Effective boundary conditions for planar quantum dot structures

Abstract

The effective-boundary condition method is extended to nanoscale mesoscopic systems. The ~EBC's ! appear as a result of the two-dimensional ~2D!-homogenization procedure and have the form of two-side anisotropic impedance boundary conditions stated on the structure surface. It has been shown that, unlike to macroscopic electrodynamics, the surface impedance tensor exhibits sharp oscillations at frequencies of optical transitions. The EBC method supplemented with well-developed mathematical techniques of classical electrodynamics creates unified basis for solution of boundary-value problems in electrodynamics of nanostructures. We have shown that the radiative lifetime of 2D array of spherical quantum dots ~QD's ! drastically changes its depen- dence on QD radius in comparison with the case of a single QD in the range of radii smaller than Bohr radius. Fascinating electronic and optical properties of spatially confined nanostructures irreducible to properties of bulk me- dia, and great potentiality of such structures in engineering applications has motivated permanent extension of their study. Among a variety of new results in this field, the recent progress in the synthesis of sheets of nanoscale three- dimensional ~3D! confined narrow-gap insertions in a host semiconductor, quantum dots ~QD's !, is of a special interest. Indeed, QD-based structures provide practical realization of the idea proposed by Dingle and Henry 1 to use structures with size quantization of charge carriers in one or more di- rections as active media of double heterostructure laser. Such a laser will show radically changed characteristics as com- pared to conventional quantum-well ~QW! lasers. 2,3 For InxGa12xAs QD's on GaAs substrates, an exceptionally bright luminescence at 1.36 mm was realized at room temperature 4 in a spectral range far beyond those available for conventional strained InxGa12xAs-GaAs QW's. The large body of recent results on physical properties of QDs and their utilization for the QD laser design has been accumu- lated in Ref. 5. The key peculiarities of QD heterostructures are related to spatial confinement of the charge-carrier motion and intrinsic spatial inhomogeneity. Since the inhomogeneity scale is much less than the optical wavelength, inclusions ~QD's ! can be treated as electrically small objects and electromagnetic response of such heterogeneous structures, composites, can be evaluated by means of effective-medium theory. 6 Appli- cation of effective-medium approach to 3D arrays of QD's has been presented in Refs. 7 and 8. In many cases, however, a planar array of QD's with intrinsic 2D periodicity of char- acteristic period much less than the optical wavelength, can be treated as more adequate and realistic model. 5 In this paper we present a general method for evaluation of electromagnetic response of planar arrays of QD's. This method, conventionally referred to as the effective- boundary-condition ~EBC! method, has been originally de- veloped for microwaves and antenna theory, 9-11 and has found a wide application in these fields. Similar approaches have also been developed in acoustics, hydrodynamics, elas- ticity theory. Recently, the EBC method has been extended to