Dielectric function of quasi-2D semiconductor nanostructures
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DIMENSIONAL SYSTEMS
Dielectric Function of Quasi-2D Semiconductor Nanostructures N. L. Bazhenov^, K. D. Mynbaev, and G. G. Zegrya Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia ^e-mail: [email protected] Submitted June 19, 2006; accepted for publication June 26, 2006
Abstract—The spatial and temporal dispersion of the dielectric function of the electron gas in quasi-2D quantum nanostructures has been studied. Analytical expressions for the dielectric function for a quantum well in the form of a δ function and a rectangular well of finite depth are derived for the first time. A criterion for transition to strictly 2D and strictly 3D cases was obtained. PACS numbers: 71.10.Ge, 71.45.Gm, 73.21.Fg DOI: 10.1134/S1063782607020145
1. INTRODUCTION At present, considerable attention is being given to the fabrication and study of semiconductor nanostructures and, in particular, structures in the form of quantum wells (QWs), quantum wires, and quantum dots. The single-electron model cannot always be used to theoretically evaluate the parameters of structures of this kind. An analysis of the interaction between particles requires that the screening of the electric field of charges be correctly taken into account [1–5]. It is known that, in the 2D case, the screening of the potential of charges leads to a strongly different spatial dependence of the dielectric function, compared with the 3D case. For example, the dependence of the Fourier component of the dielectric function on the wave vector q has in the 3D case the form [6]: ε(q) ∝ const/q2. This leads to the following spatial dependence of the potential of a probe charge on the distance r: e exp ( – r/r D ) -; ϕ ( r ) = ----------------------------r
(1)
i.e., the potential exponentially decreases as the distance r increases (rD is the screening length). For a nondegenerate carrier gas, rD coincides with the Debye screening length. At the same time, a similar dependence of the dielectric function for the 2D case has the form ε(q) ∝ const/q and the following expression has been derived for the potential within the film of thickness a [7]: ⎧1 π 2r 2r ⎫ ϕ ( r ) = e ⎨ --- – --- H 0 ⎛ -----⎞ – N 0 ⎛ -----⎞ ⎬, ⎝ ⎠ ⎝ a a⎠ ⎭ a ⎩r
(2)
where H0(x) and N0(x) are the Struve and Neumann functions, respectively. At very large distances (r a/2), the potential takes the form 2
ea ϕ ( r ) = -------3- . 4r
(3)
It can be seen that, in this case, there is no screening length and the potential decreases in proportion to the cubed distance. Thus, it is important to correctly take into account the screening effect when studying electronic effects in quantum structures. In particular, it is known [8–10] that the gain of QW semiconductor laser structures can be directly expressed in terms of the dielectric function and, therefore, a knowledge of its spatial dispersion is important for adequate calculation of the parameters of structures of this kind. The difficulty of the problem consists in the fact that real semiconductor QWs are not
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