## Dispersion relation for surface plasmon polaritons on a Schottky junction |

Optics Express, Vol. 20, Issue 7, pp. 7151-7164 (2012)

http://dx.doi.org/10.1364/OE.20.007151

Acrobat PDF (1374 KB)

### 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

## 1. Introduction

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

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]

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]

11. D. Y. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express **19**, 12524–12531 (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]

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]

## 2. Characterization of the surface plasmon field at the Schottky junction

### 2.1. Description of the metal-semiconductor junction

*E*

*is the Fermi energy level and*

_{f}*E*

*,*

_{c}*E*

*are the conduction and valence band edges, respectively. The built in potential across the depletion layer*

_{v}*ϕ*

*is given by, where*

_{bi}*ϕ*

*,*

_{bn}*N*

*,*

_{c}*N*

*and*

_{d}*V*

*denote Schottky barrier height, effective density of states in the conduction band, doping concentration of semiconductor and thermal voltage. We can apply full depletion approximation to find the width of the space charge region when an external voltage*

_{T}*V*

*is applied across the junction [19], Here*

_{A}*τ*is a parameter which depends on the surface characteristics of the semiconductor. Accordingly, for Si-Au and Si-Ag contacts, width of the space charge layer assumed to be in the range of 0.1 ∼ 1

*μ*

*m*for doping concentration around 10

^{28}

*m*

^{−3}.

### 2.2. Analytical estimation of the plasmonic field distribution

*et al.*under the approximation in which the space charge layer is represented by a piecewise linear variation [16,20

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]

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]

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]

*m*and

*s*refer to metals and semiconductors, respectively.

*ɛ*

_{H}*,*

_{ζ}*ζ*∈ {

*m,s*} is the static dielectric constant of the medium and

*ω*

_{p}*,*

_{ζ}*ζ*∈ {

*m,s*} is the plasma frequency of the medium. It is important to note that for a metal

*ɛ*

*= 1 and for a semiconductor*

_{Hm}*n*

*is a reference cross-sectional carrier density to be defined later and*

_{b}*q*and

*m*

^{*}are the magnitude of electron charge and the effective mass of charge carrier, respectively and

*n*(

*z*) is the position dependent free charge carrier density. Depending on the value of

*n*(

*z*), the junction can be divided into four region as shown in Fig. 2. In the regions

**A**,

**B**and

**D**,

*n*(

*z*) is spatially constant. In the region

**C**,

*n*(

*z*) is approximated by a linear variation as given below [20

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]

*r*and

*ς*denote the slope and vertical intersect of the linear function. Substitution of Eq. (3) to Eq. (2) enables us to write the dielectric constant in the region

**C**as where

**A**,

**B**,

**C**and

**D**sections of Fig. 2 as follows: Even though there is a possibility to have both TE and TM SPP modes to propagate along the junction in

*x*axis of the Fig. 2, application of the Maxwell’s equations with proper boundary conditions to the bounded mode at the junction shows that only the TM modes are supported [4]. For such a TM mode, only the components

*E*

*,*

_{x}*H*

*and*

_{y}*E*

*are non-zero. Noting that these components propagate along the +*

_{z}*x*direction, we can write where

*k*is the longitudinal wave number along the direction x and

*E*

*(*

_{ν}*z*) is the amplitude of

*E*

*(*

_{ν}*x,z,t*) where

*ν*∈ {

*x,z*}. For these TM modes, the Maxwell’s equations give the following set of partial differential equations for the electric field components if the external charges and currents are absent in the medium, where

**x̂**and

**ẑ**are the unit vectors in the x, z directions, respectively and

*ɛ*(

*z,*

*ω*) is the local permittivity of the medium as specified in Eq. (4). It is much more useful if these equations are written using their component form. Collecting and matching terms in

**x̂**and

**ẑ**directions gives To solve these simultaneous partial differential equations, it is better to eliminate one dependent variable. We choose to eliminate the

*E*

*(*

_{x}*z*) from the Eq. (5a). This can be done by taking the derivative of Eq. (5b) with respect to

*z*and substitution of the resulting equation in Eq. (5a). The resulting second order partial differential equation reads Once found, this solution can be used to calculate the

*E*

*(*

_{x}*z*) component from the following relation obtained using Eq. (5a) and Eq. (5b). This equation assumes different forms depending on the permittivity of regions

**A**,

**B**,

**C**and

**D**given in Eq. (4). Because the solution and the format of the equation change in each of these sections, we solve the above equation for each section separately below. It is important to notice that we need these solutions at each section because the dispersion relation applicable to Schottky junction is obtained by ensuring the continuity conditions of the field components as described later.

*E*

*(*

_{x}*z*) using the Eq. (7) where the second term goes to zero because of the permittivity is constant in the region

**A**. The solution of the Eq. (8) is well-known and can be written directly as where

*E*

*is a constant and the longitudinal wave vector is given by*

_{A}*β*

_{A}*z*),

*β*

*∈ *

_{A}_{>0}solution because it diverges in the region

*z*≤ 0. Substitution of Eq. (10) to Eq. (9) gives

**B**, and thus the permittivity is constant and given by the Eq. (4b). Therefore, as in the region

**A**, we get the following simplified version of Eq. (6), which has the following general solution where

*E*

_{B}_{1}and

*E*

_{B}_{2}are constants. We keep the most general solution here because the boundary conditions are finite and hence the general solution never diverges within the region of interest. Substitution of Eq. (11) to Eq. (9) gives

*d*

_{1}<

*z*≤

*d*

_{2}, the inequality

*E*

_{C}_{1}and

*E*

_{C}_{2}are constants, Ai′(

*z*) is the first derivative of the Airy function with respect to the variable

*z*[22], Bi′(

*z*) is the first derivative of the Bairy function with respect to the variable z and

**A**the solution of the Eq. (15) is well-known and can be written directly as where

*E*

*is a constant and the longitudinal wave vector is given by*

_{D}*β*

_{D}*z*),

*β*

*∈ *

_{D}_{>0}solution because it diverges in the region

*z*>

*d*

_{2}. Substitution of Eq. (16) to Eq. (9) gives

### 2.3. Surface plasmon dispersion relation

*ω*of the SPP field to its wave vector magnitude,

*k*. The allowed values of this relationship determine the SPP modes that are supported on the Schottky junction. The dispersion relation can be found by applying self-consistent boundary conditions at

*z*= 0,

*z*=

*d*

_{1}and

*z*=

*d*

_{2}to ensure the continuity of electromagnetic field components

*E*

*and*

_{x}*H*

*, as required by Maxwell’s equations. As in the Fig. 2, the carrier density slope and vertical intersect in region*

_{y}**C**can be derived from

*d*

_{1}and

*d*

_{2}where

*r*= 1/(

*d*

_{2}–

*d*

_{1}) and

*ς*= −

*d*

_{1}/(

*d*

_{2}−

*d*

_{1}). The application of the boundary conditions leads to where

**M**

_{6×6}(

*ω*

*,k*) is a 6 × 6 square–matrix,

**E**

_{6×1}= [

*E*

*,*

_{A}*E*

_{B1},

*E*

_{B2},

*E*

_{C1},

*E*

_{C2},

*E*

*]*

_{D}*is a 6 × 1 column–matrix containing undermined coefficients of the solutions given in Eqs. (10), (11), (14) and (16). The matrix elements,*

^{T}*z*), derivative of Airy function Ai′(

*z*), Airy function of the second kind, Bi(

*z*) and its derivative Bi′(

*z*) as follows:

**E**

_{6×1}for permissible {

*ω*

*,k*} values, the determinant of the matrix,

**M**

_{6×6}(

*ω*

*,k*) must be equal to zero. This condition results in a secular equation relating

*ω*and

*k*; which is the dispersion relation for the Schottky junction: The compact form of this dispersion relation enable us to carry out normally computationally expensive studies involving the Schottky junction with much ease. For example, the solutions of the dispersion relation can be used to study plasmonic pulses propagating along the Schottky junction by integrating the corresponding wave equation. Another use of such a solution is to use the values as a initial guess for a high accuracy numerical polishing routine with a nonlinear carrier density profile in the vicinity of the Schottky junction interface.

**C**. The solutions we previously obtained for regions

**A**,

**B**and

**D**are exact and hence do not need any numerical refinement. Based on the work of Frobenius [23

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

*E*

_{CF}_{1}and

*E*

_{CF}_{2}are constants to be determined using boundary conditions. Substituting Eq. (17) to Eq. (12) and equating coefficients of the (

*z*−

*z*

_{0})

*terms to 0, we arrive at following recurve relations for the coefficients of the infinite series:*

^{n}**10**, 3342–3355 (1974). [CrossRef]

*ω*

*, between two solutions is shown in the inset of Fig. 3, which is below 1.0%. This confirms the high accuracy of the analytical dispersion relation and we use it exclusively in the analysis below.*

_{diff}## 3. Behaviour of SPP modes on the Schottky junction

*ω*

*. The high frequency mode is more similar to the SPP dispersion of a bi-metallic system, where this mode only supports in between bulk plasma frequencies of the metal and the semiconductor. The mode starts from plasma frequency of the metal rising parallel to Au light line and reaches the asymptotic value*

_{sp,L}*ω*

*thereafter showing a slight negative slope. The limiting conditions for the two modes can be defined when*

_{sp,U}*d*

_{2}→ 0 and

*d*

_{1}→ ∞. When

*d*

_{2}→ 0, the space charge layer vanishes and creates a bi-metallic interface. Therefore the asymptotic value of upper mode (high frequency mode) is always

*ω*

*≤*

_{sp,U}*ω*′

*When*

_{sp,U}*d*

_{1}→ ∞, the metal-dielectric interface dominates. Hence asymptotic value of lower mode (low frequency mode) is always

*ω*

*≤*

_{sp,L}*ω*′

*. Here*

_{sp,L}*ω*′

*and*

_{sp,U}*ω*′

*are defined as [25*

_{sp,L}25. P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B **12**, 4032–4035 (1975). [CrossRef]

*d*

_{1}and

*d*

_{2}values of the junction (cf. Fig. 2). As in Fig. 5(a), the negative slope of the upper mode can be controlled by changing

*d*

_{1}and

*d*

_{2}. The negative slope increases when

*d*

_{1}is increased while

*d*

_{2}is decreased. In contrast, the upper level of low frequency mode decreases when

*d*

_{1}is increased and

*d*

_{2}is decreased (see Fig. 5(b)). Also an externally applied potential can be used to control the width of the space charge layer according to Eq. (1), which is considered to be the major advantage of our model. By increasing the reverse biased potential, values of

*d*

_{1}and

*d*

_{2}can be shifted by similar amount and accordingly the maximum level of two modes can be shifted along the frequency axis (see Figs. 6(a) and 6(b)). Moreover, the externally applied potential can cause to bend the energy band at the semiconductor surface downward or upward creating accumulation or inversion layer depending on forward or reverse biased conditions, respectively. The dispersion characteristics for these conditions can also be found using a similar approach where accumulation layer reduces our model into three layers while inversion layer replaces it with five layers. The plasma frequency of metal decides frequency margin values for the upper mode and also controls the maximum level of the lower mode as illustrated in Figs. 7(a) and 7(b). Altogether, it is clear that the space charge layer formed in a Schottky junction plays a significant role on its SPP dispersion characteristics with more controllability compared to the SPP modes of conventional metal-dielectric interfaces.

*ɛ*

*of the metal is complex. The imaginary part of the dielectric constant is responsible for the losses seen by the SPPs, makes the SPP wave vector complex valued. The imaginary part of this SPP wave vector*

_{mA}*k*

*and associated mode propagation length*

_{img}*L*

*can be found using attenuation coefficient of the metal*

_{spp}*α*

*and optical gain coefficient of the semiconductor*

_{m}*g*(

*n*), where

*n*denotes the minority carrier concentration: Here

*g*(

*n*) =

*g*

*(*

_{o}*n*−

*n*

*) where*

_{t}*g*

*and*

_{o}*n*

*are the gain constant and the transparency density. These two constants are calculated numerically by solving the one-electron model which defines the optical gain with band to band transition of the semiconductor [11*

_{t}11. D. Y. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express **19**, 12524–12531 (2011). [CrossRef] [PubMed]

*Ga*

_{0.47}

*In*

_{0.53}

*As*for the semiconductor material for easy comparison of the results obtained with reference [11

11. D. Y. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express **19**, 12524–12531 (2011). [CrossRef] [PubMed]

*ω*= 1.1 × 10

^{15}

*rad/s*, the attenuation coefficients of Au and Ag are 7.953 × 10

^{5}

*cm*

^{−1}and 7.542 × 10

^{5}

*cm*

^{−1}while the constants

*g*

*and*

_{o}*n*

*are found to be 8.76 × 10*

_{t}^{−16}

*cm*

^{2}and 3.7 × 10

^{16}

*cm*

^{−3}. So when the forward biased voltage across the junction increases, the width of space charge layer decreases (as in Eq. (1)) and under certain conditions set by the energy bands, it causes an inversion layer to be formed. It means the minority carrier concentration near the contact exceeds the majority carrier concentration leading to reduction of losses seen by the SPP field (Ref. [11

**19**, 12524–12531 (2011). [CrossRef] [PubMed]

## 4. Conclusion

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]

**19**, 12524–12531 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. Premaratne and G. P. Agrawal, |

2. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics |

3. | D. Sarid and W. Challener, |

4. | S. A. Maier, |

5. | G. V. Naik and A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) |

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 |

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. |

10. | D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. |

11. | D. Y. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express |

12. | I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex– |

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 |

14. | M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. |

15. | A. Yariv and R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. |

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. |

18. | L. Solymar and D. Walsh, |

19. | K. F. Brennan, |

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 |

21. | C. C. Kao and E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B |

22. | N. Lebedev and R. A. Silverman |

23. | S. S. Bayin, |

24. | E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. Lett. |

25. | P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B |

26. | H. C. Casey and M. B. Panish |

27. | D. Y. Fedyanin, “Toward an electrically pumped spaser,” Opt. Lett. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 18, 2012

Revised Manuscript: February 28, 2012

Manuscript Accepted: March 5, 2012

Published: March 13, 2012

**Citation**

Thamani Wijesinghe and Malin Premaratne, "Dispersion relation for surface plasmon polaritons on a Schottky junction," Opt. Express **20**, 7151-7164 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7151

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### References

- M. Premaratne, G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).
- W. L. Barnes, A. Dereux, T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).
- D. Sarid, 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).
- G. V. Naik, A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010). [CrossRef]
- R. T. Holm, E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).
- A. V. Krasavin, A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18, 11791–11799 (2010). [CrossRef] [PubMed]
- K. H. Aharonian, D. R. Tilley, “Propagating electromagnetic modes in thin semiconductor films,” J. Phys.: Condens. Matter I, 5391–5401 (1989). [CrossRef]
- S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006). [CrossRef]
- D. Handapangoda, I. D. Rukhlenko, M. Premaratne, C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35, 4190–4192 (2010). [CrossRef] [PubMed]
- D. Y. Fedyanin, A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011). [CrossRef] [PubMed]
- I. B. Udagedara, I. D. Rukhlenko, 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, 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]
- M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41, 1–416 (2001). [CrossRef]
- A. Yariv, R. C. C. Leite, “Dielectric waveguide mode of light propagation in p-n junctions,” Appl. Phys. Lett. 2, 55–57 (1963). [CrossRef]
- R. F. Wallis, J. J. Brion, E. Burstein, A. Hartstein, “Theory of surface polaritons in semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors, (Elsevier1972) 1448–1453.
- 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]
- L. Solymar, 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).
- S. L. Cunningham, A. A. Maradudin, 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, E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976). [CrossRef]
- N. Lebedev, R. A. SilvermanSpecial Functions and Their Applications (Dover Publication, 1972).
- S. S. Bayin, Mathematical Methods in Science and Engineering (Wiley–Interscience, 2006). [CrossRef]
- E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. Lett. 182, 539–554 (1969).
- P. Halevi, “Electromagnetic wave propagation at the interface between two conductors,” Phys. Rev. B 12, 4032–4035 (1975). [CrossRef]
- H. C. Casey, M. B. PanishHeterostructure Lasers, Part A: Fundamental Principles (Academic, 1978).
- D. Y. Fedyanin, “Toward an electrically pumped spaser,” Opt. Lett. 37, 404–406 (2012). [CrossRef] [PubMed]

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