An Analytical Approach for Prognosis of Mass Transport during Manufacturing of a Limiting Amplifier Circuit. On Increasing of Integration Rate if Elements of the Amplifier

Introduction Abstract: In this paper we introduce an analytical approach for prognosis of mass transport during manufacturing of a limiting amplifier circuit. Based on this approach we obtain conditions to increase density of elements of this circuit, manufactured by diffusion or ion implantation with optimized annealing time of dopant and/or radiation defects. The above analytical approach gives a possibility to take into account nonlinearity of the mass transport, dependences of parameters of the transport on spatial coordinate and time.


Introduction
Abstract: In this paper we introduce an analytical approach for prognosis of mass transport during manufacturing of a limiting amplifier circuit. Based on this approach we obtain conditions to increase density of elements of this circuit, manufactured by diffusion or ion implantation with optimized annealing time of dopant and/or radiation defects. The above analytical approach gives a possibility to take into account nonlinearity of the mass transport, dependences of parameters of the transport on spatial coordinate and time.
Keywords: Partial differential equation; analytical approach for solution; taking into account nonlinearity of the mass transport, dependences of parameters of the transport on spatial coordinate and time; prognosis of mass transport; manufacturing of a limiting amplifier circuit; increasing of density of elements of this circuit.
An actual and intensively solving problems of solid state electronics is increasing of integration rate of elements of integrated circuits (p-n-junctions, their systems et al) [1][2][3][4][5][6][7][8]. Increasing of the integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are using several approaches.
They are widely using laser and microwave types of annealing of infused dopants. These types of annealing are also widely using for annealing of radiation defects, generated during ion implantation [9][10][11][12][13][14][15][16][17]. Using the approaches gives a possibility to increase integration rate of elements of integrated circuits through inhomogeneity of technological parameters due to generating inhomogenous distribution of temperature. In this situation one can obtain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law [1][2][3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is doping of heterostructure by diffusion or ion implantation [1][2][3]. However in this case optimization of dopant and/or radiation defects is required [18].
In this paper we consider a heterostructure. The heterostructure consist of a substrate and several epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further we consider doping of these sections by diffusion or ion implantation. The doping gives a possibility to manufacture field-effect transistors framework a limiting amplifier circuit so as it is shown on Figs. 1. The manufacturing gives a possibility to increase density of elements of the integrator circuit [4]. After the considered doping dopant and/or radiation defects should be annealed. Framework the paper we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. We introduce an approach to decrease dimensions of the element. However it is necessary to complicate technological process.

Method of Solution
In this section we determine spatio-temporal distributions of concentrations of infused and implanted dopants. To determine these distributions we calculate appropriate solutions of the second Fick's law [1,3,18].
Boundary and initial conditions for the equations are The function C(x,y,z,t) describes the spatiotemporal distribution of concentration of dopant; T is the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient could be changed with changing materials of heterostructure, with changing temperature of materials (including annealing), with changing concentrations of dopant and radiation defects. We approximate dependences of dopant diffusion coefficient on parameters by the following relation with account results in Refs. [20][21][22]. Here the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due to Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes the limit of solubility of dopant. Parameter   [1][2][3] describes average quantity of charged defects interacted with atom of dopant [20]. The function V (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation vacancies.
Parameter V * describes the equilibrium distribution of concentration of vacancies.
The considered concentrational dependence of dopant diffusion coefficient has been described in details in [20]. It should be noted, that using diffusion type of doping did not generation radiation defects. In this situation 1= 2= 0. We determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations [21][22].
Here  =I,V. The function I (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation interstitials; D  (x,y,z,T) are the diffusion coefficients of point radiation defects; terms V 2 (x,y,z,t) and I 2 (x,y,z,t) correspond to generation divacancies and diinterstitials; Further we determine distributions in space and time of concentrations of divacancies V(x,y,z,t) and diinterstitials I(x,y,z,t) by solving the following system of equations [21][22].
Boundary and initial conditions for these equations are Here D  (x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects; kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes.
We determine solutions of Eqs. (8) t  z  y  x   I  I   I  I  I  I   I   ,  ,  ,  ,  ,  ,   ,  ,  ,  ,  ,  ,  1   Now we used the series (11) into Eqs. (6) and appropriate boundary and initial conditions. The using gives the possibility to obtain equations for initial-order approximations of concentrations of complexes of defects  0 (x,y,z,t), corrections for them  i (x,y,z,t) (for them i 1) and boundary and initial conditions for them. We remove equations and conditions to the Appendix. Solutions of the equations have been calculated by standard approaches [24][25] and presented in the Appendix.
Using the relation into Eq.(1) and conditions (2) leads to obtaining equations for the functions Cij(x,y,z,t) (i

Conclusions
In this paper we introduce an approach to increase integration rate of element of an limiting amplifier circuit. The approach gives us possibility to decrease area of the elements with smaller increasing of the element's thickness.