Modeling at the nanometric scale of interfacial defects of a semiconductor heterostructure in the isotropic and anisotropic cases for the study of the influence of stresses.
الموضوعات :Ahmed Boussaha 1 , Rafik Makhloufi 2 , Rachid Benbouta 3 , Mourad Brioua 4
1 - Laboratory LAMSM, Mechanical Engineering Department, Faculty of Technology, University of Batna 2 Mostafa Ben Boulaid, Batna, Algeria
2 - Mechanical Engineering Department, University of Batna2, Algeria
3 - Department of Mechanical Engineering, Faculty of Technology, University of Batna2, Algeria
4 - Faculty of Technology, University of Batna 2, Algeria
الکلمات المفتاحية: Interface, GaAs/Si, Isotropic elasticity, Anisotropic elasticity, Elastic fields,
ملخص المقالة :
This work aims to determine the effect of stresses caused by dislocation networks placed at the interface of a semiconductor heterostructure of the thin GaAs / Si system. In this case, we use a mathematical modeling by Fourier series expansion to numerically simulate the stresses for the two cases of isotropic and anisotropic elasticity in order to predict the mechanical behavior of the heterostructure in the presence of interfacial dislocations while respecting well-defined stress boundary conditions. After establishing the hypotheses of the chosen model, which is a thin bimetallic strip, representing the GaAs / Si semiconductor heterostructure, and the boundary conditions relating to the problem posed, we obtained results of the stress distribution around a dislocation showing that the deformation is greater near the core of the dislocation. The elastic stress relaxation is reached for a layer thickness threshold of the GaAs deposit on the Si substrate not exceeding 5 nm.
[1] Fang S.F., Adomi K., Iyer S., Morkoc H., Zabel H., Choi C., Otsuka N., 1990, Gallium arsenide and other compound semiconductors on silicon, Journal of Applied Physics, 68(7): 31-58.
[2] Nakajima K., 1992, Calculation of stresses in GaAs/Si strained heterostructures. Journal of crystal growth, 121(3): 278-296.
[3] Mao E.W., Zhao W.Q., Zhang H.R., Li A.Z., Chen J.M., Fang G.P., 1988, The influence of strain and dislocations on transport properties of GaAs/Si strained-layer heterojunctions. Phys. Stat. Sol. (a), 110(2): 515-520.
[4] Ünlü H., 2022, Strain in Microscale and Nanoscale Semiconductor Heterostructures, Progress in Nanoscale and Low-Dimensional Materials and Devices, 144: 65-115.
[5] Makhloufi R., Boussaha A., Benbouta R., Baroura L. 2021, Anisotropic and isotropic elasticity applied for the study of elastic fields generated by interfacial dislocations in a heterostructure of InAs/(001)GaAs semiconductors. Journal of Solid Mechanics, 13(4): 503-512.
[6] Boussaha A., Makhloufi R., Madani S., 2019, Displacement Fields Influence Analysis caused by dislocation networks at a three layer system interfaces on the surface topology. Journal of Solid Mechanics, 11(3): 606-614.
[7] Vincent T., 2019, Nanomécanique des champs de défauts cristallins. Mécanique des matériaux [physics.class-ph]. Université de Lorraine; Ecole Doctorale C2MP.
[8] Gutkin M.Y., Romanov A.E., 1991, Straight Edge Dislocation in a Thin Two-Phase Plate I. Elastic Stress Fields, Physica Status Solidi (a), 125(1): 107-125.
[9] Bonnet R., Verger-Gaugry J.L., 1992, Couche épitaxique mince sur un substrat semi-infini: Rôle du désaccord paramétrique et de l'épaisseur sur les distortions élastiques, Philosophical Magazine A, 66(5): 849-871.
[10] Kim, Y., Madarang, M. A., Ju, E., Laryn, T., Chu, R. J., Kim, T. S., & Jung, D. (2023). GaAs/Si Tandem Solar Cells with an Optically Transparent InAlAs/GaAs Strained Layer Superlattices Dislocation Filter Layer. Energies, 16(3), 1158.
[11] Lovergine, N., Miccoli, I., Tapfer, L., &Prete, P. (2023). GaAs hetero-epitaxial layers grown by MOVPE on exactly-oriented and off-cut (111) Si: Lattice tilt, mosaicity and defects content. Applied Surface Science, 634 : 157627.
[12] Bonnet, R., & Morton, A. J. (1987). Contraste en MET à deux ondes d'une dislocation rectiligne parallèle à la surface libre d'un cristal anisotrope. Philosophical Magazine A, 56(6), 815-830.
[13] Nye, J. F. (1985). Physical properties of crystals. Oxford: Clarendon Press.
Journal of Solid Mechanics Vol. 16, No. 1 (2024) pp. 65-73 DOI: 10.60664/jsm.2024.3031774 |
Research Paper Modeling at The Nanometric Scale of Interfacial Defects of A Semiconductor Heterostructure in The Isotropic And Anisotropic Cases For The Study of The Influence of Stresses |
A. Boussaha, R. Makhloufi, R. Benbouta 1 , M. Brioua | |
Mechanical Engineering Department, Faculty of Technology, University of Batna 2 Mostafa Ben Boulaid, Batna, Algeiria | |
Received 26 March 2023; Received in revised form 29 April 2024; Accepted 24 May 2024 | |
| ABSTRACT |
| This work aims to determine the effect of stresses caused by dislocation networks placed at the interface of a semiconductor heterostructure of the thin GaAs/Si system. In this case, we use a mathematical modeling by Fourier series expansion to numerically simulate the stresses for the two cases of isotropic and anisotropic elasticity in order to predict the mechanical behavior of the heterostructure in the presence of interfacial dislocations while respecting well-defined stress boundary conditions. The elastic stress relaxation is reached for a layer thickness threshold of the GaAs deposit on the Si substrate not exceeding 5 nm.
|
| Keywords: Nanometric; Heterostructure; GaAs/Si; Isotropic; Anisotropic; Elastic fields. |
1 INTRODUCTION
T
HE evaluation of elastic fields generated by dislocation networks in semiconductor heterostructures has become essential. To design and manufacture better semiconductors, the electronics industry uses heterostructures with greater reliability in their various properties. Among the heterostructures which have been the object of study of the mechanical behavior of thin films of nanometric thickness deposited on a substrate to solve the problems of the constraints observed in the manufacture of components used in microelectronics, mention may be made of semiconductor heterostructures GaAs/Si and InAs/GaAs which are very interesting in research for their physical and optoelectronic properties [1].
Nakajima [2] calculated for the GaAs/Si heterostructure by proposing a theoretical model while considering that the interface is coherent between the thin layers.
Mao et al. [3] treated the effect of dislocations on the deformation of GaAs/Si samples by varying the thickness deposited by means of Raman scattering and hall measurement at different temperatures.
Ünlü [4] to model semiconductor heterostructures at the microscopic and nanometric scale to assess the effect of deformation on electronic and optical properties using GaAs/Si (001) as a model.
Makhloufi et al. [5] studied the possibility of epitaxy of thin layers of InAs on GaAs.
Boussaha et al. [6] determined the fields of the displacements as well as the iso values for an anisotropic three-layer CdTe/GaAs/GaAs(001) material under the effect of two dislocation networks placed at the interfaces.
Vincent [7] presented a development of theoretical models and numerical methods for the study at the nanometric scale of plastic deformation assisted by dislocations and by grain boundary type interfaces.
An isotropic domain calculation of the stress fields, for which analytical expressions have already been proposed by Gutkin and al. [8] for the case of the InAs / (001) GaAs system.
Our study is reserved for the GaAs/Si heterostructure in order to determine the effect of tensile and compressive stresses on its mechanical behavior.
Kim et al. [10] considered in their study the potential effect for GaAs/Si tandem cells showed a 1.28 V open circuit voltage.
Lovegine et al. [11] studied the integration of III.V GaAs semiconductors on Si for the fabrication of tandem solar cells obtained by the ether oepitaxy mode of GaAS on Si.
The integration of GaAs on Si without defects is one of the challenges for researchers in order to have an association with a very good quality interface unlike the interface of the InAs/GaAs system [5] and to combine the many advantages of Si, with the high mobility and direct gap properties of GaAs, to increase the speed of processors, to add new optical functionalities in microsystems, but also to produce photovoltaic cells with high efficiency and low cost.
2 GEOMETRY OF PROBLEM
The geometry of the problem represented in figure 1 below, shows the unidirectional network of dislocations generated between the layers of the GaAs/Si heterostructure.
Fig. 1
Geometry of the thin GaAs/Si bicrystal: 1/g is the period. The crystal stiffness’s are 
and
, with thicknesses h+ and h-.
In the isotropic case, the classical differential equation of elasticity is written [9]:
(1)
λ and μ are the Lamé coefficients of the deformed medium.
The deformation is assumed to be plane and periodic along the Ox1 axis. The expression of the stress field in the case of plane strain is:
(2)
After elimination of the parameter λ, using the classical relation (Hirth and Loth):
(3)
With the Poisson's ratio υ.
The stress equations necessary for the calculations are:
(4)
(5)
(6)
In the anisotropic case we must switch from the starting system to the working system by the passage matrix
:
Fig. 2
Diagram of the two starting and working markers.
aij is the matrix which allows the transition from the starting frame to the working frame.
The matrix of the elastic constants Cij for an anisotropic material having 36 elements in the reference of the crystal is written:
The matrix Cij is symmetric; therefore, the linear behavior of a material is then described in the general case by 21 independent coefficients.
After transformation, one obtains the matrix of the elastic constants 
in the work reference:
(7)
"T" is a (6x6) matrix obtained after transformation of the passage matrix aij
And:
3 EXPRESSION OF STRESSES AND BOUNDARY CONDITIONS
The mathematical formulation that deals with the anisotropic case is different from the isotropic case, we must consider that the two media are supposed to obey the general Hooke's law:
(8)
Ou :
(9)
After substituting (9) for (8), we get:
(10)
Since the dumb indices k and l will take the same values, therefore the two terms 
and 
are equal. We obtain:
(11)
The state of equilibrium of the stresses in the region of the distortions is written:
(12)
(13)
This field of displacements can be written in this form:
(14)
uk must satisfy Hooke's generalized law, linking stresses and deformations:
(15)
can be written as follows:
(16)
Where 
and 
represent complex constants which will be determined using the boundary conditions related to the problem.
So, by substituting (16) in (14) we have the equation of the field of displacements
(17)
By deriving the field from displacement, we get the stress field and the fact that a periodic series of intrinsic dislocations produces in each medium a stress field 
whose components can be developed in Fourier series which is written:
(18)
g: network period
n: harmonic number
x1: the periodicity axis
x2: the axis of heteroepitaxy
Represent the complex roots where:
Real part of 
imaginary part of
where 
is a fourth order tensor. To pass to a second order tensor we use the simplified notation of Voigt:
(19)
Complex constants which represent the solutions of the fields of displacements and stresses
Complex constants which represent the solutions of the fields of displacements and stresses
Complex constants
conjugate of 
Complex constants
The determination of the complex constants and 
for the positive layer deposited on the negative layer representing the substrate is done by applying the boundary conditions (Fig. 3) to the field suitable constraints which are:
- The continuity of the normal stresses at the interface:
(20)
k=1, 2 and 3
- The free surfaces of the bimetallic thin strip being in equilibrium:
k=1, 2 and 3 (21)
Fig. 3
Boundary conditions in the stress field.
4 APPLICATIONS AND RESULTS
4.1 GaAs/(001)Si system
In this work we have chosen the materials Gallium Arsenide (GaAs) and Silicon (Si) which are the subject of several studies in the field of optoelectronics.
Table1
Parameters of GaAs/ Si materials [12,13]
Parameters | GaAs | Si | |
Lattice parameters a(nm) | 0.56533 | 0.5428 | |
| 0.25 | 0.23 | |
| 46.27 | 66.28 | |
Burgers vector | 0.3838 | 0.3997 | |
Burgers vector of network | 0.3917 | ||
Period of dislocation network g (nm) | 9.7 | ||
| 0.65 | ||
Anisotropic elastic constants Cij (Gpa) | C11 = 118 C12 = 53.5 C44 = 59.4 | C11 = 165.7 C12 = 63.9 C44 = 79.6 | |
We take : 
In the following, we present the results of the stress simulation in the upper free layer of the GaAs/Si heterostructure in both isotropic and anisotropic cases.
4.2 Isotropic GaAs/Si case
Figures 4, 5, 6 and 7 illustrate, in the isotropic elastic case, the iso- constraints curves of a network of edge dislocations located at the interface of two GaAs and Si facets.
The representation of the iso- constraints σ11 and σ22 in 2D and 3D varying between 5 Mpa and 10 Mpa clearly shows the importance of the deformation around the dislocation along the X2 axis for a total thickness of the heterostructure of 10 nm and a vector of burgers b = 0.3917 nm oriented along the direction of periodicity of the X1 dislocations.
|
| |
Fig. 4 Iso-constraints σ22 in 2D GaAs / (001) Si: isotropic case. | Fig. 5 Iso-constraints σ22 in 3D GaAs / (001) Si: isotropic case. | |
|
| |
Fig. 6 Iso-constraints σ11 in 2D GaAs/(001)Si: isotropic case. | Fig. 7 Iso-constraints σ11 in 3D GaAs/(001)Si: isotropic case. | |
4.3 Anisotropic GaAs/Si case
|
|
Fig. 8 Iso- constraints σ22 in 2D GaAs/(001)Si anisotropic case. | Fig. 9 Iso- constraints σ22 in 3D GaAs/(001)Si anisotropic case. |
|
|
Fig. 10 Iso- constraints σ11 in 2D GaAs/(001)Si anisotropic case. | Fig. 11 Iso- constraints σ11 in 3D GaAs/(001)Si anisotropic case. |
Fig. 12 Fig. 13
Iso- constraints σ12 in 2D GaAs/(001)Si anisotropic case. Iso- constraints σ12 in 3D GaAs/(001)Si anisotropic case.
The results obtained from the theory and the simulation calculation shown in figures 8 to 13 in the form of 2D and 3D stress maps are to be understood. The 3D relief of the deformation of the interface between GaAs/(001)Si is due to the unidirectional dislocation network of Misfit which is a function of the deposited thickness for a vector of burgers oriented along Ox1. The deformation peak is clearly significant in the vicinity of the core of the dislocation which propagates tensile and compressive stresses influencing the mechanical behavior of the heterostructure.
The importance of the elastic quantities on the surface caused by the network of interfacial dislocations causing the phenomenon of undulation represents a determining index on the possibility of using the surface for a possible 3D growth of nanometric layers. This same phenomenon allows an elastic relaxation of the heterostructure being under tensile and compressive stress for a total thickness of 10 nm.
The symmetry of the stress fields in the isotropic case is quite visible contrary to the anisotropic case because of the anisotropy effect.
5 CONCLUSIONS
This work allowed us to examine and simulate, at the nanometric scale, the stress fields generated by unidirectional Misfit dislocation networks in the cases of isotropic and anisotropic elasticity.
After establishing the hypotheses of the chosen model, which is a thin bimetallic strip, representing the GaAs/Si semiconductor heterostructure, and the boundary conditions relating to the problem posed, we obtained results of the stress distribution around a dislocation showing that the deformation is greater near the core of the dislocation.
The importance of the elastic quantities on the surface caused by the network of interfacial dislocations causing the phenomenon of undulation represents a determining index on the possibility of using the surface for a possible 3D growth of nanometric layers. This same phenomenon allows an elastic relaxation of the heterostructure being under tensile and compressive stress for a total thickness of 10 nm.
REFERENCES
[1] Fang S.F., Adomi K., Iyer S., Morkoc H., Zabel H., Choi C., Otsuka N., 1990, Gallium arsenide and other compound semiconductors on silicon, Journal of Applied Physics, 68(7): 31-58.
[2] Nakajima K., 1992, Calculation of stresses in GaAs/Si strained heterostructures. Journal of crystal growth, 121(3): 278-296.
[3] Mao E.W., Zhao W.Q., Zhang H.R., Li A.Z., Chen J.M., Fang G.P., 1988, The influence of strain and dislocations on transport properties of GaAs/Si strained-layer heterojunctions. Phys. Stat. Sol. (a), 110(2): 515-520.
[4] Ünlü H., 2022, Strain in Microscale and Nanoscale Semiconductor Heterostructures, Progress in Nanoscale and Low-Dimensional Materials and Devices, 144: 65-115.
[5] Makhloufi R., Boussaha A., Benbouta R., Baroura L. 2021, Anisotropic and isotropic elasticity applied for the study of elastic fields generated by interfacial dislocations in a heterostructure of InAs/(001)GaAs semiconductors. Journal of Solid Mechanics, 13(4): 503-512.
[6] Boussaha A., Makhloufi R., Madani S., 2019, Displacement Fields Influence Analysis caused by dislocation networks at a three layer system interfaces on the surface topology. Journal of Solid Mechanics, 11(3): 606-614.
[7] Vincent T., 2019, Nanomécanique des champs de défauts cristallins. Mécanique des matériaux [physics.class-ph]. Université de Lorraine; Ecole Doctorale C2MP.
[8] Gutkin M.Y., Romanov A.E., 1991, Straight Edge Dislocation in a Thin Two-Phase Plate I. Elastic Stress Fields, Physica Status Solidi (a), 125(1): 107-125.
[9] Bonnet R., Verger-Gaugry J.L., 1992, Couche épitaxique mince sur un substrat semi-infini: Rôle du désaccord paramétrique et de l'épaisseur sur les distortions élastiques, Philosophical Magazine A, 66(5): 849-871.
[10] Kim, Y., Madarang, M. A., Ju, E., Laryn, T., Chu, R. J., Kim, T. S., & Jung, D. (2023). GaAs/Si Tandem Solar Cells with an Optically Transparent InAlAs/GaAs Strained Layer Superlattices Dislocation Filter Layer. Energies, 16(3), 1158.
[11] Lovergine, N., Miccoli, I., Tapfer, L., &Prete, P. (2023). GaAs hetero-epitaxial layers grown by MOVPE on exactly-oriented and off-cut (111) Si: Lattice tilt, mosaicity and defects content. Applied Surface Science, 634 : 157627.
[12] Bonnet, R., & Morton, A. J. (1987). Contraste en MET à deux ondes d'une dislocation rectiligne parallèle à la surface libre d'un cristal anisotrope. Philosophical Magazine A, 56(6), 815-830.
[13] Nye, J. F. (1985). Physical properties of crystals. Oxford: Clarendon Press.
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[1] Corresponding author. Tel.: +213 33 81 21 43, Fax: +213 33 81 21 43.
E-mail address: r_benbouta@yahoo.fr (Prof. R. Benbouta)
