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

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References

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  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]

2007

1995

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).
[CrossRef]

Chang, Y.-J.

Dehan, E.

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).
[CrossRef]

Gaylord, T. K.

Henda, R.

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).
[CrossRef]

Pedroviejo, J. J.

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).
[CrossRef]

Scheid, E.

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).
[CrossRef]

Temple-Boyer, P.

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).
[CrossRef]

Appl. Opt.

Thin Solid Films

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).
[CrossRef]

Other

H. A. Lorentz, The Theory of Electrons (Teubner, 1909).

E. M. Purcell, Electricity and Magnetism, 2nd ed. (McGraw-Hill, 1985).

D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, 1989).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).

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Figures (11)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

The x component of the normalized induced-dipole-electric field for an applied field in 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 the atom positions in the x and y directions, respectively.

Fig. 3.
Fig. 3.

The y component of the normalized induced-dipole-electric field for an applied field in 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 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.

Fig. 4.
Fig. 4.

The z component of the normalized induced-dipole-electric field for an applied field in 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 the atom positions in the x and y directions, respectively.

Fig. 5.
Fig. 5.

The z component of the normalized induced-dipole-electric field for an applied field in 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 the atom positions in the x and y directions, respectively.

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

Relative permittivity x and z 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).

Fig. 8.
Fig. 8.

Relative permittivity x and z 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).

Fig. 9.
Fig. 9.

Relative permittivity x and z 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.

Fig. 10.
Fig. 10.

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

Fig. 11.
Fig. 11.

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

Equations (17)

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P = lim Δ v 0 j = 1 N Δ v p j Δ v ,
p j = α j E local , j = α j ( E appl , j + E ind , j ) ,
Δv P _ = j = 1 N Δ v α ( E _ appl , j + E _ ind , j ) ,
P _ = [ P x P y P z ] ; E _ appl , j = [ ( E appl , j ) x ( E appl , j ) y ( E appl , j ) z ] ; E _ ind , j = [ ( E ind , j ) x ( E ind , j ) y ( E ind , j ) z ] .
E ind , j = i = 1 , i j N Δ v 3 ( p i r i ) r i r i 2 p i 4 π ε 0 r i 5 ,
E _ ind , j = 1 4 π ε 0 d 3 [ ( 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 ] 1 4 π ε 0 d 3 γ͇ j p _ j ,
D _ = ε 0 ε͇ E _ appl = ε 0 E _ appl + P _ ,
Δ v P _ = α N ( Δ v ) E _ appl + α j = 1 N Δ v E _ ind , j .
E _ appl = [ ε 0 ( ε͇ ) ] 1 P _ 1 P _ ,
α N ( Δ v ) 1 P _ = Δ v P _ α j = 1 N Δ v E _ ind , j .
j = 1 N Δ v E _ ind , j = 1 4 π ε 0 d 3 { γ͇ 1 p _ 1 + γ͇ 2 p _ 2 + γ͇ N Δ v p _ N Δ v } 1 4 π ε 0 d 3 γ͇ p _ ,
γ͇ = ( γ͇ 1 + γ͇ 2 + + γ͇ N Δ v ) = ( γ xx γ xy γ xz γ yx γ yy γ yz γ zx γ zy γ zz )
p _ = ( p x p y p z )
j = 1 N Δ v E _ ind , j N 4 π ε 0 γ͇ p _ = 1 4 π ε 0 γ͇ N p _ .
α N ( Δ v ) 1 P _ = Δ v P _ α 4 π ε 0 γ͇ P ̲ .
ε͇ = + ( 4 π α N Δ v ) [ 4 π ε 0 Δ v α γ͇ ] 1 .
n x = ε x , n y = ε y , n z = ε z ,

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