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Using inhomogeneity of heterostructure and optimization of annealing to decrease dimensions multyemitter heterotransistors

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This paper presents a method for manufacturing multiemitter heterotransistors utilizing doping techniques such as diffusion and ion implantation, aimed at optimizing transistor dimensions. The authors analyze the dynamics of dopant redistribution and suggest configurations for the heterostructure to enhance p-n-junction sharpness. Key points include the influence of inhomogeneous temperature distributions and the need for specific annealing processes to manage dopant placement effectively.

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International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 DOI:10.5121/ijcsa.2014.4303 25 USING INHOMOGENEITY OF HETEROSTRUCTURE AND OPTIMIZATION OF ANNEALING TO DE- CREASE DIMENSIONS MULTYEMITTER HETERO- TRANSISTORS E.L. Pankratov1 and E.A. Bulaeva 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 ABSTRACT Framework this paper we discussed an approach to manufacture a multiemitter heterotransistor. The in- troduced approach is a branch of recently introduced approach and based on doping by diffusion or by ion implantation of required part of heterostructure and optimization of dopant and/or radiation defects. The heterostructure should has special configuration. KEYWORDS Multiemitter heterotransistor, optimization of manufacturing of transistor, analytical approach to model technological process 1. INTRODUCTION Logical elements often include into itself multiemitter transistors [1-4]. To manufacture bipolar transistors it could be used dopant diffusion, ion doping or epitaxial layer [1-8]. It is attracted an interest increasing of sharpness of p-n-junctions, which include into bipolar transistors and de- creasing of dimensions of the transistors, which include into integrated circuits [1-4]. Dimensions of transistors will be decreased by using inhomogenous distribution of temperature during laser or microwave types of annealing [9,10], using defects of materials (for example, defects could be generated during radiation processing of materials) [11-14], native inhomogeneity (multilayer property) of heterostructure [12-14]. We consider manufacture of multiemitter heterotransistor based on a hetero structure, which con- sist of a substrate and three epitaxial layers (see Fig. 1). Some dopants are infused in the epitaxial layers to manufacture p and n types of conductivities during manufacture a multiemitter transis- tor. Let us consider two cases: infusion of dopant by diffusion or ion implantation. Farther in the first case we consider annealing of dopant so long, that after the annealing the dopants should achieve interfaces between epitaxial layers. In this case one can obtain increasing of sharpness of p-n-junctions [12-14]. The increasing of sharpness could be obtained under conditions, when val- ues of dopant diffusion coefficients in last epitaxial layers is larger, than values of dopant diffu- sion coefficient in average epitaxial layer. At the same time homogeneity of dopant distributions in doped areas increases. It is attracted an interest higher doping of last epitaxial layers, than dop-
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 26 ing of average epitaxial layer, because the higher doping gives us possibility to increase sharpness of p-n-junctions. It is also practicably to choose materials of epitaxial layers and substrate so; those values of dopant diffusion coefficients in the substrate should be smaller, than in the epitax- ial layer. In this situation one can obtain thinner transistor. In the case of ion doping of hetero- structure it should be done annealing of radiation defects. It is practicably to choose so regimes of annealing of radiation defects, that dopant should achieves interface between epitaxial layers. At the same time the dopant could not diffuse into another epitaxial layer. If dopant did not achieves interface between epitaxial layers during annealing of radiation defects, additional annealing of dopant required. Main aims of the present paper are analysis of dynamics of redistribution of dopants in considered heterostructure (Figs. 1) and optimization of annealing times of dopants. In some recent papers analogous analysis have been done [4,10,12-14]. However we can not find in literature any simi- lar heterostructures as we consider in the paper. Substrate Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3 n-type dopant p-type dopant n-type dopant Fig. 1a. Heterostructure, which consist of a substrate and three epitaxial layers. View from above Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3 n-type dopantp-type dopant n-type dopant 1 n-type dopant 2 n-type dopant 3 n-type dopant 4 Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers. View from one side
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 27 2. Method of solution We analyzed spatio-temporal distribution of dopant based on the second Fick’s law [1-4] ( ) ( ) ( ) ( )       +      +      = z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC               ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equation are ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLy y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLz z tzyxC , C (x,y,z,0)=fC (x,y,z). (2) We designate spatio-temporal distribution of concentration of dopant as C(x,y,z,t); as the current coordinates and times x, y, z and t; the dopant diffusion coefficient as DС. Dependences of dopant diffusion coefficient on coordinates, temperature of annealing (with account Arrhenius low), spa- tio-temporal distributions of concentrations of dopants and radiation defects could be approx- imated by the following relation [15,16] ( ) ( ) ( ) ( ) ( ) ( )       ++      += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC    . (3) Factor DL (x,y,z,T) describes inhomogeneity of heterostructure and temperature dependence of dopant diffusion coefficient due to Arrhenius law. We designate the limit of solubility of dopant as P (x,y,z,T); the spatio-temporal distribution of concentration of radiation vacancies V (x,y,z,t); the equilibrium distribution of vacancies V* . Parameter  depends of properties of materials and could be integer in the interval  ∈[1,3] [15]. Dependence of dopant diffusion coefficient on do- pant concentration has been discussed in details in [15]. It should be noted, that radiation damage absents in the case of diffusion doping of haterostructure. In this situation 1= 2= 0. Spatio- temporal distributions of concentrations of point radiation defects could be determine by solution of the following system of equations [17-19] ( ) ( ) ( ) ( ) ( ) ( )×−      ∂ ∂ ∂ ∂ +      ∂ ∂ ∂ ∂ = ∂ ∂ tzyxI y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ,,, ,,, ,,, ,,, ,,, ,,, 2 ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxI TzyxD z Tzyxk VIIII ,,,,,,,,, ,,, ,,,,,, ,, −      ∂ ∂ ∂ ∂ +× (4) ( ) ( ) ( ) ( ) ( ) ( )×−      ∂ ∂ ∂ ∂ +      ∂ ∂ ∂ ∂ = ∂ ∂ tzyxV y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ,,, ,,, ,,, ,,, ,,, ,,, 2 ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxV TzyxD z Tzyxk VIVVV ,,,,,,,,, ,,, ,,,,,, ,, −      ∂ ∂ ∂ ∂ +× with initial (x,y,z,0)=f (x,y,z) (5a) and boundary conditions
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 28 ( ) 0 ,,, = ∂ ∂ = yLy y tzyx , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyx , ( ) 0 ,,, = ∂ ∂ = zLz z tzyx . (5b) We designate the spatio-temporal distribution of concentration of interstitials as I(x,y,z,t); =I,V; the diffusion coefficients of interstitials and vacancies as D(x,y,z,T); terms V2 (x,y, z,t) and I2 (x,y,z,t) correspond to generation of divacancies and analogous complexes of interstitials; kI,V(x, y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) parameters of recombination of point radiation defects and generation of complexes. Spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and analogous com- plexes of interstitials I(x,y,z,t) will be determined by solving the following systems of equations [17-19] ( ) ( ) ( ) ( ) ( ) +      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     with boundary and initial conditions ( ) 0 ,,, 0 = ∂ Φ∂ =x x tzyx , ( ) 0 ,,, = ∂ Φ∂ = xLx x tzyx , ( ) 0 ,,, 0 = ∂ Φ∂ =y y tzyx , ( ) 0 ,,, = ∂ Φ∂ = yLy y tzyx , ( ) 0 ,,, 0 = ∂ Φ∂ =z z tzyx , ( ) 0 ,,, = ∂ Φ∂ = zLz z tzyx , I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV (x,y,z). (7) Here DI(x,y,z,T) and DV(x,y,z,T) are diffusion coefficients of complexes of point radiation de- fects; kI(x,y,z,T) and kV(x,y,z,T) are parameters of decay of the complexes. We solved the above boundary problems by using method of averaging of function corrections [13-15,20] with decreased quantity of radiation defects [21]. Framework the approach we consid- er the initial-order approximations of concentrations of dopant and radiation defects as solutions of Eqs. (1), (4) and (6) with averaged values of diffusion coefficient D0L, D0I, D0V, D0I, D0V and zero values of parameters of recombination of defects, generation and decay of their complexes ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 0 1 2 ,,, n nCnnnnC zyxzyx C tezcycxcF LLLLLL F tzyxC , ( ) ×+= zyxzyx I LLLLLL F tzyxI 2 ,,, 0 1 ( ) ( ) ( ) ( )∑× ∞ =1n nInnnnI tezcycxcF , ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 0 1 2 ,,, n nVnnnnC zyxzyx C tezcycxcF LLLLLL F tzyxV ,
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 29 ( ) ( ) ( ) ( ) ( )∑+=Φ ∞ = ΦΦ Φ 1 0 1 2 ,,, n nnnnn zyxzyx I tezcycxcF LLLLLL F tzyx II I , ( ) +=Φ Φ zyx V LLL F tzyx V0 1 ,,, ( ) ( ) ( ) ( )∑+ ∞ = ΦΦ 1 2 n nnnnn zyx tezcycxcF LLL VV , where ( )                 ++−= 2220 22 111 exp zyx n LLL tDnte   ; ( ) ( ) ( ) ( )∫ ∫ ∫= x y zL L L nnnn udvdwdwvufvcvcucF 0 0 0 ,, ; cn()=cos(n/L); D0L, D0I, D0V, D0I, D0V are the averaged values of diffusion coeffi- cients. Standard procedure of method of averaging of function corrections [13-15,20,21] gives us possibility to calculate approximations of the second- and higher orders of do- pant and radiation defect concentrations. The approach based on replacement of the re- quired functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), I(x,y,z,t), V(x,y,z,t) on the sum of aver- age value of the n-th order approximations and approximations of the n-1-th orders, i.e. n+n-1(x,y,z,t) to calculate the n-th order approximations of the above functions. In this situation relations of the second-order approximations of the required concentrations could be written as ( ) ( ) ( ) ( ) ( )[ ] ( )    ×         + +         ++= ∗∗ TzyxP tzyxC V tzyxV V tzyxV xt tzyxC C ,,, ,,, 1 ,,,,,, 1 ,,, 12 2 2 21 2         ( ) ( ) ( ) ( ) ( ) ( )    ×         +++   × ∗∗ 2 2 21 1 ,,,,,, 1,,, ,,, ,,, V tzyxV V tzyxV TzyxD yx tzyxC TzyxD LL      ( )[ ] ( ) ( ) ( ) ( )    ×+             + +× z tzyxC TzyxD zy tzyxC TzyxP tzyxC L C          ,,, ,,, ,,, ,,, ,,, 1 1112 ( ) ( ) ( ) ( )[ ] ( )             + +         ++× ∗∗ TzyxP tzyxC V tzyxV V tzyxV C ,,, ,,, 1 ,,,,,, 1 12 2 2 21     (8) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ]                 +−× ×++−      + +      +      = +−× ×++−      + +      +      = 2 12,, 1212 1 112 2 12,, 1212 1 112 ,,,,,,,,, ,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,,,,,,,, ,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, tzyxVTzyxkTzyxk tzyxVtzyxI z tzyxV TzyxD z y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV tzyxITzyxkTzyxk tzyxVtzyxI z tzyxI TzyxD z y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI VVVVI VIV VV IIIVI VII II                                 (9)
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 30 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                  −× ×+      Φ +  Φ ×     ×+      Φ = Φ −× ×+      Φ +  Φ ×     ×+      Φ = Φ Φ ΦΦ Φ ΦΦ tzyxVTzyxktzyxV Tzyxk z tzyx TzyxD zy tzyx TzyxD yx tzyx TzyxD xt tzyx tzyxITzyxktzyxI Tzyxk z tzyx TzyxD zy tzyx TzyxD yx tzyx TzyxD xt tzyx V VV VI VV I II II II V VV I II ,,,,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,,,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, 2 , 11 12 2 , 11 12                             (10) We obtain the second-order approximations of the required concentrations by integration of the left and right sides of the above relations. The relations could be written as ( ) ( ) ( ) ( ) ( )[ ] ( )    ∫ ×         + +         ++= ∗∗ t C TzyxP zyxC V zyxV V zyxV x tzyxC 0 12 2 2 212 ,,, ,,, 1 ,,,,,, 1,,,           ( ) ( ) ( ) ( ) ( ) ( )    ∫ ×         ++++   × ∗∗ t CL V zyxV V zyxV y zyxfd x zyxC TzyxD 0 2 2 21 1 ,,,,,, 1,, ,,, ,,,          ( ) ( )[ ] ( ) ( ) ( )   ∫ ×+             + +× t L C L TzyxD z d y zyxC TzyxP zyxC TzyxD 0 112 ,,, ,,, ,,, ,,, 1,,,         ( ) ( ) ( ) ( )[ ] ( ) ( )             + +         ++× ∗∗           d z zyxC TzyxP zyxC V zyxV V zyxV C ,,, ,,, ,,, 1 ,,,,,, 1 112 2 2 21 (8a) ( ) ( ) ( ) ( ) ( ) +      ∫+      ∫= t I t I d y zyxI TzyxD y d x zyxI TzyxD x tzyxI 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( )[ ] −∫ +−      ∫+ t III t I dzyxITzyxkd z zyxI TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,,      ( ) ( )[ ] ( )[ ] ( )zyxfdzyxVzyxITzyxk I t VIVI ,,,,,,,,,,, 0 1212, +∫ ++−  (9a) ( ) ( ) ( ) ( ) ( ) +      ∫+      ∫= t V t V d y zyxV TzyxD y d x zyxV TzyxD x tzyxV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( )[ ] −∫ +−      ∫+ t IVV t V dzyxVTzyxkd z zyxV TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,,     
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 31 ( ) ( )[ ] ( )[ ] ( )zyxfdzyxVzyxITzyxk V t VIVI ,,,,,,,,,,, 0 1212, +∫ ++−  ( ) ( ) ( ) ( ) ( ) +      ∫ Φ +      ∫ Φ =Φ ΦΦ t I t I I d y zyx TzyxD y d x zyx TzyxD x tzyx II 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( ) ( )−+∫+      ∫ Φ + ΦΦ zyxfdzyxITzyxkd z zyx TzyxD z II t II t I ,,,,,,,, ,,, ,,, 0 2 , 0 1      ( ) ( )∫− t I dzyxITzyxk 0 ,,,,,,  (10a) ( ) ( ) ( ) ( ) ( ) +      ∫ Φ +      ∫ Φ =Φ ΦΦ t V t V V d y zyx TzyxD y d x zyx TzyxD x tzyx VV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( ) ( )−+∫+      ∫ Φ + ΦΦ zyxfdzyxVTzyxkd z zyx TzyxD z VV t VV t V ,,,,,,,, ,,, ,,, 0 2 , 0 1      ( ) ( )∫− t V dzyxVTzyxk 0 ,,,,,,  . We determined average values of the second-orders approximations  2 of the determining func- tions by using the standard relation [13-15,20,21] ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 x y zL L L zyx tdxdydzdtzyxtzyx LLL   . (11) To obtain the required relations for the values  2 we substitute the relations (8a)-(10a) into the relation (11) ( )∫ ∫ ∫= x y zL L L C zyx С xdydzdzyxf LLL 0 0 0 2 ,, 1  , (12) ( ) [{ −+−−+++= 112010200 2 0021001 00 2 41 2 1 IVIIIVVIIIVVIIIV II I AAAAAAA A  ( ) 00 0021001 2 1 0 0 0 2 1 ,, 1 II IVVIIIV L L L I zyx A AAA xdydzdzyxf LLL x y z +++ −         ∫ ∫ ∫− (13a) ( ) 4 313 4 2 3 4 2 4 4 42 1 B AB A ByB yB AB B V + −      − +− + = , (13b) where ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ Θ = Θ 0 0 0 0 11, ,,,,,,,,, 1 x y zL L L ji ba zyx abij tdxdydzdtzyxVtzyxITzyxkt LLL A , ( )2 0000 2 00 2 00 2 004 2 VVIIIVIVIV AAAAAB −−= , −++= 2 001000 3 0001 2 00003 IVIIIVIVIVIVIV AAAAAAAB
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 32 ( ) ( ) ( )[ ]−++−+++−− 121224 10100010010000010000 2 00 VVIVIIIIIVIVIVIVVVIIIV AAAAAAAAAAA 2 000100 2 000010 24 IVIVIVIVIIIV AAAAAA +− , ( ){ ×++++= 00 2 01 2 00 2 1001 2 002 1 IVIVIVIIIVIV AAAAAAB ( )     ×−−−−++× zyx IIIVIIIIIVIIIVIVIVIVIV LLL AAAAAAAAAAA 1 442 2011000010100001000000 ( ) ( ) ([{ ++−+++        ∫ ∫ ∫× 12122,, 10001001000001 0 0 0 IVIIIIIVIVIVIV L L L I AAAAAAAxdydzdzyxf x y z )] ( ) ( ) (     −−∫ ∫ ∫+++++ 2000 0 0 0 100101 2 10 2,, 2 12 VVII L L L V zyx IIIVIVVV AAxdydzdzyxf LLL AAAA x y z ) ( )] ( ) ([ ++−++++++− 122121 1000000110010010010111 IVIIIVIVIIIVIVIIIVIVIV AAAAAAAAAAA )]}10VVA+ , ( ) ( )   ×∫ ∫ ∫−−++= x y zL L L I x IVIIIVIVIV xdydzdzyxf L AAAAAB 0 0 0 11 2 100101001 ,, 1 812 ( )−−++++     −× 00101000010000 2 0100010020 4 1 IIIVIIIVIVIVIVIVIIIVIVII zy AAAAAAAAAAAA LL ( ) ( ) ( )     +−−+++∫ ∫ ∫− 112000100101 0 0 0 00 21,, 2 2 IVVVIIIIIVIV L L L I zyx II AAAAAAxdydzdzyxf LLL A x y z ( )] ( ) ( )[ ]0001001010100100100101 212121 IVIVIIVVIVIIIVIVIIIVIV AAAAAAAAAAA +++−+++++ , ( ) ( +++         −∫ ∫ ∫+= 1001 2 0111 0 0 0 20 2 01000 ,, 1 4 IIIVIVIV L L L I zyx IIIVII AAAAxdydzdzyxf LLL AAAB x y z ) ( ) ( ) ( )     +−−+++∫ ∫ ∫−+ 112000100101 0 0 0 002 21,, 2 1 IVVVIIIIIVIV L L L V zyx II AAAAAAxdydzdzyxf LLL A x y z ( )]2 100101 1 IIIVIV AAA +++ , 6 23 323 32 B qpqqpqy +++−−+= , ( )×−= 031 82 BBBq ( ) 8 4 21648 2 1 2 320 3 22 BBBBBB −− ++× , ( )[ ] 722823 2 2031 BBBBp −−= , 2 2 3 48 BByA −+= , ( ) ( ) ( ) +∫ ∫ ∫ ∫−Θ Θ −= Θ Φ 0 0 0 0 202 ,,,,,, 1 x y z I L L L I zyx II tdxdydzdtzyxITzyxkt LLL A ( )∫ ∫ ∫+ Φ x y zL L L I zyx xdydzdzyxf LLL 0 0 0 ,, 1 (14) ( ) ( ) ( ) +∫ ∫ ∫ ∫−Θ Θ −= Θ Φ 0 0 0 0 202 ,,,,,, 1 x y z V L L L V zyx VV tdxdydzdtzyxVTzyxkt LLL A ( )∫ ∫ ∫+ Φ x y zL L L V zyx xdydzdzyxf LLL 0 0 0 ,, 1 . The considered substitution gives us possibility to obtain equation for parameter 2C. Solution of the equation depends on value of parameter . We analyzed spatio-temporal distributions of con- centrations of dopant and radiation defects by using their the second-order approximations, which have been calculated framework the method of averaged of function corrections with decreased quantity of iterative steps. The calculated approximations are usually enough good approxima-
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 33 tions to make qualitative analysis and obtain some quantitative results. Results of analytical cal- culation have been checked by comparison with results of numerical simulation. 3. Discussion In this section we analyzed dynamic of redistribution of dopant and radiation defects in the consi- dered heterostructure during their annealing by using calculated in the previous section relations. Typical distributions of concentrations of dopant in heterostructures are presented on Figs. 2 and 3 for diffusion and ion types of doping, respectively. These 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 sharpness of p-n- junctions. At the same time one can find increasing homogeneity of dopant distribution in doped part of epitaxial layer. Increasing of sharpness of p-n-junctions leads to decreasing their switching time. The second effect leads to decreasing local heating of materials during function- ing of p-n-junction or decreasing of dimensions of the p-n-junction for fixed maximal value of local overheat. In the considered situation we shall optimize of annealing to choose compromise annealing time of infused dopant. If annealing time is small, the dopant has no time to achieve nearest interface between layers of heterostructure. In this situation distribution of concentration of dopant is not changed. If annealing time is large, distribution of concentration of dopant is too homogenous. Fig.2. Distributions of concentration of infused dopant in heterostructure from Fig. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. The distributions have been calculated under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffu- sion coefficient in substrate. Increasing of number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 34 x 0.0 0.5 1.0 1.5 2.0 C(x,Θ) 2 34 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 Ion doping of materials leads to generation radiation defects. After finishing this process radiation defects should be annealed. The annealing leads to spreading of distribution of concentration of dopant. In the ideal case dopant achieves nearest interface between materials. If dopant has no time to achieve nearest interface, it is practicably to use additional annealing of dopant. We con- sider optimization of annealing time framework recently introduced criterion [17,24-30]. Frame- work the criterion we approximate real distributions of concentrations by step-wise functions (see Figs. 4 and 5). Farther we determine optimal values of annealing time by minimization the fol- lowing mean-squared error 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 describing real distributions of dopant witgh different values of annealing time (increasing of number of curve corresponds to increasing of annealing time)
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 35 x C(x,Θ) 1 2 3 4 0 L Fig. 5. Spatial distributions of dopant in heterostructure after ion implantation. Curves 2-4 describing real distributions of dopant witgh different values of annealing time (increasing of number of curve corresponds to increasing of annealing time) 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 ==0 for equal to each other values of dopant dif- fusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal an- nealing time on value of parameter  for a/L=1/2 and ==0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0. Curve 4 is the dependence of di- mensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 36 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 ==0 for equal to each other values of do- pant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optim- al annealing time on value of parameter  for a/L=1/2 and ==0. Curve 3 is the dependence of dimension- less optimal annealing time on value of parameter  for a/L=1/2 and ==0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0 ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1  , (15) where (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. Optimal value of time of additional annealing of implanted dopant is smaller, than optimal value of time of annealing of infused dopant due to preliminary annealing of radiation defects. 4. Conclusion In this paper we consider an approach to manufacture more compact double base heterobipolar transistor. The approach based on manufacturing a heterostructure with required configuration, doping of required areas of the heterostructure by diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects. The introduced approach to manufactur the hete- robipolar multiemittertransistor gives us possibility to increase sharpness of p-n-junctions frame- work the transistor. In this situation one have a possibility to increase compactness of the transis- tor. Acknowledgments This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, grant of Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research. REFERENCES [1] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980). [2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). [3] V.G. Gusev, Yu.M. Gusev. Electronics (Moscow: Vysshaya shkola, 1991, in Russian). [4] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [5] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 37 [6] A. Kerentsev, V. Lanin. "Constructive-technological features of MOSFET-transistors" Power Elec- tronics. Issue 1. P. 34-38 (2008). [7] A.N. Andronov, N.T. Bagraev, L.E. Klyachkin, S.V. Robozerov. "Ultrashallow p+−n junctions in silicon (100): electron-beam diagnostics of sub-surface region" Semiconductors. Vol.32 (2). P. 137- 144 (1998). [8] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p−n junctions in Si prepared by pulse pho- ton annealing" Semiconductors. Vol.36 (5). P. 611-617 (2002). [9] V.I. Mazhukin, V.V. Nosov, U. Semmler. "Study of heat and thermoelastic fields in semiconductors at pulsed processing" Mathematical modelling. Vol. 12 (2), 75 (2000). [10] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. "Dopant distribution in the re- crystallization transient at the maximum melt depth induced by laser annealing" Appl. Phys. Lett. Vol. 89 (17), 172111 (2006). [11] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A. Foad, F. Cristiano and P. F. Fazzini. "Deactivation of ultrashallow boron implants in preamorphized silicon after nonmelt laser annealing with multiple scans" Appl.Phys. Lett. Vol. 89, 192105 (2006). [12] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. "Diffusion processes in semiconductor structures during microwave anneal- ing" Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). [13] E.L. Pankratov. "Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure by optimized laser annealing" J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010). [14] E.L. Pankratov, E.A. Bulaeva. "Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of dis- tributions of dopants" Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [15] E.L. Pankratov, E.A. Bulaeva. "An approach to decrease dimensions of field-effect transistors" Uni- versal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013). [16] A.M. Ivanov, N.B. Strokman, V.B. Shuman. "Properties of p+-n-structures with a buried layer con- taining radiation defects" Semiconductors. Vol. 32 (3). P. 359-365 (1998). [17] Yu.B. Bolkhvityanov, A.K. Gutakovsky, A.S. Deryabin, L.V. Sokolov. "Edge dislocations inconsis- tencies in heterostructures GexSi1-x/Si(001) (x ~1): the role of intermediate buffer layer in their edu- cation GeySi1-y (y< x)" Solid state Physics. Vol. 53 (9). P. 1699-1705 (2011). [18] S.A. Kukushkin, A.V. Osipov. "Anisotropy of solid-phase epitaxy of silicon carbide on silicon" Semiconductors. Vol. 47 (12). P. 1575-1579 (2013). [19] Y. Tanaka, H. Tanoue, K. Arai. "Electrical activation of the ion-implanted phosphorus in 4H-SiC by excimer laser annealing" J. Appl. Phys. Vol.93 (10). P. 5934-5936 (2003). [20] Yu.D. Sokolov. "About determination of dynamic forces in the mine hoisting ropes" Applied Me- chanics. Vol.1 (1). P. 23-35 (1955). [21] E.L. Pankratov. "Dynamics of delta-dopant redistribution during heterostructure growth" The Euro- pean Physical Journal B. 2007. Vol. 57, №3. P. 251-256. Authors Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary school in Nizhny Novgorod. 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 Ra- diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio- physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro- structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo- rod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. He has 96 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- 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. E.A. Bulaeva was a contributor of grant of President of Russia (grant № MK-548.2010.2). She has 29 published papers in area of her researches.

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Using inhomogeneity of heterostructure and optimization of annealing to decrease dimensions multyemitter heterotransistors

  • 1. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 DOI:10.5121/ijcsa.2014.4303 25 USING INHOMOGENEITY OF HETEROSTRUCTURE AND OPTIMIZATION OF ANNEALING TO DE- CREASE DIMENSIONS MULTYEMITTER HETERO- TRANSISTORS E.L. Pankratov1 and E.A. Bulaeva 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 ABSTRACT Framework this paper we discussed an approach to manufacture a multiemitter heterotransistor. The in- troduced approach is a branch of recently introduced approach and based on doping by diffusion or by ion implantation of required part of heterostructure and optimization of dopant and/or radiation defects. The heterostructure should has special configuration. KEYWORDS Multiemitter heterotransistor, optimization of manufacturing of transistor, analytical approach to model technological process 1. INTRODUCTION Logical elements often include into itself multiemitter transistors [1-4]. To manufacture bipolar transistors it could be used dopant diffusion, ion doping or epitaxial layer [1-8]. It is attracted an interest increasing of sharpness of p-n-junctions, which include into bipolar transistors and de- creasing of dimensions of the transistors, which include into integrated circuits [1-4]. Dimensions of transistors will be decreased by using inhomogenous distribution of temperature during laser or microwave types of annealing [9,10], using defects of materials (for example, defects could be generated during radiation processing of materials) [11-14], native inhomogeneity (multilayer property) of heterostructure [12-14]. We consider manufacture of multiemitter heterotransistor based on a hetero structure, which con- sist of a substrate and three epitaxial layers (see Fig. 1). Some dopants are infused in the epitaxial layers to manufacture p and n types of conductivities during manufacture a multiemitter transis- tor. Let us consider two cases: infusion of dopant by diffusion or ion implantation. Farther in the first case we consider annealing of dopant so long, that after the annealing the dopants should achieve interfaces between epitaxial layers. In this case one can obtain increasing of sharpness of p-n-junctions [12-14]. The increasing of sharpness could be obtained under conditions, when val- ues of dopant diffusion coefficients in last epitaxial layers is larger, than values of dopant diffu- sion coefficient in average epitaxial layer. At the same time homogeneity of dopant distributions in doped areas increases. It is attracted an interest higher doping of last epitaxial layers, than dop-
  • 2. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 26 ing of average epitaxial layer, because the higher doping gives us possibility to increase sharpness of p-n-junctions. It is also practicably to choose materials of epitaxial layers and substrate so; those values of dopant diffusion coefficients in the substrate should be smaller, than in the epitax- ial layer. In this situation one can obtain thinner transistor. In the case of ion doping of hetero- structure it should be done annealing of radiation defects. It is practicably to choose so regimes of annealing of radiation defects, that dopant should achieves interface between epitaxial layers. At the same time the dopant could not diffuse into another epitaxial layer. If dopant did not achieves interface between epitaxial layers during annealing of radiation defects, additional annealing of dopant required. Main aims of the present paper are analysis of dynamics of redistribution of dopants in considered heterostructure (Figs. 1) and optimization of annealing times of dopants. In some recent papers analogous analysis have been done [4,10,12-14]. However we can not find in literature any simi- lar heterostructures as we consider in the paper. Substrate Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3 n-type dopant p-type dopant n-type dopant Fig. 1a. Heterostructure, which consist of a substrate and three epitaxial layers. View from above Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3 n-type dopantp-type dopant n-type dopant 1 n-type dopant 2 n-type dopant 3 n-type dopant 4 Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers. View from one side
  • 3. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 27 2. Method of solution We analyzed spatio-temporal distribution of dopant based on the second Fick’s law [1-4] ( ) ( ) ( ) ( )       +      +      = z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC CCC               ,,,,,,,,,,,, . (1) Boundary and initial conditions for the equation are ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLy y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLz z tzyxC , C (x,y,z,0)=fC (x,y,z). (2) We designate spatio-temporal distribution of concentration of dopant as C(x,y,z,t); as the current coordinates and times x, y, z and t; the dopant diffusion coefficient as DС. Dependences of dopant diffusion coefficient on coordinates, temperature of annealing (with account Arrhenius low), spa- tio-temporal distributions of concentrations of dopants and radiation defects could be approx- imated by the following relation [15,16] ( ) ( ) ( ) ( ) ( ) ( )       ++      += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC    . (3) Factor DL (x,y,z,T) describes inhomogeneity of heterostructure and temperature dependence of dopant diffusion coefficient due to Arrhenius law. We designate the limit of solubility of dopant as P (x,y,z,T); the spatio-temporal distribution of concentration of radiation vacancies V (x,y,z,t); the equilibrium distribution of vacancies V* . Parameter  depends of properties of materials and could be integer in the interval  ∈[1,3] [15]. Dependence of dopant diffusion coefficient on do- pant concentration has been discussed in details in [15]. It should be noted, that radiation damage absents in the case of diffusion doping of haterostructure. In this situation 1= 2= 0. Spatio- temporal distributions of concentrations of point radiation defects could be determine by solution of the following system of equations [17-19] ( ) ( ) ( ) ( ) ( ) ( )×−      ∂ ∂ ∂ ∂ +      ∂ ∂ ∂ ∂ = ∂ ∂ tzyxI y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ,,, ,,, ,,, ,,, ,,, ,,, 2 ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxI TzyxD z Tzyxk VIIII ,,,,,,,,, ,,, ,,,,,, ,, −      ∂ ∂ ∂ ∂ +× (4) ( ) ( ) ( ) ( ) ( ) ( )×−      ∂ ∂ ∂ ∂ +      ∂ ∂ ∂ ∂ = ∂ ∂ tzyxV y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ,,, ,,, ,,, ,,, ,,, ,,, 2 ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxV TzyxD z Tzyxk VIVVV ,,,,,,,,, ,,, ,,,,,, ,, −      ∂ ∂ ∂ ∂ +× with initial (x,y,z,0)=f (x,y,z) (5a) and boundary conditions
  • 4. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 28 ( ) 0 ,,, = ∂ ∂ = yLy y tzyx , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyx , ( ) 0 ,,, = ∂ ∂ = zLz z tzyx . (5b) We designate the spatio-temporal distribution of concentration of interstitials as I(x,y,z,t); =I,V; the diffusion coefficients of interstitials and vacancies as D(x,y,z,T); terms V2 (x,y, z,t) and I2 (x,y,z,t) correspond to generation of divacancies and analogous complexes of interstitials; kI,V(x, y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) parameters of recombination of point radiation defects and generation of complexes. Spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and analogous com- plexes of interstitials I(x,y,z,t) will be determined by solving the following systems of equations [17-19] ( ) ( ) ( ) ( ) ( ) +      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     with boundary and initial conditions ( ) 0 ,,, 0 = ∂ Φ∂ =x x tzyx , ( ) 0 ,,, = ∂ Φ∂ = xLx x tzyx , ( ) 0 ,,, 0 = ∂ Φ∂ =y y tzyx , ( ) 0 ,,, = ∂ Φ∂ = yLy y tzyx , ( ) 0 ,,, 0 = ∂ Φ∂ =z z tzyx , ( ) 0 ,,, = ∂ Φ∂ = zLz z tzyx , I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV (x,y,z). (7) Here DI(x,y,z,T) and DV(x,y,z,T) are diffusion coefficients of complexes of point radiation de- fects; kI(x,y,z,T) and kV(x,y,z,T) are parameters of decay of the complexes. We solved the above boundary problems by using method of averaging of function corrections [13-15,20] with decreased quantity of radiation defects [21]. Framework the approach we consid- er the initial-order approximations of concentrations of dopant and radiation defects as solutions of Eqs. (1), (4) and (6) with averaged values of diffusion coefficient D0L, D0I, D0V, D0I, D0V and zero values of parameters of recombination of defects, generation and decay of their complexes ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 0 1 2 ,,, n nCnnnnC zyxzyx C tezcycxcF LLLLLL F tzyxC , ( ) ×+= zyxzyx I LLLLLL F tzyxI 2 ,,, 0 1 ( ) ( ) ( ) ( )∑× ∞ =1n nInnnnI tezcycxcF , ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 0 1 2 ,,, n nVnnnnC zyxzyx C tezcycxcF LLLLLL F tzyxV ,
  • 5. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 29 ( ) ( ) ( ) ( ) ( )∑+=Φ ∞ = ΦΦ Φ 1 0 1 2 ,,, n nnnnn zyxzyx I tezcycxcF LLLLLL F tzyx II I , ( ) +=Φ Φ zyx V LLL F tzyx V0 1 ,,, ( ) ( ) ( ) ( )∑+ ∞ = ΦΦ 1 2 n nnnnn zyx tezcycxcF LLL VV , where ( )                 ++−= 2220 22 111 exp zyx n LLL tDnte   ; ( ) ( ) ( ) ( )∫ ∫ ∫= x y zL L L nnnn udvdwdwvufvcvcucF 0 0 0 ,, ; cn()=cos(n/L); D0L, D0I, D0V, D0I, D0V are the averaged values of diffusion coeffi- cients. Standard procedure of method of averaging of function corrections [13-15,20,21] gives us possibility to calculate approximations of the second- and higher orders of do- pant and radiation defect concentrations. The approach based on replacement of the re- quired functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), I(x,y,z,t), V(x,y,z,t) on the sum of aver- age value of the n-th order approximations and approximations of the n-1-th orders, i.e. n+n-1(x,y,z,t) to calculate the n-th order approximations of the above functions. In this situation relations of the second-order approximations of the required concentrations could be written as ( ) ( ) ( ) ( ) ( )[ ] ( )    ×         + +         ++= ∗∗ TzyxP tzyxC V tzyxV V tzyxV xt tzyxC C ,,, ,,, 1 ,,,,,, 1 ,,, 12 2 2 21 2         ( ) ( ) ( ) ( ) ( ) ( )    ×         +++   × ∗∗ 2 2 21 1 ,,,,,, 1,,, ,,, ,,, V tzyxV V tzyxV TzyxD yx tzyxC TzyxD LL      ( )[ ] ( ) ( ) ( ) ( )    ×+             + +× z tzyxC TzyxD zy tzyxC TzyxP tzyxC L C          ,,, ,,, ,,, ,,, ,,, 1 1112 ( ) ( ) ( ) ( )[ ] ( )             + +         ++× ∗∗ TzyxP tzyxC V tzyxV V tzyxV C ,,, ,,, 1 ,,,,,, 1 12 2 2 21     (8) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( )[ ]                 +−× ×++−      + +      +      = +−× ×++−      + +      +      = 2 12,, 1212 1 112 2 12,, 1212 1 112 ,,,,,,,,, ,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,,,,,,,, ,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, tzyxVTzyxkTzyxk tzyxVtzyxI z tzyxV TzyxD z y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV tzyxITzyxkTzyxk tzyxVtzyxI z tzyxI TzyxD z y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI VVVVI VIV VV IIIVI VII II                                 (9)
  • 6. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 30 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                  −× ×+      Φ +  Φ ×     ×+      Φ = Φ −× ×+      Φ +  Φ ×     ×+      Φ = Φ Φ ΦΦ Φ ΦΦ tzyxVTzyxktzyxV Tzyxk z tzyx TzyxD zy tzyx TzyxD yx tzyx TzyxD xt tzyx tzyxITzyxktzyxI Tzyxk z tzyx TzyxD zy tzyx TzyxD yx tzyx TzyxD xt tzyx V VV VI VV I II II II V VV I II ,,,,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,,,,,,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, ,,, 2 , 11 12 2 , 11 12                             (10) We obtain the second-order approximations of the required concentrations by integration of the left and right sides of the above relations. The relations could be written as ( ) ( ) ( ) ( ) ( )[ ] ( )    ∫ ×         + +         ++= ∗∗ t C TzyxP zyxC V zyxV V zyxV x tzyxC 0 12 2 2 212 ,,, ,,, 1 ,,,,,, 1,,,           ( ) ( ) ( ) ( ) ( ) ( )    ∫ ×         ++++   × ∗∗ t CL V zyxV V zyxV y zyxfd x zyxC TzyxD 0 2 2 21 1 ,,,,,, 1,, ,,, ,,,          ( ) ( )[ ] ( ) ( ) ( )   ∫ ×+             + +× t L C L TzyxD z d y zyxC TzyxP zyxC TzyxD 0 112 ,,, ,,, ,,, ,,, 1,,,         ( ) ( ) ( ) ( )[ ] ( ) ( )             + +         ++× ∗∗           d z zyxC TzyxP zyxC V zyxV V zyxV C ,,, ,,, ,,, 1 ,,,,,, 1 112 2 2 21 (8a) ( ) ( ) ( ) ( ) ( ) +      ∫+      ∫= t I t I d y zyxI TzyxD y d x zyxI TzyxD x tzyxI 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( )[ ] −∫ +−      ∫+ t III t I dzyxITzyxkd z zyxI TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,,      ( ) ( )[ ] ( )[ ] ( )zyxfdzyxVzyxITzyxk I t VIVI ,,,,,,,,,,, 0 1212, +∫ ++−  (9a) ( ) ( ) ( ) ( ) ( ) +      ∫+      ∫= t V t V d y zyxV TzyxD y d x zyxV TzyxD x tzyxV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( )[ ] −∫ +−      ∫+ t IVV t V dzyxVTzyxkd z zyxV TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,,     
  • 7. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 31 ( ) ( )[ ] ( )[ ] ( )zyxfdzyxVzyxITzyxk V t VIVI ,,,,,,,,,,, 0 1212, +∫ ++−  ( ) ( ) ( ) ( ) ( ) +      ∫ Φ +      ∫ Φ =Φ ΦΦ t I t I I d y zyx TzyxD y d x zyx TzyxD x tzyx II 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( ) ( )−+∫+      ∫ Φ + ΦΦ zyxfdzyxITzyxkd z zyx TzyxD z II t II t I ,,,,,,,, ,,, ,,, 0 2 , 0 1      ( ) ( )∫− t I dzyxITzyxk 0 ,,,,,,  (10a) ( ) ( ) ( ) ( ) ( ) +      ∫ Φ +      ∫ Φ =Φ ΦΦ t V t V V d y zyx TzyxD y d x zyx TzyxD x tzyx VV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,,           ( ) ( ) ( ) ( ) ( )−+∫+      ∫ Φ + ΦΦ zyxfdzyxVTzyxkd z zyx TzyxD z VV t VV t V ,,,,,,,, ,,, ,,, 0 2 , 0 1      ( ) ( )∫− t V dzyxVTzyxk 0 ,,,,,,  . We determined average values of the second-orders approximations  2 of the determining func- tions by using the standard relation [13-15,20,21] ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 x y zL L L zyx tdxdydzdtzyxtzyx LLL   . (11) To obtain the required relations for the values  2 we substitute the relations (8a)-(10a) into the relation (11) ( )∫ ∫ ∫= x y zL L L C zyx С xdydzdzyxf LLL 0 0 0 2 ,, 1  , (12) ( ) [{ −+−−+++= 112010200 2 0021001 00 2 41 2 1 IVIIIVVIIIVVIIIV II I AAAAAAA A  ( ) 00 0021001 2 1 0 0 0 2 1 ,, 1 II IVVIIIV L L L I zyx A AAA xdydzdzyxf LLL x y z +++ −         ∫ ∫ ∫− (13a) ( ) 4 313 4 2 3 4 2 4 4 42 1 B AB A ByB yB AB B V + −      − +− + = , (13b) where ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ Θ = Θ 0 0 0 0 11, ,,,,,,,,, 1 x y zL L L ji ba zyx abij tdxdydzdtzyxVtzyxITzyxkt LLL A , ( )2 0000 2 00 2 00 2 004 2 VVIIIVIVIV AAAAAB −−= , −++= 2 001000 3 0001 2 00003 IVIIIVIVIVIVIV AAAAAAAB
  • 8. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 32 ( ) ( ) ( )[ ]−++−+++−− 121224 10100010010000010000 2 00 VVIVIIIIIVIVIVIVVVIIIV AAAAAAAAAAA 2 000100 2 000010 24 IVIVIVIVIIIV AAAAAA +− , ( ){ ×++++= 00 2 01 2 00 2 1001 2 002 1 IVIVIVIIIVIV AAAAAAB ( )     ×−−−−++× zyx IIIVIIIIIVIIIVIVIVIVIV LLL AAAAAAAAAAA 1 442 2011000010100001000000 ( ) ( ) ([{ ++−+++        ∫ ∫ ∫× 12122,, 10001001000001 0 0 0 IVIIIIIVIVIVIV L L L I AAAAAAAxdydzdzyxf x y z )] ( ) ( ) (     −−∫ ∫ ∫+++++ 2000 0 0 0 100101 2 10 2,, 2 12 VVII L L L V zyx IIIVIVVV AAxdydzdzyxf LLL AAAA x y z ) ( )] ( ) ([ ++−++++++− 122121 1000000110010010010111 IVIIIVIVIIIVIVIIIVIVIV AAAAAAAAAAA )]}10VVA+ , ( ) ( )   ×∫ ∫ ∫−−++= x y zL L L I x IVIIIVIVIV xdydzdzyxf L AAAAAB 0 0 0 11 2 100101001 ,, 1 812 ( )−−++++     −× 00101000010000 2 0100010020 4 1 IIIVIIIVIVIVIVIVIIIVIVII zy AAAAAAAAAAAA LL ( ) ( ) ( )     +−−+++∫ ∫ ∫− 112000100101 0 0 0 00 21,, 2 2 IVVVIIIIIVIV L L L I zyx II AAAAAAxdydzdzyxf LLL A x y z ( )] ( ) ( )[ ]0001001010100100100101 212121 IVIVIIVVIVIIIVIVIIIVIV AAAAAAAAAAA +++−+++++ , ( ) ( +++         −∫ ∫ ∫+= 1001 2 0111 0 0 0 20 2 01000 ,, 1 4 IIIVIVIV L L L I zyx IIIVII AAAAxdydzdzyxf LLL AAAB x y z ) ( ) ( ) ( )     +−−+++∫ ∫ ∫−+ 112000100101 0 0 0 002 21,, 2 1 IVVVIIIIIVIV L L L V zyx II AAAAAAxdydzdzyxf LLL A x y z ( )]2 100101 1 IIIVIV AAA +++ , 6 23 323 32 B qpqqpqy +++−−+= , ( )×−= 031 82 BBBq ( ) 8 4 21648 2 1 2 320 3 22 BBBBBB −− ++× , ( )[ ] 722823 2 2031 BBBBp −−= , 2 2 3 48 BByA −+= , ( ) ( ) ( ) +∫ ∫ ∫ ∫−Θ Θ −= Θ Φ 0 0 0 0 202 ,,,,,, 1 x y z I L L L I zyx II tdxdydzdtzyxITzyxkt LLL A ( )∫ ∫ ∫+ Φ x y zL L L I zyx xdydzdzyxf LLL 0 0 0 ,, 1 (14) ( ) ( ) ( ) +∫ ∫ ∫ ∫−Θ Θ −= Θ Φ 0 0 0 0 202 ,,,,,, 1 x y z V L L L V zyx VV tdxdydzdtzyxVTzyxkt LLL A ( )∫ ∫ ∫+ Φ x y zL L L V zyx xdydzdzyxf LLL 0 0 0 ,, 1 . The considered substitution gives us possibility to obtain equation for parameter 2C. Solution of the equation depends on value of parameter . We analyzed spatio-temporal distributions of con- centrations of dopant and radiation defects by using their the second-order approximations, which have been calculated framework the method of averaged of function corrections with decreased quantity of iterative steps. The calculated approximations are usually enough good approxima-
  • 9. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 33 tions to make qualitative analysis and obtain some quantitative results. Results of analytical cal- culation have been checked by comparison with results of numerical simulation. 3. Discussion In this section we analyzed dynamic of redistribution of dopant and radiation defects in the consi- dered heterostructure during their annealing by using calculated in the previous section relations. Typical distributions of concentrations of dopant in heterostructures are presented on Figs. 2 and 3 for diffusion and ion types of doping, respectively. These 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 sharpness of p-n- junctions. At the same time one can find increasing homogeneity of dopant distribution in doped part of epitaxial layer. Increasing of sharpness of p-n-junctions leads to decreasing their switching time. The second effect leads to decreasing local heating of materials during function- ing of p-n-junction or decreasing of dimensions of the p-n-junction for fixed maximal value of local overheat. In the considered situation we shall optimize of annealing to choose compromise annealing time of infused dopant. If annealing time is small, the dopant has no time to achieve nearest interface between layers of heterostructure. In this situation distribution of concentration of dopant is not changed. If annealing time is large, distribution of concentration of dopant is too homogenous. Fig.2. Distributions of concentration of infused dopant in heterostructure from Fig. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. The distributions have been calculated under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffu- sion coefficient in substrate. Increasing of number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure
  • 10. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 34 x 0.0 0.5 1.0 1.5 2.0 C(x,Θ) 2 34 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 Ion doping of materials leads to generation radiation defects. After finishing this process radiation defects should be annealed. The annealing leads to spreading of distribution of concentration of dopant. In the ideal case dopant achieves nearest interface between materials. If dopant has no time to achieve nearest interface, it is practicably to use additional annealing of dopant. We con- sider optimization of annealing time framework recently introduced criterion [17,24-30]. Frame- work the criterion we approximate real distributions of concentrations by step-wise functions (see Figs. 4 and 5). Farther we determine optimal values of annealing time by minimization the fol- lowing mean-squared error 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 describing real distributions of dopant witgh different values of annealing time (increasing of number of curve corresponds to increasing of annealing time)
  • 11. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 35 x C(x,Θ) 1 2 3 4 0 L Fig. 5. Spatial distributions of dopant in heterostructure after ion implantation. Curves 2-4 describing real distributions of dopant witgh different values of annealing time (increasing of number of curve corresponds to increasing of annealing time) 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 ==0 for equal to each other values of dopant dif- fusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal an- nealing time on value of parameter  for a/L=1/2 and ==0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0. Curve 4 is the dependence of di- mensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0
  • 12. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014 36 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 ==0 for equal to each other values of do- pant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optim- al annealing time on value of parameter  for a/L=1/2 and ==0. Curve 3 is the dependence of dimension- less optimal annealing time on value of parameter  for a/L=1/2 and ==0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0 ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1  , (15) where (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. Optimal value of time of additional annealing of implanted dopant is smaller, than optimal value of time of annealing of infused dopant due to preliminary annealing of radiation defects. 4. Conclusion In this paper we consider an approach to manufacture more compact double base heterobipolar transistor. The approach based on manufacturing a heterostructure with required configuration, doping of required areas of the heterostructure by diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects. The introduced approach to manufactur the hete- robipolar multiemittertransistor gives us possibility to increase sharpness of p-n-junctions frame- work the transistor. In this situation one have a possibility to increase compactness of the transis- tor. Acknowledgments This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, grant of Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research. REFERENCES [1] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980). [2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). [3] V.G. Gusev, Yu.M. Gusev. Electronics (Moscow: Vysshaya shkola, 1991, in Russian). [4] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [5] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
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