![© 2021, AJCSE. All Rights Reserved 27
REVIEW ARTICLE
On Optimization of Manufacturing of Field-effect Transistors to Increase Their
Integration Rate in the Framework of a Double-tail Dynamic Comparator
E. L. Pankratov1,2
1
Department of Mathematical and Natural Sciences, Nizhny Novgorod State University, Nizhny Novgorod, Russia,
2
Department of Higher Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia
Received on: 29-09-2021; Revised on: 30-10-2021; Accepted on: 30-11-2021
ABSTRACT
In this paper, we introduce an approach to increase integration rate of elements of a double-tail dynamic
comparator. Framework the approach we consider a heterostructure with special configuration. Several
specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of
dopant and/or radiation defects should be optimized.
Key words: Accounting of missmatch induced stress and porosity of materials, analytical approach for
modeling, double-tail dynamic comparator, optimization of manufacturing
INTRODUCTION
An actual and intensively solving problem of solid state electronics is increasing of integration rate of
elements of integrated circuits (p-n-junctions, their systems et al.).[1-8]
Increasing of the integration rate
of elements of integrated circuits could be obtain during decreasing of their dimensions. To decrease the
dimensions are using several approaches. They are widely using laser and microwave types of annealing
of infused dopants. These types of annealing are also widely using for annealing of radiation defects,
generated during ion implantation.[9-17]
Using the approaches gives a possibility to increase integration rate
of elements of integrated circuits through inhomogeneity of technological parameters due to generating
in homogenous distribution of temperature. In this situation, one can obtain decreasing dimensions
of elements of integrated circuits[18]
with account Arrhenius law.[1,3]
Another approach to manufacture
elements of integrated circuits with smaller dimensions is doping of heterostructure by diffusion or ion
implantation.[1-3]
However, in this case, optimization of dopant and/or radiation defects is required.[18]
In this paper, we consider a heterostructure. The heterostructure consist of a substrate and several
epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further, we consider
doping of these sections by diffusion or ion implantation. The doping gives a possibility to manufacture
field-effect transistors framework a double-tail dynamic comparator so as it is shown on Figure 1. The
manufacturing gives a possibility to increase density of elements of the double-tail dynamic comparator.[4]
After the considered doping, dopant and/or radiation defects should be annealed. Framework the paper,
we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. We
introduce an approach to decrease dimensions of the element. However, it is necessary to complicate
technological process.
METHOD OF SOLUTION
In this section, we determine spatio-temporal distributions of concentrations of infused and implanted
dopants.Todeterminethesedistributions,wecalculateappropriatesolutionsofthesecondFick’slaw[1,3,18,19]
Address for correspondence:
E. L. Pankratov,
E-mail: elp2004@mail.ru
Available Online at www.ajcse.info
Asian Journal of Computer Science Engineering 2021;6(4):27-45
ISSN 2581 – 3781](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-1-320.jpg)
![Pankratov: On optimization of of a double-tail dynamic comparator
AJCSE/Oct-Dec-2021/Vol 6/Issue 4 28
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
S
G
D
Figure 1: The considered comparator[4]
C x y z t
t x
D
C x y z t
x y
D
C x y z t
y
C C
, , , , , , , , ,
( )
==
( )
+
( )
+
( )
z
D
C x y z t
z
C
, , ,
(1)
Boundary and initial conditions for the equations are
∂ ( )
∂
=
∂ ( )
∂
=
∂ ( )
∂
=
= = =
C x y z t
x
C x y z t
x
C x y z t
y
x x L y
x
, , ,
,
, , ,
,
, , ,
,
0 0
0 0 0
∂
∂ ( )
∂
=
∂ ( )
∂
=
∂ ( )
∂
=
=
= =
C x y z t
y
C x y z t
z
C x y z t
z
x L
z x L
y
z
, , ,
, , ,
,
, , ,
0
0 0
0
,
, ( , , , ) ( , , )
C x y z f x y z
0 =
,(2)](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-2-320.jpg)
![Pankratov: On optimization of of a double-tail dynamic comparator
AJCSE/Oct-Dec-2021/Vol 6/Issue 4 29
The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is the
temperature of annealing; DС
is the dopant diffusion coefficient. Value of dopant diffusion coefficient
could be changed with changing materials of heterostructure, with changing temperature of materials
(including annealing), with changing concentrations of dopant and radiation defects. We approximate
dependences of dopant diffusion coefficient on parameters by the following relation with account results
in Refs.[20-22]
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
.(3)
Here, the function DL
(x,y,z,T) describes the spatial (in heterostructure) and temperature (due toArrhenius
law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes the limit of
solubility of dopant. Parameter γ ∈[1,3]
describes average quantity of charged defects interacted with
atom of dopant.[20]
The function V(x,y,z,t) describes the spatio-temporal distribution of concentration of
radiation vacancies. Parameter V* describes the equilibrium distribution of concentration of vacancies.
The considered concentrational dependence of dopant diffusion coefficient has been described in details
in.[20]
It should be noted, that using diffusion type of doping did not generation radiation defects. In this
situation ζ1
= ζ2
= 0, we determine spatio-temporal distributions of concentrations of radiation defects by
solving the following system of equations[21,22,23]
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂
+
∂
∂
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
I
( )
∂ ( )
∂
+
+
∂
∂
( )
∂ ( )
∂
− ( ) ( ) ( )−
− ( )
k x y z T I x y z t V x y z t
k x y z T I x y z
I V
I I
,
,
, , , , , , , , ,
, , , , ,
2
,
,t
( )
(4)
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂
+
∂
∂
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
z
D x y z T
V x y z t
z
V
( )
∂ ( )
∂
+
+
∂
∂
( )
∂ ( )
∂
− ( ) ( ) ( )+
+ ( )
k x y z T I x y z t V x y z t
k x y z T V x y z
I V
V V
,
,
, , , , , , , , ,
, , , , ,
2
,
, .
t
( )
Boundary and initial conditions for these equations are
∂ ( )
∂
=
∂ ( )
∂
=
∂ ( )
∂
=
= = =
x y z t
x
x y z t
x
x y z t
y
x x L y
x
, , ,
,
, , ,
,
, , ,
,
0 0
0 0 0
∂
∂ ( )
∂
=
∂ ( )
∂
=
∂ ( )
∂
=
=
= =
x y z t
y
x y z t
z
x y z t
z
y L
z z L
y
z
, , ,
, , ,
,
, , ,
0
0 0
0
,
, ( , , , ) ( , , ).
r x y z f x y z
r
0 =
,(5)
Here ρ=I,V, the function I (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation
interstitials; Dρ
(x,y,z,T) are the diffusion coefficients of point radiation defects; terms V2
(x,y,z,t) and
I2
(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V
(x,y,z,T) is the parameter of
recombination of point radiation defects; kI,I
(x,y,z,T) and kV,V
(x,y,z,T) are the parameters of generation of
simplest complexes of point radiation defects.](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-3-320.jpg)
![Pankratov: On optimization of of a double-tail dynamic comparator
AJCSE/Oct-Dec-2021/Vol 6/Issue 4 30
Further, we determine distributions in space and time of concentrations of divacancies ΦV
(x,y,z,t) and
diinterstitials ΦI
(x,y,z,t) by solving the following system of equations[21,22]
Φ Φ
Φ Φ
I
I
I
I
x y z t
t x
D x y z T
x y z t
x y
D x
, , ,
, , ,
, , ,
( )
= ( )
( )
+ ,
, , ,
, , ,
y z T
x y z t
y
I
( )
( )
+
Φ
+ ( )
( )
+ ( )
z
D x y z T
x y z t
z
k x y z T I x y z
I
I
I I
Φ
Φ
, , ,
, , ,
, , , , , ,
,
2
t
t k x y z T I x y z t
I
( )− ( ) ( )
, , , , , , (6)
Φ Φ
Φ Φ
V
V
V
V
x y z t
t x
D x y z T
x y z t
x y
D x
, , ,
, , ,
, , ,
( )
= ( )
( )
+ ,
, , ,
, , ,
y z T
x y z t
y
V
( )
( )
+
Φ
+ ( )
( )
+ ( )
z
D x y z T
x y z t
z
k x y z T V x y z
V
V
V V
Φ
Φ
, , ,
, , ,
, , , , , ,
,
2
t
t k x y z T V x y z t
V
( )− ( ) ( )
, , , , , , .
Boundary and initial conditions for these equations are
∂ ( )
∂
=
∂ ( )
∂
=
∂ ( )
∂
= = =
Φ Φ Φ
x y z t
x
x y z t
x
x y z t
y
x x L y
x
, , ,
,
, , ,
,
, , ,
0 0
0 0 =
=
∂ ( )
∂
=
∂ ( )
∂
=
∂ ( )
∂
=
=
0 0
0
0
,
, , ,
, , ,
,
, , ,
Φ
Φ Φ
x y z t
y
x y z t
z
x y z t
z
y L
z
y
z
z Lz
=
= 0
ΦI
(x,y,z,0)=fΦI
(x,y,z), ΦV
(x,y,z,0)=fΦV
(x,y,z).(7)
Here, DΦρ
(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects; kI
(x,y,z,T)
and kV
(x,y,z,T) are the parameters of decay of these complexes.
We calculate distributions of concentrations of point radiation defects in space and time by recently
elaborated approach.[18]
The approach based on transformation of approximations of diffusion coefficients
in the following form: Dρ
(x,y,z,T)=D0ρ
[1+ερ
gρ
(x,y,z,T)], where D0ρ
are the average values of diffusion
coefficients, 0≤ερ
1, |gρ
(x, y,z,T)|≤1, ρ =I,V. We also used analogs transformation of approximations of
parameters of recombination of point defects and parameters of generation of their complexes:
kI,V
(x,y,z,T)=k0I,V
[1+εI,V
gI,V
(x,y,z,T)], kI,I
(x,y,z,T)=k0I,I
[1+εI,I
gI,I
(x,y,z,T)] and kV,V
(x,y,z,T)=k0V,V
[1+εV,V
gV,V
(x,y,z,T)], where k0ρ1,ρ2
are the their average values, 0≤εI,V
1, 0≤εI,I
1, 0≤εV,V
1, | gI,V
(x,y,z,T)|≤1, |
gI,I
(x,y,z,T)|≤1, |gV,V
(x,y,z,T)|≤1. Let us introduce the following dimensionless variables:
I x y z t I x y z t I V x y z t V x y z t V L k I
, , , , , , , , , , , , , ,
* *
( ) = ( ) ( ) == ( ) =
ω 2
0 ,
,
,
V I V
I V
D D
L k D D
0 0
2
0 0 0
Ωρ ρ ρ
=
,
χ = x/Lx
, η = y/Ly
, φ = z/Lz
. The introduction leads to transformation of Eqs.(4) and conditions (5) to the
following form
∂ ( )
∂
=
∂
∂
+ ( )
∂ ( )
I D
D D
g T
I
I
I V
I I
χ η ϕ ϑ
ϑ χ
ε χ η ϕ
χ η ϕ ϑ
, , ,
, , ,
, , ,
0
0 0
1
∂
∂
+
∂
∂
+ ( )
×
{
χ η
ε χ η ϕ
1 I I
g T
, , ,
×
∂ ( )
∂
+
∂
∂
+ ( )
I D
D D
D
D D
g T
I
I V
I
I V
I I
χ η ϕ ϑ
η ϕ
ε χ η ϕ
, , ,
, , ,
0
0 0
0
0 0
1
∂ ( )
∂
− ( ) ×
I
I
χ η ϕ ϑ
ϕ
χ η ϕ ϑ
, , ,
, , ,
× + ( )
( )− + ( )
ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
1 1
I V I V I I I I I
g T V g T
, , , ,
, , , , , , , , ,
Ω
( )
I 2
χ η ϕ ϑ
, , , (8)](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-4-320.jpg)
![Pankratov: On optimization of of a double-tail dynamic comparator
AJCSE/Oct-Dec-2021/Vol 6/Issue 4 31
∂ ( )
∂
=
∂
∂
+ ( )
∂ ( )
V D
D D
g T
V
V
I V
V V
χ η ϕ ϑ
ϑ χ
ε χ η ϕ
χ η ϕ ϑ
, , ,
, , ,
, , ,
0
0 0
1
∂
∂
+
∂
∂
+ ( )
×
{
χ η
ε χ η ϕ
1 V V
g T
, , ,
×
∂ ( )
∂
+
∂
∂
+ ( )
V D
D D
D
D D
g T
V
I V
V
I V
V V
χ η ϕ ϑ
η ϕ
ε χ η ϕ
, , ,
, , ,
0
0 0
0
0 0
1
∂ ( )
∂
− ( ) ×
V
I
χ η ϕ ϑ
ϕ
χ η ϕ ϑ
, , ,
, , ,
× + ( )
( )− + ( )
ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
1 1
I V I V V V V V V
g T V g T
, , , ,
, , , , , , , , ,
Ω
( )
V 2
χ η ϕ ϑ
, , ,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
χ χ
, , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
χ χ
, , ,
1
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
η η
, , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
η η
, , ,
1
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
ϕ ϕ
, , ,
0
0 ,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
ϕ ϕ
, , ,
1
0 ,
ρ χ η ϕ ϑ
χ η ϕ ϑ
ρ
ρ
, , ,
, , ,
*
( ) =
( )
f
.(9)
We determine solutions of Eqs.(8) with conditions (9) framework recently introduced approach,[18]
that
is, as the power series
ρ χ η ϕ ϑ ε ω ρ χ η ϕ ϑ
ρ ρ
, , , , , ,
( ) = ( )
=
∞
=
∞
=
∞
∑
∑
∑ i j k
ijk
k
j
i
Ω
0
0
0
.(10)
Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain equations for
initial-order approximations of concentration of point defects
I000 χ η ϕ ϑ
, , ,
( ) and
V000 χ η ϕ ϑ
, , ,
( ) and
corrections for them
Iijk χ η ϕ ϑ
, , ,
( ) and
Vijk χ η ϕ ϑ
, , ,
( ), i ≥1, j ≥1, k ≥1. The equations are presented in
the Appendix. Solutions of the equations could be obtained by standard Fourier approach.[24,25]
The
solutions are presented in the Appendix.
Now, we calculate distributions of concentrations of simplest complexes of point radiation defects in
space and time. To determine the distributions we transform approximations of diffusion coefficients in
the following form: DΦρ
(x,y,z,T)=D0Φρ
[1+ εΦρ
gΦρ
(x,y,z,T)], where D0Φρ
are the average values of diffusion
coefficients. In this situation, the Eqs.(6) could be written as
∂
∂
∂
∂
ε
∂
∂
Φ Φ
Φ Φ Φ
I
I I I
I
x y z t
t
D
x
g x y z T
x y z t
x
, , ,
, , ,
, , ,
( )
= + ( )
( )
0 1
+ ( ) ( )+
k x y z T I x y z t
I I
, , , , , , ,
2
+ + ( )
( )
+
D
y
g x y z T
x y z t
y
D
z
I I I
I
I
0 0
1 1
Φ Φ Φ Φ
Φ
∂
∂
ε
∂
∂
∂
∂
, , ,
, , ,
+
+ ( )
( )
−
ε
∂
∂
Φ Φ
Φ
I I
I
g x y z T
x y z t
z
, , ,
, , ,
− ( ) ( )
k x y z T I x y z t
I , , , , , ,
∂
∂
∂
∂
ε
∂
∂
Φ Φ
Φ Φ Φ
V
V V V
V
x y z t
t
D
x
g x y z T
x y z t
x
, , ,
, , ,
, , ,
( )
= + ( )
( )
0 1
+ ( ) ( )+
k x y z T I x y z t
I I
, , , , , , ,
2
+ + ( )
( )
+
D
y
g x y z T
x y z t
y
D
z
V V V
V
V
0 0
1 1
Φ Φ Φ Φ
Φ
∂
∂
ε
∂
∂
∂
∂
, , ,
, , ,
+
+ ( )
( )
−
ε
∂
∂
Φ Φ
Φ
V V
V
g x y z T
x y z t
z
, , ,
, , ,
− ( ) ( )
k x y z T I x y z t
I , , , , , , .](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-5-320.jpg)
![Pankratov: On optimization of of a double-tail dynamic comparator
AJCSE/Oct-Dec-2021/Vol 6/Issue 4 32
Farther, we determine solutions of above equations as the following power series
Φ Φ
Φ
ρ ρ ρ
ε
x y z t x y z t
i
i
i
, , , , , ,
( ) = ( )
=
∞
∑
0
.(11)
Now, we used the series (11) into Eqs.(6) and appropriate boundary and initial conditions. The using
gives the possibility to obtain equations for initial-order approximations of concentrations of complexes
of defects Φρ0
(x,y,z,t), corrections for them Φρi
(x,y,z,t) (for them i ≥1) and boundary and initial conditions
for them. We remove equations and conditions to the Appendix. Solutions of the equations have been
calculated by standard approaches[24,25]
and presented in the Appendix.
Now, we calculate distribution of concentration of dopant in space and time using the approach, which
was used for analysis of radiation defects. To use the approach we consider following transformation
of approximation of dopant diffusion coefficient: DL
(x,y,z,T)=D0L
[1+ εL
gL
(x,y,z,T)], where D0L
is the
average value of dopant diffusion coefficient, 0≤εL
1, |gL
(x,y,z,T)|≤1. Farther, we consider solution of
Eq.(1) as the following series:
C x y z t C x y z t
L
i j
ij
j
i
, , , , , ,
( ) = ( )
=
∞
=
∞
∑
∑ε ξ
1
0
.
Using the relation into Eq.(1) and conditions (2) leads to obtaining equations for the functions Cij
(x,y,z,t)
(i ≥1, j ≥1), boundary and initial conditions for them. The equations are presented in the Appendix.
Solutions of the equations have been calculated by standard approaches (see, for example,[24,25]
). The
solutions are presented in the Appendix.
We analyzed distributions of concentrations of dopant and radiation defects in space and time analytically
using the second-order approximations on all parameters, which have been used in appropriate series.
Usually, the second-order approximations are enough good approximations to make qualitative analysis
and to obtain quantitative results. All analytical results have been checked by numerical simulation.
DISCUSSION
In this section, we analyzed spatio-temporal distributions of concentrations of dopants. Figure 2 shows
typical spatial distributions of concentrations of dopants in neighborhood of interfaces of heterostructures.
We calculate these distributions of concentrations of dopants under the following condition: value of
dopant diffusion coefficient in doped area is larger, than value of dopant diffusion coefficient in nearest
Figure 2: (a) Dependences of concentration of dopant, infused in heterostructure from Figure 1, on coordinate in direction,
which is perpendicular to interface between epitaxial layer substrate. Difference between values of dopant diffusion
coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient
in the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate. (b) Dependences of concentration
of dopant, implanted in heterostructure from Figure 1, on coordinate in direction, which is perpendicular to interface
between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure
increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than
value of dopant diffusion coefficient in the substrate. Curve 1 corresponds to homogenous sample and annealing time Θ
= 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
. Curve 2 corresponds to homogenous sample and annealing time Θ = 0.0057 (Lx
2
+Ly
2
+Lz
2
)/D0
.
Curves 3 and 4 correspond to heterostructure from Figure 1; annealing times Θ = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
and Θ = 0.0057
(Lx
2
+Ly
2
+Lz
2
)/D0
, respectively
b
a](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-6-320.jpg)
![Pankratov: On optimization of of a double-tail dynamic comparator
AJCSE/Oct-Dec-2021/Vol 6/Issue 4 33
areas. In this situation, one can find increasing of compactness of field-effect transistors with increasing
of homogeneity of distribution of concentration of dopant at one time. Changing relation between values
of dopant diffusion coefficients leads to opposite result [Figure 3].
It should be noted, that framework the considered approach one shall optimize annealing of dopant and/
or radiation defects. To do the optimization we used recently introduced criterion.[26-34]
The optimization
based on approximation real distribution by step-wise function ψ (x,y,z) [Figure 4]. Farther, the required
values of optimal annealing time have been calculated by minimization 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 .(12)
We show optimal values of annealing time as functions of parameters on Figure 5. It is known, that
standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal case,
after finishing the annealing dopant achieves interface between layers of heterostructure. If the dopant
has no enough time to achieve the interface, it is practicably to anneal the dopant additionally. The
Figure 5b shows the described dependences of optimal values of additional annealing time for the same
parameters as for Figure 5a. Necessity to anneal radiation defects leads to smaller values of optimal
annealing of implanted dopant in comparison with optimal annealing time of infused dopant.
Figure 3: (a) Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostructure from
Figs. 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure. Difference between values
of dopant diffusion coefficients increases with increasing of number of curves. Value of dopant diffusion coefficient in this
section is smaller, than value of dopant diffusion coefficient in nearest sections. (b) Calculated distributions of implanted
dopant in epitaxial layers of heterostructure. Solid lines are spatial distributions of implanted dopant in system of two
epitaxial layers. Dushed lines are spatial distributions of implanted dopant in one epitaxial layer. Annealing time increases
with increasing of number of curves
b
a
Figure 4: (a) Distributions of concentration of infused dopant in depth of heterostructure from Figure 1 for different values
of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of number of curve corresponds
to increasing of annealing time. (b) Distributions of concentration of implanted dopant in depth of heterostructure from
Figure 1 for different values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of
number of curve corresponds to increasing of annealing time
b
a](https://image.slidesharecdn.com/02ajcse10321new-240309050719-b39721b9/85/On-Optimization-of-Manufacturing-of-Field-effect-Transistors-to-Increase-Their-Integration-Rate-in-the-Framework-of-a-Double-tail-Dynamic-Comparator-7-320.jpg)
On Optimization of Manufacturing of Field-effect Transistors to Increase Their Integration Rate in the Framework of a Double-tail Dynamic Comparator
- 1. © 2021, AJCSE. All Rights Reserved 27 REVIEW ARTICLE On Optimization of Manufacturing of Field-effect Transistors to Increase Their Integration Rate in the Framework of a Double-tail Dynamic Comparator E. L. Pankratov1,2 1 Department of Mathematical and Natural Sciences, Nizhny Novgorod State University, Nizhny Novgorod, Russia, 2 Department of Higher Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia Received on: 29-09-2021; Revised on: 30-10-2021; Accepted on: 30-11-2021 ABSTRACT In this paper, we introduce an approach to increase integration rate of elements of a double-tail dynamic comparator. Framework the approach we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized. Key words: Accounting of missmatch induced stress and porosity of materials, analytical approach for modeling, double-tail dynamic comparator, optimization of manufacturing INTRODUCTION An actual and intensively solving problem of solid state electronics is increasing of integration rate of elements of integrated circuits (p-n-junctions, their systems et al.).[1-8] Increasing of the integration rate of elements of integrated circuits could be obtain during decreasing of their dimensions. To decrease the dimensions are using several approaches. They are widely using laser and microwave types of annealing of infused dopants. These types of annealing are also widely using for annealing of radiation defects, generated during ion implantation.[9-17] Using the approaches gives a possibility to increase integration rate of elements of integrated circuits through inhomogeneity of technological parameters due to generating in homogenous distribution of temperature. In this situation, one can obtain decreasing dimensions of elements of integrated circuits[18] with account Arrhenius law.[1,3] Another approach to manufacture elements of integrated circuits with smaller dimensions is doping of heterostructure by diffusion or ion implantation.[1-3] However, in this case, optimization of dopant and/or radiation defects is required.[18] In this paper, we consider a heterostructure. The heterostructure consist of a substrate and several epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further, we consider doping of these sections by diffusion or ion implantation. The doping gives a possibility to manufacture field-effect transistors framework a double-tail dynamic comparator so as it is shown on Figure 1. The manufacturing gives a possibility to increase density of elements of the double-tail dynamic comparator.[4] After the considered doping, dopant and/or radiation defects should be annealed. Framework the paper, we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. We introduce an approach to decrease dimensions of the element. However, it is necessary to complicate technological process. METHOD OF SOLUTION In this section, we determine spatio-temporal distributions of concentrations of infused and implanted dopants.Todeterminethesedistributions,wecalculateappropriatesolutionsofthesecondFick’slaw[1,3,18,19] Address for correspondence: E. L. Pankratov, E-mail: elp2004@mail.ru Available Online at www.ajcse.info Asian Journal of Computer Science Engineering 2021;6(4):27-45 ISSN 2581 – 3781
- 2. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 28 S G D S G D S G D S G D S G D S G D S G D S G D S G D S G D S G D S G D Figure 1: The considered comparator[4] C x y z t t x D C x y z t x y D C x y z t y C C , , , , , , , , , ( ) == ( ) + ( ) + ( ) z D C x y z t z C , , , (1) Boundary and initial conditions for the equations are ∂ ( ) ∂ = ∂ ( ) ∂ = ∂ ( ) ∂ = = = = C x y z t x C x y z t x C x y z t y x x L y x , , , , , , , , , , , , 0 0 0 0 0 ∂ ∂ ( ) ∂ = ∂ ( ) ∂ = ∂ ( ) ∂ = = = = C x y z t y C x y z t z C x y z t z x L z x L y z , , , , , , , , , , 0 0 0 0 , , ( , , , ) ( , , ) C x y z f x y z 0 = ,(2)
- 3. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 29 The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient could be changed with changing materials of heterostructure, with changing temperature of materials (including annealing), with changing concentrations of dopant and radiation defects. We approximate dependences of dopant diffusion coefficient on parameters by the following relation with account results in Refs.[20-22] 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 .(3) Here, the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due toArrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes the limit of solubility of dopant. Parameter γ ∈[1,3] describes average quantity of charged defects interacted with atom of dopant.[20] The function V(x,y,z,t) describes the spatio-temporal distribution of concentration of radiation vacancies. Parameter V* describes the equilibrium distribution of concentration of vacancies. The considered concentrational dependence of dopant diffusion coefficient has been described in details in.[20] It should be noted, that using diffusion type of doping did not generation radiation defects. In this situation ζ1 = ζ2 = 0, we determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations[21,22,23] ∂ ( ) ∂ = ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ 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 I ( ) ∂ ( ) ∂ + + ∂ ∂ ( ) ∂ ( ) ∂ − ( ) ( ) ( )− − ( ) k x y z T I x y z t V x y z t k x y z T I x y z I V I I , , , , , , , , , , , , , , , , 2 , ,t ( ) (4) ∂ ( ) ∂ = ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ 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 z D x y z T V x y z t z V ( ) ∂ ( ) ∂ + + ∂ ∂ ( ) ∂ ( ) ∂ − ( ) ( ) ( )+ + ( ) k x y z T I x y z t V x y z t k x y z T V x y z I V V V , , , , , , , , , , , , , , , , 2 , , . t ( ) Boundary and initial conditions for these equations are ∂ ( ) ∂ = ∂ ( ) ∂ = ∂ ( ) ∂ = = = = x y z t x x y z t x x y z t y x x L y x , , , , , , , , , , , , 0 0 0 0 0 ∂ ∂ ( ) ∂ = ∂ ( ) ∂ = ∂ ( ) ∂ = = = = x y z t y x y z t z x y z t z y L z z L y z , , , , , , , , , , 0 0 0 0 , , ( , , , ) ( , , ). r x y z f x y z r 0 = ,(5) Here ρ=I,V, the function I (x,y,z,t) describes the spatio-temporal distribution of concentration of radiation interstitials; Dρ (x,y,z,T) are the diffusion coefficients of point radiation defects; terms V2 (x,y,z,t) and I2 (x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V (x,y,z,T) is the parameter of recombination of point radiation defects; kI,I (x,y,z,T) and kV,V (x,y,z,T) are the parameters of generation of simplest complexes of point radiation defects.
- 4. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 30 Further, we determine distributions in space and time of concentrations of divacancies ΦV (x,y,z,t) and diinterstitials ΦI (x,y,z,t) by solving the following system of equations[21,22] Φ Φ Φ Φ I I I I x y z t t x D x y z T x y z t x y D x , , , , , , , , , ( ) = ( ) ( ) + , , , , , , , y z T x y z t y I ( ) ( ) + Φ + ( ) ( ) + ( ) z D x y z T x y z t z k x y z T I x y z I I I I Φ Φ , , , , , , , , , , , , , 2 t t k x y z T I x y z t I ( )− ( ) ( ) , , , , , , (6) Φ Φ Φ Φ V V V V x y z t t x D x y z T x y z t x y D x , , , , , , , , , ( ) = ( ) ( ) + , , , , , , , y z T x y z t y V ( ) ( ) + Φ + ( ) ( ) + ( ) z D x y z T x y z t z k x y z T V x y z V V V V Φ Φ , , , , , , , , , , , , , 2 t t k x y z T V x y z t V ( )− ( ) ( ) , , , , , , . Boundary and initial conditions for these equations are ∂ ( ) ∂ = ∂ ( ) ∂ = ∂ ( ) ∂ = = = Φ Φ Φ x y z t x x y z t x x y z t y x x L y x , , , , , , , , , , , 0 0 0 0 = = ∂ ( ) ∂ = ∂ ( ) ∂ = ∂ ( ) ∂ = = 0 0 0 0 , , , , , , , , , , , Φ Φ Φ x y z t y x y z t z x y z t z y L z y z z Lz = = 0 ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z).(7) Here, DΦρ (x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects; kI (x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes. We calculate distributions of concentrations of point radiation defects in space and time by recently elaborated approach.[18] The approach based on transformation of approximations of diffusion coefficients in the following form: Dρ (x,y,z,T)=D0ρ [1+ερ gρ (x,y,z,T)], where D0ρ are the average values of diffusion coefficients, 0≤ερ 1, |gρ (x, y,z,T)|≤1, ρ =I,V. We also used analogs transformation of approximations of parameters of recombination of point defects and parameters of generation of their complexes: kI,V (x,y,z,T)=k0I,V [1+εI,V gI,V (x,y,z,T)], kI,I (x,y,z,T)=k0I,I [1+εI,I gI,I (x,y,z,T)] and kV,V (x,y,z,T)=k0V,V [1+εV,V gV,V (x,y,z,T)], where k0ρ1,ρ2 are the their average values, 0≤εI,V 1, 0≤εI,I 1, 0≤εV,V 1, | gI,V (x,y,z,T)|≤1, | gI,I (x,y,z,T)|≤1, |gV,V (x,y,z,T)|≤1. Let us introduce the following dimensionless variables: I x y z t I x y z t I V x y z t V x y z t V L k I , , , , , , , , , , , , , , * * ( ) = ( ) ( ) == ( ) = ω 2 0 , , , V I V I V D D L k D D 0 0 2 0 0 0 Ωρ ρ ρ = , χ = x/Lx , η = y/Ly , φ = z/Lz . The introduction leads to transformation of Eqs.(4) and conditions (5) to the following form ∂ ( ) ∂ = ∂ ∂ + ( ) ∂ ( ) I D D D g T I I I V I I χ η ϕ ϑ ϑ χ ε χ η ϕ χ η ϕ ϑ , , , , , , , , , 0 0 0 1 ∂ ∂ + ∂ ∂ + ( ) × { χ η ε χ η ϕ 1 I I g T , , , × ∂ ( ) ∂ + ∂ ∂ + ( ) I D D D D D D g T I I V I I V I I χ η ϕ ϑ η ϕ ε χ η ϕ , , , , , , 0 0 0 0 0 0 1 ∂ ( ) ∂ − ( ) × I I χ η ϕ ϑ ϕ χ η ϕ ϑ , , , , , , × + ( ) ( )− + ( ) ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ 1 1 I V I V I I I I I g T V g T , , , , , , , , , , , , , Ω ( ) I 2 χ η ϕ ϑ , , , (8)
- 5. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 31 ∂ ( ) ∂ = ∂ ∂ + ( ) ∂ ( ) V D D D g T V V I V V V χ η ϕ ϑ ϑ χ ε χ η ϕ χ η ϕ ϑ , , , , , , , , , 0 0 0 1 ∂ ∂ + ∂ ∂ + ( ) × { χ η ε χ η ϕ 1 V V g T , , , × ∂ ( ) ∂ + ∂ ∂ + ( ) V D D D D D D g T V I V V I V V V χ η ϕ ϑ η ϕ ε χ η ϕ , , , , , , 0 0 0 0 0 0 1 ∂ ( ) ∂ − ( ) × V I χ η ϕ ϑ ϕ χ η ϕ ϑ , , , , , , × + ( ) ( )− + ( ) ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ 1 1 I V I V V V V V V g T V g T , , , , , , , , , , , , , Ω ( ) V 2 χ η ϕ ϑ , , , ∂ ( ) ∂ = = ρ χ η ϕ ϑ χ χ , , , 0 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ χ χ , , , 1 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ η η , , , 0 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ η η , , , 1 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ ϕ ϕ , , , 0 0 , ∂ ( ) ∂ = = ρ χ η ϕ ϑ ϕ ϕ , , , 1 0 , ρ χ η ϕ ϑ χ η ϕ ϑ ρ ρ , , , , , , * ( ) = ( ) f .(9) We determine solutions of Eqs.(8) with conditions (9) framework recently introduced approach,[18] that is, as the power series ρ χ η ϕ ϑ ε ω ρ χ η ϕ ϑ ρ ρ , , , , , , ( ) = ( ) = ∞ = ∞ = ∞ ∑ ∑ ∑ i j k ijk k j i Ω 0 0 0 .(10) Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain equations for initial-order approximations of concentration of point defects I000 χ η ϕ ϑ , , , ( ) and V000 χ η ϕ ϑ , , , ( ) and corrections for them Iijk χ η ϕ ϑ , , , ( ) and Vijk χ η ϕ ϑ , , , ( ), i ≥1, j ≥1, k ≥1. The equations are presented in the Appendix. Solutions of the equations could be obtained by standard Fourier approach.[24,25] The solutions are presented in the Appendix. Now, we calculate distributions of concentrations of simplest complexes of point radiation defects in space and time. To determine the distributions we transform approximations of diffusion coefficients in the following form: DΦρ (x,y,z,T)=D0Φρ [1+ εΦρ gΦρ (x,y,z,T)], where D0Φρ are the average values of diffusion coefficients. In this situation, the Eqs.(6) could be written as ∂ ∂ ∂ ∂ ε ∂ ∂ Φ Φ Φ Φ Φ I I I I I x y z t t D x g x y z T x y z t x , , , , , , , , , ( ) = + ( ) ( ) 0 1 + ( ) ( )+ k x y z T I x y z t I I , , , , , , , 2 + + ( ) ( ) + D y g x y z T x y z t y D z I I I I I 0 0 1 1 Φ Φ Φ Φ Φ ∂ ∂ ε ∂ ∂ ∂ ∂ , , , , , , + + ( ) ( ) − ε ∂ ∂ Φ Φ Φ I I I g x y z T x y z t z , , , , , , − ( ) ( ) k x y z T I x y z t I , , , , , , ∂ ∂ ∂ ∂ ε ∂ ∂ Φ Φ Φ Φ Φ V V V V V x y z t t D x g x y z T x y z t x , , , , , , , , , ( ) = + ( ) ( ) 0 1 + ( ) ( )+ k x y z T I x y z t I I , , , , , , , 2 + + ( ) ( ) + D y g x y z T x y z t y D z V V V V V 0 0 1 1 Φ Φ Φ Φ Φ ∂ ∂ ε ∂ ∂ ∂ ∂ , , , , , , + + ( ) ( ) − ε ∂ ∂ Φ Φ Φ V V V g x y z T x y z t z , , , , , , − ( ) ( ) k x y z T I x y z t I , , , , , , .
- 6. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 32 Farther, we determine solutions of above equations as the following power series Φ Φ Φ ρ ρ ρ ε x y z t x y z t i i i , , , , , , ( ) = ( ) = ∞ ∑ 0 .(11) Now, we used the series (11) into Eqs.(6) and appropriate boundary and initial conditions. The using gives the possibility to obtain equations for initial-order approximations of concentrations of complexes of defects Φρ0 (x,y,z,t), corrections for them Φρi (x,y,z,t) (for them i ≥1) and boundary and initial conditions for them. We remove equations and conditions to the Appendix. Solutions of the equations have been calculated by standard approaches[24,25] and presented in the Appendix. Now, we calculate distribution of concentration of dopant in space and time using the approach, which was used for analysis of radiation defects. To use the approach we consider following transformation of approximation of dopant diffusion coefficient: DL (x,y,z,T)=D0L [1+ εL gL (x,y,z,T)], where D0L is the average value of dopant diffusion coefficient, 0≤εL 1, |gL (x,y,z,T)|≤1. Farther, we consider solution of Eq.(1) as the following series: C x y z t C x y z t L i j ij j i , , , , , , ( ) = ( ) = ∞ = ∞ ∑ ∑ε ξ 1 0 . Using the relation into Eq.(1) and conditions (2) leads to obtaining equations for the functions Cij (x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions for them. The equations are presented in the Appendix. Solutions of the equations have been calculated by standard approaches (see, for example,[24,25] ). The solutions are presented in the Appendix. We analyzed distributions of concentrations of dopant and radiation defects in space and time analytically using the second-order approximations on all parameters, which have been used in appropriate series. Usually, the second-order approximations are enough good approximations to make qualitative analysis and to obtain quantitative results. All analytical results have been checked by numerical simulation. DISCUSSION In this section, we analyzed spatio-temporal distributions of concentrations of dopants. Figure 2 shows typical spatial distributions of concentrations of dopants in neighborhood of interfaces of heterostructures. We calculate these distributions of concentrations of dopants under the following condition: value of dopant diffusion coefficient in doped area is larger, than value of dopant diffusion coefficient in nearest Figure 2: (a) Dependences of concentration of dopant, infused in heterostructure from Figure 1, on coordinate in direction, which is perpendicular to interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate. (b) Dependences of concentration of dopant, implanted in heterostructure from Figure 1, on coordinate in direction, which is perpendicular to interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate. Curve 1 corresponds to homogenous sample and annealing time Θ = 0.0048 (Lx 2 +Ly 2 +Lz 2 )/D0 . Curve 2 corresponds to homogenous sample and annealing time Θ = 0.0057 (Lx 2 +Ly 2 +Lz 2 )/D0 . Curves 3 and 4 correspond to heterostructure from Figure 1; annealing times Θ = 0.0048 (Lx 2 +Ly 2 +Lz 2 )/D0 and Θ = 0.0057 (Lx 2 +Ly 2 +Lz 2 )/D0 , respectively b a
- 7. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 33 areas. In this situation, one can find increasing of compactness of field-effect transistors with increasing of homogeneity of distribution of concentration of dopant at one time. Changing relation between values of dopant diffusion coefficients leads to opposite result [Figure 3]. It should be noted, that framework the considered approach one shall optimize annealing of dopant and/ or radiation defects. To do the optimization we used recently introduced criterion.[26-34] The optimization based on approximation real distribution by step-wise function ψ (x,y,z) [Figure 4]. Farther, the required values of optimal annealing time have been calculated by minimization 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 .(12) We show optimal values of annealing time as functions of parameters on Figure 5. It is known, that standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal case, after finishing the annealing dopant achieves interface between layers of heterostructure. If the dopant has no enough time to achieve the interface, it is practicably to anneal the dopant additionally. The Figure 5b shows the described dependences of optimal values of additional annealing time for the same parameters as for Figure 5a. Necessity to anneal radiation defects leads to smaller values of optimal annealing of implanted dopant in comparison with optimal annealing time of infused dopant. Figure 3: (a) Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostructure from Figs. 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure. Difference between values of dopant diffusion coefficients increases with increasing of number of curves. Value of dopant diffusion coefficient in this section is smaller, than value of dopant diffusion coefficient in nearest sections. (b) Calculated distributions of implanted dopant in epitaxial layers of heterostructure. Solid lines are spatial distributions of implanted dopant in system of two epitaxial layers. Dushed lines are spatial distributions of implanted dopant in one epitaxial layer. Annealing time increases with increasing of number of curves b a Figure 4: (a) Distributions of concentration of infused dopant in depth of heterostructure from Figure 1 for different values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of number of curve corresponds to increasing of annealing time. (b) Distributions of concentration of implanted dopant in depth of heterostructure from Figure 1 for different values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of number of curve corresponds to increasing of annealing time b a
- 8. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 34 CONCLUSION In this paper, we introduce an approach to increase integration rate of element of a double-tail dynamic comparator. The approach gives us possibility to decrease area of the elements with smaller increasing of the element’s thickness. REFERENCES 1. Lachin VI, Savelov NS. Electronics. Rostov-Na-Donu: Phoenix; 2001. 2. Alexenko AG, Shagurin II. Microcircuitry. Moscow: Radio and Communication; 1990. 3. Avaev NA, Naumov YE, Frolkin VT. Basis of Microelectronics. Moscow: Radio and Communication; 1991. 4. Wang Z, Duan Q, Roh J. A 0.03mm2 Delta-sigma modulator with cascaded inverter amplifier. Analog Integr Circ Sig Process 2014;81:495-501. 5. Fathi D, Forouzandeh B, Masoumi N. New enhanced noise analysis in active mixers in nanoscale technologies. Nano. 2009;4:233-8. 6. Chachuli SA, Fasyar PN, Soin N, Kar NM, Yusop N. Pareto ANOVA analysis for CMOS. 0.18 µm two-stage Op-amp. Mat Sci Sem Proc 2014;24:9-14. 7. Ageev AO, Belyaev AE, Boltovets NS, Ivanov VN, Konakova RV, Kudrik YY, et al. Technologies dependencies. Semiconductors 2009;43:897-903. 8. Li Z, Waldron J, Detchprohm T, Wetzel C, Karlicek RF, Chow TP Jr. Monolithic integration of light-emitting diodes and power metal-oxide-semiconductor channel high-electron-mobility transistors for light-emitting power integrated circuits in GaN on sapphire substrate. Appl Phys Lett 2013;102:192107-9. 9. Tsai JH, Chiu SY, Lour WS, Guo DF. High-performance InGaP/GaAs pnp δ-doped heterojunction bipolar transistor. Semiconductors 2009;43:971-4. 10. Alexandrov OV, Zakhar’in AO, Sobolev NA, Shek EI, Makoviychuk MM, et al. Formation of donor centers after annealing of dysprosium and holmium implanted silicon. Semiconductors 1998;32:1029-32. 11. Kumar MJ, Singh TV. Quantum confinement effect in strained silicon MOSFET. Int J Nanosci 2008;7:81-4. 12. Sinsermsuksakul P, Hartman K, Kim SB, Heo J, Sun L, Park HH, et al. Enhancing the efficiency of SnS solar cells via band-offset engineering with a zinc oxysulfide buffer layer. Appl Phys Lett 2013;102:053901-5. 13. Reynolds JG, Reynolds CL, Mohanta A Jr., Muth JF, Rowe JE, Everitt HO, et al. Shallow acceptor complexes in p-type ZnO. Appl Phys Lett 2013;102:152114-8. 14. Ong KK, Pey KL, Lee PS, Wee AT, Wang XC, Chong YF. Dopant distribution in the recrystallization transient at the maximum melt depth induced by laser annealing. Appl Phys Lett 2006;89:172111-4. 15. Wang HT, Tan LS, Chor EF. Pulsed laser annealing of Be-implanted GaN. J Appl Phys 2006;98:094901-5. 16. Shishiyanu ST, Shishiyanu TS, Railyan SK. Shallow p-n-junctions in Si prepared by pulse photon annealing. Semiconductors 2020;36:611-7. 17. Bykov YV, Yeremeev AG, Zharova NA, Plotnikov IV, Rybakov KI, Drozdov MN, et al. Diffusion processes in Figure 5: (a) Dimensionless optimal annealing time of infused dopant as a function of several parameters. Curve 1 describes the dependence of the 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 describes the dependence of the annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0. (b) Dimensionless optimal annealing time of implanted dopant as a function of several parameters. Curve 1 describes the dependence of the 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 describes the dependence of the annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0 b a
- 9. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 35 semiconductor structures during microwave annealing. Radiophys Quantum Electron 2003;43:836-43. 18. Pankratov EL, Bulaeva EA. Doping of materials during manufacture p-n-junctions and bipolar transistors.Analytical approaches to model technological approaches and ways of optimization of distributions of dopants. Rev Theor Sci 2013;1:58-82. 19. Erofeev YN. Pulse Devices. Moscow: Higher School; 1989. 20. Kozlivsky VV. Modification of Semiconductors by Proton Beams. Saint-Petersburg: Nauka; 2003. 21. Gotra ZY. Technology of Microelectronic Devices. Moscow: Radio and Communication; 1991. 22. Vinetskiy VL, Kholodar GA. Radiative Physics of Semiconductors. Kiev: Naukova Dumka; 1979. 23. Fahey PM, Griffin PB, Plummer JD. Point defects and dopent diffusion in silicon. Rev Mod Phys 1989;61:289-388. 24. Tikhonov AN, Samarskii AA. The Mathematical Physics Equations. Nauka: Moscow; 1972. 25. Carslaw HS, Jaeger JC. Conduction of Heat in Solids. Oxford, United Kingdom: Oxford University Press; 1964. 26. PankratovEL.Dopantdiffusiondynamicsandoptimaldiffusiontimeasinfluencedbydiffusion-coefficientnonuniformity. Russ Microelectron 2007;36:33-9. 27. Pankratov EL. Redistribution of a dopant during annealing of radiation defects in a multilayer structure by laser scans for production of an implanted-junction rectifier. Int J Nanosci 2008;7:187-97. 28. Pankratov EL. On approach to optimize manufacturing of bipolar heterotransistors framework circuit of an operational amplifier to increase their integration rate. Influence mismatch-induced stress. J Comp Theor Nanosci 2017;14:4885-99. 29. Pankratov EL. On optimization of manufacturing of two-phase logic circuit based on heterostructures to increase density of their elements. Influence of miss-match induced stress. Adv Sci Eng Med 2017;9:787-801. 30. Pankratov EL, Bulaeva EA. On increasing of density of transistors in a hybrid cascaded multilevel inverter. Multidiscipline Model Mater Struct 2017;13:664-77. 31. Pankratov EL, Bulaeva EA. An approach to manufacture of bipolar transistors in thin film structures. On the method of optimization. Int J Micronano Scale Transp 2014;4:17-31. 32. Pankratov EL, Bulaeva EA. An analytical approach for analysis and optimization of formation of field-effect heterotransistors. Multidiscipline Model Mater Struct 2016;12:578-604. 33. Pankratov EL, Bulaeva EA. An approach to increase the integration rate of planar drift heterobipolar transistors. Mater Sci Semiconductor Process 2015;34:260-8. APPENDIX Equations for the functions Iijk χ η ϕ ϑ , , , ( ) and Vijk χ η ϕ ϑ , , , ( ), i ≥0, j ≥0, k ≥0 and conditions for them ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 000 0 0 2 000 2 2 000 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ η χ η ϕ ϑ ϕ 2 2 000 2 I , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 000 0 0 2 000 2 2 000 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ η χ η ϕ ϑ ϕ 2 2 000 2 V , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ I D D I I i I V i i 00 0 0 2 00 2 2 00 χ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ η , , , , , , , 2 2 2 00 2 0 0 + ∂ ( ) ∂ + × I D D i I V χ η ϕ ϑ ϕ , , , × ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( ) ∂ − χ χ η ϕ χ η ϕ ϑ χ η χ η ϕ g T I g T I I i I , , , , , , , , , 100 i i− ( ) ∂ + 100 χ η ϕ ϑ η , , , + ∂ ∂ ( ) ∂ ( ) ∂ − ϕ χ η ϕ χ η ϕ ϑ ϕ g T I I i , , , , , , 100 , i ≥1, ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ V D D V V i V I i i 00 0 0 2 00 2 2 00 χ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ η , , , , , , , 2 2 2 00 2 + ∂ ( ) ∂ + ∂ ∂ ( ) × V g T i V χ η ϕ ϑ ϕ χ χ η ϕ , , , , , , × ∂ ( ) ∂ + ∂ ∂ ( ) ∂ − − V D D D D g T V i V I V I V i 100 0 0 0 0 100 χ η ϕ ϑ χ η χ η ϕ , , , , , , χ χ η ϕ ϑ η ϕ χ η ϕ , , , , , , ( ) ∂ + ∂ ∂ ( ) × g T V × ∂ ( ) ∂ − V D D i V I 100 0 0 χ η ϕ ϑ ϕ , , , , i ≥1,
- 10. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 36 ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 010 0 0 2 010 2 2 010 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 010 2 I , , , − + ( ) ( ) ( ) 1 000 000 ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ I V I V g T I V , , , , , , , , , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 010 0 0 2 010 2 2 010 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 010 2 V , , , − + ( ) ( ) ( ) 1 000 000 ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ I V I V g T I V , , , , , , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 020 0 0 2 020 2 2 020 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 020 2 I , , , − + ( ) ( ) ( )+ 1 010 000 000 ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ I V I V g T I V I , , , , , , , , , , , χ χ η ϕ ϑ χ η ϕ ϑ , , , , , , ( ) ( ) V010 ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V I V 020 0 0 2 020 2 2 020 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 020 2 V , , , − + ( ) ( ) ( )+ 1 010 000 000 ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ I V I V g T I V I , , , , , , , , , , , χ χ η ϕ ϑ χ η ϕ ϑ , , , , , , ( ) ( ) V010 ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 001 0 0 2 001 2 2 001 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 001 2 I , , , − + ( ) ( ) 1 000 2 ε χ η ϕ χ η ϕ ϑ I I I I g T I , , , , , , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 001 0 0 2 001 2 2 001 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 001 2 V , , , − + ( ) ( ) 1 000 2 ε χ η ϕ χ η ϕ ϑ I I I I g T V , , , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 110 0 0 2 110 2 2 110 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ + × η χ η ϕ ϑ ϕ 2 2 110 2 0 0 I D D I V , , , × ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( ) ∂ χ χ η ϕ χ η ϕ ϑ χ η χ η ϕ g T I g T I I I , , , , , , , , , 010 010 0 χ η ϕ ϑ η ϕ χ η ϕ , , , , , , ( ) ∂ + ∂ ∂ ( ) × g T I × ∂ ( ) ∂ − ( ) ( )+ I I V I 010 100 000 χ η ϕ ϑ ϕ χ η ϕ ϑ χ η ϕ ϑ , , , , , , , , , 0 000 100 χ η ϕ ϑ χ η ϕ ϑ , , , , , , ( ) ( ) × V × + ( ) 1 ε χ η ϕ I I I I g T , , , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 110 0 0 2 110 2 2 110 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ + η χ η ϕ ϑ ϕ 2 2 110 2 V , , , + ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( D D g T V g T V I V V 0 0 010 χ χ η ϕ χ η ϕ ϑ χ η χ η ϕ , , , , , , , , , ) ) ∂ ( ) ∂ + V010 χ η ϕ ϑ η , , ,
- 11. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 37 + ∂ ∂ ( ) ∂ ( ) ∂ − + ϕ χ η ϕ χ η ϕ ϑ ϕ ε χ η g T V g V V V V V , , , , , , , , , , 010 1 ϕ ϕ,T ( ) × × ( ) ( )+ ( ) ( ) V I V I 100 000 000 100 χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ , , , , , , , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 002 0 0 2 002 2 2 002 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 002 2 I , , , − + ( ) ( ) ( ) 1 001 000 ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ I I I I g T I I , , , , , , , , , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 002 0 0 2 002 2 2 002 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − η χ η ϕ ϑ ϕ 2 2 002 2 V , , , − + ( ) ( ) ( ) 1 001 000 ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ V V V V g E V V , , , , , , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 101 0 0 2 101 2 2 101 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ + η χ η ϕ ϑ ϕ 2 2 101 2 I , , , + ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( D D g T I g T I V I I 0 0 001 χ χ η ϕ χ η ϕ ϑ χ η χ η ϕ , , , , , , , , , ) ) ∂ ( ) ∂ + I001 χ η ϕ ϑ η , , , + ∂ ∂ ( ) ∂ ( ) ∂ − + ( ϕ χ η ϕ χ η ϕ ϑ ϕ ε χ η ϕ g T I g T I I I , , , , , , , , , 001 1 ) ) ( ) ( ) I V 100 000 χ η ϕ ϑ χ η ϕ ϑ , , , , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 101 0 0 2 101 2 2 101 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ + η χ η ϕ ϑ ϕ 2 2 101 2 V , , , + ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( D D g T V g T V I V V 0 0 001 χ χ η ϕ χ η ϕ ϑ χ η χ η ϕ , , , , , , , , , ) ) ∂ ( ) ∂ + V001 χ η ϕ ϑ η , , , + ∂ ∂ ( ) ∂ ( ) ∂ − + ( ϕ χ η ϕ χ η ϕ ϑ ϕ ε χ η ϕ g T V g T V V V , , , , , , , , , 001 1 ) ) ( ) ( ) I V 000 100 χ η ϕ ϑ χ η ϕ ϑ , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ I D D I I I V 011 0 0 2 011 2 2 011 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − ( ) × η χ η ϕ ϑ ϕ χ η ϕ ϑ 2 2 011 2 010 I I , , , , , , × + ( ) ( )− + ( ) 1 1 000 ε χ η ϕ χ η ϕ ϑ ε χ η ϕ I I I I I V I V g T I g T , , , , , , , , , , , , , ( ) ( ) I V 001 000 χ η ϕ ϑ χ η ϕ ϑ , , , , , , ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ V D D V V V I 011 0 0 2 011 2 2 011 χ η ϕ ϑ ϑ χ η ϕ ϑ χ χ η ϕ ϑ , , , , , , , , , ( ( ) ∂ + ∂ ( ) ∂ − ( ) × η χ η ϕ ϑ ϕ χ η ϕ ϑ 2 2 011 2 010 V V , , , , , , × + ( ) ( )− + ( ) 1 1 000 ε χ η ϕ χ η ϕ ϑ ε χ η ϕ V V V V I V I V g T V g t , , , , , , , , , , , , , ( ) ( ) I V 000 001 χ η ϕ ϑ χ η ϕ ϑ , , , , , , ; ∂ ( ) ∂ = = ρ χ η ϕ ϑ χ ijk x , , , 0 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ χ ijk x , , , 1 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ η η ijk , , , 0 0 , ∂ ( ) ∂ = = ρ χ η ϕ ϑ η η ijk , , , 1 0 , ∂ ( ) ∂ = = ρ χ η ϕ ϑ ϕ ϕ ijk , , , 0 0, ∂ ( ) ∂ = = ρ χ η ϕ ϑ ϕ ϕ ijk , , , 1 0 (i ≥0, j ≥0, k ≥0);
- 12. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 38 ρ χ η ϕ χ η ϕ ρ ρ 000 0 , , , , , * ( ) = ( ) f , ρ χ η ϕ ijk , , ,0 0 ( ) = (i ≥1, j ≥1, k ≥1). Solutions of the above equations could be written as ρ χ η ϕ ϑ χ η ϕ ϑ ρ ρ 000 1 1 2 , , , ( ) = + ( ) ( ) ( ) ( ) = ∞ ∑ L L F c c c e n n n , where F nu nv n w f u v w d wd vd u n n ρ ρ ρ π π π = ( ) ( ) ( ) ( ) ∫ ∫ ∫ 1 0 1 0 1 0 1 * cos cos cos , , , cn (χ) = cos (π n χ), e n D D nI V I ϑ π ϑ ( ) = − ( ) exp 2 2 0 0 , e n D D nV I V ϑ π ϑ ( ) = − ( ) exp 2 2 0 0 ; I D D nc c c e e s u c i I V n nI nI n n 00 0 0 2 χ η ϕ ϑ π χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) v v I u v w u i n ( ) ∂ ( ) ∂ × − = ∞ ∫ ∫ ∫ ∫ ∑ 100 0 1 0 1 0 1 0 1 , , ,τ ϑ × ( ) ( ) − ( ) ( ) ( ) ( ) − c w g u v w T d wd vd u d D D nc c c e e n I I V n nI nI , , , τ π χ η ϕ ϑ 2 0 0 τ τ ϑ ( ) ( ) ( ) × ∫ ∫ ∫ ∑ = ∞ c u s v n n n 0 1 0 1 0 1 × ( ) ( ) ∂ ( ) ∂ − − ∫c w g u v w T I u v w v d wd vd u d D D nc n I i I V , , , , , , 100 0 1 0 0 2 τ τ π n n nI nI n c c e e χ η ϕ ϑ τ ϑ ( ) ( ) ( ) ( ) − ( ) × ∫ ∑ = ∞ 0 1 × ( ) ( ) ( ) ( ) ∂ ( ) ∂ − ∫ c u c v s w g u v w T I u v w w d wd vd u d n n n I i , , , , , , 100 0 1 0 1 1 0 1 ∫ ∫ , i ≥1, V D D nc c c e e s u c i V I n nV nI n n 00 0 0 2 χ η ϕ ϑ π χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) v v g u v w T V n ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ , , , 0 1 0 1 0 1 0 1 ϑ × ( ) ∂ ( ) ∂ − ( ) ( ) ( ) ( ) − c w V u u d wd vd u d D D nc c c e e n i V I n nV n 100 0 0 ,τ τ χ η ϕ ϑ I I n n n c u s v − ( ) ( ) ( ) × ∫ ∫ ∫ ∑ = ∞ τ ϑ 0 1 0 1 0 1 × ( ) ( ) ∂ ( ) ∂ − − ∫ 2 2 100 0 1 0 0 π τ τ π χ c w g u v w T V u v d wd vd u d D D nc n V i V I n , , , , ( ( ) ( ) ( ) ( ) × = ∞ ∑ c c enV n η ϕ ϑ 1 × − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ − e c u c v s w g u v w T V u w d wd vd u d nI n n n V i τ τ τ , , , , 100 0 1 1 0 1 0 1 0 ∫ ∫ ∫ ∫ ϑ , i ≥1, where sn (χ) = sin (π n χ); ρ χ η ϕ ϑ χ η ϕ ϑ τ ρ ρ 010 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) ( ) c c c e e c u c v c w n n n n n n n n ( ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ 0 1 0 1 0 1 0 1 ϑ n × + ( ) ( ) ( ) 1 000 000 ε τ τ I V I V g u v w T I u v w V u v w d wd vd u , , , , , , , , , , , d d τ ; ρ χ η ϕ ϑ χ η ϕ ϑ τ ρ ρ 020 0 0 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) D D c c c e e c u c I V n n n n n n n v v c w n I V n ( ) ( ) + × ∫ ∫ ∫ ∫ ∑ = ∞ 1 0 1 0 1 0 1 0 1 ε ϑ , × ( ) ( ) ( )+ ( g u v w T I u v w V u v w I u v w I V , , , , , , , , , , , , , 010 000 000 ) ) ( ) V u v w d wd vd u d 010 , , , ; ρ χ η ϕ ϑ χ η ϕ ϑ τ ρ ρ 001 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) ( ) c c c e e c u c v c w n n n n n n n n ( ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ 0 1 0 1 0 1 0 1 ϑ n × + ( ) ( ) 1 000 2 ε ρ τ τ ρ ρ ρ ρ , , , , , , , , g u v w T u v w d wd vd u d ; ρ χ η ϕ ϑ χ η ϕ ϑ τ ρ ρ 002 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) ( ) c c c e e c u c v c w n n n n n n n n ( ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ 0 1 0 1 0 1 0 1 ϑ n
- 13. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 39 × + ( ) ( ) ( ) 1 001 000 ε ρ τ ρ τ ρ ρ ρ ρ , , , , , , , , , , , g u v w T u v w u v w d wd vd u d d τ ; I D D nc c c e e s u I V n n n nI nI n 110 0 0 2 χ η ϕ ϑ π χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) c c v c u n n n ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ 0 1 0 1 0 1 0 1 ϑ × ( ) ∂ ( ) ∂ − ( ) ( − g u v w T I u v w u d wd vd u d D D nc c I i I V n n , , , , , , 100 0 0 2 τ τ π χ η) ) ( ) ( ) × = ∞ ∑ c e n nI n ϕ ϑ 1 × − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ − e c u s v c u g u v w T I u v w v d wd vd nI n n n I i τ τ , , , , , , 100 u u d D D I V τ π ϑ 0 1 0 1 0 1 0 0 0 2 ∫ ∫ ∫ ∫ − × × ( ) − ( ) ( ) ( ) ( ) ( ) ∂ ( ) − n e e c u c v s u g u v w T I u v w nI nI n n n I i ϑ τ τ , , , , , , 100 ∂ ∂ × ∫ ∫ ∫ ∫ ∑ = ∞ w d wd vd u d n τ ϑ 0 1 0 1 0 1 0 1 × ( ) ( ) ( )− ( ) ( ) ( ) ( ) − ( ) ( ) ( ) c c c c e c c e c u c v c n n n n nI n n nI n n n χ η ϕ χ ϑ η ϕ τ 2 v v I V n ( ) + × ∫ ∫ ∫ ∫ ∑ = ∞ 1 0 1 0 1 0 1 0 1 ε ϑ , × ( ) ( ) ( )+ ( g u v w T I u v w V u v w I u v w I V , , , , , , , , , , , , , 100 000 000 ) ) ( ) V u v w d wd vd u d 100 , , , V D D nc c c e e s u V I n n n nV nV n 110 0 0 2 χ η ϕ ϑ π χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) c c v c u n n n ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ 0 1 0 1 0 1 0 1 ϑ × ( ) ∂ ( ) ∂ − ( ) ( − g u v w T V u v w u d wd vd u d D D nc c V i V I n n , , , , , , 100 0 0 2 τ τ π χ η) ) ( ) ( ) × = ∞ ∑ c e n nV n ϕ ϑ 1 × − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ − e c u s v c u g u v w T V u v w v d wd vd nV n n n V i τ τ , , , , , , 100 u u d D D V I τ π ϑ 0 1 0 1 0 1 0 0 0 2 ∫ ∫ ∫ ∫ − × × ( ) − ( ) ( ) ( ) ( ) ( ) ∂ ( ) − ne e c u c v s u g u v w T V u v w nV nV n n n V i ϑ τ τ , , , , , , 100 ∂ ∂ × ∫ ∫ ∫ ∫ ∑ = ∞ w d wd vd u d n τ ϑ 0 1 0 1 0 1 0 1 × ( ) ( ) ( )− ( ) ( ) ( ) ( ) − ( ) ( ) ( ) + c c c c e c c e c u c v n n n n nI n n nV n n χ η ϕ χ ϑ η ϕ τ 2 1 ε ε ϑ I V I V n g u v w T , , , , , ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ 0 1 0 1 0 1 0 1 × ( ) ( ) ( )+ ( ) c w I u v w V u v w I u v w V u v n 100 000 000 100 , , , , , , , , , , , w w d wd vd u d , ( ) ; I D D nc c c e e s u I V n n n nI nI n 101 0 0 2 χ η ϕ ϑ π χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) c c v g u v w T n I n ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ , , , 0 1 0 1 0 1 0 1 ϑ × ( ) ∂ ( ) ∂ − ( ) ( ) ( ) c w I u v w u d wd vd u d D D nc c c e n I V n n n n 001 0 0 2 , , ,τ τ π χ η ϕ I I n ϑ ( ) × = ∞ ∑ 1 × ( ) ( ) ( ) ∂ ( ) ∂ − ∫ ∫s v c w g u v w T I u v w v d wd vd u d D n n I , , , , , , 001 0 1 0 1 0 2 τ τ π I I V nI n n n n D ne c c c 0 1 ϑ χ η ϕ ( ) ( ) ( ) ( ) × = ∞ ∑ × − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ e c u c v s w g u v w T I u v w w d wd vd u d nI n n n I τ τ , , , , , , 001 τ τ χ η ϕ ϑ 0 1 0 1 0 1 0 1 2 ∫ ∫ ∫ ∫ ∑ − ( ) ( ) ( ) × = ∞ c c c n n n n × ( ) − ( ) ( ) ( ) ( ) + ( ) e e c u c v c w g u v w T I u nI nI n n n I V I V ϑ τ ε 1 100 , , , , , , , , , , , , v w V u v w d wd vd u d τ τ τ ϑ ( ) ( ) ∫ ∫ ∫ ∫ 000 0 1 0 1 0 1 0 V D D nc c c e e s u V I n n n nV nV n 101 0 0 2 χ η ϕ ϑ π χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) c c v g u v w T n V n ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ , , , 0 1 0 1 0 1 0 1 ϑ
- 14. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 40 × ( ) ∂ ( ) ∂ − ( ) ( ) ( ) c w V u v w u d wd vd u d D D nc c c e n V I n n n n 001 0 0 2 , , ,τ τ π χ η ϕ I I nV n n e c u ϑ τ ϑ ( ) − ( ) ( ) × ∫ ∫ ∑ = ∞ 0 1 0 1 × ( ) ( ) ( ) ∂ ( ) ∂ − ∫ ∫s v c w g u v w T I u v w v d wd vd u d D n n I , , , , , , 001 0 1 0 1 0 2 τ τ π I I V nI n n n n D ne c c c 0 1 ϑ χ η ϕ ( ) ( ) ( ) ( ) × = ∞ ∑ × − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ e c u c v s w g u v w T V u v w w d wd vd u d nV n n n V τ τ , , , , , , 001 τ τ χ η ϕ ϑ 0 1 0 1 0 1 0 1 2 ∫ ∫ ∫ ∫ ∑ − ( ) ( ) ( ) × = ∞ c c c n n n n × ( ) − ( ) ( ) ( ) ( ) + ( ) e e c u c v c w g u v w T I u nV nV n n n I V I V ϑ τ ε 1 100 , , , , , , , , , , , , ; v w V u v w d wd vd u d τ τ τ ϑ ( ) ( ) ∫ ∫ ∫ ∫ 000 0 1 0 1 0 1 0 I c c c e e c u c v c w n n n nI nI n n n 011 2 χ η ϕ ϑ χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) ( ) ( ( ) ( ) { × ∫ ∫ ∫ ∫ ∑ = ∞ I u v w n 000 0 1 0 1 0 1 0 1 , , ,τ ϑ × + ( ) ( )+ + ( ) 1 1 010 ε τ ε I I I I I V I V g u v w T I u v w g u v w T , , , , , , , , , , , , , ( ) ( )} I u v w V u v w d wd vd u d 001 000 , , , , , , τ τ τ V c c c e e c u c v c w n n n nV nV n n n 011 2 χ η ϕ ϑ χ η ϕ ϑ τ , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( ) ( ) ( ( ) ( ) { × ∫ ∫ ∫ ∫ ∑ = ∞ I u v w n 000 0 1 0 1 0 1 0 1 , , ,τ ϑ × + ( ) ( )+ + ( ) 1 1 010 ε τ ε I I I I I V I V g u v w T I u v w g u v w T , , , , , , , , , , , , , ( ) ( )} I u v w V u v w d wd vd u d 001 000 , , , , , , . τ τ τ Equations for functions Φρi (x,y,z,t), i ≥0 to describe concentrations of simplest complexes of radiation defects. Φ Φ Φ Φ I I I I x y z t t D x y z t x x y z t y 0 0 2 0 2 2 0 2 2 , , , , , , , , , ( ) = ( ) + ( ) + Φ ΦI x y z t z 0 2 , , , ( ) + + ( ) ( )− ( ) ( ) k x y z T I x y z t k x y z T I x y z t I I I , , , , , , , , , , , , , 2 Φ Φ Φ Φ V V V V x y z t t D x y z t x x y z t y 0 0 2 0 2 2 0 2 2 , , , , , , , , , ( ) = ( ) + ( ) + Φ ΦV x y z t z 0 2 , , , ( ) + + ( ) ( )− ( ) ( ) k x y z T V x y z t k x y z T V x y z t V V V , , , , , , , , , , , , , 2 ; Φ Φ Φ Φ I i I I i I i x y z t t D x y z t x x y z t y , , , , , , , , , ( ) = ( ) + ( ) + 0 2 2 2 2 2 Φ ΦI i x y z t z , , , ( ) + 2 + ( ) ( ) + ( − D x g x y z T x y z t x y g x y z T I I I i I 0 1 Φ Φ Φ Φ , , , , , , , , , ) ) ( ) + − ΦI i x y z t y 1 , , , + ( ) ( ) − z g x y z T x y z t z I I i Φ Φ , , , , , , 1 , i≥1, Φ Φ Φ Φ V i V V i V i x y z t t D x y z t x x y z t y , , , , , , , , , ( ) = ( ) + ( ) + 0 2 2 2 2 2 Φ ΦV i x y z t z , , , ( ) + 2 + ( ) ( ) + ( − D x g x y z T x y z t x y g x y z T V V V i V 0 1 Φ Φ Φ Φ , , , , , , , , , ) ) ( ) + − ΦV i x y z t y 1 , , , + ( ) ( ) − z g x y z T x y z t z V V i Φ Φ , , , , , , 1 , i≥1;
- 15. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 41 Boundary and initial conditions for the functions takes the form ∂ ( ) ∂ = = 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 , i≥0; Φρ 0 (x,y,z,0)=fΦρ (x,y,z), Φρ i (x,y,z,0)=0, i≥1. Solutions of the above equations could be written as Φ Φ Φ 0 1 1 2 x y z t L L L L L L F c x c y c z e t x y z x y z n n n n n n , , , ( ) = + ( ) ( ) ( ) ( ) = ∞ ∑ ∑ ∑ + ( ) ( ) ( ) × = ∞ 2 1 L n c x c y c z n n n n × ( ) − ( ) ( ) ( ) ( ) ( ) ( )− e t e c u c v c w k u v w T I u v w n n n n n I I Φ Φ ρ ρ τ τ , , , , , , , 2 0 0 0 0 0 L L L t z y x ∫ ∫ ∫ ∫ − ( ) ( ) k u v w T I u v w d wd vd u d I , , , , , , , where F c u c v c w f u v w d wd vd u n n n n L L L z y x Φ Φ = ( ) ( ) ( ) ( ) ∫ ∫ ∫ , , 0 0 0 , e t n D t L L L n x y z Φ Φ ρ ρ π ( ) = − + + ( ) − − − exp 2 2 0 2 2 2 , cn (x) = cos (π n x/Lx ); Φ Φ Φ ρ π τ ρ ρ i x y z n n n n n n x y z t L L L nc x c y c z e t e s u , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) ( 2 2 ) ) ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ c v g u v w T n L L L t n z y x Φρ , , , 0 0 0 0 1 × ( ) ( ) − ( ) ( ) ( ) − c w u v w u d wd vd u d L L L nc x c y c z n I i x y z n n n ∂ τ ∂ τ π ρ Φ 1 2 2 , , , e e t e n n t n Φ Φ ρ ρ τ ( ) − ( ) × ∫ ∑ = ∞ 0 1 × − ( ) ( ) ( ) ( ) ( ) ( ) − e c u s v c w g u v w T u v w v d wd v n n n n I i Φ Φ Φ ρ ρ ρ τ ∂ τ ∂ , , , , , , 1 d d u d L L L n L L L t x y z n z y x τ π 0 0 0 0 2 1 2 ∫ ∫ ∫ ∫ ∑ − × = ∞ × ( ) − ( ) ( ) ( ) ( ) ( ) − e t e c u c v s w u v w w g u v w n n n n n I i Φ Φ Φ Φ ρ ρ ρ ρ τ ∂ τ ∂ 1 , , , , , ,T T d wd vd u d L L L t z y x ( ) × ∫ ∫ ∫ ∫ τ 0 0 0 0 × ( ) ( ) ( ) c x c y c z n n n , i ≥1, where sn (x) = sin (π n x/Lx ). Equations for the functions Cij (x,y,z,t) (i ≥0, j ≥0), boundary and initial conditions could be written as ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ + C x y z t t D C x y z t x D C x y z t y L L 00 0 2 00 2 0 2 00 2 , , , , , , , , , D D C x y z t z L 0 2 00 2 ∂ ( ) ∂ , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ + ∂ C x y z t t D C x y z t x C x y z t y C i L i i 0 0 2 0 2 2 0 2 2 , , , , , , , , , i i x y z t z 0 2 , , , ( ) ∂ + + ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( − D x g x y z T C x y z t x D y g x y z T L L i L L 0 10 0 , , , , , , , , , ) ) ∂ ( ) ∂ + − C x y z t y i 10 , , , + ∂ ∂ ( ) ∂ ( ) ∂ − D z g x y z T C x y z t z L L i 0 10 , , , , , , , i ≥1;
- 16. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 42 ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ + C x y z t t D C x y z t x D C x y z t y L L 01 0 2 01 2 0 2 01 2 , , , , , , , , , D D C x y z t z L 0 2 01 2 ∂ ( ) ∂ + , , , + ∂ ∂ ( ) ( ) ∂ ( ) ∂ + ∂ ∂ D x C x y z t P x y z T C x y z t x D L L 0 00 00 0 , , , , , , , , , y y C x y z t P x y z T C x y z t y 00 00 , , , , , , , , , ( ) ( ) ∂ ( ) ∂ + + ∂ ∂ ( ) ( ) ∂ ( ) ∂ D z C x y z t P x y z T C x y z t z L 0 00 00 , , , , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ + C x y z t t D C x y z t x D C x y z t y L L 02 0 2 02 2 0 2 02 2 , , , , , , , , , D D C x y z t z L 0 2 02 2 ∂ ( ) ∂ + , , , + ∂ ∂ ( ) ( ) ( ) ∂ ( ) − D x C x y z t C x y z t P x y z T C x y z t L 0 01 00 1 00 , , , , , , , , , , , , ∂ ∂ + ∂ ∂ ( ) ( ) ( ) × − x y C x y z t C x y z t P x y z T 01 00 1 , , , , , , , , , × ∂ ( ) ∂ + ∂ ∂ ( ) ( ) − C x y z t y z C x y z t C x y z t P x y z 00 01 00 1 , , , , , , , , , , , , T T C x y z t z D x C x y z t P x y z L ( ) ∂ ( ) ∂ + ∂ ∂ ( ) 00 0 00 , , , , , , , , , T T ( ) × × ∂ ( ) ∂ + ∂ ∂ ( ) ( ) ∂ C x y z t x y C x y z t P x y z T C x y z t 01 00 01 , , , , , , , , , , , , ( ( ) ∂ + ∂ ∂ ( ) ( ) ∂ ( ) ∂ y z C x y z t P x y z T C x y z t z 00 01 , , , , , , , , , ; ∂ ( ) ∂ = ∂ ( ) ∂ + ∂ ( ) ∂ + C x y z t t D C x y z t x D C x y z t y L L 11 0 2 11 2 0 2 11 2 , , , , , , , , , D D C x y z t z L 0 2 11 2 ∂ ( ) ∂ + , , , + ∂ ∂ ( ) ( ) ( ) ∂ ( ) ∂ − x C x y z t C x y z t P x y z T C x y z t x 10 00 1 00 , , , , , , , , , , , , + ∂ ∂ ( ) ( ) ( ) × − y C x y z t C x y z t P x y z T 10 00 1 , , , , , , , , , × ∂ ( ) ∂ + ∂ ∂ ( ) ( ) − C x y z t y z C x y z t C x y z t P x y z 00 10 00 1 , , , , , , , , , , , , T T C x y z t z D L ( ) ∂ ( ) ∂ + 00 0 , , , + ∂ ∂ ( ) ( ) ∂ ( ) ∂ + ∂ ∂ D x C x y z t P x y z T C x y z t x y C L 0 00 10 0 , , , , , , , , , 0 0 10 x y z t P x y z T C x y z t y , , , , , , , , , ( ) ( ) ∂ ( ) ∂ + + ∂ ∂ ( ) ( ) ∂ ( ) ∂ + z C x y z t P x y z T C x y z t z D L 00 10 0 , , , , , , , , , ∂ ∂ ∂ ( ) ∂ ( ) ∂ + x g x y z T C x y z t x L , , , , , , 01 + ∂ ∂ ( ) ∂ ( ) ∂ + ∂ ∂ ( ) ∂ y g x y z T C x y z t y z g x y z T C x y L L , , , , , , , , , , 01 01 , , , z t z ( ) ∂ ; C x y z t x ij x , , , ( ) = =0 0 , C x y z t x ij x Lx , , , ( ) = = 0 , C x y z t y ij y , , , ( ) = =0 0 , C x y z t y ij y Ly , , , ( ) = = 0 , C x y z t z ij z , , , ( ) = =0 0 , C x y z t z ij z Lz , , , ( ) = = 0, i ≥0, j ≥0; C00 (x,y,z,0)=fC (x,y,z), Cij (x,y,z,0)=0, i ≥1, j ≥1. Functions Cij (x,y,z,t) (i ≥0, j ≥0) could be approximated by the following series during solutions of the above equations
- 17. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 43 C x y z t F L L L L L L F c x c y c z e t C x y z x y z nC n n n nC n 00 0 1 2 , , , ( ) = + ( ) ( ) ( ) ( ) = ∞ ∑ ∑ . Here ( ) 2 2 0 2 2 2 1 1 1 exp nC C x y z e t n D t L L L = − + + , F c u c v f u v w c w d wd vd u nC n n C n L L L z y x = ( ) ( ) ( ) ( ) ∫ ∫ ∫ , , ; 0 0 0 C x y z t L L L n F c x c y c z e t e s u i x y z nC n n n nC nC n 0 2 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) π τ ( ( ) ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ c v g u v w T n L L L L t n z y x , , , 0 0 0 0 1 × ( ) ∂ ( ) ∂ − ( ) ( ) − c w C u v w u d wd vd u d L L L n F c x c y c z n i x y z nC n n n 10 2 2 , , ,τ τ π ( ( ) ( ) − ( ) × ∫ ∑ = ∞ e t e nC nC t n τ 0 1 × ( ) ( ) ( ) ( ) ∂ ( ) ∂ − ∫ c u s v c v g u v w T C u v w v d wd vd u d n n n L i L L z , , , , , , 10 0 0 τ τ y y x L x y z nC nC n L L L n F e t ∫ ∫ ∑ − ( ) × = ∞ 0 2 1 2π × ( ) ( ) ( ) − ( ) ( ) ( ) ( ) ( ) ∂ − c x c y c z e c u c v s v g u v w T C u v n n n nC n n n L i , , , , 10 , , , w w d wd vd u d L L L t z y x ( ) ∂ ∫ ∫ ∫ ∫ 0 0 0 0 , i ≥1; C x y z t L L L n F c x c y c z e t e s u x y z nC n n n nC nC n 01 2 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) π τ ( ( ) ( ) ( )× ∫ ∫ ∫ ∫ ∑ = ∞ c v c w n n L L L t n z y x 0 0 0 0 1 × ( ) ( ) ∂ ( ) ∂ − C u v w P u v w T C u v w u d wd vd u d L L L x y z 00 00 2 2 γ γ τ τ τ π , , , , , , , , , n n F c x c y c z e t nC n n n nC n ( ) ( ) ( ) ( ) × = ∞ ∑ 1 × − ( ) ( ) ( ) ( ) ( ) ( ) ∂ e c u s v c w C u v w P u v w T C u v w nC n n n τ τ τ γ γ 00 00 , , , , , , , , , ( ( ) ∂ − ( )× ∫ ∫ ∫ ∫ ∑ = ∞ v d wd vd u d L L L n e t L L L t x y z nC n z y x τ π 0 0 0 0 2 1 2 × ( ) ( ) ( ) − ( ) ( ) ( ) ( ) ( ) F c x c y c z e c u c v s w C u v w P u nC n n n nC n n n τ τ γ γ 00 , , , ,v v w T C u v w w d wd vd u d L L L t z y x , , , , , ( ) ∂ ( ) ∂ ∫ ∫ ∫ ∫ 00 0 0 0 0 τ τ ; C x y z t L L L n F c x c y c z e t e s u x y z nC n n n nC nC n 02 2 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) π τ ( ( ) ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ c v c w n n L L L t n z y x 0 0 0 0 1 × ( ) ( ) ( ) ∂ ( ) ∂ − C u v w C u v w P u v w T C u v w u d wd v 01 00 1 00 , , , , , , , , , , , , τ τ τ γ γ d d u d L L L F c x c y x y z nC n n n τ π − ( ) ( )× = ∞ ∑ 2 2 1 × ( ) ( ) − ( ) ( ) ( ) ( ) ( ) − nc z e t e c u s v C u v w C u v w P n nC nC n n τ τ τ γ 01 00 1 , , , , , , γ γ τ u v w T C u v w v L L L t z y x , , , , , , ( ) ∂ ( ) ∂ × ∫ ∫ ∫ ∫ 00 0 0 0 0 × ( ) − ( ) ( ) ( ) ( ) − ( ) c w d wd vd u d L L L n F c x c y c z e t e c n x y z nC n n n nC nC n τ π τ 2 2 u u c v n L L t n y x ( ) ( ) × ∫ ∫ ∫ ∑ = ∞ 0 0 0 1 × ( ) ( ) ( ) ( ) ∂ ( ) ∂ − s w C u v w C u v w P u v w T C u v w n 01 00 1 00 , , , , , , , , , , , , τ τ τ γ γ w w d wd vd u d L L L n c x L x y z n n z τ π 0 2 1 2 ∫ ∑ − ( ) × = ∞ × ( ) ( ) ( ) − ( ) ( ) ( ) ( ) ( ) ∂ F c y c z e t e s u c v c w C u v w C nC n n nC nC n n n 01 00 , , , u u v w u L L L t z y x , , , ( ) ∂ × ∫ ∫ ∫ ∫ 0 0 0 0 × ( ) ( ) − ( ) ( − C u v w P u v w T d wd vd u d L L L n F c x c y x y z nC n n 00 1 2 2 γ γ τ τ π , , , , , , ) ) ( ) ( ) − ( ) ( ) × ∫ ∫ ∑ = ∞ c z e t e c u n nC nC n L t n x τ 0 0 1
- 18. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 44 × ( ) ( ) ( ) ( ) ( ) ∂ − s v c w C u v w C u v w P u v w T C u v w n n 01 00 1 00 , , , , , , , , , , , τ τ γ γ , ,τ τ π ( ) ∂ − × ∫ ∫ ∑ = ∞ v d wd vd u d L L L n L L x y z n z y 0 0 2 1 2 × ( ) ( ) ( ) ( ) − ( ) ( ) ( ) ( ) ( F c x c y c z e t e c u c v s w C u v w nC n n n nC nC n n n τ τ 01 , , , ) ) ( ) ( ) × − ∫ ∫ ∫ ∫ C u v w P u v w T L L L t z y x 00 1 0 0 0 0 γ γ τ , , , , , , × ∂ ( ) ∂ − ( ) ( ) ( ) ( ) C u v w w d wd vd u d L L L F c x c y c z e t x y z nC n n n nC 00 2 2 , , ,τ τ π e e s u nC n L t n x − ( ) ( ) × ∫ ∫ ∑ = ∞ τ 0 0 1 × ( ) ( ) ( ) ( ) ∂ ( ) ∂ n c v c w C u v w P u v w T C u v w u d wd vd u d n n 00 01 γ γ τ τ , , , , , , , , , τ τ π 0 0 2 1 2 L L x y z n nC n z y L L L c x e t ∫ ∫ ∑ − ( ) ( ) × = ∞ × ( ) − ( ) ( ) ( ) ( ) ( ) ( ) ∂ F c y e c u s v c w C u v w P u v w T C nC n nC n n n τ τ γ γ 00 0 , , , , , , 1 1 0 0 0 0 u v w v d wd v d u d L L L t z y x , , ,τ τ ( ) ∂ × ∫ ∫ ∫ ∫ × ( )− ( ) ( ) ( ) ( ) − ( ) ( ) ( n c z L L L n F c x c y c z e t e c u c v n x y z nC n n n nC nC n n 2 2 π τ ) ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ s w n L L L t n z y x 0 0 0 0 1 × ( ) ( ) ∂ ( ) ∂ C u v w P u v w T C u v w w d wd vd u d 00 01 γ γ τ τ τ , , , , , , , , , ; C x y z t L L L n F c x c y c z e t e s u x y z nC n n n nC nC n 11 2 2 , , , ( ) = − ( ) ( ) ( ) ( ) − ( ) π τ ( ( ) ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ c v c w n n L L L t n z y x 0 0 0 0 1 × ( ) ∂ ( ) ∂ − ( ) ( g u v w T C u v w u d wd vd u d L L L n F c x c y L x y z nC n n , , , , , , 01 2 2 τ τ π ) ) ( ) ( ) × = ∞ ∑ c z e t n nC n 1 × − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ e c u s v c w g u v w T C u v w v d wd vd u d nC n n n L τ τ τ , , , , , , 01 0 L L L L t x y z z y x L L L ∫ ∫ ∫ ∫ − × 0 0 0 2 2π × ( ) − ( ) ( ) ( ) ( ) ( ) ∂ ( ) ∂ n e t e c u c v s w g u v w T C u v w w d w nC nC n n n L , , , , , , 01 d d vd u d L L L t n z y x 0 0 0 0 1 ∫ ∫ ∫ ∫ ∑ = ∞ × × ( ) ( ) ( )− ( ) ( ) ( ) ( ) − F c x c y c z L L L F c x c y c z e t e nC n n n x y z nC n n n nC nC 2 2 π τ ( ( ) ( ) ( ) × ∫ ∫ ∫ ∑ = ∞ s u c v n n L L t n y x 0 0 0 1 × ( ) ( ) ( ) ∂ ( ) ∂ ∫ n c w C u v w P u v w T C u v w u d wd vd u d n Lz 00 10 0 γ γ τ τ τ , , , , , , , , , − − ( ) ( )× = ∞ ∑ 2 2 1 π L L L n F c x c y x y z nC n n n × ( ) ( ) − ( ) ( ) ( ) ( ) ( ) ( ) c z e t e c u s v c w C u v w P u v w T n nC nC n n n τ τ γ γ 00 , , , , , , ∂ ∂ ( ) ∂ − ∫ ∫ ∫ ∫ C u v w v d wd vd u d L L L t z y x 10 0 0 0 0 , , ,τ τ − ( ) ( ) ( ) ( ) − ( ) ( ) ( ) ( ) 2 2 π τ L L L n F c x c y c z e t e c u c v s w C x y z nC n n n nC nC n n n 0 00 0 0 0 0 1 γ γ τ u v w P u v w T L L L t n z y x , , , , , , ( ) ( ) × ∫ ∫ ∫ ∫ ∑ = ∞ × ∂ ( ) ∂ − ( ) ( ) ( ) ( C u v w w d wd v d u d L L L n F c x c y c z e t x y z nC n n n nC 10 2 2 , , ,τ τ π ) ) − ( ) ( ) × ∫ ∫ ∑ = ∞ e s u nC n L t n x τ 0 0 1 × ( ) ( ) ( ) ( ) ( ) ∂ − c v c w C u v w C u v w P u v w T C u v w n n 10 00 1 00 , , , , , , , , , , , τ τ γ γ , ,τ τ π ( ) ∂ − × ∫ ∫ ∑ = ∞ u d wd vd u d L L L n L L x y z n z y 0 0 2 1 2
- 19. Pankratov: On optimization of of a double-tail dynamic comparator AJCSE/Oct-Dec-2021/Vol 6/Issue 4 45 × ( ) ( ) ( ) ( ) − ( ) ( ) ( ) ( ) − F c x c y c z e t e c u s v c w C u v w nC n n n nC nC n n n τ γ 00 1 , , , , , , , , , , τ τ γ ( ) ( ) ∂ ( ) ∂ × ∫ ∫ ∫ ∫ P u v w T C u v w v L L L t z y x 00 0 0 0 0 × ( ) − ( ) ( ) ( ) ( ) C u v w d wd vd u d L L L n F c x c y c z e t e x y z nC n n n nC n 10 2 2 , , ,τ τ π C C n L t n c u x − ( ) ( ) × ∫ ∫ ∑ = ∞ τ 0 0 1 × ( ) ( ) ( ) ( ) ( ) ∂ − c v s w C u v w C u v w P u v w T C u v w n n 10 00 1 00 , , , , , , , , , , , τ τ γ γ , ,τ τ ( ) ∂ ∫ ∫ w d wd vd u d L L z y 0 0 .