Boltzmann-Bloch Equation Approach to the Theory of the Optical Inter- and Intraband Response in Noble Metals

Abstract

In this paper we introduce momentum-resolved metal Boltzmann-Bloch equations (MBBE) for the combined description of electronic intra- and interband processes in noble metals. This microscopic framework incorporates a full treatment of many-body electron-electron and electron-phonon interactions, relevant for relaxation and dephasing processes after optical excitation. For the example of gold, we calculate the linear optical response for near-infrared and visible energies. This provides insight into the interplay of microscopic processes hidden in phenomenological Drude-Lorentz models. The complex geometry of the Fermi surface is treated by an anisotropic electronic dispersion model, which is necessary to explain the temperature dependent spectrum over the whole frequency range of intra- and interband transitions.

Figures (11)

• Figure 1: Schematic illustration of the anisotropic dispersion relation $\varepsilon^{\lambda P}_{k_{\perp}k_{\parallel}}$ in the vicinity of high symmetry points $P = X,L$ of the valence band (orange, $\lambda = v$) and conduction band (green, $\lambda = c$) with Fermi energy $E_F$. The light blue circles and the purple line respectively depict the electron occupations $f_{\mathbf{k}}^{\lambda\sigma}$ and the interband transitions $p_{\mathbf{k}}^{vc\sigma}$. The blue arrows account for excitation processes from occupied states into unoccupied states either within the conduction band (intraband processes) or from the valence into the conduction band (interband processes) induced by an external electrical field $\mathbf{E}$, cp. discussion in Sec. \ref{['sec:theo_background']}. Many-body relaxation ($\gamma_{\mathbf{k}}^{\lambda}$) of the occupations and dephasing processes ($\gamma_{\mathbf{k}}^{p}$) of the interband transitions by electron-phonon and electron-electron scattering processes, are schematically indicated by the red arrows. The Fermi surface of the anisotropic dispersion model is shown in the upper left corner, assembled from the contributions of all fourteen $X$ and $L$ sections of the fcc Brillouin zone.
• Figure 2: The electronic band structure of the $5d$ valence- and $6sp$ conduction band in gold, calculated ab initio by Rangel et al. rangel_band_2012 (dotted lines in (i-iii)), is quadratically expanded in the vicinities of the $X$ and $L$ high symmetry points in both directions towards the $\Gamma$ ($k_{\parallel}$) and the $W$ point ($k_{\perp}$). Three different wave number ranges of optimal fit $\Delta k$, indicated by the shaded areas, are compared. In (i) we consider the entirety of the dispersion along the $PW$ paths ($P \in \{X,L\}$) such that it holds $\Delta k = \overline{PW}$. This allows for the fit to be in good global agreement with the dispersion. In (ii) we choose $\Delta k = 0.6\overline{PW}$ and in (iii) $\Delta k = 0.2\overline{PW}$, resulting in a more accurate fit in the vicinities of the $X$ and $L$ points at the cost of a worse overall agreement with the band structure.
• Figure 3: The first Brillouin zone is approximated by fourteen high symmetry cones $C^P$ with circular base of radius $R_P$ (shown here for one hexagonal $L$-face of the Brillouin zone), such that the area of the circular base is equal to the area of each face. The coordinates of the momentum $\mathbf{k} = (k_{\parallel},k_{\perp},\phi)$ are defined in Eq. (\ref{['eq:dispersion_coordinates']}), relative to the $X$ or $L$ high symmetry point. The rotation of the red shaded triangle $T^{L}$ around the $\Gamma L$ axis generates the cone $C^L$.
• Figure 4: The electronic DOS of the anisotropic two-band model for the wave number ranges of optimal fit in Fig. \ref{['fig_dispersion']}(i-iii) is compared to the free electron DOS as well as to experimental data from Smith et al. smith_photoemission_1974. The inset shows a zoom-in for energies relevant for this paper.
• Figure 5: Temperature dependence of the (a,b) electron-phonon and (c,d) electron-electron relaxation rates. In (a) and (c) the rates are shown from 75 K to 700 K as function of the wave number $k$. In (b) and (d) the rates are evaluated at the Fermi wavenumber $k_F=12.0~\text{nm}^{-1}$ and shown as a function of temperature (orange lines). The electron-phonon and electron-electron rates are respectively linearly and quadratically fitted (blue dashed lines).