OPTIMIZATION OF MANUFACTURE OF FIELDEFFECT HETEROTRANSISTORS WITHOUT P-NJUNCTIONS TO DECREASE THEIR...

International Journal of Recent advances in Physics (IJRAP) Vol.3, No.3, August 2014

DOI : 10.14810/ijrap.2014.3301 1

OPTIMIZATION OF MANUFACTURE OF FIELD-

EFFECT HETEROTRANSISTORS WITHOUT P-N-JUNCTIONS TO DECREASE THEIR DIMENSIONS

E.L. Pankratov1, E.A. Bulaeva

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

It has been recently shown, that manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors

in a multilayer structure by diffusion or ion implantation under condition of optimization of dopant and/or

radiation defects leads to increasing of sharpness of p-n-junctions (both single p-n-junctions and p-n-

junctions, which include into their system). In this situation one can also obtain increasing of homogeneity

of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without p-

n-junction. Optimization of technological process with using inhomogeneity of heterostructure give us

possibility to manufacture the transistors as more compact.

KEYWORDS

Field-effect heterotransistors; Decreasing of Dimensions of Transistors

1.INTRODUCTION

The development of electronics makes it necessary to reduce the size of integrated circuit

elements and their discrete analogs [1-7]. To reduce the required sizes formed by diffusion and

implantation of p-n-junctions and their systems (such transistors and thyristors) it could be used

several approaches. First of all it should be considered laser and microwave types of annealing [8-

14]. Using this types of annealing leads to generation inhomogenous distribution of temperature

gives us possibility to increase sharpness of p-n-junctions with simultaneous increasing of

homogeneity of distribution of dopant concentration in doped area [8-14]. In this situation one

can obtain more shallow p-n-junctions and at the same time to decrease dimensions of transistors,

which include into itself the p-n-junctions. The second way to decrease dimensions of elements of

integrated circuits is using of native inhomogeneity of heterostructure and optimization of

annealing [12-18]. In this case one can obtain increasing of sharpness of p-n-junctions and at the

same time increasing of homogeneity of distribution of dopant concentration in doped area [12-

18]. Distribution of concentration of dopant could be also changed under influence of radiation

processing [19]. In this situation radiation processing could be also used to increase sharpness of

single p-n-junction and p-n-junctions, which include into their system [20,21].

In this paper we consider a heterostructure, which consist of a substrate and a multi-section

epitaxial layer (see Figs. 1 and 2). Farther we consider manufacturing a field-effect transistor


2

without p-n-junction in this heterostructure. Appropriate contacts are presented in the Figs. 1 and

2. A dopant has been infused or implanted in areas of source and drain before manufacturing the

contacts to produce required types of conductivity. Farther we consider annealing of dopant

and/or radiation defects to infuse the dopant on required depth (in the first

case) or to decrease quantity of radiation defects (in the second case). Annealing of radiation

defects leads to spreading of distribution of concentration of dopant. To restrict the dopant

diffusion into substrate and to manufacture more thin structure it is practicably to choose

properties of heterostructure so, that dopant diffusion coefficient in the substrate should be as

smaller as it is possible, than dopant diffusion coefficient in epitaxial layer [12-18]. In this case it

is practicably to optimize annealing time [12-18]. If dopant did not achieve interface between

layers of heterostructure during annealing of radiation defects it is practicably to choose addition

annealing of dopant. Main aims of the present paper are (i) modeling of redistribution of dopant

and radiation defects; (ii) optimization of annealing time.

Fig. 1. Heterostructure with a substrate and multi-section epitaxial layer. View from top

Fig. 2. Heterostructure with a substrate and multi-section epitaxial layer. View from one side

2. Method of solution

To solve our aims we determine spatio-temporal distribution of concentration of dopant. We

determine the distribution by solving the second Fick’s law [1-4]

Substrate

GateSource Drain

Epitaxial layer


3

( ) ( )

=

x

txCD

xt

tzyxCC

∂

∂

∂

∂

∂

∂ ,,,, (1)

with boundary and initial conditions

( )0

,

0

=∂

∂

=xx

txC,

( )0

,=

∂

∂

=Lxx

txC, C (x,0)=fС (x). (2)

Here C(x,t) is 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 depends

on properties of materials of heterostructure, speed of heating and cooling of heterostructure (with

account Arrhenius law). Dependences of dopant diffusion on parameters could be approximated

by the following function [22-25]

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,1

,

,1,

V

txV

V

txV

TxP

txCTxDD LC ςςξ

γ

γ

, (3)

where DL (x,T) is the spatial (due to native inhomogeneity of heterostructure) and temperature

(due to Arrhenius law) dependences of dopant diffusion coefficient; P (x, T) is the limit of

solubility of dopant; parameter γ depends on properties of materials and could be integer in the

following interval γ ∈[1,3] [25]; V (x,t) is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibrium distribution of vacancies. Concentrational dependence

of dopant diffusion coefficient has been described in details in [25]. It should be noted, that using

of doping of materials by diffusion leads to absents of radiation damage ζ1= ζ2= 0. We determine

spatio-temporal distributions of concentrations of radiation defects by solving the following

system of equations [23,24]

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

( )( )

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

−−

=

−−

=

txVTxktxVtxITxkx

txVTxD

xt

txV

txITxktxVtxITxkx

txITxD

xt

txI

VVVIV

IIVII

,,,,,,

,,

,,,,,,

,,

2

,,

2

,,

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

(4)

with initial

ρ (x,0)=fρ (x) (5a)

and boundary conditions

( )0

,

0

=∂

∂

=xx

txI,

( )0

,=

∂

∂

=Lxx

txI,

( )0

,

0

=∂

∂

=xx

txV,

( )0

,=

∂

∂

=Lxx

txV, I(x,0)=fI (x), V(x,0)=fV (x). (5b)


4

Here ρ =I,V; I (x,t) is the spatio-temporal distribution of concentration of interstitials; Dρ(x,T) is

the diffusion coefficient of point radiation defects (vacancies and interstitials); terms V2(x,t) and

I2(x,t) corresponds to generation of divacancies and diinterstitials; kI,V(x,T), kI,I(x,T) and kV,V(x,T)

are parameters of recombination of point radiation defects and generation their simplest

complexes (divacancies and diinterstitials).

We determine spatio-temporal distributions of concentrations of divacancies ΦV (x,t) and

diinterstitials ΦI (x,t) by solving of the following system of equations [23,24]

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

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

−+

Φ=

Φ

−+

Φ=

Φ

Φ

Φ

txVTxktxVTxkx

txTxD

xt

tx

txITxktxITxkx

txTxD

xt

tx

VVV

V

V

V

III

I

I

I

,,,,,

,,

,,,,,

,,

2

,

2

,

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

(6)

with boundary and initial conditions

( )0

,

0

=∂

∂

=xx

txI,

( )0

,=

∂

∂

=Lxx

txI,

( )0

,

0

=∂

∂

=xx

txV,

( )0

,=

∂

∂

=Lxx

txV, I(x,0)=fI (x), V(x,0)=fV (x). (7)

Here DΦI(x,T) and DΦV(x,T) are diffusion coefficients of divacancies and diinterstitials; kI(x,T) and

kV (x,T) are parameters of decay of complexes of radiation defects.

To determine spatio-temporal distributions of concentrations of point radiation defects we used

recently introduced approach [16,18]. Framework the approach we transform approximations of

diffusion coefficients of point radiation defects to the following form: Dρ(x,T)=D0ρ[1+ερgρ(x,T)],

where D0ρ is the average value of the diffusion coefficients, 0≤ερ< 1, |gρ(x,T)|≤1, ρ =I,V. We also

transform parameters of recombination of point defects and generation their complexes in the

same form: kI,V(x, T)=k0I,V[1+εI,V gI,V(x,T)], kI,I(x,T)=k0I,I[1+εI,I gI,I(x,T)] and kV,V(x,T) = k0V,V [1+εV,V

gV,V(x,T)], where k0ρ1,ρ2 is the appropriate average values of the above parameters, 0≤εI,V<1, 0≤εI,I

< 1, 0≤εV,V< 1, | gI,V (x,T)|≤1, | gI,I(x,T)|≤1, |gV,V(x,T)|≤1. We introduce the following dimensionless

variables: 2

00 LtDDVI

=ϑ , VIVI

DDkL 00,0

2=ω , ( ) ( ) *,,~

ItxItxI = , χ = x/Lx, η = y/Ly, φ =

z/Lz, ( ) ( ) *,,~

VtxVtxV = , VI,

DDkL000

2

ρρρΩ = . The introduction transforms Eq.(4) and

conditions (5) in the following form

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

( )[ ] ( ) ( )

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

( )[ ] ( ) ( )

+−

−+Ω−

∂

∂+

∂

∂=

∂

∂

+−

−+Ω−

∂

∂+

∂

∂=

∂

∂

ϑχϑχχεω

ϑχχεχ

ϑχχε

χϑ

ϑχ

ϑχϑχχεω

ϑχχεχ

ϑχχε

χϑ

ϑχ

,~

,~

,1

,~

,1,

~

,1,

~

,~

,~

,1

,~

,1,

~

,1,

~

,,

2

,,

00

0

,,

2

,,

00

0

VITg

VTgV

TgDD

DV

VITg

ITgI

TgDD

DI

VIVI

VVVVVVV

VI

V

VIVI

IIIIIII

VI

I

(8)


5

( )

0,~

0

=∂

∂

=χχ

ϑχρ,

( )0

,~

1

=∂

∂

=χχ

ϑχρ, ( )

( )*

,,~

ρ

ϑχϑχρ ρf

= (9)

We determine solutions of Eqs.(8) and conditions (9) framework recently introduce [16,18]

approach as the power series

( ) ( )∑ ∑ ∑Ω=∞

=

∞

=

∞

=0 0 0

,~,~

i j kijk

kji ϑχρωεϑχρ ρρ . (10)

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

equations for the zero-order approximations of concentrations of point defects ( )ϑχ ,~

000I and

( )ϑχ ,~

000V , corrections for them ( )ϑχ ,~

ijkI and ( )ϑχ ,

~ijk

V (i ≥1, j ≥1, k ≥1) and conditions for all

functions ( )ϑχ ,~

ijkI and ( )ϑχ ,

~ijk

V (i ≥0, j ≥0, k ≥0)

( ) ( )

( ) ( )

∂

∂=

∂

∂

∂

∂=

∂

∂

2

000

2

0

0000

2

000

2

0

0000

,~

,~

,~

,~

χ

ϑχ

ϑ

ϑχ

χ

ϑχ

ϑ

ϑχ

V

D

DV

I

D

DI

I

V

V

I

;

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂=

∂

∂

∂

∂

∂

∂+

∂

∂=

∂

∂

−

−

χ

ϑχχ

χχ

ϑχ

ϑ

ϑχ

χ

ϑχχ

χχ

ϑχ

ϑ

ϑχ

,~

,,

~,

~

,~

,,

~,

~

100

0

0

2

00

2

0

000

100

0

0

2

00

2

0

000

i

V

I

Vi

I

Vi

i

I

V

Ii

V

Ii

VTg

D

DV

D

DV

ITg

D

DI

D

DI

, i ≥1;

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

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

+−∂

∂=

∂

∂

+−∂

∂=

∂

∂

ϑχϑχχεχ

ϑχ

ϑ

ϑχ

ϑχϑχχεχ

ϑχ

ϑ

ϑχ

,~

,~

,1,

~,

~

,~

,~

,1,

~,

~

000000,,2

010

2

0

0010

000000,,2

010

2

0

0010

VITgV

D

DV

VITgI

D

DI

VIVI

I

V

VIVI

V

I

;

( ) ( )

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

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

++−

−∂

∂=

∂

∂

++−

−∂

∂=

∂

∂

ϑχϑχϑχϑχχε

χ

ϑχ

ϑ

ϑχ


χ

ϑχ

ϑ

ϑχ

,~

,~

,~

,~

,1

,~

,~

,~

,~

,~

,~

,1

,~

,~

010000000010,,

2

020

2

0

0020

010000000010,,

2

020

2

0

0020

VIVITg

V

D

DV

VIVITg

I

D

DI

VIVI

V

I

VIVI

V

I

;


6

( ) ( ) ( )[ ] ( )

( ) ( ) ( )[ ] ( )

+−∂

∂=

∂

∂

+−∂

∂=

∂

∂

ϑχχεχ

ϑχ

ϑ

ϑχ

ϑχχεχ

ϑχ

ϑ

ϑχ

,~

,1,

~,

~

,~

,1,

~,

~

2

000,,2

001

2

0

0001

2

000,,2

001

2

0

0001

VTgV

D

DV

ITgI

D

DI

IIII

I

V

IIII

V

I

;

( ) ( )( )

( )

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

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

++−

−

∂

∂

∂

∂+

∂

∂=

∂

∂

++−

−

∂

∂

∂

∂+

∂

∂=

∂

∂


χ

ϑχχ

χχ

ϑχ

ϑ

ϑχ


χ

ϑχχ

χχ

ϑχ

ϑ

ϑχ

,~

,~

,~

,~

,1

,~

,,

~,

~

,~

,~

,~

,~

,1

,~

,,

~,

~

100000000100,,

010

0

0

2

110

2

0

0110

100000000100,,

010

0

0

2

110

2

0

0110

IVIVTg

VTg

D

DV

D

DV

VIVITg

ITg

D

DI

D

DI

VVVV

V

I

V

I

V

IIII

I

V

I

V

I

;

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

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

+−∂

∂=

∂

∂

+−∂

∂=

∂

∂

ϑχϑχχεχ

ϑχ

ϑ

ϑχ

ϑχϑχχεχ

ϑχ

ϑ

ϑχ

,~

,~

,1,

~,

~

,~

,~

,1,

~,

~

000001,,2

002

2

0

0002

000001,,2

002

2

0

0002

VVTgV

D

DV

IITgI

D

DI

IIII

I

V

IIII

V

I

;

( ) ( ) ( ) ( )

( )[ ] ( ) ( )

( ) ( ) ( ) ( )

( )[ ] ( ) ( )

+−

−

∂

∂

∂

∂+

∂

∂=

∂

∂

+−

−

∂

∂

∂

∂+

∂

∂=

∂

∂

ϑχϑχχε

χ

ϑχχ

χχ

ϑχ

ϑ

ϑχ

ϑχϑχχε

χ

ϑχχ

χχ

ϑχ

ϑ

ϑχ

,~

,~

,1

,~

,,

~,

~

,~

,~

,1

,~

,,

~,

~

100000

001

0

0

2

101

2

0

0101

000100

001

0

0

2

101

2

0

0101

VITg

VTg

D

DV

D

DV

VITg

ITg

D

DI

D

DI

VV

V

I

V

I

V

II

I

V

I

V

I

;

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

( )[ ] ( ) ( )

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

( )[ ] ( ) ( )

+−

−+−∂

∂=

∂

∂

+−

−+−∂

∂=

∂

∂

ϑχϑχχε

ϑχϑχχεχ

ϑχ

ϑ

ϑχ

ϑχϑχχε

ϑχϑχχεχ

ϑχ

ϑ

ϑχ

,~

,~

,1

,~

,~

,1,

~,

~

,~

,~

,1

,~

,~

,1,

~,

~

001000,,

010000,,2

011

2

0

0011

000001,,

010000,,2

011

2

0

0011

VITg

VVTgV

D

DV

VITg

IITgI

D

DI

VIVI

VVVV

I

V

VIVI

IIII

V

I

;

( )0

,~

0

=∂

∂

=x

ijk

χ

ϑχρ,

( )0

,~

1

=∂

∂

=x

ijk

χ

ϑχρ, (i ≥0, j ≥0, k ≥0);

( ) ( ) *

0000,~ ρχχρ ρf= , ( ) 00,~ =χρ

ijk (i ≥1, j ≥1, k ≥1).


7

Solutions of the obtained equations could be obtained by standard approaches (see, for example,

[26,27]). The solutions with account appropriate boundary and initial conditions could be written

as

( ) ( ) ( )∑+=∞

=1000

21,~

nnn

ecFLL

ϑχϑχρ ρρ ,

where ( ) ( )IVnI

DDne 00

22exp ϑπϑ −= , ( ) ( )

VInVDDne 00

22exp ϑπϑ −= , cn(χ) = cos (π n χ),

( ) ( )∫=1

0*

cos1

udufunFnn ρρ π

ρ;

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

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

∑ ∫ ∫∂

∂−−=

∑ ∫ ∫∂

∂−−=

∞

=

−

∞

=

−

1 0

1

0

100

0

0

00

1 0

1

0

100

0

0

00

,~

,2,~

,~

,2,~

n

i

VnnVnVn

I

V

i

n

i

InnInIn

V

I

i

dudu

uVTuguseecn

D

DV

dudu

uITuguseecn

D

DI

ϑ

ϑ

ττ

τϑχπϑχ

ττ

τϑχπϑχ

, i ≥1,

where sn(χ) = sin (π n χ);

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

=1 0

1

0000000,,010 ,

~,

~,1,2,~

nVIVInnn

duduVuITuguceecϑ

ρρ τττετϑφχϑχρ ;

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

=1 0

1

0,,

0

0

020 ,12,~n

VIVInnn

V

I TuguceecD

D ϑ

ρρ ετϑχϑχρ

( ) ( ) ( ) ( )[ ] τϑχϑχϑχϑχ dudVIVI ,~

,~

,~

,~

010000000010 +× ;

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

=1 0

1

0

2

000,,001 ,~,12,~n

nnnnduduTuguceec

ϑ

ρρρρρρ ττρετϑχϑχρ ;

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

=1 0

1

0000001,,002 ,~,~,12,~

nnnnn

duduuTuguceecϑ

ρρρρρρ ττρτρετϑχϑχρ ;

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

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

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

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

∫ ∫ ++−×

×∑−∑ ∫ ∫∂

∂−−=

∫ ∫ ++−×

×∑−∑ ∫ ∫∂

∂−−=

∞

=

∞

=

∞

=

∞

=

ϑ

ϑ

ϑ

ϑ

τετττττϑ

χττ

τϑχπϑχ

τετττττϑ

χττ

τϑχπϑχ

0

1

0,,000100100000

11 0

1

0

010

0

0

110

0

1

0,,100000000100

11 0

1

0

010

0

0

110

,1,~

,~

,~

,~

2,

~

,2,~

,1,~

,~

,~

,~

2,

~

,2,~

dudTuguVuIuVuIucee

cdudu

uVTuguseecn

D

DV

dudTuguVuIuVuIucee

cdudu

uITuguseecn

D

DI

VIVInnVnV

nn

nVnnVnVn

I

V

VIVInnInI

nn

nInnInIn

V

I

;


8

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

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

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

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

∑ ∫ ∫ +−−

−∑ ∫ ∫∂

∂−−=

∑ ∫ ∫ +−−

−∑ ∫ ∫∂

∂−−=

∞

=

∞

=

∞

=

∞

=

1 0

1

0100000,,

1 0

1

0

001

0

0

101

1 0

1

0000100,,

1 0

1

0

001

0

0

101

,~

,~

,12

,~

,2,~

,~

,~

,12

,~

,2,~

nVIVInnVnVn

nVnnVnVn

I

V

nVIVInnInIn

nInnInIn

V

I

duduVuITuguceec

dudu

uVTuguseecn

D

DV

duduVuITuguceec

dudu

uITuguseecn

D

DI

ϑ

ϑ

ϑ

ϑ

τττετϑχ

ττ

τϑχπϑχ

τττετϑχ

ττ

τϑχπϑχ

;

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

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

( )[ ] ( ) ( )

++

∑ ∫ ∫ ++−−=

++

∑ ∫ ∫ ++−−=

∞

=

∞

=

.,~

,~

,1

,~

,~

,12,~

,~

,~

,1

,~

,~

,12,~

000001,,

1 0

1

0010000,,011

000001,,

1 0

1

0010000,,011

τττε

ττετϑχϑχ

τττε

ττετϑχϑχ

ϑ

ϑ

duduVuITug

uVuVTuguceecV

duduVuITug

uIuITuguceecI

VIVI

nVVVVnnVnVn

VIVI

nIIIInnInIn

Farther we determine spatio-temporal distribution of concentration of complexes of point

radiation defects. To calculate the distribution we transform approximation of diffusion

coefficient to the following form: DΦρ(x,T)=D0Φρ[1+εΦρgΦρ(x,T)], where D0Φρ are the average

values of diffusion coefficient. After this transformation Eqs.(6) transforms to the following form

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

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

−+

Φ

+=Φ

−+

Φ

+=Φ

ΦΦΦ

ΦΦΦ

txVTxktxVTxkx

txTxg

xD

t

tx

txITxktxITxkx

txTxg

xD

t

tx

VVV

V

VVV

V

III

I

III

I

,,,,,

,1,

,,,,,

,1,

2

,0

2

,0

∂

∂ε

∂

∂

∂

∂

∂

∂ε

∂

∂

∂

∂

Let us determine solutions of the above equations as the following power series

( ) ( )∑ Φ=Φ∞

=Φ

0

,,i

i

itxtx ρρρ ε . (11)

Substitution of the series (11) into Eqs.(6) and conditions for them gives us possibility to obtain

equations for initial-order approximations of concentrations of complexes, corrections for them

and conditions for all equations in the following form

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

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

−+Φ

=Φ

−+Φ

=Φ

Φ

Φ

txVTxktxVTxkx

txD

t

tx

txITxktxITxkx

txD

t

tx

VVV

V

V

V

III

I

I

I

,,,,,,

,,,,,,

2

,2

0

2

0

0

2

,2

0

2

0

0

∂

∂

∂

∂

∂

∂

∂

∂

;


9

( ) ( )( )

( )

( ) ( )( )

( )

Φ+

Φ=

Φ

Φ+

Φ=

Φ

−

ΦΦΦ

−

ΦΦΦ

x

txTxg

xD

x

txD

t

tx

x

txTxg

xD

x

txD

t

tx

iV

VV

iV

V

iV

iI

II

iI

I

iI

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

,,

,,

,,

,,

1

02

2

0

1

02

2

0

, i≥1;

( )0

,

0

=∂

Φ∂

=x

i

x

txρ,

( )0

,=

∂

Φ∂

=Lx

i

x

txρ, i≥0; Φρ0(x,0)=fΦρ (x), Φρi(x,0)=0, i≥1.

Solutions of the above equations could be written as

( ) ( ) ( )+∑+=Φ∞

=ΦΦ

10

21,

nnn

texcFLL

txρρρ

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

=ΦΦ

1 0 0

2

,,,,,

2

n

t L

IIInnnnduduITukuITukucetexcn

Lττττ

ρρ,

where ( ) ( )2

0

22exp LtDnten ρρ

π ΦΦ −= , ( ) ( )∫= ΦΦ

L

nnudufucF

0ρρ

, cn(x) = cos (π n x/L);

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

∑ ∫ ∫Φ

−−=Φ∞

=

−

ΦΦΦ1 0 0

1

2

,,

2,

n

t LiI

nnnnidud

u

uTugusetexcn

Ltx τ

∂

τ∂τ

π ρ

ρρρρ , i≥1,

where sn(x) = sin (π n x/L)

Spatio-temporal distribution of concentration of dopant we determine framework the recently introduced approach. Framework the approach we transform approximation of dopant diffusion

coefficient to the following form: DL(x,T)=D0L[1+εLgL(x,T)], where D0L is the average value of

dopant diffusion coefficient, 0≤εL< 1, |gL(x,T)|≤1. Farther we determine solution of the Eq.(1) as the following power series

( ) ( )∑ ∑=∞

=

∞

=0 1

,,i j

ij

ji

LtxCtxC ξε .

Substitution of the series into Eq.(1) and conditions (2) gives us possibility to obtain zero-order

approximation of dopant concentration C00(x,t), corrections for them Cij(x,t) (i ≥1, j ≥1) and

conditions for all functions. The equations and conditions could be written as

( ) ( )2

00

2

0

00,,

x

txCD

t

txCL

∂

∂=

∂

∂;

( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂=

∂

∂ −

x

txCTxg

xD

x

txCD

t

txCi

LL

i

L

i,

,,,

10

02

0

2

0

0 , i ≥1;

( ) ( ) ( )( )

( )

∂

∂

∂

∂+

∂

∂=

∂

∂

x

txC

TxP

txC

xD

x

txCD

t

txCLL

,

,

,,,0000

02

01

2

0

01

γ

γ

;


10

( ) ( )( )

( )( )

( )+

∂

∂

∂

∂+

∂

∂=

∂

∂ −

x

txC

TxP

txCtxC

xD

x

txCD

t

txCLL

,

,

,,

,,00

1

00

0102

02

2

0

02

γ

γ

( )( )

( )

∂

∂

∂

∂+

x

txC

TxP

txC

xD

L

,

,

,0100

0 γ

γ

;

( ) ( ) ( ) ( )( )

( )+

∂

∂

∂

∂+

∂

∂=

∂

∂ −

x

txC

TxP

txCtxC

xD

x

txCD

t

txCLL

,

,

,,

,, 00

1

00

1002

11

2

0

11

γ

γ

( )( )

( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

x

txCTxg

xD

x

txC

TxP

txC

xD

LLL

,,

,

,

,01

0

1000

0 γ

γ

;

( )0

,

0

==x

ij

x

txC

∂

∂,

( )0

,=

=Lx

ij

x

txC

∂

∂, i ≥0, j ≥0; C00(x,0)=fC (x), Cij(x,0)=0, i ≥1, j ≥1.

Solutions of the equations with account appropriate boundary and initial conditions could be

obtained by standard approaches (see, for example, [26,27]). The solutions are

( ) ( ) ( )∑+=∞

=100

21,

nnCnC

texcFLL

txC ,

where ( ) ( )2

0

22exp LtDnteCnC

π−= , ( ) ( )∫=L

CnnCudufucF

0

;

( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫

∂

∂−−=

∞

=

−

1 0 0

10

20

,,

2,

n

t Li

LnnCnCnnCidud

u

uCTugusetexcFn

LtxC τ

ττ

π, i ≥1;

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

( )∑ ∫ ∫

∂

∂−−=

∞

=1 0 0

0000

201

,

,

,2,

n

t L

nnCnCnnCdud

u

uC

TuP

uCusetexcFn

LtxC τ

τττ

πγ

γ

;

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

( )−∑ ∫ ∫

∂

∂−−=

∞

=

−

1 0 0

00

1

00

01202

,

,

,,

2,

n

t L

nnCnCnnCdud

u

uC

TuP

uCuCusetexcFn

LtxC τ

ττττ

πγ

γ

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

( )∑ ∫ ∫

∂

∂−−

∞

=1 0 0

0100

2

,

,

,2

n

t L

nnCnCnnCdud

u

uC

TuP

uCusetexcFn

Lτ

τττ

πγ

γ

;

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∑ ∫ ∫

∂

∂−−=

∞

=

∞

= 12

1 0 0

01

211

2,,

2,

nnC

n

t L

LnnCnCnnCFn

Ldud

u

uCTugusetexcFn

LtxC

πτ

ττ

π

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

( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫∂

∂−×

∞

=1 02

0 0

1000 2,

,

,

n

t

nCnCnnC

t L

nnCnCnetexcFn

Ldud

u

uC

TuP

uCusetexc τ

πτ

τττ

γ

γ

( ) ( )( )

( )( )

∫∂

∂×

−L

ndud

u

uC

TuP

uCuCus

0

00

1

00

10

,

,

,, τ

τττ

γ

γ

.

Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has

been done analytically by using the second-order approximations framework recently introduced

power series. The approximation is enough good approximation to make qualitative analysis and

to obtain some quantitative results. All obtained analytical results have been checked by

comparison with results, calculated by numerical simulation.


11

3. Discussion

In this section based on obtained in the previous section relations we analysed dynamics of

redistribution of dopant, which was infused in heterostructure from Figs. 1 and 2 by diffusion or

ion implantation, during annealing of the dopant and/or radiation defects. We take into account

radiation damage during consideration dopant redistribution after ion doping of heterostructure.

The Fig. 3 shows typical distributions of concentrations of infused dopant in heterostructure in

direction, which is perpendicular to interface between epitaxial layer and substrate. The

distributions have been calculated under condition, when value of dopant diffusion coefficient in

epitaxial layer is larger, than value of dopant diffusion coefficient in substrate. Analogous

distributions of dopant concentrations for ion doping are presented in Fig. 4. The Figs. 3 and 4

shows, that presents of interface between layers of heterostructure under above condition gives us

possibility to obtain more thin field-effect transistor. At the same time one can find increasing of

homogeneity of dopant distribution in doped area.

Fig.3. Distributions of concentration of infused dopant in heterostructure from Figs. 1 and 2 in

direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of

number of curve corresponds to increasing of difference between values of dopant diffusion

coefficient in layers of heterostructure under condition, when value of dopant diffusion

coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate

Fig.4. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 and 2 in

direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3

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


12

corresponds to annealing time Θ = 0.0048(Lx2+Ly

2+Lz

2)/D0. Curves 2 and 4 corresponds to

annealing time Θ = 0.0057(Lx2+Ly

2+Lz2)/D0. Curves 1 and 2 corresponds to homogenous sample.

Curves 3 and 4 corresponds to heterostructure under condition, when value of dopant diffusion

coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate

Fig.5. Spatial distributions of concentration of infused dopant in heterostructure from Figs. 1 and

2. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of dopant

for different values of annealing time. Increasing of number of curve correspondsincreasing of

annealing time It should be noted, that in this situation optimization of annealing time required.

Reason of this optimization is following. If annealing time is small, dopant did not achieve

interface between layers of heterostructure. In this situation inhomogeneity of dopant

concentration increases. If annealing time is large, dopant will infused too deep. Optimal

annealing time we determine framework recently introduce approach [12-18,19,20]. Framework

the criterion we approximate real distribution of dopant concentration by stepwise function (see

Figs. 5 and 6). Farther we determine the following mean-squared error between real distribution

of dopant concentration and the stepwise approximation

( ) ( )[ ]∫ −Θ=L

xdxxCL

U0

,1

ψ , (12)

where ψ (x) is the approximation function, which presented in Figs. 5 and 6 as curve 1. We

determine optimal annealing time by minimization mean-squared error (12). Dependences of

optimal annealing time on parameters for diffusion a ion types of doping are presented on Figs. 7

and 8. Optimal annealing time, which corresponds to ion doping, is smaller, than the same time

for doping by diffusion. Reason of this difference is necessity to anneal radiation defects.

Optimization of annealing time for ion doping of materials should be done only in this case, when

dopant did not achieves interface between layers of heterostructure during annealing of radiation

defects.

C(x

,Θ)

0 Lx

2

13

4


13

Fig.6. Spatial distributions of concentration of implanted dopant in heterostructure from Figs. 1

and 2. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of

dopant for different values of annealing time. Increasing of number of curve corresponds to

increasing of annealing time

Fig. 7. Dependences of dimensionless optimal annealing time for doping by diffusion, which have

been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the

dependence of dimensionless optimal 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 is the

dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and ξ =

γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of

parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal

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

x

C(x

,Θ)

1

23

4

0 L

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


14

Fig.8. Dependences of dimensionless optimal annealing time for doping by ion implantation,

which have been obtained by minimization of mean-squared error, on several parameters. Curve 1

is the dependence of dimensionless optimal 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 is

the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and

ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of

parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal

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

4. CONCLUSIONS

In this paper we consider an approach to manufacture more thin field-effect transistors. The

approach based on doping of heterostructure by diffusion or ion implantation and optimization of

annealing of dopant and/or radiation defects.

Acknowledgments

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,

grant of Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific

research.

REFERENCES [1]I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980).

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

[3]V.G. Gusev, Yu.M. Gusev. Electronics. (Moscow: Vysshaya shkola, 1991, in Russian).

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

[5]A. Kerentsev, V. Lanin. Power Electronics. Issue 1. P. 34-38 (2008).

[6]A.N. Andronov, N.T. Bagraev, L.E. Klyachkin, S.V. Robozerov. Semiconductors. Vol.32 (2). P. 137-

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[7]S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p-n-junctions in Si prepared by pulse

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[8]V.I. Mazhukin, V.V. Nosov, U. Semmler. "Study of heat and thermoelastic fields in semiconductors at

pulsed processing" Mathematical modelling. 12 (2), 75 (2000).

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0.00

0.04

0.08

0.12

Θ D

0 L

-2

3

2

4

1


15

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

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Drozdov, V.D. Skupov. "Diffusion processes in semiconductor structures during microwave

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[12] E.L. Pankratov. " Redistribution of dopant during microwave annealing of a multilayer structure for

production p-n-junction " J. Appl. Phys. Vol. 103 (6). P. 064320-064330 (2008).

[13] E.L. Pankratov. " Optimization of near-surficial annealing for decreasing of depth of p-n-junction in

semiconductor heterostructure " Proc. of SPIE. Vol. 7521, 75211D (2010).

[14] E.L. Pankratov. " Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure

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

[15] E.L. Pankratov. "Influence of mechanical stress in semiconductor heterostructure on density of p-n-

junctions" Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012).

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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

Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in

Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of

Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny

Novgorod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor

course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in

area of his researches.

Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of

village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school

“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of


16

Architecture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same

time she is a student of courses “Translator in the field of professional communication” and “Design

(interior art)” in the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant

MK-548.2010.2). She has 29 published papers in area of her researches.

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