Random Piezoelectric Field in Real -Oriented Strain-Relaxed SemiconductorMathematics and Statistics
Random Piezoelectric Field in Real -Oriented Strain-Relaxed Semiconductor
It is well known  that the elastic lattice deformation due to lattice mismatch between different layers of a semi-conductor heterostructure can lead both to microscopic effects, e.g., change of the band gap and lift of the degen-eracy of the valence band, and to macroscopic effects, e.g., piezoelectric field. An exact knowledge of the lattice mis-match-induced strain field and its impacts is, therefore, of vital importance to the interpretation of observed phenom-ena as well as the design and development of new elec-tronic and optoelectronic devices. Strain fields in two-dimensional (2D) epilayers have intensively been studied both experimentally and theoretically .
The strain field in a system is closely related to its boundaries. It was shown that because of strain release via additional free surfaces in a semiconductor structure with reduced spatial dimensions, e.g., 1D quantum wire [2-4] and 0D quantum dot [2,4], the strain field is consid-erably changed. Recently, it has been pointed out [5,6] thatthe roughness of an interface between lattice-mismatched layers in a 2D semiconductor structure is an important mechanism for strain relaxation. This means the roughness may bring about a drastic modification of the stress field.
This in turn implies according to Hooke’s law  a radical redistribution of the strain field.
Thus, the aim of this Letter is to present a theory of the strain field and its effects, especially the piezoelectric field, in a real zinc-blende heterostructure grown especially on a (001) substrate, taking adequate account of strain release by interface roughness. It is to be noted that the roughness-induced piezoelectric field has been addressed in [8,9]. In Ref.  the piezoelectric polarization was, however, as-sumed to exist only with a non- oriented substrate and to take a constant flat-interface value.
As a model system, we are examining a quantum well composed of a strained layer sandwiched between two unstrained layers, where the well thickness is smaller but the barrier ones larger than the critical thickness. It was proved [1,7] that if the substrate has been oriented along a high symmetry direction, e.g.,  and the interface is supposed to be ideal, i.e., absolutely flat, the stress field within the well is uniform and biaxial. Accordingly, the strain field is uniform and has no shear components, i.e., with zero off-diagonal components.
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