About Manufacturing More Compact Bipolar Heterotransistors

In the present time one of the actual questions of is increasing of density of elements of integrated circuits (p-n-junctions, transistors, etc.) [1-6]. It is also attracted an interest manufacturing more thin integrated circuits [1,5]. One of approaches to solve the questions is increasing of sharpness of diffusive-junction and implanted-junction rectifiers (both single rectifiers and rectifiers in their systems: transistors, thyristors, etc.), using epitaxial layers with smaller thickness in epitaxial p-njunctions [7-9]. To increase sharpness of diffusive-junction and implanted-junction rectifiers are could be used inhomogenous distribution of temperature during laser and microwave annealing [10-14], radiation processing [15], native defects in materials (such as dislocations of discrepancy in heterostructures) [16]. In this paper based on recently introduce approach [17-23] we consider an alternative approach to increase sharpness of diffusive-junction and implanted-junction rectifiers. Framework the approach we consider heterostructure, which consist of a substrate and an epitaxial layer (see Fig. 1). The epitaxial layer consists of some materials. Some dopants have been infused into the epitaxial layer by the way, which has been presented in the Fig. 1. Further we consider annealing of the dopants with optimal continuance [17-23] to achievement nearest interfaces of the heterostructure by the dopant. Main aim of the present paper is analysis of dynamics of redistribution of dopant in the heterostructure and estimation of the optimal annealing time.


Introduction
In the present time one of the actual questions of is increasing of density of elements of integrated circuits (p-n-junctions, transistors, etc.) [1][2][3][4][5][6]. It is also attracted an interest manufacturing more thin integrated circuits [1,5]. One of approaches to solve the questions is increasing of sharpness of diffusive-junction and implanted-junction rectifiers (both single rectifiers and rectifiers in their systems: transistors, thyristors, etc.), using epitaxial layers with smaller thickness in epitaxial p-n-junctions [7][8][9]. To increase sharpness of diffusive-junction and implanted-junction rectifiers are could be used inhomogenous distribution of temperature during laser and microwave annealing [10][11][12][13][14], radiation processing [15], native defects in materials (such as dislocations of discrepancy in heterostructures) [16]. In this paper based on recently introduce approach [17][18][19][20][21][22][23] we consider an alternative approach to increase sharpness of diffusive-junction and implanted-junction rectifiers. Framework the approach we consider heterostructure, which consist of a substrate and an epitaxial layer (see Fig. 1). The epitaxial layer consists of some materials. Some dopants have been infused into the epitaxial layer by the way, which has been presented in the Fig. 1. Further we consider annealing of the dopants with optimal continuance [17][18][19][20][21][22][23] to achievement nearest interfaces of the heterostructure by the dopant. Main aim of the present paper is analysis of dynamics of redistribution of dopant in the heterostructure and estimation of the optimal annealing time.

Method of Solution
Let us described redistribution of dopant during annealing in presented on Fig. 1 heterostructure by the second Fick's low in the following form [3,24] where C (x,y,z,t) is the spatio-temporal distribution of dopant; DC is the dopant diffusion coefficient. Value of the dopant diffusion coefficient depends on properties of materials of heterostructure, speed of heating and cooling of heterostructure. The dependences could be approximated by the following function [24] ( where DL(x,y,z,T) is the dopant diffusion coefficient for low level of doping; P(x,y,z,T) is the limit of solubility of dopant; parameter γ depends on properties of materials of heterostructure and could be integer in the following interval γ ∈ [1,3] [24]. Spatio-temporal distribution of temperature, which leads to variation of dopant diffusion coefficient, we describe by the second low of Fourier [25] ( ) ( where c (T) is the heat capacitance of heterostructure. Temperature distribution of the capacitance could be approximated by the following relation: c (T) = cass{1-ω⋅exp[-ζ ⋅T(x,y,z,t)/Td]} (see, for example, [25]). In the most interest interval of temperature, when current temperature T(x,y,z,t) is larger, than Debye temperature Td, one can consider the following limiting case c (T) ≈ cass [25]; p(x,y,z,t) is the volumetric density of power, which the considered heterostructure absorbing during annealing of dopant; λ (x,y,z,T) is the heat conduction coefficient. Temperature dependence of the heat conduction coefficient could be approximated by the following polynomial function: λ (x,y,z,T) = λass(x,y,z)[1+µ⋅Tdϕ/Tϕ(x,y,z,t)].
Equations (1) and (3) should be complemented by the boundary and initial conditions where Tr is the equilibrium distribution of temperature, which coincides with it's equilibrium distribution.
First of all we estimate spatiotemporal distribution of temperature. In the common case exact solution of Eq. (3) is unknown. To obtain an approximate solution we transform the Eq. (3) to the following integro-differential form , , We solved the Eq.(5) by method of averaging of function corrections [26] with decreased quantity of iteration steps [27]. To decreased quantity of iteration steps with fixed value of exactness of solution we used exacter initial-order approximation of spatiotemporal distribution of temperature. We used solution of the Eq.(3) with averaged value of thermal diffusivity α0ass as the exacter initial-order approximation of spatiotemporal distribution of temperature. The solution could be obtained by standard approaches [28,29] and could be approximated by the following sum where where sn(x)=sin(π n x/L). The second-order approximation of temperature T2(x,t) framework modified method of averaged of function correction could be determined by using standard procedure (see, for example, [26]). Framework the approach we replace the function T(x,t) in the right side of Eq.(5) on the following sum: T(x,y,z,t)→ α2T +T1(x,y,z,t). The replacement leads to the following result , , We determine the parameter α2T by the following relation [26] Θ (7) and (8) into the relation (9) gives us possibility to obtain the following equation for the parameter α2T 2  1  2  1  1   , , , , , , , , ,

z T x y z t d z d y d x d t L y L z T x y z t d z d y L
, , , 2  (   1  2  2  1  1  0  0   , , ,  , , , , , ,  1  2  2  2  1  0   , , ,  , ,  3 , , , 2 To make qualitative analysis and to obtain some quantitative results it is usually enough to use the second-order approximation framework the method of averaging of function corrections (see, for example, [26,27]). In this situation we used only the second-order approximations of temperature and concentration of dopant. Farther we transform the Eq.(1) into the integro-differential form , , , 1 , , , To determine analytical solution of the Eq.(11) we used method of averaging of function corrections [26] with decreased quantity of iteration steps [27]. Framework the approach to determine the first-order approximation of dopant concentration we replace the function C (x,y,z,t) in the right side of the Eq.(11) on the solution of the Eq. (1) with averaged value of diffusion coefficient D0L. The solution could be written as where  The second-order approximation of dopant concentration could be calculated by standard iteration procedure [26,27], i.e. by replacement of the function C(x,y,z,t) in the right side of the Eq.(11) on the following sum: C(x,y,z,t)→α2C +C1(x,y,z,t). The replacement gives us the following result Parameter α2C could be determined by using the relation (9) [26,27]. Substitution of the relation (14) into the relation (9) gives us possibility to obtain the equation for calculation of the parameter α2C in the following form , , , , , Farther we consider some examples of values of parameter α2C for different vales of parameter γ.

Discussion
In this section based on relations, calculated in the previous section, we analyzed dynamics of temperature and redistribution of concentration of dopant in presented on the Fig. 1 heterostructure. The Fig. 2 shows typical distributions in the neighborhood of an interface between layers of heterostructure under the conditions, when source of dopant is situated in the left layer and the dopant diffusion coefficient of the left layer is larger, than the dopant diffusion coefficient of the right layer. The Fig. 2 also shows, that interface between layers of heterostructure gives us possibility to increase sharpness of p-n-junction and at the same time to increase homogeneity of dopant distribution in doped area. Increasing of sharpness of p-n-junction gives us possibility to decrease switching time of the junction. Increasing of homogeneity of dopant distribution gives us possibility to decrease local overheat during switching of the devices or to decrease dimensions of p-n-junctions and transistors with fixed value of local overheat. Similar constructions of transistors have been described in [6,30]. However, in these references it has been considered manufacturing of transistors in homogenous materials. Manufacturing transistors in heterostructures by diffusion and implantation of ions of dopant gives us possibility to obtain more compact transistors. Another way to obtain field-effect and bipolar transistors is epitaxial growth [31,32]. However it is more easy to use diffusion and implantation of ions of dopant for local doping of materials for manufacturing elements of integrated circuits.  It should be noted, that after annealing with small continuance dopant did not achieved interface between layers of heterostructure and distribution of concentration of dopant is not enough homogenous (see Fig. 3). Increasing of annealing time leads to increasing of homogeneity of dopant distribution. At large values of annealing time one can obtain too homogenous dopant distribution (see Fig. 3). In this situation one shall to determine compromise annealing time. We determine the compromise annealing time framework recently introduced criterion [17][18][19][20][21][22][23]. Framework the criterion we approximate real distribution of dopant by step-wise function (see Fig. 3). Farther we minimized the following mean-squared error where ψ (x,y,z) is the step-wise approximation function, which presented on Fig. 3 as curve 1. Dependences of optimal annealing time, which has been obtain by minimization by the mean-squared error (15), on several parameters are presented on Fig. 4

Conclusion
In the present paper we introduce an approach to manufacture more compact bipolar heterotransistors. This approach based on using of inhomogeneity of heterostructure and optimization of annealing time.