Aluminum plasmonics: optimization of plasmonic properties using liquid-prism-coupled ellipsometry

Kenneth Diest; Vladimir Liberman; Donna M. Lennon; Paul B. Welander; Mordechai Rothschild

doi:10.1364/OE.21.028638

1. Introduction

The rapidly expanding field of plasmonics relies heavily on optimal combinations of materials and nanopatterning. Suitably engineered nanostructures possess surface plasmon resonances (SPRs) at certain wavelengths, and both linear and nonlinear phenomena are observed at or near these resonances. However, the broad application of plasmonics is hampered by two interrelated factors: the availability of low-loss metals in the UV, visible or near-IR; and the ability to perform large area patterning of nanoscale structures. To date, the choice of material has been the primary driver in the demonstration of plasmonic effects, since the list of metals with low ohmic losses and long dephasing times is limited mostly to gold and silver [1]. The shortness of this list has broad implications. First, the SPRs of Au and Ag are at visible and near IR wavelengths; however, at blue or ultraviolet wavelengths, surface plasmons become unstable in gold and silver and their performance deteriorates due to the generation of electron-hole pairs. As a result, applications such as surface enhanced Raman spectroscopy (SERS) of trace chemicals cannot generally take advantage of additional molecular resonance enhancement, because molecular resonances often occur at UV wavelengths. Second, large-scale fabrication of Au or Ag nanostructures cannot benefit from the broadly installed base of fabrication equipment devoted to silicon microelectronics because these materials both form deep-trap states in silicon [2]. This presents a considerable barrier to accessing advanced tools used in photolithography, thin film deposition and etching, or chemical-mechanical polishing.

These drawbacks can however be overcome with the implementation of a largely unexplored plasmonic material, namely aluminum. Throughout most of the near-UV and visible spectrum, aluminum exhibits very low losses [3], and the electron configuration of Al is such that the plasmon resonance occurs at much shorter wavelengths than in Au or Ag [4]. Furthermore, Al is widely used in microelectronics fabrication, and processes are well known for Al deposition, patterning and etching. These considerations have led us to explore the feasibility of Al plasmonics. One limitation to this approach is that the material properties of Al films used in microelectronics are significantly different from those desired in plasmonics. Specifically, in microelectronics applications, Al thin films are deposited with very fine grain structure to enable planar films and high-resolution patterning for plasmonic structures; however, this nanostructure is undesirable for plasmonic applications, where electron scattering at grain boundaries needs to be minimized in order to maintain coherent surface plasmon behavior over long distances.

In this paper we describe a systematic approach to optimize the nanostructure of Al films for plasmonic applications. We demonstrate a novel method to experimentally determine the plasmonic figure of merit ( $FOM = ε_{real}^{2} (ω) / ε_{imag} (ω)$ ), and identify thin film deposition and planarization processes of Al that yield the best plasmonic behavior. Our experimental method couples spectroscopic ellipsometry with surface plasmon polaritons (SPPs), by coupling through a liquid prism cell developed by J.A. Woollam Co. [5]. This approach allows any index matching fluid to be confined in a prism geometry and to directly measure aluminum samples deposited on different substrates under varying deposition conditions. The flexibility of this approach is evident in that the coupling conditions for SPR measurements can now be varied by changing the index of the prism itself simply by removing one liquid and replacing it with another. We also determine the optical properties of each sample using the Ashcroft and Sturm theory of optical conductivity [6]. This approach allows us to extract the electron scattering times for both Drude and effective interband transitions, interband transition energies, and the optical mass of electrons.

The FOM values obtained from these measurements were compared with literature values of the more traditional plasmonic materials Au, Ag, and Cu. We show that at blue and UV frequencies, optimized Al has the highest FOM of the four materials, thus providing a quantitative basis for the advantage of Al plasmonics at these frequencies.

2. Experiment

Five aluminum samples were analyzed using liquid prism ellipsometry: a (111) single crystal substrate from MTI Crystal [7]; a 100 nm thick film deposited on a (100) silicon substrate using electron beam evaporation with no active control of the substrate temperature at a base pressure of 1.2 × 10⁻⁸ Torr and a rate of 0.5Å/s; and two films deposited using DC magnetron sputtering in an argon-hydrogen environment on (100) silicon substrates. For the two sputtered films, the deposition power was 1 kW, the Ar flow rate was 10 sccm, and the 90%:10% Ar:H flow rate was 15 sccm. The first sputtered film was 200 nm thick and deposited at room temperature, while the second sputtered sample was 500 nm thick and deposited at 350°C. For the latter sample, deposition was followed by a chemical mechanical polishing (CMP) step to thin the aluminum to 100 nm. The combination of high deposition temperature and CMP was used to produce an aluminum film with both large grain size and low surface roughness. The final film was deposited 100 nm thick using molecular beam epitaxy (MBE). To minimize the lattice mismatch between the aluminum film and the substrate, the film was grown on C-plane sapphire rather than silicon, with a (111) orientation at a substrate temperature of 175°C. All five samples were measured using a 1 μm × 1 μm atomic force microscopy (AFM) scan. The surface roughness of each sample was extracted directly from the measurement; however, the median grain size was determined by manually outlining and measuring the grain boundaries within each image. A summary of these results is shown in Table 1. In the case of the single crystal sample, no grain boundaries were observed within the scan range of the tool. For all samples, no attempt was made to remove the native oxide layer

Table 1. AFM analysis of the five aluminum samples studied.

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To perform liquid prism ellipsometry, each sample was first coated with a 200 nm thick layer of magnesium fluoride, MgF₂, to serve as the dielectric interface that supports surface plasmons. The sample was then mounted to the bottom Viton gasket of a Woollam 500 μL Horizontal Liquid Cell [8]. Cargille 5610 laser liquid [9] was chosen as the index matching fluid used in the liquid prism with optical properties modeled as a Cauchy layer [10]

n = A + \frac{B}{λ^{2}} + \frac{C}{λ^{4}}

where A = 1.509271, B = 646533, C = 2.917535e12, and λ is the wavelength in Angstroms [9]. Each sample was measured from 300 – 1000 nm. As a reference, standard spectroscopic ellipsometry was performed on a layer of MgF₂ deposited on bare silicon. The measured data was fit as a Cauchy layer with A = 1.3747, B = 219650, C = 0, and this value was used for all subsequent modeling. A schematic of the entire experimental setup is shown in Fig. 1.

Fig. 1 Liquid prism coupling schematic with the thin film layer stack used for each aluminum sample studied. The modified Otto coupling configuration enables coupling into SPPs on as-deposited samples.

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2.1. Liquid prism coupling to surface plasmons

The method of using attenuated total reflection (ATR) to measure the plasmonic quality of metal surfaces relies on prism coupling using either the Kretschmann-Raether [11] or Otto [12] configuration to efficiently match the wave vector of incident light with the wave vector of surface plasmon polaritons (SPPs) supported at a metal / dielectric interface. By coupling light into SPPs, ATR is able to probe the microstructure of the metal while the SPP propagates along the metal dielectric interface. To measure the five aluminum samples listed in Section 2, we combine ATR with spectroscopic ellipsometry in the form of total internal reflection ellipsometry (TIRE) [16, 17]. This approach combines the accuracy and phase information of spectroscopic ellipsometry with the high sensitivity of surface plasmon resonance (SPR) coupling, and provides a highly sensitive method for probing the SPP quality of plasmonic films. Combining TIRE with the liquid prism configuration described above, we were able to couple into surface plasmons on the surface of the aluminum films by tuning the refractive index of the prism, the refractive index and thickness of the MgF₂ spacer layer, and the angle of incidence so that the k_x component of the incident light matches that of the surface plasmon. Thus, the liquid-prism approach takes advantage of the Otto configuration’s ability to measure the plasmonic properties of as-deposited films while avoiding the challenge of maintaining a uniform, planar separation between the prism and the underlying film.

3. Measurements

3.1. Ellipsometry modeling

All Fresnel modeling for this study was done using the WVASE32® software [18] with the corresponding optical constants determined in Section 2. The ellipsometry model for all five samples consisted of an aluminum base layer, a 3 nm native aluminum oxide AlO_x layer, a 200 nm MgF₂ layer, and Cargille 5610 laser liquid as the top, ambient layer. In each model, the aluminum layer was represented as either a 3- or 4-oscillator system including a Drude, Tauc-Lorentz, and Lorentz oscillators [18]. Also, the thickness of the AlO_x layer was confirmed independently using spectroscopic ellipsometry. To obtain an accurate fit for samples with an RMS roughness > 1.63nm, an additional intermix layer was needed between the AlO_x and MgF₂ layer. This was modeled as a Maxwell-Garnett effective medium material with a depolarization factor of 1/3, and was made up of 25% aluminum, 25% MgF₂, and 50% AlO_x. For each sample, the only terms in the model that were allowed to vary were the intensities and positions of the oscillators, as well as the thickness of the intermix layer, if present. The results of each fit are shown in Table 2.

Table 2. Modeling parameters used in the ellipsometry analysis of each aluminum sample. In the bottom row, the terms D, T-L, and L refer to the Drude, Tauc-Lorentz, and Lorentz oscillators needed to accurately model each sample.

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In Fig. 2(a), we model Ψ and Δ curves for the layered structure shown in Fig. 1 with the materials constants listed in Section 2 and aluminum values from the Palik Handbook of Optical Constants [19]. These curves correspond to an angle of incidence of 69° which gave the strongest SPR resonance. We see from these curves that the dip in the Ψ spectrum corresponds to the minima in reflection resulting from coupling to surface plasmons on the aluminum sample. This resonance occurs at 390 nm and is indicated by the vertical dashed line. At the same resonance wavelength, we also note an even stronger shift in Δ. The steepness of this curve clearly indicates the increase in sensitivity of this approach over standard reflectometry or conventional external ellipsometry. The additional minimum in the Ψ spectrum seen at 825 nm corresponds to the (200) interband transition in aluminum. For comparison, Fig. 2(b) shows experimentally measured Ψ and Δ curves for the layered structure shown in Fig. 1 with the room temperature sputtered sample. Measurements were performed from 69° to 71°, and we see that the strongest coupling occurs at 70°. Also, the sensitivity of the curves to coupling angle is evident in how quickly the dip in the Ψ spectrum recedes when the coupling angle shifts by a single degree.

Fig. 2 Modeled Ψ and Δ curves for the layered structure shown in Fig. 1 using the materials constants listed in Section 2 and aluminum values from the WVASE32® database, (a). The vertical dashed line corresponds to the minima in reflection spectra resulting from coupling to surface plasmons on an aluminum sample. In (b), experimentally measured Ψ and Δ curves for the layered structure shown in Fig. 1 with the room temperature sputtered sample.

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3.2. Ellipsometry results

Experimentally measured Ψ and Δ curves for the five samples are shown in Fig. 3. For each plot, the y-axis on the left corresponds to Ψ values and are shown in blue. The y-axis on the right corresponds to Δ values and are shown in green. For each curve, the experimentally measured data points are shown as open circles, and the ellipsometry fits are shown as solid lines. A dashed, vertical line is shown on each plot and corresponds to the minima in the Ψ spectrum. For all five materials, we see that in addition to the reflection data from the Ψ curve, the region around the reflection minima corresponds to the region of highest sensitivity in the Δ curve.

Fig. 3 Ψ and Δ curves for the five aluminium samples measured. Experimentally measured data is shown as open circles and model fits are shown as solid lines. The vertical, dashed black line corresponds to the wavelength of strongest coupling. Measurements were performed from 69° to 71° for each sample, and the curve shown represents the angle of strongest SPR coupling.

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For each plot, measurements were taken from 69° to 71°, and the angle that provided the strongest SPR coupling is shown in Fig. 3. For both the sputtered and MBE aluminum, experimental measurements and subsequent modeling show that the strongest SPR coupling was at 69°. The high temperature sputtered and CMP’d film showed the strongest coupling at 70°, while the single crystal sample and evaporated film had optimal coupling at 71°. This indicates that the smoothest samples, with no intermix layer, coupled most efficiently at sharper angles to the film, while the presence of larger surface roughnesses and the resulting intermix layer caused the strongest coupling to occur at more shallow angles.

For all of the aluminum samples except the evaporated film, strong coupling occurred between 380 and 420 nm, while the evaporated film showed SPR coupling at a much longer wavelength, 515 nm. This change in the critical coupling condition results from the significant surface roughness of the evaporated film, and the additional effective medium intermix layer that is required to fit the experimental data. The resulting broadband permittivity for each sample is plotted in Fig. 4, where the real part of the permittivity is shown in (a) and the imaginary part of the permittivity is shown in (b).

Fig. 4 For each sample, the real and imaginary parts of the permittivity are shown in (a) and (b), respectively. The figure of merit for the five aluminum samples is plotted in (c). In (d), the high temperature sputtered and CMP’d aluminum film is compared with the Johnson and Christy data for gold, silver, and copper [23].

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In Fig. 4(c), we incorporate the quality factor FOM Q_SPP of the local-field enhancement at the aluminum surface that takes into account both the loss within the metal as well as the real part of the permittivity [20, 21]

Q_{SPP} (ω) = \frac{ε_{r}^{2} (ω)}{ε_{i} (ω)}

where ε_r and ε_i are the real and imaginary components of the dielectric constant, respectively. These values are plotted for the five aluminum samples in Fig. 4(a) and 4(b). From Fig. 4(c) we see that the single crystal sample has the largest FOM for all wavelengths measured. While this result is to be expected, we note that the high temperature sputtered and CMP’d film has the second largest FOM, and films deposited using standard electron beam evaporation have a lower quality factor than all other samples studied. This can be explained by the higher surface roughness in evaporated films, which decreases the overall coupling efficiency, and smaller grain sizes, which increase the overall scattering [22]. Additionally, as we saw in the ellipsometry spectrum, interband transitions at ∼ 825 nm introduce a dip in the FOM for all five samples.

In Fig. 4(d) we plot the FOM for the high temperature sputtered and CMP’d film with the Johnson and Christy data from three other well-studied plasmonic materials: gold, silver, and copper [23]. From this plot, we see that in the blue and ultraviolet region of the spectrum, the SPR in aluminum is actually stronger than those in gold, silver, or copper; and this is in good agreement with results reported elsewhere [20]. Across most of the visible and infrared spectrum, silver is clearly the dominant material; however, it is important to note that aluminum outperforms both gold and copper for wavelengths below 642 nm.

4. Interband absorption

To understand the physical mechanisms that influence the real and imaginary components of the permittivity of each aluminum sample shown in Fig. 4(a) and 4(b), we analyze the complex optical conductivity based on both the Drude and two interband terms that contribute to the electron relaxation times. It has been previously shown that these effects dominate the optical properties of the samples due to the “parallel-band” effect [3], and this corresponds to the hexagonal (111) and (200) Brillouin-zone faces. For aluminum, this type of analysis is straightforward because its absorption spectrum in the range studied here is dominated by a single (200) parallel band transition at 1.55 eV [6, 24, 3]. This allows accurate fitting of both the Drude and interband terms.

Previously, Ashcroft and Sturm [6] showed that for polyvalent metals including aluminum, the bulk properties can be expressed as a function of the photon-energy independent parameters [25, 26]. Following this approach, we split the dielectric constant model into two parts, the Drude contributions and the interband contributions, which are given by

ε_{r} = 1 + \frac{4 π σ_{i}}{ω}

ε_{i} = \frac{4 π σ_{r}}{ω}

σ_{r} = 8 σ_{r, 111}^{(I B)} + 6 σ_{r, 200}^{(I B)} + σ_{r}^{(D)}

σ_{i} = 8 σ_{i, 111}^{(I B)} + 6 σ_{i, 200}^{(I B)} + σ_{i}^{(D)}

where

σ_{r}^{(D)}

and

σ_{i}^{(D)}

are the real and imaginary components of the Drude contribution to the optical conductivity; and

σ_{r}^{(I B)}

and

σ_{i}^{(I B)}

are the real and imaginary components of the interband contribution to the optical conductivity. The full expression for the four optical conductivity terms is listed in Appendix 6.

The broadband real and imaginary permittivity for each of the aluminum samples was fit to the model in Equations 7 - 6 using the nonlinear optimization with mesh adaptive direct search (NOMAD) algorithm [27, 28, 29, 30]. This is a gradient-free approach that varied the Drude electron relaxation time (τ^(D)), the electron relaxation time due to interband transitions (τ^(IB)), the Drude optical mass relative to the electron mass (m_opt/m), and the Fourier coefficients (U_K) of the pseudo potential associated with the transitions for the reciprocal lattice vector K. This was done to minimize the sum of the RMS error, at all wavelengths, between the model and the ε_r and ε_i curves simultaneously. The results from each optimization are shown in Table 3.

Table 3. Optical properties for the five aluminum samples extracted by fitting the ellipsometry data in Fig. 4 to the Ashcroft and Sturm model of optical conductivity. For comparison, the last row lists grain size measurements from Table 1, and the last column lists experimental and theoretical predictions in the literature. The order in which the numbers are listed corresponds to the order in which the references are listed.

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For each of the five variables in Table 3, we also list experimental and theoretical predictions in the literature. For the literature values of each variable, the order in which the numbers are listed corresponds to the order in which the references are listed. Also, the order in which the five aluminum samples are listed left to right, corresponds to a decreasing FOM from Fig. 4, with single crystal aluminum having the highest FOM. We also note the direct correlation between the scattering constant due to interband transitions τ^(IB), and the microstructure of the samples in the form of the median grain diameter. This is also the case for the relative FOM’s. For τ^(IB), the single crystal sample has a value of τ^(IB) = 0.484 × 10⁻¹⁴ s which is very close to the theoretically calculated value of τ^(IB) = 0.5 × 10⁻¹⁴ s used in the original work by Ashcroft and Sturm [6]; while the evaporated film has a value of τ^(IB) = 0.227 × 10⁻¹⁴ s which is very close to evaporated films studied by Shinya et al. who reported τ^(IB) = 0.5 × 10⁻¹⁴ s [32].

We also note an inverse correlation between the Drude optical mass relative to the electron mass m_opt/m, and the FOM’s. We see that the lower ratios agree with the theoretically reported values of Brust [33] or Ashcroft and Sturm [6] while the higher values in the present work corresponding to reported values by Shinya et al. [32].

Both trends correspond to a modification of the band-structure in each sample, when compared with single crystal values. This agrees with previous work which established that microstructure can strongly affect the optical properties of metal films [34], and that for decreasing grain size, the intrinsic optical response of a film is modified such that the band-structure is broader and less well-defined [3, 35, 36, 37]. These results indicate that the deposition of aluminum at elevated temperature with subsequent polishing produces a significant improvement in the plasmonic properties of aluminum films when compared with room temperature films deposited using either sputtering or evaporation.

During the fitting procedure, it was observed that τ^(D) did not have a strong influence on the overall fits for each sample. To obtain a more accurate fit of τ^(D), ellipsometric measurements at longer wavelengths are required. For each optimization, τ^(D) would consistently run to the upper bound (3.00 × 10⁻¹⁴ s) to maximize the quality of the overall fit. This trend is in agreement with results previously reported by Shinya et al. [32], and decreasing the upper bound to 1.00 × 10⁻¹⁴ s maintained the same quality of fit between 300 and ∼ 900 nm, with small deviations in the fit between ∼ 900 and 1000 nm.

Previous efforts to study the optical conductivity of aluminum have noted that at wavelengths shorter than 1000 nm, the (111) parallel-band contribution to the permittivity is weak and optical measurements at longer wavelengths were required to accurately determine the value of U₁₁₁ [26, 32]. As a result, literature values of 0.19 eV [32] and 0.25 eV [26, 38] were set as constants, and each optimization allowed U₁₁₁ to vary between 0.19 and 0.25. This showed a significant improvement in the overall quality of the final fits.

Finally, for every optimization, the (200) interband transition energy, U₂₀₀, was allowed to vary between 0.1 and 1 eV, and for all five samples the optimal value was between 0.729 and 0.744 eV, which agrees with results reported elsewhere. This is a direct result of the fact that the (200) parallel band transition at 825 nm was accurately measured within the range of each ellipsometry measurement.

5. Summary

As an essential step toward developing effective plasmonic devices with Al, we have established a method to quantify and optimize the plasmonic behavior of Al thin films. We have developed an experimental procedure using liquid prism-coupled spectroscopic ellipsometry, complemented by the extraction of the plasmonic $FOM = ε_{real}^{2} (ω) / ε_{imag} (ω)$ and its analysis in terms of underlying material properties. This procedure was applied to Al thin films deposited by four different methods, as well as to single crystal Al. The best performance was achieved with Al films that have been sputter-deposited at high temperatures of 350°C, followed by CMP. The high-temperature deposition yielded large grains of ∼ 1.4 μm in-plane diameter, while the CMP provided for planar interfaces with 1.8 nm RMS roughness. The wavelength of strongest SPR coupling of optimized Al films occurred between 380 and 420 nm. The plasmonic FOM for the polished Al was compared to the literature values of Au, Ag, and Cu. At wavelengths shorter than 420 nm, Al exhibited the highest FOM, while at longer wavelengths Ag had the highest FOM.

The Ashcroft and Sturm theory of optical conductivity was fit to both the real and imaginary permittivities for each of the five samples to extract the electron scattering times for both Drude and effective interband transitions, interband transition energies, and the optical mass of electrons. The results of this approach are shown to be consistent with theoretical and experimental predictions in the literature. We observe a direct correlation between the FOM for each sample and the scattering time due to interband transitions. Additionally, the Drude optical mass of electrons in samples scales inversely with the samples FOM. This is attributed to a modification of the band structure in each sample with microstructure.

These results indicate that Al-based plasmonics is a promising field, which can extend many applications to the UV-blue wavelength range, while benefiting from the availability of fabrication tools currently used in advanced microelectronics.

6. Appendix A

The full Ashcroft and Sturm expression for the complex optical conductivity of aluminum, based on both Drude and interband transitions is given by

ε_{r} = 1 + \frac{4 π σ_{i}}{ω}

ε_{i} = \frac{4 π σ_{r}}{ω}

σ_{r} = 8 σ_{r, 111}^{(I B)} + 6 σ_{r, 200}^{(I B)} + σ_{r}^{(D)}

σ_{i} = 8 σ_{i, 111}^{(I B)} + 6 σ_{i, 200}^{(I B)} + σ_{i}^{(D)}

σ_{r}^{(I B)} = σ_{a} (a_{0} K) | \frac{2 U_{K}}{ℏ ω} | {{[1 - {(\frac{2 U_{K}}{ℏ ω})}^{2} - {(\frac{1}{ω τ^{(I B)}})}^{2}]}^{2} + \frac{4}{{(ω τ^{(I B)})}^{2}}}^{- 1 / 4} \frac{{(ω τ^{(I B)})}^{2}}{1 + {(ω τ^{(I B)})}^{2}} J (ω)

where the function J(ω) is defined as

\begin{array}{l} J (ω) & = \frac{4 z b ρ}{π (z^{2} + b^{2})} {tan}^{- 1} t_{0} + \frac{1}{2 π} (\frac{z^{2} - b^{2}}{z^{2} + b^{2}} cos ϕ + \frac{2 z b}{z^{2} + b^{2}} sin ϕ) ln (\frac{t_{0}^{2} + 2 t_{0} ρ sin ϕ + ρ^{2}}{t_{0}^{2} - 2 t_{0} ρ sin ϕ + ρ^{2}}) \\ + \frac{1}{π} (\frac{z^{2} - b^{2}}{z^{2} + b^{2}} sin ϕ - \frac{2 z b}{z^{2} + b^{2}} cos ϕ) [{tan}^{- 1} (\frac{t_{0} - ρ sin ϕ}{ρ cos ϕ})] \end{array}

and

\begin{array}{l} σ_{i}^{(I B)} & = σ_{a} a_{0} K \frac{1}{2 b π ρ} (\frac{1}{2} sin ϕ_{1} ln (\frac{z_{0}^{2} - 1 + 2 {(z_{0}^{2} - 1)}^{1 / 2} ρ cos ϕ_{1} + ρ^{2}}{z_{0}^{2} - 1 - 2 {(z_{0}^{2} - 1)}^{1 / 2} ρ cos ϕ_{1} + ρ^{2}}) \\ + cos ϕ_{1} [{tan}^{- 1} (\frac{{(z_{0}^{2} - 1)}^{1 / 2} + ρ cos ϕ_{1}}{ρ sin ϕ_{1}}) + {tan}^{- 1} (\frac{{(z_{0}^{2} - 1)}^{1 / 2} - ρ cos ϕ_{1}}{ρ sin ϕ_{1}})] \\ + \frac{b^{2} - z^{2}}{b^{2} + z^{2}} {\frac{4 b z ρ}{b^{2} + z^{2}} {tan}^{- 1} {(z_{0}^{2} - 1)}^{1 / 2} + \frac{1}{2} (\frac{z^{2} - b^{2}}{z^{2} + b^{2}} cos ϕ + \frac{2 z b}{z^{2} + b^{2}} sin ϕ) \\ \times ln (\frac{z_{0}^{2} - 1 + 2 {(z_{0}^{2} - 1)}^{1 / 2} ρ sin ϕ + ρ^{2}}{z_{0}^{2} - 1 - 2 {(z_{0}^{2} - 1)}^{1 / 2} ρ sin ϕ + ρ^{2}}) + \frac{z^{2} - b^{2}}{z^{2} + b^{2}} (sin ϕ - \frac{2 z b}{z^{2} + b^{2}} cos ϕ) \\ \times [{tan}^{- 1} (\frac{{(z_{0}^{2} - 1)}^{1 / 2} + ρ sin ϕ}{ρ cos ϕ}) + {tan}^{- 1} (\frac{{(z_{0}^{2} - 1)}^{1 / 2} - ρ sin ϕ}{ρ cos ϕ})]}) \end{array}

σ_{r}^{(D)} = σ_{a} (2 a_{0} k_{F}) \frac{8}{π} \frac{E_{F} τ^{(D)}}{ℏ [1 + {(ω τ^{(D)})}^{2}]} \frac{m}{m_{opt}}

σ_{i}^{(D)} = - σ_{r}^{(D)} ω τ^{(D)}

where

ϕ = \frac{1}{2} [\frac{π}{2} - {tan}^{- 1} (\frac{1 + b^{2} - z^{2}}{2 z b})], ϕ_{1} = \frac{1}{2} [\frac{π}{2} + {tan}^{- 1} (\frac{1 + b^{2} - z^{2}}{2 z b})]

ρ = {[{(1 + b^{2} - z^{2})}^{2} + 4 z^{2} b^{2}]}^{1 / 4}

z_{0} = ℏ ω_{0} / 2 | U_{K} |, z = ℏ ω / 2 | U_{K} |

ω_{0} = \frac{1}{ℏ} [2 {(\frac{ℏ^{4} k_{F}^{2} K^{2}}{4 m^{2}} + U_{K}^{2})}^{1 / 2} - \frac{ℏ^{2} K^{2}}{2 m}]

b = ℏ / (2 τ^{(I B)} | U_{K} |), b / z = 1 / ω τ^{(I B)}

t_{0} = {(z_{0}^{2} - 1)}^{1 / 2}, σ_{a} = \frac{e^{2}}{24 π ℏ a_{0}}

Here, a₀ is the Bohr radius, m_opt/m is the Drude optical mass relative to the electron mass, k_F and E_f are the Fermi wave vector and Fermi energy, τ^(D) is the Drude electron relaxation time, τ^(IB) is the electron relaxation time due to interband transitions, and ħ is the reduced Planck constant. U_K is the Fourier coefficient of the pseudo potential associated with the transitions for the reciprocal lattice vector K. For

σ_{r, 111}^{(I B)}

and

σ_{i, 111}^{(I B)}

,

K = 2 π \sqrt{3} / a

and U_K = U₁₁₁. Similarly, for

σ_{r, 200}^{(I B)}

and

σ_{i, 200}^{(I B)}

, K = 4π/a and U_K = U₂₀₀.

Acknowledgments

This work was sponsored by the Assistant Secretary of Defense for Research & Engineering under Air Force Contract #FA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Government.

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	Crystal	CMP	MBE	RT Sputtered	Evaporated
RMS Roughness (nm)	2.1	1.80	0.43	1.63	6.9
Median Grain Diameter (nm)	> 100, 000	1438	939	124.2	52.2

	Crystal	CMP	MBE	RT Sputtered	Evaporated
RMS Roughness (nm)	2.1	1.80	0.43	1.63	6.9
Intermix Thickness (nm)	2.10	2.45	0	0	3.32
Intermix Fraction (%)	25	25	0	0	25
Oscillators	D,T-L,L,L	D,T-L,L,L	D,T-L,L	D,T-L,L	D,T-L,L

	Crystal	CMP	MBE	RT Sputtered	Evaporated	Literature
τ^(IB) × 10⁻¹⁴ (s)	0.484	0.420	0.353	0.334	0.227	0.5, 0.38, 0.26 [6, 31, 32]
τ^(D) × 10⁻¹⁴ (s)	2.64	3.00	3.00	3.00	3.00	≥ 1.00 [32, 26]
m_opt/m	1.30	1.40	1.56	1.56	1.92	1.45, 1.55, 1.76 [33, 6, 32]
U₂₀₀ (eV)	0.744	0.738	0.729	0.736	0.744	0.73, 0.74 [32, 31]
U₁₁₁ (eV)	0.25	0.19	0.19	0.19	0.19	0.25, 0.19 [26, 32]
Median Grain Diameter (nm)	> 100, 000	1438	939	124.2	52.2

	Crystal	CMP	MBE	RT Sputtered	Evaporated
RMS Roughness (nm)	2.1	1.80	0.43	1.63	6.9
Median Grain Diameter (nm)	> 100, 000	1438	939	124.2	52.2

	Crystal	CMP	MBE	RT Sputtered	Evaporated
RMS Roughness (nm)	2.1	1.80	0.43	1.63	6.9
Intermix Thickness (nm)	2.10	2.45	0	0	3.32
Intermix Fraction (%)	25	25	0	0	25
Oscillators	D,T-L,L,L	D,T-L,L,L	D,T-L,L	D,T-L,L	D,T-L,L

Aluminum plasmonics: optimization of plasmonic properties using liquid-prism-coupled ellipsometry

Abstract

1. Introduction

2. Experiment

2.1. Liquid prism coupling to surface plasmons

3. Measurements

3.1. Ellipsometry modeling

3.2. Ellipsometry results

4. Interband absorption

5. Summary

6. Appendix A

Acknowledgments

References and links

Cited By

Figures (4)

Tables (3)

Equations (21)

Optics Express