Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the framework of self-assembled deformation-diffusion model

The model of spatial-temporal distribution of point defects in a three-layer stressed nanoheterosystem GaAs/In$_x$Ga$_{1 - x}$As/GaAs considering the self-assembled deformation-diffusion interaction is constructed. Within the framework of this model, the profile of spatial-temporal distribution of vacancies (interstitial atoms) in the stressed nanoheterosystem GaAs/In$_x$Ga$_{1 - x}$As/GaAs is calculated. It is shown that in the case of a stationary state ($t>5\tau _d^{(2)}$), the concentration of vacancies in the inhomogeneously compressed interlayer is smaller relative to the initial average value $N_{d0}^{(2)}$ by 16%


Introduction
Intensive development of nanotechnologies has provided an opportunity to create nanoelectronic devices on the basis of stressed nanoheterosystems GaAs/In x Ga 1−x As/GaAs (ZnTe/Zn 1−x Cd x Te/ZnTe). The active region of such structures are layers In x Ga 1−x As, Zn 1−x Cd x Te, in which the electron-hole gas is localized being bounded on two sides of the potential barriers GaAs (ZnTe).
It is known that optical and electric properties of such devices depend significantly on both the lattice deformation of the contacting systems and the spatial distribution of point defects.
Such defects can penetrate from the surface or arise in the process of epitaxial growth. Besides, diffusion processes play an important role in the technology of fabricating optoelectronic devices. They are related with the redistribution of impurities in a semiconductor structure caused by both the ordinary gradient concentration of defects and the gradient of deformation tensor.
The interaction of defects with the deformation field, created by both the mismatch of the crystal lattice of the contacting materials and the point defects, causes a spatial redistribution of the latter. It can lead both to accumulation and to a decrease of the number of defects in the active region (In x Ga 1−x As, Zn 1−x Cd x Te) of the operating element depending on the character of the deformation created both by on the mismatch between the lattice parameters of contacting layers of heterostructure, the temperature of growth, molecular fluxes Ga and As, concentration and chemical nature of the doped impurities.
The strain caused by the mismatch between the lattice of the epitaxial layer and the substrate can be elastic when the thickness of the layer does not exceed a defined critical value [5]. Otherwise, mismatch dislocations are formed accompanied by a sharp worsening of both the optical and the electric characteristics of devices. However, in the layers In x Ga 1−x As with the mismatch less than critical there is a significant decline of the mobility and the intensity of photoluminescence at certain terms [5], which is related to the increased number of point defects and a corresponding increase of the diffusion barrier to the atoms of the third group.
In experimental work [6], it is shown that in a heterostructure GaAs/In x Ga 1−x As, the stressed quantum-size heterolayers hamper the diffusion of hydrogen and defects into the bulk of the material which leads to a substantial difference of their spatial distribution in a heterostructure and homogeneous layers. Theoretical research of the stationary distribution of defects within the framework of the self-assembled deformation-diffusion model has been considered in the work [7]. Therefore, in order to create devices with prescribed physical properties, it is necessary to construct a spatial-temporal deformation-diffusion model that describes the self-assembled deformation-diffusion processes in stressed nanoheterostructures having their own point defects and impurities.
The aim of this work is to construct a spatial-temporal deformation-diffusion model and calculate the spatial-temporal profile distribution of point defects (interstitial atoms and vacancies) in three-layer stressed nanoheterosystems GaAs/In x Ga 1−x As/GaAs (ZnTe/Zn 1−x Cd x Te/ZnTe).

The model of spatial-temporal redistribution of defects in a threelayer stressed nanoheterosystem
Let us consider stressed nanoheterosystems GaAs/In x Ga 1−x As/GaAs (ZnTe/Zn 1−x Cd x Te/ZnTe) having interlayers InAs (CdTe) of the thickness 2a, that include three layers ( The mechanical deformation that occurs due to the mismatch between the lattice parameters of contacting materials of a heterosystem is approximated by the function [7]: where ε 0 = ε xx + ε y y + ε zz < 0 is the relative change of the elementary cell volume of the grown layer on heteroboundaries z = |a|; ε y y = ε zz = (a i +1 − a i )/a i , ε xx = −(2C (2) 12 /C (2) 11 )ε y y , where i = 1, 3 corresponds to the layers GaAs (ZnTe), i = 2 corresponds to the interlayers In x Ga 1−x As (Zn 1−x Cd x Te); a i are the crystal lattice parameters of the contacting materials GaAs (ZnTe) and InAs (CdTe) of the heterostructure, respectively; C (2) 11 and C (2) 12 are the elastic constants of the material In x Ga 1−x As (Zn 1−x Cd x Te). Epitaxial growth on a substrate with the mismatch between the lattice parameters takes place simultaneously with the diffusion process, which is caused by both the concentration gradient of point defects [gradN (i ) d l (z, t )] and the gradient parameter of deformation [gradU i (z, t )]. The latter induces an additional diffusion flux of defects, which is opposite to the ordinary gradient concentration flux of defects. Therefore, in the basis of this model it is necessary to put a self-assembled system of non-stationary equations for the parameter deformation U i (z, t ) and concentration of impurities heterosystem, the redistribution of which is performed similarly to an ordinary diffusion flux, thus, by the deformation component of the flux Let the point defects be distributed with the initial average concentration N (i ) d 0 in the i -th layer in a particular heterosystem. As a result of their self-assembled interaction through the deformation field, created by both the mismatch between lattice parameters of contacting materials of a heterosystem and the presence of defects, there is a variation of the concentration profile of point defects and of the character of deformation. The mechanical stress in epitaxial layers created by both the point defects and the mismatch between lattice parameters of contacting materials is described by the expression: where ρ i , c i are the density of the i -th medium and the longitudinal speed of the sound, respectively. The wave equation for the deformation parameter U i (z, t ) is of the form: Taking into account (2), equation (3) for the renormalized deformation, U i (z, t ) looks as follows: The equation for the defect concentration (interstitial atoms and vacancies) is of the form [7]: where D i is the diffusion coefficient of point defects in the i -th layer, G (i ) d is the generation rate of the defects, τ (i ) d is the lifetime of the defects in the i -th layer that is determined by the frequency and the amplitude of mechanical fluctuations in the megahertz range (ω 10 6 Hz, τ (i ) d ∼ 1 µs) that arise in the process of the formation of heteroboundaries in stressed nanoheterostructures and in the process of the occurrence of defects (acoustic emission) [8]. As a result, a self-assembled system of equations (4), (5) is received for determination of the spatialtemporal distribution of the concentration of defects N (i ) d (z, t ) and the deformation parameter U i (z, t ) in the different regions of the three-layer nanoheterostructure.

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The defect concentration can be written in the form: where N (i ) d l (z, t ) is the spatially inhomogeneous component of the defect concentration. Taking into account the presentation (6) where where L d i is the diffusion length of the defect in the i -th layer.
In approximation (8), from equation (4), there will be found and it will be put into the equation (7). As a result, differential equation for determination of the spatial-temporal distribution of defects in the stressed heterosystem is received where N (i ) d c = k B T ρc 2 i /θ (i ) d is the critical defect concentration, which being exceeded results in the selforganization of the defects [9].
In addition, the conditions of the equality of concentration of impurities and their fluxes must be satisfied on the boundary layer of the heterostructure shown in figure 1: At the primary moment of time Entering the following dimensionless variables equations (9) take the form: , and boundary conditions (10) can be written: As seen from equation (12), parameter β describes the nature of the deformation effect caused both by the action of the stressed heteroboundary and the action of point defects of the type of compression or tension centers. This parameter can take both the positive values β > 0 (ε 0 > 0, ∆Ω (2) > 0; ε 0 < 0, ∆Ω (2) < 0) and the negative values β < 0 (ε 0 > 0, ∆Ω (2) < 0; ε 0 < 0, ∆Ω (2) > 0). (14) is searched in the form:

The solution of equations (13) with boundary conditions
where Z i ( z, θ) satisfy the following equations:

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with boundary conditions: Solutions of equations (16) with boundary conditions (17) are presented in the appendix at

Analysis of the numerical results and discussion
In As shown in figures 2 and 3, the profile of the spatial-temporal distribution of the defect concentration of the type of compression (vacancies, figure 2) or tension (figure 3) centers in a three-layer stressed nanoheterosystem is of a nonmonotonous character. If an internal epitaxial layer undergoes an inhomogeneous compression deformation (ε 0 < 0) due to the mismatch between the lattice parameter of the contacting epitaxial layers, then a decrease (an increase) of the concentration of vacancies (interstitial atoms) in the interlayer of the three-layer nanoheterostructures will be observed.
If the epitaxial layer undergoes the tension deformation due to a mismatch between the lattice parameter of the epitaxial layer and the substrate (a s > a 0 , ε 0 > 0, where a s is the lattice parameter of the substrate; a 0 is the lattice parameter of the stackable layer), the opposite effect will be observed: near the heteroboundary there will be accumulation of vacancies and a decrease of the concentration of interstitial atoms. This, in turn, will lead to a decrease of the tension deformation in the epitaxial layer near the heteroboundary.
The effect of impoverishment (enrichment) in the interlayer of vacancies (interstitial atoms) has been observed in experimental works [6,10] after the growth (decline) of the intensity of photoluminescence in stressed nanoheterostructures.

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N (2) d 0 by 16 %. If the ratio a/L = 0.1, then the established concentration of the vacancies in the middle layer is larger than the initial average concentration N (2) d 0 . Such a reduction of the established concentration of the vacancies in the workspace of a nanoheterosystem is correlated with the experimental results of the work [10].