Using inhomogeneity of heterostructure and optimization of annealing to decrease dimensions...

International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.3, June 2014

DOI:10.5121/ijcsa.2014.4303 25

USING INHOMOGENEITYOFHETEROSTRUCTUREANDOPTIMIZATIONOF ANNEALING TODE-CREASEDIMENSIONSMULTYEMITTERHETERO-

TRANSISTORS

E.L. Pankratov1 and E.A. Bulaeva 2

1Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,Russia

2Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'inskystreet, Nizhny Novgorod, 603950, Russia

ABSTRACT

Framework this paper we discussed an approach to manufacture a multiemitter heterotransistor. The in-troduced approach is a branch of recently introduced approach and based on doping by diffusion or by ionimplantation of required part of heterostructure and optimization of dopant and/or radiation defects. Theheterostructure should has special configuration.

KEYWORDS

Multiemitter heterotransistor, optimization of manufacturing of transistor, analytical approach to modeltechnological process

1. INTRODUCTION

Logical elements often include into itself multiemitter transistors [1-4]. To manufacture bipolartransistors it could be used dopant diffusion, ion doping or epitaxial layer [1-8]. It is attracted aninterest increasing of sharpness of p-n-junctions, which include into bipolar transistors and de-creasing of dimensions of the transistors, which include into integrated circuits [1-4]. Dimensionsof transistors will be decreased by using inhomogenous distribution of temperature during laser ormicrowave types of annealing [9,10], using defects of materials (for example, defects could begenerated during radiation processing of materials) [11-14], native inhomogeneity (multilayerproperty) of heterostructure [12-14].

We consider manufacture of multiemitter heterotransistor based on a hetero structure, which con-sist of a substrate and three epitaxial layers (see Fig. 1). Some dopants are infused in the epitaxiallayers to manufacture p and n types of conductivities during manufacture a multiemitter transis-tor. Let us consider two cases: infusion of dopant by diffusion or ion implantation. Farther in thefirst case we consider annealing of dopant so long, that after the annealing the dopants shouldachieve interfaces between epitaxial layers. In this case one can obtain increasing of sharpness ofp-n-junctions [12-14]. The increasing of sharpness could be obtained under conditions, when val-ues of dopant diffusion coefficients in last epitaxial layers is larger, than values of dopant diffu-sion coefficient in average epitaxial layer. At the same time homogeneity of dopant distributionsin doped areas increases. It is attracted an interest higher doping of last epitaxial layers, than dop-


26

ing of average epitaxial layer, because the higher doping gives us possibility to increase sharpnessof p-n-junctions. It is also practicably to choose materials of epitaxial layers and substrate so;those values of dopant diffusion coefficients in the substrate should be smaller, than in the epitax-ial layer. In this situation one can obtain thinner transistor. In the case of ion doping of hetero-structure it should be done annealing of radiation defects. It is practicably to choose so regimes ofannealing of radiation defects, that dopant should achieves interface between epitaxial layers. Atthe same time the dopant could not diffuse into another epitaxial layer. If dopant did not achievesinterface between epitaxial layers during annealing of radiation defects, additional annealing ofdopant required.

Main aims of the present paper are analysis of dynamics of redistribution of dopants in consideredheterostructure (Figs. 1) and optimization of annealing times of dopants. In some recent papersanalogous analysis have been done [4,10,12-14]. However we can not find in literature any simi-lar heterostructures as we consider in the paper.

Substrate

Epitaxial layer 1

Epitaxial layer 2

Epitaxial layer 3

n-type dopant p-type dopant n-type dopant

Fig. 1a. Heterostructure, which consist of a substrate and three epitaxial layers. View from above

Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3

n-type dopantp-type dopant

n-type dopant 1

n-type dopant 2

n-type dopant 3n-type dopant 4

Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers. View from one side


27

2. Method of solution

We analyzed spatio-temporal distribution of dopant based on the second Fick’s law [1-4]

( ) ( ) ( ) ( )

+

+

=

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

tzyxCCCC

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

. (1)

Boundary and initial conditions for the equation are

( )0

,,,

0

=∂

∂

=xx

tzyxC,

( )0

,,, =∂

∂

= xLxx

tzyxC,

( )0

,,,

0

=∂

∂

=yy

tzyxC,

( )0

,,, =∂

∂

= yLyy

tzyxC,

( )0

,,,

0

=∂

∂

=zz

tzyxC,

( )0

,,, =∂

∂

= zLzz

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

We designate spatio-temporal distribution of concentration of dopant as C(x,y,z,t); as the currentcoordinates and times x, y, z and t; the dopant diffusion coefficient as DС. Dependences of dopantdiffusion coefficient on coordinates, temperature of annealing (with account Arrhenius low), spa-tio-temporal distributions of concentrations of dopants and radiation defects could be approx-imated by the following relation [15,16]

( ) ( )( )

( ) ( )( )

++

+= 2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD LC

. (3)

Factor DL (x,y,z,T) describes inhomogeneity of heterostructure and temperature dependence ofdopant diffusion coefficient due to Arrhenius law. We designate the limit of solubility of dopantas P (x,y,z,T); the spatio-temporal distribution of concentration of radiation vacancies V (x,y,z,t);the equilibrium distribution of vacancies V*. Parameter depends of properties of materials andcould be integer in the interval ∈[1,3] [15]. Dependence of dopant diffusion coefficient on do-pant concentration has been discussed in details in [15]. It should be noted, that radiation damageabsents in the case of diffusion doping of haterostructure. In this situation 1= 2= 0. Spatio-temporal distributions of concentrations of point radiation defects could be determine by solutionof the following system of equations [17-19]

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂∂∂+

∂

∂∂∂=

∂∂

tzyxIy

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII ,,,

,,,,,,

,,,,,,

,,, 2

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

tzyxITzyxD

zTzyxk VIIII ,,,,,,,,,

,,,,,,,,, ,, −

∂

∂∂∂+× (4)

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂∂∂+

∂

∂∂∂=

∂∂

tzyxVy

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV ,,,

,,,,,,

,,,,,,

,,, 2

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

tzyxVTzyxD

zTzyxk

VIVVV,,,,,,,,,

,,,,,,,,, ,, −

∂

∂∂∂+×

with initial

(x,y,z,0)=f (x,y,z) (5a)

and boundary conditions


28

( )0

,,, =∂

∂

= yLyy

tzyx,

( )0

,,,

0

=∂

∂

=zz

tzyx,

( )0

,,, =∂

∂

= zLzz

tzyx. (5b)

We designate the spatio-temporal distribution of concentration of interstitials as I (x,y,z,t); =I,V;the diffusion coefficients of interstitials and vacancies as D(x,y,z,T); terms V2(x,y, z,t) andI2(x,y,z,t) correspond to generation of divacancies and analogous complexes of interstitials; kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) parameters of recombination of point radiation defects andgeneration of complexes.

Spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and analogous com-plexes of interstitials I(x,y,z,t) will be determined by solving the following systems of equations[17-19]

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyx I

I

I

I

I

,,,

,,,,,,

,,,,,,

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

tzyxTzyxD

z III

I

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

,,, 2

, −+

Φ+ Φ

(6)

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyx V

V

V

V

V

,,,

,,,,,,

,,,,,,

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

tzyxTzyxD

z VVV

V

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

,,, 2

, −+

Φ+ Φ

with boundary and initial conditions

( )0

,,,

0

=∂

Φ∂

=xx

tzyx ,( )

0,,,

=∂

Φ∂

= xLxx

tzyx ,( )

0,,,

0

=∂

Φ∂

=yy

tzyx ,( )

0,,,

=∂

Φ∂

= yLyy

tzyx ,

( )0

,,,

0

=∂

Φ∂

=zz

tzyx ,( )

0,,,

=∂

Φ∂

= zLzz

tzyx , I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV (x,y,z). (7)

Here DI(x,y,z,T) and DV(x,y,z,T) are diffusion coefficients of complexes of point radiation de-fects; kI(x,y,z,T) and kV(x,y,z,T) are parameters of decay of the complexes.

We solved the above boundary problems by using method of averaging of function corrections[13-15,20] with decreased quantity of radiation defects [21]. Framework the approach we consid-er the initial-order approximations of concentrations of dopant and radiation defects as solutionsof Eqs. (1), (4) and (6) with averaged values of diffusion coefficient D0L, D0I, D0V, D0I, D0V andzero values of parameters of recombination of defects, generation and decay of their complexes

( ) ( ) ( ) ( ) ( )∑+=∞

=1

01

2,,,

nnCnnnnC

zyxzyx

C tezcycxcFLLLLLL

FtzyxC , ( ) ×+=

zyxzyx

I

LLLLLL

FtzyxI

2,,, 0

1

( ) ( ) ( ) ( )∑×∞

=1nnInnnnI tezcycxcF , ( ) ( ) ( ) ( ) ( )∑+=

∞

=1

01

2,,,

nnVnnnnC

zyxzyx

C tezcycxcFLLLLLL

FtzyxV ,


29

( ) ( ) ( ) ( ) ( )∑+=Φ∞

=ΦΦ

Φ

1

01

2,,,

nnnnnn

zyxzyxI tezcycxcF

LLLLLL

Ftzyx

II

I , ( ) +=Φ Φ

zyxV LLL

Ftzyx V0

1 ,,,

( ) ( ) ( ) ( )∑+∞

=ΦΦ

1

2

nnnnnn

zyx

tezcycxcFLLL VV

,

where ( )

++−=

222022 111

expzyx

nLLL

tDnte ; ( ) ( ) ( ) ( )∫ ∫ ∫=x y zL L L

nnnn udvdwdwvufvcvcucF0 0 0

,, ;

cn()=cos (n/L); D0L, D0I, D0V, D0I, D0V are the averaged values of diffusion coeffi-cients. Standard procedure of method of averaging of function corrections [13-15,20,21]gives us possibility to calculate approximations of the second- and higher orders of do-pant and radiation defect concentrations. The approach based on replacement of the re-quired functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), I(x,y,z,t), V(x,y,z,t) on the sum of aver-age value of the n-th order approximations and approximations of the n-1-th orders, i.e.n+n-1(x,y,z,t) to calculate the n-th order approximations of the above functions. In thissituation relations of the second-order approximations of the required concentrationscould be written as

( ) ( ) ( )( )

( )[ ]( )

×

+

+

++=

∗∗ TzyxP

tzyxC

V

tzyxV

V

tzyxV

xt

tzyxC C

,,,

,,,1

,,,,,,1

,,, 122

2

212

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

×

+++

×

∗∗ 2

2

211 ,,,,,,

1,,,,,,

,,,V

tzyxV

V

tzyxVTzyxD

yx

tzyxCTzyxD LL

( )[ ]( )

( ) ( ) ( )

×+

++×

z

tzyxCTzyxD

zy

tzyxC

TzyxP

tzyxCL

C

,,,,,,

,,,

,,,

,,,1 1112

( ) ( )( )

( )[ ]( )

++

++×

∗∗ TzyxP

tzyxC

V

tzyxV

V

tzyxV C

,,,

,,,1

,,,,,,1 12

2

2

21

(8)

( ) ( ) ( ) ( ) ( )

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

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

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

( ) ( ) ( )[ ]

+−×

×++−

+

+

+

=

+−×

×++−

+

+

+

=

212,,

12121

112

212,,

12121

112

,,,,,,,,,

,,,,,,,,,

,,,

,,,,,,

,,,,,,

,,,

,,,,,,,,,

,,,,,,,,,

,,,

,,,,,,

,,,,,,

,,,

tzyxVTzyxkTzyxk

tzyxVtzyxIz

tzyxVTzyxD

z

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxV

tzyxITzyxkTzyxk

tzyxVtzyxIz

tzyxITzyxD

z

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxI

VVVVI

VIV

VV

IIIVI

VII

II

(9)


30

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

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

( ) ( ) ( ) ( )

( ) ( ) ( )

−×

×+

Φ+

Φ×

×+

Φ=Φ

−×

×+

Φ+

Φ×

×+

Φ=Φ

Φ

ΦΦ

Φ

ΦΦ

tzyxVTzyxktzyxV

Tzyxkz

tzyxTzyxD

zy

tzyx

TzyxDyx

tzyxTzyxD

xt

tzyx

tzyxITzyxktzyxI

Tzyxkz

tzyxTzyxD

zy

tzyx

TzyxDyx

tzyxTzyxD

xt

tzyx

V

VVVI

VV

I

IIII

II

V

VV

I

II

,,,,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

2

,11

12

2

,11

12

(10)

We obtain the second-order approximations of the required concentrations by integration of theleft and right sides of the above relations. The relations could be written as

( ) ( ) ( )( )

( )[ ]( )

∫ ×

++

++=

∗∗

tC

TzyxP

zyxC

V

zyxV

V

zyxV

xtzyxC

0

122

2

212,,,

,,,1

,,,,,,1,,,

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

∫ ×

++++

×

∗∗

t

CLV

zyxV

V

zyxV

yzyxfd

x

zyxCTzyxD

02

2

211 ,,,,,,

1,,,,,

,,,

( ) ( )[ ]( )

( ) ( )

∫ ×+

++×

t

LC

L TzyxDz

dy

zyxC

TzyxP

zyxCTzyxD

0

112 ,,,,,,

,,,

,,,1,,,

( ) ( )( )

( )[ ]( )

( )

+

+

++×

∗∗

dz

zyxC

TzyxP

zyxC

V

zyxV

V

zyxV C ,,,

,,,

,,,1

,,,,,,1 112

2

2

21 (8a)

( ) ( ) ( ) ( ) ( )+

∫+

∫=

t

I

t

I dy

zyxITzyxD

yd

x

zyxITzyxD

xtzyxI

0

1

0

12

,,,,,,

,,,,,,,,,

( ) ( ) ( ) ( )[ ] −∫ +−

∫+

t

III

t

I dzyxITzyxkdz

zyxITzyxD

z 0

212,

0

1 ,,,,,,,,,

,,,

( ) ( )[ ] ( )[ ] ( )zyxfdzyxVzyxITzyxk I

t

VIVI ,,,,,,,,,,,0

1212, +∫ ++− (9a)

( ) ( ) ( ) ( ) ( )+

∫+

∫=

t

V

t

V dy

zyxVTzyxD

yd

x

zyxVTzyxD

xtzyxV

0

1

0

12

,,,,,,

,,,,,,,,,

( ) ( ) ( ) ( )[ ] −∫ +−

∫+

t

IVV

t

V dzyxVTzyxkdz

zyxVTzyxD

z 0

212,

0

1 ,,,,,,,,,

,,,


31

( ) ( )[ ] ( )[ ] ( )zyxfdzyxVzyxITzyxk V

t

VIVI ,,,,,,,,,,,0

1212, +∫ ++−

( ) ( ) ( ) ( ) ( ) +

∫

Φ+

∫

Φ=Φ ΦΦ

tI

tI

I dy

zyxTzyxD

yd

x

zyxTzyxD

xtzyx

II0

1

0

12

,,,,,,

,,,,,,,,,

( ) ( ) ( ) ( ) ( )−+∫+

∫

Φ+ ΦΦ zyxfdzyxITzyxkdz

zyxTzyxD

z II

t

II

tI ,,,,,,,,

,,,,,,

0

2,

0

1

( ) ( )∫−t

I dzyxITzyxk0

,,,,,, (10a)

( ) ( ) ( ) ( ) ( )+

∫

Φ+

∫

Φ=Φ ΦΦ

tV

tV

V dy

zyxTzyxD

yd

x

zyxTzyxD

xtzyx

VV0

1

0

12

,,,,,,

,,,,,,,,,

( ) ( ) ( ) ( ) ( )−+∫+

∫

Φ+ ΦΦ zyxfdzyxVTzyxkdz

zyxTzyxD

z VV

t

VV

tV ,,,,,,,,

,,,,,,

0

2,

0

1

( ) ( )∫−t

V dzyxVTzyxk0

,,,,,, .

We determined average values of the second-orders approximations 2 of the determining func-tions by using the standard relation [13-15,20,21]

( ) ( )[ ]∫ ∫ ∫ ∫ −Θ

=Θ

0 0 0 0122 ,,,,,,

1 x y zL L L

zyx

tdxdydzdtzyxtzyxLLL

. (11)

To obtain the required relations for the values 2 we substitute the relations (8a)-(10a) into therelation (11)

( )∫ ∫ ∫=x y zL L L

Czyx

С xdydzdzyxfLLL 0 0 0

2 ,,1 , (12)

( ) [{ −+−−+++= 1120102002

002100100

2 412

1IVIIIVVIIIVVIIIV

III AAAAAAA

A

( )00

00210012

1

0 0 0 2

1,,

1

II

IVVIIIVL L L

Izyx A

AAAxdydzdzyxf

LLL

x y z +++−

∫ ∫ ∫− (13a)

( )4

3134

23

42 4

442

1

B

AB

A

ByByB

AB

BV

+−

−+−+= , (13b)

where ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−ΘΘ

=Θ

0 0 0 011, ,,,,,,,,,

1 x y zL L Lji

bazyx

abij tdxdydzdtzyxVtzyxITzyxktLLL

A ,

( )2

00002

002

002

004 2 VVIIIVIVIV AAAAAB −−= , −++= 2001000

30001

200003 IVIIIVIVIVIVIV AAAAAAAB


32

( ) ( ) ( )[ ]−++−+++−− 121224 101000100100000100002

00 VVIVIIIIIVIVIVIVVVIIIV AAAAAAAAAAA2

0001002

000010 24 IVIVIVIVIIIV AAAAAA +− , ( ){ ×++++= 002

012

002

10012

002 1 IVIVIVIIIVIV AAAAAAB

( )

×−−−−++×

zyxIIIVIIIIIVIIIVIVIVIVIV LLL

AAAAAAAAAAA1

442 2011000010100001000000

( ) ( ) ([{ ++−+++

∫ ∫ ∫× 12122,, 100010010000010 0 0

IVIIIIIVIVIVIV

L L L

I AAAAAAAxdydzdzyxfx y z

)] ( ) ( ) (

−−∫ ∫ ∫+++++ 2000

0 0 0100101

210 2,,

212 VVII

L L L

Vzyx

IIIVIVVV AAxdydzdzyxfLLL

AAAAx y z

) ( )] ( ) ([ ++−++++++− 122121 1000000110010010010111 IVIIIVIVIIIVIVIIIVIVIV AAAAAAAAAAA

)]}10VVA+ , ( ) ( )

×∫ ∫ ∫−−++=

x y zL L L

Ix

IVIIIVIVIV xdydzdzyxfL

AAAAAB0 0 0

112

100101001 ,,1

812

( )−−++++

−× 00101000010000

20100010020 4

1IIIVIIIVIVIVIVIVIIIVIVII

zy

AAAAAAAAAAAALL

( ) ( ) ( )

+−−+++∫ ∫ ∫− 112000100101

0 0 0

00 21,,2

2 IVVVIIIIIVIV

L L L

Izyx

II AAAAAAxdydzdzyxfLLL

A x y z

( )] ( ) ( )[ ]0001001010100100100101 212121 IVIVIIVVIVIIIVIVIIIVIV AAAAAAAAAAA +++−+++++ ,

( ) ( +++

−∫ ∫ ∫+= 1001

20111

0 0 020

201000 ,,

14 IIIVIVIV

L L L

Izyx

IIIVII AAAAxdydzdzyxfLLL

AAABx y z

) ( ) ( ) ( )

+−−+++∫ ∫ ∫−+ 112000100101

0 0 0

002 21,,2

1 IVVVIIIIIVIV

L L L

Vzyx

II AAAAAAxdydzdzyxfLLL

A x y z

( )]2100101 1 IIIVIV AAA +++ ,

623 323 32 B

qpqqpqy +++−−+= , ( )×−= 031 82 BBBq

( )8

4

21648

21

2320

322 BBBBBB −−++× , ( )[ ] 722823 2

2031 BBBBp −−= , 223 48 BByA −+= ,

( ) ( ) ( ) +∫ ∫ ∫ ∫−ΘΘ

−=Θ

Φ0 0 0 0

202 ,,,,,,1 x y z

I

L L L

Izyx

II tdxdydzdtzyxITzyxktLLL

A

( )∫ ∫ ∫+ Φ

x y zL L L

Izyx

xdydzdzyxfLLL 0 0 0

,,1

(14)

( ) ( ) ( ) +∫ ∫ ∫ ∫−ΘΘ

−=Θ

Φ0 0 0 0

202 ,,,,,,1 x y z

V

L L L

Vzyx

VV tdxdydzdtzyxVTzyxktLLL

A

( )∫ ∫ ∫+ Φ

x y zL L L

Vzyx

xdydzdzyxfLLL 0 0 0

,,1 .

The considered substitution gives us possibility to obtain equation for parameter 2C. Solution ofthe equation depends on value of parameter . We analyzed spatio-temporal distributions of con-centrations of dopant and radiation defects by using their the second-order approximations, whichhave been calculated framework the method of averaged of function corrections with decreasedquantity of iterative steps. The calculated approximations are usually enough good approxima-


33

tions to make qualitative analysis and obtain some quantitative results. Results of analytical cal-culation have been checked by comparison with results of numerical simulation.

3. Discussion

In this section we analyzed dynamic of redistribution of dopant and radiation defects in the consi-dered heterostructure during their annealing by using calculated in the previous section relations.Typical distributions of concentrations of dopant in heterostructures are presented on Figs. 2 and3 for diffusion and ion types of doping, respectively. These distributions have been calculated forthe case, when value of dopant diffusion coefficient in doped area is larger, than in nearest areas.The figures show, that inhomogeneity of heterostructure gives us possibility to increase sharpnessof p-n- junctions. At the same time one can find increasing homogeneity of dopant distribution indoped part of epitaxial layer. Increasing of sharpness of p-n-junctions leads to decreasing theirswitching time. The second effect leads to decreasing local heating of materials during function-ing of p-n-junction or decreasing of dimensions of the p-n-junction for fixed maximal value oflocal overheat. In the considered situation we shall optimize of annealing to choose compromiseannealing time of infused dopant. If annealing time is small, the dopant has no time to achievenearest interface between layers of heterostructure. In this situation distribution of concentrationof dopant is not changed. If annealing time is large, distribution of concentration of dopant is toohomogenous.

Fig.2. Distributions of concentration of infused dopant in heterostructure from Fig. 1 in direction, which isperpendicular to interface between epitaxial layer substrate. The distributions have been calculated under

condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffu-sion coefficient in substrate. Increasing of number of curve corresponds to increasing of difference between

values of dopant diffusion coefficient in layers of heterostructure


34

x0.0

0.5

1.0

1.5

2.0

C(x,Θ

)

234

1

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

Epitaxial layer Substrate

Fig.3. Distributions of concentration of implanted dopant in heterostructure from Fig. 1 in direction, whichis perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 corresponds to annealing

time Θ = 0.0048(Lx2+Ly

2+Lz2)/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 diffusioncoefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate

Ion doping of materials leads to generation radiation defects. After finishing this process radiationdefects should be annealed. The annealing leads to spreading of distribution of concentration ofdopant. In the ideal case dopant achieves nearest interface between materials. If dopant has notime to achieve nearest interface, it is practicably to use additional annealing of dopant. We con-sider optimization of annealing time framework recently introduced criterion [17,24-30]. Frame-work the criterion we approximate real distributions of concentrations by step-wise functions (seeFigs. 4 and 5). Farther we determine optimal values of annealing time by minimization the fol-lowing mean-squared error

C(x,Θ

)

0 Lx

2

13

4

Fig. 4. Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distribu-tion of dopant. Curves 2-4 describing real distributions of dopant witgh different values of annealing time

(increasing of number of curve corresponds to increasing of annealing time)


35

x

C(x,Θ

)

1

234

0 L

Fig. 5. Spatial distributions of dopant in heterostructure after ion implantation. Curves 2-4 describing realdistributions of dopant witgh different values of annealing time (increasing of number of curve corresponds

to increasing of annealing time)

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

ΘD

0L-2

3

2

4

1

Fig.6. Dependences of dimensionless optimal annealing time for doping by diffusion, which have been ob-tained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence of dimen-sionless optimal annealing time on the relation a/L and ==0 for equal to each other values of dopant dif-

fusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optimal an-nealing time on value of parameter for a/L=1/2 and ==0. Curve 3 is the dependence of dimensionlessoptimal annealing time on value of parameter for a/L=1/2 and ==0. Curve 4 is the dependence of di-

mensionless optimal annealing time on value of parameter for a/L=1/2 and ==0


36

0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ

0.00

0.04

0.08

0.12

ΘD

0L-2

3

2

4

1

Fig.7. Dependences of dimensionless optimal annealing time for doping by ion implantation, which havebeen obtained by minimization of mean-squared error, on several parameters. Curve 1 is the dependence ofdimensionless optimal annealing time on the relation a/L and ==0 for equal to each other values of do-

pant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless optim-al annealing time on value of parameter for a/L=1/2 and ==0. Curve 3 is the dependence of dimension-less optimal annealing time on value of parameter for a/L=1/2 and ==0. Curve 4 is the dependence of

dimensionless optimal annealing time on value of parameter for a/L=1/2 and ==0

( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L

zyx

xdydzdzyxzyxCLLL

U0 0 0

,,,,,1 , (15)

where (x,y,z) is the approximation function. Dependences of optimal values of annealing timeon parameters are presented on Figs. 6 and 7 for diffusion and ion types of doping, respectively.Optimal value of time of additional annealing of implanted dopant is smaller, than optimal valueof time of annealing of infused dopant due to preliminary annealing of radiation defects.

4. Conclusion

In this paper we consider an approach to manufacture more compact double base heterobipolartransistor. The approach based on manufacturing a heterostructure with required configuration,doping of required areas of the heterostructure by diffusion or ion implantation and optimizationof annealing of dopant and/or radiation defects. The introduced approach to manufactur the hete-robipolar multiemittertransistor gives us possibility to increase sharpness of p-n-junctions frame-work the transistor. In this situation one have a possibility to increase compactness of the transis-tor.

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 scientificresearch.

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Authors

Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondaryschool 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 Doctorcourse in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers inarea of his researches.

Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school ofvillage 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 sheis a student of courses “Translator in the field of professional communication” and “Design (interior art)” inthe 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.

Using inhomogeneity of heterostructure and optimization of annealing to decrease dimensions...

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