First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials

At a Glance

Metadata	Details
Publication Date	2020-01-10
Journal	Reports on Progress in Physics
Authors	Samuel Poncé, Wenbin Li, Sven Reichardt, Feliciano Giustino
Citations	277
Analysis	Full AI Review Included

Executive Summary

This review provides a comprehensive analysis of the state-of-the-art in ab initio (first-principles) calculations of charge carrier mobility (µ) and conductivity in advanced semiconductors and two-dimensional (2D) materials.

Predictive Accuracy Achieved: Modern methods, primarily based on solving the linearized Boltzmann Transport Equation (BTE) using Density Functional Perturbation Theory (DFPT) for electron-phonon interactions (EPI), yield mobilities for bulk materials (e.g., Si, GaAs) that agree with experiments typically within 20%.
Methodological Advancement: The key to predictive accuracy is the parameter-free calculation and interpolation of EPI matrix elements onto ultra-dense momentum grids (k- and q-points), often utilizing Maximally Localized Wannier Functions (MLWFs).
Iterative BTE Superiority: The iterative solution of the BTE is crucial for accuracy, especially in polar or anisotropic materials (GaAs, GaN), as simpler approximations like the Self-Energy Relaxation Time Approximation (SERTA) often significantly underestimate the true mobility.
2D Material Challenges: Transport in 2D materials is complex due to strong anisotropy (Phosphorene), the dominance of flexural (ZA) phonons (Silicene), and strong layer dependence (InSe), leading to greater discrepancies between calculation and measurement compared to 3D bulk materials.
Future Opportunities: Research is moving toward incorporating many-body effects (GW, DMFT) to improve band structure accuracy, and extending the methodology to quantum materials, including topological insulators and systems leveraging Berry phase effects for spintronics.

Technical Specifications

Calculated and measured intrinsic carrier mobilities at room temperature (300 K) for key materials, alongside critical computational parameters.

Parameter	Value	Unit	Context
Si Electron Mobility (Calc.)	1366	cm²/Vs	BTE/MLWF, Phonon-limited
Si Hole Mobility (Calc.)	658	cm²/Vs	BTE/MLWF, SOC included
GaAs Electron Mobility (Calc.)	8340	cm²/Vs	Iterative BTE, 15% higher than SERTA
GaAs Electron Mobility (Exp. Range)	7200 to 9000	cm²/Vs	Measured intrinsic mobility
w-GaN Electron Mobility (Calc.)	905	cm²/Vs	BTE, SOC included
w-GaN Hole Mobility (Calc.)	44	cm²/Vs	BTE, Low mobility due to heavy mass/acoustic phonons
β-Ga₂O₃ Electron Mobility (Calc.)	115 to 155	cm²/Vs	Limited by LO phonons (~21 meV mode)
MAPbI₃ Average Mobility (Calc.)	80	cm²/Vs	Dominated by polar LO phonons
Monolayer MoS₂ Mobility (Calc. Range)	130 to 410	cm²/Vs	Calculated intrinsic mobility range
Monolayer Silicene Mobility (Calc.)	~1200	cm²/Vs	Free-standing, without ZA phonon coupling
Phosphorene Hole Mobility (Armchair)	586	cm²/Vs	Anisotropic transport, high carrier density
Phosphorene Hole Mobility (Zigzag)	44	cm²/Vs	Anisotropic transport, high carrier density
Computational Scaling (3D BTE)	O(N⁶)	N/A	N = number of grid points along one reciprocal vector
Required k-point Density (GaAs)	400 x 400 x 400	k-points	Required for converged iterative BTE solution

Key Methodologies

The standard ab initio workflow for calculating charge carrier mobility relies on solving the linearized Boltzmann Transport Equation (BTE) and requires several computationally intensive steps:

Electronic Structure Determination:
- Calculate the electronic band structure (eigenvalues and velocities) using Density Functional Theory (DFT).
- For improved accuracy, especially in materials with underestimated band gaps, use many-body corrections like GW perturbation theory.
Phonon Spectrum Calculation:
- Determine the phonon frequencies and eigenvectors using Density Functional Perturbation Theory (DFPT).
Electron-Phonon Interaction (EPI) Matrix Elements:
- Calculate the EPI matrix elements (g_mnv(k, q)) on a coarse grid of electron (k) and phonon (q) momenta using DFPT.
- For polar materials (e.g., GaN, GaAs), explicitly separate and treat the long-range (Fröhlich) interaction component, which diverges as 1/|q| near the Brillouin zone center.
Wannier Interpolation and Grid Densification:
- Employ Maximally Localized Wannier Functions (MLWFs) to efficiently interpolate the electronic structure and the EPI matrix elements from the coarse DFPT grid onto ultra-dense k- and q-point grids (e.g., 10⁸ points). This is essential for converging the scattering integrals.
Solving the Boltzmann Transport Equation (BTE):
- Calculate the total decay rate (scattering rate) τ^-1, which includes scattering out of the state (out-scattering) and scattering back into the state (in-scattering).
- Solve the linearized BTE iteratively to obtain the linear response coefficients (dε_βf_nk), which fully account for the momentum dependence of the scattering processes.
Transport Coefficient Calculation:
- Compute the conductivity tensor (σ_αβ) and subsequently the drift mobility (µ) and Hall factor (r^H) from the converged BTE solution.

Commercial Applications

The predictive ab initio methods reviewed are critical for the rational design and optimization of materials across several high-technology sectors:

High-Power and High-Frequency Devices:
- Materials: Gallium Nitride (GaN), Gallium Oxide (Ga₂O₃), Diamond.
- Application: Designing next-generation power electronics, high-frequency radio frequency (RF) amplifiers, and high-voltage switches where high mobility and large breakdown fields are essential.
Optoelectronics and Energy Conversion:
- Materials: Methylammonium Lead Triiodide Perovskites (MAPbI₃), Gallium Arsenide (GaAs).
- Application: Optimizing charge extraction and transport in high-efficiency solar cells, photodetectors, and light-emitting diodes (LEDs).
Flexible and Nanoscale Electronics:
- Materials: Graphene, Molybdenum Disulfide (MoS₂), Phosphorene, Indium Selenide (InSe).
- Application: Developing ultra-thin transistors, flexible displays, and sensors where intrinsic mobility limits device switching speed and power consumption.
Spintronics and Quantum Materials:
- Materials: Topological Insulators (TIs), materials exhibiting Rashba splitting.
- Application: Calculating spin relaxation rates and designing materials that exploit spin-momentum locking or Berry phase effects to create low-dissipation spin-polarized currents for quantum computing and memory.
Materials Informatics and High-Throughput Screening:
- Application: Integrating predictive mobility calculations into High-Throughput (HT) computational databases (e.g., Materials Project, C2DB) to rapidly screen thousands of candidate materials for specific transport requirements, accelerating the discovery pipeline.

View Original Abstract

One of the fundamental properties of semiconductors is their ability to support highly tunable electric currents in the presence of electric fields or carrier concentration gradients. These properties are described by transport coefficients such as electron and hole mobilities. Over the last decades, our understanding of carrier mobilities has largely been shaped by experimental investigations and empirical models. Recently, advances in electronic structure methods for real materials have made it possible to study these properties with predictive accuracy and without resorting to empirical parameters. These new developments are unlocking exciting new opportunities, from exploring carrier transport in quantum matter to in silico designing new semiconductors with tailored transport properties. In this article, we review the most recent developments in the area of ab initio calculations of carrier mobilities of semiconductors. Our aim is threefold: to make this rapidly-growing research area accessible to a broad community of condensed-matter theorists and materials scientists; to identify key challenges that need to be addressed in order to increase the predictive power of these methods; and to identify new opportunities for increasing the impact of these computational methods on the science and technology of advanced materials. The review is organized in three parts. In the first part, we offer a brief historical overview of approaches to the calculation of carrier mobilities, and we establish the conceptual framework underlying modern ab initio approaches. We summarize the Boltzmann theory of carrier transport and we discuss its scope of applicability, merits, and limitations in the broader context of many-body Green’s function approaches. We discuss recent implementations of the Boltzmann formalism within the context of density functional theory and many-body perturbation theory calculations, placing an emphasis on the key computational challenges and suggested solutions. In the second part of the article, we review applications of these methods to materials of current interest, from three-dimensional semiconductors to layered and two-dimensional materials. In particular, we discuss in detail recent investigations of classic materials such as silicon, diamond, gallium arsenide, gallium nitride, gallium oxide, and lead halide perovskites as well as low-dimensional semiconductors such as graphene, silicene, phosphorene, molybdenum disulfide, and indium selenide. We also review recent efforts toward high-throughput calculations of carrier transport. In the last part, we identify important classes of materials for which an ab initio study of carrier mobilities is warranted. We discuss the extension of the methodology to study topological quantum matter and materials for spintronics and we comment on the possibility of incorporating Berry-phase effects and many-body correlations beyond the standard Boltzmann formalism.

Tech Support

Original Source

DOI: https://doi.org/10.1088/1361-6633/ab6a43