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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7151–7164
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Dispersion relation for surface plasmon polaritons on a Schottky junction

Thamani Wijesinghe and Malin Premaratne  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7151-7164 (2012)
http://dx.doi.org/10.1364/OE.20.007151


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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

Surface plasmon polartions (SPPs) are hybrid transverse magnetic optical waves that are bounded and propagate along metal–dielectric interfaces and oscillate collectively with free electrons in the metal [1

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

]. The ability of metal–dielectric interfaces and metal nano structures to localize and manipulate SPP modes at nano scale have opened up intensive research avenues in areas such as photonics and integrated optics, biomedical imaging, molecular sensing and spectroscopy [2

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

4

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

]. The new capabilities of SPP are being explored using various structures ranging from simple metal–dielectric interfaces to complex nano structures consisting of semiconductor-dielectric interfaces, enabling inter-operability and integration with existing semiconductor technology. The enhanced performance of plasmonic structures are achieved by modifying a simple metal–dielectric interface into multi layered structures or by modifying the metal geometry itself, which include metal grooves, ridges or changing the properties of the medium [5

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

8

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

].

SPP has been widely studied for many decades to understand the physics behind the phenomena and to extend the knowledge towards device applications. But it was only recently the interest grew towards gain assisted SPP amplification to overcome high propagation losses experienced in plasmonic structures [9

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

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]

]. Among plethora of technologies available for providing gain for SPPs, one promising approach is the replacement of passive dielectric material by a semiconductor and use electrical pumping to achieve gain [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]

]. So, there is great need to have a clear understanding of the behaviour of SPPs on semiconductor– metal junctions. Even though there are a large number of papers analysing the SPP behaviour on metal–dielectric interfaces [1

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

, 4

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

], the associated results cannot be directly ported to semiconductor–metal interfaces. This is mainly due the reason that the conventional analysis of surface plasmon modes on metal–dielectric interfaces rely on knowing the exact value of permittivity at either side of the interface [1

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

]. A pivotal assumption of such an analysis is that the formation of the metal–dielectric interface does not change the material properties and the materials forming the interface have identical permittivities before and after the formation of the interface [4

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

].

However, this assumption breaks down if an interface is made between a metal and a semiconductor, which is commonly known as a Schottky junction. The properties of Schottky junction interface predominantly depends on the size and shape of the interface, doping density of impurities in the semiconductor medium and type and quality of the metal [14

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

]. Interestingly, some of these interface properties of the Schottky junction can be changed by applying an external potential difference across the junction. It is this potent property of Schottky junctions makes it very attractive for modern plasmonic applications. Later we show that under certain conditions, such an interface can sustain one or two surface plasmon polariton (SPP) mode/s. 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 [1

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

, 4

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

]. The analysis demands special care because, unlike in a metal–dielectric interface, a region of nonuniform free charge carrier density profile develops surrounding a metal–semiconductor interface. The width of this region is in the range of hundreds of nanometers. This substantial variations in the free charge density in the vicinity of the metal–semiconductor interface influence the permittivity seen by the SPP wave propagating along the interface. Moreover, such charge variations may give rise to energy confinement at the junction and severely constrain the features of electromagnetic waves that can collectively oscillate with the free electron plasma [15

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

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]

].

In this paper, we carry out a detailed analysis to derive an analytical expression for the dispersion relation for a Schottky junction and show that a very good agreement exists between analytical and numerical dispersion relations. Our analysis takes into account the space charge layer formed due to the charge distribution across the Schottky junction and resulting new boundary conditions. Unlike in a dielectric–metal interface where a single SPP mode is supported, we show that the solution of the new analytical dispersion relation correctly predicts two SPP modes of low energy and high energy, which have already observed in bi-metallic systems experimentally. Moreover, SPP mode behaviour under various charge density profiles, varying external potential and different metals (e.g. Au, Ag, Al and Cu) are presented. This paper is organized as follows. In Section 2, we outline an analytical description of Schottky junction and derive the characteristic equations for the existence of plasmon modes in a model four layered system introducing new boundary conditions for space charge layer. Section 3 describes the numerical solution for the SPP dispersion relation of the junction. The effect of system parameters on dispersion characteristics are also examined. Section 4 concludes this paper after summarising the main result and emphasizing the importance of this study.

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

The metal–semiconductor contact creates a hetero-interface (widely called a Schottky junction) and consists of a space charge region with a varying free charge carrier density profile formed according to the band structure of the semiconductor and the metal [18

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

, 19

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

]. In electronics engineering, junctions play an important role because of their ability to rectify currents, which is a direct result of having a space charge layer on the semiconductor side of the interface. In this section, we briefly describe the junction properties and associated charge distribution before deriving an analytical expression for the dispersion relation for the SPP modes propagating along the vicinity of the Schottky junction interface.

2.1. Description of the metal-semiconductor junction

To introduce the notation used and the major parameters considered in our model, this section briefly defines the intrinsic and extrinsic parameters of the Schottky junction. The band structure for a metal and n-Semiconductor contact is shown in Fig. 1. Here Ef is the Fermi energy level and Ec, Ev are the conduction and valence band edges, respectively. The built in potential across the depletion layer ϕbi is given by,
qϕbi=qϕbn(EcEF)=qϕbnVTlnNcNd
where ϕbn, Nc, Nd and VT 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 VA is applied across the junction [19

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

],
d=τ2ɛs(ϕbiVA)qNd.
(1)
Here 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 μm for doping concentration around 1028m−3.

Fig. 1 The energy band diagram of a metal and n–type semiconductor junction in equilibrium. The work functions are denoted by ϕM ad ϕs where ϕM > ϕs and χs denotes electron affinity of the semiconductor.

2.2. Analytical estimation of the plasmonic field distribution

The dispersion characteristics of plasmon modes at a semiconductor–vacuum interface has been studied by Wallis et al. under the approximation in which the space charge layer is represented by a piecewise linear variation [16

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.

,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]

]. Even though several attempts have been made to replace this linear variation assumption by much smoother transition (e.g. for exponential profile see [21

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]

].), no substantial improvements can be gained but lead to complex set of equations that are hard to solve [21

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]

]. As our prime aim of this paper is to obtain an analytical result for the dispersion relation, without loss of generality, we assume a piecewise linear variation of the charge profile in the subsequent analysis [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]

]. However, it must be emphasised that any complex profile can always be approximated by a reasonably chosen piecewise linear variation and thus the fundamental does not allude a limitation of our analysis.

In this context, the Schottky Junction is represented by four layers as shown in Fig. 2. The structure consisting a semi-infinite semiconductor film in contact with a semi-infinite metal. The dielectric function for both media is defined by the lossless Drude model in the absence of interband transitions as given below,
ɛζ(z,ω)=ɛHζ[1ωpζ2ω2],ζ{m,s}
(2)
where the indexes m and s refer to metals and semiconductors, respectively. ɛHζ, ζ ∈ {m,s} is the static dielectric constant of the medium and ωpζ, ζ ∈ {m,s} is the plasma frequency of the medium. It is important to note that for a metal ɛHm = 1 and for a semiconductor ωps2(z)=4πq2n(z)/(ɛHsm*)ωps02n(z)/nb, where nb is a reference cross-sectional carrier density to be defined later and ωps02=4πq2nb/(ɛHsm*). Here 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]

].
n(z)=nb(rz+ς).
(3)
Here 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
ɛs(z,ω)=ɛHsrωps02ω2(zz0)
where z0=(ω2/ωps02ς)/r. With the definition of these new variables, we can summarise the permittivity of the A, B, C and D sections of Fig. 2 as follows:
region A:z0,ɛmA(z,ω)=1ωpm2ω2
(4a)
region B:0<zd1ɛsB(z,ω)=ɛHs
(4b)
region C:d1<zd2ɛsC(z,ω)=ɛHsrωps02ω2(zz0)
(4c)
region D:d2<zɛsD(z,ω)=ɛHs[1ωps02ω2]
(4d)
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

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

]. For such a TM mode, only the components Ex, Hy and Ez are non-zero. Noting that these components propagate along the +x direction, we can write
Eν(x,z,t)=Eν(z)exp(ikxiωt),ν{x,z},
where k is the longitudinal wave number along the direction x and Eν (z) is the amplitude of Eν (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,
××(Ex(x,z,t)x^+Ez(x,z,t)z^)=ɛ(z,ω)c22t2(Ex(x,z,t)x^+Ez(x,z,t)z^)
where and 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 and directions gives
ikEz(z)z2Ex(z)z2=ω2ɛ(z,ω)c2Ex(z)
(5a)
ikEx(z)z+k2Ez(z)=ω2ɛ(z,ω)c2Ez(z)
(5b)
To solve these simultaneous partial differential equations, it is better to eliminate one dependent variable. We choose to eliminate the Ex(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
2Ez(z)z2+ln(ɛ(z,ω))xEz(z)z[k2ω2ɛ(z,ω)c21ɛ(z,ω)2ɛ(z,ω)z2+(1ɛ(z,ω)ɛ(z,ω)z)2]Ez(z)=0
(6)
Once found, this solution can be used to calculate the Ex(z) component from the following relation obtained using Eq. (5a) and Eq. (5b).
Ex(z)=ik(Ez(z)z+ln(ε(z,ω))zEz(z))
(7)
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.

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.

(i) Solution of Eq. (6) in region A where z ≤ 0 :

Here the dielectric function is spatially constant (see Eq. (4a)) and hence Eq. (6) assumes the following simple form:
2Ezz2+(k02εmAk2)=0
(8)
Once found this solution, we can get Ex(z) using the Eq. (7) where the second term goes to zero because of the permittivity is constant in the region A.
Ex(z)=ikEzz
(9)
The solution of the Eq. (8) is well-known and can be written directly as
Ez(z)=EAexp(βAz),βA>0
(10)
where EA is a constant and the longitudinal wave vector is given by βA2=k2k02ɛmA. We have ignored the exp(−βAz), βA ∈ 𝕉>0 solution because it diverges in the region z ≤ 0. Substitution of Eq. (10) to Eq. (9) gives
Ez(z)=iβAkEAexp(βz),βA>0.

(ii) Solution of Eq. (6) in region B where 0 < zd1 :

The presence of the electrical field in the Schottky junction has completely depleted the free carriers in the region 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),
2Ezz2+(k02ɛsBk2)=0,
which has the following general solution
Ez(z)=EB1cos(βBz)+EB2sin(βBz),βB0
(11)
where βB2=k2k02ɛsB and EB1 and EB2 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
Ex(z)=iβBk(EB1sin(βBz)+EB2cos(βBz)).

(iii) Solution of Eq. (6) in region C where d1 < zd2 :

The carrier density is assumed to vary linearly in this section as shown in Fig. 2. This carrier density dependency translates to the variation of the permittivity as shown in Eq. (4c). Substitution of the permittivity variation to Eqs. (6) and (7) result in the following equations:
2Ez(z)z2+1(zz0)Ez(z)z[k2+ɛHsrωps02c2(zz0)+1(zz0)2]Ez(z)=0
(12)
Ex(z)=ik[Ez(z)z+Ez(z)(zz0)]
(13)
Noting that when d1 < zd2, the inequality ɛHsrωps02c2(zz0)3+1k2(zz0)2, we get
2Ez(z)z2+1(zz0)Ez(z)z[ɛHsrωps02c2(zz0)+1(zz0)2]Ez(z)=0
This equation has the following analytical solution
Ez(z)=EC1zz0Ai(zz0ξ)+EC2zz0Bi(zz0ξ)
(14)
where EC1 and EC2 are constants, Ai′(z) is the first derivative of the Airy function with respect to the variable z [22

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

], Bi′(z) is the first derivative of the Bairy function with respect to the variable z and ξ3=c2/εHsrωps02. Substitution of Eq. (14) to Eq. (13) gives,
Ex(z)=ik[EC1ξ2Ai(zz0ξ)+EC2ξ2Bi(zz0ξ)]

(iv) Solution of Eq. (6) in region D where z > d2 :

Here the dielectric function is spatially constant (see Eq. (4d)) and hence Eq. (6) assumes the following simple form:
2Ezz2+(k02εsDk2)=0
(15)
As in the region A the solution of the Eq. (15) is well-known and can be written directly as
Ez(z)=EDexp(βDz),βD>0,
(16)
where ED is a constant and the longitudinal wave vector is given by βD2=k2k02εsD. We have ignored the exp(βDz), βD ∈ 𝕉>0 solution because it diverges in the region z > d2. Substitution of Eq. (16) to Eq. (9) gives
Ex(z)=iβDkEDexp(βDz),βD>0.

2.3. Surface plasmon dispersion relation

The dispersion relation for the Schottky junction relates the angular frequency, 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 = d1 and z = d2 to ensure the continuity of electromagnetic field components Ex and Hy, as required by Maxwell’s equations. As in the Fig. 2, the carrier density slope and vertical intersect in region C can be derived from d1 and d2 where r = 1/(d2d1) and ς = −d1/(d2d1). The application of the boundary conditions leads to
M6×6(ω,k)E6×1=0
where M6×6(ω,k) is a 6 × 6 square–matrix,
M6×6(ω,k)=[εmA(ω)εHs0000βA0βB0000cos(βBd1)sin(βBd1)Ez1c(d1)Ez2c(d1)00βBsin(βBd1)βBcos(βBd1)Ex1c(d1)Ex2c(d1)0000Ez1c(d2)Ez2c(d2)exp(βDd2)000Ex1c(d2)Ex2c(d2)βDexp(βDd2)]
and E6×1 = [EA, EB1, EB2, EC1, EC2, ED]T is a 6 × 1 column–matrix containing undermined coefficients of the solutions given in Eqs. (10), (11), (14) and (16). The matrix elements, Ez1c(z), Ez2c(z), Ex1c(z) and Ex2c(z) are defined using the Airy function, Ai(z), derivative of Airy function Ai′(z), Airy function of the second kind, Bi(z) and its derivative Bi′(z) as follows:
Ez1c(z)=1zz0Ai(zz0ξ),Ez2c(z)=1zz0Bi(zz0ξ),Ex1c(z)=ik[1ξ2Ai(zz0ξ)],Ex2c(z)=ik[1ξ2Bi(zz0ξ)].

In order to obtain a non trivial solution for the coefficient matrix E6×1 for permissible {ω,k} values, the determinant of the matrix, M6×6(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:
Dispersion Relation of the Schottky Junction:Det(M6×6(ω,k))=0
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.

It is instructive to look at the accuracy of this dispersion relation against a detailed numerical solution constructed using an infinite series for the region 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]

], we assume that the Eq. (12) has an infinite series in the following form
Ez(z)=(ECF1+ECF2ln|zz0|)n=0un(zz0)n1+ECF2n=0vn(zz0)n1
(17)
where ECF1 and ECF2 are constants to be determined using boundary conditions. Substituting Eq. (17) to Eq. (12) and equating coefficients of the (zz0)n terms to 0, we arrive at following recurve relations for the coefficients of the infinite series:
u0=0,u1=0,u2=k22,un=k2un2+εHsrωps02c2un3n(n2),n3v0=1,v1=0,v2=k24,vn=2(n1)un+k2vn2+εHsrωps02c2vn3n(n2),n3.
Substitution of Eq. (17) to Eq. (13) gives,
Ex(z)=ik(ECF1+ECF2ln|zz0|+ECF2zz0)n=1nun(zz0)n2+ikECF2n=1nvn(zz0)n2.

Figure 3 shows the plasmonic dispersion curves calculated using our Airy function based method and the Frobenius series solution above. We obtained the same result for semiconductor–vacuum interface given in Ref. [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]

] based on the Frobenius solution. The result in Fig. 3 is obtained for the metal-semiconductor interface for easy comparison. The difference, ωdiff, 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.

Fig. 3 Plasmonic dispersion curves for a metal–semiconductor interface using our Airy function method and Frobenius series solution method. The difference ωdiff between the curves is shown in the inset, which is below 1.0%.

3. Behaviour of SPP modes on the Schottky junction

To gain some insight into the types of modes supported by the Schottky junction depicted in Fig. 2, a set of interesting scenarios are considered in this section. Unless otherwise stated, we assume that the Schottky junction is made by interfacing Gold (Au) to doped Silicon (n–Si) (see Fig. 1). It is known that SPP propagation in multi-layer systems support multiple modes depending on layer widths [24

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

]. Here we noted two surface plasmon modes for the Schottky junction at high frequency and low frequency regions. The two modes are plotted in Fig. 4 where the low frequency mode is magnified in the inset. The lower mode behaves similar to the SPP dispersion of metal-dielectric system where it stays right to the bulk Si light line and level off at asymptotic value ωsp,L. 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 ωsp,U thereafter showing a slight negative slope. The limiting conditions for the two modes can be defined when d2 → 0 and d1 → ∞. When d2 → 0, the space charge layer vanishes and creates a bi-metallic interface. Therefore the asymptotic value of upper mode (high frequency mode) is always ωsp,Uωsp,U When d1 → ∞, the metal-dielectric interface dominates. Hence asymptotic value of lower mode (low frequency mode) is always ωsp,Lωsp,L. Here ωsp,U and ωsp,L are defined as [25

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

],
ωsp,U'=ωpm2+εHsωps02(1+εHs)ωsp,L'=ωpm2(1+εHs)

Fig. 4 The dispersion curves of Schottky junction for ωps0 = 2ωpm. 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 ɛHs = 11.9, ωpm = 1.36 × 1016rad/s and d1, d2 values are taken for doping concentration of 10−28m−3.

A set of dispersion curves are depicted in Figs. 57 to illustrate the effect of free carrier charge density profiles, applied voltage and plasma frequency for the SPP dispersion in a Schottky junction. The carrier density profile decides the d1 and d2 values of the junction (cf. Fig. 2). As in Fig. 5(a), the negative slope of the upper mode can be controlled by changing d1 and d2. The negative slope increases when d1 is increased while d2 is decreased. In contrast, the upper level of low frequency mode decreases when d1 is increased and d2 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 d1 and d2 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.

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, ωps0 and d1, d2 values are normalised by c/ωps0.
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, ωps0 and d1, d2 values are normalised by c/ωps0.
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, ωps0.

Now consider the scenario where the dielectric function ɛmA 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 kimg and associated mode propagation length Lspp can be found using attenuation coefficient of the metal αm and optical gain coefficient of the semiconductor g(n), where n denotes the minority carrier concentration:
Lspp=12kimg=1αmg(n)
Here g(n) = go(nnt) where go and nt 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

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

, 26

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

]. Also the variation of SPP propagation length with the externally applied voltage is analysed for two different metals as depicted in Fig. 8. As is apparent, the propagation length increases with the biased voltage which has defined below the breakdown voltage of the semiconductor. Here we used n-type Ga0.47In0.53As 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]

]. At ω = 1.1 × 1015rad/s, the attenuation coefficients of Au and Ag are 7.953 × 105 cm−1 and 7.542 × 105 cm−1 while the constants go and nt are found to be 8.76 × 10−16cm2 and 3.7 × 1016cm−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

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

, 26

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

]). In other words, we can control the losses seen by the SPP field by changing the space charge layer width using an external voltage and thus effectively gaining the ability to control the mode propagation length.

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.

4. Conclusion

Acknowledgments

References and links

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]

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

  1. M. Premaratne, G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).
  2. W. L. Barnes, A. Dereux, T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. Photonics 424, 824–830 (2003).
  3. D. Sarid, 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, A. Boltasseva, “Semiconductors for Plasmonics and Metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010). [CrossRef]
  6. R. T. Holm, E. D. Palik, “Surface plasmons in semiconductor-insulator multilayers,” CRC Crit. Rev. Sol. State Mat. Sci., 397–404 (2006).
  7. A. V. Krasavin, A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18, 11791–11799 (2010). [CrossRef] [PubMed]
  8. K. H. Aharonian, 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, C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35, 4190–4192 (2010). [CrossRef] [PubMed]
  11. D. Y. Fedyanin, 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, 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, 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, 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, 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, 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, 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, E. M. Conmell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14, 2464–2479 (1976). [CrossRef]
  22. N. Lebedev, 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, 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]

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