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arxiv: 2501.15992 · v2 · submitted 2025-01-27 · ❄️ cond-mat.mtrl-sci · physics.app-ph

Band gap renormalization, carrier mobility, and transport in Mg₂Si and Ca₂Si: textit{Ab initio} scattering and Boltzmann transport equation study

Vinod Kumar Solet , Sudhir K. Pandey This is my paper

Pith reviewed 2026-05-23 05:22 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.app-ph
keywords Mg2SiCa2Sielectron-phonon interactionBoltzmann transport equationthermoelectric figure of meritband gap renormalizationcarrier mobilityrelaxation time approximation
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0 comments X

The pith

Electron-phonon interactions must be included explicitly to obtain accurate thermoelectric transport properties in Mg2Si and Ca2Si

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

The paper performs first-principles electron-phonon interaction calculations based on many-body perturbation theory and solves the Boltzmann transport equation under several relaxation-time approximations for Mg2Si and Ca2Si. It shows that constant-relaxation-time models produce zT values of 0.35 and 0.38, but explicit scattering lowers these to 0.08 and 0.085. Temperature-dependent band-gap renormalizations of tens of meV are obtained, and mobilities from the different approximations are reported with convergence checks. The results align more closely with available experimental mobilities when electron-phonon effects are treated in detail rather than approximated as constant.

Core claim

For PBE band gaps of 0.21 eV in Mg2Si and 0.56 eV in Ca2Si, zero-point renormalization corrections of 29-33 meV and 37-51 meV are found, yielding 300 K gaps of 0.15-0.154 eV and 0.46-0.5 eV. At 300 K the electron mobilities are 351, 573 and 524 cm2 V-1 s-1 for Mg2Si under SERTA, MRTA and IBTE, with correspondingly lower values for Ca2Si. Thermoelectric coefficients depend strongly on the chosen relaxation-time approximation, and the maximum zT falls from 0.35 (0.38) under constant relaxation time to 0.08 (0.085) under MRTA once EPI is included; lattice thermal conductivities from phonon-phonon scattering are 22.7 and 7.2 W m-1 K-1.

What carries the argument

The Boltzmann transport equation solved under self-energy relaxation-time approximation (SERTA), momentum relaxation-time approximation (MRTA) and iterative BTE (IBTE), with scattering rates supplied by ab initio electron-phonon interactions from many-body perturbation theory.

If this is right

  • SERTA agrees better with IBTE at higher temperatures while MRTA agrees better at lower temperatures.
  • SERTA-derived mobilities for Mg2Si closely match available experimental values.
  • Thermoelectric transport coefficients obtained with SERTA or MRTA show improved agreement with experiment compared with constant relaxation time over electron concentrations from 10^17 to 10^20 cm^-3.
  • Lattice thermal conductivity at 300 K is 22.7 W m^-1 K^-1 for Mg2Si and 7.2 W m^-1 K^-1 for Ca2Si due to phonon-phonon interactions.
  • Reducing lattice thermal conductivity through nanostructuring and mass-difference scattering is identified as a route to higher zT even after EPI is included.

Where Pith is reading between the lines

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

  • The same EPI-inclusive BTE workflow could be applied to other thermoelectric silicides to test whether constant-relaxation-time overestimates of zT are common.
  • Optimal device temperatures might be selected where the difference between SERTA and MRTA is smallest, reducing uncertainty in performance predictions.
  • Further calculations at doping levels beyond 10^20 cm^-3 could reveal whether the EPI corrections remain dominant at high carrier densities.

Load-bearing premise

The PBE starting band gaps together with the chosen many-body perturbation corrections and BTE convergence settings produce quantitatively reliable mobilities and zT values over the examined temperature and doping range.

What would settle it

An experimental electron mobility for Ca2Si at 300 K that lies well outside the calculated range of roughly 100-197 cm2 V-1 s-1 under the three relaxation-time approximations would indicate that the EPI treatment or starting electronic structure needs revision.

Figures

Figures reproduced from arXiv: 2501.15992 by Sudhir K. Pandey, Vinod Kumar Solet.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) DFT, ZPR correction in band gap at 0 K [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Convergence of the electron mobility at 300 K in (a) Mg [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Temperature-dependent electron mobility of (a) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Seebeck coefficient as a function of temperature at differe [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Electrical conductivity ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (a) Calculated lattice thermal conductivity [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: For [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Temperature dependence of (a) lattice thermal [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Temperature dependence of (a) lattice thermal [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
read the original abstract

We perform first-principles electron-phonon interaction (EPI) calculations based on many-body perturbation theory to study the temperature-dependent band-gap and charge-carrier transport properties for Mg$_{2}$Si and Ca$_{2}$Si using the Boltzmann transport equation (BTE) under different relaxation-time approximations (RTAs). For a PBE band gap of 0.21 (0.56) eV in Mg$_{2}$Si (Ca$_{2}$Si), a zero-point renormalization correction of 29-33 (37-51) meV is obtained using various approaches, while the gap at 300 K is 0.15-0.154 (0.46-0.5) eV. The electron mobility ($\mu_{e}$), with a detailed convergence study at 300 K, is evaluated using linearized (self-energy and momentum RTA, or SERTA and MRTA) and iterative BTE (IBTE) solutions. At 300 K, the $\mu_{e}$ values are 351 (100), 573 (197), and 524 (163) cm$^{2}V^{-1}s^{-1}$ from SERTA, MRTA, and IBTE, respectively, for Mg$_{2}$Si (Ca$_{2}$Si). SERTA (MRTA) provides results in better agreement with IBTE at higher (lower) temperatures, while SERTA-derived $\mu_{e}$ closely matches experimental $\mu_{e}$ values for Mg$_{2}$Si. Thermoelectric (TE) transport coefficients significantly influenced by the choice of RTA, with SERTA and MRTA yielding improved agreement with experimental results compared to constant RTA (CRTA) for Mg$_{2}$Si over an electron concentration range of $10^{17}$ to $10^{20}$ cm$^{-3}$. The lattice thermal conductivity ($\kappa_{ph}$) at 300 K due to phonon-phonon interactions is estimated to be 22.7 (7.2) W m$^{-1}K^{-1}$ for Mg$_{2}$Si (Ca$_{2}$Si). The highest calculated figure of merit (zT) under CRTA is 0.35 (0.38), which decreases to 0.08 (0.085) when EPI is included using MRTA. This study clearly identifies the critical role of EPI in accurate transport predictions of TE silicides. Finally, we explore strategies to enhance zT by reducing $\kappa_{ph}$ through nanostructuring and mass-difference scattering.

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

1 major / 2 minor

Summary. The manuscript reports first-principles many-body perturbation theory calculations of electron-phonon interactions in Mg₂Si and Ca₂Si. It computes zero-point and temperature-dependent band-gap renormalization starting from PBE gaps, then evaluates electron mobility at 300 K (and its temperature dependence) by solving the Boltzmann transport equation under SERTA, MRTA, and IBTE. Lattice thermal conductivity from phonon-phonon scattering is also obtained. The central numerical claim is that the thermoelectric figure of merit drops sharply when EPI is included via MRTA (zT = 0.08 for Mg₂Si, 0.085 for Ca₂Si) relative to CRTA (0.35 and 0.38), while SERTA mobilities match experiment for Mg₂Si and both SERTA/MRTA improve agreement with measured transport coefficients over CRTA. Strategies for further zT enhancement via reduced κ_ph are discussed.

Significance. If the reported mobilities and zT values are robust, the work supplies concrete, quantitative evidence that constant-RTA overestimates zT in these silicides and that EPI must be treated with momentum- or self-energy-dependent relaxation times. The detailed 300 K convergence study for mobility and the direct experimental comparison for μ_e are clear strengths. The manuscript also supplies first-principles κ_ph values that can be used in future device modeling.

major comments (1)
  1. [Abstract] Abstract: the headline result that zT falls from 0.35 (0.38) under CRTA to 0.08 (0.085) under MRTA is presented as the effect of including EPI. However, the same abstract states that SERTA μ_e = 351 cm² V⁻¹ s⁻¹ closely matches experiment while MRTA = 573 cm² V⁻¹ s⁻¹, and that SERTA agrees better with IBTE at higher T. No zT values are given for SERTA (or IBTE). Because thermoelectric coefficients are stated to be “significantly influenced by the choice of RTA,” the quantitative reduction claimed for EPI cannot be assessed under the approximation that reproduces the measured mobility; this is load-bearing for the central assertion that EPI must be included for accurate predictions.
minor comments (2)
  1. [Abstract] Abstract: the ranges given for ZPR (29-33 meV, 37-51 meV) and 300 K gaps (0.15-0.154 eV, 0.46-0.5 eV) are presented without identifying which many-body approach produces each endpoint; a short table or explicit mapping would improve clarity.
  2. [Abstract] Abstract: the statement that “SERTA and MRTA yielding improved agreement with experimental results compared to CRTA” is made for TE transport coefficients, yet only zT under CRTA and MRTA is quantified; the sentence should specify which coefficients were compared.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline result that zT falls from 0.35 (0.38) under CRTA to 0.08 (0.085) under MRTA is presented as the effect of including EPI. However, the same abstract states that SERTA μ_e = 351 cm² V⁻¹ s⁻¹ closely matches experiment while MRTA = 573 cm² V⁻¹ s⁻¹, and that SERTA agrees better with IBTE at higher T. No zT values are given for SERTA (or IBTE). Because thermoelectric coefficients are stated to be “significantly influenced by the choice of RTA,” the quantitative reduction claimed for EPI cannot be assessed under the approximation that reproduces the measured mobility; this is load-bearing for the central assertion that EPI must be included for accurate predictions.

    Authors: We agree that the abstract would benefit from reporting zT under SERTA (and IBTE) to enable direct comparison with the mobility that matches experiment. In the revised manuscript we will add the SERTA zT values for both compounds across the doping range, together with a brief statement clarifying why MRTA was used for the headline EPI-included result (momentum relaxation is treated more completely). This revision directly addresses the concern that the quantitative EPI effect on zT cannot be assessed under the RTA matching measured mobility. revision: yes

Circularity Check

0 steps flagged

Minor self-citation not load-bearing; derivation from first-principles EPI+BTE inputs

full rationale

The paper derives band-gap renormalizations, mobilities (SERTA/MRTA/IBTE), lattice thermal conductivity, and zT values directly from PBE starting points plus many-body perturbation theory corrections and linearized/iterative BTE solutions. No quoted equation reduces a reported mobility or zT to a parameter fitted from the same dataset; results are compared to external experiment. Any self-citations are peripheral and do not carry the central claim that EPI inclusion lowers zT.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculations rest on standard DFT and many-body perturbation theory approximations whose accuracy for these materials is taken as given; no new entities or ad-hoc fitted parameters are introduced beyond the choice of PBE functional and the various RTA schemes.

axioms (2)
  • domain assumption PBE DFT band structures provide a suitable starting point for subsequent EPI corrections
    Initial gaps of 0.21 eV (Mg2Si) and 0.56 eV (Ca2Si) are taken directly from PBE.
  • domain assumption The linearized and iterative BTE solutions under the stated RTAs converge to physically meaningful mobilities
    Detailed convergence study at 300 K is claimed but not inspectable from abstract.

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