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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015 DOI : 10.5121/ijaceee.2015.3301 1 ON VERTICAL INTEGRATION FRAMEWORK ELEMENT OF TRANSISTOR-TRANSISTOR LOGIC E.L. Pankratov 1 , E.A. Bulaeva 1,2 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 increase vertical integration of elements of transistor-transistor logic with function AND-NOT. Framework the approach we consider a heterostructure with special confi- guration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized. KEYWORDS Transistor-transistor logic; optimization of manufacturing; decreasing of dimensions of transistor; analyti- cal approach for modelling 1. INTRODUCTION An actual and intensively solving problems of solid state electronics is increasing of integration rate of elements of integrated circuits (p-n-junctions, their systems et al) [1-8]. Increasing of the integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are using several approaches. They are widely using laser and microwave types of annealing of in- fused dopants. These types of annealing are also widely using for annealing of radiation defects, generated during ion implantation [9-17]. Using the approaches gives a possibility to increase integration rate of elements of integrated circuits through inhomogeneity of technological para- meters due to generating inhomogenous distribution of temperature. In this situation one can ob- tain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law [1,3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is doping of heterostructure by diffusion or ion implantation [1-3]. However in this case optimiza- tion of dopant and/or radiation defects is required [18]. In this paper we consider a heterostructure presented in Figs. 1. The heterostructure consist of a substrate and several epitaxial layers (see Figs. 1). Some sections have been manufactured in the epitaxial layers so as it is shown on Figs. 1. Further we consider doping of these sections by dif- fusion or ion implantation. The doping gives a possibility to manufacture transistors and p-n- junction so as it is shown on Figs. 1. The manufacturing gives a possibility to prepare element of transistor-transistor logic on Fig. 1a. After the considered doping dopant and/or radiation defects should be annealed. Framework the paper we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. Similar logical element has been considered in [19]. We introduce an approach to decrease dimensions of the element. However it is necessary to complicate technological process.
23

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Page 1: ON VERTICAL INTEGRATION FRAMEWORK ELEMENT OF TRANSISTOR TRANSISTOR LOGIC · transistor-transistor logic on Fig. 1 a. After the considered doping dopant and/or radiation defects should

International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

DOI : 10.5121/ijaceee.2015.3301 1

ON VERTICAL INTEGRATION FRAMEWORK

ELEMENT OF TRANSISTOR-TRANSISTOR LOGIC

E.L. Pankratov1, E.A. Bulaeva

1,2

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 increase vertical integration of elements of transistor-transistor

logic with function AND-NOT. Framework the approach we consider a heterostructure with special confi-

guration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation.

Annealing of dopant and/or radiation defects should be optimized.

KEYWORDS

Transistor-transistor logic; optimization of manufacturing; decreasing of dimensions of transistor; analyti-

cal approach for modelling

1. INTRODUCTION

An actual and intensively solving problems of solid state electronics is increasing of integration

rate of elements of integrated circuits (p-n-junctions, their systems et al) [1-8]. Increasing of the

integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are

using several approaches. They are widely using laser and microwave types of annealing of in-

fused dopants. These types of annealing are also widely using for annealing of radiation defects,

generated during ion implantation [9-17]. Using the approaches gives a possibility to increase

integration rate of elements of integrated circuits through inhomogeneity of technological para-

meters due to generating inhomogenous distribution of temperature. In this situation one can ob-

tain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law

[1,3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is

doping of heterostructure by diffusion or ion implantation [1-3]. However in this case optimiza-

tion of dopant and/or radiation defects is required [18].

In this paper we consider a heterostructure presented in Figs. 1. The heterostructure consist of a

substrate and several epitaxial layers (see Figs. 1). Some sections have been manufactured in the

epitaxial layers so as it is shown on Figs. 1. Further we consider doping of these sections by dif-

fusion or ion implantation. The doping gives a possibility to manufacture transistors and p-n-

junction so as it is shown on Figs. 1. The manufacturing gives a possibility to prepare element of

transistor-transistor logic on Fig. 1a. After the considered doping dopant and/or radiation defects

should be annealed. Framework the paper we analyzed dynamics of redistribution of dopant

and/or radiation defects during their annealing. Similar logical element has been considered in

[19]. We introduce an approach to decrease dimensions of the element. However it is necessary to

complicate technological process.

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

2

Fig. 1a. Composition element transistor-transistor logic. View from above. Black marked transistors and p-

n-junction manufactured by using doping of appropriate sections of the epitaxial layer. Dimensions of these

devices are decreased. Transistor 1 is a multiemitter transistor. Emitters have been marked by using letter E.

The index indicates their number in the multiemitter transistor. D1 and D2 mean dopants of p and n types in

p-n-junction. Red marked resistors (Ri) and wires have no decreasing of their dimensions

Fig. 1b. Heterostructure, which consist of a substrate and epitaxial layer with sections, manufactured by

using another materials. The figure shows integration of a multiemitter and homoemitter transistors. Dashed

lines are illustrated wires

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

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Substrate

Base

Collector

Emitter

Base

Collector

Emitter

Fig. 1c. Heterostructure, which consist of a substrate and epitaxial layer with sections, manufactured by

using another materials. The figure shows integration of two homoemitter transistors. Dashed lines are illu-

strated wires

2. METHOD OF SOLUTION

In this section we determine spatio-temporal distributions of concentrations of infused and im-

planted dopants. To determine these distributions we calculate appropriate solutions of the second

Fick's law [1,3,18]

( ) ( ) ( ) ( )

+

+

=

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxCCCC

∂ ,,,,,,,,,,,,. (1)

Boundary and initial conditions for the equations are

( )0

,,,

0

=∂

=xx

tzyxC,

( )0

,,,=

= xLxx

tzyxC,

( )0

,,,

0

=∂

=yy

tzyxC,

( )0

,,,=

= yLxy

tzyxC,

( )0

,,,

0

=∂

=zz

tzyxC,

( )0

,,,=

= zLxz

tzyxC, C (x,y,z,0)=f (x,y,z). (2)

The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is

the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion

coefficient could be changed with changing materials of heterostructure, with changing tempera-

ture of materials (including annealing), with changing concentrations of dopant and radiation de-

fects. We approximate dependences of dopant diffusion coefficient on parameters by the follow-

ing relation with account results in Refs. [20-22]

( ) ( )( )

( ) ( )

( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD LC ςςξ

γ

γ

. (3)

Here the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due to

Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes

the limit of solubility of dopant. Parameter γ ∈[1,3] describes average quantity of charged defects

interacted with atom of dopant [20]. The function V (x,y,z,t) describes the spatio-temporal distri-

bution of concentration of radiation vacancies. Parameter V* describes the equilibrium distribution

of concentration of vacancies. The considered concentrational dependence of dopant diffusion

coefficient has been described in details in [20]. It should be noted, that using diffusion type of

doping did not generation radiation defects. In this situation ζ1= ζ2= 0. We determine spatio-

temporal distributions of concentrations of radiation defects by solving the following system of

equations [21,22]

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

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( ) ( ) ( ) ( ) ( ) ( ) ×−

∂+

∂=

∂Tzyxk

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIIII

,,,,,,

,,,,,,

,,,,,,

,

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxITzyxD

ztzyxI

VII,,,,,,,,,

,,,,,,,,, ,

2 −

∂+× (4)

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂+

∂=

∂Tzyxk

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVVV ,,,

,,,,,,

,,,,,,

,,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxVTzyxD

ztzyxV

VIV,,,,,,,,,

,,,,,,,,,

,

2 −

∂+× .

Boundary and initial conditions for these equations are

( )0

,,,

0

=∂

=xx

tzyxρ,

( )0

,,,=

= xLxx

tzyxρ,

( )0

,,,

0

=∂

=yy

tzyxρ,

( )0

,,,=

= yLyy

tzyxρ,

( )0

,,,

0

=∂

=zz

tzyxρ,

( )0

,,,=

= zLzz

tzyxρ, ρ (x,y,z,0)=fρ (x,y,z). (5)

Here ρ =I,V. The function I (x,y,z,t) describes the spatio-temporal distribution of concentration of

radiation interstitials; Dρ(x,y,z,T) are the diffusion coefficients of point radiation defects; terms

V2(x,y,z,t) and I

2(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the

parameter of recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parame-

ters of generation of simplest complexes of point radiation defects.

Further we determine distributions in space and time of concentrations of divacancies ΦV(x,y,z,t)

and diinterstitials ΦI(x,y,z,t) by solving the following system of equations [21,22]

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkz

tzyxTzyxD

zIII

I

I,,,,,,,,,,,,

,,,,,, 2

,−+

Φ+ Φ

∂ (6)

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkz

tzyxTzyxD

zVVV

V

V,,,,,,,,,,,,

,,,,,, 2

,−+

Φ+ Φ ∂

∂.

Boundary and initial conditions for these equations are

( )0

,,,

0

=∂

Φ∂

=xx

tzyxρ,

( )0

,,,=

Φ∂

= xLxx

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=yy

tzyxρ,

( )0

,,,=

Φ∂

= yLyy

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=zz

tzyxρ,

( )0

,,,=

Φ∂

= zLzz

tzyxρ, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7)

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

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Here DΦρ(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects;

kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes.

We calculate distributions of concentrations of point radiation defects in space and time by re-

cently elaborated approach [18]. The approach based on transformation of approximations of dif-

fusion coefficients in the following form: Dρ(x,y,z,T)=D0ρ[1+ερ gρ(x,y,z,T)], where D0ρ are the av-

erage values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z,T)|≤1, ρ =I,V. We also used analogous

transformation of approximations of parameters of recombination of point defects and parameters

of generation of their complexes: kI,V(x,y,z,T)=k0I,V[1+εI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I [1+εI,I

gI,I(x,y,z,T)] and kV,V (x,y,z,T) = k0V,V [1+εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are the their average values,

0≤εI,V <1, 0≤εI,I <1, 0≤εV,V<1, | gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |gV,V(x,y,z,T)|≤1. Let us introduce

the following dimensionless variables: ( ) ( ) *,,,,,,~

ItzyxItzyxI = , χ = x/Lx, η = y /Ly,

( ) ( ) *,,,,,,~

VtzyxVtzyxV = , 2

00LtDD

VI=ϑ ,

VIVIDDkL

00,0

2=ω , VI

DDkL 00,0

2

ρρρ =Ω , φ =

z/Lz. The introduction leads to transformation of Eqs.(4) and conditions (5) to the following form

( ) ( )[ ] ( ) ( )[ ] ×+∂

∂+

∂+

∂=

∂Tg

ITg

DD

DIIIII

VI

I ,,,1,,,

~

,,,1,,,

~

00

0 φηχεηχ

ϑφηχφηχε

χϑ

ϑφηχ

( ) ( )[ ] ( ) ( ) ×−

∂+

∂+

∂× ϑφηχ

φ

ϑφηχφηχε

φη

ϑφηχ,,,

~,,,~

,,,1,,,

~

00

0

00

0 II

TgDD

D

DD

DIII

VI

I

VI

I

( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,~

,,,1,,,~

,,,1 2

,,,,ITgVTg

IIIIIVIVI+Ω−+× (8)

( ) ( )[ ] ( ) ( )[ ] ×+∂

∂+

∂+

∂=

∂Tg

VTg

DD

DVVVVV

VI

V,,,1

,,,~

,,,1,,,

~

00

0φηχε

ηχ

ϑφηχφηχε

χϑ

ϑφηχ

( ) ( )[ ] ( ) ( ) ×−

∂+

∂+

∂× ϑφηχ

φ

ϑφηχφηχε

φη

ϑφηχ,,,

~,,,~

,,,1,,,

~

00

0

00

0 IV

TgDD

D

DD

DVVV

VI

V

VI

V

( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,~

,,,1,,,~

,,,1 2

,,,,VTgVTg

VVVVVVIVI+Ω−+×

( )0

,,,~

0

=∂

=χχ

ϑφηχρ,

( )0

,,,~

1

=∂

=χχ

ϑφηχρ,

( )0

,,,~

0

=∂

=ηη

ϑφηχρ,

( )0

,,,~

1

=∂

=ηη

ϑφηχρ,

( )0

,,,~

0

=∂

=φφ

ϑφηχρ,

( )0

,,,~

1

=∂

=φφ

ϑφηχρ, ( )

( )*

,,,,,,~

ρ

ϑφηχϑφηχρ ρf

= . (9)

We determine solutions of Eqs.(8) with conditions (9) framework recently introduced approach

[18], i.e. as the power series

( ) ( )∑ ∑ ∑Ω=∞

=

=

=0 0 0

,,,~,,,~i j k

ijk

kji ϑφηχρωεϑφηχρ ρρ . (10)

Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain equa-

tions for initial-order approximations of concentration of point defects ( )ϑφηχ ,,,~

000I and

( )ϑφηχ ,,,~

000V and corrections for them ( )ϑφηχ ,,,

~ijk

I and ( )ϑφηχ ,,,~

ijkV , i ≥1, j ≥1, k ≥1. The equa-

tions are presented in the Appendix. Solutions of the equations could be obtained by standard

Fourier approach [24,25]. The solutions are presented in the Appendix.

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

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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 coef-

ficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the average

values of diffusion coefficients. In this situation the Eqs.(6) could be written as

( )( )[ ] ( )

( ) ( )++

Φ

+=Φ

ΦΦΦ tzyxITzyxkx

tzyxTzyxg

xD

t

tzyxII

I

III

I ,,,,,,,,,

,,,1,,, 2

,0∂

∂ε

( )[ ] ( )( )[ ] ( )

Φ

++

Φ

++ ΦΦΦΦΦΦz

tzyxTzyxg

zD

y

tzyxTzyxg

yD I

III

I

III∂

∂ε

∂ε

∂ ,,,,,,1

,,,,,,1 00

( ) ( )tzyxITzyxkI

,,,,,,−

( )( )[ ]

( )( ) ( ) ++

Φ

+=Φ

ΦΦΦ tzyxITzyxkx

tzyxTzyxg

xD

t

tzyxII

V

VVV

V ,,,,,,,,,

,,,1,,, 2

,0∂

∂ε

( )[ ] ( ) ( )[ ] ( )−

Φ

++

Φ

++ ΦΦΦΦΦΦz

tzyxTzyxg

zD

y

tzyxTzyxg

yD V

VVV

V

VVV∂

∂ε

∂ε

∂ ,,,,,,1

,,,,,,1

00

( ) ( )tzyxITzyxkI

,,,,,,− .

Farther we determine solutions of above equations as the following power series

( ) ( )∑ Φ=Φ∞

0

,,,,,,i

i

itzyxtzyx ρρρ ε . (11)

Now we used the series (11) into Eqs.(6) and appropriate boundary and initial conditions. The

using gives the possibility to obtain equations for initial-order approximations of concentrations

of complexes of defects Φρ0(x,y,z,t), corrections for them Φρi(x,y,z,t) (for them i ≥1) and boundary

and initial conditions for them. We remove equations and conditions to the Appendix. Solutions

of the equations have been calculated by standard approaches [24,25] and presented in the Ap-

pendix.

Now we calculate distribution of concentration of dopant in space and time by using the ap-

proach, which was used for analysis of radiation defects. To use the approach we consider follow-

ing transformation of approximation of dopant diffusion coefficient: DL(x,y,z,T)=D0L[1+

εLgL(x,y,z,T)], where D0L is the average value of dopant diffusion coefficient, 0≤εL< 1,

|gL(x,y,z,T)|≤1. Farther we consider solution of Eq.(1) as the following series:

( ) ( )∑ ∑=∞

=

=0 1

,,,,,,i j

ij

ji

LtzyxCtzyxC ξε .

Using the relation into Eq.(1) and conditions (2) leads to obtaining equations for the functions

Cij(x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions for them. The equations are presented in

the Appendix. Solutions of the equations have been calculated by standard approaches (see, for

example, [24,25]). The solutions are presented in the Appendix.

We analyzed distributions of concentrations of dopant and radiation defects in space and time

analytically by using the second-order approximations on all parameters, which have been used in

appropriate series. Usually the second-order approximations are enough good approximations to

make qualitative analysis and to obtain quantitative results. All analytical results have been

checked by numerical simulation.

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

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3. DISCUSSION

In this section we analyzed spatio-temporal distributions of concentrations of dopants. Figs. 2

shows typical spatial distributions of concentrations of dopants in neighborhood of interfaces of

heterostructures. We calculate these distributions of concentrations of dopants under the follow-

ing condition: value of dopant diffusion coefficient in doped area is larger, than value of dopant

diffusion coefficient in nearest areas. In this situation one can find increasing of sharpness of p-n-

junctions with increasing of homogeneity of distribution of concentration of dopant at one time.

These both effects could be obtained in both situations, when p-n-junctions are single and frame-

work their systems (transistors, thyristors). Changing relation between values of dopant diffusion

coefficients leads to opposite result (see Figs. 3).

Fig. 2a. Dependences of concentration of dopant, infused in heterostructure from Figs. 1, on coordinate in

direction, which is perpendicular to interface between epitaxial layer substrate. Difference between values

of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves.

Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coeffi-

cient in the substrate

x

0.0

0.5

1.0

1.5

2.0

C(x

,Θ)

23

4

1

0 L/4 L/2 3L/4 L

Epitaxial layer Substrate

Fig. 2b. Dependences of concentration of dopant, implanted in heterostructure from Figs. 1, on coordinate

in direction, which is perpendicular to interface between epitaxial layer substrate. Difference between val-

ues of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of

curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion

coefficient in the substrate. Curve 1 corresponds to homogenous sample and annealing time Θ = 0.0048

(Lx2+Ly

2+Lz

2)/D0. Curve 2 corresponds to homogenous sample and annealing time Θ = 0.0057 (Lx

2+Ly

2+

Lz2)/D0. Curves 3 and 4 correspond to heterostructure from Figs. 1; annealing times Θ = 0.0048 (Lx

2+Ly

2+

Lz2)/D0 and Θ = 0.0057 (Lx

2+Ly2+ Lz

2)/D0, respectively

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

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Fig.3a. Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostruc-

ture from Figs. 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure.

Difference between values of dopant diffusion coefficients increases with increasing of number of curves.

Value of dopant diffusion coefficient in this section is smaller, than value of dopant diffusion coefficient in

nearest sections

x

0.00000

0.00001

0.00010

0.00100

0.01000

0.10000

1.00000

C(x

, Θ)

fC(x)

L/40 L/2 3L/4 Lx0

1

2

Substrate

Epitaxial layer 1

Epitaxial layer 2

Fig.3b. Calculated distributions of implanted dopant in epitaxial layers of heterostructure. Solid lines are

spatial distributions of implanted dopant in system of two epitaxial layers. Dushed lines are spatial distribu-

tions of implanted dopant in one epitaxial layer. Annealing time increases with increasing of number of

curves

It should be noted, that framework the considered approach one shall optimize annealing of do-

pant and/or radiation defects. To do the optimization we used recently introduced criterion [26-

34]. The optimization based on approximation real distribution by step-wise function ψ (x,y, z)

(see Figs. 4). Farther the required values of optimal annealing time have been calculated by mi-

nimization the following mean-squared error

( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L

zyx

xdydzdzyxzyxCLLL

U0 0 0

,,,,,1

ψ . (12)

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

9

C(x

,Θ)

0 Lx

2

13

4

Fig.4a. Distributions of concentration of infused dopant in depth of heterostructure from Fig. 1 for different

values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of num-

ber of curve corresponds to increasing of annealing time

x

C(x

,Θ)

1

23

4

0 L

Fig.4b. Distributions of concentration of implanted dopant in depth of heterostructure from Fig. 1 for dif-

ferent values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of

number of curve corresponds to increasing of annealing time

We show optimal values of annealing time as functions of parameters on Figs. 5. It is known, that

standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal

case after finishing the annealing dopant achieves interface between layers of heterostructure. If

the dopant has no enough time to achieve the interface, it is practicably to anneal the dopant addi-

tionally. The Fig. 5b shows the described dependences of optimal values of additional annealing

time for the same parameters as for Fig. 5a. Necessity to anneal radiation defects leads to smaller

values of optimal annealing of implanted dopant in comparison with optimal annealing time of

infused dopant.

0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ

0.0

0.1

0.2

0.3

0.4

0.5

Θ D

0 L

-2

3

2

4

1

Fig.5a. Dimensionless optimal annealing time of infused dopant as a function of several parameters. Curve

1 describes the dependence of the annealing time on the relation a/L and ξ = γ = 0 for equal to each other

values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence of

the annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of

the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 describes the dependence of

the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

10

0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ

0.00

0.04

0.08

0.12

Θ D

0 L

-2

3

2

4

1

Fig.5b. Dimensionless optimal annealing time of implanted dopant as a function of several parameters.

Curve 1 describes the dependence of the annealing time on the relation a/L and ξ = γ = 0 for equal to each

other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence

of the annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence

of the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 describes the dependence of

the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0

4. CONCLUSIONS

In this paper we introduce an approach of vertical integration framework element of transistor-

transistor logic. The approach gives us possibility to decrease area of the elements with smaller

increasing of the element’s thickness.

ACKNOWLEDGEMENTS

This work is supported by the agreement of August 27, 2013 02.В.49.21.0003 between The

Ministry of education and science of the Russian Federation and Lobachevsky State University of

Nizhni Novgorod and educational fellowship for scientific research of Government of Russian

and of Nizhny Novgorod State University of Architecture and Civil Engineering.

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[14] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the re-

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Russian).

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[26] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion-

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[27] E.L. Pankratov. Redistribution of dopant during annealing of radiative defects in a multilayer struc-

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P. 187–197 (2008).

[28] E.L. Pankratov. Decreasing of depth of implanted-junction rectifier in semiconductor heterostruc-

ture by optimized laser annealing. J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010).

[29] 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).

[30] E.L. Pankratov, E.A. Bulaeva. Optimization of doping of heterostructure during manufacturing of p-

i-n-diodes. Nanoscience and Nanoengineering. Vol. 1 (1). P. 7-14 (2013).

[31] 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).

[32] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture a heterobipolar transistors in thin film

structures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31

(2014).

[33] E.L. Pankratov, E.A. Bulaeva. Application of native inhomogeneities to increase compactness of

vertical field-effect transistors. J. Nanoengineering and Nanomanufacturing. Vol. 2 (3). P. 275-280

(2012).

[34] E.L. Pankratov, E.A. Bulaeva. Influence of drain of dopant on distribution of dopant in diffusion-

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

12

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 110 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 74 published papers in area of her researches.

APPENDIX

Equations for the functions ( )ϑφηχ ,,,~

ijkI and ( )ϑφηχ ,,,

~ijk

V , i ≥0, j ≥0, k ≥0 and conditions for

them

( ) ( ) ( ) ( )2

000

2

0

0

2

000

2

0

0

2

000

2

0

0000,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ

∂+

∂+

∂=

∂ I

D

DI

D

DI

D

DI

V

I

V

I

V

I

( ) ( ) ( ) ( )2

000

2

0

0

2

000

2

0

0

2

000

2

0

0000,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ

∂+

∂+

∂=

∂ V

D

DV

D

DV

D

DV

I

V

I

V

I

V ;

( ) ( ) ( ) ( )( )

×

∂+

∂+

∂+

∂=

∂Tg

III

D

DII

iii

V

Ii ,,,,,,

~,,,

~,,,

~,

~

2

00

2

2

00

2

2

00

2

0

000 φηχχφ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑχ

( )( )

( )( )

×

∂+

∂+

∂× −− Tg

ITg

D

D

D

DII

i

I

V

I

V

Ii ,,,,,,

~

,,,,,,

~100

0

0

0

0100 φηχφη

ϑφηχφηχ

ηχ

ϑφηχ

( )

V

Ii

D

DI

0

0100,,,

~

∂× −

φ

ϑφηχ, i ≥1,

( ) ( ) ( ) ( )( )

×

∂+

∂+

∂+

∂=

∂Tg

VVV

D

DVV

iii

I

Vi ,,,,,,

~,,,

~,,,

~,

~

2

00

2

2

00

2

2

00

2

0

000 φηχχφ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑχ

( )( )

( )( )

×

∂+

∂+

∂× −− Tg

VTg

D

D

D

DVV

i

V

I

V

I

Vi ,,,,,,

~

,,,,,,

~100

0

0

0

0100 φηχφη

ϑφηχφηχ

ηχ

ϑφηχ

( )

I

Vi

D

DV

0

0100,,,

~

∂× −

φ

ϑφηχ, i ≥1,

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

13

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

010

2

2

010

2

2

010

2

0

0010,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~

,,,~

,,,1 000000,, VITgVIVI

+−

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

010

2

2

010

2

2

010

2

0

0010,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~

,,,~

,,,1000000,,

VITgVIVI

+− ;

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

020

2

2

020

2

2

020

2

0

0020 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,~

,,,~

,,,~

,,,~

,,,1010000000010,,

VIVITgVIVI

++−

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

020

2

2

020

2

2

020

2

0

0020,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

V

I

( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,~

,,,~

,,,~

,,,~

,,,1010000000010,,

VIVITgVIVI

++− ;

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

001

2

2

001

2

2

001

2

0

0001,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( )ϑφηχφηχε ,,,~

,,,1 2

000,,ITg

IIII+−

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

001

2

2

001

2

2

001

2

0

0001 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( )[ ] ( )ϑφηχφηχε ,,,~

,,,1 2

000,,VTg

IIII+− ;

( ) ( ) ( ) ( )×+

∂+

∂+

∂=

V

I

V

I

D

DIII

D

DI

0

0

2

110

2

2

110

2

2

110

2

0

0110 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ

( )( )

( )( )

( )[

×∂

∂+

∂+

∂× Tg

ITg

ITg III ,,,

,,,~

,,,,,,

~

,,, 010010 φηχφη

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

( )( ) ( ) ( ) ( )[ ] ×+−

∂× ϑφηχϑφηχϑφηχϑφηχ

φ

ϑφηχ,,,

~,,,

~,,,

~,,,

~,,,~

100000000100

010 VIVII

( )[ ]Tg IIII ,,,1 ,, φηχε+×

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

14

( ) ( ) ( ) ( )×+

∂+

∂+

∂=

I

V

I

V

D

DVVV

D

DV

0

0

2

110

2

2

110

2

2

110

2

0

0110 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ

( )( )

( )( )

( )[

×∂

∂+

∂+

∂× Tg

VTg

VTg

IVV,,,

,,,~

,,,,,,

~

,,, 010010 φηχφη

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

( )( ) ( ) ( ) ( )[ ] ×+−

∂× ϑφηχϑφηχϑφηχϑφηχ

φ

ϑφηχ,,,

~,,,

~,,,

~,,,

~,,,~

100000000100

010 IVIVV

( )[ ]Tg VVVV ,,,1 ,, φηχε+× ;

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

002

2

2

002

2

2

002

2

0

0002,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~

,,,~

,,,1000001,,

IITgIIII

+−

( ) ( ) ( ) ( )−

∂+

∂+

∂=

∂2

002

2

2

002

2

2

002

2

0

0002 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~

,,,~

,,,1000001,,

VVЕgVVVV

+− ;

( ) ( ) ( ) ( )+

∂+

∂+

∂=

∂2

101

2

2

101

2

2

101

2

0

0101,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( ) ( ) ( ) ( )

+

∂+

∂+

η

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

,,,~

,,,,,,

~

,,, 001001

0

0I

TgI

TgD

DII

V

I

( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεφ

ϑφηχφηχ

φ,,,

~,,,

~,,,1

,,,~

,,,000100

001 VITgI

TgIII

+−

∂+

( ) ( ) ( ) ( )+

∂+

∂+

∂=

∂2

101

2

2

101

2

2

101

2

0

0101,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( ) ( ) ( ) ( )

+

∂+

∂+

η

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

,,,~

,,,,,,

~

,,, 001001

0

0V

TgV

TgD

DVV

I

V

( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεφ

ϑφηχφηχ

φ,,,

~,,,

~,,,1

,,,~

,,,100000

001 VITgV

TgVVV

+−

∂+ ;

( ) ( ) ( ) ( )( ) ×−

∂+

∂+

∂=

∂ϑφηχ

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ,,,

~,,,~

,,,~

,,,~

,,,~

0102

011

2

2

011

2

2

011

2

0

0011 IIII

D

DI

V

I

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

15

( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,,~

,,,~

,,,1,,,~

,,,1000001,,000,,

VITgITgVIVIIIII

+−+×

( ) ( ) ( ) ( )( ) ×−

∂+

∂+

∂=

∂ϑφηχ

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ,,,

~,,,~

,,,~

,,,~

,,,~

0102

011

2

2

011

2

2

011

2

0

0011 VVVV

D

DV

I

V

( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,,~

,,,~

,,,1,,,~

,,,1001000,,000,,

VItgVTgVIVIVVVV

+−+× ;

( )0

,,,~

0

=∂

=x

ijk

χ

ϑφηχρ,

( )0

,,,~

1

=∂

=x

ijk

χ

ϑφηχρ,

( )0

,,,~

0

=∂

=ηη

ϑφηχρijk

, ( )

0,,,~

1

=∂

=ηη

ϑφηχρijk

,

( )0

,,,~

0

=∂

=φφ

ϑφηχρijk

, ( )

0,,,~

1

=∂

=φφ

ϑφηχρijk

(i ≥0, j ≥0, k ≥0);

( ) ( ) *

000,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρ

ijk (i ≥1, j ≥1, k ≥1).

Solutions of the above equations could be written as

( ) ( ) ( ) ( ) ( )∑+=∞

=1000

21,,,~

nnn

ecccFLL

ϑφηχϑφηχρ ρρ ,

where ( ) ( ) ( ) ( )∫ ∫ ∫=1

0

1

0

1

0*

,,coscoscos1

udvdwdwvufwnvnunFnn ρρ πππ

ρ, ( ) ( )

IVnIDDne 00

22exp ϑπϑ −= ,

cn(χ) = cos (π n χ), ( ) ( )VInV

DDne00

22exp ϑπϑ −= ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫∂

∂−−=

=

1 0

1

0

1

0

1

0

100

0

0

00

,,,~

2,,,~

n

i

nnnInIn

V

I

iu

wvuIvcuseecccn

D

DI

ϑ ττϑφηχπϑφηχ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞

=1 0

1

0

1

00

02,,,n

nnnInIn

V

I

Invsuceecccn

D

DdudvdwdTwvugwc

ϑ

τϑφηχπτ

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫∂

∂×

=

1 00

01

0

100 2,,,

~

,,,n

nInIn

V

Ii

In eecccnD

Ddudvdwd

v

wvuITwvugwc

ϑ

τϑφηχπττ

( ) ( ) ( ) ( )( )

∫ ∫ ∫∂

∂× −

1

0

1

0

1

0

100,,,

~

,,, ττ

dudvdwdw

wvuITwvugwsvcuc i

Innn, i ≥1,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

00,,,2,,,

~

nVnnnInVn

I

V

iTwvugvcuseecccn

D

DV

ϑ

τϑφηχπϑφηχ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−∂

∂×

=

1 0

1

0

1

00

0100,

~

nnnnInVn

I

Vi

nvsuceecccn

D

Ddudvdwd

u

uVwc

ϑ

τϑφηχττ

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

16

( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫∂

∂×

=

10

01

0

100 2,

~

,,,2n

nVn

I

Vi

Vnecccn

D

Ddudvdwd

v

uVTwvugwc ϑφηχπτ

τπ

( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

∂−× −

ϑ

ττ

τ0

1

0

1

0

1

0

100,

~

,,, dudvdwdw

uVTwvugwsvcuce i

VnnnnI, i ≥1,

where sn(χ) = sin (π n χ);

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞

=1 0

1

0

1

0

1

0010

2,,,~n

nnnnnnnnwcvcuceeccc

ϑ

ρρ τϑφηχϑφηχρ

( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvugVIVI

,,,~

,,,~

,,,1000000,,

+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ×∑ ∫ ∫ ∫ ∫ +−−=∞

=1 0

1

0

1

0

1

0,

0

0

02012,,,~

nVInnnnnnnn

V

I wcvcuceecccD

D ϑ

ρρ ετϑφηχϑφηχρ

( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvugVI

,,,~

,,,~

,,,~

,,,~

,,,010000000010,

+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

0001 2,,,~

nnnnnnnnn

wcvcuceecccϑ

ρρ τϑφηχϑφηχρ

( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 2

000,,+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=∞

=1 0

1

0

1

0

1

0002

2,,,~n

nnnnnnnnwcvcuceeccc

ϑ

ρρ τϑφηχϑφηχρ

( )[ ] ( ) ( ) ττρτρε ρρρρ dudvdwdwvuwvuTwvug ,,,~,,,~,,,1000001,,

+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

1102,,,

~

nnnnnInInnn

V

I ucvcuseecccnD

DI

ϑ

τϑφηχπϑφηχ

( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

=

10

0100 2,,,

~

,,,n

nInnn

V

Ii

Iecccn

D

Ddudvdwd

u

wvuITwvug ϑφηχπτ

τ

( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂−× −

V

Ii

InnnnID

Ddudvdwd

v

wvuITwvugucvsuce

0

0

0

1

0

1

0

1

0

100 2,,,

~

,,, πττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∂−×

=

1 0

1

0

1

0

1

0

100,,,

~

,,,n

i

InnnnInIdudvdwd

w

wvuITwvugusvcuceen

ϑ

ττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×∞

=1 0

1

0

1

0

1

0,

12n

VInnnnInnnInnnnvcvcuceccecccc

ϑ

ετφηϑχφηχ

( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvugVI

,,,~

,,,~

,,,~

,,,~

,,,100000000100,

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

17

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

1102,,,

~

nnnnnVnVnnn

I

V ucvcuseecccnD

DV

ϑ

τϑφηχπϑφηχ

( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

=

10

0100 2,,,

~

,,,n

nVnnn

I

Vi

Vecccn

D

Ddudvdwd

u

wvuVTwvug ϑφηχπτ

τ

( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂−× −

I

Vi

VnnnnVD

Ddudvdwd

v

wvuVTwvugucvsuce

0

0

0

1

0

1

0

1

0

100 2,,,

~

,,, πττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∂−×

=

1 0

1

0

1

0

1

0

100,,,

~

,,,n

i

VnnnnVnVdudvdwd

w

wvuVTwvugusvcuceen

ϑ

ττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−−×∞

=1 0

1

0

1

0

1

0,,

,,,12n

VIVInnnVnnnInnnnTwvugvcuceccecccc

ϑ

ετφηϑχφηχ

( ) ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuIwcn

,,,~

,,,~

,,,~

,,,~

100000000100+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

101,,,2,,,

~

nInnnInInnn

V

I TwvugvcuseecccnD

DI

ϑ

τϑφηχπϑφηχ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

=1 0

1

00

0001 2,,,

~

nnnInInnn

V

I

nuceecccn

D

Ddudvdwd

u

wvuIwc

ϑ

τϑφηχπττ

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ×∑−∫ ∫∂

∂×

=10

01

0

1

0

001 2,,,

~

,,,n

nnnnI

V

I

Inncccen

D

Ddudvdwd

v

wvuITwvugwcvs φηχϑπτ

τ

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫∂

∂−×

=10

1

0

1

0

1

0

001 2,,,

~

,,,n

nnnInnnnIcccdudvdwd

w

wvuITwvugwsvcuce φηχτ

ττ

ϑ

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−×ϑ

τττετϑ0

1

0

1

0

1

0000100,,

,,,~

,,,~

,,,1 dudvdwdwvuVwvuITwvugwcvcuceeVIVInnnnInI

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

101,,,2,,,

~

nVnnnVnVnnn

I

V TwvugvcuseecccnD

DV

ϑ

τϑφηχπϑφηχ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

=1 0

1

00

0001 2,,,

~

nnnVnInnn

I

V

nuceecccn

D

Ddudvdwd

u

wvuVwc

ϑ

τϑφηχπττ

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ×∑−∫ ∫∂

∂×

=10

01

0

1

0

001 2,,,

~

,,,n

nnnnV

I

V

Vnncccen

D

Ddudvdwd

v

wvuVTwvugwcvs φηχϑπτ

τ

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫∂

∂−×

=10

1

0

1

0

1

0

001 2,,,

~

,,,n

nnnVnnnnVcccdudvdwd

w

wvuVTwvugwsvcuce φηχτ

ττ

ϑ

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

18

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−×ϑ

τττετϑ0

1

0

1

0

1

0000100,,

,,,~

,,,~

,,,1 dudvdwdwvuVwvuITwvugwcvcuceeVIVInnnnVnV

;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞

=1 0

1

0

1

0

1

0000011

,,,~

2,,,~

nnnnnInInnn

wvuIwcvcuceecccIϑ

ττϑφηχϑφηχ

( )[ ] ( ) ( )[ ] ( ) ( ) τττετε dudvdwdwvuVwvuITwvugwvuITwvugVIVIIIII

,,,~

,,,~

,,,1,,,~

,,,1000001,,010,,

+++×

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞

=1 0

1

0

1

0

1

0000011

,,,~

2,,,~

nnnnnVnVnnn

wvuIwcvcuceecccVϑ

ττϑφηχϑφηχ

( )[ ] ( ) ( )[ ] ( ) ( ) τττετε dudvdwdwvuVwvuITwvugwvuITwvugVIVIVVVV

,,,~

,,,~

,,,1,,,~

,,,1000001,,010,,

+++× .

Equations for functions Φρi(x,y,z,t), i ≥0 to describe concentrations of simplest complexes of radi-

ation defects.

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

0

2

2

0

2

2

0

2

0

0,,,,,,,,,,,,

z

tzyx

y

tzyx

x

tzyxD

t

tzyxIII

I

I

( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkIII

,,,,,,,,,,,, 2

,−+

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

0

2

2

0

2

2

0

2

0

0,,,,,,,,,,,,

z

tzyx

y

tzyx

x

tzyxD

t

tzyxVVV

V

V

( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkVVV

,,,,,,,,,,,, 2

,−+ ;

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

2

2

2

2

2

0

,,,,,,,,,,,,

z

tzyx

y

tzyx

x

tzyxD

t

tzyxiIiIiI

I

iI

( )( )

( )( )

+

Φ+

Φ+

Φ

ΦΦy

tzyxTzyxg

yx

tzyxTzyxg

xD

iI

I

iI

II∂

∂ ,,,,,,

,,,,,,

11

0

( )( )

Φ+

Φz

tzyxTzyxg

z

iI

I∂

∂ ,,,,,,

1, i≥1,

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

2

2

2

2

2

0

,,,,,,,,,,,,

z

tzyx

y

tzyx

x

tzyxD

t

tzyxiViViV

V

iV

( )( )

( )( )

+

Φ+

Φ+

Φ

ΦΦy

tzyxTzyxg

yx

tzyxTzyxg

xD

iV

V

iV

VV∂

∂ ,,,,,,

,,,,,,

11

0

( )( )

Φ+

Φz

tzyxTzyxg

z

iV

V∂

∂ ,,,,,,

1, i≥1;

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

19

Boundary and initial conditions for the functions takes the form

( )0

,,,

0

=∂

Φ∂

=x

i

x

tzyxρ,

( )0

,,,=

Φ∂

= xLx

i

x

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=y

i

y

tzyxρ,

( )0

,,,=

Φ∂

= yLy

i

y

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=z

i

z

tzyxρ,

( )0

,,,=

Φ∂

= zLz

i

z

tzyxρ, i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.

Solutions of the above equations could be written as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ∞

=

=ΦΦ

110

221,,,

nnnn

nnnnnn

zyxzyx

zcycxcnL

tezcycxcFLLLLLL

tzyxρρρ

( ) ( ) ( ) ( ) ( ) ( ) ( )[∫ ∫ ∫ ∫ −−× ΦΦ

t L L L

IInnnnn

x y z

wvuITwvukwcvcucete0 0 0 0

2

, ,,,,,, ττρρ

( ) ( )] ττ dudvdwdwvuITwvukI

,,,,,,− ,

where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ

x y zL L L

nnnnudvdwdwvufwcvcucF

0 0 0

,,ρρ

, ( )

++−= ΦΦ 2220

22 111exp

zyx

nLLL

tDnteρρ

π ,

cn(x) = cos (π n x/Lx);

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ∞

=ΦΦΦ

1 0 0 0 02

,,,2

,,,n

t L L L

nnnnnnn

zyx

i

x y z

TwvugvcusetezcycxcnLLL

tzyxρρρ

τπ

ρ

( )( )

( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−Φ

×∞

=ΦΦ

1 02

1 2,,,

n

t

nnnnn

zyx

iI

netezcycxcn

LLLdudvdwd

u

wvuwc τ

πτ

τ∂ρρ

ρ

( ) ( ) ( ) ( ) ( )( )

×∑−∫ ∫ ∫ ∫Φ

−×∞

=

ΦΦ1

20 0 0 0

1 2,,,,,,

nzyx

t L L LiI

nnnnn

LLLdudvdwd

v

wvuTwvugwcvsuce

x y z πτ

τ∂τ ρ

ρρ

( ) ( ) ( ) ( ) ( )( )

( ) ×∫ ∫ ∫ ∫Φ

−× Φ

ΦΦ

t L L LiI

nnnnn

x y z

dudvdwdTwvugw

wvuwsvcucete

0 0 0 0

1

,,,,,,

τ∂

τ∂τ

ρ

ρ

ρρ

( ) ( ) ( )zcycxcnnn

× , i ≥1,

where sn(x) = sin (π n x/Lx).

Equations for the functions Cij(x,y,z,t) (i ≥0, j ≥0), boundary and initial conditions could be writ-

ten as

( ) ( ) ( ) ( )2

00

2

02

00

2

02

00

2

0

00 ,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

∂+

∂+

∂=

∂;

( ) ( ) ( ) ( )+

∂+

∂+

∂=

∂2

0

2

2

0

2

2

0

2

0

0,,,,,,,,,,,,

z

tzyxC

y

tzyxC

x

tzyxCD

t

tzyxCiii

L

i

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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

20

( )( )

( )( )

+

∂+

∂+ −−

y

tzyxCTzyxg

yD

x

tzyxCTzyxg

xD i

LL

i

LL

,,,,,,

,,,,,, 10

0

10

0

( ) ( )

∂+ −

z

tzyxCTzyxg

zD i

LL

,,,,,, 10

0, i ≥1;

( ) ( ) ( ) ( )+

∂+

∂+

∂=

∂2

01

2

02

01

2

02

01

2

0

01 ,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

( )( )

( ) ( )( )

( )+

∂+

∂+

y

tzyxC

TzyxP

tzyxC

yD

x

tzyxC

TzyxP

tzyxC

xD

LL

,,,

,,,

,,,,,,

,,,

,,,0000

0

0000

0 γ

γ

γ

γ

( )( )

( )

∂+

z

tzyxC

TzyxP

tzyxC

zD

L

,,,

,,,

,,, 0000

0 γ

γ

;

( ) ( ) ( ) ( )+

∂+

∂+

∂=

∂2

02

2

02

02

2

02

02

2

0

02 ,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

( )( )

( )( )

( )( )

( )

×

∂+

∂+

−−

TzyxP

tzyxCtzyxC

yx

tzyxC

TzyxP

tzyxCtzyxC

xD

L,,,

,,,,,,

,,,

,,,

,,,,,,

1

00

01

00

1

00

010 γ

γ

γ

γ

( )( )

( )( )

( )+

∂+

∂×

z

tzyxC

TzyxP

tzyxCtzyxC

zy

tzyxC ,,,

,,,

,,,,,,

,,, 00

1

00

01

00

γ

γ

( )( )

( )( )

( ) ( )( )

×

∂+

∂+

∂×

TzyxP

tzyxC

xD

z

tzyxC

TzyxP

tzyxCtzyxC

zy

tzyxCL

,,,

,,,,,,

,,,

,,,,,,

,,, 00

0

00

1

00

01

00

γ

γ

γ

γ

( ) ( )( )

( ) ( )( )

( )

∂+

∂+

∂×

z

tzyxC

TzyxP

tzyxC

zy

tzyxC

TzyxP

tzyxC

yx

tzyxC ,,,

,,,

,,,,,,

,,,

,,,,,,0100010001

γ

γ

γ

γ

;

( ) ( ) ( ) ( )+

∂+

∂+

∂=

∂2

11

2

02

11

2

02

11

2

0

11,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

( ) ( )( )

( ) ( ) ( )( )

×

∂+

∂+

−−

TzyxP

tzyxCtzyxC

yx

tzyxC

TzyxP

tzyxCtzyxC

x ,,,

,,,,,,

,,,

,,,

,,,,,,

1

00

10

00

1

00

10 γ

γ

γ

γ

( ) ( ) ( )( )

( )+

∂+

∂×

LD

z

tzyxC

TzyxP

tzyxCtzyxC

zy

tzyxC0

00

1

00

10

00 ,,,

,,,

,,,,,,

,,,γ

γ

( )( )

( ) ( )( )

( )

+

∂+

∂+

y

tzyxC

TzyxP

tzyxC

yx

tzyxC

TzyxP

tzyxC

xD

L

,,,

,,,

,,,,,,

,,,

,,,10001000

0 γ

γ

γ

γ

Page 21: ON VERTICAL INTEGRATION FRAMEWORK ELEMENT OF TRANSISTOR TRANSISTOR LOGIC · transistor-transistor logic on Fig. 1 a. After the considered doping dopant and/or radiation defects should

International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

21

( )( )

( ) ( ) ( )

+

∂+

∂+

x

tzyxCTzyxg

xD

z

tzyxC

TzyxP

tzyxC

zLL

,,,,,,

,,,

,,,

,,,01

0

1000

γ

γ

( ) ( ) ( ) ( )

∂+

∂+

z

tzyxCTzyxg

zy

tzyxCTzyxg

yLL

,,,,,,

,,,,,, 0101 ;

( )0

,,,

0

==x

ij

x

tzyxC

∂,

( )0

,,,=

= xLx

ij

x

tzyxC

∂,

( )0

,,,

0

==y

ij

y

tzyxC

∂,

( )0

,,,=

= yLy

ij

y

tzyxC

∂,

( )0

,,,

0

==z

ij

z

tzyxC

∂,

( )0

,,,=

= zLz

ij

z

tzyxC

∂, i ≥0, j ≥0;

C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i ≥1, j ≥1.

Functions Cij(x,y,z,t) (i ≥0, j ≥0) could be approximated by the following series during solutions

of the above equations

( ) ( ) ( ) ( ) ( )∑+=∞

=100

21,,,

nnCnnnnC

zyxzyx

tezcycxcFLLLLLL

tzyxC .

Here ( )

++−=

2220

22 111exp

zyx

CnCLLL

tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=x y zL L L

nCnnnCudvdwdwcwvufvcucF

0 0 0

,, ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 020

,,,2

,,,n

t L L L

LnnnCnCnnnnC

zyx

i

x y z

TwvugvcusetezcycxcFnLLL

tzyxC τπ

( )( )

( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∂

∂×

=

1 02

10 2,,,

n

t

nCnCnnnnC

zyx

i

netezcycxcFn

LLLdudvdwd

u

wvuCwc τ

πτ

τ

( ) ( ) ( ) ( )( )

( )∑ ×−∫ ∫ ∫∂

∂×

=

12

0 0 0

10 2,,,,,,

nnCnC

zyx

L L Li

LnnnteFn

LLLdudvdwd

v

wvuCTwvugvcvsuc

x y z πτ

τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

∫ ∫ ∫ ∫∂

∂−× −

t L L Li

LnnnnCnnn

x y z

dudvdwdw

wvuCTwvugvsvcucezcycxc

0 0 0 0

10 ,,,,,, τ

ττ , i ≥1;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0201

2,,,

n

t L L L

nnnnCnCnnnnC

zyx

x y z

wcvcusetezcycxcFnLLL

tzyxC τπ

( )( )

( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

=12

0000 2,,,

,,,

,,,

nnCnnnnC

zyx

tezcycxcFnLLL

dudvdwdu

wvuC

TwvuP

wvuC πτ

ττγ

γ

( ) ( ) ( ) ( )( )( )

( )( )×∑−∫ ∫ ∫ ∫

∂−×

=12

0 0 0 0

0000 2,,,

,,,

,,,

nnC

zyx

t L L L

nnnnCten

LLLdudvdwd

v

wvuC

TwvuP

wvuCwcvsuce

x y z πτ

τττ

γ

γ

Page 22: ON VERTICAL INTEGRATION FRAMEWORK ELEMENT OF TRANSISTOR TRANSISTOR LOGIC · transistor-transistor logic on Fig. 1 a. After the considered doping dopant and/or radiation defects should

International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

22

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )

( )∫ ∫ ∫ ∫

∂−×

t L L L

nnnnCnnnnC

x y z

dudvdwdw

wvuC

TwvuP

wvuCwsvcucezcycxcF

0 0 0 0

0000,,,

,,,

,,,τ

τττ

γ

γ

;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0202

2,,,

n

t L L L

nnnnCnCnnnnC

zyx

x y z

wcvcusetezcycxcFnLLL

tzyxC τπ

( ) ( )( )

( ) ( ) ( )×∑−∂

∂×

=

12

00

1

00

01

2,,,

,,,

,,,,,,

nnnnC

zyx

ycxcFLLL

dudvdwdu

wvuC

TwvuP

wvuCwvuC

πτ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )×∫ ∫ ∫ ∫

∂−×

−t L L L

nnnCnCn

x y z

v

wvuC

TwvuP

wvuCwvuCvsucetezcn

0 0 0 0

00

1

00

01

,,,

,,,

,,,,,,

ττττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞

=1 0 0 02

2

n

t L L

nnnCnCnnnnC

zyx

n

x y

vcucetezcycxcFnLLL

dudvdwdwc τπ

τ

( ) ( ) ( )( )

( ) ( ) ×∑−∫∂

∂×

=

12

0

00

1

00

01

2,,,

,,,

,,,,,,

nn

zyx

L

nxcn

LLLdudvdwd

w

wvuC

TwvuP

wvuCwvuCws

z πτ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫

∂−×

t L L L

nnnnCnCnnnC

x y z

u

wvuCwvuCwcvcusetezcycF

0 0 0 0

00

01

,,,,,,

τττ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×∞

=

1 0 02

1

00 2

,,,

,,,

n

t L

nnCnCnnnnC

zyx

x

ucetezcycxcFnLLL

dudvdwdTwvuP

wvuCτ

πτ

τγ

γ

( ) ( ) ( ) ( )( )

( )×∑−∫ ∫

∂×

=

12

0 0

00

1

00

01

2,,,

,,,

,,,,,,

nzyx

L L

nnn

LLLdudvdwd

v

wvuC

TwvuP

wvuCwvuCwcvs

y z πτ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∫ ∫ ∫ ∫−×−t L L L

nnnnCnCnnnnC

x y z

TwvuP

wvuCwvuCwsvcucetezcycxcF

0 0 0 0

1

00

01,,,

,,,,,,

γ

γ τττ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

=1 0 02

00 2,,,

n

t L

nnCnCnnnnC

zyx

x

usetezcycxcFLLL

dudvdwdw

wvuCτ

πτ

τ

( ) ( ) ( )( )

( ) ( ) ( )∑ ×−∫ ∫∂

∂×

=12

0 0

0100 2,,,

,,,

,,,

nnCn

zyx

L L

nntexc

LLLdudvdwd

u

wvuC

TwvuP

wvuCwcvcn

y z πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( )( )

( )×∫ ∫ ∫ ∫

∂−×

t L L L

nnnnCnnC

x y z

dudvdwdv

wvuC

TwvuP

wvuCwcvsuceycF

0 0 0 0

0100 ,,,

,,,

,,,τ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞

=1 0 0 0 02

2

n

t L L L

nnnnCnCnnnnC

zyx

n

x y z

wsvcucetezcycxcFnLLL

zcn τπ

( )( )

( )τ

ττγ

γ

dudvdwdw

wvuC

TwvuP

wvuC

∂×

,,,

,,,

,,,0100 ;

Page 23: ON VERTICAL INTEGRATION FRAMEWORK ELEMENT OF TRANSISTOR TRANSISTOR LOGIC · transistor-transistor logic on Fig. 1 a. After the considered doping dopant and/or radiation defects should

International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015

23

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0211

2,,,

n

t L L L

nnnnCnCnnnnC

zyx

x y z

wcvcusetezcycxcFnLLL

tzyxC τπ

( )( )

( ) ( ) ( ) ( ) ×∑−∂

∂×

=12

01 2,,,,,,

nnCnnnnC

zyx

LtezcycxcFn

LLLdudvdwd

u

wvuCTwvug

πτ

τ

( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂−×

20 0 0 0

01 2,,,,,,

zyx

t L L L

LnnnnCLLL

dudvdwdv

wvuCTwvugwcvsuce

x y z πτ

ττ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∂−×

=1 0 0 0 0

01 ,,,,,,

n

t L L L

LnnnnCnC

x y z

dudvdwdw

wvuCTwvugwsvcuceten τ

ττ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞

=1 0 0 02

2

n

t L L

nnnCnCnnnnC

zyx

nnnnC

x y

vcusetezcycxcFLLL

zcycxcF τπ

( ) ( )( )

( ) ( ) ( )×∑−∫∂

∂×

=12

0

1000 2,,,

,,,

,,,

nnnnC

zyx

L

nycxcFn

LLLdudvdwd

u

wvuC

TwvuP

wvuCwcn

z πτ

ττγ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )−∫ ∫ ∫ ∫

∂−×

t L L L

nnnnCnCn

x y z

dudvdwdv

wvuC

TwvuP

wvuCwcvsucetezc

0 0 0 0

1000 ,,,

,,,

,,,τ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

×∑ ∫ ∫ ∫ ∫−−∞

=1 0 0 0 0

00

2,,,

,,,2

n

t L L L

nnnnCnCnnnnC

zyx

x y z

TwvuP

wvuCwsvcucetezcycxcFn

LLLγ

γ ττ

π

( )( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−

∂×

=1 0 02

10 2,,,

n

t L

nnCnCnnnnC

zyx

x

usetezcycxcFnLLL

dudvdwdw

wvuCτ

πτ

τ

( ) ( ) ( ) ( )( )

( )×∑−∫ ∫

∂×

=

12

0 0

00

1

00

10

2,,,

,,,

,,,,,,

nzyx

L L

nnn

LLLdudvdwd

u

wvuC

TwvuP

wvuCwvuCwcvc

y z πτ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )×∫ ∫ ∫ ∫

∂−×

−t L L L

nnnnCnCnnnnC

x y z

v

wvuC

TwvuP

wvuCwcvsucetezcycxcF

0 0 0 0

00

1

00,,,

,,,

,,, τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×∞

=1 0 0210

2,,,

n

t L

nnCnCnnnnC

zyx

x

ucetezcycxcFnLLL

dudvdwdwvuC τπ

ττ

( ) ( ) ( ) ( )( )

( )∫ ∫

∂×

−y zL L

nndudvdwd

w

wvuC

TwvuP

wvuCwvuCwsvc

0 0

00

1

00

10

,,,

,,,

,,,,,, τ

τττ

γ

γ

.