Birefringence characteristics of nanoscale dielectrics ... - OSA Publishing

Birefringence characteristics of nanoscale dielectrics with cubic and tetragonal lattices Yin-Jung Chang1 * and Thomas K. Gaylord2 1 Department

of Optics and Photonics, National Central University Chung-Li, Taiwan, R.O.C. 2 School of Electrical and Computer Engineering, Georgia Institute of Technology Atlanta, Georgia 30332, U.S.A. [email protected]

Abstract: The birefringence in nanometer-scale dielectrics with the largest dimensions ranging from about 3 nm to 20 nm has been quantified by evaluating directly the summation of induced-dipole-electric-field contributions from all individual atoms within the entire dielectric volume. Various configurations in representative cubic and tetragonal systems are investigated by varying the ratio of lattice constants and the number of atoms in various directions to illustrate the chain-like and plane-like behavior regimes. The dielectric properties of the finite cubic crystal lattices change from isotropic to birefringent (uniaxial or biaxial) when the entire dielectric volume is changed from a cube to a rectangular parallelepiped in shape. In finite tetragonal crystals the birefringence increases with the increasing lattice constant ratios. The largest uniaxial birefringence occurs for non-cube dielectric volume with tetragonal lattices. © 2010 Optical Society of America OCIS codes: (160.4760) Optical properties; (160.1190) Anisotropic optical materials; (160.3130) Integrated optics materials.

References and links 1. 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 all aspects of our daily lives. The development of nano-fabrication technologies has made the concept of nanophotonics possible where new photonic devices and dielectric materials are used to overcome the conventional diffraction limit to manipulate light at the nanoscale. In nanophotonics, nanoscale dielectric materials are ubiquitous and have a direct impact on device #117020 - $15.00 USD

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Received 10 Sep 2009; revised 8 Dec 2009; accepted 28 Dec 2009; published 6 Jan 2010

18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 809

functionality and performance. Understanding the optical properties, in particular, the indices of refraction and the birefringence, of these nanoscale dielectrics is fundamentally important. This knowledge is needed, for example, to provide various indices of refraction and birefringences for the design of future nanophotonic integrated circuits. In this paper, the birefringence characteristics of representative nanometer-scale dielectric volumes with cubic and tetragonal crystal structures are quantified by summing the induceddipole-electric-field contributions from the individual atoms in the dielectric volume. The induced-dipole-electric field experienced by each atom (or dipole) is a function of its atomic position relative to all other atoms in the dielectric volume. In other words, instead of using the Lorentz spherical surface [1] to separate the nearby dipoles and the rest of the material, all induced dipoles in the dielectric volume are explicitly included which is needed when modeling the limited number of atoms in nanoscale structures. The relative permittivities of nanometerscale dielectrics in cubic and tetragonal crystal systems are quantified with an emphasis on chain-like and plane-like behavior regimes. Birefringence is shown to occur if the nanometerscale dielectric volume is of rectangular parallelepiped shape, regardless of whether its primitive lattice is cubic or non-cubic. Moreover, at small scales the total volume of a rectangular parallelepiped dielectric plays a crucial role that determines the degree of electric polarization and 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 optical properties over the nanoscale volume. We note that the motions of electrons in atoms and molecules are characterized by periods on the order of 10−16 second [3], corresponding to frequencies close to those of the visible light wave. Strictly nonpolar materials behave practically the same from zero frequency up to frequencies of visible light since electrons are able to follow the time-varying field. Because of this, it is possible to ignore the time dependence. For simplicity, the treatment developed here is for the nonpolar crystalline materials of identical atoms arranged in cubic or tetragonal lattices. To analyze the macroscopic average optical properties, in particular the relative permittivity and birefringence, the electric polarization vector P is first considered. It is the macroscopic electric dipole moment per unit volume in the material and is given by [4, 5] NΔv

∑ pj

j=1

, (1) Δv where N is the number of atoms per unit volume and the numerator represents the vector sum of the induced dipole moments p j within a small volume Δv. If we assume that all the induced dipoles are directly proportional to, and in the same direction as, their corresponding local field Elocal, j , then P = lim

Δv→0

p j = α j Elocal, 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 induced electric fields acting on the j-th atom, respectively [2]. At optical frequency, α j is mainly contributed by the electronic polarizability and is assumed to be a constant for a given type of spherical atoms. Using Eq. (2), Eq. (1) can be expressed in its column vector form as ΔvP =

NΔv

∑α

Eappl, j + Eind, j ,

(3)

j=1

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Fig. 1. Schematic of the induced dipole moments in a nanoscale parallelepiped dielectric slab due to an x-directed applied field.

where ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ Px Eappl, j x Eind, j x P = ⎣Py ⎦ ; Eappl, j = ⎣Eappl, j y ⎦ ; Eind, j = ⎣Eind, j y ⎦ . Pz Eappl, j z Eind, j z

(4)

The induced-dipole-electric field at the j-th atom location Eind, j is the vector sum of individual induced-dipole-electric-fields from all other atoms within Δv (taken as the volume of the entire nano-scale dielectric) at that location Eind, j =

3(pi · ri )ri − ri2 pi , ∑ 4πε0 ri5 i=1,i= j NΔv

(5)

where pi is the induced dipole moment associated with the i-th atom, ri is the position vector from the i-th atom to the observation point (i.e. j-th atom) and ri = |ri | is the magnitude of the position vector ri (Fig.1). Since, for example, an applied electric field in the x direction would produce induced-dipole-electric field in the x-, y-, and z-directions, Eq. (5) may be rewritten as ⎡ ⎤⎡ ⎤ (γ j )xy (γ j )xz (p j )x (γ ) 1 1 ⎣ j xx (γ j )yx (γ j )yy (γ j )yz ⎦ ⎣(p j )y ⎦ ≡ Eind, j = γ p, (6) 3 4πε0 d 4πε0 d 3 j j (γ j )zx (γ j )zy (γ j )zz (p j )z where d is the smallest lattice constant, γ is the (de)polarizing factor tensor at the j-th atom j

location, and p j is 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 notation of p j is used for representing the varying small displacement between electronic cloud and nucleus 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 γ j tensor is interpreted as the (de)polarizing factor of the j-th atom in the u direction due to an applied electric field (and thus the dipole moment) in the v direction. The (de)polarizing factors are calculated using the equations given in Appendices A and B in [2]. The constitutive relation between the macroscopic electric displacement D and the electric polarization vector P is given by [4–6] D = ε0 ε Eappl = ε0 Eappl + P, #117020 - $15.00 USD

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(7)



where ε is the 3 × 3 relative permittivity tensor. Assuming the applied electric field is a constant across the entire dielectric volume, then Eappl, j = Eappl . Consequently, Eq. (3) becomes ΔvP = α N(Δv)Eappl + α

NΔv

∑ Eind, j .

(8)

j=1

Using Eq. (7), the externally applied field can be expressed in terms of the polarization vector P

−1 Eappl = ε0 ε − I P ≡ Q−1 P,

(9)

where I is the 3 × 3 identity matrix. Substitute Eq. (9) into Eq. (8) yields

α N(Δv)Q−1 P = ΔvP − α

NΔv

∑ Eind, j .

(10)

j=1

The summation of the induced-dipole-electric field experienced at each atomic position may be expressed and further approximated as NΔv

∑ Eind, j

=

1 γ p + γ p + · · · γ p 2 2 NΔv NΔv 4πε0 d 3 1 1

≈

1 γ p , 4πε0 d 3

j=1

(11)

where

γ = γ + γ +···+ γ 1

2

NΔv

⎛ γxx = ⎝γyx γzx

γxy γyy γzy

⎞ γxz γyz ⎠ γzz

which may be interpreted as the macroscopic (de)polarizing tensor and ⎛ ⎞ px p = ⎝ py ⎠ pz

(12)

(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-like crystal and d = c for the chain-like crystal). This is a direct result from normalizing the other two basis vectors with respect to d. Furthermore, the volume of one primitive cell can always be expressed in terms of the smallest primitive basis vector. Therefore the number of atoms per unit volume N is the reciprocal of the smallest volume unit of the dielectric and Eq. (11) becomes NΔv

N

∑ Eind, j ≈ 4πε0 γ

j=1

p =

1 γN p . 4πε0

(14)

Furthermore, since the polarization vector P is defined as the volume density of electric dipole moment, 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

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α N(Δv)Q−1 P = ΔvP −

α γ P. 4πε0

(15)

The relative permittivity tensor ε can then be obtained as follows: −1 ε = I + (4πα NΔv) I 4πε0 ΔvI − αγγ .

(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 permittivity averaged over the dielectric volume. The averaging is inherent in the macroscopic field quantities D, Eappl , and P and also in the expression P = N p . The resultant equation of ε in Eq. (16) applies to any dielectric material of identical atoms and of any size, not necessarily at nanoscale, provided the (de)polarizing factors (and thus the (de)polarizing tensor they construct) can be obtained. By properly choosing the orientation of the coordinate system, the macroscopic refractive indices 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 subscript notation 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 is negative 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 finite cubic lattice array of atoms is also examined by setting a = c in the tetragonal crystal system. In both 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 = (Mx+ +Mx− +1), My = (My+ +My− +1), and Mz = (Mz+ + Mz− + 1), where Mu+ and Mu− denote the respective number of atoms in the ±u directions with respect to the reference plane. The polarizability α used in the calculations is 2.9 × 10−24 cm3 , typical of SiO2 [7]. The lattice constant is chosen to be 2.62 A˚ which is approximately the distance between oxygen atoms in SiO2 . It should be emphasized that the calculations presented in this work are for dielectric volumes whose largest dimensions range from 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 would require increased computational resources since the induced-dipole-electric field at each atomic position is the superposition of the contribution from every other atom within the dielectric volume. 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). They x are normalized to p/4πε0 c3 , where p is the magnitude of the dipole moment. The Eind,x fields of the upper and lower planes are identical because of the reflection symmetry with respect to y z and Eind,z fields. Although not shown, the the central plane. This property also holds for Eind,y y x appearance and magnitude of Eind,y and Eind,x are identical, except for a 90◦ difference in the azmuthal angle. This is due to the 4-fold symmetry about the z-axis. The physical reasoning of Figs. 2 - 5 is summarized as follows:

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x Normalized Induced Eind,x

Upper(Lower) Plane

0

−0.5

−1

−1.5

Central Plane

−2 20

20

15

y/a

15

10 10

5

5 0

x/a

0

Fig. 2. The x component of the normalized induced-dipole-electric field for an applied field x , across the central plane of a triple 400-atom plane (Mx = My = in the x direction, Eind,x 20, Mz = 3) dielectric volume with a/c = 2. Integer values of x/a and y/a represent the atom positions in the x and y directions, respectively.

x Normalized Induced Eind,y

0.5 Centeral Plane

0

−0.5 20 20

15

y/a

15

10 10

5

5 0

x/a

0

Fig. 3. The y component of the normalized induced-dipole-electric field for an applied field x , across the central plane of a triple 400-atom plane (Mx = My = in the x direction, Eind,y 20, Mz = 3) dielectric volume with a/c = 2. For the upper and lower planes, the deviation is less than 8.8%. Integer values of x/a and y/a represent the atom positions in the x and y directions, respectively.

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Fig. 4. The z component of the normalized induced-dipole-electric field for an applied field x , across the central plane of a triple 400-atom plane (Mx = My = in the x direction, Eind,z 20, Mz = 3) dielectric volume with a/c = 2. Integer values of x/a and y/a represent the atom positions in the x and y directions, respectively.

z Normalized Induced Eind,z

Central Plane 3.5 3 2.5 2 1.5 1 0.5 0 20 15

y/a

10

20

Upper(Lower) Plane

15 10

5

5 0

x/a

0

Fig. 5. The z component of the normalized induced-dipole-electric field for an applied field z , across the central plane of a triple 400-atom plane (Mx = My = in the z direction, Eind,z 20, Mz = 3) dielectric volume with a/c = 2. Integer values of x/a and y/a represent the atom positions in the x and y directions, respectively.

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z x 1. The magnitudes of Eind,x and Eind,z of the central plane are larger since they are reinforced by the upper and lower planes. x at x/a = 10 has a maximum since there is a strong reinforcement from dipoles at 2. Eind,x larger and lower values of x/a. Correspondingly, it has minima at x/a = 1 and x/a = 20 due to a one-sided dipole reinforcement only. x along y/a = 1 (y/a = 20) is due to zero cancellation effect 3. The turned-up edge of Eind,x from dipoles at y/a < 1 (y/a > 20) for no dipoles are present there. x cancels at the center of each of the three planes due to symmetry. 4. Eind,y x cancels along x/a = 10 of each of the three planes due to symmetry. 5. Eind,z x in the upper plane decreases (increases) moving from (x/a, y/a) = (10, 10) toward 6. Eind,z (x/a, y/a) = (1, 10) [(x/a, y/a) = (20, 10)] due to the missing cancellation effect from dipoles at smaller (larger) values of x/a since they are absent. x in the lower plane decreases (increases) moving from (x/a, y/a) = (10, 10) toward 7. Eind,z (x/a, y/a) = (20, 10) [(x/a, y/a) = (1, 10)] due to the missing cancellation effect from dipoles at larger (smaller) values of x/a since they are absent. x in the upper plane decreases slightly moving from (x/a, y/a) = (20, 10) toward 8. Eind,z the corner at (x/a, y/a) = (20, 1) [(x/a, y/a) = (20, 20)] due to weak reinforcement (+z component) from dipoles at y/a < 1 (y/a > 20) since they are absent. Similarly, the slight x in the upper plane at the corner (x/a, y/a) = (1, 1) [(x/a, y/a) = (1, 20)] increase of Eind,z is due to weak reinforcement (−z component) from dipoles at y/a < 1 (y/a > 20) since they are absent. z peaks at four corners due to weak reinforcement (−z component) from dipoles 9. Eind,z inside the dielectric volume. There are no dipoles in the rest of seven octants relative to the corner to contribute to the −z-component reinforcement.

Because of the position-dependent reinforcement/cancellation effect, the induced-dipolefield contributions become more uniform across the entire plane except near the edges/corners x as the number of atoms in one plane (Mx × My ) increases. Although variations do exist in Eind,y x x (in all three planes) and Eind,z (in the upper and lower planes), the total sum of Eind,y and the x within the dielectric volume are zero. Mathematically, this corresponds to total sum of Eind,z γyx = 0 and γzx = 0 in Eq. (12). Similarly, γxy = γzy = 0 and γxz = γyz = 0 for a y-directed and a zdirected applied electric field, respectively. Thus the macroscopic induced-dipole-electric-field contributions 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 coordinate system with the primitive cell axes and would lead to zero off-diagonal terms in the relative permittivity tensor. Before further calculations were conducted, the convergence of the relative permittivity as a function 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 number of atoms of 45 is assumed for all three orthogonal directions, corresponding to the largest dimension of 19.97 nm for the diagonal of the dielectric volume. The value of ε converges with the 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

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4.8

c/a = 1, Mx = My = Mz Isotropic Relative Permittivity, ε

4.6 4.4 4.2 4 3.8 3.6

Isotropic ε

3.4 3.2 3 0

1

2

3

4

5

6

7

8

Number of Atoms in a Cubic Crystal, Mx × My × Mz

9

10 4 x 10

Fig. 6. Relative permittivity ε versus various total number of atoms (Mx × My × Mz ) in a cubic crystal (Mx = My = Mz , c/a = 1).

of the relative permittivity may be explained physically by the decrease in both equivalent polarization surface charge density and the equivalent polarization volume charge density as the total number of atoms in the dielectric volume increases. It should be noted that the dimension or total number of atoms at which the ε value starts to converge depends on the polarizability and the lattice constant. The plane-like (linear chain-like) structures can be achieved by conceptually compressing the array of lattices in the plane of the a and b axes (along the c axis) or by constructing the dielectric volume in a plane-like (chain-like) shape with differing number of atoms in three orthogonal directions. The application of one approach does not preclude the other. For instance, a stack of many rectangular layers along the c axis would show a linear chain-like shape (even though each 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 linear chain-like dielectric rods. To investigate the birefringence characteristics in the plane-like and chain-like behavior regimes, calculations were made with a fixed primitive lattice volume and a fixed number of atoms as c/a or a/c varies. This ensures the volume of the dielectric structure remains unchanged while its shape, when viewed as a whole, changes. For cases where conceptually compressing primitive lattice structure along c axis or in the plane of the a and b axes, the number of atoms in all three orthogonal directions is assumed identical (i.e. Mx = My = Mz ). This would exclude any contribution introduced by varying the number of atoms in different directions for shaping the nanoscale dielectric. Figure 7 gives the calculation 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 ). This is due to the crystalline symmetry, regardless of whether the dielectric volume is an infinite or finite cubic crystal lattice. With the increase of c/a, the structure is effectively more compact in the plane of the a and b axes and exhibits a plane-like shape for each layer. Accordingly, the z x (Eind,z ) introduces a polarizing (depolarizing) effect, thus positive (negative) normalized Eind,x z increasing (decreasing) the relative permittivity εx (εz ). For the chain-like structure, the Eind,z component contributes to the polarizing factor, resulting in a larger εz compared to εx . This effect remains provided the lattice is non-cubic (c = a = b) in the tetragonal crystal system. #117020 - $15.00 USD

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Ratio of Lattice Constants, a/c 7

Mx = 20, My = 20, Mz = 20 6.5

Relative Permittivity, ε

6 5.5 5

εz (chain-like) 4.5 4

εx (plane-like)

3.5

εx (chain-like) 3 2.5 2 1

εz (plane-like) 1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Ratio of Lattice Constants, c/a

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).

70

Mz = 3


60

50

Mx = My = 50

εx

40

Mx = My = 40 30

Mx = My = 30 20

Mx = My = 20 Mx = My = 10

10

εz 0 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


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 900atom 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 atoms in 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-like or chain-like shape for both cubic and tetragonal crystals. For instance, a nanoscale dielectric crystal consisting of a stack of triple 400-atom layers (Mx = My = 20 and Mz = 3) would

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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 the cubic and tetragonal lattice systems were analyzed to provide a better physical picture of these configurations. Figure 8 shows the relative permittivity εx and εz versus c/a for various planelike configurations. For these cases εx (obtained when the applied electric field is parallel to the triple (Mx × My )-atom plane) is larger than εz (obtained when the applied electric field is perpendicular 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 polarizing and 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 unequal owing to differing number of atoms in the x and z directions. Likewise for a 100-atom crosssection chain with 30 atoms in length (Mx = My = 10 and Mz = 30), the atoms are arranged to form a chain-like structure; thus εz > εx due to a larger polarizing effect created by induced dipoles oriented along the c axis (Fig. 9). As is the case in plane-like configurations, the increase in the chain length by adding more atoms along the c axis (those with Mz = {40, 50, 60, 70} in Fig. 9) increases εz and lowers εx at a given ratio of lattice constants, creating greater positive uniaxial birefringence. Thus, as opposed to bulk crystalline materials, birefringence may exist in a nanoscale dielectric that is of rectangular parallelepiped shape (i.e. plane-like or chain-like) even if primitive lattice is cubic. 25

Mx = My = 10 Mz = 70


20

Mz = 60 Mz = 50 Mz = 40

15

εz

Mz = 30

10

5

εx

0 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Ratio of Lattice Constants, a/c

Fig. 9. Relative permittivity εx and εz versus a/c ratio for five 100-atom-cross-section chains 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 tetragonal lattices are shown in Figs. 10 and 11. In this representative case, the number of atoms in the x, y, and z directions is Mx = 20, My = 10, and Mz = 5, respectively. Since the number of atoms along principal axes differ, the induced-dipole-electric-field contributions γxx , γyy , and γzz vary accordingly, resulting in different polarizing (γxx > γyy > 0) and depolarizing (γzz < 0) effects when the applied electric field is along different principal axes. Thus there is no surprise that εx > 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

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14

Mx = 20, My = 10, Mz = 5 12


εx 10

8

6

εy

4

εz 2 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


Fig. 10. Relative permittivity ε versus c/a (plane-like) ratio for a rectangular parallelepiped dielectric volume with Mx = 20, My = 10, and Mz = 5.

9

Mx = 20, My = 10, Mz = 5


8

7

6

εx

5

4

εy εz

3

2 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Ratio of Lattice Constants, a/c

Fig. 11. Relative permittivity ε versus a/c (chain-like) ratio for a rectangular parallelepiped dielectric volume with Mx = 20, My = 10, and Mz = 5.

dielectric volume still resembles a planar thin slab that consists of many 5-atom chains and the relation εx > εy > εz remain unchanged. As a/c increases further, the normalized induceddipole-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 in Fig. 11. However, since there are not many planes in the z direction, εz is inherently small even at a/c = 2 compared to the largest εx and εy values at a/c = 1. #117020 - $15.00 USD

(C) 2010 OSA



4.

Conclusions

The inherent uniaxial and biaxial birefringence in nanometer-scale dielectrics has been quantified by evaluating directly the relative permittivity via the summation of induced-dipoleelectric-field contributions at all atomic positions within the dielectric volume. The cubic and tetragonal crystal systems have been used as the representative cases. Varying either the ratio of lattice constants or the number of atoms in three orthogonal directions or both illustrates the chain-like or plane-like behaviors under the condition of fixed primitive lattice volume. A strong polarizing effect and thus higher relative permittivity occurs when the applied electric field is parallel to the dielectric volume orientation of larger dimensions (i.e., the length of a chainlike dielectric or the lateral extent of a plane-like dielectric). For cubic crystals, the material is isotropic for both finite and infinite arrays of atoms; however, it becomes uniaxial or biaxial 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 remains unchanged). For equal numbers of atoms along the a and b axes, both cubic and tetragonal crystal systems are positive uniaxial when they are in chain-like shapes and negative uniaxial when in plane-like shapes. Furthermore, birefringence is shown to occur in a rectangular parallelepiped 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

(C) 2010 OSA

