Thermoelectric coefficients of $n$-doped silicon from first principles via the solution of the Boltzmann transport equation

We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearized Boltzmann transport equation, also including the effect of nonequilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system, including electron-phonon interactions; carriers' scattering rates are computed using standard perturbation theory. We exploit Wannier interpolation (both for electronic bands and electron-phonon matrix elements) for an efficient sampling of the Brillouin zone, and the solution of the Boltzmann equation is achieved via a fast and stable conjugate gradient scheme. We discuss the application of this approach to $n$-doped silicon. In particular, we discuss a number of thermoelectric properties such as the thermal and electrical conductivities of electrons, the Lorenz number and the Seebeck coefficient, including the phonon drag effect, in a range of temperatures and carrier concentrations. This approach gives results in good agreement with experimental data and provides a detailed characterization of the nature and the relative importance of the individual scattering mechanisms. Moreover, the access to the exact solution of the Boltzmann equation for a realistic system provides a direct way to assess the accuracy of different flavors of relaxation time approximation, as well as of models that are popular in the thermoelectric community to estimate transport coefficients.

Figure 1

Electron mobility as a function of carrier concentration at $T=300\mathrm{K}$. The solid red line is obtained by solving the BE that includes both impurity and phonon scattering. The dashed green line (dot-dashed blue line) is from the solution of the BE with just phonon (charged impurity) scattering. The experimental data are taken from Ref. [37].

Figure 2

Electron mobility as a function temperature $\mathrm{at}n={10}^{14} {\mathrm{cm}}^{-3}$. The solid red line is obtained from the solution of the BE. The experimental data are from Canali et al. [40] (diamonds), Logan et al. [41] (squares), and Norton et al. [42] (triangles).

Figure 3

Electrical resistivity as a function of carrier concentration at $T=300\mathrm{K}$. Experimental data from Ref. [43].

Figure 4

Relative contribution of the different scattering mechanisms to electrical resistivity at $T=300\mathrm{K}$. Carriers' concentrations are $2.4\times {10}^{14}$ (upper panel) and $1.3\times {10}^{19}{\mathrm{cm}}^{-3}$ (lower panel). The charged impurities' contribution (Imp), intra-valley phonon scattering (Intra) and intervalley phonon scattering (f and g stand for f processes and g processes, described in Ref. [20]) are shown. The phonon scattering are separated in acoustic (dark red) and optical (light yellow) branches, and different patterns are used for the modes: diagonal lines for TA1 and TO1, dense diagonal lines for TA2 and TO2 and vertical lines for LA and LO. The phonon energies of the different modes are also shown.

Figure 5

Seebeck coefficient as a function of carrier concentration at $T=300\mathrm{K}$. The solid red line is the total Seebeck coefficient, the red dashed line is the diffusive contribution, the orange dotted line is the CRTA result, the green dot-dash-dashed (blue dot-dot-dashed) line is the diffusive $S$ computed including only phonon (impurity) scattering. The solid black diamonds are the experimental results [14], while the open diamonds represent the diffusive Seebeck extracted from the experimental data [14, 15].

Figure 6

Contribution of phonon modes to the phonon drag part of the Seebeck coefficient at $T=300$ K and for $n=1.75\times {10}^{14}{\mathrm{cm}}^{-3}$. The solid red line is the contribution from all the modes; the blue long-dashed (green short-dashed) line is the contribution from TA (LA) phonons.

Figure 7

Seebeck coefficient as a function of temperature for $n=1.75\times {10}^{14}{\mathrm{cm}}^{-3}$. The red open circles are the total theoretical Seebeck coefficient while the dashed red line is the diffusive part of $S$. The black diamonds are the experimental data [14].

Figure 8

Electronic thermal conductivity as a function of carrier concentration at $T=300\mathrm{K}$. The red solid line is from the solution of the BE that includes all the scattering mechanisms and the phonon drag. The blue long-dashed line is the same as the red solid line but without the phonon drag.

Figure 9

Lorenz number $L$ as a function of carrier concentration at $T=300\mathrm{K}$. The red solid line is from the solution of the BE that includes all the scattering mechanisms and the phonon drag. The red long-dashed line is the same as the red solid line but without the phonon drag. The green dot-dash-dashed (blu dot-dot-dashed) line is from the BE including only phonon (impurity) scattering. The orange short-dashed line is the CRTA. The black dotted horizontal line is the Wiedemann-Franz result.

Figure 10

Lorenz number $L$ as a function of Seebeck coefficient $S$ at $T=300\mathrm{K}$. The red solid line (dashed blue line) is from the solution of the BE with (without) the phonon drag. The green dot-dashed line is the model proposed in Ref. [48]. The black dotted horizontal line is the Wiedemann-Franz result.

Figure 11

Performance of the CG algorithm. (Top) Resistivity (normalized to the converged result ${\rho }_C$) as a function of the number of iterations. (Bottom) Deviation from the Kelvin relation as a function of the number of iterations. The dashed blue (solid red) line is the result of the CG (with preconditioning). The results are for $n=1.75\times {10}^{14} {\mathrm{cm}}^{-3},T=300$ K and with phonons in thermal equilibrium.

Figure 12

Electron mobility at room temperature and for $n=1.6\times {10}^{15}{\mathrm{cm}}^{-3}$ as a function of the number of $k$ points used to solve the BE. Different electron-phonon coupling grids are used: $N_{\mathrm{EPC}}=70$ (red empty symbols), 90 (blue patterned symbols), and 110 (green full symbols). The values of ${\sigma }_g$ used are 1 (triangles), 4 (circles), and 7 meV (squares). The results obtained with less (more) than ${10}^4 \left(5\times {10}^4\right) k$ points on the $x$ axis are for $p=1 \left(p=3\right)$; the remaining points are for $p=2$. The energy window used is 0.15 eV.

Figure 13

$L$ and $S$ as a function of the energy window used to compute the integrals of the transport coefficients in Eq. (7). (Top) Relative accumulation of $S$ (solid blue line) and $L$ (dashed red line) computed under the assumption of phonons in thermal equilibrium. (Bottom) $L$ as a function of the energy window for different scattering mechanisms. Dashed red line corresponds to $L$ computed with all scattering terms, while the solid green line (dotted blu line) is for the impurity (phonon) contribution. In all cases, the BE at room temperature and for $n={10}^{14} {\mathrm{cm}}^{-3}$ is solved for all the states up to 0.32 eV above the bottom of the conduction band.

Figure 14

Intrinsic electron mobility as a function of temperature computed with different approaches: the solid red line is from solution of the BE (CG), the dashed green line is the relaxation time approximation (RTA), the dot-dashed blu line (dotted orange line) is obtained using ${\chi }_m\left(\mathit{k}\right)\propto v_m^{\left|\right|}\left(\mathit{k}\right){\tau }_{m\mathit{k}}^{\mathrm{RTA}} \left({\chi }_m\left(\mathit{k}\right)\propto v_m^{\left|\right|}\left(\mathit{k}\right)\right)$ as a trial function in the variational formula Eq. (14). The symbols are the experimental data as in Fig. 2.