1.
Introduction

In the present time, the influence of different types of radiation processing on semiconductors is being intensively analyzed^[1–3]. One actual reason of the processing is doping of semiconductor materials to manufacture p–n junctions, transistors and other electronic devices in different technological ways. Another reason for analysis of influence of radiation processing on semiconductor materials is random impacts of radiation particles from cosmos irradiation or from other sources. Based on the analysis several recommendations to increase radiation resistance have been formulated^[4–6]. One of them is choosing of materials with higher radiation resistance. Another one is using overlayers to decrease influence of undesirable irradiation. The third one is using annealing of radiation defects to decrease their quantity.



In this paper we analyze redistribution of point radiation defects, interaction between themselves and redistribution of their simplest complexes in multilayer structures. The analysis shows nonmonotonicity of distributions of concentrations of radiation defects, which recently was found experimentally^[7]. To make the analysis we improve the model of analysis in comparison with previous references and used a recently introduced analytical approach to analyze these processes without crosslinking of solutions in layers of these structures.

2.
Method of solution

We determine spatio-temporal distributions of concentrations of point defects by solution of the following system of equations^{[8, 9]}.











Here I (r, z, t) and V (r, z, t) are the spatio-temporal distributions of concentrations of interstitials and vacancies. The first terms of these equations describe diffusion of radiation defects. One can find dependence of diffusion coefficients D_ρ (in diffusion terms of Eq. (1)) of defects on coordinate z due to presence of several layers framework of the considered multilayer structure. The second and the third terms of Eq. (1) describe generation of the simplest complexes of radiation defects and recombination of point radiation defects, respectively. The Eq. (1) should be complemented by boundary and initial conditions. We consider such boundary and initial conditions, which corresponds to absence of flow of defects through external boundary of the considered multilayer structure and distributions of concentrations of defects before starting of their stabilization. These conditions could be written as:











To simplify solution of Eq. (1) we transform these differential equations to the following integral form:










Now we will solve Eq. (1a) by method of averaging of function corrections^[10] with decreasing quantity of iteration steps^[11]. For the framework of the approach we used solutions of linear Eq. (1) with averaged diffusion coefficients D_0IV as initial-order approximations of solutions of Eq. (1a). These initial-order approximations could be solved by standard Fourier approach^{[12, 13]} and could be written as:










Here ${F_{nI}} = frac{2}{{{R^2}L}}int_0^R {r,{J_0}left( {nfrac{u}{R}}
ight)int_0^L {I,left( {u,v,0}
ight),,cos ,left( {frac{{pi ,n,v}}{L}}
ight){text{d}}v{text{d}}u} ,}$, ${F_{nV}} = frac{2}{{{R^2}L}}int_0^R {r,{J_0}left( {nfrac{u}{R}}
ight)} int_0^L {V,left( {u,v,0}
ight)} ,,cos ,left( {frac{{pi ,n,v}}{L}}
ight){text{d}}v{text{d}}u$, J₀(nr/R) is the Bessel function of the first type and zeroth-order. Substitution of the above series into Eq. (1a) gives us a possibility to obtain the first-order approximations of concentrations of point radiation defects in the following form

















We determine the second- and highest-orders of approximations of concentrations of point radiation defects framework standard iterative procedure of method of averaging of function corrections^[10]. In this case n-th-order approximations of concentrations of defects will be determined by following replacement I (r, z, t)→α_nI + I_n?1 (r, z, t), V (r, z, t)→α_nV + V_n?1 (r, z, t) in right sides of Eq. (1a), where α_nI and α_nV are the average values of the considered approximations. In this situation the second-order approximations of concentrations of point defects could be written as:










Average values of the second-order approximations of the considered concentrations could be determined by the following standard relations










Substitution of the appropriate approximations of concentrations of point defects into above relations leads to the following relations










where










Θ is the continuance of observation of variation in time of the concentrations of these defects.　 Equations for concentrations of simplest complexes of point defects (divacancies Φ_V(r, z, t) and diinterstitials Φ_I(r, z, t)) could be written as











with boundary and initial conditions











Here D_Φρ(z, T) are the diffusion coefficients of the above complexes of radiation defects; k_ρ(z, T) are the parameters of decay of the above complexes. To simplify solution of the above equations we transform them to the following integral form










Now we solve systems of Eq. (3a) by method of averaging of function corrections with decreasing quantity of iteration steps. For the framework of the approach we used solutions of Eq. (3) with averaged diffusion coefficients D_0IV and without any nonlinearity on concentration of defects as initial-order approximations of solutions of Eq. (3a). These initial-order approximations could be solved by standard Fourier approach and could be written as










Here ${F_{n,{varPhi _I}}} = frac{2}{{{R^2}L}}int_0^R {u{kern 1pt} {J_0}left( {nfrac{u}{R}}
ight){kern 1pt} } int_0^L {{varPhi _I}left( {u,v,0}
ight){kern 1pt} {kern 1pt} cos {kern 1pt} left( {frac{{pi {kern 1pt} n{kern 1pt} v}}{L}}
ight){kern 1pt} {text{d}}v{text{d}}u} $, ${F_{n,{varPhi _V}}} = frac{2}{{{R^2}L}}int_0^R {u,{J_0}left( {nfrac{u}{R}}
ight)} int_0^L {{varPhi _V},left( {u,v,0}
ight),cos ,left( {frac{{pi ,n,v}}{L}}
ight){text{d}}v{text{d}}u} $, J₀(nr/R) is the Bessel function of the first type and zeroth-order. Substitution of the above series into Eq. (1a) gives us a possibility to obtain the first-order approximations of concentrations of point radiation defects in the following form

















We determine the second- and highest-orders of approximations of concentrations of simplest complexes of point radiation defects framework standard iterative procedure of method of averaging of function corrections^[10]. In this case n-th-order approximations of concentrations of complexes of defects will be determined by following replacement Φ_I(r, z, t)→α_nΦI +Φ_In-1(r, z, t), Φ_V(r, z, t)→α_nΦV + Φ_Vn-1(r, z, t) in right sides of Eq. (3a), where α_nI and α_nV are the average values of the considered approximations. In this situation the second-order approximations of concentrations of complexes of point defects could be written as










We determine spatio-temporal distribution of temperature of local heating, generated during generation of radiation defects due to radiation processing, by solution of the following equation











Temperature dependence of heat capacitance $c(T) $ could be approximated by the following function: $ c(T)={c_0}left[ {1 - exp left( { - T/{T_{
m{d}}}}
ight)}
ight]$, where T_d is the Debye temperature. If current temperature is larger, than the Debye temperature, heat conduction coefficient could be approximated by the following function: $lambda left( {z,T}
ight){
m{ }} = {lambda _1}left( z
ight)left{ {1 + mu {{left[ {{T_{
m{d}}}/Tleft( {r,z,t}
ight)}
ight]}^varphi }}
ight}$. In the same area of temperatures one can consider the following relation: c(T) ≈ c₀. The Eq. (5) is complemented by the following boundary and initial conditions











Here T_r is the room temperature. With account dependence of heat diffusion coefficient on temperature one can transform the Eq. (5) to the following differential (7a) or integral (7b) form

















We solved the Eq. (7b) by method of averaging of function corrections with decreased quantity of iterative steps. For the framework of the approach we used solutions of Eq. (5) with averaged heat diffusion coefficients λ₀ without any nonlinearity on temperature as initial-order approximations of solutions of Eq. (7b). These initial-order approximations could be solved by standard Fourier approach and could be written as










Here $ {F_n} = frac{1}{{{R^2}L}}int_0^R {r,{J_0}left( {nfrac{u}{R}}
ight)int_0^L {Tleft( {u,v,0}
ight),cos ,left( {frac{{pi ,n,v}}{L}}
ight){text{d}}v{text{d}}u} } $, J₀(nr/R) is the Bessel function of the first type and zeroth-order. Substitution of the above series into Eq. (7b) gives us a possibility to obtain the first-order approximation of temperature in the following form










We determine the second- and highest-orders of approximation of temperature framework standard iterative procedure of method of averaging of function corrections^[10]. In this case n-th-order approximation of temperature will be determined by following replacement T(r, z, t) → α_nT + T_n-1(r, z, t) in right sides of Eq. (7b), where α_nT is the average value of the considered approximation. In this situation the second-order approximation of temperature could be written as










Not yet known average value α_2T by solution of the following equation

3.
Discussion

In this paper we analyzed spatio-temporal distributions of concentrations of radiation defects in a multilayer structure with account of spatio-temporal distribution of local heating, generated during generation of radiation defects due to radiation processing. Fig. 1 shows distribution of concentration of implanted dopant (curve 1), distribution of concentration of interstitials (curve 2) and distribution of concentration of vacancies (curve 3). In this figure, to the right of the maximum of the given concentrations, one can find a local minimum concentration of defects. The local minimum is presented on interface between layers of the considered multilayer structure. Recently the minimum has been experimentally obtained and discussed in Ref. [7]. Probably the interface between layers of multilayer structure could be considered as a drain of radiation defects. To take into account the drain we introduce last terms in Eq. (1).






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Figure1.
(Color online) Distribution of concentration for the implanted dopant (curve 1), interstitials (curve 2) and vacancies (curve 3). Solid lines are analytical results, and dashed lines are results from cluster dynamics model.

4.
Conclusion

In this paper we introduce a model of redistribution of point radiation defects, interaction between themselves and redistribution of their simplest complexes in multilayer structures. The model gives a possibility to qualitatively describe nonmonotonicity of distributions of concentrations of radiation defects, which recently was found experimentally. To take into account the nonmonotonicity we complement the model for analysis of distribution of concentration of radiation defects, which recently was used in literature. To analyze the model we used an approach of solution of boundary problems, which could be used without crosslinking of solutions on interfaces between layers of the considered multilayer structures.