Anharmonic vibrational properties in periodic systems: energy, electron-phonon coupling, and stress

A unified approach is used to study vibrational properties of periodic systems with first-principles methods and including anharmonic effects. Our approach provides a theoretical basis for the determination of phonon-dependent quantities at finite temperatures. The low-energy portion of the Born-Oppenheimer energy surface is mapped and used to calculate the total vibrational energy including anharmonic effects, electron-phonon coupling, and the vibrational contribution to the stress tensor. We report results for the temperature dependence of the electronic band gap and the linear coefficient of thermal expansion of diamond, lithium hydride, and lithium deuteride.

Figure 1

Harmonic and anharmonic BO energy surfaces per unit cell for an optical $\Gamma$-point phonon in (a) diamond and (b) H${}^7$Li. The leading anharmonic term for diamond is cubic, and that for H${}^7$Li is quartic. For a harmonic ground-state wave function (Gaussian), the one-sigma amplitude is $9.1$ a.u. for diamond and $13.7$ a.u. for H${}^7$Li. In real space, the one-sigma amplitudes correspond to 0.043 a.u. for diamond and 0.299 a.u. for H${}^7$Li.

Figure 2

Anharmonic energy correction as a function of the number of normal modes (supercell size) for diamond, H${}^7$Li, and D${}^7$Li. The anharmonic correction is larger for the lighter elements.

Figure 3

Anharmonic two-dimensional BO energy surface per unit cell for a pair ($q_1,q_2$) of optical modes at the $\Gamma$ point for H${}^7$Li. We plot (a) the exact energy surface and (b) the approximate energy surface for independent phonons only. For a harmonic ground-state wave function (Gaussian), the one-sigma amplitude is $13.7$ a.u.

Figure 4

Temperature dependence of the thermal band gap $E_{\mathrm{g}}$ of diamond. The DFT result (red solid curve) is offset to match experimental data (black diamonds) at zero temperature. The experimental data are from Ref. 44.

Figure 5

Temperature dependence of the thermal band gap $E_{\mathrm{g}}$ of lithium hydride for the isotopes H${}^7$Li (red) and D${}^7$Li (green) calculated within DFT. The black double-dashed-dotted line indicates the value of the static band gap. Dashed lines refer to the renormalized gaps due to electron-phonon interactions, dashed-dotted lines refer to the renormalization due to thermal expansion with temperature, and the solid lines are the sum of the two contributions.

Figure 6

Temperature dependence of the coefficient of linear expansion $\alpha$ for diamond. The black diamonds are experimental results from Ref. 55. The inset shows the temperature dependence of the lattice parameter.

Figure 7

Temperature dependence of the coefficient of linear expansion $\alpha$ for lithium hydride with isotopic compositions H${}^7$Li and D${}^7$Li. The experimental data are from Ref. 56. The inset shows the temperature dependence of the lattice parameters.