1. Introduction

Surface plasmon polaritons represent a kind of hybridization of different electromagnetic modes and a particular case study of the interaction of light with matter [1, 2, 3, 4]. Indeed, the word polaritons often stands for a combination of photons with phonons or plasmons, which are respectively charge oscillations of ions in a multipolar lattice or simply of the electron density. Bulk polaritons are the modes propagating inside an infinite medium without considering boundary condition and are usually due to a mixing between photons and the transverse charge oscillations of the material. In the presence of an interface the emergence of a surface mode is also possible resulting from the coupling between photons and longitudinal oscillations, as in the case of a surface plasmon polariton (SPP) for example. The frequency regions in which bulk polaritons exist are associated to the possibility of waves propagating inside the medium, and thus to its partial (at least) transparency. Differently, it is inside the gap between these regions, where the material should be totally reflecting because of a negative dielectric function, that the surface modes can be excited. Surface modes are important for different reasons. For example, they contribute to the build up of the surface energies of materials. For applications, being spatially described by an exponential decay away from the interface, they represent a confinement tool for the field and can be used for dielectric constant monitoring or fluorescence enhancing in chemical- or bio-sensing [5]. They can be engineered as fast communication paths in opto-electronic devices and must be taken in consideration in conventional or organic light emitting diodes to limit the huge loss channel they may give rise to [6, 7]. Recently they have been thought also, inversely, as a possible tool for achieving a more efficient light coupling in electroluminescent devices [8], or in traditional [9] and organic solar cells [10, 11].

Many applications and studies of SPPs concern their propagation at metal/dielectric interfaces, being associated to the well known oscillations of free electrons in metals. Tipically, homogeneous films or dispersed nanoparticles of gold and silver are considered as good plasmonic materials [12, 13]. In principle, SPPs can exist in the suitable (ω,k) regions on many conductive or even dielectric materials [14], and a growing number of studies are being published concerning single films or more complex multilayered heterostructures [13, 15, 16, 17, 18].

Transparent conducting oxides (TCO) are important materials for many opto-electronic devices and include the most commonly used indium tin oxide (ITO), fluorine doped tin oxide (FTO or SFO), and aluminium doped zinc oxide (AZO). In general, due to their lower carrier concentration (10^20–22 cm ^-3) compared to metals (10²³ cm ^-3), TCOs posses a smaller plasma frequency which lies in the infrared region and consequently a much larger skin depth. While SPPs confined below the visible range may be of interest in devices that make use of infrared radiation, other considerations can be drawn [19]. A grating coupler to excite SPP at infrared wavelengths for example should not require the higher resolutions as in the case of SPPs in metals [20]. Moreover, a SPP based 2D optical tweezers application [21] implemented on a transparent oxide could take advantage of the transparency of the substrate utlising a configuration with two beams well separated in frequency, one for optical microscopy and one for infrared enhanced trapping. Further, the larger skin depth converts that the hybridization of the two independent surfaces modes into the symmetric and antisymmetric modes, should be taken into account even for larger film thicknesses [22] respect to the metal case [1, 23, 24]. Also, the same long-range plasmons attained in lossy metal films of very small thickness [13] may be obtained for TCO films. The experimental and theoretical literature of SPPs in TCOs, in particular ITO [20, 25, 26, 27, 28, 29], ITO and AZO [22, 23], and ITO/metal interfaces [30], is growing and quite ample. However, works on FTO are still lacking in the literature even if at least two works may be cited as preliminary stage [31, 32].

2. FTO films preliminary characterization

The samples used for the experiments were FTO films coated on 3mm thick sodalime substrates with a sheet resistance of about 8Ω/sqr., purchased from ManSolar. The thickness of the films was estimated to be about 510nm by means of ellipsometry measurements. Transmittance and reflectance spectra were measured at quasi normal incidence in the range λ ∊ [0.5µm,2.5µm] by means of a Perkin Elmer λ 900 spectrophotometer. The spectra were fitted by assuming a Drude dispersion for the refractive index n and extinction coefficient κ.

The standard Drude law states that the complex dielectric constant due to free carriers is:

where ω_p is the angular plasma frequency, Γ the damping and ε _∞ is the asymptotic dielectric constant due to all the resonances located at higher frequencies and screening in the considered range. In terms of material parameters: ω ² _p=ne ²/mε ₀ and Γ=1/τ, with n, e and m density, charge and effective mass of the free charge carriers, while τ is their scattering time.

To fit the reflectance and transmittance measurements at quasi normal incidence we used the Drude relation with the real and imaginary parts expressed in terms of the wavelength:

 

Fig. 1. Quasi-normal incidence R and T measurements in the λ ∊ [0.5µm,2.5µm] range. Good fits according to a Drude model are obtained in the λ ∊ [1.0µm,2.5µm] interval and are used to retrieve the parameters and the n and κ dispersion of Tab. 1 and Fig. 2.

Download Full Size | PDF

In Fig. 1 we show the experimental transmittance and reflectance spectra together with the fitting curves. Fits are limited to the λ ∊ [1.0µm,2.5µm] range because of some uncertainity in the knowledge of the refractive index of the substrate outside such interval. The values of the ε _∞, A₁ and A₂ parameters retrieved from the fits, together with the correspondent values of the angular plasma frequency ω_p, damping Γ and asymptotic value ε _∞, are listed in Table 1. The associated dispersions for the n and κ of FTO are shown in Fig. 2.

Table 1. Drude resonance parameters obtained from the fit of the quasi-normal incidence reflectance and transmittance spectrophotometer measurements in the range λ ∊ [1.0µm,2.5µm].

View Table

 

Fig. 2. Refractive index (solid) and extinction coefficient (dashed) dispersions with wavelength, according to the Drude parameters of Tab. 1 obtained from fits of quasi-normal incidence reflectance and transmittance measurements in the λ ∊ [1.0µm,2.5µm] range. Points are from attenuated total reflection (ATR) angular scans at discrete wavelengths λ=633nm,830nm,1300nm and 1550nm. Small vertical bars at the bottom indicate the range of the tunable infrared laser for the ATR measurements. The ticks on the top axis indicate the plasma, screened plasma and surface plasmon frequencies.

Download Full Size | PDF

It is convenient to note that the real part of the permittivity crosses zero (n ²-κ ²=ε_R=0) at the screened plasma frequency ω̅_p. The value of ω̅_p can be derived from the plasma frequency ω_p considering the equivalent expression:

where ${\bar{\omega }}_p=\omega \left({\epsilon }_R=0\right)=\sqrt{{\omega }_p^2⁄{\epsilon }_{\infty }-{\Gamma }^2}\approx 1.04·{10}^{15}$ rad/s with a corresponding λ̅_p≈1.8µm. ω̅_p represents a lower frequency limit for the existence of the bulk mode, since the permittivity is negative below it, and at the same time an upper limit for the surface one (in the case of an interface between FTO and a material with positive permittivity). The last consideration derives from the well known requirement for a true surface mode (i.e. the field exponentially decaying away from the interface) that the dielectric functions on the two sides of the interface must have a different sign. Instead, the requirement for which SPP propagation is expected to appear at the FTO/air interface is more strict, being ε_R<-ε_air=-1. Such a condition is verified for frequencies lower than the value called surface plasmon (polariton) frequency, ${\omega }_{\mathrm{sp}}=\omega \left({\epsilon }_R=-1\right)=\sqrt{{\omega }_p^2⁄\left({\epsilon }_{\infty }+1\right)-{\Gamma }^2}\approx 0.89·{10}^{15}$ rad/s in the case of air and λ_sp≈2.1µm.

In Fig. 2 the small vertical ticks at the top indicate the measured plasma, screened plasma and surface plasmon frequencies. Following our previous considerations, bulk modes should appear on the left of ω̅_p and surface modes at the interface with air on the right of ω_sp. The complex permittivity (Eq. 1 with parameters from Tab. 1) will be used in the following sections to compute the bulk and surface modes dispersions in a frequency vs wavevector plot together with a film reflectance map, in order to highlight the feature corresponding to SPPs resonance and its location in the reflectance pattern.

3. Computation of bulk and surface dispersion curves

It is in principle possible to use the dielectric function of any pair of media to preliminarly derive the light-dispersion curve for the propagating modes inside the bulk (ω,q) of both media:

and at their interface (ω,k)

where q refers to the total wavevector and k to the in-plane component. Boundary conditions across the interface are translated into the conservation of angular frequency ω, but not of the total wavevector q whose orthogonal component may receive an impulsive contribution during the impinging on the discontinuity represented by the interface itself. In the ideal case of smooth interfaces, the in-plane component k of the wavevector doesn’t receive any contribution and is conserved.

In Fig. 3 we report the computed dispersion curves for the bulk and surface modes, in the case of FTO/vacuum and FTO/glass interfaces. We extended the wavelength range in which the simulation was carried out from the 2.5 µm limit up to 10 µm. This last extension must be taken with caution, but it appears feasible as FTIR measurements done in this range [32] do not show any other resonance.

The surface mode dispersion associated to a single interface between a dielectric and a conductor with only a real part of the Drude dielectric function is simple and well known, bending below the light line for small momentum (photonic character) and being asymptotically horizontal for large momentum (plasmonic character) [3]. Instead, the presence of damping (i.e. an imaginary part of the dielectric permittivity) causes a backbending both of the bulk polariton and the SPP dispersion curves. The complex value of the permittivity means that also the wavevector k is complex. In this case in the (ω,k) plots, k represents the real part of the in-plane complex wavevector k̃. As can be seen in Fig. 3, in the case of the bulk mode in FTO (solid), the dispersion undergoes a backbending in the bottom-left small momentum-small wavelength region ((ω,q)⇒ zero) where the imaginary part of q rises, corresponding to an absorption associated to ε_I. In the case of the surface modes, a complex permittivity causes a backbending of the dispersion curves (FTO/air long dash; FTO/glass dadot) in a region just below ω̅_p and a corresponding attenuation, associated to an imaginary part of the in-plane wavevector k. In both bulk and surface dispersions, bottom and upper branches connect to each other.

 

Fig. 3. Curves corresponding to the dispersion relations of Eq. (4) and (5). Bulk mode in FTO (solid); SPP at the FTO/glass (dadot) and SPP at the FTO/air (long dash) interfaces. Straight lines are the bulk dispersions (light lines) in air (n=1, solid) and inside glass (n=1.515, dot). True surface modes are points on an SPP dispersion curve when this is on the right of both the FTO bulk curve and the corresponding dielectric light line. Dispersions are plotted versus the real part of the in-plane wavevector k. The small trapezoidal area centered at k ≈0.4×10⁷ rad/m and λ ≈1.5µm represents the accessible experimental window with our angular ATR set-up and tunable IR laser.

Download Full Size | PDF

Not all points along a surface mode curve corresponds to Fano modes, meaning that they are exponentially decaying away from the interface in both media. Only the branch of the curve below ω̅_p and on the right of the associated dielectric light line correspond to Fano modes. Those parts of the dispersion curve situated below ω̅_p but on the left of the straight light line are called more properly Evanescent (or quasi bound modes, QBM). The Brewster modes are above ω̅_p and lying on the left of both the dielectric light line and the FTO bulk curve. These are also called the radiative plasmon polariton RPP, upper branch, and are associated to a lack of reflectance from the FTO/dielectric interface for TM waves.

4. Simulation of film reflectivity maps

The previous SPP dispersions are explicitly computed only in the case of a single interface between FTO and a dielectric, and can serve as underlying guides for the reflectance features in layered structures. A further step in the modellization consists in computing a reflectance map using Fresnel coefficients for single interfaces embedded in a well known three layer model that considers multiple reflections and infinite summation over them, as originally introduced by Airy [14, 33] (also known as the three phase Fresnel equations of reflection). At this stage, even in the case of a merely real dielectric function for the TCO, the finite film thickness causes the deformation of the SPP curves when the two interfaces (in the specific, glass/FTO and FTO/air) become closer and the associated independent modes get hybridized in the two, symmetric and antisymmetric, modes (short and long range SPPs). For sake of completeness, we just cite here the fact that apart from the three layers model, valid for a single film between two semi-infinite media, there is also a more general transfer matrix method TMM that can be used for an arbitrary multilayered planar structure [34].

In Fig. 4 we show a reflectance map computed for a 510nm thick FTO film with the Drude dielectric function represented in Fig. 2 (Eq. 1 with the parameters in Tab. 1). The entrance medium is a BK7 glass prism with n ₀=1.515 and the exit one is air, n_air=1, as shown in the inset of Fig. 4. The incident angle inside the prism is θ ∊ [0°,90°] and the light polarization is TM. High reflectance zones are shown as brighter, while low ones are darker. The map is complex and exhibits sharp edges together with smooth gradients. Thus it would be challenging to identify the proper modes types without the curves computed in Fig. 3 as guides. Numerically, it could be possible to decrease Γ, by a factor of 10 or 100, just to compute a map that displays a thinner SPP feature. Furthermore, the model allows to vary the thickness of the FTO film, to see how the modes’ dispersion evolves and to investigate their nature more deeply. In the movie Fig. 4 (Media 1) the reflectance map is shown for increasing values of the FTO thickness d ∊ [0, 1 µm] [35]. Such an approach can be exploited to some extent for predicting the behaviour of films with different thickness.

In the left upper region of the Fig. 4, the shadowed area corresponds to a low reflectivity of the film because of increased transmittivity due to the existence of bulk polariton modes capable of transporting electromagnetic oscillations. On the right hand side of the air light line, the region appears grey because of the condition of total reflection attained at the exit TCO/air interface. Absorption across the film decreases the total reflection somewhat. If the thickness of the film increases, the level of absorption rises and the region’s reflectivity falls down.

The dark band starting from the air light line on the left and crossing the prism light line on the right (i.e. the boundary of the work domain) is associated to the bulk polariton curve theoretically derived for FTO and shown in Fig. 3. The band is dark because it corresponds to grazing angles (propagation close to 90°) inside the TCO and so to very long optical paths. Consequently, high losses are achieved even for the low extinction coefficient present in this ω range.

On the bottom of the triangular domain, between the two air and glass light lines, a shallow curve can be observed representing the true surface or Fano mode, i.e. the SPP at the FTO/air interface. The SPP at the FTO/glass interface cannot be accessed when impinging from inside the glass itself. The computed SPP mode at FTO/air interface is tangent to the light line in air at low k, because of the retardation effect (finite speed of light), while it asymptotically tends to ω_sp ≈ 2.1µm for larger k. It tends to vanish on the high momentum side and near ω_sp, because the distance from the FTO bulk mode is around its maximum (in terms of k_SPP-q_bulk, see Fig. 3). This converts in a large imaginary orthogonal component γ of the wavevector q and so in power extinction before arriving to excite the SPP on the FTO/air side.

The thicker the film, the less the power at disposal to couple with the SPP on the exit interface, FTO/air. This is one of the reasons why an optimal thickness for the observance of surface plasmon exists. On the other hand, a layer which is too thin gives hybridization between the modes from the two interfaces. The hybridization in the film is associated to deformations of the dispersion as is well known in literature [3, 14]. We should keep in mind that such deformations are different from the backbending due to the complex permittivity. Actually, when considering both the damping together with an increasing film thickness starting from zero, simulations allow us to observe a transition region between ω _p and ω_sp, firstly connecting and then evolving into the distinctive bulk and surface features as represented in Fig. 4. Such a transition region, ascribed to the back-bended and distorted SPP, strongly depends on the film thickness d, in terms of the λ_p/d ratio.

 

Fig. 4. Computed reflectance map for the three layer model with BK7 glass, 510 nm thick FTO film, air. Light is incident from the glass side, in the domain λ ∊ [0.5µm,10µm]-θ ∊ [0°,90°]. The dark area on the left is separated from total reflection brighter area on the right by the air light line. For thinner films the SPP feature is better resolved, and it is also possible to observe its formation and evolution as the thickness increases (Media 1).

Download Full Size | PDF

From this point of view, TCOs, having a greater λ_p than metals, represent a better system to observe and resolve SPP variations vs thickness. Such a characteristic, together with the fact that the Drude resonance is further away from other resonances (inter-band transitions associated to the visible - UV energy gap) [29], make ITO and FTO good systems for the observation of SPPs. The first time that the evolution of this transition region was clearly observed in a TCO to our knowledge is for ITO in two previous works [22, 29]. Here we report the same behaviour computed for an FTO film, modeled with a dielectric permittivity derived from measurements, for the first time. A transition region between the bulk mode edge ω̅_p and the surface one ω_sp is shown to be strongly absorbing for thin layers, d<200nm, e.g. see Fig. 4 (Media 1). The predicted range for surface plasmon polariton in the near-infrared is for λ above 2100 nm, while the optimal film thickness for observation of SPP is estimated to be around 200nm, which for λ≈3.0µm gives a reflectance minimum inside the SPP dip equal to zero. In Fig. 5 we show as an example of the reflectance profiles associated to the SPP feature computed for a FTO thickness of 200nm and at three wavelengths, λ=2.5µm (dashed), 3.0µm (solid) and 3.5µm (dadot).

We underline the apparently counter-intuitive nature of the phenomenon that happens in the transition region. The absorption (power can only be absorbed by the FTO film in the present case, since it is the only medium with an extinction coefficient) is greater for thinner films. This behaviour is simply explained by considering that for thicker films, the light doesn’t reach the exit FTO/air interface and doesn’t excite any SPPs, so it is reflected after undergoing just a weak absorption. For thinner films, the excitation of SPP propagating along FTO/air interface provokes a longer path and a stronger absorption.

 

Fig. 5. Computed reflectance profile for the three layer model with a 200 nm thick FTO film. The three curves are for λ=2.5µm (dashed), 3.0µm (solid) and 3.5µm (dadot). Minimum of the SPP feature is zero, while for thinner or thicker films the minimum tends to increase.

Download Full Size | PDF

5. ATR measurements

An experimental investigation of the light dispersion features of FTO samples was performed by an Attenuated Total Reflection (ATR) method, implemented in a Kretschmann-Raether (or prism-metal-air, PMA) configuration on a previously described setup [22]. The angular region θ ∊ [27°,67°] is accessed by means of a rotational stage. Four single wavelength laser diodes or a tunable laser diode (Nettest, Tunics-Plus) are used as light sources, alternatively. The collimated and TM-polarized laser beams are focused on the sample by means of a low numerical aperture lens (f=250 mm) through the input facet of a 45° BK7 glass coupling prism (n _BK7=1.515). θ is the angle between the normal to the FTO layer and the direction of the incident beam inside the prism. Reflectance R(θ) is obtained by means of a (θ,2θ) detection scheme in which the sample is rotated with respect to the incident beam at fixed wavelength. A lens (f=50 mm) collects and focuses the reflected light onto a photodiode. The collection lens and the photodiode are rotated according to the angular position of the sample. Measured reflected light is then normalised to the bare prism reflectance, accounting for Fresnel reflections from its interfaces.

 

Fig. 6. ATR angular scans obtained at fixed wavelengths. Attenuated total reflectance just on the right of the edge is maximum for the 633nm curve (O), then for the 830nm (×), 1300nm ([]) and 1550nm (∆) ones. The fits shown are obtained by means of the commercial software TFCalc.

Download Full Size | PDF

In Fig. 6, angular scans performed at the fixed wavelengths of 633, 830, 1300 and 1550 nm are shown. For each curve the total reflection edge occurs at θ=42° (this angle corresponds to the air light line of the previous Figs. 2 and 3). The reflection level just at the right of the edge increases with photon energy. Such level is different from unity for each curve, so the term total reflection (referred to the FTO/air interface) does not apply to the whole layered structure, because of absorption inside the FTO layer. Moreover, each curve is not flat on the right of the edge θ=42°. It is possible to note the presence of a valley, that is on the left in the 1550nm curve, in the centre of the 1300nm one, on the right and almost out respectively in the 830nm and 633nm curves. Such valley may be well identified with the dispersion of the bulk polariton, if considering the positions of the actual discrete wavelengths in Fig. 4. The solid curves shown in Fig. 6 are obtained from accurate fits of the angular scans by means of the commercial software TFCalc. In such fits, the matching oil (n_oil=1.66) between the prism and the glass substrate is also taken into account. The refractive index and extinction coefficient of the FTO derived from such fits at the four wavelengths we used are all reported in Fig. 2 as discrete points, showing a good agreement with the Drude dispersion obtained by the independent spectrophotometer measurements at quasi normal incidence.

An ATR R(θ,λ) map is obtained by means of the tunable laser scanning the wavelength over the entire accessible range λ ∈ [1.45µm,1.59µm] for each angle of incidence. Such map is shown in Fig. 7, where the wavelength on the y-axis is inverted in order to be consistent with the k-ω domain of the previous figures. It is to be noted that the vertical light line for total reflection appearing at θ=42°, coincides with the diagonal straight air light line of Fig. 3 and Fig. 4. In the present case, maximum reflectivity (white) corresponds to R=0.40 and minimum (black) to R=0.00. The scale is not linear to enhance picture contrast. Due to the limited extension of the experimental λ -region at disposal, just a tail coming out from the light line is present. From a comparison with the computed reflectance map of Fig. 4, it is possible to exclude the presence of a surface plasmon polariton in the present NIR range and to identify the observed feature with the bulk polariton. Differently from our previous work on ITO [22], neither the SPP nor the transitional region can be observed since they are in a lower energy range not accessible with our tunable IR laser, because ω_p lies lower in FTO compared to ITO.

 

Fig. 7. ATR map obtained in Kretschmann configuration for a 510 nm thick FTO film over a sodalime glass sample. The maximum reflectance value corresponds to 0.40 (brightest) and the lowest to zero (darkest). The vertical edge where total reflection occurs is located at θ=42° and corresponds to the light line in air, n=1.

Download Full Size | PDF

6. Conclusions

In the present work we report on the investigation of both bulk and surface plasmon polaritons on a fluorine doped tin oxide film. The transmittance and reflectance spectra measured at quasi normal incidence by a spectrophotometer represented a good tool for the evaluation of the refractive index and extinction coefficient in a extended wavelength range. Indeed, in such a TCO, as already reported for ITO, the free electrons plasma frequency is quite far from other resonances allowing a good fit by a generalized Drude model in the considered range. Moreover, this means that the surface plasmon feature is easily observed making FTO together with ITO good study systems for such surface modes. The retrieved dielectric function has been used to compute the dispersion of both bulk and surface modes inside the FTO and at its interfaces with dielectric constant media, i.e. air and glass. Furthermore, a reflectivity map in a selected wavevector (associated to experimental incidence angle) and wavelength bidimensional domain was performed. Such an experimental/computational procedure allows the evaluation of the reflectivity and modes’ features for different film thicknesses. Rather than as an applicative tool for some specific purpose, here we just highlight the general behaviour of thin and absorbing conducting films. Results show an interesting thickness dependent transition region between the bulk polariton and the surface plasmon branches, as recently observed in indium tin oxide. Also, they predict that range of existence of a surface plasmon polariton at the FTO/air interface is well below the visible and in the near infrared at λ≥2100 nm. An optimal thickness for its observation is also estimated to be around 200nm. Finally, an attenuated total reflection technique is shown to be an experimentally equivalent tool to obtain the same bidimensional reflectivity map. In the present case, the IR laser can be tuned in the λ ∈ [1.45µm,1.59µm] range, allowing us to observe a feature that can be attributed to bulk polariton and not to a surface plasmon on the basis of the previous characterization. In order to directly observe the SPP feature (or to better characterize the bulk polariton dispersion) in a R(θ,λ) map, tools capable to explore different or more extended wavelength regions could be used, as for example implementing varying angle reflectivity measurements on spectrophotometer, FTIR or ellipsometer set-ups.