Abstract: The birefringence in nanometer-scale dielectrics with thelargest dimensions ranging from about 3 nm to 20 nm has been quantifiedby evaluating directly the summation of induced-dipole-electric-fieldcontributions from all individual atoms within the entire dielectric volume.Various configurations in representative cubic and tetragonal systemsare investigated by varying the ratio of lattice constants and the numberof atoms in various directions to illustrate the chain-like and plane-likebehavior regimes. The dielectric properties of the finite cubic crystallattices change from isotropic to birefringent (uniaxial or biaxial) whenthe entire dielectric volume is changed from a cube to a rectangularparallelepiped in shape. In finite tetragonal crystals the birefringenceincreases with the increasing lattice constant ratios. The largest uniaxialbirefringence occurs for non-cube dielectric volume with tetragonal lattices.
References and links1. H. A. Lorentz, The Theory of Electrons (Teubner, 1909).2. T. K. Gaylord and Y.-J. Chang, “Induced-dipole-electric-field contribution of atomic chains and atomic planes to
the refractive index and birefringence of nanoscale crystalline delectrics,” Appl. Opt. 46, 6476–6482 (2007).3. E. M. Purcell, Electricity and Magnetism, 2nd ed. (McGraw-Hill, 1985).4. D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, 1989).5. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).6. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).7. E. Dehan, P. Temple-Boyer, R. Henda, J.J. Pedroviejo, and E. Scheid, “Optical and structural properties of SiOx
and SiNx, materials,” Thin Solid Films 266, 14–19 (1995).
1. Introduction
Nanoelectronics is poised to enter into information technologies that will affect virtually allaspects of our daily lives. The development of nano-fabrication technologies has made theconcept of nanophotonics possible where new photonic devices and dielectric materials areused to overcome the conventional diffraction limit to manipulate light at the nanoscale. Innanophotonics, nanoscale dielectric materials are ubiquitous and have a direct impact on device
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(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 809
functionality and performance. Understanding the optical properties, in particular, the indices ofrefraction and the birefringence, of these nanoscale dielectrics is fundamentally important. Thisknowledge is needed, for example, to provide various indices of refraction and birefringencesfor the design of future nanophotonic integrated circuits.
In this paper, the birefringence characteristics of representative nanometer-scale dielectricvolumes with cubic and tetragonal crystal structures are quantified by summing the induced-dipole-electric-field contributions from the individual atoms in the dielectric volume. Theinduced-dipole-electric field experienced by each atom (or dipole) is a function of its atomicposition relative to all other atoms in the dielectric volume. In other words, instead of usingthe Lorentz spherical surface [1] to separate the nearby dipoles and the rest of the material, allinduced dipoles in the dielectric volume are explicitly included which is needed when modelingthe limited number of atoms in nanoscale structures. The relative permittivities of nanometer-scale dielectrics in cubic and tetragonal crystal systems are quantified with an emphasis onchain-like and plane-like behavior regimes. Birefringence is shown to occur if the nanometer-scale dielectric volume is of rectangular parallelepiped shape, regardless of whether its prim-itive lattice is cubic or non-cubic. Moreover, at small scales the total volume of a rectangularparallelepiped dielectric plays a crucial role that determines the degree of electric polarizationand thus the relative permittivity as the lattice-constant ratio increases or decreases.
2. Method of Analysis
The present work extends the analysis of characteristics at a point [2] to the average opti-cal properties over the nanoscale volume. We note that the motions of electrons in atoms andmolecules are characterized by periods on the order of 10−16 second [3], corresponding to fre-quencies close to those of the visible light wave. Strictly nonpolar materials behave practicallythe same from zero frequency up to frequencies of visible light since electrons are able to fol-low the time-varying field. Because of this, it is possible to ignore the time dependence. Forsimplicity, the treatment developed here is for the nonpolar crystalline materials of identicalatoms arranged in cubic or tetragonal lattices.
To analyze the macroscopic average optical properties, in particular the relative permittivityand birefringence, the electric polarization vector P is first considered. It is the macroscopicelectric dipole moment per unit volume in the material and is given by [4, 5]
P = limΔv→0
NΔv
∑j=1
p j
Δv, (1)
where N is the number of atoms per unit volume and the numerator represents the vector sumof the induced dipole moments p j within a small volume Δv. If we assume that all the induceddipoles are directly proportional to, and in the same direction as, their corresponding local fieldElocal, j, then
p j = α jElocal, j = α j(Eappl, j +Eind, j
), (2)
where α j is the polarizability of the j-th atom and Eappl, j and Eind, j are the applied and inducedelectric fields acting on the j-th atom, respectively [2]. At optical frequency, α j is mainly con-tributed by the electronic polarizability and is assumed to be a constant for a given type ofspherical atoms. Using Eq. (2), Eq. (1) can be expressed in its column vector form as
ΔvP =NΔv
∑j=1
α(Eappl, j +Eind, j
), (3)
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(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 810
Fig. 1. Schematic of the induced dipole moments in a nanoscale parallelepiped dielectricslab due to an x-directed applied field.
where
P =
⎡
⎣Px
Py
Pz
⎤
⎦ ; Eappl, j =
⎡
⎣
(Eappl, j
)x(
Eappl, j)
y(Eappl, j
)z
⎤
⎦ ; Eind, j =
⎡
⎣
(Eind, j
)x(
Eind, j)
y(Eind, j
)z
⎤
⎦ . (4)
The induced-dipole-electric field at the j-th atom location Eind, j is the vector sum of individualinduced-dipole-electric-fields from all other atoms within Δv (taken as the volume of the entirenano-scale dielectric) at that location
Eind, j =NΔv
∑i=1,i�= j
3(pi · ri)ri − r2i pi
4πε0r5i
, (5)
where pi is the induced dipole moment associated with the i-th atom, ri is the position vectorfrom the i-th atom to the observation point (i.e. j-th atom) and ri = |ri| is the magnitude of theposition vector ri (Fig.1). Since, for example, an applied electric field in the x direction wouldproduce induced-dipole-electric field in the x-, y-, and z-directions, Eq. (5) may be rewritten as
Eind, j =1
4πε0d3
⎡
⎣(γ j)xx (γ j)xy (γ j)xz
(γ j)yx (γ j)yy (γ j)yz
(γ j)zx (γ j)zy (γ j)zz
⎤
⎦
⎡
⎣(p j)x
(p j)y
(p j)z
⎤
⎦ ≡ 14πε0d3 γγγ
jp
j, (6)
where d is the smallest lattice constant, γj
is the (de)polarizing factor tensor at the j-th atom
location, and pjis the induced dipole moment (in column vector form) at the j-th atom location
with (p j)x, (p j)y, and (p j)z being its x-, y-, and z-component, respectively. The explicit notationof p
jis used for representing the varying small displacement between electronic cloud and nu-
cleus throughout the dielectric volume. In a cubic crystal, d = a = c. For the chain-like (c < a)tetragonal structure, d = c while for the plane-like (a < c) structure, d = a. The element (γ j)uv
in the γγγ j tensor is interpreted as the (de)polarizing factor of the j-th atom in the u direction dueto an applied electric field (and thus the dipole moment) in the v direction. The (de)polarizingfactors are calculated using the equations given in Appendices A and B in [2].
The constitutive relation between the macroscopic electric displacement D and the electricpolarization vector P is given by [4–6]
D = ε0εεε Eappl = ε0Eappl +P, (7)
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(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 811
where εεε is the 3×3 relative permittivity tensor. Assuming the applied electric field is a constantacross the entire dielectric volume, then Eappl, j = Eappl . Consequently, Eq. (3) becomes
ΔvP = αN(Δv)Eappl +αNΔv
∑j=1
Eind, j. (8)
Using Eq. (7), the externally applied field can be expressed in terms of the polarization vectorP
Eappl =[ε0
(εεε − I
)]−1P ≡ Q−1P, (9)
where I is the 3×3 identity matrix. Substitute Eq. (9) into Eq. (8) yields
αN(Δv)Q−1P = ΔvP−αNΔv
∑j=1
Eind, j. (10)
The summation of the induced-dipole-electric field experienced at each atomic position may beexpressed and further approximated as
NΔv
∑j=1
Eind, j =1
4πε0d3
{γγγ
1p
1+ γγγ
2p
2+ · · ·γγγ
NΔvp
NΔv
}
≈ 14πε0d3 γγγ
⟨p⟩, (11)
where
γγγ =(
γγγ1+ γγγ
2+ · · ·+ γγγ
NΔv
)=
⎛
⎝γxx γxy γxz
γyx γyy γyz
γzx γzy γzz
⎞
⎠ (12)
which may be interpreted as the macroscopic (de)polarizing tensor and
⟨p⟩
=
⎛
⎝px
py
pz
⎞
⎠ (13)
is the average dipole moment averaged over the dielectric volume.Now consider the smallest volume unit of the dielectric. The lattice constant d in Eq. (11) is
chosen to be the smallest primitive basis vector of the crystal structure (d = a for the plane-likecrystal and d = c for the chain-like crystal). This is a direct result from normalizing the othertwo basis vectors with respect to d. Furthermore, the volume of one primitive cell can alwaysbe expressed in terms of the smallest primitive basis vector. Therefore the number of atomsper unit volume N is the reciprocal of the smallest volume unit of the dielectric and Eq. (11)becomes
NΔv
∑j=1
Eind, j ≈N
4πε0γγγ⟨p⟩
=1
4πε0γγγN
⟨p⟩. (14)
Furthermore, since the polarization vector P is defined as the volume density of electric dipolemoment, P may be also written, in a macroscopic average sense, as P = N
⟨p⟩
[5]. Eq. (10)may thus be rewritten as
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 812
αN(Δv)Q−1P = ΔvP− α4πε0
γγγP. (15)
The relative permittivity tensor εεε can then be obtained as follows:
εεε = I+(4παNΔv)I[4πε0ΔvI−αγγγ
]−1. (16)
Note that the polarizability α is in units of F·m2, N in m−3, Δv in m3, and ε0 in F/m. Eq. (16)is therefore dimensionless as expected.
The calculated relative permittivity [Eq.(16)] is the macroscopic value of the relative permit-tivity averaged over the dielectric volume. The averaging is inherent in the macroscopic fieldquantities D, Eappl , and P and also in the expression P = N
⟨p⟩. The resultant equation of εεε
in Eq. (16) applies to any dielectric material of identical atoms and of any size, not necessar-ily at nanoscale, provided the (de)polarizing factors (and thus the (de)polarizing tensor theyconstruct) can be obtained.
By properly choosing the orientation of the coordinate system, the macroscopic refractiveindices are given by
nx =√
εx, ny =√εy, nz =
√εz, (17)
for the applied electric field along the x, y, and z directions. Note that the reduced subscriptnotation εu = εuu; u = {x,y,z} is used for simplicity. The birefringence is then given by (Δn) =nz − nx. If nz > nx = ny, then it is positive uniaxial birefringence. If nz < nx = ny, then it isnegative unaxial birefringence. If nz �= nx �= ny, then it is biaxial birefringence.
3. Results and Discussions
To quantify the birefringence of a dielectric volume at nanoscale, a finite tetragonal lattice(a = b �= c,α = β = γ = 90◦) array of identical atoms is treated as a representative case. A finitecubic lattice array of atoms is also examined by setting a = c in the tetragonal crystal system. Inboth cases, a finite array of identical atoms can be described by Mx, My, and Mz atoms in the x,y, and z directions, respectively. More specifically, Mx = (M+
x +M−x +1), My = (M+
y +M−y +1),
and Mz = (M+z + M−
z + 1), where M+u and M−
u denote the respective number of atoms in the±u directions with respect to the reference plane. The polarizability α used in the calculationsis 2.9× 10−24 cm3, typical of SiO2 [7]. The lattice constant is chosen to be 2.62 A which isapproximately the distance between oxygen atoms in SiO2. It should be emphasized that thecalculations presented in this work are for dielectric volumes whose largest dimensions rangefrom about 3 nm to 20 nm, corresponding to a total number of atoms from 300 (Mx = My =10,Mz = 3) to 91125 (Mx = My = Mz = 45). Calculations for larger dielectric volumes wouldrequire increased computational resources since the induced-dipole-electric field at each atomicposition is the superposition of the contribution from every other atom within the dielectricvolume.
Figures 2 - 5 illustrate the normalized induced-dipole-electric fields across the upper, central,and lower planes of atoms in a triple 400-atom plane at a/c = 2 (Mx = My = 20, Mz = 3). Theyare normalized to p/4πε0c3, where p is the magnitude of the dipole moment. The Ex
ind,x fieldsof the upper and lower planes are identical because of the reflection symmetry with respect tothe central plane. This property also holds for Ey
ind,y and Ezind,z fields. Although not shown, the
appearance and magnitude of Eyind,y and Ex
ind,x are identical, except for a 90◦ difference in theazmuthal angle. This is due to the 4-fold symmetry about the z-axis. The physical reasoning ofFigs. 2 - 5 is summarized as follows:
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 813
0
5
10
15
20
05
1015
20
−2
−1.5
−1
−0.5
0
x/ay/a
Nor
mal
ized
Indu
ced
Ex in
d,x
Upper(Lower) Plane
Central Plane
Fig. 2. The x component of the normalized induced-dipole-electric field for an applied fieldin the x direction, Ex
ind,x, across the central plane of a triple 400-atom plane (Mx = My =20,Mz = 3) dielectric volume with a/c = 2. Integer values of x/a and y/a represent theatom positions in the x and y directions, respectively.
0
5
10
15
20
05
1015
20
−0.5
0
0.5
x/ay/a
Nor
mal
ized
Indu
ced
Ex in
d,y Centeral Plane
Fig. 3. The y component of the normalized induced-dipole-electric field for an applied fieldin the x direction, Ex
ind,y, across the central plane of a triple 400-atom plane (Mx = My =20,Mz = 3) dielectric volume with a/c = 2. For the upper and lower planes, the deviationis less than 8.8%. Integer values of x/a and y/a represent the atom positions in the x and ydirections, respectively.
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 814
Fig. 4. The z component of the normalized induced-dipole-electric field for an applied fieldin the x direction, Ex
ind,z, across the central plane of a triple 400-atom plane (Mx = My =20,Mz = 3) dielectric volume with a/c = 2. Integer values of x/a and y/a represent theatom positions in the x and y directions, respectively.
0
5
10
15
20
05
1015
20
0
0.5
1
1.5
2
2.5
3
3.5
x/ay/a
Nor
mal
ized
Indu
ced
Ez in
d,z
Upper(Lower) Plane
Central Plane
Fig. 5. The z component of the normalized induced-dipole-electric field for an applied fieldin the z direction, Ez
ind,z, across the central plane of a triple 400-atom plane (Mx = My =20,Mz = 3) dielectric volume with a/c = 2. Integer values of x/a and y/a represent theatom positions in the x and y directions, respectively.
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(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 815
1. The magnitudes of Exind,x and Ez
ind,z of the central plane are larger since they are reinforcedby the upper and lower planes.
2. Exind,x at x/a = 10 has a maximum since there is a strong reinforcement from dipoles at
larger and lower values of x/a. Correspondingly, it has minima at x/a = 1 and x/a = 20due to a one-sided dipole reinforcement only.
3. The turned-up edge of Exind,x along y/a = 1 (y/a = 20) is due to zero cancellation effect
from dipoles at y/a < 1 (y/a > 20) for no dipoles are present there.
4. Exind,y cancels at the center of each of the three planes due to symmetry.
5. Exind,z cancels along x/a = 10 of each of the three planes due to symmetry.
6. Exind,z in the upper plane decreases (increases) moving from (x/a,y/a) = (10,10) toward
(x/a,y/a) = (1,10) [(x/a,y/a) = (20,10)] due to the missing cancellation effect fromdipoles at smaller (larger) values of x/a since they are absent.
7. Exind,z in the lower plane decreases (increases) moving from (x/a,y/a) = (10,10) toward
(x/a,y/a) = (20,10) [(x/a,y/a) = (1,10)] due to the missing cancellation effect fromdipoles at larger (smaller) values of x/a since they are absent.
8. Exind,z in the upper plane decreases slightly moving from (x/a,y/a) = (20,10) toward
the corner at (x/a,y/a) = (20,1) [(x/a,y/a) = (20,20)] due to weak reinforcement (+zcomponent) from dipoles at y/a < 1 (y/a > 20) since they are absent. Similarly, the slightincrease of Ex
ind,z in the upper plane at the corner (x/a,y/a) = (1,1) [(x/a,y/a) = (1,20)]is due to weak reinforcement (−z component) from dipoles at y/a < 1 (y/a > 20) sincethey are absent.
9. Ezind,z peaks at four corners due to weak reinforcement (−z component) from dipoles
inside the dielectric volume. There are no dipoles in the rest of seven octants relative tothe corner to contribute to the −z-component reinforcement.
Because of the position-dependent reinforcement/cancellation effect, the induced-dipole-field contributions become more uniform across the entire plane except near the edges/cornersas the number of atoms in one plane (Mx×My) increases. Although variations do exist in Ex
ind,y(in all three planes) and Ex
ind,z (in the upper and lower planes), the total sum of Exind,y and the
total sum of Exind,z within the dielectric volume are zero. Mathematically, this corresponds to
γyx = 0 and γzx = 0 in Eq. (12). Similarly, γxy = γzy = 0 and γxz = γyz = 0 for a y-directed and a z-directed applied electric field, respectively. Thus the macroscopic induced-dipole-electric-fieldcontributions vanish in the directions that are, for the rectangular parallelepiped case presented,orthogonal to the applied electric field. This is a direct consequence of aligning the coordinatesystem with the primitive cell axes and would lead to zero off-diagonal terms in the relativepermittivity tensor.
Before further calculations were conducted, the convergence of the relative permittivity as afunction of the total number of atoms involved was investigated to validate the present approach.Figure (6) gives the plot of ε versus the total number of atoms (Mx×My×Mz) for a cubic crystal(Mx = My = Mz and c/a = 1). Due to limited computational resources, the largest numberof atoms of 45 is assumed for all three orthogonal directions, corresponding to the largestdimension of 19.97 nm for the diagonal of the dielectric volume. The value of ε converges withthe increasing total number of atoms in the crystal. Moreover, since Mx = My = Mz and c/a = 1,εx = εy = εz. Consequently, the crystal is isotropic as expected. The initial precipitous decrease
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 816
0 1 2 3 4 5 6 7 8 9 10
x 104
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
Number of Atoms in a Cubic Crystal, Mx × My × Mz
Isot
ropi
cR
elat
ive
Per
mittivi
ty,ε
Isotropic ε
c/a = 1, Mx = My = Mz
Fig. 6. Relative permittivity ε versus various total number of atoms (Mx ×My ×Mz) in acubic crystal (Mx = My = Mz, c/a = 1).
of the relative permittivity may be explained physically by the decrease in both equivalentpolarization surface charge density and the equivalent polarization volume charge density as thetotal number of atoms in the dielectric volume increases. It should be noted that the dimensionor total number of atoms at which the ε value starts to converge depends on the polarizabilityand the lattice constant.
The plane-like (linear chain-like) structures can be achieved by conceptually compressing thearray of lattices in the plane of the a and b axes (along the c axis) or by constructing the dielec-tric volume in a plane-like (chain-like) shape with differing number of atoms in three orthogonaldirections. The application of one approach does not preclude the other. For instance, a stackof many rectangular layers along the c axis would show a linear chain-like shape (even thougheach constituent layer is plane-like) provided c(Mz−1) is larger than a(Mx−1) and a(My−1).Conversely, a dielectric volume that is plane-like in shape may consist of many short linearchain-like dielectric rods. To investigate the birefringence characteristics in the plane-like andchain-like behavior regimes, calculations were made with a fixed primitive lattice volume and afixed number of atoms as c/a or a/c varies. This ensures the volume of the dielectric structureremains unchanged while its shape, when viewed as a whole, changes.
For cases where conceptually compressing primitive lattice structure along c axis or in theplane of the a and b axes, the number of atoms in all three orthogonal directions is assumedidentical (i.e. Mx = My = Mz). This would exclude any contribution introduced by varying thenumber of atoms in different directions for shaping the nanoscale dielectric. Figure 7 gives thecalculation results for (Mx,My,Mz) = (20,20,20) in plane-like and chain-like behavior regimes.At c/a = a/c = 1 the crystal is a cube in shape and behaves isotropically (εx = εy = εz). Thisis due to the crystalline symmetry, regardless of whether the dielectric volume is an infinite orfinite cubic crystal lattice. With the increase of c/a, the structure is effectively more compactin the plane of the a and b axes and exhibits a plane-like shape for each layer. Accordingly, thepositive (negative) normalized Ex
ind,x (Ezind,z) introduces a polarizing (depolarizing) effect, thus
increasing (decreasing) the relative permittivity εx (εz). For the chain-like structure, the Ezind,z
component contributes to the polarizing factor, resulting in a larger εz compared to εx. Thiseffect remains provided the lattice is non-cubic (c �= a = b) in the tetragonal crystal system.
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 817
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 22
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Ratio of Lattice Constants, c/a
Rel
ativ
ePer
mit
tivi
ty,ε
εx (plane-like)
εx (chain-like)
εz (chain-like)
Mx = 20,My = 20,Mz = 20
Ratio of Lattice Constants, a/c
εz (plane-like)
Fig. 7. Relative permittivity εx and εz versus c/a (negative-uniaxial plane-like) and a/c(positive-uniaxial chain-like) ratios for a finite cubic array of atoms with (Mx,My,Mz) =(20,20,20).
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
10
20
30
40
50
60
70
Ratio of Lattice Constants, c/a
Rel
ativ
ePer
mit
tivi
ty,ε
Mx = My = 40
Mx = My = 30
Mx = My = 20
Mx = My = 10
Mz = 3
Mx = My = 50
εx
εz
Fig. 8. Relative permittivity εx and εz versus c/a ratio for a triple 100-atom plane(Mx,My,Mz) = (10,10,3), a triple 400-atom plane (Mx,My,Mz) = (20,20,3), a triple 900-atom plane (Mx,My,Mz) = (30,30,3), a triple 1600-atom plane (Mx,My,Mz) = (40,40,3),and a triple 2500-atom plane (Mx,My,Mz) = (50,50,3).
The effects of changing the ratio of lattice constants with a fixed and equal number of atomsin three orthogonal directions (so that c > a or a > c is achieved) have been described above.We then turn our attention to the case where the entire dielectric volume shows a plane-likeor chain-like shape for both cubic and tetragonal crystals. For instance, a nanoscale dielectriccrystal consisting of a stack of triple 400-atom layers (Mx = My = 20 and Mz = 3) would
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 818
resemble a plane-like structure. Similarly, a nanoscale dielectric with Mx = My = 10 and Mz =30, a stack of thirty 100-atom layers, would resemble a chain-like structure. In either case thecubic and tetragonal lattice systems were analyzed to provide a better physical picture of theseconfigurations. Figure 8 shows the relative permittivity εx and εz versus c/a for various plane-like configurations. For these cases εx (obtained when the applied electric field is parallel tothe triple (Mx ×My)-atom plane) is larger than εz (obtained when the applied electric field isperpendicular to the triple (Mx×My)-atom plane), regardless of their primitive lattice structures.The increase in the total number of atoms in each constituent plane enhances both the polarizingand depolarizing effects, thus raising εx and reducing εz further. Note that at c/a = a/c = 1, εx �=εz since the x-directed and z-directed induced-dipole-electric-field contributions are unequalowing to differing number of atoms in the x and z directions. Likewise for a 100-atom cross-section chain with 30 atoms in length (Mx = My = 10 and Mz = 30), the atoms are arrangedto form a chain-like structure; thus εz > εx due to a larger polarizing effect created by induceddipoles oriented along the c axis (Fig. 9). As is the case in plane-like configurations, the increasein the chain length by adding more atoms along the c axis (those with Mz = {40,50,60,70} inFig. 9) increases εz and lowers εx at a given ratio of lattice constants, creating greater positiveuniaxial birefringence. Thus, as opposed to bulk crystalline materials, birefringence may existin a nanoscale dielectric that is of rectangular parallelepiped shape (i.e. plane-like or chain-like)even if primitive lattice is cubic.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
5
10
15
20
25
Ratio of Lattice Constants, a/c
Rel
ativ
ePer
mit
tivi
ty,ε
Mz = 40
Mz = 50
Mz = 30
Mz = 60
εz
εx
Mz = 70
Mx = My = 10
Fig. 9. Relative permittivity εx and εz versus a/c ratio for five 100-atom-cross-sectionchains with 30, 40, 50, 60, and 70 atoms in length denoted by (Mx,My,Mz) = (10,10,30),(Mx,My,Mz) = (10,10,40), (Mx,My,Mz) = (10,10,50), (Mx,My,Mz) = (10,10,60), and(Mx,My,Mz) = (10,10,70), respectively.
The biaxial birefringence behaviors of a nanoscale dielectric volume with cubic and tetrago-nal lattices are shown in Figs. 10 and 11. In this representative case, the number of atoms in thex, y, and z directions is Mx = 20, My = 10, and Mz = 5, respectively. Since the number of atomsalong principal axes differ, the induced-dipole-electric-field contributions γxx, γyy, and γzz varyaccordingly, resulting in different polarizing (γxx > γyy > 0) and depolarizing (γzz < 0) effectswhen the applied electric field is along different principal axes. Thus there is no surprise thatεx > εy > εz at c/a = 1 in Fig. 10. Moreover, increasing c/a ratio monotonically enhances εx
and εy but slowly reduces εz (Fig. 10). In contrast, at smaller values of a/c in Fig. 11, the entire
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 819
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 22
4
6
8
10
12
14
Ratio of Lattice Constants, c/a
Rel
ativ
ePer
mit
tivi
ty,ε
εz
εx
εy
Mx = 20,My = 10,Mz = 5
Fig. 10. Relative permittivity ε versus c/a (plane-like) ratio for a rectangular parallelepipeddielectric volume with Mx = 20, My = 10, and Mz = 5.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 22
3
4
5
6
7
8
9
Ratio of Lattice Constants, a/c
Rel
ativ
ePer
mit
tivi
ty,ε
εy
εz
Mx = 20,My = 10,Mz = 5
εx
Fig. 11. Relative permittivity ε versus a/c (chain-like) ratio for a rectangular parallelepipeddielectric volume with Mx = 20, My = 10, and Mz = 5.
dielectric volume still resembles a planar thin slab that consists of many 5-atom chains andthe relation εx > εy > εz remain unchanged. As a/c increases further, the normalized induced-dipole-electric-field contributions γzz increases from negative to positive values but γxx and γyy
decrease from positive to negative values. More specifically, γzz (and thus εz) surpasses γyy (εy)at a/c = 1.28 and γxx (εx) at a/c = 1.39. Hence εz can eventually surpass εy and εx, as shown inFig. 11. However, since there are not many planes in the z direction, εz is inherently small evenat a/c = 2 compared to the largest εx and εy values at a/c = 1.
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 820
4. Conclusions
The inherent uniaxial and biaxial birefringence in nanometer-scale dielectrics has been quan-tified by evaluating directly the relative permittivity via the summation of induced-dipole-electric-field contributions at all atomic positions within the dielectric volume. The cubic andtetragonal crystal systems have been used as the representative cases. Varying either the ratioof lattice constants or the number of atoms in three orthogonal directions or both illustratesthe chain-like or plane-like behaviors under the condition of fixed primitive lattice volume. Astrong polarizing effect and thus higher relative permittivity occurs when the applied electricfield is parallel to the dielectric volume orientation of larger dimensions (i.e., the length of achainlike dielectric or the lateral extent of a plane-like dielectric). For cubic crystals, the ma-terial is isotropic for both finite and infinite arrays of atoms; however, it becomes uniaxial orbiaxial when the geometry of the entire dielectric volume has a rectangular parallelepiped shape(by varying the number of atoms in the x, y, and/or z directions while the cubic lattice remainsunchanged). For equal numbers of atoms along the a and b axes, both cubic and tetragonalcrystal systems are positive uniaxial when they are in chain-like shapes and negative uniax-ial when in plane-like shapes. Furthermore, birefringence is shown to occur in a rectangularparallelepiped dielectric volume with unequal number of atoms along the primitive cell axes,regardless of whether the crystal lattice is cubic or tetragonal.
#117020 - $15.00 USD Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010
(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 821
