![International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2
41
AN APPROACH TO OPTIMIZE OF MANUFACTURING
OF A VOLTAGE REFERENCE BASED ON
HETEROSTRUCTURES TO INCREASE DENSITY OF
THEIR ELEMENTS. ANALYSIS OF INFLUENCE OF
MISS-MATCH INDUCED STRESS AND POROSITY OF
MATERIALS ON TECHNOLOGICAL PROCESS
E.L. Pankratov
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
ABSTRACT
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage
reference. Framework the approach we consider manufacturing the inverter in heterostructure with specif-
ic configuration. Several required areas of the heterostructure should be doped by diffusion or ion implan-
tation. After that dopant and radiation defects should by annealed framework optimized scheme. We also
consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We
introduce an analytical approach to analyze mass and heat transport in heterostructures during manufac-
turing of integrated circuits with account mismatch-induced stress.
KEYWORDS
Field-effect heterotransystors, voltage reference, decreasing of dimensions, optimization of manufacturing,
miss-match induced stress, porosity of materials.
1. INTRODUCTION
In the present time several actual problems of the solid state electronics (such as increasing of
performance, reliability and density of elements of integrated circuits: diodes, field-effect and
bipolar transistors) are intensively solving [1-6]. To increase the performance of these devices it
is attracted an interest determination of materials with higher values of charge carriers mobility
[7-10]. One way to decrease dimensions of elements of integrated circuits is manufacturing them
in thin film heterostructures [3-5,11]. In this case it is possible to use inhomogeneity of
heterostructure and necessary optimization of doping of electronic materials [12] and develop-
ment of epitaxial technology to improve these materials (including analysis of mismatch induced
stress) [14-16]. An alternative approaches to increase dimensions of integrated circuits are using
of laser and microwave types of annealing [17-19].
Framework the paper we introduce an approach to manufacture field-effect transistors. The ap-
proach gives a possibility to decrease their dimensions with increasing their density framework a
voltage reference [20]. We also consider possibility to decrease mismatch-induced stress to de-
crease quantity of defects, generated due to the stress. In this paper we consider a heterostructure,
which consist of a substrate and an epitaxial layer (see Fig. 1). We also consider a porous buffer
layer between the substrate and the epitaxial layer. The epitaxial layer includes into itself several](https://image.slidesharecdn.com/1218jares06-200320100726/85/AN-APPROACH-TO-OPTIMIZE-OF-MANUFACTURING-OF-A-VOLTAGE-REFERENCE-BASED-ON-HETEROSTRUCTURES-TO-INCREASE-DENSITY-OF-THEIR-ELEMENTS-ANALYSIS-OF-INFLUENCE-OF-MISS-MATCH-INDUCED-STRESS-AND-POROSITY-OF-MATERIALS-ON-TECHNOLOGICAL-PROCESS-1-320.jpg)
![International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2
42
sections, which were manufactured by using other materials. These sections have been doped by
diffusion or ion implantation to manufacture the required types of conductivity (p or n). These
areas became sources, drains and gates (see Fig. 1). After this doping it is required annealing of
dopant and/or radiation defects. Main aim of the present paper is analysis of redistribution of do-
pant and radiation defects to determine conditions, which correspond to decreasing of elements of
the considered voltage reference and at the same time to increase their density. At the same time
we consider a possibility to decrease mismatch-induced stress.
Fig. 1a. Structure of the considered voltage reference [20]
Fig. 1b. Heterostructure with a substrate, epitaxial layers and buffer layer (view from side)
2. METHOD OF SOLUTION
To solve our aim we determine and analyzed spatio-temporal distribution of concentration of do-
pant in the considered heterostructure. We determine the distribution by solving the second Fick's
law in the following form [1,21-24]
( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
( ) ( ) ( ) ( ) +
∫∇
∂
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Ω+
∫∇
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zL
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WdtWyxCtzyx
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,,,,,,,,,,,, µµ (1)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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AN APPROACH TO OPTIMIZE OF MANUFACTURING OF A VOLTAGE REFERENCE BASED ON HETEROSTRUCTURES TO INCREASE DENSITY OF THEIR ELEMENTS. ANALYSIS OF INFLUENCE OF MISS-MATCH INDUCED STRESS AND POROSITY OF MATERIALS ON TECHNOLOGICAL PROCESS
- 1. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 41 AN APPROACH TO OPTIMIZE OF MANUFACTURING OF A VOLTAGE REFERENCE BASED ON HETEROSTRUCTURES TO INCREASE DENSITY OF THEIR ELEMENTS. ANALYSIS OF INFLUENCE OF MISS-MATCH INDUCED STRESS AND POROSITY OF MATERIALS ON TECHNOLOGICAL PROCESS E.L. Pankratov Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia ABSTRACT In this paper we introduce an approach to increase density of field-effect transistors framework a voltage reference. Framework the approach we consider manufacturing the inverter in heterostructure with specif- ic configuration. Several required areas of the heterostructure should be doped by diffusion or ion implan- tation. After that dopant and radiation defects should by annealed framework optimized scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures during manufac- turing of integrated circuits with account mismatch-induced stress. KEYWORDS Field-effect heterotransystors, voltage reference, decreasing of dimensions, optimization of manufacturing, miss-match induced stress, porosity of materials. 1. INTRODUCTION In the present time several actual problems of the solid state electronics (such as increasing of performance, reliability and density of elements of integrated circuits: diodes, field-effect and bipolar transistors) are intensively solving [1-6]. To increase the performance of these devices it is attracted an interest determination of materials with higher values of charge carriers mobility [7-10]. One way to decrease dimensions of elements of integrated circuits is manufacturing them in thin film heterostructures [3-5,11]. In this case it is possible to use inhomogeneity of heterostructure and necessary optimization of doping of electronic materials [12] and develop- ment of epitaxial technology to improve these materials (including analysis of mismatch induced stress) [14-16]. An alternative approaches to increase dimensions of integrated circuits are using of laser and microwave types of annealing [17-19]. Framework the paper we introduce an approach to manufacture field-effect transistors. The ap- proach gives a possibility to decrease their dimensions with increasing their density framework a voltage reference [20]. We also consider possibility to decrease mismatch-induced stress to de- crease quantity of defects, generated due to the stress. In this paper we consider a heterostructure, which consist of a substrate and an epitaxial layer (see Fig. 1). We also consider a porous buffer layer between the substrate and the epitaxial layer. The epitaxial layer includes into itself several
- 2. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 42 sections, which were manufactured by using other materials. These sections have been doped by diffusion or ion implantation to manufacture the required types of conductivity (p or n). These areas became sources, drains and gates (see Fig. 1). After this doping it is required annealing of dopant and/or radiation defects. Main aim of the present paper is analysis of redistribution of do- pant and radiation defects to determine conditions, which correspond to decreasing of elements of the considered voltage reference and at the same time to increase their density. At the same time we consider a possibility to decrease mismatch-induced stress. Fig. 1a. Structure of the considered voltage reference [20] Fig. 1b. Heterostructure with a substrate, epitaxial layers and buffer layer (view from side) 2. METHOD OF SOLUTION To solve our aim we determine and analyzed spatio-temporal distribution of concentration of do- pant in the considered heterostructure. We determine the distribution by solving the second Fick's law in the following form [1,21-24] ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, ( ) ( ) ( ) ( ) + ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zL S S zL S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x 0 1 0 1 ,,,,,,,,,,,, µµ (1)
- 3. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 43 ( ) ( ) ( ) + + + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SCSCSC ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ,,,,,,,,, 222 with boundary and initial conditions ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , C(x,y,z,0)=fC(x,y,z), ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC . Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic vol- ume of dopant; ∇s is the symbol of surficial gradient; ( )∫ zL zdtzyxC 0 ,,, is the surficial concentration of dopant on interface between layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface between layers of heterostructure); µ1(x,y,z,t) and µ2(x,y,z,t) are the chemical potential due to the presence of mismatch-induced stress and porosity of material; D and DS are the coefficients of volumetric and surficial diffusions. Values of dopant diffusions coeffi- cients depends on properties of materials of heterostructure, speed of heating and cooling of mate- rials during annealing and spatio-temporal distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be approximated by the following relations [25-27] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ , ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD SLSS ςςξ γ γ . (2) Here DL (x,y,z,T) and DLS (x,y,z,T) are the spatial (due to accounting all layers of heterostruicture) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; P (x,y,z,T) is the limit of solubility of dopant; parameter γ depends on properties of materials and could be integer in the following interval γ ∈[1,3] [25]; V (x,y,z,t) is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibrium dis- tribution of vacancies. Concentrational dependence of dopant diffusion coefficient has been de- scribed in details in [25]. Spatio-temporal distributions of concentration of point radiation defects have been determined by solving the following system of equations [21-24,26,27] ( ) ( ) ( ) ( ) ( ) + + = y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ×−− + tzyxITzyxktzyxITzyxk z tzyxI TzyxD z VIIII ,,,,,,,,,,,, ,,, ,,, , 2 , ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∫∇ ∂ ∂ Ω+× x tzyx TkV D x WdtWyxItzyx Tk D x tzyxV SI zL S IS ,,, ,,,,,,,,, 2 0 µ µ (3) ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + zL S ISSISI WdtWyxItzyx Tk D yz tzyx TkV D zy tzyx TkV D y 0 22 ,,,,,, ,,,,,, µ µµ
- 4. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 44 ( ) ( ) ( ) ( ) ( ) + + = y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ×−− + tzyxITzyxktzyxVTzyxk z tzyxV TzyxD z VIVVV ,,,,,,,,,,,, ,,, ,,, , 2 , ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∫∇ ∂ ∂ Ω+× x tzyx TkV D x WdtWyxVtzyx Tk D x tzyxV SV zL S VS ,,, ,,,,,,,,, 2 0 µ µ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∫∇ ∂ ∂ Ω+ y tzyx TkV D y WdtWyxVtzyx Tk D y SV zL S VS ,,, ,,,,,, 2 0 µ µ ( ) ∂ ∂ ∂ ∂ + z tzyx TkV D z SV ,,,2 µ with boundary and initial conditions ( ) 0 ,,, 0 = =x x tzyxI ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxI ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =z z tzyxI ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =x x tzyxV ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxV ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =z z tzyxV ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxV ∂ ∂ , ( ) ++ +=+++ ∞ 2 1 2 1 2 1 111 2 1,,, zyxTk VttVztVytVxV nnn ωλ , I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (4) Here I(x,y,z,t) is the spatio-temporal distribution of concentration of radiation interstitials; I* is the equilibrium distribution of interstitials; DI(x,y,z,T), DV(x,y,z,T), DIS(x,y, z,T), DVS(x,y,z,T) are the coefficients of volumetric and surficial diffusions of interstitials and vacancies, respectively; terms V2 (x,y,z,t) and I2 (x,y,z,t) correspond to generation of divacancies and diinterstitials, respec- tively (see, for example, [27] and appropriate references in this book); kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parameters of recombination of point radiation defects and generation of their complexes; k is the Boltzmann constant; ω = a3 , a is the interatomic distance; λ is the specific surface energy. To account porosity of buffer layers we assume, that porous are approximately cylindrical with average values 2 1 2 1 yxr += and z1 before annealing [24]. With time small pores decomposing on vacancies. The vacancies absorbing by larger pores [28]. With time large pores became larger due to absorbing the vacancies and became more spherical [28]. Distribution of concentration of vacancies in heterostructure, existing due to porosity, could be determined by summing on all pores, i.e. ( ) ( )∑ ∑ ∑ +++= = = = l i m j n k p tkzjyixVtzyxV 0 0 0 ,,,,,, χβα , 222 zyxR ++= . Here α, β and χ are the average distances between centers of pores in directions x, y and z; l, m and n are the quantity of pores inappropriate directions. Spatio-temporal distributions of divacancies ΦV (x,y,z,t) and diinterstitials ΦI (x,y,z,t) could be de- termined by solving the following system of equations [26,27]
- 5. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 45 ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ zL IS SII I WdtWyxtzyx Tk D xz tzyx TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )++ ∫ Φ∇ ∂ ∂ Ω+ Φ tzyxITzyxkWdtWyxtzyx Tk D y II zL IS SI ,,,,,,,,,,,, 2 , 0 1 µ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ,,,,,,,,, 222 µµµ ( ) ( )tzyxITzyxkI ,,,,,,+ (5) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ zL VS SVV V WdtWyxtzyx Tk D xz tzyx TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )++ ∫Φ∇ ∂ ∂ Ω+ Φ tzyxVTzyxkWdtWyxtzyx Tk D y VV zL VS SV ,,,,,,,,,,,, 2 , 0 1 µ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SVSVSV ,,,,,,,,, 222 µµµ ( ) ( )tzyxVTzyxkV ,,,,,,+ with boundary and initial conditions ( ) 0 ,,, 0 = Φ =x I x tzyx ∂ ∂ , ( ) 0 ,,, = Φ = xLx I x tzyx ∂ ∂ , ( ) 0 ,,, 0 = Φ =y I y tzyx ∂ ∂ , ( ) 0 ,,, = Φ = yLy I y tzyx ∂ ∂ , ( ) 0 ,,, 0 = Φ =z I z tzyx ∂ ∂ , ( ) 0 ,,, = Φ = zLz I z tzyx ∂ ∂ , ( ) 0 ,,, 0 = Φ =x V x tzyx ∂ ∂ , ( ) 0 ,,, = Φ = xLx V x tzyx ∂ ∂ , ( ) 0 ,,, 0 = Φ =y V y tzyx ∂ ∂ , ( ) 0 ,,, = Φ = yLy V y tzyx ∂ ∂ , ( ) 0 ,,, 0 = Φ =z V z tzyx ∂ ∂ , ( ) 0 ,,, = Φ = zLz V z tzyx ∂ ∂ , (6) ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). Here DΦI(x,y,z,T), DΦV(x,y,z,T), DΦIS(x,y,z,T) and DΦVS(x,y,z,T) are the coefficients of volumetric and surficial diffusions of complexes of radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are the pa- rameters of decay of complexes of radiation defects. Chemical potential µ1 in Eq.(1) could be determine by the following relation [21] µ1=E(z)Ωσij [uij(x,y,z,t)+uji(x,y,z,t)]/2, (7)
- 6. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 46 where E(z) is the Young modulus, σij is the stress tensor; ∂ ∂ + ∂ ∂ = i j j i ij x u x u u 2 1 is the deformation tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector ( )tzyxu ,,, ρ ; xi, xj are the coordinate x, y, z. The Eq. (3) could be transform to the following form ( ) ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ = i j j i i j j i x tzyxu x tzyxu x tzyxu x tzyxu tzyx ,,,,,, 2 1,,,,,, ,,,µ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )zETtzyxTzzK x tzyxu z z ij k kij ij 2 ,,,3 ,,, 21 000 Ω −− − ∂ ∂ − +− δβε σ δσ δε , where σ is Poisson coefficient; ε0 = (as-aEL)/aEL is the mismatch parameter; as, aEL are lattice dis- tances of the substrate and the epitaxial layer; K is the modulus of uniform compression; β is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for our case) with room temperature. Components of displacement vector could be obtained by solution of the following equations [22] ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z xzxyxxx ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z yzyyyxy ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ (8) ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z zzzyzxz ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ , where ( ) ( )[ ] ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + = k k ij k kij i j j i ij x tzyxu x tzyxu x tzyxu x tzyxu z zE ,,,,,, 3 ,,,,,, 12 δ δ σ σ ( ) ( ) ( ) ( )[ ]r TtzyxTzKzzK −−× ,,,β , ρ(z) is the density of materials of heterostructure, δij Is the Kronecker symbol. Conditions for the system of Eq. (8) could be written in the form ( ) 0 ,,,0 = ∂ ∂ x tzyu ρ ; ( ) 0 ,,, = ∂ ∂ x tzyLu x ρ ; ( ) 0 ,,0, = ∂ ∂ y tzxu ρ ; ( ) 0 ,,, = ∂ ∂ y tzLxu y ρ ; ( ) 0 ,0,, = ∂ ∂ z tyxu ρ ; ( ) 0 ,,, = ∂ ∂ z tLyxu z ρ ; ( ) 0 0,,, uzyxu ρρ = ; ( ) 0 ,,, uzyxu ρρ =∞ . We determine spatio-temporal distributions of concentrations of dopant and radiation defects by solving the Eqs.(1), (3) and (5) framework standard method of averaging of function corrections [29]. Previously we transform the Eqs.(1), (3) and (5) to the following form with account initial distributions of the considered concentrations ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1a)
- 7. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 47 ( ) ( ) ( ) + + + + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SCSCSC ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ,,,,,,,,, 222 ( ) ( ) ( ) + ∫∇ ∂ ∂ Ω+ + zL S SSC WdtWyxCtzyx Tk D xz tzyx TkV D z 0 2 ,,,,,, ,,, µ ∂ µ∂ ∂ ∂ ( ) ( ) ∫∇ ∂ ∂ Ω+ zL S S WdtWyxCtzyx Tk D y 0 ,,,,,,µ ( ) ( ) ( ) ( ) ( ) + + = y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫∇ ∂ ∂ Ω+ + zL S IS I WdtWyxItzyx Tk D xz tzyxI TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )−− ∫∇ ∂ ∂ Ω+ tzyxITzyxkWdtWyxItzyx Tk D y II zL S IS ,,,,,,,,,,,, 2 , 0 1 µ ( ) ( ) ( ) ( ) ( )tzyxftzyxVtzyxITzyxk IVI δ,,,,,,,,,,,, +− (3a) ( ) ( ) ( ) ( ) ( ) + + = y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫∇ ∂ ∂ Ω+ + zL S VS V WdtWyxVtzyx Tk D xz tzyxV TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )−− ∫∇ ∂ ∂ Ω+ tzyxITzyxkWdtWyxItzyx Tk D y II zL S IS ,,,,,,,,,,,, 2 , 0 1 µ ( ) ( ) ( ) ( ) ( )tzyxftzyxVtzyxITzyxk VVI δ,,,,,,,,,,,, +− ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫ Φ∇ ∂ ∂ Ω+ Φ + Φ Φ zL IS SII I WdtWyxtzyx Tk D xz tzyx TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )++ ∫ Φ∇ ∂ ∂ Ω+ Φ tzyxITzyxkWdtWyxtzyx Tk D y I zL IS SI ,,,,,,,,,,,, 0 1 µ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ,,,,,,,,, 222 µµµ ( ) ( ) ( ) ( )tzyxftzyxITzyxk III δ,,,,,,,, 2 , Φ ++ (5a) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ + Φ Φ zL VS SVV V WdtWyxtzyx Tk D xz tzyx TzyxD z 0 1 ,,,,,, ,,, ,,, µ ∂ ∂ ∂ ∂
- 8. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 48 ( ) ( ) ( ) ( )++ ∫Φ∇ ∂ ∂ Ω+ Φ tzyxITzyxkWdtWyxtzyx Tk D y I zL IS SI ,,,,,,,,,,,, 0 1 µ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SVSVSV ,,,,,,,,, 222 µµµ ( ) ( ) ( ) ( )tzyxftzyxVTzyxk VVV δ,,,,,,,, 2 , Φ ++ . Farther we replace concentrations of dopant and radiation defects in right sides of Eqs. (1a), (3a) and (5a) on their not yet known average values α1ρ. In this situation we obtain equations for the first-order approximations of the required concentrations in the following form ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= ∂ ∂ tzyx Tk D z y tzyx Tk D z xt tzyxC S S CS S C ,,,,,, ,,, 1111 1 µαµα ( ) ( ) ( ) ( ) + + ++ y tzyx TkV D yx tzyx TkV D x tzyxf SCSC C ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ δ ,,,,,, ,, 22 ( ) + z tzyx TkV D z SC ∂ µ∂ ∂ ∂ ,,,2 (1b) ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= tzyx Tk D z y tzyx Tk D x z t tzyxI S IS IS IS I ,,,,,, ,,, 11 1 µαµα ∂ ∂ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ,,,,,,,,, 222 µµµ ( ) ( ) ( ) ( )TzyxkTzyxktzyxf VIVIIIII ,,,,,,,, ,11, 2 1 αααδ −−+ (3b) ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= tzyx Tk D z y tzyx Tk D x z t tzyxV S VS VS VS V ,,,,,, ,,, 1111 1 µαµα ∂ ∂ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SVSVSV ,,,,,,,,, 222 µµµ ( ) ( ) ( ) ( )TzyxkTzyxktzyxf VIVIVVVV ,,,,,,,, ,11, 2 1 αααδ −−+ ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= Φ Φ Φ Φ Φ tzyx Tk D y ztzyx Tk D x z t tzyx S SI IS SI I I ,,,,,, ,,, 1111 1 µαµα ∂ ∂ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ,,,,,,,,, 222 µµµ ( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxktzyxf IIII ,,,,,,,,,,,,,, 2 , +++ Φ δ (5b) ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= Φ Φ Φ Φ Φ tzyx Tk D y ztzyx Tk D x z t tzyx S SV VS SV V V ,,,,,, ,,, 1111 1 µαµα ∂ ∂ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SVSVSV ,,,,,,,,, 222 µµµ ( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxktzyxf VVVV ,,,,,,,,,,,,,, 2 , +++ Φ δ .
- 9. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 49 Integration of the left and right sides of the Eqs. (1b), (3b) and (5b) on time gives us possibility to obtain relations for above approximation in the final form ( ) ( ) ( ) ( ) ( ) ( )∫ × ++∇ ∂ ∂ Ω= t SLS V zyxV V zyxV zyx Tk z TzyxD x tzyxC 0 2* 2 2*111 ,,,,,, 1,,,,,,,,, τ ς τ ςτµ ( ) ( ) ( ) ( )∫ + +∇ ∂ ∂ Ω+ +× t CS LSSC CS C TzyxP TzyxDzyx y d TzyxP 0 1 11 1 1 ,,, 1,,,,,, ,,, 1 γ γ γ γ αξ τµατ αξ α ( ) ( ) ( ) ( ) ( ) +∫++ ++× t SC C d x zyx TkV D x zyxfd V zyxV V zyxV Tk z 0 2 2* 2 2*1 ,,, ,, ,,,,,, 1 τ ∂ τµ∂ ∂ ∂ τ τ ς τ ς ( ) ( ) ∫+∫+ t SC t SC d z zyx TkV D z d y zyx TkV D y 0 2 0 2 ,,,,,, τ ∂ τµ∂ ∂ ∂ τ ∂ τµ∂ ∂ ∂ (1c) ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω= t S IS I t S IS I dzyx Tk D y zdzyx Tk D x ztzyxI 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + t SI t SI t SI d z tzyx TkV D z d y tzyx TkV D y d x tzyx TkV D x 0 2 0 2 0 2 ,,,,,,,,, τ µ τ µ τ µ ( ) ( ) ( )∫−∫−+ t VIVI t IIII dTzyxkdTzyxkzyxf 0 ,11 0 , 2 1 ,,,,,,,, ταατα (3c) ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω= t S IS V t S IS V dzyx Tk D y zdzyx Tk D x ztzyxV 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + t SV t SV t SV d z tzyx TkV D z d y tzyx TkV D y d x tzyx TkV D x 0 2 0 2 0 2 ,,,,,,,,, τ µ τ µ τ µ ( ) ( ) ( )∫−∫−+ t VIVI t VVVV dTzyxkdTzyxkzyxf 0 ,11 0 , 2 1 ,,,,,,,, ταατα ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω=Φ Φ Φ Φ Φ t S SI I t S SI II dzyx Tk D x zdzyx Tk D x ztzyx 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) ∫ × ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ ++ ΦΦΦ Φ t SI t SI t SI I TkV D z d y zyx TkV D y d x zyx TkV D x zyxf 00 2 0 2 ,,,,,, ,, τ τµ τ τµ (5c) ( ) ( ) ( ) ( ) ( )∫+∫+ ∂ ∂ × t II t I dzyxITzyxkdzyxITzyxkd z zyx 0 2 , 0 2 ,,,,,,,,,,,, ,,, τττττ τµ ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω=Φ Φ Φ Φ Φ t S SV V t S SV VV dzyx Tk D x zdzyx Tk D x ztzyx 0 11 0 111 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) ×∫ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ ++ ΦΦΦ Φ t SV t SV t SV V TkV D z d y zyx TkV D y d x zyx TkV D x zyxf 00 2 0 2 ,,,,,, ,, τ τµ τ τµ ( ) ( ) ( ) ( ) ( )∫+∫+ ∂ ∂ × t VV t V dzyxVTzyxkdzyxVTzyxkd z zyx 0 2 , 0 2 ,,,,,,,,,,,, ,,, τττττ τµ . We determine average values of the first-order approximations of concentrations of dopant and radiation defects by the following standard relation [29]
- 10. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 50 ( )∫ ∫ ∫ ∫ Θ = Θ 0 0 0 0 11 ,,, 1 xL yL zL zyx tdxdydzdtzyx LLL ρα ρ . (9) Substitution of the relations (1c), (3c) and (5c) into relation (9) gives us possibility to obtain re- quired average values in the following form ( )∫ ∫ ∫= xL yL zL C zyx C xdydzdzyxf LLL 0 0 0 1 ,, 1 α , ( ) 4 3 4 1 2 3 2 4 2 3 1 4 4 4 a Aa a aLLLBa B a Aa zyx I + − Θ+Θ +− + =α , ( ) Θ−−∫ ∫ ∫ Θ = zyxIII xL yL zL I IIV V LLLSxdydzdzyxf S 001 0 0 0 100 1 ,, 1 α α α , where ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ 0 0 0 0 11, ,,,,,,,,, xL yL zL ji ij tdxdydzdtzyxVtzyxITzyxktS ρρρρ , ( )0000 2 00004 VVIIIVII SSSSa −= , 0000 2 0000003 VVIIIVIIIV SSSSSa −+= , ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫= xL yL zL I xL yL zL VIVIV xdydzdzyxfxdydzdzyxfSSa 0 0 00 0 0 2 00002 ,,2,, ( )∫ ∫ ∫−Θ−Θ+× xL yL zL IIVVVzyxzyxIVIIVV xdydzdzyxfSSLLLLLLSSS 0 0 0 2 0000 222222 000000 ,, , ×= 001 IV Sa ( )∫ ∫ ∫× xL yL zL I xdydzdzyxf 0 0 0 ,, , ( ) 2 0 0 0 000 ,, ∫ ∫ ∫= xL yL zL IVV xdydzdzyxfSa , 4 2 2 4 2 32 48 a a a a yA Θ−Θ+= , 3 323 32 4 2 6 qpqqpq a a B ++−−++ Θ = , × Θ−Θ− Θ− Θ = 4 2 32 2 4 31 02 4 2 3 44 24 a a a a aa LLLa a a q zyx 2 4 2 1 4 222 3 4 3 2 3 2 4 2 0 8548 a a LLL a a a a zyx Θ − Θ − Θ × , 4 2 2 4 31402 1812 4 a a a aaLLLaa p zyx Θ − Θ− Θ= , ( )∫ ∫ ∫+ Θ + Θ = ΦΦ xL yL zL I zyxzyx II zyx I I xdydzdzyxf LLLLLL S LLL R 0 0 0 201 1 ,, 1 α ( )∫ ∫ ∫+ Θ + Θ = ΦΦ xL yL zL V zyxzyx VV zyx V V xdydzdzyxf LLLLLL S LLL R 0 0 0 201 1 ,, 1 α , where ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ 0 0 0 0 1 ,,,,,, xL yL zL i Ii tdxdydzdtzyxITzyxktRρ . We determine approximations of the second and higher orders of concentrations of dopant and radiation defects framework standard iterative procedure of method of averaging of function cor- rections [29]. Framework this procedure to determine approximations of the n-th order of concen- trations of dopant and radiation defects we replace the required concentrations in the Eqs. (1c), (3c), (5c) on the following sum αnρ+ρn-1(x,y,z,t). In this case the second-order approximations of required concebtrations could be written as ( ) ( )[ ] ( ) ( ) ( ) ( )∫ × ++ + + ∂ ∂ = t C V zyxV V zyxV TzyxP zyxC x tzyxC 0 2* 2 2*1 12 2 ,,,,,, 1 ,,, ,,, 1,,, τ ς τ ς τα ξ γ γ ( ) ( ) ( ) ( ) ( ) ( )∫ × ++ ∂ ∂ + ∂ ∂ × t LL V zyxV V zyxV TzyxD y d x zyxC TzyxD 0 2* 2 2*1 1 ,,,,,, 1,,, ,,, ,,, τ ς τ ςτ τ ( ) ( )[ ] ( ) ( ) ( ) ( )∫ × ++ ∂ ∂ + + + ∂ ∂ × t C V zyxV V zyxV zTzyxP tzyxC y zyxC 0 2* 2 2*1 121 ,,,,,, 1 ,,, ,,, 1 ,,, τ ς τ ς α ξ τ γ γ
- 11. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 51 ( ) ( ) ( )[ ] ( ) ( )++ + + ∂ ∂ × zyxfd TzyxP zyxC z zyxC TzyxD C C L ,, ,,, ,,, 1 ,,, ,,, 121 τ τα ξ τ γ γ ( ) ( )[ ] ( ) ×∫ ∇ ∂ ∂ Ω+∫ ∫ +∇ ∂ ∂ Ω+ t S S t zL CS S Tk D zyx y dWdWyxCzyx Tk D x 00 0 12 ,,,,,,,,, τµττατµ ( )[ ] ( ) ( ) + + +∫ +× y tzyx TkV D yx tzyx TkV D x dWdWyxC SCSC zL C ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ττα ,,,,,, ,,, 22 0 12 ( ) ( ) + + z tzyx TkV D zy tzyx TkV D y SCSC ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ,,,,,, 22 (1e) ( ) ( ) ( ) ( ) ( ) +∫+∫= t I t I d y zyxI TzyxD y d x zyxI TzyxD x tzyxI 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] −∫ +−∫+ t III t I dzyxITzyxkd z zyxI TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,, ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )[ ] ( )×∫∇ ∂ ∂ +∫ ++− t S t VIVI zyx x dzyxVzyxITzyxk 00 1212, ,,,,,,,,,,,, τµττατα ( )[ ] ( ) ( )[ ]×∫ ∫ +∇ ∂ ∂ +∫ +Ω× t zL IS zL I IS WyxIzyx y dWdWyxI Tk D 0 0 12 0 12 ,,,,,,,,, τατµττα ( ) ( ) + + +Ω× y tzyx TkV D yx tzyx TkV D x dWd Tk D SISIIS ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ τ ,,,,,, 22 ( ) ( )zyxf z tzyx TkV D z I SI ,, ,,,2 + + ∂ µ∂ ∂ ∂ (3e) ( ) ( ) ( ) ( ) ( ) +∫+∫= t V t V d y zyxV TzyxD y d x zyxV TzyxD x tzyxV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] −∫ +−∫+ t VVV t V dzyxVTzyxkd z zyxV TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,, ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )[ ] ( )×∫∇ ∂ ∂ +∫ ++− t S t VIVI zyx x dzyxVzyxITzyxk 00 1212, ,,,,,,,,,,,, τµττατα ( )[ ] ( ) ( )[ ]×∫ ∫ +∇ ∂ ∂ +∫ +Ω× t zL VS zL V VS WyxVzyx y dWdWyxV Tk D 0 0 12 0 12 ,,,,,,,,, τατµττα ( ) ( ) + + +Ω× y tzyx TkV D yx tzyx TkV D x dWd Tk D SVSVVS ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ τ ,,,,,, 22 ( ) ( )zyxf z tzyx TkV D z V SV ,, ,,,2 + + ∂ µ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( )∫ + Φ +∫ Φ =Φ ΦΦ t I I t I II dTzyxD y zyx y d x zyx TzyxD x tzyx 0 1 0 1 2 ,,, ,,,,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( )[ ] ( ) ×∫ ∇∫ Φ+ ∂ ∂ Ω+∫ Φ + ΦΦ t S zL II t I I zyxWdWyx x d z zyx TzyxD z 0 0 12 0 1 ,,,,,, ,,, ,,, τµτατ ∂ τ∂ ∂ ∂
- 12. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 52 ( ) ( )[ ] ( ) ×∫+∫ ∫ Φ+∇ ∂ ∂ Ω+× Φ ΦΦ tt zL IIS SISI zyxIdWdWyxzyx Tk D y d Tk D 0 2 0 0 12 ,,,,,,,,, τττατµτ ( ) ( ) ( ) +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +× ΦΦ t SI t SI II d y zyx TkV D y d x zyx TkV D x dTzyxk 0 2 0 2 , ,,,,,, ,,, τ τµ τ τµ τ ( ) ( ) ( ) ( )∫++∫ ∂ ∂ ∂ ∂ + Φ Φ t II t SI dzyxITzyxkzyxfd z zyx TkV D z 00 2 ,,,,,,,, ,,, τττ τµ (5e) ( ) ( ) ( ) ( ) ( ) +∫ Φ +∫ Φ =Φ ΦΦ t V V t V VV dTzyxD y zyx y d x zyx TzyxD x tzyx 0 1 0 1 2 ,,, ,,,,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] ×∫ ∫ Φ+∇ ∂ ∂ Ω+∫ Φ + ΦΦ t zL VVS t V V WdWyxzyx x d z zyx TzyxD z 0 0 12 0 1 ,,,,,, ,,, ,,, τατµτ ∂ τ∂ ∂ ∂ ( )[ ] ( ) ( )++∫ ∇∫ Φ+ ∂ ∂ Ω+× ΦΦ ΦΦ zyxfdzyxWdWyx Tk D y d Tk D V t S zL VV SVSV ,,,,,,,, 0 0 12 ττµτατ ( ) ( ) ( ) ( ) +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫+ ΦΦ t SV t SV t VV d y zyx TkV D y d x zyx TkV D x dzyxVTzyxk 0 2 0 2 0 2 , ,,,,,, ,,,,,, τ τµ τ τµ ττ ( ) ( ) ( )∫+∫ ∂ ∂ ∂ ∂ + Φ t V t SV dzyxVTzyxkd z zyx TkV D z 00 2 ,,,,,, ,,, τττ τµ . Average values of the second-order approximations of required approximations by using the fol- lowing standard relation [29] ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 xL yL zL zyx tdxdydzdtzyxtzyx LLL ρρα ρ . (10) Substitution of the relations (1e), (3e), (5e) into relation (10) gives us possibility to obtain rela- tions for required average values α2ρ α2C=0, α2ΦI =0, α2ΦV =0, ( ) 4 3 4 1 2 3 2 4 2 3 2 4 4 4 b Eb b bLLLFa F b Eb zyx V + − Θ+Θ +− + =α , ( ) 00201 11021001200 2 2 2 2 IVVIV VIVVzyxVIVVVVVVV I SS SSLLLSSSC α αα α + −−Θ++−− = , where 00 2 0000 2 004 11 IIVV zyx VVIV zyx SS LLL SS LLL b Θ − Θ = , ( )+Θ++ Θ −= zyxIVVV zyx VVII LLLSS LLL SS b 1001 0000 3 2 ( ) ( ) 3333 10 2 00 1001 2 00 011001 0000 22 zyx IVIV zyxIVVV zyx IV zyxIVIIIV zyx VVIV LLL SS LLLSS LLL S LLLSSS LLL SS Θ −Θ++ Θ +Θ+++ Θ + , ( ) ( ) ( ) × Θ ++Θ+Θ+−−++ Θ = x IV IVIIzyxzyxVVIVVIVVV zyx VVII L S SSLLLLLLSSCSS LLL SS b 01 0110 2 01101102 0000 2 22 ( ) ( ) ( − Θ −+Θ+Θ+++ Θ +× V zyx IV IVzyxVVzyxIVIIIV zyx IV zy VV C LLL S SLLLSLLLSSS LLL S LL S 2 00 1001011001 0000 222 ) zyx IVIV IV zyx IVI IVVV LLL SS S LLL SC SS Θ − Θ +−− 0100 102222 2 00 1102 2 , ( ×Θ++ Θ ++ = yxIVVV zyx VVVIV LLSS LLL CSS b 1001 0211 1 2
- 13. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 53 ) ( )( ) × Θ − Θ −Θ++++Θ Θ +× zyx IV zyx IVIV zyIVVVIVIIzyx zyx IV IIz LLL S LLL SS LLLSSSSLLL LLL S SL 00 2 0110 10010110 01 00 22 ( )( ) 010011021001 223 IVIVIIVVVVzyxIIIV SSCSSCLLLSS +−−Θ++× , ( ) −+ Θ = 2 0200 00 0 VVIV zyx II SS LLL S b ( )( ) × Θ −− −+−−++Θ Θ − zyx IVVVV IVIVIIVVVVIVIIzyx zyx IV LLL SSC SSCSSCSSLLL LLL S 1102 01 2 0111020110 01 22 ( )011001 2 IVIIzyxIV SSLLLS ++Θ× , zyx IV zyx IIII zyx III IV zyx VI I LLL S LLL SS LLL S S LLL C Θ − Θ − Θ + Θ = 11202000 2 1 00 11 ααα , 110200 2 10011 IVVVVVVIVVIV SSSSC −−+= ααα , 4 2 2 4 2 32 48 a a a a yE Θ−Θ+= , −−++ Θ = 3 32 4 2 6 rsr a a F 3 32 rsr ++− , − Θ − Θ−Θ Θ − Θ− Θ = 2 4 2 1 4 222 4 2 32 22 4 2 0 4 31 02 4 2 3 8 4 8 4 24 b b LLL b b b b b b bb LLLb b b r zyxzyx 3 4 3 2 3 54b bΘ − , 4 2 2 4 31402 1812 4 b b b bbLLLbb s zyx Θ − Θ− Θ= . Farther we determine solutions of Eqs.(8), i.e. components of displacement vector. We used the same method of averaging of function correction to solve the Eqs.(8). Framework this paper we determine concentration of dopant, concentrations of radiation defects and components of dis- placement vector by using the second-order approximation framework method of averaging of function corrections. This approximation is usually enough good approximation to make qualita- tive analysis and to obtain some quantitative results. All obtained results have been checked by comparison with results of numerical simulations. 3. DISCUSSION In this section we analyzed dynamics of redistributions of dopant and radiation defects during annealing and under influence of mismatch-induced stress and modification of porosity. Typical distributions of concentrations of dopant in heterostructures are presented on Figs. 2 and 3 for diffusion and ion types of doping, respectively. The-se distributions have been calculated for the case, when value of dopant diffusion coefficient in doped area is larger, than in nearest areas. The figures show, that inhomogeneity of heterostructure gives us possibility to increase compactness of concentrations of dopants and at the same time to increase homogeneity of dopant distribution in doped part of epitaxial layer. However framework this approach of manufacturing of bipolar transistor it is necessary to optimize annealing of dopant and/or radiation defects. Reason of this optimization is following. If annealing time is small, the dopant did not achieve any interfaces between materials of heterostructure. In this situation one cannot find any modifications of distri- bution of concentration of dopant. If annealing time is large, distribution of concentration of do- pant is too homogenous. We optimize annealing time framework recently introduces approach [30-38]. Framework this criterion we approximate real distribution of concentration of dopant by step-wise function (see Figs. 4 and 5). Farther we determine optimal values of annealing time by minimization of the following mean-squared error
- 14. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 54 Fig.2. Distributions of concentration of infused dopant in heterostructure from Fig. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffu- sion coefficient in substrate x 0.0 0.5 1.0 1.5 2.0 C(x,Θ) 2 3 4 1 0 L/4 L/2 3L/4 L Epitaxial layer Substrate Fig.3. Distributions of concentration of implanted dopant in heterostructure from Fig. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 corresponds to annealing time Θ = 0.0048(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(Lx 2 + Ly 2 +Lz 2 )/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate
- 15. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 55 C(x,Θ) 0 Lx 2 1 3 4 Fig. 4. Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distribu- tion of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. Increasing of number of curve corresponds to increasing of annealing time x C(x,Θ) 1 2 3 4 0 L Fig. 5. Spatial distributions of dopant in heterostructure after ion implantation. Curve 1 is idealized distribu- tion of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. Increasing of number of curve corresponds to increasing of annealing time ( ) ( )[ ]∫ ∫ ∫ −Θ= xL yL zL zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ , (15) where ψ(x,y,z) is the approximation function. Dependences of optimal values of annealing time on parameters are presented on Figs. 6 and 7 for diffusion and ion types of doping, respectively. It should be noted, that it is necessary to anneal radiation defects after ion implantation. One could find spreading of concentration of distribution of dopant during this annealing. In the ideal case distribution of dopant achieves appropriate interfaces between materials of heterostructure during annealing of radiation defects. If dopant did not achieve any interfaces during annealing of radia- tion defects, it is practicably to additionally anneal the dopant. In this situation optimal value of additional annealing time of implanted dopant is smaller, than annealing time of infused dopant.
- 16. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 56 0.0 0.1 0.2 0.3 0.4 0.5 a/L, ξ, ε, γ 0.0 0.1 0.2 0.3 0.4 0.5 ΘD0L -2 3 2 4 1 Fig.6. Dependences of dimensionless optimal annealing time for doping by diffusion, which have been ob- tained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence of dimen- sionless optimal annealing time on the relation a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the dependence of dimension- less optimal annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0 0.0 0.1 0.2 0.3 0.4 0.5 a/L, ξ, ε, γ 0.00 0.04 0.08 0.12 ΘD0L -2 3 2 4 1 Fig.7. Dependences of dimensionless optimal annealing time for doping by ion implantation, which have been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless op- timal annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the dependence of di- mensionless optimal annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the de- pendence of dimensionless optimal annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0 Far- ther we analyzed influence of relaxation of mechanical stress on distribution of dopant in doped areas of heterostructure. Under following condition ε0< 0 one can find compression of distribution of concentration of dopant near interface between materials of heterostructure. Contrary (at ε0>0) one can find spreading of distribution of concentration of dopant in this area. This changing of distribution of concentration of dopant could be at least partially compensated by using laser annealing [38]. This type of annealing gives us possi- bility to accelerate diffusion of dopant and another processes in annealed area due to inhomogenous distri- bution of temperature and Arrhenius law. Accounting relaxation of mismatch-induced stress in heterostructure could leads to changing of optimal values of annealing time. At the same time modification of porosity gives us possibility to decrease value of mechanical stress. On the one hand mismatch-induced stress changing of optimal values of annealing time. At the same time modification of porosity gives us possibility to decrease value of mechanical stress. On the one hand mismatch-induced stress could be used to increase density of elements of integrated circuits. On the other hand could leads to generation disloca- tions of the discrepancy. Figs. 8 and 9 show distributions of concentration of vacancies in porous materials
- 17. International Journal of Advance Robotics & Expert Systems (JARES) Vol.1, No.2 57 and component of displacement vector, which is perpendicular to interface between layers of heterostructure. z 0.0 0.2 0.4 0.6 0.8 1.0 Uz 1 2 0.0 a Fig. 8. Normalized dependences of component uz of displacement vector on coordinate z for nonporous (curve 1) and porous (curve 2) epitaxial layers z V(z) 1 2 0.4 0.2 0.0 0.0 a Fig. 9. Normalized dependences of vacancy concentrations on coordinate z in unstressed (curve 1) and stressed (curve 2) epitaxial layers 4. CONCLUSIONS In this paper we model redistribution of infused and implanted dopants with account relaxation mismatch-induced stress during manufacturing field-effect heterotransistors framework a voltage reference. We formulate recommendations for optimization of annealing to decrease dimensions of transistors and to increase their density. We formulate recommendations to decrease mismatch- induced stress. Analytical approach to model diffusion and ion types of doping with account con- current changing of parameters in space and time has been introduced. At the same time the ap- proach gives us possibility to take into account nonlinearity of considered processes.
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