Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model
The model of spatial-temporal distribution of point defects in a three-layer stressed nanoheterosystem GaAs/InxGa₁-xAs/GaAs considering the self-assembled deformation-diffusion interaction is constructed.
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irk-123456789-1535672019-08-29T23:19:58Z Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model Peleshchak, R.M. Kulyk, N.Ya. Doroshenko, M.V. The model of spatial-temporal distribution of point defects in a three-layer stressed nanoheterosystem GaAs/InxGa₁-xAs/GaAs considering the self-assembled deformation-diffusion interaction is constructed. Побудовано модель просторово-часового розподiлу точкових дефектiв у тришаровiй напруженiй наногетеросистемi GaAs/InxGa₁-xAs/GaAs з врахуванням самоузгодженої деформацiйно-дифузiйної взаємодiї. 2015 Article Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model / R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko // Condensed Matter Physics. — 2015. — Т. 18, № 2. — С. 23602: 1–12. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 67.80.Mg, 68.55.Ln DOI:10.5488/CMP.18.23602 arXiv:1506.03967 http://dspace.nbuv.gov.ua/handle/123456789/153567 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The model of spatial-temporal distribution of point defects in a three-layer stressed nanoheterosystem GaAs/InxGa₁-xAs/GaAs considering the self-assembled deformation-diffusion interaction is constructed. |
| format |
Article |
| author |
Peleshchak, R.M. Kulyk, N.Ya. Doroshenko, M.V. |
| spellingShingle |
Peleshchak, R.M. Kulyk, N.Ya. Doroshenko, M.V. Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model Condensed Matter Physics |
| author_facet |
Peleshchak, R.M. Kulyk, N.Ya. Doroshenko, M.V. |
| author_sort |
Peleshchak, R.M. |
| title |
Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model |
| title_short |
Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model |
| title_full |
Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model |
| title_fullStr |
Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model |
| title_full_unstemmed |
Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model |
| title_sort |
spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model |
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Інститут фізики конденсованих систем НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/153567 |
| citation_txt |
Spatial-temporal redistribution of point defects in three-layer stressed nanoheterosystems within the limits of self-assembled deformation-diffusion model / R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko // Condensed Matter Physics. — 2015. — Т. 18, № 2. — С. 23602: 1–12. — Бібліогр.: 10 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT peleshchakrm spatialtemporalredistributionofpointdefectsinthreelayerstressednanoheterosystemswithinthelimitsofselfassembleddeformationdiffusionmodel AT kulyknya spatialtemporalredistributionofpointdefectsinthreelayerstressednanoheterosystemswithinthelimitsofselfassembleddeformationdiffusionmodel AT doroshenkomv spatialtemporalredistributionofpointdefectsinthreelayerstressednanoheterosystemswithinthelimitsofselfassembleddeformationdiffusionmodel |
| first_indexed |
2025-07-14T05:01:29Z |
| last_indexed |
2025-07-14T05:01:29Z |
| _version_ |
1837597238574972928 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 2, 23602: 1–12
DOI: 10.5488/CMP.18.23602
http://www.icmp.lviv.ua/journal
Spatial-temporal redistribution of point defects in
three-layer stressed nanoheterosystems within the
framework of self-assembled deformation-diffusion
model
R.M. Peleshchak, N.Ya. Kulyk∗, M.V. Doroshenko
Drohobych Ivan Franko State Pedagogical University, 24 Franko St., 82100 Drohobych, Ukraine
Received November 20, 2014, in final form February 10, 2015
The model of spatial-temporal distribution of point defects in a three-layer stressed nanoheterosystem
GaAs/InxGa1−xAs/GaAs considering the self-assembled deformation-diffusion interaction is constructed. Withinthe framework of this model, the profile of spatial-temporal distribution of vacancies (interstitial atoms) in the
stressed nanoheterosystem GaAs/InxGa1−xAs/GaAs is calculated. It is shown that in the case of a stationarystate (t > 5τ(2)
d ), the concentration of vacancies in the inhomogeneously compressed interlayer is smaller rela-
tive to the initial average value N (2)
d0 by 16%.
Key words: spatial-temporal distribution, vacancies, interstitial atoms
PACS: 67.80.Mg, 68.55.Ln
1. Introduction
Intensive development of nanotechnologies has provided an opportunity to create nanoelectronic
devices on the basis of stressed nanoheterosystems GaAs/InxGa1−xAs/GaAs (ZnTe/Zn1−xCdxTe/ZnTe). Theactive region of such structures are layers InxGa1−xAs, Zn1−xCdxTe, in which the electron-hole gas islocalized 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, dif-
fusion 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 lat-
tice 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 (InxGa1−xAs,Zn1−xCdxTe) of the operating element depending on the character of the deformation created both bythe mismatch between parameters of contacting crystal lattices (ε0 = 7% (4%) for GaAs/InxGa1−xAs/GaAs(ZnTe/Zn1−xCdxTe/ZnTe), respectively [1, 2]), and by the action of defects. In particular, it is known thatthe gallium arsenide grown using themethod of themolecular-beam epitaxy at low temperature contains
an excess of arsenic [3, 4]. Introduction of excessive arsenic causes a tetragonal distortion of the lattice
material GaAs and the generation of point defects in it: interstitial atoms (As), vacancies (Ga) and anti-
structural defects (AsGa), which, in turn, leads to their spatial redistribution. The lattice deformation andconcentration of point defects generated under the action of the gradient of deformation tensor depend
∗E-mail:delenkonadia@mail.ru
© R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko, 2015 23602-1
http://dx.doi.org/10.5488/CMP.18.23602
http://www.icmp.lviv.ua/journal
R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko
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, mis-
match dislocations are formed accompanied by a sharp worsening of both the optical and the electric
characteristics of devices. However, in the layers InxGa1−xAs with the mismatch less than critical thereis 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/InxGa1−xAs, the stressedquantum-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 homoge-
neous 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/InxGa1−xAs/GaAs (ZnTe/Zn1−xCdxTe/ZnTe).
2. The model of spatial-temporal redistribution of defects in a three-
layer stressed nanoheterosystem
Let us consider stressed nanoheterosystems GaAs/InxGa1−xAs/GaAs (ZnTe/Zn1−xCdxTe/ZnTe) havinginterlayers InAs (CdTe) of the thickness 2a, that include three layers (figure 1), where N (1)
d0 , N (2)
d0 , N (3)
d0 arethe initial average defect concentrations, respectively, and D1, D2, D3 are diffusion coefficients. Supposethat the external layers GaAs (ZnTe) are of the thickness that considerably exceeds the width of the in-
terlayer of the heterostructure (2a/L ¿ 1), so deformation of these layers can be neglected [εi (z) = 0,
i = 1,3].
The mechanical deformation that occurs due to the mismatch between the lattice parameters of con-
tacting materials of a heterosystem is approximated by the function [7]:
εi (z) =
{
ε0
z2
a2 , i = 2,
0, i = 1,3,
(1)
where ε0 = εxx +εy y +εzz < 0 is the relative change of the elementary cell volume of the grown layer on
heteroboundaries z = |a|; εy y = εzz = (ai+1 −ai )/ai , εxx = −(2C (2)
12
/C (2)
11
)εy y , where i = 1,3 corresponds
Figure 1. The model of stressed nanoheterostructures GaAs/InxGa1−xAs/GaAs (ZnTe/Zn1−xCdxTe/ZnTe).
23602-2
Spatial-temporal redistribution of point defects . . .
to the layers GaAs (ZnTe), i = 2 corresponds to the interlayers InxGa1−xAs (Zn1−xCdxTe); ai are the crys-tal 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 InxGa1−xAs (Zn1−xCdxTe).Epitaxial growth on a substrate with the mismatch between the lattice parameters takes place si-
multaneously with the diffusion process, which is caused by both the concentration gradient of point
defects [gradN (i )
dl (z, t )] and the gradient parameter of deformation [gradUi (z, t )]. The latter induces an
additional diffusion flux of defects, which is opposite to the ordinary gradient concentration flux of de-
fects. Therefore, in the basis of this model it is necessary to put a self-assembled system of non-stationary
equations for the parameter deformationUi (z, t ) and concentration of impurities N (i )
d (z, t ) in a stressed
heterosystem, the redistribution of which is performed similarly to an ordinary diffusion flux,
J (i )dif (z, t ) =−Di
∂N (i )
d (z, t )
∂z
,
thus, by the deformation component of the flux
J (i )def (z, t ) =−Di
θ(i )
d
kBT
N (i )
d (z, t )
∂Ui (z, t )
∂z
,
where θ(i )
d = K (i )∆Ω(i ) is the mechanical deformation potential, K (i ) = (C (i )
11 +2C (i )
12 )/3 is the module of
uniform compression of the i -th material,∆Ω(i ) is the variation of elementary cell volume at the presence
of a defect in the i -th layer.
Let the point defects be distributed with the initial average concentration N (i )
d0 in the i -th layer in a
particular heterosystem. As a result of their self-assembled interaction through the deformation field, cre-
ated 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:
σi (z, t ) = ρi c2
i Ui (z, t )−θ(i )
d N (i )
d (z, t )−ρi c2
i εi (z), (2)
where ρi ,ci are the density of the i -th medium and the longitudinal speed of the sound, respectively.
The wave equation for the deformation parameterUi (z, t ) is of the form:
ρi
∂
2
Ui
∂t 2 = ∂2σi
∂z2 . (3)
Taking into account (2), equation (3) for the renormalized deformation,Ui (z, t ) looks as follows:
1
c2
i
∂2Ui (z, t )
∂t 2 = ∂2Ui (z, t )
∂z2 −
θ(i )
d
ρi c2
i
∂2N (i )
d (z, t )
∂z2 − ∂2εi (z)
∂z2 . (4)
The equation for the defect concentration (interstitial atoms and vacancies) is of the form [7]:
∂N (i )
d (z, t )
∂t
= Di
∂2N (i )
d (z, t )
∂z2 −Di
θ(i )
d
kBT
∂
∂z
[
N (i )
d (z, t )
∂Ui (z, t )
∂z
]
+G (i )
d −
N (i )
d (z, t )
τ(i )
d
, (5)
where Di 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 Ê 106 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 spatial-
temporal distribution of the concentration of defects N (i )
d (z, t ) and the deformation parameterUi (z, t ) in
the different regions of the three-layer nanoheterostructure.
23602-3
R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko
The defect concentration can be written in the form:
N (i )
d (z, t ) = N (i )
d0 +N (i )
dl (z, t ), (6)
where N (i )
dl (z, t ) is the spatially inhomogeneous component of the defect concentration.
Taking into account the presentation (6) in the approximation N (i )
dl ¿ N (i )
d0 , the equation of diffusionis written as follows:
∂N (i )
dl (z, t )
∂t
= Di
∂2N (i )
dl (z, t )
∂z2 −Di N (i )
d0
θ(i )
d
kBT
∂2Ui (z, t )
∂z2 +G ′(i )
d −
N (i )
dl (z, t )
τ(i )
d
, (7)
where G ′(i )
d = G (i )
d − N (i )
d0(z, t )/τ(i )
d is the generation rate of point defects under the effect of mechani-
cal fluctuations (ω ∼ 106 Hz) that arise in the process of the formation of heteroboundaries in stressed
nanoheterostructures.
A further solution of self-assembled systems of equations (4), (5) will be searched in the approxima-
tion
(
L(i )
d
)2
2Di
2
∂2Ui (z, t )
∂t 2 ¿ ε0,
namely
∂2Ui (z, t )
∂t 2 ≈ 0, (8)
where Ldi is the diffusion length of the defect in the i -th layer.
In approximation (8), from equation (4), there will be found ∂2Ui (z,t )
∂z2 and it will be put into the equa-
tion (7). As a result, differential equation for determination of the spatial-temporal distribution of defects
in the stressed heterosystem is received
∂N (i )
dl (z, t )
∂t
=
[
Di
(
1−
N (i )
d0
N (i )
dc
)]
∂2N (i )
dl (z, t )
∂z2 −Di
N (i )
d0
N (i )
dc∆Ω
∂2εi (z)
∂z2 +G ′(i )
d −
N (i )
dl (z, t )
τ(i )
d
, (9)
where N (i )
dc = kBTρc2
i /θ(i )
d is the critical defect concentration, which being exceeded results in the self-
organization 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:
∂N (1)
dl (z, t )
∂z
∣∣∣∣
z→−∞
= 0,
∂N (3)
dl (z, t )
∂z
∣∣∣∣
z→∞
= 0,
N (1)
dl (−a, t ) = N (2)
dl (−a, t ), N (2)
dl (a, t ) = N (3)
dl (a, t ), (10)
J1 (−a, t ) = J2 (−a, t ) , J2 (a, t ) = J3 (a, t ) ,
where Ji (z, t ) =− ∂
∂z
{
Di N (i )
dl (z, t )[1−εi (z)]
}. At the primary moment of time
N (i )
dl (z,0) = 0. (11)
Entering the following dimensionless variables
θ = t
τ(2)
d
, L(i )
d =
√
Diτ
(i )
d ,
N (i )
dl (z, t ) = Yi (z, t )G ′(i )
d τ(i )
d ,
_
z = z
L
,
_
L = L
a
, N (i )
d0 =G ′(i )
d τ(i )
d ,
23602-4
Spatial-temporal redistribution of point defects . . .
β= D2
N (2)
d0
N (2)
dc
2ε0
∆Ω(2)a2G (2)
d
, (12)
equations (9) take the form:
∂Y1(
_
z ,θ)
∂θ
= D1
D2
(
L(2)
d
L
)2 (
1−
N (1)
d0
N (1)
dc
)
∂2Y1(
_
z ,θ)
∂
_
z
2 −λ1
[
Y1(
_
z ,θ)−1
]
,
∂Y2(
_
z ,θ)
∂θ
=
(
L(2)
d
L
)2 (
1−
N (2)
d0
N (2)
dc
)
∂2Y2(
_
z ,θ)
∂
_
z
2 +β−λ2
[
Y2(
_
z ,θ)−1
]
, (13)
∂Y3(
_
z ,θ)
∂θ
= D3
D2
(
L(2)
d
L
)2 (
1−
N (3)
d0
N (3)
dc
)
∂2Y3(
_
z ,θ)
∂
_
z
2 −λ3
[
Y3(
_
z ,θ)−1
]
,
where λ1 = D1
D2
(
L(2)
d /L(1)
d
)2, λ2 = 1, λ3 = D3
D2
(
L(2)
d /L(3)
d
)2, and boundary conditions (10) can be written:
∂Y1(
_
z ,θ)
∂
_
z
∣∣∣_
z →−_L
= 0,
Y1(−_
a,θ) =
G (2)
d τ(2)
d
G (1)
d τ(1)
d
Y2(−_
a,θ),
Y2(
_
a,θ) =
G (3)
d τ(3)
d
G (2)
d τ(2)
d
Y3(
_
a,θ), (14)
D1
D2
(
L(2)
d
L
)2 (
−1+
N (1)
d0
N (1)
dc
)
∂Y1(−_
a,θ)
∂
_
z
=
(
L(2)
d
L
)2 (
−1+
N (2)
d0
N (2)
dc
)
∂Y2(−_
a,θ)
∂
_
z
−β_a,
(
L(2)
d
L
)2 (
−1+
N (2)
d0
N (2)
dc
)
∂Y2(
_
a,θ)
∂
_
z
+β_a = D3
D2
(
L(2)
d
L
)2 (
−1+
N (3)
d0
N (3)
dc
)
∂Y3(
_
a,θ)
∂
_
z
,
∂Y3(
_
z ,θ)
∂
_
z
∣∣∣_
z →_
L
= 0.
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 (ε0 > 0, ∆Ω(2) > 0; ε0 < 0,
∆Ω(2) < 0) and the negative values β< 0 (ε0 > 0, ∆Ω(2) < 0; ε0 < 0, ∆Ω(2) > 0).
The solution of equations (13) with boundary conditions (14) is searched in the form:
Yi (
_
z ,θ) = e−λiθZi (
_
z ,θ), (15)
where Zi (
_
z ,θ) satisfy the following equations:
∂Z1(
_
z ,θ)
∂θ
= D1
D2
(
L(2)
d
L
)2 (
1−
N (1)
d0
N 1
dc
)
∂2Z1(
_
z ,θ)
∂
_
z
2 +λ1eλ1θ ,
∂Z2(
_
z ,θ)
∂θ
=
(
L(2)
d
L
)2 (
1−
N (2)
d0
N (2)
dc
)
∂2Z2(
_
z ,θ)
∂
_
z
2 +eθ
(
β+1
)
, (16)
∂Z3(
_
z ,θ)
∂θ
= D3
D2
(
L(2)
d
L
)2 (
1−
N (3)
d0
N (3)
dc
)
∂2Z3(
_
z ,θ)
∂
_
z
2 +λ3eλ3θ ,
23602-5
R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko
with boundary conditions:
∂Z1(
_
z ,θ)
∂
_
z
∣∣∣_
z →−_L
= 0,
e−λ1θZ1(−_
a,θ) =
G (2)
d τ(2)
d
G (1)
d τ(1)
d
e−θZ2(−_
a,θ),
e−θZ2(
_
a,θ) =
G (3)
d τ(3)
d
G (2)
d τ(2)
d
e−λ3θZ3(
_
a,θ), (17)
D1
D2
(
L(2)
d
L
)2
e−λ1θ
(
−1+
N (1)
d0
N (1)
dc
)
∂Z1(−_
a,θ)
∂
_
z
=
(
L(2)
d
L
)2
e−θ
(
−1+
N (2)
d0
N (2)
dc
)
∂Z2(−_
a,θ)
∂
_
z
−β_a,
(
L(2)
d
L
)2
e−θ
(
−1+
N (2)
d0
N (2)
dc
)
∂Z2(
_
a, t )
∂
_
z
+β_a = D3
D2
(
L(2)
d
L
)2
e−λ3θ
(
−1+
N (3)
d0
N (3)
dc
)
∂Z3(
_
a, t )
∂
_
z
,
∂Z3(
_
z ,θ)
∂
_
z
∣∣∣_
z →_
L
= 0.
Solutions of equations (16) with boundary conditions (17) are presented in the appendix at
λ1 =λ2 =λ3 = 1,
G (2)
d τ(2)
d
G (1)
d τ(1)
d
= 1,
G (2)
d τ(2)
d
G (3)
d τ(3)
d
= 1,
D1
D2
(
L(2)
d
a
)2
= 1,
D3
D2
(
L(2)
d
a
)2
= 1.
3. Analysis of the numerical results and discussion
In figures 2 and 3) there is shown the spatial-temporal redistribution of vacancies Y (
_
z ,θ) (figure 2)
and interstitial atoms (figure 3) in a three-layer stressed nanoheterosystem GaAs/InxGa1−xAs/GaAs un-der the effect of the deformation caused by both the mismatch between parameters of the contacting
lattices (ε0 = ∆a/a = 7%) and by the action of a point defect. Calculations were carried out for the fol-
lowing values of the parameters: ε0 = 0.07; a = 0.05L, 0.1L (L — the thickness of nanoheterostructure);
N (1)
d0 /N (1)
dc = 0.5; N (2)
d0 /N (2)
dc = 0.8; N (3)
d0 /N (3)
dc = 0.6; C (2)
11 = 0.833Mbar; C (2)
12 = 0.453Mbar; C (1)
11 = 1.223Mbar;
C (1)
12 = 0.571Mbar; T = 300 K; θ(i )
d = 5 eV [9]; Di = 10−5 cm2/s; τ(2)
d = 1 µs.
Figure 2. (Color online) Profile of the spatial-temporal distribution of the vacancies concentration in a
three-layer stressed nanoheterosystem having inhomogeneous-compressed interlayer N (2)
d0 /N (2)
dc = 0.5;
N (2)
d0 /N (2)
dc = 0.8; N (2)
d0 /N (2)
dc = 0.6; β= 10.2 [formula (12)].
23602-6
Spatial-temporal redistribution of point defects . . .
Figure 3. (Color online) Profile of the spatial-temporal distribution of the concentration of intersti-
tial atoms in a three-layer stressed nanoheterosystem having inhomogeneous-compressed interlayer
N (2)
d0 /N (2)
dc = 0.5; N (2)
d0 /N (2)
dc = 0.8; N (2)
d0 /N (2)
dc = 0.6; β=−10.2 [formula (12)].
As shown in figures 2 and 3, the profile of the spatial-temporal distribution of the defect concentra-
tion 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 inho-
mogeneous compression deformation (ε0 < 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 pa-
rameter of the epitaxial layer and the substrate (as > a0, ε0 > 0, where as is the lattice parameter of thesubstrate; a0 is the lattice parameter of the stackable layer), the opposite effect will be observed: near theheteroboundary there will be accumulation of vacancies and a decrease of the concentration of intersti-
tial 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.
Figure 4. The cut of the spatial-temporal distribution of the concentration of vacancies along the growth
axis at different times: 1 — at the moment t = τ(2)
d ; 2 — t = 5τ(2)
d ; N (1)
d0 /N (1)
dc = 0.5; N (2)
d0 /N (2)
dc = 0.8;
N (3)
d0 /N (3)
dc = 0.6; β= 10.2 [formula (12)]; a = 0.05L.
23602-7
R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko
Figure 5. The cut of the spatial-temporal distribution of the concentration of interstitial atoms along the
growth axis at different times: 1— at the moment t = τ(2)
d ; 2— t = 5τ(2)
d ; N (1)
d0 /N (1)
dc = 0.5; N (2)
d0 /N (2)
dc = 0.8;
N (3)
d0 /N (3)
dc = 0.6; β= 10.2 [formula (12)]; a = 0.1L.
In figures 4 and 5, numerical calculations of the cut of the spatial-temporal distribution of the con-
centration of vacancies along the growth axis (OX ) of heterosystem at different times are presented:
t = 0; τ(2)
d ; 5τ(2)
d (τ(2)
d is the average time of finding the defect in one of the equilibrium positions in the
interlayer nanoheterosystem, namely a settled life) and for the different thicknesses of the interlayer of
a nanoheterostructure (a = 0.05L, figure 4 and a = 0.1L, figure 5).
As seen from figures 4 and 5, during the time interval 0 É t É 5τ(2)
d there occurs a spatial-temporal
redistribution of the defects, so that in the inhomogeneous-compressed interlayer [figure 1, region 2,
formula (1)] they become smaller relative to their initial average value N (2)
d0 by ≈ 13.7%,16% at different
times τ(2)
d , 5τ(2)
d , respectively. Starting from the time t > 5τ(2)
d , there practically establishes a stationarystate of the distribution of the defects in a three-layer stressed nanoheterosystem. Thus, the deformation
field of the interlayer (−a É z É a) clears away the workspace from the defects which finally makes the
material of the workspace having a greater intensity of photoluminescence [10]. In exterlayers (figure 1,
regions 1, 3) of the heterosystem, the concentration of the defects asymmetricallymonotonously increases
from the boundary of the contacting materials and becomes larger than their average value N (1)
d0 , N (3)
d0 .If there are no defects (N (i )
d0 = 0) in the contacting materials, inhomogeneous deformation is created
only due to a mismatch between the lattice parameters of contacting materials [εi (z) = ε0
z2
a2 , i = 2],
and in the absence of the mismatch between the lattice parameters (ε0 = 0), the deformation [U (i ) =
(θ(i )
d /K (i ))N (i )
d ] is caused only by the point defects.
4. Conclusions
• It has been established that the concentration profile of point defects at N (i )
d0 < N (i )
dc is of non-monotonous character with a minimum in the middle of the interlayer InxGa1−xAs which is im-poverished by the point defect of the type of compression centers when the interlayer of the het-
erostructure GaAs/InxGa1−xAs/GaAs undergoes an inhomogeneous compression, while in the caseof inhomogeneous tension the opposite effect takes place. Thus, the deformation field and the num-
ber of the defects in the workspace of nanooptoelectronic devices can be controlled.
• It is shown that if the ratio of the thickness of the middle layer to the thicknesses of the external lay-
ers of a nanoheterosystem GaAs/InxGa1−xAs/GaAs is a/L = 0.05, the ratio of the initial average con-
centrations of the vacancies in the layers of a nanoheterosystem is N (1)
d0 /N (1)
dc = 0.5, N (2)
d0 /N (2)
dc = 0.8,
N (3)
d0 /N (3)
dc = 0.6 and the value of the deformation parameter is β = 10.2, then the established con-
centration of vacancies in the middle layer Y (
_
z ,5τ(2)
d ) is less than the initial average concentration
23602-8
Spatial-temporal redistribution of point defects . . .
N (2)
d0 by 16 %. If the ratio a/L = 0.1, then the established concentration of the vacancies in the mid-
dle layer is larger than the initial average concentration N (2)
d0 . Such a reduction of the establishedconcentration of the vacancies in the workspace of a nanoheterosystem is correlated with the ex-
perimental results of the work [10].
Appendix
To find the solution of differential equations (16) with boundary conditions (17), the integral Laplace
transformation is used:
Xi (
_
z , p) =
∞∫
0
Zi (
_
z ,θ)e−pθdθ. (18)
Then, the differential equation (16) and boundary conditions (17) take up the form:
H1X ′′
1
(
ẑ, p
)−p X1
(
ẑ, p
)+ 1
p −1
= 0,
H2X ′′
2
(
ẑ, p
)−p X2
(
ẑ, p
)+ β+1
p −1
= 0, (19)
H3X ′′
3
(
ẑ, p
)−p X3
(
ẑ, p
)+ 1
p −1
= 0.
X1
(
−_
a, p
)
= X2
(
−_
a, p
)
,
−H1X ′
1
(
−_
a, p
)
=−H2X ′
2
(
−_
a, p
)
− β
_
a
p −1
,
X2
(
_
a, p
)
= X3
(
_
a, p
)
,
−H2X ′
2
(
_
a, p
)
+ β
_
a
p −1
=−H3X ′
3
(
_
a, p
)
,
(20)
where Hi = 1−N (i )
d0/N (i )
dc .Analytical solutions of differential equations (19) in each layer are as follows:
X1(
_
z , p) =C1exp
(√
p
H1
ẑ
)
+ 1
p
(
p −1
) , −_
a É _
z É−_
l ,
X2(
_
z , p) =C2exp
(√
p
H2
ẑ
)
+C3exp
(√
p
H2
ẑ
)
+ β+1
p
(
p −1
) , −_
a É _
z É _
a , (21)
X3(
_
z , p) =C4exp
(√
p
H2
ẑ
)
+ 1
p
(
p −1
) ,
_
a É _
z É−_
l .
Integration constantsC1,C2,C3,C4 are determined from the following system of algebraic equations:
C1exp
(
−
√
p
H1
_
a
)
−C2exp
(
−
√
p
H2
_
a
)
−C3exp
(√
p
H2
_
a
)
= β
p
(
p −1
) ,
−C1
√
H1exp
(
−
√
p
H1
_
a
)
+C2
√
H2exp
(
−
√
p
H2
_
a
)
−C3
√
H2exp
(√
p
H2
_
a
)
= β
_
ap
p
(
p −1
) ,
C2exp
(√
p
H2
_
a
)
+C3exp
(
−
√
p
H2
_
a
)
−C4exp
(
−
√
p
H3
_
a
)
=− β
p
(
p −1
) ,
−C2
√
H2exp
(√
p
H2
_
a
)
+C3
√
H2exp
(
−
√
p
H2
_
a
)
+C4
√
H3exp
(
−
√
p
H3
_
a
)
= β
_
ap
p
(
p −1
) .
(22)
23602-9
R.M. Peleshchak, N.Ya. Kulyk, M.V. Doroshenko
By carrying out the inverse Laplace transformation we obtain the spatial-temporal redistribution of
the point defects in the first, second and third stressed layers, respectively:
Y1(
_
z ,θ) = e−θZ1(
_
z ,θ)−e−θ+1,
Y2(
_
z ,θ) = Z2(
_
z ,θ)
(
1−e−θ
)(
1+β)
, (23)
Y3(
_
z ,θ) = e−θZ3(
_
z ,θ)−e−θ+1,
where functions Z1(
_
z ,θ), Z2(
_
z ,θ) and Z3(
_
z ,θ) are the solutions of differential equations (16) with bound-
ary conditions (17), respectively:
Z1(
_
z ,θ) = 1
b
[
eθ−1+eθ
_
ap
H2
Erf(pθ)]
l3 +
2∑
i=1
∞∑
m=0
(−1)m am
bm+1
×
∞∑
k=1
l2i−1r2i−1θ
kΦ1(
_
z ,θ,k)+
∞∑
m=0
(−1)m am
bm+1
∞∑
k=1
l2r2θ
kΦ2(
_
z ,θ,k), (24)
where
Φ1(
_
z ,θ,k) =
1F1
(
1
2 −k, 3
2 ,− r 2
2i−1
4θ
)
p
θΓ
(
k + 1
2
) +
1F1
(
−k, 1
2 ,− r 2
2i−1
4θ
)
r2i−1Γ (k +1)
− (−1)i
1F1
(
1
2 −k, 1
2 ,− r 2
2i−1
4θ
)
p
θr2i−1Γ
(
k + 1
2
) +
1F1
(
1−k, 3
2 ,− r 2
2i−1
4θ
)
θΓ
(
k + 1
2
) , (25)
Φ2(
_
z ,θ,k) =−
1F1
(
1
2 −k, 3
2 ,− r 2
2
4θ
)
p
θΓ
(
k + 1
2
) +
1F1
(
−k, 1
2 ,− r 2
2
4θ
)
r2Γ (k +1)
, (26)
a =
(√
H1 −
√
H2
)(√
H2 −
√
H3
)
, b =
(√
H1 +
√
H2
)(√
H2 +
√
H3
)
,
l1 =β
√
H2
(√
H3 −
√
H2
)
, l2 =− l1
_
ap
H2
,
l3 =β
√
H2
(√
H3 +
√
H2
)
, l4 = l3
_
ap
H2
,
r1 =−
_
z +_
ap
H1
+ 4
_
a (m +1)p
H2
, r2 =−
_
z +_
ap
H1
+ 2
_
a (2m +1)p
H2
, r3 =−
_
z +_
ap
H1
+ 4
_
amp
H2
.
Z2(
_
z ,θ) = 1
b
[
(eθ−1)(p3 +p4)+ (p5 +p6)eθErf(pθ)
]
+
4∑
i=1
∞∑
m=0
(−1)m am
bm+1
∞∑
k=1
pi eiθ
kΦ3(
_
z ,θ,k), (27)
where
Φ3(
_
z ,θ,k) = −
1F1
(
1
2 −k, 3
2 ,− e2
i
4θ
)
p
θΓ
(
k + 1
2
) +
1F1
(
−k, 1
2 ,− e2
i
4θ
)
eiΓ (k +1)
+
_
ap
H2
1F1
(
1
2 −k, 1
2 ,− e2
i
4θ
)
p
θeiΓ
(
k + 1
2
) +
1F1
(
1−k, 3
2 ,− e2
i
4θ
)
θΓ
(
k + 1
2
) , (28)
23602-10
Spatial-temporal redistribution of point defects . . .
p1 =β
√
H2
(√
H3 −
√
H2
)
, p2 =β_a
(√
H2 −
√
H3
)
, p3 =− p2
_
ap
H2
,
p4 =−β
√
H3
(√
H1 +
√
H2
)
, p5 =− p4
_
ap
H3
, p6 =β
√
H1
(√
H2 −
√
H3
)
,
e1 =−
_
z +_
ap
H2
+ 4
_
a (m +1)p
H2
, e2 =−
_
z +_
ap
H2
+ 2
_
a (2m +1)p
H2
,
e3 =
_
z −_
ap
H2
+ 4
_
a (m +1)p
H2
, e4 =
_
z −_
ap
H2
+ 2
_
a (2m +1)p
H2
.
Z3(
_
z ,θ) = 1
b
[
eθ−1+eθ
_
ap
H2
Erf(pθ)
]
l3 +
2∑
i=1
∞∑
m=0
(−1)m am
bm+1
×
∞∑
k=1
l2i−1r2i−1θ
kΦ4(
_
z ,θ,k)+
∞∑
m=0
(−1)m am
bm+1
∞∑
k=1
l2r2θ
kΦ5(
_
z ,θ,k), (29)
where
Φ4(
_
z ,θ,k) = −
1F1
(
1
2 −k, 3
2 ,− r 2
2i−1
4θ
)
p
θΓ
(
k + 1
2
) +
1F1
(
−k, 1
2 ,− r 2
2i−1
4θ
)
r2i−1Γ (k +1)
+
_
ap
H2
1F1
(
1
2 −k, 1
2 ,− r 2
2i−1
4θ
)
p
θr2i−1Γ
(
k + 1
2
) +
1F1
(
1−k, 3
2 ,− r 2
2i−1
4θ
)
θΓ
(
k + 1
2
) , (30)
Φ5(
_
z ,θ,k) =−
1F1
(
1
2 −k, 3
2 ,− r 2
2
4θ
)
p
θΓ
(
k + 1
2
) +
1F1
(
−k, 1
2 ,− r 2
2
4θ
)
r2Γ (k +1)
, (31)
l1 =β
√
H2
(√
H1 −
√
H2
)
, l2 =− l1
_
ap
H2
,
l3 =β
√
H2
(√
H2 +
√
H1
)
, l4 = l3
_
ap
H2
,
r1 =
_
z −_
ap
H1
+ 4
_
a (m +1)p
H2
, r2 =
_
z −_
ap
H1
+ 2
_
a (2m +1)p
H2
, r3 =
_
z −_
ap
H1
+ 4
_
amp
H2
.
References
1. Ledentsov N.N., Ustinov V.M., Shchukin V.A., Kop’ev P.S., Alferov Zh.I., Bimberg D., Fiz. Tekh. Poluprovodn., 1998,
32, No. 4, 385 [Semiconductors, 1998, 32, No. 4, 343; doi:10.1134/1.1187396].
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Просторово-часовий перерозподiл точкових дефектiв у
тришарових напружених наногетеросистемах у межах
самоузгодженої деформацiйно-дифузiйної моделi
Р.М. Пелещак, Н.Я. Кулик,М.В. Дорошенко
Дрогобицький державний педагогiчний унiверситет iм. Iвана Франка,
вул. I. Франка, 24, 82100 Дрогобич, Україна
Побудовано модель просторово-часового розподiлу точкових дефектiв у тришаровiй напруженiй наноге-
теросистемi GaAs/InxGa1−xAs/GaAs з врахуванням самоузгодженої деформацiйно-дифузiйної взаємодiї.
У межах цiєї моделi розраховано профiль просторово-часового розподiлу вакансiй (мiжвузлових атомiв)
у напруженiй наногетеросистемi GaAs/InxGa1−xAs/GaAs та показано, що у випадку встановлення стацiо-нарного стану (t > 5τ(2)
d ), концентрацiя вакансiй в неоднорiдно стиснутому внутрiшньому шарi є меншою
вiдносно вихiдного середнього значення N (2)
d0 на 16%.
Ключовi слова: просторово-часовий перерозподiл, вакансiї, мiжвузловi атоми
23602-12
http://dx.doi.org/10.1063/1.354472
Introduction
The model of spatial-temporal redistribution of defects in a three-layer stressed nanoheterosystem
Analysis of the numerical results and discussion
Conclusions
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