🔬 applied physics

Revisiting Theoretical Modeling of Seebeck Coefficient of Semiconductors

This paper identifies thermodynamic flaws in existing Seebeck coefficient models and proposes a theoretically sound, low-cost reversible model based on the drift-diffusion equation with the Soret effect, which accurately predicts experimental data for highly n-doped silicon while offering a concise alternative for thermoelectric property evaluation.

Original authors: Yuchao Hua

Published 2026-06-23

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Yuchao Hua

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long, narrow hallway (a semiconductor) connecting a hot room and a cold room. You want to know how much "push" or voltage is generated just by the temperature difference between the two ends. In the world of physics, this push is called the Seebeck coefficient. It's the engine behind thermoelectric generators, which turn heat directly into electricity.

For a long time, scientists have tried to write a perfect math formula to predict this push. However, the author of this paper, Yu-Chao Hua, argues that the old formulas have some serious logical "bugs" in their design.

Here is a simple breakdown of what the paper says, using everyday analogies:

1. The Problem: The Old Maps Were Wrong

Think of the old methods for calculating this voltage as using a map with three major errors:

Mixing Up the Currency: Imagine trying to pay for a coffee using your bank balance (electrochemical potential) but the cashier only accepts cash in your pocket (electrical potential). The old formulas often confused these two distinct things, leading to incorrect calculations.
Ignoring the Moving Floor: Imagine walking on a treadmill that is slowly changing speed and height as you walk. The old formulas assumed the floor was flat and static, ignoring how the "bottom of the energy hill" (the conduction band) and the "water level" of electrons (the Fermi level) actually shift as you move from the hot end to the cold end.
The "Perfect" Heat Assumption: Some methods tried to calculate heat flow by assuming the process was perfectly smooth and reversible, like a frictionless slide. But in reality, heat flow is messy and full of friction. Assuming it's perfect creates a logical paradox.

2. The New Solution: A Better Way to Walk

The author proposes a new way to calculate this voltage using a "Drift-Diffusion" approach. Think of this as watching how a crowd of people (electrons) moves through the hallway.

The Soret Effect: This is like a crowd of people naturally spreading out from a hot, crowded area to a cooler, less crowded area. The author uses this natural spreading behavior to figure out how the voltage builds up without needing to assume the process is perfectly frictionless.
The "Reversible" Shortcut: The author found that in many common situations (near-equilibrium), you can use a very simple, "reversible" model.
- The Analogy: Imagine you want to know the average speed of cars on a highway. The complex way (BTE) involves measuring every car's speed, how long they brake, and how they swerve (scattering/relaxation). The author's "Reversible Model" is like looking at the traffic flow as a whole fluid. It ignores the individual braking and swerving, yet surprisingly, it still gives a very accurate estimate of the average speed.
- The Benefit: This simple model acts as a lower bound. It tells you the minimum voltage you can expect. It's fast, easy to calculate, and doesn't get bogged down in complex details.

3. The Test Drive: Silicon Highways

To prove this new method works, the author tested it on highly-doped Silicon (a very common material used in electronics). They compared three different "maps":

The Reversible Map (New): Simple and fast.
The Ballistic Map: Assumes electrons fly through like bullets without hitting anything.
The BTE Map: The complex, traditional method that tries to account for every collision.

The Results:

The Reversible Model was a "fairly good" guess. It consistently predicted the lowest possible voltage, and the actual experimental data was usually just a bit higher than this prediction. It was accurate enough to be useful and much faster to compute.
The Ballistic Model was hit-or-miss. Sometimes it was too high, sometimes too low.
The Traditional BTE Model was a disaster in this specific test. It wildly overestimated the voltage (by up to 90% in some cases). This happened because it relied on a specific assumption about how electrons collide that didn't fit the real-world data well.

4. The Takeaway

The paper concludes that while the complex, traditional methods are popular, they are often flawed or computationally expensive. The author's new "Reversible Model" is a simpler, theoretically sound alternative.

In plain English: If you need to quickly estimate how much electricity a material will generate from heat, don't bother with the overly complicated math that tries to track every single electron collision. Use the author's simpler formula. It gives you a reliable "floor" for the answer, is much faster to calculate, and avoids the logical traps of the old methods. It's like using a reliable compass instead of a broken GPS that tries to calculate every single turn you might make.

Technical Summary: Revisiting Theoretical Modeling of Seebeck Coefficient of Semiconductors

Problem Statement
The paper identifies fundamental thermodynamic flaws in existing methodologies used to model the Seebeck coefficient ( $\alpha_S$ ) in semiconductors, particularly those relying on the Boltzmann Transport Equation (BTE) and specific heat flux definitions. The author highlights three primary issues in conventional approaches:

Potential Confusion: A frequent conflation of electrochemical potential (Fermi level, $E_F$ ) and electrostatic potential ( $\varphi$ ) in definitions, leading to incorrect derivations.
Arbitrary Neglect: In BTE derivations, the spatial gradients of the conduction band bottom ( $E_C$ ) and Fermi level ( $E_F$ ) are often arbitrarily neglected when calculating the distribution function's gradient, despite these variables changing across the sample even under open-circuit (OC) conditions.
Improper Heat Flux: The definition of local heat flux ( $J_Q = T J_S = J_E - \mu J_N$ ) relies on a strong assumption of local quasi-static (reversible) processes with zero entropy production. This contradicts fundamental non-equilibrium thermodynamics principles yet is widely used in BTE, Landauer-Büttiker, and Kubo formula applications.

Methodology
To address these flaws, the author proposes an alternative methodology based on the drift-diffusion equation incorporating the Soret effect under open-circuit conditions. This approach avoids explicit heat flux presentations and the ambiguity between electrochemical and electrostatic potentials.

The derivation proceeds in three stages:

Near-Equilibrium (Band Term): Assuming Local Equilibrium (LEA) and local electrical neutrality, the author derives a closed-form model for the band term of $\alpha_S$ . By setting the net charge flux to zero and utilizing generalized Einstein relations, a "reversible model" is recovered. This model eliminates explicit electron velocity and relaxation time terms, relying instead on the spatial gradient of the Fermi level.
Phonon-Drag Term: The methodology is extended to include the phonon-drag effect by adding a charge flux term ( $J_d$ ) arising from momentum exchange with non-equilibrium phonons. The resulting expression for the phonon-drag term is shown to be identical in form to the conventional BTE-based model.
Ballistic Case: For the ballistic regime (where LEA is invalid), the author utilizes the Landauer formulation to calculate charge fluxes from hot and cold terminals, deriving the conventional ballistic Seebeck coefficient expression.

The paper unifies these models (BTE, Reversible, and Ballistic) into a single integral form dependent on an energy exponent $r$ , which varies based on scattering mechanisms and material dimensionality.

Key Contributions

Thermodynamic Correction: The paper rigorously critiques the thermodynamic inconsistencies in standard BTE derivations and heat flux definitions.
Derivation of the Reversible Model: It demonstrates that the reversible model (previously presented in the author's prior work) is valid for the band term of the Seebeck coefficient in near-equilibrium cases, despite dropping velocity and relaxation terms.
Theoretical Bounds: The study establishes that the reversible model provides a theoretical lower bound for the Seebeck coefficient band term (corresponding to $r=1/2$ for 3D materials), while the BTE model with weakly-screened ionized impurity scattering provides an upper bound ( $r=3$ ).
Unified Framework: It presents a unified mathematical framework to compare the reversible, ballistic, and BTE models based on the energy dependence of the density of states, velocity, and relaxation time.

Results
A case study was performed on highly n-doped Silicon ( $n_d > 1 \times 10^{19} \text{ cm}^{-3}$ ), comparing the three models against experimental data from Geballe et al. and Sadhu et al.:

Reversible Model: Provided "fairly-good" predictions with low computational cost. It consistently served as the lower bound for experimental data, with mean relative deviations of 13%, 18%, and 5% for doping concentrations of $2.7 \times 10^{19}$ , $6.0 \times 10^{19}$ , and $7.0 \times 10^{19} \text{ cm}^{-3}$ , respectively.
Ballistic Model: Showed mixed performance, with deviations of 3%, 6%, and 16%. It overestimated the data for the nanowire case ( $7.0 \times 10^{19} \text{ cm}^{-3}$ ).
BTE Model: Using weakly-screened ionized impurity scattering (the upper bound assumption), the model significantly overestimated the experimental values, with mean relative deviations reaching 60% to 90%. This highlights the sensitivity of BTE results to the choice of relaxation time formulation.

Significance and Claims
The paper claims that the proposed reversible model offers a concise, theoretically sound, and computationally efficient alternative for evaluating thermoelectric properties, particularly in near-equilibrium and reversible cases. By eliminating the need for complex scattering time calculations and avoiding thermodynamic paradoxes, the model is presented as a viable tool for high-throughput calculations and material screening. The author emphasizes that while the model neglects irreversible mechanisms (scattering), it accurately captures the lower bound of the Seebeck coefficient, making it a robust baseline for thermoelectric property evaluation.

1. The Problem: The Old Maps Were Wrong

2. The New Solution: A Better Way to Walk

3. The Test Drive: Silicon Highways

4. The Takeaway

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