An Approach to Model Optimization of Man-Ufacturing of Emitter-Coupled Logic

International Journal of Recent advances in Physics (IJRAP) Vol.4, No.2, May 2015
DOI : 10.14810/ijrap.2015.4201 1
AN APPROACH TO MODEL OPTIMIZATION OF MAN-
UFACTURING OF EMITTER-COUPLED LOGIC
E.L. Pankratov1,3
, E.A. Bulaeva1,2
1
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod,
603950, Russia
2
Nizhny Novgorod State University of Architecture and Civil Engineering, 65
Il'insky street, Nizhny Novgorod, 603950, Russia
3
Nizhny Novgorod Academy of the Ministry of Internal Affairs of Russia, 3 Ankudi-
novskoe Shosse, Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we consider an approach to optimization manufacturing elements of emitter-coupled logic.
The optimization gives us possibility to decrease dimensions of these elements. The technological part
based on manufacture a heterostructure with required configuration, doping by diffusion or ion implanta-
tion of required areas of heterostructure and optimization of annealing of dopant and/or radiation defects.
The modelling part based on modified method of functional corrections, which using without crosslinking
of solutions on interfaces between layers of heterostructure.
KEYWORDS
analytical approach for modelling; emitter-coupled logic; optimization of manufacturing; decreasing of
dimensions; de-creasing of overheats
1. INTRODUCTION
Intensive development of electronic technique leads to increasing of performance of elements of
integrated circuits ant to increasing of degree of their integration (p-n-junctions, bipolar and field-
effect transistors, thyristors, ...) [1-6]. Increasing of the performance could be obtain by develop-
ment of new and optimization of existing technological processes. Another way to increase the
performance in determination of materials with higher speed of transport of charge carriers [7-
10]. Increasing of degree of integration of elements of integrated circuits leads to necessity to de-
crease dimensions of the elements. To decrease the dimensions it could be used different ap-
proaches. Two of them are laser and microwave types of annealing [11-13]. Using this types of
annealing leads to generation of inhomogenous distribution of temperature. The inhomogeneity
leads to decreasing of dimensions of elements of integrated circuits due to Arrhenius law. Anoth-
er way to decrease the dimensions is doping of epitaxial layers of heterostructures by diffusion
and ion implantation [14-17]. However in this case it is practicably to optimize annealing of do-
pant and/or radiation defects [17-24]. It is known, that using radiation processing of materials of
heterostructure and homogeneous samples leads to changing of distributions of concentrations of
dopants in them [25].
In this paper as a development of works [17-24] we consider an approach to manufacture more
compact elements of emitter-coupled logic. To illustrate the approach we consider a heterostruc-
ture, which is illustrated on Fig. 1. The heterostructure consist of a substrate and three epitaxial

2
layers. Some sections have been manufactured in the epitaxial layers such as it shown on the fig-
ure. After finishing of the nearest to the substrate epitaxial layer we consider doping of the sec-
tions of the layer by diffusion or ion implantation. The sections will have role of collectors during
functioning of the manufacturing device. Farther we consider annealing of dopant and/or radia-
tion defects. After finishing this annealing we consider two another epitaxial layers with appro-
priate sections (see Fig. 1). After finishing of manufacturing of each epitaxial layer all sections of
the last layer have been doped by diffusion or ion implantation. Farther we consider microwave
annealing of dopant and/or radiation defects. Frequency of electro-magnetic radiation should be
so, that thickness of scin-layer should be larger, than thickness of external epitaxial layer and
smaller, than sum of thicknesses of external and average epitaxial layers. Sections of the average
and external epitaxial layers will have roles of bases and emitter, respectively, during functioning
of the considered device. Main aim of the present paper is analysis of dynamics of redistribution
of infused and implanted dopants and/or radiation defects during annealing.
Fig. 1. Heterostructure with four layers, which consist of a substrate and three epitaxial layer with sections,
manufactured by using another materials
2. METHOD OF SOLUTION
To solve our aims we determine spatio-temporal distribution of concentration of dopant. We de-
termine the required distribution by solving the second Fick's law [1,14 -17]
       



































z
t
z
y
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C ,
,
,
,
,
,
,
,
,
,
,
,
(1)
with initial and boundary conditions
C(x,y,z,0)=fC (x,y,z),
  0
,
,
,
0




x
x
t
z
y
x
C
,
  0
,
,
,



 x
L
x
x
t
z
y
x
C
,
  0
,
,
,
0




y
y
t
z
y
x
C
,
  0
,
,
,



 y
L
x
y
t
z
y
x
C
,
  0
,
,
,
0




z
z
t
z
y
x
C
,
  0
,
,
,



 z
L
x
z
t
z
y
x
C
.
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; x, y and z are current
coordinates, t is the current time; T is the temperature of annealing; DС is the dopant diffusion
coefficient. Value of dopant diffusion coefficient depends on properties of materials of hetero-
structure, speed of heating and cooling of the heterostructure (with account Arrhenius law). De-
pendences of dopant diffusion coefficient on parameters could be approximated by the following
relation [26-28]

3
   
 
   
  
















 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
V
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
D L
C 

 

, (3)
where DL(x,y,z,T) is the spatial (due to presents several layers with different properties framework
heterostructure) and temperature (due to Arrhenius law) dependences of dopant diffusion coeffi-
cient; P (x,y,z,T) is the limit of solubility of dopant; parameter  depends on properties of materi-
als and could be integer in the following interval [1,3] [26]; V (x,y,z,t) is the spatio- temporal
distribution of radiation vacancies; V*
is the equilibrium distribution of vacancies. Concentration-
al dependence of dopant diffusion coefficient is describes in details in [26]. It should be noted,
that diffusion type of doping did not leads to generation of radiation defects. In this situation 1=
2=0. We determine spatio-temporal distributions of concentrations of point radiation defects as
solutions of the following system of equations [27,28]
         
         
   
         
         
   



































































t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
V
V
V
I
V
V
V
I
I
V
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2
,
,
2
,
,




























(3)
with boundary and initial conditions
  0
,
,
,
0


x
x
t
z
y
x
I


,
  0
,
,
,

 x
L
x
x
t
z
y
x
I


,
  0
,
,
,
0


y
y
t
z
y
x
I


,
  0
,
,
,

 y
L
y
y
t
z
y
x
I


,
  0
,
,
,
0


z
z
t
z
y
x
I


,
  0
,
,
,

 z
L
z
z
t
z
y
x
I


,
  0
,
,
,
0


x
x
t
z
y
x
V


,
  0
,
,
,

 x
L
x
x
t
z
y
x
V


,
  0
,
,
,
0


y
y
t
z
y
x
V


,
  0
,
,
,

 y
L
y
y
t
z
y
x
V


,
  0
,
,
,
0


z
z
t
z
y
x
V


,
  0
,
,
,

 z
L
z
z
t
z
y
x
V


,
I(x,y,z,0)=fI (x,y,z), V(x,y,z,0)=fV (x,y,z). (4)
Here =I,V; I(x,y,z,t) is the spatio-temporal distribution of concentration of radiation interstitials;
D(x,y,z,T) are the diffusion coefficients of vacancies and interstitials; terms V2
(x,y,z,t) and
I2
(x,y,z,t) correspond to generation of divacancies and diinterstitials; kI,V(x,y,z,T), kI,I(x,y,z,T) and
kV,V(x,y,z,T) are parameters of recombination of point radiation defects and generation their com-
plexes. Spatio-temporal distributions of concentrations of simplest complexes (divacancies
V(x,y,z,t) and diinterstitials I(x,y,z,t)) of radiation defects have been determined by solving the
following system of equations [27,28]
          





 






 




y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x I
I
I
I
I









 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,

4
           
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
I
I
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, 






 
 




(6)
          





 






 




y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x V
V
V
V
V









 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
           
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
V
V
V
V
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, 






 
 




 
0
,
,
,
0





x
x
t
z
y
x

,
 
0
,
,
,




 x
L
x
x
t
z
y
x

,
 
0
,
,
,
0





y
y
t
z
y
x

,
 
0
,
,
,




 y
L
y
y
t
z
y
x

,
 
0
,
,
,
0





z
z
t
z
y
x

,
 
0
,
,
,




 z
L
z
z
t
z
y
x

,
I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (7)
Here D(x,y,z,T) are the diffusion coefficients of complexes of radiation defects; kI(x,y,z,T) and
kV (x,y,z,T) are the parameters of decay of complexes of radiation defects.
We determine spatio-temporal distribution of temperature by solving of the second law of Fourier
[29]
            
























y
t
z
y
x
T
T
z
y
x
y
x
t
z
y
x
T
T
z
y
x
x
t
t
z
y
x
T
T
c
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,


     
t
z
y
x
p
z
t
z
y
x
T
T
z
y
x
z
,
,
,
,
,
,
,
,
, 










  , (8)
  0
,
,
,
0




x
x
t
z
y
x
T
,
  0
,
,
,



 x
L
x
x
t
z
y
x
T
,
  0
,
,
,
0




y
y
t
z
y
x
T
,
  0
,
,
,



 y
L
x
y
t
z
y
x
T
,
  0
,
,
,
0




z
z
t
z
y
x
T
,
  0
,
,
,



 z
L
x
z
t
z
y
x
T
, T(x,y,z,0)=fT (x,y,z), (9)
where T(x,y,z,t) is the spatio-temporal distribution of temperature of annealing; c (T)= cass[1-
exp(-T(x,y,z,t)/Td)] is the heat capacitance (in the most interest case, when current temperature is
approximately equal or larger, than Debye temperature Td, we have possibility to consider the
following limiting case c (T)cass) [29];  is the heat conduction coefficient, which depends on
properties of materials and temperature (dependence on temperature of the heat conduction coef-
ficient could be approximated by the function (x,y,z,T)=ass(x,y,z)[1+ (Td/T(x,y,z,t))
] in the
most interesting interval of temperature [29]); p(x,y,z,t) is the volumetric density of power of
heating; (x,y,z,T)=(x, y,z,T)/c (T) is the thermal diffusivity.
First of all we calculate spatio-temporal distribution of temperature. To make the calculation we
used recently considered approach [17,20,30]. Framework the approach we transform approxima-
tion of thermal diffusivity to the following form: ass(x,y,z)=ass(x,y,z)/cass=0ass[1+T gT(x,y, z)].
Farther we determine solution of the Eq.(8) as the following power series [17,20,30]

5
   
 





0 0
,
,
,
,
,
,
i j
ij
j
i
T t
z
y
x
T
t
z
y
x
T 
 . (10)
Substitution of the series into Eq.(8) gives us possibility to obtain system of equations for initial-
order approximation of temperature T00(x,y,z,t) and corrections for them Tij(x,y,z,t) (i1, j1). The
equations are presented in the Appendix. Substitution of the series (10) into boundary and initial
conditions (9) gives us possibility to obtain boundary and initial conditions for all functions
Tij(x,y,z,t) (i0, j0). The solutions are presented in the Appendix. Solutions of the equations for
the functions Tij(x,y,z,t) (i0, j 0) have been calculated by standard approaches [31,32] frame-
work the second-order approximation on parameters  and . The solutions are presented in the
Appendix. To make qualitative analysis and to obtain some quantitative results the second-order
approximation is usually enough good approximation (see, for example, [17,20,30]). Analytical
results give us possibility to obtain and demonstrably. All analytical results have been checked by
comparison with numerical one.
To determine spatio-temporal distributions of concentrations of point defects we used recently
introduces approach [17,20,30]. Framework the approach we transform approximation of their
diffusion coefficients to the following form D(x,y,z,T)=D0[1+g(x,y,z,T)], where D0 are the
average values of the diffusion coefficients, 0<1, |g(x,y,z,T)|1, =I,V. Analogous transfor-
mation of approximations of parameters of recombination of point defects and generation their
complexes has been used: 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 appropriate 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
following dimensionless variables:     *
,
,
,
,
,
,
~
I
t
z
y
x
I
t
z
y
x
I  ,     *
,
,
,
,
,
,
~
V
t
z
y
x
V
t
z
y
x
V  , =
x/Lx, = y/Ly, = z/Lz, 2
0
0 L
t
D
D V
I

 , V
I
V
I D
D
k
L 0
0
,
0
2

 , V
I D
D
k
L 0
0
,
0
2


 
 . The
introduction leads to following transformation of Eqs.(4) and conditions (5)
   
    















V
I
I
I
I
V
I
I
D
D
D
I
T
g
D
D
D
I
0
0
0
0
0
0 ,
,
,
~
,
,
,
1
,
,
,
~















 
     
    












































,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1
0
0
0 I
T
g
D
D
D
I
T
g I
I
V
I
I
I
I
 
         
 
T
g
I
V
I
T
g I
I
I
I
I
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 ,
,
2
,
, 



















 



 (11)
   
    















V
I
V
V
V
V
I
V
D
D
D
V
T
g
D
D
D
V
0
0
0
0
0
0 ,
,
,
~
,
,
,
1
,
,
,
~















 
     
    
























V
I
V
V
V
V
V
D
D
D
V
T
g
V
T
g
0
0
0
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1




















 
         
 
T
g
V
V
I
T
g V
V
V
V
V
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 ,
,
2
,
, 



















 




  0
,
,
,
~
0











,
  0
,
,
,
~
1











,
  0
,
,
,
~
0











,
  0
,
,
,
~
1











,
  0
,
,
,
~
0











,
  0
,
,
,
~
1











,  
 
*
,
,
,
,
,
,
~









 
f
 . (12)

6
We solved the Eqs.(11) with conditions (12) by using the approach, considered in [17,20,30].
Framework the approach we determine solutions of the equations as the following power series
   
  







0 0 0
,
,
,
~
,
,
,
~
i j k
ijk
k
j
i











 
 . (13)
Substitution of the series (13) into Eqs.(11) and conditions (12) gives us possibility to obtain
equations for initial-order approximations of concentrations of point radiation defects
 




 ,
,
,
~
000 and corrections for them  




 ,
,
,
~
ijk , i1, j1, k1. The equations and condi-
tions for them are presented in the Appendix. The equations with account the conditions have
been solved by standard approaches (see, for example, Fourier approach [31,32]). The solutions
are presented in the Appendix.
Farther we calculate spathio-temporal distributions of concentrations of complexes of radiation
defects. To determine these distributions we transform approximations of diffusion coefficients to
the following form: D(x,y,z,T)=D0[1+g(x,y,z,T)], where D0 are the average values of
the coefficients. In this situation the Eqs. (6) have been transform to the following form
   
    






 






 I
I
I
I
D
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x I
I
0
0
,
,
,
,
,
,
1
,
,
,







 
     
    





 







 

 



z
t
z
y
x
T
z
y
x
g
z
y
t
z
y
x
T
z
y
x
g
y
I
I
I
I
I
I









 ,
,
,
,
,
,
1
,
,
,
,
,
,
1
       
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
D I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
, 2
,
0 

 
   
    






 






 V
V
V
V
D
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x V
V
0
0
0
0
,
,
,
,
,
,
1
,
,
,







 
     
    





 







 

 



z
t
z
y
x
T
z
y
x
g
z
y
t
z
y
x
T
z
y
x
g
y
V
V
V
V
V
V









 ,
,
,
,
,
,
1
,
,
,
,
,
,
1
       
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
D V
V
V
V
,
,
,
,
,
,
,
,
,
,
,
, 2
,
0 

  .
Let us determine solutions of the above equations in the following form
   
 





0
,
,
,
,
,
,
i
i
i
t
z
y
x
t
z
y
x 

  . (14)
Substitution of the series (14) into Eqs.(6) and appropriate boundary and initial conditions gives
us possibility to obtain equations for all functions i(x,y,z,t), i 1, boundary and initial condi-
tions for them. The equations and conditions are presented in the Appendix. The equation have
been solved by standard approaches [31,32]. The obtained solutions are presented in the Appen-
dix.
To calculate spatio-temporal distribution of concentration of dopant we transform approximation
of dopant diffusion coefficient in the following form: 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 determine
solution of Eq.(1) as the following power series
   
 





0 1
,
,
,
,
,
,
i j
ij
j
i
L t
z
y
x
C
t
z
y
x
C 
 . (15)

7
Substitution of the series in the Eq.(1) and condition (2) gives us possibility to obtain equation for
initial-order approximation of dopant concentration C00(x,y,z,t), equations for corrections for them
Cij(x,y,z,t) (i 1, j 1), boundary and initial conditions for all functions Cij(x,y,z,t) (i0, j 0). The
equations and conditions are presented in the Appendix. The equations have been solved by
standard approaches (see, for example, [31,32]). The solutions are presented in the Appendix.
Analysis of spatiotemporal distributions of concentrations of dopant and radiation defects and
temperature have been done analytically by using the second-order approximations on all consid-
ered parameters. The approximations are usually enough good approximations to make qualita-
tive analysis and to obtain some quantitative results (see, for example, [17,20,30]). Analytical
results give us possibility to obtain and demonstrably. All analytical results have been checked by
comparison with numerical one.
3. Discussion
In this section we analyzed dynamic of redistribution of dopant and radiation defects in the con-
sidered heterostructure during their annealing by using calculated in the previous section rela-
tions. 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 calcu-
lated for the case, when value of dopant diffusion coefficient in doped area is larger, than in near-
est 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 of homogeneity of dopant
distribution in doped part of epitaxial layer. Increasing of sharpness of p-n-junctions gives us pos-
sibility to decrease their switching time. Increasing of homogeneity of dopant distribution in
doped are gives us possibility to decrease local overheat of doped material during functioning of
p-n-junction or decreasing of dimensions of the p-n-junction for fixed maximal value of local
overheat. On the other hand inhomogeneity of doping of base of transistor leads to generation an
electric field in the base. The electric field gives us possibility to accelerate transport of charge
carriers. The acceleration gives us possibility to increase performance of the transistor.
Fig.2. Distributions of concentration of infused dopant in heterostructure from Fig. 1 in 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 diffu-
sion coefficient in substrate

8
x
0.0
0.5
1.0
1.5
2.0
C(x,

)
2
3
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig.3. Distributions of concentration of implanted dopant in heterostructure from Fig. 1 in direction, which
is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 corresponds to annealing
time =0.0048(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 2 and 4 corresponds to annealing time =0.0057(Lx
2
+Ly
2
+Lz
2
)/D0.
Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to heterostructure under
condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffu-
sion coefficient in substrate
To help in manufacturing of the drift transistors it could be used microwave annealing. Using this
type of annealing give us possibility to generate inhomogenous distribution of temperature. The
inhomogeneity gives us possibility to form more inhomogenous distribution of concentration of
dopant due to Arrhenius law [33]. Using microwave annealing leads to necessity to choose fre-
quency of electro-magnetic irradiation so, that thickness of scin-layer should be larger, than
thickness of external epitaxial layer, and smaller, than sum of thicknesses of external and average
epitaxial layers.
It should be noted, that increasing of annealing time leads to increasing of homogeneity of distri-
bution of concentration of dopant. In this situation the distribution became too large. If annealing
time is too small, dopant has not time to achieve nearest interface between materials of hetero-
structure. In this situation distribution of concentration of dopant has not any changing. In this
situation it is attracted an interest optimization of annealing. We consider the optimization of an-
nealing framework recently introduced criterion [17-24,33]. Framework the criterion we approx-
imate real distributions of concentrations of dopants by step-wise function  (x,y,z). Farther we
minimize the following mean-squared error to optimize annealing time
   
 
   


x y z
L L L
z
y
x
x
d
y
d
z
d
z
y
x
z
y
x
C
L
L
L
U
0 0 0
,
,
,
,
,
1
 . (16)

9
0.0 0.1 0.2 0.3 0.4 0.5
a/L, , , 
0.0
0.1
0.2
0.3
0.4
0.5

D
0
L
-2
3
2
4
1
Fig.4. Dependences of dimensionless optimal annealing time for doping by diffusion, which have been ob-
tained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence of dimen-
sionless optimal annealing time on the relation a/L and ==0 for equal to each other values of dopant dif-
fusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal an-
nealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 3 is the dependence of dimensionless
optimal annealing time on value of parameter  for a/L=1/2 and  =  =0. Curve 4 is the dependence of di-
mensionless optimal annealing time on value of parameter  for a/L=1/2 and ==0
0.0 0.1 0.2 0.3 0.4 0.5
a/L, , , 
0.00
0.04
0.08
0.12

D
0
L
-2
3
2
4
1
Fig.5. 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  =  = 0 for equal to each other values of
dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless op-
timal annealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 3 is the dependence of di-
mensionless optimal annealing time on value of parameter  for a/L=1/2 and  =  = 0. Curve 4 is the de-
pendence of dimensionless optimal annealing time on value of parameter  for a/L=1/2 and  =  = 0
Dependences of optimal values of annealing time are presented on Figs. 4 and 5. Optimal values
of annealing time of implanted dopant is smaller, than optimal values of annealing time of infused
dopant. The difference existing due to necessity of annealing of radiation defects. During anneal-
ing radiation defects one can find spreading of distribution of concentration of dopant. In the ideal

10
case during annealing of radiation defects should achieves interface between materials of hetero-
structure. If dopant has no time to achieve the interface, additional annealing of dopant attracted
an interest. The Fig. 5 shows exactly dependences of time of additional annealing.
4. CONCLUSIONS
In this paper we consider an approach to manufacture more compact elements of emitter-coupled
logic. The approach based on manufacturing a heterostructure with required configuration, doping
of required areas of the heterostructure by diffusion or ion implantation and optimization of an-
nealing of dopant and/or radiation defects.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
grant of Scientific School of Russia, the agreement of August 27, 2013 № 02.В.49.21.0003 be-
tween The Ministry of education and science of the Russian Federation and Lobachevsky State
University of Nizhni Novgorod and educational fellowship for scientific research of Nizhny Nov-
gorod State University of Architecture and Civil Engineering.
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Appendix
Equations for initial-order approximation of temperature T00(x,y,z,t), corrections for the approxi-
mation Tij(x,y,z,t) (i0, j0), boundary and initial conditions for them have been obtain by substi-
tution of the series (10) into Eqs.(8) and conditions (9) and equating of coefficients at equal pow-
ers of parameters, which have been used in the series (10). The equations and conditions could be
written as
         
ass
ass
t
z
y
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p
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12
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Conditions for the functions Tij(x,y,z,t) (i0, j0)
 
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
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,
,



 x
L
x
ij
x
t
z
y
x
T
,
 
0
,
,
,
0




y
ij
y
t
z
y
x
T
,
 
0
,
,
,



 y
L
x
ij
y
t
z
y
x
T
,
 
0
,
,
,
0




z
ij
z
t
z
y
x
T
,
 
0
,
,
,



 z
L
x
ij
z
t
z
y
x
T
, T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i1, j1.

13
Solutions of the above equations with account boundary and initial conditions for them have been
calculated by using standard Fourier approach [31,32]. Using the approach leads to the following
result
           


  


1
0 0 0
00
2
,
,
1
,
,
,
n
nT
n
n
n
z
y
x
L L L
T
z
y
x
t
e
z
c
y
c
x
c
L
L
L
u
d
v
d
w
d
w
v
u
f
L
L
L
t
z
y
x
T
x y z
          
   

  

t L L L
ass
z
y
x
L L L
T
n
n
n
x y z
x y z
d
u
d
v
d
w
d
w
v
u
p
L
L
L
u
d
v
d
w
d
w
v
u
f
w
c
v
c
u
c
0 0 0 0
0 0 0
,
,
,
1
,
, 


                 
    



1 0 0 0 0
,
,
,
2
n
t L L L
ass
n
n
n
nT
nT
n
n
n
z
y
x
x y z
d
u
d
v
d
w
d
w
v
u
p
w
c
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L



 ,
where cn()=cos(n/L),  



















 2
2
2
0
2
2 1
1
1
exp
z
y
x
ass
nT
L
L
L
t
n
t
e 
 ;
                 
    



1 0 0 0 0
2
0
0 ,
,
,
2
,
,
,
n
t L L L
T
n
n
nT
nT
n
n
n
z
y
x
ass
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
t
z
y
x
T 


             
  







1 0
2
0
10
2
,
,
,
n
t
nT
nT
n
n
n
z
y
x
ass
i
n e
t
e
z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
T
w
c 




              


  






1
2
0
0 0 0
10
2
,
,
,
,
,
,
n
n
n
z
y
x
ass
L L L
i
n
T
n
n y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
T
w
c
T
w
v
u
g
v
s
u
c
x y z 



               
   



 
t L L L
i
T
n
n
n
nT
nT
n
x y z
d
u
d
v
d
w
d
w
w
v
u
T
T
w
v
u
g
w
s
v
c
u
c
e
t
e
z
c
0 0 0 0
10 ,
,
,
,
,
, 

 , i1,
where sn()=sin(n/L);
                 
    



1 0 0 0 0
2
0
01
2
,
,
,
n
t L L L
n
n
n
nT
nT
n
n
n
z
y
x
d
ass
x y z
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
T
t
z
y
x
T 



 
 
         
  





1 0
2
0
00
2
00
2
2
,
,
,
,
,
,
n
t
nT
nT
n
n
n
z
y
x
d
ass e
t
e
z
c
y
c
x
c
n
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
u
w
v
u
T





 

       
 
     


  




1
2
0
0 0 0 00
2
00
2
2
,
,
,
,
,
,
n
n
n
n
z
y
x
d
ass
L L L
n
n
n z
c
y
c
x
c
n
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
v
w
v
u
T
w
c
v
s
u
c
x y z 






           
 
 


   





1
0
0 0 0 0 00
2
00
2
2
,
,
,
,
,
,
n
n
z
y
x
ass
t L L L
n
n
n
nT
nT x
c
L
L
L
w
v
u
T
d
u
d
v
d
w
d
u
w
v
u
T
w
s
v
c
u
c
e
t
e
x y z 




 
               
 

    








 
t L L L
n
n
n
nT
nT
n
n
d
x y z
w
v
u
T
d
u
d
v
d
w
d
u
w
v
u
T
w
c
v
c
u
c
e
t
e
z
c
y
c
T
0 0 0 0
1
00
2
00
,
,
,
,
,
,



 

                  
     










1 0 0 0 0
2
00
0 ,
,
,
2
n
t L L L
n
n
n
nT
nT
n
n
n
z
y
x
d
ass
x y z
v
w
v
u
T
w
c
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T 




 
             
   






1 0 0 0
0
1
00
2
,
,
, n
t L L
n
n
nT
nT
n
n
n
z
y
x
ass
d
x y
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d




 

   
 
 







 
z
L
n
w
v
u
T
d
u
d
v
d
w
d
w
w
v
u
T
w
c
0
1
00
2
00
,
,
,
,
,
,




;

14
                 
    



1 0 0 0 0
2
0
02
2
,
,
,
n
t L L L
n
n
n
nT
nT
n
n
n
z
y
x
d
ass
x y z
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
T
t
z
y
x
T 



 
 
           
  






1 0 0
2
0
00
2
01
2
2
,
,
,
,
,
,
n
t L
n
nT
nT
n
n
n
z
y
x
d
ass
x
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
u
w
v
u
T





 

     
 
     
 

 




1
2
0
0 0 00
2
01
2
2
,
,
,
,
,
,
n
nT
n
n
z
y
x
d
ass
L L
n
n t
e
y
c
x
c
n
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
v
w
v
u
T
w
c
v
s
n
y z







           
 


   




z
y
x
ass
d
t L L L
n
n
n
nT
n
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
w
w
v
u
T
w
s
v
c
u
c
e
z
c
x y z
2
0
0 0 0 0 00
2
01
2
2
,
,
,
,
,
, 




 

                 
    







1 0 0 0 0
01
00 ,
,
,
,
,
,
n
t L L L
n
n
nT
nT
n
n
n
x y z
u
w
v
u
T
u
w
v
u
T
v
c
u
c
e
t
e
z
c
y
c
x
c



 
 
           
  






1 0 0
2
0
1
00
2
,
,
, n
t L
n
nT
nT
n
n
n
z
y
x
ass
d
n
x
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
w
c 



 

       
 
 


 








1
2
0
0 0
1
00
01
00
2
,
,
,
,
,
,
,
,
,
n
n
z
y
x
ass
d
L L
n
n x
c
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
v
w
v
u
T
v
w
v
u
T
w
c
v
c
y z 




 

                 
 
   





 
t L L L
n
n
n
nT
nT
n
n
x y z
w
v
u
T
d
u
d
v
d
w
d
w
w
v
u
T
w
w
v
u
T
w
c
v
c
u
c
e
t
e
z
c
y
c
0 0 0 0
1
00
01
00
,
,
,
,
,
,
,
,
,




 
;
                   
    









1 0 0 0 0
2
00
2
0
11
,
,
,
2
,
,
,
n
t L L L
n
n
n
nT
nT
n
n
n
z
y
x
ass
x y z
w
w
v
u
T
w
c
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
t
z
y
x
T



           
 




















 




d
u
d
v
d
w
d
w
v
u
T
w
v
u
T
w
w
v
u
T
w
g
w
v
w
v
u
T
T
w
v
u
g
T
w
v
u
g T
T
T
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
00
01
00
2
00
2
                   
    












1 0 0 0 0
2
01
2
2
01
2
0 ,
,
,
,
,
,
2
n
t L L L
n
n
n
nT
nT
n
n
n
z
y
x
ass
x y z
u
w
v
u
T
u
w
v
u
T
w
c
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L




 
   
        





















 


d
u
d
v
d
w
d
w
w
v
u
T
T
w
v
u
g
w
v
w
v
u
T
T
w
v
u
g
T
w
v
u
g T
T
T
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
,
1 01
2
01
2
               
 
 
     











1 0 0 0 0
2
00
2
1
00
10
0 ,
,
,
,
,
,
,
,
,
2
n
t L L L
n
n
nT
nT
n
n
n
z
y
x
d
ass
x y z
u
w
v
u
T
w
v
u
T
w
v
u
T
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T 






 
 
   
 
    









 

z
y
x
d
ass
n
L
L
L
T
d
u
d
v
d
w
d
w
c
w
w
v
u
T
w
v
u
T
w
v
u
T
v
w
v
u
T
w
v
u
T
w
v
u
T 









 0
2
00
2
1
00
10
2
00
2
1
00
10
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
               
 
    
















1 0 0 0 0
2
10
2
2
10
2
2
10
2
,
,
,
,
,
,
,
,
,
n
t L L L
n
n
n
nT
nT
x y z
w
w
v
u
T
v
w
v
u
T
u
w
v
u
T
w
c
v
c
u
c
e
t
e




 
                   
   




1 0 0 0
0
00
2
2
,
,
, n
t L L
n
n
nT
nT
n
n
n
z
y
x
d
ass
n
n
n
x y
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T
z
c
y
c
x
c
w
v
u
T
d
u
d
v
d
w
d



 

           
























z
L
T
T
n
v
w
v
u
T
w
w
v
u
T
T
w
v
u
g
w
u
w
v
u
T
T
w
v
u
g
w
c
0
2
00
2
00
2
00
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,



 
 
              
   




1 0 0 0
0
00
2
,
,
,
,
,
,
n
t L L
n
n
nT
nT
n
n
n
z
y
x
d
ass
T
x y
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
T
w
v
u
g 



 


15
            
 















z
L
n
w
w
v
u
T
v
w
v
u
T
v
w
v
u
T
u
w
v
u
T
u
w
v
u
T
w
c
0
10
00
10
00
10 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 




 
 
             
   











1 0 0 0
0
1
00
00
2
,
,
,
,
,
,
n
t L L
n
n
nT
nT
n
n
n
z
y
x
d
ass
x y
v
c
u
c
e
t
e
z
c
y
c
x
c
L
L
L
T
w
v
u
T
d
u
d
v
d
w
d
w
w
v
u
T





 

       
 





































 
z
L
n
w
v
u
T
d
u
d
v
d
w
d
w
w
v
u
T
v
w
v
u
T
u
w
v
u
T
w
c
0
1
00
2
00
2
00
2
00
,
,
,
,
,
,
,
,
,
,
,
,






.
Equations for initial-order approximation of concentrations of point radiation defects
 



 ,
,
,
~
000
I and  



 ,
,
,
~
000
V , corrections for the approximation  



 ,
,
,
~
ijk
I and
 



 ,
,
,
~
ijk
V (i1, j1, k1), boundary and initial conditions for them have been obtain by sub-
stitution of the series (13) into Eqs.(4) and conditions (5) and equating of coefficients at equal
powers of parameters, which have been used in the series (13). The equations and conditions
could be written as
       
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0
000 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~






























 I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
       
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0
000 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~






























 V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V ;
        















 
















 ,
,
,
~
,
,
,
,
,
,
~
,
~
100
0
0
2
00
2
0
0
00 i
I
V
I
i
V
I
i I
T
g
D
D
I
D
D
I
      













 













 ,
,
,
~
,
,
,
,
,
,
~
100
0
0
2
00
2
0
0 i
I
V
I
i
V
I I
T
g
D
D
I
D
D
     













 













 ,
,
,
~
,
,
,
,
,
,
~
100
0
0
2
00
2
0
0 i
I
V
I
i
V
I I
T
g
D
D
I
D
D
        















 
















 ,
,
,
~
,
,
,
,
,
,
~
,
~
100
0
0
2
00
2
0
0
00 i
V
I
V
i
I
V
i V
T
g
D
D
V
D
D
V
      













 













 ,
,
,
~
,
,
,
,
,
,
~
100
0
0
2
00
2
0
0 i
V
I
V
i
I
V V
T
g
D
D
V
D
D
     













 













 ,
,
,
~
,
,
,
,
,
,
~
100
0
0
2
00
2
0
0 i
V
I
V
i
I
V V
T
g
D
D
V
D
D
, i1;
     








2
010
2
0
0
2
010
2
0
0
010 ,
,
,
~
,
,
,
~
,
,
,
~














 I
D
D
I
D
D
I
V
I
V
I
   
     

















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
000
000
,
,
2
010
2
0
0
V
I
T
g
I
D
D
V
I
V
I
V
I





     








2
010
2
0
0
2
010
2
0
0
010 ,
,
,
~
,
,
,
~
,
,
,
~














 V
D
D
V
D
D
V
I
V
I
V
   
     

















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
000
000
,
,
2
010
2
0
0
V
I
T
g
V
D
D
V
I
V
I
I
V




 ;
       











2
020
2
0
0
2
020
2
0
0
2
020
2
0
0
020 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I

16
 
         
 



















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
, V
I
V
I
T
g V
I
V
I 


       











2
020
2
0
0
2
020
2
0
0
2
020
2
0
0
020 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
D
D
V
D
D
V
D
D
V
V
I
V
I
V
I
 
         
 



















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
, V
I
V
I
T
g V
I
V
I 

 ;
     








2
001
2
0
0
2
001
2
0
0
001 ,
,
,
~
,
,
,
~
,
,
,
~














 I
D
D
I
D
D
I
V
I
V
I
   
   













,
,
,
~
,
,
,
1
,
,
,
~
2
000
,
,
2
001
2
0
0
I
T
g
I
D
D
I
I
I
I
V
I





     








2
001
2
0
0
2
001
2
0
0
001 ,
,
,
~
,
,
,
~
,
,
,
~














 V
D
D
V
D
D
V
I
V
I
V
   
   













,
,
,
~
,
,
,
1
,
,
,
~
2
000
,
,
2
001
2
0
0
V
T
g
V
D
D
I
I
I
I
I
V




 ;
        

















V
I
I
V
I
V
I
D
D
I
T
g
D
D
I
D
D
I
0
0
010
0
0
2
110
2
0
0
110 ,
,
,
~
,
,
,
,
,
,
~
,
,
,
~



















       
















 2
110
2
0
0
010
0
0
2
110
2
,
,
,
~
,
,
,
~
,
,
,
,
,
,
~


















 I
D
D
I
T
g
D
D
I
V
I
I
V
I
     
 












 T
g
I
T
g
D
D
I
I
I
I
I
V
I
,
,
,
1
,
,
,
~
,
,
, ,
,
010
0
0













       
 















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100 V
I
V
I 

        

















I
V
V
I
V
I
V
D
D
V
T
g
D
D
V
D
D
V
0
0
010
0
0
2
110
2
0
0
110 ,
,
,
~
,
,
,
,
,
,
~
,
,
,
~



















       
















 2
110
2
0
0
010
0
0
2
110
2
,
,
,
~
,
,
,
~
,
,
,
,
,
,
~


















 V
D
D
V
T
g
D
D
V
I
V
V
I
V
     
 












 T
g
V
T
g
D
D
V
V
V
V
V
I
V
,
,
,
1
,
,
,
~
,
,
, ,
,
010
0
0













       
 















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100 I
V
I
V 
 ;
       











2
002
2
0
0
2
002
2
0
0
2
002
2
0
0
002 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, I
I
T
g I
I
I
I


       











2
002
2
0
0
2
002
2
0
0
2
002
2
0
0
002 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, V
V
Е
g V
V
V
V

 ;
        

















V
I
V
I
I
V
I
D
D
D
D
I
T
g
I
D
D
I
0
0
0
0
001
2
101
2
0
0
101 ,
,
,
~
,
,
,
,
,
,
~
,
,
,
~



















       
















 2
101
2
0
0
001
0
0
2
101
2
,
,
,
~
,
,
,
~
,
,
,
,
,
,
~


















 I
D
D
I
T
g
D
D
I
V
I
I
V
I

17
     
     





















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 000
100
001
0
0
V
I
T
g
I
T
g
D
D
I
I
I
V
I













        

















I
V
I
V
V
I
V
D
D
D
D
V
T
g
V
D
D
V
0
0
0
0
001
2
101
2
0
0
101 ,
,
,
~
,
,
,
,
,
,
~
,
,
,
~



















       
















 2
101
2
0
0
001
0
0
2
101
2
,
,
,
~
,
,
,
~
,
,
,
,
,
,
~


















 V
D
D
V
T
g
D
D
V
I
V
V
I
V
     
     





















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 100
000
001
0
0
V
I
T
g
V
T
g
D
D
V
V
V
I
V












 ;
       











2
011
2
0
0
2
011
2
0
0
2
011
2
0
0
011 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
 
     

 










 ,
,
,
~
,
,
,
~
,
,
,
1 010
000
,
, I
I
T
g I
I
I
I
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, V
I
T
g V
I
V
I


       











2
011
2
0
0
2
011
2
0
0
2
011
2
0
0
011 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
 
     

 










 ,
,
,
~
,
,
,
~
,
,
,
1 010
000
,
, V
V
T
g V
V
V
V
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 001
000
,
, V
I
t
g V
I
V
I

 ;
 
0
,
,
,
~
0




x
ijk






,
 
0
,
,
,
~
1




x
ijk






,
 
0
,
,
,
~
0










ijk
,
 
0
,
,
,
~
1










ijk
,
 
0
,
,
,
~
0










ijk
,
 
0
,
,
,
~
1










ijk
(i0, j0, k0);     *
000 ,
,
0
,
,
,
~ 






 
f
 ,
  0
0
,
,
,
~ 



ijk (i0, j0, k0).
result
         




1
000
2
1
,
,
,
~
n
n
n e
c
c
c
F
L
L








 
 ,
where        
  

1
0
1
0
1
0
*
,
,
cos
cos
cos
1
u
d
v
d
w
d
w
v
u
f
w
n
v
n
u
n
F n
n 
 



,    
I
V
nI D
D
n
e 0
0
2
2
exp 

 
 ,
   
V
I
nV D
D
n
e 0
0
2
2
exp 

 
 , cn()=cos(n);
                 
    




1 0
1
0
1
0
1
0
0
0
00 ,
,
,
2
,
,
,
~
n
I
n
n
nI
nI
n
V
I
i T
w
v
u
g
v
c
u
s
e
e
c
c
c
D
D
I











             
  







1 0
0
0
100
2
,
,
,
~
n
nI
nI
n
V
I
i
n e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
w
c
n









           


  






1
0
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
n
V
I
i
I
n
n
n c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c
v
s
u
c 



                 
   



 







0
1
0
1
0
1
0
100 ,
,
,
~
,
,
, d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e
e
c
c i
I
n
n
n
nI
nI

18
                 
 
   




1 0
1
0
1
0
1
0
0
0
00 ,
,
,
2
,
,
,
~
n
V
n
n
nI
nV
n
I
V
i T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V











               
 
 








1 0
1
0
0
0
100
2
,
~
n
n
nI
nV
n
I
V
i
n u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
u
V
w
c
n









               


 






1
0
0
1
0
1
0
100
2
,
~
,
,
,
n
nV
n
I
V
i
V
n
n e
c
c
c
D
D
d
u
d
v
d
w
d
v
u
V
T
w
v
u
g
w
c
v
s 






           
   



 




0
1
0
1
0
1
0
100 ,
~
,
,
, d
u
d
v
d
w
d
w
u
V
T
w
v
u
g
w
s
v
c
u
c
e
n i
V
n
n
n
nI , i1,
where sn()=sin(n);
                 
     




1 0
1
0
1
0
1
0
010 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c


 









 
      


 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
V
I ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,

 ;
             
 
     




1 0
1
0
1
0
1
0
,
,
0
0
020 ,
,
,
1
2
,
,
,
~
n
V
I
V
I
n
n
n
n
n
V
I
T
w
v
u
g
e
e
c
c
c
D
D 

 










             
  



 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
w
c
v
c
u
c n
n
n ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
010
000
000
010 
 ;
                 
    




1 0
1
0
1
0
1
0
001 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c


 









 
    


 


 d
u
d
v
d
w
d
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
1 2
000
,
,

 ;
                 
   
 




1 0
1
0
1
0
1
0
002 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c


 









 
      




 


 d
u
d
v
d
w
d
w
v
u
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
,

 ;
                 
    




1 0
1
0
1
0
1
0
0
0
110 2
,
,
,
~
n
n
n
n
nI
nI
n
n
n
V
I
u
c
v
c
u
s
e
e
c
c
c
n
D
D
I











           








1
0
0
100
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
i
I e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
T
w
v
u
g 






            

   



 
V
I
i
I
n
n
n
nI
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
u
c
v
s
u
c
e
0
0
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
, 




              
    







1 0
1
0
1
0
1
0
100 ,
,
,
~
,
,
,
n
i
I
n
n
n
nI
nI d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
u
s
v
c
u
c
e
e
n





                     
 
    




1 0
1
0
1
0
1
0
100 ,
,
,
~
2
n
n
n
nI
n
n
nI
n
n
n
n w
v
u
I
v
c
u
c
e
c
c
e
c
c
c
c










       
    



 d
u
d
v
d
w
d
v
c
T
w
v
u
g
w
v
u
V
w
v
u
I
w
v
u
V n
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
~
,
,
100
000
000 


                 
    




1 0
1
0
1
0
1
0
0
0
110 2
,
,
,
~
n
n
n
n
nV
nV
n
n
n
I
V
u
c
v
c
u
s
e
e
c
c
c
n
D
D
V











           








1
0
0
100
2
,
,
,
~
,
,
,
n
nV
n
n
n
I
V
i
V e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
V
T
w
v
u
g 






            

   



 
I
V
i
V
n
n
n
nV
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
u
c
v
s
u
c
e
0
0
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
, 





19
              
    







1 0
1
0
1
0
1
0
100 ,
,
,
~
,
,
,
n
i
V
n
n
n
nV
nV d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
u
s
v
c
u
c
e
e
n





                     

     




1 0
1
0
1
0
1
0
100 ,
,
,
~
2
n
n
n
nV
nI
n
n
n
n
n
n w
v
u
I
v
c
u
c
e
e
c
c
c
c
c
c










       
    



 d
u
d
v
d
w
d
v
c
T
w
v
u
g
w
v
u
V
w
v
u
I
w
v
u
V n
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
~
,
,
100
000
000 

 ;
                 
    




1 0
1
0
1
0
1
0
0
0
101 2
,
,
,
~
n
n
n
n
nI
nI
n
n
n
V
I
w
c
v
c
u
s
e
e
c
c
c
n
D
D
I











           






1
0
0
001
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
I e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
T
w
v
u
g 






            

   




V
I
I
n
n
n
nI
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c
v
s
u
c
e
0
0
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
, 




              
    





1 0
1
0
1
0
1
0
001 ,
,
,
~
,
,
,
n
I
n
n
n
nI
nI d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e
e
n





                 
 
   




1 0
1
0
,
, ,
,
,
1
2
n
V
I
V
I
nI
nI
n
n
n
n
n
n T
w
v
u
g
e
e
c
c
c
c
c
c










      

 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
u
cn ,
,
,
~
,
,
,
~
000
100

                 
    




1 0
1
0
1
0
1
0
0
0
101 2
,
,
,
~
n
n
n
n
nV
nV
n
n
n
I
V
w
c
v
c
u
s
e
e
c
c
c
n
D
D
V











           






1
0
0
001
2
,
,
,
~
,
,
,
n
nV
n
n
n
I
V
V e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
V
T
w
v
u
g 






            

   




I
V
V
n
n
n
nV
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
w
c
v
s
u
c
e
0
0
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
, 




              
    





1 0
1
0
1
0
1
0
001 ,
,
,
~
,
,
,
n
V
n
n
n
nV
nV d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
w
s
v
c
u
c
e
e
n





                     
    




1 0
1
0
1
0
1
0
000 ,
,
,
~
2
n
n
n
nV
nV
n
n
n
n
n
n w
v
u
I
v
c
u
c
e
e
c
c
c
c
c
c










   
    

 d
u
d
v
d
w
d
w
v
u
V
T
w
v
u
g
w
c V
I
V
I
n ,
,
,
~
,
,
,
1 100
,
,

 ;
                   
 

     





1 0
1
0
1
0
1
0
,
,
011 ,
,
,
1
2
,
,
,
~
n
I
I
I
I
n
n
n
nI
nI
n
n
n T
w
v
u
g
w
c
v
c
u
c
e
e
c
c
c
I











     
      




 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
I
w
v
u
I V
I
V
I ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
~
000
001
,
,
010
000 


                   
 

     





1 0
1
0
1
0
1
0
,
,
011 ,
,
,
1
2
,
,
,
~
n
V
V
V
V
n
n
n
nV
nV
n
n
n T
w
v
u
g
w
c
v
c
u
c
e
e
c
c
c
V











     
      




 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
V
w
v
u
V V
I
V
I ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
~
001
000
,
,
010
000 

 .
Equations for initial-order approximation of concentrations of simplest complexes of radiation
defects 0(x,y,z,t), corrections for the approximation i(x,y,z,t) (i1), boundary and initial con-
ditions for them have been obtain by substitution of the series (14) into Eqs.(6) and conditions (7)
and equating of coefficients at equal powers of parameters, which have been used in the series
(14). The equations and conditions could be written as

20
       









 2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x I
I
I
I
I
I
I








       
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k I
I
I ,
,
,
,
,
,
,
,
,
,
,
, 2
, 

       









 2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x V
V
V
V
V
V
V








       
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k V
V
V ,
,
,
,
,
,
,
,
,
,
,
, 2
, 
 ;
       










 2
2
0
2
2
0
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x i
I
I
i
I
I
i
I
I
i
I








 
 
 
 






 






 







y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
i
I
I
I
i
I
I
I







 ,
,
,
,
,
,
,
,
,
,
,
,
1
0
1
0
 
 





 




z
t
z
y
x
T
z
y
x
g
z
D
i
I
I
I



 ,
,
,
,
,
,
1
0
       










 2
2
0
2
2
0
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x i
V
V
i
V
V
i
V
V
i
V








 
 
 
 






 






 







y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
i
V
V
V
i
V
V
V







 ,
,
,
,
,
,
,
,
,
,
,
,
1
0
1
0
 
 





 




z
t
z
y
x
T
z
y
x
g
z
D
i
V
V
V



 ,
,
,
,
,
,
1
0 , i1;
 
0
,
,
,
0





x
i
x
t
z
y
x

,
 
0
,
,
,




 x
L
x
i
x
t
z
y
x

,
 
0
,
,
,
0





y
i
y
t
z
y
x

,
 
0
,
,
,




 y
L
y
i
y
t
z
y
x

,
 
0
,
,
,
0





z
i
z
t
z
y
x

,
 
0
,
,
,




 z
L
z
i
z
t
z
y
x

, i>0; 0(x,y,z,0)=f (x,y,z), i(x,y,z,0)=0, i>1.
result
               












1
1
0
2
2
1
,
,
,
n
n
n
n
n
n
n
n
n
n
z
y
x
z
y
x
z
c
y
c
x
c
n
L
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
t
z
y
x 


               
 
    


 

t L L L
I
I
I
n
n
n
n
x y z
w
v
u
I
T
w
v
u
k
w
v
u
I
T
w
v
u
k
v
c
u
c
e
t
e
0 0 0 0
2
, ,
,
,
,
,
,
,
,
,
,
,
, 




  
d
u
d
v
d
w
d
w
cn
 ,
where  



















 
 2
2
2
0
2
2 1
1
1
exp
z
y
x
n
L
L
L
t
D
n
t
e 

 ,      
   
 

x y z
L L L
n
n
n w
v
u
f
v
c
u
c
F
0 0 0
,
,


  u
d
v
d
w
d
w
cn
 , cn(x)=cos(nx/Lx);
                 
    









1 0 0 0 0
2
,
,
,
2
,
,
,
n
t L L L
n
n
n
n
n
n
n
z
y
x
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
t
z
y
x 





 
 
            
  









1 0 0
2
1 2
,
,
,
n
t L
n
n
n
n
n
n
z
y
x
i
I
n
x
u
c
e
t
e
z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
w
c 









21
     
 
     


 







1
2
0 0
1 2
,
,
,
,
,
,
n
n
n
n
z
y
x
L L
i
I
n
n t
e
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
T
w
v
u
g
w
c
v
s
y z








           
 
   






t L L L
i
I
n
n
n
n
n
x y z
d
u
d
v
d
w
d
w
w
v
u
T
w
v
u
g
w
s
v
c
u
c
e
z
c
0 0 0 0
1 ,
,
,
,
,
, 



 


, i1,
where sn(x)=sin(nx/Lx).
Equations for initial-order approximation of dopant concentration C00(x,y,z,t), corrections for the
approximation Cij(x,y,z,t) (i1, j1), boundary and initial conditions for them have been obtain by
substitution of the series (15) into Eqs.(1) and conditions (2) and equating of coefficients at equal
powers of parameters, which have been used in the series (15). The equations and conditions
could be written as
       
2
00
2
0
2
00
2
0
2
00
2
0
00 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L











;
       











2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C i
L
i
L
i
L
i
        





















 

y
t
z
y
x
C
T
z
y
x
g
y
D
x
t
z
y
x
C
T
z
y
x
g
x
D i
L
L
i
L
L
,
,
,
,
,
,
,
,
,
,
,
, 10
0
10
0
   










 
z
t
z
y
x
C
T
z
y
x
g
z
D i
L
L
,
,
,
,
,
, 10
0 , i1;
       











2
01
2
0
2
01
2
0
2
01
2
0
01 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
 
 
   
 
  






















y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
D
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
D L
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
00
0
00
00
0 



 
 
 











z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
D L
,
,
,
,
,
,
,
,
, 00
00
0 

;
       











2
02
2
0
2
02
2
0
2
02
2
0
02 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
   
 
     
 




















T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
x
D L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
00
01
00
1
00
01
0 



     
 
  



















L
L
L D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
D
D
y
t
z
y
x
C
0
00
1
00
01
0
0
00 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,


 
 
   
 
  






















y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
D
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 01
00
0
01
00




 
 
 











z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
D L
,
,
,
,
,
,
,
,
, 01
00
0 

;
       











2
11
2
0
2
11
2
0
2
11
2
0
11 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L

22
   
 
     
 




















T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
x
D L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
00
10
00
1
00
10
0 



     
 
  



















L
L
L D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
D
D
y
t
z
y
x
C
0
00
1
00
10
0
0
00 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,


 
 
   
 
  






















y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
D
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 10
00
0
10
00




 
 
      






















x
t
z
y
x
C
T
z
y
x
g
x
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
D L
L
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 01
0
10
00
0 

       






















z
t
z
y
x
C
T
z
y
x
g
z
D
y
t
z
y
x
C
T
z
y
x
g
y
D L
L
L
L
,
,
,
,
,
,
,
,
,
,
,
, 01
0
01
0 ;
 
0
,
,
,
0


x
ij
x
t
z
y
x
C


,
 
0
,
,
,

 x
L
x
ij
x
t
z
y
x
C


,
 
0
,
,
,
0


y
ij
y
t
z
y
x
C


,
 
0
,
,
,

 y
L
y
ij
y
t
z
y
x
C


,
 
0
,
,
,
0


z
ij
z
t
z
y
x
C


,
 
0
,
,
,

 z
L
z
ij
z
t
z
y
x
C


, i0, j0;
C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i1, j1.
result
         




1
00
2
1
,
,
,
n
nC
n
n
n
nC
z
y
x
z
y
x
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
t
z
y
x
C ,
where  



















 
 2
2
2
0
2
2 1
1
1
exp
z
y
x
n
L
L
L
t
D
n
t
e 

 ,      
   

x y z
L L L
C
n
n
nC w
v
u
f
v
c
u
c
F
0 0 0
,
,
  u
d
v
d
w
d
w
cn
 ;
                 
    




1 0 0 0 0
2
0 ,
,
,
2
,
,
,
n
t L L L
L
n
n
nC
nC
n
n
n
nC
z
y
x
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C 

               
  








1 0 0
2
10 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
i
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
v
c 



              


 






1
2
0 0
10 2
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
i
L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
v
c
v
s
y z 


             
   



 
t L L L
i
L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
v
s
v
c
u
c
e
t
e
0 0 0 0
10 ,
,
,
,
,
, 

 , i1;
                 
 

    




1 0 0 0 0
00
2
01
,
,
,
,
,
,
2
,
,
,
n
t L L L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
T
w
v
u
P
w
v
u
C
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C 




                
  






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23
           
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24
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, 
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 

.
Authors:
Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary
school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:
from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra-
diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio-
physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro-
structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo-
rod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor
course in Radiophysical Department of Nizhny Novgorod State University. He has 105 published papers in
area of his researches.
Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of
village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school
“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec-
ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she
is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in
the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant № MK-548.2010.2).
She has 52 published papers in area of her researches.

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An Approach to Model Optimization of Man-Ufacturing of Emitter-Coupled Logic