{"id":"52e2b0c5-9f9c-4de6-a081-522285dabbc8","arxiv_id":"2501.00088","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper derives effective superconducting pairing terms for a proximitized germanium two-dimensional hole gas, predicting anisotropic, momentum-dependent singlet and triplet pairings, f-type superconductivity, and Bogoliubov Fermi surfaces.","lead":"This paper develops a theory of how superconductivity is induced in a two-dimensional hole gas in germanium, going beyond simple models to include multiple hole bands and spin-orbit effects. It predicts unusual superconducting pairings and magnetic-field-driven features that could be tested in tunneling and microwave experiments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The low-|μ| heavy-hole predictions (hard gaps near Δ_parent, van Hove singularities, Bogoliubov Fermi surfaces) assume a direct HH pairing Δ_HH of order 200 μeV; the microscopic derivation in Appendix C shows Δ_HH vanishes unless interface tunneling into p_x/p_y orbitals is nonzero, a condition…","rationale":"The reader's weakest assumption was the constant-pairing approximation. I largely agree but want to single out the direct HH pairing amplitude, because the paper's most distinctive experimental signatures (f-type pairing in Eq. (13) at low μ, in-plane gap tilt, and BFS in Fig. 9) are computed with Δ_HH = 200 μeV. Appendix C relates these to p_x/p_y tunneling, whose symmetry selection is not quantified. This is not an internal inconsistency: the 4KP diagonalization and the Schrieffer-Wolff reduction are careful, and the code is available. The concern is external, about the microscopic input Δ_HH. It is load-bearing because if the interface is approximately planar, Δ_HH vanishes and the low-|μ| results revert to a much weaker CB-only induced gap, changing both quantitative and qualitative claims. The paper explicitly marks the direct-valence scenario as possible rather than established, so the paper is appropriately CONDITIONAL rather than wrong; I therefore keep the reader's verdict unchanged while proposing a concrete interface calculation as the decisive check.","tokens_in":30172,"tokens_out":10905,"duration_ms":116569,"concrete_test":"Compute the Ge(001)/Al interface tunneling matrix elements t_s, t_px, t_py, t_pz from an ab initio or atomistic tight-binding calculation of the hybrid interface, then evaluate Δ_HH/Δ_s and Δ_LH/Δ_s with Eq. (C4). If |t_px|, |t_py| are much smaller than |t_pz| such that |Δ_HH|/Δ_s < 0.05, recompute Figs. 6-9 with only Δ_LH and CB pairing; a shift of the gap-closing field by more than the plotted range, or loss of the Bogoliubov Fermi surfaces, would confirm that the headline predictions depend on an unverified interface assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is not the mere existence of momentum-dependent pairing—that follows already from CB-mediated pairing in Eq. (12)—but the specific strong-gap regime used for experimental signatures in Secs. VA-VB. That regime is built on H_Δ^v in Eq. (6) with Δ_HH and Δ_LH treated as independent constants. Appendix C derives these constants from a spin-preserving tunneling Hamiltonian: 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). For an ideal planar interface, only t_pz survives by symmetry, giving Δ_HH = Δ_R = 0 and only a LH pairing. The paper's low-|μ| scenario (Figs. 6-9) instead takes Δ_HH = 200 μeV and Δ_LH = 0, i.e., it assumes the symmetry-broken interface produces sizable t_px and t_py tunneling. No estimate, calculation, or experimental constraint is provided for t_px and t_py relative to t_pz. If t_px and t_py are suppressed, the HH-dominated gap and all in-plane-field consequences (gap tilt, DOS diamond structure, BFS in Fig. 9) disappear; the remaining CB-only channel yields gaps at most ~10% of Δ_CB (Sec. IVB), qualitatively changing the predicted field scales and possibly the visibility of the features. The paper acknowledges interface symmetry breaking only as a possibility ('may introduce'), so the predictions are conditional on an unverified microscopic input.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30559,"tokens_out":7694,"duration_ms":84839,"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":[{"comment":"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.","section":"Appendix C, Eq. (C4); Secs. IVC, VA, VB"},{"comment":"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.","section":"Appendix D, Fig. D.1"},{"comment":"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).","section":"Sec. IVD, Eq. (14) and Fig. 5"}],"minor_comments":[{"comment":"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.","section":"Sec. IVA, Eq. (12)"},{"comment":"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.","section":"Abstract and Sec. I"},{"comment":"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.","section":"Fig. 7 caption"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central derivation is self-contained and the analytical results are valuable, but the paper's most striking predictions depend on a parameter regime that the authors' own Appendix C shows to be unattainable or at best unconstrained. I would not reject the manuscript, because the issue can be addressed by computing or bounding the interface tunneling amplitudes, or by recasting the predictions as a function of those amplitudes. I would ask the editor to require that revision before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the general pairing structure in the Rashba basis for a 4-band Kohn-Luttinger Hamiltonian, Eq. (10), with explicit analytical expressions for CB-mediated and direct valence-band pairings, Eqs. (12) and (13). The authors also give a clean exact diagonalization of the 4-band model, benchmark it against the parent 8KP model, and connect the resulting anisotropic pairing to observable DOS features: logarithmic van Hove singularities, gap tilts, and Bogoliubov Fermi surfaces. That is a real step beyond adding constant pairing to the lowest hole band, and the paper is honest about the partial overlap with Ref. [67]. The code is available, which is good practice.\n\nThe soft spots are proportionate to how much weight they carry. The main one, which the stress-test note correctly identifies, is the microscopic input for the direct heavy-hole pairing. Appendix C derives Δ_HH ∝ (t_px^2 + t_py^2), which vanishes for an ideal planar interface where only t_pz survives. Yet the headline experimental regime in Figs. 6-9 sets Δ_HH = 200 μeV and Δ_LH = 0, with no estimate or constraint on the symmetry-breaking tunneling amplitudes t_px and t_py. So the hard-gap, HH-dominated predictions are conditional on an interface property that the paper does not characterize. This is not fatal: the CB-only channel already produces momentum-dependent, anisotropic pairing, and the framework is general enough to accommodate whatever the interface actually does. But the authors should either provide a microscopic estimate for the off-normal tunneling amplitudes or explicitly frame the HH regime as an illustration of the formalism rather than a definite prediction.\n\nTwo smaller issues. The neglect of mixed conduction-valence pairing terms, acknowledged in Appendix C, may produce linear-in-k corrections that could shift the quantitative gap values, and the variational confinement wavefunction is not benchmarked against a more exact treatment. Neither undermines the central claim that proximity pairing in a 2DHG is non-trivial and momentum-dependent; that claim survives even in the CB-only limit.\n\nOverall, this is a careful derivation with useful analytical results and experimentally relevant signatures. The reader’s conditional verdict is fair. I’d send it out, and ask the authors to address the interface-tunneling conditionality before publication.","headline":"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.","tokens_in":31117,"tokens_out":1739,"would_cite":true,"duration_ms":21026,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["74.45.+c","73.21.-b","71.70.Ej","74.78.Fk"],"model":"deepseek-v4-flash","headline":"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…","keywords":["superconducting proximity effect","two-dimensional hole gas","germanium","Kohn-Luttinger Hamiltonian","spin-orbit coupling","Bogoliubov Fermi surface","van Hove singularity","singlet-triplet pairing"],"falsifier":"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.","tokens_in":29947,"feed_emoji":"🧲","tokens_out":6821,"duration_ms":65280,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hole-gas superconductivity is not simple s-wave","feed_subtitle":"Germanium holes inherit anisotropic, momentum-dependent pairing with distinctive magnetic-field signatures.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\Delta_{\\rm CB}$."],"supporting_citations":[{"why":"Supplies the Schrieffer-Wolff transformation and the 4-band Kohn-Luttinger framework used to derive the effective proximitized hole theory.","marker":"[24]"},{"why":"Prior model adding constant pairings to both heavy- and light-hole bands; the paper extends and benchmarks against it.","marker":"[52]"},{"why":"Established that coupling to the conduction band exports superconductivity into the valence band; the paper builds on this to derive the momentum-dependent induced pairing from $\\Delta_{\\rm CB}$.","marker":"[54]"},{"why":"Experimental detection of induced pairing via kinetic inductance in an Al-InAs hybrid, cited as the method to probe Bogoliubov-Fermi surfaces in the hole system.","marker":"[55]"},{"why":"Experimental observation of segmented Fermi surface induced by Cooper pair momentum in a proximitized topological insulator, supporting the gapless-regime interpretation.","marker":"[56]"},{"why":"Provides the Zeeman-induced gapless superconductivity and partial-Fermi-surface framework used to interpret the large-field behavior.","marker":"[57]"},{"why":"Proximitized 2D Rashba model whose density-of-states signatures the hole-gas results generalize with anisotropy and tilting.","marker":"[58]"},{"why":"Parallel derivation of the tunneling-induced constant pairings and justification that integrating out the superconductor yields the constant pairing terms used here.","marker":"[67]"},{"why":"Supplies the Ge band-structure parameters used in the numerics and analytical estimates.","marker":"[73]"}],"fun_headline_variants":["Hole-gas proximity: no simple s-wave","Germanium holes get f-type pairing via proximity","Anisotropic cubic pairing emerges in hole-gas hybrids","Momentum-dependent pairing from hole proximity effect","Bogoliubov Fermi surfaces predicted for hole-gas proximity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hole-gas proximity: no simple s-wave","Germanium holes get f-type pairing via proximity","Anisotropic cubic pairing emerges in hole-gas hybrids","Momentum-dependent pairing from hole proximity effect","Bogoliubov Fermi surfaces predicted for hole-gas proximity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000978,"raw_usage":{"total_tokens":4190,"prompt_tokens":1018,"completion_tokens":3172,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":3097}},"tokens_in":634,"tokens_out":3172,"duration_ms":22961,"temperature":1.0,"reasoning_tokens":3097,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:59:00.259720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dimoulas, P","cited_arxiv_id":null,"evidence_quote":"Supplies the Schrieffer-Wolff transformation and the 4-band Kohn-Luttinger framework used to derive the effective proximitized hole theory."},{"cited_title":"Laubscher, J","cited_arxiv_id":null,"evidence_quote":"Prior model adding constant pairings to both heavy- and light-hole bands; the paper extends and benchmarks against it."},{"cited_title":"Moghaddam, T","cited_arxiv_id":null,"evidence_quote":"Experimental detection of induced pairing via kinetic inductance in an Al-InAs hybrid, cited as the method to probe Bogoliubov-Fermi surfaces in the hole system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Experimental observation of segmented Fermi surface induced by Cooper pair momentum in a proximitized topological insulator, supporting the gapless-regime interpretation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Zeeman-induced gapless superconductivity and partial-Fermi-surface framework used to interpret the large-field behavior."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proximitized 2D Rashba model whose density-of-states signatures the hole-gas results generalize with anisotropy and tilting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Parallel derivation of the tunneling-induced constant pairings and justification that integrating out the superconductor yields the constant pairing terms used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Ge band-structure parameters used in the numerics and analytical estimates."}],"review_version":1}