Abstract

The conventional analysis of surface plasmon modes on dielectric–metal interfaces requires clearly defining the permittivity discontinuity at the interface. A pivotal assumption of such an analysis is that the formation of the dielectric-metal interface does not change the material properties and the materials forming the interface have identical permittivities before and after the formation of the interface. However, this assumption breaks down if an interface is made between a metal and a semiconductor which is commonly known as a Schottky junction. Under certain conditions, such an interface can sustain a surface plasmon polariton (SPP) mode. It is also possible to change the properties of the media surrounding the Schottky junction interface by applying an external potential difference across the junction. Central to the understanding of the SPP mode behaviour in such a complex morphological interface is the dispersion relation which defines the feasible SPP modes and their characteristics. Here, we carry out a detailed analysis to derive an analytical expression for the dispersion relation for a Schottky junction. Our analysis takes into account the space charge layer formed due to the charge distribution across the Schottky junction and resulting new boundary conditions.

© 2012 OSA

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References

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  1. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).
  3. D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling and Applications (Cambridge University Press, 2010).
  4. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science, 2007).
  5. G. V. Naik and A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).
    [CrossRef]
  6. R. T. Holm and E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).
  7. A. V. Krasavin and A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18, 11791–11799 (2010).
    [CrossRef] [PubMed]
  8. K. H. Aharonian and D. R. Tilley, “Propagating electromagnetic modes in thin semiconductor films,” J. Phys.: Condens. Matter I, 5391–5401 (1989).
    [CrossRef]
  9. S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
    [CrossRef]
  10. D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35, 4190–4192 (2010).
    [CrossRef] [PubMed]
  11. D. Y. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011).
    [CrossRef] [PubMed]
  12. I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
    [CrossRef]
  13. I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Surface plasmon–polariton propagation in piecewise linear chains of nanospheres: The role of optical gain and chain layout,” Opt. Express 19, 19973–19986 (2011).
    [CrossRef] [PubMed]
  14. M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1–416 (2001).
    [CrossRef]
  15. A. Yariv and R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. 2, 55–57 (1963).
    [CrossRef]
  16. R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.
  17. J. C. Inkson, “Many-body effects at metal-semiconductor junctions. I. Surface plasmons and the electron-electron screened interaction,” J. Phys. C: Solid State Phys. 5, 2599–2610 (1972).
    [CrossRef]
  18. L. Solymar and D. Walsh, Electrical Properties of Materials (Oxford University Press, 2004).
  19. K. F. Brennan, Introduction to Semiconductor Devices: For Computing and Telecommunications Applications (Cambridge University Press, 2005).
  20. S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface-plasmon-polariton dispersion curve,” Phys. Rev. B 10, 3342–3355 (1974).
    [CrossRef]
  21. C. C. Kao and E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976).
    [CrossRef]
  22. N. Lebedev and R. A. SilvermanSpecial Functions and Their Applications (Dover Publication, 1972).
  23. S. S. Bayin, Mathematical Methods in Science and Engineering (Wiley–Interscience, 2006).
    [CrossRef]
  24. E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. Lett. 182, 539–554 (1969).
  25. P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B 12, 4032–4035 (1975).
    [CrossRef]
  26. H. C. Casey and M. B. PanishHeterostructure Lasers, Part A: Fundamental Principles (Academic, 1978).
  27. D. Y. Fedyanin, “Toward an electrically pumped spaser,” Opt. Lett. 37, 404–406 (2012).
    [CrossRef] [PubMed]

2012 (1)

2011 (3)

2010 (3)

2006 (2)

R. T. Holm and E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[CrossRef]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).

2001 (1)

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1–416 (2001).
[CrossRef]

1989 (1)

K. H. Aharonian and D. R. Tilley, “Propagating electromagnetic modes in thin semiconductor films,” J. Phys.: Condens. Matter I, 5391–5401 (1989).
[CrossRef]

1976 (1)

C. C. Kao and E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976).
[CrossRef]

1975 (1)

P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B 12, 4032–4035 (1975).
[CrossRef]

1974 (1)

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface-plasmon-polariton dispersion curve,” Phys. Rev. B 10, 3342–3355 (1974).
[CrossRef]

1972 (1)

J. C. Inkson, “Many-body effects at metal-semiconductor junctions. I. Surface plasmons and the electron-electron screened interaction,” J. Phys. C: Solid State Phys. 5, 2599–2610 (1972).
[CrossRef]

1969 (1)

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. Lett. 182, 539–554 (1969).

1963 (1)

A. Yariv and R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. 2, 55–57 (1963).
[CrossRef]

Agrawal, G. P.

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).

Aharonian, K. H.

K. H. Aharonian and D. R. Tilley, “Propagating electromagnetic modes in thin semiconductor films,” J. Phys.: Condens. Matter I, 5391–5401 (1989).
[CrossRef]

Arsenin, A. V.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).

Bayin, S. S.

S. S. Bayin, Mathematical Methods in Science and Engineering (Wiley–Interscience, 2006).
[CrossRef]

Boltasseva, A.

G. V. Naik and A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).
[CrossRef]

Brennan, K. F.

K. F. Brennan, Introduction to Semiconductor Devices: For Computing and Telecommunications Applications (Cambridge University Press, 2005).

Brion, J. J.

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.

Burstein, E.

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.

Casey, H. C.

H. C. Casey and M. B. PanishHeterostructure Lasers, Part A: Fundamental Principles (Academic, 1978).

Challener, W.

D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling and Applications (Cambridge University Press, 2010).

Conmell, E. M.

C. C. Kao and E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976).
[CrossRef]

Cunningham, S. L.

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface-plasmon-polariton dispersion curve,” Phys. Rev. B 10, 3342–3355 (1974).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).

Economou, E. N.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. Lett. 182, 539–554 (1969).

Fedyanin, D. Y.

Halevi, P.

P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B 12, 4032–4035 (1975).
[CrossRef]

Handapangoda, D.

Hartstein, A.

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.

Holm, R. T.

R. T. Holm and E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).

Inkson, J. C.

J. C. Inkson, “Many-body effects at metal-semiconductor junctions. I. Surface plasmons and the electron-electron screened interaction,” J. Phys. C: Solid State Phys. 5, 2599–2610 (1972).
[CrossRef]

Jagadish, C.

Kao, C. C.

C. C. Kao and E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976).
[CrossRef]

Krasavin, A. V.

Kushwaha, M. S.

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1–416 (2001).
[CrossRef]

Lebedev, N.

N. Lebedev and R. A. SilvermanSpecial Functions and Their Applications (Dover Publication, 1972).

Leite, R. C. C.

A. Yariv and R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. 2, 55–57 (1963).
[CrossRef]

Maier, S. A.

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science, 2007).

Maradudin, A. A.

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface-plasmon-polariton dispersion curve,” Phys. Rev. B 10, 3342–3355 (1974).
[CrossRef]

Naik, G. V.

G. V. Naik and A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).
[CrossRef]

Palik, E. D.

R. T. Holm and E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).

Panish, M. B.

H. C. Casey and M. B. PanishHeterostructure Lasers, Part A: Fundamental Principles (Academic, 1978).

Premaratne, M.

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Surface plasmon–polariton propagation in piecewise linear chains of nanospheres: The role of optical gain and chain layout,” Opt. Express 19, 19973–19986 (2011).
[CrossRef] [PubMed]

D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35, 4190–4192 (2010).
[CrossRef] [PubMed]

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).

Rukhlenko, I. D.

Sarid, D.

D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling and Applications (Cambridge University Press, 2010).

Silverman, R. A.

N. Lebedev and R. A. SilvermanSpecial Functions and Their Applications (Dover Publication, 1972).

Solymar, L.

L. Solymar and D. Walsh, Electrical Properties of Materials (Oxford University Press, 2004).

Tilley, D. R.

K. H. Aharonian and D. R. Tilley, “Propagating electromagnetic modes in thin semiconductor films,” J. Phys.: Condens. Matter I, 5391–5401 (1989).
[CrossRef]

Udagedara, I. B.

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Surface plasmon–polariton propagation in piecewise linear chains of nanospheres: The role of optical gain and chain layout,” Opt. Express 19, 19973–19986 (2011).
[CrossRef] [PubMed]

Wallis, R. F.

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface-plasmon-polariton dispersion curve,” Phys. Rev. B 10, 3342–3355 (1974).
[CrossRef]

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.

Walsh, D.

L. Solymar and D. Walsh, Electrical Properties of Materials (Oxford University Press, 2004).

Yariv, A.

A. Yariv and R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. 2, 55–57 (1963).
[CrossRef]

Zayats, A. V.

Appl. Phys. Lett. (1)

A. Yariv and R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. 2, 55–57 (1963).
[CrossRef]

CRC Crit. Rev. Sol. State Mat. Sci. (1)

R. T. Holm and E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).

J. Phys. C: Solid State Phys. (1)

J. C. Inkson, “Many-body effects at metal-semiconductor junctions. I. Surface plasmons and the electron-electron screened interaction,” J. Phys. C: Solid State Phys. 5, 2599–2610 (1972).
[CrossRef]

J. Phys.: Condens. Matter I (1)

K. H. Aharonian and D. R. Tilley, “Propagating electromagnetic modes in thin semiconductor films,” J. Phys.: Condens. Matter I, 5391–5401 (1989).
[CrossRef]

Nat. Photonics (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).

Opt. Commun. (1)

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. B (4)

P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B 12, 4032–4035 (1975).
[CrossRef]

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface-plasmon-polariton dispersion curve,” Phys. Rev. B 10, 3342–3355 (1974).
[CrossRef]

C. C. Kao and E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. Lett. 182, 539–554 (1969).

Phys. Status Solidi (RRL) (1)

G. V. Naik and A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).
[CrossRef]

Surf. Sci. Rep. (1)

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1–416 (2001).
[CrossRef]

Other (9)

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).

D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling and Applications (Cambridge University Press, 2010).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science, 2007).

N. Lebedev and R. A. SilvermanSpecial Functions and Their Applications (Dover Publication, 1972).

S. S. Bayin, Mathematical Methods in Science and Engineering (Wiley–Interscience, 2006).
[CrossRef]

L. Solymar and D. Walsh, Electrical Properties of Materials (Oxford University Press, 2004).

K. F. Brennan, Introduction to Semiconductor Devices: For Computing and Telecommunications Applications (Cambridge University Press, 2005).

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.

H. C. Casey and M. B. PanishHeterostructure Lasers, Part A: Fundamental Principles (Academic, 1978).

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Figures (8)

Fig. 1
Fig. 1

The energy band diagram of a metal and n–type semiconductor junction in equilibrium. The work functions are denoted by ϕ M ad ϕ s where ϕ M > ϕ s and χ s denotes electron affinity of the semiconductor.

Fig. 2
Fig. 2

A Schematic illusion of the Schottky Junction including the division of the junction to sections A, B, C, and D based on the carrier density distribution.

Fig. 3
Fig. 3

Plasmonic dispersion curves for a metal–semiconductor interface using our Airy function method and Frobenius series solution method. The difference ω diff between the curves is shown in the inset, which is below 1.0%.

Fig. 4
Fig. 4

The dispersion curves of Schottky junction for ω ps 0 = 2ω pm . The frequencies marked in the figure are normalized to bulk plasma frequency of the semiconductor. Two dispersion modes are shown where lower mode is magnified in the inset of the figure. The calculations are carried out for ɛ Hs = 11.9, ω pm = 1.36 × 1016rad/s and d1, d2 values are taken for doping concentration of 10−28m−3.

Fig. 5
Fig. 5

Dispersion curves for varying carrier density profiles shown in the inset. (a) variation of upper modes with slope r and (b) variation of lower modes with slope r. The frequencies marked in the figure are normalised to bulk plasma frequency of the semiconductor, ω ps 0 and d1, d2 values are normalised by c/ω ps 0.

Fig. 6
Fig. 6

Dispersion curves for different space charge layer widths by varying externally applied potential. (a) Upper modes for different d1 and d2 values as in the inset. (b) Lower modes for different d1 and d2 values as in the inset of (a) The values of d1 and d2 vary by 0.1 steps starting from 0.05 and 0.1 respectively. The frequencies marked in the figure are normalised to bulk plasma frequency of the semiconductor, ω ps 0 and d1, d2 values are normalised by c/ω ps 0.

Fig. 7
Fig. 7

Dispersion curves obtained using different metals for the Schottky junction: (a) Upper modes and (b) Lower modes. Each upper mode curve starts from the bulk plasma frequency of the particular metal. The frequencies marked in the figure are normalised to bulk plasma frequency of the semiconductor, ω ps 0.

Fig. 8
Fig. 8

Dependence of effective propagation length of the plasmon mode with varying forward biased voltage across the Schottky junction. The attenuation coefficients of Au and Ag at ω = 1.1 × 1015 rad/s are 7.953 × 105 cm−1 and 7.542 × 105 cm−1.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

q ϕ b i = q ϕ b n ( E c E F ) = q ϕ b n V T ln N c N d
d = τ 2 ɛ s ( ϕ b i V A ) q N d .
ɛ ζ ( z , ω ) = ɛ H ζ [ 1 ω p ζ 2 ω 2 ] , ζ { m , s }
n ( z ) = n b ( r z + ς ) .
ɛ s ( z , ω ) = ɛ H s r ω p s 0 2 ω 2 ( z z 0 )
region  A : z 0 , ɛ m A ( z , ω ) = 1 ω p m 2 ω 2
region  B : 0 < z d 1 ɛ s B ( z , ω ) = ɛ H s
region  C : d 1 < z d 2 ɛ s C ( z , ω ) = ɛ H s r ω p s 0 2 ω 2 ( z z 0 )
region  D : d 2 < z ɛ s D ( z , ω ) = ɛ H s [ 1 ω p s 0 2 ω 2 ]
E ν ( x , z , t ) = E ν ( z ) exp ( i k x i ω t ) , ν { x , z } ,
× × ( E x ( x , z , t ) x ^ + E z ( x , z , t ) z ^ ) = ɛ ( z , ω ) c 2 2 t 2 ( E x ( x , z , t ) x ^ + E z ( x , z , t ) z ^ )
i k E z ( z ) z 2 E x ( z ) z 2 = ω 2 ɛ ( z , ω ) c 2 E x ( z )
i k E x ( z ) z + k 2 E z ( z ) = ω 2 ɛ ( z , ω ) c 2 E z ( z )
2 E z ( z ) z 2 + ln ( ɛ ( z , ω ) ) x E z ( z ) z [ k 2 ω 2 ɛ ( z , ω ) c 2 1 ɛ ( z , ω ) 2 ɛ ( z , ω ) z 2 + ( 1 ɛ ( z , ω ) ɛ ( z , ω ) z ) 2 ] E z ( z ) = 0
E x ( z ) = i k ( E z ( z ) z + ln ( ε ( z , ω ) ) z E z ( z ) )
2 E z z 2 + ( k 0 2 ε m A k 2 ) = 0
E x ( z ) = i k E z z
E z ( z ) = E A exp ( β A z ) , β A > 0
E z ( z ) = i β A k E A exp ( β z ) , β A > 0 .
2 E z z 2 + ( k 0 2 ɛ s B k 2 ) = 0 ,
E z ( z ) = E B 1 cos ( β B z ) + E B 2 sin ( β B z ) , β B 0
E x ( z ) = i β B k ( E B 1 sin ( β B z ) + E B 2 cos ( β B z ) ) .
2 E z ( z ) z 2 + 1 ( z z 0 ) E z ( z ) z [ k 2 + ɛ H s r ω p s 0 2 c 2 ( z z 0 ) + 1 ( z z 0 ) 2 ] E z ( z ) = 0
E x ( z ) = i k [ E z ( z ) z + E z ( z ) ( z z 0 ) ]
2 E z ( z ) z 2 + 1 ( z z 0 ) E z ( z ) z [ ɛ H s r ω p s 0 2 c 2 ( z z 0 ) + 1 ( z z 0 ) 2 ] E z ( z ) = 0
E z ( z ) = E C 1 z z 0 A i ( z z 0 ξ ) + E C 2 z z 0 B i ( z z 0 ξ )
E x ( z ) = i k [ E C 1 ξ 2 Ai ( z z 0 ξ ) + E C 2 ξ 2 Bi ( z z 0 ξ ) ]
2 E z z 2 + ( k 0 2 ε s D k 2 ) = 0
E z ( z ) = E D exp ( β D z ) , β D > 0 ,
E x ( z ) = i β D k E D exp ( β D z ) , β D > 0 .
M 6 × 6 ( ω , k ) E 6 × 1 = 0
M 6 × 6 ( ω , k ) = [ ε m A ( ω ) ε H s 0 0 0 0 β A 0 β B 0 0 0 0 cos ( β B d 1 ) sin ( β B d 1 ) E z 1 c ( d 1 ) E z 2 c ( d 1 ) 0 0 β B sin ( β B d 1 ) β B cos ( β B d 1 ) E x 1 c ( d 1 ) E x 2 c ( d 1 ) 0 0 0 0 E z 1 c ( d 2 ) E z 2 c ( d 2 ) exp ( β D d 2 ) 0 0 0 E x 1 c ( d 2 ) E x 2 c ( d 2 ) β D exp ( β D d 2 ) ]
E z 1 c ( z ) = 1 z z 0 A i ( z z 0 ξ ) , E z 2 c ( z ) = 1 z z 0 B i ( z z 0 ξ ) , E x 1 c ( z ) = i k [ 1 ξ 2 Ai ( z z 0 ξ ) ] , E x 2 c ( z ) = i k [ 1 ξ 2 Bi ( z z 0 ξ ) ] .
Dispersion   Relation   of   the   Schottky   Junction : Det( M 6 × 6 ( ω , k )) = 0
E z ( z ) = ( E C F 1 + E C F 2 ln | z z 0 | ) n = 0 u n ( z z 0 ) n 1 + E C F 2 n = 0 v n ( z z 0 ) n 1
u 0 = 0 , u 1 = 0 , u 2 = k 2 2 , u n = k 2 u n 2 + ε H s r ω p s 0 2 c 2 u n 3 n ( n 2 ) , n 3 v 0 = 1 , v 1 = 0 , v 2 = k 2 4 , v n = 2 ( n 1 ) u n + k 2 v n 2 + ε H s r ω p s 0 2 c 2 v n 3 n ( n 2 ) , n 3.
E x ( z ) = i k ( E C F 1 + E C F 2 ln | z z 0 | + E C F 2 z z 0 ) n = 1 n u n ( z z 0 ) n 2 + i k E C F 2 n = 1 n v n ( z z 0 ) n 2 .
ω s p , U ' = ω p m 2 + ε H s ω p s 0 2 ( 1 + ε H s ) ω s p , L ' = ω p m 2 ( 1 + ε H s )
L spp = 1 2 k img = 1 α m g ( n )

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