ISSN 10637826, Semiconductors, 2013, Vol. 47, No. 13, pp. 1667–1673. © Pleiades Publishing, Ltd., 2013. Original Russian Text © I.V. Lavrov, 2013, published in Izvestiya vysshikh uchebnykh zavedenii. Elektronika, 2012, Vol. 96, No. 4, pp. 3–12.
BASIC RESEARCH
Effective Conductivity of a Polycrystalline Medium with Weak Macroscopic Anisotropy I. V. Lavrov^ National Research University of Electronic Technology ^email:
[email protected] Received September 14, 2011
Abstract—In the effective medium approximation, an analytical solution is obtained for calculating the tensor of the effective conductivity of a polycrystalline medium with weak uniaxial texture. The medium is assumed to con sist of a single type of biaxial spherical crystallites. The orientation of the crystallites is taken into account by means of the theory of rotation group representations. A solution generalized to triaxial textures is obtained. Keywords: effective medium approximation, tensor of the effective conductivity, crystallyte, texture, poly crystalline medium, rotation group, orientation, generalized spherical functions. DOI: 10.1134/S1063782613130113
The problem of calculating the effective conductiv ity of polycrystalline media with uniaxial texture was solved for two specific cases: weakly anisotropic crys tallites and small spread in the crystallite orientations [1, 2]. In study [1], a medium consisting of uniaxial crystallites was considered; in [2], a medium consist ing of biaxial crystallites was considered. The aim of this study is to obtain an analytical solu tion for the case of a weak macroscopic anisotropy of a polycrytal, i.e., for almost uniformly distributed crystallite orientations. Similar to study [2], the crys tallites are assumed to be biaxial. The texture is con sidered to be initially uniaxial; however, for the case of weak macroscopic anisotropy, a solution in the linear approximation by a parameter characterizing the non uniformity of the crystalliteorientation distribution allows simple generalization to a certain class of triax ial textures, which is of great practical importance. To overcome the computational complexity caused by the need to take into account crystallite orientations, in this study, similar to [2, 3], we use the theory of rota tion group representations (SO(3) group) [4].
The crystallites possess ohmic contacts to one another and can have different sizes and orientations of the principal axes ξηζ of their conductivity tensors. We introduce the system of coordinates xyz, associ ated with the sample, with the axis z being directed along that of sample texture (assumed to be uniaxial) and the x and y axes being directed perpendicular to it. Then, the orientation of a specific crystallite in the system xyz is determined by the rotation g(ψ, ϑ, ϕ) from xyz to system ξηζ of the principle axes of this crystallite. Here, ψ, ϑ, and ϕ are the Euler angles. As in study [2], the density of the crystalliteorientation distribution in the system xyz is taken in the form
PROBLEM STATEMENT
In f(ϑ), we took into account factor sinϑ of the invariant measure of the SO(3) group [4]. We state the problem of calculating the effective conductivity tensor σe of the medium sample, which is defined by the equation 〈j〉 = σe〈E〉, where 〈j〉 is the sampleaveraged electric current density and 〈E〉 = E0. For uniaxial texture, σe in the system xyz is
π/2
∫ f ( ϑ ) dϑ = 1.
xx
σe
σ1 0 0 0 σ2 0 .
(3)
0
Let the uniform electric field E0 be applied to boundary S of a conducting polycrystalline sample with volume V. The medium is assumed to consist of singletype spherical crystallites with a biaxial con ductivity tensor with a matrix in the system ξηζ of its principal axes
σ' =
2 –1
p ( ψ, ϑ, ϕ ) = ( 8π ) f ( ϑ ), (2) 0 ≤ ψ ≤ 2π, 0 ≤ ϑ ≤ π, 0 ≤ ϕ ≤ 2π, where f(ϑ) is the density of the distribution of angles ϑ between the axis z of the texture and the axes ζ of the crystallites, which satisfies the relation f(π – ϑ) = f(ϑ) due to the linearity of the medium and has the normal ization condition
(1)
0 0 σ3
σe =
xx
0 σe 0
1667
0
0 0 zz
0 σe
.
(4)
1668
LAVROV
The effective medium approximation yields the equation for σe [5]: –1
〈 ( I – ( σ – σ e )Γ ) ( σ – σ e )〉 = 0,
Γ Γ at
zz
3
which can be reduced to the form 0 3
0
( σ e ) – 0.25σ e ( σ 1 σ 2 + σ 2 σ 3 + σ 1 σ 3 )
Γ Γ
zz
(6a)
0
For almost uniform distribution, σe = σ e I + δσe, Γ = Γ0I + δΓ where, with regard to (2),
zz
ε = 1 – σ e /σ e ;
δσ e 0
2
ε – ε – ln ( – ε + 1 – ε ) = , xx zz 2 σe σe ε –ε
δσ e =
δΓ
0
x δσ e
0
δΓ =
,
0 z
0
Equation (5) is averaged over all crystallite orienta tions in the system xyz with density (2).
–1
A 0 = ( I – ( σ – σ e I )Γ 0 ) ,
0
Ignoring the terms with a smallness order higher than the first as compared with δσe, δΓ, we have
0
( 2σ e + σ 1 ) 0
0
–1
0
0
0
( 2σ e + σ 2 )
–1
–1
,
0 0
( 2σ e + σ 3 )
0
–1
0 0
0 0
( σ 2 – σ e ) ( 2σ e + σ 2 )
0
Note that, in virtue of (7), SpB0 = 0. We denote the components of tensor A0 in the sys tems ξηζ and xyz as a 'lj and alj, respectively. Analo gously, we denote the components B0 in these systems
(8)
where A0 and B0 in the system ξηζ are
0 ( σ 1 – σ e ) ( 2σ e + σ 1 )
0
〈 A 0 δσ e〉 + Γ 0 〈 A 0 δσ e B 0〉 – 〈 B 0 δΓB 0〉 = 〈 B 0〉 ,
0
B '0 =
z
and Eq. (5) acquires the form
Let σ e be the effectiveconductivity tensor of a medium for the case of uniformly distributed crystal
0 3σ e
0 δΓ
B 0 = A 0 ( σ – σ e I ).
–1
0
,
0
( I – ( σ – σ e )Γ ) ≈ ( I – A 0 ( Γ 0 δσ e – ( σ – σ e I )δΓ ) )A 0
SOLUTION FOR THE CASE OF WEAK MACROSCOPIC ANISOTROPY
0
0 x
k
0
A '0 =
0
with |δσ e | Ⰶ 1, |δΓk| Ⰶ 1, k = x, z. We introduce the second rank tensors
ε < 0.
0 3σ e
x
0 δΓ
0 δσ e
0
(6b)
1 – 1/ε ln ( – ε + 1 – ε ) – 1 = zz σe ε at
(7b)
For uniform distribution, tensor Γ is equal to 0 Γ0 = Γ0I, where, according to [5], Γ0 = –(3 σ e )–1.
x
xx
(7a)
– 0.25σ 1 σ 2 σ 3 = 0.
1/ε – 1 arcsin ε – 1 = zz σe ε xx
0
0
∑
2
ε – ε – arcsin ε = , xx zz 2 σe σe ε ε
ε > 0,
0
σi – σe = 0, 0 2σ + σ e i i=1
(5)
where I is the secondrank unit tensor and Γ is the ten sor that is uniaxial under condition (4) and diagonal with the components Γxx = Γyy in the system xyz: xx
0
lite orientations. Then, σ e = σ e I, where σ e satisfies the equation [6]
0
–1
.
0 0
0
( σ 3 – σ e ) ( 2σ e + σ 3 )
–1
by b 'lj and blj. Then, according to the formulas of trans formation of the components of the symmetric sec ondrank tensors by means of irreducible representa tions of the SO(3) group [3, 7], the components of ten sors A0 and B0 in xyz are SEMICONDUCTORS
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No. 13
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EFFECTIVE CONDUCTIVITY OF A POLYCRYSTALLINE MEDIUM 0⎛ k a kk = 3σ e ⎜ D/3 – 0.5 ( – 1 ) ⎝
×
∑ a˜ ' [ T s
2 – 2, s ( g )
2
+ T 2, s ( g ) + ( – 1 )
σ2 – σ1 , a˜ '±2 = 1 0 2 ( 2σ e + σ 1 ) ( 2σ 0e + σ 2 )
∑
⎞ 2 2/3T 0, s ( g ) ] ⎟ , ⎠
2
k
s = –2
1669
1 1 1 2 a˜ '0 = ⎛ – – ⎞ , (12) 0 0 0 ⎝ 6 ( 2σ e + σ 3 ) ( 2σ e + σ 1 ) ( 2σ e + σ 2 )⎠ 0 a˜ ' = 0, b˜ 's = – 3σ a˜ ' , s = – 2, …, 2. ±1
e s
x
δΓx and δΓz from (6a) and (6b) are decomposed by δσ e
k = 1, 2,
z
and δσ e accurate to the linear terms:
⎞ 0⎛ 2 a 33 = 3σ e ⎜ D/3 + 2/3 a˜ 's T 0, s ( g )⎟ , ⎝ ⎠ s = –2 2
∑
a 12 = a 21
a 13 = a 31
∑
2 a˜ 's [ T 2, s ( g )
–
s = –2
∑
In virtue of condition (2), the tensor equation (8) is reduced to two scalar equations for components with the indices 11 and 33. By calculating these compo nents of the terms in Eq. (8) and collecting the coeffi x z cients at δσ e and δσ e , with regard to (13) we obtain the two scalar equations
2 2 a˜ 's [ T –1, s ( g ) – T 1, s ( g ) ],
s = –2 0
3σ a 23 = a 32 = – i e 2
2
∑ a˜ ' [ T s
2 – 1, s ( g )
2
+ T 1, s ( g ) ],
s = –2
x
z
⎧ c 11 δσ e + c 13 δσ e = 〈 b 11〉 ⎨ ⎩ c 31 δσ xe + c 33 δσ ze = 〈 b 33〉 ,
k+1 3 0 b kk = σ e ( – 1 ) 2
(14)
where
2
×
(13)
2 x 3 z 1 δΓ ≈ 2 δσ e + δσ e . 0 5 3 ( σe ) 5 z
2 T –2, s ( g ) ],
2
0
3σ = e 2
x 1 4 δσ x + 1 δσ z , δΓ ≈ e e 0 2 5 3 ( σe ) 5
2
0
3σ = i e 2
(9)
∑ b˜ ' [ T s
2 – 2, s ( g )
2
+ T 2, s ( g ) + ( – 1 )
k
2
2/3T 0, s ( g ) ],
0 –1
c 11 = 〈 a 11〉 – ( 3σ e ) ( 〈 a 11 b 11〉 + 〈 a 12 b 12〉 )
s = –2
0 2 –1
k = 1, 2, 2 0, s ( g ),
0 2 –1
0
(10)
2
∑ b˜ ' [ T s
2 2, s ( g )
2
2
2
– [ 3 ( σ e ) ] ( 0.2 [ 〈 b 11〉 + 〈 b 12〉 ] + 0.6 〈 b 13〉 ),
s = –2
3σ b 12 = b 21 = i e 2
2
0 –1
∑ b˜ ' T s
2
c 13 = – ( 3σ e ) 〈 a 13 b 13〉
2
0
b 33 = 3σ e 2/3
2
– [ 3 ( σ e ) ] ( 0.8 ( 〈 b 11〉 + 〈 b 12〉 ) + 0.4 〈 b 13〉 ),
0 –1
2
– T –2, s ( g ) ],
(15)
c 31 = – ( 3σ e ) ( 〈 a 13 b 13〉 + 〈 a 23 b 23〉 ) 0 2 –1
2
2
2
– [ 3 ( σ e ) ] ( 0.8 [ 〈 b 13〉 + 〈 b 23〉 ] + 0.4 〈 b 33〉 ),
s = –2
0 –1
b 13 = b 31 = 2
∑ b˜ ' [ T s
2 – 1, s ( g )
2
– T 1, s ( g ) ],
0 2 –1
2
2
2
– [ 3 ( σ e ) ] ( 0.2 [ 〈 b 13〉 + 〈 b 23〉 ] + 0.6 〈 b 33〉 ).
s = –2 0
b 23 = b 32
c 33 = 〈 a 33〉 – ( 3σ e ) 〈 a 33 b 33〉
2
0 3σ e
3σ = – i e 2
The average values of the tensor components A0 and B0 and their products are expressed, as can be seen from (9) and (10), via the average values of the gener 2 alized spherical functions 〈 T mn ( g )〉 and their products
2
∑ b˜ ' [ T s
2 – 1, s ( g )
2
+ T 1, s ( g ) ],
s = –2
l
where T ms ( g ) , l = 0, 1, 2, …; m, s = –2, …, 2 are the generalized spherical functions [4]; a '11 + a '22 + a '33 D = = 0 3σ e SEMICONDUCTORS
2
π/2
3
∑
2
〈 T mn ( g )T m'n' ( g )〉 , which, in turn, at orientation distri bution (2) are expressed via the integrals [2]
0 ( 2σ e
–1
+ σk ) ;
k=1
Vol. 47
(11)
I1 =
∫ cos ϑf ( ϑ ) dϑ, 0
No. 13
2013
π/2 2
I2 =
∫ cos ϑf ( ϑ ) dϑ. 4
0
(16)
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LAVROV
For the case of almost uniform distribution I 1 = 1/3 + ΔI 1 ,
I 2 = 1/5 + ΔI 2 ,
(17)
where |ΔI1| Ⰶ 1, |ΔI2| Ⰶ 1. For 〈b11〉 and 〈b33〉, we have 9 3 0 2 〈 b 11〉 ≈ ( σ e ) a˜ '0 ΔI 1 , 2 2
xx
z
2 2 0 0 2 c 11 ≈ σ e D – 201 ( σ e ) [ ( a˜ '0 ) + 2 ( a˜ '2 ) ], 50 2 2 99 0 2 c 13 ≈ c 31 ≈ – ( σ e ) [ ( a˜ '0 ) + 2 ( a˜ '2 ) ], 25
(18)
where D is determined from (11) and a˜ '0 and a˜ '2 , from (12); therefore σ1 – σ3 2 2 2 ( a˜ '0 ) + 2 ( a˜ '2 ) = 3 ( 2σ 0 + σ ) 2 ( 2σ 0 + σ ) e 3 e 1
(19)
σ3 – σ2 σ2 – σ1 + . + 2 2 0 0 0 0 ( 2σ e + σ 1 ) ( 2σ e + σ 2 ) ( 2σ e + σ 2 ) ( 2σ e + σ 3 ) Solving system (14), we find the final expressions x z for δσ e and δσ e 2 2 x 0 9 3 0 δσ e ≈ σ e a˜ '0 [ D – 6σ e ( ( a˜ '0 ) + 2 ( a˜ '2 ) ) ]ΔI 1 , Δ 8 (20) x
δσ e ≈ – 2δσ e , where 2 2 2 303 0 Δ = D – σ e D [ ( a˜ '0 ) + 2 ( a˜ '2 ) ] 50
× 29 × 41 ( σ 0 ) 2 [ a˜ ' 2 + 2 a˜ ' 2 ] 2 ; + 9 ( 0) ( 2) e 2 50 2
zz
0
z
σ e ≈ σ e + δσ e ,
(22)
where σ e is the effective conductivity of the polycrystal at the uniform distribution of crystallite orientations, x which is the only positive root of Eq. (7b) and δσ e and z
δσ e are calculated from (20) and (21). The dependence of the components of the effective conductivity of the medium on the distribution of the crystallite orientations is expressed via quantity ΔI1 which depends on the parameter determining the degree of nonuniformity of the crystalliteorientation distribution. For example, if we take the expression β
– 1 β cos ϑ
fβ ( ϑ ) = β ( e – 1 ) e
sin ϑ,
0 ≤ ϑ ≤ π/2
0 51 ( σ 0 ) 2 [ a˜ ' 2 + 2 a˜ ' 2 ], c 33 ≈ σ e D – ( 0) ( 2) e 25
2
x
0
〈 b 33〉 ≈ – 2 〈 b 11〉 .
by small values δσ e , δσ e . Thus, for clk (l, k = 1, 3) we obtain
z
0
σ e ≈ σ e + δσ e ,
In calculating the remaining averages, it is suffi cient to limit consideration to the zero approximation I1 ≈ 1/3, I2 ≈ 1/5 since in Eqs. (14) they are multiplied x
Thus, in the case of weak macroscopic anisotropy with crystalliteorientation distribution (2), the effec tiveconductivity tensor of the polycrystal in the sys tem of coordinates xyz of the sample has the form (4) with the components
(21)
( a˜ '0 ) + 2 ( a˜ '2 ) is calculated from (19); ΔI1 is the incre ment of integral I1 (see (16) and (17)) for the case of almost uniform distribution of the crystallite orienta tions.
(23)
as a model of the almost uniform distribution with parameter β determining the degree of nonunifor mity of the distribution, then ΔI1 ≈ β/12. Physically, the density fβ(ϑ) of the distribution of angles ϑ corre sponds to the Boltzmann distribution of dipoles in a uniform electric field and the difference between the highest and lowest densities of the orientation distri bution per unit volume of the SO(3) group amounts approximately to β/(4π2). PARTICULAR CASE OF A POLYCRYSTAL CONSISTING OF UNIAXIAL CRYSTALLITES In this case, the conductivity tensor of a crystal in the system ξηζ of its principle axes has the form (σ0 ≡ σ1) 10 0 σ' = σ 0 0 1 0 , 00α
(24)
where α is the coefficient of crystallite anisotropy. The orientation of the uniaxial tensor is specified by two scalar parameters, e.g., the spherical angles ϑ and ϕ. The density of the crystalliteorientation distri bution corresponding to uniaxial texture in the system xyz of the sample is always –1
p ( ϑ, ϕ ) = ( 2π ) f ( ϑ ), 0 ≤ ϑ ≤ π/2, 0 ≤ ϕ ≤ 2π,
(25)
with normalization condition (3) for f(ϑ). The effec tiveconductivity tensor in the system xyz has the SEMICONDUCTORS
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2013
EFFECTIVE CONDUCTIVITY OF A POLYCRYSTALLINE MEDIUM
form (4) and their components are calculated from the expressions [7] 0
xx
σ e = σ 0 ( 1 – u 0 – k ΔI 1 ), zz
(26)
0
σ e = σ 0 ( 1 – u 0 + 2k ΔI 1 ), where u 0 = 0.25 ( 3 – 9 + 8 ( α – 1 ) ),
(27)
1671
0 (α – 1) k = 1 – ( α – 1 ) 2
(28) 3 – 0.04u 0 + 2.64 ( α – 1 ) × 2 . 27 + 24 (α – 1) – 18u 0 – 12.04 (α – 1)u 0 + 2.64 (α – 1) The coefficient of anisotropy of the effective con zz xx ductivity αe ≈ σ e /σ e is
9 – 6u 0 + 4 ( α – 1 ) ΔI α e = 1 + 9 ( α – 1 ) 1. 2 2 27 + 24 ( α – 1 ) – 18u 0 – 12.04 ( α – 1 )u 0 + 2.64 ( α – 1 ) We present asymptotic estimations for u0, k0, and αe for the cases of weakly and strongly anisotropic crys tallites. At small values of (α – 1), i.e., for weakly anisotro pic crystallites, we have u0 ≈ –(α – 1)/3; therefore, (28) and (29) are reduced to the expressions 0 2 k ≈ 1 ( α – 1 ) – 1 ( α – 1 ) , 2 9 2 3 2 α e ≈ 1 + ( α – 1 ) – ( α – 1 ) ΔI 1 . 2 3
At (α – 1) Ⰷ 1, i.e. for strongly anisotropic crystal lites with σ|| Ⰷ σ⊥, where σ|| and σ⊥ are the conductiv ities along and perpendicular to the crystallite axis, respectively, we have 3 1 u 0 ≈ – α – 1 + , 4 2 0 1 , k ≈ 25 2 α – 1 1 – 169 2 22 132 α – 1
75 101 2 1 α e ≈ 1 + ⎛ 1 – ⎞ ΔI 1 . 11 ⎝ 66 α – 1⎠ At the limit of very large α, we may consider that 75 α e ≈ 1 + ΔI 1 . 11 At 0 < α Ⰶ 1, i.e., for strongly anisotropic crystal lites with σ|| Ⰶ σ⊥, 2 1 u 0 ≈ ( 1 – 2α + 4α ), 2
0 75 135 k ≈ – 1 – α , 133 133
450 401 α e ≈ 1 – 1 – α ΔI 1 . 133 133
Vol. 47
˜ zz σ e was estimated by formulas (26)–(28) with the use of the relative error in the form zz ˜ zz zz σ e – σe . ε rel = zz xx σe – σe xx
zz
The calculation showed that ε rel and ε rel depend on both the degree of nonuniformity of the orientation
450 α e ≈ 1 – ΔI 1 . 133 SEMICONDUCTORS
RESULTS OF THE NUMERICAL SIMULATION Using the MATLAB package, a program complex was developed and calculations for some polycrystal line media with uniaxial crystallites were made. The values of the effectiveconductivity tensor σe were determined by solving system of equations (5) using the Newton method and then compared with analogous val ues obtained by analytical approximation (26)–(28). The density of the crystalliteorientation distribution was taken in the form (25) with the distribution density f(ϑ) of angles ϑ between the texture axis and the crys tallite axes in the form (23) at different values of parameter β. For such a distribution, all the crystallite orientations are distributed over the surface of a unit hemisphere with the difference between the orienta tion distribution density per unit sphere area at the pole (ϑ = 0) and equator (ϑ = π/2) being approxi mately equal to β/(2π). Figure 1 shows a comparison of the dependences of xx zz components σ e and σ e of a manganese polycrystal (σ0 = 2.37 × 105 Ω–1 cm–1, α = 1.21 [8]) on the degree of nonuniformity of the crystalliteorientation distri bution. The dependences were obtained by numerical solution of system (5) and analytical approximation by formulas (26)–(28). Figure 2 presents analogous depen dences for an artificial graphite polycrystal with strongly anisotropic crystallites (σ0 = 2.26 × 104 Ω–1 cm–1, α = 2.6 × 10–4). For the case of weak macroscopic anisotropy of a xx zz polycrystal, the values of σ e and σ e are similar and ˜ xx the accuracy of their analytical approximation σ e and
xx ˜ xx xx e – σe , ε rel = σ zz xx σe – σe
At the limit of very small α, we have
(29)
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2013
1672
LAVROV
σexx, σezz × 105 Ω−1 cm−1 2.62 2.60 2.58 2.56 2.54 2.52 2.50 2.48 2.46
σexx, σezz × 105 Ω−1 cm−1 0.13 0.12 0.11 0.10 0.09 0.08 0.07
1
2 0
0.5
1.0
1.5
2.0
2.5
xx
zz
β
xx
GENERALIZATION TO TRIAXIAL TEXTURES Let a sample of a macroscopically isotropic poly crystal be subjected to n external influences with the measure parameters q1, q2, …, qn. We assume that each of these influences results in uniaxial texture with its axis. This can be, e.g., stress along a certain axis, which leads to corresponding sample strain. If the external influences are small, then any macroscopic property of the sample in the vicinity of the isotropic state can be decomposed by the values of these effects, limiting consideration to the linear approximation. In particu lar, for the effectiveconductivity tensor of the sample, we have
∑
k=1
zz
xx
zz
tions of the numerical solution are (⋅⋅⋅䊐⋅⋅⋅ σ e , ⋅⋅⋅䊊⋅⋅⋅ σ e )
zz
distribution specified by parameter β and crystallite anisotropy coefficient α. Figure 3 shows the depen dence of boundaries for β on α when the relative error of the analytical approximation (26)–(28), as com pared with the data of numerical calculation, is no more than 1%.
σ e ( q 1, …, q n ) ≈ σ e ( 0, …, 0 ) +
zz
the analytical approximation (1—σ e , 2— σ e ); Designa
zz
of the numerical solution are (⋅⋅⋅䊐⋅⋅⋅ σ e , ⋅⋅⋅䊊⋅⋅⋅ σ e ).
n
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 β xx
analytical approximation (1—σ e , 2— σ e ). Designations xx
2
Fig. 2. Dependences of components σ e and σ e of an arti ficial graphite polycrystal on parameter β: solid lines show
Fig. 1. Dependences of components σ e and σ e of a man ganese polycrystal on parameter β: solid lines show the xx
1
∂σ q k e ( 0, …, 0 ). (30) ∂q k
It is convenient to consider the result of each influ ence in the system of coordinates associated with the influence. Therefore, we introduce the systems xkykzk, k = 1, …, n, as follows: the axis zk is directed along the axis of texture forming under the kth influence on the sample and the axes xk and yk, perpendicular to the axis zk and one another. The crystallite orientation in the system xkykzk is specified by the rotation gk(ψk, ϑk, ϕk), where ψk, ϑk, ϕk are its Euler angles. Let pk(ψk, ϑk, ϕk) = (8π2)–1fk(ϑk) be the density of the crystalliteorientation distribution in the system xkykzk resulting from the kth influence. We introduce also
the laboratory system XYZ. Let Ck, k = 1, …, n be the matrices of rotations from XYZ to xkykzk. As a measure of the external influences, parameters β1, …, βn can be used which describe the degrees of nonuniformity of the crystalliteorientation distributions 1 n resulting from each influence and quantities ΔI 1 , …, ΔI 1 , k
k
where ΔI 1 is the increment of the integral ΔI 1 =
∫
π/2 0
2
cos ϑ k f k ( ϑ k ) dϑ k (see (16) and (17)) under the kth
influence; i.e., (30) can be written in the form 0 (σe(0, …, 0) = σ e I ) n
0
σe ≈ σe I +
∂σ e
( 0, …, 0 ). ∑ ΔI ∂ ( ΔI ) k 1
k=1
(31)
k 1
As can be seen from (4), (20), and (31), the tensors ∂σ e ( 0, …, 0 ) , k = 1, …, n in their systems xkykzk k ∂ ( ΔI 1 ) have the same form β1, β2 2.15 1.90 1.65 1.40 1.15 0.90 β2 0.65 0.40 0.15 −0.10 β1 −0.35 −0.60 −4 −3 −2 −1 10 10 10 10 100 101 102 103
α
Fig. 3. Dependence of the boundaries β1 and β2 of the range with an error of no more than 1% on the crystallite anisotropy coefficient α. SEMICONDUCTORS
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EFFECTIVE CONDUCTIVITY OF A POLYCRYSTALLINE MEDIUM
∂σ 'e ( 0, …, 0 ) k ∂ ( ΔI 1 )
1
10 0 (32) 2 2 0 9 3 0 = σ e a˜ '0 [ D – 6σ e ( ( a˜ '0 ) + 2 ( a˜ '2 ) ) ] 0 1 0 . Δ 8 0 0 –2 k = 1, …, n. Upon the transition from system xkykzk to XYZ, these tensors are transformed according to the formula ∂σ e ∂σ 'e –1 = C k C k . k k ∂ ( ΔI 1 ) ∂ ( ΔI 1 )
(33)
Substituting (32) and (33) into (31), we obtain the final expression for the tensor of σe in the system XYZ in the presence of n external axial influences: 0 9 3 σ 0 a˜ ' [ – 0 ( a˜ ' 2 + 2 a˜ ' 2 ) ] σ e ≈ σ e I + 6σ e ( 0 ) ( 2) e 0 D Δ 8 n
×
∑
k ΔI 1 C k
k=1
(34)
10 0 –1 0 1 0 Ck , 0 0 –2
where D is determined from (11), a˜ 0' and a˜ 2' , from (12), and Δ, from (21). Let us consider the particular case when there are three axial influences on a medium sample with mutu ally perpendicular axes. We choose the system XYZ such that the axis z1 of the first influence coincides with the axis X and the axes z2 and z3 of the second and third influences, with the axes Y and Z, respectively. Then, the matrices of rotations from XYZ to xkykzk, k = 1, 2, 3, are
C1 =
001 100 , 010
C3 =
C2 =
100 010 001
010 001 , 100
= I.
As a result, tensor σe in the system XYZ will be diag onal with the components 2 2 kk 0 0 9 3 0 σ e ≈ σ e + σ e a˜ '0 [ D – 6σ e ( ( a˜ '0 ) + 2 ( a˜ '2 ) ) ] Δ 8
⎛ 3 j k⎞ × ⎜ ΔI 1 – 3ΔI 1⎟ , ⎝j = 1 ⎠
∑
SEMICONDUCTORS
(35) k = 1, 2, 3,
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2
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3
where quantities ΔI 1 , ΔI 1, ΔI 1 are determined by the degree of each influence. Expressions (35) can be used for estimating the val ues of the external influences, since, measuring the components of the sample conductivity tensor, one 11 22 33 can obtain its main values σ e , σ e , and σ e and using 1
2
3
system (35), calculate the values of ΔI 1 , ΔI 1, ΔI 1 directly related to the external influences. Thus, we have demonstrated the principle possibil ity of using polycrystalline macroscopically isotropic materials in the fabrication of strain sensors, since the parameters of the texture acquired by a material upon external stresses can be estimated by measuring the electrical characteristics of a material. The main result of this study is the obtained analyt ical dependence of components of the effectivecon ductivity tensor σe of a polycrystalline medium with weak uniaxial texture on the parameters describing the conducting properties of crystallites and on the crys talliteorientation distribution. This dependence is expressed by formulas (20)–(22) and, in the particular case of uniaxial crystallites, by formulas (26)–(28). Generalization of the analytical dependence to the case of triaxial texture presented by expression (34) is of practical importance, since in most cases material textures are triaxial. ACKNOWLEDGMENTS This study was supported by the Ministry of Educa tion and Science of the Russian Federation, Federal Targeted Program Scientific and Pedagogical Person nel of Innovative Russia, 2009–2013 (State contract no. 16.740.11.0491) and the Russian Foundation for Basic Research, projects nos. 090801232a and 10 0801163a. REFERENCES 1. I. V. Lavrov, Izv. Vyssh. Uchebn. Zaved., Elektron., No. 1, 3 (2008). 2. I. V. Lavrov, Izv. Vyssh. Uchebn. Zaved., Elektron., No. 3, 3 (2010). 3. I. V. Lavrov, Ekol. Vestn. Nauch. Tsentrov Cherno morsk. Ekonom. Sotrudn., No. 1, 52 (2009). 4. 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). 5. D. Stroud, Phys. Rev. B 12, 3368 (1975). 6. J. Helsing and A. Helte, J. Appl. Phys. 69, 3583 (1991). 7. I. Lavrov, Dielectric and Conductive Properties of Inho mogeneous Media with Texture (LAP Lambert Aca demic Publishing, Saarbrucken, 2011). 8. Physical Values, The Handbook, Ed. by I. S. Grigor’ev and E. Z. Meilikhov (Energoatomizdat, Moscow, 1991) [in Russian].
Translated by E. Bondareva No. 13
2013
