About influence of buffer porous layers between epitaxial layers of heterostructure on distributions of concentrations of dopants in heterobipolar transistors

In this paper we introduce an approach to manufacture a heterobipolar transistors. Framework this approach we consider doping by diffusion or by ion implantation of required parts of a heterostructure with special configuration and optimization of annealing of dopant and/or radiation defects. In this case one have possibility to manufacture bipolar transistors, which include into itself p-n-junctions with higher sharpness and smaller dimensions. We also consider influence of presents of buffer porous layers between epitaxial layers of heterostructure on distributions of concentrations of dopants in the considered transistors. An approach to decrease value of mismatch-induced stress has been considered.


INTRODUCTION
Presently the performance of devices of solid-state electronics intensively increased (p-njunctions, field-effect and bipolar transistors, ...) [1][2][3][4][5][6].One way to increase performance is using materials with higher values of charge carriers mobility [7][8][9][10].Another way to increase the performance is elaboration new and optimization of existing technological processes.Framework the approach we introduce an approach to manufacture a bipolar heterotransistor.The approach gives us possibility to decrease switching time of p-njunctions, which included in the bipolar heterotransistor.At the same time using this approach gives us possibility to manufacture the transistors as more compact.At the same time we consider possibility to decrease influence of mismatch-induced stress on distributions of concentrations of dopants in considered heterostructure.In this paper we consider a heterostructure, which consist of a substrate and three epitaxial layers (see Fig. 1).Between these epitaxial layers one have been manufactured buffer porous layers.These epitaxial layers have been doped by diffusion or ion implantation after finishing of their epitaxial growth.After finishing of manufacturing and doping of the heterostructure annealing of dopant and/or radiation defects have been done.Main aim of the present paper is analysis of dynamic of redistribution of dopant and radiation defects with account modification of porosity and relaxation of mechanical stress.

METHOD OF SOLUTION
To solve our aim we determine and analyzed spatio-temporal distribution of concentration of dopant in the considered heterostructure.We determine the distribution by solving the second Fick's law in the following form [1,[11][12][13][14] (1) with boundary and initial conditions x y z t C x y W t dW , , , , , , Here C(x, y, z, t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic volume of dopant; ∇ s is the symbol of surficial gradient; is the surficial concentration of dopant on interface between layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface between layers of heterostructure); μ 1 (x, y, z, t) and μ 2 (x, y, z, t) are the chemical potential due to the presence of mismatch-induced stress and porosity of material; D and D S are the coefficients of volumetric and surficial diffusions.Values of dopant diffusions coefficients depends on properties of materials of heterostructure, speed of heating and cooling of materials during annealing and spatio-temporal distribution of concentration of dopant.Dependences of dopant diffusions coefficients on parameters could be approximated by the following relations [15][16][17] , Here D L (x, y, z, T) and D LS (x, y, z, T) are the spatial (due to accounting all layers of heterostruicture) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; 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] [15]; V (x, y, z, t) is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibrium distribution of vacancies.Concentrational dependence of dopant diffusion coefficient has been described in details in [3].Spatiotemporal distributions of concentration of point radiation defects have been determined by solving the following system of equations [11-14, 16, 17] C x y z t dz , , , with boundary and initial conditions , , , , , , .
Here I (x, y, z, t) is the spatio-temporal distribution of concentration of radiation interstitials; I* is the equilibrium distribution of interstitials; D I (x, y, z, T), D V (x, y, z, T), D IS (x, y, z, T), D VS (x, y, z, T) are the coefficients of volumetric and surficial diffusions of interstitials and vacancies, respectively; terms V 2 (x, y, z, t) and I 2 (x, y, z, t) correspond to generation of , , , , , , x y z t z , , , , , , divacancies and diinterstitials, respectively (see, for example, [17] and appropriate references in this book); k I, V (x, y, z, T ), k I, I (x, y, z, T) and k V, V (x, y, z, T) are the parameters of recombination of point radiation defects and generation of their complexes; k is the Boltzmann constant; ω = a 3 , a is the interatomic distance;  is the specific surface energy.To account porosity of buffer layers we assume, that porous are approximately cylindrical with average values and z 1 before annealing [18].With time small pores decomposing on vacancies.The vacancies absorbing by larger pores [14].With time large pores became larger due to absorbing the vacancies and became more spherical [14].Distribution of concentration of vacancies in heterostructure, existing due to porosity, could be determined by summing on all pores, i.e.
Here α, β and χ are the average distances between centers of pores in directions x, y and z; l, m and n are the quantity of pores inappropriate directions.Spatio-temporal distributions of divacancies Φ V (x, y, z, t) and diinterstitials Φ I (x, y, z, t) could be determined by solving the following system of equations [16,17] (5) with boundary and initial conditions , , , , , , Here D ΦI (x, y, z, T), D ΦV (x, y, z, T), D ΦIS (x, y, z, T) and D ΦVS (x, y, z, T) are the coefficients of volumetric and surficial diffusions of complexes of radiation defects; k I (x, y, z, T) and k V (x, y, z, T) are the parameters of decay of complexes of radiation defects.
Chemical potential μ 1 in Eq.( 1) could be determine by the following relation [11] where E(z) is the Young modulus, σ ij is the stress tensor; is the deformation tensor; u i , u j are the components u x (x, y, z, t), u y (x, y, z, t) and u z (x, y, z, t) of the displacement vector x i , x j are the coordinate x, y, z.The Eq. ( 3) could be transform to the following form where σ is Poisson coefficient; ε 0 = (a sa EL )/a EL is the mismatch parameter; a s , a EL are lattice distances of the substrate and the epitaxial layer; K is the modulus of uniform compression; β is the coefficient of thermal expansion; T r is the equilibrium temperature, which coincide (for our case) with room temperature.Components of displacement vector could be obtained by solution of the following equations [27] ∂ ∂ ( ) x y z t y , , , 0  .
Conditions for the system of Eq. ( 8) could be written in the form ; ; ; ;

;
We determine spatio-temporal distributions of concentrations of dopant and radiation defects by solving the Eqs.( 1), ( 3) and ( 5) framework standard method of averaging of function corrections [19].Previously we transform the Eqs.( 1), ( 3) and ( 5) to the following form with account initial distributions of the considered concentrations , , , . .
Farther we replace concentrations of dopant and radiation defects in right sides of Eqs.(1a), (3a) and (5a) on their not yet known average values α 1ρ .In this situation we obtain equations for the first-order approximations of the required concentrations in the following form .
Integration of the left and right sides of the Eqs.(1b), (3b) and (5b) on time gives us possibility to obtain relations for above approximation in the final form , , , , , , , , , , , , , , , , , ,    We determine average values of the first-order approximations of concentrations of dopant and radiation defects by the following standard relation [19] . ( Substitution of the relations (1c), (3c) and (5c) into relation (9) gives us possibility to obtain required average values in the following form , where x y z T I x y z t V x y z t dz dy dx dt a S , , , , x y z where .
We determine approximations of the second and higher orders of concentrations of dopant and radiation defects framework standard iterative procedure of method of averaging of function corrections [19].Framework this procedure to determine approximations of the n-th order of concentrations of dopant and radiation defects we replace the required concentrations in the Eqs.(1c), (3c), (5c) on the following sum α nρ + ρ n-1 (x, y, z, t).The replacement leads to the following transformation of the appropriate equations (1d)

y z t I x y W t dW x
x y z t x , , , , , , , , , , , , .
Average values of the second-order approximations of required approximations by using the following standard relation [19] (10) Substitution of the relations (1e), (3e), (5e) into relation (10) gives us possibility to obtain relations for required average values α 2ρ  ) ( ) Farther we determine solutions of Eqs.(8), i.e. components of displacement vector.To determine the first-order approximations of the considered components framework method of averaging of function corrections we replace the required functions in the right sides of the equations by their not yet known average values α i .The substitution leads to the following result , , .
Integration of the left and the right sides of the above relations on time t leads to the following result ( x y z x y z 2 0 4 1 3 ρ ( ) ( )  ( ) ( ) Approximations of the second and higher orders of components of displacement vector could be determined by using standard replacement of the required compo-nents on the following sums α i + u i (x, y, z, t) [19].The replacement leads to the following result , , , 3 1 , , , , , , Integration of the left and right sides of the above relations on time t leads to the following result , , , , , , , , , , , , , , , , , , , , , , 1 x y ,,, , , , Framework this paper we determine concentration of dopant, concentrations of radiation defects and components of displacement vector by using the second-order approximation framework method of averaging of function corrections.This approximation is usually enough good approximation to make qualitative analysis and to obtain some quantitative results.All obtained results have been checked by comparison with results of numerical simulations.

DISCUSSION
In this section we analyzed dynamics of redistributions of dopant and radiation defects during annealing and under influence of mismatch-induced stress and modification of porosity.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-junction gives us possibility to decrease switching time.The second effect leads to decreasing local heating of materials during functioning of p-n-junction or decreasing of dimensions of the p-n-junction for fixed maximal value of local overheat.However framework this approach of manufacturing of bipolar transistor it is necessary to optimize annealing of dopant and/or radiation defects.Reason of this optimization is following.If annealing time is small, the dopant did not achieve any interfaces between materials of heterostructure.In this situation one cannot find any modifications of diffusion coefficient in all parts of heterostructure.Curve 2 is the dependence of dimensionless optimal an-nealing 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 ξ = 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 distribution of concentration of dopant.If annealing time is large, distribution of concentration of dopant is too homogenous.We optimize annealing time framework recently introduces approach [20][21][22][23][24][25][26][27].Framework this criterion we approximate real distribution of concentration of dopant by step-wise function (see Figs. 4 and 5).Farther we determine optimal values of annealing time by minimization of the following mean-squared error (15) where ψ (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.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 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.It should be noted, that in the case, when value of dopant diffusion coefficient in the average epitaxial layer is larger, than dopant diffusion coefficient into another layers, it is practicably to consider higher level of doping of the average epitaxial layer in comparison with another layers.Reason of this doping is following.Dopants after infusion or ion 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 ξ = 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 implantation in the first and the second epitaxial layers will be accelerated after achievements nearest interface.In this situation one can find at least partial compensation of dopant in the average epitaxial layer.If value of dopant diffusion coefficient in the average epitaxial layer is smaller, than in other epitaxial layer, it is practicably to use higher level of doping of another epitaxial layers.It should be noted, that using ion implantation leads to generation of radiation defects in doped materials of heterostructure.Presents of radiation defects in doped materials leads to diffusion stimulated by radiation.In this situation we obtain increasing of homogeneity of distribution of concentration of dopant.Farther we analyzed influence of relaxation of mechanical stress on distribution of dopant 134 About influence of buffer porous layers between epitaxial layers of heterostructure on distributions of concentrations of dopants in heterobipolar transistors in doped areas of heterostructure.Under following condition ε 0 < 0 one can find compression of distribution of concentration of dopant near interface between materials of heterostructure.Contrary (at ε 0 > 0) one can find spreading of distribution of concentration of dopant in this area.This changing of distribution of concentration of dopant could be at least partially compensated by using laser annealing [24].This type of annealing gives us possibility to accelerate diffusion of dopant and another processes in annealed area due to inhomogenous distribution of temperature and Arrhenius law.Accounting relaxation of mismatch-induced stress in heterostructure could leads to changing of optimal values of annealing time.At the same time modification of porosity gives us possibility to decrease value of mechanical stress.On the one hand mismatch-induced stress could be used to increase density of elements of integrated circuits.On the other hand could leads to generation dislocations of the discrepancy.Figs. 8 and 9 show distributions of concentration of vacancies in porous materials and component of displacement vector, which is perpendicular to interface between layers of heterostructure.

CONCLUSION
In this paper we introduce an approach to manufacture heterobipolar transistors with more sharper p-n-junctions.Framework the approach of manufacturing of transistors we introduce an approach to decrease mismatch-induced stress.

Figure 1 :
Figure 1: Heterostructure with a substrate, three epitaxial layers and three buffer porous layers

Figure 2 :Figure 3 :
Figure2: Distributions of concentration of infused dopant in heterostructure from Figure1in 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

Figure 4 :Figure 5 :Figure 6 :
Figure 4: Spatial distributions of dopant in heterostructure after dopant infusion.Curve 1 is idealized distribution of dopant.Curves 2-4 are real distributions of dopant for different values of annealing time.Increasing of number of curve corresponds to increasing of annealing time

Figure 7 :
Figure 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 ξ = γ = 0 for equal to each other values of dopant

Figure 8 :Figure 9 :
Figure 8: Normalized dependences of component u z of displacement vector on coordinate z for nonporous (curve 1) and porous (curve 2) epitaxial layers