content_image/357017.jpg content_image/357018.jpg
Interestingly, for low values of B , when many Landau levels are filled, both the systems have the same flat band condition[19]. 

The dynamic and static response properties of an electron system are all embodied in the structure of the density-density correlation function. We employ the Ehrenreich-Cohen Self- Consistent Field (SCF) approach[22] to calculate the density-density correlation function. The SCF treatment presented here is by its nature a high density approximation which has been successful in the study of collective excitations in low-dimensional systems[10– 13, 17, 23, 24] (semiconductor superlattices and quantum wire structures). Such success has been convincingly attested by the excellent agreement of SCF predictions of plasmon spectra with experiments. 

Following the SCF approach, the density-density correlation function of the interacting system can be expressed as 

<img src='content_image/357017.jpg'>

with Π ± 0 (¯ q, ω ) the density-density correlation function of the non-interacting system, v$_{c}$ (¯ q ) = 2 πe 2 k ¯ q the 2-D Coulomb potential, k being the dielectric constant and ¯ q is the two-dimensional wave number. Making use of the transformation k$_{y}$ → − k$_{y}$ , with the fact that ε ± ( n, k$_{y}$ ) is an even function of k$_{y}$, and at the same time interchanging n ↔ n ′ we write for the non-interacting density-density correlation function appearing in equation (9) 

<img src='content_image/357018.jpg'>

In writing the above equation we converted the k$_{y}$ -sum into an integral over x$_{0}$ . f ( ε ( n, x$_{0}$ )) is the Fermi-Dirac distribution function, C$_{nn}$ ′ ( x ) = n 2 ! n ′ $_{$_{1}$!}$e − $^{x}$xn 1 − n 2 [ L n 1 − n 2 2 ( x )] $^{2}$with x = l $^{2}$¯ q 2 2 , n$_{1}$ = max( n , n ′ ), n$_{2}$ = min( n , n ′ ), and L$_{n}$ ( x ) an associated Laguerre polynomial x$_{0}$ = − ℏ k$_{y}$ m ∗ $_{ω$_{c}$}$, and x ′ 0 = − ℏ q$_{y}$ m ∗ $_{ω$_{c}$}$. Without modulation ( V ± n = 0), the above expression reduces to the result in [13]. 

The above equations (9, 10) will be the starting point of our examination of the intra- Landau band plasmons. These correlation functions are the essential ingredients for theo- retical considerations of such diverse problems as high frequency and steady state transport, static and dynamic screening and correlation phenomena. 

