# A model for prognosis of influence of radiation dose on value of charge carrier mobility: an analytical approach for analysis of the introduced model – Journal of Electrical Systems and Information Technology

#### ByEvgeny L. Pankratov

Aug 11, 2022

To solve our aim, we calculate and analyzed the spatiotemporal distribution of concentration of the considered dopant in the above multilayer structure. We calculate the above distribution of dopant by solving the second Fick’s law in the following form [6,7,8]

$$frac{{partial ,C,left( {x,t} right)}}{partial ,t} = frac{partial ,}{{partial ,x}}left[ {D_{C} frac{{partial ,C,left( {x,t} right)}}{partial ,x}} right]$$

(1)

Boundary and initial conditions for the above equation could be written as

$$left. {frac{{partial ,C,left( {x,t} right)}}{partial ,x}} right|_{x = 0} = 0,,,left. {frac{{partial ,C,left( {x,t} right)}}{partial ,x}} right|_{x = L} = 0,,,Cleft( {x,0} right) = f_{C } left( x right).$$

Here C(x, t) is the spatiotemporal distribution of concentration of dopant; D is the dopant diffusion coefficient. Values of dopant diffusion coefficient depend on properties of materials of multilayer structure, speed of heating and cooling of materials during annealing and spatiotemporal distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be approximated by the following relation [6,7,8]

$$D_{C} = D_{L} left( {x,T} right),left[ {1 + xi frac{{C^{gamma } left( {x,t} right)}}{{P^{gamma } left( {x,T} right)}}} right],left[ {1 + varsigma_{1} frac{{V,left( {x,t} right)}}{{V^{*} }} + varsigma_{2} frac{{V^{2} left( {x,t} right)}}{{left( {V^{*} } right)^{2} }}} right]$$

(2)

Here DL (x,T) is the spatial (due to accounting all layers of multilayer structure) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficient; T is the temperature of annealing; P (x,T) is the limit of solubility of dopant; parameter γ depends on properties of materials and could be integer usially in the following interval γ [1, 3] [6]; and V (x,t) is the spatiotemporal distribution of concentration of radiation vacancies with the equilibrium distribution V*. Concentrational dependence of dopant diffusion coefficient has been described in detail in [6]. Spatiotemporal distributions of concentration of point radiation defects have been determined by solving the following system of equations [6,7,8]

left{ begin{aligned} frac{{partial I,left( {x,t} right)}}{partial t} = & ,frac{partial }{partial x}left[ {D_{I} left( {x,T} right)frac{{partial I,left( {x,t} right)}}{partial ,x}} right] – k_{I,I} left( {x,T} right),I^{2} left( {x,t} right) \ & – k_{I,V} left( {x,T} right),I,left( {x,t} right),,V,left( {x,t} right) \ frac{{partial V,left( {x,t} right)}}{partial t} = &, frac{partial }{partial x}left[ {D_{V} left( {x,T} right)frac{{partial V,left( {x,t} right)}}{partial x}} right] – k_{V,V} left( {x,T} right),V^{2} left( {x,t} right) \ & – k_{I,V} left( {x,T} right),I,left( {x,t} right),,V,left( {x,t} right) \ end{aligned} right.

(3)

Boundary and initial conditions for the above equations could be written as

begin{aligned} & left. {frac{{partial I,left( {x,t} right)}}{partial x}} right|_{x = 0} = 0,,,left. {frac{{partial I,left( {x,t} right)}}{partial x}} right|_{x = L} = 0,,,left. {frac{{partial V,left( {x,t} right)}}{partial x}} right|_{x = 0} = 0,,,left. {frac{{partial V,left( {x,t} right)}}{partial x}} right|_{x = L} = 0,, \ & Ileft( {x,0} right) = f_{I } left( x right),Vleft( {x,0} right) = f_{V } left( x right). \ end{aligned}

(4)

Here I (x,t) is the spatiotemporal distribution of concentration of radiation interstitials with the equilibrium distribution I*; DI(x,T) and DV(x,T) are the diffusion coefficients of interstitials and vacancies, respectively; terms V2(x,t) and I2(x,t) decribe generation of divacancies and diinterstitials, respectively (see, for example, [8] and appropriate references in this book); and kI,V(x,T), kI,I(x,T) and kV,V(x,T) are the parameters of recombination of point radiation defects and generation of their complexes. Spatiotemporal distributions of divacancies ΦV (x,t) and diinterstitials ΦI (x,t) could be determined by solving the following system of equations [7, 8]

left{ begin{aligned} frac{{partial Phi_{I} left( {x,t} right)}}{partial t} = & ,frac{partial }{partial x}left[ {D_{{Phi_{I} }} left( {x,T} right)frac{{partial Phi_{I} left( {x,t} right)}}{partial x}} right] + k_{I} left( {x,T} right),I,left( {x,t} right) \ & + k_{I,I} left( {x,T} right),I^{2} left( {x,t} right) \ frac{{partial Phi_{V} left( {x,t} right)}}{partial t} = &, frac{partial }{partial x}left[ {D_{{Phi_{V} }} left( {x,T} right)frac{{partial Phi_{V} left( {x,t} right)}}{partial x}} right] + k_{V} left( {x,T} right),V,left( {x,t} right) \ & + k_{V,V} left( {x,T} right),V^{2} left( {x,t} right) \ end{aligned} right.

(5)

Boundary and initial conditions for the above equations could be written as

begin{aligned} & left. {frac{{partial Phi _{I} ,left( {x,t} right)}}{{partial x}}} right|_{{x = 0}} = 0,,,left. {frac{{partial Phi _{I} ,left( {x,t} right)}}{{partial x}}} right|_{{x = L}} = 0,,,left. {frac{{partial Phi _{V} ,left( {x,t} right)}}{{partial x}}} right|_{{x = 0}} = 0,,,left. {frac{{partial Phi _{V} ,left( {x,t} right)}}{{partial x}}} right|_{{x = L}} = 0, \ & Phi _{I} left( {x,0} right) = f_{{Phi I~}} left( x right),,,Phi _{V} left( {x,0} right) = f_{{Phi V~}} left( x right). \ end{aligned}

(6)

Here DΦI(x,T) and DΦV(x,T) are the diffusion coefficients of simplest complexes of radiation defects; kI(x,T) and kV(x,T) are the parameters of decay of these complexes of radiation defects.

We calculate spatiotemporal distributions of the considered concentrations of dopant and radiation defects by solving Eqs. (1), (3) and (5) in the framework of method of averaging of function corrections [9,10,11]. Previously let us transform Eqs. (1), (3) and (5) to the following form with account initial distributions of the considered concentrations

$$frac{{partial ,C,left( {x,t} right)}}{partial ,t} = frac{partial ,}{{partial ,x}}left[ {D_{C} frac{{partial ,C,left( {x,t} right)}}{partial ,x}} right] + f_{C} ,left( x right),delta left( t right)$$

(1a)

left{ {begin{array}{*{20}l} begin{aligned} frac{{partial Ileft( {x,t} right)}}{{partial t}} = & ,frac{partial }{{partial x}}left[ {D_{I} left( {x,T} right)frac{{partial I,left( {x,t} right)}}{{partial x}}} right] – k_{{I,I}} left( {x,T} right),I^{2} left( {x,t} right) \ & – k_{{I,V}} left( {x,T} right),,I,left( {x,t} right),,V,left( {x,t} right) + f_{I} ,left( x right),delta left( t right) \ end{aligned} hfill \ begin{aligned} frac{{partial Vleft( {x,t} right)}}{{partial t}} = & ,frac{partial }{{partial x}}left[ {D_{V} left( {x,T} right)frac{{partial V,left( {x,t} right)}}{{partial ,x}}} right] – k_{{V,V}} left( {x,T} right),V^{2} left( {x,t} right) \ & – k_{{I,V}} left( {x,T} right),I,left( {x,t} right),,V,left( {x,t} right) + f_{V} ,left( x right),delta left( t right) \ end{aligned} hfill \ end{array} } right.

(3a)

left{ begin{aligned} frac{{partial Phi_{I} left( {x,t} right)}}{partial t} = & ,frac{partial }{partial x}left[ {D_{{Phi_{I} }} left( {x,T} right)frac{{partial Phi_{I} left( {x,t} right)}}{partial x}} right] + k_{I} left( {x,T} right),I,left( {x,t} right) \ & , + k_{I,I} left( {x,T} right),I^{2} left( {x,t} right) + f_{{Phi_{I} }} left( x right),delta ,left( t right) \ frac{{partial Phi_{V} left( {x,t} right)}}{partial t} = & ,frac{partial }{partial x}left[ {D_{{Phi_{V} }} left( {x,T} right)frac{{partial Phi_{V} left( {x,t} right)}}{partial x}} right] + k_{V} left( {x,T} right),V,left( {x,t} right) \ & , + k_{V,V} left( {x,T} right),V^{2} left( {x,t} right) + f_{{Phi_{V} }} left( x right),delta ,left( t right) \ end{aligned} right.

(5a)

Now in the framework of the method let us replace the concentrations of dopant and radiation defects in the right sides of Eqs. (1a), (3a) and (5a) on their not yet known average values α1ρ. In this situation, we obtain equations to calculate the first-order approximations of the required concentrations in the following form

$$frac{{partial C_{1} left( {x,t} right)}}{partial t} = f_{C} ,left( {x,} right),delta left( t right)$$

(1b)

$$left{ begin{gathered} frac{{partial I_{1} left( {x,t} right)}}{partial t} = f_{I} ,left( x right),delta left( t right) – alpha_{1I}^{2} k_{I,I} left( {x,T} right) – alpha_{1I} alpha_{1V} k_{I,V} left( {x,T} right) hfill \ frac{{partial V_{1} left( {x,t} right)}}{partial t} = f_{V} ,left( x right),delta left( t right) – alpha_{1V}^{2} k_{V,V} left( {x,T} right) – alpha_{1I} alpha_{1V} k_{I,V} left( {x,T} right) hfill \ end{gathered} right.$$

(3b)

$$left{ begin{gathered} frac{{partial Phi_{1I} left( {x,t} right)}}{partial t} = f_{{Phi_{I} }} ,left( x right),delta left( t right) + k_{I} left( {x,T} right),I,left( {x,t} right) + k_{I,I} left( {x,T} right),I^{2} left( {x,t} right) hfill \ frac{{partial Phi_{1V} left( {x,t} right)}}{partial t} = f_{{Phi_{V} }} ,left( x right),delta left( t right) + k_{V} left( {x,T} right),I,left( {x,t} right) + k_{V,V} left( {x,T} right),V^{2} left( {x,t} right) hfill \ end{gathered} right.$$

(5b)

Integration of the left and right sides of the above Eqs. (1b), (3b) and (5b) on time gives us possibility to obtain relations for above approximation in the final form

$$C_{1} left( {x,t} right) = f_{C} ,left( x right)$$

(1c)

$$left{ begin{gathered} I_{1} left( {x,t} right) = f_{I} ,left( x right) – alpha_{1I}^{2} intlimits_{0}^{t} {k_{I,I} left( {x,T} right),{text{d}}tau } – alpha_{1I} alpha_{1V} intlimits_{0}^{t} {k_{I,V} left( {x,T} right),{text{d}}tau } hfill \ V_{1} left( {x,t} right) = f_{V} ,left( x right) – alpha_{1V}^{2} intlimits_{0}^{t} {k_{V,V} left( {x,T} right),{text{d}}tau } – alpha_{1I} alpha_{1V} intlimits_{0}^{t} {k_{I,V} left( {x,T} right),{text{d}}tau } hfill \ end{gathered} right.$$

(3c)

$$left{ begin{gathered} Phi_{1I} left( {x,t} right) = f_{{Phi_{I} }} ,left( x right) + intlimits_{0}^{t} {k_{I} left( {x,T} right),I,left( {x,tau } right),{text{d}}tau } + intlimits_{0}^{t} {k_{I,I} left( {x,T} right),I^{2} left( {x,tau } right),{text{d}}tau } hfill \ Phi_{1V} left( {x,t} right) = f_{{Phi_{V} }} ,left( x right) + intlimits_{0}^{t} {k_{V} left( {x,T} right),V,left( {x,tau } right),{text{d}}tau } + intlimits_{0}^{t} {k_{V,V} left( {x,T} right),V^{2} left( {x,tau } right),{text{d}}tau } hfill \ end{gathered} right.$$

(5c)

We calculate average values of the first-order approximations of concentrations of dopant and radiation defects by using the following standard relation [9,10,11]

$$alpha_{1rho } = frac{1}{Theta L}intlimits_{0}^{Theta } {intlimits_{0}^{L} {rho_{1} left( {x,t} right),,{text{d}}x,{text{d}}t} }$$

(7)

Substitution of the relations (1c), (3c) and (5c) in the relation (7) gives us the possibility to obtain required average values in the following form

begin{aligned} alpha_{1C} = & frac{1}{L}intlimits_{0}^{L} {f_{C} left( x right),{text{d}}x} ,,,,alpha_{1I} = sqrt {frac{{left( {a_{3} + A} right)^{2} }}{{4,a_{4}^{2} }} – 4,,left( {B + frac{{Theta ,a_{3} B + Theta^{2} L,a_{1} }}{{a_{4} }}} right)} – frac{{a_{3} + A}}{{4,a_{4} }},, \ alpha_{1V} = & frac{1}{{S_{IV,00} }}left[ {frac{Theta }{{alpha_{1I} }}intlimits_{0}^{{L_{x} }} {f_{I} ,left( x right),,{text{d}}x} – alpha_{1I} S_{II,00} – Theta L} right] \ end{aligned}

where (S_{rho rho ,,ij} = intlimits_{0}^{Theta } {left( {Theta – t} right)intlimits_{0}^{L} {,k_{rho ,rho } left( {x,T} right),I_{1}^{i} left( {x,t} right),V_{1}^{j} left( {x,t} right),,{text{d}}x,{text{d}}t} },,)(a_{4} = S_{II,00} left( {S_{IV,00}^{2} – S_{II,00} S_{VV,00} } right),,)(a_{3} = S_{IV00} S_{II00} + S_{IV00}^{2} – S_{II00} S_{VV00},,)(a_{2} = S_{IV00} S_{IV00}^{2} intlimits_{0}^{L} {f_{V} ,left( x right),,{text{d}}x} + 2,S_{VV00} S_{II00} intlimits_{0}^{L} {f_{I} ,left( {x,} right),{text{d}}x},+, S_{IV00} Theta ,L^{2} – Theta ,L^{2} – S_{IV00}^{2} intlimits_{0}^{L} {f_{I} ,left( x right),,{text{d}}x},,) (a_{1} = S_{IV00} intlimits_{0}^{L} {f_{I} ,left( {x,} right),,{text{d}}x},,)(a_{0} = S_{VV00} left[ {intlimits_{0}^{L} {f_{I} ,left( {x,} right),{text{d}}x} } right]^{2},,)(A = sqrt {8,y + Theta^{2} frac{{a_{3}^{2} }}{{a_{4}^{2} }} – 4,Theta ,frac{{a_{2} }}{{a_{4} }}},,)( B = frac{{Theta ,a_{2} }}{{6,a_{4} }} + sqrt[3]{{sqrt {q^{2} + p^{3} } – q}} – sqrt[3]{{sqrt {q^{2} + p^{3} } + q}} ,,)(q = frac{{Theta^{3} a_{2} }}{{,24,a_{4}^{2} }}) (times ,,left( {4a_{0} – Theta ,Lfrac{{a_{1} a_{3} }}{{a_{4} }}} right) – Theta^{3} frac{{a_{0} }}{{8a_{4}^{2} }}left( {4,a_{2} – Theta frac{{a_{3}^{2} }}{{a_{4} }}} right) – frac{{Theta^{3} a_{2}^{3} }}{{54,a_{4}^{3} }} – L^{2} frac{{Theta^{4} a_{1}^{2} }}{{8,a_{4}^{2} }},,)(p = Theta^{2} frac{{4a_{0} a_{4} – Theta ,L,a_{1} a_{3} }}{{12,a_{4}^{2} }}) (- {{Theta ,a_{2} } mathord{left/ {vphantom {{Theta ,a_{2} } {18,a_{4} }}} right. kern-nulldelimiterspace} {18,a_{4} }},,)

begin{aligned} alpha_{{1Phi_{I} }} = & ,,frac{{R_{I1} }}{Theta ,L} + frac{{S_{II,20} }}{Theta ,L} + frac{1}{L}intlimits_{0}^{L} {f_{{Phi_{I} }} ,left( x right),{text{d}}x} ,,,alpha_{{1Phi_{V} }} \ = &, ,frac{{R_{V1} }}{Theta ,L} + frac{{S_{VV,20} }}{Theta ,L} + frac{1}{L}intlimits_{0}^{L} {f_{{Phi_{V} }} ,left( x right),,{text{d}}x} , \ end{aligned}

where (R_{rho ,,i} = intlimits_{0}^{Theta } {left( {Theta – t} right)intlimits_{0}^{L} {k_{I} left( {x,T} right),I_{1}^{i} left( {x,t} right),{text{d}}x,{text{d}}t} }).

Now let us calculate approximations of the second and the higher orders of the considered concentrations of dopant and radiation defects in the framework of the standard iterative procedure of the method of averaging of function corrections [9,10,11]. In the framework of the procedure to calculate approximations of the n-th order of concentrations of dopant and radiation defects, we replace the required concentrations in Eqs. (1c), (3c), (5c) on the following sum α + ρn1 (x,t). The replacement leads to the following transformation of the appropriate equations

begin{aligned} frac{{partial ,C_{2} left( {x,t} right)}}{partial ,t} = &, frac{partial ,}{{partial ,x}}left( {D_{,L} left( {x,T} right),left{ {1 + xi frac{{left[ {alpha_{2C} + C_{1} left( {x,t} right)} right]^{gamma } }}{{P^{gamma } left( {x,T} right)}}} right},,left[ {1 + varsigma_{1} frac{{V,left( {x,t} right)}}{{V^{*} }} + varsigma_{2} frac{{V^{2} left( {x,t} right)}}{{left( {V^{*} } right)^{2} }}} right],frac{{partial ,C_{1} left( {x,t} right)}}{partial ,x}} right) \ & +, f_{C} ,left( x right),,delta left( t right) \ end{aligned}

(1d)

left{ begin{aligned} frac{{partial I_{2} left( {x,t} right)}}{partial t} = &, frac{partial }{partial x}left[ {D_{I} left( {x,T} right)frac{{partial I_{1} left( {x,t} right)}}{partial x}} right] – k_{I,I} left( {x,T} right),left[ {alpha_{1I} + I_{1} left( {x,t} right)} right]^{2} \ & – k_{I,V} left( {x,T} right),left[ {alpha_{1I} + I_{1} left( {x,t} right)} right],left[ {alpha_{1V} + V_{1} left( {x,t} right)} right] \ frac{{partial V_{2} left( {x,t} right)}}{partial t} = &, frac{partial }{partial x}left[ {D_{V} left( {x,T} right)frac{{partial V_{1} left( {x,t} right)}}{partial x}} right] – k_{V,V} left( {x,T} right),left[ {alpha_{1V} + V_{1} left( {x,t} right)} right]^{2} \ & – k_{I,V} left( {x,T} right),left[ {alpha_{1I} + I_{1} left( {x,t} right)} right],left[ {alpha_{1V} + V_{1} left( {x,t} right)} right] \ end{aligned} right.

(3d)

left{ begin{aligned} frac{{partial Phi_{2I} left( {x,t} right)}}{partial t} = &, frac{partial }{partial x}left[ {D_{{Phi_{I} }} left( {x,T} right)frac{{partial Phi_{1I} left( {x,t} right)}}{partial x}} right] + k_{I,I} left( {x,T} right),I^{2} left( {x,t} right) \ &quad+ k_{I} left( {x,T} right),I,left( {x,t} right) + f_{{Phi_{I} }} ,left( x right),delta left( t right) \ frac{{partial Phi_{2V} left( {x,t} right)}}{partial t} = &, frac{partial }{partial x}left[ {D_{{Phi_{V} }} left( {x,T} right)frac{{partial Phi_{1V} left( {x,t} right)}}{partial x}} right] + k_{V,V} left( {x,T} right),V^{2} left( {x,t} right) \ &quad+ k_{V} left( {x,T} right),V,left( {x,t} right) + f_{{Phi_{V} }} ,left( x right),delta left( t right) \ end{aligned} right.

(5d)

Integration of the left and the right sides of the above Eqs. (1d), (3d) and (5d) gives us possibility to obtain relations for the required concentrations in the final form

begin{aligned} C_{2} left( {x,t} right) = & ,f_{C} ,left( x right) + frac{partial }{{partial x}}intlimits_{0}^{t} {D_{{,L}} left( {x,T} right),,left{ {1 + xi frac{{left[ {alpha _{{2C}} + C_{1} left( {x,tau } right)} right]^{gamma } }}{{P^{gamma } left( {x,T} right)}}} right}} \ & ,left[ {1 + varsigma _{1} frac{{V,left( {x,tau } right)}}{{V^{*} }} + varsigma _{2} frac{{V^{2} left( {x,tau } right)}}{{left( {V^{*} } right)^{2} }}} right],,frac{{partial C_{1} left( {x,tau } right)}}{{partial x}},{text{d}}tau \ end{aligned}

(1e)

left{ begin{aligned} I_{2} left( {x,t} right) = &, frac{partial }{partial x}intlimits_{0}^{t} {D_{I} left( {x,T} right)frac{{partial I_{1} left( {x,tau } right)}}{partial x}{text{d}},tau } + f_{I} left( x right) – intlimits_{0}^{t} {k_{I,I} left( {x,T} right),,left[ {alpha_{2I} + I_{1} left( {x,tau } right)} right]^{2} {text{d}}tau } \ & – intlimits_{0}^{t} {k_{I,V} left( {x,T} right),,left[ {alpha_{2I} + I_{1} left( {x,tau } right)} right],left[ {alpha_{2V} + V_{1} left( {x,tau } right)} right],,{text{d}}tau } \ V_{2} left( {x,t} right) = &, frac{partial }{partial x}intlimits_{0}^{t} {D_{V} left( {x,T} right)frac{{partial ,V_{1} left( {x,tau } right)}}{partial ,x}{text{d}}tau } + f_{V} left( x right) – intlimits_{0}^{t} {k_{V,V} left( {x,T} right),,left[ {alpha_{2V} + V_{1} left( {x,tau } right)} right]^{2} {text{d}}tau } \ & – intlimits_{0}^{t} {k_{I,V} left( {x,T} right),,left[ {alpha_{2I} + I_{1} left( {x,tau } right)} right],left[ {alpha_{2V} + V_{1} left( {x,tau } right)} right],,{text{d}}tau } \ end{aligned} right.

(3e)

left{ begin{aligned} Phi_{2I} left( {x,t} right) = &, frac{partial }{partial x}intlimits_{0}^{t} {D_{{Phi_{I} }} left( {x,T} right)frac{{partial Phi_{1I} left( {x,tau } right)}}{partial x}{text{d}}tau } + intlimits_{0}^{t} {k_{I,I} left( {x,T} right),I^{2} left( {x,tau } right),,{text{d}}tau } \ & + intlimits_{0}^{t} {k_{I} left( {x,T} right),I,left( {x,tau } right),,dtau } + f_{{Phi_{I} }} ,left( x right) \ Phi_{2V} left( {x,t} right) = &, frac{partial }{partial x}intlimits_{0}^{t} {D_{{Phi_{V} }} left( {x,T} right)frac{{partial Phi_{1V} left( {x,tau } right)}}{partial x}{text{d}}tau } + intlimits_{0}^{t} {k_{V,V} left( {x,T} right),V^{2} left( {x,tau } right),,{text{d}}tau } + \ & + intlimits_{0}^{t} {k_{V} left( {x,T} right),V,left( {x,tau } right),,{text{d}}tau } + f_{{Phi_{V} }} ,left( x right) \ end{aligned} right.

(5e)

Average values of the second-order approximations of considered approximations by using the following standard relation [9,10,11]

$$alpha_{2rho } = frac{1}{Theta L}intlimits_{0}^{Theta } {intlimits_{0}^{L} {left[ {rho_{2} left( {x,t} right) – rho_{1} left( {x,t} right)} right],{text{d}}x,{text{d}}t} }$$

(8)

Substitution of the relations (1e), (3e), (5e) in the relation (8) gives us the possibility to obtain relations for required average values α 2ρ

$$a_{2C} = 0,,,a_{2FI} = 0,,,a_{2FV} = 0,,,alpha_{2V} = sqrt {frac{{left( {b_{3} + E} right)^{2} }}{{4,b_{4}^{2} }} – 4,,left( {F + frac{{Theta ,a_{3} F + Theta^{2} L,,b_{1} }}{{b_{4} }}} right)} – frac{{b_{3} + E}}{{4,b_{4} }},$$

$$alpha_{2I} = frac{{C_{V} – alpha_{2V}^{2} S_{VV00} – alpha_{2V} left( {2,S_{VV01} + S_{I,V10} + Theta ,L} right),, – S_{V,V02} – S_{I,V11} }}{{S_{IV01} + alpha_{2V} S_{IV00} }},$$

where (b_{4} = frac{1}{Theta ,L}left( {S_{IV00}^{2} S_{V,V00} – S_{VV00}^{2} S_{II00} } right),, b_{3} = – frac{{S_{{II00}} S_{{VV00}} }}{{Theta ,L}}left( {2S_{{VV01}} + S_{{IV10}} + Theta ,L} right) + frac{{S_{{IV00}} }}{{Theta ,L}} ) (times ,,S_{VV00} ,left( {S_{IV01} + 2S_{II10} + S_{IV01} + Theta ,L} right) + frac{{S_{IV00}^{2} }}{Theta ,L}left( {2S_{VV01} + S_{IV10} + Theta ,L} right) – frac{{S_{IV00}^{2} S_{IV10} }}{{Theta^{3} L^{3} }},,)(b_{2} = frac{{S_{II00} }}{Theta ,L},times ,S_{VV00} left( {S_{VV02} + S_{IV11} + C_{V} } right) – left( {S_{IV10} – 2S_{VV01} + Theta ,L} right)^{2} + left( {Theta ,L + 2S_{II10} + S_{IV01} } right),S_{IV01} frac{{S_{VV00} }}{Theta ,L}) (+ frac{{S_{IV00} }}{Theta ,L}left( {S_{IV01} + 2S_{II10} + 2S_{IV01} + Theta ,L} right),left( {2S_{VV01} + Theta ,L + S_{IV10} } right) – frac{{S_{IV00}^{2} }}{Theta ,L}left( {C_{V} – S_{VV02} – S_{IV11} } right))(+ C_{I} frac{{S_{IV00}^{2} }}{{Theta^{2} L^{2} }} – frac{{2,S_{IV10} }}{Theta ,L}S_{IV00} S_{IV01},,b_{1} = frac{{S_{II00} }}{Theta ,L}left( {S_{IV11} + S_{VV02} + C_{V} } right),left( {2S_{VV01} + S_{IV10} + Theta ,L} right) + frac{{S_{IV01} }}{Theta ,L}) (times ,,left( {Theta ,L + 2S_{II10} + S_{IV01} } right),left( {2S_{VV01} + S_{IV10} + Theta ,L} right) – S_{IV10} frac{{S_{IV01}^{2} }}{Theta ,L} – frac{{S_{IV00} }}{Theta ,L}left( {3S_{IV01} + 2S_{II10} + Theta ,L} right))(times ,,left( {C_{V} – S_{VV02} – S_{IV11} } right) + 2,C_{I} S_{IV00} S_{IV01},,b_{0} = frac{{S_{II00} }}{Theta ,L}left( {S_{IV00} + S_{VV02} } right)^{2} – left( {Theta ,L + 2,S_{II10} + S_{IV01} } right).,)(times ,,left( {C_{V} – S_{VV02} – S_{IV11} } right)frac{{S_{IV01} }}{Theta ,L} + 2C_{I} S_{IV01}^{2} – frac{{S_{IV01} }}{Theta ,L}left( {C_{V} – S_{VV02} – S_{IV11} } right),left( {Theta ,L + 2,S_{II10} + S_{IV01} } right),,)(C_{I} = frac{{alpha_{1I} alpha_{1V} S_{IV00} + alpha_{1I}^{2} S_{II00} – S_{II20} S_{II20} – S_{IV11} }}{Theta ,L},,)(C_{V} = alpha_{1I} alpha_{1V} S_{IV00} + alpha_{1V}^{2} S_{VV00} – S_{VV02} – S_{IV11}) (E = sqrt {8,y + Theta^{2} frac{{a_{3}^{2} }}{{a_{4}^{2} }} – 4,Theta ,frac{{a_{2} }}{{a_{4} }}},,)( F = frac{{Theta ,a_{2} }}{{6,a_{4} }} + sqrt[3]{{sqrt {r^{2} + s^{3} } – r}} – sqrt[3]{{sqrt {r^{2} + s^{3} } + r}} ,,)(r = frac{{Theta^{3} b_{2} }}{{,24,b_{4}^{2} }},) ( times ,,left( {4b_{0} – Theta ,Lfrac{{b_{1} b_{3} }}{{b_{4} }}} right) – frac{{Theta ^{3} b_{2}^{3} }}{{54,b_{4}^{3} }} – frac{{b_{0} Theta ^{3} }}{{8b_{4}^{2} }},left( {4,,b_{2} – Theta frac{{b_{3}^{2} }}{{b_{4} }}} right) – L^{2} frac{{Theta ^{4} b_{1}^{2} }}{{8,b_{4}^{2} }} ,,)(s = Theta^{2} frac{{4b_{0} b_{4} – Theta ,L,b_{1} b_{3} }}{{12,b_{4}^{2} }}) (- {{Theta ,b_{2} } mathord{left/ {vphantom {{Theta ,b_{2} } {18,b_{4} }}} right. kern-nulldelimiterspace} {18,b_{4} }}.)

In the framework of the paper, we calculate the considered concentrations of dopant and radiation defects by using the second-order approximation in the framework of the method of averaging of function corrections. The approximation is usually enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained results were checked by comparison with results of numerical simulations.