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AN APPROACH TO OPTIMIZE REGIMES OF MANUFACTURING OF COMPLEMENTARY HORIZONTAL FIELD-EFFECT TRANSISTOR

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In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect heterotransistor. Based on analytical solution of the considered boundary problems some recommendations have been formulated to optimize
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  International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014 DOI : 10.14810/ijrap.2014.3205 55                     E.L. Pankratov 1  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  A  BSTRACT     In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect heterotransistor. Based on analytical solution of the considered boundary problems some recommendations have been formulated to optimize technological processes.  K   EYWORDS    Horizontal field-effect transistor, modelling of manufacturing of transistor, recommendations for optimisation of manufacturing of transistor 1.   I NTRODUCTION   In the present time it is intensively increasing degree of integration of elements of integrated circuits [1-8]. At the same time it is obtaining decreasing of dimensions of the elements. To decrease dimensions of elements of integrated circuits it is traditionally using some approaches. Two of them are laser and microwave types of annealing of dopants and/or radiation defects during manufacturing  p - n -junctions, field-effect and bipolar transistors, thyristors [9-15]. Another way to increase degree of integration of elements of integrated circuits is using of inhomogeneity of heterostructures on the basis of which integrated circuits are manufactured [13-19]. However in this case it is practicably to optimize annealing of dopant and/or radiation defects. It is known, that distribution of concentrations of dopants in elements of integrated circuits and their discrete analogs will be changed under influence of radiation processing (for example, during ion implantation) [20]. Because of this to decrease dimensions of elements of integrated circuits and their discrete it is attracted an interest radiation processing of materials [21,22]. In this paper we consider manufacturing of complementary field-effect heterotransistor. Structure of the heterotransistor is presented on the Fig. 1. The heterostructure consist of a substrate and epitaxial layer. The epitaxial layer has several sections, which have been manufactured by using another materials. Some dopants have been infused or implanted in the sections to manufacture required types of conductivity (  p  or n ). Farther we consider annealing of dopant (for doping by diffusion) and/or radiation defects (during ion doping). Main aim of the present paper we analyzed dynamics of redistribution of dopant and radiation defects to formulate conditions, which correspond to manufacture more thin heterotransistor with smaller dimensions into another dimensions.  International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014 56 SubstrateDrain SourceSource DrainGate Gate  p p n n   Fig.1. Heterostructure with a substrate and epitaxial layer with several sections 2.   METHOD OF SOLUTION To solve our aims we determine spatio-temporal distribution of concentration of dopant. We determine the distributions by solving the second Fick’s law [1,3-5] ( ) ( ) ( ) ( )  +  +  =  zt  z y xC   D z yt  z y xC   D y xt  z y xC   D xt t  z y xC  C C C  ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ,,,,,,,,,,,,  (1) with boundary and initial conditions ( ) 0,,, 0 =∂∂ =  x  xt  z y xC  , ( ) 0,,, =∂∂ =  x  L x  xt  z y xC  , ( ) 0,,, 0 =∂∂ =  y  yt  z y xC  ,  ( ) 0,,, =∂∂ =  y  L x  yt  z y xC  , (2) ( ) 0,,, 0 =∂∂ =  z  zt  z y xC  , ( ) 0,,, =∂∂ =  z  L x  zt  z y xC  , C  (  x ,  y ,  z ,0) =f    (  x ,  y ,  z ). Here   C  (  x ,  y ,  z , t  )   is   the   spatio-temporal   distribution of concentration of dopant; T   is the temperature of    annealing;    D     is   the   dopant   diffusion   coefficient.   Value   of dopant diffusion coefficient depends on properties of materials in layers of heterostructure, speed of heating and cooling of hetero-structure (with account Arrhenius law). Dependences of dopant diffusion coefficient on parameters could be approximated by the following relation [23-25] ( ) ( )( )( ) ( ) ( )   ++  += 2*22*1 ,,,,,,1,,,,,,1,,, V t  z y xV  V t  z y xV  T  z y xP t  z y xC  T  z y x D D  LC   ς ς ξ  γ  γ   , (3) where  D  L (  x ,  y ,  z , T  ) is the spatial (due to inhomogeneity of heterostructure) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficient; 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] [23]; V    (  x ,  y ,  z , t  ) is the spatio-temporal distribution of concentration of vacancies;   V  *   is   the   equilibrium   distribution   of    concentration   of    vacancies.   Concentrational   depen-dence of dopant diffusion coefficient has been discussed in details in the Ref. [23]. It should be noted that doping of heterostructure by diffusion did not leads to generation of radiation damage and ζ  1 =   ζ  2 =  International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014 57 0. Spatio-temporal distributions of concentrations of point radiation defects we determine by solving the following system equations [24,25] ( )( ) ( )( ) ( ) +  ∂∂∂∂+  ∂∂∂∂=∂∂  yt  z y x I  T  z y x D  y xt  z y x I  T  z y x D  xt t  z y x I   I  I  ,,,,,,,,,,,,,,,  (4)   ( ) ( )( ) ( ) ( ) ( ) ( ) t  z y x I T  z y xk t  z y xV t  z y x I T  z y xk   zt  z y x I  T  z y x D  z  I  I V  I  I  ,,,,,,,,,,,,,,, ,,,,,, 2,,  −−  ∂∂∂∂+   ( )( ) ( )( ) ( ) +  ∂∂∂∂+  ∂∂∂∂=∂∂  yt  z y xV  T  z y x D  y xt  z y xV  T  z y x D  xt t  z y xV  V V  ,,,,,,,,,,,,,,,   ( ) ( )( ) ( ) ( ) ( ) ( ) t  z y xV T  z y xk t  z y xV t  z y x I T  z y xk   zt  z y xV  T  z y x D  z  V V V  I V  ,,,,,,,,,,,,,,, ,,,,,, 2,,  −−  ∂∂∂∂+   with initial  ρ  (  x ,  y ,  z ,0) =f   ρ    (  x ,  y ,  z ) (5 a ) and boundary conditions ( ) 0,,, 0 =∂∂ =  x  xt  z y x  ρ  , ( ) 0,,, =∂∂ =  x  L x  xt  z y x  ρ  , ( ) 0,,, 0 =∂∂ =  y  yt  z y x  ρ  , ( ) 0,,, =∂∂ =  y  L y  yt  z y x  ρ  , ( ) 0,,, 0 =∂∂ =  z  zt  z y x  ρ  , ( ) 0,,, =∂∂ =  z  L z  zt  z y x  ρ  . (5 b ) Here  ρ  =  I  , V  ;  I (  x ,  y ,  z , t  ) are the spatio-temporal distributions of concentrations of radiation interstitials and radiation vacancies;  D  ρ  (  x ,  y ,  z , T  ) are the diffusion coefficients of the interstitials and vacancies; terms V  2 (  x ,  y ,  z , t  ) and  I  2 (  x ,  y ,  z , t  ) correspond to generation of divacancies and diinterstitials, respectively; k   I  , V  (  x ,  y ,  z , T  ), k   I  ,  I  (  x ,  y ,  z , T  ) and k  V  , V  (  x ,  y ,  z , T  ) are parameters of recombination of point defects and generation of their complexes, respectively. Spatio-temporal distributions of concentrations of divacansies Φ  V    (  x ,  y ,  z , t  ) and diinterstitials Φ   I    (  x ,  y ,  z , t  ) we determine by solution the following system of equations [24,25] ( )( ) ( )( ) ( ) +   Φ+   Φ=Φ ΦΦ  yt  z y x T  z y x D  y xt  z y x T  z y x D  xt t  z y x  I  I  I  I  I  ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ,,,,,,,,,,,,,,,   ( ) ( )( ) ( ) ( ) ( ) t  z y x I T  z y xk t  z y x I T  z y xk   zt  z y x T  z y x D  z  I  I  I  I  I  ,,,,,,,,,,,, ,,,,,, 2,  −+   Φ+  Φ ∂ ∂ ∂ ∂   (6) ( )( ) ( )( ) ( ) +   Φ+   Φ=Φ ΦΦ  yt  z y x T  z y x D  y xt  z y x T  z y x D  xt t  z y x V V V V V  ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  ,,,,,,,,,,,,,,,   ( ) ( )( ) ( ) ( ) ( ) t  z y xV T  z y xk t  z y xV T  z y xk   zt  z y x T  z y x D  z  V V V V V  ,,,,,,,,,,,, ,,,,,, 2,  −+   Φ+ Φ ∂ ∂ ∂ ∂    with boundary and initial conditions  International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014 58 ( ) 0,,, 0 =∂Φ∂ =  x  xt  z y x  ρ  , ( ) 0,,, =∂Φ∂ =  x  L x  xt  z y x  ρ  , ( ) 0,,, 0 =∂Φ∂ =  y  yt  z y x  ρ  , ( ) 0,,, =∂Φ∂ =  y  L y  yt  z y x  ρ  , ( ) 0,,, 0 =∂Φ∂ =  z  zt  z y x  ρ  , ( ) 0,,, =∂Φ∂ =  z  L z  zt  z y x  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)   Here  D Φ   I  (  x ,  y ,  z , T  ) and  D Φ  V  (  x ,  y ,  z , T  ) are diffusion coefficients of complexes of point radiation defects; k   I  (  x ,  y ,  z , T  ) and k  V   (  x ,  y ,  z , T  ) are parameters of decay of complexes of point radiation defects. To determine spatio-temporal distributions of concentrations of point radiation defects we used recently elaborated approach [16,19,22]. Framework the approach we transform approximations of diffusion coefficients of point radiation defects to the following form:  D  ρ  (  x ,  y ,  z , T  )=  D 0  ρ  [1+ ε   ρ  g  ρ  (  x ,  y ,  z , T  )], where  D 0  ρ   are the average values of the diffusion coefficients, 0 ≤ ε   ρ  < 1, | g  ρ  (  x ,  y ,  z , T  )| ≤ 1,  ρ   =  I  , V  . We used the same transformation for approximations of parameters of recombination of point radiation defects and generation of their complexes: k   I  , V  (  x ,  y ,  z , T  )= k  0  I  , V  [1+ ε   I  , V g  I  , V  (  x ,  y ,  z , T  )], k   I  ,  I  (  x ,  y ,  z , T  )= k  0  I  ,  I  [1+ ε   I  ,  I g  I  ,  I  (  x ,  y ,  z , T  )]    k  V  , V  (  x ,  y ,  z , T  ) = k  0 V  , V [1+ ε  V  , V g V  , V  (  x ,  y ,  z , T  )], where k  0  ρ  1,  ρ  2  are the appropriate average values, 0 ≤ ε   I  , V  < 1, 0 ≤ ε   I  ,  I < 1, 0 ≤ ε  V  , V  <\1, |   g  I  , V  (  x ,  y ,  z , T  )| ≤ 1, |   g  I  ,  I  (  x ,  y ,  z , T  )| ≤ 1, | g V  , V  (  x ,  y ,  z , T  )| ≤ 1. Let us introduce the following dimensionless   variables:  χ  =    x  /   L  x , η    =    y  /   L  y , φ  =    z  /   L  z , ( ) ( ) * ,,,,,, ~  I t  z y x I t  z y x I   = , ( ) ( ) * ,,,,,, ~ V t  z y xV t  z y xV   = , 200  Lt  D D V  I  = ϑ  , V  I V  I   D Dk  L 00,0 2 = ω  , V  I   D Dk  L 00,0 2  ρ  ρ  ρ   =Ω . The introduction leads to modification of Eqs.(4) and conditions (5) ( )( ) [ ]  ( )( ) [ ]  ×+∂∂+  ∂∂+∂∂=∂∂ T g I T g D D D I   I  I  I  I  V  I  I  ,,,1 ,,,~,,,1 ,,,~ 000 φ η  χ ε  η  χ ϑ φ η  χ  φ η  χ ε   χ ϑ ϑ φ η  χ    ( )( ) [ ]  ( )( )  ×−  ∂∂+∂∂+  ∂∂×  ϑ φ η  χ  η ϑ φ η  χ  φ η  χ ε  φ φ ϑ φ η  χ  ,,,~,,,~,,,1 ,,,~ 000000  I  I T g D D D D D D I   I  I V  I  I V  I  I    ( ) [ ]  ( ) ( ) ( ) [ ] T g I V T g  I  I  I  I  I V  I V  I  ,,,1,,, ~,,,~,,,1 ,,2,,  φ η  χ ε ϑ φ η  χ ϑ φ η  χ φ η  χ ε ω   +Ω−+×  (8) ( )( ) [ ]  ( )( ) [ ]  ×+∂∂+  ∂∂+∂∂=∂∂ T gV T g D D DV  V V V V  V  I V  ,,,1 ,,,~,,,1 ,,,~ 000 φ η  χ ε  η  χ ϑ φ η  χ  φ η  χ ε   χ ϑ ϑ φ η  χ    ( )( ) [ ]  ( )( )  ×−  ∂∂+∂∂+  ∂∂×  ϑ φ η  χ  η ϑ φ η  χ  φ η  χ ε  φ φ ϑ φ η  χ  ,,,~,,,~,,,1 ,,,~ 000000  I V T g D D D D D DV  V V V  I V V  I V    ( ) [ ]  ( ) ( ) ( ) [ ] T gV V T g V V V V V V  I V  I  ,,,1,,, ~,,,~,,,1 ,,2,,  φ η  χ ε ϑ φ η  χ ϑ φ η  χ φ η  χ ε ω   +Ω−+×   ( ) 0,,,~ 0 =∂∂ =  χ   χ ϑ φ η  χ  ρ  , ( ) 0,,,~ 1 =∂∂ =  χ   χ ϑ φ η  χ  ρ  , ( ) 0,,,~ 0 =∂∂ = η  η ϑ φ η  χ  ρ  , ( ) 0,,,~ 1 =∂∂ = η  η ϑ φ η  χ  ρ  , ( ) 0,,,~ 0 =∂∂ = φ  φ ϑ φ η  χ  ρ  , ( ) 0,,,~ 1 =∂∂ = φ  φ ϑ φ η  χ  ρ  , ( ) ( ) * ,,,,,,~  ρ ϑ φ η ϑ φ η  χ  ρ   ρ   f  = . (9) We determine solution of Eqs.(8) and conditions (9) by approach from Refs. [16,19,22], i.e. as the following power series ( ) ( )    Ω= ∞=∞=∞= 000 ,,,~,,,~ i j k ijk k  ji ϑ φ η  χ  ρ ω ε ϑ φ η  χ  ρ   ρ  ρ  . (10)  International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014 59 Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain equations for initial-order approximations of concentrations of point radiation defects ( ) ϑ φ η  χ  ,,,~ 000  I   and ( ) ϑ φ η  χ  ,,,~ 000 V   and corrections for them ( ) ϑ φ η  χ  ,,,~ ijk   I   and ( ) ϑ φ η  χ  ,,,~ ijk  V  , i ≥ 1,  j ≥ 1, k ≥ 1. The equations and conditions for them are presented in the Appendix. Solutions of them have been obtained by standard approaches (see, for example, [26,27]). The solutions have been obtained in the Appendix. Farther we determine spatio-temporal distributions of concentrations of complexes of point radiation defects. To obtain the concentrations we transform approximations of diffusion coefficients to the following form:  D Φρ  (  x ,  y ,  z , T  )=  D 0 Φρ  [1+ ε  Φρ  g Φρ  (  x ,  y ,  z , T  )], where  D 0 Φρ   are the average values of diffusion coefficients. After this transformation the Eqs.(6) takes the form ( )( ) [ ]  ( )( ) [ ]  ( )( ) [ ] ( )( ) ( ) ( ) ( )( )( ) [ ]  ( )( ) [ ]  ( )( ) [ ] ( )( ) ( ) ( ) ( )  −+  Φ×  ×++   Φ+××+   Φ+=Φ−+  Φ×  ×++   Φ+××+   Φ+=Φ ΦΦΦΦΦ ΦΦΦΦ ΦΦΦΦΦ ΦΦΦΦ t  z y xV T  z y xk t  z y xV T  z y xk  D  zt  z y x T  z y xg  z yt  z y x T  z y xg  y D xt  z y x T  z y xg  x Dt t  z y x t  z y x I T  z y xk t  z y x I T  z y xk  D  zt  z y x T  z y xg  z yt  z y x T  z y xg  y D xt  z y x T  z y xg  x Dt t  z y x V V V V  V V V V V V V V V V V V  I  I  I  I   I  I  I  I  I  I  I  I  I  I  I  I  ,,,,,,,,,,,, ,,,,,,1 ,,,,,,1 ,,,,,,1 ,,,,,,,,,,,,,,, ,,,,,,1 ,,,,,,1 ,,,,,,1 ,,, 2,0002,000 ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂    We determine solutions of the above equations as the following power series ( ) ( )   Φ=Φ ∞=Φ 0 ,,,,,, iii t  z y xt  z y x  ρ  ρ  ρ   ε  . (11) Substitution of the series (11) into Eqs.(6) and appropriate boundary and initial conditions gives us possibility to obtain equations for initial-order approximations of concentrations of complexes of point of radiation defects Φ   ρ  0 (  x ,  y ,  z , t  ), corrections for them Φ   ρ  i (  x ,  y ,  z , t  ), i   ≥ 1, boundary and initial conditions for all functions Φ   ρ  i (  x ,  y ,  z , t  ), i   ≥ 0. The equations and conditions are presented in the Appendix. Solutions of the equations have been solved by standard approaches [26,27] and presented in the Appendix. Spatio-temporal distribution of dopant concentration we determine framework the same approach as for determination of concentrations of radiation defects. Framework the approach we transform approximation of dopant diffusion coefficient in the following form:  D  L (  x ,  y ,  z , T  )=  D 0  L [1+ ε   L g  L (  x ,  y ,  z , T  )],  D 0  L  is the average value of dopant diffusion coefficient, 0 ≤ ε   L <   1, | g  L (  x ,  y ,  z , T  )| ≤ 1. We determine solution of Eq.(1) as the following power series
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