and coherent transport (Zhu et al., 2021, Jauregui et al., 2017).
• Nanowire Josephson junctions, where photon-assisted MAR (PAT-MAR) under microwave irradiation directly reveals quantized sidebands at energies matching the effective charge $q_n = n e$ for the $n$th order process (Carrad et al., 2022).

MAR signatures extend to multiterminal and array geometries, producing novel SGS features at voltages beyond the canonical $V_n$, including linear combinations of biases and fractional steps in SNS arrays (Houzet et al., 2010, Chtchelkatchev, 2010).

A summary of generic MAR resonance conditions under diverse conditions:

System Type	MAR Resonance Voltage Condition	Reference
Two-terminal SNS	$V_n = 2\Delta/(n e)$	(Zhu et al., 2021, Jauregui et al., 2017)
Asymmetric Gaps	$V_n = (\Delta_L+\Delta_R)/(n e)$	(Liao et al., 14 Jan 2026)
Multiterminal	$e\,V_{\mathrm{eff}} = 2\Delta/n$ (linear combinations)	(Houzet et al., 2010, Mélin et al., 2015)
Dirty SNS arrays	$eV_n = 2\Delta/k$ and fractional values	(Chtchelkatchev, 2010)
Topological systems	Additional splitting/halving at $B>B_c$	(San-Jose et al., 2013, Golub, 2014)

3. Effects of Junction Properties and Material Context

3.1 Transparency and Junction Symmetry

• For low transparency ($D \ll 1$), MAR steps in $I$–$V$ scale as $D^n$ and appear as sharp onsets at $|eV|=(\Delta_L+\Delta_R)$. As $D$ increases, MAR transitions to a continuous subgap current, and more subharmonic steps become visible (Liao et al., 14 Jan 2026).
• Asymmetric gaps shift and split MAR features, leading to SGS steps at both $eV = \Delta_L/n$, $eV = \Delta_R/n$, and $eV = (\Delta_L+\Delta_R)/n$ (Liao et al., 14 Jan 2026).
• Multichannel junctions can be described by summing MAR currents in each channel, with effective parameters for transparency and channel number derived from fits to experimental data (Liao et al., 14 Jan 2026).

3.2 Coherence vs. Incoherence; Ballistic and Diffusive Regimes

• In phase-coherent, ballistic junctions (short compared to coherence length), subgap current and SGS are determined by the formation of Andreev bound states and phase-sensitive resonance ladders (Zhu et al., 2021, Olthof et al., 2023).
• In long, diffusive (incoherent) SNS or SINIS junctions, MAR persists but is described by coupled kinetic equations (typically Usadel-type), producing SGS at predicted subharmonics, but without phase coherence between reflections (Polkin et al., 2022, Chtchelkatchev, 2010).

3.3 Magnetic, Topological, and Spin-Active Junctions

• Magnetic precession (nanomagnet coupled point contacts) leads to spin-polarized MAR, with even-odd effects: only odd-$n$ MAR steps survive due to destructive interference between spin-flip pathways. Shapiro-like spin steps may appear at voltages proportional to precession frequency, $eV = (\omega_L)/(2en)$ (Holmqvist et al., 2013).
• Topological junctions (e.g., proximitized nanowires in the topological phase) exhibit unique features: gap threshold halving ($eV = \Delta_-/e$), persistent MAR oscillations, and Majorana-bound-state-induced current even for vanishing bias (San-Jose et al., 2013, Golub, 2014).
• Junctions with spin-active interlayers or broken time-reversal symmetry experience modified MAR resonance conditions and strong nonreciprocity in angle-resolved $I(V;\theta)$, traceable to the chirality and degeneracy of subgap states (Olthof et al., 2023).

4. Advanced Topics: Multiterminal MAR, Photon-Assisted MAR, and Parameter Extraction

4.1 Multiterminal and Array Structures

• In multi-superconductor junctions (e.g., three-terminal), MAR processes occur between all terminal pairs. Subharmonic features arise at voltages satisfying $e[pV_{21} + qV_{31}] = 2\Delta$, with $(p, q)$ denoting numbers of reflections at each interface (Houzet et al., 2010, Mélin et al., 2015). These are observed as a dense fan of SGS lines in the voltage space.
• In arrays of diffusive SNS junctions, MAR-induced SGS may appear at fractional voltages, depending on details of the array geometry and interface parameters (Chtchelkatchev, 2010).

4.2 Photon-Assisted MAR and Spectroscopic Applications

• Under microwave irradiation, MAR features acquire photon sidebands, with energy spacings scaling as $\Delta E_n = \hbar \omega / n$ for the $n$th MAR order. This enables direct identification of the effective charge transferred in each process ($q_n = n e$), and is a robust spectroscopic tool for probing hybrid device spectra and distinguishing MAR from other subgap excitations (Carrad et al., 2022).

4.3 Quantitative Fitting and Parameter Extraction

• Fits to asymmetric MAR models allow extraction of superconducting gap magnitudes, interface transparency, and the number of conduction channels in STM and other specialized geometries (Liao et al., 14 Jan 2026). MAR also provides access to equilibrium and nonequilibrium supercurrent dynamics, with theoretical predictions for critical and switching currents anchored by the MAR framework (Liao et al., 14 Jan 2026, Zhu et al., 2021).

5. MAR in the Context of Superconducting Order and Exotic Phenomena

MAR-based spectroscopy is a principal tool in the resolution of multiband superconductivity, gap anisotropy, and the coupling of bosonic modes to quasiparticles:

• Planar break-junction MAR in pnictide superconductors reveals multiple gaps, temperature dependence, and anisotropy, with over-gap fine structure interpreted as signatures of bosonic mode coupling (e.g., spin-resonance) (Kuzmicheva et al., 2021).
• Gate-tunable MAR in nanoribbon-topological insulator devices allows phase space mapping of proximity-induced superconductivity, high interface transparency, and signatures consistent with Majorana physics (Jauregui et al., 2017).
• In half-metal-superconductor contacts with non-uniform magnetization, MAR formalism provides the connection between microscopically-calculated Andreev amplitudes, symmetry constraints from quasiclassical Green function theory, and observable transport features—including selection rules for spin-triplet proximity effects (Kupferschmidt et al., 2010).

6. Impact and Future Directions

MAR stands as a central mechanism in the understanding of nonequilibrium superconducting hybrid structures. Its subharmonic features encode microscopic information about transparency, coherence, the presence of topological modes, and the symmetry of order parameters. MAR provides diagnostic access to phenomena including spin transport under dynamic magnetization (Holmqvist et al., 2013), unconventional pairing (Kuzmicheva et al., 2021), the diode effect (Zazunov et al., 2023), and multipair/quartet currents in multiterminal circuits (Mélin et al., 2015).

Current trends exploit MAR for:

• Quantum spectroscopies involving photon-assisted MAR and time-resolved techniques (Carrad et al., 2022).
• Disambiguation of topological states (e.g., Majorana zero modes) via field-insensitive MAR step positions (Golub, 2014, San-Jose et al., 2013).
• Nanoscale superconducting circuit engineering—MAR governs fundamental performance and operating signature in superconducting quantum bits, diodes, and hybrid topological devices (Jauregui et al., 2017, Zazunov et al., 2023).

MAR thus continues to be an indispensable theoretical and experimental axis in the physics of superconducting hybrids, topological materials, and quantum technologies.