Analytical Approach to Model and Diagnostic Distribution of Dopant in an Implanted-Heterojunction Rectifier Accounting for Mechanical Stress

We calculate spatiotemporal distributions of dopant in an implanted-heterojunction rectifier. We analyzed the influence of inhomogeneity of heterostructure on dopant distribution.The influence of radiation processing of materials of the heterostructure, which has been done during ion implantation, on properties of the heterostructure has been also analyzed. It has been shown that radiation processing of materials of heterostructure leads to a decrease in mechanical stress in heterostructure. Our calculations have been done by using analytical approach, which gives us the possibility to obtain all results without joining solutions on all interfaces of heterostructure.


Introduction
In the present time one can find intensive development of devices of solid state electronic devices.One way to this development is increasing of frequency of switching of p-n-junctions [1][2][3].To solve this problem it could be new search materials with higher charge carriers mobility [4][5][6][7].Another way to decrease switching time is increasing sharpness of p-n-junctions [8,9].It has been recently shown that manufacturing diffusing-or implanted-junction rectifiers in a heterostructure () and optimization time of annealing of dopant and/or radiation defects give us possibility to increase sharpness of p-n-junctions [10][11][12].It is known that in any  one could find mechanical stress.The strain arising due to mismatch of lattice distance in layers of  [13,14].Let us consider the following situation for simultaneous decreasing of mechanical stress and increasing of sharpness of p-n-junctions.We consider a  with two layers, which consist of a substrate () and an epitaxial layer (EL), with known type of conductivity: n or p (see Figure 1).A dopant has been implanted in the  through the EL.The dopant produces the type of conductivity of , which reverses in comparison with type of conductivity of EL.In this situation we consider such conditions of implantation, under which major portion of dopant will be implanted in the .Further annealing of radiation defects has been considered.The main aim of the present paper is to determine the conditions, under which the mechanical stress in the  will be decreased and at the same time sharpness of the p-n-junction will be increased.

Method of Solution
To solve our aims let us determine spatiotemporal distribution of dopant.We determine the distribution by solving the second Fick's law [1][2][3] ( Here ( ⃗ , ) is the spatiotemporal distribution of dopant; Ω is the atomic volume; symbols ∇ and ∇  denote volumetric and surface gradients; ⃗  is the vector with components , , and ; ∫   0 ( ⃗ , ) is the surface concentration of dopant on interface between layers of ;  ( ⃗ , ) is the chemical potential;   and   are the coefficients of volumetric and surface diffusions, respectively.The first term of (2) describes the thermal diffusion of dopant.The second term of the equation describes the surficial diffusion under influence of mechanical stress.Values of the diffusion coefficients depend on properties of materials of layers of , rates of heating and cooling of  spatiotemporal distributions of concentrations of dopant, and radiation defects.The dependence can be approximated by the following relations [2]: In the relations we used the following notations:   ( ⃗ , ) and   ( ⃗ , ) are the spatial (due to inhomogeneity of heterostructure and radiation damage of materials) and temperature (due to Arrhenius law) dependence of dopant diffusion coefficients;  is the annealing temperature; ( ⃗ , ) is the limit of solubility of dopant; parameter  depends on properties of materials and could be integer in the following interval  ∈ [1, 3] [17]; ( ⃗ , ) is the spatiotemporal distribution of vacancies;  * is the equilibrium distribution of vacancies.Dependence of dopant diffusion coefficients on concentration of dopant is discussed in detail in [17].Dependence of dopant diffusion coefficients on concentrations of vacancies is generalization of analogous relation in [18].The generalization accounting for the generation of di-vacancies.The generation accounting by quadratic terms of the approximation [19].We determine spatiotemporal distributions of concentrations of radiation defects by solving the following system of equations [19,20]: with boundary and initial conditions Here  is  or ; ( ⃗ , ) is the spatiotemporal distribution of interstitials;  * is the equilibrium distribution of interstitials;   ( ⃗ , ),   ( ⃗ , ) are the volumetric and surface diffusion coefficients of interstitials and vacancies, respectively;  , ( ⃗ , ),  , ( ⃗ , ), and  , ( ⃗ , ) are the parameters of recombination of point defects (first of them) and generation of their complexes, respectively.The first term of (4) describes the thermal diffusion of point radiation defects.The second term of the equations describes the surficial diffusion under influence of mechanical stress.Terms with  2 ( ⃗ , ) and  2 ( ⃗ , ) correspond to generation of divacancies and di-interstitials (see, e.g., [19] and appropriate references in the work).The last terms of the equations correspond to recombination of point defects.
We determined spatiotemporal distributions of concentrations of divacancies Φ  ( ⃗ , ) and di-interstitials Φ  ( ⃗ , ) as solution of the following system of equations [19][20][21]: with boundary and initial conditions Here  Φ ( ⃗ , ) and  Φ ( ⃗ , ) are the diffusion coefficients of volumetric and surface diffusions of divacancies and di-interstitials, respectively;   ( ⃗ , ) and   ( ⃗ , ) are the parameters of decay of complexes of defects (divacancies and di-interstitials).The first term of (6) describes the thermal diffusion of complexes of point radiation defects.The second term of the equations describes the surficial diffusion under influence of mechanical stress.The third terms of the equations correspond to decay of complexes of radiation defects.The last terms of (6) correspond to generation of new complexes of radiation defects.
Chemical potential in (1), (4), and (6) could be determined by the following relation [13,14]: where First of all let us estimate components of displacement vector.To make this procedure we used method of averaging of function corrections [11,12,21,23].Let us previously transform the equations of the system (11) to the integrodifferential form.The transformation is based on integration of the both sides of (11) on time .After that we determine both integration constants by using both appropriate conditions.Furthermore we made analogous procedure with Physics Research International integration on coordinate .The main aim of this transformation is to remove derivative on coordinate , because many parameters ((), (), (), . ..) depend on coordinate .Differentiation of step-wise functions leads to unphysical results.At the same time we were forced to make integration of both sides of (11) on time .It is necessary to calculate an approximate solution of (11).The necessity is more clear for calculation of concentrations of dopant and radiation defects.The obtained integrodifferential equations are bulky and as being due to have been removed into Appendix.Framework the approach we replace components of displacement vector   ( ⃗ , ) in the right sides of integro-differential equations on their not yet known average values  1 , where  = , , .The average values we determined as where  =       ,  =   .
The replacement gives us the possibility to obtain the first-order approximation of the components of displacement vector as follows: Here ⃗  1 is the vector with components , , and ;  = 3 √ /Θ 2  0 ;  0 is the average value of Young modulus.Substitution of the first-order approximations into relation (13) gives us the possibility to determine average values  1 .Simple mathematical transformations give us the average values in the following final form: where Approximation of components of displacement vector of the second-order  2 ( ⃗ , ) and approximation of the components with higher order  (i.e.,   ( ⃗ , )) could be calculated by replacement of the components in the following sums   +  −1 ( ⃗ , ), where Results of the replacement for  = 2 and results of calculations of the parameters  2 are presented in the Appendix, because the parameters are bulky.
Furthermore we determine spatiotemporal distributions of concentrations of point radiation defects.To calculate the distributions we also used method of averaging of function corrections.In the framework of the approach we replace the concentrations ( ⃗ , ) on their average values  1 .After the replacement we obtained equations for the first-order approximations of concentrations of point radiation defects Integration on time of the left and right sides of the above equations gives us the possibility to obtain the functions  1 ( ⃗ , ),  = ,  in the final form Average values of the first-order approximations of concentrations of point radiation defects could be calculated by standard relation where  = , .Substitution of ( 16) into the relation (19) gives us the possibility to obtain the following equations for the average values  1 : where

Solution of the system of equations is as follows:
Approximations of the second and higher orders of concentrations of point radiation defects within the framework method of averaging of function corrections could be obtained by using standard iteration procedure, that is, by replacing of the functions ( ⃗ , ) on the following sums   +  −1 ( ⃗ , ), where  is the order of approximation.In the present paper we consider only the second-order approximations of the functions ( ⃗ , ).The above substitution and integration on time of the left and right sides give us the possibility to obtain the second-order approximations of concentrations of defects  2 ( ⃗ , ) in the following form: Average values of the second-order approximation and approximations of the higher orders of concentrations of point radiation defects   could be calculated by using standard relation [11,12,21,23] Substitution of the first-and second-order approximations of the functions ( ⃗ , ) into (23) gives us the possibility to obtain equations for the average values  2 Solution of the above system of equations could be written as where System of (6) we solve by using method of averaging of function corrections in the same form as for system of (4).Within the framework of the approach we replace the functions Φ  ( ⃗ , ) in the right sides of (6) on their average values  1Φ .After the replacement we obtain the equations for the first-order approximations of concentrations of simplest complexes of radiation defects Φ 1 ( ⃗ , ) in the following form: Integration of left and right parts of the equations on time gives us the possibility to obtain the first-order approximations of concentrations of divacancies and di-interstitials in the final form Average values  1Φ and of the first-order approximations of concentrations of divacancies and di-interstitials can be calculated by standard relation, which analogous to relations ( 16) and (23).By calculation we obtain the following result: where The second-order approximations of concentrations of divacancies and di-interstitials could be calculated by standard iteration procedure of method of averaging of function corrections.The approximations could be written as Average values  2Φ can be determined by standard relation.The relation is similar with relations ( 16) and (23).

Results of calculation of the average values could be written as
where We calculate the distribution of dopant concentration by the same approach, which was used to calculate distributions of concentrations of defects.The first-order approximation of dopant concentration could be written as Average value  1 of the first-order approximation of dopant concentration  1 ( ⃗ , ) can be calculated by using standard relation, which is analogous to relations ( 13) and (19).Substituting of the function  1 ( ⃗ , ) into appropriate relation we obtain equation for the average value in the final form  1 where The solution of the equation depends on value of parameter .
We determine the second-order approximation of dopant concentration  2 (, , , ) within the framework of the same approach, which has been used for calculation of the secondorder approximations of concentrations of radiation defects.
After simple mathematical transformation we obtain the function  2 (, , , ) in the final form We determine the parameter  2 by standard relation, which is analogous to relations ( 16) and (23).Substitution of the second-order approximation of dopant concentration  2 ( ⃗ , ) into appropriate relation and simple mathematical transformations leads to the following equation for the parameter  2 : where Further we used the obtained approximations of components of displacement vector, radiation defects, and dopant concentrations.The obtained relations give us the possibility to analyze demonstratively relaxations of dopant and radiation defects concentrations and components of displacement vector.Using numerical approaches gives us the possibility to amend the obtained results.In this situation we used both analytical and numerical approaches to solve (1), ( 4), (6), and (11).

Discussion
In this paragraph we analyzed dynamics of redistribution of implanted dopant in a heterostructure accounting for dynamics of redistribution of point radiation defects, redistribution of simplest complexes of point radiation defects, and relaxation of mechanical stress in the heterostructure.Some calculated dopant distributions are presented in Figure 2. Some of the distributions have been calculated for homogenous sample, and other distributions have been calculated for heterostructure.The figure shows that inhomogeneity of the heterostructure leads to increasing sharpness of p-njunctions and homogeneity of dopant distribution in area, which was enriched by the dopant.Both effects will be increased with increasing difference between values of diffusion coefficients of dopant in layers of .Implantation of ion of dopant leads to radiation processing of materials of .
The radiation processing leads to a decrease in the mechanical stress in the .Dependences of component of displacement vector   on coordinate  is presented in Figure 3.The coordinate  is perpendicular to interface between layers of H (see Figure 1).Figure 3 shows that the value of component of displacement vector   decreases after radiation processing (such as ion implantation).Probably, reason of the decreasing is dislodging of several atoms from matrix of materials of .The dislodging leads to generation of radiation vacancies.Farther one can obtain compacting of area of  under influence of mechanical stress.Ulterior annealing leads to balancing of concentration of interstitials in .

Conclusion
In this paper we introduce an approach to increase sharpness of implanted-junction rectifier in a heterostructure and at the same time to increase homogeneity of dopant distribution in area, enriched by the dopant.At the same time we introduce an approach to decrease value of mechanical stress in the heterostructure due to radiation processing of materials during ion implantation.As accompanying results it could be considered as an analytical approach to calculate spatiotemporal distributions of dopant and radiation defects (point defects and their simplest complexes) concentrations, as well as relaxation of mechanical stress in heterostructure, which was taken into account during calculation of distributions concentrations.

Appendix
Integrodifferential form of equations for components of displacement vector ⃗ ( ⃗ , ) is as follows: Calculation of parameters  2 leads to the following results: (A.5)

Figure 3 :
Figure 3: Normalized dependence of displacement vector   on coordinate  for epitaxial layer before (curve 1) and after (curve 2) radiation processing.
() is the Young modulus, which depends on kind of layer of  (EL or );   is the stress tensor;      = (1/ 2)(   /  +    /  ) is the deformation tensor;   ,   are  0 = (  −  EL )/ EL is the mismatch parameter;   ,  EL are lattice distances of  and EL; () is the modulus of uniform compression; () is the coefficient of thermal expansion;   is the equilibrium