Decreasing of quantity of radiation de fects in

International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.2, April 2014

DOI:10.5121/ijcsa.2014.4203 21

Decreasing Of Quantity Of Radiation De-Fects InImplanted-Junction Heterorec-Tifiers By Using

Overlayers

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

Recently we introduced an approach to increase sharpness of diffusion-junction and implanted-junctionheterorectifiers. The heterorectifiers could by single and as a part of heterobipolar transistors. Howevermanufacturing p-n-junctions by ion implantation leads to generation of radiation defects in materials ofheterostructure. In this paper we introduce an approach to use an overlayer and optimization of annealingof radiation defects to decrease quantity of radiation defects.

KEYWORDS

Implanted-junction heterorectifiers, decreasing of quantity of radiation defects, overlayers

1. INTRODUCTION

In the present time degree of integration of elements of integrated circuits (such as p-n-junctions,field-effect and bipolar transistors, thyristors) increases [1-8]. The increasing of degree of inte-gration leads to decreasing of dimensions of elements of integrated circuits. To decrease dimen-sions of elements of integrated circuits could be used some approaches [9-15]. At the same timewith decreasing of dimensions of elements of integrated circuits attracted an interest increasing ofquality of solid state electronic materials [16-20].

It has been recently shown (see, for example, [13-15,21]), that manufactured by diffusion or ionimplantation p-n-junctions and bipolar transistors in heterostructures gives us possibility to in-crease sharpness of the p-n-junctions. However manufacturing implanted-junction rectifiers leadsto generation of radiation defects. In this paper we consider described in the Ref. [21] heterostruc-ture, which consist of a substrate and one or two epitaxial layers (see Figs. 1 and 2).

We assume that type of conductivity of the substrates is known (n or p). We also assume that adopant is implanted into one of the epitaxial layers for generation into the layer required type ofconductivity (p or n). In the first case (see Fig. 1) the dopant is implanted into the single epitaxiallayer. In the second case (see Fig. 2) the dopant is implanted into the nearest to the substrate layerof heterostructure. At the same time we assume, that type of conductivity of the external epitaxiallayer (see Fig. 2) coincides with type of conductivity of the substrate. Farther we consider anneal-ing of radiation defects. This paper is about analytical modeling redistribution of dopant and radi-ation defects and analysis the distribution. We solved our aim by analytical calculation distribu-tions of concentrations of dopant and radiation defects. We also consider a modification of recent-


22

ly introduce analytical approach to solve boundary problems to decrease quantity of iterationsteps.

2. Method of solution

To model redistribution of dopant let us consider the second Fick’s law [1,3-5]

( ) ( )

=

x

txCD

xt

txCC

,,

(1)

x

D(x

) D1

D2

0 La

fC(x)

D0

Fig. 1. Heterostructure, which consist of a substrate and an epitaxial layer. The figure also shows initialdistribution (before starting of annealing of radiation defects) if implanted dipant

x

D(x

)

D1

D2

0 La1

fC(x)

D0

D3

a2

Fig. 2. Heterostructure, which consist of a substrate and two epitaxial layers. The figure alsoshows initial distribution (before starting of annealing of radiation defects) if implanted dopant

with boundary and initial conditions

( )0

,

0

=∂

∂

=xx

txC,

( )0

, =∂

∂

=Lxx

txC, C(x,0)=f (x). (2)

Here C(x,t) is the spatio-temporal distribution of concentration of dopant; T is the temperature ofannealing; DС is the dopant diffusion coefficient. Values of dopant diffusion coefficient are differfrom each other in different materials of heterostructure. Values of dopant diffusion coefficient


23

also changing with variation of temperature (with account Arrhenius law), after radiationprocessing and in high-doping case. We approximate dependences of dopant diffusion coefficienton parameters by the following function based on the following references [22-24]

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,1

,

,1,

V

txV

V

txV

TxP

txCTxDD LC

, (3)

Here DL (x,T) and P (x,T) are the spatial and temperature dependences of dopant diffusion coeffi-cient and limit solubility of dopant. Reason of spatial dependence of the above functions ispresents one or more epitaxial layers in the heterostructure. Reason of temperature dependence ofthe diffusion coefficient and limit of solubility is the Arrhenius law. Values of parameter aredifferent in different materials and could be integer in the following interval ∈[1,3] [22]. V (x,t)is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibriumdistribution of vacancies. Concentrational dependence of dopant diffusion coefficient is describedin details in [22]. We determine spatio-temporal distributions of concentrations of point radiationdefects by solving the following system of equations [23,24]

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

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

−−

∂

∂∂∂=

∂∂

−−

∂

∂∂∂=

∂∂

txVtxITxktxVTxkx

txVTxD

xt

txV

txVtxITxktxITxkx

txITxD

xt

txI

VIVVV

VIIII

,,,,,,

,,

,,,,,,

,,

,2

,

,2

,

(4)

with initial

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

and boundary conditions

( )0

,

0

=∂

∂

=xx

tx,

( )0

, =∂

∂

=Lxx

tx. (5b)

Here =I,V. We denote the spatio-temporal distribution of concentration of interstitials as I (x,t).We denote the diffusion coefficients of point radiation defects D(x,T). Terms V2(x,t) and I2(x,t)correspond to generation of divacancies and diinterstitials. We denote the parameters of recombi-nation of point radiation defects and generation of their complexes as kI,V(x,T), kI,I(x,T) andkV,V(x,T).

We determine spatio-temporal distributions of concentrations of divacancies V(x,t) and diinters-titials I (x,t) by solution of the following system of equations [23,24]

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

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

−+

Φ=Φ

−+

Φ=Φ

Φ

Φ

txVTxktxVTxkx

txTxD

xt

tx

txITxktxITxkx

txTxD

xt

tx

VVVV

VV

IIII

II

,,,,,

,,

,,,,,

,,

2,

2,

(6)

with boundary and initial conditions

( )0

,

0

=∂

Φ∂

=xx

tx ,( )

0,

=∂

Φ∂

=Lxx

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


24

We denote the diffusion coefficients of simplest complexes of point radiation defects D(x,T).We denote the parameters of decay of complexes of point defects k(x,T).

We calculate spatio-temporal distribution of concentration of dopant by using method of averag-ing of function corrections [13-15,21,25] with decreased quantity of iteration steps [26]. We usedthe following functions as initial-order approximations of considered concentrations

Framework the approach we used distributions of concentrations of dopant and radiation defectswith averaged values of dopant diffusion coefficient D0L, D0I, D0V, D0I, D0V and zero values ofparameters of recombination of defects, generation and decay of their complexes as the initial-order approximations of solutions of Eqs. (1), (4) and (6). The initial-order approximations couldbe written as

( ) ( ) ( )∑+=∞

=11

21,

nnCnnC texcF

LLtxC , ( ) ( ) ( )∑+=

∞

=11

21,

nnInnI texcF

LLtxI , ( ) ( ) ( )∑+=

∞

=11

21,

nnVnnV texcF

LLtxV ,

( ) ( ) ( )∑+=Φ∞

=ΦΦ

11

21,

nnnnI texcF

LLtx

II, ( ) ( ) ( )∑+=Φ

∞

=ΦΦ

11

21,

nnnnV texcF

LLtx

VV,

where ( ) ( )∫=L

nn udufucF0

; ( ) ( )20

22exp LtDnten −= ; cn() = cos (n/L); D0L, D0I, D0V,

D0I, D0V are the averaged values of diffusion coefficients. Approximations of thesecond- and higher orders of dopant and defect concentrations we determine frameworkstandard procedure of method of averaging of function corrections [13-15, 21,25,26].Framework the approach to determine the n-th order approximations of the required con-centrations we replace the functions C(x,t), I(x,t), V(x,t), I(x,t) and V(x,t) on the sum ofaverage value of the n-th order approximations and approximations of the n-1-th orders,i.e. n+n-1(x,t). In this situation relations of the second-order approximations of the re-quired concentrations could be written as

( ) ( ) ( ) ( )( )

( )[ ]( )

( )

++

++=

∗∗ x

txC

TxP

txC

V

txV

V

txVTxD

xt

txC CL

,

,

,1

,,1,

, 1122

2

212 (8)

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

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

( ) ( )[ ]

+−

−++−

=

+−

−++−

=

2

12,

1212,12

2

12,

1212,12

,,

,,,,

,,

,,

,,,,

,,

txVTxk

txVtxITxkx

txVTxD

xt

txV

txITxk

txVtxITxkx

txITxD

xt

txI

IVV

VIVIV

III

VIVII

(9)

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

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

−+

Φ=Φ

−+

Φ=Φ

Φ

Φ

txVTxktxVTxkx

txTxD

xt

tx

txITxktxITxkx

txTxD

xt

tx

VVVV

VV

IIII

II

,,,,,

,,

,,,,,

,,

2,

12

2,

12

(10)


25

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

( ) =tzyxC ,,,2

( ) ( ) ( )( )

( )[ ]( )

( )

∫

+

+

++=

∗∗

tC

L dx

xC

TxP

xC

V

xV

V

xVTxD

x 0

112

2

2

21

,

,

,1

,,1,

(8a)

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

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

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

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

+∫ ++−

+∫ +−

∫=

+∫ ++−

−∫ +−

∫=

xfdxVxITxk

dxVTxkdx

xVTxD

xtxV

xfdxVxITxk

dxITxkdx

xITxD

xtxI

V

t

VIVI

t

VVV

t

V

I

t

VIVI

t

III

t

I

01212,

0

2

12,0

12

01212,

0

2

12,0

12

,,,

,,,

,,

,,,

,,,

,,

(9a)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

+∫−

−∫+

∫

Φ=Φ

+∫−

−∫+

∫

Φ=Φ

Φ

Φ

Φ

Φ

xfdxVTxk

dxVTxkdx

xTxD

xtx

xfdxITxk

dxITxkdx

xTxD

xtx

V

t

V

t

VV

tV

VV

I

t

I

t

II

tI

II

0

0

2,

0

12

0

0

2,

0

12

,,

,,,

,,

,,

,,,

,,

(10a)

Average values of the second-orders approximations 2 should be determined by the standardrelation [13-15,21,25,26]

( ) ( )[ ]∫ ∫ −Θ

=Θ

0 0122 ,,

1 L

tdxdtxtxL

. (11)

Substitution of the relations (8a)-(10a) into the relation (11) gives us possibility to obtain the re-quired relations for the values 2

( )∫=L

CС xdxfL 0

2

1 , (12)

( ) ( )

}

( )

+−

−+−+=

+++−×

×

∫−+−−+++=

,4

442

1

2

1

2

1

141

4

3134

2

3

4

2

00

0021001

00

2

1

00

01120102

2

00210012

B

AB

A

ByByB

AB

B

A

AAA

AA

xdxfL

AAAAAA

V

II

IVVIIIV

II

II

L

IIVIIIVVIVVIIIVI

(13)


26

where ( ) ( ) ( ) ( )∫ ∫−ΘΘ

=Θ

0 011, ,,,

1 Lji

baabij tdxdtxVtxITxktL

A , ( )2

00002

002

002

004 2 VVIIIVIVIV AAAAAB −−= ,

( ×−−−++= 002

002

0001002

0000102

0010003

00012

00003 424 IIIVIVIVIVIVIIIVIVIIIVIVIVIVIV AAAAAAAAAAAAAAAB

) ( ) ( )[ ]12122 101000100100000100 ++−+++× VVIVIIIIIVIVIVIVVV AAAAAAAAA , { += 201

200

2002 IVIVIV AAAB

( ) ( ) ( ×+++

∫−−−+++ 0001000001000

201100

2

1001 21

41 IVIVIVIVIVIV

L

IIIIVIIIIIV AAAAAAxdxfL

AAAAA

)} ( ) ( )[ ]{ ×+++−+++−×

LAAAAAAAAAAA VVIVIIIVIVIIIVIVIIIVII

22122124 2

1010000001100100001010

( ) ( ) ( ) ( ) ×+++−−+++∫× 100101112000100101

000 121 IIIVIVIVVVIIIIIVIV

L

VII AAAAAAAAAxdxfA

( ) ( )[ ]]}0001101000100100 21212 IVIVVVIVIIIIIVIV AAAAAAAA +++−++× , ( ) ×++= 2

10011 12 IIIV AAB

( ) ( −+++

−∫−−× 100001000020

0110001000100

18 IIIVIVIVIVII

L

IIVIIIVIVIVIV AAAAAAxdxfL

AAAAAA

) ( ) ( )[ −++−−+++− 100101001010000100002

010010 1244 IIIVIVIIIVIIIVIVIVIVIVIIIV AAAAAAAAAAAAA

( ) ( ) ( ) ( )[ ++++++∫+−− 100100100101

000112000 121

22 IIIVIVIIIVIV

L

VIIIVVVII AAAAAAxdxfAL

AAA

( )]122 1010000001 ++−+ VVIVIIIVIV AAAAA , ( ) +

−∫+= 11

020

201000

14 IV

L

IIIIVII AxdxfL

AAAB

( ) ( ) ( ) ([ ++−−++−+++ 0101112000100101

2

10012

01 211 IVIVIVVVIIIIIVIVIIIVIV AAAAAAAAAAA

) ( )2

00010

21

∫+++L

VIIII xdxfAL

A , 223 48 BByA −+= , −+−+=

623 32 B

qpqy

3 32 qpq ++− , ( )[ ] 3643 22031 BBBBp −−= ,

( )×+−−+=

248

4

2162

21

2320

32 BBBBBB

q

( )031 4BBB −× ,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

∫+∫ ∫−ΘΘ

−=

∫+∫ ∫−ΘΘ

−=

Φ

Θ

Φ

Φ

Θ

Φ

L

V

L

VVV

L

I

L

III

xdxfL

tdxdtxVTxktL

A

xdxfL

tdxdtxITxktL

A

V

I

00 0202

00 0202

1,,

1

1,,

1

. (14)

The considered substitution gives us possibility to obtain equation for parameter 2C. Solution ofthe equation depends on value of parameter . Recently we obtain, that second-order approxima-tion is enough good approximation to obtain main qualitative and some quantitative results. Inthis situation we consider second-order approximations only. Results of analytical modeling havebeen checked numerically.

3. Discussion

In this section we analyzed redistribution of dopant and radiation defects during their annealingbased on distributions of their concentrations, calculated in the previous section. Figs. 3, 4 and 5show spatial distributions of concentrations of dopant in heterostructure with one epitaxial layer(see Fig. 3) and in heterostructure with two epitaxial layers (see Figs. 4 and 5) for equal values ofannealing time framework every group of curves and under condition, when value dopant diffu-sion coefficient in the substrate is smaller, than in doped epitaxial layer. The Figs. 4 and 5 show


27

distributions of concentration of dopant for the case, when value of dopant diffusion coefficient inexternal epitaxial layer is smaller and larger in comparison with value of dopant diffusion coeffi-cient in internal epitaxial layer, respectively. One can find by consideration the figures, thatpresents interface between layers of heterostructure leads to increasing sharpness of p-n-junctionand at the same time to increase homogeneity of distribution of concentration of dopant in dopedarea. Using heterostructure with two epitaxial layers gives us possibility to manufacture a bipolartransistor (see Fig. 4) or to shift from damaged during ion implantation area an implanted-junction rectifier under condition, when quantity of implanted dopant is enough large to changetype of conductivity in required areas (see Fig. 5).

x0.0

0.5

1.0

1.5

2.0C(

x,Θ)

34

56

12

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

SubstrateEpitaxial layer 1

Fig. 3. Spatial distributions of implanted dopant after annealing with continuance Θ =0.0048L2/D0 (curves1, 3, 5) and Θ =0.0057L2/D0 (curves 2, 4, 6). Curves 1 and 2 are experimental data for homogenous struc-

tures (see [27,28]). Curves 3 and 4 are analogous calculated distributions of concentrations of dopant.Curves 5 and 6 are calculated distributions of concentrations of dopant in heterostructure with two layers

for Dsubstrate<Depitaxial layer and =0.6. Coordinate of interface is a=L/2

x0.0

0.5

1.0

1.5

2.0

C(x,Θ

)

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

1

2

SubstrateEpitaxial layer 1

Epitaxial layer 2

Fig. 4. Calculated spatial distributions of implanted dopant in homogenous material (curve 1) and in hetero-structure with two epitaxial layers (curve 2) after annealing with continuance Θ =0.005L2/D0 for Depitaxial

layer2=Dsubstrate<Depitaxial layer1 and =0.6. Coordinates of interfaces are a1=L/4 and a2=3L/4


28

x0.00000

0.00001

0.00010

0.00100

0.01000

0.10000

1.00000

C(x,Θ

)

fC(x)

L/40 L/2 3L/4 Lx0

1

2

Substrate

Epitaxial layer 1

Epitaxial layer 2

Fig. 5. Distributions of concentrations of implanted dopant in heterostructure with two epitaxial layers (sol-id lines) and with one epitaxial layer (dushed lines) for different values of annealing time. Increasing of

number of curves corresponds to increasing of annealing time

In the ideal case implanted dopant achieves interface between layers of heterostructure duringannealing. After that diffusion of dopant finishing. In this case one can obtain maximal increasingof sharpness of p-n-junction [13-15,21]. If the dopant did not achieves the interface during theannealing it is practicably to make additional annealing of dopant. In this case it is attracted aninterest optimization of annealing time. We optimize annealing time framework recently intro-duced criterion [13-15,21]. Framework the criterion we approximate distribution of concentrationof dopant by step-wise function (see Fig. 6). Farther we minimize the mean-squared error to cal-culate optimal value of annealing time

( ) ( )[ ]∫ −Θ=L

xdxxCL

U0

,1 . (15)

The Fig. 7 shows dependences of optimal annealing time on several parameters.

x

C(x,Θ

)

1

234

0 L

Fig.6. Spatial distributions of concentration of implanted dopant in heterostructure from Figs. 1and 2. Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of do-

pant for different values of annealing time. Curve 2 corresponds to minimal annealing time.Curve 3 corresponds to average annealing time. Curve 4 corresponds to maximal annealing time


29

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 of

dimensionless optimal annealing time on the relation a/L and = = 0 for equal to each other values ofdopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of dimensionless op-timal annealing time on value of parameter for a/L=1/2 and = = 0. Curve 3 is the dependence of di-mensionless optimal annealing time on value of parameter for a/L=1/2 and = = 0. Curve 4 is the de-

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

4. Conclusion

In this paper we consider approaches to manufacture p-n-heterojunctions and bipolar hetero-transistors. It has been shown, that using an overlayer, which has been grown over damaged area,could leads to shifting of implanted-junction rectifier from damaged area.

Acknowledgments

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,grant of Scientific School of Russia and educational fellowship for scientific research.

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 schoolof 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 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.

Technology

Decreasing of quantity of radiation de fects in