On Approach for Estimation of Maximal Continuance of Diffusion Nand Ion Type of Doping

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 10, No.1/2, June 2024
DOI: 10.5121/antj.2024.10201 1
ON APPROACH FOR ESTIMATION OF MAXIMAL
CONTINUANCE OF DIFFUSION
AND ION TYPE OF DOPING
E.L. Pankratov
1
Nizhny Novgorod State University, 23 Gagarin avenue,
Nizhny Novgorod, 603950, Russia
2
Nizhny Novgorod State Agrotechnical University, 97 Gagarin avenue,
Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper an analytical approach for estimation of maximal continuance of manufacturing of integrated
circuit elements by dopant diffusion and ion implantation has been introduced. We analyzed influence of
parameters of considered technological processes on the value of it's maximal continuance.
KEYWORDS
Manufacturing of integrated circuit elements; dopant diffusion; ion implantation; maximal continuance of
technological processes; analytical approach for prognosis.
1. INTRODUCTION
One of the intensively solved problems for production of solid-state electronics devices is
increasing of the integration rate of elements of integrated circuit (p-n- junctions; field-effect and
bipolar transistors; ...), as well as increasing of their performance [1-8]. Different methods are
using for manufacture of elements of integrated circuits. Some of them are ion and diffusion
types of doping of required sections of electronic materials, epitaxial growth of multilayer
structures, fusion of materials [9-17]. Main aim of the present paper is estimation of maximal
continuance of ion and diffusion types of doping. The accompanying of the present paper is
development of analytical approach for analysis of the considered continuance.
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 [1,3,18,19]
   







x
t
x
C
D
x
t
t
x
C
C





 ,
,
(1)
Boundary and initial conditions for the equations are for finite source of dopant

2
  0
,
0




x
x
t
x
C
,
  0
,



L
x
x
t
x
C
, C (x,0)=f (x); (2a)
for infinite source of dopant
C (0,t)=C0,
  0
,



L
x
x
t
x
C
, C (x>0,0)=0. (2b)
The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is
the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion
coefficient could be changed with changing materials of heterostructure, with changing
temperature of materials (including annealing), with changing concentrations of dopant and
radiation defects. We approximate dependences of dopant diffusion coefficient on parameters by
the following relation with account results in Refs. [19-21]
   
 
   
  
















 2
*
2
2
*
1
,
,
1
,
,
1
,
V
t
x
V
V
t
x
V
T
x
P
t
x
C
T
x
D
D L
C 

 

. (3)
Here the function DL (x,T) describes the spatial (in heterostructure) and temperature (due to
Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,T) describes the
limit of solubility of dopant. Parameter  [1,3] describes average quantity of charged defects
interacted with atom of dopant [19]. The function V (x,t) describes the spatio-temporal
distribution of concentration of radiation vacancies with equilibrium distribution V*
. The
considered concentrational dependence of dopant diffusion coefficient has been described in
details in [19]. It should be noted, that using diffusion type of doping did not generation radiation
defects. In this situation 1= 2= 0. We determine spatio-temporal distributions of concentrations
of radiation defects by solving the following system of equations [20,21]
               
               





































t
x
V
t
x
I
T
x
k
t
x
V
T
x
k
x
t
x
V
T
x
D
x
t
t
x
V
t
x
V
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
I
T
x
D
x
t
t
x
I
V
I
V
V
V
V
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2
,
,
2
,
(4)
Boundary and initial conditions for these equations are
  0
,
0




x
x
t
x

,
  0
,



L
x
x
t
x

,  (x,0)=f (x). (5)
Here  =I,V. The function I (x,t) describes the spatio-temporal distribution of concentration of
radiation interstitials with equilibrium distribution I*
; D(x,T) are the diffusion coefficients of
point radiation defects; terms V2
(x,t) and I2
(x,t) correspond to generation divacancies and
diinterstitials; kI,V(x,T) is the parameter of recombination of point radiation defects; kI,I(x,T) and
kV,V(x,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,t) and
diinterstitials I(x,t) by solving the following system of equations [20,21]

3
             
             














 









 




t
x
V
T
x
k
t
x
V
T
x
k
x
t
x
T
x
D
x
t
t
x
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
T
x
D
x
t
t
x
V
V
V
V
V
V
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2
,
2
,












(6)
Boundary and initial conditions for these equations are
 
0
,
0





x
x
t
x

,
 
0
,




L
x
x
t
x

,  (x,0)=f (x). (7)
Here D(x,T) are the diffusion coefficients of the above complexes of radiation defects; kI(x,T)
and kV(x,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 [22]. The approach based on transformation of approximations of
diffusion coefficients in the following form: D(x,T)=D0[1+ g(x, T)], where D0 are the
average values of diffusion coefficients, 0<1, |g(x,T)|1. We also used analogous
transformation of approximations of parameters of recombination of point defects and parameters
of generation of their complexes: kI,V(x,T)=k0I,V [1+I,VgI,V(x,T)], kI,I(x,T)=k0I,I[1+I,I gI,I(x,T)] and
kV,V(x,T)=k0V,V[1+V,V gV,V(x,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,T)|1, | gI,I(x, T)|1, |gV,V(x,T)|1. Let us introduce the following dimensionless
variables:     *
,
,
~
I
t
x
I
t
x
I  ,     *
,
,
~
V
t
x
V
t
x
V  , V
I
V
I D
D
k
L 0
0
,
0
2

 ,  = x/Lx,
V
I D
D
k
L 0
0
,
0
2


 
 , 2
0
0 L
t
D
D V
I

 . The introduction leads to transformation of Eqs. (4) and
conditions (5) to the following form
   
   
 
       
   
   
   
 
       
   



























































































,
~
,
1
,
~
,
~
,
1
,
~
,
1
,
~
,
~
,
1
,
~
,
~
,
1
,
~
,
1
,
~
2
,
,
,
,
0
0
0
2
,
,
,
,
0
0
0
V
T
g
V
I
T
g
V
T
g
D
D
D
V
I
T
g
V
I
T
g
I
T
g
D
D
D
I
V
V
V
V
V
V
I
V
I
V
V
V
I
V
I
I
I
I
I
V
I
V
I
I
I
V
I
I
(8)
  0
,
~
0









,
  0
,
~
1









,  
 
*
,
,
~





 
f
 . (9)
We determine solutions of Eqs. (8) with conditions (9) framework recently introduced approach
[22], 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 and
corrections for them  


 ,
~
ijk , i 1, j 1, k 1. The equations are presented in the Appendix.

4
Solutions of the equations could be obtained by standard Fourier approach [23,24]. 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: D(x,T)=D0[1+ g(x,T)], where D0 are the average
values of diffusion coefficients. In this situation the Eqs.(6) could be written as
   
           
   
           














 










 









t
x
V
T
x
k
t
x
V
T
x
k
x
t
x
T
x
g
x
D
t
t
x
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
T
x
g
x
D
t
t
x
V
V
V
V
V
V
V
V
I
I
I
I
I
I
I
I
,
,
,
,
,
,
1
,
,
,
,
,
,
,
1
,
2
,
0
2
,
0














Farther we determine solutions of above equations as the following power series
   
 





0
,
,
i
i
i
t
x
t
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 concentrations
of complexes of defects 0(x,t), corrections for them i(x,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 [23,24] and presented in the
Appendix.
Now we calculate distribution of concentration of dopant in space and time by using the
approach, which was used for analysis of radiation defects. To use the approach we consider
following transformation of approximation of dopant diffusion coefficient: DL(x,T)=D0L[1+
LgL(x,T)], where D0L is the average value of dopant diffusion coefficient, 0L< 1, |gL(x,T)|1.
Farther we consider solution of Eq. (1) as the following series:
   
 





0 1
,
,
,
,
i j
ij
j
i
L t
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,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, [23,24]). The solutions are presented in the Appendix. We analyzed distributions of
concentrations of dopant and radiation defects in space and time analytically by using the second-
order approximations on all parameters, which have been used in appropriate series. Usually the
second-order approximations are enough good approximations to make qualitative analysis and to
obtain quantitative results. All analytical results have been checked by numerical simulation.
Let us to use recently introduce criterion to estimate maximal value of continuance of
technological process [25]. In the framework of the criterion let us approximate changing of
considered concentrations in time by the following step-wise function (see Figs. 1-4)
 (x,t) = a0+a1[1(t)-1(t-)], (12)

5
where 1(t) is the single step-wise function [26]. Not yet known parameters a0, a1 and  could
have different values in different points of the considered material. Values of these parameters
were determined were determined by minimization of the following the mean-squared error
   
 
 

N
t
dt
t
x
t
x
C
U
0
2
,
,  , (14)
where tN is the observation time of transition process. Minimization of the mean- square error
(14) gives a possibility to obtain the following relations for calculation of the considered
parameters
  


 1
0
0
, a
t
a
t
d
t
x
C N
tN
, (15a)
   




1
0
0
, a
a
t
d
t
x
C , (15b)
C (x,)=a0+0.5a1. (15c)
The criterion is optimal. However the approach did not leads to obtaining analytical relations for
calculation of the considered maximal value of continuance of technological process. To obtain
analytical relations for the considered relations it is attracted an interest asymptotically optimal
criteria. To obtain transition to the criteria one shall consider the following limiting case tN.
In this case one can obtain the following relations: a0 = C(x,) and a1= C(x,0)-C(x,). Before
consideration of the following limiting transition one shall the transform relation (15a) to the
following form
 
  

  1
0
0
, a
t
d
a
t
x
C
N
t
.
Further obtaining of time of step-wise changing of approximation function (13) under condition
of the limiting case tN one can obtain the following criterion to estimate time scales, which
known as rectangle with equal square
 
   
 
   


 




,
0
,
,
,
0
x
C
x
C
t
d
x
C
t
x
C
x . (16)
Monotonous in time concentrations of dopant (see Figs. 1 and 2) could be approximated by the
following functions
C(x,t) =  [1-exp(-t/)], C(x,t) =  exp(-t/). (17)
Substitution of the above relations into the relation (16) at fixed value of observation time of the
diffusion process tN gives a possibility to obtain the following relation for the considered time
 =  [1-exp (-tN/)].

6
t
C(x,t)
C(x,)
tN
opt(x) (x)
a0(x)
a1(x)
2
3
C(x,0)
1
Fig. 1. Monotonic increasing of concentration of dopant (curve 1); optimal approximation of transition
process, which was obtained by minimization of mean-squared error (14) (curve 2); asymptotically optimal
approximation of transition process (curve 3)
t
C(x,t)
C(x,0)
a1(x)
a0(x)
tN
opt(x)(x)
1
2
3
C(x,)
Fig. 2. Monotonic decreasing of concentration of dopant (curve 1); optimal approximation of transition
Consideration limiting case tN leads to equality of single time scale of monotonous variation
in time of dopant concentration and time scale, which was determined by relation (16). It should
be noted, that relation (15c) at the limiting case tN  takes the form of another asymptotically
optimal criterion. In the framework the second asymptotically optimal criterion maximal value
maximal value of diffusion doping could be estimated as time of changed of the considered
concentration in two times, i.e.
C (x,)=[C(x,0) +C(x,)]/2. (18)
Fig. 3. Non-monotonic decreasing of concentration of dopant (curve 1); optimal approximation of
transition process, which was obtained by minimization of mean-squared error (14) (curve 2);
asymptotically optimal approximation of transition process (curve 3)

7
t
C(x,t)
C(x,)
a1(x)
a0(x)
tN
opt(x)
C(x,0)
1
2
Fig. 4. Non-monotonic increasing of concentration of dopant (curve 1); optimal approximation of transition
However the first asymptotically optimal criterion (15) is nonlinear. Nonlinear criterion leads to
obtaining smaller quantity of analytical relations for the considered maximal value of
technological processes in comparison with criterion (16). In this situation we will use criterion
(16) in future. However the criterion (16) has own disadvantage: the criterion could be used for
monotonous in time concentrations of dopant. For non-monotonous in time concentrations of
dopant the criterion (16) leads to underestimated values of the considered time. In this situation
the considered time could takes negative values. It is attracted an interest maximal value of the
considered time will be achieves, when initial distribution of concentration of infused dopant is
presents near one boundary of the considered structure (i.e. f (x)= (x)) and point of observation
of this concentration is presented on other boundary of the considered structure (i.e. x = L), which
should be doped. If the observation time on diffusion doping tN is large in comparison with
limiting time of technological process , than transitions processes are absent at times t >.
3. DISCUSSION
In this section we analyzed limiting continuance of technological process for different profiles of
diffusion coefficients without any variations in time (for example, annealing temperature is
constant). Wide using have different multilayer structures. In this situation we will consider
several normalized profiles of dopant diffusion coefficient g (x), which are presented on Fig. 5.
Analysis of limiting continuance of technological process shows, that in the case of infusion of
dopant from finite source maximal variation of the considered continuance could be find in
symmetrical structure with respect to it's middle (see Fig. 6). In the case of infusion of dopant
from infinite source maximal variation of the considered continuance could be find in
asymmetrical structure with respect to it's middle (see Fig. 7). Multilayer structures, which were
presented on Figs. 6a and 7a, correspond to maximal increasing of the considered limiting
continuance of technological processes (at fixed average value of dopant diffusion coefficient
D0). Multilayer structures, which were presented on Figs. 6b and 7b, correspond to maximal
decreasing of the considered limiting continuance of technological processes (at fixed average
value of dopant diffusion coefficient D0).

8
g0(x)
0
g1(x)
g2(x)
0
g3(x)
0
0 L
0
g4(x)
g5(x)
g6(x)
0
0
0
Fig. 5. Normalized profiles of dopant diffusion coefficients
We analyzed dependences of the considered limiting continuance on thicknesses of layers of
multilayer structures. Variation of thicknesses of layers of multilayer structures not gives a
possibility to find profiles of dopant diffusion coefficient, which correspond to larger influence
on the considered continuance, in comparison with profiles, which were presented on Figs. 6 and
7. Increasing of quantity of layers of the considered multilayer structures leads to decreasing of
influence of variation of dopant diffusion coefficient on the limiting continuance of technological
process. Figs. 8 show dependences of the considered continuance on the value of parameter  for
profiles of dopant diffusion coefficient, which were presented on Figs. 6 and 7. These figures
show, that the considered continuance could be decreased on several percents and increased on
several orders in comparison with continuance 0 for averaged value of dopant diffusion
coefficient D0. The continuance 0 for averaged value of dopant diffusion coefficient D0 is equal
to 0=L2
/6D0 for finite source of dopant and 0=L2
/2D0 for infinite source of dopant.
0 L/4 L/2 3L/4 L
2
1
0
-1
-2
Fig. 6a. Normalized profiles of dopant diffusion coefficient, which correspond to maximal increasing of
the limiting continuance of diffusion doping from finite source of dopant

9
0 L/4 L/2 3L/4 L
2
1
0
-1
-2
Fig. 6b. Normalized profiles of dopant diffusion coefficient, which correspond to maximal decreasing of
the limiting continuance of diffusion doping from finite source of dopant
0 L/4 L/2 3L/4 L
2
1
0
-1
-2
Fig. 7a. Normalized profiles of dopant diffusion coefficient, which correspond to maximal increasing of
the limiting continuance of diffusion doping from infinite source of dopant
0 L/4 L/2 3L/4 L
2
1
0
-1
-2
Fig. 7b. Normalized profiles of dopant diffusion coefficient, which correspond to maximal decreasing of
the limiting continuance of diffusion doping from infinite source of dopant

10
Fig. 8a. Dependences of limiting continuance of diffusion doping for finite source of dopant on value of
parameter . Curve 1 corresponds to profile of dopant diffusion coefficient with decreased limiting
continuance of technological process (see Fig. 6b). Curve 2 corresponds to profile of dopant diffusion
coefficient with increased limiting continuance of technological process (see Fig. 6a)
Fig. 8b. Dependences of limiting continuance of diffusion doping for infinite source of dopant on value of
parameter . Curve 1 corresponds to profile of dopant diffusion coefficient with decreased limiting
continuance of technological process (see Fig. 7b). Curve 2 corresponds to profile of dopant diffusion
coefficient with increased limiting continuance of technological process (see Fig. 7a)
x
D(x)
D1
D2
0 L
a1
fC(x)
D0
D3
a2
Fig. 9. Profile of dopant diffusion coefficient, which corresponds to maximal increasing of limiting
continuance of ion doping. Profile of dopant diffusion coefficient, which corresponds to maximal
decreasing of limiting continuance of ion doping, has the same difference with the above profile as for
profiles 6b и 6a

11
Now let us consider influence of temporal variations of dopant diffusion coefficient on value of
limiting continuance of technological process in homogenous material. The considered situation
could be consider, for example, for nonstationary annealing of dopant and/or radiation defects,
which are presents in homogenous material. In this case (as for multilayer structure) increasing of
the considered limiting continuance is essentially smaller, than decreasing one at fixed value of
averaged diffusion coefficient. The same conclusion could be obtained during analysis of joint
changing of dopant diffusion coefficient in space and time.
Analogous conclusions about influence of variations of dopant diffusion coefficient on limiting
continuance of technological process could be obtained for ion type of doping. At the same time
one can find changing of thicknesses of layers of multilayer structures (see Fig. 9). This changing
taking into account presents of maximal value of concentration of dopant in depth of the
multilayer structure, but not on it's external boundary as for diffusion doping. Also qualitatively
similar influence of spatial and temporal variations of dopant diffusion coefficients of radiation
defects and accounted other parameters (parameters of recombination of point radiation defects;
parameters of generation of complexes of point defects; parameters of decay of complexes of
radiation defects) on limiting value of continuance of annealing time.
4. CONCLUSIONS
In this paper we introduce an analytical approach to estimate limiting value of continuance of
technological process during doping (doping by diffusion; ion doping) of materials to
manufacture elements of integrated circuits. We analyzed influence of parameters on value of the
considered limiting continuance.
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12
[16] Shishiyanu, S.T., Shishiyanu, T.S., Railyan, S.K., 2002, Semiconductors, Vol. 36, № 5,, pp. 611-
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Company.
APPENDIX
Equations for functions  

,
~
ijk
I andи  

,
~
ijk
V , i0, j0, k0 and conditions for them could be
written as
   
2
000
2
0
0
000 ,
~
,
~










 I
D
D
I
V
I
,
   
2
000
2
0
0
000 ,
~
,
~










 V
D
D
V
I
V
;
       
       































































,
~
,
,
~
,
~
,
~
,
,
~
,
~
100
0
0
2
00
2
0
0
00
100
0
0
2
00
2
0
0
00
i
V
I
V
i
I
V
i
i
I
V
I
i
V
I
i
V
T
g
D
D
V
D
D
V
I
T
g
D
D
I
D
D
I
, i 1;
     
     
     
     













































,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
1
,
~
,
~
000
000
,
,
2
010
2
0
0
010
000
000
,
,
2
010
2
0
0
010
V
I
T
g
V
D
D
V
V
I
T
g
I
D
D
I
V
I
V
I
I
V
V
I
V
I
V
I
;
   
 
         
 
   
 
         
 





























































,
~
,
~
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
~
,
~
,
1
,
~
,
~
010
000
000
010
,
,
2
020
2
0
0
020
010
000
000
010
,
,
2
020
2
0
0
020
V
I
V
I
T
g
V
D
D
V
V
I
V
I
T
g
I
D
D
I
V
I
V
I
V
I
V
I
V
I
V
I
;

13
     
   
     
   









































,
~
,
1
,
~
,
~
,
~
,
1
,
~
,
~
2
000
,
,
2
001
2
0
0
001
2
000
,
,
2
001
2
0
0
001
V
T
g
V
D
D
V
I
T
g
I
D
D
I
V
V
I
I
I
V
I
I
I
I
V
I
;
       
 
         
 
       
 
         
 




























































































,
~
,
~
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
~
,
~
,
~
,
1
,
~
,
,
~
,
~
100
100
100
0100
,
,
010
0
0
2
110
2
0
0
110
100
000
000
100
,
,
010
0
0
2
110
2
0
0
110
V
I
V
I
T
g
V
T
g
D
D
V
D
D
V
V
I
V
I
T
g
I
T
g
D
D
I
D
D
I
V
V
V
V
V
I
V
I
V
I
I
I
I
I
V
I
V
I
;
     
     
     
     













































,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
1
,
~
,
~
000
001
,
,
2
002
2
0
0
002
000
001
,
,
2
002
2
0
0
002
V
V
T
g
V
D
D
V
I
I
T
g
I
D
D
I
V
V
V
V
I
V
I
I
I
I
V
I
;
   
 
 
 
     
   
 
 
 
     























































































,
~
,
~
,
1
,
~
,
,
~
,
~
,
~
,
~
,
1
,
~
,
,
~
,
~
100
000
001
0
0
2
101
2
0
0
101
000
100
001
0
0
2
101
2
0
0
101
V
I
T
g
V
T
g
D
D
V
D
D
V
V
I
T
g
I
T
g
D
D
I
D
D
I
V
V
V
I
V
I
V
I
I
I
V
I
V
I
;
   
 
     
 
     
   
 
     
 
     



































































,
~
,
~
,
1
,
~
,
~
,
1
,
~
,
~
,
~
,
~
,
1
,
~
,
~
,
1
,
~
,
~
001
000
,
,
010
000
,
,
2
011
2
0
0
011
000
001
,
,
010
000
,
,
2
011
2
0
0
011
V
I
t
g
V
V
T
g
V
D
D
V
V
I
T
g
I
I
T
g
I
D
D
I
V
I
V
I
V
V
V
V
I
V
V
I
V
I
I
I
I
I
V
I
;
 
0
,
~
0




x
ijk




,
 
0
,
~
1




x
ijk




(i 0, j 0, k 0);  
 
*
000 0
,
~



 
f
 ,
  0
0
,
~ 

ijk (i 1, j 1, k 1).
Solutions of the above equations with appropriate boundary and initial conditions could be
written as
     




1
0
000
2
,
~
n
n
n e
c
F
L
L
F




 


,
where    


1
0
*
cos
1
u
d
u
f
u
n
F n
n 
 

,    
I
V
nI D
D
n
e 0
0
2
2
exp 

 
 , cn() = cos ( n ),
   
V
I
nV D
D
n
e 0
0
2
2
exp 

 
 ;

14
             
             







  





  











1 0
1
0
100
0
0
00
1 0
1
0
100
0
0
00
,
~
,
2
,
~
,
~
,
2
,
~
n
i
V
n
nI
nV
n
I
V
i
n
i
I
n
nI
nI
n
V
I
i
d
u
d
u
u
V
T
u
g
u
s
e
e
c
n
D
D
V
d
u
d
u
u
I
T
u
g
u
s
e
e
c
n
D
D
I


















, i 1,
where sn() = sin ( n );
                 
     




1 0
1
0
1
0
1
0
010 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c


 









 
      


 d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
V
I ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,

 ;
           
 
   




1 0
1
0
,
,
0
0
020 ,
1
2
,
~
n
V
I
V
I
n
n
n
n
V
I
T
u
g
u
c
e
e
c
D
D 

 






       
  



 d
u
d
u
V
u
I
u
V
u
I ,
~
,
~
,
~
,
~
010
000
000
010 
 ;
           
   
   




1 0
1
0
2
000
,
,
001 ,
~
,
1
2
,
~
n
n
n
n
n d
u
d
u
T
u
g
u
c
e
e
c






 








 ;
           
     
   




1 0
1
0
000
001
,
,
002 ,
~
,
~
,
1
2
,
~
n
n
n
n
n d
u
d
u
u
T
u
g
u
c
e
e
c






 










 ;
             
     
         
 
   
             
     
         
 
   



















   




  







   




  





















































nV
n
n
V
I
V
I
n
nV
n
i
V
n
nV
nV
n
I
V
n
nI
n
V
I
V
I
n
nI
n
i
I
n
nI
nI
n
V
I
e
c
d
u
d
u
V
u
I
u
V
u
I
T
u
g
u
c
e
d
u
d
u
u
V
T
u
g
u
s
e
e
c
n
D
D
V
c
e
d
u
d
u
V
u
I
u
V
u
I
T
u
g
u
c
e
d
u
d
u
u
I
T
u
g
u
s
e
e
c
n
D
D
I
10
1
0
000
100
100
000
,
,
1 0
1
0
100
0
0
110
10
1
0
100
000
000
100
,
,
1 0
1
0
100
0
0
110
,
~
,
~
,
~
,
~
,
1
2
,
~
,
2
,
~
,
~
,
~
,
~
,
~
,
1
2
,
~
,
2
,
~
;
             
         
     
             
         
     













   



  





   



  













1 0
1
0
100
000
,
,
1 0
1
0
001
0
0
101
1 0
1
0
000
100
,
,
1 0
1
0
001
0
0
101
,
~
,
~
,
1
2
,
~
,
2
,
~
,
~
,
~
,
1
2
,
~
,
2
,
~
n
V
I
V
I
n
nV
nV
n
n
V
n
nV
nV
n
I
V
n
V
I
V
I
n
nI
nI
n
n
I
n
nI
nI
n
V
I
d
u
d
u
V
u
I
T
u
g
u
c
e
e
c
d
u
d
u
u
V
T
u
g
u
s
e
e
c
n
D
D
V
d
u
d
u
V
u
I
T
u
g
u
c
e
e
c
d
u
d
u
u
I
T
u
g
u
s
e
e
c
n
D
D
I


































;
           
     

 
     
           
     

 
     











   






   


































d
u
d
u
V
u
I
T
u
g
u
V
u
V
T
u
g
u
c
e
e
c
V
d
u
d
u
V
u
I
T
u
g
u
I
u
I
T
u
g
u
c
e
e
c
I
V
I
V
I
n
V
V
V
V
n
nV
nV
n
V
I
V
I
n
I
I
I
I
n
nI
nI
n
,
~
,
~
,
1
,
~
,
~
,
1
2
,
~
,
~
,
~
,
1
,
~
,
~
,
1
2
,
~
001
000
,
,
1 0
1
0
010
000
,
,
011
000
001
,
,
1 0
1
0
010
000
,
,
011
Equations for functions i(x,t), i 0, boundary and initial conditions takes the form

15
           
           



















t
x
V
T
x
k
t
x
V
T
x
k
x
t
x
D
t
t
x
t
x
I
T
x
k
t
x
I
T
x
k
x
t
x
D
t
t
x
V
V
V
V
V
V
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
2
,
2
0
2
0
0
2
,
2
0
2
0
0








;
   
 
 
   
 
 












 









 












x
t
x
T
x
g
x
D
x
t
x
D
t
t
x
x
t
x
T
x
g
x
D
x
t
x
D
t
t
x
i
V
V
V
i
V
V
i
V
i
I
I
I
i
I
I
i
I
















,
,
,
,
,
,
,
,
1
0
2
2
0
1
0
2
2
0
, i1;
 
0
,
0





x
i
x
t
x

,
 
0
,




L
x
i
x
t
x

, i0; 0(x,0)=f (x), i(x,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 t
e
x
c
n
L
t
e
x
c
F
L
L
F
t
x 




           
 
  

 
t L
I
I
I
n
n d
u
d
u
I
T
u
k
u
I
T
u
k
u
c
e
0 0
2
, ,
,
,
, 




,
where    
2
0
2
2
exp L
t
D
n
t
en 

 
 
 ,    

 

x
L
n
n u
d
u
f
u
c
F
0


;
           
 
  











1 0 0
1
2
,
,
2
,
,
,
n
t L i
I
n
n
n
n
i d
u
d
u
u
T
u
g
u
s
e
t
e
x
c
n
L
t
z
y
x 




 



 , i1.
Equations for initial-order approximation of concentration of dopant C00(x,t) and corrections for
them Cij(x,t) (i 1, j 1), boundary and initial conditions for the above equations could be written
as
   
2
00
2
0
00 ,
,
x
t
x
C
D
t
t
x
C
L





;
       















 
x
t
x
C
T
x
g
x
D
x
t
x
C
D
t
t
x
C i
L
L
i
L
i ,
,
,
, 10
0
2
0
2
0
0
, i 1;
     
 
 
















x
t
x
C
T
x
P
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
, 00
00
0
2
01
2
0
01


;
   
 
 
 
 
















 
x
t
x
C
T
x
P
t
x
C
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
,
, 00
1
00
01
0
2
02
2
0
02


 
 
 











x
t
x
C
T
x
P
t
x
C
x
D L
,
,
, 01
00
0 

;
       
 
  















 
x
t
x
C
T
x
P
t
x
C
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
,
, 00
1
00
01
0
2
02
2
0
02


 
 
 











x
t
x
C
T
x
P
t
x
C
x
D L
,
,
, 01
00
0 

;
   
 
 
 
 
















 
x
t
x
C
T
x
P
t
x
C
t
x
C
x
D
x
t
x
C
D
t
t
x
C
L
L
,
,
,
,
,
, 00
1
00
10
0
2
11
2
0
11



16
 
 
     






















x
t
x
C
T
x
g
x
D
x
t
x
C
T
x
P
t
x
C
x
D L
L
L
,
,
,
,
, 01
0
10
00
0 

;
for finite source of dopant
 
0
,
0


x
ij
x
t
x
C


,
 
0
,

L
x
ij
x
t
x
C


, i 0, j 0; C00(x,0) = fC (x), Cij(x,0) = 0, i 1, j 1.
C00(0,t)=C0,
 
0
,

L
x
ij
x
t
x
C


, Cij(x>0,t)=0, C00(0,0) = C0, Cij(x >0,0)=0, i 0, j 0.
Solutions of the above equations with account boundary and initial conditions by standard
Fourier approach takes the form
for finite source of dopant
     




1
0
00
2
,
n
nC
n
nC
C
t
e
x
c
F
L
L
F
t
x
C ,
where    
2
0
2
2
exp L
t
D
n
t
e C
nC 

 ,    


L
C
n
nC u
d
u
f
u
c
F
0
;
             
  








1 0 0
10
2
0
,
,
2
,
,
,
n
t L
i
L
n
nC
nC
n
nC
i d
u
d
u
u
C
T
u
g
u
s
e
t
e
x
c
F
n
L
t
z
y
x
C 



, i 1;
           
 
 
  






1 0 0
00
00
2
01
,
,
,
2
,
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
s
e
t
e
x
c
F
n
L
t
x
C 






;
             
 
  
  








1 0 0
00
1
00
01
2
02
,
,
,
,
2
,
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
C
u
s
e
t
e
x
c
F
n
L
t
x
C 







           
 
  
  







1 0 0
00
1
00
01
2
,
,
,
,
2
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
C
u
s
e
t
e
x
c
F
n
L








         
 
 
  





1 0 0
01
00
2
,
,
,
2
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
s
e
t
e
x
c
F
n
L







;
              
  






1 0 0
01
2
11
,
,
2
,
n
t L
L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
g
u
s
e
t
e
x
c
F
n
L
t
x
C 



         
 
  
  





1 0 0
10
00
2
,
,
,
2
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
s
e
t
e
x
c
F
n
L







           
 
 
  







1 0 0
00
1
00
10
2
,
,
,
,
2
n
t L
n
nC
nC
n
nC d
u
d
u
u
C
T
u
P
u
C
u
C
u
s
e
t
e
x
c
F
n
L








.
     






 




 






0
5
.
0
0
00
5
.
0
sin
5
.
0
1
2
1
,
n
n t
e
L
x
n
n
C
t
x
C


,
           
  






 









0 00
5
.
0
5
.
0
5
.
0
3
0
0
10
5
.
0
sin
5
.
0
2
,
n m
t
m
n
n u
e
u
e
t
e
L
x
n
n
L
C
D
t
x
C


   
  u
d
u
H
u
H m
n
m
n 

 
 1 ,
           
  

 




 











0 10
5
.
0
2
0
5
.
0
6
0
3
2
0
20 5
.
0
5
.
0
sin
5
.
0
2
,
n m
t
k
k
k u
e
n
t
e
L
x
k
k
L
C
D
t
x
C


     
         
 
 


 








u
m
n
m
n
m
n
k
n
k
n
n u
d
d
H
H
e
e
u
H
u
H
u
e
0
1
5
.
0
5
.
0
1
5
.
0 



 ,

17
where      
 




 

L
n y
d
L
y
n
t
y
t
H
0
5
.
0
sin
,

 ,
  2
1
01 , 

 


t
x
C ,
where      
   

  









 










0 0 0
5
,
0
5
.
0
2
3
0
1
5
.
0
1
5
.
0
sin
5
.
0
2
n k m
n
km
n
kk
mm
nn
t
e
t
e
k
t
e
L
x
n
n
C 


      












 2
2
1
5
.
0
1
1
k
n
m
k
n
mm
, ekm(t)=(t) ek+0.5(t) em+0.5(t),
     





 
  










 


 










0 0 0 0
0
5
.
0
3
5
0
2
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
sin
5
.
0
3
,
0
n k l j
i
n
j
i
l
k
t
e
L
x
n
n
C 



         





















 2
2
2
2
2
1
5
.
0
1
5
.
0
1
1
j
i
l
k
n
j
i
l
k
n
j
i
l
k
nn
       














 2
2
2
2
1
5
.
0
1
1
5
.
0
1
j
i
l
k
n
j
i
l
k
n
           























 2
2
2
2
2
2
5
.
0
1
2
5
.
0
1
1
5
.
0
1
l
k
j
i
n
l
k
j
i
n
l
k
j
i
n
   
          

















2
2
2
2
2
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0 j
i
l
k
n
t
e
t
e n
klij
,  = 3,
  4
3
2
02 , 

 


t
x
C ,
where
   
   
   
   
  
 

















 









0 1 0 1
2
2
2
2
3
5
0
3
1
5
.
0
1
5
.
0
5
.
0
5
.
0
sin
4
n k l m k
n
l
k
n
k
n
k
n
l
l
L
x
n
C 


     
         
    
 
  























0 0
2
2
2
2
2
2
2
2
2
2
5
.
0
1
5
.
0
1
5
.
0
5
.
0
5
.
0
1
5
.
0
5
.
0
i j i
l
j
i
l
j
j
i
l
i
l
i
l
k
n
     
       
   
      





















 


2
2
2
5
.
0
2
2
2
2
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0 l
k
n
t
e
t
e
j
i
k
n
t
e
t
e
i
t
e n
kl
n
klj
n
,
       
   





   













 



 






0 0 0
2
2
2
5
.
0
4
7
0
4
5
.
0
1
5
.
0
5
.
0
5
.
0
sin
5
.
0
3
2
,
1
,
0
n i j
n
j
i
n
i
j
t
e
L
x
n
n
C 




       
   





























0 0 0 0
2
2
1
2
1
2
2
1 2 5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
1
5
.
0
1
k l m m m
m
l
k
j
m
l
k
j
i
n
       













 2
2
1
2
2
2
1
2
5
.
0
1
1
5
.
0
1
m
m
l
k
j
m
m
l
k
j
       













 2
2
1
2
2
2
1
2
1
5
.
0
1
5
.
0
1
m
m
l
k
j
m
m
l
k
j
       













 2
2
1
2
2
2
1
2
5
.
0
1
1
5
.
0
1
m
m
l
k
j
m
m
l
k
j
   
   
     






















 2
2
2
2
2
2
1
2
5
.
0
5
.
0
5
.
0
5
.
0
1
2
5
.
0
1
j
i
n
t
e
t
e
m
m
m
l
k
j
n
ij

18
   
             



















 2
2
2
1
2
2
2
2
2
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
2
1
m
m
l
k
j
i
n
t
e
t
e n
m
iklm
 
 
   
  

















 








0 0 0
2
5
4
3
2
2
1
2
1
3
1
7
0
1 2 3 1
5
.
0
1
5
.
0
1
5
.
0
sin
5
.
0
4
n n n n
n
n
n
n
n
L
x
n
n
C 

       














 2
5
4
3
2
2
1
2
5
4
3
2
2
1 1
5
.
0
1
1
5
.
0
1
n
n
n
n
n
n
n
n
n
n
       













 2
5
4
3
2
2
1
2
5
4
3
2
2
1 5
.
0
1
1
5
.
0
1
n
n
n
n
n
n
n
n
n
n
         

 





















0 0
2
2
2
5
3
2
5
4
3
2
2
1 5
.
0
5
.
0
5
.
0
1
5
.
0
1
2
5
.
0
1
k m m
k
n
n
n
n
n
n
n
       
   
   



























 2
5
2
1
2
5
2
2
5
2
5
.
0
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1 1
5
n
n
t
e
t
e
k
n
m
k
n
m
k
n
n
   
            















 2
2
2
4
2
3
2
2
2
1 5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
1
4
3
2
m
k
n
n
n
n
t
e
t
e n
km
n
n
n
.
C11(x,t)=5+26+27-28+29+10+411,
where         
  







 









0 0 0
3
5
.
0
3
0
0
5
5
.
0
1
5
.
0
5
.
0
sin
5
.
0
2
l n k
l
k
n
t
e
L
x
l
l
L
C
D 



             

 




















0
2
2
2
2
2
2
2
1
5
.
0
1
5
.
0
1
5
.
0
5
.
0
5
.
0
1
m k
n
m
k
n
m
n
k
n
       
     
 
 



 






t
n
l
n
l
n
m
k
l u
d
u
H
u
H
u
e
u
e
u
e
u
e
0
1
5
.
0
5
.
0
5
.
0
5
.
0 ,
     
       
 
  


















 









0 0 0
2
2
2
2
5
.
0
2
5
0
2
0
6
1
5
.
0
1
5
.
0
1
5
.
0
sin
5
.
0
n k l
n
k
n
l
k
n
l
t
e
L
x
n
n
L
C
D 


                 
 
  

 













0 0
1
5
.
0
5
.
0
0
5
.
0
5
.
0
5
.
0
3
2
5
.
0
5
.
0
m
u
m
l
m
l
m
l
t
l
k
n u
d
d
H
H
e
e
u
e
u
e
u
e
k
l 



 ,
             
 
  

 




 












00
1
5
.
0
5
.
0
0
5
.
0
3
0
0
7
5
.
0
sin
5
.
0
m
t
n
m
n
m
m
n
n
n u
d
u
H
u
H
u
e
u
e
L
x
n
t
e
n
L
C
D 


,
           

    







 













0 0 0 0
5
.
0
5
.
0
5
.
0
3
0
0
8
5
.
0
1
5
.
0
sin
5
.
0
n m k
t
m
k
n
km
n
n u
H
u
e
u
e
k
t
e
L
x
n
n
L
C
D 



       u
d
u
H
u
H
u
H m
k
n
m
k
n
m
k
n 5
.
1
5
.
0
5
.
0 







 

 ,
           






     








 


 











0 0 0 00
5
.
0
3
0
0
9
5
.
0
1
5
.
0
1
5
.
0
sin
5
.
0
1
,
0
n k l m
t
m
n
l
k
klm
m
n u
I
u
e
u
e
l
k
L
x
n
t
e
n
L
C
D 




+Ik-l+n-m(u)+Ik-l-n-m-1(u)+Ik-l+n+m+1(u)-Ik-l-n-m-1(u)-Ik+l+n-m+1(u)-Im-k-l+n-(u)-Im+k+l+n+2(u)] d u,  >2,
     
 
 
 
L
n
y
d
L
y
n
t
y
t
I
0
1
5
.
0
cos
, 
 ,

19
           
 





    








 


 









0 0 0 0
1
2
5
.
0
3
3
0
0
10
5
.
0
1
5
.
0
5
.
0
sin
5
.
0
2
3
,
0
i n
t
k
n
i
n
i
i
k
u
I
u
I
n
L
x
i
t
e
i
L
C
D 



       
 






















0 0
2
2
1
2
2
2
1
2
1
1 2 5
.
0
1
5
.
0
1
5
.
0
1
m m m
m
l
k
n
m
m
l
k
n
m
       














 2
2
1
2
2
2
1
2
1
5
.
0
1
1
5
.
0
1
m
m
l
k
n
m
m
l
k
n
       














 2
2
1
2
2
2
1
2
1
5
.
0
1
1
5
.
0
1
m
m
l
k
n
m
m
l
k
n
       






















0
2
2
2
1
2
2
2
1
2
5
.
0
1
5
.
0
1
2
5
.
0
1
5
.
0
1
l l
m
m
m
l
k
n
m
m
l
k
n
       
         
u
d
m
m
l
k
n
u
e
u
e
u
e
u
e n
i
m
klm
i
2
2
2
1
2
2
2
5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
2
1












 ,  =3,
       





     









 


 










0 0 0 0 0
5
.
0
3
5
3
0
2
0
01
5
.
0
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
sin
5
.
0
3
,
0
i j k l n
i n
l
k
j
L
x
i
t
e
i
L
C
D 



           






















 2
2
2
2
2
2
1
5
.
0
1
5
.
0
1
5
.
0
1
n
l
k
j
i
n
l
k
j
i
n
l
k
j
i
       





















 2
2
2
2
1
1
1
1
5
.
0
1
n
l
k
j
ii
n
l
k
j
ii
n
l
k
j
i
           
 
   










00 0
1
5
,
0
ln
5
.
0
m
t u
m
n
m
n
m
n
jk
i u
d
d
I
I
e
e
u
e
u
e 



 ,  =3,
       





     
 








 


 











0 0 0 0 0 0
5
.
0
5
.
0
3
2
0
2
0
11
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
sin
5
.
0
3
,
0
n k l m i
t
n
n u
e
m
l
k
L
x
n
t
e
n
L
C
D 



   
        




 























 u
H
u
H
u
H
u
H
u
H
u
e m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
klmi 5
.
0
5
.
0
5
.
0
5
.
0
5
.
0
         




 























 u
H
u
H
u
H
u
H
u
H m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
m
l
i
n
k 5
.
0
5
.
1
5
.
1
5
.
0
5
.
1
         




 























 u
H
u
H
u
H
u
H
u
H m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
m
l
i
n
k
m
l
i
n
k 5
.
1
5
.
0
5
.
0
5
.
1
5
.
0
  u
d
u
H m
l
i
n
k 5
.
2





 .

On Approach for Estimation of Maximal Continuance of Diffusion Nand Ion Type of Doping

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On Approach for Estimation of Maximal Continuance of Diffusion Nand Ion Type of Doping