Out-of-phase plasmons in double-layer electron systems

It is well known that a double-layer electron system supports an out-of-phase plasmon in which the densities of the two layers oscillate out of phase with each other. In RPA this mode exhibits acoustic behavior (i.e., a linear dispersion in $q$ at small $q$). It has been suggested that at low density it may acquire an ``optical'' character , i.e., a gap at $q=0$$^1$. Here we investigate the long-wavelength dispersion relation of out-of-phase plasmons using dynamic exchange-correlation (xc) kernels within the density-density response function for a double-layer system. Starting from the recently formulated exact expression for the dynamic xc-kernel at long-wavelengths $ f_{{\rm xc},\alpha\beta}(q,\omega)= [2\delta_{\alpha\beta}-1]{n^2\over n_\alpha n_\beta}\,{A(\omega) \over q^2} +B_{\alpha\beta}(\omega)+{\cal O}(q^2) $ where $\alpha,\,\beta$ are the layer indices, $n_\alpha+n_\beta=2n$, we obtain a workable formula for the frequency dependent function $A (\omega)$. A mode-decoupling form for $\hbox{Im}[A(\omega)]$ together with the Kramers-Kronig relation and exact high and low frequency limits allows us to construct $A(\omega)$. Solving for the zeros of the dielectric function of the double- layer system we obtain the long-wavelength form of the out-of- phase plasmon dispersion and discuss the possibility of a gap developing at low density.\\ $^1$G. Kalman, V. Valtchinov, and K. I. Golden, Phys. Rev. Lett. {\bf 82}, 3124 (1999)