One-shot calculation of temperature-dependent optical spectra and phonon-induced band-gap renormalization

Recently, Zacharias et al. [Phys. Rev. Lett. 115, 177401 (2015)] developed an ab initio theory of temperature-dependent optical absorption spectra and band gaps in semiconductors and insulators. In that work, the zero-point renormalization and the temperature dependence were obtained by sampling the nuclear wave functions using a stochastic approach. In the present work, we show that the stochastic sampling of Zacharias et al. can be replaced by fully deterministic supercell calculations based on a single optimal configuration of the atomic positions. We demonstrate that a single calculation is able to capture the temperature-dependent band-gap renormalization including quantum nuclear effects in direct-gap and indirect-gap semiconductors, as well as phonon-assisted optical absorption in indirect-gap semiconductors. In order to demonstrate this methodology, we calculate from first principles the temperature-dependent optical absorption spectra and the renormalization of direct and indirect band gaps in silicon, diamond, and gallium arsenide, and we obtain good agreement with experiment and with previous calculations. In this work we also establish the formal connection between the Williams-Lax theory of optical transitions and the related theories of indirect absorption by Hall, Bardeen, and Blatt, and of temperature-dependent band structures by Allen and Heine. The present methodology enables systematic ab initio calculations of optical absorption spectra at finite temperature, including both direct and indirect transitions. This feature will be useful for high-throughput calculations of optical properties at finite temperature and for calculating temperature-dependent optical properties using high-level theories such as $GW$ and Bethe-Salpeter approaches.

Figure 1

Absorption coefficient of (a) Si, (b) C, and (c) GaAs at room temperature. Calculations with the atoms clamped at their equilibrium positions are shown as blue dashed lines. Calculations using the WL method in the atomic configuration specified by Eq. (5) are shown as red solid lines. The experimental data for Si are from Ref. [29] (grey discs), those for C are from Refs. [30] (grey discs) and [33] (grey circles). Experimental data for GaAs are from Refs. [34] (grey discs) and [35] (grey circles). The thin vertical lines indicate the direct and indirect band gaps calculated for nuclei in their equilibrium positions. The calculations were performed using $8\times 8\times 8$ supercells, using a Gaussian broadening of 30 meV for Si and C and of 50 meV for GaAs.

Figure 2

Convergence of the WL dielectric function as a function of supercell size: (a) Si, (b) C, and (c) GaAs. The grey curves are for calculations using $2\times 2\times 2$ supercells; blue solid lines indicate calculations using $4\times 4\times 4$ supercells; thick red solid lines are for $8\times 8\times 8$ supercells. Each curve was obtained using a single calculation, in the configuration specified by Eq. (5). For Si and C the electronic Brillouin zone was sampled using 1920, 240, and 30 k points for the $2\times 2\times 2,4\times 4\times 4$, and $8\times 8\times 8$ supercells, respectively. In the case of GaAs, a finer sampling was required, and we used 6400, 800, and 100 k points, respectively. The calculations were performed using a Gaussian broadening of 30 meV for Si and C and of 50 meV for GaAs. The insets show the band gap extracted from the Tauc plots. These values automatically include zero-point renormalization.

Figure 3

Effect of using multiple atomic configurations in the evaluation of the WL dielectric function of silicon. (a) Comparison between calculations performed using a single atomic configuration as specified by Eq. (5) (red solid lines), and calculations using this configuration and its antithetic pair from Eq. (12) (blue solid line). We report calculations for increasing supercell size. (b) Comparison between calculations performed using a single configuration as in (a) (red solid lines), and those performed using four configurations, as specified by Eqs. (5), (12), and (13). The curves corresponding to the $8\times 8\times 8$ supercell have been rigidly shifted by 0.1 eV: without such a shift the curves would overlap with those obtained using a $4\times 4\times 4$ supercell.

Figure 4

(a) Comparison between measured and calculated optical absorption coefficients of silicon at 300 K. The grey dots are experimental data from Ref. [29], the red line is a calculation using the Hall-Bardeen-Blatt theory, and the blue line is a calculation using the Williams-Lax theory. (b) Comparison between measured absorption coefficient of diamond at 300 K [data from Refs. [30] (discs) and [33] (circles)] and our calculations using the HBB theory or the WL theory. The color code is the same as in (a). In both panels, the calculations were performed on $4\times 4\times 4$ supercells, using 65 random points in the electronic Brillouin zone (i.e., $>4000$ points in the Brillouin zone of the crystalline unit cell), and a Gaussian smearing of 50 meV. The thin vertical lines indicate the energy of the band gaps calculated at the equilibrium geometries.

Figure 5

(a) Tauc plot for determining the indirect band gap of silicon as a function of temperature. The red lines show ${\left({\omega }^2{\varepsilon }_2\right)}^{1/2}$, and the thin black lines are the corresponding linear fits at each temperature. The indirect gap is obtained from the intercept with the horizontal axis. (b) Temperature-dependence of the indirect band gap of silicon: present theory (red discs) and experimental data from Ref. [48] (grey discs). The solid line is a guide to the eye. The theoretical values were corrected to match the zero-point renormalization calculated as discussed in Appendix  pp3. (c) Convergence of the zero-point renormalization of the indirect gap of silicon with respect to the Brillouin-zone sampling of an $8\times 8\times 8$ supercell. (d) Calculated second-derivatives of the real part of the dielectric function of silicon. (e) Temperature-dependence of the direct gap of silicon: present calculations (red discs) and experimental data from Ref. [49]. The solid line is a guide to the eye. (f) Convergence of the zero-point renormalization of the direct gap of silicon with respect to the Brillouin-zone sampling. All calculations in this figure were performed using an $8\times 8\times 8$ supercell.

Figure 6

(a) Tauc plot to determine the indirect gap of diamond vs temperature. The red/black lines indicate ${\left({\omega }^2{\varepsilon }_2\right)}^{1/2}$ and the corresponding linear fits at each temperature. The indirect gap is obtained from the intercept with the horizontal axis. In this case the linear fits were performed in the range of photon energies $\hslash \omega =0$-5.6 eV. (b) Temperature dependence of the indirect gap of diamond: this work (red discs) and experimental data from Ref. [33] (grey discs). The solid line is a guide to the eye. The theoretical values were corrected to match the zero-point renormalization calculated as discussed in Appendix  pp3. (c) Convergence of the zero-point renormalization of the indirect gap of diamond with respect to the Brillouin-zone sampling (for an $8\times 8\times 8$ supercell). (d) Calculated ${\partial }^2{\varepsilon }_1(\omega ;T)/\partial {\omega }^2$ for diamond at two different temperatures. The $E_0^{\prime }$ transition is identified using the deep minimum, following Ref. [50]. (e) Temperature-dependent direct band gap of diamond: current calculations (red discs) and experimental data from Ref. [50]. We report the experimental data corresponding to both diamond samples $\mathrm{II}a$ (grey circles) and $\mathrm{II}b$ (grey discs) used in Ref. [50]. (f) Convergence of the zero-point renormalization of the direct band gap of diamond with respect to Brillouin-zone sampling. All calculations were performed using an $8\times 8\times 8$ supercell.

Figure 7

Density of states near the valence band edge of diamond at 0 K. Calculations were performed on $2\times 2\times 2$ (green), $4\times 4\times 4$ (red), and $8\times 8\times 8$ (blue) supercells using the one-shot WL method. We used 2560, 320 and 40 k points, respectively, and a Gaussian broadening of 15 meV. The broken degeneracy of the valence band top, which can be seen for the $2\times 2\times 2$ supercell, tends to vanish with increasing supercell size.

Figure 8

(a) Direct optical absorption onset in GaAs for various temperatures, as calculated using the WL theory in a $8\times 8\times 8$ supercell. The calculations were performed in the quasiharmonic approximation. The red curves correspond to ${\left({\omega }^2{\varepsilon }_2\right)}^2$ and the thin black lines are the corresponding linear fits (within the range of photon energies 1.42–1.62 eV). (b) Temperature dependence of the band gap of GaAs: calculations in the quasiharmonic approximation (red discs), calculations without considering the lattice thermal expansion (blue discs), and experimental data from Ref. [51] (grey discs). The lines are guides to the eye. (c) Convergence of the calculated zero-point renormalization with respect to the number of $\mathbf{k}$ points, in a $8\times 8\times 8$ supercell.

Figure 9

Square-root of the joint density of states of (a) silicon, (b) diamond, and (c) gallium arsenide. The calculations were performed with nuclei clamped in their equilibrium positions (blue lines), and using the one-shot WL method at $T=0$ K (red lines). The horizontal offset between blue and red curves corresponds to the zero-point renormalization of the band gap in each case. The calculations for the equilibrium structures were performed in the primitive unit cells, using 20 480, 20 480, and 51 200 random k points for Si, C, and GaAs, respectively. The WL calculations were performed on $8\times 8\times 8$ supercells, using 40, 40, and 100 random k points for Si, C, and GaAs, respectively. A Gaussian broadening of 30 meV was used in all plots.

Figure 10

Density of states of diamond close to the valence and conduction band edges: calculation at the equilibrium geometry (green line), one-shot WL calculation (red line), and AH calculation (blue line). The latter two calculations were performed for $T=300$ K, using a $4\times 4\times 4$ supercell and 65 k points in the Brillouin zone of the supercell. The calculation with the atoms at their equilibrium geometry was carried out within the primitive unit cell, using 4160 k points. A Gaussian smearing of 30 meV was used for all curves.