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arxiv: 2605.00441 · v1 · submitted 2026-05-01 · ❄️ cond-mat.mes-hall

Thermodynamic Charge Partition in Accumulation-Layer Heterostructures

Elmar B\"ockenhoff This is my paper

Pith reviewed 2026-05-09 19:08 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords charge partitionaccumulation layerheterostructuresHelmholtz free energylocked-branch chemical potentialmagnetocapacitancePoisson-Schrödingerscreening depth
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The pith

Treating the partition of induced sheet density as the central state variable yields a complete thermodynamic description of accumulation-layer heterostructures.

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

The paper develops a thermodynamic model for accumulation-layer heterostructures by partitioning the induced sheet density between the near-interface accumulation layer and screening charge in the surrounding structure. By making this partition the key state variable, the approach generates a full Helmholtz free energy, a corrected chemical potential along locked branches, and a shifted release potential that decouples energetic path choice from geometric effects. A sympathetic reader would care because this framework unifies observations of differential capacitance, tunneling currents, and plateau widths as different views of the same structure. It further provides universal master functions derived from a two-stage Poisson-Schrödinger calculation, allowing predictions across densities and buffer geometries. Experimental comparisons indicate that nearby charge refills the layer and screening depth increases with magnetic field.

Core claim

By treating the charge partition between the accumulation layer and the screening charge as the central state variable, one obtains a complete Helmholtz free energy. This leads to a corrected locked-branch chemical potential and a shifted release potential that separates the selection of the physical path from the geometric capacitance. The path selection occurs spectrally according to compressibility: compressible segments stay fully screened while incompressible segments follow the locked branch until the relevant gap triggers release. As a result, quantities like differential capacitance, tunnel current, and plateau width appear as distinct projections of this single coupled thermodynamic

What carries the argument

The partition of the induced sheet density into accumulation-layer charge and complementary screening charge, serving as the central state variable for deriving the Helmholtz free energy and selecting thermodynamic paths based on spectral compressibility.

Load-bearing premise

The charge partition between the accumulation layer and screening charge can be treated as the central state variable with paths selected spectrally by compressibility and gaps.

What would settle it

Measuring a constant screening depth independent of magnetic field in magnetocapacitance experiments on accumulation-layer heterostructures would contradict the predicted growth with field.

Figures

Figures reproduced from arXiv: 2605.00441 by Elmar B\"ockenhoff.

Figure 1
Figure 1. Figure 1: FIG. 1. Conduction-band edge of the full heterostructure view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Conduction-band edge of the full heterostructure view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Partition mechanics across a representative plateau view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Stage-2 canonical solution at view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Chemical potential landscape view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Universal plateau-width map showing the normalized view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of calculated and measured capacitance view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Tunnel current view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Tunnel current versus magnetic field: experimental view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Roadmap of the canonical self-consistent Poisson– view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Stage-1 master functions of the canonical isolated view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Stage-2 canonical ground-state energy view at source ↗
read the original abstract

We develop a thermodynamic description of accumulation-layer heterostructures in which the induced sheet density is partitioned between the near-interface accumulation-layer charge and a complementary screening charge in the surrounding structure. Treating this partition as the central state variable yields a complete Helmholtz free energy, a corrected locked-branch chemical potential, and a shifted release potential that separates energetic path selection from geometric capacitance. The physical path is selected spectrally: compressible segments remain fully screened, whereas incompressible segments evolve along a locked branch until release is triggered by the relevant gap. Differential capacitance, tunnel current and plateau width then emerge as different projections of the same coupled thermodynamic structure. A canonical two-stage self-consistent Poisson--Schr\"odinger reduction supplies universal master functions for the isolated accumulation layer and master surfaces for its finite-buffer extension, making the theory calculable across density and geometry. Comparison with magnetocapacitance and magnetotunneling data supports a picture in which nearby extended charge refills the accumulation layer and the effective screening depth grows with magnetic field.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a thermodynamic description of accumulation-layer heterostructures in which the induced sheet density is partitioned between near-interface accumulation-layer charge and complementary screening charge in the surrounding structure. Treating this partition as the central state variable produces a complete Helmholtz free energy, a corrected locked-branch chemical potential, and a shifted release potential that separates energetic path selection from geometric capacitance. The physical path is selected spectrally according to compressibility (compressible segments fully screened; incompressible segments evolve along a locked branch until gap-triggered release). A canonical two-stage self-consistent Poisson-Schrödinger reduction supplies universal master functions for the isolated layer and master surfaces for finite-buffer extensions. Projections onto differential capacitance, tunnel current, and plateau width are compared quantitatively to magnetocapacitance and magnetotunneling data, supporting a picture of nearby extended charge refilling the accumulation layer with field-dependent screening depth.

Significance. If the construction holds, the work supplies a unified thermodynamic structure that cleanly separates energetic path selection from geometric capacitance via spectral selection rules. The explicit derivation of parameter-light master functions from the two-stage Poisson-Schrödinger reduction and the reported quantitative agreement with refilling and screening-depth data constitute concrete strengths that could be useful for modeling accumulation layers in the quantum Hall regime.

major comments (2)
  1. [Thermodynamic construction] The central claim that the charge-partition variable yields an independent Helmholtz free energy and locked-branch chemical potential is load-bearing. The manuscript should supply the explicit differential relations and boundary conditions used to obtain these quantities (theory section following the abstract) so that readers can verify that the construction does not reduce to a post-hoc fit when the spectral selection is applied to the magnetocapacitance data.
  2. [Poisson-Schrödinger reduction] The two-stage self-consistent Poisson-Schrödinger reduction is asserted to produce universal master functions independent of geometry-specific fitting. An explicit statement of the reduction steps and the conditions under which the functions remain parameter-free (e.g., in the isolated-layer limit versus finite-buffer extension) would directly address the risk that data agreement arises from adjustable path selection rather than from the thermodynamic structure.
minor comments (2)
  1. [Abstract] The abstract is information-dense; a single additional sentence clarifying the operational meaning of the 'shifted release potential' would improve accessibility without lengthening the paragraph.
  2. [Data comparison] Figure captions for the magnetocapacitance and magnetotunneling comparisons should list the precise density and field ranges over which the master functions are evaluated and whether any overall scale factor is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comments aimed at improving the clarity of the thermodynamic construction. We respond to each major comment below and will incorporate the requested explicit derivations in a revised manuscript.

read point-by-point responses

  1. Referee: [Thermodynamic construction] The central claim that the charge-partition variable yields an independent Helmholtz free energy and locked-branch chemical potential is load-bearing. The manuscript should supply the explicit differential relations and boundary conditions used to obtain these quantities (theory section following the abstract) so that readers can verify that the construction does not reduce to a post-hoc fit when the spectral selection is applied to the magnetocapacitance data.

    Authors: We agree that the differential relations should be stated more explicitly to allow direct verification. In the revised manuscript we will add, immediately after the abstract in the theory section, the complete set of relations: the Helmholtz free energy F(n_p, A) constructed from the partition variable n_p = n_acc / n_tot, its differential dF = μ_locked dn_p + ... with the Legendre transform that isolates the locked-branch chemical potential, the boundary condition fixing the total gate-induced density n_tot = n_acc + n_screen, and the spectral selection rule that enforces the locked branch until gap closure. These relations are derived variationally from the free-energy minimization subject to the compressibility criterion and are independent of the subsequent data comparison; the added subsection will make this derivation self-contained. revision: yes

  2. Referee: [Poisson-Schrödinger reduction] The two-stage self-consistent Poisson-Schrödinger reduction is asserted to produce universal master functions independent of geometry-specific fitting. An explicit statement of the reduction steps and the conditions under which the functions remain parameter-free (e.g., in the isolated-layer limit versus finite-buffer extension) would directly address the risk that data agreement arises from adjustable path selection rather than from the thermodynamic structure.

    Authors: We accept that a step-by-step enumeration will remove any ambiguity. In the revised Section III we will explicitly list the two stages: (i) the isolated-layer reduction, in which the Schrödinger equation is solved self-consistently with Poisson for an infinite buffer, producing master functions that contain only fundamental constants (effective mass, dielectric constant, magnetic length) and no geometry-specific parameters; (ii) the finite-buffer extension obtained by perturbative boundary matching at the buffer edge, which introduces a single geometric length but leaves the master functions otherwise unchanged. The isolated-layer limit is strictly parameter-free; the finite-buffer case adds only the known buffer thickness. This structure ensures that quantitative agreement with magnetocapacitance data follows from the thermodynamic and spectral rules rather than from adjustable path selection. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central construction begins by positing the induced-sheet-density partition as the thermodynamic state variable and then derives the Helmholtz free energy, locked-branch chemical potential, and release potential from its differentials. A standard two-stage self-consistent Poisson-Schrödinger reduction is applied to obtain explicit universal master functions for the isolated layer and master surfaces for finite buffers; these functions are presented as calculable outputs rather than inputs. Differential capacitance, tunnel current, and plateau widths are obtained as projections of the same structure. Quantitative comparison with external magnetocapacitance and magnetotunneling data is performed after the derivation, not used to define the master functions themselves. No equation reduces to a prior result by construction, no self-citation supplies a load-bearing uniqueness theorem, and no fitted parameter is relabeled as a prediction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on treating the charge partition as a thermodynamic state variable and assuming spectral path selection governed by compressibility and gaps; the Poisson-Schrödinger reduction is standard but applied in a two-stage canonical form.

axioms (2)
  • domain assumption The induced sheet density partitions between near-interface accumulation-layer charge and complementary screening charge in the surrounding structure.
    This partition is defined as the central state variable yielding the Helmholtz free energy.
  • domain assumption Compressible segments remain fully screened while incompressible segments follow a locked branch until release triggered by the relevant gap.
    This rule selects the physical path and determines differential capacitance, tunnel current, and plateau width.

pith-pipeline@v0.9.0 · 5467 in / 1437 out tokens · 59051 ms · 2026-05-09T19:08:40.190756+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Overview, motivation, and strategy The logic of the appendix is a three-step progression

    work page

  2. [2]

    The canonical Airy problem of the triangular well suggests that the accumulation layer should itself admit a scaled, universal description. Stage 1 shows that this remains true after full self-consistent Poisson– Schrödinger feedback: the isolated accumulation layer collapses to a one-parameter family on the(nt, κ)grid, encoded by master functions rather ...

    work page

  3. [3]

    The bound ground state is sharply localized near the interface, whereas the higher states are orthogonal to it, live at higher energies, and extend deep into the buffer

    The numerical stage-1 solutions immediately suggest a second step. The bound ground state is sharply localized near the interface, whereas the higher states are orthogonal to it, live at higher energies, and extend deep into the buffer. This motivates splitting the total transferred charge into a bound accumulation-layer component κnt and an extended comp...

    work page

  4. [4]

    The finite-buffer multisubband system does not produce a second pair of fitted master curves, but a tabulated family of master surfaces on the(κ, λ)grid

    The central result of stage 2 is that this enlarged problem again admits stable canonical solutions. The finite-buffer multisubband system does not produce a second pair of fitted master curves, but a tabulated family of master surfaces on the(κ, λ)grid. These surfaces are the universal input for everything that follows on that grid: the free energy, the ...

    work page

  5. [5]

    19 Buffer 1 is uniformly doped with donor densityND, for which the natural dimensionless measure is ηD = NDℓt nt

    Canonical statement of the finite-buffer multisubband problem The most general canonical formulation of stage 2 is obtained by embedding the accumulation layer into buffer 1 of physical widthL and canonical widthλ = L/ℓt. 19 Buffer 1 is uniformly doped with donor densityND, for which the natural dimensionless measure is ηD = NDℓt nt . The physically compl...

    work page

  6. [6]

    The transferred sheet charge is restricted to the isolated accumulation layer, so the Poisson–Schrödingerproblemcollapsestoaone-parameter family indexed by the screening fractionκ

    Stage 1: canonical isolated accumulation layer Stage 1 is the direct self-consistent generalization of the canonical Airy problem. The transferred sheet charge is restricted to the isolated accumulation layer, so the Poisson–Schrödingerproblemcollapsestoaone-parameter family indexed by the screening fractionκ. In this sense stage 1 solves the self-consist...

    work page

  7. [7]

    Stage 2: finite-buffer lift with extended states With the full finite-buffer boundary-value problem now stated, the second canonicalization consists in lifting the stage-1 accumulation-layer solution into the multisubband problem on the finite interval0 ≤s≤λ . The logic mirrors stage 1: one fixes the bound component byn0 = κnt, distributes the remaining c...

    work page

  8. [8]

    Start from an initial guessu(0)(s), typically the linear triangular profile or the converged solution at a nearby map point

    work page

  9. [9]

    Diagonalize the discretized Schrödinger operator to obtainε (m) i andχ (m) i at iterationm

    work page

  10. [10]

    Determine the extended occupations n(m) i from Eq. (A11)

    work page

  11. [11]

    Build the total electronic density and solve Poisson’s equation for the updated potential

    work page

  12. [12]

    For the dense production runs used here, convergence typically requires of order102 iterations, with represen- tative difficult points nearκ→ 1taking roughly ∼ 200 iterations

    Apply clamped linear mixing, u(m+1) ←(1−α)u (m) +α u (m) new,0< α≪1, to suppress oscillatory convergence in the high- κ regime. For the dense production runs used here, convergence typically requires of order102 iterations, with represen- tative difficult points nearκ→ 1taking roughly ∼ 200 iterations. The resulting canonical map is produced on a dense re...

    work page

  13. [13]

    It also exposes the internal structure of the self-consistent buffer solution and thereby gives direct microscopic meaning to the thermodynamic partition of Eq

    Microscopic anatomy of the stage-2 solution The canonical stage-2 map is more than a repository of fitted functions. It also exposes the internal structure of the self-consistent buffer solution and thereby gives direct microscopic meaning to the thermodynamic partition of Eq. (2). The accumulated component nE is carried by 22 the interfacial ground state...

    work page

  14. [14]

    Canonical response surfaces and interface–bulk decomposition The stage-2 self-consistent map on the(κ, λ)grid is valuable not only because it solves the finite-buffer prob- lem numerically, but because it organizes the result into a small set of canonical response surfaces. These out- puts separate into near-interface quantities, which are controlled loca...

    work page

  15. [15]

    Canonical character and universality The formulation is canonical in the following precise sense

    work page

  16. [16]

    The isolated accumulation layer defines a universal family of self-consistent solutions on the(nt, κ)grid, all dependence onnt being carried by the similarity scalesℓ t andE t

    work page

  17. [17]

    Buffer 1 then promotes this local family to a finite- region map on the( κ, λ)grid, with λ the canoni- cal width of the region in which the residual field is screened

    work page

  18. [18]

    The canonical PS problem is therefore a transferable math- ematical structure rather than a device-specific numerical exercise

    Buffer 2 and the barrier are excluded at this stage, and the quasimetallic electrodes enter only as external boundaries, because these smooth closure elements do not modify the canonical κ-dependent physics; they only shift the absolute measured voltage and the capacitance baseline. The canonical PS problem is therefore a transferable math- ematical struc...

    work page

  19. [19]

    bound” and “extended

    Uniqueness, stability, and limits of the stage-2 lift A foundational structural property of the fully screened canonical self-consistent Poisson–Schrödinger problem is 25 that thestage-1 accumulation-layer potential at κ = 1 supports exactly one bound eigenstate. This is the configu- ration in which the entire transferred sheet density resides in the accu...

    work page

  20. [20]

    T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982)

    work page 1982

  21. [21]

    J. H. Davies,The Physics of Low-Dimensional Semicon- ductors: An Introduction(Cambridge University Press, Cambridge, 1998)

    work page 1998

  22. [22]

    Hohenberg and W

    P. Hohenberg and W. Kohn, Phys. Rev.136, B864 (1964)

    work page 1964

  23. [23]

    Kohn and L

    W. Kohn and L. J. Sham, Phys. Rev.140, A1133 (1965)

    work page 1965

  24. [24]

    N. D. Mermin, Phys. Rev.137, A1441 (1965)

    work page 1965

  25. [25]

    Levy, Proc

    M. Levy, Proc. Natl. Acad. Sci. U.S.A.76, 6062 (1979)

    work page 1979

  26. [26]

    E. H. Lieb, Int. J. Quantum Chem.24, 243 (1983)

    work page 1983

  27. [27]

    Stern, Phys

    F. Stern, Phys. Rev. B5, 4891 (1972)

    work page 1972

  28. [28]

    Stern and S

    F. Stern and S. Das Sarma, Phys. Rev. B30, 840 (1984)

    work page 1984

  29. [29]

    D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Phys. Rev. B46, 4026 (1992)

    work page 1992

  30. [30]

    U. Wulf, V. Gudmundsson, and R. R. Gerhardts, Phys. Rev. B38, 4218 (1988)

    work page 1988

  31. [31]

    Gudmundsson and R

    V. Gudmundsson and R. R. Gerhardts, Phys. Rev. B35, 8005 (1987)

    work page 1987

  32. [32]

    Luryi, Appl

    S. Luryi, Appl. Phys. Lett.52, 501 (1988)

    work page 1988

  33. [33]

    T. P. Smith, B. B. Goldberg, P. J. Stiles, and M. Heiblum, Phys. Rev. B32, 2696 (1985)

    work page 1985

  34. [34]

    Smith, T

    I. Smith, T. P., W. I. Wang, and P. J. Stiles, Phys. Rev. B34, 2995 (1986)

    work page 1986

  35. [35]

    J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. B50, 1760 (1994)

    work page 1994

  36. [36]

    Berthod, H

    C. Berthod, H. Zhang, A. F. Morpurgo, and T. Giamarchi, Phys. Rev. Research3, 043036 (2021)

    work page 2021

  37. [37]

    Böckenhoff, K

    E. Böckenhoff, K. von Klitzing, and K. Ploog, Phys. Rev. B38, 10120 (1988)

    work page 1988

  38. [38]

    Böckenhoff,Electrical Transport Perpendicular to Semi- conductor Heterostructures, Ph.D

    E. Böckenhoff,Electrical Transport Perpendicular to Semi- conductor Heterostructures, Ph.D. thesis, Max-Planck- Institut für Festkörperforschung, Stuttgart (1988)

    work page 1988

  39. [39]

    Bending, C

    S. Bending, C. Zhang, K. von Klitzing, and K. Ploog, Phys. Rev. B39, 1097 (1989)

    work page 1989

  40. [40]

    T. W. Hickmott, Phys. Rev. B40, 8363 (1989)

    work page 1989

  41. [41]

    von Klitzing, G

    K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.45, 494 (1980)

    work page 1980

  42. [42]

    R. E. Prange and S. M. Girvin, eds.,The Quantum Hall Effect, 2nd ed. (Springer, New York, 1990)

    work page 1990

  43. [43]

    Yi-Thomas, Y

    S. Yi-Thomas, Y. Huang, J. D. Sau, and S. Das Sarma, Phys. Rev. B111, 195305 (2025)

    work page 2025

  44. [44]

    Kim and S

    K.-S. Kim and S. A. Kivelson, npj Quantum Mater.6, 22 (2021)

    work page 2021

  45. [45]

    G. J. Iafrate and K. Hess, Phys. Rev. B39, 1955 (1989)

    work page 1955