Role of electron-phonon interactions versus electron-electron interactions in the broadening mechanism of the electron and hole linewidths in bulk Be

We report on ab initio study of electron-phonon $\left(e\text{−}\mathrm{ph}\right)$ and electron-electron $\left(e\text{−}e\right)$ interactions in bulk Be. The calculations show that the $e\text{−}\mathrm{ph}$ coupling parameter $\lambda$ varies from 0.01 to 1.02 as a function of electron energy and momentum, $\lambda =0.21$ at the Fermi level as averaged over momenta. The $e\text{−}\mathrm{ph}$ contribution ${\Gamma }_{e\text{−}\mathrm{ph}}$ to the electrons and holes lifetime broadening also manifests clear dependence on the momentum and energy of an electron state. We demonstrate that the $e\text{−}\mathrm{ph}$ coupling matrix elements strongly affect the Eliashberg function especially for low phonon frequencies. By using the Debye model relation between ${\Gamma }_{e\text{−}\mathrm{ph}}$ and $\lambda$ the characteristic Debye frequency ${\omega }_D$ is obtained in good agreement with the experimental one. The $e\text{−}e$ contribution to the lifetime broadening is evaluated from the imaginary part of the electron self-energy computed within a GW approximation.

Figure 1

LDA energy bands along the symmetry directions and density of electron states (right panel) for hcp Be.

Figure 2

Calculated phonon dispersion curves along the high-symmetry directions for hcp Be. The filled circles indicate the experimentally measured frequencies (Ref. 48).

Figure 3

Phonon density of states $F\left(w\right)$ (dashed line) and electron-phonon spectral function ${\alpha }^{2}F\left(w\right)$ averaged over the Fermi surface (solid line) for Be. The coupling function ${\alpha }^2$ [defined as the ratio ${\alpha }^{2}F\left(w\right)∕F\left(w\right)$] is shown by dotted line.

Figure 4

The electron-phonon spectral function ${\alpha }^2{F}_{\mathbf{k},i}\left(w\right)$ is shown for some electron states at the $\Gamma$ (a), $H$ (c), $K$ (d) symmetry points and at the Fermi level (b) in the $\Gamma M$ direction.

Figure 5

The $T=0$ phonon induced lifetime broadening ${\Gamma }_{e\text{−}\mathrm{ph}}$ as a function of the energy for ${\Sigma }_3$ (a) and ${\Sigma }_1$ (b) energy bands in the $\Gamma M$ direction and the energy dependence of the electron-phonon coupling parameter $\lambda$ (c), (d) as well as of the characteristic Debye frequency ${\omega }_D$ (e), (f) calculated from equation ${\Gamma }_{e\text{−}\mathrm{ph}}=2\pi \lambda \hslash {\omega }_D∕3$ for these states.

Figure 6

The temperature dependence of the phonon induced lifetime broadening for the ${\Gamma }_3^{+}$, $K_5^1$, $M_4^{-}$ (solid lines) and the ${\Gamma }_4^{-}$, $K_5^2$, $M_2^{-}$ (dashed lines) electron states.

Figure 7

The linewidth ${\Gamma }_{e\text{−}e}$ of holes and excited electrons for Be is given versus the square of the energy. In the left panel ${\Gamma }_{e\text{−}e}$ is shown (diamonds) for states located within $4\mathrm{eV}$ of $E_F$ and in the right panel ${\Gamma }_{e\text{−}e}$ is given for states within $8\mathrm{eV}$ of $E_F$. The double-dotted-dashed curve is the result of Eq. (9) for a Wigner-Seitz radius of $r_s=1.87$. The solid line is the result of a full jellium $GW$ calculation for the same value of $r_s$.

Figure 8

Linewidth of holes and excited electrons due to electron-electron scattering in Be plotted versus the energy measured from the Fermi energy for wave vectors from the different directions in the BZ. Each plot contains the data (filled diamonds) for the first four bands. The dashed lines are a guide to the eye. The solid lines denote the result of a full jellium $GW$ calculation for $r_s=1.87$.

Figure 9

As Fig. 8 but here the linewidth is plotted linearly versus the energy.

Figure 10

The lifetime of holes and excited electrons for the bands ${\Sigma }_1$ (diamonds) and ${\Sigma }_3$ (circles) in fs plotted versus the energy measured from the Fermi energy. The solid line denotes the result for a full jellium $GW$ calculation for $r_s=1.87$.