Abstract

Physical properties of materials are known to be different from the bulk at the nanometer scale. In this context, the dependence of optical properties of nanometric gold thin films with respect to film thickness is studied using density functional theory (DFT). We find that the in-plane plasma frequency of the gold thin film decreases with decreasing thickness and that the optical permittivity tensor is highly anisotropic as well as thickness dependent. Quantitative knowledge of planar metal film permittivity’s thickness dependence can improve the accuracy and reliability of the designs of plasmonic devices and electromagnetic metamaterials. The strong anisotropy observed may become an alternative method of realizing indefinite media.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  36. X. Wang, K.-P. Chen, M. Zhao, and D. D. Nolte, “Refractive index and dielectric constant transition of ultra-thin gold from cluster to films,” Opt. Express18(24), 24859–24867 (2010).
    [CrossRef] [PubMed]
  37. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72(7), 075405 (2005).
    [CrossRef]
  38. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics2(2), 131–138 (2013).

2013

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics2(2), 131–138 (2013).

2012

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science336(6078), 205–209 (2012).
[CrossRef] [PubMed]

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature483(7390), 421–427 (2012).
[CrossRef] [PubMed]

2011

C. Dellagiacoma, T. Lasser, O. J. F. Martin, A. Degiron, J. J. Mock, and D. R. Smith, “Simulation of complex plasmonic circuits including bends,” Opt. Express19(20), 18979–18988 (2011).
[CrossRef] [PubMed]

J. Yan, K. W. Jacobsen, and K. Y. Thygesen, “First principles study of surface plasmons on Ag(111) and H/Ag(111),” Phys. Rev. B84(23), 235430 (2011).
[CrossRef]

M. Hövel, B. Gompf, and M. Dressel, “Electrodynamics of ultrathin gold films at the insulator-to-metal transition,” Thin Solid Films519(9), 2955–2958 (2011).
[CrossRef]

2010

X. Wang, K.-P. Chen, M. Zhao, and D. D. Nolte, “Refractive index and dielectric constant transition of ultra-thin gold from cluster to films,” Opt. Express18(24), 24859–24867 (2010).
[CrossRef] [PubMed]

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Y. He and T. Zeng, “First-principle study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C114(42), 18023–18030 (2010).
[CrossRef]

K. Glantschnig and C. Ambrosch-Draxl, “Relativistic effects on the linear optical properties of Au, Pt, Pb and W,” New J. Phys.12(10), 103048 (2010).
[CrossRef]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9(3), 205–213 (2010).
[CrossRef] [PubMed]

2009

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: The effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem C.Lett113(8), 3041–3045 (2009).
[CrossRef]

2008

B. J. Wiley, D. J. Lipomi, J. Bao, F. Capasso, and G. M. Whitesides, “Fabrication of surface plasmon resonators by nanoskiving single-crystalline gold microplates,” Nano Lett.8(9), 3023–3028 (2008).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

2007

J. Harl, G. Kresse, L. D. Sun, M. Hohage, and L. Zeppenfeld, “Ab initio reflectance difference spectra of the bare and adsorbate covered Cu(110) surfaces,” Phys. Rev. B76(3), 035436 (2007).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

2006

2005

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72(7), 075405 (2005).
[CrossRef]

2004

2003

J. L. West and N. J. Halas, “Engineered nanomaterials for biophotonics applications: Improving sensing, imaging, and therapeutics,” Annu. Rev. Biomed. Eng.5(1), 285–292 (2003).
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett.90(7), 077405 (2003).
[CrossRef] [PubMed]

2002

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys.74(2), 601–659 (2002).
[CrossRef]

2001

G. Onida, W. G. Schmidt, O. Pulci, M. Palummo, A. Marini, C. Hogan, and R. Del Sole, “Theory for modeling the optical properties of surfaces,” Phys. Status Solidi188(4), 1233–1242 (2001) (a).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

2000

S. Link and M. A. El-Sayed, “Shape and size-dependence of radiative, non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem.19(3), 409–453 (2000).
[CrossRef]

1999

G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B59(3), 1758–1775 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999).
[CrossRef]

1996

J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett.77(18), 3865–3868 (1996).
[CrossRef] [PubMed]

G. Kresse and J. Furthmuller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” J. Comput. Mater. Sci6(1), 15–50 (1996).
[CrossRef]

G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B Condens. Matter54(16), 11169–11186 (1996).
[CrossRef] [PubMed]

1976

H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Phys. Rev. B13(12), 5188–5192 (1976).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1969

U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys.224(4), 307–323 (1969).
[CrossRef]

1968

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp.10(4), 509–514 (1968).
[CrossRef]

1904

J. C. M. Garnett, “Colours in metal glasses and in metallic films,” Phil. Trans. Roy. Soc. London A203(359-371), 385–420 (1904).
[CrossRef]

Alekseyev, L. V.

Ambrosch-Draxl, C.

K. Glantschnig and C. Ambrosch-Draxl, “Relativistic effects on the linear optical properties of Au, Pt, Pb and W,” New J. Phys.12(10), 103048 (2010).
[CrossRef]

Arnold, M. D.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: The effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem C.Lett113(8), 3041–3045 (2009).
[CrossRef]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9(3), 205–213 (2010).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72(7), 075405 (2005).
[CrossRef]

Baehr-Jones, T.

Bao, J.

B. J. Wiley, D. J. Lipomi, J. Bao, F. Capasso, and G. M. Whitesides, “Fabrication of surface plasmon resonators by nanoskiving single-crystalline gold microplates,” Nano Lett.8(9), 3023–3028 (2008).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Biagioni, P.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Blaber, M. G.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: The effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem C.Lett113(8), 3041–3045 (2009).
[CrossRef]

Brüning, C.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Burke, K.

J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett.77(18), 3865–3868 (1996).
[CrossRef] [PubMed]

Burrows, A.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics2(2), 131–138 (2013).

Callegari, V.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Capasso, F.

B. J. Wiley, D. J. Lipomi, J. Bao, F. Capasso, and G. M. Whitesides, “Fabrication of surface plasmon resonators by nanoskiving single-crystalline gold microplates,” Nano Lett.8(9), 3023–3028 (2008).
[CrossRef] [PubMed]

Chen, K.-P.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Degiron, A.

Del Sole, R.

G. Onida, W. G. Schmidt, O. Pulci, M. Palummo, A. Marini, C. Hogan, and R. Del Sole, “Theory for modeling the optical properties of surfaces,” Phys. Status Solidi188(4), 1233–1242 (2001) (a).
[CrossRef]

Dellagiacoma, C.

Dionne, J. A.

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature483(7390), 421–427 (2012).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72(7), 075405 (2005).
[CrossRef]

Dressel, M.

M. Hövel, B. Gompf, and M. Dressel, “Electrodynamics of ultrathin gold films at the insulator-to-metal transition,” Thin Solid Films519(9), 2955–2958 (2011).
[CrossRef]

El-Sayed, M. A.

S. Link and M. A. El-Sayed, “Shape and size-dependence of radiative, non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem.19(3), 409–453 (2000).
[CrossRef]

Ernzerhof, M.

J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett.77(18), 3865–3868 (1996).
[CrossRef] [PubMed]

Feichtner, T.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Fischer, S. V.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics2(2), 131–138 (2013).

Forchel, A.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Ford, M. J.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: The effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem C.Lett113(8), 3041–3045 (2009).
[CrossRef]

Fragstein, C. V.

U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys.224(4), 307–323 (1969).
[CrossRef]

Furthmuller, J.

G. Kresse and J. Furthmuller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” J. Comput. Mater. Sci6(1), 15–50 (1996).
[CrossRef]

Furthmüller, J.

G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B Condens. Matter54(16), 11169–11186 (1996).
[CrossRef] [PubMed]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in metal glasses and in metallic films,” Phil. Trans. Roy. Soc. London A203(359-371), 385–420 (1904).
[CrossRef]

Geisler, P.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Glantschnig, K.

K. Glantschnig and C. Ambrosch-Draxl, “Relativistic effects on the linear optical properties of Au, Pt, Pb and W,” New J. Phys.12(10), 103048 (2010).
[CrossRef]

Gompf, B.

M. Hövel, B. Gompf, and M. Dressel, “Electrodynamics of ultrathin gold films at the insulator-to-metal transition,” Thin Solid Films519(9), 2955–2958 (2011).
[CrossRef]

Halas, N. J.

J. L. West and N. J. Halas, “Engineered nanomaterials for biophotonics applications: Improving sensing, imaging, and therapeutics,” Annu. Rev. Biomed. Eng.5(1), 285–292 (2003).
[CrossRef] [PubMed]

Harl, J.

J. Harl, G. Kresse, L. D. Sun, M. Hohage, and L. Zeppenfeld, “Ab initio reflectance difference spectra of the bare and adsorbate covered Cu(110) surfaces,” Phys. Rev. B76(3), 035436 (2007).
[CrossRef]

He, Y.

Y. He and T. Zeng, “First-principle study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C114(42), 18023–18030 (2010).
[CrossRef]

Hecht, B.

J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
[CrossRef] [PubMed]

Hochberg, M.

Hogan, C.

G. Onida, W. G. Schmidt, O. Pulci, M. Palummo, A. Marini, C. Hogan, and R. Del Sole, “Theory for modeling the optical properties of surfaces,” Phys. Status Solidi188(4), 1233–1242 (2001) (a).
[CrossRef]

Hohage, M.

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J. S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nat Commun1(9), 150 (2010), doi:.
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Figures (9)

Fig. 1
Fig. 1

The real (a) and imaginary (b) part of the permittivity of bulk gold as predicted by the adopted DFT method, in comparison with data from standard literature.

Fig. 2
Fig. 2

The interband and intraband contributions to the imaginary (a) and real (b) parts of the permittivity of bulk gold, as evaluated from the adopted DFT method. The electronic band structure of bulk gold along the ΓX and ΓL directions as calculated from the adopted DFT method (c). The red dotted line in (c) indicates the Fermi-level. The labels s,p,d represent the dominant orbital characters of the bands at their high-symmetry points.

Fig. 3
Fig. 3

Variations in (a) In-plane plasma frequency, (b) the real part of the in-plane permittivity, and (c) imaginary part of the in-plane permittivity as a function of lateral film-thickness. Note that the real part of the permittivity for all cases decreases monotonically below 2 eV and, hence, is not highlighted in (b). (d) represents the interband contributions to the imaginary part of the in-plane permittivity for the indicated select thin-film systems.

Fig. 4
Fig. 4

The band structure for (a) 7.02 nm, thick and (b) 3.63 nm (111) gold thin-films in the ΓX direction.

Fig. 5
Fig. 5

(a) Imaginary part and (b) real part of the out-of-plane permittivity as a function of the film-thickness.

Fig. 6
Fig. 6

An illustration of the surface plasmon guided mode being investigated at the interface between a metal and a dielectric material

Fig. 7
Fig. 7

(a) Real part of the wave number of the of guided modes at the surface of an isotropic metal (blue squares) and an anisotropic metal (red dots); (b) imaginary part of the wave number of the guided modes at the surface of an isotropic metal (blue squares) and an anisotropic metal (red dots).

Fig. 8
Fig. 8

An illustration of the two surface plasmon guided modes on a finite thickness metal film sandwiched between two identical dielectric media. The red and blue curves illustrate the electrical field profile of the symmetric (even) and antisymmetric (odd) modes, respectively.

Fig. 9
Fig. 9

Dispersion characteristics for the two guided modes in the structure illustrated in Fig. 8. The empty squares plot the solutions using the isotropic permittivity in the metal. The solid red circles plot the solutions using the anisotropic permittivity in the metal. (a) and (b) show the antisymmetric mode. (c) and (d) show the symmetric mode. (a) and (c) show the real part of the wave number component along the interface; (b) and (d) show its imaginary part.

Equations (6)

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Im[ ε x 1 x 2 inter ( ω ) ]= 8 π 2 e 2 V lim q0 1 | q | 2 m,n 2 f nk u m,k+q x ^ 1 | u n,k u n,k | u m,k+q x ^ 2 ×  [ δ( ξ m,k+q ξ n,k +ω )δ( ξ m,k+q ξ n,k ω ) ]
Re[ ε intra ( ω ) ]=1 ω p 2 ω 2 + Γ 2 ;      Im[ ε intra ( ω ) ]= Γ ω p 2 ω 3 +ω Γ 2
ω p 2 ( x 1 , x 2 )= 4π e 2 V n,k 2 f( ξ n ) ξ n ( ξ n (k) k x ^ 1 )( ξ n (k) k x ^ 2 )
ε m =[ ε xx 0 0 0 ε yy 0 0 0 ε zz ]
ε xx itan( i ε xx ε zz k x 2 ε xx k 0 2 L 2 ) ε d ε xx ε zz k x 2 ε xx k 0 2 k x 2 ε d k 0 2 =0
ε xx +icot( i ε xx ε zz k x 2 ε xx k 0 2 L 2 ) ε d ε xx ε zz k x 2 ε xx k 0 2 k x 2 ε d k 0 2 =0

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