ON DECREASING OF DIMENSIONS OF FIELDEFFECT TRANSISTORS WITH SEVERAL SOURCES

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016

DOI:10.5121/msej.2016.3301 1

ON DECREASING OF DIMENSIONS OF FIELD-

EFFECT TRANSISTORS WITH SEVERAL

SOURCES

E.L. Pankratov, E.A. Bulaeva

Nizhny Novgorod State University, 2 Gagarin avenue, Nizhny Novgorod, 603950, Russia

ABSTRACT

We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several

sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture

heterostructure with specific configuration. After that it is necessary to dope required areas of the hetero-

structure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After

the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass

and heat transport during technological processes. Using the approach leads to take into account nonlineari-

ty of mass and heat transport and variation in space and time (at one time) physical parameters of these

processes

KEYWORDS

Field-effect transistor, transistor with several channels, increasing of compactness of transistors

1. INTRODUCTION

Now several problems of solid state electronic intensively solving. The problems are increasing

of density of elements of integrated circuits and at the same time decreasing of dimensions of

these elements [1-4], increasing performance [5-7] and increasing reliability [8,9]. Now one can

find intensive development of both power electronic devices and logical elements. In this paper

we consider an approach to manufacture more compact field-effect heterotransistor with several

sources. Framework the approach it is necessary to manufacture a heterostructure. The hetero-

structure consist of a substrate and an epitaxial layer (see Fig. 1). Several sections have been

manufactured into the epitaxial layer. These sections manufactured by using other materials (see

Fig. 1). The sections have been doped by diffusion or ion implantation to obtain required type of

conductivity (n or p). After the doping one can manufacture a field-effect transistor framework

the considered heterostructure so as it is shown on the Fig. 1. The doping should be finished by

annealing of dopant and/or radiation defects. The annealing should be optimized. The optimiza-

tion attracted an interest to manufacture more compact distributions of concentrations of dopant.

Framework the paper we formulate conditions to increase compactness and at the same time to

increase homogeneity of distribution of concentration of dopant in enriched by the dopant area.

Fig. 1. Structure of a field-effect heterotransistor. Top side of the structure


2

2. METHOD OF SOLUTION

To solve our aim we determine distribution of concentration of dopant C(x,y,z,t) in space and

time. To determine the distribution we solve the following boundary problem

( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxC ,,,,,,,,,,,, (1)

( )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)=fC (x,y,z). (2)

Here T is the temperature of annealing; DС is the dopant diffusion coefficient. Dopant diffusion

coefficient takes another value in other materials. Heating and cooling of heterostructure (see

Arrhenius law) also leads to changing of value of diffusion coefficient. We consider following

approximation of concentrational dependences of dopant diffusion coefficient [10-12]

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

LCςςξ

γ

γ

. (3)

Here function DL (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate

and temperature; function P (x,y,z,T) describes dependences of limit of solubility on coordinate

and temperature; parameter γ ∈[1,3] is integer and depends on properties of materials; function V

(x,y,z,t) describes distribution of concentration of radiation vacancies on space and time with

equilibrium distribution V*. Dependence of dopant diffusion coefficient on concentration of do-

pant has been investigated and described in details in [10]. Diffusion of dopant gives a possibility

to dope materials without generation radiation defects. In this situation ζ1= ζ2= 0. Ion doping of

dopant leads to generation radiation defects. Distributions of concentrations of point radiation

defects have been determined by solving the following boundary problem [11,12]

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂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 −

∂

∂

∂

∂+×

( )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)


3

Here ρ =I,V; distribution of concentration of radiation interstitials in space and time describes by

the function I (x,y,z,t); terms of Eqs.(4) with quadric concentrations of point radiation defects

(V2(x,y,z,t) and I2(x,y,z,t)) correspond to generation of simplest complexes of radiation defects

(divacancies and diinterstitials); temperature and spatial dependences of diffusion coefficients of

point radiation defects describe by functions Dρ(x,y,z,T); temperature and spatial dependences of

parameter of recombination of point radiation defects describe by function kI,V(x,y,z,T); functions

kI,I(x,y,z,T) and kV,V(x,y,z,T) describe spatial and temperature dependences of parameters of gener-

ation of simplest complexes of point radiation defects.

Concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI (x,y,z,t) have been calculated by

solution of the following boundary problem [11,12]

( ) ( ) ( ) ( ) ( )+

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 ∂

∂

∂

∂

( )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)

Functions DΦρ(x,y,z,T) describe spatial and temperature dependences of diffusion coefficients of

simplest complexes of point radiation defects; functions kI(x,y,z,T) and kV (x,y,z,T) describe spatial

and temperature dependences of parameters of decay of the above complexes.

Now let us consider equivalent integro-differential form of Eq.(1)

( ) ( ) ( ) ( )( )

×∫ ∫ ∫

++=∫ ∫ ∫

t y

L

z

LL

x

L

y

L

z

Lzyx y zx y z V

wvxV

V

wvxVTwvxDudvdwdtwvuC

LLL

zyx

02*

2

2*1

,,,,,,1,,,,,,

τς

τς

( )( )

( ) ( ) ( )×∫ ∫ ∫+

+×

t x

L

z

LL

zxzy x z y

wyuCTwyuD

LL

zx

LL

zyd

x

wvxC

TwvxP

wvxC

0

,,,,,,

,,,

,,,

,,,1

∂

τ∂τ

∂

τ∂τξ

γ

γ

( ) ( )( )

( )( )

( )∫ ∫ ∫ ×+

+

++×

t x

L

y

LL

yx x y

TzvuDLL

yxd

TzyxP

wyuC

V

wyuV

V

wyuV

02*

2

2*1 ,,,,,,

,,,1

,,,,,,1 τ

τξ

τς

τς

γ

γ

( ) ( )( )

( )( )

( )×+

+

++×

zyxLLL

zyxd

z

zvuC

TzyxP

zvuC

V

zvuV

V

zvuVτ

∂

τ∂τξ

τς

τς

γ

γ,,,

,,,

,,,1

,,,,,,1

2*

2

2*1

( )∫ ∫ ∫×x

L

y

L

z

Lx y z

udvdwdwvuf ,, . (1a)


4

We used the Bubnov-Galerkin approach [13] to calculate solution of the above equation. To use

the approach we consider of the Eq.(1a) as the following series

( ) ( ) ( ) ( ) ( )∑==

N

nnCnnnnC

tezcycxcatzyxC0

0 ,,, .

Here ( ) ( )[ ]222

0

22exp −−− ++−=zyxCnC

LLLtDnte π , cn(χ) = cos (π n χ/Lχ). Number of terms N of the

above series is finite. The series is almost coincides with solution of Eq.(1) in the linear case (i.e.

with ξ = 0) and average value of dopant diffusion coefficient D0. Framework the approach we

substitute the above series into Eq.(1a). After the substitution we obtain

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

∑+−=∑

==

t y

L

z

L

N

nnCnnnnC

N

nnCnnn

C

y z TwvxPewcvcxcatezsysxs

n

azyx

0 1132 ,,,

1γ

γ ξτ

π

( ) ( )( )

( ) ( ) ( ) ( ) ( ) −∑

++×

=zy

N

nnCnnnnCL

LL

zydewcvcxsanTwvxD

V

wvxV

V

wvxV

12*

2

2*1 ,,,,,,,,,

1 τττ

ςτ

ς

( ) ( )( )

( ) ( ) ( ) ( )∫ ∫ ∫

×

∑+

++−

=

t x

L

z

L

N

mmCmmmmC

zx x z

ewcycucaV

wyuV

V

wyuV

LL

zx

0 12*

2

2*1 1,,,,,,

1γ

ττ

ςτ

ς

( )( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∑

×=

t x

L

y

LL

yx

N

nnCnnnnCL

x y

TzvuDLL

yxdewcysucanTwyuD

TwyuP 01

,,,,,,,,,

ττξ

γ

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

( )×

++

∑+×

=2*

2

2*11

,,,,,,1

,,,1

V

zvuV

V

zvuVezcvcuca

TzvuP

N

nnCnnnnC

τς

τςτ

ξ γ

γ

( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×=

x

L

y

L

z

Lzyx

N

nnCnnnnC

x y z

udvdwdwvufLLL

zyxdezsvcucan ,,

1

ττ . (8)

Here sn(χ) = sin (π n χ/Lχ). To determine coefficients an it is necessary to use orthogonality condi-

tion of terms of the above series framework scale of heterostructure. Using the condition leads to

the following equations to calculate of coefficients an for any quantity of terms N

( ) ( ) ( ) ( ) ( ) ( )( )∫ ∫ ∫ ∫ ×

∑+−=∑−

==

t L L L N

nnCnnnnCL

N

nnC

nCzyxx y z

TzyxPezcycxcaTzyxDte

n

aLLL

0 0 0 0 1165

222

,,,1,,,

γ

γ ξτ

π

( ) ( )( )

( ) ( ) ( ) ( )[ ] ×∑

−+

++×

=

N

nn

y

nnn

nCzyyc

n

Lysyycxs

n

a

V

zyxV

V

zyxVLL

12*

2

2*1212

,,,,,,1

2 π

τς

τς

π

( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫

∑−

−+×=

t L L L N

nnCnnnnCn

z

nnCn

x y z

ezcycxcadxdydzdzcn

Lzszezc

0 0 0 0 1

1γ

ττπ

τ

( )( )

( ) ( )( )

( )

++

++

+×*12*

2

2*1

,,,1

,,,,,,1,,,1

,,, V

zyxV

V

zyxV

V

zyxVTzyxD

TzyxPL

τς

τς

τς

ξγ

( )( )

( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )∑ ×

−+

−+

+

=

N

nnnnn

z

nn

x

n

znC zcysxczcn

Lzszxc

n

Lxsx

n

La

V

zyxV

122*

2

2 2112

,,,

πππ

τς

( ) ( ) ( ) ( ) ( )( )∫ ∫ ∫ ∫ ×

∑+−×

=

t L L L N

nnCnnnnC

yx

nC

x y z

TzyxPezcycxca

LLdxdydzde

0 0 0 0 12 ,,,

12 γ

γ ξτ

πττ


5

( ) ( )( )

( )( ) ( ) ( )[ ] ×∑

−+

++×

=

N

nn

x

nn

nC

Lxc

n

Lxsxxc

n

a

V

zyxV

V

zyxVTzyxD

1*12*

2

2 1,,,,,,

1,,,π

τς

τς

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ ×

−++

−+×=

N

n

L

n

x

nnCn

y

nnn

x

xcn

Lxsxdxdydzdeyc

n

Lysyzsyc

1 0

11π

ττπ

( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫

−+

−+×y z

L L

n

z

nn

y

nxdydzdzyxfzc

n

Lzszyc

n

Lysy

0 0

,,11ππ

. (9)

Now we consider several examples. For γ = 0 we obtain

( ) ( )[ ] ( ) ( )[ ] ( ) ( )∫ ∫ ∫

×+

−+

−+=x y zL L L

x

nn

y

nn

y

nnCn

Lxsxydzdzyxfzc

n

Lzszyc

n

Lysya

0 0 0

,,11πππ

( )[ ]} ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){

∫ ∫ ∫ ∫ +

−+−×t L L L

nn

y

nLnnnn

x y z

ysyzcn

LzszTzyxDzcycxs

nxdxc

0 0 0 0

1,,,22

1π

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

( ) ×

+

++

−+ xdydzdzcTzyxPV

zyxV

V

zyxVyc

n

Lnn

y

,,,1

,,,,,,11

2*

2

2*1 γ

ξτς

τς

π

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫

−+

−++×t L L L

n

y

nnnn

y

nnnCnC

x y z

zcn

Lzszzcysxc

n

Lxsxxcede

0 0 0 0

121ππ

τττ

( )( )

( ) ( )( )

+

++

+× τ

τς

τς

ξγ

dxdydzdV

zyxV

V

zyxV

TzyxPTzyxD

L 2*

2

2*1

,,,,,,1

,,,1,,,

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∫ ∫

−+

−++t L L L

Ln

y

nnn

x

nnnC

x y z

TzyxDycn

Lysyycxc

n

Lxsxxce

0 0 0 0

,,,11ππ

τ

( )( )

( )( )

( )1

65

222

*12*

2

2

,,,,,,1

,,,12

−

−

++

+×

n

LLLdxdydzd

V

zyxV

V

zyxV

TzyxPzs zzz

nπ

ττ

ςτ

ςξ

γ.(10)

For γ = 1 calculation of parameters an leads to the following results

( ) ( ) ( ) ( )∫ ∫ ∫+±−=x y zL L L

nnnnn

n

n

nCxdydzdzyxfzcycxca

0 0 0

2 ,,42

αβα

β. (11)

Here ( ) ( )( )

( )( ) ( )

( )∫ ∫ ∫ ∫ ×

++=

t L L LL

nn

zy

n

x y z

V

zyxV

V

zyxV

TzyxP

TzyxDycxs

n

LL

0 0 0 02*

2

2*12

,,,,,,1

,,,

,,,2

2

τς

τς

π

ξα

( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×+

−+

−+×n

LLdexdydzdzc

n

Lzszyc

n

Lysyzc zx

nCn

z

nn

y

nn 2211

π

ξττ

ππ

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

( )( ) ( )[ ] ×∫ ∫ ∫ ∫

−−

−+×t L L L

n

z

n

L

nn

x

nnnC

x y z

zcn

Lzsz

TzyxP

TzyxDysxc

n

Lxsxxce

0 0 0 0

1,,,

,,,21

ππτ

( ) ( ) ( )( )

( ) ( ) ( ) ×∫ ∫ ∫+

++×

t L L

nnnC

yx

n

x y

ycxcen

LLdxdydzd

V

zyxV

V

zyxVzc

0 0 022*

2

2*12

,,,,,,1 τ

π

ξτ

τς

τς


6

( )( )

( )( ) ( )

( )( ) ( )[ ] ( ){∫ ×

−+

++×

zL

nn

x

n

L

nysxc

n

Lxsx

V

zyxV

V

zyxV

TzyxP

TzyxDzs

02*

2

2*1 1,,,,,,

1,,,

,,,2

π

τς

τς

( )[ ] τπ

dxdydzdycn

Ly n

y

−+× 1 , ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫=t L L L

LnnnnC

zy

n

x y z

TzyxDzcycxsen

LL

0 0 0 02

,,,22

τπ

β

( ) ( )( )

( ) ( )[ ] ( )[ ] ( ) ×

+−

−+

++× ysyyc

n

Lzdzc

n

Lzsz

V

zyxV

V

zyxVnn

y

n

z

n11

,,,,,,1

2*

2

2*1ππ

τς

τς

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )∫ ∫ ∫ ∫

++

−++×t L L L

nnn

x

nnnC

zxx y z

V

zyxVzcysxc

n

Lxsxxce

n

LLdxdyd

0 0 0 0*12

,,,121

2

τς

πτ

πτ

( )( )

( ) ( ) ( )[ ] ( )∫ ×+

−+

+

t

nC

yx

n

z

nLe

n

LLdxdydzdzc

n

LzszTzyxD

V

zyxV

022*

2

22

1,,,,,,

τπ

τπ

τς

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

∫ ∫ ∫ ×

++

−+

−+×x y zL L L

n

y

nn

x

n

V

zyxV

V

zyxVyc

n

Lysyxc

n

Lxsx

0 0 02*

2

2*1

,,,,,,111

τς

τς

ππ

( ) ( ) ( ) ( ) ( ) 65222,,,2 nteLLLdxdxcydyczdTzyxDzs nCzyxnnLn πτ −× .

It could be used the same approach to calculate values of parameters an for larger values of the

parameterγ. However the relations became more bulky and will not be present in the paper. The

considered approach gives a possibility to calculate distributions of concentrations of dopant and

radiation defects without joining of the above concentration on interfaces of the considered hete-

rostructure.

We solved equations of the system (4) by using Bubnov-Galerkin approach. To use the approach we previously transform the differential equations to the following integro-differential form

( ) ( )( )

×+∫ ∫ ∫∂

∂=∫ ∫ ∫

zx

t y

L

z

LI

zy

x

L

y

L

z

LzyxLL

zxdvdwd

x

wvxITwvxD

LL

zyudvdwdtwvuI

LLL

zyx

y zx y z 0

,,,,,,,,, τ

τ

( ) ( ) ( ) ( )−∫ ∫ ∫

∂

∂+∫ ∫ ∫

∂

∂×

t x

L

y

LI

yx

t x

L

z

LI

x yx z

dudvdz

zvuITzvuD

LL

yxdudwd

x

wyuITwyuD

00

,,,,,,

,,,,,, τ

ττ

τ

( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫−x

L

y

L

z

LII

zyx

x

L

y

L

z

LVI

zyx x y zx y z

TwvukLLL

zyxudvdwdtwvuVtwvuITwvuk

LLL

zyx,,,,,,,,,,,,

,,

( ) ( )∫ ∫ ∫+×x

L

y

L

z

LI

zyx x y z

udvdwdwvufLLL

zyxudvdwdtwvuI ,,,,,

2 (4a)

( ) ( )( )

×+∫ ∫ ∫∂

∂=∫ ∫ ∫

zx

t y

L

z

LV

zy

x

L

y

L

z

LzyxLL

zxdvdwd

x

wvxVTwvxD

LL

zyudvdwdtwvuV

LLL

zyx

y zx y z 0

,,,,,,,,, τ

τ

( ) ( ) ( ) ( )−∫ ∫ ∫

∂

∂+∫ ∫ ∫

∂

∂×

t x

L

y

LV

yx

t x

L

z

LV

x yx z

dudvdz

zvuVTzvuD

LL

yxdudwd

x

wyuVTwyuD

00

,,,,,,

,,,,,, τ

ττ

τ

( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫−x

L

y

L

z

LVV

zyx

x

L

y

L

z

LVI

zyx x y zx y z

TwvukLLL

zyxudvdwdtwvuVtwvuITwvuk

LLL

zyx,,,,,,,,,,,,

,,

( ) ( )∫ ∫ ∫+×x

L

y

L

z

LV

zyx x y z

udvdwdwvufLLL

zyxudvdwdtwvuV ,,,,,

2 .

Now we consider solutions of the above integro-differential equations as the following series


7

( ) ( ) ( ) ( ) ( )∑==

N

nnnnnn

tezcycxcatzyx1

0 ,,, ρρρ .

Here anρ are coefficients, which should be determined. After substitution of the series into Eqs.

(4a) one can obtain

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

N

n

t y

L

z

LInnnInI

zyx

N

nnInnn

nI

y z

dvdwdTwvxDzcyceaLLL

zytezsysxs

n

azyx

1 0133

,,, ττπ

π

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

N

nnnI

zyx

N

n

t x

L

z

LInnnInnI

zyx

nzsa

LLL

yxdudwdTwyuDzcxceysa

LLL

zxxs

x z11 0

,,,π

ττπ

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

∑−∫ ∫ ∫×

=

x

L

y

L

z

L

N

nnInnnnI

zyx

t x

L

y

LInnnI

x y zx y

tewcvcucaLLL

zyxdudvdTzvuDycxce

2

10

,,, ττ

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

x

L

y

L

z

L

N

n

N

nnnnnVnInnnnI

zyx

II

x y z

wcvcucatewcvcucaLLL

zyxudvdwdTvvuk

1 1,

,,,

( ) ( ) ( )∫ ∫ ∫+×x

L

y

L

z

LI

zyx

VInV

x y z

udvdwdwvufLLL

zyxudvdwdTvvukte ,,,,,, (12)

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

N

n

t y

L

z

LVnnnVnV

zyx

N

nnVnnn

nV

y z

dvdwdTwvxDzcyceaLLL

zytezsysxs

n

azyx

1 0133

,,, ττπ

π

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

N

nnnV

zyx

N

n

t x

L

z

LVnnnVnnV

zyx

nzsa

LLL

yxdudwdTwyuDzcxceysa

LLL

zxxs

x z11 0

,,,π

ττπ

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

∑−∫ ∫ ∫×

=

x

L

y

L

z

L

N

nnVnnnnV

zyx

t x

L

y

LVnnnV

x y zx y

tewcvcucaLLL

zyxdudvdTzvuDycxce

2

10

,,, ττ

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

x

L

y

L

z

L

N

n

N

nnnnnVnInnnnI

zyx

VV

x y z

wcvcucatewcvcucaLLL

zyxudvdwdTvvuk

1 1, ,,,

( ) ( ) ( )∫ ∫ ∫+×x

L

y

L

z

LV

zyx

VInV

x y z

udvdwdwvufLLL

zyxudvdwdTvvukte ,,,,,

,.

We use orthogonality condition of functions in the above series on scale of the heterostructure to

calculate coefficients anρ. Using the condition gives a possibility to obtain equations for calcula-

tion the above coefficients for any quantity N of terms of considered series

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

nynnI

nI

x

N

nnI

nIzyxx y

ycn

LysyLxce

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πτ

ππ

( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫

++−−∫

−+×=

N

n

t L

xn

xnI

y

L

n

z

nI

xz

Lxcn

L

n

a

Ldxdydzdzc

n

LzszTzyxD

1 0 02

0

122

11

2,,,

ππτ

π

( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ −

−+++y z

L L

nInn

z

nzIndexdydyczdzc

n

LzszLTzyxDxsx

0 0

21122

2,,,2 ττπ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

nyn

x

nxnI

nI

z

x y

ycn

LysyLxc

n

LxsxLe

n

a

L 1 0 0 02

122

2122

22

1

ππτ

π


8

( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫

+−+−∫ −×=

N

n

L

nn

x

xnInI

L

In

xz

xsxxcn

LLteadxdydzdTzyxDzc

1 0

2

0

2122

2,,,21π

τ

( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫

+−+

−++×y z

L L

nn

z

zIIn

y

nyxdydzdzszzc

n

LLTzyxkyc

n

LysyL

0 0, 212

2,,,12

22

ππ

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫

−++

−++−=

N

n

L L

n

y

nyn

x

nxnVnInVnI

x y

ycn

LysyLxc

n

LxsxLteteaa

1 0 0

122

2122

2ππ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫

−++∫

−++×=

N

n

L

n

x

n

L

n

z

nzVI

xz

xcn

Lxsxxdydzdzc

n

LzszLTzyxk

1 00, 112

22,,,

ππ

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫

−++

−+×y z

L L

n

z

nzIn

y

nxdydzdzc

n

LzszLTzyxfyc

n

Lysy

0 0

122

2,,,1ππ

(13)

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

nynnV

nV

x

N

nnV

nVzyxx y

ycn

LysyLxce

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πτ

ππ

( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫

++−−∫

−+×=

N

n

t L

xn

xnV

y

L

n

z

nV

xz

Lxcn

L

n

a

Ldxdydzdzc

n

LzszTzyxD

1 0 02

0

122

11

2,,,

ππτ

π

( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ −

−+++y z

L L

nVnn

z

nzVndexdydyczdzc

n

LzszLTzyxDxsx

0 0

21122

2,,,2 ττπ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

nyn

x

nxnV

nV

z

x y

ycn

LysyLxc

n

LxsxLe

n

a

L 1 0 0 02

122

2122

22

1

ππτ

π

( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫

+−+−∫ −×=

N

n

L

nn

x

xnVnV

L

Vn

xz

xsxxcn

LLteadxdydzdTzyxDzc

1 0

2

0

2122

2,,,21π

τ

( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫

+−+

−++×y z

L L

nn

z

zVVn

y

nyxdydzdzszzc

n

LLTzyxkyc

n

LysyL

0 0,

2122

,,,122

2ππ

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫

−++

−++−=

N

n

L L

n

y

nyn

x

nxnVnInVnI

x y

ycn

LysyLxc

n

LxsxLteteaa

1 0 0

122

2122

2ππ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫

−++∫

−++×=

N

n

L

n

x

n

L

n

z

nzVI

xz

xcn

Lxsxxdydzdzc

n

LzszLTzyxk

1 00,

1122

2,,,ππ

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫

−++

−+×y z

L L

n

z

nzVn

y

nxdydzdzc

n

LzszLTzyxfyc

n

Lysy

0 0

122

2,,,1ππ

.

Final relations for the above parameters takes the form

( )

−+−

+±

+−=

A

ybyb

Ab

b

Aba nInV

nI

2

3

4

2

3

4

3 444

λγ,

nInI

nInInInInI

nVa

aaa

χ

λδγ ++−=

2

. (14)

Here ( ) ( ) ( ) ( )[ ] ( )∫ ∫ ∫

×++

−++=x y zL L L

y

ynn

x

nxnnn

LLysyxc

n

LxsxLTzyxkte

0 0 0,

2212

22,,,2

ππγ ρρρρ


9

( )[ ]} ( ) ( )[ ] xdydzdzcn

LzszLyc

n

z

nzn

−++−× 122

212π

, ( ) ( ){∫ ∫ ∫ +=t L L

nn

x

n

x y

ysyenL 0 0 0

22

1τ

πδ ρρ

( )[ ] ( ) ( )[ ] ( ) ( )[ ] ×+∫ −

−+

−+y

L

nn

z

nn

y

LdxdxcydzdTzyxDzc

n

Lzszyc

n

L z

πτ

ππρ

2

121,,,1

21

2 0

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

+−−

−++×t L L L

n

z

nn

x

nxn

x y z

zcn

LTzyxDycxc

n

LxsxLe

n 0 0 0 02

122

,,,211222

1

ππτ ρρ

( ) } ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ +

−+++++t L L

nn

x

xnn

z

zn

x y

ysyxcn

LLxsxe

nLdxdydzdLzsz

0 0 02

1222

12

πτ

πτ ρ

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

LLLdxdydzdTzyxDzcyc

n

LL

n

zyxL

nn

y

y

z

ρρπ

τπ 65

222

0

,,,2112

−∫ −

−++ , ( ) ×= tenInIV

χ

( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ){∫ ∫ ∫ ++

+−+

−+×x y zL L L

nzVInn

y

yn

x

nnV zszLTzyxkysyycn

LLxc

n

Lxsxte

0 0 0, 2,,,212

21

ππ

( )[ ] xdydzdzcn

Ln

z

−+ 122π

, ( ) ( )[ ] ( ) ( )[ ] {∫ ∫ ∫ ×

−+

−+=x y zL L L

n

y

nn

x

nnzyc

n

Lysyxc

n

Lxsx

0 0 0

11ππ

λ ρ

( ) ( )[ ] ( ) xdydzdTzyxfzcn

Lzs

n

z

n,,,1 ρ

π

−+× , 22

4 nInInInVb χγγγ −= , −−= 2

3 2nInInInInV

b χδδγγ

nInInVγχδ− , ( ) 22

2 2 nInInVnInInVnInVnInInVb χλλδχδγγλδγ −+−+= , nInInVnInVnI

b λχδδγλ −= 21,

2

2

348 bbyA −+= ,

2

4

2

342

9

3

b

bbbp

−= ,

3

4

2

4132

3

3

54

2792

b

bbbbbq

+−= , −++−−+= 3 323 32 qpqqpqy

43 3bb− .

Now we will calculate distributions of concentrations of simplest complexes of radiation defects

as the following functional series

( ) ( ) ( ) ( ) ( )∑=Φ=

Φ

N

nnnnnn

tezcycxcatzyx1

0 ,,, ρρρ (15)

with not yet known coefficients anΦρ. To calculate these coefficient we transform the Eqs.(6) to

the following integro-differential form

( ) ( )( )

+∫ ∫ ∫Φ

=∫ ∫ ∫Φ Φ

t y

L

z

L

I

I

zy

x

L

y

L

z

LI

zyx y zx y z

dvdwdx

wvxTwvxD

LL

zyudvdwdtwvu

LLL

zyx

0

,,,,,,,,, τ

∂

τ∂

( )( )

( )( )

∫ ∫ ∫ ×Φ

+∫ ∫ ∫Φ

+ ΦΦ

t x

L

y

L

I

I

t x

L

z

L

I

I

zx x yx z

dudvdz

zvuTzvuDdudwd

y

wyuTwyuD

LL

zx

00

,,,,,,

,,,,,, τ

∂

τ∂τ

∂

τ∂

( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+×x

L

y

L

z

LI

zyx

x

L

y

L

z

LII

zyxyx x y zx y z

TwvukLLL

zyxudvdwdwvuITwvuk

LLL

zyx

LL

yx,,,,,,,,, 2

,τ

( ) ( )∫ ∫ ∫+× Φ

x

L

y

L

z

LI

zyx x y z

udvdwdwvufLLL

zyxudvdwdwvuI ,,,,, τ (6a)

( ) ( )( )

+∫ ∫ ∫Φ

=∫ ∫ ∫Φ Φ

t y

L

z

L

V

V

zy

x

L

y

L

z

LV

zyx y zx y z

dvdwdx

wvxTwvxD

LL

zyudvdwdtwvu

LLL

zyx

0

,,,,,,,,, τ

∂

τ∂

( )( )

( )( )

∫ ∫ ∫ ×Φ

+∫ ∫ ∫Φ

+ ΦΦ

t x

L

y

L

V

V

t x

L

z

L

V

V

zx x yx z

dudvdz

zvuTzvuDdudwd

y

wyuTwyuD

LL

zx

00

,,,,,,

,,,,,, τ

∂

τ∂τ

∂

τ∂


10

( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+×x

L

y

L

z

LV

zyx

x

L

y

L

z

LVV

zyxyx x y zx y z

TwvukLLL

zyxudvdwdwvuVTwvuk

LLL

zyx

LL

yx,,,,,,,,,

2

, τ

( ) ( )∫ ∫ ∫+× Φ

x

L

y

L

z

LV

zyx x y z

udvdwdwvufLLL

zyxudvdwdwvuV ,,,,, τ .

Further we substitute the previously considered series in the Eqs.(6a). In this situation we obtain

the following equation

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

ΦΦ=

ΦN

n

t y

L

z

LInnInIn

zyx

N

nnInnn

In

y z

TwvxDvctexsanLLL

zytezsysxs

n

azyx

1 0133

,,,π

π

( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

LInnInnIn

zyx

n

x z

dudwdTwvuDwcucteysnaLLL

zxdvdwdwc

1 0

,,, τπ

τ

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

ΦΦΦ

x

L

y

L

z

LII

N

n

t x

L

y

LInnInnIn

zyx x y zx y

TwvukdudvdTzvuDvcuctezsanLLL

yx,,,,,, ,

1 0

τπ

( ) ( ) ( ) ×+∫ ∫ ∫−×zyx

x

L

y

L

z

LI

zyxzyxLLL

zyxudvdwdwvuITwvuk

LLL

zyx

LLL

zyxudvdwdwvuI

x y z

ττ ,,,,,,,,,2

( )∫ ∫ ∫× Φ

x

L

y

L

z

LI

x y z

udvdwdwvuf ,, (16)

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

ΦΦ=

ΦN

n

t y

L

z

LVnnVnVn

zyx

N

nnVnnn

Vn

y z

TwvxDvctexsanLLL

zytezsysxs

n

azyx

1 0133

,,,π

π

( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

LVnnVnnVn

zyx

n

x z

dudwdTwvuDwcucteysnaLLL

zxdvdwdwc

1 0

,,, τπ

τ

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

ΦΦΦ

x

L

y

L

z

LVV

N

n

t x

L

y

LVnnVnnVn

zyx x y zx y

TwvukdudvdTzvuDvcuctezsanLLL

yx,,,,,,

,1 0

τπ

( ) ( ) ( ) ×+∫ ∫ ∫−×zyx

x

L

y

L

z

LV

zyxzyxLLL

zyxudvdwdwvuVTwvuk

LLL

zyx

LLL

zyxudvdwdwvuV

x y z

ττ ,,,,,,,,,2

( )∫ ∫ ∫× Φ

x

L

y

L

z

LV

x y z

udvdwdwvuf ,, .

We use orthogonality condition of functions in the above series on scale of the heterostructure to

calculate coefficients anΦρ. Using the condition gives a possibility to obtain equations for calcula-

tion the above coefficients for any quantity N of terms of considered series

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−=

Φ

=Φ

ΦN

n

t L L

n

y

nyn

In

x

N

nIn

Inzyxx y

ycn

LysyLxc

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫

−+×=

Φ

ΦΦ

N

n

t L

xn

y

InL

Inn

z

nI

xz

LxsxLn

adexdydzdzc

n

LzszTzyxD

1 0 02

0

22

11

2,,,

πττ

π

( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫

−+−−

+ ΦΦ

x

L L

Inn

z

nIn

n

xL

dexdydzdzcn

LzszTzyxDyc

n

xcL

y z

πττ

ππ

11

2,,,21

2

12

0 0


11

( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫ ∫

+−+

−+×=

ΦΦΦ

N

n

t L L L

Iyn

y

nn

x

nIn

Inx y z

TzyxDLycn

Lysyxc

n

Lxsxe

n

a

1 0 0 0 02

,,,122

2122 ππ

τ

( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ×∑∫ ∫ ∫

−+

+−+−×=

Φ

N

n

t L L

n

y

nnn

x

Inn

x y

ycn

Lysyxsxxc

n

Ledxdydzdyc

10 0 0

12

12

21ππ

ττ

( ) ( ) ( )[ ] ( ) ( ) ( ){∑ ∫ ∫ +−∫

+−×=

ΦΦ

N

n

t L

nIn

L

nn

z

II

Inxz

xsxexdydzdzszzcn

LTzyxktzyxI

n

a

10 00,

2

331

2,,,,,, τ

ππ

( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫

+−

+−

−+ Φy z

L L

n

z

Inn

yIn

n

x zcn

LtzyxITzyxkysyyc

n

L

n

axc

n

L

0 033

12

,,,,,,12

12 ππππ

( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−+

−+++=

Φ

ΦN

n

t L L

n

y

nn

x

nIn

In

n

x y

ycn

Lysyxc

n

Lxsxe

n

axdydzdzsz

1 0 0 033

12

12 ππ

τπ

( )[ ] ( ) ( )∫

+−× Φ

zL

Inn

z xdydzdzyxfzszzcn

L

0

,,12π

(17)

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−=

Φ

=Φ

ΦN

n

t L L

n

y

nyn

Vn

x

N

nVn

Vnzyxx y

ycn

LysyLxc

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫

−+×=

Φ

ΦΦ

N

n

t L

xn

y

VnL

Vnn

z

nV

xz

LxsxLn

adexdydzdzc

n

LzszTzyxD

1 0 02

0

22

11

2,,,

πττ

π

( )( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫

−+−−

+ ΦΦ

x

L L

Vnn

z

nVn

n

xL

dexdydzdzcn

LzszTzyxDyc

n

xcL

y z

πττ

ππ

11

2,,,21

2

12

0 0

( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫ ∫

+−+

−+×=

ΦΦ

ΦN

n

t L L L

Vyn

y

nn

x

nVn

Vnx y z

TzyxDLycn

Lysyxc

n

Lxsxe

n

a

1 0 0 0 02

,,,122

2122 ππ

τ

( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ×∑∫ ∫ ∫

−+

+−+−×=

Φ

N

n

t L L

n

y

nnn

x

Vnn

x y

ycn

Lysyxsxxc

n

Ledxdydzdyc

10 0 0

12

12

21ππ

ττ

( ) ( ) ( )[ ] ( ) ( ) ( ){∑ ∫ ∫ +−∫

+−×=

Φ

ΦN

n

t L

nVn

L

nn

z

VV

Vnxz

xsxexdydzdzszzcn

LTzyxktzyxV

n

a

10 00,

2

331

2,,,,,, τ

ππ

( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫

+−

+−

−+ Φy z

L L

n

z

Vnn

yVn

n

x zcn

LtzyxVTzyxkysyyc

n

L

n

axc

n

L

0 033

12

,,,,,,12

12 ππππ

( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−+

−+++=

Φ

ΦN

n

t L L

n

y

nn

x

nVn

Vn

n

x y

ycn

Lysyxc

n

Lxsxe

n

axdydzdzsz

1 0 0 033

12

12 ππ

τπ

( )[ ] ( ) ( )∫

+−× Φ

zL

Vnn

z xdydzdzyxfzszzcn

L

0

,,12π

.


12

Fig.2a. Spatial distributions of concentration of dopant after infusion and annealing with the same anneal-

ing time before and after interface between layers of heterostructure. Curve 1 is the distribution of concen-

tration of dopant in homogenous sample with averaged dopant diffusion coefficient D0. Curves 2-6 are the

distribution of concentration of dopant in heterostructure with increasing difference between values of dif-

fusion coefficient. Value of dopant diffusion coefficient in the substrate is smaller, than in epitaxial the

layer

3. DISCUSSION

In this section analysis of distributions of concentration of dopant, infused (see Fig. 2a) or im-

planted (see Fig. 2b) into epitaxial layer have been done. Annealing time is the same for all

curves of these figures. One can find from these figures, that absolute value of gradient of con-

centration of dopant increases due to presents an interface between layers of heterostructure. In

this situation dimensions of the considered transistors decreases. At the same time homogeneity of

concentration of dopant in enriched area increases.

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. Spatial distributions of concentration of dopant after implantation and annealing with the same an-

nealing time before and after interface between layers of heterostructure. Annealing time for curves 1 and 3

is equal to Θ = 0.0048(Lx2+Ly

2+Lz

2)/D0. Annealing time for curves 2 and 4 is equal to Θ = 0.0057(Lx

2

+Ly2+Lz

2)/D0. Curves 1 and 2 are the distributions of concentration of dopant in homogenous sample.

Curves 3 and 4 are distributions of concentration of dopant in heterostructure under condition, when value

of dopant diffusion coefficient in the substrate is smaller, than in epitaxial the layer


13

To estimate optimal annealing time we estimate decreasing of absolute value of gradient of con-centration of dopant near interface between layers of the heterostructure with increasing of an-

nealing time. Decreasing of annealing time leads to increasing inhomogeneity of distribution of

concentration of dopant. Estimation of the compromise value of annealing time has been done by using recently introduced criterion [14,15]. To use the criterion we approximate real spatial dis-

tribution of concentration of dopant by idealized step-wise function ψ (x,y,z). After that we esti-

mate the required optimal annealing time by minimization of mean-square error [16-20]

( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L

zyx

xdydzdzyxzyxCLLL

U0 0 0

,,,,,1

ψ . (18)

Minimization of the above mean-squared error leads to dependences of optimal annealing time on

parameters, which are presented on Figs. 3. It should be noted, that radiation defects, generated

during ion implantation, should be annealed. After ideal optimization of annealing time the im-

planted dopant should achieve the interface between layers of heterostructure. If annealing time is

smaller, it is attracted an interest to make additional annealing to achieve the interface. The Fig.

3b shows dependences of additional annealing time.

The figures shows, that optimal annealing time of implanted dopant is smaller in comparison with

optimal annealing time of infused dopant. If the considered heterostructure have been doped by

diffusion, any radiation damage of materials of layers is absent. On the other hand radiation processing of materials of heterostructure (including of ion implantation) leads to decreasing of

mismatch-induced stress in the processed heterostructure [20].

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. 3a. Dependences of optimal annealing time of infused dopant on several parameters. Dependence of

the optimal annealing time on normalized thickness of the epitaxial layer a/L describes by curve 1 for ξ = γ =

0 for equal to each other values of dopant diffusion coefficient in both parts of heterostructure. Curve 2

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

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

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


14

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.3b. Dependences of optimal annealing time of implanted dopant on several parameters. Dependence of

the optimal annealing time on normalized thickness of the epitaxial layer a/L describes by curve 1 for ξ = γ =

0 for equal to each other values of dopant diffusion coefficient in both parts of heterostructure. Curve 2

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

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

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

4. CONCLUSIONS

In this paper we formulate several recommendations to optimize manufacture heterotransistor with several source based on prognosis of time varying of spatial distributions of concentrations

of infused and implanted dopants in specific heterostructure. We also introduce analytical ap-

proach to prognosis diffusion and ion types of doping with account variation in space and time

parameters of technological parameters and nonlinearity of mass and heat transport.

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, educational fellowship for scientific research of Government of Russian and

educational fellowship for scientific research of Government of Nizhny Novgorod region of Rus-

sia.

REFERENCES

[1] Z. Ramezani, A.A. Orouji. A silicon-on-insulator metal semiconductor field-effect transistor with an L-

shaped buried oxide for high output-power density. Mat. Sci. Sem. Proc. Vol. 19. P. 124-129 (2014).

[2] Ch. Dong, J. Shi, J. Wu, Y. Chen, D. Zhou, Z. Hu, H. Xie, R. Zhan, Zh. Zou. Improvements in passiva-

tion effect of amorphous InGaZnO thin film transistors. Mat. Sci. Sem. Proc. Vol. 20. P. 7-11 (2014).

[3] V. Papageorgiou, A. Khalid, C. Li, M.J. Steer, D.R.S. Cumming. Integration techniques of pHEMTs

and planar Gunn diodes on GaAs substrates. Solid-State Electronics. Vol. 102. P. 87-92 (2014).

[4] R. Grassi, A. Gnudi, V. Di Lecce, E. Gnani, S. Reggiani, G. Baccarani. Boosting the voltage gain of

graphene FETs through a differential amplifier scheme with positive feedback. Solid-State Electronics.

Vol. 100. P. 54-60 (2014).

[5] A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M. Litvin,

V.V. Milenin, A.V. Sachenko. Au-TiBx-n-6H-SiC Schottky barrier diodes: Specific features of charge

transport in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 897-903 (2009).

[6] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).


15

[7] T.T.A. Tuan, D.-H. Kuo, C.-C. Li, G.-Z. Li. Effect of temperature dependence on electrical characteri-

zation of p-n GaN diode fabricated by RF magnetron sputtering. Mat. Sci. Appl. Vol. 6. P. 809-817

(2015).

[8] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).

[9] R. Pal, R. Pandey, N. Pandey, R.Ch. Tiwari. Single CDBA Based Voltage Mode Bistable Multivibrator

and Its Applications. Circuits and Systems. Vol. 6 (11). P. 237-251 (2015).

[10] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).

[11] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,

1979, in Russian).

[12] P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon. Rev. Mod. Phys.

Vol. 61 (2). P. 289-388 (1989).

[13] M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Integral equations ("Science", Moscow, 1976).

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

coefficient nonuniformity. Russian Microelectronics. Vol. 36 (1). P. 33-39 (2007).

[15] E.L. Pankratov. Redistribution of dopant during annealing of radiative defects in a multilayer structure

by laser scans for production an implanted-junction rectifiers. Int. J. Nanoscience. Vol. 7 (4-5). P. 187–

197 (2008).

[16] 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).

[17] E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to decrease

surface of chip. International Journal of Modern Physics B. Vol. 29 (5). P. 1550023-1-1550023-12

(2015).

[18] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture of bipolar transistors in thin film struc-

tures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014).

[19] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using

radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1250035 (2012).

[20] E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure by

radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (2014).

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. 2012-2015 Full Doctor course in Radiophysical

Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of

Nizhny Novgorod State University. He has 155 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. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny

Novgorod State University. She has 103 published papers in area of her researches.

ON DECREASING OF DIMENSIONS OF FIELDEFFECT TRANSISTORS WITH SEVERAL SOURCES

Science

Transcript of ON DECREASING OF DIMENSIONS OF FIELDEFFECT TRANSISTORS WITH SEVERAL SOURCES