Imaging geometry- and phase-controlled spectra in a surface-state Andreev cavity

Adrian Greichgauer; Jens Brede; Yoichi Ando

arxiv: 2606.28189 · v1 · pith:UJRWBOXEnew · submitted 2026-06-26 · ❄️ cond-mat.supr-con · cond-mat.mes-hall

Imaging geometry- and phase-controlled spectra in a surface-state Andreev cavity

Adrian Greichgauer , Yoichi Ando , Jens Brede This is my paper

Pith reviewed 2026-06-29 02:04 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mes-hall

keywords Andreev cavitysurface statesscanning tunneling spectroscopysuperconducting proximity effectsemiclassical trajectoriesvortex phase

0 comments

The pith

Geometry of Andreev trajectories controls the magnetic-field scale and zero-field energies of low-energy spectra in surface-state superconducting cavities.

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

The paper shows that confined Cu(111) surface states on superconducting Nb(110) form Andreev cavities where electron-hole trajectories are shaped by the island geometry. Scanning tunneling spectra reveal that the in-plane field strength at which resolved low-energy features collapse scales with the transverse width available to those trajectories, while the zero-field excitation energy scales with the characteristic path length. Spatial maps inside islands and the effect of vortex phase textures both follow the same trends. A minimal semiclassical model of phase accumulation along the trajectories accounts for all these observations.

Core claim

In these Andreev cavities the in-plane magnetic-field scale for the collapse of the resolved low-energy spectrum is set by the transverse extent available to Andreev trajectories, while the zero-field excitation energy evolves with the characteristic trajectory length; both trends, together with intra-island spatial variations and the response to vortex phase textures, are reproduced by a minimal semiclassical phase-accumulation picture.

What carries the argument

Minimal semiclassical phase-accumulation picture along geometry-defined Andreev trajectories

Load-bearing premise

The measured spectra are produced mainly by phase-coherent electron-hole motion along trajectories whose lengths and widths are fixed by the island shape, with little contribution from disorder or full quantum interference.

What would settle it

Finding that the collapse field or zero-field energy shows no systematic dependence on measured island dimensions, or deviates strongly from the semiclassical phase-accumulation prediction across multiple islands.

Figures

Figures reproduced from arXiv: 2606.28189 by Adrian Greichgauer, Jens Brede, Yoichi Ando.

**Figure 1.** Figure 1: Island geometry and the magnetic-field evolution of a proximitised Cu(111) surface state. a, Threedimensional rendering of the STM topography of a Cu(111) island on Nb(110). The wavy arrow marks boundary surface–bulk scattering, giving the confined surface state an effective gap scale Δeff. The dashed black line illustrates a trajectory associated with the length scale ℓ. An in-plane field 𝐵𝑦 induces a su… view at source ↗

**Figure 2.** Figure 2: Dynamical and magnetic-field-induced phase scales across different Cu(111) islands. a–c, Representative background-subtracted topographies of Cu(111) islands of different size and shape for magnetic field applied along 𝐵𝑦, shown with a common scale bar. Crosses mark the spectroscopy positions. Grey shading represents the field-induced superconducting phase gradient as in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 3.** Figure 3: a shows the largest island studied, with two linespectroscopy paths probing complementary aspects of the local geometry. This provides an internal test of the geometric picture: within a single island, the local trajectory ensemble changes with tip position while the material parameters and global island boundary remain fixed. For each tip position 𝑟, the local ray-traced ensemble is recalculated from th… view at source ↗

**Figure 4.** Figure 4: Local spectral response to a vortex phase texture. a, Zoom into the lower region of the Cu(111) island shown in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

Andreev cavities provide a setting in which superconducting proximity spectra are shaped by phase-coherent electron-hole motion along extended trajectories. While such Andreev physics is well established in transport, local spectra in two-dimensional cavities remain largely unexplored in real space. Here we use scanning tunnelling spectroscopy to study confined Cu(111) surface states coupled to superconducting Nb(110). The in-plane magnetic-field scale for the collapse of the resolved low-energy spectrum is controlled by the transverse extent available to Andreev trajectories, while the zero-field excitation energy evolves with the characteristic trajectory length. These trends, together with spatial variations within individual islands and the response to vortex phase textures, are captured by a minimal semiclassical phase-accumulation picture. Our results identify geometry-defined Andreev trajectories as a design principle for phase-coherent superconducting cavities accessible by local spectroscopy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

STM maps show geometry setting Andreev spectra in Cu islands on Nb via semiclassical trajectories, but the data-to-model link looks only qualitative so far.

read the letter

The core result is real-space STM spectra on Cu(111) surface states proximitized by Nb(110) islands where the in-plane field scale for low-energy spectrum collapse tracks the island width and the zero-field energies track the characteristic Andreev path length. Spatial maps inside single islands and the response to vortex phase shifts also line up with the same minimal phase-accumulation picture.

The experiment itself is the clear advance: transport studies of Andreev cavities exist, but local spectroscopy that directly images how geometry sets the spectrum in two-dimensional surface states has not been shown before. The ability to vary island size and see both the field scale and the zero-field energy move accordingly is useful for anyone trying to design phase-coherent proximity structures.

The soft spot is the quantitative support for the semiclassical interpretation. The abstract states that trends are captured by the model, yet without the actual spectra, error bars, or explicit checks against disorder broadening or full BdG interference, it is not obvious that geometry-defined trajectories are the dominant mechanism rather than one possible description. The stress-test concern about alternative contributions is fair; if the match remains visual and no parameter-free predictions are tested, the design-principle claim stays under-constrained.

This is for people working on mesoscopic superconductivity and local probes of proximity effects. A reader who wants concrete examples of geometry-controlled Andreev states will get something from the figures even if the model discussion needs tightening.

It deserves peer review because the experimental platform is new for this problem and the claims are specific enough that referees can ask for the missing quantitative tests.

Referee Report

1 major / 0 minor

Summary. The manuscript presents scanning tunneling spectroscopy (STS) measurements on Cu(111) surface states proximitized by Nb(110) islands, forming Andreev cavities. It claims that the in-plane magnetic-field scale for collapse of the resolved low-energy spectrum is set by the transverse extent of Andreev trajectories, while the zero-field excitation energy scales with characteristic trajectory length. Spatial variations within islands and responses to vortex phase textures are also reported to follow these trends. All observations are stated to be captured by a minimal semiclassical phase-accumulation model, establishing geometry-defined Andreev trajectories as a design principle for phase-coherent superconducting cavities accessible via local spectroscopy.

Significance. If the central claim holds, the work demonstrates real-space control of proximity-induced spectra through island geometry in a 2D surface-state system, with direct imaging of trajectory-dependent features. The experimental use of STS to resolve both field scales and spatial variations, combined with a parameter-free semiclassical interpretation, provides a concrete route to designing Andreev cavities without relying on transport averaging. This could impact efforts to engineer phase-coherent superconducting nanostructures.

major comments (1)

[Abstract and semiclassical model section] Abstract and semiclassical model section: the load-bearing claim is that observed scalings (B-field collapse set by transverse extent; zero-field energy set by trajectory length) arise predominantly from geometry-defined phase-coherent Andreev trajectories under the minimal semiclassical picture, with negligible contributions from disorder, interface scattering, or full BdG quantum interference. The manuscript must provide a concrete argument or quantitative test showing that alternative mechanisms cannot reproduce the same geometry-dependent trends; without this, the interpretation that geometry is the dominant design principle remains under-constrained.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading and the constructive major comment. We respond point-by-point below.

read point-by-point responses

Referee: [Abstract and semiclassical model section] Abstract and semiclassical model section: the load-bearing claim is that observed scalings (B-field collapse set by transverse extent; zero-field energy set by trajectory length) arise predominantly from geometry-defined phase-coherent Andreev trajectories under the minimal semiclassical picture, with negligible contributions from disorder, interface scattering, or full BdG quantum interference. The manuscript must provide a concrete argument or quantitative test showing that alternative mechanisms cannot reproduce the same geometry-dependent trends; without this, the interpretation that geometry is the dominant design principle remains under-constrained.

Authors: We agree that the manuscript would be strengthened by an explicit discussion of why alternative mechanisms are unlikely to reproduce the observed geometry-dependent scalings. The Cu(111) surface states are known from prior STM work to be exceptionally clean, with mean free paths exceeding several hundred nanometers; the islands studied here are smaller than this length scale, and the spectra remain sharp and spatially uniform within each island, inconsistent with strong disorder broadening. Interface scattering at the Nb/Cu boundary would be expected to produce additional subgap states or broadening that varies with island perimeter rather than the specific trajectory length and transverse extent reported. Full BdG quantum interference is approximated by the semiclassical phase accumulation when the Fermi wavelength is much shorter than the trajectory length, which holds for the micron-scale cavities. In the revised manuscript we will add a dedicated paragraph in the semiclassical model section that makes these arguments and cites the relevant surface-state literature. A direct numerical comparison to full BdG simulations for the experimental island sizes remains computationally prohibitive at present. revision: partial

standing simulated objections not resolved

A quantitative test against full numerical BdG calculations for the specific experimental island geometries and sizes cannot be performed with currently available computational resources.

Circularity Check

0 steps flagged

No circularity: semiclassical interpretation applied to data without self-referential reduction

full rationale

The provided abstract and context describe experimental STM spectra interpreted via a minimal semiclassical phase-accumulation picture that qualitatively captures geometry-dependent trends in B-field collapse and zero-field energies. No equations, parameter fits to the target data, or self-citations are shown that would make any claimed prediction equivalent to its inputs by construction. The model is presented as an interpretive framework rather than a closed derivation chain, satisfying the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of a semiclassical phase-accumulation model to the confined surface-state trajectories; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)

domain assumption Semiclassical phase accumulation along Andreev trajectories accurately reproduces the observed spectral features in the 2D cavities.
Invoked in the abstract to capture magnetic-field scale, zero-field energy, spatial variations, and vortex response.

pith-pipeline@v0.9.1-grok · 5682 in / 1277 out tokens · 37976 ms · 2026-06-29T02:04:33.878719+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Imaging geometry- and phase-controlled spectra in a surface-state Andreev cavity

Kohsaka, Y. SIDAM: Analysis tools for spectroscopic imaging scanning tunneling mi- croscopy/spectroscopy.https://github.com /yuksk/SIDAM(2025). GitHub repository, ver- sion v9.8.8; accessed 2026-04-28. 10 + -20 0Δz (nm) a b 30 20 10 0 P (%) 250200150100500 ℓ n (nm) ℓ ≈ 274 nm 100 nm + e f Trajectory ensembleE 0-defining ℓ bundle d Bx + B0,x-defining w bun...

2025

[1] [1]

Nanotechnol.17, 39–44 (2022)

Baumgartner, C.et al.Supercurrent rectification andmagnetochiraleffectsinsymmetricJosephson junctions.Nat. Nanotechnol.17, 39–44 (2022). doi: 10.1038/s41565-021-01009-9

work page doi:10.1038/s41565-021-01009-9 2022

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doi: 10.1038/s41586-019-1148-9

Ren, H.et al.Topological superconductivity in a phase-controlled Josephson junction.Nature569, 93–98 (2019). doi: 10.1038/s41586-019-1148-9

work page doi:10.1038/s41586-019-1148-9 2019

[3] [3]

Pankratova, N.et al.Multiterminal Josephson effect.Phys. Rev. X10, 031051 (2020). doi: 10.1103/PhysRevX.10.031051

work page doi:10.1103/physrevx.10.031051 2020

[4] [4]

& Catelani, G

Riwar, R.-P. & Catelani, G. Efficient quasiparti- cle traps with low dissipation through gap engi- neering.Phys. Rev. B100, 144514 (2019). doi: 10.1103/PhysRevB.100.144514

work page doi:10.1103/physrevb.100.144514 2019

[5] [5]

de Gennes, P. G. & Saint-James, D. Elemen- tary excitations in the vicinity of a normal metal- superconductingmetalcontact.Phys. Lett.4,151– 152(1963). doi: 10.1016/0031-9163(63)90148-3

work page doi:10.1016/0031-9163(63)90148-3 1963

[6] [6]

Andreev, A. F. Electron spectrum of the interme- diate state of superconductors.Sov. Phys. JETP 22, 455–458 (1966)

1966

[7] [7]

Rev.175,537–542 (1968)

McMillan,W.L.Tunnelingmodelofthesupercon- ductingproximityeffect.Phys. Rev.175,537–542 (1968). doi: 10.1103/PhysRev.175.537

work page doi:10.1103/physrev.175.537 1968

[8] [8]

Kulik, I. O. Macroscopic quantization and the proximity effect in S-N-S junctions.Sov. Phys. JETP30, 944–950 (1970)

1970

[9] [9]

Spectralflow,Magnusforce,andmutual friction via the geometric optics limit of Andreev reflection.Phys

Stone,M. Spectralflow,Magnusforce,andmutual friction via the geometric optics limit of Andreev reflection.Phys. Rev. B54, 13222–13229 (1996). doi: 10.1103/PhysRevB.54.13222

work page doi:10.1103/physrevb.54.13222 1996

[10] [10]

Beenakker, C. W. J. Random-matrix theory of quantum transport.Rev. Mod. Phys.69, 731–808 (1997). doi: 10.1103/RevModPhys.69.731

work page doi:10.1103/revmodphys.69.731 1997

[11] [11]

Mandal, P.et al.Interfering trajectories in a bal- listic Andreev cavity.Phys. Rev. B113, 104517 (2026). doi: 10.1103/hls3-jqzj

work page doi:10.1103/hls3-jqzj 2026

[12] [13]

Phys.11, 332–337 (2015)

Roditchev, D.et al.Direct observation of Joseph- son vortex cores.Nat. Phys.11, 332–337 (2015). doi: 10.1038/NPHYS3240

work page doi:10.1038/nphys3240 2015

[13] [15]

I., Bergeret, F

Ortuzar, J., Pascual, J. I., Bergeret, F. S. & Cazalilla, M. A. Theory of a single magnetic impurity on a thin metal film in proximity to a superconductor.Phys. Rev. B108,024511(2023). doi: 10.1103/PhysRevB.108.024511

work page doi:10.1103/physrevb.108.024511 2023

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doi: 10.1038/s41586-023-06312-0

Schneider,L.et al.Proximitysuperconductivityin atom-by-atom crafted quantum dots.Nature621, 60–65(2023). doi: 10.1038/s41586-023-06312-0

work page doi:10.1038/s41586-023-06312-0 2023

[15] [18]

F., Lutz, C

Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to quantum corrals on a metal surface.Science262, 218–220 (1993). doi: 10.1126/science.262.5131.218

work page doi:10.1126/science.262.5131.218 1993

[16] [19]

Francica, F

Reinthaler, R. W., Tkachov, G. & Hankiewicz, E. M. Superconducting quantum spin Hall sys- tems with giant orbital𝑔factors.Phys. Rev. B92, 161303 (2015). doi: 10.1103/Phys- RevB.92.161303

work page doi:10.1103/phys- 2015

[17] [20]

CreatingMajoranamodesfrom segmented Fermi surface.Nat

Papaj,M.&Fu,L. CreatingMajoranamodesfrom segmented Fermi surface.Nat. Commun.12, 577 (2021). doi: 10.1038/s41467-020-20690-3

work page doi:10.1038/s41467-020-20690-3 2021

[18] [21]

K., Bruder, C., Schön, G

Belzig, W., Wilhelm, F. K., Bruder, C., Schön, G. & Zaikin, A. D. Quasiclassical Green’s function approachtomesoscopicsuperconductivity.Super- lattices Microstruct.25, 1251–1288 (1999). doi: 10.1006/spmi.1999.0710

work page doi:10.1006/spmi.1999.0710 1999

[19] [22]

Adv.11, eadw4898 (2025)

Nikodem,E.et al.Tunablesuperconductingdiode effect in a topological nano-SQUID.Sci. Adv.11, eadw4898 (2025). doi: 10.1126/sciadv.adw4898

work page doi:10.1126/sciadv.adw4898 2025

[20] [23]

& Zirnbauer, M

Altland, A. & Zirnbauer, M. R. Random ma- trix theory of a chaotic Andreev quantum dot. Phys. Rev. Lett.76, 3420–3423 (1996). doi: 10.1103/PhysRevLett.76.3420

work page doi:10.1103/physrevlett.76.3420 1996

[21] [24]

Piquero-Zulaica, I.et al.Engineering quantum states and electronic landscapes through surface molecular nanoarchitectures.Rev. Mod. Phys. 94, 045008 (2022). doi: 10.1103/RevMod- Phys.94.045008

work page doi:10.1103/revmod- 2022

[22] [25]

Razinkin, A. S. & Kuznetsov, M. V. Scanning tunneling microscopy (STM) of low-dimensional NbO structures on the Nb(110) surface.Phys. Met. Metallogr.110, 531–541 (2010). doi: 10.1134/S0031918X10120033

work page doi:10.1134/s0031918x10120033 2010

[23] [26]

W.et al.Field-directed sput- ter sharpening for tailored probe materials and atomic-scale lithography.Nat

Schmucker, S. W.et al.Field-directed sput- ter sharpening for tailored probe materials and atomic-scale lithography.Nat. Commun.3, 935 (2012). doi: 10.1038/ncomms1907

work page doi:10.1038/ncomms1907 2012

[24] [27]

Imaging geometry- and phase-controlled spectra in a surface-state Andreev cavity

Kohsaka, Y. SIDAM: Analysis tools for spectroscopic imaging scanning tunneling mi- croscopy/spectroscopy.https://github.com /yuksk/SIDAM(2025). GitHub repository, ver- sion v9.8.8; accessed 2026-04-28. 10 + -20 0Δz (nm) a b 30 20 10 0 P (%) 250200150100500 ℓ n (nm) ℓ ≈ 274 nm 100 nm + e f Trajectory ensembleE 0-defining ℓ bundle d Bx + B0,x-defining w bun...

2025