ISSN 1063-7826, Semiconductors, 2016, Vol. 50, No. 13, pp. 1708–1715. © Pleiades Publishing, Ltd., 2016. Original Russian Text © M.I. Zavgorodnyaya, I.V. Lavrov, 2016, published in Izvestiya vysshikh uchebnykh zavedenii. Elektronika, 2015, Vol. 20, No. 6, pp. 565–575.
MATERIALS FOR ELECTRONIC ENGINEERING
Methods of Accounting for Inclusion-Shape Randomness in Calculating the Effective Dielectric Characteristics of Heterogeneous Textured Materials M. I. Zavgorodnyayaa and I. V. Lavrovb* a
MEPhI National Research Nuclear University, Moscow, 115409 Russia National Research University of Electronic Technology, Ploschad’ Shokina 1, Zelenograd, Moscow, 124498 Russia * e-mail:
[email protected]
b
Submitted April 13, 2015
Abstract—Two methods of accounting for the inclusion-shape randomness, an analytical method and a method for simulating a medium with several inclusion types, are considered for calculating the effective permittivity tensor of a textured heterogeneous matrix-type medium with inclusions of a random ellipsoidal shape. The methods are based on the generalized Maxwell–Garnett model. The rotation group representations are used to consider the distribution of inclusion orientations. The results of calculations by these methods of the effective dielectric characteristics of porous silicon models in an alternating electromagnetic field in the frequency range of 103–108 Hz are compared. Keywords: heterogeneous material, composite, effective permittivity tensor, texture, orientation, random ellipsoidal shape, Maxwell–Garnett approximation. DOI: 10.1134/S1063782616130121
INTRODUCTION The problem of calculating the effective physical characteristics of heterogeneous materials has attracted the attention of researchers for one and a half centuries, despite the large number of papers on this subject. First, this is explained by the fact that heterogeneous materials, including composites, nanocomposites, polycrystals, and their films, as well as various porous structures, are widely used in science and engineering. Second, the Earth’s lithosphere is also a heterogeneous medium and the prediction of the properties of heterogeneous materials depending on their composition and structure is of great importance for geophysics (mineral exploration, and the prediction of earthquakes and volcanic eruptions). Third, the variety of such materials is so great and their structure is so complex that there is no universal method for solving this problem that would be equally appropriate for heterogeneous materials of all types. A fundamental review of heterogeneous materials and methods for determining their effective properties was given in [1]. A review of methods for calculating the effective characteristics of inhomogeneous media was also given in [2]. Most heterogeneous materials are textured, it is manifested in the anisotropy of their macroscopic properties. The causes of the texture formation are different. As an example, in developing artificial materi-
als, crystallites are intentionally arranged into structures with preferred directions; in natural materials, the texture is formed under external stresses and fields. The texture formation process in a polycrystalline material was simulated in [3]. The estimation of the characteristics of textured materials is associated with significant problems because of computation complexities in the consideration of inhomogeneity grain orientations and the insufficient accuracy of the existing methods for studying the textures of real materials. Examples of the analytical approach to the texture consideration in polycrystalline and matrix-type materials can be found in [4–6] and [7–10], respectively. One of the texture types, the shape texture, is associated with the preferred orientation of crystallites if they have a nonspherical shape. At the same time, taking into account the crystallite shape itself is an extremely difficult problem. In most cases, it is assumed that crystallites have a regular geometrical shape: ellipsoidal, cylindrical, etc. However, even in the development of materials with a certain fixed crystallite (inclusion) shape, it is impossible to achieve an ideal shape of all crystallites at the current state of the technology. More adequate prediction of heterogeneous material properties requires the development of models that consider the statistical variance of the crystallite shape. In most studies where the crystallite
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METHODS OF ACCOUNTING FOR INCLUSION-SHAPE RANDOMNESS
shape randomness is taken into account, the geometrical parameters that define crystallite sizes within the chosen regular geometrical shape are varied. In particular, if crystallites are spheroidal, the so-called aspect ratio, which is the ratio of the spheroid semiaxes, is varied according to a certain statistical law, and then averaging is performed by integrating over this parameter, taking its distribution density into account. As an example, in [11], it was accepted that the aspect ratio is uniformly distributed over a certain segment; in [12], it is Gaussian distributed with specified average and dispersion. The objective of this study is to compare two methods for calculating effective permittivity tensor components of a textured matrix composite with inclusions of a random ellipsoidal shape with a small deviation from the spheroidal shape. The first method is analytical, based on the expansion of ellipsoid geometrical factors in a power series of relative deviations of ellipsoid semiaxes [13]. The second method is simulation where the Maxwell−Garnett model is generalized to the case of a matrix medium with several inclusion (crystallite) types. In this case, the continuous distribution of relative deviations of ellipsoid semiaxes is approximated by the discrete one [14]. The methods are compared via the results of simulation of effective dielectric characteristics of porous silicon in an alternating electromagnetic field in the frequency range of 1 kHz−100 MHz. STATEMENT OF THE PROBLEM OF CALCULATING THE EFFECTIVE PERMITTIVITY OF A MATRIX HETEROGENEOUS MEDIUM AND ITS SOLUTION IN THE GENERALIZED MAXWELL−GARNETT APPROXIMATION Let there be a macroscopically homogeneous sample of a heterogeneous material of volume V, consisting of a homogeneous isotropic matrix with same-type anisotropic inclusions from the viewpoint of their physical characteristics. The total inclusion fraction is d. The principal axes of the inclusion permittivity tensors are assumed to coincide with the axes of the corresponding ellipsoids. Inclusion orientations are distributed according to a certain probability law; the inclusion shape is random ellipsoidal. The inclusion orientation and shape are considered independently of each other. Let a dc electric field E0 be applied to the boundary S of a given sample. The material permittivity tensor ε(r) is a random piecewise function of coordinates,
⎧ε mI, r ∈ V m, (1) ε(r) = ⎨ ⎩ε k , r ∈ V k , k = 1, N , where ε m is the matrix permittivity, V m is the volume occupied by the matrix, I is the second-rank unit tensor, ε k and V k are the permittivity tensor and volume of SEMICONDUCTORS
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the kth inclusion, respectively, and N is the number of inclusions in the sample. The electrostatic equations div D = 4πρ, rot E = 0 and the constitutive equation D = εE lead to the boundary-value problem for the potential ϕ(r) in a given medium (E = −∇ϕ )
∇ ⋅ ε(r)∇ϕ(r) = − 4πρ(r),
ϕ S = −(E 0 ⋅ r),
(2)
where ρ(r) is the volume charge density. The problem is posed of calculating the effective permittivity tensor ε e of a heterogeneous material, relating the average electric-induction and electricfield strength vectors D = ε e E . In the generalized Maxwell−Garnett approximation for ε e , expression [9] was derived
ε e = [(1 − d )ε mI + d κ k
] [(1 − d )I + d
λk
] −1 ,
(3)
where the tensors λ k and κ k related to a particular inclusion are introduced, −1
−1
λ k = [I + (ε m ) L k (ε k − ε mI)] , κ k = ε kλ k,
k = 1, N ,
where L k is the tensor of the ellipsoid geometrical factors. To simplify the expressions, the subscript k referred to the inclusion number wiil be omitted below. Due to the coincidence of the tensor principal axes ε and ellipsoid axes, the tensor principal values λ and κ are calculated by the formulas (ε 'j are the tensor ε principal values)
λ 'j = ε m(ε m + L j (ε 'j − ε m )) κ 'j = ε 'j λ 'j ,
−1
(4)
j = 1,2,3,
where L j are the tensor L principal values (a1, a2, a3 are the ellipsoid semiaxes),
L j (a1, a2, a3 ) = ∞
×
∫ (a 0
a1a2a3 2
dq 2 j
+ q) ⎡⎣(a1 + q)(a2 2 + q)(a3 2 + q)⎤⎦ 2
1/2
,
(5)
j = 1, 2, 3.
The tensors λ and κ in Eq. (3) are averaged over all sample inclusions. In this case, this is averaging over all orientations and all inclusion shapes. Assuming that the inclusion shape and orientation are independent, averaging can be performed sequentially over the inclusion orientations and shapes,
λ = λ
o f
,
κ = κ
o f
,
where ⋅ f is averaging over shapes and ⋅ o is averaging over inclusion orientations.
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where F (ψ ', θ ', ϕ ') is the inclusion distribution function over orientations (DFO) [15]; the integral over the SO(3) group is written as
AVERAGING OVER INCLUSION ORIENTATIONS Let the xyz coordinate system be related to the composite sample. To each inclusion, we introduce the system of ξηζ axes of its ellipsoid; then, the inclusion orientation g ' ≡ g '(ψ ', θ ', ϕ ') in the system xyz (ψ ', θ ', ϕ ' are the Euler angles) is the rotation from xyz to ξηζ . The tensors λ and κ are averaged over inclusion orientations in the system xyz by the formulas
λ
o
=
∫
κ
λF ( g ')dg ',
o
∫
=
SO(3)
2π π 2π
∫
(⋅) dg ' ≡
SO(3)
∫ ∫ ∫ (⋅) sin θ'd ψ 'd θ'd ϕ'. 0 0 0
Orientation-averaged component of the tensors λ and κ in the system xyz were calculated in [9] and after averaging over shapes takes the form (numbering of subscripts l, j = 1, 2, 3 of tensor components in the system xyz corresponds to x1 ≡ x, x 2 ≡ y, x3 ≡ z ) [13]
κ F ( g ')dg ' ,
SO(3)
2
α jj
o f
= A
f
/3 + (− 1)
j +1
∑ 〈α ' 〉 ∫
dg 'F ( g ')[T−2,s ( g ') + T2,s ( g ') + (− 1) 2
s f
s =− 2
2
j
j = 1,2,
2
2/3T0,s ( g ')],
SO(3) 2
α 33
o f
= A
f
/3 + 8 3
∑ 〈α ' 〉 ∫ s f
s =− 2
dg ′ F ( g ')T0,s ( g ') , 2
(6)
SO(3)
2
α12
=i
o f
∑ 〈α ' 〉 ∫ s f
s =− 2
dg 'F ( g ')[T2,2s ( g ') − T−22,s ( g ')],
SO(3)
2
α j3
o f
= (−i )
j −1
∑ 〈α ' 〉 ∫ s f
s =− 2
o f
2
m = − 2,… ,2) are in essence λ lj , λ 'j , λ 'm , if λ lj
λ '0 = (2λ '3 − λ1' − λ '2 )/ 6,
λ '−1 = λ 1' = 0,
κ '0 = (2κ '3 − κ1' − κ '2 )/ 6,
κ '−1 = κ 1' = 0;
Let us consider the axisymmetric distribution of orientations under the additional condition of independence on angle ϕ ' (θ′ is the angle between the z and ζ axes), i.e., the DFO such as
(
0
are cal-
θ ' f (θ ') sin θ ' d θ ' .
(7)
f (θ ') sin θ ' d θ ' = 1 . Then,
λ
o f
in the system
)
o f
(
π /2
for the components of tensor xyz , we have the expressions
= 〈λ1' 〉 f + 〈λ '2 〉 f (1 − I 1)/2 + 〈λ '3 〉 f I 1,
f
∫
0
= 〈λ 22 〉 o
= 〈λ1' 〉 f + 〈λ '2 〉 f (1 + I 1)/4 + 〈λ '3 〉 f (1 − I 1) / 2,
)
π 2 2
normalization condition
o f
where
∫ cos
o f
(8) F ( g '(ψ ', θ ', ϕ ')) = (8π 2 ) −1 f (θ '). The one-dimensional density f (θ ') satisfies the
l l Tms ( g ') ≡ Tms (ψ ', θ ', ϕ '), l = 0,1,… ; m, s = −l, l are generalized spherical functions [16].
I1 =
j = 1,2,
2
are calculated, which are defined by the formulas
⎧λ ' = λ ' = (λ ' − λ ' )/2, ⎪ −2 2 1 2 ⎨ ⎪⎩κ '−2 = κ '2 = (κ1' − κ '2 )/2,
⎧ λ ⎪ 11 ⎨ ⎪ λ 33 ⎩
j
SO(3)
where A = α1' + α '2 + α '3 ; α lj , α 'j , α 'm (l, j = 1, 2, 3, culated, or κ lj , κ 'j , κ 'm , if κ lj
dg 'F ( g ')[T−1,s ( g ′) + (− 1) T1,s ( g ')],
(10)
(9)
For the components of tensor κ o f , similar expressions occur. Substituting the obtained formulas into expression (3), we will find the expressions for tensor ε e components in the system xyz for the orientation distribution such as (8), SEMICONDUCTORS
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METHODS OF ACCOUNTING FOR INCLUSION-SHAPE RANDOMNESS
(
1711
)
⎧ 4(1 − d )ε m + d [〈κ1' 〉 f + 〈κ '2 〉 f ](1 + I 1) + 2〈κ '3 〉 f (1 − I 1) ⎪ , ⎪( ε e )11 = ( ε e ) 22 = 4(1 − d ) + d [〈λ1' 〉 f + 〈λ '2 〉 f ](1 + I 1) + 2〈λ '3 〉 f (1 − I 1) ⎪⎪ ⎨ ⎪ 2(1 − d )ε m + d [〈κ1' 〉 f + 〈κ '2 〉 f ](1 − I 1) + 2〈κ '3 〉 f I 1 ⎪( ε e ) = , (ε e )ij = 0, i ≠ j. 33 ⎪ ' ' ' 2(1 − d ) + d [〈λ1〉 f + 〈λ 2 〉 f ](1 − I 1) + 2〈λ 3 〉 f I 1 ⎪⎩
(
(
(
)
As a model of the axisymmetric distribution, the Rayleigh-type distribution with one-dimensional density is considered, 2 ⎛ ⎞ f (θ ') = 12 13 exp ⎜ − tan 2θ ' ⎟ , s cos θ′ ⎝ 2s ⎠
〈λ 'j 〉 f ≈ λ 'j [1 + z j σ1 + y j σ 2 ], 0
to be small: e j 2 ≡ σ j 2 1, j = 1, 2 ; it is also considered that e1 and e2 are independent; therefore e1e2 = 0 . As seen from Eqs. (6) and (9), to average the tensor components λ and κ in the system over all inclusions, the principal components of tensors λ and κ averaged over the shape should be calculated: 〈λ 'j 〉 f , 〈κ 'j 〉 f , j = 1,2,3. Let us consider and compare two methods for calculating such averages, i.e., the analytical and simulation methods. An Analytical Method for Considering the Inclusion-Shape Randomness This method was proposed in [13] and consists in the expansion of ellipsoid geometrical factors L1, L2, L3 , and then the quantities λ 'j , κ 'j ( j = 1, 2, 3) SEMICONDUCTORS
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2
〈 κ 'j 〉 f = ε 'j 〈λ 'j 〉 f ,
2
(13)
j = 1, 2, 3,
where
z j = q j 2 A j1 2 − q j B j1, λ 'j 0 =
METHODS FOR AVERAGING OVER INCLUSION SHAPES The inclusion shape is taken to be random ellipsoidal. The spheroidal shape with the semiaxis ratio a : a : c is considered as the average shape. The third semiaxis a3 of all inclusions is fixed: a3 = c = fixe , a1 and a2 semiaxes randomly diverge from the average value equal to a , i.e., a1 = a2 = a . In this case, the shape of each inclusion is defined by a random vector, whose components are relative deviations e1, e2 of a1, a2 semiaxes from their average values: e j = ( a j − a ) a, j = 1, 2. It is clear that e1 = e2 = 0 . The variances of e1, e2 are considered
)
(11)
in powers to the second order inclusive, followed by averaging. Finally, for 〈λ 'j 〉 f , 〈 κ 'j 〉 f , we obtain the expressions
0 ≤ θ ' ≤ π , (12) 2
where the parameter s 2 plays the role of the variance in the ζ -axis orientations of inclusions; thus 2 2 2 tan θ′ = π 2 s , tan θ ' = 2s , D[tan θ′] = 4 − π s 2 [10].
)
εm ε m + L j 0(ε 'j − ε m )
y j = q j 2 A j 2 2 − q j B j 2, ,
qj =
(14)
ε 'j − ε m
, ε m + L j 0(ε 'j − ε m ) (15)
L j 0 = L j (a, a, c), 2
4 A11 = A22 = a c J 2 − 3 a c J 3, 2 2 2 A12 = A21 = a c J 2 − 1 a 4c J 3, 2 2 2 A31 = A32 = a c J1 − 1 a 4c J2, 2 2 B11 = B22 = 15 a 6c J 4 − 9 a 4c J 3, 4 4 B12 = B21 = 3 a 4c (a 2 J 4 − J 3 ), 4 B31 = B32 = 3 a 4c (a 2 J3 − J2 ), 4
(16)
∞
Jn =
∫ (a
2
dq , 12 + q) (c 2 + q)
∫ (a
2
dq . 32 2 + q) (c + q)
0 ∞
Jn =
0
n
n
A Method of Simulating the Inclusion-Shape Randomness The method for taking into consideration the shape randomness is based on simulating a heterogeneous material with several inclusion types. If a material contains M inclusion types with specific parameters and orientation distributions, the tensor ε e will be cal-
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culated by the formula that generalizes Eq. (3) to this case [16]
⎡ ε e = ⎢(1 − ⎢⎣
M
∑d
⎤ κp ⎥ ⎥⎦
M
p )ε mI
+
p =1
∑d
p
p =1
(17) −1 ⎡ ⎤ p ⋅ ⎢(1 − d p )I + dp λ ⎥ , ⎢⎣ ⎥⎦ p =1 p =1 where the subscript p shows that the quantity relates to a given inclusion type; angle brackets mean averaging over all p-type inclusions. In this method, the continuous distribution of e1, e2 is approximated by the discrete distribution by dividing the square − 1 < e1, e2 < 1 into (2n + 1)2 cells and setting a relative inclusion fraction for each cell, with deviations of the semiaxes lying within this cell. In this case, if the cell (k1, k 2 ) contains a relative inclusion fraction d k1,k2 , it is accepted that all inclusions of this cell have relative deviations equal to the cell center coordinates: M
∑
e1 = 2k1 (2n + 1),
⎧ λ k1,k2 ⎪ 11 ⎨ k1,k 2 ⎪ λ 33 ⎩
The sum of relative inclusion fractions over all cells
∑
o
o
k1,k 2 = λ 22
=
(
o
(
n
∑ ∑
∑
(18)
∑
n n ⎡ ε e = ⎢(1 − d )ε mI + d d k1,k2 κ k1,k2 ⎢⎣ k1 =−n k 2 =−n
M
e2 = 2k 2 (2n + 1) .
n
is unity: d = 1. Let the total volume k1 =−n k 2 =−n k1,k 2 fraction of all inclusions be d , then formula (17) can be applied to this model in the form [14]
n n ⎡ k ,k ⋅ ⎢(1 − d )I + d d k1,k2 λ 1 2 ⎢⎣ k1 =−n k 2 =−n
∑ ∑
−1
⎤ ⎥ o ⎥⎦
⎤ ⎥ , o ⎥⎦
(19)
where κ k1,k2 and λ k1,k2 are tensors of inclusions with relative semiaxis deviations (18). Averaging in (19) is performed over orientations of a given inclusion type. If all inclusion types have an orientation distribution with the DFO such as (8), the tensors λ k1,k2 o averaged over orientations are uniaxial with the principal values
)
= (λ1' ) k1,k2 + (λ '2 ) k1,k2 (1 + I 1)/4 + (λ '3 ) k1,k2 (1 − I 1)/2,
)
(20)
(λ1' ) k1,k2 + (λ′2 ) k1,k2 (1 − I 1)/2 + (λ '3 ) k1,k2 I 1,
where (λ 'j ) k1,k2 , j = 1,2,3 are the principal values of tensors λ k1,k2 ; I 1 is defined by formula (10). For components of tensors κ k1,k2 , similar formulas occur. o
COMPARISON OF THE RESULTS OF THE SIMULATION OF THE FREQUENCY DIELECTRIC CHARACTERISTICS OF POROUS SILICON OBTAINED BY TWO METHODS Using formulas (11), (13)–(16) (the analytical method) and (19), (20) (a method of simulating a material with several inclusion types), the frequency dielectric characteristics of porous silicon with fibrous or layered structure in the xyz system were calculated in the frequency range of 103–108 Hz. Due to the small volume fraction of silicon in a material, it was considered as inclusions, and air was considered as a matrix. The dependence of the silicon permittivity on the electromagnetic field frequency at low frequencies is given by ε(ω) = ε s + i 4πσ s ω , where σ s = 0.435 × 10 −3 Ω −1m −1 is the silicon static conductivity and ε s = 11.7 is the static permittivity [17]. Porous silicon with fibrous structure was simulated by inclusions of a random ellipsoidal shape with
a small variance around the average highly prolate spheroidal shape. A material with a layered structure was simulated by inclusions with small deviations from the average highly oblate spheroidal shape. The inclusion orientations were considered to be axisymmetrically distributed with the DFO such as (8) and onedimensional density (12) at various variances, s 2 . The distribution of relative deviations e1, e2 of inclusion semiaxes was considered to be normal with variances σ1 2, σ 2 2 . The binding partitions between fibers or layers in a material were disregarded. The objective of simulation was to compare the results obtained by two methods and to estimate the limits of applicability of the analytical method depending on the variances e1, e2 and the parameters that characterize the average shapes of the inclusions. Some calculated results are shown in Figs. 1−4. We can see from the curves in Figs. 1, 2, 4 that the analytical and simulation methods yield very similar results for small variances in inclusion shapes (σ1, σ 2 ≤ 0.08). At any aspect ratios of the average inclusion shape, the relative difference in the calculation of effective dielectric characteristics in the frequency range of 1 kHz–100 MHz does not exceed 0.1%. The disagreements in the results obtained by SEMICONDUCTORS
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(a) σ = 0.3
1.30 1.28
1.24
σ = 0.2 σ = 0.1
Reε33
Reε11
1.26
d = 0.1 a:c = 1:20 s2 = 0.002
1.22 1.20 1.18 103
104
105
106
107
108 ν, Hz
(b) 22 σ = 0.3 d = 0.1 20 18 a:c = 1:20 16 s2 = 0.002 14 σ = 0.2 σ = 0.1 12 10 8 6 4 2 0 3 10 104 105 106 107 108 ν, Hz
(c)
Imε11
0.020
σ = 0.3
d = 0.1 a:c = 1:20 s2 = 0.002 σ = 0.2 σ = 0.1
0.015
Imε33
0.025
0.010 0.005 0 103
104
105
106
107
108 ν, Hz
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(d) 9 8 σ = 0.2 7 6 5 σ = 0.1 4 σ = 0.3 3 2 1 0 3 10 104 105 106
d = 0.1 a:c = 1:20 s2 = 0.002
107
108 ν, Hz
δ(Reεjj)
(a) 0.02 σ = 0.1 σ = 0.2 0.01 0 −0.01 −0.02 σ = 0.3 −0.03 −0.04 −0.05 −0.06 −0.07 103 104 105 106 ×10–3 1.5 σ = 0.1 0
−6 −8
−10 103
104
108 ν, Hz
(c)
σ = 0.2
−4
107
105
σ = 0.3 d = 0.1 a:c = 20:1 s2 = 0.002
106
107
108 ν, Hz
δ(Imεjj)
δ(Reεjj)
−2
d = 0.1 a:c = 1:20 s2 = 0.002
δ(Imεjj)
Fig. 1. The frequency characteristics of the (a, b) real and (c, d) imaginary parts of components (a, c) (εe)11 and (b, d) (εe)33 of the effective dielectric characteristic of the model of porous silicon with fibrous structure at various inclusion-shape variances: solid and dashed curves correspond to the analytical method and simulation, respectively.
0.05 σ = 0.1 0 −0.05 −0.10 σ = 0.2 −0.15 −0.20 −0.25 −0.30 −0.35 −0.40 3 10 104
(b)
σ = 0.3
105
106
d = 0.1 a:c = 1:20 s2 = 0.002
107
108 ν, Hz
(d) 0 −0.01 σ = 0.1 −0.02 σ = 0.2 σ = 0.3 −0.03 d = 0.1 −0.04 a:c = 20:1 −0.05 s2 = 0.002 −0.06 −0.07 3 10 104 105 106 107 108 ν, Hz
Fig. 2. The frequency dependences of the relative difference between the analytical method and simulation results in the calculation of the (a, c) real and (b, d) imaginary parts of the components (εe)11 (solid curves) and (εe)33 (dashed curves) for the porous silicon models at various inclusion-shape variances. SEMICONDUCTORS
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×10−3 (a) 4 1:100 3 2 5:1 1 0 −1 −2 20:1 1:5 1:20 −3 −4 −5 −6 103 104 105
×10−3 0 5:1
(b)
−5
d = 0.1 σ = 0.18 s2 = 0.002 106
107
108 ν, Hz
δ(Imεjj)
δ(Reεjj)
1714
−10 20:1
−15
1:1
−20 −25 −30 103
1:20
1:100 104
105
1:5 d = 0.1 σ = 0.18 s2 = 0.002 106
107
108 ν, Hz
Fig. 3. The frequency dependences of the relative difference between the analytical method and simulation results in the calculation of the (a) real and (b) imaginary parts of the components (εe)11 (solid curves) and (εe)33 (dashed curves) for the porous silicon models at various aspect ratios a:c of the average inclusion shape.
(a)
σ2
1% 0.35 0.1%
0.25
0.12 0.10
0.20
0
0.1%
0.08
0.15
0.05
1%
0.14
0.30
0.10
(b)
σ2 0.16
d = 0.1 a:c = 1:20 s2 = 0.002 0.05 0.10 0.15 0.20 0.25 0.30 0.35 σ1
0.06 0.04 0.02 0
a:c = 1:20 s2 = 0.002 0.04
0.08
0.12
0.16 σ1
Fig. 4. The regions on the σ1,σ2 parameter plane in which the relative deviations of (a) real and (b) imaginary parts of the effective permittivity component (εe)11 of the porous silicon model with a fibrous structure in the range of 1 kHz–100 MHz, calculated by the analytical method do not exceed 0.1 or 1%.
different methods increase with the shape variance. This is explained by an increase in the terms in powers for which the expansion is performed in the procedure of the analytical method; hence, the remainder of the series increases, which is neglected. The difference between the results of the two considered methods can be interpreted as the error of the analytical method because of the above mentioned fact along with what the method of simulation can be arbitrarily precise in the limits of the accuracy of the quasi-static Maxwell– Garnett approximation via increasing the number of cells into which the range of relative deviations of inclusion semiaxes is divided. One of the characteristic features of the frequency dependences shown in Figs. 2 and 3 is that the relative error of the analytical method is at a maximum at low frequencies. This is explained by the fact that the silicon permittivity at low frequencies contains a very
large imaginary part: Im(ε(ω)) = 4πσ s ω, since the conductivity σs is taken in the CGS units in this formula: the value 0.435 × 10 −3 Ω −1m −1 should be multiplied by (4πε 0 ) −1 ≈ 9 × 10 9 . From this it follows that the term in which the expansion is performed in the analytical method procedure at low frequencies tends to its maximum (at fixed shape variances). Hence, the maximum value also has an error because of the neglect of the remainder of the series. Analyzing the frequency dependences shown in Figs. 3 and 4, we note that the relative error of the analytical method depends on the aspect ratio a : c of the average inclusion shapes. The largest difference in the results takes place at a highly prolate average shape of the inclusions: at a : c ≤ 1 : 20; in this case, the imagSEMICONDUCTORS
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METHODS OF ACCOUNTING FOR INCLUSION-SHAPE RANDOMNESS
inary parts of dielectric characteristics differ to a greater extent than the real parts. CONCLUSIONS To estimate the effect of the inclusion shape randomness on dielectric characteristics of heterogeneous materials in an alternating electromagnetic field, both proposed methods can be used, with either having specific advantages and disadvantages. The disadvantage of the analytical method is an increase in its error with the variance of the inclusion shape. Nevertheless, the relative error does not exceed 1% in the range of 1 kHz–100 MHz at the shape variance σ1, σ 2 ≤ 0.14 . One advantage of the analytical method is the very low computational resource intensity from the viewpoint of the required time and memory. In particular, in calculating the characteristics of a material with very prolate inclusions, the simulation method requires a computational time that is several hundred times longer than that for the analytical method. The simulation method can be used at arbitrary geometrically allowable shape variances. The methods considered here can be used in the development of materials with desirable physical properties and in geophysics in the analysis of dielectric spectroscopy results. It should also be noted that, in contrast to the analytical method, the simulation method can be applied to calculate the optical characteristics of composites with a dielectric matrix and randomly shaped metal inclusions. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research, projects nos. 13-08-00672-a and 14-08-00654-a.
SEMICONDUCTORS
Vol. 50
No. 13
2016
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