Plasmon dispersions in simple metals and Heusler compounds

We present a comprehensive study of plasmon dispersions in simple metals and Heusler compounds based on an accurate ab initio evaluation of the momentum- and frequency-dependent dielectric function $\varepsilon (\vec{q},\omega )$ in the random-phase approximation. Using a momentum-dependent tetrahedron method for the computation of the dielectric function, we extract and analyze “full” and “intraband” plasmon dispersions: The “full” plasma dispersion is obtained by including all bands, the intraband plasma dispersion by including only intraband transitions. For the simple metals silver and aluminum, we show that the intraband plasmon dispersion has an unexpected downward slope and is therefore markedly different from the results of an effective-mass electron-gas model and the full plasmon dispersion. For the two Heusler compounds ${\mathrm{Co}}_2\mathrm{Fe}\mathrm{Si}$ and ${\mathrm{Co}}_2\mathrm{Mn}\mathrm{Si}$, we present spectra for the dielectric function, their loss functions, and plasmon dispersions. The latter exhibit the same negative intraband plasmon dispersion as found in the simple metals. We also discuss the influence of spin mixing on the plasmon dispersion.

Figure 1

Computed real (black solid line) and imaginary (black dashed line) part of the dielectric function $\varepsilon (\vec{q},\omega )$ for $\left|\vec{q}\right|=0.44{\mathrm{nm}}^{-1}$. (a) Results for Al in comparison to optical measurements (thin blue lines) (Refs. 7 and 26). (b) Results for Ag in comparison to optical measurements (Ref. 8) (thin blue lines). The red dotted line shows the intraband $\Re \varepsilon (\vec{q},\omega )$ whose zero determines the intraband plasmon resonance.

Figure 2

Imaginary part of the dielectric function of (a) aluminum and (b) silver for small energies and different $\vec{q}$. The black solid lines correspond to $|\vec{q}|=0.44{\mathrm{nm}}^{-1}$ ($|\vec{q}|=0.44{\mathrm{nm}}^{-1}$), the red dashed lines to $|\vec{q}|=0.88{\mathrm{nm}}^{-1}$ ($|\vec{q}|=0.87{\mathrm{nm}}^{-1}$), and the blue dotted lines to $|\vec{q}|=1.32{\mathrm{nm}}^{-1}$ ($\left|\vec{q}\right|=1.31{\mathrm{nm}}^{-1}$) in the aluminum (silver) calculation. Also shown are $q=0$ dielectric functions obtained directly from the elk code for two different broadenings $\hslash \gamma =0.01$ Ha (thin gray line) and $\hslash \gamma =0.005$ Ha (gray thin-dashed line).

Figure 3

Loss function of aluminum on a large (a) and a small (b) energy scale. The arrows mark the peaks due to the acoustic plasmon resonance. The wave vectors are the same as in Fig. 2.

Figure 4

Loss function of silver on a large (a) and a small (b) energy scale. The arrow in (a) marks the optical plasmon and the arrows in (b) correspond to the edges of the acoustic plasmon signature. The wave vectors are the same as in Fig. 2.

Figure 5

Different plasmon dispersions (red markers) and top of the intraband electron-hole continuum (blue crosses) of aluminum. Shown are the effective plasma frequency (diamonds), intraband plasma frequency (open circles) and intraband plasma frequency without transitions between Kramers pairs of states (open squares) using ${17}^3$ and ${61}^3$ $\vec{k}$ points in the 1. BZ. The scatter of the continuum and the plasma frequencies is due to the different $\vec{q}$ vectors with the same modulus. The black bar at $q=0$ denotes the range of the published values for ${\omega }_{\mathrm{Pl}}\left(\vec{q}\to 0\right)$ (Refs. 7, 38, and 39), and the thin solid line corresponds to the experimental results of Ref. 37. Acoustic plasmons in the $\Gamma \mathrm{L}$ direction are indicated by red “+”.

Figure 6

Same as Fig. 5 for silver. The first two roots of ${\varepsilon }_1(\vec{q},\omega )$ (full calculation) in the $\Gamma \mathrm{L}$ direction are plotted as open diamonds. These roots give rise to the optical plasmon resonance marked in Fig. 4a. Published theoretical results for ${\omega }_{\mathrm{Pl}}\left(\vec{q}\right)$ are indicated for $q=0$ (black bar on the energy axis) (Refs. 39, 42, 43, 44). The measured curves for $q>0$ from Refs. 16 and 41 (black lines) are indistinguishable on this energy scale.

Figure 7

Computed real (black solid line) and imaginary (black dashed line) part of the dielectric function of CFS (a) and CMS (b) for $q=0.55{\mathrm{nm}}^{-1}$, where we used $35\times 35\times 35$ $\vec{k}$ points in the (full) 1. BZ and included the first 62 (60) bands in the CFS (CMS) calculation.

Figure 8

Computed loss function for CFS (a) and CMS (b) for the parameters given in Fig. 7.

Figure 9

Same as Fig. 6 for CFS (a) and CMS (b) using ${13}^3$ and ${35}^3$ $\vec{k}$ points in the (full) 1. BZ. The black bars at $q=0$ correspond to the range of values obtained by the DFT code (Ref. 19) and recent calculations (Refs. 49, 50, 51).