Electron-Phonon Renormalization of the Direct Band Gap of Diamond

We calculate from first principles the temperature-dependent renormalization of the direct band gap of diamond arising from electron-phonon interactions. The calculated temperature dependence is in good agreement with spectroscopic ellipsometry measurements, and the zero-point renormalization of the band gap is found to be as large as 0.6 eV. We also calculate the temperature-dependent broadening of the direct absorption edge and find good agreement with experiment. Our work calls for a critical revision of the band structures of other carbon-based materials calculated by neglecting electron-phonon interactions.

Figure 1

Electron-phonon diagrams leading to the temperature dependence of the electron energy: self-energy diagram (left) and Debye-Waller diagram (right). Solid and curved lines represent the electron and phonon Green’s function, respectively. The circles in the SE diagram represent the standard electron-phonon matrix element, the circle in the DW diagram represents a one electron-two phonons matrix element.

Figure 2

Temperature dependence of the direct band gap of diamond: calculated gap (black disks), and experimental data from Ref. 10. The calculated band gap is obtained by adding the SE and DW thermal shifts to our $GW$ gap $E_g^{GW}=7.715\mathrm{eV}$. The $GW$ gap is calculated using the method of Ref. 1 as implemented in the BerkeleyGW code [33]. The experimental data are from the first-derivative line shape analysis of the type IIA sample [(red) circles] and from the second-derivative line shape analysis of the same sample [(blue) squares].

Figure 3

Breakdown of the temperature shift of the direct band gap of diamond in terms of SE [(blue) dotted line] and DW [(red) dashed line] contributions. The circles are the total shift as in Fig. 2 and the solid black line is the fit with a Bose-Einstein law $\Delta E_g(T)=-a[1+2(e^{\Theta /T}-1{)}^{-1}]$, with $a=615\mathrm{meV}$ and $k_B\Theta =118\mathrm{meV}$. The nonmonotonic behavior of the SE term results from the slightly different temperature dependence of the ${\Gamma }_{25v}^{\prime }$ and the ${\Gamma }_{15c}$ states, and from the fact that the gap renormalization is obtained through their difference. The different temperature dependence can be traced back to different electron-phonon couplings in Eq. (2), and hence to the different character of the ${\Gamma }_{25v}^{\prime }$ and ${\Gamma }_{15c}$ states in diamond.

Figure 4

Temperature-dependent broadening of the direct absorption edge of diamond. The black disks are our data calculated using Eq. (7). The (red) circles and (blue) squares are the experimental data from the first- and second-derivative line shape analysis of the type IIA sample in Ref. 10, respectively.