Dispersion Relation for Surface Plasmon Polaritons on a Schottky Junction

The conventional analysis of surface plasmon modes on dielectric–metal interfaces requires clearly defining the permittivity dis-continuity 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. Optimization of gain–assisted waveguid-ing in metal–dielectric nanowires, " Opt. Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres, " Phys. Surface plasmon–polariton propagation in piecewise linear chains of nanospheres: The role of optical gain and chain layout, " Opt. Effect of a charge layer on the surface-plasmon-polariton dispersion curve , " Phys. Surface plasmon dispersion of semiconductors with depletion or accumulation layers, " Phys.


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].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][3][4].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][6][7][8].
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][10][11][12][13].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].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 [1,4], 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].A pivotal assumption of such an analysis is that the formation of the metaldielectric 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].
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].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 [1,4].The analysis demands special care because, unlike in a metal-dielectric interface, a region of nonuniform free charge carrier density profile develops surrounding a metalsemiconductor 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][16][17].
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.

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 [18,19].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.

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 E f is the Fermi energy level and E c , E v are the conduction and valence band edges, respectively.The built in potential across the depletion layer φ bi is given by, where φ bn , N c , N d and V T 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 V A is applied across the junction [19], 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 10 28 m −3 .

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 [16,20].Even though several attempts have been made to replace this linear variation assumption by much smoother transition (e.g. for exponential profile see [21].), no substantial improvements can be gained but lead to complex set of equations that are hard to solve [21].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].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, where the indexes m and s refer to metals and semiconductors, respectively.ε Hζ , ζ ∈ {m, s} is the static dielectric constant of the medium and ω pζ , ζ ∈ {m, s} is the plasma frequency of the medium.It is important to note that for a metal ε Hm = 1 and for a semiconductor where n b is a reference cross-sectional carrier density to be defined later and ω 2 ps0 = 4πq 2 n b /(ε Hs m * ).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].
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 where z 0 = (ω 2 /ω 2 ps0 − ς )/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: 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 y and E z are non-zero.Noting that these components propagate along the +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 x and ẑ are the unit vectors in the x, z directions, respectively and ε(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 x 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).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.
(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: Once found this solution, we can get 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 A is a constant and the longitudinal wave vector is given by β 2 A = k 2 − k 2 0 ε mA .We have ignored the exp(−β A z), β A ∈ R >0 solution because it diverges in the region z ≤ 0. Substitution of Eq. ( 10) to Eq. ( 9) gives (ii) Solution of Eq. (6) in region B where 0 < z ≤ d 1 : 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), which has the following general solution where β 2 B = k 2 − k 2 0 ε sB and E B1 and E B2 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 (iii) Solution of Eq. ( 6) in region C where d 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: Noting that when d 1 < z ≤ d 2 , the inequality This equation has the following analytical solution where E C1 and E C2 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 ξ 3 = c 2 /ε Hs rω 2 ps0 .Substitution of Eq. ( 14) to Eq. ( 13) gives, Here the dielectric function is spatially constant (see Eq. (4d)) and hence Eq. ( 6) assumes the following simple form: As in the region A the solution of the Eq. ( 15) is well-known and can be written directly as where E D is a constant and the longitudinal wave vector is given by β 2 D = k 2 − k 2 0 ε sD .We have ignored the exp(β D z), β D ∈ R >0 solution because it diverges in the region z > d 2 .Substitution of Eq. ( 16) to Eq. ( 9) gives

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 = d 1 and z = d 2 to ensure the continuity of electromagnetic field components E x and H y , as required by Maxwell's equations.As in the where M 6×6 (ω, k) is a 6 × 6 square-matrix, and T is a 6 × 1 column-matrix containing undermined coefficients of the solutions given in Eqs. ( 10), ( 11), ( 14) and ( 16).The matrix elements, E c z1 (z), E c z2 (z), E c x1 (z) and E c x2 (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: In order to obtain a non trivial solution for the coefficient matrix 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: Dispersion Relation of 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.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], we assume that the Eq. ( 12) has an infinite series in the following form where E CF1 and E CF2 are constants to be determined using boundary conditions.Substituting Eq. ( 17) to Eq. ( 12) and equating coefficients of the (z − z 0 ) n terms to 0, we arrive at following recurve relations for the coefficients of the infinite series: Substitution of Eq. ( 17) to Eq. ( 13) gives, 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] based on the Frobenius solution.The result in Fig. 3 is obtained for the metal-semiconductor interface for easy comparison.The difference, ω di f f , 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.

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].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 ω sp,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 of the metal rising parallel to Au light line and reaches the value ω sp,U thereafter showing a slight negative slope.The limiting conditions for the two modes can be defined when 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 sp,U .When d 1 → ∞, the metal-dielectric interface dominates.Hence asymptotic value of lower mode (low frequency mode) is always ω sp,L ≤ ω sp,L .Here ω sp,U and ω sp,L are defined as [25], A set of dispersion curves are depicted in Figs.5-7 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 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.Now consider the scenario where the dielectric function ε mA 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 k img and associated mode propagation length L spp can be found using attenuation coefficient of the metal α m and optical gain coefficient of the semiconductor g(n), where n denotes the minority carrier concentration: Here g(n) = g o (n − n t ) where g o and n t 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 [11,26].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 Ga 0.47 In 0.53 As for the semiconductor material for easy comparison of the results obtained with reference [11].At 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 o and n t are found to be 8.76 × 10 −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.[11,26]).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.

Conclusion
We derived a highly accurate (i.e.within 1% of the full numerical solution) analytical dispersion relation for surface plasmon modes on a Schottky junction interface.Our derivation takes into account the appearance of the depletion region in the vicinity of the junction and the resulting nonuniform permittivity along the depletion region.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 an initial guess for a high accuracy numerical polishing routine with a nonlinear carrier density profile in the vicinity of the Schottky junction interface.Unlike a metal-dielectric interface which supports a single SPP mode, we that the Schottky junction is able to support two basic SPP modes, that are termed upper mode and lower mode.The upper mode behaves very similar to a SPP mode on a bi-metallic interface.In contrast to this, the lower mode behaves much more like a SPP mode on a metal-dielectric interface.It is interesting to note that the two modes obtained in this paper for the plasmon dispersion in a Schottky junction are qualitatively in agreement with the results of Inkson [17].Moreover, the effects of charge density profile, plasma frequency of metal and externally applied potential on dispersion characteristics were also discussed.The charge density profile controls the negative slope of the upper mode curve and bulk plasma frequency controls the frequency margins of the two modes.It was found that using an externally applied potential across the junction, the maximum level of two modes can be shifted along the frequency axis.Thus, it is clear that the Schottky junctions provides a greater controllability over SPP modes on its interface compared with the conventional metal-dielectric interfaces.Especially given that a population inversion can be made in the vicinity of the Schottky junction (cf.[11]), it is very likely that SPP waveguides will be made incorporating Schottky junctions for next generation integrated optics/plasmonics applications requiring long propagation distances.

Fig. 1 .
Fig. 1.The energy band diagram of a metal and n-type semiconductor junction in equilibrium.The work functions are denoted by φ M ad φ s where φ M > φ s and χ s denotes electron affinity of the semiconductor.

Fig. 2 .
Fig. 2. A Schematic illusion of the Schottky Junction including the division of the junction to sections A, B, C, and D based on the carrier density distribution.
#161732 -$15.00USD Received 18 Jan 2012; revised 28 Feb 2012; accepted 5 Mar 2012; published 13 Mar 2012 (C) 2012 OSA This equation assumes different forms depending on the permittivity of regions A, B, C and D given in Eq. ( iv) Solution of Eq. (6) in region D where z > d 2 :

Fig. 3 .(Fig. 4 .
Fig.3.Plasmonic dispersion curves for a metal-semiconductor interface using our Airy function method and Frobenius series solution method.The difference ω di f f between the curves is shown in the inset, which is below 1.0%.

Fig. 5 .Fig. 6 .
Fig. 5. Dispersion curves for varying carrier density profiles shown in the inset.(a) variation of upper modes with slope r and (b) variation of lower modes with slope r.The frequencies marked in the figure are normalised to bulk plasma frequency of the semiconductor, ω ps0 and d 1 , d 2 values are normalised by c/ω ps0 .

Fig. 7 .Fig. 8 .
Fig. 7. Dispersion curves obtained using different metals for the Schottky junction: (a) Upper modes and (b) Lower modes.Each upper mode curve starts from the bulk plasma frequency of the particular metal.The frequencies marked in the figure are normalised to bulk plasma frequency of the semiconductor, ω ps0 .