and annealing, Semiconductor manufacturing techniques.

An actual and intensively solving problems of solid state electronics is increasing of integration rate of elements of integrated circuits (p-n-junctions, their systems et al) [1]–[8]. Increasing of the integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are using several approaches. They are widely using laser and microwave types of annealing of infused dopants. These types of annealing are also widely using for annealing of radiation defects, generated during ion implantation [9]–[17]. Using the approaches gives a possibility to increase integration rate of elements of integrated circuits through inhomogeneity of technological parameters due to generating inhomogenous distribution of temperature. In this situation one can obtain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law [1], [3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is doping of heterostructure by diffusion or ion implantation [1]–[3]. However in this case optimization of dopant and/or radiation defects is required [18].

In this paper we consider a heterostructure. The heterostructure consist of a substrate and several epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further we consider doping of these sections by diffusion or ion implantation. The doping gives a possibility to manufacture bipolar heterotransistors framework a single leg N-level cascaded H-bridge multilevel inverter so as it is shown on

In this section we determine spatio-temporal distributions of concentrations of infused and implanted dopants. To determine these distributions we calculate appropriate solutions of the second Fick's law [1], [3], [18].

Boundary and initial conditions for the equations are

The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient could be changed with changing materials of heterostructure, with changing temperature of materials (including annealing), with changing concentrations of dopant and radiation defects. We approximate dependences of dopant diffusion coefficient on parameters by the following relation with account results in Refs [19]–[21].

. (3)

Here the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due to Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes the limit of solubility of dopant. Parameter  [1,3] describes average quantity of charged defects interacted with atom of dopant [19]. The function V (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation vacancies. Parameter V* describes the equilibrium distribution of concentration of vacancies. The considered concentrational dependence of dopant diffusion coefficient has been described in details in [19]. It should be noted, that using diffusion type of doping did not generation radiation defects. In this situation 1= 2= 0. We determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations [20], [21].

Boundary and initial conditions for these equations are

Here  =I,V. The function I (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation interstitials; D(x,y,z,T) are the diffusion coefficients of point radiation defects; terms V2(x,y,z,t) and I2(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the parameter of recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parameters of generation of simplest complexes of point radiation defects.

Further we determine distributions in space and time of concentrations of divacancies V(x,y,z,t) and diinterstitials I(x,y,z,t) by solving the following system of equations [20], [21].

( , , , ) ( , , , ) ( , , , ) = [ ( , , , ) ] + [ ( , , , ) ] +

( , , , ) + [ ( , , , ) ] + , ( , , , ) ( , , , ) − ( , , , ) ( , , , )

( , , , ) ( , , , ) ( , , , ) = [ ( , , , ) ] + [ ( , , , ) ] +

( , , , ) 2

+ [ ( , , , ) ] + , ( , , , ) ( , , , ) − ( , , , ) ( , , , ).

Boundary and initial conditions for these equations are Here D(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are the parameters of decay of these complexes.

We calculate distributions of concentrations of point radiation defects in space and time by recently elaborated approach [18]. The approach based on transformation of approximations of diffusion coefficients in the following form: D(x,y,z,T)=D0[1+ g(x,y,z,T)], where D0 are the average values of diffusion coefficients, 0<1, |g(x, y,z,T)|1,  =I,V. We also used analogous transformation of approximations of parameters of recombination of point defects and parameters of generation of their complexes: kI,V(x,y,z,T)=k0I,V[1+I,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I [1+I,I gI,I(x,y,z,T)] and kV,V(x,y,z,T)=k0V,V [1+V,V gV,V(x,y,z,T)], where k01,2 are the their average values, 0I,V <1, 0I,I <1, 0V,V<1, | gI,V(x,y,z,T)|1, | gI,I(x,y,z,T)|1, |gV,V(x,y,z,T)|1. Let us introduce the following dimensionless variables: , , , , ,  = x/Lx,  = y /Ly,  = z/Lz. The introduction leads to transformation of Eq.(4) and Conditions (5) to the following form

, , , ,

, , .

We determine solutions of Eq.(8) with Conditions (9) framework recently introduced approach [18], i.e. as the power series

. (10)

Substitution of the Series (10) into Eq.(8) and Conditions (9) gives us possibility to obtain equations for initial-order approximations of concentration of point defects and and corrections for them and , i 1, j 1, k 1. The equations are presented in the Appendix. Solutions of the equations could be obtained by standard Fourier approach [22], [23]. The solutions are presented in the

Now we calculate distributions of concentrations of simplest complexes of point radiation defects in space and time. To determine the distributions we transform approximations of diffusion coefficients in the following form: D(x,y,z,T)=D0[1+ g(x,y,z,T)], where D0 are the average values of diffusion coefficients. In this situation the Eq.(6) could be written as:

Farther we determine solutions of above equations as the following power series

(11)

Now we used the Series (11) into Eq. (6) and appropriate boundary and initial conditions. The using gives the possibility to obtain equations for initial-order approximations of concentrations of complexes of defects 0(x,y,z,t), corrections for them i(x,y,z,t) (for them i 1) and boundary and initial conditions for them. We remove equations and conditions to the Appendix. Solutions of the equations have been calculated by standard approaches [22], [23] and presented in the Appendix.

Now we calculate distribution of concentration of dopant in space and time by using the approach, which was used for analysis of radiation defects. To use the approach we consider following transformation of approximation of dopant diffusion coefficient: DL(x,y,z,T)=D0L[1+ LgL(x,y,z,T)], where D0L is the average value of dopant diffusion coefficient, 0L< 1, |gL(x,y,z,T)|1. Farther we consider solution of Eq.(1) as the following series:

Using the relation into Eq. (1) and Conditions (2) leads to obtaining equations for the functions Cij(x,y,z,t) (i 1, j 1), boundary and initial conditions for them. The equations are presented in the Appendix. Solutions of the equations have been calculated by standard approaches (see, for example, [22], [23]). The solutions are presented in the Appendix.

We analyzed distributions of concentrations of dopant and radiation defects in space and time analytically by using the second-order approximations on all parameters, which have been used in appropriate series. Usually the second-order approximations are enough good approximations to make qualitative analysis and to obtain quantitative results. All analytical results have been checked by numerical simulation.

In this section we analyzed spatio-temporal distributions of concentrations of dopants. Fig. 3 and Fig. 4 shows typical spatial distributions of concentrations of dopants in neighborhood of interfaces of heterostructures. We calculate these distributions of concentrations of dopants under the following condition: value of dopant diffusion coefficient in doped area is larger, than value of dopant diffusion coefficient in nearest areas. In this situation one can find increasing of compactness of field-effect transistors with increasing of homogeneity of distribution of concentration of dopant at one time. Changing relation between values of dopant diffusion coefficients leads to opposite result (see Fig. 4 and Fig. 5).

Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate.

Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate. Curve 1 corresponds to homogenous sample and annealing time  = 0.0048 (Lx2+Ly2+Lz2)/D0. Curve 2 corresponds to homogenous sample and annealing time  = 0.0057 (Lx2+Ly2+Lz2)/D0. Curves 3 and 4 correspond to heterostructure from Figs. 1; annealing times  = 0.0048 (Lx2+Ly2+Lz2)/D0 and  = 0.0057 (Lx2 +Ly2+Lz2)/D0, respectively.

Difference between values of dopant diffusion coefficients increases with increasing of number of curves. Value of dopant diffusion coefficient in this section is smaller, than value of dopant diffusion coefficient in nearest sections.

It should be noted, that framework the considered approach one shall optimize annealing of dopant and/or radiation defects. To do the optimization we used recently introduced criterion [24]–[31]. The optimization based on approximation real distribution by step-wise function  (x,y,z) (see Fig. 7 and Fig. 8). Farther the required values of optimal annealing time have been calculated by minimization the following mean- squared error.

(12)

Solid lines are spatial distributions of implanted dopant in system of two epitaxial layers. Dushed lines are spatial distributions of implanted dopant in one epitaxial layer. Annealing time increases with increasing of number of curves.

Increasing of number of curve corresponds to increasing of annealing time.

We show optimal values of annealing time as functions of parameters on Fig. 8 and Fig. 9. It is known, that standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal case after finishing the annealing dopant achieves interface between layers of heterostructure. If the dopant has no enough time to achieve the interface, it is practicably to anneal the dopant additionally. The Fig. 9 shows the described dependences of optimal values of additional annealing time for the same parameters as for Fig. 8. Necessity to anneal radiation defects leads to smaller values of optimal annealing of implanted dopant in comparison with optimal annealing time of infused dopant.

Increasing of number of curve corresponds to increasing of annealing time.

Curve 1 describes the dependence of the annealing time on the relation a/L and  =  = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 3 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 4 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =  = 0.

Curve 1 describes the dependence of the annealing time on the relation a/L and  =  = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 3 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 4 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =  = 0.

In this paper we introduce an approach to increase integration rate of bipolar heterotransistors framework single leg N-level cascaded H-bridge multilevel inverter. The approach gives us possibility to decrease area of the elements with smaller increasing of the element’s thickness.

Equations for the functions and , i 0, j 0, k 0 and conditions for them

, i1,

, i1, Solutions of the above equations could be written as

where where sn() = sin ( n ); Equations for functions i(x,y,z,t), i 0 to describe concentrations of simplest complexes of radiation defects.

Boundary and initial conditions for the functions takes the form

Solutions of the above equations could be written as

where where sn(x) = sin ( n x/Lx).

Equations for the functions Cij(x,y,z,t) (i 0, j 0), boundary and initial conditions could be written as Functions Cij(x,y,z,t) (i 0, j 0) could be approximated by the following series during solutions of the above equations

Here