neous media with a texture. Attempts to construct such theories were made in. [2, 5â8]. In [2, 5, 6], the effective permittivity tensor of a composite consisting of a ...
ISSN 10637826, Semiconductors, 2011, Vol. 45, No. 13, pp. 1621–1627. © Pleiades Publishing, Ltd., 2011. Original Russian Text © I.V. Lavrov, 2011, published in Izvestiya vysshikh uchebnykh zavedenii. Elektronika, 2010, Vol. 83, No. 3, pp. 3–12.
BASIC RESEARCH
Effective Conductivity of a Polycrystalline Medium. Uniaxial Texture and Biaxial Crystallites I. V. Lavrov Moscow Institute of Electronic Technology (Technical University), Zelenograd, Moscow Oblast, 124498 Russia email: iglavr@mail.ru Submitted January 25, 2010
Abstract—The effective conductivity tensor of a polycrystalline medium with a texture has been calculated based on the method of selfconsistent solution and the theory of rotation group representations. The medium consists of a singletype of biaxial spherical crystallites, oriented in space according to some proba bilistic law, which implies the existence of uniaxial texture. An analytical solution is obtained for two cases: (i) weakly anisotropic crystallites and (ii) a small spread in the orientations of one of the crystallite axes with respect to the texture axis. DOI: 10.1134/S106378261113015X
1. INTRODUCTION Studies concerning the influence of orientations of an inhomogeneous medium’s components (inclusions in composites or crystallites in polycrystals) on the effective electric properties of this medium meet sig nificant computational difficulties, and in many stud ies of such media the orientations of crystallites are assumed to be either specified or absolutely random (isotropic) [1, 2]. However, the results of some investi gations show that both texture and some randomness should manifest themselves in the orientations of crys tallites in real materials [3, 4]. In this context, a topical task is to construct adequate theories to explain the electric and other properties of randomly inhomoge neous media with a texture. Attempts to construct such theories were made in [2, 5–8]. In [2, 5, 6], the effective permittivity tensor of a composite consisting of a uniform isotropic matrix and a singletype of crystallites (anisotropic uniaxial spherical [2], isotropic ellipsoidal [5], and anisotropic ellipsoidal [6]), incorporated into this matrix, was cal culated using the Maxwell–Garnett approximation. In [7] the problem of determining the effective con ductivity tensor for a polycrystal composed of uniaxial spherical crystallites was solved. The crystallites were assumed to be oriented in space over some probabilis tic law. Genchev [8] considered a polycrystalline medium composed of uniaxial spheroidal crystallites of several shapes, oriented in the same direction, and derived an expression for the effective conductivity tensor of this polycrystal in the effectivemedium approximation. However, he could not generalize the result obtained to the case of distribution of crystallite axis orientations over some probabilistic law (because a serious mistake was made).
In this study the theory developed in [7], based on the selfconsistent solution [1, 9], is generalized to a polycrystalline medium composed of biaxial spherical crystallites (which have conductivity tensors with three different principal values). Crystallites are assumed to be oriented in space over some probabilis tic law, which implies the presence of a selected (tex ture) axis. This type of texture can be observed in the case of the deposition of thin films or when a material is formed under an external uniform field. The pur pose of this study is to determine the components of the effective conductivity tensor of the medium as functions of the parameters describing the conductiv ity of individual crystallites and the distribution of their orientations. The computational complexity caused by the necessity of taking into account the crys tallite orientations is overcome (as in [5, 6]) using the theory of rotation group representations [10]. 2. STATEMENT OF THE PROBLEM Let us consider a sample of a conducting polycrys talline medium of volume V, composed of biaxial crys tallites of the same type, which are in ohmic contact with each other and differ in their sizes and orienta tions in space. We assume that all crystallites are spher ical and their orientations are related to the directions of the principal axes of the conductivity tensors σ. The system of principal axes of a specific crystallite will be denoted as ξηζ; then in this system the tensor σ is described by the matrix
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1 0 0 σ' = σ 0 0 α 2 0 . 0 0 α3
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Let the polycrystalline medium under consider ation have a particular direction: texture axis. We choose a coordinate system xyz related to the texture: the z axis coincides with the texture axis, and all three axes are mutually orthogonal. Then the orientation of this crystallite in the xyz system will be set by the rota tion g(ψ, ϑ, ϕ), where ψ, ϑ, and ϕ are the Euler angles. The volume fraction dV/V of the crystallites the orien tations of which belong to the elementary volume in the space of angular parameters d3ω = [ψ; ψ + dψ][ϑ; ϑ + dϑ][ϕ; ϕ + dϕ] is
where p(ψ, ϑ, ϕ) is the distribution density of crystal lite orientations, with an allowance for the invariant measure factor sinϑ [10]. We assume, as in [6, 7], that the distribution of the crystallite orientations has rota tional symmetries with respect to the z and ζ axes; i.e., its density has the form 2 –1
(1)
∫
π/2 0
f ( ϑ ) dϑ = 1.
Below, the distribution density of crystallite orienta tions will be taken in the form (1). Let a uniform electric field E0 be applied to the boundary S of this polycrystalline sample. Then the effective conductivity tensor σe of the medium is determined by the equation 〈J〉 = σe〈E〉, where J is the current density and 〈E〉 = E0. In the xyz system of the texture the tensor σe has the form xx
σe =
σe
0
0
0
xx σe
0
.
(2)
zz
0 σe
0
The method of selfconsistent solution leads to the following equation for σe [7, 9]: –1
〈 ( I – ( σ – σ e )Γ ) ( σ – σ e )〉 = 0,
Γ
xx
zz
2
ε – ε – arcsin ε = , xx zz 2 σe σe ε ε 1/ε – 1 arcsin ε – 1 = zz σe ε at
Γ
xx
zz
(3)
where I is a unit symmetric tensor and Γ is a tensor that is diagonal in xyz; Γxx = Γyy and its components are
(4a)
ε > 0,
2
ε – ε – ln ( – ε + 1 – ε ) = , xx zz 2 σe σe ε –ε 1 – 1/ε ln ( – ε + 1 – ε ) – 1 = zz σe ε at xx
(4b)
ε < 0,
zz
where ε = 1 – σ e /σ e . The averaging in (3) is performed over all crystallite orientations in the xyz system, and, since σe has two xx
where f(ϑ) is the distribution density of angles ϑ between the texture axis and the ζ axes of crystallites. Since the medium under consideration is considered as linear, it has an inversion symmetry, due to which f(ϑ) satisfies the relation f(π – ϑ) = f(ϑ) and the nor malization condition in the form
Γ
Γ
dV/V = p ( ψ, ϑ, ϕ )dψdϑdϕ,
p ( ψ, ϑ, ϕ ) = ( 8π ) f ( ϑ ), 0 ≤ ψ < 2π, 0 ≤ ϑ ≤ π, 0 ≤ ϕ < 2π,
calculated (for spherical crystallites) using the for mulas [7, 9]
zz
independent components, σ e and σ e , (under condi tion (1)), tensor equation (3) is reduced to two scalar equations for the components with subscripts 11 and 33: 〈(I – (σ – σe)Γ)–1(σ – σe)〉kk = 0, k = 1, 3. In some cases this problem can be solved analytically in the lin ear or quadratic approximation in small parameters. In particular, if the components of the tensor –1 σ 0 ( σ – σ e ) are small, i.e., –1
σ 0 ( σ lj – ( σ e ) lj ) Ⰶ 1,
l, j = 1, 2, 3,
(5)
then (I – (σ – σe)Γ)–1 ≈ I + (σ – σe)Γ and system (3) can be rewritten in the form 〈 σ – σ e〉 kk + 〈 ( σ – σ e )Γ ( σ – σ e )〉 kk = 0,
(6)
k = 1, 3.
Here, we solved the problem for the following two cases (where condition (5) is satisfied): (i) crystallites are weakly anisotropic, i.e., |α2 – 1| Ⰶ 1 and |α3 – 1| Ⰶ 1; (ii) the distribution of the angles ϑ between the tex ture axis and the ζ axes of crystallites has a small spread, and |α2 – 1| Ⰶ 1; i.e., two of three principal val ues of the conductivity tensor σ of a crystallite differ only slightly. 3. CALCULATION OF THE COMPONENTS OF THE TENSOR σ IN THE xyz SYSTEM AND SOME AUXILIARY RELATIONS Since averaging in Eqs. (3) and (6) is performed in the xyz coordinate system, it is necessary to calculate SEMICONDUCTORS
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EFFECTIVE CONDUCTIVITY OF A POLYCRYSTALLINE MEDIUM
the components of the tensor σ for a specific crystallite in this system. Expanding σ in a sum of two tensors [6], one of which is isotropic (spherical part of the tensor) and does not change at rotations of the coordinate sys tem and the other is symmetric with zero trace (devia tor) and can be transformed via an irreducible repre sentation (with a weight of two) of the group SO(3) of 3D rotations [10], we obtain the components of the tensor σ in the xyz system:
2
〈 T mn ( g )〉 = 0, 2 2 〈 T mn ( g )T m'n' ( g )〉
σ kk
2
∑
∑
k = 1, 2,
2
2
〈( T 00 ) 〉 = 0.25 (9I 2 – 6I 1 + 1),
2
2 2 3 〈 T 0, 2 T 0, –2〉 = ( I 2 – 2I 1 + 1 ), 8 2 2 〈 T 2, 2 T –2, –2〉
2
=
2 2 〈 T 2, –2 T –2, 2〉
2
2
2
2
2
2
2
〈 T 1, 2 T –1, –2〉 = 〈 T 1, –2 T –1, 2〉 = 0.25 ( I 2 – 1 ),
∑
(7b)
where
2
∑ σ˜ ' [ T s
2 2, s ( g )
π/2
2
– T –2, s ( g ) ],
I1 =
s = –2
π/2
∫ cos ϑf ( ϑ ) dϑ, 2
I2 =
0
σ 13 = σ 31 σ 23 = σ 32
2
∑
˜ 's [ T 2–1, s ( g ) – T 21, s ( g ) ] , σ
s = –2
σ = – i 0 2
(7c)
2
∑ σ˜ ' [ T s
2 – 1, s ( g )
2
+ T 1, s ( g ) ],
4
(10)
0
4. ANALYTICAL SOLUTION FOR WEAKLY ANISOTROPIC CRYSTALLITES In this case, σe is found from the system of equa xx
l
˜ '±2 = ( 1 – α 2 )/2, σ
∫ cos ϑf ( ϑ ) dϑ.
zz
tions (6); here, σ e and σ e can be written as
s = –2
where D = 1 + α2 + α3, T ms ( g ) (m, s = –l, …, l) are ˜ 's (s = generalized spherical functions [10], and σ –2, …, 2) are calculated using the formulas
xx
σ e = σ 0 ( 1 – u x ), ( u x Ⰶ 1,
zz
σe = σ0 ( 1 – uz ) u z Ⰶ 1 ).
(11)
For Γxx and Γzz, it is sufficient to be restricted to a zero approximation:
˜ '±1 = 0, σ
˜ 0' = ( 2α 3 – 1 – α 2 )/ 6. σ
xx
zz
Γ ≈Γ ≈Γ
To continue the calculations, we need the following 2 values averaged over the group SO(3): 〈 T mn ( g )〉 and 2
(9)
1 ( I + 6I + 1 ), = 2 1 16
〈 T 1, 1 T –1, –1〉 = 〈 T 1, –1 T –1, 1〉 = 0.25 ( 4I 2 – 3I 1 + 1 ),
2 ⎛ ⎞ ˜ 's T 20, s ( g )⎟ , σ 33 = σ 0 ⎜ D/3 + 2/3 σ ⎝ ⎠ s = –2
σ = 0 2
(8)
〈 T 0, 1 T 0, –1〉 = 1.5 ( I 2 – I 1 ),
⎞ (7a) ˜ 's [ T 2–2, s ( g ) + T 22, s ( g ) + ( – 1 ) k 2/3T 20, s ( g ) ] ⎟ , σ ⎠ s = –2
σ σ 12 = σ 21 = i 0 2
m + m' + n + n' ≠ 0.
= 0,
〈T 00〉 = 0.5 (3I 1 – 1),
2
×
m + n ≠ 0,
For nonzero values of these means, we have (the argument g is omitted for brevity):
2
⎛ k = σ 0 ⎜ D/3 – 0.5 ( – 1 ) ⎝
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2
〈 T mn ( g )T m'n' ( g )〉 (m, n, m', n' = –2, …, 2). With con dition (1) satisfied, we have
–1
xx α2 = α3 = 1
= – ( 3σ 0 ) .
(12)
Substituting (2), (7a)–(7c), (11), and (12) into (6) and taking into account (8) and (9), we finally obtain (with a quadratic accuracy in (α2 – 1) and (α3 – 1)) the components of the tensor σe:
⎧ xx 2 2 1 1 ⎪ σ e ≈ σ 0 1 + 1 (α 2 – 1) (1 + I 1) + 1 (α 3 – 1) (1 – I 1) – (α 2 – 1) (1 + I 1) (3 – I 1) – (α 3 – 1) (α 3 – α 2) (1 – I 1) , 4 2 48 12 ⎪ (13) ⎨ 2 2 ⎪ zz 1 1 1 ( α 2 – 1 ) ( 1 – I 1 ) – ( α 3 – 1 ) ( α 3 – α 2 )I 1 ( 1 – I 1 ) , ⎪ σ e ≈ σ 0 1 + 2 ( α 2 – 1 ) ( 1 – I 1 ) + ( α 3 – 1 )I 1 – 12 3 ⎩
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where I1 is found from (10). In the case of uniform dis tribution (I1 = 1/3), the medium is isotropic as a whole, with the scalar conductivity 2 1 1 2 σ e ≈ σ 0 1 + ( α 2 – 1 ) + ( α 3 – 1 ) – ( α 2 – 1 ) 3 3 27
2 – ( α 3 – 1 ) ( α 3 – α 2 ) . 27 If all crystallite axes ζ are codirectional (ϑ = 0, I1 = 1), relations (13) yield (14)
zz σe
≈ α3 σ0 . For uniaxial crystallites with the axis ζ (α2 = 1, α3 ≡ α), formula (13) coincides with the result previ ously reported in [7]. 5. SOLUTION OF THE PROBLEM IN THE CASE OF WEAKLY SPREAD ORIENTATIONS OF ζ AXES OF CRYSTALLITES, PROVIDED THAT |α2 – 1| Ⰶ 1 As in the previous case, the problem is reduced to the solution of Eq. (6). Let s2 (s2 Ⰶ 1) be a half of the initial secondorder moment of a random value tanϑ [7]. xx zz We will write σ e and σ e in the form xx
zz
xx
zz
σ e = σ e0 – σ 0 δ z ,
(15)
where δx and δz are of the same order of smallness as xx
zz
max(s2, |α2 – 1|); σ e0 and σ e0 are the values of the xx
zz
components σ e and σ e at s2 = 0. In the first approx imation in (α2 – 1) (Appendix, (A5), (A6)), xx
σ e0 ≈ σ 0 [ 1 + ( α 2 – 1 )/2 ],
zz
σ e0 ≈ σ 0 α 3 .
(16)
s ≤ 0.04. xx
(17) zz
To find the components σ e and σ e in the linear approximation in (α2 – 1), s2, it is sufficient to assume xx
zz
xx
zz
that Γ ≈ Γ 0 and Γ ≈ Γ 0 , where Γ 0 and Γ 0 are the xx
zz
xx zz values of Γ and Γ at α2 = 1 and s2 = 0. xx
zz
Having calculated Γ 0 and Γ 0 from (4a) and (4b), xx
zz
we obtain ε ≡ 1 – σ e0 /σ e0 ≈ 1 – 1/α3, α 3 – 1 – α 3 arcsin 1 – 1/α 3 xx Γ 0 = , 3/2 2σ 0 ( α 3 – 1 ) zz
Γ0
arcsin 1 – 1/α 3 – α 3 – 1 = , 3/2 σ0 ( α3 – 1 )
zz
2
⎧ σ e ≈ σ 0 (0.5 (α 2 + 1) + s (α 3 – 1) [1 + σ 0 Γ 0 (α 3 – 1)]), ⎨ zz (19) ⎩ σ e ≈ σ 0 ( α 3 – 2s 2 ( α 3 – 1 ) [ 1 – σ 0 Γ xx 0 ( α 3 – 1 ) ] ), xx
zz
where Γ 0 and Γ 0 are calculated from (18a) or (18b). In the particular case of uniaxial crystallites with a ζ axis (α2 = 1, α3 ≡ α) formulas (19) coincide with the results obtained previously in [7]. Obviously, the appli cability of formulas (19) is limited by the condition (α2 – 1)2 Ⰶ s2 ≤ 0.04. These results can easily be applied to the problems of determining the tensors of effective permittivity and effective magnetic permeability or heat conductivity of an inhomogeneous textured medium, provided that this medium is linear. 6. RESULTS OF THE NUMERICAL SIMULATION Calculations in the MATLAB medium were per formed for some polycrystalline media. Specifically, we found the components of the effective conductivity tensor σe by solving the system of equations (3) using the Newton method; the distribution density for the angles ϑ between the texture axis and the ζ axes of crystallites was taken in the form –2
2
I 1 ≈ 1 – 2s ,
(18b) 1 – α 3 – ln ( 1/α 3 + 1/α 3 – 1 ) = , 3/2 σ0 ( 1 – α3 ) at 1 – α3 > (1 – α2)/2 and α2 < 1. Finally, having solved (6) taking into account (2), (7.1)–(7.3), (8), (9), and (15)–(17), we have zz Γ0
–2
–2
2
f ( ϑ ) = s cos ϑ tan ϑ exp [ – 0.5s tan ϑ ],
The following estimate is valid for I1 [7]: 2
α 3 ln ( 1/α 3 + 1/α 3 – 1 ) – 1 – α 3 xx Γ 0 = , 3/2 2σ 0 ( 1 – α 3 )
xx
xx 2 σ e ≈ σ 0 1 + 1 ( α 2 – 1 ) – 1 ( α 2 – 1 ) , 2 12
σ e = σ e0 – σ 0 δ x ,
at α3 – 1 > (α2 – 1)/2 and α2 > 1;
(18a)
0 ≤ ϑ ≤ π/2,
(20)
which corresponds to the normal distribution (with a variance s2) of Beltrami coordinates of the crystallite ζ axes [7]; the parameter s2 characterizes the spread in the orientations of the ζ axes of crystallites with respect to the texture axis. We also compared the val ues of the components of the tensor σe of polycrystals, which were obtained by solving numerically system (3), with similar values in the quadratic analytical approx imation (13). Figure 1 shows the dependences of the components xx zz σ e and σ e of a gallium polycrystal (σa = 62.5 × 103 Ω–1 cm–1, σb = 133.3 × 103 Ω–1 cm–1, and σc = 19.88 × 103 Ω–1 cm–1; a, b, and c are the axes of the crystallographic coordinate system [11]) on the parameter s2 for two versions of the distribution of the SEMICONDUCTORS
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σexx, σezz
120 110 100 90 80 70 60 50 40 30 20
9.60
1625
9.35 9.10 8.85 8.60 8.35 8.10 7.85 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 s2
0
xx
1
0
2
3
4
5
6
7
8
9
s2
Fig. 2. Comparison of the dependences of the components
zz
Fig. 1. Dependences of the components σ e and σ e of a
xx
zz
ε e and ε e of the effective permittivity tensor of a barium
gallium polycrystal on the parameter s2 for two versions of crystallite orientation distributions: cases of rotational
sulfate polycrystal on the parameter s2 obtained by xx
zz
numerical solution of system (3) (䊊) ε e and (䊐) ε e , with their analytical approximation according to formulas (13)
xx
symmetry with respect to (gray curves) the c axis (䊊) σ e zz
7.60
xx
and (䉭) σ e and (black curves) the b axis (䊏) σ e and
xx
zz
(––) ε e and (—) ε e .
zz (䉫) σ e .
gallium crystallite axes in space. In the first case, the c axis is assumed to be the ζ axis, and the a and b axes are considered as the ξ and η axes, respectively; the distribution of the crystallite orientations is assumed to be independent of the rotation angle ϕ of crystallite around the crystallographic axis c. In this case, σ0 = 62.5 × 103 Ω–1 cm–1, α2 = 2.133, and α3 = 0.318. These dependences are presented by two gray curves in Fig. 1. The black curves correspond to the second dis tribution version, where the b axis is taken to be the ζ axis, and the a and c axes are considered as the ξ and η axes, respectively. In this case, σ0 = 62.5 × 103 Ω–1 cm–1, α2 = 0.318, and α3 = 2.133. All four curves intersect at the same point at s2 ≈ 2.02; i.e., at this s2 value the polycrystalline medium is on the whole isotropic. xx
In Fig. 2 the dependences of the components ε e zz
and ε e of the effective permittivity tensor of a barium sulfate polycrystal (εa = 7.65, εb = 12.2, εc = 7.7) on the parameter s2, which were obtained by the numerical solution of system (3), are compared with the result obtained by an analytical approximation using formu las (13). The ζ axis was taken to be the c axis; i.e., in this case, ε0 = 7.65, α2 = 1.595, and α3 = 1.0065 (all conductivity components in the formulas must be replaced by the corresponding permittivity compo nents). SEMICONDUCTORS
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Figure 3 shows an area in the plane of parameters α2 and α3, in which the relative error δσe = xx
zz
xx
max(| δσ e |, | δσ e |) in calculating the components σ e zz
and σ e from formulas (13) (in comparison with the numerical solution of system (3)) does not exceed 1% at different spreads of the orientations of the ζ axes of crystallites (the calculations were performed at six val ues of the parameter s2 from different ranges: 0.1, 0.6, 1.3, 2.0, 4.0, and 20.0). In addition, the maximum val ues of the relative error δαe in calculating the anisot zz xx ropy coefficient αe = σ e / σ e of the polycrystal are indicated at some boundary points. As follows from Fig. 3, the analytical approximation (13) (which was derived on the assumption of a weak anisotropy of the crystallites) provides an acceptable accuracy in calcu xx zz lating the components σ e and σ e for many polycrys tals composed of moderately anisotropic crystallites. The main result of our study is the explicit analyti cal dependence of the components of the effective conductivity tensor σe of a polycrystalline medium (in the coordinate system xyz related to the texture) on the parameters σ0, α2, and α3, which describe the con ducting properties of an individual crystallite, as well as on the distribution of crystallite orientations: via the integral I1 (10) or the parameter s2, which character izes the orientation spread of one of the crystallite axes (ζ axis) with respect to the texture axis. For the two cases considered above, this dependence is given by formulas (13) and (19), which generalize similar
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results, obtained in [7], to a polycrystalline medium composed of biaxial crystallites. It was found by numerical simulation that formulas (13) are not restricted to the case of weakly anisotropic crystallites but can also be applied to many polycrystals composed of moderately anisotropic crystallites (Fig. 3). The analytical dependences obtained in this study can be used, for example, to analyze the optical properties of thin films in the far IR range. Note that in the absence of spread in the orienta tions of crystallite axes ζ, the effective conductivity in the plane oriented perpendicularly to the texture axis depends on the conductivity of crystallites along this axis; this dependence manifests itself in the solution with a quadratic accuracy in (α2 – 1) (Appendix, (A6)), although it should be absent from physical consider ations. This fact is apparently related to the specific features of the method and model used by us. Never theless, the results obtained can be considered correct, because the error of the method used has a higher order of smallness than the calculation error. APPENDIX CALCULATION OF THE EFFECTIVE CONDUCTIVITY TENSOR OF A POLYCRYSTALLINE MEDIUM IN THE ABSENCE OF SPREAD IN THE ORIENTATIONS OF CRYSTALLITE AXES ζ In the case of zero orientation spread, the angle ϑ between the ζ and z axes is 0, and the orientation of an
( σ – σe ) = σ0
arbitrary crystallite can be set by the ϕ between the x and ξ axes. If the directions of the ξ and η axes of crys tallites are distributed uniformly in the xy plane, which is implied by condition (1), the distribution density of the angles ϕ has the form p(ϕ) = 1/2π, 0 ≤ ϕ < 2π. The matrix C(ϕ) of rotation from xyz to ξηζ and the tensor σ of crystallite in the xyz system can be writ ten as cos ϕ – sin ϕ 0 sin ϕ cos ϕ 0 , 0 0 1
C(ϕ) =
2
2
cos ϕ + α 2 sin ϕ ( 1 – α 2 ) sin ϕ cos ϕ 0
σ = σ0
2
2
( 1 – α 2 ) sin ϕ cos ϕ sin ϕ + α 2 cos ϕ 0 0
.
α3
0
We introduce the following designations: xx
u x = 1 – σ e /σ 0 , xx – 1
vx = ux – ( σ0 Γ ) , zz
xx
γ = Γ /Γ ,
zz
u z = 1 – σ e /σ 0 , zz – 1 v z = u z – ( σ 0 Γ ) , (A1)
β 2 = ( α 2 – 1 )/2 ,
β 3 = ( α 3 – 1 )/2.
Then, with an allowance for (2), we have
β 2 ( 1 – cos 2ϕ ) + u x
– β 2 sin 2ϕ
0
– β 2 sin 2ϕ
β 2 ( 1 + cos 2ϕ ) + u x
0
0
0
u z + 2β 3
(A2)
,
2
( I – ( σ – σ e )Γ )
–1
1 = – xx σ0 Γ
2β 2 cos ϕ + v x β 2 sin 2ϕ v x ( 2β 2 + v x ) v x ( 2β 2 + v x )
0 .
2
2β 2 sin ϕ + v x β 2 sin 2ϕ v x ( 2β 2 + v x ) v x ( 2β 2 + v x ) 0
After substituting (A2) and (A3) into (3) and aver aging over all values of the angle ϕ with a density p(ϕ), we obtain the equations for determining the compo xx zz nents σ e and σ e :
0
(A3)
0 ( γ ( v z + 2β 3 ) )
–1
zz
σe = α3 σ0 .
(A5)
(A4)
We will search for the solution to the first equation in (A4) in the case |α2 – 1| Ⰶ 1 in the form of a series in powers of (α2 – 1), restricting ourselves to quadratic terms. As a result, we have the expression
The second equation in (A4), with an allowance for (A1), yields
σ e ≈ σ 0 1 + 0.5 ( α 2 – 1 ) + 0.25σ 0 Γ 0 ( α 2 – 1 ) , (A6)
β 2 ( u x + v x ) + u x v x = 0,
u z + 2β 3 = 0.
xx
xx
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EFFECTIVE CONDUCTIVITY OF A POLYCRYSTALLINE MEDIUM δαe = 1.28%
α3 2.0
0.74%
0.71%
xx
0.95% δαe < 1%
1.2 0.8
0.71%
0.6 0.4
0
0.65%
0.86%
δαe = 1.4%
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 α2
Fig. 3. Area in the plane of parameters (α2, α3), in which xx
the relative error δσe in calculating the components σ e zz
and σ e from formulas (13) in comparison with the numerical solution of system (3) does not exceed 1%. The maximum values of the relative error δαe in calculating the anisotropy coefficient of a polycrystal are indicated at the boundary points.
xx
where Γ 0 ≡ Γ
= –1/3, formulas (A6)
REFERENCES
1.0
0.2
α3 = 1
and (A5) coincide with formulas (14), which were derived on the assumption that |α3 – 1| Ⰶ 1.
1.6 1.4
spread in the orientations of the ζ axes of crystallites and under the condition |α2 – 1| Ⰶ 1. Note that in the limiting case, at α3 = 1 and σ 0 Γ 0
1.8
1627
xx α2 = 1
is found from (18a) or (18b).
Thus, (A6) and (A5) are the desired expressions for xx zz the components σ e and σ e of the effective conductivity tensor of a polycrystalline medium in the absence of
SEMICONDUCTORS
Vol. 45
No. 13
2011
1. A. G. Fokin, Phys. Usp. 39, 1009 (1996). 2. O. Levy and D. Stroud, Phys. Rev. B 56, 8035 (1997). 3. V. I. Kolesnikov, I. I. Chekasina, B. V. Bardushkin, et al., Vestn. Yuzhn. Nauchn. Tsentra RAN 4 (3), 3 (2008). 4. S. K. Maksimov and K. S. Maksimov, Izv. Vyssh. Uchebn. Zaved., Elektron., No. 1, 49 (2008). 5. E. N. Ivanov and I. V. Lavrov, Oboron. Kompleks Nauch.Tekh. Progressu Rossii, No. 1, 73 (2007). 6. I. V. Lavrov, Ekol. Vestn. Nauch. Tsentrov ChES, No. 1, 52 (2009). 7. I. V. Lavrov, Izv. Vyssh. Uchebn. Zaved., Elektron., No. 1, 3 (2008). 8. Z. D. Genchev, Supercond. Sci. Technol. 6, 532 (1993). 9. D. Stroud, Phys. Rev. B 12, 3368 (1975). 10. I. M. Gelfand, R. A. Minlos, and Z. Ya. Shapiro, Rep resentations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963; Fizmatgiz, Moscow, 1958). 11. LandoltBörnstein, Numerical Data and Functional Relationships in Science and Technology. New Series, Ed. by K.H. Hellwege, Group III, Vol. 15a (Springer, Berlin, Heidelberg, New York, 1982).
Translated by Yu. Sin’kov
