AJCSE

1. © 2019, AJCSE. All Rights Reserved 6
RESEARCH ARTICLE
On Prognosis of Manufacturing of a Broadband Power Amplifiers based on
Heterotransistors to Increase Density of their Elements with Account Miss-match
Induced Stress and Porosity of Materials
E. L. Pankratov1,2
1
Department of Mathematival and Natural Sciences, Nizhny Novgorod State University, 23 Gagarin Avenue,
Nizhny Novgorod, 603950, Russia, 2
Department of Higher Mathenatics, Nizhny Novgorod State Technical
University, 24 Minin Street, Nizhny Novgorod, 603950, Russia
Received on: 01-12-2018; Revised on: 25-12-2018; Accepted on: 25-01-2019
ABSTRACT
In this paper, we introduce an approach to increase the density of field-effect transistors framework
a broadband power amplifier. Framework the approach, we consider manufacturing the inverter in
heterostructure with specific configuration. Several required areas of the heterostructure should be doped
by diffusion or ion implantation. After that, dopant and radiation defects should be annealed framework
optimized scheme. We also consider an approach to decrease the value of mismatch-induced stress in
the considered heterostructure. We introduce an analytical approach to analyze mass and heat transport
in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
Key words: Broadband power amplifier, optimization of manufacturing, increasing of element
integration rate
INTRODUCTION
In the present time, several actual problems of the solid-state electronics (such as increasing of
performance, reliability, and density of elements of integrated circuits: Diodes, field-effect, and bipolar
transistors) are intensively solving.[1-6]
To increase the performance of these devices, it is attracted an
interest determination of materials with higher values of charge carriers mobility.[7-10]
One way to decrease
dimensions of elements of integrated circuits is manufacturing them in thin-film heterostructures.[3-5,11]
In this case, it is possible to use inhomogeneity of heterostructure and necessary optimization of doping
of electronic materials[12]
and development of epitaxial technology to improve these materials (including
analysis of mismatch-induced stress).[13-15]
An alternative approach to increase dimensions of integrated
circuits is using of laser and microwave types of annealing.[16-18]
Framework the paper, we introduce an approach to manufacture field-effect transistors. The approach gives
a possibility to decrease their dimensions with increasing their density framework a broadband power
amplifier. We also consider possibility to decrease mismatch-induced stress to decrease the quantity of
defects, generated due to the stress. In this paper, we consider a heterostructure, which consist of a substrate
and an epitaxial layer [Figure 1]. We also consider a buffer layer between the substrate and the epitaxial
layer. The epitaxial layer includes itself several sections, which were manufactured using another material.
These sections have been doped by diffusion or ion implantation to manufacture the required types of
conductivity (p or n). These areas became sources, drains, and gates [Figure 1]. After this doping, it is
required annealing of dopant and/or radiation defects. The main aim of the present paper is analysis of
redistribution of dopant and radiation defects to determine conditions, which correspond to decrease of
elements of the considered voltage reference and at the same time to increase their density. At the same
time, we consider a possibility to decrease mismatch-induced stress.
Address of correspondence:
E. L. Pankratov
E-mail: elp2004@mail.ru
Available Online at www.ajcse.info
Asian Journal of Computer Science Engineering 2019;4(1):6-29
ISSN 2581 – 3781

2. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 7
Method of solution
To solve our aim, we determine and analyzed spatiotemporal distribution of concentration of dopant in
the considered heterostructure. We determine the distribution by solving the second Fick’s law in the
following form: [1,20-23]
∂
∂
∂
∂
∂
∂
∂
C x y z t
t x
D
C x y z t
x y
D
C x y z t
y
, , , , , , , , ,
( )
=
∂ ( )
∂





 +
( )
∂






 +
( )






∂
∂
∂
z
D
C x y z t
z
, , ,

+ ∇ ( ) ( )








∫
Ω
∂
∂ x
D
kT
x y z t C x y W t dW
S
S
Lz
µ1
0
, , , , , ,
+ ∇ ( ) ( )








∫
Ω
∂
∂ y
D
kT
x y z t C x y W t dW
S
S
Lz
µ1
0
, , , , , , (1)
+
( )





 +
( )




∂
∂
∂ µ
∂
∂
∂
∂ µ
∂
x
D
V kT
x y z t
x y
D
V kT
x y z t
y
C S C S
2 2
, , , , , ,


 +
( )






∂
∂
∂ µ
∂
z
D
V kT
x y z t
z
C S 2 , , ,
with boundary and initial conditions
∂ ( )
∂
=
=
C x y z t
x x
, , ,
0
0 ,
∂ ( )
∂
=
=
C x y z t
x x Lx
, , ,
0,
∂ ( )
∂
=
=
C x y z t
y y
, , ,
0
0 , C (x,y,z,0)=fC
(x,y,z),
Figure 1: (a) Structure of the considered amplifier[19]
(b) Heterostructure with a substrate, epitaxial layers, and buffer layer
(view from side)
b
a

3. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 8
∂ ( )
∂
=
=
C x y z t
y x Ly
, , ,
0 ,
∂ ( )
∂
=
=
C x y z t
z z
, , ,
0
0,
∂ ( )
∂
=
=
C x y z t
z x Lz
, , ,
0.
Here, C(x,y,z,t) is the spatiotemporal distribution of concentration of dopant; Ω is the atomic volume of
dopant; ∇s
is the symbol of surficial gradient; C x y z t d z
Lz
, , ,
( )
∫
0
is the surficial concentration of dopant
on interface between layers of heterostructure (in this situation, we assume that Z-axis is perpendicular
to interface between layers of heterostructure); µ1
(x,y,z,t) and µ2
(x,y,z,t) are the chemical potential due to
the presence of mismatch-induced stress and porosity of material; and D and DS
are the coefficients of
volumetric and surficial diffusions. Values of dopant diffusion coefficients depend on properties of
materialsofheterostructure,speedofheatingandcoolingofmaterialsduringannealing,andspatiotemporal
distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters
could be approximated by the following relations: [24-26]
D D x y z T
C x y z t
P x y z T
V x y z t
C L
= ( ) +
( )
( )





 +
(
, , ,
, , ,
, , ,
, , ,
1 1 1
ξ ς
γ
γ
)
)
+
( )
( )








V
V x y z t
V
* *
, , ,
ς2
2
2
,
D D x y z T
C x y z t
P x y z T
V x y z
S S L S
= ( ) +
( )
( )





 +
, , ,
, , ,
, , ,
, , ,
1 1 1
ξ ς
γ
γ
t
t
V
V x y z t
V
( )
+
( )
( )








* *
, , ,
ς2
2
2
. (2)
Here, DL
(x,y,z,T) and DLS
(x,y,z,T) are the spatial (due to accounting all layers of heterostructure) and
temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature
of annealing; P(x,y,z,T) is the limit of solubility of dopant; parameter γ depends on properties of
materials and could be integer in the following interval γ ∈[1,3];[24]
V (x,y,z,t) is the spatiotemporal
distribution of concentration of radiation vacancies; V* is the equilibrium distribution of vacancies.
Concentration dependence of dopant diffusion coefficient has been described in details in Kitayama
et al.[24]
Spatiotemporal distributions of concentration of point radiation defects have been determined
by solving the following system of equations: [20-23,25,26]








I x y z t
t x
D x y z T
I x y z t
x y
D x y z
I I
, , ,
, , ,
, , ,
, ,
( )
= ( )
( )





 + ,
,
, , ,
T
I x y z t
y
( )
( )








+ ( )
( )





 − ( ) (




z
D x y z T
I x y z t
z
k x y z T I x y z t
I I I
, , ,
, , ,
, , , , , ,
,
2
)
)− ( )
k x y z T
I V
, , , ,
( ) ( ) ( ) ( )
0
, , , , , , , , , , , ,
z
L
IS
S
D
I x y z t V x y z t x y z t I x y W t dW
x kT

 
∂
× + Ω ∇
 
∂  
 
∫
( ) ( )
( )
2
0
, , ,
, , , , , ,
z
L
I S
IS
S
D x y z t
D
x y z t I x y W t dW
y kT x V kT x


   
∂
∂ ∂
+Ω ∇ +
   
∂ ∂ ∂
   
 
∫
( ) ( )
2 2
, , , , , ,
I S I S
D D
x y z t x y z t
y V kT y z V kT z
 
   
∂ ∂
∂ ∂
+ +
   
∂ ∂ ∂ ∂
   
(3)








V x y z t
t x
D x y z T
V x y z t
x y
D x y z
V V
, , ,
, , ,
, , ,
, ,
( )
= ( )
( )





 + ,
,
, , ,
T
V x y z t
y
( )
( )









4. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 9
+ ( )
( )





 − ( ) (




z
D x y z T
V x y z t
z
k x y z T V x y z t
V V V
, , ,
, , ,
, , , , , ,
,
2
)
)− ( )
k x y z T
I V
, , , ,
( ) ( ) ( ) ( )
0
, , , , , , , , , , , ,
z
L
VS
S
D
I x y z t V x y z t x y z t V x y W t dW
x kT

 
∂
× + Ω ∇
 
∂  
 
∫
( ) ( )
( )
2
0
, , ,
, , , , , ,
z
L
V S
VS
S
D x y z t
D
x y z t V x y W t dW
y kT x V kT x


   
∂
∂ ∂
+Ω ∇ +
   
∂ ∂ ∂
   
 
∫
( ) ( )
2 2
, , , , , ,
V S V S
D D
x y z t x y z t
y V kT y z V kT z

   
∂ ∂
∂ ∂
+ +
   
∂ ∂ ∂ ∂
   
m
with boundary and initial conditions


I x y z t
x x
, , ,
( )
=
=0
0,


I x y z t
x x Lx
, , ,
( )
=
=
0 ,


I x y z t
y y
, , ,
( )
=
=0
0 ,


I x y z t
y y Ly
, , ,
( )
=
=
0 ,


I x y z t
z z
, , ,
( )
=
=0
0 ,


I x y z t
z z Lz
, , ,
( )
=
=
0 ,


V x y z t
x x
, , ,
( )
=
=0
0,


V x y z t
x x Lx
, , ,
( )
=
=
0,


V x y z t
y y
, , ,
( )
=
=0
0 , (4)


V x y z t
y y Ly
, , ,
( )
=
=
0 ,


V x y z t
z z
, , ,
( )
=
=0
0,


V x y z t
z z Lz
, , ,
( )
=
=
0 , I (x,y,z,0)=
=fI
(x,y,z), V (x,y,z,0)=fV
(x,y,z), V x V t y V t z V t t V
kT x y z
n n n
1 1 1
1
2
1
2
1
2
1
2
+ + +
( )= +
+ +






∞
, , ,

.
Here, I (x,y,z,t) is the spatiotemporal distribution of concentration of radiation interstitials; I* is the
equilibrium distribution of interstitials; DI
(x,y,z,T), DV
(x,y,z,T), DIS
(x,y, z,T), and DVS
(x,y,z,T) are the
coefficients of volumetric and surficial diffusions of interstitials and vacancies, respectively; terms
V2
(x,y,z,t) and I2
(x,y,z,t) correspond to the generation of divacancies and di-interstitials, respectively (for
example, [26]
and appropriate references in this book); kI,V
(x,y,z,T), kI,I
(x,y,z,T), and kV,V
(x,y,z,T) are the
parameters of recombination of point radiation defects and generation of their complexes; k is the
Boltzmann constant; ω = a3
, a is the interatomic distance; and  is the specific surface energy. To
account the porosity of buffer layers, we assume that porous is approximately cylindrical with average
values r x y
= +
1
2
1
2
and z1
before annealing.[23]
With time, small pores decomposing on vacancies. The
vacancies absorbing by larger pores.[27]
With time, large pores became larger due to absorbing the
vacancies and became more spherical.[27]
Distribution of concentration of vacancies in heterostructure,
existing due to porosity, could be determined by summing on all pores, i.e.,
V x y z t V x i y j z k t
p
k
n
j
m
i
l
, , , , , ,
( ) = + + +
( )
=
=
=
∑
∑
∑ α β χ
0
0
0
, R x y z
= + +
2 2 2
.
Here, α, β, and χ are the average distances between centers of pores in directions x, y, and z; l, m, and n
are the quantity of pores inappropriate directions.

5. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 10
Spatiotemporal distributions of divacancies ΦV
(x,y,z,t) and di-interstitials ΦI
(x,y,z, t) could be determined
by solving the following system of equations: [25,26]








Φ Φ
Φ Φ
I I
x y z t
t x
D x y z T
x y z t
x y
D x
I I
, , ,
, , ,
, , ,
( )
= ( )
( )





 + ,
, , ,
, , ,
y z T
x y z t
y
I
( )
( )








Φ
+ ( )
( )





 +
∂
∂
∇
∂
∂
∂
∂
µ
z
D x y z T
x y z t
z x
D
kT
x y z t
I
I
I S
S
Φ
Φ
Φ
Ω
, , ,
, , ,
, , ,
1 (
( ) ( )








∫ ΦI
L
x y W t dW
z
, , ,
0
( ) ( ) ( ) ( )
2
1 ,
0
, , , , , , , , , , , ,
z
I
L
S
S I I I
D
x y z t x y W t dW k x y z T I x y z t
y kT

Φ
 
∂
+Ω ∇ Φ +
 
∂  
 
∫
( ) ( ) ( )
2 2 2
, , , , , , , , ,
I I I
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )
k x y z T I x y z t
I , , , , , , (5)








Φ Φ
Φ Φ
V V
x y z t
t x
D x y z T
x y z t
x y
D x
V V
, , ,
, , ,
, , ,
( )
= ( )
( )





 + ,
, , ,
, , ,
y z T
x y z t
y
V
( )
( )








Φ
+ ( )
( )





 +
∂
∂
∇
∂
∂
∂
∂
µ
z
D x y z T
x y z t
z x
D
kT
x y z t
V
V
V S
S
Φ
Φ
Φ
Ω
, , ,
, , ,
, , ,
1 (
( ) ( )








∫ ΦV
L
x y W t dW
z
, , ,
0
( ) ( ) ( ) ( )
2
1 ,
0
, , , , , , , , , , , ,
z
V
L
S
S V V V
D
x y z t x y W t dW k x y z T V x y z t
y kT

Φ
 
∂
+Ω ∇ Φ +
 
∂  
 
∫
( ) ( ) ( )
2 2 2
, , , , , , , , ,
V V V
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )
k x y z T V x y z t
V , , , , , ,
with boundary and initial conditions


ΦI
x
x y z t
x
, , ,
( )
=
=0
0 ,


ΦI
x L
x y z t
x
x
, , ,
( )
=
=
0 ,


ΦI
y
x y z t
y
, , ,
( )
=
=0
0 ,


ΦI
y L
x y z t
y
y
, , ,
( )
=
=
0 ,


ΦI
z
x y z t
z
, , ,
( )
=
=0
0 ,


ΦI
z L
x y z t
z
z
, , ,
( )
=
=
0,


ΦV
x
x y z t
x
, , ,
( )
=
=0
0 ,


ΦV
x L
x y z t
x
x
, , ,
( )
=
=
0 ,


ΦV
y
x y z t
y
, , ,
( )
=
=0
0 , (6)


ΦV
y L
x y z t
y
y
, , ,
( )
=
=
0
,


ΦV
z
x y z t
z
, , ,
( )
=
=0
0,


ΦV
z L
x y z t
z
z
, , ,
( )
=
=
0 ,
ΦI
(x,y,z,0)=fΦI
(x,y,z), ΦV
(x,y,z,0)=fΦV
(x,y,z).

6. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 11
Here, DΦI
(x,y,z,T), DΦV
(x,y,z,T), DΦIS
(x,y,z,T), and DΦVS
(x,y,z,T) are the coefficients of volumetric and
surficial diffusions 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.
Chemical potential µ1
in Equation (1) could be determined by the following relation: [20]
µ1
=E(z)Ωσij
[uij
(x,y,z,t)+uji
(x,y,z,t)]/2, (7)
Where, E(z) is the Young modulus, σij
is the stress tensor; u
u
x
u
x
ij
i
j
j
i
=
∂
∂
+
∂
∂






1
2
is the deformation tensor;
ui
and uj
are the components ux
(x,y,z,t), uy
(x,y,z,t), and uz
(x,y,z,t) of the displacement vector

u x y z t
, , ,
( );
xi
and xj
are the coordinate x, y, and z. The Equation (3) could be transformed to the following form:
( )
( ) ( ) ( ) ( )
, , , , , ,
, , , , , ,
1
, , ,
2
j j
i i
j i j i
u x y z t u x y z t
u x y z t u x y z t
x y z t
x x x x


   
∂ ∂
∂ ∂

=
+ +
   

∂ ∂ ∂ ∂
   

   

− +
( )
− ( )
∂ ( )
∂
−





 − ( ) ( )
ε δ
σ δ
σ
ε β
0 0
1 2
3
ij
ij k
k
z
z
u x y z t
x
K z z T x
, , ,
, y
y z t T E z
ij
, ,
( )−

 






( )
0
2
δ
Ω
,
Where, σ is Poisson coefficient; ε0
= (as
− aEL
)/aEL
is the mismatch parameter; as
and aEL
are lattice
distances of the substrate and the epitaxial layer; K is the modulus of uniform compression; β is the
coefficient of thermal expansion; and Tr
is the equilibrium temperature, which coincide (for our case) with
room temperature. Components of displacement vector could be obtained by solution of the following
equations: [21]
ρ
σ σ σ
z
u x y z t
t
x y z t
x
x y z t
y
x
x xx xy xz
( )
( )
∂
=
( )
∂
+
( )
∂
+
∂ ∂ ∂ ∂
2
2
, , , , , , , , , ,
, , ,
y z t
z
( )
∂
ρ
σ σ σ
z
u x y z t
t
x y z t
x
x y z t
y
x
y yx yy yz
( )
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
∂
2
2
, , , , , , , , , ,
, , ,
y z t
z
( )
∂
ρ
σ σ σ
z
u x y z t
t
x y z t
x
x y z t
y
x
z zx zy zz
( )
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
∂
2
2
, , , , , , , , , ,
, , ,
y z t
z
( )
∂
,
where σ
σ
δ
ij
i
j
j
i
ij k
E z
z
u x y z t
x
u x y z t
x
u
=
( )
+ ( )

 

∂ ( )
∂
+
∂ ( )
∂
−
∂
2 1 3
, , , , , , x
x y z t
x
K z
k
ij
, , ,
( )
∂








+ ( ) ×
δ
×
∂ ( )
∂
− ( ) ( ) ( )−

 

u x y z t
x
z K z T x y z t T
k
k
r
, , ,
, , ,
 , ρ (z) is the density of materials of heterostructure and δij
is the Kronecker symbol. With account, the relation for σij
last system of equation could be written as
ρ
σ
z
u x y z t
t
K z
E z
z
u x
x x
( )
∂ ( )
∂
= ( )+
( )
+ ( )

 











∂
2
2
2
5
6 1
, , , , y
y z t
x
K z
E z
z
, ,
( )
∂
+ ( )−
( )
+ ( )

 











2
3 1 σ
×
∂ ( )
∂ ∂
+
( )
+ ( )

 

∂ ( )
∂
+
∂
2 2
2
2
2 1
u x y z t
x y
E z
z
u x y z t
y
u x y
y y z
, , , , , , ,

,
, ,
z t
z
K z
E z
z
( )
∂





 + ( )+
( )
+ ( )

 









2
3 1 
×
∂ ( )
∂ ∂
− ( ) ( )
∂ ( )
∂
2
u x y z t
x z
K z z
T x y z t
x
z , , , , , ,


7. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 12
ρ
σ
z
u x y z t
t
E z
z
u x y z t
x
u x
y y x
( )
∂ ( )
∂
=
( )
+ ( )

 

∂ ( )
∂
+
∂
2
2
2
2
2
2 1
, , , , , , ,
, , , , , ,
y z t
x y
T x y z t
y
( )
∂ ∂





 −
∂ ( )
∂
× ( ) ( )+
∂
∂
( )
+ ( )

 

∂ ( )
∂
+
∂ ( )
∂

K z z
z
E z
z
u x y z t
z
u x y z t
y
y z
β
σ
2 1
, , , , , ,
















+
∂ ( )
∂
2
2
u x y z t
y
y , , ,
(8)
×
( )
+ ( )

 

+ ( )










+ ( )−
( )
+ ( )

 





5
12 1 6 1
E z
z
K z K z
E z
z
 







∂ ( )
∂ ∂
+ ( )
∂ ( )
∂ ∂
2 2
u x y z t
y z
K z
u x y z t
x y
y y
, , , , , ,
ρ
σ
z
u x y z t
t
E z
z
u x y z t
x
u x
z z z
( )
∂ ( )
∂
=
( )
+ ( )

 

∂ ( )
∂
+
∂
2
2
2
2
2
2 1
, , , , , , ,
, , , , , ,
y z t
y
u x y z t
x z
x
( )
∂
+
∂ ( )
∂ ∂


 2
2
+
∂ ( )
∂ ∂


 +
∂
∂
( )
∂ ( )
∂
+
∂ ( )
∂
2
u x y z t
y z z
K z
u x y z t
x
u x y z t
y
y x y
, , , , , , , , ,
+
+
∂ ( )
∂
















u x y z t
z
x , , ,
+
∂
∂
( )
+ ( )
∂ ( )
∂
−
∂ ( )
∂
−
∂ (
1
6 1
6
z
E z
z
u x y z t
z
u x y z t
x
u x y z t
z x y

, , , , , , , , , )
)
∂
−
∂ ( )
∂
















y
u x y z t
z
z , , ,
− ( ) ( )
∂ ( )
∂
K z z
T x y z t
z

, , ,
.
Conditions for the system of Equation (8) could be written in the form
∂ ( )
∂
=

u y z t
x
0
0
, , ,
;
∂ ( )
∂
=

u L y z t
x
x , , ,
0 ;
∂ ( )
∂
=

u x z t
y
, , ,
0
0 ;
∂ ( )
∂
=

u x L z t
y
y
, , ,
0 ;
∂ ( )
∂
=

u x y t
z
, , ,
0
0 ;
∂ ( )
∂
=

u x y L t
z
z
, , ,
0;
 
u x y z u
, , ,0 0
( ) = ;
 
u x y z u
, , ,∞
( ) = 0 .
We determine spatiotemporal distributions of concentrations of dopant and radiation defects by solving
the Equations (1), (3), and (5) framework standard method of averaging of function corrections.[28]
Previously, we transform the Equations (1), (3), and (5) to the following form with account initial
distributions of the considered concentrations
∂ ( )
∂
=
∂
∂
∂ ( )
∂





 +
∂
∂
∂ ( )
∂



C x y z t
t x
D
C x y z t
x y
D
C x y z t
y
, , , , , , , , , 


 +
∂
∂
∂ ( )
∂





 +
z
D
C x y z t
z
, , ,
(1a)
+
( )





 +
( )




∂
∂
∂ µ
∂
∂
∂
∂ µ
∂
x
D
V kT
x y z t
x y
D
V kT
x y z t
y
C S C S
2 2
, , , , , ,


 +
( )






∂
∂
∂ µ
∂
z
D
V kT
x y z t
z
C S 2 , , ,
+
( )





 +
∂
∂
∇ ( )
∂
∂
∂ µ
∂
µ
z
D
V k T
x y z t
z x
D
kT
x y z t C x y W
C S S
S
2 , , ,
, , , , , ,
Ω t
t dW
Lz
( )








∫
0
+
∂
∂
∇ ( ) ( )








∫
Ω
y
D
kT
x y z t C x y W t dW
S
S
Lz
 , , , , , ,
0

8. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 13








I x y z t
t x
D x y z T
I x y z t
x y
D x y z
I I
, , ,
, , ,
, , ,
, ,
( )
= ( )
( )





 + ,
,
, , ,
T
I x y z t
y
( )
( )








+ ( )
( )





 +
∂
∂
∇ ( )
∂
∂
∂
∂
µ
z
D x y z T
I x y z t
z x
D
kT
x y z t I
I
IS
S
, , ,
, , ,
, , ,
Ω 1 x
x y W t dW
Lz
, , ,
( )








∫
0
( ) ( ) ( ) ( )
2
1 ,
0
, , , , , , , , , , , ,
z
L
IS
S I I
D
x y z t I x y W t dW k x y z T I x y z t
y kT

 
∂
+Ω ∇ −
 
∂  
 
∫
− ( ) ( ) ( )+ ( ) ( )
k x y z T I x y z t V x y z t f x y z t
I V I
, , , , , , , , , , , ,  (3a)








V x y z t
t x
D x y z T
V x y z t
x y
D x y z
V V
, , ,
, , ,
, , ,
, ,
( )
= ( )
( )





 + ,
,
, , ,
T
V x y z t
y
( )
( )








+ ( )
( )





 +
∂
∂
∇ ( )
∂
∂
∂
∂
µ
z
D x y z T
V x y z t
z x
D
kT
x y z t V
V
VS
S
, , ,
, , ,
, , ,
Ω 1 x
x y W t dW
Lz
, , ,
( )








∫
0
+
∂
∂
∇ ( ) ( )








−
∫
Ω
y
D
kT
x y z t I x y W t dW k x y z
IS
S
L
I I
z
1
0
, , , , , , , , ,
, T
T I x y z t
( ) ( )
2
, , ,
− ( ) ( ) ( )+ ( ) ( )
k x y z T I x y z t V x y z t f x y z t
I V V
, , , , , , , , , , , , 








Φ Φ
Φ Φ
I I
x y z t
t x
D x y z T
x y z t
x y
D x
I I
, , ,
, , ,
, , ,
( )
= ( )
( )





 + ,
, , ,
, , ,
y z T
x y z t
y
I
( )
( )








Φ
+ ( )
( )





 +
∂
∂
∇
∂
∂
∂
∂
µ
z
D x y z T
x y z t
z x
D
kT
x y z t
I
I
I S
S
Φ
Φ
Φ
Ω
, , ,
, , ,
, , ,
1 (
( ) ( )








∫ ΦI
L
x y W t dW
z
, , ,
0
( ) ( ) ( ) ( )
1
0
, , , , , , , , , , , ,
z
I
L
S
S I I
D
x y z t x y W t dW k x y z T I x y z t
y kT

Φ
 
∂
+Ω ∇ Φ +
 
∂  
 
∫
( ) ( ) ( )
2 2 2
, , , , , , , , ,
I I I
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )+ ( ) ( )
k x y z T I x y z t f x y z t
I I I
, , , , , , , , ,
2
Φ  (5a)








Φ Φ
Φ Φ
V V
x y z t
t x
D x y z T
x y z t
x y
D x
V V
, , ,
, , ,
, , ,
( )
= ( )
( )





 + ,
, , ,
, , ,
y z T
x y z t
y
V
( )
( )








Φ
+ ( )
( )





 +
∂
∂
∇
∂
∂
∂
∂
µ
z
D x y z T
x y z t
z x
D
kT
x y z t
V
V
V S
S
Φ
Φ
Φ
Ω
, , ,
, , ,
, , ,
1 (
( ) ( )








∫ ΦV
L
x y W t dW
z
, , ,
0
+
∂
∂
∇ ( ) ( )








+
∫
Ω Φ
Φ
y
D
kT
x y z t x y W t dW k x y z
I
z
S
S I
L
I
1
0
, , , , , , , , ,T
T I x y z t
( ) ( )
, , ,

9. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 14
( ) ( ) ( )
2 2 2
, , , , , , , , ,
V V V
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )+ ( ) ( )
k x y z T V x y z t f x y z t
V V V
, , , , , , , , ,
2
Φ  .
Farther, we replace concentrations of dopant and radiation defects in the right sides of Equations (1a),
(3a), and (5a) on their not yet known average values α1ρ
. In this situation, we obtain equations for the
first-order approximations of the required concentrations in the following form:
∂ ( )
∂
=
∂
∂
∇ ( )





 +
∂
∂
C x y z t
t x
z
D
kT
x y z t
y
z
D
k
C
S
S C
S
1
1 1 1
, , ,
, , ,
α µ α
Ω Ω
T
T
x y z t
S
∇ ( )






µ1 , , ,
+ ( ) ( )+
( )





 +
f x y z t
x
D
V kT
x y z t
x y
D
V kT
x
C
C S C S
, ,
, , ,
δ
∂
∂
∂ µ
∂
∂
∂
∂ µ
2 2 ,
, , ,
y z t
y
( )






∂
+
( )






∂
∂
∂ µ
∂
z
D
V k T
x y z t
z
C S 2 , , ,
(1b)
∂
∂
α µ α
I x y z t
t
z
x
D
kT
x y z t
y
z
D
I
IS
S I
IS
1
1 1
, , ,
, , ,
( )
=
∂
∂
∇ ( )





 +
∂
∂
Ω Ω
k
kT
x y z t
S
∇ ( )






µ , , ,
( ) ( ) ( )
2 2 2
, , , , , , , , ,
I S I S I S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )− ( )− ( )
f x y z t k x y z T k x y z T
I I I I I V I V
, , , , , , , ,
, ,
δ α α α
1
2
1 1 (3b)
∂
∂
α µ α
V x y z t
t
z
x
D
kT
x y z t
y
z
D
V
VS
S V
V
1
1 1 1
, , ,
, , ,
( )
=
∂
∂
∇ ( )





 +
∂
∂
Ω Ω S
S
S
kT
x y z t
∇ ( )






µ1 , , ,
( ) ( ) ( )
2 2 2
, , , , , , , , ,
V S V S V S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )− ( )− ( )
f x y z t k x y z T k x y z T
V V V V I V I V
, , , , , , , ,
, ,
δ α α α
1
2
1 1
∂
∂
α µ α
Φ
Ω Ω
Φ
Φ
Φ
1
1 1 1
I S
S
x y z t
t
z
x
D
kT
x y z t z
I
I
I
, , ,
, , ,
( )
=
∂
∂
∇ ( )





 +
∂
∂
∂
∇ ( )






y
D
kT
x y z t
I S
S
Φ
µ1 , , ,
( ) ( ) ( )
2 2 2
, , , , , , , , ,
I I I
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( ) ( )+ ( ) ( )+ ( )
f x y z t k x y z T I x y z t k x y z T I x y z
I I I I
Φ , , , , , , , , , , , , ,
,
 2
,
,t
( ) (5b)
∂
∂
α µ α
Φ
Ω Ω
Φ
Φ
Φ
1
1 1 1
V S
S
x y z t
t
z
x
D
kT
x y z t z
V
V
V
, , ,
, , ,
( )
=
∂
∂
∇ ( )





 +
∂
∂
∂
∇ ( )






y
D
kT
x y z t
V S
S
Φ
µ1 , , ,
( ) ( ) ( )
2 2 2
, , , , , , , , ,
V V V
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     

10. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 15
+ ( ) ( )+ ( ) ( )+ ( )
f x y z t k x y z T V x y z t k x y z T V x y z
V V V V
Φ , , , , , , , , , , , , ,
,
 2
,
,t
( ).
Integration of the left and right sides of the Equations (1b), (3b), and (5b) on time gives us possibility to
obtain relations for above approximation in the final form
C x y z t
x
D x y z T
z
kT
V x y z
V
V x y
C S L
1 1 1 2
2
1
, , , , , ,
, , , ,
*
( ) =
∂
∂
( ) +
( )
+
α ς
τ
ς
Ω
,
, ,
*
z
V
t
τ
( )
( )








∫ 2
0
×∇ ( ) +
( )











+
∂
∂
S
S C
C S L
x y z
P x y z T
d
y
D x
µ τ
ξ α
τ α
γ
γ
1
1
1
1
, , ,
, , ,
, y
y z T
P x y z T
S C
t
, ,
, , ,
( ) +
( )






∫ 1 1
0
ξ αγ
γ
× ∇ ( ) +
( )
+
( )
( )





Ω S x y z
z
kT
V x y z
V
V x y z
V
µ τ ς
τ
ς
τ
1 1 2
2
2
1
, , ,
, , , , , ,
* *




+ ∫
d
x
D
V kT
C S
t
τ
∂
∂ 0
×
( )
+
( )
+
∫
∂ µ τ
∂
τ
∂
∂
∂ µ τ
∂
τ
∂
∂
∂ µ
2 2
0
x y z
x
d
y
D
V kT
x y z
y
d
z
D
V kT
C S
t
C S
, , , , , , 2
2
0
x y z
z
d
t
, , ,τ
∂
τ
( )
∫
+ ( )
f x y z
C , , (1c)
I x y z t z
x
D
kT
x y z d z
y
D
kT
I
IS
S
t
I
IS
1 1 1
0
1
, , , , , ,
( ) =
∂
∂
∇ ( ) +
∂
∂
∇
∫
α µ τ τ α
Ω Ω S
S
t
x y z d
µ τ τ
1
0
, , ,
( )
∫
+
∂
∂
∂ ( )
∂
+
∂
∂
∂ ( )
∂
∫ ∫
x
D
V kT
x y z t
x
d
y
D
V kT
x y z t
y
d
I S
t
I S
t
µ
τ
µ
τ
2
0
2
0
, , , , , ,
+
+
∂
∂
∂ ( )
∂
∫
z
D
V kT
x y z t
z
d
I S
t
µ
τ
2
0
, , ,
+ ( )− ( ) − ( )
∫ ∫
f x y z k x y z T d k x y z T d
I I I I
t
I V I V
t
, , , , , , , ,
, ,
α τ α α τ
1
2
0
1 1
0
(3c)
V x y z t z
x
D
kT
x y z d z
y
D
kT
V
IS
S
t
V
IS
1 1 1
0
1
, , , , , ,
( ) =
∂
∂
∇ ( ) +
∂
∂
∇
∫
α µ τ τ α
Ω Ω S
S
t
x y z d
µ τ τ
1
0
, , ,
( )
∫
+
∂
∂
∂ ( )
∂
+
∂
∂
∂ ( )
∂
∫ ∫
x
D
V k T
x y z t
x
d
y
D
V kT
x y z t
y
d
V S
t
V S
t
µ
τ
µ
τ
2
0
2
0
, , , , , ,
+
+
∂
∂
∂ ( )
∂
∫
z
D
V kT
x y z t
z
d
V S
t
µ
τ
2
0
, , ,
+ ( )− ( ) − ( )
∫ ∫
f x y z k x y z T d k x y z T d
V V V V
t
I V I V
t
, , , , , , , ,
, ,
α τ α α τ
1
2
0
1 1
0
Φ Ω Ω
Φ
Φ Φ
1 1 1
0
I
S
S
t
S
x y z t z
x
D
kT
x y z d
x
D
kT
I
I I
, , , , , ,
( ) =
∂
∂
∇ ( ) +
∂
∂
∇
∫
α µ τ τ S
S
t
x y z d
µ τ τ
1
0
, , ,
( )
∫
× + ( )+
∂
∂
∂ ( )
∂
+
∂
∂
∫
α
µ τ
τ
1
2
0
Φ Φ
Φ Φ
I I
I I
z f x y z
x
D
V kT
x y z
x
d
y
D
V kT
S
t
S
, ,
, , , ∂
∂ ( )
∂
+
∫
µ τ
τ
2
0
x y z
y
d
t
, , ,
(5c)
+
∂
∂
∂ ( )
∂
+ ( ) ( )
∫ ∫
z
D
V k T
x y z
z
d k x y z T I x y z d
I S
t
I
t
Φ µ τ
τ τ τ
2
0 0
, , ,
, , , , , ,
+ ( ) ( )
∫k x y z T I x y z d
I I
t
, , , , , , ,
2
0
 

11. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 16
Φ Ω Ω
Φ
Φ Φ
1 1 1
0
V
S
S
t
S
x y z t z
x
D
kT
x y z d
x
D
kT
V
V V
, , , , , ,
( ) =
∂
∂
∇ ( ) +
∂
∂
∇
∫
α µ τ τ S
S
t
x y z d
µ τ τ
1
0
, , ,
( )
∫
× + ( )+
∂
∂
∂ ( )
∂
+
∂
∂
∫
α
µ τ
τ
1
2
0
Φ Φ
Φ Φ
V V
V V
z f x y z
x
D
V kT
x y z
x
d
y
D
V kT
S
t
S
, ,
, , , ∂
∂ ( )
∂
∫
µ τ
τ
2
0
x y z
y
d
t
, , ,
+
∂
∂
∂ ( )
∂
+ ( ) ( )
∫ ∫
z
D
V kT
x y z
z
d k x y z T V x y z d
V S
t
V
t
Φ µ τ
τ τ τ
2
0 0
, , ,
, , , , , ,
+ ( ) ( )
∫k x y z T V x y z d
V V
t
, , , , , , ,
2
0
  .
We determine the average values of the first-order approximations of concentrations of dopant and
radiation defects by the following standard relation: [28]
α ρ
ρ
1 1
0
0
0
0
1
= ( )
∫
∫
∫
∫
Θ
Θ
L L L
x y z t d z d y d xd t
x y z
L
L
L z
y
x
, , , . (9)
Substitution of the relations (1c), (3c), and (5c) into relation (9) gives us possibility to obtain required
average values in the following form:
1
0
0
0
1
C
x y z
C
L
L
L
L L L
f x y z d z d y d x
z
y
x
= ( )
∫
∫
∫ , , , 1
3
2
4
2
3
2
1
4
4
4
I
x y z
a A
a
B
a B L L L a
a
=
+
( ) − +
+






Θ Θ
−
+
a A
a
3
4
4
, 


1
00 1 0
0
0
1 00
1
V
IV I
I
L
L
L
I II x y z
S
f x y z d z d y d x S L L L
z
y
x
= ( ) − −
∫
∫
∫
Θ
Θ
, ,









,
where S t k x y z T I x y z t V x y z t d z d y d xd t
ij
i j
L
  
= −
( ) ( ) ( ) ( )
Θ , , , , , , , , , ,
1 1
0
z
z
y
x
L
L
∫
∫
∫
∫ 0
0
0
Θ
, a SII
4 00
=
× −
( )
S S S
IV II VV
00
2
00 00 , a S S S S S
IV II IV II VV
3 00 00 00
2
00 00
= + − , a f x y z d z d y d x
V
L
L
L z
y
x
2
0
0
0
= ( )
∫
∫
∫ , ,
× + + ( )
S S S L L L S S f x y z d z d y d x
IV IV IV x y z VV II I
Lz
00 00
2
00
2 2 2
00 00
0
2
Θ , ,
∫
∫
∫
∫ −
0
0
2 2 2
00
L
L
x y z VV
y
x
L L L S
Θ
− ( )
∫
∫
∫
S f x y z d z d y d x
IV I
L
L
L z
y
x
00
2
0
0
0
, , , a S f x y z d z d y d x
IV I
L
L
L z
y
x
1 00
0
0
0
= ( )
∫
∫
∫ , , , a SVV
0 00
=
× ( )








∫
∫
∫ f x y z d z d y d x
I
L
L
L z
y
x
, ,
0
0
0
2
, A y
a
a
a
a
= + −
8 4
2 3
2
4
2
2
4
Θ Θ , B
a
a
q p q
= + + − −
Θ 2
4
2 3
3
6
− + +
q p q
2 3
3
, q
a
a
a L L L
a a
a
a
a
a
a
a
x y z
= −





 − −



Θ
Θ Θ Θ Θ
3
2
4
2 0
1 3
4
2 0
4
2 2
2 3
2
4
24
4
8
4



 −

12. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 17
− −
Θ Θ
3
2
3
4
3
2 2 2
4
1
2
4
2
54 8
a
a
L L L
a
a
x y z , p
a a L L L a a
a
a
a
x y z
=
−
−
Θ
Θ Θ
2 0 4 1 3
4
2
2
4
4
12 18
,
1
1 20
0
0
1
Φ Φ
Θ Θ
I I
z
R
L L L
S
L L L L L L
f x y z d z d y d x
I
x y z
II
x y z x y z
L
L
= + + ( )
∫ , ,
y
y
x
L
∫
∫
0
1
1 20
0
0
1
Φ Φ
Θ Θ
V V
z
R
L L L
S
L L L L L L
f x y z d z d y d x
V
x y z
VV
x y z x y z
L
L
= + + ( )
∫ , ,
y
y
x
L
∫
∫
0
,
where R t k x y z T I x y z t d z d y d x d t
i I
i
L
L
L z
y
x
 = −
( ) ( ) ( )
∫
∫
∫
∫ Θ
Θ
, , , , , ,
1
0
0
0
0
.
We determine approximations of the second and higher orders of concentrations of dopant and radiation
defects framework standard iterative procedure of the method of averaging of function corrections.[28]
Framework this procedure to determine approximations of the n-th order of concentrations of dopant
and radiation defects, we replace the required concentrations in the Equations (1c), (3c), and (5c) on the
following sum αnρ
+ρn-1
(x,y,z,t). The replacement leads to the following transformation of the appropriate
equations
∂ ( )
∂
=
∂
∂
+
+ ( )

 

( )






C x y z t
t x
C x y z t
P x y z T
C
2 2 1
1
, , , , , ,
, , ,
ξ
α
γ
γ 




+
( )
+
( )
( )












1 1 2
2
2
ς ς
V x y z t
V
V x y z t
V
, , , , , ,
* *
× ( )
∂ ( )
∂


 +
∂
∂
+
( )
+
D x y z T
C x y z t
x y
V x y z t
V
V x y
L , , ,
, , , , , , , ,
*
1
1 2
2
1  
z
z t
V
C x y z t
y
, , , ,
*
( )
( )








∂ ( )
∂




2
1
× ( ) +
+ ( )

 

( )












D x y z T
C x y z t
P x y z T
L
C
, , ,
, , ,
, , ,
1
2 1
ξ
α
γ
γ



+
∂
∂
+
( )
+
( )
( )












z
V x y z t
V
V x y z t
V
1 1 2
2
2
ς ς
, , , , , ,
* *
× ( )
∂ ( )
∂
+
+ ( )

 

D x y z T
C x y z t
z
C x y z t
P x y z T
L
C
, , ,
, , , , , ,
, , ,
1 2 1
1 ξ
α
γ
γ
(
( )














+ ( ) ( )
f x y z t
C , , δ
+
( )





 +
( )




∂
∂
∂ µ
∂
∂
∂
∂ µ
∂
x
D
V kT
x y z t
x y
D
V kT
x y z t
y
C S C S
2 2
, , , , , ,


 +
( )






∂
∂
∂ µ
∂
z
D
V kT
x y z t
z
C S 2 , , ,
+
∂
∂
∇ ( ) + ( )

 











∫
Ω
x
D
kT
x y z t C x y W t dW
S
S C
Lz
µ α
1 2
0
, , , , , ,
+
∂
∂
∇ ( ) + ( )

 











∫
Ω
y
D
kT
x y z t C x y W t dW
S
S C
Lz
µ α
1 2
0
, , , , , , (1d)








I x y z t
t x
D x y z T
I x y z t
x y
D x y
I I
2 1
, , ,
, , ,
, , ,
,
( )
= ( )
( )





 + ,
, ,
, , ,
z T
I x y z t
y
( )
( )








1
+ ( )
( )





 − ( ) +
∂
∂
∂
∂
α
z
D x y z T
I x y z t
z
k x y z T I x y
I I I I
, , ,
, , ,
, , , ,
,
1
1 1 ,
, , , , ,
,
z t k x y z T
I V
( )

 
 − ( )
2

13. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 18
× + ( )

 
 + ( )

 
 +
∂
∂
∇ ( )
α α µ α
1 1 1 1 2
I V S
I x y z t V x y z t
x
x y z t
, , , , , , , , ,
Ω I
I
L
I x y W t dW
z
+ ( )

 






∫ 1
0
, , ,
×



+
∂
∂
∇ ( ) + ( )

 


 ∫
D
kT y
D
kT
x y z t I x y W t dW
IS IS
S I
Lz
Ω µ α
, , , , , ,
2 1
0









+
∂
∂
∂ ( )
∂
∫
x
x y z t
x
t
µ2
0
, , ,
× +
∂
∂
∂ ( )
∂
+
∂
∂
∂
∫
D
V kT
d
y
D
V kT
x y z t
y
d
z
D
V kT
x y z t
I S I S
t
I S
τ
µ
τ
µ
2
0
2
, , , , , ,
(
( )
∂
∫ z
d
t
τ
0
(3d)








V x y z t
t x
D x y z T
V x y z t
x y
D x y
V V
2 1
, , ,
, , ,
, , ,
,
( )
= ( )
( )





 + ,
, ,
, , ,
z T
V x y z t
y
( )
( )








1
+ ( )
( )





 − ( ) +
∂
∂
∂
∂
α
z
D x y z T
V x y z t
z
k x y z T V x y
V V V V
, , ,
, , ,
, , , ,
,
1
1 1 ,
, , , , ,
,
z t k x y z T
I V
( )

 
 − ( )
2
× + ( )

 
 + ( )

 
 +
∂
∂
∇ ( )
α α µ α
1 1 1 1 2
I V S
I x y z t V x y z t
x
x y z t
, , , , , , , , ,
Ω V
V
L
V x y W t dW
z
+ ( )

 






∫ 1
0
, , ,
×



+
∂
∂
∇ ( ) + ( )

 


 ∫
D
k T y
D
kT
x y z t V x y W t dW
VS VS
S V
Lz
Ω µ α
, , , , , ,
2 1
0









+
∂
∂
∂ ( )
∂
∫
x
x y z t
x
t
µ2
0
, , ,
× +
∂
∂
∂ ( )
∂
+
∂
∂
∂
∫
D
V kT
d
y
D
V kT
x y z t
y
d
z
D
V kT
x y z t
V S V S
t
V S
τ
µ
τ
µ
2
0
2
, , , , , ,
(
( )
∂
∫ z
d
t
τ
0








Φ Φ
Φ Φ
2 1
I I
x y z t
t x
D x y z T
x y z t
x y
D
I
, , ,
, , ,
, , ,
( )
= ( )
( )





 + I
I
x y z T
x y z t
y
I
, , ,
, , ,
( )
( )








Φ1
+
∂
∂
∇ ( ) + ( )

 









∫
Ω Φ
Φ
Φ
x
D
kT
x y z t x y W t dW
I
I
z
S
S I
L
µ α
, , , , , ,
2 1
0 


+ ( ) ( )
k x y z T I x y z t
I I
, , , , , , ,
2
+
∂
∂
∇ ( ) + ( )

 









∫
Ω Φ
Φ
Φ
y
D
kT
x y z t x y W t dW
I
I
z
S
S I
L
µ α
, , , , , ,
2 1
0 


+ ( ) ( )
k x y z T I x y z t
I , , , , , ,
( ) ( ) ( )
2 2 2
, , , , , , , , ,
I I I
S S S
D D D
x y z t x y z t x y z t
x V kT x y V kT y z V kT z
  
Φ Φ Φ
     
∂ ∂ ∂
∂ ∂ ∂
+ + +
     
∂ ∂ ∂ ∂ ∂ ∂
     
+ ( )
( )





 + ( ) ( )
∂
∂
∂
∂
δ
z
D x y z T
x y z t
z
f x y z t
I I
I
Φ Φ
Φ
, , ,
, , ,
, ,
1
(5d)








Φ Φ
Φ Φ
2 1
V V
x y z t
t x
D x y z T
x y z t
x y
D
V
, , ,
, , ,
, , ,
( )
= ( )
( )





 + V
V
x y z T
x y z t
y
V
, , ,
, , ,
( )
( )








Φ1
+
∂
∂
∇ ( ) + ( )












∫
Ω Φ
Φ
Φ
x
D
kT
x y z t x y W t dW
V
V
z
S
S V
L
µ α
, , , , , ,
2 1
0 


+ ( ) ( )
k x y z T V x y z t
V V
, , , , , , ,
2
+
∂
∂
∇ ( ) + ( )












∫
Ω Φ
Φ
Φ
y
D
kT
x y z t x y W t dW
V
V
z
S
S V
L
µ α
, , , , , ,
2 1
0 


+ ( ) ( )
k x y z T V x y z t
V , , , , , ,

14. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 19
+
∂
∂
∂ ( )
∂





 +
∂
∂
∂ ( )
∂


x
D
V k T
x y z t
x y
D
V kT
x y z t
y
V V
S S
Φ Φ
 
2 2
, , , , , ,




 +
∂
∂
∂ ( )
∂






z
D
V kT
x y z t
z
V S
Φ 2 , , ,
+ ( )
( )





 + ( ) ( )
∂
∂
∂
∂
δ
z
D x y z T
x y z t
z
f x y z t
V V
V
Φ Φ
Φ
, , ,
, , ,
, ,
1
.
Integration of the left and the right sides of Equations (1d), (3d), and (5d) gives us possibility to obtain
relations for the required concentrations in the final form
C x y z t
x
C x y z
P x y z T
C
2
2 1
1
, , ,
, , ,
, , ,
( ) =
∂
∂
+
+ ( )

 

( )









ξ
α τ
γ
γ


+
( )
+
( )
( )








∫ 1 1 2
2
2
0
ς
τ
ς
τ
V x y z
V
V x y z
V
t
, , , , , ,
* *
× ( )
∂ ( )
∂
+
∂
∂
( ) +
( )
D x y z T
C x y z
x
d
y
D x y z T
V x y z
V
L L
, , ,
, , ,
, , ,
, , ,
1
1
1
τ
τ ς
τ
*
* *
, , ,
+
( )
( )








∫ ς
τ
2
2
2
0
V x y z
V
t
×
∂ ( )
∂
+
+ ( )

 

( )









C x y z
y
C x y z t
P x y z T
C
1 2 1
1
, , , , , ,
, , ,
τ
ξ
α
γ
γ


+
∂
∂
+
( )
+
( )
( )








∫
z
V x y z
V
V x y z
V
t
1 1 2
2
2
0
ς
τ
ς
τ
, , , , , ,
* *
× ( )
∂ ( )
∂
+
+ ( )

 

D x y z T
C x y z
z
C x y z
P x y z T
L
C
, , ,
, , , , , ,
, , ,
1 2 1
1
τ
ξ
α τ
γ
γ
(
( )










+ ( )
d f x y z
C
τ , ,
+
∂
∂
∇ ( ) + ( )

 
 +
∂
∂
∇
∫
∫
Ω
x
D
kT
x y z C x y W dW d
y
S
S C
L
t
S
z
µ τ α τ τ µ
, , , , , ,
2 1
0
0
x
x y z
t
, , ,τ
( )
∫
0
× + ( )

 
 +
( )

∫
Ω
D
kT
C x y W dW d
x
D
V kT
x y z t
x
S
C
L
C S
z
α τ τ
∂
∂
∂ µ
∂
2 1
0
2
, , ,
, , ,






+
( )





 +
( )




∂
∂
∂ µ
∂
∂
∂
∂ µ
∂
y
D
V kT
x y z t
y z
D
V kT
x y z t
z
C S C S
2 2
, , , , , ,


 (1e)
I x y z t
x
D x y z T
I x y z
x
d
y
D x y z T
I
t
I
2
1
0
, , , , , ,
, , ,
, , ,
( ) = ( )
( )
+
∫
∂
∂
∂ τ
∂
τ
∂
∂
(
( )
( )
∫
∂ τ
∂
τ
I x y z
y
d
t
1
0
, , ,
+ ( )
( )
− ( ) +
∫
∂
∂
∂ τ
∂
τ α
z
D x y z T
I x y z
z
d k x y z T I x y
I
t
I I I
, , ,
, , ,
, , , , ,
,
1
0
2 1 z
z d
t
,τ τ
( )

 

∫
2
0
− ( ) + ( )

 
 + ( )

 
 +
∫k x y z T I x y z V x y z d
I V I V
t
, , , , , , , , , ,
α τ α τ τ
2 1 2 1
0
∂
∂
∂
∇ ( )
∫
x
x y z
S
t
µ τ
, , ,
0
× + ( )

 
 +
∂
∂
∇ ( ) +
∫
Ω
D
kT
I x y W dW d
y
x y z I x
IS
I
L
S I
z
α τ τ µ τ α
2 1
0
2 1
, , , , , , , y
y W
L
t z
, ,τ
( )

 

∫
∫ 0
0
× +
( )





 +
Ω
D
kT
d W d
x
D
V kT
x y z t
x y
D
V kT
x y
IS I S I S
τ
∂
∂
∂ µ
∂
∂
∂
∂ µ
2 2
, , , , , z
z t
y
,
( )






∂
+
( )





 + ( )
∂
∂
∂ µ
∂
z
D
V k T
x y z t
z
f x y z
I S
I
2 , , ,
, , (3e)

15. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 20
V x y z t
x
D x y z T
V x y z
x
d
y
D x y z T
V
t
V
2
1
0
, , , , , ,
, , ,
, , ,
( ) = ( )
( )
+
∫
∂
∂
∂ τ
∂
τ
∂
∂
(
( )
( )
∫
∂ τ
∂
τ
V x y z
y
d
t
1
0
, , ,
+ ( )
( )
− ( ) +
∫
∂
∂
∂ τ
∂
τ α
z
D x y z T
V x y z
z
d k x y z T V x y
V
t
V V V
, , ,
, , ,
, , , , ,
,
1
0
2 1 z
z d
t
,τ τ
( )

 

∫
2
0
− ( ) + ( )

 
 + ( )

 
 +
∫k x y z T I x y z V x y z d
I V I V
t
, , , , , , , , , ,
α τ α τ τ
2 1 2 1
0
∂
∂
∂
∇ ( )
∫
x
x y z
S
t
µ τ
, , ,
0
× + ( )

 
 +
∂
∂
∇ ( ) +
∫
Ω
D
kT
V x y W dW d
y
x y z V x
VS
V
L
S V
z
α τ τ µ τ α
2 1
0
2 1
, , , , , , , y
y W
L
t z
, ,τ
( )

 

∫
∫ 0
0
× +
( )





 +
Ω
D
kT
d W d
x
D
V kT
x y z t
x y
D
V kT
x y
VS V S V S
τ
∂
∂
∂ µ
∂
∂
∂
∂ µ
2 2
, , , , , z
z t
y
,
( )






∂
+
( )





 + ( )
∂
∂
∂ µ
∂
z
D
V k T
x y z t
z
f x y z
V S
V
2 , , ,
, ,
Φ
Φ Φ
Φ
2
1
0
1
I
I
t
I
x y z t
x
D x y z T
x y z
x
d
y
x
I
, , , , , ,
, , , ,
( ) = ( )
( )
+
∫
∂
∂
∂ τ
∂
τ
∂
∂
∂ y
y z
y
t
, ,τ
∂
( )
∫
0
× ( ) + ( )
( )
+
∂
∂
∇
∫
D x y z T d
z
D x y z T
x y z
z
d
x
I I
I
t
S
Φ Φ
Φ
Ω
, , , , , ,
, , ,
τ
∂
∂
∂ τ
∂
τ
1
0
µ
µ τ
x y z
t
, , ,
( )
∫
0
× + ( )

 
 +
∂
∂
+
∫
D
kT
x y W dW d
y
D
kT
x y
I
I
z
I
I
S
I
L
S
I
Φ
Φ
Φ
Φ
Φ Ω Φ
α τ τ α
2 1
0
2 1
, , , , ,
, ,
W dW
L
t z
τ
( )

 

∫
∫ 0
0
× ∇ ( ) + ( ) ( ) +
∂
∂
∂
∫
S I I
t
S
x y z d k x y z T I x y z d
x
D
V kT
I
µ τ τ τ τ
µ
, , , , , , , , ,
,
2
0
Φ 2
2
0
x y z
x
d
t
, , ,τ
τ
( )
∂
∫
+
∂
∂
∂ ( )
∂
+
∂
∂
∂ ( )
∂
∫
y
D
V kT
x y z
y
d
z
D
V kT
x y z
z
d
I I
S
t
S
Φ Φ
µ τ
τ
µ τ
τ
2
0
2
0
, , , , , ,
t
t
f x y z
I
∫ + ( )
Φ , ,
+ ( ) ( )
∫k x y z T I x y z d
I
t
, , , , , , 
0
(5e)
Φ
Φ Φ
Φ
2
1
0
1
V
V
t
V
x y z t
x
D x y z T
x y z
x
d
y
x
V
, , , , , ,
, , , ,
( ) = ( )
( )
+
∫
∂
∂
∂ τ
∂
τ
∂
∂
∂ y
y z
y
t
, ,τ
∂
( )
∫
0
× ( ) + ( )
( )
+
∂
∂
∇
∫
D x y z T d
z
D x y z T
x y z
z
d
x
V V
V
t
S
Φ Φ
Φ
Ω
, , , , , ,
, , ,
τ
∂
∂
∂ τ
∂
τ
1
0
µ
µ τ
x y z
t
, , ,
( )
∫
0
× + ( )



 +
∂
∂
+
∫
D
kT
x y W dW d
y
D
kT
x y
V
V
z
V
V
S
V
L
S
V
Φ
Φ
Φ
Φ
Φ Ω Φ
α τ τ α
2 1
0
2 1
, , , , ,
, ,
W dW
L
t z
τ
( )




∫
∫ 0
0
× ∇ ( ) + ( ) ( ) +
∂
∂
∂
∫
S V V
t
S
x y z d k x y z T V x y z d
x
D
V kT
V
µ τ τ τ τ
µ
, , , , , , , , ,
,
2
0
Φ 2
2
0
x y z
x
d
t
, , ,τ
τ
( )
∂
∫

16. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 21
+
∂
∂
∂ ( )
∂
+
∂
∂
∂ ( )
∂
∫
y
D
V k T
x y z
y
d
z
D
V kT
x y z
z
d
V V
S
t
S
Φ Φ
µ τ
τ
µ τ
τ
2
0
2
0
, , , , , ,
t
t
f x y z
V
∫ + ( )+
Φ , ,
+ ( ) ( )
∫k x y z T V x y z d
V
t
, , , , , , 
0
.
Average values of the second-order approximations of required approximations using the following
standard relation: [28]
α ρ ρ
ρ
2 2 1
0
0
0
1
= ( )− ( )

 

∫
∫
ΘL L L
x y z t x y z t d z d y d xd t
x y z
L
L
L z
y
x
, , , , , ,
∫
∫
∫
0
Θ
(10)
Substitution of the relations (1e), (3e), and (5e) into relation (10) gives us possibility to obtain relations
for required average values α2ρ
α2C
=0, α2ΦI
=0, α2ΦV
=0, 2
3
2
4
2
3
2
1
4
3
4
4
4
4
V
x y z
b E
b
F
a F L L L b
b
b E
b
=
+
( ) − +
+





 −
+
Θ Θ
,

 
2
2
2
00 2 01 10 02 11
0
2
I
V V VV V VV IV x y z V V IV
IV
C S S S L L L S S
S
=
− − + +
( ) − −
Θ
1
1 2 00
+ V IV
S
,
where b
L L L
S S
L L L
S S
x y z
IV V V
x y z
VV II
4 00
2
00 00
2
00
1 1
= −
Θ Θ
, b
S S
L L L
S S
II VV
x y z
VV IV
3
00 00
01 10
2
= − +
(
Θ
+ )+ + + +
( )+
Θ
Θ
Θ
L L L
S S
L L L
S S S L L L
S
x y z
IV VV
x y z
IV II IV x y z
I
00 00
01 10 01
2 V
V
x y z
VV IV
L L L
S S
00
2
01 10
2
Θ
+
(
+ )−
Θ
Θ
L L L
S S
L L L
x y z
IV IV
x y z
00
2
10
3 3 3 3
, b
S S
L L L
S S C S S L L
II VV
x y z
VV IV V IV VV x y
2
00 00
02 11 10 01
2
= + +
( )− − +
(
Θ
Θ
× ) + + +
( )+
L
S S
L L L
L L L S S
S
L L L
z
IV VV
x y z
x y z II IV
IV
x y z
2 01 00
10 01
00
2
Θ
Θ
Θ
S
S S S L L
IV II IV x y
01 10 01
2 2
+ + +
( Θ
× ) + +
( )− − −
( )+
L S L L L S
S
L L L
C S S
C
z VV x y z IV
IV
x y z
V VV IV
2 01 10
00
2
02 11
Θ
Θ
I
I IV
x y z
IV
x y z
S
L L L
S
L L L
00
2
2 2 2 2
10
2
Θ Θ
−
×S S
IV IV
00 01 , b S
S S C
L L L
S S L L L
S
II
IV VV V
x y z
VV IV x y z
IV
1 00
11 02
01 10
01
2
=
+ +
+ +
( )+
Θ
Θ
ΘL
L L L
L L
x y z
x y
Θ
(
× + + ) + +
( )− −
L S S S S LL L
S S
L L L
z II IV VV IV y z
IV IV
x y z
2 2
10 01 01 10
10 01
2
Θ
Θ
S
S
L L L
S S
IV
x y z
IV II
00
01 10
3 2
Θ
+
(
+ ) − −
( )+
ΘL L L C S S C S S
x y z V VV IV I IV IV
02 11 00 01
2 , b
S
L L L
S S
S
L L L
II
x y z
IV VV
IV
x y z
0
00
00 02
2 01
= +
( ) −
Θ
× + +
( ) − −
( )+ −
1
2 2
10 01 02 11 01
2
01
Θ
ΘL L L S S C S S C S S
C
x y z II IV V VV IV I IV IV
V
V VV IV
x y z
S S
L L L
− −
02 11
Θ
× + +
( ) = +
Θ
Θ Θ
L L L S S C
L L L
S
S
L L
x y z II IV I
I V
x y z
IV
I II
x
2 10 01
1 1
00
1
2
00
  
y
y z
II II
x y z
IV
x y z
L
S S
L L L
S
L L L
− −
20 20 11
Θ Θ
,

17. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 22
C S S S S
V I V IV V VV VV IV
= + − −
  
1 1 00 1
2
00 02 11 , E y
a
a
a
a
= + −
8 4
2 3
2
4
2
2
4
Θ Θ , F
a
a
=
Θ 2
4
6
+ + − − + +
r s r r s r
2 3
3 2 3
3
, r
b
b
b L L L
b b
b
b
b
b
b
x y z
= −





 − −
Θ
Θ
Θ Θ
3
2
4
2 0
1 3
4
3
2
3
4
3 0
2
4
2
24
4
54 8
× −





 −
4
8
2
2 3
2
4
2 2 2
4
1
2
4
2
Θ Θ
Θ
b
b
b
L L L
b
b
x y z , s
b b L L L b b
b
b
b
x y z
=
−
−
Θ
Θ Θ
2 0 4 1 3
4
2
2
4
4
12 18
.
Farther, we determine solutions of Equation (8), i.e. components of displacement vector. To determine
the first-order approximations of the considered components framework method of averaging of function
corrections, we replace the required functions in the right sides of the equations by their not yet known
average values αi
. The substitution leads to the following result:
ρ β
z
u x y z t
t
K z z
T x y z t
x
x
( )
∂ ( )
∂
= − ( ) ( )
∂ ( )
∂
2
1
2
, , , , , ,
,  z
u x y z t
t
y
( )
∂ ( )
∂
=
2
1
2
, , ,
= − ( ) ( )
∂ ( )
∂
K z z
T x y z t
y

, , ,
, ρ β
z
u x y z t
t
K z z
T x y z t
z
z
( )
∂ ( )
∂
= − ( ) ( )
∂ ( )
∂
2
1
2
, , , , , ,
.
Integration of the left and the right sides of the above relations on time t leads to the following result:
u x y z t u K z
z
z x
T x y z d d
x x
t
1 0
0
0
, , , , , ,
( ) = + ( )
( )
( )
∂
∂
( ) −
∫
∫
β
ρ
τ τ ϑ
ϑ
− ( )
( )
( )
∂
∂
( )
∫
∫
∞
K z
z
z x
T x y z d d
β
ρ
τ τ ϑ
ϑ
, , ,
0
0
,
u x y z t u K z
z
z y
T x y z d d
y y
t
1 0
0
0
, , , , , ,
( ) = + ( )
( )
( )
∂
∂
( )
∫
∫
β
ρ
τ τ ϑ
ϑ
− ( )
( )
( )
∂
∂
( )
∫
∫
∞
K z
z
z y
T x y z d d
β
ρ
τ τ ϑ
ϑ
, , ,
0
0
,
u x y z t u K z
z
z z
T x y z d d
z z
t
1 0
0
0
, , , , , ,
( ) = + ( )
( )
( )
∂
∂
( )
∫
∫
β
ρ
τ τ ϑ
ϑ
− ( )
( )
( )
∂
∂
( )
∫
∫
∞
K z
z
z z
T x y z d d
β
ρ
τ τ ϑ
ϑ
, , ,
0
0
.
Approximations of the second and higher orders of components of displacement vector could be
determined using standard replacement of the required components on the following sums αi
+ui
(x,y,z,t).[28]
The replacement leads to the following result:
ρ
σ
z
u x y z t
t
K z
E z
z
u
x x
( )
∂ ( )
∂
= ( )+
( )
+ ( )

 











∂
2
2
2
2
1
5
6 1
, , , x
x y z t
x
K z
E z
z
, , ,
( )
∂
+ ( )−
( )
+ ( )

 











2
3 1 σ

18. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 23
×
∂ ( )
∂ ∂
+
( )
+ ( )

 

∂ ( )
∂
+
∂
2
1
2
1
2
2
1
2 1
u x y z t
x y
E z
z
u x y z t
y
u
y y z
, , , , , ,

x
x y z t
z
T x y z t
x
, , , , , ,
( )
∂





 −
∂ ( )
∂
2
× ( ) ( )+ ( )+
( )
+ ( )

 











∂ ( )
∂ ∂
K z z K z
E z
z
u x y z t
x z
z
β
σ
3 1
2
1 , , ,
ρ
σ
z
u x y z t
t
E z
z
u x y z t
x
u
y y
( )
∂ ( )
∂
=
( )
+ ( )

 

∂ ( )
∂
+
∂
2
2
2
2
1
2
2
2 1
, , , , , , 1
1x x y z t
x y
T x y z t
y
, , , , , ,
( )
∂ ∂





 −
∂ ( )
∂
× ( ) ( )+
∂
∂
( )
+ ( )

 

∂ ( )
∂
+
∂ ( )
∂
K z z
z
E z
z
u x y z t
z
u x y z t
y z
β
σ
2 1
1 1
, , , , , ,
y
y
u x y z t
y
y
















+
∂ ( )
∂
2
1
2
, , ,
×
( )
+ ( )

 

+ ( )










+ ( )−
( )
+ ( )

 





5
12 1 6 1
E z
z
K z K z
E z
z
 







∂ ( )
∂ ∂
+ ( )
∂ ( )
∂ ∂
2
1
2
1
u x y z t
y z
K z
u x y z t
x y
y y
, , , , , ,
ρ
σ
z
u x y z t
t
E z
z
u x y z t
x
u
z z
( )
∂ ( )
∂
=
( )
+ ( )

 

∂ ( )
∂
+
∂
2
2
2
2
1
2
2
2 1
, , , , , , 1
1
2
2
1
z x
x y z t
y
u x y z t
x z
, , , , , ,
( )
∂
+
∂ ( )
∂ ∂



+
∂ ( )
∂ ∂


 +
∂
∂
( )
∂ ( )
∂
+
∂ (
2
1 1 1
u x y z t
y z z
K z
u x y z t
x
u x y z t
y x y
, , , , , , , , , )
)
∂
+
∂ ( )
∂
















y
u x y z t
z
x
1 , , ,
+
( )
+ ( )

 

∂
∂
∂ ( )
∂
−
∂ ( )
∂
−
∂
E z
z z
u x y z t
z
u x y z t
x
u x
z x y
6 1
6 1 1 1

, , , , , , , y
y z t
y
u x y z t
z
z
, , , , ,
( )
∂
−
∂ ( )
∂






1
−
∂ ( )
∂
−
∂ ( )
∂
−
∂ ( )
∂








u x y z t
x
u x y z t
y
u x y z t
z
E z
x y z
1 1 1
, , , , , , , , , (
( )
+ ( )
− ( ) ( )
∂ ( )
∂
1 σ
β
z
K z z
T x y z t
z
, , ,
.
Integration of the left and right sides of the above relations on time t leads to the following result:
u x y z t
z
K z
E z
z x
u x
x x
2
2
2 1
1 5
6 1
, , ,
( ) =
( )
( )+
( )
+ ( )

 











∂
∂
ρ σ
,
, , ,
y z d d
z
K z
t
τ τ ϑ
ρ
ϑ
( ) +
( )
( )
{
∫
∫ 0
0
1
−
( )
+ ( )

 






∂
∂ ∂
( ) +
( )
(
∫
∫
E z
z x y
u x y z d d
E z
z
y
t
3 1 2
2
1
0
0
σ
τ τ ϑ
ρ
ϑ
, , ,
)
)
∂
∂
( )


 ∫
∫
2
2 1
0
0
y
u x y z d d
y
t
, , ,τ τ ϑ
ϑ
+
∂
∂
( )



+ ( )
+
( )
∂
∂ ∂
∫
∫
2
2 1
0
0
2
1
1
1
1
z
u x y z d d
z z x z
u x y z
z
t
z
, , , , ,
τ τ ϑ
σ ρ
ϑ
,
,τ τ ϑ
ϑ
( ) ( )
{
∫
∫ d d K z
t
0
0
+
( )
+ ( )

 






− ( )
( )
( )
∂
∂
( ) −
∫
∫
E z
z
K z
z
z x
T x y z d d
t
3 1 0
0
σ
β
ρ
τ τ ϑ
ϑ
, , ,
∂
∂
∂
( )
∫
∫
∞
2
2 1
0
0
x
u x y z d d
x , , ,τ τ ϑ
ϑ
×
( )
( )+
( )
+ ( )

 











− ( )−
( )
+ ( )

 

1 5
6 1 3 1
ρ σ σ
z
K z
E z
z
K z
E z
z











∂
∂ ∂
( )
∫
∫
∞
2
1
0
0
x y
u x y z d d
y , , ,τ τ ϑ
ϑ
×
( )
−
( )
( ) + ( )

 

∂
∂
( ) +
∂
∂
∫
∫
∞
1
2 1
2
2 1
0
0
2
ρ ρ σ
τ τ ϑ
ϑ
z
E z
z z y
u x y z d d
z
y , , , 2
2 1
0
0
u x y z d d
z , , ,τ τ ϑ
ϑ
( )






∫
∫
∞

19. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 24
−
( )
( )+
( )
+ ( )

 











∂
∂ ∂
( )
1
3 1
2
1
ρ σ
τ τ ϑ
z
K z
E z
z x z
u x y z d d
z , , ,
0
0
0
0
ϑ
β
ρ
∫
∫
∞
+ + ( )
( )
( )
u K z
z
z
x
×
∂
∂
( )
∫
∫
∞
x
T x y z d d
, , ,τ τ ϑ
ϑ
0
0
u x y z t
E z
z z x
u x y z d d
y x
t
2
2
2 1
0
0
2 1
, , , , , ,
( ) =
( )
( ) + ( )

 

∂
∂
( )
∫
ρ σ
τ τ ϑ
ϑ
∫
∫ ∫
∫
+
∂
∂ ∂
( )






2
1
0
0
x y
u x y z d d
x
t
, , ,τ τ ϑ
ϑ
×
+ ( )
+
( )
( )
∂
∂ ∂
( ) +
( )
( )
∫
∫
1
1
1 5
12
2
1
0
0
σ ρ
τ τ ϑ
ρ
ϑ
z
K z
z x y
u x y z d d
z
E z
y
t
, , ,
1
1+ ( )

 

+ ( )










σ z
K z
×
∂
∂
( ) +
( )
∂
∂
( )
+ ( )
∂
∂
∫
∫
2
2 1
0
0
1
1
2 1
y
u x y z d d
z z
E z
z z
u x y
x
t
y
, , , , ,
τ τ ϑ
ρ σ
ϑ
z
z d d
t
,τ τ ϑ
ϑ
( )








∫
∫ 0
0
+
∂
∂
( )








− ( )
( )
( )
( )
∫
∫
y
u x y z d d K z
z
z
T x y z d
z
t
1
0
0
, , , , , ,
τ τ ϑ
β
ρ
τ
ϑ
τ
τ ϑ
σ
ϑ
d
E z
z
t
0
0
6 1
∫
∫ −
( )
+ ( )

 






− ( )




 ( )
∂
∂ ∂
( ) −
( )
( )
∂
∂
∫
∫
K z
z y z
u x y z d d
E z
z x
y
t
1
2
2
1
0
0
2
2
ρ
τ τ ϑ
ρ
ϑ
, , , u
u x y z d d
x
1
0
0
, , ,τ τ ϑ
ϑ
( )


 ∫
∫
∞
+
∂
∂ ∂
( )



+ ( )
− ( )
( )
( )
∫
∫
∞
2
1
0
0
1
1
x y
u x y z d d
z
K z
z
z
T x y z
x , , , , ,
τ τ ϑ
σ
β
ρ
ϑ
,
,τ τ ϑ
ρ
ϑ
( ) −
( )
( )
∫
∫
∞
d d
K z
z
0
0
×
∂
∂ ∂
( ) −
( )
∂
∂
( )
∫
∫
∞
2
1
0
0
2
2 1
0
1
x y
u x y z d d
z y
u x y z d d
y x
, , , , , ,
τ τ ϑ
ρ
τ τ ϑ
ϑ ϑ
∫
∫
∫
∞
( )
+ ( )

 






0
5
12 1
E z
z
σ
+ ( )



−
∂
∂
( )
+ ( )
∂
∂
( ) +
∂
∂
∫
∫
∞
K z
z
E z
z z
u x y z d d
y
u x y
y z
1
1
0
0
1
σ
τ τ ϑ
ϑ
, , , , , z
z d d
,τ τ ϑ
ϑ
( )
















∫
∫
∞
0
0
×
( )
−
( )
( )−
( )
+ ( )

 











∂
∂ ∂
1
2
1
6 1
2
1
ρ ρ σ
z z
K z
E z
z y z
u x y z
y , , ,τ
τ τ ϑ
ϑ
( ) +
∫
∫
∞
d d u y
0
0
0
u x y z t
E z
z x
u x y z d d
z z
, , , , , ,
( ) =
( )
+ ( )

 

∂
∂
( ) +
∂
∂
∫
∫
∞
2 1
2
2 1
0
0
2
σ
τ τ ϑ
ϑ
y
y
u x y z d d
z
2 1
0
0
, , ,τ τ ϑ
ϑ
( )


 ∫
∫
∞
+
∂
∂ ∂
( ) +
∂
∂ ∂
( )
∫
∫ ∫
∫
∞ ∞
2
1
0
0
2
1
0
0
x z
u x y z d d
y z
u x y z d d
x y
, , , , , ,
τ τ ϑ τ τ ϑ
ϑ ϑ




( )
+
( )
1 1
ρ ρ
z z
×
∂
∂
( )
∂
∂
( ) +
∂
∂
( )
∫
∫ ∫
∞ ∞
z
K z
x
u x y z d d
y
u x y z d d
x x
1
0
0
1
0
0
, , , , , ,
τ τ ϑ τ τ ϑ
ϑ ϑ
∫
∫








+
∂
∂
( )








+
( )
∂
∂
( )
+ ( )
∂
∂
∫
∫
∞
z
u x y z d d
z z
E z
z z
x
1
0
0
1
6 1
6
, , ,τ τ ϑ
ρ σ
ϑ
u
u x y z d d
z
1
0
0
, , ,τ τ ϑ
ϑ
( )








∫
∫
∞
−
∂
∂
( ) −
∂
∂
( ) −
∂
∂
∫
∫ ∫
∫
∞ ∞
x
u x y z d d
y
u x y z d d
z
u
x y
1
0
0
1
0
0
1
, , , , , ,
τ τ ϑ τ τ ϑ
ϑ ϑ
z
z x y z d d
, , ,τ τ ϑ
ϑ
( )








∫
∫
∞
0
0

20. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 25
− ( )
( )
( )
∂
∂
( ) +
∫
∫
∞
K z
z
z z
T x y z d d u z
β
ρ
τ τ ϑ
ϑ
, , ,
0
0
0 .
Framework this paper, we determine the concentration of dopant, concentrations of radiation defects,
and components of displacement vector using the second-order approximation framework method of
averaging of function corrections. This approximation is usually enough good approximation to make
qualitative analysis and to obtain some quantitative results. All obtained results have been checked by
comparison with the results of numerical simulations.
DISCUSSION
Inthissection,weanalyzedthedynamicsofredistributionsofdopantandradiationdefectsduringannealing
and under influence of mismatch-induced stress and modification of porosity. Typical distributions of
concentrations of dopant in heterostructures are presented in Figures 2 and 3 for diffusion and ion types
of doping, respectively. These distributions have been calculated for the case when the value of dopant
diffusion coefficient in doped area is larger than in nearest areas. The figures show that inhomogeneity
of heterostructure gives us possibility to increase compactness of concentrations of dopants and at the
same time to increase homogeneity of dopant distribution in doped part of epitaxial layer. However,
framework this approach of manufacturing of bipolar transistor, it is necessary to optimize annealing
of dopant and/or radiation defects. Reason of this optimization is the following. If annealing time is
small, the dopant did not achieve any interfaces between materials of heterostructure. In this situation,
one cannot find any modifications of distribution of concentration of dopant. If annealing time is large,
distribution of concentration of dopant is too homogenous. We optimize annealing time framework
recently introduces approach.[12,29-36]
Framework this criterion, we approximate the real distribution of
concentration of dopant by step-wise function [Figures 4 and 5].
Farther, we determine the optimal values of annealing time by minimization of the following mean
squared error:
U
L L L
C x y z x y z d z d y d x
x y z
L
L
L z
y
x
= ( )− ( )

 

∫
∫
∫
1
0
0
0
, , , , ,
Θ  , (15)
Where, ψ (x,y,z) is the approximation function. Dependences of optimal values of annealing time on
parameters are presented in Figures 6 and 7 for diffusion and ion types of doping, respectively. It should
Figure 2: Distributions of concentration of infused dopant in heterostructure from Figure 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 the value of
dopant diffusion coefficient in epitaxial layer is larger than the value of dopant diffusion coefficient in substrate

21. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 26
be noted that it is necessary to anneal radiation defects after ion implantation. One could find spreading
of concentration of distribution of dopant during this annealing. In the ideal case, distribution of dopant
Figure 3: Distributions of concentration of implanted dopant in heterostructure from Figure 1 in direction, which
is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 correspond to annealing time
Θ = 0.0048(Lx
2
+Ly
2
+Lz
2
)/D0
. Curves 2 and 4 correspond to annealing time Θ = 0.0057(Lx
2
+Ly
2
+Lz
2
)/D0
. Curves 1 and 2
correspond to homogenous sample. Curves 3 and 4 correspond to heterostructure under condition when the value of dopant
diffusion coefficient in epitaxial layer is larger than the value of dopant diffusion coefficient in substrate
Figure 4: Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distribution of
dopant. Curves 2–4 are real distributions of dopant for different values of annealing time. Increasing of number of curve
corresponds to increasing of annealing time
Figure 5: Spatial distributions of dopant in heterostructure after ion implantation. Curve 1 is idealized distribution of
dopant. Curves 2–4 are real distributions of dopant for different values of annealing time. Increasing of the number of curve
corresponds to increase of annealing time

22. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 27
achieves appropriate interfaces between materials of heterostructure during annealing of radiation
defects. If dopant did not achieve any interfaces during annealing of radiation defects, it is practicably to
additionally anneal the dopant. In this situation, optimal value of additional annealing time of implanted
dopant is smaller than annealing time of infused dopant.
Farther, we analyzed the influence of relaxation of mechanical stress on distribution of dopant in doped
areas of heterostructure. Under the following condition ε0
0, one can find compression of distribution
of concentration of dopant near interface between materials of heterostructure. Contrary (at ε0
0), one
can find spreading of distribution of concentration of dopant in this area. This changing of distribution
of concentration of dopant could be at least partially compensated using laser annealing.[36]
This type of
annealing gives us possibility to accelerate diffusion of dopant and another process in annealed area due
to inhomogeneous distribution of temperature and Arrhenius law. Accounting relaxation of mismatch-
induced stress in heterostructure could lead to change of optimal values of annealing time. At the same
time, modification of porosity gives us possibility to decrease the value of mechanical stress. On the one
Figure 6: Dependences of dimensionless optimal annealing time for doping by diffusion, 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 ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of
heterostructure. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2
and ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ for a/L=1/2
and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter γ for a/L=1/2 and
ε = ξ = 0
Figure 7: 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 ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of
heterostructure. Curve 2 is the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2
and ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ for a/L=1/2
and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time on value of parameter γ for a/L=1/2 and
ε = ξ = 0

23. Pankratov: On manufacturing of a broadband power amplifiers
AJCSE/Jan-Mar-2019/Vol 4/Issue 1 28
hand, mismatch-induced stress could be used to increase density of elements of integrated circuits, on the
other hand, could lead to generation dislocations of the discrepancy. Figures 8 and 9 show distributions
of concentration of vacancies in porous materials and component of displacement vector, which are
perpendicular to interface between layers of heterostructure.
CONCLUSION
In this paper, we model redistribution of infused and implanted dopants with account relaxation
mismatch-induced stress during manufacturing field-effect heterotransistors framework a broadband
power amplifier. We formulate recommendations for optimization of annealing to decrease dimensions of
transistors and to increase their density. We formulate recommendations to decrease mismatch-induced
stress. Analytical approach to model diffusion and ion types of doping with account concurrent changing
of parameters in space and time has been introduced. At the same time, the approach gives us possibility
to take into account nonlinearity of considered processes.
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of displacement vector on coordinate z for non-porous (curve 1) and
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Figure 9: Normalized dependences of vacancy concentrations on coordinate z in unstressed (curve 1) and stressed (curve 2)
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AJCSE/Jan-Mar-2019/Vol 4/Issue 1 29
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