Modeling of manufacturing of a field effect transistor to determine conditions to decrease length of channel
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This document presents a modeling approach for manufacturing a field-effect heterotransistor to optimize the technological process, particularly to reduce the channel length through mechanical stress. The authors analyze the influence of mechanical stress on the dopant concentration distribution in the heterostructure, using mathematical modeling based on Fick's laws. Key outcomes include reductions in transistor channel length, thickness, and enhancements in density for integrated circuits.
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
DOI:10.5121/ijcsa.2014.4105 43
MODELING OF MANUFACTURING OF A FIELD-EFFECT
TRANSISTOR TO DETERMINE CONDITIONS TO
DECREASE LENGTH OF CHANNEL
E.L. Pankratov1
, E.A. Bulaeva2
1
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia
2
Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky street,
Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we introduce an approach to model technological process of manufacture of a field-effect
heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length
of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing
of thickness of the heterotransistors and increasing of their density, which were comprised in integrated
circuits.
KEYWORDS
Modelling of manufacture of a field-effect heterotransistor, Accounting mechanical stress; Optimization the
technological process, Decreasing length of channel
1. INTRODUCTION
One of intensively solved problems of the solid-state electronics is decreasing of elements of in-
tegrated circuits and their discrete analogs [1-7]. To solve this problem attracted an interest wide-
ly used laser and microwave types of annealing [8-14]. These types of annealing leads to genera-
tion of inhomogeneous distribution of temperature. In this situation one can obtain increasing of
sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area
[8-14]. The first effects give us possibility to decrease switching time of p-n-junction and value of
local overheats during operating of the device or to manufacture more shallow p-n-junction with
fixed maximal value of local overheat. An alternative approach to laser and microwave types of
annealing one can use native inhomogeneity of heterostructure and optimization of annealing of
dopant and/or radiation defects to manufacture diffusive –junction and implanted-junction rectifi-
ers [12-18]. It is known, that radiation processing of materials is also leads to modification of dis-
tribution of dopant concentration [19]. The radiation processing could be also used to increase of
sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area
[20, 21]. Distribution of dopant concentration is also depends on mechanical stress in heterostruc-
ture [15].
In this paper we consider a field-effect heterotransistor. Manufacturing of the transistor based on
combination of main ideas of Refs. [5] and [12-18]. Framework the combination we consider a
heterostructure with a substrate and an epitaxial layer with several sections. Some dopants have
been infused or implanted into the section to produce required types of conductivity. Farther we](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-1-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
44
consider optimal annealing of dopant and/or radiation defects. Manufacturing of the transistor has
been considered in details in Ref. [17]. Main aim of the present paper is analysis of influence of
mechanical stress on length of channel of the field-effect transistor.
Fig. 1. Heterostructure with substrate and epitaxial layer with two or three sections
a1a2-a1Lx-a2-a1
c1 c1
c2
oxide
source drain
gate
c3
Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view
2. Method of solution
To solve our aim we determine spatio-temporal distribution of concentration of dopant in the con-
sidered heterostructure and make the analysis. We determine spatio-temporal distribution of con-
centration of dopant with account mechanical stress by solving the second Fick’s law [1-4,22,23].
We assume to simplify illustration our main results, that we have possibility to use rectangular
coordinate system. In this situation the second Fick’s law could be written as](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-2-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
45
( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
zz L
S
S
L
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x 00
,,,,,,,,,,,,
with boundary and initial conditions
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=fC(x,y,z).
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic vo-
lume; ∇S is the operators of surficial gradient; ( )∫
zL
zdtzyxC
0
,,, is the surficial concentration of
dopant on interface between layers of heterostructure; (x,y,z,t) is the chemical potential; D and
DS are coefficients of volumetric and surficial (due to mechanical stress) of diffusion. Zero values
of derivatives on external boundaries of heterostructure corresponds to absence of dopant flow
through the boundaries. The first, the second and the third terms of the Eq.(1) correspond to free
diffusion of atoms of dopant in three directions: x, y, and z, respectively [1,3-6,12-21]. Two last
terms of the Eq.(1) correspond to moving of atoms of dopant under influence of mechanical stress
[22]. Values of the diffusion coefficients depend on properties of materials of heterostructure,
speed of heating and cooling of heterostructure, spatio-temporal distribution of concentration of
dopant. Concentraional dependence of dopant diffusion coefficients have been approximated by
the following relations [24]
( ) ( )
( )
+=
TzyxP
tzyxC
TzyxDD L
,,,
,,,
1,,,
, ( ) ( )
( )
+=
TzyxP
tzyxC
TzyxDD SLSS
,,,
,,,
1,,,
. (2)
Here DL (x,y,z,T) and DLS (x,y,z,T) are the spatial (due to inhomogeneity of heterostructure) and
the temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the
temperature of annealing; P (x,y,z,T) is the limit of solubility of dopant; parameter could be in-
teger in the following interval ∈[1,3] [24]. Concentrational dependence of dopant diffusion
coefficients has been discussed in details in [24]. Chemical potential could be determine by the
following relation [22]
=E(z)Ωij[uij(x,y,z,t)+uji(x,y,z,t)]/2, (3)
where E is the Young modulus; ij is the stress tensor;
∂
∂
+
∂
∂
=
i
j
j
i
ij
x
u
x
u
u
2
1
is the deformation
tensor [22,23]; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displace-
ment vector ( )tzyxu ,,,
; xi, xj are the coordinates x, y, z. Relation (3) could be transform to the
following form [23]](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-3-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
46
( ) ( ) ( ) ( ) ( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
=
i
j
j
i
i
j
j
i
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
tzyx
,,,,,,
2
1,,,,,,
,,,
( )
( )
( )
( ) ( ) ( ) ( )[ ]
−−
−
∂
∂
−
+−
Ω
× ij
k
kij
ij TtzyxTzzK
x
tzyxu
z
z
zE
000 ,,,3
,,,
212
, (3a)
where is the Poisson coefficient; 0=(as-aEL)/aEL is the mismatch strain; as, aEL are the lattice
spacings for substrate and epitaxial layer, respectively; K is the modulus of uniform compression;
is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for
our case) with room temperature. Components of displacement vector could be obtained by solv-
ing of the following equations [23]
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
zzzyzxz
yzyyyxy
xzxyxxx
,,,,,,,,,,,,
,,,,,,,,,,,,
,,,,,,,,,,,,
2
2
2
2
2
2
where
( )
( )[ ]
( ) ( ) ( ) ( )
×
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
+
=
k
k
ij
k
kij
i
j
j
i
ij
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
z
zE ,,,,,,
3
,,,,,,
12
( ) ( ) ( ) ( )[ ]rTtzyxTzKzzK −−× ,,, , (z) is the density of materials, ij is the Kronecker symbol.
Accounting the relation (3a) into the above equations gives us possibility to transform the equa-
tions to the following form
( )
( )
( ) ( )
( )[ ]
( )
( ) ( )
( )[ ]
( )
+
∂∂
∂
+
−+
∂
∂
+
+=
∂
∂
yx
tzyxu
z
zE
zK
x
tzyxu
z
zE
zK
t
tzyxu
z
yxx
,,,
13
,,,
16
5,,,
2
2
2
2
2
( )
( )[ ]
( ) ( )
( ) ( )
( )[ ]
( )
( ) ×−
∂∂
∂
+
++
∂
∂
+
∂
∂
+
+ zK
zx
tzyxu
z
zE
zK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
2
2
2
2
2
( ) ( )
x
tzyxT
z
∂
∂
×
,,,
( )
( ) ( )
( )[ ]
( ) ( )
( ) ( ) ( ) +
∂
∂
−
∂∂
∂
+
∂
∂
+
=
∂
∂
y
tzyxT
zKz
yx
tzyxu
x
tzyxu
z
zE
t
tzyxu
z xyy ,,,,,,,,,
12
,,, 2
2
2
2
2
( )
( )[ ]
( ) ( ) ( )
( )[ ]
( )
( )
+
∂
∂
+
+
+
∂
∂
+
∂
∂
+∂
∂
+ 2
2
,,,
112
5,,,,,,
12 y
tzyxu
zK
z
zE
y
tzyxu
z
tzyxu
z
zE
z
yzy
](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-4-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
47
( ) ( )
( )[ ]
( )
( )
( )
yx
tzyxu
zK
zy
tzyxu
z
zE
zK
yy
∂∂
∂
+
∂∂
∂
+
−+
,,,,,,
16
22
(4)
( )
( ) ( ) ( ) ( ) ( )
×
∂∂
∂
+
∂∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
t
tzyxu
z
yxzzz
,,,,,,,,,,,,,,,
22
2
2
2
2
2
2
( )
( )[ ]
( )
( ) ( ) ( ) ( )×
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
+
×
z
tzyxT
z
tzyxu
y
tzyxu
x
tzyxu
zK
zz
zE xyx ,,,,,,,,,,,,
12
( ) ( ) ( )
( )
( ) ( ) ( ) ( )
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
+∂
∂
+×
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zE
z
zzK zyxz ,,,,,,,,,,,,
6
16
1
.
Conditions for the displacement vector could be written as
( ) 0
,,,0
=
∂
∂
x
tzyu
;
( ) 0
,,,
=
∂
∂
x
tzyLu x
;
( ) 0
,,0,
=
∂
∂
y
tzxu
;
( ) 0
,,,
=
∂
∂
y
tzLxu y
;
( ) 0
,0,,
=
∂
∂
z
tyxu
;
( ) 0
,,,
=
∂
∂
z
tLyxu z
; ( ) 00,,, uzyxu
= ; ( ) 0,,, uzyxu
=∞ .
Zero values of derivatives on external boundaries of heterostructure corresponds to absence of
correspond to the same situation as in the conditions for dopant. Farther let us analyze spatio-
temporal distribution of temperature during annealing of dopant. We determine spatio-temporal
distribution of temperature as solution of the second law of Fourier [25]
( ) ( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxT
zy
tzyxT
yx
tzyxT
xt
tzyxT
Tc
,,,,,,,,,,,,
( )tzyxp ,,,+ (5)
with boundary and initial conditions
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxT
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxT
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxT
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxT
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxT
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxT
, T(x,y,z,0)=fT (x,y,z).
Here is the heat conduction coefficient. Value of the coefficient depends on materials of hetero-
structure and temperature. Temperature dependence of heat conduction coefficient in most inter-
est area could be approximated by the following function: (x,y,z,T)= ass(x,y,z)[1+
Td
/T
(x,y,z,t)] (see, for example, [25]). c(T)=cass[1- exp(-T(x,y,z,t)/Td)] is the heat capacitance;
Td is the Debye temperature [25]. The temperature T(x,y,z,t) is approximately equal or larger, than
Debye temperature Td for most interesting for us temperature interval. In this situation one can
write the following approximate relation: c(T)≈cass. p(x,y,z,t) in the last term of Eq.(4) is the](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-5-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
48
volumetric density of heat, which escapes in the heterostructure. The previous terms in the right
side of the equation correspond to free conduction of heat. Zero values of derivatives on external
boundaries of heterostructure corresponds to absence of correspond to the same situation as in the
conditions for dopant. Framework the paper it is attracted an interest microwave annealing of do-
pant. This approach leads to generation inhomogenous distribution of temperature [12-14,26],
because the volumetric density of heat generating in near-surficial area with scin-layer thickness.
Such distribution of temperature gives us possibility to increase sharpness of p-n-junctions with
increasing of homogeneity of dopant distribution in enriched area due to Arrhenius law. In this
situation it is practicably to choose necessary frequency of electro-magnetic field. After this
choosing thickness of scin-layer should be approximately equal to thickness of epitaxial layer.
First of all we calculate spatio-temporal distribution of temperature. We calculate spatio-temporal
distribution of temperature by using recently introduce approach [15, 17,18]. Framework the ap-
proach we transform approximation of thermal diffusivity ass(x,y,z) =ass(x,y,z)/cass =0ass[1+T
gT(x,y,z)]. Farther we determine solution of Eqs. (5) as the following power series
( ) ( )∑ ∑=
∞
=
∞
=0 0
,,,,,,
i j
ij
ji
T tzyxTtzyxT . (6)
Substitution of the series into Eq.(6) gives us possibility to obtain system of equations for the ini-
tial-order approximation of temperature T00(x,y,z,t) and corrections for them Tij(x,y,z,t) (i≥1, j≥1).
The procedure has been described in details in [12-14,18]. In this situation we will not present the
procedure in this paper. Substitution of the series (6) in boundary and initial conditions for spatio-
temporal distribution of temperature gives us possibility to obtain boundary and initial conditions
for the functions Tij(x,y,z,t) (i≥0, j≥0). The procedure has been described in details in [12-14,18]
and will not be presented here. Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0)
have been obtain as the second-order approximation on the parameters ε and µ by standard ap-
proaches [28,29] and will not be presented in the paper. Recently we obtain, that the second-order
approximation is enough good approximation to make qualitative analysis and to obtain some
quantitative results (see, for example, [15,17,18]). Analytical results have allowed to identify and
to illustrate the main dependence. To check our results obtained we used numerical approaches.
Farther let us estimate components of the displacement vector. The components could be obtained
by using the same approach as for calculation distribution of temperature. However, relations for
components could by calculated in shorter form by using method of averaging of function correc-
tions [12-14,16,27]. It is practicably to transform differential equations of system (4) to integro-
differential form. The integral equations are presented in the Appendix. We determine the first-
order approximations of components of the displacement vector by replacement the required
functions u(x,y,z,t) on their average values u1. The average values u1 have been calculated by
the following relations
us1=Ms1/4LΘ, (7)
where ( )∫ ∫ ∫ ∫
Θ
− −
=
0 0
,,,
x
x
y
y
zL
L
L
L
L
isis tdxdydzdtzyxuM . The replacement leads to the following
results
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
−∫ ∫
∂
∂
−−∫ ∫
∂
∂
=
∞ t zz
x dwd
x
wyxT
wwKtdwd
x
wyxT
wwKttzyxu
0 00 0
1
,,,,,,
,,,
](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-6-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
49
( )] 101 ,,, uxxux twyx +Φ− ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
−∫ ∫
∂
∂
−−∫ ∫
∂
∂
=
∞ t zz
y dwd
y
wyxT
wwKtdwd
y
wyxT
wwKttzyxu
0 00 0
1
,,,,,,
,,,
( )] 101 ,,, uyyuy twyx +Φ− ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
−∫ ∫
∂
∂
−∫ ∫
∂
∂
=
∞ t zz
z dwd
w
wyxT
wwKtdwd
w
wyxT
wwKttzyxu
0 00 0
1
,,,,,,
,,,
( )] 101 ,,, zzuz twyx +Φ− .
Substitution of the above relations into relations (7) gives us possibility to obtain values of the
parameters u1. Appropriate relations could be written as
( ) ( )[ ]
( ) ( )∫ −
ΘΧ−∞Χ
Θ= zL
z
xxz
ux
zdzzLL
L
0
3
20
1
8
,
( ) ( )[ ]
( ) ( )∫ −
ΘΧ−∞Χ
Θ= zL
z
yyz
uy
zdzzLL
L
0
3
20
1
8
,
( ) ( )[ ]
( ) ( )∫ −
ΘΧ−∞ΧΘ
= zL
z
zzz
uz
zdzzLL
L
0
3
20
1
8
,
where ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−
Θ
+=ΘΧ
Θ
− −0 0
,,,
1
x
x
y
y
zL
L
L
L
L
z
i
is dxdydzd
s
zyxT
zzKzL
t
.
The second-order approximations of components of the displacement vector by replacement of
the required functions u(x,y,z,t) on the following sums us2+us1(x,y,z,t), where us2=(Mus2-Mus1)/
4L3
Θ. Results of calculations of the second-order approximations us2(x,y,z,t) and their average
values αus2 are will not presented in this paper, because the procedure is standard.
Spatio-temporal distribution of concentration of dopant we obtain by solving the Eq.(1). To solve
the equations we used recently introduce approach [15,17,18] and transform approximations of
dopant diffusion coefficients DL(x,y,z,T) and DSL(x,y,z,T) to the following form: DL(x,y,z,T)=
D0L[1+LgL(x,y,z,T)] and DSL(x,y,z,T)=D0SL[1+SLgSL(x,y,z,T)]. We also introduce the following
dimensionless parameter: =D0SL/D0L. In this situation the Eq.(1) takes the form
( ) ( )[ ] ( )
( )
( ) ( )
×
∂
∂
∂
∂
+
∂
∂
++
∂
∂
=
∂
∂
y
tzyxC
yx
tzyxC
TzP
tzyxC
Tzg
x
D
t
tzyxC
LLL
,,,,,,
,
,,,
1,1
,,,
0
( )[ ] ( )
( )
( )[ ] ( )
( )
×
++
∂
∂
+
++×
TzyxP
tzyxC
Tzyxg
z
DD
TzyxP
tzyxC
Tzg LLLLLL
,,,
,,,
1,,,1
,,,
,,,
1,1 00
( ) ( )[ ] ( )
( )
( )
×∫
++
∂
∂
Ω+
∂
∂
×
zL
SLSLSL WdtWyxC
TzP
tzyxC
Tzg
x
D
z
tzyxC
0
0 ,,,
,
,,,
1,1
,,,
(8)](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-7-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
50
( ) ( )
( )
( )
( )
∫
∇
+
∂
∂
Ω+
∇
×
zL
S
SLL
S
WdtWyxC
Tk
tzyx
TzP
tzyxC
y
DD
Tk
tzyx
0
00 ,,,
,,,
,
,,,
1
,,,
.
We determine solution of Eq.(1) as the following power series
( ) ( )∑ ∑ ∑=
∞
=
∞
=
∞
=0 0 0
,,,,,,
i j k
ijk
kji
L tzyxCtzyxC . (9)
Substitution of the series into Eq.(8) gives us possibility to obtain systems of equations for initial-
order approximation of concentration of dopant C000(x,y,z,t) and corrections for them Cijk(x,
y,z,t) (i ≥1, j ≥1, k ≥1).The equations will not presented in this paper. Substitution of the series
into appropriate boundary and initial conditions gives us possibility to obtain boundary and initial
conditions for all functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0). Solutions of equations for the functions
Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0) have been calculated by standard approaches [28,29] and will not
presented in this paper.
Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has
been done analytically by using the second-order approximations framework recently introduced
power series. The approximation is enough good approximation to make qualitative analysis and
to obtain some quantitative results. All obtained analytical results have been checked by compari-
son with results, calculated by numerical simulation.
x, y
C(x,Θ),C(y,Θ)
2
3
1
Fig. 3. Distributions of concentration of dopant in directions, which is perpendicular to interface
between layers of heterostructure. Curve 1 is a dopant distribution in a homogenous sample.
Curve 2 corresponds to negative value of the mismatch strain. Curve 3 corresponds to positive
value of the mismatch strain
3. Discussion
In this section we analyzed redistribution of dopant under influence of mechanical stress based on
relations calculated in previous section. Fig. 3 show spatio-temporal distribution of concentration
of dopant in a homogenous sample (curve 1) and in heterostructure with negative and positive
value of the mismatch strain 0 (curve 2 and 3, respectively). The figure shows, that manufactur-
ing field-effect heterotransistors gives us possibility to make more compact field-effect transistors
in direction, which is parallel to interface between layers of heterostructure. As a consequence of
the increasing of compactness one can obtain decreasing of length of channel of field- effect tran-
sistor and increasing of density of the transistors in integrated circuits. It should be noted, that
presents of interface between layers of heterostructure give us possibility to manufacture more
thin transistors [17].](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-8-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
51
4. Conclusion
In this paper we consider possibility to decrease length of channel of field-effect transistor by us-
ing mechanical stress in heterostructure. At the same time with decreasing of the length it could
be increased density of transistors in integrated circuits.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research.
REFERENCES
[1] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980).
[2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).
[3] V.G. Gusev, Yu.M. Gusev. Electronics. (Moscow: Vysshaya shkola, 1991, in Russian).
[4] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
[5] A. Kerentsev, V. Lanin. "Constructive-technological features of MOSFET-transistors" Power Elec-
tronics. Issue 1. P. 34-38 (2008).
[6] A.N. Andronov, N.T. Bagraev, L.E. Klyachkin, S.V. Robozerov. "Ultrashallow p+−n junc-tions in
silicon (100): electron-beam diagnostics of sub-surface region" Semiconductors. Vol.32 (2). P. 137-
144 (1998).
[7] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p−n junctions in Si prepared by pulse pho-
ton annealing" Semiconductors. Vol.36 (5). P. 611-617 (2002).
[8] V.I. Mazhukin, V.V. Nosov, U. Semmler. "Study of heat and thermoelastic fields in semiconductors
at pulsed processing" Mathematical modeling. Vol. 12 (2), 75 (2000).
[9] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. "Dopant distribution in the re-
crystallization transient at the maximum melt depth induced by laser annealing" Appl. Phys. Lett.
Vol. 89 (17), 172111 (2006).
[10] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A.
Foad, F. Cristiano, P. F. Fazzini. "Deactivation of ultrashallow boron implants in preamorphized sili-
con after nonmelt laser annealing with multiple scans" Appl.Phys. Lett. Vol. 89, 192105 (2006).
[11] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N.
Drozdov, V.D. Skupov. "Diffusion processes in semiconductor structures during microwave anneal-
ing" Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).
[12] E.L. Pankratov. "Redistribution of dopant during microwave annealing of a multilayer structure for
production p-n-junction" J. Appl. Phys. Vol. 103 (6). P. 064320-064330 (2008).
[13] E.L. Pankratov. "Optimization of near-surficial annealing for decreasing of depth of p-n-junction in
semiconductor heterostructure" Proc. of SPIE. Vol. 7521, 75211D (2010).
[14] E.L. Pankratov. "Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure
by optimized laser annealing" J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010).
[15] E.L. Pankratov. "Influence of mechanical stress in semiconductor heterostructure on density of p-n-
junctions" Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012).
[16] E.L. Pankratov, E.A. Bulaeva. "Application of native inhomogeneities to increase compactness of
vertical field-effect transistors" J. Comp. Theor. Nanoscience. Vol. 10 (4). P. 888-893 (2013).
[17] E.L. Pankratov, E.A. Bulaeva. "An approach to decrease dimensions of field-effect transistors " Uni-
versal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013).
[18] E.L. Pankratov, E.A. Bulaeva. "Doping of materials during manufacture p-n-junctions and bipolar
transistors. Analytical approaches to model technological approaches and ways of optimization of dis-
tributions of dopants" Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013).
[19] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in
Russian).
[20] E.L. Pankratov. "Decreasing of depth of p-n-junction in a semiconductor heterostructure by serial
radiation processing and microwave annealing" J. Comp. Theor. Nanoscience. Vol. 9 (1). P. 41-49
(2012).](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-9-320.jpg)
![International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014
52
[21] E.L. Pankratov, E.A. Bulaeva. "Increasing of sharpness of diffusion-junction heterorectifier by using
radiation processing" Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012).
[22] Y.W. Zhang, A.F. Bower. "Numerical simulations of islands formation in a coherent strained epitaxial
thin film system" Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999).
[23] L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001,
in Russian).
[24] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).
[25] K.V. Shalimova. Physics of semiconductors (Moscow: Energoatomizdat, 1985, in Rus-sian).
[26] V.E. Kuz’michev. Laws and formulas of physics. Kiev: Naukova Dumka, 1989 (in Russian).
[27] Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting Applied Mechanics. Vol.1
(1). P. 23-35 (1955).
[28] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in
Russian).
[29] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
Authors
Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary
school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:
from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra-
diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio-
physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro-
structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo-
rod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor
course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in
area of his researches.
Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of
village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school
“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec-
ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she
is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in
the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant № MK-548.2010.2).
She has 29 published papers in area of her researches.](https://image.slidesharecdn.com/modelingofmanufacturingofafield-effecttransistortodetermineconditionstodecreaselengthofchannel-140312020409-phpapp01/85/Modeling-of-manufacturing-of-a-field-effect-transistor-to-determine-conditions-to-decrease-length-of-channel-10-320.jpg)
Modeling of manufacturing of a field effect transistor to determine conditions to decrease length of channel
- 1. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 DOI:10.5121/ijcsa.2014.4105 43 MODELING OF MANUFACTURING OF A FIELD-EFFECT TRANSISTOR TO DETERMINE CONDITIONS TO DECREASE LENGTH OF CHANNEL E.L. Pankratov1 , E.A. Bulaeva2 1 Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky street, Nizhny Novgorod, 603950, Russia ABSTRACT In this paper we introduce an approach to model technological process of manufacture of a field-effect heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing of thickness of the heterotransistors and increasing of their density, which were comprised in integrated circuits. KEYWORDS Modelling of manufacture of a field-effect heterotransistor, Accounting mechanical stress; Optimization the technological process, Decreasing length of channel 1. INTRODUCTION One of intensively solved problems of the solid-state electronics is decreasing of elements of in- tegrated circuits and their discrete analogs [1-7]. To solve this problem attracted an interest wide- ly used laser and microwave types of annealing [8-14]. These types of annealing leads to genera- tion of inhomogeneous distribution of temperature. In this situation one can obtain increasing of sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area [8-14]. The first effects give us possibility to decrease switching time of p-n-junction and value of local overheats during operating of the device or to manufacture more shallow p-n-junction with fixed maximal value of local overheat. An alternative approach to laser and microwave types of annealing one can use native inhomogeneity of heterostructure and optimization of annealing of dopant and/or radiation defects to manufacture diffusive –junction and implanted-junction rectifi- ers [12-18]. It is known, that radiation processing of materials is also leads to modification of dis- tribution of dopant concentration [19]. The radiation processing could be also used to increase of sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area [20, 21]. Distribution of dopant concentration is also depends on mechanical stress in heterostruc- ture [15]. In this paper we consider a field-effect heterotransistor. Manufacturing of the transistor based on combination of main ideas of Refs. [5] and [12-18]. Framework the combination we consider a heterostructure with a substrate and an epitaxial layer with several sections. Some dopants have been infused or implanted into the section to produce required types of conductivity. Farther we
- 2. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 44 consider optimal annealing of dopant and/or radiation defects. Manufacturing of the transistor has been considered in details in Ref. [17]. Main aim of the present paper is analysis of influence of mechanical stress on length of channel of the field-effect transistor. Fig. 1. Heterostructure with substrate and epitaxial layer with two or three sections a1a2-a1Lx-a2-a1 c1 c1 c2 oxide source drain gate c3 Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view 2. Method of solution To solve our aim we determine spatio-temporal distribution of concentration of dopant in the con- sidered heterostructure and make the analysis. We determine spatio-temporal distribution of con- centration of dopant with account mechanical stress by solving the second Fick’s law [1-4,22,23]. We assume to simplify illustration our main results, that we have possibility to use rectangular coordinate system. In this situation the second Fick’s law could be written as
- 3. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 45 ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1) ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zz L S S L S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x 00 ,,,,,,,,,,,, with boundary and initial conditions ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC , C(x,y,z,0)=fC(x,y,z). Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; Ω is the atomic vo- lume; ∇S is the operators of surficial gradient; ( )∫ zL zdtzyxC 0 ,,, is the surficial concentration of dopant on interface between layers of heterostructure; (x,y,z,t) is the chemical potential; D and DS are coefficients of volumetric and surficial (due to mechanical stress) of diffusion. Zero values of derivatives on external boundaries of heterostructure corresponds to absence of dopant flow through the boundaries. The first, the second and the third terms of the Eq.(1) correspond to free diffusion of atoms of dopant in three directions: x, y, and z, respectively [1,3-6,12-21]. Two last terms of the Eq.(1) correspond to moving of atoms of dopant under influence of mechanical stress [22]. Values of the diffusion coefficients depend on properties of materials of heterostructure, speed of heating and cooling of heterostructure, spatio-temporal distribution of concentration of dopant. Concentraional dependence of dopant diffusion coefficients have been approximated by the following relations [24] ( ) ( ) ( ) += TzyxP tzyxC TzyxDD L ,,, ,,, 1,,, , ( ) ( ) ( ) += TzyxP tzyxC TzyxDD SLSS ,,, ,,, 1,,, . (2) Here DL (x,y,z,T) and DLS (x,y,z,T) are the spatial (due to inhomogeneity of heterostructure) and the temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; P (x,y,z,T) is the limit of solubility of dopant; parameter could be in- teger in the following interval ∈[1,3] [24]. Concentrational dependence of dopant diffusion coefficients has been discussed in details in [24]. Chemical potential could be determine by the following relation [22] =E(z)Ωij[uij(x,y,z,t)+uji(x,y,z,t)]/2, (3) where E is the Young modulus; ij is the stress tensor; ∂ ∂ + ∂ ∂ = i j j i ij x u x u u 2 1 is the deformation tensor [22,23]; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displace- ment vector ( )tzyxu ,,, ; xi, xj are the coordinates x, y, z. Relation (3) could be transform to the following form [23]
- 4. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 46 ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ = i j j i i j j i x tzyxu x tzyxu x tzyxu x tzyxu tzyx ,,,,,, 2 1,,,,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] −− − ∂ ∂ − +− Ω × ij k kij ij TtzyxTzzK x tzyxu z z zE 000 ,,,3 ,,, 212 , (3a) where is the Poisson coefficient; 0=(as-aEL)/aEL is the mismatch strain; as, aEL are the lattice spacings for substrate and epitaxial layer, respectively; K is the modulus of uniform compression; is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for our case) with room temperature. Components of displacement vector could be obtained by solv- ing of the following equations [23] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ z tzyx y tzyx x tzyx t tzyxu z z tzyx y tzyx x tzyx t tzyxu z z tzyx y tzyx x tzyx t tzyxu z zzzyzxz yzyyyxy xzxyxxx ,,,,,,,,,,,, ,,,,,,,,,,,, ,,,,,,,,,,,, 2 2 2 2 2 2 where ( ) ( )[ ] ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + = k k ij k kij i j j i ij x tzyxu x tzyxu x tzyxu x tzyxu z zE ,,,,,, 3 ,,,,,, 12 ( ) ( ) ( ) ( )[ ]rTtzyxTzKzzK −−× ,,, , (z) is the density of materials, ij is the Kronecker symbol. Accounting the relation (3a) into the above equations gives us possibility to transform the equa- tions to the following form ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) + ∂∂ ∂ + −+ ∂ ∂ + += ∂ ∂ yx tzyxu z zE zK x tzyxu z zE zK t tzyxu z yxx ,,, 13 ,,, 16 5,,, 2 2 2 2 2 ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ×− ∂∂ ∂ + ++ ∂ ∂ + ∂ ∂ + + zK zx tzyxu z zE zK z tzyxu y tzyxu z zE zzy ,,, 13 ,,,,,, 12 2 2 2 2 2 ( ) ( ) x tzyxT z ∂ ∂ × ,,, ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zKz yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 2 2 2 2 2 ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) + ∂ ∂ + + + ∂ ∂ + ∂ ∂ +∂ ∂ + 2 2 ,,, 112 5,,,,,, 12 y tzyxu zK z zE y tzyxu z tzyxu z zE z yzy
- 5. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 47 ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 22 (4) ( ) ( ) ( ) ( ) ( ) ( ) × ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxu t tzyxu z yxzzz ,,,,,,,,,,,,,,, 22 2 2 2 2 2 2 ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )× ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + + × z tzyxT z tzyxu y tzyxu x tzyxu zK zz zE xyx ,,,,,,,,,,,, 12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ +∂ ∂ +× z tzyxu y tzyxu x tzyxu z tzyxu z zE z zzK zyxz ,,,,,,,,,,,, 6 16 1 . Conditions for the displacement vector could be written as ( ) 0 ,,,0 = ∂ ∂ x tzyu ; ( ) 0 ,,, = ∂ ∂ x tzyLu x ; ( ) 0 ,,0, = ∂ ∂ y tzxu ; ( ) 0 ,,, = ∂ ∂ y tzLxu y ; ( ) 0 ,0,, = ∂ ∂ z tyxu ; ( ) 0 ,,, = ∂ ∂ z tLyxu z ; ( ) 00,,, uzyxu = ; ( ) 0,,, uzyxu =∞ . Zero values of derivatives on external boundaries of heterostructure corresponds to absence of correspond to the same situation as in the conditions for dopant. Farther let us analyze spatio- temporal distribution of temperature during annealing of dopant. We determine spatio-temporal distribution of temperature as solution of the second law of Fourier [25] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxT zy tzyxT yx tzyxT xt tzyxT Tc ,,,,,,,,,,,, ( )tzyxp ,,,+ (5) with boundary and initial conditions ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxT , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxT , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxT , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxT , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxT , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxT , T(x,y,z,0)=fT (x,y,z). Here is the heat conduction coefficient. Value of the coefficient depends on materials of hetero- structure and temperature. Temperature dependence of heat conduction coefficient in most inter- est area could be approximated by the following function: (x,y,z,T)= ass(x,y,z)[1+ Td /T (x,y,z,t)] (see, for example, [25]). c(T)=cass[1- exp(-T(x,y,z,t)/Td)] is the heat capacitance; Td is the Debye temperature [25]. The temperature T(x,y,z,t) is approximately equal or larger, than Debye temperature Td for most interesting for us temperature interval. In this situation one can write the following approximate relation: c(T)≈cass. p(x,y,z,t) in the last term of Eq.(4) is the
- 6. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 48 volumetric density of heat, which escapes in the heterostructure. The previous terms in the right side of the equation correspond to free conduction of heat. Zero values of derivatives on external boundaries of heterostructure corresponds to absence of correspond to the same situation as in the conditions for dopant. Framework the paper it is attracted an interest microwave annealing of do- pant. This approach leads to generation inhomogenous distribution of temperature [12-14,26], because the volumetric density of heat generating in near-surficial area with scin-layer thickness. Such distribution of temperature gives us possibility to increase sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area due to Arrhenius law. In this situation it is practicably to choose necessary frequency of electro-magnetic field. After this choosing thickness of scin-layer should be approximately equal to thickness of epitaxial layer. First of all we calculate spatio-temporal distribution of temperature. We calculate spatio-temporal distribution of temperature by using recently introduce approach [15, 17,18]. Framework the ap- proach we transform approximation of thermal diffusivity ass(x,y,z) =ass(x,y,z)/cass =0ass[1+T gT(x,y,z)]. Farther we determine solution of Eqs. (5) as the following power series ( ) ( )∑ ∑= ∞ = ∞ =0 0 ,,,,,, i j ij ji T tzyxTtzyxT . (6) Substitution of the series into Eq.(6) gives us possibility to obtain system of equations for the ini- tial-order approximation of temperature T00(x,y,z,t) and corrections for them Tij(x,y,z,t) (i≥1, j≥1). The procedure has been described in details in [12-14,18]. In this situation we will not present the procedure in this paper. Substitution of the series (6) in boundary and initial conditions for spatio- temporal distribution of temperature gives us possibility to obtain boundary and initial conditions for the functions Tij(x,y,z,t) (i≥0, j≥0). The procedure has been described in details in [12-14,18] and will not be presented here. Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) have been obtain as the second-order approximation on the parameters ε and µ by standard ap- proaches [28,29] and will not be presented in the paper. Recently we obtain, that the second-order approximation is enough good approximation to make qualitative analysis and to obtain some quantitative results (see, for example, [15,17,18]). Analytical results have allowed to identify and to illustrate the main dependence. To check our results obtained we used numerical approaches. Farther let us estimate components of the displacement vector. The components could be obtained by using the same approach as for calculation distribution of temperature. However, relations for components could by calculated in shorter form by using method of averaging of function correc- tions [12-14,16,27]. It is practicably to transform differential equations of system (4) to integro- differential form. The integral equations are presented in the Appendix. We determine the first- order approximations of components of the displacement vector by replacement the required functions u(x,y,z,t) on their average values u1. The average values u1 have been calculated by the following relations us1=Ms1/4LΘ, (7) where ( )∫ ∫ ∫ ∫ Θ − − = 0 0 ,,, x x y y zL L L L L isis tdxdydzdtzyxuM . The replacement leads to the following results ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∂ ∂ −−∫ ∫ ∂ ∂ = ∞ t zz x dwd x wyxT wwKtdwd x wyxT wwKttzyxu 0 00 0 1 ,,,,,, ,,,
- 7. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 49 ( )] 101 ,,, uxxux twyx +Φ− , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∂ ∂ −−∫ ∫ ∂ ∂ = ∞ t zz y dwd y wyxT wwKtdwd y wyxT wwKttzyxu 0 00 0 1 ,,,,,, ,,, ( )] 101 ,,, uyyuy twyx +Φ− , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = ∞ t zz z dwd w wyxT wwKtdwd w wyxT wwKttzyxu 0 00 0 1 ,,,,,, ,,, ( )] 101 ,,, zzuz twyx +Φ− . Substitution of the above relations into relations (7) gives us possibility to obtain values of the parameters u1. Appropriate relations could be written as ( ) ( )[ ] ( ) ( )∫ − ΘΧ−∞Χ Θ= zL z xxz ux zdzzLL L 0 3 20 1 8 , ( ) ( )[ ] ( ) ( )∫ − ΘΧ−∞Χ Θ= zL z yyz uy zdzzLL L 0 3 20 1 8 , ( ) ( )[ ] ( ) ( )∫ − ΘΧ−∞ΧΘ = zL z zzz uz zdzzLL L 0 3 20 1 8 , where ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ − Θ +=ΘΧ Θ − −0 0 ,,, 1 x x y y zL L L L L z i is dxdydzd s zyxT zzKzL t . The second-order approximations of components of the displacement vector by replacement of the required functions u(x,y,z,t) on the following sums us2+us1(x,y,z,t), where us2=(Mus2-Mus1)/ 4L3 Θ. Results of calculations of the second-order approximations us2(x,y,z,t) and their average values αus2 are will not presented in this paper, because the procedure is standard. Spatio-temporal distribution of concentration of dopant we obtain by solving the Eq.(1). To solve the equations we used recently introduce approach [15,17,18] and transform approximations of dopant diffusion coefficients DL(x,y,z,T) and DSL(x,y,z,T) to the following form: DL(x,y,z,T)= D0L[1+LgL(x,y,z,T)] and DSL(x,y,z,T)=D0SL[1+SLgSL(x,y,z,T)]. We also introduce the following dimensionless parameter: =D0SL/D0L. In this situation the Eq.(1) takes the form ( ) ( )[ ] ( ) ( ) ( ) ( ) × ∂ ∂ ∂ ∂ + ∂ ∂ ++ ∂ ∂ = ∂ ∂ y tzyxC yx tzyxC TzP tzyxC Tzg x D t tzyxC LLL ,,,,,, , ,,, 1,1 ,,, 0 ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) × ++ ∂ ∂ + ++× TzyxP tzyxC Tzyxg z DD TzyxP tzyxC Tzg LLLLLL ,,, ,,, 1,,,1 ,,, ,,, 1,1 00 ( ) ( )[ ] ( ) ( ) ( ) ×∫ ++ ∂ ∂ Ω+ ∂ ∂ × zL SLSLSL WdtWyxC TzP tzyxC Tzg x D z tzyxC 0 0 ,,, , ,,, 1,1 ,,, (8)
- 8. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 50 ( ) ( ) ( ) ( ) ( ) ∫ ∇ + ∂ ∂ Ω+ ∇ × zL S SLL S WdtWyxC Tk tzyx TzP tzyxC y DD Tk tzyx 0 00 ,,, ,,, , ,,, 1 ,,, . We determine solution of Eq.(1) as the following power series ( ) ( )∑ ∑ ∑= ∞ = ∞ = ∞ =0 0 0 ,,,,,, i j k ijk kji L tzyxCtzyxC . (9) Substitution of the series into Eq.(8) gives us possibility to obtain systems of equations for initial- order approximation of concentration of dopant C000(x,y,z,t) and corrections for them Cijk(x, y,z,t) (i ≥1, j ≥1, k ≥1).The equations will not presented in this paper. Substitution of the series into appropriate boundary and initial conditions gives us possibility to obtain boundary and initial conditions for all functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0). Solutions of equations for the functions Cijk(x,y,z,t) (i ≥0, j ≥0, k ≥0) have been calculated by standard approaches [28,29] and will not presented in this paper. Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has been done analytically by using the second-order approximations framework recently introduced power series. The approximation is enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained analytical results have been checked by compari- son with results, calculated by numerical simulation. x, y C(x,Θ),C(y,Θ) 2 3 1 Fig. 3. Distributions of concentration of dopant in directions, which is perpendicular to interface between layers of heterostructure. Curve 1 is a dopant distribution in a homogenous sample. Curve 2 corresponds to negative value of the mismatch strain. Curve 3 corresponds to positive value of the mismatch strain 3. Discussion In this section we analyzed redistribution of dopant under influence of mechanical stress based on relations calculated in previous section. Fig. 3 show spatio-temporal distribution of concentration of dopant in a homogenous sample (curve 1) and in heterostructure with negative and positive value of the mismatch strain 0 (curve 2 and 3, respectively). The figure shows, that manufactur- ing field-effect heterotransistors gives us possibility to make more compact field-effect transistors in direction, which is parallel to interface between layers of heterostructure. As a consequence of the increasing of compactness one can obtain decreasing of length of channel of field- effect tran- sistor and increasing of density of the transistors in integrated circuits. It should be noted, that presents of interface between layers of heterostructure give us possibility to manufacture more thin transistors [17].
- 9. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 51 4. Conclusion In this paper we consider possibility to decrease length of channel of field-effect transistor by us- ing mechanical stress in heterostructure. At the same time with decreasing of the length it could be increased density of transistors in integrated circuits. ACKNOWLEDGEMENTS This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research. REFERENCES [1] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980). [2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). [3] V.G. Gusev, Yu.M. Gusev. Electronics. (Moscow: Vysshaya shkola, 1991, in Russian). [4] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [5] A. Kerentsev, V. Lanin. "Constructive-technological features of MOSFET-transistors" Power Elec- tronics. Issue 1. P. 34-38 (2008). [6] A.N. Andronov, N.T. Bagraev, L.E. Klyachkin, S.V. Robozerov. "Ultrashallow p+−n junc-tions in silicon (100): electron-beam diagnostics of sub-surface region" Semiconductors. Vol.32 (2). P. 137- 144 (1998). [7] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p−n junctions in Si prepared by pulse pho- ton annealing" Semiconductors. Vol.36 (5). P. 611-617 (2002). [8] V.I. Mazhukin, V.V. Nosov, U. Semmler. "Study of heat and thermoelastic fields in semiconductors at pulsed processing" Mathematical modeling. Vol. 12 (2), 75 (2000). [9] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. "Dopant distribution in the re- crystallization transient at the maximum melt depth induced by laser annealing" Appl. Phys. Lett. Vol. 89 (17), 172111 (2006). [10] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A. Foad, F. Cristiano, P. F. Fazzini. "Deactivation of ultrashallow boron implants in preamorphized sili- con after nonmelt laser annealing with multiple scans" Appl.Phys. Lett. Vol. 89, 192105 (2006). [11] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. "Diffusion processes in semiconductor structures during microwave anneal- ing" Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). [12] E.L. Pankratov. "Redistribution of dopant during microwave annealing of a multilayer structure for production p-n-junction" J. Appl. Phys. Vol. 103 (6). P. 064320-064330 (2008). [13] E.L. Pankratov. "Optimization of near-surficial annealing for decreasing of depth of p-n-junction in semiconductor heterostructure" Proc. of SPIE. Vol. 7521, 75211D (2010). [14] E.L. Pankratov. "Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure by optimized laser annealing" J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010). [15] E.L. Pankratov. "Influence of mechanical stress in semiconductor heterostructure on density of p-n- junctions" Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012). [16] E.L. Pankratov, E.A. Bulaeva. "Application of native inhomogeneities to increase compactness of vertical field-effect transistors" J. Comp. Theor. Nanoscience. Vol. 10 (4). P. 888-893 (2013). [17] E.L. Pankratov, E.A. Bulaeva. "An approach to decrease dimensions of field-effect transistors " Uni- versal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013). [18] E.L. Pankratov, E.A. Bulaeva. "Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of dis- tributions of dopants" Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [19] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in Russian). [20] E.L. Pankratov. "Decreasing of depth of p-n-junction in a semiconductor heterostructure by serial radiation processing and microwave annealing" J. Comp. Theor. Nanoscience. Vol. 9 (1). P. 41-49 (2012).
- 10. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.1, February 2014 52 [21] E.L. Pankratov, E.A. Bulaeva. "Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing" Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012). [22] Y.W. Zhang, A.F. Bower. "Numerical simulations of islands formation in a coherent strained epitaxial thin film system" Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999). [23] L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001, in Russian). [24] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [25] K.V. Shalimova. Physics of semiconductors (Moscow: Energoatomizdat, 1985, in Rus-sian). [26] V.E. Kuz’michev. Laws and formulas of physics. Kiev: Naukova Dumka, 1989 (in Russian). [27] Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting Applied Mechanics. Vol.1 (1). P. 23-35 (1955). [28] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in Russian). [29] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964). Authors Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University: from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra- diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio- physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro- structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo- rod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in area of his researches. Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school “Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec- ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant № MK-548.2010.2). She has 29 published papers in area of her researches.