Charge Waves

In the previous sections, the dispersion equation for collective oscillations of the dipole and the 2DEG was derived and analyzed. Its solutions are the characteristic frequencies (eigenvalues) of oscillations. In this section, we are going to construct eigenfunctions, which correspond to those eigenvalues and are the solutions of the integral equation (LABEL:integal_equation). The analysis of these solutions will allow one to understand the behavior of the electron subsystem at collective oscillations.
\begin{equation} \par {I_{z}=K_{x}I_{x}=K_{z}^{-1}I_{x},}\par \nonumber \\ \end{equation}
Let us recall that the integral equation (LABEL:integal_equation) for the charge carrier concentration \(n_{k}\) must be specified for the corresponding type of collective oscillations, after the solutions of the system of algebraic equations (LABEL:integal_equation_system) for \(I_{x}\), \(\,I_{y}\), and \(\,I_{z}\) (they are functionals of \(n_{k}\)) have been found. For two mixed \(x-z\) orientations of the dipole, the quantity \(I_{y}\) is nullified. Then, using the first and third equations of this system, it is easy to obtain
where the notations \({K_{x}=B_{0}S_{0}/(1-B_{0}S_{z})}\) and \({K_{z}=B_{0}S_{0}/(1-B_{0}S_{x})}\) are introduced (in these notations, the dispersion equation for the \(x\)- and \(z\)-branches reads \({K_{x}K_{z}=1}\)). Then, the solution of the linear integral equation (LABEL:integal_equation), which corresponds to the \(x\)-branch of frequencies, is
\begin{equation} \label{rok}n_{k}^{(x)}=CB_{0}\frac{k\left(k_{x}+K_{x}k\right)}{\Delta_{e}(\omega_{x},k)}e^{-kh}\,,\\ \end{equation}
where the superscript in parentheses in the notation \(n_{k}^{(x)}\) means that the frequency entering into the quantities \(B_{0}\), \(K_{x}\), and \(\Delta_{e}\) belongs to the \(x\)-branch. Generally speaking, the constant \(C\) is an arbitrary complex value characterized by an amplitude and a phase. The latter is insignificant, because it can always be zeroed by shifting the start of time counting. Therefore, \(C\) is assumed real-valued below.
Substituting Eq. (\ref{rok}) into the formula of the Fourier transformation, we obtain the space-time distribution of a concentration perturbation,
\begin{equation} \par {J_{x}(\mathbf{r})=CB_{0}\int d^{2}k\frac{kk_{x}}{\Delta_{e}(\omega_{j},k)}e^{-kh+i\mathbf{kr}}}\,,\par \nonumber \\ \end{equation} \begin{equation} \label{nrok}n^{(x)}(\mathbf{r},t)=CB_{0}\int d^{2}k\frac{k\left(k_{x}+K_{x}k\right)}{\Delta_{e}(\omega_{x},k)}e^{-kh+i\mathbf{kr}-i\omega_{x}t}.\\ \end{equation}
It is the real part of expression (\ref{nrok}) that has a physical meaning. Let
\begin{equation} \par {J_{z}(\mathbf{r})=CB_{0}\int d^{2}k\frac{k^{2}}{\Delta_{e}(\omega_{j},k)}e^{-kh+i\mathbf{kr}}}\par \nonumber \\ \end{equation} \begin{equation} \par {n^{\prime\>(x)}(\mathbf{r},\tau_{x})=}\par \nonumber \\ \end{equation}
and \({\tau_{j}=\omega_{j}^{\prime}t}\), where \(j=x\) for the \(x\)-branch. Extracting the real part of the concentration perturbation (\ref{nrok}), we obtain
\begin{equation} \par =\displaystyle{e^{\frac{\omega_{x}^{\prime\prime}}{\omega_{x}^{\prime}}\tau_{x}}\biggl{[}(J_{x}^{\prime}(\mathbf{r})+K_{x}^{\prime}J_{z}^{\prime}(\mathbf{r})-K_{x}^{\prime\prime}J_{z}^{\prime\prime}(\mathbf{r}))\cos{\tau_{x}}+}\par \nonumber \\ \end{equation} \begin{equation} \label{nrokRe}{+(J_{x}^{\prime\prime}(\mathbf{r})+K_{x}^{\prime}J_{z}^{\prime\prime}(\mathbf{r})+K_{x}^{\prime\prime}J_{z}^{\prime}(\mathbf{r}))\sin{\tau_{x}}\biggr{]}}.\\ \end{equation}