OPTIMIZATION OF TECHNOLOGICAL PROCESS TO DECREASE DIMENSIONS OF CIRCUITS XOR, MANUFECTURED BASED ON FIELD-EFFECT HETEROTRANSISTORS

International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 DOI:10.5121/ijfcst.2016.6101 1 O PTIMIZATION OF TECHNOLOGICAL PROCESS TO DECREASE DIMENSIONS OF CIRCUITS XOR, MANUFECTURED BASED ON FIELD - EFFECT HETEROTRANSISTORS E.L. Pankratov 1 , E.A. Bulaeva 1,2 1 Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod, 603950, Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky Street, Nizhny Novgorod, 603950, Russia A BSTRACT The paper describes an approach of increasing of integration rate of elements of integrated circuits. The approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one should manufacture a heterostructure with specific configuration. After that several special areas of the heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical approaches of modeling of technological process. K EYWORDS Circuits XOR; increasing of density of elements; optimization of technological process. 1. I NTRODUCTION One of intensively solving problems of solid state electronics is improvement of frequency characteristics of electronic devices and their reliability. Another intensively solving problem is increasing of integration rate of integrated circuits with decreasing of their dimensions [1-9]. To solve these problems they were used searching materials with higher values of charge carriers motilities, development new and elaboration existing technological approaches [1-14]. In the present paper we consider circuit XOR from [15]. Based on recently considered approaches [1623] we consider an approach to decrease dimensions of the circuit. The approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer. The epitaxial layer manufactured with several sections. To manufacture these sections another materials have been used. These sections have been doped by diffusion or ion implantation. The doping gives a possibility to generate another type of conductivity (p or n). After finishing of manufacturing of the circuit XOR these sections will be used by sources, drains and gates (see Fig. 1). Dopant and radiation defects should be annealed after finishing the dopant diffusion and the ion implantation. Our aim framework the paper is analysis of redistribution of dopant and redistribution of radiation defects to prognosis technological process. The accompanying aim of the present paper is development of analytical approach for prognosis technological processes with account all required influenced factors. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 2 2. M ETHOD OF S OLUTION We solve our aim by calculation and analysis distribution of concentrations of dopants in space and time. The required distribution has been determined by solving the second Fick's law in the following form [24,25] ( ) ( ) ( ) ( )       +       +       =z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equations are Fig. 1. Structure of epitaxial layer. View from top International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 3 ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = x Lx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = y Lx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = z Lx z tzyxC , C (x,y,z,0)=f (x,y,z). (2) Here the function C(x,y,z,t) describes the distribution of concentration of dopant in space and time. D С describes distribution the dopant diffusion coefficient in space and as a function of temperature of annealing. Dopant diffusion coefficient will be changed with changing of materials of heterostructure, heating and cooling of heterostructure during annealing of dopant or radiation defects (with account Arrhenius law). Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of annealing and concentrations of dopant and radiation defects could be written as [26-28] ( ) ( ) ( ) ( ) ( ) ( )        ++      += 2 * 2 2 * 1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ . (3) Here function D L (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate and temperature of annealing T. Function P (x,y,z,T) describes the same dependences of the limit of solubility of dopant. The parameter γ is integer and usually could be varying in the following interval γ ∈ [1,3]. The parameter describes quantity of charged defects, which interacting (in average) with each atom of dopant. Ref.[26] describes more detailed information about dependence of dopant diffusion coefficient on concentration of dopant. Spatio-temporal distribution of concentration of radiation vacancies described by the function V (x,y,z,t). The equilibrium distribution of concentration of vacancies has been denoted as V * . It is known, that doping of materials by diffusion did not leads to radiation damage of materials. In this situation ζ 1 = ζ 2 = 0. We determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations [27,28] ( ) ( ) ( ) ( ) ( ) ( ) ×−       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ = ∂ ∂Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI IIII ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( ) tzyxVtzyxITzyxk z tzyxI TzyxD z tzyxI VII ,,,,,,,,, ,,, ,,,,,, , 2 −       ∂ ∂ ∂ ∂ +× (4) ( ) ( ) ( ) ( ) ( ) ( ) ×−       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ = ∂ ∂Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVVV ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( ) tzyxVtzyxITzyxk z tzyxV TzyxD z tzyxV VIV ,,,,,,,,, ,,, ,,,,,, , 2 −       ∂ ∂ ∂ ∂ +× . Boundary and initial conditions for these equations are ( ) 0 ,,, 0 = ∂ ∂ =x x tzyx ρ , ( ) 0 ,,, = ∂ ∂ = x Lx x tzyx ρ , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyx ρ , ( ) 0 ,,, = ∂ ∂ = y Ly y tzyx ρ , International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 4 ( ) 0 ,,, 0 = ∂ ∂ =z z tzyx ρ , ( ) 0 ,,, = ∂ ∂ = z Lz z tzyx ρ , ρ (x,y,z,0)=f ρ (x,y,z). (5) Here ρ =I,V. We denote spatio-temporal distribution of concentration of radiation interstitials as I (x,y,z,t). Dependences of the diffusion coefficients of point radiation defects on coordinate and temperature have been denoted as D ρ (x,y,z,T). The quadric on concentrations terms of Eqs. (4) describes generation divacancies and diinterstitials. Parameter of recombination of point radiation defects and parameters of generation of simplest complexes of point radiation defects have been denoted as the following functions k I,V (x,y,z,T), k I,I (x,y,z,T) and k V,V (x,y,z,T), respectively. Now let us calculate distributions of concentrations of divacancies Φ V (x,y,z,t) and diinterstitials Φ I (x,y,z,t) in space and time by solving the following system of equations [27,28] ( ) ( ) ( ) ( ) ( ) +      Φ +      Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) tzyxITzyxktzyxITzyxk z tzyx TzyxD z III I I ,,,,,,,,,,,, ,,, ,,, 2 , −+      Φ + Φ ∂ ∂ ∂ ∂ (6) ( ) ( ) ( ) ( ) ( ) +      Φ +      Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) tzyxVTzyxktzyxVTzyxk z tzyx TzyxD z VVV V V ,,,,,,,,,,,, ,,, ,,, 2 , −+      Φ + Φ ∂ ∂ ∂ ∂ . Boundary and initial conditions for these equations are ( ) 0 ,,, 0 = ∂ Φ∂ =x x tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ = x Lx x tzyx ρ , ( ) 0 ,,, 0 = ∂ Φ∂ =y y tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ = y Ly y tzyx ρ , ( ) 0 ,,, 0 = ∂ Φ∂ =z z tzyx ρ , ( ) 0 ,,, = ∂ Φ∂ = z Lz z tzyx ρ , Φ I (x,y,z,0)=f Φ I (x,y,z), Φ V (x,y,z,0)=f Φ V (x,y,z). (7) The functions D Φρ (x,y,z,T) describe dependences of the diffusion coefficients of the above complexes of radiation defects on coordinate and temperature. The functions k I (x,y,z,T) and k V (x,y,z, T) describe the parameters of decay of these complexes on coordinate and temperature. To describe physical processes they are usually solving nonlinear equations with space and time varying coefficients. In this situation only several limiting cases have been analyzed [29-32]. One way to solve the problem is solving the Eqs. (1), (4), (6) by the Bubnov-Galerkin approach [33] after appropriate transformation of these transformation. To determine the spatio-temporal distribution of concentration of dopant we transform the Eq.(1) to the following integrodifferential form ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫      ++= ∫ ∫ ∫ ty L z L L x L y L z L zyx y zx y z V wvxV V wvxV TwvxDudvdwdtwvuC LLL zyx 02 * 2 2 * 1 ,,, ,,, 1,,,,,, τ ς τ ς International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 5 ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫      ++      +× t x L z L L zy x z TzyxP wyuC TwyuD LL zy d x wvxC TwvxP wvxC 0 ,,, ,,, 1,,, ,,, ,,, ,,, 1 γ γ γ γ τ ξτ ∂ τ∂ τ ξ ( ) ( ) ( ) ( ) ( ) × ∫∫∫ +      ++× t x L y L L zx x y TzvuD LL zx d y wyuC V wyuV V wyuV 0 2 * 2 2 * 1 ,,, ,,, ,,, ,,, 1 τ ∂ τ∂ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) +      +      ++× yx LL yx d z zvuC TzyxP zvuC V zvuV V zvuV τ ∂ τ∂ τ ξ τ ς τ ς γ γ ,,, ,,, ,,, 1 ,,, ,,, 1 2 * 2 2 * 1 ( ) ∫ ∫ ∫ + x L y L z L zyx x y z udvdwdwvuf LLL zyx ,, . (1a) Now let us determine solution of Eq.(1a) by Bubnov-Galerkin approach [33]. To use the approach we consider solution of the Eq.(1a) as the following series ( ) ( ) ( ) ( ) ( ) ∑ = = N nnCnnnnC tezcycxcatzyxC 0 0 ,,, . Here ( ) ( ) [ ] 222 0 22 exp −−− ++−= zyxCnC LLLtDnte π , c n ( χ ) = cos ( π n χ /L χ ). Number of terms N in the series is finite. The above series is almost the same with solution of linear Eq.(1) (i.e. for ξ = 0) and averaged dopant diffusion coefficient D 0 . Substitution of the series into Eq.(1a) leads to the following result ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫  ×      ∑ +−= ∑ == ty L z L N nnCnnnnC zy N nnCnnn C y z ewcvcxca LL zy tezsysxs n a zyx 011 32 1 γ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∑      ++    × = N nnnnCL vcxsaTwvxD V wvxV V wvxV TwvxP 1 2 * 2 2 * 1 ,,, ,,,,,, 1 ,,, τ ς τ ς ξ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ×            ∑ +−× = t x L z L N mmCmmmmC zx nCn x z TwyuP ewcycuca LL zx dewcn 01 ,,, 1 γ γ ξ τττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∑      ++× = ττ τ ς τ ς dewcysucn V wyuV V wyuV TwyuD N nnCnnnL 1 2 * 2 2 * 1 ,,,,,, 1,,, ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫            ∑ +−× = t x L y L N nnCnnnnCL yx nC x y ezcvcuca TzvuP TzvuD LL yx a 01 ,,, 1,,, γ γ τ ξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×+ ∑      ++× =zyx N nnCnnnnC LLL zyx dezsvcucan V zvuV V zvuV ττ τ ς τ ς 1 2 * 2 2 * 1 ,,,,,, 1 ( ) ∫ ∫ ∫ × x L y L z L x y z udvdwdwvuf ,, , International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 6 where s n ( χ ) = sin ( π n χ /L χ ). We used condition of orthogonality to determine coefficients a n in the considered series. The coefficients a n could be calculated for any quantity of terms N. In the common case the relations could be written as ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫   ×      ∑ +−= ∑ − == tLLLN nnCnnnnCL zy N nnC nC zyx xyz ezcycxcaTzyxD LL te n a LLL 0 0 0 0 1 2 165 222 1,,, 2 γ τ ππ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∑      ++    × = N nnCnnn nC ezcycxs n a V zyxV V zyxV TzyxP 1 2 * 2 2 * 1 2 ,,,,,, 1 ,,, τ τ ς τ ς ξ γ ( ) ( ) [ ] ( ) ( ) [ ] ( ) × ∫ ∫ ∫ ∫ −      −+      −+× tLLL Ln z nn y n xyz TzyxDdxdydzdzc n L zszyc n L ysy 0 0 0 0 ,,,11 τ ππ ( ) ( ) ( ) ( ) ( ) ( ) ( )   ++            ∑ +× =* 1 1 ,,, 1 ,,, 1,,, V zyxV TzyxP ezcycxcaTzyxD N nnCnnnnCL τ ς ξ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ∑×      −+      ++    + = N n nC n x nn a xc n L xsx V zyxV V zyxV V zyxV 1 2 * 2 2 * 1 2 * 2 21 ,,,,,, 1 ,,, π τ ς τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) [ ] ×−      −+× 22 2 12 2 π τ π τ π yx n z nnCnnn zx LL dxdydzdzc n L zszezcysxc LL ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫   ++            ∑ +× = tLLLN nnCnnnnC xyz V zyxV TzyxP ezcycxca 0 0 0 0 2 * 2 2 1 ,,, 1 ,,, 1 τ ς ξ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] × ∑     −+    + = N nn x nnnn nC L xc n L xsxzsycxc n a TzyxD V zyxV 1 * 1 1,,, ,,, π τ ς ( ) ( ) [ ] ( ) ( ) ( ) [ ] ∑∫×      −++      −+× = N n L n x nnCn y n x xc n L xsxdxdydzdeyc n L ysy 10 11 π ττ π ( ) ( ) [ ] ( ) ( ) [ ] ( ) ∫ ∫      −+      −+× yz LL n z nn y n xdydzdzyxfzc n L zszyc n L ysy 0 0 ,,11 ππ . As an example for γ = 0 we obtain ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ){ ∫ ∫ ∫ +      −+      −+= xyz LLL nn y nn y nnC xsxydzdzyxfzc n L zszyc n L ysya 0 0 0 ,,11 ππ ( ) [ ] ( ) ( ) ( ) ( ) [ ] ( )       ∫ ∫ ∫ ∫ ×      −+    −× tLLL Ln y nnn x n xyz TzyxDyc n L ysyycxs n xd n L xc 0 0 0 0 ,,,12 2 1 ππ International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 7 ( ) ( ) [ ] ( ) ( ) ( ) ( ) ×      +      ++      −+× TzyxP V zyxV V zyxV zc n L zsz n y n ,,, 1 ,,,,,, 11 2 * 2 2 * 1 γ ξτ ς τ ς π ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) × ∫ ∫ ∫ ∫      −++× tLLL nnn y nnnCnCn xyz zcysxc n L xsxxcedexdydzdzc 0 0 0 0 21 π τττ ( ) ( ) [ ] ( ) ( ) ( ) ( ) ×      ++      +      −+× 2 * 2 2 * 1 ,,,,,, 1 ,,, 11 V zyxV V zyxV TzyxP zc n L zsz n y n τ ς τ ς ξ π γ ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ){ ∫ ∫ ∫ ×      −++× tLL nnn x nnnCL xy ysycxc n L xsxxcedxdydzdTzyxD 0 0 0 1,,, π ττ ( ) [ ] ( ) ( ) ( ) ( ) ( ) ∫  ++      +    −+× z L Lnn y V zyxV TzyxP TzyxDzsyc n L y 0 2 * 2 2 ,,, 1 ,,, 1,,,21 τ ς ξ π γ ( ) ( ) 1 65 222 * 1 ,,, −     −       +te n LLL dxdydzd V zyxV nC zzz π τ τ ς . For γ = 1 one can obtain the following relation to determine required parameters ( ) ( ) ( ) ( ) ∫ ∫ ∫ +±−= xyz LLL nnnnn n n nC xdydzdzyxfzcycxca 0 0 0 2 ,,4 2 αβ α β , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ×      ++= tLLL nnnnC zy n xyz V zyxV V zyxV zcycxse n LL 0 0 0 0 2 * 2 2 * 1 2 ,,, ,,, 12 2 τ ς τ ςτ π ξ α ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ×+      −+      −+× n LL dxdydzdzc n L zszyc n L ysy TzyxP TzyxD zx n z nn y n L 2 2 11 ,,, ,,, π ξ τ ππ ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] × ∫ ∫ ∫ ∫      −−      −+× tLLL n z n L nn x nnnC xyz zc n L zsz TzyxP TzyxD zcxc n L xsxxce 0 0 0 0 1 ,,, ,,, 1 ππ τ ( ) ( ) ( ) ( ) ( ) ( ) ×+      ++× n LL dxdydyszd V zyxV V zyxV TzyxP TzyxD yx n L 2 2 * 2 2 * 1 2 2 ,,, ,,, 1 ,,, ,,, π ξ τ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ ∫      ++× tLLL L nnnnC xyz V zyxV V zyxV TzyxP TzyxD zsycxce 0 0 0 0 2 * 2 2 * 1 ,,,,,, 1 ,,, ,,, 2 τ ς τ ςτ ( ) ( ) [ ] ( ) ( ) [ ] τ ππ dxdydzdyc n L ysyxc n L xsx n y nn x n      −+      −+× 11 , ( ) × ∫ = t nC zy ne n LL 0 2 2 τ π β ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ∫ ∫ ∫ ×      ++      −+× xyz LLL nn y nnn V zyxV V zyxV zcyc n L ysyycxs 0 0 0 2 * 2 2 * 1 ,,, ,,, 112 τ ς τ ς π International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 8 ( ) ( ) ( ) [ ] ( ) ( ) ( ) × ∫ ∫ ∫ +      −+× tLL nnnC zx n z nL xy ysxce n LL dxdydzdzc n L zszTzyxD 0 0 0 2 2 2 1,,, τ π τ π ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) × ∫     ++      −+× z L nLn x nV zyxV V zyxV zcTzyxDxc n L xsx 0 2 * 2 2 * 1 ,,, ,,, 1,,,1 τ ς τ ς π ( ) ( ) [ ] ( ) ( ) ( ) [ ] × ∫ ∫      −++      −+× tL n x nnC yx n z n x xc n L xsxe n LL dxdydzdzc n L zsz 0 0 2 1 2 1 π τ π τ π ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ∫ ∫ ×      ++      −+× yz LL Ln y nn V zyxV V zyxV TzyxDyc n L ysyxc 0 0 2 * 2 2 * 1 ,,, ,,, 1,,,1 τ ς τ ς π ( ) ( ) ( ) 65222 2nteLLLdxdydyczdzs nCzyxnn π τ − × . The same approach could be used for calculation parameters a n for different values of parameter γ . However the relations are bulky and will not be presented in the paper. Advantage of the approach is absent of necessity to join dopant concentration on interfaces of heterostructure. The same Bubnov-Galerkin approach has been used for solution the Eqs.(4). Previously we transform the differential equations to the following integrodifferential form ( ) ( ) ( ) + ∫ ∫ ∫ ∂ ∂ = ∫ ∫ ∫ ty L z L I zy x L y L z L zyx y zx y z dvdwd x wvxI TwvxD LL zy udvdwdtwvuI LLL zyx 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ( ) × ∫ ∫ ∫ − ∫ ∫ ∫ ∂ ∂ + x L y L z L VI zyx t x L z L I zx x y zx z twvuITwvuk LLL zyx dudwd x wyuI TwyuD LL zx ,,,,,, ,,, ,,, , 0 τ τ ( ) ( ) ( ) ×− ∫ ∫ ∫ ∂ ∂ +× zyx t x L y L I yx LLL zyx dudvdTzvuD z zvuI LL yx udvdwdtwvuV x y 0 ,,, ,,, ,,, τ τ ( ) ( ) ( ) ∫ ∫ ∫ + ∫ ∫ ∫ × x L y L z L I zyx x L y L z L II x y zx y z udvdwdwvuf LLL zyx udvdwdtwvuITwvuk ,,,,,,,, 2 , (4a) ( ) ( ) ( ) + ∫ ∫ ∫ ∂ ∂ = ∫ ∫ ∫ ty L z L V zy x L y L z L zyx y zx y z dvdwd x wvxV TwvxD LL zy udvdwdtwvuV LLL zyx 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) × ∫ ∫ ∫ ∂ ∂ + ∫∫∫ ∂ ∂ + t x L y L yx t x L z L V zx x yx z z zvuV LL yx dudwd x wyuV TwyuD LL zx 00 ,,,,,, ,,, τ τ τ ( ) ( ) ( ) ( ) − ∫ ∫ ∫ −× x L y L z L VI zyx V x y z udvdwdtwvuVtwvuITwvuk LLL zyx dudvdTzvuD ,,,,,,,,,,,, , τ ( ) ( ) ( ) ∫ ∫ ∫ + ∫ ∫ ∫ − x L y L z L V zyx x L y L z L VV zyx x y zx y z udvdwdwvuf LLL zyx udvdwdtwvuVTwvuk LLL zyx ,,,,,,,, 2 ,. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 9 We determine spatio-temporal distributions of concentrations of point defects as the same series ( ) ( ) ( ) ( ) ( ) ∑ = = N nnnnnn tezcycxcatzyx 1 0 ,,, ρρ ρ . Parameters a n ρ should be determined in future. Substitution of the series into Eqs.(4a) leads to the following results ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∑∫ ∫ ∫ −= ∑ == N n ty L z L InnnI zyx N nnInnn nI y z vdwdTwvxDzcyca LLL zy tezsysxs n a zyx 10 133 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) − ∑∫ ∫ ∫ −× = N n t x L z L InnnInnI zyx nnI x z dudwdTwyuDzcxceysa LLL zx xsde 10 ,,, ττ π ττ ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ − ∑∫ ∫ ∫ − = x L y L z L II N n t x L y L InnnInnI zyx x y zx y TvvukdudvdTzvuDycxcezsa LLL yx ,,,,,, , 10 ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ ∑ −      ∑ × == x L y L z L N nnnnnI zyxzyx N nnInnnnI x y z wcvcuca LLL zyx LLL zyx udvdwdtewcvcuca 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ + ∑ × = x L y L z L I N nVInVnnnnVnI x y z udvdwdwvufudvdwdTvvuktewcvcucate ,,,,, 1, zyx LLLzyx × ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∑∫ ∫ ∫ −= ∑ == N n ty L z L VnnnV zyx N nnVnnn nV y z vdwdTwvxDzcyca LLL zy tezsysxs n a zyx 10 133 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) − ∑∫ ∫ ∫ −× = N n t x L z L VnnnVnnV zyx nnV x z dudwdTwyuDzcxceysa LLL zx xsde 10 ,,, ττ π ττ ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ − ∑∫ ∫ ∫ − = x L y L z L VV N n t x L y L VnnnVnnV zyx x y zx y TvvukdudvdTzvuDycxcezsa LLL yx ,,,,,, , 10 ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ ∑ −      ∑ × == x L y L z L N nnnnnI zyxzyx N nnInnnnV x y z wcvcuca LLL zyx LLL zyx udvdwdtewcvcuca 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∫ ∫ ∫ + ∑ × = x L y L z L V N nVInVnnnnVnI x y z udvdwdwvufudvdwdTvvuktewcvcucate ,,,,, 1, zyx LLLzyx × . We used orthogonality condition of functions of the considered series framework the heterostructure to calculate coefficients a n ρ . The coefficients a n could be calculated for any quantity of terms N. In the common case equations for the required coefficients could be written as ( ) ( ) [ ] ( ) ( ) [ ] × ∑∫ ∫ ∫      −++−−= ∑ − == N n tLL n y nyn nI x N nnI nI zyx xy yc n L ysyLxc n a L te n a LLL 10 0 0 2 165 222 12 2 221 2 1 πππ International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 16 x 0.0 0.5 1.0 1.5 2.0 C ( x , Θ ) 2 3 4 1 0 L /4 L /2 3 L /4 L Epitaxial layer Substrate Fig.2b. Spatial distributions of infused dopant concentration in the considered heterostructure. Curves 1 and 3 corresponds to annealing time Θ = 0.0048(L x 2 +L y 2 +L z 2 )/D 0 . Curves 2 and 4 corresponds to annealing time Θ = 0.0057(L x 2 +L y 2 +L z 2 )/D 0 . Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to the considered heterostructure. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves x 0.00000 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 C ( x , Θ ) f C ( x ) L /4 0 L /2 3 L /4 L x 0 1 2 Substrate Epitaxial layer 1 Epitaxial layer 2 Fig.3. Implanted dopant distributions in heterostructure in heterostructure with two epitaxial layers (solid lines) and with one epitaxial layer (dushed lines) for different values of annealing time. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves Increasing of annealing time leads to acceleration of diffusion. In this situation one can find increasing quantity of dopant in materials near doped sections. If annealing time is small, the dopant can not achieves nearest interface between layers of heterostructure. These effects are shown by Figs. 5 and 6. We used recently introduced criterion [16-23] to estimate compromise value of annealing time. Framework the criterion we approximate real distribution of concentration of dopant by idealized step-wise distribution ψ (x,y,z), which would be better to use for minimization dimensions of elements of the considered circuit XOR [19-26]. Farther the required compromise annealing time has been calculated by minimization the following mean-squared error International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 17 ( ) ( ) [ ] ∫ ∫ ∫ −Θ= xyz LLL zyx xdydzdzyxzyxC LLL U 000 ,,,,, 1 ψ . (8) C ( x , Θ ) 0 L x 2 1 3 4 Fig. 4. Distributions of concentrations of infused dopant in the considered heterostructure. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for different values of annealing time for increasing of annealing time with increasing of number of curve x C ( x , Θ ) 1 23 4 0 L Fig. 5. Distributions of concentrations of implanted dopant in the considered heterostructure. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for different values of annealing time for increasing of annealing time with increasing of number of curve We analyzed optimal value of annealing time. The analysis shows, that optimal value of annealing time for ion type of doping is smaller, than optimal value of annealing time for diffusion type of doping. It is known, that ion doping of materials leads to radiation damage of doped materials. In this situation radiation defects should be annealed. The annealing leads to the above difference between optimal values of annealing time of dopant. The annealing of implanted dopant is necessary in the case, when the dopant did not achieved nearest interface between layers of heterostructure during annealing of radiation defects. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 18 It should be noted, that using diffusion type of doping did not leads to radiation damage of materials. However radiation damage of materials during ion doping gives a possibility to decrease mismatch-induced stress in heterostructure [34]. 4. C ONCLUSIONS In this paper we introduced an approach to decrease dimensions of a circuit XOR. The approach based on optimization of manufacturing field-effect heterotransistors, which includes into itself the considered circuit. A CKNOWLEDGEMENTS This work is supported by the agreement of August 27, 2013 № 02. 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From 1995 to 2004 he was educated in Nizhny Novgorod State University: from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgorod State University of Architecture and Civil Engineering. 2012-2015 Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of Nizhny Novgorod State University. He has 135 published papers in area of his researches. Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school “Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec- International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016 20 ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in the University. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny Novgorod State University. She has 90 published papers in area of her researches.