Optimization of Manufacturing of Operational Amplifier Manufactured by Using Field-effect Heterotransistor to Decrease Their Dimensions

In this paper we introduce an approach to decrease dimensions of operational amplifier based on field-effect heterotransistors. Dimensions of the elements will be decreased due to manufacture heterostructure with specific structure, doping of required areas of the heterostruc-ture by diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects.


Introduction
In the present time density of elements of integrated circuits and their performance intensively increasing. Simultaneously with increasing of the density of these elements of integrated circuits their dimensions decreases. One way to decrease dimensions of these elements of these integrated circuit is manufacturing of these elements in thin-film heterostructures [1][2][3][4]. An alternative approach to decrease dimensions of the elements of integrated circuits is using laser and microwave types annealing [5][6][7]. Using these types of annealing leads to generation inhomogeneous distribution of temperature. Due to Arrhenius law the inhomogeneity of the diffusion coefficient and other parameters of process. The inhomogeneity gives us possibility to decrease dimensions of elements of integrated circuits. Changing of properties of electronic materials could be obtain by using radiation processing of these materials [8,9].
In this paper we consider an operational amplifier based on field-effect heterotransistors described in Ref.
[10] (see Fig.1). We assume, that the considered element has been manufactured in heterostructure from Fig. 1. The heterostructure consist of a substrate and an epitaxial layer. The epitaxial layer includes into itself several sections manufactured by using another materials. The sections have been doped by diffusion or ion implantation to generation into these sections required type of conductivity (n or p). In this paper we analyzed redistribution of dopant during annealing of dopant and/ or radiation defects to formulate conditions for decreasing of dimensions of the considered amplifier.

Method of solution
We determine spatio-temporal distribution of concentration of dopant by solving the following boundary problem (1) with boundary and initial conditions , , , , , , 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 С is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends on properties of materials, speed of heating and cooling of heterostructure (with account Arrhenius law). Dependences of dopant diffusion coefficients could be approximated by the following function [9,11,12] , (3) where D L (x,y,z,T) is the spatial (due to existing several layers wit different properties in 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 could be integer framework the following interval γ ∈ [1,3] [9]; V (x,y,z,t) is the spatio-temporal distribution of concentration of radiation vacancies; V * is the equilibrium distribution of concentration of vacancies. Concentrational dependence of dopant diffusion coefficient have been discussed in details in [9]. It should be noted, that using diffusion type of doping did not leads to generation radiation defects and ζ 1 = ζ 2 = 0. We determine spatio-temporal distributions of concentrations of point defects have been determine by solving the following system of equations [11,12] (4) z  y  x  I   I  I   ,  ,  ,  ,  ,  ,  ,  ,  ,  , t  z  y  x  I  T  z  y  x  k  z   t  z  y  x  I  T  z  y  x  D  z   V  I  I   ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  , z  y  x  I  T  z  y  x  k I   I   ,  ,  , Here ρ =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentration of radiation interstitials; D ρ (x,y,z,T) is the diffusion coefficients of radiation interstitials and vacancies; terms V 2 (x,y,z,t) and I 2 (x,y,z,t) correspond to generation of divacancies and diinterstitials; k I,V (x,y,z,T) is the parameter of recombination of point radiation defects; k ρ,ρ (x,y,z,T) are the parameters of generation of simplest complexes of point radiation defects.
Here D Φρ (x,y,z,T) are the diffusion coefficients of complexes of radiation defects; k ρ (x,y,z,T) are the parameters of decay of complexes of radiation defects.
We determine spatio-temporal distributions of concentrations of dopant and radiation defects by using method of averaging of function corrections [13] with decreased quantity of iteration steps [14]. Framework the approach we used solutions of Eqs. (1), (4) and (6)  With the second-order approximations and higher orders approximations of concentrations of dopant and radiation defects we determine framework for standard iterative procedure [13,14]. Framework of this procedure to calculate approximations with the n-th-order one shall replace the functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), Φ I (x,y,z,t), Φ V (x,y, z,t) in the right sides of the Eqs. (1), (4) and (6) on the following sums α nρ +ρ n-1 (x,y,z, t). As an example we present equations for the second-order approximations of the considered concentrations   0   1  1  2   ,  ,  ,  ,  ,  ,  ,  ,  ,   ,  ,  ,  1 Optimization of Manufacturing of Operational Amplifier Manufactured by Using Field-effect Heterotransistor to Decrease Their Dimensions We determine average values of the second-orders approximations of the considered concentrations by using the following standard relations [13,14] . (11) Substitution of relations (8a)-(10a) into relation (11) gives us possibility to obtain relations for the required average where ,     II  IV  II  II  IV  II  IV  IV  IV  IV .  1  2  2  1  2  1   10  00  00  01  10  01  00  10  01  01  11  IV  II  IV  IV  II  IV  IV  II  IV  IV  IV 1   II  IV  II  IV  IV  IV  IV  IV  II  IV  IV  II 1   IV  IV  II  VV  IV  II  IV  IV  II  IV  IV Optimization of Manufacturing of Operational Amplifier Manufactured by Using Field-effect Heterotransistor to Decrease Their Dimensions The considered substitution gives us possibility to obtain equation for parameter α 2C . Solution of the equation depends on value of parameter γ. Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has been done by using their second-order approximations framework the method of averaged of function corrections with decreased quantity of iterative steps. The second-order approximation is usually good enough approximation to make qualitative analysis and obtain some quantitative results. Results of analytical calculation have been checked by comparison with results of numerical simulation.

Discussion
In this section we analyzed the spatio-temporal distribution of concentration of dopant in the considered heterostructure during annealing. Figs

Figure2a.
Distributions of concentration of infused dopant in heterostructure from Figs. 1 and 2 in direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate Figure2b. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 and 2 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 It should be noted, that it is necessary to anneal radiation defects after ion implantation. One could find spreading of concentration of distribution of dopant during this annealing. In the ideal case distribution of dopant achieves appropriate interfaces between materials of heterostructure during annealing of radiation defects. If dopant did not achieves any interfaces during annealing of radiation defects, it is practicably to additionally anneal the dopant. In this situation optimal value of additional annealing time of implanted dopant is smaller, than annealing time of infused dopant. At the same time ion type of doping gives us possibility to decrease mismatch-induced stress in heterostructure [21].

Figure4a.
Dependences of dimensionless optimal annealing time for doping by diffusion, which have been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter ε for a/L= 1/2 and ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ for a/L =1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter γ for a/L= 1/2 and ε = ξ = 0 Figure4b. Dependences of dimensionless optimal annealing time for doping by ion implantation, which have been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter ε for a/L= 1/2 and ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ for a/L =1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter γ for a/L= 1/2 and ε = ξ = 0

Conclusions
In this paper we model redistribution of infused and implanted dopants during manufacture of operational amplifier based on field-effect heterotransistors. Several recommendations to optimize manufacturing process of the heterotransistors have been formulated. Analytical approach to model diffusion and ion types of doping with account changing of parameters in space and at the same time in time has been introduced. At the same time the approach gives us possibility to take into account nonlinearity of doping processes.