Band gap renormalization, carrier mobility, and transport in Mg$_{2}$Si and Ca$_{2}$Si: \textit{Ab initio} scattering and Boltzmann transport equation study

This study employs first-principles electron-phonon interaction calculations and the Boltzmann transport equation to comprehensively analyze the temperature-dependent band gaps, carrier mobilities, and thermoelectric transport properties of Mg$_2$Si and Ca$_2$Si, demonstrating that accurately accounting for electron-phonon scattering is critical for predicting experimental mobility and thermoelectric performance while identifying nanostructuring as a viable strategy to enhance their figure of merit.

The Gist

Imagine you are trying to build a better thermoelectric generator. Think of this device as a magical toaster that doesn't just toast bread, but actually turns the heat from a fire directly into electricity to power your lights. The efficiency of this "magic toaster" depends on a material's ability to conduct electricity well while blocking heat.

The scientists in this paper are investigating two specific materials, Mg₂Si (Magnesium Silicide) and Ca₂Si (Calcium Silicide), to see if they are good candidates for this job. They are essentially acting as high-tech detectives, using supercomputers to simulate how these materials behave at the atomic level.

Here is a breakdown of their investigation using simple analogies:

1. The "Traffic Jam" Problem (Electron Mobility)

Imagine electrons (the carriers of electricity) as cars driving on a highway.

• The Old Way (CRTA): Previous studies often assumed the highway was perfectly smooth and that every car drove at the same speed, ignoring traffic lights or potholes. They used a "Constant Relaxation Time" (CRTA), which is like saying, "All cars take the same amount of time to get stuck, no matter what."
• The New Way (EPI & RTAs): This paper says, "No, that's not realistic!" In reality, electrons crash into vibrating atoms (phonons), creating a traffic jam. The authors used a more advanced method called Electron-Phonon Interaction (EPI).
  • They tested three different "traffic rules" (approximations):
    1. SERTA: A simple rule where cars just slow down when they hit a bump.
    2. MRTA: A smarter rule that accounts for cars changing direction after a crash.
    3. IBTE: The "God's eye view" simulation that tracks every single car's path perfectly, but it's incredibly slow and expensive to run.

The Discovery: For Mg₂Si, the simple rule (SERTA) actually matched real-world experiments best! It was like finding a shortcut that worked perfectly. However, for Ca₂Si, the rules behaved differently, showing that you can't use a "one-size-fits-all" approach for different materials.

2. The "Shrinking Suit" (Band Gap Renormalization)

Imagine the material's "band gap" (the energy needed to get an electron moving) is like a suit of clothes.

• At Absolute Zero: The suit fits perfectly.
• As it gets Hot: The atoms inside the material start vibrating wildly (like a person shivering or dancing). This vibration changes the shape of the suit, making it tighter or looser. This is called Band Gap Renormalization.
• The Result: The scientists found that as the temperature rises, the "suit" shrinks significantly. For Mg₂Si, the gap gets much smaller (by about 30-50 meV) just by heating it up. This is crucial because if the gap changes, the material's ability to conduct electricity changes too. They found that ignoring this "shrinking suit" leads to wrong predictions.

3. The "Heat Leak" (Thermal Conductivity)

To make a good thermoelectric generator, you want the material to let electricity flow but stop heat from flowing.

• The Problem: Heat travels through the material like sound waves (phonons) bouncing around a room.
• The Solution (Nanostructuring): Imagine the material is a large, open ballroom. If the dancers (heat waves) can run across the whole room without hitting anything, heat escapes quickly.
  • The authors suggest putting up walls (nanostructures) or furniture (impurities) in the ballroom.
  • If the walls are the right size, they block the heavy heat waves (phonons) from crossing the room, but they are small enough that the electricity cars (electrons) can still weave through the gaps.
  • The Math: They calculated that by making the material "nano-structured" (adding tiny grains), they could cut the heat leakage by 55-60%. This is a huge win!

4. The Final Score (Figure of Merit, zT)

The ultimate goal is the zT score. Think of this as the "Grade" the material gets for being a thermoelectric generator.

• Old Predictions: Using the old, simple methods, the materials looked okay, with grades around 0.35.
• New Predictions: When they used the realistic "traffic jam" rules (EPI) and added the "walls" (nanostructuring), the grades dropped initially (because the realistic physics showed the materials aren't as perfect as we thought).
• The Twist: However, once they added the nanostructuring (the walls), the grades shot up again!
  • For Ca₂Si, they predicted a potential grade of 0.4 if made into a nanostructured material.
  • They also found that Ca₂Si is a "dual-purpose" superhero: it's great for solar cells (turning light to electricity) and thermoelectrics (turning heat to electricity).

Summary: What does this all mean?

This paper is a reality check. It tells us that to predict how well a material works, we can't use simple, lazy math. We have to simulate the chaotic "traffic jams" of electrons and the "shrinking suits" of atoms.

The Takeaway:

• Mg₂Si is a solid, proven performer, but we need to be careful with how we calculate its efficiency.
• Ca₂Si is a new, exciting contender that might be even better, especially if we build it with tiny "nano-walls" to stop heat from escaping.
• The Future: The authors are calling for real-world labs to actually build these Ca₂Si materials to see if the computer predictions hold true. If they do, we might have a new, cheap, non-toxic material to power our future devices using waste heat!

Technical Summary

1. Problem Statement

Thermoelectric (TE) materials require a high figure of merit ($zT$), which depends on optimizing the power factor ($S^2\sigma$) while minimizing thermal conductivity ($\kappa$). A major challenge in predicting TE properties is the accurate modeling of charge carrier transport.

• Limitations of Current Methods: Most studies rely on the Constant Relaxation Time Approximation (CRTA), which assumes carrier lifetime is independent of energy and scattering mechanisms. This often fails for semiconductors where electron-phonon interactions (EPI) are strong and carrier lifetimes depend heavily on band dispersion.
• Specific Gap: While Mg$_2$Si is a well-studied TE material, Ca$_2$Si has received less attention. Previous theoretical studies on these silicides often lacked rigorous treatment of EPI, leading to discrepancies between calculated and experimental transport coefficients (mobility, Seebeck coefficient, conductivity).
• Objective: To perform a rigorous first-principles study of Mg$_2$Si and Ca$_2$Si, incorporating EPI to calculate temperature-dependent band gap renormalization, carrier mobility, and TE transport coefficients, comparing different relaxation-time approximations (RTAs) against experimental data.

2. Methodology

The authors employed a comprehensive first-principles approach combining Density Functional Theory (DFT) with Many-Body Perturbation Theory (MBPT) and the Boltzmann Transport Equation (BTE).

• Electronic Structure & Renormalization:

  • Calculations were performed using the Abinit software with PBE functionals and norm-conserving pseudopotentials.
  • Band Gap Renormalization: The electron-phonon self-energy ($\Sigma_{ep}$) was calculated using the Fan-Migdal and Debye-Waller terms. Two approaches were used to determine the quasiparticle (QP) energy:
    1. On-the-Mass-Shell (OTMS): Standard approximation.
    2. Linearized-QP-Equation (LQE): Includes the renormalization factor ($Z$-factor) for higher accuracy.
  • Zero-point renormalization (ZPR) and temperature-dependent band gap shifts were analyzed up to 800 K.

• Carrier Transport (Electrons):

  • Scattering Mechanisms: EPI was treated as the primary scattering mechanism.
  • RTA Variants: The study compared three methods for solving the BTE:
    1. SERTA (Self-Energy RTA): Ignores momentum conservation in scattering.
    2. MRTA (Momentum RTA): Accounts for momentum loss (back-scattering).
    3. IBTE (Iterative BTE): The exact solution, computationally expensive but highly accurate.
  • Convergence: A detailed convergence study was performed on $k$-point (electron) and $q$-point (phonon) grids, utilizing a Double-Grid (DG) technique to reduce computational cost while maintaining accuracy.

• Lattice Transport (Phonons):

  • Lattice thermal conductivity ($\kappa_{ph}$) was calculated using the Phono3py code based on phonon-phonon interactions (PPI) within the Single-Mode Relaxation Time Approximation (SMRTA).
  • Strategies to enhance $zT$ were modeled, including nanostructuring (phonon-boundary scattering) and mass-difference scattering (doping with Bi and Sb).

3. Key Results

A. Band Gap Renormalization

• Zero-Point Renormalization (ZPR): The band gaps were significantly reduced by ZPR.
  • Mg$_2$Si: Reduced by $\sim$29–33 meV (from 0.21 eV to $\sim$0.18 eV).
  • Ca$_2$Si: Reduced by $\sim$37–51 meV (from 0.56 eV to $\sim$0.51 eV).
• Temperature Dependence: The band gap decreases with increasing temperature. At 300 K, the gaps are $\sim$0.15–0.154 eV (Mg$_2$Si) and $\sim$0.46–0.5 eV (Ca$_2$Si).
• Z-Factor Impact: The inclusion of the $Z$-factor (LQE) had opposite effects on the two materials: it increased the band gap variation with temperature for Mg$_2$Si but reduced the sensitivity for Ca$_2$Si.

B. Carrier Mobility ($\mu_e$)

• Convergence: IBTE required extremely dense grids ($144^3$) to converge, whereas SERTA and MRTA converged faster ($48^3$ to $60^3$).
• 300 K Values (Mg$_2$Si):
  • SERTA: $\sim$351 cm$^2$ V$^{-1}$ s$^{-1}$ (Matches experimental $\sim$350 cm$^2$ V$^{-1}$ s$^{-1}$).
  • MRTA: $\sim$573 cm$^2$ V$^{-1}$ s$^{-1}$ (Overestimates).
  • IBTE: $\sim$524 cm$^2$ V$^{-1}$ s$^{-1}$.
• 300 K Values (Ca$_2$Si):
  • SERTA: $\sim$100 cm$^2$ V$^{-1}$ s$^{-1}$.
  • MRTA: $\sim$197 cm$^2$ V$^{-1}$ s$^{-1}$.
  • IBTE: $\sim$163 cm$^2$ V$^{-1}$ s$^{-1}$.
• Trend: Mobility decreases with temperature for both materials due to increased phonon scattering. SERTA generally aligns better with experiments for Mg$_2$Si, while MRTA shows better agreement with IBTE at lower temperatures.

C. Thermoelectric Transport Coefficients

• Seebeck Coefficient ($S$) & Conductivity ($\sigma$):
  • CRTA results deviated significantly from experiments, especially at high carrier concentrations.
  • Including EPI (via SERTA/MRTA) significantly improved agreement with experimental data for both $S$ and $\sigma$.
  • For Mg$_2$Si, SERTA/MRTA results matched experimental trends better than CRTA, particularly at concentrations of $10^{17}$–$10^{19}$ cm$^{-3}$.
• Thermal Conductivity ($\kappa$):
  • Lattice ($\kappa_{ph}$): Calculated as $\sim$22.7 W m$^{-1}$ K$^{-1}$ for Mg$_2$Si and $\sim$7.2 W m$^{-1}$ K$^{-1}$ for Ca$_2$Si at 300 K. Ca$_2$Si has lower $\kappa_{ph}$ due to heavier Ca atoms.
  • Electronic ($\kappa_e$): Follows the Wiedemann-Franz law; decreases with temperature when EPI is included.

D. Figure of Merit ($zT$) and Enhancement Strategies

• Bulk $zT$:
  • Using CRTA, $zT$ was overestimated ($\sim$0.35–0.38).
  • Using MRTA (more accurate), the optimal $zT$ for Mg$_2$Si at 900 K was $\sim$0.08, and for Ca$_2$Si at 700 K was $\sim$0.085 (at optimal doping $n_e \approx 10^{19}$ cm$^{-3}$).
• Nanostructuring:
  • Phonon Mean Free Path (MFP) analysis showed that phonons with MFPs $<30$ nm (Mg$_2$Si) and $<20$ nm (Ca$_2$Si) contribute significantly to heat transport.
  • Introducing grain boundaries of 30 nm (Mg$_2$Si) and 20 nm (Ca$_2$Si) reduced $\kappa_{ph}$ by ~55–60% at 300 K.
  • This nanostructuring boosted $zT$ by more than 2-fold at room temperature and 1.5-fold at high temperatures.
• Doping (Mass-Difference Scattering):
  • Doping Mg$_2$Si with Bi and Sb reduced $\kappa_{ph}$ significantly.
  • For Ca$_2$Si, 2% Sb doping was predicted to yield a $zT$ of $\sim$0.17, potentially reaching 0.4 when combined with nanostructuring.

4. Significance and Conclusions

• Validation of Theory: The study demonstrates that SERTA provides the most accurate prediction for electron mobility in Mg$_2$Si when compared to experimental data, while MRTA is crucial for understanding back-scattering effects in Ca$_2$Si.
• Beyond CRTA: The work highlights that ignoring energy-dependent scattering (EPI) leads to significant errors in predicting TE performance. Accurate modeling requires moving beyond the constant relaxation time approximation.
• Ca$_2$Si Potential: Ca$_2$Si is identified as a promising, low-cost, non-toxic dual-purpose material for both solar cells and thermoelectric generators. Its lower lattice thermal conductivity compared to Mg$_2$Si makes it particularly attractive for TE applications.
• Strategic Optimization: The combination of nanostructuring (to suppress $\kappa_{ph}$) and optimal doping ($10^{19}$ cm$^{-3}$) offers a viable pathway to achieve high $zT$ values ($>0.3$) in these silicides, validating their potential for practical energy harvesting devices.

This paper provides a critical benchmark for theoretical transport studies in silicides, emphasizing the necessity of rigorous electron-phonon coupling calculations for reliable materials design.