MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSISTOR FOR MODIFICATION ENE by journalmsej

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSISTOR FOR MODIFICATION ENERGY BAND DIAGRAM E.L. Pankratov1, E.A. Bulaeva1,2 1

Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky street, Nizhny Novgorod, 603950, Russia

ABSTRACT In this paper we consider an approach of manufacturing more compact field-effect heterotransistors. The approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce an approach of modification of energy band diagram by additional doping of channel of the transistors. We also consider an analytical approach to model and optimize technological process.

KEYWORDS Modification of profile of dopant; decreasing of dimension of field-effect transistor; modification of energy band diagram

1. INTRODUCTION Development of solid state electronic leads to increasing performance of the appropriate electronic devices [1-11]. At the same time one can find increasing integration rate of integrated circuits [1-3,5,7]. In this situation dimensions of elements of integrated circuits decreases. To increase performance of solid-state electronics devices are now elaborating new technological processes of manufacturing of solid state electronic devices. Another ways to increase the performance are optimization of existing technological processes and determination new materials with higher values of charge carriers mobilities. To decrease dimensions of elements of integrated circuits they are elaborating new and optimizing existing technological processes. Framework this paper we introduce an approach to decrease dimensions of field-effect heterotransistors. At the same time we introduce an approach of modification of energy band diagram for regulation of transport of charge carriers. The approach based on manufacturing a field-effect transistor in the heterostructure from Fig. 1. The heterostructure consists of a substrate and an epitaxial layer. They are have been considered four sections in the epitaxial layer. The sections have been doped by diffusion or ion implantation. Left and right sections will be considered in future as source and drain, respectively. Both average sections became as channel of transistor. Using one section instead two sections leads to simplification of structure of transistor. However using additional doped section framework the channel of the considered transistor gives us possibility to modify energy band diagram in the structure. After finishing of the considered doping annealing of dopant and/or radiation defects should be done. Several conditions for achievement of decreasing of dimensions of the field-effect transistor have been formulated.

DOI:10.5121/msej.2016.3101

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

2. METHOD OF SOLUTION To solve our aim we determine spatio-temporal distributions of concentrations of dopants. We calculate the required distributions by solving the second Fick's law in the following form [12,13]

∂ C (x, y, z, t ) ∂  ∂ C (x, y, z, t )  ∂  ∂ C (x, y, z, t )  ∂  ∂ C (x, y, z, t )  = DC +  DC +  DC  . (1) ∂t ∂ x  ∂ x ∂ y ∂ z  ∂ y  ∂ z  Boundary and initial conditions for the equations are

Fig. 1. Heterostructure, which consist of a substrate and an epitaxial layer.

View from side ∂ C (x, y, z, t ) ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, =0, = 0, =0, ∂x ∂ x ∂ y ∂y x=0 x = Lx y =0 x=Ly ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, = 0 , C (x,y,z,0)=f (x,y,z). ∂z ∂z z =0 x = Lz

(2)

Here the function C(x,y,z,t) describes the distribution of concentration of dopant in space and time. DС describes distribution the dopant diffusion coefficient in space and as a function of temperature of annealing. Dopant diffusion coefficient will be changed with changing of materials of heterostructure, heating and cooling of heterostructure during annealing of dopant or radiation defects (with account Arrhenius law). Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of annealing and concentrations of dopant and radiation defects could be written as [14-16]  C γ ( x, y , z , t )   V ( x, y , z , t ) V 2 ( x, y , z , t )  DC = DL ( x, y, z , T ) 1 + ξ γ 1 + ς + ς  . 1 2  P (x, y , z , T )   V* (V * )2  

(3)

Here function DL (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate and temperature of annealing T. Function P (x,y,z,T) describes the same dependences of the limit of solubility of dopant. The parameter γ is integer and usually could be varying in the following interval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in aver2

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

age) with each atom of dopant. Ref.[14] describes more detailed information about dependence of dopant diffusion coefficient on concentration of dopant. Spatio-temporal distribution of concentration of radiation vacancies described by the function V (x,y,z,t). The equilibrium distribution of concentration of vacancies has been denoted as V*. It is known, that doping of materials by diffusion did not leads to radiation damage of materials. In this situation ζ1= ζ2= 0. We determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations [15,16]

∂ I (x, y, z, t ) ∂  ∂ I ( x, y, z, t )  ∂  ∂ I (x, y, z, t )  =  DI (x, y, z, T ) +  DI ( x, y, z, T )  − k I , I (x, y, z, T ) × ∂t ∂x ∂x ∂ y ∂y    × I 2 ( x, y , z , t ) +

∂ I (x, y , z , t )  ∂   D I ( x, y , z , T )  − k I ,V ( x, y , z, T ) I (x, y , z, t ) V ( x, y , z, t ) ∂z  ∂z 

(4)

∂ V (x, y, z, t ) ∂  ∂ V (x, y, z, t ) ∂  ∂ V (x, y, z, t )  = DV (x, y, z, T ) +  DV (x, y, z, T )  − kV ,V (x, y, z, T ) × ∂t ∂x ∂x ∂y  ∂y  × V 2 ( x, y , z , t ) +

∂ V ( x, y , z , t )  ∂   DV (x, y , z , T )  − k I ,V ( x, y, z, T ) I ( x, y , z , t ) V ( x, y , z , t ) . ∂z  ∂z 

Boundary and initial conditions for these equations are ∂ ρ (x, y, z, t ) ∂ ρ ( x, y , z , t ) ∂ ρ (x, y , z , t ) ∂ ρ ( x, y , z , t ) =0, = 0, =0, =0, ∂x ∂x ∂y ∂y x=0 x = Lx y =0 y= Ly ∂ ρ ( x, y , z , t ) ∂ ρ ( x, y , z , t ) =0, = 0 , ρ (x,y,z,0)=fρ (x,y,z). ∂z ∂z z =0 z = Lz

(5)

Here ρ =I,V. We denote spatio-temporal distribution of concentration of radiation interstitials as I (x,y,z,t). Dependences of the diffusion coefficients of point radiation defects on coordinate and temperature have been denoted as Dρ(x,y,z,T). The quadric on concentrations terms of Eqs. (4) describes generation divacancies and diinterstitials. Parameter of recombination of point radiation defects and parameters of generation of simplest complexes of point radiation defects have been denoted as the following functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T), respectively. Now let us calculate distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials ΦI(x,y,z,t) in space and time by solving the following system of equations [15,16]

∂ Φ I ( x, y, z, t ) ∂  ∂ Φ I ( x, y, z, t )  ∂  ∂ Φ I ( x, y, z, t )  =  DΦI ( x, y, z,T ) +  DΦI ( x, y, z, T ) + ∂t ∂x ∂x ∂y  ∂y  +

∂ Φ I (x, y, z, t )  ∂  2 DΦI (x, y, z, T )  + k I , I (x, y, z, T ) I (x, y, z, t ) − k I (x, y, z, T ) I (x, y, z, t ) ∂ z  ∂z 

(6)

∂ ΦV (x, y, z, t ) ∂  ∂ ΦV (x, y, z, t )  ∂  ∂ ΦV (x, y, z, t ) = DΦV (x, y, z,T ) + DΦV (x, y, z,T )    + ∂t ∂x ∂x ∂y  ∂y  +

∂ ΦV (x, y, z, t )  ∂  2  DΦV (x, y, z, T )  + kV ,V (x, y, z, T )V (x, y, z, t ) − kV (x, y, z, T )V (x, y, z, t ) . ∂z ∂z  3

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

Boundary and initial conditions for these equations are ∂ Φ ρ ( x, y , z , t ) ∂x

=0,

=0, z =0

=0,

∂x

x =0

∂ Φ ρ ( x, y , z , t ) ∂z

∂ Φ ρ ( x, y , z , t ) x = Lx

∂ Φρ ( x, y, z, t ) ∂z

∂ Φ ρ ( x, y , z , t ) ∂y

= 0, y =0

∂ Φρ (x, y, z, t ) ∂y

= 0, y=Ly

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

The functions DΦρ(x,y,z,T) describe dependences of the diffusion coefficients of the above complexes of radiation defects on coordinate and temperature. The functions kI(x,y,z,T) and kV(x,y,z, T) describe the parameters of decay of these complexes on coordinate and temperature. To determine spatio-temporal distribution of concentration of dopant we transform the Eq.(1) to the following integro-differential form t y z xyz x y z ∫ ∫ ∫ C (u , v, w, t ) d w d v d u = ∫ ∫ ∫ D L ( x, v, w, T ) 0 L y Lz L x L y L z Lx L y Lz

 V ( x, v, w, τ ) V 2 ( x, v, w, τ )  ς ς 1 + + ×  1 2 V* (V * )2  

t x z   yz C γ ( x, v, w,τ )  ∂ C ( x, v, w,τ ) C γ (u, y, w,τ )  τ ξ × 1 + ξ γ d + D ( u , y , w , T ) 1 + × ∫∫∫ L  ∂ x P ( x, v, w, T )  L y Lz 0 Lx Lz P γ ( x, y, z, T )   

t x y  V (u , y, w,τ ) xz V 2 (u , y, w,τ )  ∂ C (u , y, w,τ ) ς × 1 + ς 1 + d τ + ∫ ∫ ∫ D L (u , v, z , T ) ×  2 V* Lx Lz 0 Lx L y ∂ y (V * )2  

 V (u, v, z ,τ ) xy V 2 (u , v, z ,τ )  C γ (u , v, z ,τ )  ∂ C (u , v, z ,τ ) ς ξ × 1 + ς 1 + 1 + dτ +  2 2   γ * * V Lx L y ∂ z (V )   P (x, y, z, T )  +

xyz x y z ∫ ∫ ∫ f (u , v, w) d w d v d u . L x L y L z Lx L y L z

(1a)

Now let us determine solution of Eq.(1a) by Bubnov-Galerkin approach [17]. To use the approach we consider solution of the Eq.(1a) as the following series N

C0 ( x, y, z , t ) = ∑ anC cn ( x ) cn ( y ) cn ( z ) enC (t ) . n=0

[

(

)]

Here enC (t ) = exp − π 2 n 2 D0 C t L−x2 + L−y2 + L−z2 , cn(χ) = cos (π n χ/Lχ). Number of terms N in the series is finite. The above series is almost the same with solution of linear Eq.(1) (i.e. for ξ = 0) and averaged dopant diffusion coefficient D0. Substitution of the series into Eq.(1a) leads to the following result

xyz

π2 ×

γ aC yz t y z N  × ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s x s y s z e t = − 1 + a c x c v c w e τ ∑ ∫ ∫ ∫  n n n nC nC n n n nC 3  n =1 n L y Lz 0 L y Lz   n =1 N

∑

N  V ( x, v, w,τ ) V 2 ( x, v, w,τ )  1 + + D ( x , v , w , T ) a nC s n ( x ) c n (v ) × ς ς    ∑ 1 2 L 2 P γ (x, v, w, T )   V* n =1 V* 

( )

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× n cn (w) enC (τ ) d τ −

γ  xz t x z  N ξ ∫ ∫ ∫ 1 +  ∑ amC cm (u ) cm ( y ) cm (w) emC (τ ) γ ×  P (u, y, w, T )  Lx Lz 0 Lx Lz  m=1

 V (u , y, w,τ ) V 2 (u , y, w,τ ) N × DL (u , y, w, T ) 1 + ς 1 + ς ∑ n c (u ) s ( y ) c (w) e (τ ) d τ × 2 V* (V * )2  n=1 n n n nC 

× a nC −

γ N  xy t x y ξ ∑  × ( ) ( ) ( ) ( ) a ∫ ∫ ∫ DL (u , v, z , T ) 1 + γ nC c n u c n v c n z enC τ   L x L y 0 Lx L y  P (u , v, z , T )  n =1

 V (u , v, z ,τ ) V 2 (u , v, z,τ )  N xyz × 1 + ς 1 + ς ×  n∑=1 n a nC cn (u ) cn (v ) s n ( z ) enC (τ ) d τ + 2 * * 2 V Lx L y L z (V )   x y z

× ∫ ∫ ∫ f (u , v, w) d w d v d u , L x L y Lz

where sn(χ) = sin (π n χ/Lχ). We used condition of orthogonality to determine coefficients an in the considered series. The coefficients an could be calculated for any quantity of terms N. In the common case the relations could be written as

−

L2x L2y L2z

π5

γ L y L z t L x L y Lz a nC  N  × ( ) ( ) ( ) ( ) ( ) ( ) e t = − D x , y , z , T 1 + a c x c y c z e τ ∑ ∫ ∫ ∫ ∫  nC L nC n n n nC 6  n =1 n 2π 2 0 0 0 0  n=1 N

∑

 V ( x, y, z ,τ ) V 2 ( x, y, z,τ )  N anC 1 + ς + ς s (2 x ) cn ( y ) cn ( z ) enC (τ )× ∑   1 2 P γ ( x, y , z , T )   V* (V * )2  n=1 n n

t Lx L y Lz Ly  [cn ( y ) − 1] z sn (z ) + Lz [cn (z ) − 1] d z d y d x d τ − ∫ ∫ ∫ ∫ DL (x, y, z,T )× ×  y sn ( y ) + 0 0 0 0 πn πn   

γ N   ξ V ( x, y, z ,τ ) × DL (x, y, z , T ) 1 +  ∑ anC cn (x ) cn ( y ) cn (z ) enC (τ ) γ 1+ ς1 +      P ( x, y , z , T )   V*   n =1

+ς2

Lx V 2 (x, y, z ,τ )   V ( x, y, z ,τ ) V 2 ( x, y, z,τ )  N  [cn (x ) − 1] anC × ς ς 1 + +   n∑=1  x sn ( x ) + 1 2 * * 2 * 2 V πn (V )   (V )    n

LL   Lx L z L cn ( x ) sn (2 y ) cn ( z ) enC (τ )  z sn ( z ) + z [cn ( z ) − 1] d z d y d x d τ − x 2y × 2 2π πn 2π  

γ t Lx L y Lz  N  ξ V 2 ( x, y, z,τ ) × ∫ ∫ ∫ ∫ 1 +  ∑ a nC cn (x ) cn ( y ) cn (z ) enC (τ ) γ 1+ ς 2 +    P ( x, y , z , T )   0 0 0 0 (V * )2  n =1

+ ς1

N a   L V ( x, y, z ,τ )  DL (x, y, z , T )∑ nC cn ( x ) cn ( y ) s n ( z )  x sn (x ) + x [cn ( x ) − 1] × *  n =1 n V πn   

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 N Lx  L    L ×  y sn ( y ) + y [cn ( y ) − 1] enC (τ ) d z d y d x d τ + ∑ ∫  x s n ( x ) + x [cn ( x ) − 1] × n =1 0 πn πn    

Ly L   Lz   L × ∫  y s n ( y ) + y [cn ( y ) − 1] ∫  z s n ( z ) + z [cn ( z ) − 1] f ( x, y, z ) d z d y d x . 0 πn πn  0 

As an example for γ = 0 we obtain Lx Ly L L  Lz   anC = ∫ ∫  y s n ( y ) + y [cn ( y ) − 1] ∫  z sn ( z ) + y [cn (z ) − 1] f (x, y, z )d z d y { x sn ( x ) + 0 0 πn πn  0 

× [cn ( x ) − 1]

Ly Lz Ly  Lx   n  t Lx [cn ( y ) − 1] ∫ DL (x, y, z,T ) ×  d x  ∫ ∫ sn (2 x ) ∫ cn ( y ) y sn ( y ) + 0 π n   2 0 0 πn  0

L    V ( x, y, z,τ ) V 2 ( x, y, z ,τ )   ξ ×  z sn (z ) + y [cn ( z ) − 1]1 + ς 1 + × ς  1 + γ 2 * * 2 V πn (V )   P (x, y, z,T )   Lx Lz t L   Ly × cn ( z ) d z d y d x enC (τ ) d τ + ∫ enC (τ ) ∫ cn ( x ) x sn ( x ) + y [cn ( x ) − 1] ∫ sn (2 y ) ∫ cn (z ) × 0 0 0 πn  0

L    ξ V ( x, y, z ,τ ) V 2 (x, y, z ,τ )  1 ×  z sn (z ) + y [cn ( z ) − 1]1 + γ + + × ς ς 1 2  V* πn (V * )2     P ( x, y , z , T )   Lx t  Ly L × DL (x, y, z , T ) d z d y d x d τ + ∫ enC (τ ) ∫ cn ( x ) x sn ( x ) + x [cn ( x ) − 1] ∫ cn ( y ) { sn ( y ) × 0 0 πn  0

× y+

[c ( y ) − 1] ∫ s (2 z ) D (x, y, z,T ) 1 + Lz

πn

0



 V 2 ( x, y, z ,τ ) 1 + + ς  2 P γ ( x, y, z, T )   (V * )2

−1

  L2z L2z L2z V ( x, y, z,τ ) + ς1 d z d y d x d τ  − 5 6 enC (t ) . *  V   π n  For γ = 1 one can obtain the following relation to determine required parameters

a nC = −

where α n =

L L L βn ± β n2 + 4α n ∫ cn ( x ) ∫ cn ( y ) ∫ cn ( z ) f ( x, y, z ) d z d y d x , 0 0 0 2α n y

L L L ξ L y Lz t  V (x, y, z,τ ) V 2 (x, y, z,τ ) ( ) ( ) ( ) ( ) τ ς ς e s 2 x c y c z 1 × + + ∫ ∫ ∫ ∫  nC n n n 1 2 0 0 0 V* 2π 2 n 0 (V * )2   x

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Ly  DL ( x, y, z, T )  [cn ( y ) − 1]z sn (z ) + Lz [cn (z ) − 1] d z d y d x d τ + ξ Lx2Lz ×  y sn ( y ) + P(x, y, z, T )  2π n πn πn  

Lx t   L y Lz L D ( x, y , z , T )  Lz × ∫ enC (τ ) ∫ c n ( x )  x s n ( x ) + x [cn ( x ) − 1] ∫ ∫ cn ( z ) L [cn (z ) − 1] ×  z s n (z ) − 0 0 πn P ( x, y , z , T )  πn  0 0 

ξ Lx L y D L ( x, y , z , T )  V ( x, y , z , τ ) V 2 ( x, y , z ,τ )  ς + × 1 + ς 1  d z s n (2 y ) d y d x d τ + 2 * * 2 2π 2 n P ( x, y , z , T )  V (V )  t

× ∫ enC (τ ) ∫ cn ( x ) ∫ cn ( y ) ∫ s n (2 z )

DL ( x, y , z , T )  V ( x, y, z ,τ ) V 2 ( x, y, z ,τ )  1 + + × ς ς  1 2 P ( x, y , z , T )  V* (V * )2 

L y Lz t L    L ×  x s n ( x ) + x [cn ( x ) − 1] y s n ( y ) + y [cn ( y ) − 1] d z d y d x d τ , β n = ∫ enC (τ ) × πn πn 2 nπ 2 0    2 Ly Lx Ly    L z × ∫ s n (2 x ) ∫ cn ( y )  y s n ( y ) + [cn ( y ) − 1] ∫ cn (z ) 1 + ς 1 V (x, y*, z,τ ) + ς 2 V (x, *y,2z,τ ) × 0 0 πn V (V )   0  Ly Lx   LL t L × DL ( x, y, z , T )  z s n ( z ) + z [c n ( z ) − 1] d z d y d x d τ + x z2 ∫ enC (τ ) ∫ c n ( x ) ∫ s n (2 y ) × 0 0 πn 2 nπ 0  

   Lz L V ( x, y , z ,τ ) V 2 ( x, y , z , τ )  ς ×  x s n ( x ) + x [c n ( x ) − 1] ∫ DL (x, y, z , T ) c n ( z ) 1 + ς 1 + × 2 πn V* (V * )2   0  Lx Lx L y t  L    L ×  z s n ( z ) + z [c n (z ) − 1] d z d y d x d τ + e (τ ) ∫  x s n (x ) + x [c n ( x ) − 1] × 2 ∫ nC 0 2 nπ 0 πn πn    

2 Ly Ly    Lz × cn ( x ) ∫  y s n ( y ) + [cn ( y ) − 1] ∫ DL (x, y, z, T ) 1 + ς 1 V (x, y*, z,τ ) + ς 2 V (x, *y,2z,τ ) × 0 πn V (V )   0 

× s n (2 z ) d z c n ( y ) d y d x d τ − L2x L2y L2z enC (t ) π 5 n 6 . The same approach could be used for calculation parameters an for different values of parameter γ. However the relations are bulky and will not be presented in the paper. Advantage of the approach is absent of necessity to join dopant concentration on interfaces of heterostructure. The same Bubnov-Galerkin approach has been used for solution the Eqs.(4). Previously we transform the differential equations to the following integro- differential form

xyz x y z yz t y z ∂ I ( x, v, w,τ ) d wd v dτ + ∫ ∫ ∫ I (u , v, w, t ) d w d v d u = ∫ ∫ ∫ DI ( x, v, w, T ) Lx L y Lz Lx L y Lz L y L z 0 L y Lz ∂x

∂ I (u, y, w,τ ) xz t x z xyz x y z d wd u dτ − ∫ ∫ ∫ DI (u, y, w, T ) ∫ ∫ ∫ k I ,V (u, v, w, T ) I (u, v, w, t ) × Lx Lz 0 L x L z ∂x Lx L y L z L x L y L z

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

× V (u , v, w, t ) d w d v d u +

x y t x y ∂ I (u , v, z ,τ ) xyz × D I (u , v, z , T ) d v d u d τ − ∫∫∫ ∂z L x L y 0 Lx L y Lx L y L z

x y z

× ∫ ∫ ∫ k I , I (u , v, w, T ) I 2 (u , v, w, t ) d w d v d u + Lx L y L z

xyz x y z ∫ ∫ ∫ f I (u, v, w) d w d v d u (4a) L x L y L z L x L y Lz

xyz x y z yz t y z ∂ V ( x, v, w,τ ) d wd vdτ + ∫ ∫ ∫ V (u , v, w, t ) d w d v d u = ∫ ∫ ∫ DV ( x, v, w, T ) L x L y L z Lx L y Lz L y L z 0 L y Lz ∂x

xz t x z ∂ V (u , y, w,τ ) x y t x y ∂ V (u , v, z ,τ ) d wd u dτ + × ∫ ∫ ∫ DV (u , y, w, T ) ∫∫∫ L x L z 0 Lx Lz ∂x L x L y 0 Lx L y ∂z

× DV (u, v, z, T ) d v d u d τ −

−

xyz x y z ∫ ∫ ∫ k I ,V (u, v, w, T ) I (u, v, w, t ) V (u, v, w, t ) d w d v d u − L x L y L z Lx L y Lz

xyz x y z xyz x y z 2 ∫ ∫ ∫ kV ,V (u, v, w, T ) V (u, v, w, t ) d w d v d u + ∫ ∫ ∫ f V (u, v, w) d w d v d u . Lx L y L z Lx L y Lz L x L y Lz Lx L y Lz

We determine spatio-temporal distributions of concentrations of point defects as the same series N

ρ 0 ( x, y, z, t ) = ∑ a nρ cn ( x ) cn ( y ) cn ( z ) enρ (t ) . n =1

Parameters anρ should be determined in future. Substitution of the series into Eqs.(4a) leads to the following results

xyz

t y z anI y zπ N s n (x ) s n ( y ) s n ( z ) enI (t ) = − ∑ anI ∫ ∫ cn ( y ) ∫ cn ( z ) DI (x, v, w, T ) d w d v × 3 n =1 n Lz Lx L y Lz n=1 0 L y N

∑

× enI (τ ) d τ s n ( x ) −

−

t x z x zπ N ∑ a nI s n ( y ) ∫ enI (τ ) ∫ c n (x ) ∫ cn ( z ) DI (u, y, w, T ) d w d u d τ − 0 Lx Lz Lx L y Lz n =1

y t x x y z x yπ N ∑ a nI s n ( z ) ∫ enI (τ ) ∫ cn ( x ) ∫ c n ( y ) DI (u , v, z , T ) d v d u d τ − ∫ ∫ ∫ k I , I (u , v, v, T ) × 0 Lx Ly Lx L y Lz Lx L y Lz n =1

2 N xyz xyz x y z N × ∑ anI cn (u ) cn (v ) cn (w) enI (t ) d w d v d u − ∫ ∫ ∫ ∑ anI cn (u ) cn (v ) cn (w) × n=1  Lx L y Lz Lx Ly Lz Lx Ly Lz n =1

x y z

n =1

Lx L y Lz

× enI (t ) ∑ anV cn (u ) cn (v ) cn (w) enV (t ) k I ,V (u, v, v, T ) d w d v d u + ∫ ∫ ∫ f I (u, v, w) d w d v d u ×

× x y z Lx L y L z xyz

π3

t y z anV y zπ N s ( x ) s ( y ) s ( z ) e ( t ) = − a c ( y ) cn (z ) DV (x, v, w, T ) d w d v × ∑ ∫ ∫ ∫ n n n nV nV n 3 n =1 n 0 Ly Lz Lx L y Lz n=1 N

∑

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× enV (τ ) d τ s n ( x ) −

−

t x z x zπ N ∑ anV s n ( y ) ∫ enV (τ ) ∫ cn ( x ) ∫ cn ( z ) DV (u, y, w, T ) d w d u d τ − 0 Lx Lz Lx L y Lz n=1

y t x x y z x yπ N ∑ a nV sn ( z ) ∫ enV (τ ) ∫ cn (x ) ∫ cn ( y ) DV (u , v, z, T ) d v d u d τ − ∫ ∫ ∫ kV ,V (u , v, v, T ) × 0 Lx Ly Lx L y L z Lx L y Lz n=1

2 N xyz xyz x y z N × ∑ anV cn (u ) cn (v ) cn (w) enI (t ) d w d v d u − ∫ ∫ ∫ ∑ anI cn (u ) cn (v ) cn (w) × n=1  Lx Ly Lz Lx Ly Lz Lx Ly Lz n=1

x y z

n =1

Lx L y Lz

× enI (t ) ∑ anV cn (u ) cn (v) cn (w) enV (t ) k I ,V (u, v, v,T ) d w d v d u + ∫ ∫ ∫ fV (u, v, w) d w d v d u × × x y z Lx L y L z . We used orthogonality condition of functions of the considered series framework the heterostructure to calculate coefficients anρ. The coefficients an could be calculated for any quantity of terms N. In the common case equations for the required coefficients could be written as

−

L2x L2y L2z

π5

Ly Ly   a nI 1 N a nI t Lx ( ) [ ( ) ] [ e t = − 1 − c 2 x cn (2 y ) − 1] × ∑ ∫ ∫ ∫  L y + y sn (2 y ) + nI n 6 2 n =1 n 0 2π Lx n=1 n 0 0 2π n   N

∑

Lz   L 1 N a nI t Lx × ∫ DI (x, y , z , T ) z s n ( z ) + z [c n (z ) − 1] d z d y d x enI (τ ) d τ − ∑ ∫ ∫ { x s n (2 x ) + 0 2π n 2π L y n =1 n 2 0 0  

+ Lx +

L y Lz Lx [cn (2 x ) − 1]  ∫ ∫ DI (x, y, z,T )Lz + z sn (2z ) + Lz [cn (2z ) − 1] d z [1 − cn (2 y )] × πn 2π n 0 0  

Lz   L × d y d x enI (τ ) d τ ∫ DI ( x, y, z, T )Lz + z sn (2 z ) + z [c n (2 z ) − 1] d z d y d x enI (τ ) d τ − 0 2π n  

−

Ly Ly   L 1 N anI t Lx  [ cn (2 y ) − 1] × ∑ 2 ∫ ∫  Lx + x s n (2 x ) + x [cn (2 x ) − 1] ∫  Ly + y sn (2 y ) + 2π Lz n=1 n 0 0  2π n 2π n 0 

Lz Lx N  L × ∫ [1 − cn (2 z )] DI ( x, y, z, T ) d z d y d x enI (τ ) d τ − ∑ anI2 enI (2 t ) ∫  Lx + x [cn (2 x ) − 1] + n =1 0 0 2π n 

Ly L   Lz  L + x sn (2 x )} ∫  L y + y s n (2 y ) + y [cn (2 y ) − 1] ∫ k I , I ( x, y, z , T ) Lz + z [cn (2 z ) − 1] + 0 2π n 2π n  0 

Lx N  Ly L + z s n (2 z )}d z d y d x − ∑ a nI a nV enI (t ) enV ( t ) ∫  Lx + x s n (2 x ) + x [c n (2 x ) − 1] ∫ { L y + n =1 0 2π n  0

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

+ y sn (2 y ) +

Ly 2π n

[c (2 y ) − 1] ∫ k (x, y, z,T )L Lz

0

I ,V



+ z s n (2 z ) +

Lz [cn (2 z ) − 1] d z × 2π n 

N Lx  L Ly   Lz L × d y d x + ∑ ∫  x s n ( x ) + x [cn (x ) − 1] ∫  y sn ( y ) + y [cn ( y ) − 1] ∫ f I ( x, y, z, T ) × n =1 0 πn πn   0 0

  L ×  Lz + z s n (2 z ) + z [c n (2 z ) − 1] d z d y d x 2π n  

−

L2x L2y L2z

π5

Ly Ly   anV 1 N anV t Lx e ( t ) = − [ 1 − c ( 2 x ) ] [ cn (2 y ) − 1] × ∑ ∫ ∫ ∫ Ly + y sn (2 y ) + nV n 6 2 n =1 n 0 2π Lx n=1 n 0 0 2π n   N

∑

Lz   L 1 N anV t Lx × ∫ DV (x, y, z, T ) z sn (z ) + z [cn (z ) − 1] d z d y d x enV (τ ) d τ − ∑ ∫ ∫ { x s n (2 x ) + 0 2π n 2π Ly n =1 n 2 0 0  

+ Lx +

L y Lz Lx [cn (2 x ) − 1]  ∫ ∫ DV (x, y, z,T )Lz + z sn (2 z ) + Lz [cn (2 z ) − 1] d z [1 − cn (2 y )] × πn 2π n 0 0  

Lz   L × d y d x enV (τ ) d τ ∫ DV ( x, y, z, T )Lz + z sn (2 z ) + z [cn (2 z ) − 1] d z d y d x enV (τ ) d τ − 0 2π n  

−

Ly Ly  L 1 N a nV t Lx  [cn (2 y ) − 1] × ∑ 2 ∫ ∫  Lx + x sn (2 x ) + x [cn (2 x ) − 1] ∫  L y + y sn (2 y ) + n = 1 2π Lz n 0 0  2π n 2π n 0 

Lz Lx N  L × ∫ [1 − cn (2 z )] DV ( x, y , z, T ) d z d y d x enV (τ ) d τ − ∑ anV2 enV (2 t ) ∫  Lx + x [cn (2 x ) − 1] + n =1 0 0 2π n 

Ly L   Lz  L + x sn (2 x )} ∫  L y + y s n (2 y ) + y [cn (2 y ) − 1] ∫ kV ,V ( x, y, z , T ) Lz + z [cn (2 z ) − 1] + 0 2π n 2π n  0 

Lx N  Ly L + z sn (2 z )}d z d y d x − ∑ a nI a nV enI (t )enV ( t ) ∫  Lx + x sn (2 x ) + x [cn (2 x ) − 1] ∫ { L y + n =1 0 2π n  0

+ y sn (2 y ) +

Ly 2π n

[c (2 y ) − 1] ∫ k (x, y, z,T )L Lz

0

I ,V



+ z s n (2 z ) +

Lz [cn (2 z ) − 1] d z × 2π n 

N Lx  L Ly   Lz L × d y d x + ∑ ∫  x sn ( x ) + x [cn ( x ) − 1] ∫  y s n ( y ) + y [cn ( y ) − 1] ∫ f V ( x, y, z , T ) × n =1 0 πn πn  0 0

  L ×  Lz + z sn (2 z ) + z [cn (2 z ) − 1] d z d y d x . 2π n   10

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

In the final form relations for required parameters could be written as

a nI = −

b3 + A ± 4 b4

+ A) b y − γ nV λ2nI  − 4 b4  y + 3 A 4  2

Lx L y L z



0 0 0



where γ nρ = enρ (2 t ) ∫ ∫ ∫ k ρ , ρ ( x, y, z , T )  Lx + x sn (2 x ) +

Ly 2π n

[c (2 y ) − 1]L n



+ z s n (2 z ) +

γ a 2 + δ nI anI + λnI   , anV = − nI nI , χ nI anI 

Lx [cn (2 x ) − 1]{y sn (2 y ) + Ly + 2π n 

t Lz [cn (2 z ) − 1] d z d y d x , δ nρ = 1 2 ∫ enρ (τ )× 2π n 2π Lx n 0 

Lx L y L   Lz   L × ∫ ∫  y sn ( y ) + y [cn ( y ) − 1] ∫  z sn ( z ) + z [cn ( z ) − 1] Dρ (x, y, z , T ) d z d y [1 − 0 0 2π n 2π n  0 

− cn (2 x )] d x d τ + + z sn (2 z ) +

+ Lx +

Lx Lz t  Ly Lx 1 ( ) ( ) [ ( ) ] τ e L + x s 2 x + c 2 x − 1 ∫ x  ∫ [1 − cn (2 y )] ∫ {Lz + n n 2 ∫ nρ 0 0 2π L y n 0 πn  0

Lx t Lz [cn (2 z ) − 1]Dρ (x, y, z,T )d z d y d x d τ + 1 2 ∫ enρ (τ ) ∫ {x sn (2 x ) + 0 2π n 2 π Lz n 0 

Ly Lz L Lx [cn (2 x ) − 1] ∫ Ly + y sn ( y ) + y [cn ( y ) − 1] ∫ [1 − cn (2 z )]Dρ (x, y, z,T ) d z × 2π n πn 0 0

× d y d x dτ −

L2x L2y L2z

π 5n6

Lx Ly  Ly  L enρ (t ) , χ nIV = ∫  x sn ( x ) + x [cn ( x ) − 1] ∫ Ly + [cn (2 y ) − 1] + 0 πn 2π n  0

Lz   L + y sn (2 y )}∫ k I ,V ( x, y, z , T )  Lz + z s n (2 z ) + z [cn (2 z ) − 1] d z d y d x enI (t ) enV ( t ) , 0 2π n   Ly Lx L L     z  L L λnρ = ∫ x sn (x) + x [cn ( x) − 1] ∫  y sn ( y ) + y [cn ( y ) − 1] ∫ z sn (z ) + z [cn (z ) − 1] × 0 πn πn πn  0 0 

× f ρ (x, y, z, T ) d z d y d x , b4 = γ nV γ nI2 − γ nI χ nI2 , b3 = 2γ nV γ nI δ nI − δ nI χ nI2 − δ nV χ nI γ nI , A = 8 y + b32 − 4b2 , b2 = γ nV δ nI2 + 2λnI γ nV γ nI − δ nV χ nI δ nI + (λnV − λnI )χ nI2 , b1 = 2λnI × × γ nV δ nI − δ nV χ nI λnI , y = 3

q 2 + p3 − q − 3

q = (2b33 − 9b2 b3 + 27b1b42 ) 54 b43 .

q2 + p3 + q −

b3 3b b − b 2 , p= 2 4 2 3 , 3b4 9 b4

We determine distributions of concentrations of simplest complexes of radiation defects in space and time as the following functional series N

Φ ρ 0 (x, y, z , t ) = ∑ a nΦρ cn ( x ) cn ( y ) cn (z ) enρ (t ) . n =1

Here anΦρ are the coefficients, which should be determined. Let us previously transform the Eqs. (6) to the following integro-differential form

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 t y z ∂ Φ I ( x, v, w,τ ) xyz x y z d wd v dτ × ∫ ∫ ∫ Φ I (u, v, w, t ) d w d v d u = ∫ ∫ ∫ DΦI ( x, v, w, T ) 0 L y Lz ∂x Lx L y Lz Lx L y Lz

∂ Φ I (u , y, w,τ ) yz xz t x z xy t x y + d wd u dτ + ∫ ∫ ∫ DΦI (u , y, w, T ) ∫ ∫ ∫ DΦI (u , v, z, T ) × ∂y L y Lz L x L z 0 Lx L z Lx L y 0 Lx L y

∂ Φ I (u , v, z,τ ) xyz x y z 2 d vd udτ + ∫ ∫ ∫ k I , I (u , v, w, T ) I (u , v, w,τ ) d w d v d u − (6a) ∂z Lx L y Lz L L L x y z

−

xyz x y z xyz x y z ∫ ∫ ∫ k I (u, v, w, T ) I (u, v, w,τ ) d w d v d u + ∫ ∫ ∫ f ΦI (u, v, w) d w d v d u L x L y L z L x L y Lz Lx L y Lz Lx L y Lz

t y z ∂ Φ V (x, v, w,τ ) xyz x y z d wd v dτ × ∫ ∫ ∫ Φ V (u , v, w, t ) d w d v d u = ∫ ∫ ∫ DΦV ( x, v, w, T ) 0 L y Lz ∂x Lx L y Lz Lx L y Lz

∂ Φ V (u , y, w,τ ) yz xz t x z xy t x y + d wd u dτ + ∫ ∫ ∫ DΦV (u , y, w, T ) ∫ ∫ ∫ DΦV (u , v, z, T ) × ∂y L y L z L x L z 0 Lx L z Lx L y 0 Lx L y

∂ Φ V (u, v, z ,τ ) xyz x y z 2 d vd udτ + ∫ ∫ ∫ kV ,V (u, v, w, T )V (u, v, w,τ ) d w d v d u − ∂z Lx L y Lz L L L x y z

−

xyz x y z xyz x y z ∫ ∫ ∫ kV (u , v, w, T ) V (u , v, w,τ ) d w d v d u + ∫ ∫ ∫ f ΦV (u , v, w) d w d v d u . Lx L y Lz Lx L y Lz L x L y L z Lx L y L z

Substitution of the previously considered series in the Eqs.(6a) leads to the following form t y z a nΦI y zπ N s n ( x ) s n ( y ) s n ( z ) enI (t ) = − ∑ n a nΦI s n ( x )enI (t )∫ ∫ ∫ cn (v )cn (w) × 3 3 n =1 π n 0 L y Lz Lx L y L z n=1 N

− x y z∑

× DΦI ( x, v, w, T ) d w d v d τ −

× n s n ( y )enΦI (t ) −

t x z x zπ N ∑ a nΦI ∫ ∫ ∫ c n (u ) cn (w) DΦI (u , v, w, T ) d w d u d τ × 0 L x Lz Lx L y Lz n =1

t x y x yπ N ∑ n anΦI s n ( z )enΦI (t )∫ ∫ ∫ cn (u )cn (v ) DΦI (u, v, z , T ) d v d u d τ + 0 Lx L y Lx L y Lz n=1

x y z xyz x y z 2 ( ) ( ) k u , v , w , T I u , v , w , τ d w d v d u + ∫ ∫ ∫ I ,I ∫ ∫ ∫ f ΦI (u, v, w)d w d v d u × Lx L y Lz Lx Ly Lz Lx Ly Lz

xyz xyz x y z − ∫ ∫ ∫ k I (u, v, w, T ) I (u, v, w,τ ) d w d v d u L x L y L z L x L y L z L x L y Lz

t y z a nΦV y zπ N ( ) ( ) ( ) ( ) ( ) ( ) s x s y s z e t = − n a s x e t ∑ ∫ ∫ ∫ c n (v ) c n (w) × n n n nV nΦV n nV 3 3 n =1 π n 0 L y Lz L x L y L z n =1 N

− x y z∑

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× DΦV (x, v, w, T ) d w d v d τ − × a nΦV sn ( y )enΦV (t ) −

× a nΦV +

x zπ N t x z ∑ n ∫ ∫ ∫ cn (u )c n (w) DΦV (u, v, w, T ) d w d u d τ × Lx L y Lz n =1 0 Lx Lz

t x y x yπ N ∑ n s n ( z )enΦV (t )∫ ∫ ∫ cn (u )cn (v ) DΦV (u , v, z , T ) d v d u d τ × 0 Lx L y Lx L y Lz n =1

x y z xyz x y z 2 ∫ ∫ ∫ kV ,V (u, v, w, T )V (u , v, w,τ ) d w d v d u + ∫ ∫ ∫ f ΦV (u , v, w) d w d v d u × L x L y Lz Lx L y Lz Lx L y Lz

xyz xyz x y z − ∫ ∫ ∫ kV (u, v, w, T ) V (u, v, w,τ ) d w d v d u . Lx L y Lz Lx L y Lz Lx L y Lz

We used orthogonality condition of functions of the considered series framework the heterostructure to calculate coefficients anΦρ. The coefficients anΦρ could be calculated for any quantity of terms N. In the common case equations for the required coefficients could be written as

−

L2x L2y L2z

π5

Ly Ly  a nΦI 1 N t Lx ( ) [ ( ) ] ( ) [cn (2 y ) − 1] × e t = − 1 − c 2 x L + y s 2 y + ∑ ∫ ∫ ∫  n Φ I n y n 6 n =1 n n = 1 0 2π Lx 0 0 2π n   N

∑

  anΦI Lz Lz 1 N t Lx D ( x , y , z , T ) z s ( z ) + [ c ( z ) − 1 ] d z d y d x e ( ) d − τ τ ∑ ∫ ∫ {x sn (2 x ) + ∫   n n nΦI ΦI n2 0 2π n 2π n=1 0 0  

+ Lx +

× anΦI

Ly Lz Lx [cn (2x) − 1] ∫ [1 − cn (2 y )] ∫ DΦI (x, y, z,T )z sn (z ) + Lz [cn (z ) − 1] d z d y d x × 0 2π n 2π n  0 

Ly L y  enΦI (τ ) Lx 1 N anΦI t Lx  d − x s ( x ) + [ c ( x ) − 1 ] y s ( 2 y ) + [cn (2 y ) − 1] + τ ∑ ∫ ∫ ∫    n n n n 2 Ly 2π Lx n=1 n 2 0 0  2π n 2π n 0 Lz

+ Ly }∫ [1 − cn (2 y )] DΦI ( x, y, z, T ) d z d y d x enΦI (τ ) d τ + 0

Lx  Lx anΦI t e ( ) [cn (x) −1] + τ ∫ ∫ nΦI 3 3 0 2π n π n=1 n 0 

∑

Ly L   Lz  L + x sn (x )} ∫  y sn ( y ) + y [cn ( y ) − 1] ∫ I 2 ( x, y, z , t ) k I , I ( x, y, z , T )  z [cn ( z ) − 1] + 0 2π n  0  2π n

+ z sn ( z )} d z d y d x −

Lx  Ly  L y a n ΦI t Lx ( ) ( ) [ ( ) ] [cn ( y ) − 1] + τ e x s x + c x − 1 ∫ ∫  ∫ n Φ I n n 0 π 3 n =1 n 3 0 2π n   0  2π n

∑

Lz   L 1 N a + y sn ( y )}∫  z s n ( z ) + z [cn ( z ) − 1] k I ( x, y, z , T ) I (x, y, z , t ) d z d y d x + 3 ∑ nΦ3 I × 0 2π n π n =1 n  

Lx t L  Ly   Lz  L L × ∫ enΦI (τ ) ∫  x sn (x ) + x [cn ( x ) − 1] ∫  y sn ( y ) + y [cn ( y ) − 1] ∫  z [cn ( z ) − 1] + 0 0 2π n 2π n   0  0  2π n

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

+ z s n ( z )} f ΦI ( x, y, z ) d z d y d x

−

L2x L2y L2z

π5

Ly Ly  a nΦV 1 N t Lx ( ) [ ( ) ] ( ) [cn (2 y ) − 1] × e t = − 1 − c 2 x L + y s 2 y + ∑ ∫ ∫ ∫  n Φ V n y n 6 n =1 n n = 1 0 2π Lx 0 0 2π n   N

∑

  anΦV Lz Lz 1 N t Lx ( ) ( ) ( ) ( ) D x , y , z , T z s z + [ c z − 1 ] d z d y d x e d − τ τ ∑ ∫ ∫ {x sn (2x ) + ∫   ΦV n n nΦV n2 0 2π n 2π n=1 0 0   + Lx +

× anΦV

Ly Lz Lx [cn (2x) −1] ∫ [1 − cn (2 y)] ∫ DΦV (x, y, z,T ) z sn (z ) + Lz [cn (z ) −1] d z d y d x × 0 2π n 2π n 0  

Ly Ly  enΦV (τ ) Lx 1 N anΦV t Lx  d − x s ( x ) + [ c ( x ) − 1 ] [cn (2 y ) −1] + τ ∑ ∫ ∫ ∫  y sn (2 y ) +   n n 2 2 n Ly 2π Lx n=1 n 0 0  2π n 2π n 0 Lz

+ Ly }∫ [1 − cn (2 y )] DΦV (x, y, z,T ) d z d y d x enΦV (τ ) d τ + 0

Lx  Lx anΦV t ( ) τ e [cn (x) − 1] + ∫ ∫ nΦV 3 3 0 2π n π n=1 n 0 

∑

Ly L   Lz  L + x sn (x )} ∫  y sn ( y ) + y [cn ( y ) − 1] ∫ V 2 ( x, y, z, t ) kV ,V ( x, y, z, T )  z [cn ( z ) − 1] + 0 2π n  0  2π n

+ z sn (z )} d z d y d x −

Lx  L y  L y anΦV t Lx e ( ) x s ( x ) + [ c ( x ) − 1 ] [cn ( y ) − 1] + τ ∫ ∫  ∫ n Φ V n n 0 π 3 n =1 n 3 0 2π n   0  2π n

∑

Lz   L 1 N a + y sn ( y )}∫  z s n ( z ) + z [cn (z ) − 1] kV (x, y, z , T )V ( x, y, z , t ) d z d y d x + 3 ∑ nΦ3V × 0 2π n π n=1 n  

Lx t L  L y   Lz  L L × ∫ enΦV (τ ) ∫  x sn ( x ) + x [cn (x ) − 1] ∫  y s n ( y ) + y [cn ( y ) − 1] ∫  z [cn ( z ) − 1] + 0 0 2π n 2π n  0  0  2π n

+ z sn ( z )} f ΦV ( x, y, z ) d z d y d x .

3. DISCUSSION In the present paper we analyzed redistribution of infused and implanted dopants in heterostructure, which have been presented on Fig. 1. The analysis has been done by using relations, calculated in the previous section. First of all we consider situation, when dopant diffusion coefficient in doped area is larger, than in nearest areas. It has been shown, that in this case distribution of concentration of dopant became more compact in comparison with wise versa situation (see Figs. 2 and 3 for diffusion and ion types of doping). In the wise versa situation (when dopant diffusion coefficient in doped area is smaller, than in nearest areas) we obtained spreading of distribution of concentration of dopant. In this case outside of doping material one can find higher spreading in comparison with wise versa situation (see Fig. 4). It should be noted, that properties of layers of multilayer structure varying in space: varying layers with larger and smaller values of the diffusion coefficient. In this situation with account results, which shown on Figs. 2-4, layers of the considered heterostructure with smaller value of 14

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

dopant should has smaller level of doping in comparison with nearest layers to exclude changing of type of doping in the nearest layers. For a more complete doping of each section and at the same to decrease diffusion of dopant into nearest sections it is attracted an interest optimization of annealing of dopant and /or radiation defects. Let us optimize annealing of dopant and/or radiation defects by using recently introduce criterion [18-23]. Framework the criterion we approximate real distribution of concentration of dopant by idealized step-wise distribution of concentration ψ (x,y,z) (see Figs. 5 and 6 for diffusive or ion types of doping). Farther we determine optimal annealing time by minimization of mean-squared error

Fig.2. Spatial distributions of infused dopant concentration in the considered heterostructure.

The considered direction perpendicular to the interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves 2.0 1

C(x,Θ)

1.5

2 3

1.0 Epitaxial layer

0.5

Substrate

0.0 L L/2 3L/4 x Fig.3. Spatial distributions of infused dopant concentration in the considered heterostructure. 0

L/4

Curves 1 and 3 corresponds to annealing time Θ = 0.0048(Lx2+Ly2+Lz2)/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(Lx2+Ly2+Lz2)/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to the considered heterostructure. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

Fig.4. Implanted dopant distributions in heterostructure in heterostructure with two epitaxial layers (solid lines) and with one epitaxial layer (dushed lines) for different values of annealing time.

Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves U=

1 Lx L y Lz ∫ ∫ ∫ [C ( x, y, z , Θ ) − ψ (x, y , z )] d z d y d x . Lx Ly Lz 0 0 0

(8)

Dependences of optimal annealing time are shown on Figs. 7 and 8 for diffusion and ion types of doping. It should be noted, that after finishing of ion doping of material it is necessary to anneal radiation defects. In the ideal case after finishing of the annealing dopant achieves nearest interface between materials of heterostructure. If the dopant has no time to achieve the interface, it is practicably to additionally anneal the dopant. The Fig. 8 shows dependences of the exactly additional annealing time of implanted dopant. Necessity of annealing of radiation defects leads to smaller values of optimal annealing time for ion doping in comparison with values of optimal annealing time for diffusion type of doping. Using diffusion type of doping did not leads to so large damage in comparison with damage during ion type of doping.

C(x,Θ)

2 3 1

Fig. 5. Distributions of concentrations of infused dopant in the considered heterostructure.

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 2 3 4 C(x,Θ)

Fig. 6. Distributions of concentrations of implanted dopant in the considered heterostructure.

Curve 1 is the idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for different values of annealing time for increasing of annealing time with increasing of number of curve Dependences of optimal annealing time are shown on Figs. 7 and 8 for diffusion and ion types of doping. It should be noted, that after finishing of ion doping of material it is necessary to anneal radiation defects. In the ideal case after finishing of the annealing dopant achieves nearest interface between materials of heterostructure. If the dopant has no time to achieve the interface, it is practicably to additionally anneal the dopant. The Fig. 8 shows dependences of the exactly additional annealing time of implanted dopant. Necessity of annealing of radiation defects leads to smaller values of optimal annealing time for ion doping in comparison with values of optimal annealing time for diffusion type of doping. Using diffusion type of doping did not leads to so large damage in comparison with damage during ion type of doping. 0.5

Θ D0 L

-2

0.4

0.3 3

0.2

1 0.1 0.0 0.0

0.1

0.2 0.3 a/L, ξ, ε, γ

0.4

0.5

Fig.7. Optimal annealing time of infused dopant as dependences of several parameters.

Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epitaxial layer 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 the considered annealing time on the parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε = ξ = 0

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016 0.12 2 0.08 Θ D0 L

-2

0.04

0.00 0.0

0.1

0.2 0.3 a/L, ξ, ε, γ

0.4

0.5

Fig.8. Optimal annealing time of implanted dopant as dependences of several parameters.

4. CONCLUSIONS In this paper we introduced an approach to manufacture of field-effect of transistors which gives a possibility to decrease their dimensions. The decreasing based on manufacturing the transistors in a heterostructure with specific configuration, doping of required areas of the heterostructure by diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects. Framework the approach we introduce an approach of additional doping of channel. The additional doping gives us possibility to modify energy band diagram. We also consider an analytical approach to model and optimize technological process.

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 Russia, educational fellowship for scientific research of Government of Nizhny Novgorod region of Russia and educational fellowship for scientific research of Nizhny Novgorod State University of Architecture and Civil Engineering.

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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 Radio physics, from 1999 to 2001 it was master course in Radiophysics with specialization in Statistical Radio-physics, 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. 2012-2015 Full Doctor course in Radio physical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of Nizhny Novgorod State University. He has 135 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 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. Since 2014 E.A. Bulaeva is in a PhD program in Radio-physical Department of Nizhny Novgorod State University. She has 90 published papers in area of her researches. 19