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MOSCOW INSTITUTE OF ELECTRONIC TECHNOLOGY Zelenograd

Igor V. Lavrov

CALCULATION OF THE EFFECTIVE DIELECTRIC AND TRANSPORT PROPERTIES OF INHOMOGENEOUS

TEXTURED MATERIALS

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I. PERMITTIVITY OF TEXTURED COMPOSITES

,m ε – permittivities of matrix and inclusions

rd – volume fraction of r - type inclusions

eε – effective permittivity of a composite sample:

Fig.1. Composite with ellipsoidal inclusions

, eD ε E (1.1)

– distribution density of r -type inclusions;

– average electric displacement and electric field of composite sample

( )rw q

,,m r qE E – average electric fields in matrix and in the inclusion with parameters

( , )r q – set of inclusion’s parameters

V – volume of a composite sample

1 1( ) , ( )V V

V dV V dV D D x E E x (1.2)

,(1 ) ( ) ,r m r r r qr rd d w q dq E E E

,(1 ) ( ) ,r m r r r qr rd d w q dq D D D

(1.3)

(1.4)

( , )r q

,, ,,m m m r qr q r q D E D ε E (1.5)

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Maxwell-Garnett approximation:

1

(1 ) (1 ) ,r m r r rr rr r r r

d d d d

eε I κ I λ (1.8)

, 1,2,3;j j j j

(1.9)

, ,mr q E λ E (1.6)

1[ ( )] , 1,2,3,j m j j m mL j (1.7)

,j jL – principal values of tensors of depolarization and permittivity of inclusion with parameters

, ,( ) , ( )r r q r r qr rw q dq w q dq κ κ λ λ

( , )r q

,r qλ λ – tensor related with -inclusion and having the principal values( , )r q

I – identity tensor; κ ελ – tensor with principal values

averaged on all inclusions of r –type ,κ λtensors

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Matrix system with ellipsoidal inclusions:

1. One-type inclusions of similar form :

2. Inclusions with a casual form:

3. Composite with a complex texture:

11, ,r d d : : ,a b c const( , , ):q g xyz

1( ) ( )w q w g

Fig.2. Description of orientation of inclusion

2 2

0 0 0( ) ( , , ) sin 1.w g dg w d d d

2 21 1 2 2 1 21, 1,2; () ( ) ( ) .i i f fff

e i w e w e de de

1 2( , , ),q g e e 1 2 1 2 1 2( ) , ( ) , 0,ff f

e a c c e b c c e e e e

1, , ;r n , , ( ),r r rd w gε( , , ): .q g XYZ

(1.10)

(sample) (inclusion)

– semiaxes of inclusions;, ,a b c

– distribution density of orientations of inclusions,

XYZ– laboratory system;

1 1 2 2( ) ( ) ( ) ( ).f fw q w g w e w e

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1 2

22 2 22, 2, 0,

22

33 0,2

22 2

12 2, 2,2

13

( 1) ( )3 2

( ) ( ) ( 1) 2 3 ( ) , 1,2,

2 3 ( ) ( ),3

( ) ( ) ( ) ,21( ) ( )2

l

ll ss

ls s s

s ss

s s ss

ll s

D dg w g

T g T g T g lD dg w g T g

i dg w g T g T g

i dg w g T

2

2 21, 1,

2( ) ( 1) ( ) , 1,2,l

s ss

g T g l

(1.12)

1 2 3 2 1 2 1 0 3 1 2; 2, 0, 2 6D (1.13)

,lj lj xyzin systemComponents

, ( )lm sT g – generalized spherical functions,

1. One-type inclusions of similar form

1(1 ) (1 ) ,md d d d eε I κ I λ (1.11)

( , ,lj j lj j , ):lj j or

0,1,2, ; , , , ;l m s l l

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2. Composite with a casual ellipsoidal formof inclusions, close to the sphere form

1 2( ) , ( ) ,e a c c e b c c (1.14)

2 2 2 21 2 1 2 1 1 2 20, 1, 1.

f f f f fe e e e e e

1 1 2 2( ) ( , , ) ( ) ( ),f fw q w w e w e

(1.15)

(1.16)

Form is a random vector with components:

1 1 2 2( ), ( )f fw e w e – distribution densities of 1 2, .e e

,o of f λ λ κ κ

1(1 ) (1 ) ,md d d d eε I κ I λ (1.11)

– consecutive averaging on orientations and forms of inclusions

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3. Composite with a complex texture

1 1( ) ( ), 1.

n nr r r

r rw g A w g A

(1.17)

1

1 1 1 1(1 ) (1 )

n n n nr m r r rr rr r r r

d d d d

eε I κ I λ (1.18)

: , 1, ,r r r rXYZ x y z r n C 1 1, .r r r rr r r r

λ C λ C κ C κ C (1.19)

, , ( )r r rd w gε – volume fraction, tensor of permittivity and distribution density of

orientations of r –type inclusions

Superposition of distributions:

( 1, )r r rx y z r n – coordinate system, related with orientations’ distribution of r -type inclusions

XYZ – laboratory coordinate system;

– matrices of rotations,

(1.12)

,r rλ κ ,r r λ κ– tensors in laboratory system; – tensors in system .r r rx y z

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II. EFFECTIVE CONDUCTIVITY OF TEXTURED POLYCRYSTALS

iσ

eσ

– conductivity tensor of i -th crystallite

0E – electric field applied at the boundary of polycrystalline sample of volume

– tensor of effective conductivity of polycrystalline sample :

,e j σ E (2.1)

0, 0, ( ) δ ( )e j E σ x σ σ x

0

( ) ( ), ,( ) ( ) , .

e VS

0

σ x σ x xx x E x x

(2.2)

Ellipsoidal crystallites, effective-medium approximation:

1[ ( ) ] ( ) 0,e e I σ σ Γ σ σ

1 2 3G( , ) , , 1,2,3; , , ,i

kli l

kSn dS k l x x x y x z

x

x x

(2.3)

(2.4)

, j E – average current density and electric field of polycrystalline sample

G( , )x x – Green function of problem (2.2);

n – external normal to iS

( ) x scalar potential

VS

iS – surface of i-th crystallite;

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Spherical one-type crystallites, axial texture Kind of orientations’ distribution density:

1. Uniaxial crystallites :

2. Biaxial crystallites :

( , , ):g xyz

xyzTensors in system of sample :

2 1( ) (8 ) ( ) sin ,w g f

0 00 0 ,0 0

xxe

xxe e

zze

σ

0 00 0 .0 0

xxxx

zz

Γ

0

1 0 00 1 0 .0 0

σConductivity tensor of crystallite in system ofprincipal axes :

0 23

1 0 00 0 .0 0

σConductivity tensor of crystallite in system ofprincipal axes :

(2.6)

(2.7)

(2.8)

(2.5)

(sample) (crystallite)

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1. Uniaxial crystallites, analitical decisions:

1. Poorly anisotropic crystallites : 1 1

2. Small disorder in orientations of crystallites:

(2.9)

2 1s

(2.11b)

2 20 1 1

20 1 1 1

1 11 ( 1)(1 ) ( 1) (1 ) ,2 12

11 ( 1) ( 1) (1 ) ,3

xxe

zze

I I

I I I

2

21

0cos ( ) .I f d

2 10

2 1 2 10

1 1arcsin 1 ,

( 1)(1 ( 1) arcsin 1 ) ;

xxe

zze

s

s

(2.11a)1:

1:

2 1 2 1 20

2 1 2 1 2 1 20

1 1 ln[ ( 1)] ,

(1 ) 1 (1 ) ln[ ( 1)] .

xxe

zze

s

s

3. Weak macroscopic anisotropy of the polycrystal:0 0

0 0 1 0 0 11 1(1 ), (1 2 ),xx zze eu k I u k I (2.12)

0 1 10,25[3 9 8( 1) ], 1 3;u I I 00

1 20 0

75 66( 1)( 1) ( 1)1 .2 25 27 24( 1) 18 12,04( 1) 2,64( 1)

uku u (2.13)

(2.10)

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2. Biaxial crystallites, analitical decisions :

1. Poorly anisotropic crystallites : 2 31 1, 1 1

2. Small disorder in orientations of axes of crystallites and affinity of two principal values of conductivity tensor of crystallites:

0 2 1 3 1

2 22 1 1 3 3 2 1

2 20 2 1 3 1 2 1

3 3 2 1 1

1 11 ( 1)(1 ) ( 1)(1 )4 2

1 1( 1) (1 )(3 ) ( 1)( )(1 ) ,48 12

1 11 ( 1)(1 ) ( 1) ( 1) (1 )2 12

1( 1)( ) (1 ) .3

xx

zz

I I

I I I

I I I

I I

e

e

(2.14)

2

21, 1 1s

22

22 0 3 0 0 30

20 3 3 0 0 30

0 1, 0

0,5( 1) ( 1) [ 1 ( 1) ] ,

2 ( 1) [ 1 ( 1) ] ,

, , .

xx zz

zz xx

kk kks

s

s

k x z

e

e(2.15)

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0 0,5 1 1,50

0.5

1

1.5

Линия уровня ошибки

e

e

Some results of numerical simulation

Fig.3. Dependences of effective conductivitycomponents of gallium polycrystal on disorder value for two variants of distribution of crystallites’ orientations. Light-green and dark-greencurves correspond to a case of rotational symmetry to crystallographic axis c, blue and violet curves – to axis b

2s

2

2 2 2tg tg( ) expcos 2

fs s

Distribution density of angles :

Fig.4. The area on the plane of parameters , in which relative distortion of analitical decision (2.14) is less then 1% in comparison with the numerical decision of system (2.3).

2 3,

,eσ

(2.16)

0 1 2 3 4 5 6 7 820

30

40

50

60

70

80

90

100

Поликристалл галлия

Дисперсия

Eff

. co

nduc

tivity

xx-comp.(rot.axis c)

zz-comp.(rot.axis c)xx-comp.(rot.axis b)zz-comp.(rot.axis b)

10(Ohmcm)

3

-1

Polycrystal of gallium

Dispersion

Level curve of 1% - distortion