ON APPROACH TO INCREASE INTEGRATION RATE OF FIELD-EFFECT HETEROTRANSISTORS IN THE FRAMEWORK OF A BOOTSTRAP SWITCH

International Journal of Microelectronics Engineering (IJME), Vol.10, No.1, January 2024
DOI: 10.5121/ijme.2024.10101 1
ON APPROACH TO INCREASE INTEGRATION
RATE OF FIELD-EFFECT HETEROTRANSISTORS
IN THE FRAMEWORK OF A BOOTSTRAP SWITCH
E.L. Pankratov
Nizhny Novgorod State University, 23 Gagarin avenue,
Nizhny Novgorod, 603950, Russia
Nizhny Novgorod State Agrotechnical University, 97 Gagarin avenue,
Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we introduce an approach to increase integration rate of field-effect heterotransistors in
the framework of a bootstrap switch. In the framework of 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.
KEYWORDS
heterotransistors; bootstrap switch; optimization of manufacturing; analytical approach for prognosis.
1. INTRODUCTION
An actual and intensively solving problems of solid state electronics is increasing of integra-
tion rate of elements of integrated circuits (p-n-junctions, their systems et al) [1-8]. Increasing
of the integration rate leads to necessity to decrease their dimensions. To decrease the dimen-
sions are using several approaches. They are widely using laser and microwave types of an-
nealing 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 pos-
sibility to increase integration rate of elements of integrated circuits through inhomogeneity
of technological parameters due to generating inhomogenous distribution of temperature. In
this situation one can obtain decreasing dimensions of elements of integrated circuits [18]
with account Arrhenius law [1,3]. Another approach to manufacture elements of integrated
circuits with smaller dimensions is doping of heterostructure by diffusion or ion implantation
[1-3]. However in this case optimization of dopant and/or radiation defects is required [18].
In this paper we consider a heterostructure. The heterostructure consist of a substrate and sev-
eral 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 heterotransistors in the framework of a bootstrap switch
so as it is shown on Figs. 1. The manufacturing gives a possibility to increase density of ele-
ments of the integrator circuit. After the considered doping dopant and/or radiation defects
should be annealed. In the framework of 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.

2
Fig. 1. The considered switch
2. METHOD OF SOLUTION
In this section we determine spatio-temporal distributions of concentrations of infused and
implanted dopants. To determine these distributions we calculate appropriate solutions of the
second Fick's law [1,3,18]
       





















z
t
z
y
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C
C
C
C













 ,
,
,
,
,
,
,
,
,
,
,
,
.
(1)
Boundary and initial conditions for the equations are
  0
,
,
,
0




x
x
t
z
y
x
C
,
  0
,
,
,



 x
L
x
x
t
z
y
x
C
,
  0
,
,
,
0




y
y
t
z
y
x
C
,
  0
,
,
,



 y
L
x
y
t
z
y
x
C
,
  0
,
,
,
0




z
z
t
z
y
x
C
,
  0
,
,
,



 z
L
x
z
t
z
y
x
C
, C (x,y,z,0)=f (x,y,z). (2)
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 dif-
fusion 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]
   
 
   
  














 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
V
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
D L
C


 

. (3)
Here the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due to
Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) de-
scribes the limit of solubility of dopant. Parameter  [1,3] describes average quantity of
charged defects interacted with atom of dopant [20]. The function V(x,y,z,t) describes the spa-
tio-temporal distribution of concentration of radiation vacancies. Parameter V*
describes the
equilibrium distribution of concentration of vacancies. The considered concentrational de-
pendence of dopant diffusion coefficient has been described in details in [20]. It should be

3
noted, that using diffusion type of doping did not generation radiation defects. In this situation
1= 2= 0. We determine spatio-temporal distributions of concentrations of radiation defects
by solving the following system of equations [21,22]
          
























y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
         











 t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
V
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
   
t
z
y
x
I
T
z
y
x
k I
I
,
,
,
,
,
, 2
,
 (4)
          
























y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
         











 t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
V
I
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
   
t
z
y
x
V
T
z
y
x
k V
V
,
,
,
,
,
, 2
,
 .
Boundary and initial conditions for these equations are
  0
,
,
,
0




x
x
t
z
y
x

,
  0
,
,
,



 x
L
x
x
t
z
y
x

,
  0
,
,
,
0




y
y
t
z
y
x

,
  0
,
,
,



 y
L
y
y
t
z
y
x

,
  0
,
,
,
0




z
z
t
z
y
x

,
  0
,
,
,



 z
L
z
z
t
z
y
x

,  (x,y,z,0)=f (x,y,z). (5)
Here  =I,V. The function I (x,y,z,t) describes the spatio-temporal distribution of concentra-
tion of radiation interstitials; D(x,y,z,T) are the diffusion coefficients of point radiation de-
fects; terms V2
(x,y,z,t) and I2
(x,y,z,t) correspond to generation divacancies and diinterstitials;
kI,V(x,y,z,T) is the parameter of recombination of point radiation defects; kI,I(x,y,z,T) and
kV,V(x,y,z,T) are the parameters of generation of simplest complexes of point radiation defects.
Further we determine distributions in space and time of concentrations of divacancies
V(x,y,z,t) and diinterstitials I(x,y,z,t) by solving the following system of equations [21,22]
          





 






 




y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x I
I
I
I
I









 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
           
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,







 
 




(6)
          





 






 




y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x V
V
V
V
V









 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
           
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
V
V
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,







 
 




.
Boundary and initial conditions for these equations are
 
0
,
,
,
0





x
x
t
z
y
x

,
 
0
,
,
,




 x
L
x
x
t
z
y
x

,
 
0
,
,
,
0





y
y
t
z
y
x

,
 
0
,
,
,




 y
L
y
y
t
z
y
x

,

4
 
0
,
,
,
0





z
z
t
z
y
x

,
 
0
,
,
,




 z
L
z
z
t
z
y
x

, I (x,y,z,0)=fI (x,y,z), V (x,y,z,0)=fV (x,y,z). (7)
Here D(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects;
kI(x,y,z,T) and kV(x,y,z,T) are the parameters of decay of these complexes.
We calculate distributions of concentrations of point radiation defects in space and time by
recently elaborated approach [18]. The approach based on transformation of approximations
of diffusion coefficients in the following form: D(x,y,z,T)=D0[1+ g(x,y,z,T)], where D0
are the average values of diffusion coefficients, 0<1, |g(x, y,z,T)|1,  =I,V. We also used
analogous transformation of approximations of parameters of recombination of point defects
and parameters of generation of their complexes: kI,V(x,y,z,T)=k0I,V[1+I,V gI,V(x,y,z,T)],
kI,I(x,y,z,T)=k0I,I [1+I,I gI,I(x,y,z,T)] and kV,V(x,y,z,T)=k0V,V [1+V,V gV,V(x,y,z,T)], where k01,2 are
the their average values, 0I,V <1, 0I,I <1, 0V,V<1, | gI,V(x,y,z,T)|1, | gI,I(x,y,z,T)|1,
|gV,V(x,y,z,T)|1. Let us introduce the following dimensionless variables:  = x/Lx,
    *
,
,
,
,
,
,
~
I
t
z
y
x
I
t
z
y
x
I  ,     *
,
,
,
,
,
,
~
V
t
z
y
x
V
t
z
y
x
V  , V
I
V
I
D
D
k
L 0
0
,
0
2

 ,
V
I
D
D
k
L 0
0
,
0
2




 , 2
0
0
L
t
D
D V
I

 ,  = y /Ly,  = z/Lz. The introduction leads to transfor-
mation of Eqs.(4) and conditions (5) to the following form
   
     
 
 


















T
g
I
T
g
D
D
D
I
I
I
I
I
V
I
I
,
,
,
1
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0




















   
     


















 


















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
0
0
0
I
I
T
g
D
D
D
D
D
D
I
I
I
V
I
I
V
I
I
 
     
   
















 ,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1 2
,
,
,
,
I
T
g
V
T
g I
I
I
I
I
V
I
V
I




 (8)
   
     
 
 


















T
g
V
T
g
D
D
D
V
V
V
V
V
V
I
V
,
,
,
1
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0




















   
     


















 


















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
0
0
0
I
V
T
g
D
D
D
D
D
D
V
V
V
V
I
V
V
I
V
 
     
   
















 ,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1 2
,
,
,
,
V
T
g
V
T
g V
V
V
V
V
V
I
V
I





  0
,
,
,
~
0











,
  0
,
,
,
~
1











,
  0
,
,
,
~
0











,
  0
,
,
,
~
1











,
  0
,
,
,
~
0











,
  0
,
,
,
~
1











,  
 
*
,
,
,
,
,
,
~









 
f
 . (9)
We determine solutions of Eqs.(8) with conditions (9) framework recently introduced ap-
proach [18], i.e. as the power series
   
  







0 0 0
,
,
,
~
,
,
,
~
i j k
ijk
k
j
i











 
 . (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  



 ,
,
,
~
000
I and
 



 ,
,
,
~
000
V and corrections for them  



 ,
,
,
~
ijk
I and  



 ,
,
,
~
ijk
V , i 1, j 1, k 1. The

5
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 dif-
fusion coefficients in the following form: D(x,y,z,T)=D0[1+ g(x,y,z,T)], where D0
are the average values of diffusion coefficients. In this situation the Eqs.(6) could be written
as
   
       






 






t
z
y
x
I
T
z
y
x
k
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x
I
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
, 2
,
0







 
     
    





 







 

 





z
t
z
y
x
T
z
y
x
g
z
D
y
t
z
y
x
T
z
y
x
g
y
D I
I
I
I
I
I
I
I









 ,
,
,
,
,
,
1
,
,
,
,
,
,
1 0
0
   
t
z
y
x
I
T
z
y
x
kI
,
,
,
,
,
,

   
       






 






t
z
y
x
I
T
z
y
x
k
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x
I
I
V
V
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
, 2
,
0







 
     
   






 







 

 





z
t
z
y
x
T
z
y
x
g
z
D
y
t
z
y
x
T
z
y
x
g
y
D V
V
V
V
V
V
V
V









 ,
,
,
,
,
,
1
,
,
,
,
,
,
1 0
0
   
t
z
y
x
I
T
z
y
x
kI
,
,
,
,
,
,
 .
Farther we determine solutions of above equations as the following power series
   
 





0
,
,
,
,
,
,
i
i
i
t
z
y
x
t
z
y
x 


 . (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 concentra-
tions of complexes of defects 0(x,y,z,t), corrections for them i(x,y,z,t) (for them i 1) and
boundary and initial conditions for them. We remove equations and conditions to the Appen-
dix. Solutions of the equations have been calculated by standard approaches [24,25] and pre-
sented in the Appendix.
Now we calculate distribution of concentration of dopant in space and time by using the ap-
proach, which was used for analysis of radiation defects. To use the approach we consider
following transformation of approximation of dopant diffusion coefficient:
DL(x,y,z,T)=D0L[1+ LgL(x,y,z,T)], where D0L is the average value of dopant diffusion coeffi-
cient, 0L< 1, |gL(x,y,z,T)|1. Farther we consider solution of Eq.(1) as the following series:
   
 





0 1
,
,
,
,
,
,
i j
ij
j
i
L
t
z
y
x
C
t
z
y
x
C 
 .
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 by using the second-order approximations on all parameters, which have been
used in appropriate series. Usually the second-order approximations are enough good approx-

6
imations to make qualitative analysis and to obtain quantitative results. All analytical results
have been checked by numerical simulation.
3. DISCUSSION
In this section we analyzed spatio-temporal distributions of concentrations of dopants. Figs. 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 areas. In this situation one can find increasing of
compactness of field-effect transistors with increasing of homogeneity of distribution of con-
centration of dopant at one time. Changing relation between values of dopant diffusion coef-
ficients leads to opposite result (see Figs. 3).
It should be noted, that in the framework of the considered approach one shall optimize an-
nealing of dopant and/or radiation defects. To do the optimization we used recently intro-
duced criterion [26-34]. The optimization based on approximation real distribution by step-
wise function  (x,y, z) (see Figs. 4). Farther the required values of optimal annealing time
have been calculated by minimization the following mean- squared error
   
 
   


x
L y
L z
L
z
y
x
x
d
y
d
z
d
z
y
x
z
y
x
C
L
L
L
U
0 0 0
,
,
,
,
,
1
 . (12)
Fig. 2a. Dependences of concentration of dopant, infused in heterostructure from Figs. 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

7
x
0.0
0.5
1.0
1.5
2.0
C(x,

)
2
3
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig. 2b. Dependences of concentration of dopant, implanted in heterostructure from Figs. 1, on coordi-
nate in direction, which is perpendicular to interface between epitaxial layer substrate.
Difference between values of dopant diffusion coefficient in layers of heterostructure increas-
es with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial
layer is larger, than value of dopant diffusion coefficient in the substrate. Curve 1 corresponds
to homogenous sample and annealing time  = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0. Curve 2 corresponds
to homogenous sample and annealing time  = 0.0057 (Lx
2
+Ly
2
+Lz
2
)/D0. Curves 3 and 4 cor-
respond to heterostructure from Figs. 1; annealing times  = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0 and  =
0.0057 (Lx
2
+Ly
2
+Lz
2
)/D0, respectively
Fig.3a. Distributions of concentration of dopant, infused in average section of epitaxial layer of hetero-
structure from Figs. 1 in direction parallel to interface between epitaxial layer and substrate of hetero-
structure.
Difference between values of dopant diffusion coefficients increases with increasing of num-
ber of curves. Value of dopant diffusion coefficient in this section is smaller, than value of
dopant diffusion coefficient in nearest sections

8
x
0.00000
0.00001
0.00010
0.00100
0.01000
0.10000
1.00000
C(x,

)
fC(x)
L/4
0 L/2 3L/4 L
x0
1
2
Substrate
Epitaxial
layer 1
Epitaxial
layer 2
Fig.3b. 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
C(x,

)
0 Lx
2
1
3
4
Fig.4a. Distributions of concentration of infused dopant in depth of heterostructure from Fig. 1 for dif-
ferent values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increas-
ing of number of curve corresponds to increasing of annealing time
x
C(x,

)
1
2
3
4
0 L
Fig.4b. Distributions of concentration of implanted dopant in depth of heterostructure from Fig. 1 for
different values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). In-
creasing of number of curve corresponds to increasing of annealing time

9
0.0 0.1 0.2 0.3 0.4 0.5
a/L, , , 
0.0
0.1
0.2
0.3
0.4
0.5

D
0
L
-2
3
2
4
1
Fig. 5a. 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  =  = 0 for
equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve
2 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =
 = 0. Curve 3 describes the dependence of the annealing time on value of parameter  for
a/L=1/2 and  =  = 0. Curve 4 describes the dependence of the annealing time on value of
parameter  for a/L=1/2 and  =  = 0
0.0 0.1 0.2 0.3 0.4 0.5
a/L, , , 
0.00
0.04
0.08
0.12

D
0
L
-2
3
2
4
1
Fig.5b. 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  =  = 0 for
equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve
2 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and  =
 = 0. Curve 3 describes the dependence of the annealing time on value of parameter  for
a/L=1/2 and  =  = 0. Curve 4 describes the dependence of the annealing time on value of
parameter  for a/L=1/2 and  =  = 0
We show optimal values of annealing time as functions of parameters on Figs. 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 hetero-
structure. If the dopant has no enough time to achieve the interface, it is practicably to anneal
the dopant additionally. The Fig. 5b shows the described dependences of optimal values of
additional annealing time for the same parameters as for Fig. 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.

10
4. CONCLUSIONS
In this paper we introduce an approach to increase integration rate of field-effect heterotran-
sistors in the framework of a bootstrap switch. The approach gives us possibility to decrease
area of the elements with smaller increasing of the element’s thickness.
REFERENCES
[1] V.I. Lachin, N.S. Savelov. Electronics. Phoenix, Rostov-na-Donu, 2001.
[2] A.G. Alexenko, I.I. Shagurin. Microcircuitry. Radio and communication, Moscow, 1990.
[3] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics. Radio and communica-
tion, Moscow, 1991.
[4] C.M. Ippolito, A. Italia, R. Guerra, G. Palmisano. Analog. Integr. Circ. Sig. Process. Vol. 80. P.
459-471 (2014).
[5] D. Fathi, B. Forouzandeh, N. Masoumi. Nano. Vol. 4 (4). P. 233-238 (2009).
[6] S.A. Chachuli, P.N.A. Fasyar, N. Soin, N.M. Kar, N. Yusop. Mat. Sci. Sem. Proc. Vol. 24. P. 9-
14 (2014).
[7] A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M.
Litvin, V.V. Milenin, A.V. Sachenko. Semiconductors. Vol. 43 (7). P. 897-903 (2009).
[8] Z. Li, J. Waldron, T. Detchprohm, C. Wetzel, R.F. Karlicek, Jr.T.P. Chow. Appl. Phys. Lett.
Vol. 102 (19). P. 192107-192109 (2013).
[9] Jung-Hui Tsai, Shao-Yen Chiu, Wen-Shiung Lour, Der-Feng Guo. Semiconductors. Vol. 43 (7).
P. 971-974 (2009).
[10] O.V. Alexandrov, A.O. Zakhar'in, N.A. Sobolev, E.I. Shek, M.M. Makoviychuk, E.O. Parshin.
Semiconductors. Vol. 32 (9). P. 1029-1032 (1998).
[11] M.J. Kumar, T.V. Singh. Int. J. Nanoscience. Vol. 7 (2-3). P. 81-84 (2008).
[12] P. Sinsermsuksakul, K. Hartman, S.B. Kim, J. Heo, L. Sun, H.H. Park, R. Chakraborty, T.
Buonassisi, R.G. Gordon. Appl. Phys. Lett. Vol. 102 (5). P. 053901-053905 (2013).
[13] J.G. Reynolds, C.L. Reynolds, Jr.A. Mohanta, J.F. Muth, J.E. Rowe, H.O. Everitt, D.E. Aspnes.
Appl. Phys. Lett. Vol. 102 (15). P. 152114-152118 (2013).
[14] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Appl. Phys. Lett. Vol. 89
(17). P. 172111-172114 (2006).
[15] S. Kundu, S. Sarkar, P. Mandal, A. Islam. Analog Integrated Circuits and Signal Processing.
Vol. 99 (1). P. 95-109 (2019).
[16] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. Semiconductors. Vol. 36 (5). P. 611-617
(2002).
[17] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov,
Yu.N. Drozdov, V.D. Skupov. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843
(2003).
[18] E.L. Pankratov, E.A. Bulaeva. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013).
[19] Yu.N. Erofeev. Pulse devices. Moscow: Higher School, 1989.
[20] V.V. Kozlivsky. Modification of semiconductors by proton beams. Sant- Peterburg: Nauka,
2003.
[21] Z.Yu. Gotra. Technology of microelectronic devices. Moscow: Radio and communication, 1991.
[22] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. Kiev: Naukova Dumka,
Kiev.
[23] P.M. Fahey, P.B. Griffin, J.D. Plummer. Rev. Mod. Phys. Vol. 61 (2). P. 289-388 (1989).
[24] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations. Moscow: Nauka, 1972.
[25] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids. London: Oxford University Press, 1964.
[26] E.L. Pankratov. Russian Microelectronics. Vol. 36 (1). P. 33-39 (2007).
[27] E.L. Pankratov. Int. J. Nanoscience. Vol. 7 (4-5). P. 187-197 (2008).
[28] E.L. Pankratov. J. Comp. Theor. Nanoscience. Vol. 14 (10). P. 4885-4899 (2017).
[29] E.L. Pankratov. Advanced science, engineering and medicine. Vol. 9 (9). P. 787-801 (2017).
[30] E.L. Pankratov, E.A. Bulaeva. Materials science in semiconductor processing. Vol. 34. P. 260-
268 (2015).
[31] E.L. Pankratov, E.A. Bulaeva. Multidiscipline Modeling in Materials and Structures. Vol. 13
(4). P. 664-677 (2017).

11
[32] E.L. Pankratov. Journal of coupled systems and multiscale dynamics. Vol. 6 (1). P. 36-52
(2018).
[33] E.L. Pankratov. Journal of computational and theoretical nanoscience. Vol. 15 (6/7). P. 1898-
1914 (2018).
APPENDIX
Equations for the functions  



 ,
,
,
~
ijk
I and  



 ,
,
,
~
ijk
V , i 0, j 0, k 0 and conditions
for them
       

















2
000
2
2
000
2
2
000
2
0
0
000
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
I
I
D
D
I
V
I
       

















2
000
2
2
000
2
2
000
2
0
0
000
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
I
V
;
        


















V
I
i
i
i
V
I
i
D
D
I
I
I
D
D
I
0
0
2
00
2
2
00
2
2
00
2
0
0
00
,
,
,
~
,
,
,
~
,
,
,
~
,
~


















       



























 



















,
,
,
~
,
,
,
,
,
,
~
,
,
, 100
100 i
I
i
I
I
T
g
I
T
g
   















 









,
,
,
~
,
,
, 100
i
I
I
T
g , i 1,
       
 


























T
g
V
V
V
D
D
V
V
i
i
i
I
V
i
,
,
,
,
,
,
~
,
,
,
~
,
,
,
~
,
~
2
00
2
2
00
2
2
00
2
0
0
00






















 
 
 
 


























 

T
g
V
T
g
D
D
D
D
V
V
i
V
I
V
I
V
i
,
,
,
,
,
,
~
,
,
,
,
,
,
~
100
0
0
0
0
100


















 
I
V
i
D
D
V
0
0
100
,
,
,
~






 





, i 1,
       




















2
010
2
2
010
2
2
010
2
0
0
010
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
I
I
D
D
I
V
I
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,
V
I
T
g V
I
V
I


        

















2
010
2
2
010
2
2
010
2
0
0
010
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
I
V
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,
V
I
T
g V
I
V
I

 ;
        

















2
020
2
2
020
2
2
020
2
0
0
020
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
I
I
D
D
I
V
I
 
         
 



















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
,
V
I
V
I
T
g V
I
V
I



        

















2
020
2
2
020
2
2
020
2
0
0
020
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
V
I
 
         
 



















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
,
V
I
V
I
T
g V
I
V
I


 ;

12
       




















2
001
2
2
001
2
2
001
2
0
0
001
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
I
I
D
D
I
V
I
 
   







 ,
,
,
~
,
,
,
1 2
000
,
,
I
T
g I
I
I
I


       




















2
001
2
2
001
2
2
001
2
0
0
001
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
I
V
 
   







 ,
,
,
~
,
,
,
1 2
000
,
,
V
T
g I
I
I
I

 ;
        


















V
I
V
I
D
D
I
I
I
D
D
I
0
0
2
110
2
2
110
2
2
110
2
0
0
110
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~




















 
 
 
 
 



































 T
g
I
T
g
I
T
g I
I
I ,
,
,
,
,
,
~
,
,
,
,
,
,
~
,
,
, 010
010






















         
 













 




















,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100
010
V
I
V
I
I
 
 
T
g I
I
I
I
,
,
,
1 ,
,






        

















2
110
2
2
110
2
2
110
2
0
0
110
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
I
V
       














































,
,
,
~
,
,
,
,
,
,
~
,
,
, 010
010
0
0
V
T
g
V
T
g
D
D
V
V
I
V
     
 

















 T
g
V
T
g V
V
V
V
V
,
,
,
1
,
,
,
~
,
,
, ,
,
010













       
 















 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100
I
V
I
V 
 ;
        

















2
002
2
2
002
2
2
002
2
0
0
002
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
I
I
D
D
I
V
I
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
,
I
I
T
g I
I
I
I


        

















2
002
2
2
002
2
2
002
2
0
0
002
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
I
V
 
     











 ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
,
V
V
Е
g V
V
V
V

 ;
        

















2
101
2
2
101
2
2
101
2
0
0
101
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 I
I
I
D
D
I
V
I
       














































,
,
,
~
,
,
,
,
,
,
~
,
,
, 001
001
0
0
I
T
g
I
T
g
D
D
I
I
V
I
     
     





















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 000
100
001
V
I
T
g
I
T
g I
I
I


















        

















2
101
2
2
101
2
2
101
2
0
0
101
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~



















 V
V
V
D
D
V
I
V

13
       














































,
,
,
~
,
,
,
,
,
,
~
,
,
, 001
001
0
0
V
T
g
V
T
g
D
D
V
V
I
V
     
     





















,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 100
000
001
V
I
T
g
V
T
g V
V
V

















 ;
         










































,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
010
2
011
2
2
011
2
2
011
2
0
0
011
I
I
I
I
D
D
I
V
I
 
     
     



















 ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1 000
001
,
,
000
,
,
V
I
T
g
I
T
g V
I
V
I
I
I
I
I




         










































,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
010
2
011
2
2
011
2
2
011
2
0
0
011
V
V
V
V
D
D
V
I
V
 
     
     



















 ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1 001
000
,
,
000
,
,
V
I
t
g
V
T
g V
I
V
I
V
V
V
V



 ;
 
0
,
,
,
~
0




x
ijk






,
 
0
,
,
,
~
1




x
ijk






,
 
0
,
,
,
~
0










ijk
,
 
0
,
,
,
~
1










ijk
,
 
0
,
,
,
~
0










ijk
,
 
0
,
,
,
~
1










ijk
(i 0, j 0, k 0);
    *
000
,
,
0
,
,
,
~ 






 
f
 ,   0
0
,
,
,
~ 



ijk
(i 1, j 1, k 1).
Solutions of the above equations could be written as
         




1
000
2
1
,
,
,
~
n
n
n
e
c
c
c
F
L
L








 

,
where        
  

1
0
1
0
1
0
*
,
,
cos
cos
cos
1
u
d
v
d
w
d
w
v
u
f
w
n
v
n
u
n
F n
n 





, cn() = cos ( n ),
   
I
V
nI
D
D
n
e 0
0
2
2
exp 

 
 ,    
V
I
nV
D
D
n
e 0
0
2
2
exp 

 
 ;
                  
    








1 0
1
0
1
0
1
0
100
0
0
00
,
,
,
~
2
,
,
,
~
n
i
n
n
nI
nI
n
V
I
i
u
w
v
u
I
v
c
u
s
e
e
c
c
c
n
D
D
I
 










                 
   




1 0
1
0
1
0
0
0
2
,
,
,
n
n
n
nI
nI
n
V
I
I
n
v
s
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
T
w
v
u
g
w
c








               
  








1 0
0
0
1
0
100
2
,
,
,
~
,
,
,
n
nI
nI
n
V
I
i
I
n
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c









         
  


 
1
0
1
0
1
0
100
,
,
,
~
,
,
, 

d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c i
I
n
n
n , i 1,
                 
 
   




1 0
1
0
1
0
1
0
0
0
00
,
,
,
2
,
,
,
~
n
V
n
n
nI
nV
n
I
V
i
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V











                 
 
  








1 0
1
0
1
0
0
0
100
,
~
n
n
n
nI
nV
n
I
V
i
n
v
s
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
u
V
w
c








             
 








1
0
0
1
0
100
2
,
~
,
,
,
2
n
nV
n
I
V
i
V
n
e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
u
V
T
w
v
u
g
w
c 







           
   



 




0
1
0
1
0
1
0
100
,
~
,
,
, d
u
d
v
d
w
d
w
u
V
T
w
v
u
g
w
s
v
c
u
c
e i
V
n
n
n
nI , i 1,

14
where sn() = sin ( n );
                 
     




1 0
1
0
1
0
1
0
010
2
,
,
,
~
n
n
n
n
n
n
n
n
n
w
c
v
c
u
c
e
e
c
c
c













 
      


 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
V
I
,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,

 ;
                  
     




1 0
1
0
1
0
1
0
,
0
0
020
1
2
,
,
,
~
n
V
I
n
n
n
n
n
n
n
n
V
I
w
c
v
c
u
c
e
e
c
c
c
D
D 













         
  



 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
, 010
000
000
010
,

 ;
                  
    




1 0
1
0
1
0
1
0
001
2
,
,
,
~
n
n
n
n
n
n
n
n
n
w
c
v
c
u
c
e
e
c
c
c













 
    


 



d
u
d
v
d
w
d
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
1 2
000
,
,

 ;
                 
   
 




1 0
1
0
1
0
1
0
002
2
,
,
,
~
n
n
n
n
n
n
n
n
n
w
c
v
c
u
c
e
e
c
c
c













 
      




 



d
u
d
v
d
w
d
w
v
u
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
,

 ;
                 
    




1 0
1
0
1
0
1
0
0
0
110
2
,
,
,
~
n
n
n
n
nI
nI
n
n
n
V
I
u
c
v
c
u
s
e
e
c
c
c
n
D
D
I











           








1
0
0
100
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
i
I
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
T
w
v
u
g 






            

   



 
V
I
i
I
n
n
n
nI
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
u
c
v
s
u
c
e
0
0
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
, 




              
    







1 0
1
0
1
0
1
0
100
,
,
,
~
,
,
,
n
i
I
n
n
n
nI
nI
d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
u
s
v
c
u
c
e
e
n





                     
     





1 0
1
0
1
0
1
0
,
1
2
n
V
I
n
n
n
nI
n
n
nI
n
n
n
n
v
c
v
c
u
c
e
c
c
e
c
c
c
c










         
  



 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
, 100
000
000
100
,


                 
    




1 0
1
0
1
0
1
0
0
0
110
2
,
,
,
~
n
n
n
n
nV
nV
n
n
n
I
V
u
c
v
c
u
s
e
e
c
c
c
n
D
D
V











           








1
0
0
100
2
,
,
,
~
,
,
,
n
nV
n
n
n
I
V
i
V
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
V
T
w
v
u
g 






            

   



 
I
V
i
V
n
n
n
nV
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
u
c
v
s
u
c
e
0
0
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
, 




              
    







1 0
1
0
1
0
1
0
100
,
,
,
~
,
,
,
n
i
V
n
n
n
nV
nV
d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
u
s
v
c
u
c
e
e
n





                     
 
     





1 0
1
0
1
0
1
0
,
,
,
,
,
1
2
n
V
I
V
I
n
n
nV
n
n
nI
n
n
n
n
T
w
v
u
g
v
c
u
c
e
c
c
e
c
c
c
c










         
  



 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
w
cn
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100

 ;
                 
    




1 0
1
0
1
0
1
0
0
0
101
,
,
,
2
,
,
,
~
n
I
n
n
nI
nI
n
n
n
V
I
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
I











           






1
0
0
001
2
,
,
,
~
n
nI
n
n
n
V
I
n
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
w
c 







15
               


 




1
0
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
n
nI
V
I
I
n
n
c
c
c
e
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c
v
s 






                  


   





1
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
n
I
n
n
n
nI
c
c
c
d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e 






           
     
    









0
1
0
1
0
1
0
000
100
,
,
,
,
,
~
,
,
,
~
,
,
,
1 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g
w
c
v
c
u
c
e
e V
I
V
I
n
n
n
nI
nI
                 
    




1 0
1
0
1
0
1
0
0
0
101
,
,
,
2
,
,
,
~
n
V
n
n
nV
nV
n
n
n
I
V
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V











               
  






1 0
1
0
0
0
001
2
,
,
,
~
n
n
nV
nI
n
n
n
I
V
n
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
V
w
c









               


 




1
0
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
n
nI
V
I
I
n
n
c
c
c
e
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c
v
s 






                  


   





1
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
n
V
n
n
n
nV
c
c
c
d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
w
s
v
c
u
c
e 






           
     
    









0
1
0
1
0
1
0
000
100
,
,
,
,
,
~
,
,
,
~
,
,
,
1 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g
w
c
v
c
u
c
e
e V
I
V
I
n
n
n
nV
nV
;
                   

     




1 0
1
0
1
0
1
0
000
011
,
,
,
~
2
,
,
,
~
n
n
n
n
nI
nI
n
n
n
w
v
u
I
w
c
v
c
u
c
e
e
c
c
c
I











 
     
      




 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
I
T
w
v
u
g V
I
V
I
I
I
I
I
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1 000
001
,
,
010
,
,




                   

     




1 0
1
0
1
0
1
0
000
011
,
,
,
~
2
,
,
,
~
n
n
n
n
nV
nV
n
n
n
w
v
u
I
w
c
v
c
u
c
e
e
c
c
c
V











 
     
      




 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
I
T
w
v
u
g V
I
V
I
I
I
I
I
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
1 000
001
,
,
010
,
,



 .
Equations for functions i(x,y,z,t), i 0 to describe concentrations of simplest complexes of radiation
defects.
        





 






 2
0
2
2
0
2
2
0
2
0
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
y
t
z
y
x
x
t
z
y
x
D
t
t
z
y
x I
I
I
I
I








       
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k I
I
I
,
,
,
,
,
,
,
,
,
,
,
, 2
,


        





 






 2
0
2
2
0
2
2
0
2
0
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
y
t
z
y
x
x
t
z
y
x
D
t
t
z
y
x V
V
V
V
V








       
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k V
V
V
,
,
,
,
,
,
,
,
,
,
,
, 2
,

 ;
       






 






 2
2
2
2
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
y
t
z
y
x
x
t
z
y
x
D
t
t
z
y
x i
I
i
I
i
I
I
i
I








 
 
 
 









 






 
 




y
t
z
y
x
T
z
y
x
g
y
x
t
z
y
x
T
z
y
x
g
x
D i
I
I
i
I
I
I







 ,
,
,
,
,
,
,
,
,
,
,
, 1
1
0
 
 








 
 

z
t
z
y
x
T
z
y
x
g
z
i
I
I



 ,
,
,
,
,
, 1
, i1,
       






 






 2
2
2
2
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
y
t
z
y
x
x
t
z
y
x
D
t
t
z
y
x i
V
i
V
i
V
V
i
V









16
 
 
 
 









 






 
 




y
t
z
y
x
T
z
y
x
g
y
x
t
z
y
x
T
z
y
x
g
x
D i
V
V
i
V
V
V







 ,
,
,
,
,
,
,
,
,
,
,
, 1
1
0
 
 








 
 

z
t
z
y
x
T
z
y
x
g
z
i
V
V



 ,
,
,
,
,
, 1
, i1;
Boundary and initial conditions for the functions takes the form
 
0
,
,
,
0





x
i
x
t
z
y
x

,
 
0
,
,
,




 x
L
x
i
x
t
z
y
x

,
 
0
,
,
,
0





y
i
y
t
z
y
x

,
 
0
,
,
,




 y
L
y
i
y
t
z
y
x

,
 
0
,
,
,
0





z
i
z
t
z
y
x

,
 
0
,
,
,




 z
L
z
i
z
t
z
y
x

, i0; 0(x,y,z,0)=f (x,y,z), i(x,y,z,0)=0, i1.
Solutions of the above equations could be written as
                













1
1
0
0
2
2
,
,
,
n
n
n
n
n
n
n
n
n
n
z
y
x
z
y
x
z
c
y
c
x
c
n
L
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
F
t
z
y
x 



             

    

 

t x
L y
L z
L
I
I
n
n
n
n
n
w
v
u
I
T
w
v
u
k
w
c
v
c
u
c
e
t
e
0 0 0 0
2
,
,
,
,
,
,
, 



    
 d
u
d
v
d
w
d
w
v
u
I
T
w
v
u
kI
,
,
,
,
,
,
 ,
where        
  
 

x
L y
L z
L
n
n
n
n
u
d
v
d
w
d
w
v
u
f
w
c
v
c
u
c
F
0 0 0
,
,


,    
 
2
2
2
0
2
2
exp 







 z
y
x
n
L
L
L
t
D
n
t
e 

 , cn(x) =
cos ( n x/Lx);
                    
    









1 0 0 0 0
2
,
,
,
2
,
,
,
n
t x
L y
L z
L
n
n
n
n
n
n
n
n
z
y
x
i
T
w
v
u
g
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
t
z
y
x 





 
            
   










1 0 0
2
1 2
,
,
,
n
t t
n
n
n
n
n
n
z
y
x
i
I
e
e
t
e
z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u











         
 



  

 






1
2
0 0 0
1
0
2
,
,
,
,
,
,
n
z
y
x
x
L y
L z
L
i
I
n
n
t
n
n
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
T
w
v
u
g
w
c
v
s
e
u
c





 


           
 
  
   


 



t x
L y
L z
L
i
I
n
n
n
n
n
n
d
u
d
v
d
w
d
T
w
v
u
g
w
w
v
u
w
s
v
c
u
c
e
x
c
t
e
0 0 0 0
1
,
,
,
,
,
,




 



   
z
c
y
c n
n
 , i 1,
where sn(x) = sin ( n x/Lx). Equations for the functions Cij(x,y,z,t) (i 0, j 0), boundary and initial
conditions could be written as
       
2
00
2
0
2
00
2
0
2
00
2
0
00
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L











;
        

















2
0
2
2
0
2
2
0
2
0
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C i
i
i
L
i
        





















 

y
t
z
y
x
C
T
z
y
x
g
y
D
x
t
z
y
x
C
T
z
y
x
g
x
D i
L
L
i
L
L
,
,
,
,
,
,
,
,
,
,
,
, 10
0
10
0

17
   










 
z
t
z
y
x
C
T
z
y
x
g
z
D i
L
L
,
,
,
,
,
, 10
0
, i 1;
       











2
01
2
0
2
01
2
0
2
01
2
0
01
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
 
 
   
 
  






















y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
D
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
D L
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
00
0
00
00
0 



 
 
 











z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
D L
,
,
,
,
,
,
,
,
, 00
00
0 

;
       











2
02
2
0
2
02
2
0
2
02
2
0
02
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
   
 
     
 























T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
x
D L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
00
01
00
1
00
01
0 



     
 
  





















z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
y
t
z
y
x
C ,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
1
00
01
00


     
 
   
 































T
z
y
x
P
t
z
y
x
C
x
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
y
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
0
00
1
00
01
00




   
 
   
 
 































z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
x
t
z
y
x
C ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 01
00
01
00
01




;
       











2
11
2
0
2
11
2
0
2
11
2
0
11
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
   
 
     
 























T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
x ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
00
10
00
1
00
10 



     
 
  





















L
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
y
t
z
y
x
C
0
00
1
00
10
00
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,


 
 
   
 
 


























y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
D L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 10
00
10
00
0 



 
 
     





























x
t
z
y
x
C
T
z
y
x
g
x
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
L
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 01
0
10
00


       

























z
t
z
y
x
C
T
z
y
x
g
z
y
t
z
y
x
C
T
z
y
x
g
y
L
L
,
,
,
,
,
,
,
,
,
,
,
, 01
01
;
 
0
,
,
,
0


x
ij
x
t
z
y
x
C


,
 
0
,
,
,

 x
L
x
ij
x
t
z
y
x
C


,
 
0
,
,
,
0


y
ij
y
t
z
y
x
C


,
 
0
,
,
,

 y
L
y
ij
y
t
z
y
x
C


,
 
0
,
,
,
0


z
ij
z
t
z
y
x
C


,
 
0
,
,
,

 z
L
z
ij
z
t
z
y
x
C


, i 0, j 0; C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0,
i 1, j 1.
Functions Cij(x,y,z,t) (i 0, j 0) could be approximated by the following series during solutions of the
above equations

18
         




1
0
00
2
,
,
,
n
nC
n
n
n
nC
z
y
x
z
y
x
C
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
F
t
z
y
x
C .
Here  



















 2
2
2
0
2
2 1
1
1
exp
z
y
x
C
nC
L
L
L
t
D
n
t
e  ,        
  

x
L y
L z
L
n
C
n
n
nC
u
d
v
d
w
d
w
c
w
v
u
f
v
c
u
c
F
0 0 0
,
, ;
                 
    




1 0 0 0 0
2
0
,
,
,
2
,
,
,
n
t x
L y
L z
L
L
n
n
nC
nC
n
n
n
nC
z
y
x
i
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C 

             
  







1 0
2
10
2
,
,
,
n
t
nC
nC
n
n
n
nC
z
y
x
i
n
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c 



           
 

  






1
2
0 0 0
10
2
,
,
,
,
,
,
n
nC
nC
z
y
x
x
L y
L z
L
i
L
n
n
n
t
e
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
v
c
v
s
u
c



                 
   



 
t x
L y
L z
L
i
L
n
n
n
nC
n
n
n
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
v
s
v
c
u
c
e
z
c
y
c
x
c
0 0 0 0
10
,
,
,
,
,
, 

 , i 1;
                 
    




1 0 0 0 0
2
01
2
,
,
,
n
t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
z
y
x
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C 

 
 
         






1
2
00
00
2
,
,
,
,
,
,
,
,
,
n
nC
n
n
n
nC
z
y
x
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C 





         
 
   


   





1
2
0 0 0 0
00
00 2
,
,
,
,
,
,
,
,
,
n
nC
z
y
x
t x
L y
L z
L
n
n
n
nC
t
e
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e




 

               
 
 
   




t x
L y
L z
L
n
n
n
nC
n
n
n
nC
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
u
c
e
z
c
y
c
x
c
F
0 0 0 0
00
00
,
,
,
,
,
,
,
,
,



 

;
                 
    




1 0 0 0 0
2
02
2
,
,
,
n
t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
z
y
x
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C 

   
 
     








1
2
00
1
00
01
2
,
,
,
,
,
,
,
,
,
,
,
,
n
n
n
nC
z
y
x
y
c
x
c
F
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C




 

             
 
  
   





t x
L y
L z
L
n
n
nC
nC
n
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
v
s
u
c
e
t
e
z
c
n
0 0 0 0
00
1
00
01
,
,
,
,
,
,
,
,
,
,
,
,



 

               
   




1 0 0 0
2
2
n
t x
L y
L
n
n
nC
nC
n
n
n
nC
z
y
x
n
v
c
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
c 


     
 
   









1
2
0
00
1
00
01
2
,
,
,
,
,
,
,
,
,
,
,
,
n
n
z
y
x
z
L
n
x
c
n
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
s




 

                  
   




t x
L y
L z
L
n
n
n
nC
nC
n
n
nC
u
w
v
u
C
w
v
u
C
w
c
v
c
u
s
e
t
e
z
c
y
c
F
0 0 0 0
00
01
,
,
,
,
,
,



 
 
           
  






1 0 0
2
1
00
2
,
,
,
,
,
,
n
t x
L
n
nC
nC
n
n
n
nC
z
y
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
T
w
v
u
P
w
v
u
C






       
 
  


 






1
2
0 0
00
1
00
01
2
,
,
,
,
,
,
,
,
,
,
,
,
n
z
y
x
y
L z
L
n
n
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
c
v
s




 

                   
 

   



t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
T
w
v
u
P
w
v
u
C
w
v
u
C
w
s
v
c
u
c
e
t
e
z
c
y
c
x
c
F
0 0 0 0
1
00
01
,
,
,
,
,
,
,
,
, 




             
  






1 0 0
2
00
2
,
,
,
n
t x
L
n
nC
nC
n
n
n
nC
z
y
x
u
s
e
t
e
z
c
y
c
x
c
F
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C





19
     
 
     
 

 




1
2
0 0
01
00
2
,
,
,
,
,
,
,
,
,
n
nC
n
z
y
x
y
L z
L
n
n
t
e
x
c
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
c
n






           
 
  
   




t x
L y
L z
L
n
n
n
nC
n
nC
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e
y
c
F
0 0 0 0
01
00
,
,
,
,
,
,
,
,
,



 

                 
    




1 0 0 0 0
2
2
n
t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
z
y
x
n
w
s
v
c
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
z
c
n 

 
 
  




d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C



,
,
,
,
,
,
,
,
, 01
00
;
                 
    




1 0 0 0 0
2
11
2
,
,
,
n
t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
z
y
x
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C 

           






1
2
01
2
,
,
,
,
,
,
n
nC
n
n
n
nC
z
y
x
L
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
g



            

   



 2
0 0 0 0
01
2
,
,
,
,
,
,
z
y
x
t x
L y
L z
L
L
n
n
n
nC
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
w
c
v
s
u
c
e




              
    





1 0 0 0 0
01
,
,
,
,
,
,
n
t x
L y
L z
L
L
n
n
n
nC
nC
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
w
s
v
c
u
c
e
t
e
n 


                   
   




1 0 0 0
2
2
n
t x
L y
L
n
n
nC
nC
n
n
n
nC
z
y
x
n
n
n
nC
v
c
u
s
e
t
e
z
c
y
c
x
c
F
L
L
L
z
c
y
c
x
c
F 

   
 
     







1
2
0
10
00
2
,
,
,
,
,
,
,
,
,
n
n
n
nC
z
y
x
z
L
n
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
n






             
 
  
   




t x
L y
L z
L
n
n
n
nC
nC
n
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e
t
e
z
c
0 0 0 0
10
00
,
,
,
,
,
,
,
,
,



 

                 
 

    



1 0 0 0 0
00
2
,
,
,
,
,
,
2
n
t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
z
y
x
T
w
v
u
P
w
v
u
C
w
s
v
c
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L 




             
  






1 0 0
2
10
2
,
,
,
n
t x
L
n
nC
nC
n
n
n
nC
z
y
x
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C




       
 
  


 






1
2
0 0
00
1
00
10
2
,
,
,
,
,
,
,
,
,
,
,
,
n
z
y
x
y
L z
L
n
n
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
c
v
c




 

                 
 
  
   





t x
L y
L z
L
n
n
n
nC
nC
n
n
n
nC
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e
t
e
z
c
y
c
x
c
F
0 0 0 0
00
1
00
,
,
,
,
,
,
,
,
, 

 

             
  




1 0 0
2
10
2
,
,
,
n
t x
L
n
nC
nC
n
n
n
nC
z
y
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
v
u
C 



       
 
 
 




y
L z
L
n
n
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
s
v
c
0 0
00
1
00
10
,
,
,
,
,
,
,
,
,
,
,
, 


 

.

ON APPROACH TO INCREASE INTEGRATION RATE OF FIELD-EFFECT HETEROTRANSISTORS IN THE FRAMEWORK OF A BOOTSTRAP SWITCH

More Related Content

Similar to ON APPROACH TO INCREASE INTEGRATION RATE OF FIELD-EFFECT HETEROTRANSISTORS IN THE FRAMEWORK OF A BOOTSTRAP SWITCH (20)

Recently uploaded (20)

ON APPROACH TO INCREASE INTEGRATION RATE OF FIELD-EFFECT HETEROTRANSISTORS IN THE FRAMEWORK OF A BOOTSTRAP SWITCH