2.5.3.1 Plasmon Concept

The plasmon concept arises from the consideration of an interacting electron gas described by the following Hamiltonian:

$$\hat{H}_\mathrm{el.gas}=\sum_{i}\frac{\vec{p}_{i}^{2}}{2m}+\frac{1}{2}\sum_{i\ne j}\frac{e^{2}}{4\pi\varepsilon\vert\vec{r}_{i}-\vec{r}_{j}\vert},$$ (2.125)

where the first sum gives the kinetic energy and the second one arises from the Coulomb interaction between electrons. This Hamiltonian can be rewritten using the random phase approximation [31,32,33,34] as follows:

$$\hat{H}_\mathrm{el.gas}=\hat{H}_\mathrm{el.gas}^{0}+\hat{H}_\math... ...r}+\hat{H}_\mathrm{el.gas}^\mathrm{pl}+ \hat{H}_\mathrm{el.gas}^\mathrm{el-pl},$$ (2.126)

where each contribution can be represented in the second quantized form [35]. The first term $\hat{H}_\mathrm{el.gas}^{0}$ is the kinetic energy of the electron gas:

$$\hat{H}_\mathrm{el.gas}^{0}=\sum_{\vec{k}}\frac{\hbar^{2}\vec{k}^{2}}{2m}c_{\vec{k}}^{+}c_{\vec{k}},$$ (2.127)

where $c_{\vec{k}}^{+}$ and $c_{\vec{k}}$ are the electron creation and annihilation operators, respectively, and spherical and parabolic dispersion is assumed.

The second term $\hat{H}_\mathrm{el.gas}^\mathrm{el-el.scr}$ gives the contribution from a two-electron screened Coulomb interaction:

$$\hat{H}_{el.gas}^{el-el.scr}=\sum_{\vec{k}>\vec{q}_{c}}\frac{2\pi... ...vec{k}}^{+}c_{\vec{k}_{\mu}-\vec{k}}^{+}c_{\vec{k}_{\mu}}c_{\vec{k}_{\lambda}}.$$ (2.128)

As can be seen, this term accounts for scattering of two electrons with the initial quasi-momenta $\vec{k}_{\lambda}$ and $\vec{k}_{\mu}$ and the final quasi-momenta $\vec{k}_{\lambda}+\vec{k}$ and $\vec{k}_{\mu}-\vec{k}$ respectively. Screening is taken into account through the cut-off wave vector $\vec{q}_{c}$ which separates short- and long-range parts of the Coulombic term.

The third term $\hat{H}_\mathrm{el.gas}^\mathrm{pl}$ describes a non-interacting plasmon gas, that is, the quantized oscillations of the electron gas^2.34:

$$\hat{H}_\mathrm{el.gas}^\mathrm{pl}=\sum_{\vec{k}}\hbar\omega_{pl}\biggl(a_{\vec{k}}^{+}a_{\vec{k}}+\frac{1}{2}\biggr),$$ (2.129)

where $a_{\vec{k}}^{+}$, $a_{\vec{k}}$ are the plasmon creation and annihilation operators and $\hbar\omega_{pl}$ is the plasmon energy.

The forth term $\hat{H}_\mathrm{el.gas}^\mathrm{el-pl}$ represents the electron-plasmon interaction:

$$\hat{H}_\mathrm{el.gas}^\mathrm{el-pl}= \sum_{\vec{k}<\vec{q}_{c}... ...k}+\vec{q}}^{+}c_{\vec{k}}+a_{-\vec{q}}^{+}c_{\vec{k}+\vec{q}}^{+}c_{\vec{k}}),$$ (2.130)

where two terms in the second sum can conveniently be treated in terms of absorption and emission of a plasmon in the same way as it has been shown above for phonons.

The possible plasmon-phonon coupling [36,37,38] is not considered in this work as it plays an important role only in polar semiconductors where in the degenerate case the frequencies of the charge density fluctuations are comparable to the optical frequencies.

S. Smirnov: