Zero-point renormalization of the band gap of semiconductors and insulators using the projector augmented wave method
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2022-09-29 |
| Journal | Physical review. B./Physical review. B |
| Authors | Manuel Engel, Henrique Miranda, Laurent Chaput, Atsushi Togo, Carla Verdi |
| Institutions | Centre National de la Recherche Scientifique, Laboratoire Ănergies et MĂ©canique ThĂ©orique et AppliquĂ©e |
| Citations | 32 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis study provides highly accurate, first-principles calculations of the Zero-Point Renormalization (ZPR) of the electronic band gap for 28 critical semiconductors and insulators, essential for modern device engineering.
- Core Achievement: Determined the ZPR (band gap shift due to electron-phonon coupling at 0 K) for 28 materials, including Si, diamond, GaN, ZnO, and various oxides.
- Method Validation: Utilized the Projector-Augmented-Wave (PAW) method within the non-adiabatic Allen-Heine-Cardona (AHC) theory, achieving excellent agreement with recent literature.
- Computational Efficiency: Compared two PAW formulations (All-Electron (AE) and Pseudized (PS)) and demonstrated that the PS formulation converges significantly faster with respect to the number of intermediate electronic states.
- Polar Material Accuracy: Explicitly incorporated long-range electrostatic interactions using a generalized Fröhlich model, confirming its necessity for accurate ZPR calculation in polar compounds (e.g., MgO, LiF).
- Reference Data: The converged ZPR values serve as stringent reference data for validating other computational methods, including supercell-based and Wannier interpolation approaches.
- Largest Shift: The largest ZPR observed was -1231 meV for Lithium Fluoride (LiF-rs), highlighting the critical nature of this correction for certain materials.
Technical Specifications
Section titled âTechnical SpecificationsâThe following table summarizes key quantitative results and computational parameters used in the converged ZPR calculations (using the Pseudized (PS) approach).
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Materials Studied | 28 | Solids | Semiconductors and Insulators (Cubic, Rock-salt, Wurtzite, Tetragonal) |
| ZPR (Diamond, C-cd) | -323 | meV | Band gap renormalization at 0 K |
| ZPR (Silicon, Si-cd) | -58 | meV | Band gap renormalization at 0 K |
| ZPR (Zinc Oxide, ZnO-w) | -175 | meV | Band gap renormalization at 0 K |
| ZPR (Lithium Fluoride, LiF-rs) | -1231 | meV | Largest ZPR observed in the study |
| ZPR Relative Error (vs. Ref. [3]) | Less than 5% | % | For most materials, confirming high accuracy |
| Smearing Parameter (ÎŽ) | 10 | meV | Used for Lorentzian broadening of energy transitions |
| Maximum q-point Grid Density | 64x64x64 | q-points | Used for extrapolation to infinite density |
| DFT Functional Used | PBE | N/A | Perdew-Burke-Ernzerhof generalized gradient approximation |
| Supercell Size (Cubic/RS) | 4x4x4 | Primitive Cells | Used for force constant and potential derivative calculation |
Key Methodologies
Section titled âKey MethodologiesâThe ZPR calculations were performed using a sophisticated computational workflow based on Density Functional Theory (DFT) and many-body perturbation theory.
- DFT and PAW Framework: All calculations were performed using the VASP code, employing the Projector-Augmented-Wave (PAW) method to accurately describe all-electron wave functions while maintaining computational efficiency.
- Lattice Optimization: Initial lattice parameters, Born effective charges (Z*), and macroscopic ion-clamped static dielectric tensors (Δâ) were optimized by minimizing the DFT total-energy functional.
- Electron-Phonon Matrix Elements: The core quantity, gmnk,vq, was calculated using the finite atomic displacement method in large supercells (e.g., 4x4x4). The Pseudized (PS) formulation of the PAW matrix element was selected for final results due to its superior convergence rate.
- Long-Range Electrostatics: For polar materials, the long-range component of the electron-phonon potential derivative was treated explicitly using a generalized Fröhlich model, crucial for accurate results at small phonon wave vectors (q).
- ZPR Calculation: The ZPR was calculated using the non-adiabatic AHC formula, which includes both the Fan-Migdal (FM) and Debye-Waller (DW) contributions, accounting for energy transfer during phonon emission/absorption.
- Convergence and Extrapolation: Calculations were performed on increasingly dense q-point grids. The final ZPR was extrapolated to an infinitely dense grid by assuming a linear dependence on nq-1/3 at high densities.
Commercial Applications
Section titled âCommercial ApplicationsâAccurate determination of the ZPR is fundamental for predicting the temperature-dependent electronic properties of materials, directly impacting the design and performance of devices across several high-technology sectors.
- High-Power Electronics:
- Materials: SiC, GaN, AlN, ZnO.
- Relevance: Accurate band gap is essential for designing high-voltage, high-frequency power devices and rectifiers, where thermal stability and operating temperature are critical constraints.
- Optoelectronics and Photonics:
- Materials: GaN, ZnS, CdTe, AlAs.
- Relevance: Precise knowledge of the band gap determines the emission and absorption wavelengths of LEDs, laser diodes, and photodetectors, ensuring devices operate at target frequencies (e.g., UV or visible light).
- Thermal Management and Thermoelectrics:
- Materials: TiO2, SnO2, Silicates.
- Relevance: Electron-phonon coupling governs thermal transport and carrier mobility, which are key factors in thermoelectric energy conversion and heat dissipation in microprocessors.
- Insulators and Dielectrics:
- Materials: MgO, LiF, SiO2.
- Relevance: Accurate band gap data is necessary for modeling dielectric breakdown strength and leakage currents in high-k dielectrics used in capacitors and memory devices.
- Computational Materials Discovery:
- Relevance: The highly validated ZPR data provides a crucial benchmark for developing faster, less resource-intensive computational methods (like machine learning potentials or simplified interpolation schemes) used to accelerate the discovery of new functional materials.
View Original Abstract
We evaluate the zero-point renormalization (ZPR) due to electron-phonon\ninteractions of 28 solids using the projector-augmented-wave (PAW) method. The\ncalculations cover diamond, many zincblende semiconductors, rock-salt and\nwurtzite oxides, as well as silicate and titania. Particular care is taken to\ninclude long-range electrostatic interactions via a generalized Fr\âohlich\nmodel, as discussed in Phys. Rev. Lett. 115, 176401 (2015) and Phys. Rev. B 92,\n054307 (2015). The data are compared to recent calculations, npj Computational\nMaterials 6, 167 (2020), and generally very good agreement is found. We discuss\nin detail the evaluation of the electron-phonon matrix elements within the PAW\nmethod. We show that two distinct versions can be obtained depending on when\nthe atomic derivatives are taken. If the PAW transformation is applied before\ntaking derivatives with respect to the ionic positions, equations similar to\nthe ones conventionally used in pseudopotential codes are obtained. If the PAW\ntransformation is used after taking the derivatives, the full-potential spirit\nis largely maintained. We show that both variants yield very similar ZPRs for\nselected materials when the rigid-ion approximation is employed. In practice,\nwe find however that the pseudo version converges more rapidly with respect to\nthe number of included unoccupied states.\n