Pith. sign in

REVIEW 3 major objections 4 minor 74 references

Theory of superconducting proximity effect in hole-based hybrid semiconductor-superconductor devices

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that the proximity-induced superconducting pairing in a two-dimensional hole gas is not a constant s-wave term but a momentum-dependent, anisotropic mixture of singlet and triplet components, with experimentally testable…

desk verdict Solid effective-theory paper on hole proximity pairing, with a load-bearing interface assumption that should be flagged before the experimental predictions are taken at face value. read the letter →

arxiv 2501.00088 v2 pith:MQFN6GAP submitted 2024-12-30 cond-mat.mes-hall cond-mat.supr-con

D. Michel Pino , Rubén Seoane-Souto , Maria José Calderón , Ramón Aguado , José Carlos Abadillo-Uriel This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.supr-con PACS 74.45.+c73.21.-b71.70.Ej74.78.Fk
keywords superconductingproximityeffecttwo-dimensionalholegasgermaniumKohn-LuttingerHamiltonianspin-orbitcouplingBogoliubovFermisurfacevanHovesingularitysinglet-tripletpairing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

What does a hole gas actually inherit when placed in contact with a superconductor? This paper argues that the answer is much richer than the standard assumption of a constant s-wave pairing term: for a two-dimensional hole gas in germanium, the proximity-induced pairing is a momentum-dependent, anisotropic mixture of singlet and triplet components whose precise form depends on whether the superconductor couples to the conduction band, to heavy and light holes, or produces mixed heavy-light terms. The authors derive this structure analytically from an 8-band k.p model, giving compact expressions for the pairing in the basis that diagonalizes the 4-band Kohn-Luttinger Hamiltonian. If the derivation holds, it changes what experiments should look for: tunneling spectra should show logarithmic van Hove singularities instead of BCS square-root singularities, and large in-plane magnetic fields should drive anisotropic gapless phases with Bogoliubov-Fermi surfaces. These signatures would let experiments distinguish the microscopic proximity channel and test whether hole-based hybrids require physics beyond effective s-wave models.

What carries the argument

The central object is the effective pairing matrix in the basis that diagonalizes the 4-band Kohn-Luttinger Hamiltonian, Eq. (10): a block matrix with intraband blocks $\tilde{\Delta}_i \cdot \sigma$ and interband blocks. Each intraband block's $\sigma_0$ and $\sigma_z$ components are longitudinal (singlet-like, gap-opening at the Fermi surface), while $\sigma_x$ and $\sigma_y$ are transverse (triplet-like, anticrossings away from the Fermi level). The derivation combines a Schrieffer-Wolff elimination of the superconductor (yielding constant $\Delta_{\rm CB}$, $\Delta_{\rm HH}$, $\Delta_{\rm LH}$, $\Delta_R$, $\Delta_S$ pairings in the 8-band model), an integration of the conduction band to obtain the 4KP form, and an exact analytical diagonalization of the 4-band Kohn-Luttinger Hamiltonian using the unitary $U$, so that the pairing is written in the same rotated Nambu basis as the band structure. The analytical expansions Eqs. (12)-(13) then expose the momentum structure: cubic-in-momentum Rashba factors $e^{3i\phi_k}$, anisotropies set by $\gamma_- = \gamma_3 - \gamma_2$, and the distinction between conduction-mediated pairing (with both longitudinal and transverse components) and direct HH/LH pairing (purely longitudinal).

What would settle it

Measure the tunneling density of states of an Al/Ge two-dimensional hole gas as a function of in-plane magnetic field magnitude and orientation at low $|\mu|$: the model predicts an irreversible spectral-gap closure with a diamond-shaped $E=0$ region and field-splitting logarithmic van Hove singularities when the Zeeman field has a component parallel to the Rashba field, whereas a constant s-wave model would show an isotropic gap that closes and reopens without these features.

Watch

Extended reading notes

Core claim

The paper establishes that when a superconductor proximitizes a two-dimensional hole gas, the effective pairing induced in the hole bands is not a constant s-wave term. Starting from an 8-band k.p Kane model of Ge and integrating out the superconductor, the authors derive a general 4-band effective theory whose pairing block has the structure of Eq. (10): in the basis that diagonalizes the Kohn-Luttinger Hamiltonian, pairing decomposes into intraband longitudinal components ($\sigma_0$ and $\sigma_z$, which open gaps at the Fermi surface) and transverse components ($\sigma_x$ and $\sigma_y$, which anticross away from it), together with interband heavy-hole/light-hole blocks. For conduction-band-mediated proximity they give explicit momentum-dependent expressions, Eq. (12), that are anisotropic, cubic in momentum, and involve the Rashba coefficient; for direct heavy/light-hole proximity, Eq. (13), the pairing is purely longitudinal. These structures produce logarithmic van Hove singularities in the density of states, f-type (cubic) superconductivity, gate-tunable gaps, and, at sufficiently large in-plane magnetic fields, gapless regimes with Bogoliubov-Fermi surfaces.

Load-bearing premise

The load-bearing premise is that the proximity effect is well approximated by constant, momentum-independent pairing terms induced directly in the semiconductor bands through spin-preserving tunneling, so that all nontrivial momentum dependence comes from the band structure and its couplings; if the pairing itself were significantly momentum-dependent, the analytical expressions and their predicted experimental signatures would change.

Editorial extensions

If this is right

  • In a proximitized Ge 2DHG, the induced pairing is not a constant s-wave term; it contains both singlet-like longitudinal and triplet-like transverse components with a cubic-in-momentum structure in the Rashba basis.
  • The density of states replaces BCS square-root singularities with logarithmic van Hove singularities whose energy positions split and reorder with magnetic field, giving sharp tunneling-spectroscopy fingerprints.
  • Large in-plane magnetic fields tilt the spectrum and close the spectral gap irreversibly, producing anisotropic Bogoliubov-Fermi surfaces with shapes tunable by field orientation, field strength, and electric field.
  • The induced gaps are gate-tunable through the chemical potential and vertical electric field, and their different dependences can identify which proximity channel (conduction band, heavy hole, light hole, or mixed) dominates in a given device.
  • Direct heavy-hole or light-hole pairing can produce gaps close to the parent superconducting gap, whereas conduction-band-only proximity yields gaps of at most about 10% of $\Delta_{\rm CB}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the effective pairing is indeed momentum-dependent with triplet components, then topological-phase diagrams of hole-based nanowires and Josephson junctions may differ qualitatively from those drawn from effective s-wave models, with consequences for Majorana zero-mode predictions.
  • The same 8KP-to-4KP reduction could be applied to other p-orbital valence-band materials, such as silicon or strained III-V heterostructures, by substituting the appropriate Luttinger parameters; the anisotropy terms proportional to $\gamma_-$ and to the Rashba coefficient should appear generically.
  • The predicted invariance of one van Hove singularity as a function of in-plane field amplitude is a sharp, background-free spectroscopic target: searching for an unshifted peak in the DOS as $B$ varies would directly test the cubic Rashba structure of the pairing.
  • The neglect of momentum-dependent corrections to the pairing itself could be probed by comparing the 4KP predictions against a full 8KP calculation with a finite-thickness superconducting layer, particularly near $\phi_k = \pi/4$, where the effective model already differs by about a factor of two in the conduction-band-only case.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript develops a microscopic theory of the superconducting proximity effect in Ge-based two-dimensional hole gases. Starting from an 8-band k.p Kane model with constant pairing terms in the conduction and valence bands, the authors integrate out the superconductor and the conduction band to obtain an effective 4-band Kohn-Luttinger Bogoliubov-de Gennes Hamiltonian for heavy and light holes. The 4-band problem is diagonalized analytically in the vertical-confinement geometry, yielding compact analytical expressions for the effective pairings in the Rashba basis, including momentum-dependent singlet and triplet components (Eqs. (10)-(13)). These expressions are used to predict gap anisotropies, magnetic-field-dependent van Hove singularities in the density of states, f-type pairing character, gapless regimes under in-plane fields, and Bogoliubov Fermi surfaces. The derivation is benchmarked against the parent 8KP model in Appendix D, and the code is publicly available.

Significance. If the results hold, the paper provides a substantially more complete effective theory of proximity-induced superconductivity in hole gases than previous constant-pairing models, and it makes specific falsifiable predictions for tunneling spectroscopy and microwave experiments. The analytical diagonalization of the 4KP model is a genuine technical contribution, the treatment of conduction- and valence-band-mediated pairings is self-contained, and the authors provide reproducible code. The main experimental predictions, however, rest on the assumption that a direct heavy-hole pairing Δ_HH can be induced with a magnitude comparable to the parent gap; this assumption is not derived from the microscopic tunneling model in Appendix C and is in tension with it, as detailed in the major comments.

major comments (3)
  1. [Appendix C, Eq. (C4); Secs. IVC, VA, VB] The low-|μ| scenario that generates the paper's headline experimental signatures (Figs. 6-9) is built on the choice Δ_HH = 200 μeV and Δ_LH = 0. According to the authors' own tunneling derivation, Δ_HH/Δ_s ≈ -(t_px^2 + t_py^2)/(2 E_s^2) and Δ_LH/Δ_s ≈ (t_px^2 + t_py^2 + 4 t_pz^2)/(6 E_s^2). Two consequences follow. First, for any nonzero tunneling amplitudes one has Δ_LH ≠ 0 whenever Δ_HH ≠ 0, so the specific pair (Δ_HH ≠ 0, Δ_LH = 0) used in Figs. 6-9 is not attainable within the model of Appendix C. Second, for an ideal planar interface only t_pz survives by symmetry, which gives Δ_HH = 0 and a purely light-hole pairing; the statement that real interfaces 'may introduce' the needed broken symmetry is not backed by any estimate of t_px,t_py relative to t_pz. Because the heavy-hole-dominated gap is the basis for the predicted gap tilt, van Hove diamond structure, and Bogoliubov Fermi surfaces, the central experimental predictions are conditional on an unverified microscopic input. The manuscript should either compute or bound these tunneling amplitudes for a Ge/Al interface, or reformulate the predictions as functions of the tunneling amplitudes rather than as independent Δ_HH and Δ_LH values.
  2. [Appendix D, Fig. D.1] The benchmark of the effective 4KP theory against the parent 8KP model shows deviations of up to a factor of 2 in the induced gaps near φ_k = π/4 (panel d). The text attributes this to the neglect of the split-off band and argues that it is irrelevant in the strained 2DHG regime, but no quantitative test of that claim is provided for the parameter range used in Secs. VA-VB (e.g., μ = -0.01 eV, F = 0.5 MV/m). Since the analytical formulas in Sec. IVA are expansions of the same 4KP model, the factor-2 discrepancy leaves a quantitative uncertainty in the predicted field scales and in the positions of the van Hove singularities. Please either provide an explicit 4KP-vs-8KP comparison in the confined 2DHG geometry or state the expected quantitative accuracy of the predictions.
  3. [Sec. IVD, Eq. (14) and Fig. 5] The mixed heavy-light pairing terms Δ_R e^{iχ_R} and Δ_S e^{iχ_S} are presented as generic symmetry-breaking terms and used to predict directional gap rotations and additional density-of-states singularities. However, the microscopic derivation in Appendix C shows that these amplitudes are not independent degrees of freedom: from Eq. (C4), Δ_R' ∝ (t_px - i t_py)^2 and Δ_S' ∝ (t_px - i t_py) t_pz. Since the manuscript does not estimate the interface tunneling amplitudes, the magnitudes and phases used in Fig. 5 are unconstrained. A quantitative prediction, such as the extra van Hove singularities attributed to these terms, requires either a microscopic estimate or an explicit scan over the physically allowed parameter range consistent with Eq. (C4).
minor comments (4)
  1. [Sec. IVA, Eq. (12)] The symbol α_R is called an 'adimensional Rashba coefficient' but is defined as α_R = γ_3 α_0 p/(E_hl m_0), which depends on p; please rename it or clarify that it is a momentum-dependent dimensionless quantity.
  2. [Abstract and Sec. I] The text repeatedly refers to 'full numerics' but the only explicit benchmark is the bulk comparison in Appendix D; a sentence describing how the 8KP and confined 2DHG spectra are computed would help the reader reproduce the figures.
  3. [Fig. 7 caption] The dashed lines marking the expected van Hove singularity positions in panels (a) and (b) are difficult to distinguish from the color-scale features; please increase their contrast or label them explicitly.
  4. [References] Several references are incomplete or missing journal identifiers, for example Ref. [2] (the journal 'Semicond. Sci. Technol.' is missing) and the abstract contains a missing space in '92%nuclear'; please correct these presentation issues.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the analytical pairing structure is derived from the 8-band k·p Hamiltonian and benchmarked numerically; self-citations are present but not load-bearing.

full rationale

The central derivation chain is self-contained. The paper starts from the 8KP BdG Hamiltonian (Sec. II) with explicit pairing blocks, integrates out the superconductor, and obtains the 4KP and 2DHG pairings by exact diagonalization of H_4KP (Appendix F). Equations (12), (13), and (15) are outputs of that diagonalization, not quantities fitted to the DOS or Fermi-surface results in Sec. V. The CB-only channel is benchmarked against the parent 8KP model in Appendix D, with deviations of order 15%, which is an independent numerical check rather than a restatement of the input. The direct HH/LH pairings in Eq. (6) are stated inputs, not fitted parameters; the low-|mu| predictions in Secs. VA-VB are conditional consequences of choosing Delta_HH = 200 micro-eV and Delta_LH = 0. Appendix C shows Delta_HH vanishes when the p_x and p_y tunneling amplitudes vanish, so the HH-dominated signatures depend on interface symmetry breaking, but this is an acknowledged modeling assumption (Sec. IVC, Appendix C) and a possible correctness concern, not a circular reduction. No prediction is obtained by renaming an input, and no external result is replaced by a self-citation chain. Self-citations such as Refs. [36], [39], and [42] appear in supporting contexts (g-factor perturbation theory, cQED examples) and are not load-bearing for the central pairing derivation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper does not introduce new physical particles or forces. Its free parameters are the pairing amplitudes and tunneling parameters that are standard inputs in proximity-effect theories, plus variational confinement parameters. The main assumptions are the validity of the k.p model, the constant-pairing approximation, and the neglect of split-off and orbital corrections to the pairing, which are standard approximations but not rigorously justified.

free parameters (4)
  • Delta_CB, Delta_HH, Delta_LH = 200 µeV in numerical examples
    These are the parent superconducting pairing potentials assumed for each band. They are inputs, not derived, and are taken from typical Al values in the numerical examples.
  • Delta_R, Delta_S, phases chi_R, chi_S = 200 µeV and 0 or pi/4 in examples
    These parametrize symmetry-breaking mixed HH-LH pairing terms. They are introduced ad hoc in Sec. IVD to model interface disorder and are not independently constrained.
  • Tunneling amplitudes t_s, t_px, t_py, t_pz = Not specified, only ratios used in Appendix C
    The effective pairings Delta_CB, Delta_HH, etc., are expressed in terms of these tunneling amplitudes. Their magnitudes are not determined by the theory and are treated as free inputs.
  • Variational parameter beta for confinement = Minimized numerically for each mass
    The Bastard wavefunction parameter beta is chosen to minimize the vertical ground-state energy. It is a variational parameter, not a fundamental constant.
assumptions (5)
  • domain assumption The 8-band k.p Kane model accurately describes the Ge band structure.
    The starting point is the 8KP model with parameters from Ref. [73]. This is standard for Ge but is an assumed model, not an exact description.
  • ad hoc to paper The Schrieffer-Wolff transformation to second order in tunneling is valid.
    The paper uses this to integrate out the superconductor, but the small parameter is not explicitly quantified. Higher-order terms are neglected without showing they are small.
  • ad hoc to paper The superconductor semiconductor interface preserves spin and can be described by constant tunneling elements.
    This assumption is central and is discussed in Appendix C. It allows constant pairing terms but may break down for realistic interfaces with spin-orbit coupling or momentum-dependent tunneling.
  • domain assumption The split-off band can be neglected in the 2DHG regime for strained Ge.
    The split-off band is 290 meV away, which is large compared to typical confinement energies. The authors justify this, but it is still an approximation that could affect quantitative predictions.
  • domain assumption The confinement wavefunction is the ground-state Bastard wavefunction.
    The vertical confinement is described by a variational Bastard wavefunction. This neglects non-parabolicity and excited subbands, which could be relevant at high fields or densities.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Theory of superconducting proximity effect in hole-based hybrid semiconductor-superconductor devices." pith.science (2026). https://pith.science/paper/MQFN6GAP

@misc{pith2026250100088,
  author       = {Pith},
  title        = {Pith review of: Theory of superconducting proximity effect in hole-based hybrid semiconductor-superconductor devices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQFN6GAP}},
  note         = {Machine review of arXiv:2501.00088}
}
read the original abstract

Hybrid superconductor-semiconductor systems have received a great deal of attention in the last few years because of their potential for quantum engineering, including novel qubits and topological devices. The proximity effect, the process by which the semiconductor inherits superconducting correlations, is an essential physical mechanism of such hybrids. Recent experiments have demonstrated the proximity effect in hole-based semiconductors, but, in contrast to electrons, the precise mechanism by which the hole bands acquire superconducting correlations remains an open question. In addition, hole spins exhibit a complex strong spin-orbit interaction, with largely anisotropic responses to electric and magnetic fields, further motivating the importance of understanding the interplay between such effects and the proximity effect. In this work, we analyze this physics with focus on germanium-based two-dimensional gases. Specifically, we develop an effective theory supported by full numerics, allowing us to extract various analytical expressions and predict different types of superconducting correlations including non-standard forms of singlet and triplet pairing mechanisms with non-trivial momentum dependence; as well as different Zeeman and Rashba spin-orbit contributions. This, together with their precise dependence on electric and magnetic fields, allows us to make specific experimental predictions, including the emergence of f-type superconductivity, Bogoliubov Fermi surfaces, and gapless regimes caused by large in-plane magnetic fields.

Figures

Figures reproduced from arXiv: 2501.00088 by the authors.

Figure 1
Figure 1. 8KP model. (a-b) Energy dispersion of the BdG 8KP model along the k (kx = ky) in-plane direction (perpendicular to the growth direction). Bands are labeled as {e/e, hh/h ¯ h, lh/l ¯ h, sh/s ¯ h¯} respectively for electrons, heavy holes, light holes, split-off holes, and their respective particle and time-reversed counterparts. CB pairing. (c-d) Gaps opening along the kx = ky direction; at (c) k1(π/4, π/2, µ) and (d)… view at source ↗
Figure 2
Figure 2. (a) shows that, for large negative values of µ, both HH and LH bands cross the Fermi level and there are up to four different crossings. In this regime, the bands ex￾hibit strong HH-LH hybridization, as can be seen from the color code. In [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Induced pairings in the 2DHG 4KP model with SC proximity coming from the CB only. (a) Energy dispersion of the BdG 2DHG 4KP model along the kx direction. CB pairing. (b) SC pairing as a function of the polar coordinate ϕk, such as (kx, ky) = k(cos ϕk, sin ϕk); at k1,2,3,4(ϕk, µ). (c-f) Gaps opening along the kx direction; at k1,2,3,4(0, µ), respectively. (g-h) Gaps opening at ϕk = 0 as a function of (g) µ with F = 5… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Direct LH/HH pairing. Gaps ∆(k1,2,3,4) along the kx direction, as a function of µ for (a) HH and (b) LH direct pairing. The vertical dashed line indicates µ = −Ehl, where ∆(k1, k2) emerge. Parameters used: ∆LH/HH = 200 µeV; F = 5 MV/m; Table I; Appendix E. ings on the …
Figure 5
Figure 5. Figure 5: Mixed LH-HH pairing. (a) Gaps ∆(k1, k2, k3, k4) and their dependence on µ for mixed pairing ∆R = 200 µeV and χR = 0. (b) Gaps ∆(k3, k4) as a function of ϕk for µ = −0.01 eV for ∆HH = 180 µeV and ∆R = 20 µeV with χR = 0 (solid) and χR = π/4 (dashed). V. EXPERIMENTAL SIG…
Figure 6
Figure 6. Figure 6: Magnetic-field dependence of pairings with direct HH contribution. (a) Tilted bands as a function of kx around the Fermi level crossings at ki for B = 1/ √ 2(1, 1, 0) (solid blue) and ||B|| = 0 (dashed grey). The gap ∆(gap) is shown as a pink band around E = 0 and the …
Figure 7
Figure 7. Figure 7: DOS as a function of magnetic field. (a) DOS centered around E = 0 as a function of the vertical magnetic field Bz with µ = −0.01 eV. (b) DOS centered around E = 0 as a function of the in-plane magnetic field Bx. (c) DOS at low in-plane magnetic field, in the correspon…
Figure 8
Figure 8. Figure 8: Maxima, minima and van Hove singularities. (a) Energy at k3 and k4 as a function of ϕk. While maxima and minima of the oscillations in both bands coincide at the same ϕk for B = 0 (grey dotted lines), this is no longer the case at finite in-plane magnetic field (Bx = 0…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

74 extracted references · 53 canonical work pages

  1. [67]

    E.Ivchenko, A.Y.Kaminski,andU.Rössler,Heavy-light hole mixing at zinc-blende (001) interfaces under normal incidence, Physical Review B54, 5852 (1996)

    work page 1996

  2. [1]

    Exact expressions for the pairing terms The exact diagonalization of the time-reversed terms follows an identical reasoning. LetU=U 4U3U2U1 and U′ =U′ 4U′ 3U′ 2U′ 1, withU′ i being the time-reversed equiv- alent ofUi, any pairing Hamiltonian within the valence bandH ∆ v,j becomes H∆ (d) v,j =UH ∆ v,jU′†,(F7) 18 this equation allows us to extract exact ana...

    work page

  3. [2]

    Alicea, New directions in the pursuit of majorana fermions in solid state systems, Rep

    J. Alicea, New directions in the pursuit of majorana fermions in solid state systems, Rep. Prog. Phys.75, 076501 (2012)

    2012

  4. [3]

    Leijnse and K

    M. Leijnse and K. Flensberg, Introduction to topological superconductivity and Majorana fermions,27, 124003 (2012)

    work page 2012

  5. [4]

    Aguado, La Rivista del Nuovo Cimento40, 523 (2017)

    R. Aguado, La Rivista del Nuovo Cimento40, 523 (2017)

    work page 2017

  6. [5]

    R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, Nature Re- view Materials3, 52 (2018)

    work page 2018

  7. [6]

    Prada, P

    E. Prada, P. San-Jose, M. W. A. de Moor, A. Geresdi, E. J. H. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L. P. Kouwenhoven, From andreev to majorana bound states in hybrid superconductor– semiconductor nanowires, Nat. Rev. Phys.2, 575 (2020)

    work page 2020

  8. [7]

    Aguado and L

    R. Aguado and L. P. Kouwenhoven, Majorana qubits for topological quantum computing, Physics Today73, 44 (2020)

    2020

Show all 74 references
  1. [8]

    Flensberg, F

    K. Flensberg, F. von Oppen, and A. Stern, Engineered platforms for topological superconductivity and majo- rana zero modes, Nat. Rev. Mat.6, 944 (2021)

    2021

  2. [9]

    R.Aguado,Aperspectiveonsemiconductor-basedsuper- conducting qubits, Applied Physics Letters117, 240501 (2020)

    2020

  3. [10]

    Seoane Souto and R

    R. Seoane Souto and R. Aguado,Subgap States in Semiconductor-Superconductor Devices for Quantum Technologies: Andreev Qubits and Minimal Majorana Chains, chapter 3 in New Trends and Platforms for Quantum Technologies, edited by R. Aguado, R. Citro, M. Lewenstein, and M. Stern ...

    2024

  4. [11]

    Burkard, T

    G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Reviews of Modern Physics95, 025003 (2023)

    2023

  5. [12]

    Huang, C

    W. Huang, C. Yang, K. Chan, T. Tanttu, B. Hensen, R. Leon, M. Fogarty, J. Hwang, F. Hudson, K. M. Itoh, et al., Fidelity benchmarks for two-qubit gates in silicon, Nature569, 532 (2019)

    2019

  6. [13]

    Noiri, K

    A. Noiri, K. Takeda, T. Nakajima, T. Kobayashi, A. Sammak, G. Scappucci, and S. Tarucha, Fast uni- versal quantum gate above the fault-tolerance threshold in silicon, Nature601, 338 (2022)

    2022

  7. [14]

    A. R. Mills, C. R. Guinn, M. J. Gullans, A. J. Sigillito, M. M. Feldman, E. Nielsen, and J. R. Petta, Two-qubit silicon quantum processor with operation fidelity exceed- ing 99%, Science Advances8, eabn5130 (2022)

    2022

  8. [15]

    X. Xue, T. Watson, J. Helsen, D. R. Ward, D. E. Sav- age, M. G. Lagally, S. N. Coppersmith, M. Eriksson, S. Wehner, and L. M. Vandersypen, Benchmarking gate fidelities in a si/sige two-qubit device, Physical Review X9, 021011 (2019)

    2019

  9. [16]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wall- raff, Circuit quantum electrodynamics, Reviews of Mod- ern Physics93, 025005 (2021)

    2021

  10. [17]

    Serniak, T

    M.Hays, V.Fatemi, D.Bouman, J.Cerrillo, S.Diamond, K. Serniak, T. Connolly, P. Krogstrup, J. Nygård, A. L. Yeyati, A. Geresdi, and M. H. Devoret, Coherent ma- nipulation of an andreev spin qubit, Science373, 430 (2021)

    2021

  11. [18]

    Pita-Vidal, A

    M. Pita-Vidal, A. Bargerbos, R. Žitko, L. J. Splitthoff, L. Grünhaupt, J. J. Wesdorp, Y. Liu, L. P. Kouwen- hoven, R. Aguado, B. van Heck, A. Kou, and C. K. An- dersen, Direct manipulation of a superconducting spin qubit strongly coupled to a transmon qubit, Nature Physics19, ...

    2023

  12. [19]

    E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Spin-resolved andreev levels and parity crossings in hybrid superconductor– semiconductor nanostructures, Nature Nanotechnology 9, 79 (2014)

    2014

  13. [20]

    Janvier, L

    C. Janvier, L. Tosi, L. Bretheau, Ç. Ö. Girit, M. Stern, P. Bertet, P. Joyez, D. Vion, D. Esteve, M. F. Goffman, H. Pothier, and C. Urbina, Coherent manipulation of an- dreev states in superconducting atomic contacts, Science 349, 1199 (2015)

    2015

  14. [21]

    M. Hays, V. Fatemi, K. Serniak, D. Bouman, S. Dia- mond, G. de Lange, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, Continuous monitoring of a trapped superconducting spin, Nature Physics16, 1103 (2020)

    2020

  15. [22]

    Bargerbos, M

    A. Bargerbos, M. Pita-Vidal, R. Žitko, J. Ávila, L. J. Splitthoff, L. Grünhaupt, J. J. Wesdorp, C. K. Andersen, Y. Liu, L. P. Kouwenhoven, R. Aguado, A. Kou, and B. van Heck, Singlet-doublet transitions of a quantum dot josephson junction detected in a transmon circuit, PRX Qu...

    2022

  16. [23]

    Bargerbos, M

    A. Bargerbos, M. Pita-Vidal, R. Žitko, L. J. Splitthoff, L. Grünhaupt, J. J. Wesdorp, Y. Liu, L. P. Kouwen- hoven, R. Aguado, C. K. Andersen, A. Kou, and B. van Heck, Spectroscopy of spin-split andreev levels in a quan- tum dot with superconducting leads, Phys. Rev. Lett. 131,...

    2023

  17. [24]

    Dimoulas, P

    A. Dimoulas, P. Tsipas, A. Sotiropoulos, and E. Evan- gelou, Fermi-level pinning and charge neutrality level in germanium, Applied physics letters89, 10.1063/1.2410241 (2006)

    2006 doi

  18. [25]

    Winkler,Spin-orbit coupling effects in two- dimensional electron and hole systems, Springer tracts in modern physics (Springer, Berlin, 2003)

    R. Winkler,Spin-orbit coupling effects in two- dimensional electron and hole systems, Springer tracts in modern physics (Springer, Berlin, 2003)

    2003

  19. [26]

    Philippopoulos, S

    P. Philippopoulos, S. Chesi, D. Culcer, and W. Coish, Pseudospin-electric coupling for holes beyond the envelope-function approximation, Physical Review B 102, 075310 (2020)

    2020

  20. [27]

    Liu, J.-X

    Y. Liu, J.-X. Xiong, Z. Wang, W.-L. Ma, S. Guan, J.- W. Luo, and S.-S. Li, Emergent linear rashba spin-orbit coupling offers fast manipulation of hole-spin qubits in germanium, Physical Review B105, 075313 (2022)

    2022

  21. [28]

    E. A. Rodríguez-Mena, J. C. Abadillo-Uriel, G. Veste, B. Martinez, J. Li, B. Sklénard, and Y.-M. Niquet, Linear-in-momentum spin orbit interactions in planar ge/gesi heterostructures and spin qubits, Physical Re- view B108, 205416 (2023)

    2023

  22. [29]

    Crippa, R

    A. Crippa, R. Maurand, L. Bourdet, D. Kotekar-Patil, A. Amisse, X. Jehl, M. Sanquer, R. Laviéville, H. Bo- huslavskyi, L. Hutin,et al., Electrical spin driving by g- matrix modulation in spin-orbit qubits, Physical review letters120, 137702 (2018)

    2018

  23. [30]

    V. P. Michal, B. Venitucci, and Y.-M. Niquet, Longitu- dinal and transverse electric field manipulation of hole spin-orbit qubits in one-dimensional channels, Physical Review B103, 045305 (2021)

    2021

  24. [31]

    Terrazos, E

    L. Terrazos, E. Marcellina, Z. Wang, S. Coppersmith, 21 M. Friesen, A. Hamilton, X. Hu, B. Koiller, A. Saraiva, D. Culcer,et al., Theory of hole-spin qubits in strained germanium quantum dots, Physical Review B103, 125201 (2021)

    2021

  25. [32]

    Zhang, H

    T. Zhang, H. Liu, F. Gao, G. Xu, K. Wang, X. Zhang, G. Cao, T. Wang, J. Zhang, X. Hu,et al., Anisotropic g-factor and spin–orbit field in a germanium hut wire double quantum dot, Nano Letters21, 3835 (2021)

    2021

  26. [33]

    F. N. Froning, L. C. Camenzind, O. A. van der Molen, A. Li, E. P. Bakkers, D. M. Zumbühl, and F. R. Braak- man, Ultrafast hole spin qubit with gate-tunable spin– orbit switch functionality, Nature Nanotechnology16, 308 (2021)

    2021

  27. [34]

    Liles, F

    S. Liles, F. Martins, D. Miserev, A. Kiselev, I. Thorvald- son, M.Rendell, I.Jin, F.Hudson, M.Veldhorst, K.Itoh, et al., Electrical control of the g tensor of the first hole in a silicon mos quantum dot, Physical Review B104, 235303 (2021)

    2021

  28. [35]

    Niquet, Hole spin manipulation in inhomoge- neous and nonseparable electric fields, Physical Review B106, 235426 (2022)

    B.Martinez, J.C.Abadillo-Uriel, E.A.Rodríguez-Mena, and Y.-M. Niquet, Hole spin manipulation in inhomoge- neous and nonseparable electric fields, Physical Review B106, 235426 (2022)

    2022

  29. [36]

    Jirovec, P

    D. Jirovec, P. M. Mutter, A. Hofmann, A. Crippa, M. Rychetsky, D. L. Craig, J. Kukucka, F. Martins, A. Ballabio, N. Ares,et al., Dynamics of hole singlet- triplet qubits with large g-factor differences, Physical Re- view Letters128, 126803 (2022)

    2022

  30. [37]

    J. C. Abadillo-Uriel, E. A. Rodríguez-Mena, B. Mar- tinez, and Y.-M. Niquet, Hole-spin driving by strain- induced spin-orbit interactions, Physical Review Letters 131, 097002 (2023)

    2023

  31. [38]

    Maurand, X

    R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bo- huslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer,et al., A cmos silicon spin qubit, Nature communications7, 13575 (2016)

    2016

  32. [39]

    N. W. Hendrickx, W. I. Lawrie, M. Russ, F. van Rigge- len, S. L. de Snoo, R. N. Schouten, A. Sammak, G. Scap- pucci, and M. Veldhorst, A four-qubit germanium quan- tum processor, Nature591, 580 (2021)

    2021

  33. [40]

    Rambal, H

    C.X.Yu, S.Zihlmann, J.C.Abadillo-Uriel, V.P.Michal, N. Rambal, H. Niebojewski, T. Bedecarrats, M. Vinet, É. Dumur, M. Filippone,et al., Strong coupling between a photon and a hole spin in silicon, Nature Nanotechnol- ogy18, 741 (2023)

    2023

  34. [41]

    De Palma, F

    F. De Palma, F. Oppliger, W. Jang, S. Bosco, M. Janík, S. Calcaterra, G. Katsaros, G. Isella, D. Loss, and P. Scarlino, Strong hole-photon coupling in planar ge for probing charge degree and strongly correlated states, Na- ture Communications15, 10177 (2024)

    2024

  35. [42]

    Janík, K

    M. Janík, K. Roux, C. B. Espinosa, O. Sagi, A. Bagh- dadi, T. Adletzberger, S. Calcaterra, M. Botifoll, A. G. Manjón, J. Arbiol,et al., Strong charge-photon cou- pling in planar germanium enabled by granular alu- minium superinductors, arXiv preprint arXiv:2407.03079 10.48550/...

    2024 doi

  36. [43]

    Michal, J

    V. Michal, J. Abadillo-Uriel, S. Zihlmann, R. Maurand, Y.-M. Niquet, and M. Filippone, Tunable hole spin- photon interaction based on g-matrix modulation, Phys- ical Review B107, L041303 (2023)

    2023

  37. [44]

    Bosco, P

    S. Bosco, P. Scarlino, J. Klinovaja, and D. Loss, Fully tunable longitudinal spin-photon interactions in si and ge quantum dots, Physical Review Letters129, 066801 (2022)

    2022

  38. [45]

    Tosato, V

    A. Tosato, V. Levajac, J.-Y. Wang, C. J. Boor, F. Bor- soi, M. Botifoll, C. N. Borja, S. Martí-Sánchez, J. Arbiol, A. Sammak, M. Veldhorst, and G. Scappucci, Hard su- perconducting gap in germanium, Communications Ma- terials4, 23 (2023)

    2023

  39. [46]

    O. Sagi, A. Crippa, M. Valentini, M. Janik, L. Baghumyan, G. Fabris, L. Kapoor, F. Hassani, J. Fink, S. Calcaterra, D. Chrastina, G. Isella, and G. Katsaros, A gate tunable transmon qubit in planar ge, Nature Communications15, 6400 (2024)

    2024

  40. [47]

    Kiyooka, C

    E. Kiyooka, C. Tangchingchai, L. Noirot, A. Leblanc, B. Brun, S. Zihlmann, R. Maurand, V. Schmitt, E. Du- mur, J.-M. Hartmann, F. Lefloch, and S. De Franceschi, Gatemon qubit on a germanium quantum-well het- erostructure, Nano Letters 10.1021/acs.nanolett.4c05539 (2024)

    2024 doi

  41. [48]

    Valentini, O

    M. Valentini, O. Sagi, L. Baghumyan, T. de Gi- jsel, J. Jung, S. Calcaterra, A. Ballabio, J. Aguil- era Servin, K. Aggarwal, M. Janik, T. Adletzberger, R. Seoane Souto, M. Leijnse, J. Danon, C. Schrade, E. Bakkers, D. Chrastina, G. Isella, and G. Katsaros, Parity-conserving co...

    2024

  42. [49]

    Leblanc, C

    A. Leblanc, C. Tangchingchai, Z. S. Momtaz, E. Kiy- ooka, J.-M. Hartmann, F. Gustavo, J.-L. Thomassin, B. Brun, V. Schmitt, S. Zihlmann,et al., Gate and flux tunable sin(2φ) josephson element in proximitized junc- tions, arXiv preprint arXiv:2405.14695 (2024)

    2024 arXiv

  43. [50]

    Hinderling, S

    M. Hinderling, S. C. ten Kate, M. Coraiola, D. Hax- ell, M. Stiefel, M. Mergenthaler, S. Paredes, S. Be- dell, D. Sabonis, and F. Nichele, Direct microwave spec- troscopy of andreev bound states in planarGejosephson junctions, PRX Quantum5, 030357 (2024)

    2024

  44. [51]

    Lakic, W

    L. Lakic, W. I. Lawrie, D. van Driel, L. E. Stehouwer, M. Veldhorst, G. Scappucci, F. Kuemmeth, and A. Chat- terjee, A proximitized quantum dot in germanium, arXiv preprint arXiv:2405.02013 (2024)

    2024 arXiv

  45. [52]

    Laubscher, J

    K. Laubscher, J. D. Sau, and S. Das Sarma, Germanium- based hybrid semiconductor-superconductor topological quantum computing platforms: Disorder effects, Physi- cal Review B110, 155431 (2024)

    2024

  46. [53]

    Adelsberger, H

    C. Adelsberger, H. F. Legg, D. Loss, and J. Klino- vaja, Microscopic analysis of proximity-induced super- conductivity and metallization effects in superconductor- germanium hole nanowires, Physical Review B108, 155433 (2023)

    2023

  47. [54]

    Luethi, K

    M. Luethi, K. Laubscher, S. Bosco, D. Loss, and J. Kli- novaja, Planar josephson junctions in germanium: Effect of cubic spin-orbit interaction, Phys. Rev. B107, 035435 (2023)

    2023

  48. [55]

    Moghaddam, T

    A. Moghaddam, T. Kernreiter, M. Governale, and U. Zülicke, Exporting superconductivity across the gap: Proximity effect for semiconductor valence-band states due to contact with a simple-metal superconductor, Physical Review B89, 184507 (2014)

    2014

  49. [56]

    D. Phan, J. Senior, A. Ghazaryan, M. Hatefipour, W. M. Strickland, J. Shabani, M. Serbyn, and A. P. Higgin- botham, Detecting inducedp±ippairing at the al-inas interface with a quantum microwave circuit, Phys. Rev. Lett.128, 107701 (2022)

    2022

  50. [57]

    Z. Zhu, M. Papaj, X.-A. Nie, H.-K. Xu, Y.-S. Gu, X. Yang, D. Guan, S. Wang, Y. Li, C. Liu, J. Luo, Z.- A. Xu, H. Zheng, L. Fu, and J.-F. Jia, Discovery of seg- mented fermi surface induced by cooper pair momentum, 22 Science374, 1381 (2021)

    2021

  51. [58]

    N. F. Q. Yuan and L. Fu, Zeeman-induced gapless su- perconductivity with a partial fermi surface, Phys. Rev. B97, 115139 (2018)

    2018

  52. [59]

    S. S. Babkin, A. P. Higginbotham, and M. Serbyn, Proximity-induced gapless superconductivity in two- dimensional Rashba semiconductor in magnetic field, SciPost Phys.16, 115 (2024)

    2024

  53. [60]

    N. Ares, V. N. Golovach, G. Katsaros, M. Stof- fel, F. Fournel, L. I. Glazman, O. G. Schmidt, and S. De Franceschi, Nature of tunable hole g factors in quantum dots, Physical review letters110, 046602 (2013)

    2013

  54. [61]

    However, these effective masses are reversed in the in-plane motion [24]

    The terms ’heavy hole’ and ‘light hole’ refer to their re- spectively large and small effective masses for the motion in the growth direction. However, these effective masses are reversed in the in-plane motion [24]. The names HH (LH) usually refer to theirzcomponent of angula...

  55. [62]

    Scappucci, C

    G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Katsaros, and M. Veldhorst, The germanium quantum information route, Nature Reviews Materials6, 926 (2021)

    2021

  56. [63]

    Other materials, such as Si, have a much smaller split-off energy splitting and a 6KP theory is more adequate for their description

  57. [64]

    G. L. Bir and G. E. Pikus, Symmetry and strain-induced effects in semiconductors, (No Title) (1974)

    1974

  58. [65]

    Z. wang, E. Marcellina, A. R. Hamilton, J. H. Cullen, S. Rogge, J. Salfi, and D. Culcer, Optimal operation points for ultrafast, highly coherent ge hole spin-orbit qubits, npj Quantum Information7, 10.1038/s41534- 021-00386-2 (2021)

    2021 doi

  59. [66]

    Futterer, M

    D. Futterer, M. Governale, U. Zülicke, and J. König, Band-mixing-mediated andreev reflection of semiconduc- tor holes, Physical Review B—Condensed Matter and Materials Physics84, 104526 (2011)

    2011

  60. [68]

    S. S. Babkin, B. Joecker, K. Flensberg, M. Serbyn, and J. Danon, Superconducting proximity effect in two- dimensionalholegases,Phys.Rev.B111,214518(2025)

    2025

  61. [69]

    J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Generic new platform for topological quantum compu- tation using semiconductor heterostructures, Phys. Rev. Lett.104, 040502 (2010)

    2010

  62. [70]

    Alicea, Majorana fermions in a tunable semiconduc- tor device, Physical Review B—Condensed Matter and Materials Physics81, 125318 (2010)

    J. Alicea, Majorana fermions in a tunable semiconduc- tor device, Physical Review B—Condensed Matter and Materials Physics81, 125318 (2010)

    2010

  63. [71]

    R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Ma- jorana fermions and a topological phase transition in semiconductor-superconductor heterostructures, Physi- cal review letters105, 077001 (2010)

    2010

  64. [72]

    Y. Oreg, G. Refael, and F. von Oppen, Helical liquids andmajoranaboundstatesinquantumwires,Phys.Rev. Lett.105, 177002 (2010)

    2010

  65. [73]

    Github repository for the code: https://github.com/dmichelpino/superholes

  66. [74]

    Paul, 8-band k·p modelling of mid-infrared intersub- band absorption in ge quantum wells, Journal of Applied Physics120, 10.1063/1.4959259 (2016)

    D. Paul, 8-band k·p modelling of mid-infrared intersub- band absorption in ge quantum wells, Journal of Applied Physics120, 10.1063/1.4959259 (2016)

    2016 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.