Revisiting Theoretical Modeling of Seebeck Coefficient of Semiconductors

Yuchao Hua

arxiv: 2606.21879 · v1 · pith:OYLFA7I4new · submitted 2026-06-20 · ⚛️ physics.app-ph · cond-mat.mtrl-sci

Revisiting Theoretical Modeling of Seebeck Coefficient of Semiconductors

Yuchao Hua This is my paper

Pith reviewed 2026-06-26 11:15 UTC · model grok-4.3

classification ⚛️ physics.app-ph cond-mat.mtrl-sci

keywords Seebeck coefficientsemiconductorsdrift-diffusion equationSoret effectreversible modelphonon-dragthermoelectric propertiesn-doped silicon

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The pith

A reversible model recovered from the drift-diffusion equation with Soret effect supplies the theoretical lower bound on the Seebeck coefficient and matches data for highly n-doped silicon.

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

Common closed-form expressions for the Seebeck coefficient contain thermodynamic inconsistencies, such as mixing electrochemical and electrical potentials or mishandling spatial gradients of the distribution function. The paper instead starts from the drift-diffusion equation that includes the Soret effect and imposes the open-circuit condition. In the near-equilibrium limit this procedure recovers a reversible model for the band contribution that drops all electron velocity and relaxation terms while leaving the phonon-drag term identical to the Boltzmann-transport result. The resulting expression is shown to be the theoretical minimum value of the coefficient and produces close agreement with measured values on heavily doped n-type silicon.

Core claim

The reversible model recovered from the drift-diffusion equation with Soret effect in the open-circuit near-equilibrium case eliminates the electron velocity and relaxation terms, supplies the theoretical lower bound on the Seebeck coefficient, and yields fairly good predictions for experimental data on highly n-doped silicon.

What carries the argument

Reversible model recovered from drift-diffusion equation with Soret effect under open-circuit near-equilibrium conditions, which removes velocity and relaxation terms for the band contribution while retaining the phonon-drag term in BTE form.

If this is right

The reversible model constitutes the theoretical lower bound for the Seebeck coefficient.
The phonon-drag contribution retains exactly the same form as the one obtained from the Boltzmann transport equation.
For highly n-doped silicon the reversible model produces predictions that agree with published experimental values.
Because the model is thermodynamically consistent, algebraically simple, and computationally inexpensive, it offers a practical alternative for evaluating thermoelectric properties when the system remains near equilibrium and reversible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same drift-diffusion starting point could be retained without the near-equilibrium assumption to generate a velocity-dependent correction that quantifies how far real devices depart from the lower bound.
Because the phonon-drag term matches the BTE expression exactly, any future refinement of phonon-drag scattering mechanisms can be inserted directly into the reversible framework without re-deriving the entire coefficient.
The lower-bound property suggests that measured Seebeck values in other n-type semiconductors should lie above the reversible prediction once the same open-circuit near-equilibrium conditions are enforced.

Load-bearing premise

The open-circuit condition together with the near-equilibrium limit permits removal of velocity and relaxation terms while preserving thermodynamic consistency and experimental accuracy.

What would settle it

Measurement of a Seebeck coefficient in highly n-doped silicon, under open-circuit near-equilibrium conditions, that lies below the value given by the reversible model.

read the original abstract

We revisit the closed-form models of Seebeck coefficient, and identify the thermodynamic flaws in those extensively-used methodologies, essentially including the confusion of electrochemical potential and electrical potential within the definition, arbitrarily neglecting the dependence on conduction band bottom and Fermi level when calculating distribution function spatial gradient, and the improper heat flux presentation. Here, an alternative methodology is presented, which derives the Seebeck coefficient model based on the drift-diffusion equation with the Soret effect in the open circuit condition. The reversible model that eliminates the electron velocity and relaxation terms in the formulation is recovered for the band term in the near-equilibrium case, and it can give the lower bound of Seebeck coefficient in theory, while the phonon-drag term is of the identical form to the Boltzmann transport equation(BTE)-based one. A case study is performed for highly-n-doped Silicon, where the reversible, ballistic(based on Landauer formulation) and BTE models are compared with each other and with the experimental data. The reversible model gives the fairly-good predictions for the experiments. Given its theoretical-soundness, simple form and low computation cost, the reversible model can serve as a concise alternative for evaluating thermoelectric properties in both the reversible and near-equilibrium cases.

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

The paper recovers a reversible lower-bound Seebeck model from drift-diffusion plus Soret under open-circuit near-equilibrium conditions and shows it matches n-doped silicon data without parameters.

read the letter

The main takeaway is that the authors start from the drift-diffusion equation with the Soret effect, apply the open-circuit condition, and in the near-equilibrium limit drop the electron velocity and relaxation terms to recover a reversible model for the band contribution to the Seebeck coefficient. They position this as the theoretical lower bound while keeping the phonon-drag term in the same form as the BTE result. The silicon case study then compares this reversible expression, a Landauer ballistic model, and the BTE against experimental data on highly n-doped material, where the reversible version performs reasonably with no free parameters.

What is actually new is the explicit framing of the derivation as a thermodynamic correction to prior closed-form models, with the listed flaws (mixing of electrochemical and electrical potential, neglect of band-bottom and Fermi-level dependence in the distribution gradient, and improper heat flux) addressed by staying inside the drift-diffusion plus Soret framework. The work does well at keeping the algebra tied to standard transport equations and at delivering a low-cost expression that still produces usable predictions for the silicon benchmark.

The soft spots are limited. The lower-bound property is demonstrated only inside the stated open-circuit near-equilibrium limit, so it would be useful to see how far the claim extends when those conditions are relaxed. The experimental agreement is described as “fairly good,” but the abstract supplies no quantitative error numbers, which makes it harder to judge the practical improvement over existing expressions. No circularity or fitted constants appear in the description.

This is for researchers who model thermoelectric transport in semiconductors and want a concise, grounded alternative to full BTE calculations. A reader working on energy materials would find the derivation and the direct silicon comparison useful. The paper deserves a serious referee because the starting equations are standard, the silicon test is concrete and parameter-free, and the claims are checkable against existing data and derivations.

Referee Report

2 major / 1 minor

Summary. The paper identifies thermodynamic flaws in extensively-used closed-form models of the Seebeck coefficient, including confusion between electrochemical and electrical potentials, neglecting dependence on conduction band bottom and Fermi level in distribution function gradients, and improper heat flux presentation. It presents an alternative derivation based on the drift-diffusion equation with the Soret effect under open-circuit conditions. In the near-equilibrium case, it recovers a reversible model for the band term that eliminates electron velocity and relaxation terms, providing a theoretical lower bound on the Seebeck coefficient, while the phonon-drag term matches the BTE-based form. A case study on highly n-doped silicon compares the reversible, ballistic (Landauer), and BTE models to experimental data, with the reversible model providing fairly good predictions.

Significance. If the central derivation holds, this work provides a thermodynamically sound, simple, and computationally efficient model for the Seebeck coefficient that serves as a lower bound in reversible near-equilibrium cases. The parameter-free nature and favorable comparison to experimental data on n-doped silicon suggest it could be a useful concise alternative for evaluating thermoelectric properties, addressing flaws in prior models.

major comments (2)

[Abstract] Abstract: the claim that the open-circuit near-equilibrium limit recovers the reversible model by exact removal of velocity and relaxation terms from the drift-diffusion equation with Soret effect is asserted without supplying the explicit derivation steps or the final expression; this is load-bearing for the central claim and requires detailed algebra to verify thermodynamic consistency and the lower-bound property.
[Case study on highly-n-doped Silicon] Case study on highly-n-doped Silicon: quantitative error metrics (e.g., mean absolute percentage error or R²) comparing the reversible model's predictions to experimental data on n-doped Si are not supplied, preventing rigorous assessment of the 'fairly-good' agreement and direct comparison to the ballistic and BTE models.

minor comments (1)

Clarify whether the recovered reversible model applies strictly to the band term or to the full Seebeck coefficient expression (including phonon-drag).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the open-circuit near-equilibrium limit recovers the reversible model by exact removal of velocity and relaxation terms from the drift-diffusion equation with Soret effect is asserted without supplying the explicit derivation steps or the final expression; this is load-bearing for the central claim and requires detailed algebra to verify thermodynamic consistency and the lower-bound property.

Authors: We agree that the abstract states the recovery of the reversible model without the explicit algebra. The main text derives the model from the drift-diffusion equation with Soret effect under open-circuit conditions, but to allow direct verification of the term cancellation and lower-bound property we will add an appendix containing the full algebraic steps and the final expression for the reversible band term. revision: yes
Referee: [Case study on highly-n-doped Silicon] Case study on highly-n-doped Silicon: quantitative error metrics (e.g., mean absolute percentage error or R²) comparing the reversible model's predictions to experimental data on n-doped Si are not supplied, preventing rigorous assessment of the 'fairly-good' agreement and direct comparison to the ballistic and BTE models.

Authors: We agree that quantitative metrics would permit a more objective comparison. In the revised manuscript we will report mean absolute percentage error and R² values for the reversible, ballistic, and BTE predictions against the n-doped silicon experimental data. revision: yes

Circularity Check

0 steps flagged

Derivation from standard drift-diffusion/Soret equations shows no reduction to inputs or self-citation chain

full rationale

The paper states that its Seebeck model is obtained by applying the open-circuit condition and near-equilibrium limit to the drift-diffusion equation with Soret effect, which removes velocity and relaxation terms to recover the reversible band term. This step is presented as a direct consequence of the stated boundary conditions rather than a redefinition or fit. No load-bearing self-citations, uniqueness theorems from prior author work, or fitted parameters renamed as predictions appear in the abstract or described derivation. The phonon-drag term is noted to match BTE form by comparison, and the n-doped Si case study is described as parameter-free. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the drift-diffusion framework augmented by the Soret term and on the legitimacy of taking the near-equilibrium open-circuit limit to drop velocity and relaxation contributions. No free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The drift-diffusion equation augmented by the Soret effect correctly describes carrier transport in the semiconductors and conditions considered.
Invoked as the starting point for the alternative methodology.
domain assumption The open-circuit near-equilibrium limit permits elimination of electron velocity and relaxation terms without violating thermodynamic consistency.
Required to recover the reversible band term claimed to be the lower bound.

pith-pipeline@v0.9.1-grok · 5738 in / 1517 out tokens · 40267 ms · 2026-06-26T11:15:09.851822+00:00 · methodology

discussion (0)

Reference graph

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