Modeling of Phonon Dispersion in a Semiconductor Quantum Dot Crystal
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Modeling of Phonon Dispersion in a Semiconductor Quantum Dot Crystal Olga L. Lazarenkova and Alexander A. Balandin Department of Electrical Engineering University of California at Riverside Riverside, California 92521 ABSTRACT We describe a model for numerical calculation of phonon spectrum in a threedimensional regimented array of semiconductor quantum dots. Regimentation and possibility of carrier mini-band formation make this structure analogous to a crystal, e.g. quantum dot crystal. It is demonstrated that the acoustic phonon dispersion undergoes strong modification in such a structure leading to emergence of low-energy quasi-optical branches. Strong phonon spectrum modification is expected to affect carrier relaxation and transport properties. INTRODUCTION Regimented or partially regimented 2D and 3D multiple arrays of quantum dots (QDs) have already been fabricated by a variety of techniques [1-3]. Regimentation along all three directions [2] brings an analogy with bulk crystals, where the role of atoms is played by quantum dots. Formation of carrier mini-bands when the disorder is small takes this analogy even further. Thus, we refer to these structures as quantum dot crystals (QDC). Apart from the fundamental science importance of the investigation of electron (hole) and phonon spectrum in regimented quantum dot arrays, there is a significant practical interest to this problem due to a variety of proposed applications [1-3]. Despite recent achievements in self-assembly of QDC and many experimental reports on electrical and optical characterization of such structures, few theoretical papers deal with characteristics of a regimented ensemble of quantum dots [4-9]. In this paper we investigate the phonon spectrum of three-dimensional regimented quantum dot superlattice using an elastic continuum approximation. As an example of material system we consider Ge dots on Si structure. THEORETICAL MODEL We investigate a quantum dot array with a very high degree of orthorhombic threedimensional regimentation of quantum dots. We assume that all the dots are of the same size and do not have any surface defects. Although we limit our analysis to the parallelepiped shape of the dots, our numerical approach can be readily applied to the dots of arbitrary geometry. We consider acoustic phonon modes in 3D-ordered quantum dots embedded in a host material of the same crystal structure with different elastic properties. In the long-wavelength limit, acoustic phonon dispersion can be described by a continuum model in which the displacement vector u(r) of a geometrical point inside the material with r = (x,y,z) coordinates obeys the following equation of motion [10] ∂ 2u ∂Τ ∂u ∂ cijkl i ρ 2 i = å ik = ååå (1) ∂x k ∂t k ∂x k j k l ∂x j
ˆ is the stress tensor. The components of the tensor cijkl where where ρ is mass density and T are defined by 36 elastic stiffness constants cαβ . However, in semiconductors of cubic W10.4.1 Downloaded from https://www.cambridge.org/core. University of Arizona, on 26 Aug 2017 at 10:49:12, subject t
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