Ab initio electronic transport model with explicit solution to the linearized Boltzmann transport equation

Accurate models of carrier transport are essential for describing the electronic properties of semiconductor materials. To the best of our knowledge, the current models following the framework of the Boltzmann transport equation (BTE) either rely heavily on experimental data (i.e., semiempirical), or utilize simplifying assumptions, such as the constant relaxation time approximation (BTE-cRTA). While these models offer valuable physical insights and accurate calculations of transport properties in some cases, they often lack sufficient accuracy—particularly in capturing the correct trends with temperature and carrier concentration. We present here a transport model for calculating low-field electrical drift mobility and Seebeck coefficient of $n$-type semiconductors, by explicitly considering relevant physical phenomena (i.e., elastic and inelastic scattering mechanisms). We first rewrite expressions for the rates of elastic scattering mechanisms, in terms of ab initio properties, such as the band structure, density of states, and polar optical phonon frequency. We then solve the linear BTE to obtain the perturbation to the electron distribution—resulting from the dominant scattering mechanisms—and use this to calculate the overall mobility and Seebeck coefficient. Therefore, we have developed an ab initio model for calculating mobility and Seebeck coefficient using the Boltzmann transport (aMoBT) equation. Using aMoBT, we accurately calculate electrical transport properties of the compound $n$-type semiconductors, GaAs and InN, over various ranges of temperature and carrier concentration. aMoBT is fully predictive and provides high accuracy when compared to experimental measurements on both GaAs and InN, and vastly outperforms both semiempirical models and the BTE-cRTA. Therefore, we assert that this approach represents a first step towards a fully ab initio carrier transport model that is valid in all compound semiconductors.

Figure 1

Band structure of cubic GaAs and wurtzite InN, normalized so that the Fermi level is set to zero at the conduction band minimum.

Figure 2

Conduction bands expressed in terms of the average energy as a function of distance from the CBM (i.e., center of Brillouin zone, or $\mathrm{\Gamma }$ point), as calculated from semiempirical expressions (in $\mathbf{k}·\mathbf{p}$ formulation) versus ab initio. The difference at higher $k$ values has a significant impact on transport properties, especially at high temperatures. The values of $U$ for the $d$ orbitals of gallium and arsenic, respectively, are in parentheses, while those for InN are taken from the published literature [49, 50].

Figure 3

Calculated and experimental [3, 64] mobility data for GaAs at various electron concentrations and temperatures. More details on the experimental data, including donor and acceptor concentrations, are available in Table 4. (a) Comparison of model trends with varying temperature and electron concentration to experimental data. The values of U used for the $d$-orbitals of Gallium and Arsenic, respectively, are in parentheses. (b) Limitation of ”pure” GaAs mobility from each scattering mechanism.

Figure 4

Electrical conductivity of GaAs calculated using aMoBT (solid line) and the BTE-cRTA framework, and compared to experimental [3] data. The Fermi level is calculated by matching the calculated carrier concentration to $n=3\times {10}^{13}$. This has been done either at the mentioned temperature and kept constant over the whole temperature range, or in the case of “matched Fermi,” at each temperature, the Fermi level is adjusted to the given $n$. The relaxation time, $\tau$, is determined by fitting the calculated conductivity to the corresponding experimental value at 300 K. The calculated value for $\tau$ is $5\times {10}^{-13}$ s.

Figure 5

Calculated (by aMoBT) and experimental [4] GaAs Seebeck coefficient, at different ratios of the ionized impurity concentration, $N_{ii}$, to the electron concentration, $n$. For the Pisarenko plot, using Eq. (14), we have used values of $r=-\frac{1}{2}$ for acoustic phonons, and $m^{*}=0.063$. The Fermi levels for BTE-cRTA calculations were obtained by integrating the density of state to obtain the corresponding carrier concentrations; thus the same band structure and Fermi levels have been used to obtain the black and green lines.

Figure 6

Calculated (by aMoBT) and experimental transport properties of the three InN samples listed in Table 5. The dashed line is calculated by the semiempirical model used by Miller et al. [8], while the solid line is calculated by the proposed ab initio transport model (aMoBT).

Figure 7

Calculated and experimental [8] values for InN mobility at $n=9\times {10}^{17}{\text{cm}}^{-3}$ (sample B in Table 5). Each line represents the mobility if limited only by the corresponding mechanism.

Figure 8

Phonon band structures of InN and GaAs calculated by Phonopy [31].

Figure 9

Sensitivity analysis of the mobility of GaAs pure sample (see Table 4). We changed here only the static, ${\epsilon }_s$, and high frequency, ${\epsilon }_{\infty }$, dielectric constant from $-20\%$ to +20% of the base case values reported in Table 3. The results are sensitive to dielectric constant at low and high temperatures.

Figure 10

Sensitivity analysis of the mobility of GaAs pure sample (see Table 4) calculated by aMoBT. We changed here the crystal structure and subsequently the newly calculated optical phonon frequencies. The calculated mobility is sensitive to the strain at all temperatures.