Surface plasmon polaritons at linearly graded semiconductor interfaces D. Blazek,1,2 M. Cada,1,2,* and J. Pistora2 1
Department of Electrical and Computer Engineering, Dalhousie University, Halifax, B3J 2X4 Nova Scotia,Canada 2 Nanotechnology Center, VSB-Technical University of Ostrava, 17. listopadu 15, Ostrava-Poruba, 708 33, Czech Republic * [email protected]
Abstract: New results are reported on investigation of dispersion curves for surface plasmon polaritons (SPPs) at an inhomogenously doped semiconductor/dielectric interface whereby the dielectric is represented by the same undoped semiconductor. The doped semiconductor is described by its frequency-dependent permittivity that varies with the depth. It is shown that a transition layer (TL) with a linear change in carrier concentration supports one branch dispersion curve regardless of the TL thickness. The obtained dispersion curves reach a maximum at a finite frequency depending on the TL thickness, and subsequently asymptotically approach the zero frequency in the shortwave limit. Therefore two surface plasmon modes are supported at a given frequency: a long-wave mode with a positive group velocity and a short-wave mode with a negative group velocity. A condition of a zero group velocity can be satisfied by tuning the TL layer. It is shown that the conventional dispersion relation for SPPs at a TL with a zero thickness is an asymptotic solution, and the convergence of real dispersion curves is point-wise instead of an expected uniform convergence. ©2014 Optical Society of America OCIS codes: (240.5420) Polaritons; (250.5403) Plasmonics; (160.6000) Semiconductor materials.
References and links 1. 2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
A. D. Boardman, Electromagnetic Surface Modes (Wiley, 1982). V. M. Agranovich, “Effects of the transition layer and spatial dispersion in the spectra of surface polaritons,” in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, Modern Problems in Condensed Matter Sciences, V. M. Agranovich and D. L. Mills, eds. (1, 1982), pp.187–236. S. A. Maier, Plasmonics: Fundamentals and Application, (Springer Science and Business Media LLC, 2007). H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, 1986). P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009). M. Cada and J. Pistora, “Optical Plasmons in Semiconductors,” in Proceedings of International Symposium on Optical and Microwave Technologies, (ISMOT, 2011), pp. 23–29. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012). V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004). E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990). V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975). T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978). N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008). D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6264
15. J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, “Metamaterial electromagnetic concentrators with arbitrary geometries,” Opt. Express 17(22), 19656–19661 (2009). 16. P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009). 17. S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013). 18. 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(8), 3342–3355 (1974). 19. B. A. Kruger and J. K. S. Poon, “Optical guided waves at graded metal-dielectric interfaces,” Opt. Lett. 36(11), 2155–2157 (2011). 20. C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976). 21. M. Fox, Optical Properties of Solids, (Oxford University Press, 2001, 2010, 2012). 22. M. Cada, D. Blazek, J. Pistora, K. Postava, and P. Siroky, “Theoretical and experimental study of plasmonic effects in heavily dopes gallium arsenide and indium phosphide,” Opt. Mater. Express 5(2), 340–352 (2015). 23. A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009). 24. I. M. Mandel, I. Bendoym, Y. U. Jung, A. B. Golovin, and T. D. Crouse, “Dispersion engineering of surface plasmons,” Opt. Express 21(26), 31883–31893 (2013). 25. J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003). 26. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
1. Introduction SPPs are transverse magnetic (TM) surface electromagnetic waves that propagate typically along a metal–dielectric interface [1–3]. The model of a surface plasmon (SP) was introduced [4] after the discovery of bulk plasmons. SPs have been observed at many interface structures [5] including the semiconductor ones [6], and various applications have been proposed and investigated [7]. The structure of SPPs that propagates along an interface between metal and dielectric materials is well known. The excited SPPs waves consist of associated evanescent fields that penetrate both the dielectric and the metal. The field is concentrated mainly near the surface [8]. Due to its concentration, the SPPs are very sensitive to the surface properties such as surface roughness or inhomogeneity of the permittivity. The properties of a transition layer (TL) between the two media should thus be considered in a model. The problem of the TL in optics is quite an old one, but the attention of researchers has been concentrated on the problem of the reflection and transmission of light [2]. For the metals, a TL can be introduced by depositing a thin metal layer on a substrate. The dispersion curve is split into two branches with a frequency gap appearing between them if the plasma frequency of the deposited layer is much smaller than that of the plasma frequency of the substrate [9]. The transition layer can give rise to an SP dispersion curve that has a section with negative slopes, i.e. negative group velocities [2,9]. The predicted phenomena have been experimentally observed [10–12]. The existence of negative and zero group velocities have thus been known since at least the seventies. It is worth noting that today a great attention is devoted to the negative index metamaterials (NIM). The unusual properties of NIM are most prominently revealed at an interface between positive and negative index materials [13]. Only recently it has been realized that inhomogeneous metamaterials enable such fascinating functionalities as cloaking or wave concentrators [14,15]. Moreover, there exists an analytical solution based on confluent hypergeometric functions for gradually changing material properties (permittivity and permeability simultaneously) from the positive to the negative values if the profile is linear or exponential [16]. The phenomenon of localization and resonant enhancement of the field in the vicinity of the point where material properties change signs was confirmed [13,16]. Moreover, it has been shown that the heterostructure waveguide mode is an extraordinary entity where the Poynting vector points towards the + z direction in the material with positive
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6265
permittivity, while it points towards –z direction in the NIM [17]. The resulting energy velocity (which equals to the group velocity in the lossless structure) may be both parallel or antiparallel to the SPPs wave vector depending on the overall energy flow. This implies the regime with zero or near-zero group velocities when the positive and negative energy flow is balanced [17]. Our investigation of a TL in semiconductors was motivated by the surface effects that lead to the existence of depletion, accumulation and inversion layers [18]. In contrast to the metals, the permittivity of a semiconductor can be made a continuous function of spatial coordinates. An analytical solution does not exist in these situations, and series solutions based on the Frobenius method were used [18–20]. A complicated structure of branches with both positive and negative slopes was discovered [20]. In addition to solutions for complex permittivity profiles, we have found only one article dealing with the geometry where the permittivity is linearly graded from its positive value in the dielectric to the negative bulk value in the substrate [19]. It was concluded there that no bound modes exist in the idealized case of lossless materials, which is a different result than the one reported here. The studied structure is shown in Fig. 1. A TL with a linearly graded doping concentration is situated between the doped and undoped semi-infinite layers. All three layers are made of the same semiconductor. At a sufficiently low frequency the permitivity of the doped semiconductor becomes negative enough for an SP confined to the TL to exist.
Fig. 1. Studied surface structure; doping concentration and permittivity are functions of xcoordinate; SPPs propagate along z-direction.
Our investigation is thus devoted to the properties of SPPs at an inhomogeneously doped semiconductor/dielectric interface. We assume that the linear grading of the permittivity is caused by doping profile in the semiconductor ranging from zero free carrier concentration in the superstrate and continuously rising to the bulk concentration in the substrate. The local relation between the displacement vector, D, and the electric field vector, E, has been employed, where the permittivity depends only on the local plasma frequency. The validity of this assumption is discussed elsewhere [18]. In addition, the dielectric permittivity tensor is assumed, for simplicity, to be isotropic and characterized by a single variable ε. This is not generally the case in, for example, III/V semiconductors, however, the assumption here is satisfactory for the purposes of our analysis. The linear dependence of the charge density on the position has been chosen since it is the most fundamental problem while it is still simple and treatable analytically. More complex profiles, which subject is certainly interesting and worth pursuing, would complicate the study to the point that an analytical asymptotic solution would not be attainable and thus the obtained differential equation would possess a more complex singularity. The analysis would become burdened with a number of new parameters and the insightful and rather elegant solution would be lost What has transpired while considering a more general situation was that the interface position, i.e. the plane where the permittivity is equal to zero, depends on the frequency. For a zero frequency this point is closest to the dielectric (or even at the edge with the transition #234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6266
layer); it then moves with increasing frequency towards the lower edge of the transition layer, reaching it at the plasma frequency. If the permittivity is not a linear function of the position, the character of the interface depends on frequency, which complicates the analysis significantly and, again, the insightful analytical solution is lost. We have calculated SPPs dispersion curves for different TL. We have found only one nonradiative branch for each TL thickness. This branch follows the light-line at small wavenumbers, then reaches a maximum frequency that depends on the TL thickness, and finally decreases and approaches the zero frequency in the limit of large wavenumbers. The obtained dispersion curves converge point-wise to the usual dispersion curve as the TL thickness approaches zero. The tangential magnetic field H y and the perpendicular displacement field D x have a rounded peak, i.e. with a continuous derivative, at a position x 0 where the permittivity crosses the zero value, while the tangential displacement field D z (x 0 ) is zero. The confinement of the SPPs increases with rising wavenumber and finally the whole field is localized within the TL. One can also note that the nature of our results is, indeed, an opposite to an engineering approach that calls for a model with several free parameters that enable one to tailor the dispersion curve based on an application need. In [23,24], for example, such an engineering problem is addressed by creating a multilayered dielectric structure deposited on the conductive substrate. By modifying properties of individual layers the dispersion curve can be properly adjusted. It should be noted that for a practical use of presented phenomena, especially the frozen light mode, a more complex structure might have to be proposed, which would take into account the real characteristics of the used semiconductors; that would also extend the modal index bandwidth (MIB). A large MIB is neccesary to be able to couple to the slow light mode and also to enable the efficient energy accumulation without spatial spreading [17]. 2. The mathematical model The macroscopic permittivity of a substrate semiconductor with free carriers can be described by the well-known Drude model [21]:
εS = (ω ) ε ∞ 1 −
ω p2 ω 2
(1)
where the damping term is neglected, ε ∞ is the dispersion-less lattice permittivity of the undoped semiconductor. ω p is the plasma frequency which depends on the effective mass and the concentration of free carriers. We do not consider the damping term since semiconductors we study (e.g. GaAs, InP, Si) are completely transparent in the frequency region of interest. It should be noted that the relaxation time of electrons in many common semiconductors is of the order of 10−12 – 10−13 s at the room temperature irrespective of the electron mobility because the damping is inversely proportional to the product of mobility and effective mass. The second approximation in our model is an assumption of a zero plasma frequency ω p in undoped semiconductors due to the low intrinsic density. The intrinsic density is a product of the density of states and Boltzman exponential factor containing the bandgap energy and temperature. As a result the intrinsic plasma frequency (IPF) at room temperature lays in a wide range of 108 s−1 – 1013 s−1. It may be concluded that our model will work for most of the semiconductors where the IPF is smaller than the scattering frequency. Specifically for GaAs and InP, for example, we measured relaxation times to be less than 100 fs [22] (IPF are 108 s−1 and 109 s−1, respectively) while the operation frequencies are almost two orders of magnitude higher. Thus both the damping term and IPF can be neglected and the dispersion equation derived describes the problem adequately. Both the damping term and
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6267
IPF can be decreased rapidly by cooling the semiconductor to the low temperatures thus substantially improving the reliability of (1) even further. A variation of the permittivity with the semiconductor depth (x-axis) is assumed as:
ε S (ω ) for x ≤ − xN ω p2 x ε (ω= for − xN < x < 0 , x ) ε ∞ 1 + 2 ω x N ε ∞ for x > 0
(2)
where xN is the TL thickness. Figure 1 shows the permittivity’s depth variation for a given frequency. Note that the permittivity becomes negative at a position x 0 , which is required for an SPP to exist, depending on the plasma frequency and the depth. The dielectric, in this case the same yet undoped semiconductor, extends into x > 0. Since SPPs are TM polarised, only TM modes propagating in the z-direction are studied here. The spatial field distribution can be described by complex field amplitudes H y (x), D x (x) and D z (x), each of them multiplied by the time/propagation term exp[j (ωt - βz)]. Under these assumptions the set of Maxwell’s equations yields three independent equations, written in the frequency domain, as:
Dx =
β H y Dz = ω
1 dH y jω dx
(3)
d 2 H y 1 d ε dH y = + ( β 2 − µ0 ε 0 ε ω 2 ) H y ε dx dx dx 2
The first two equations in Eq. (3) relate magnetic and electric field amplitudes while the third one describes the development of the magnetic field once the appropriate boundary conditions for tangential components are applied. The equation can be rewritten into a form:
d 1 dH y dx ε ( x ) dx
β2 β2 ω2 − µ0ε 0ω 2 H y = − 2 = ε ( x) ε ( x) c
H y
(4)
which enables one to express the tangential component of the electric field via:
g= ω Ex ( x ) j=
1 dH y ε ( x ) dx
(5)
where g ( x ) is an intermediate function used for later derivations (see below). It is convenient to rescale spatial dimensions by a factor ω / c to normalize the expressions for further numerical analysis and interpretations. Let v be the dimensionless rescaled spatial coordinate v = (ω / c ) x and neff = ( c / ω ) β be the modal index [25]. Equation (4) assumes the form: 2 d 1 dH y dg ( v ) neff = − 1 H y = dv ee ( v ) dv dv ( v )
(6)
The solution to Eq. (6) is:
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6268
h ( −vN ) eiKv1 ( v + vN ) for v < −vN = −vN < v < 0 for H y (v) h (v) h ( 0 ) e − iKv 2 v for
Department of Electrical and Computer Engineering, Dalhousie University, Halifax, B3J 2X4 Nova Scotia,Canada 2 Nanotechnology Center, VSB-Technical University of Ostrava, 17. listopadu 15, Ostrava-Poruba, 708 33, Czech Republic * [email protected]
Abstract: New results are reported on investigation of dispersion curves for surface plasmon polaritons (SPPs) at an inhomogenously doped semiconductor/dielectric interface whereby the dielectric is represented by the same undoped semiconductor. The doped semiconductor is described by its frequency-dependent permittivity that varies with the depth. It is shown that a transition layer (TL) with a linear change in carrier concentration supports one branch dispersion curve regardless of the TL thickness. The obtained dispersion curves reach a maximum at a finite frequency depending on the TL thickness, and subsequently asymptotically approach the zero frequency in the shortwave limit. Therefore two surface plasmon modes are supported at a given frequency: a long-wave mode with a positive group velocity and a short-wave mode with a negative group velocity. A condition of a zero group velocity can be satisfied by tuning the TL layer. It is shown that the conventional dispersion relation for SPPs at a TL with a zero thickness is an asymptotic solution, and the convergence of real dispersion curves is point-wise instead of an expected uniform convergence. ©2014 Optical Society of America OCIS codes: (240.5420) Polaritons; (250.5403) Plasmonics; (160.6000) Semiconductor materials.
References and links 1. 2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
A. D. Boardman, Electromagnetic Surface Modes (Wiley, 1982). V. M. Agranovich, “Effects of the transition layer and spatial dispersion in the spectra of surface polaritons,” in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, Modern Problems in Condensed Matter Sciences, V. M. Agranovich and D. L. Mills, eds. (1, 1982), pp.187–236. S. A. Maier, Plasmonics: Fundamentals and Application, (Springer Science and Business Media LLC, 2007). H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, 1986). P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009). M. Cada and J. Pistora, “Optical Plasmons in Semiconductors,” in Proceedings of International Symposium on Optical and Microwave Technologies, (ISMOT, 2011), pp. 23–29. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012). V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004). E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990). V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975). T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978). N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008). D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6264
15. J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, “Metamaterial electromagnetic concentrators with arbitrary geometries,” Opt. Express 17(22), 19656–19661 (2009). 16. P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009). 17. S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013). 18. 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(8), 3342–3355 (1974). 19. B. A. Kruger and J. K. S. Poon, “Optical guided waves at graded metal-dielectric interfaces,” Opt. Lett. 36(11), 2155–2157 (2011). 20. C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976). 21. M. Fox, Optical Properties of Solids, (Oxford University Press, 2001, 2010, 2012). 22. M. Cada, D. Blazek, J. Pistora, K. Postava, and P. Siroky, “Theoretical and experimental study of plasmonic effects in heavily dopes gallium arsenide and indium phosphide,” Opt. Mater. Express 5(2), 340–352 (2015). 23. A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009). 24. I. M. Mandel, I. Bendoym, Y. U. Jung, A. B. Golovin, and T. D. Crouse, “Dispersion engineering of surface plasmons,” Opt. Express 21(26), 31883–31893 (2013). 25. J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003). 26. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
1. Introduction SPPs are transverse magnetic (TM) surface electromagnetic waves that propagate typically along a metal–dielectric interface [1–3]. The model of a surface plasmon (SP) was introduced [4] after the discovery of bulk plasmons. SPs have been observed at many interface structures [5] including the semiconductor ones [6], and various applications have been proposed and investigated [7]. The structure of SPPs that propagates along an interface between metal and dielectric materials is well known. The excited SPPs waves consist of associated evanescent fields that penetrate both the dielectric and the metal. The field is concentrated mainly near the surface [8]. Due to its concentration, the SPPs are very sensitive to the surface properties such as surface roughness or inhomogeneity of the permittivity. The properties of a transition layer (TL) between the two media should thus be considered in a model. The problem of the TL in optics is quite an old one, but the attention of researchers has been concentrated on the problem of the reflection and transmission of light [2]. For the metals, a TL can be introduced by depositing a thin metal layer on a substrate. The dispersion curve is split into two branches with a frequency gap appearing between them if the plasma frequency of the deposited layer is much smaller than that of the plasma frequency of the substrate [9]. The transition layer can give rise to an SP dispersion curve that has a section with negative slopes, i.e. negative group velocities [2,9]. The predicted phenomena have been experimentally observed [10–12]. The existence of negative and zero group velocities have thus been known since at least the seventies. It is worth noting that today a great attention is devoted to the negative index metamaterials (NIM). The unusual properties of NIM are most prominently revealed at an interface between positive and negative index materials [13]. Only recently it has been realized that inhomogeneous metamaterials enable such fascinating functionalities as cloaking or wave concentrators [14,15]. Moreover, there exists an analytical solution based on confluent hypergeometric functions for gradually changing material properties (permittivity and permeability simultaneously) from the positive to the negative values if the profile is linear or exponential [16]. The phenomenon of localization and resonant enhancement of the field in the vicinity of the point where material properties change signs was confirmed [13,16]. Moreover, it has been shown that the heterostructure waveguide mode is an extraordinary entity where the Poynting vector points towards the + z direction in the material with positive
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6265
permittivity, while it points towards –z direction in the NIM [17]. The resulting energy velocity (which equals to the group velocity in the lossless structure) may be both parallel or antiparallel to the SPPs wave vector depending on the overall energy flow. This implies the regime with zero or near-zero group velocities when the positive and negative energy flow is balanced [17]. Our investigation of a TL in semiconductors was motivated by the surface effects that lead to the existence of depletion, accumulation and inversion layers [18]. In contrast to the metals, the permittivity of a semiconductor can be made a continuous function of spatial coordinates. An analytical solution does not exist in these situations, and series solutions based on the Frobenius method were used [18–20]. A complicated structure of branches with both positive and negative slopes was discovered [20]. In addition to solutions for complex permittivity profiles, we have found only one article dealing with the geometry where the permittivity is linearly graded from its positive value in the dielectric to the negative bulk value in the substrate [19]. It was concluded there that no bound modes exist in the idealized case of lossless materials, which is a different result than the one reported here. The studied structure is shown in Fig. 1. A TL with a linearly graded doping concentration is situated between the doped and undoped semi-infinite layers. All three layers are made of the same semiconductor. At a sufficiently low frequency the permitivity of the doped semiconductor becomes negative enough for an SP confined to the TL to exist.
Fig. 1. Studied surface structure; doping concentration and permittivity are functions of xcoordinate; SPPs propagate along z-direction.
Our investigation is thus devoted to the properties of SPPs at an inhomogeneously doped semiconductor/dielectric interface. We assume that the linear grading of the permittivity is caused by doping profile in the semiconductor ranging from zero free carrier concentration in the superstrate and continuously rising to the bulk concentration in the substrate. The local relation between the displacement vector, D, and the electric field vector, E, has been employed, where the permittivity depends only on the local plasma frequency. The validity of this assumption is discussed elsewhere [18]. In addition, the dielectric permittivity tensor is assumed, for simplicity, to be isotropic and characterized by a single variable ε. This is not generally the case in, for example, III/V semiconductors, however, the assumption here is satisfactory for the purposes of our analysis. The linear dependence of the charge density on the position has been chosen since it is the most fundamental problem while it is still simple and treatable analytically. More complex profiles, which subject is certainly interesting and worth pursuing, would complicate the study to the point that an analytical asymptotic solution would not be attainable and thus the obtained differential equation would possess a more complex singularity. The analysis would become burdened with a number of new parameters and the insightful and rather elegant solution would be lost What has transpired while considering a more general situation was that the interface position, i.e. the plane where the permittivity is equal to zero, depends on the frequency. For a zero frequency this point is closest to the dielectric (or even at the edge with the transition #234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6266
layer); it then moves with increasing frequency towards the lower edge of the transition layer, reaching it at the plasma frequency. If the permittivity is not a linear function of the position, the character of the interface depends on frequency, which complicates the analysis significantly and, again, the insightful analytical solution is lost. We have calculated SPPs dispersion curves for different TL. We have found only one nonradiative branch for each TL thickness. This branch follows the light-line at small wavenumbers, then reaches a maximum frequency that depends on the TL thickness, and finally decreases and approaches the zero frequency in the limit of large wavenumbers. The obtained dispersion curves converge point-wise to the usual dispersion curve as the TL thickness approaches zero. The tangential magnetic field H y and the perpendicular displacement field D x have a rounded peak, i.e. with a continuous derivative, at a position x 0 where the permittivity crosses the zero value, while the tangential displacement field D z (x 0 ) is zero. The confinement of the SPPs increases with rising wavenumber and finally the whole field is localized within the TL. One can also note that the nature of our results is, indeed, an opposite to an engineering approach that calls for a model with several free parameters that enable one to tailor the dispersion curve based on an application need. In [23,24], for example, such an engineering problem is addressed by creating a multilayered dielectric structure deposited on the conductive substrate. By modifying properties of individual layers the dispersion curve can be properly adjusted. It should be noted that for a practical use of presented phenomena, especially the frozen light mode, a more complex structure might have to be proposed, which would take into account the real characteristics of the used semiconductors; that would also extend the modal index bandwidth (MIB). A large MIB is neccesary to be able to couple to the slow light mode and also to enable the efficient energy accumulation without spatial spreading [17]. 2. The mathematical model The macroscopic permittivity of a substrate semiconductor with free carriers can be described by the well-known Drude model [21]:
εS = (ω ) ε ∞ 1 −
ω p2 ω 2
(1)
where the damping term is neglected, ε ∞ is the dispersion-less lattice permittivity of the undoped semiconductor. ω p is the plasma frequency which depends on the effective mass and the concentration of free carriers. We do not consider the damping term since semiconductors we study (e.g. GaAs, InP, Si) are completely transparent in the frequency region of interest. It should be noted that the relaxation time of electrons in many common semiconductors is of the order of 10−12 – 10−13 s at the room temperature irrespective of the electron mobility because the damping is inversely proportional to the product of mobility and effective mass. The second approximation in our model is an assumption of a zero plasma frequency ω p in undoped semiconductors due to the low intrinsic density. The intrinsic density is a product of the density of states and Boltzman exponential factor containing the bandgap energy and temperature. As a result the intrinsic plasma frequency (IPF) at room temperature lays in a wide range of 108 s−1 – 1013 s−1. It may be concluded that our model will work for most of the semiconductors where the IPF is smaller than the scattering frequency. Specifically for GaAs and InP, for example, we measured relaxation times to be less than 100 fs [22] (IPF are 108 s−1 and 109 s−1, respectively) while the operation frequencies are almost two orders of magnitude higher. Thus both the damping term and IPF can be neglected and the dispersion equation derived describes the problem adequately. Both the damping term and
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6267
IPF can be decreased rapidly by cooling the semiconductor to the low temperatures thus substantially improving the reliability of (1) even further. A variation of the permittivity with the semiconductor depth (x-axis) is assumed as:
ε S (ω ) for x ≤ − xN ω p2 x ε (ω= for − xN < x < 0 , x ) ε ∞ 1 + 2 ω x N ε ∞ for x > 0
(2)
where xN is the TL thickness. Figure 1 shows the permittivity’s depth variation for a given frequency. Note that the permittivity becomes negative at a position x 0 , which is required for an SPP to exist, depending on the plasma frequency and the depth. The dielectric, in this case the same yet undoped semiconductor, extends into x > 0. Since SPPs are TM polarised, only TM modes propagating in the z-direction are studied here. The spatial field distribution can be described by complex field amplitudes H y (x), D x (x) and D z (x), each of them multiplied by the time/propagation term exp[j (ωt - βz)]. Under these assumptions the set of Maxwell’s equations yields three independent equations, written in the frequency domain, as:
Dx =
β H y Dz = ω
1 dH y jω dx
(3)
d 2 H y 1 d ε dH y = + ( β 2 − µ0 ε 0 ε ω 2 ) H y ε dx dx dx 2
The first two equations in Eq. (3) relate magnetic and electric field amplitudes while the third one describes the development of the magnetic field once the appropriate boundary conditions for tangential components are applied. The equation can be rewritten into a form:
d 1 dH y dx ε ( x ) dx
β2 β2 ω2 − µ0ε 0ω 2 H y = − 2 = ε ( x) ε ( x) c
H y
(4)
which enables one to express the tangential component of the electric field via:
g= ω Ex ( x ) j=
1 dH y ε ( x ) dx
(5)
where g ( x ) is an intermediate function used for later derivations (see below). It is convenient to rescale spatial dimensions by a factor ω / c to normalize the expressions for further numerical analysis and interpretations. Let v be the dimensionless rescaled spatial coordinate v = (ω / c ) x and neff = ( c / ω ) β be the modal index [25]. Equation (4) assumes the form: 2 d 1 dH y dg ( v ) neff = − 1 H y = dv ee ( v ) dv dv ( v )
(6)
The solution to Eq. (6) is:
#234941 - $15.00 USD © 2015 OSA
Received 19 Feb 2015; accepted 19 Feb 2015; published 27 Feb 2015 9 Mar 2015 | Vol. 23, No. 5 | DOI:10.1364/OE.23.006264 | OPTICS EXPRESS 6268
h ( −vN ) eiKv1 ( v + vN ) for v < −vN = −vN < v < 0 for H y (v) h (v) h ( 0 ) e − iKv 2 v for
