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While plasmonic metals can manipulate optical energy at the nanoscale, they suffer from significant losses at visible wavelengths. We investigate the potential of low temperature to decrease such losses in optically thick Ag films. We extract the complex dielectric function (or relative permittivity) from spectroscopic ellipsometry measurements for smooth single-crystalline, smooth polycrystalline, and rough polycrystalline films down to liquid-helium temperatures and fit these data to a temperature-dependent Drude model. Smooth single-crystalline films exhibited the largest improvements relative to room temperature. Below 50 K, the surface plasmon polariton propagation lengths increased by ~50% at 650 nm. In rough polycrystalline films, improvements of 10% are expected.
© 2015 Optical Society of America
1. Introduction
Surface plasmon polaritons are electromagnetic oscillations that exist at metal surfaces [1, 2]. They allow sub-diffraction confinement and nanoscale manipulation of electromagnetic energy [3], enabling applications in sensing [4, 5], spectroscopy [6], and optoelectronic circuits [7, 8]. However, the most commonly used plasmonic metals, Ag and Au, which exhibit the best properties at visible frequencies, still suffer significant energy losses that limit performance [3, 9, 10]. On the high-frequency end of the visible spectrum, large losses occur due to interband transitions [11]. These can be avoided by operating at lower frequencies, but dissipation caused by electron-electron, electron-phonon, grain-boundary, and surface-defect scattering in the metal remains [12–22]. Despite attempts to reduce these scattering processes by decreasing surface roughness and increasing grain size [23–29], additional improvements are sought.
A possible route to limit further the effect of these scattering losses is to decrease temperature. Thus, several efforts have recently studied plasmonic behavior in metallic nanoparticles or thin films at low temperatures [10, 30, 31]. Linewidth narrowing was observed in the extinction spectra of cooled Au nanorod arrays [10]. Based on these results, a doubling of the surface-plasmon-polariton (SPP) propagation length was predicted at 860 nm for planar gold surfaces cooled to 80 K. However, experiments aiming to observe such an increase on thin Ag films reported only modest improvements of the SPP propagation length [31]. The measured propagation was limited by surface and defect scattering (i.e. film quality). To date, similar experiments have yet to be performed on high-quality Ag films.
More generally, reliable low-temperature values for the experimental dielectric function of Ag are not available. Instead, tabulated room-temperature data (e.g. those of Johnson and Christy [11]) are often combined with the Drude model to predict the low-temperature plasmonic response. This approach can be inaccurate for two reasons. First, experimental films often exhibit dielectric functions (ε) that are different than those reported by Johnson and Christy [11]. They often have less negative real components (εR) and more positive imaginary components (εIm). Second, the assumption of the Drude model with room-temperature parameters may not be applicable.
To provide more reliable data while also exploring the potential of low temperature to improve plasmonic performance, here we investigate the optical properties of high-quality Ag films at cryogenic temperatures. To avoid the interband transitions, where optical losses are high, we limit our experiments to wavelengths longer than 450 nm. We report the experimental dielectric function (relative permittivity) and tabulate the relevant Drude and Drude-Lorentz parameters for optically thick Ag films over a broad range of temperatures, T. We also examine how the quality of the film affects the optical response. Finally, we measure experimental SPP propagation lengths on ultrasmooth, single-crystalline Ag films at 298 and 25 K.
2. Experiment
2.1 Film preparation
We prepared three types of 200-nm-thick Ag films of different crystallinity and surface quality. First, single-crystalline films [28] (Sample SC) were grown epitaxially by sputtering Ag onto freshly cleaved mica substrates (Highest Grade V1 AFM Mica Discs, Ted Pella). Specifically, the mica substrates were pumped down in the sputtering chamber (Kurt J. Lesker PVD 75) to a pressure below 10−6 Torr. They were then heated to 360 °C at a rate of 3 °C/min and pre-annealed for 1 h. Ag (99.99%, Kurt J. Lesker) was sputtered at an optimized rate of 110 nm/min at 470 W and a target distance of 17 cm under an Ar pressure of 6 x 10−3 Torr. After a 30 min post-anneal, the films were allowed to cool to room temperature. Atomic force microscopy (AFM) measurements over a 2.5 x 2.5 µm2 area revealed a root-mean-squared (RMS) roughness of 0.47 nm [Fig. 1(a)].
Fig. 1 Atomic force microscope (AFM) images of the three types of Ag films investigated: (a) single-crystalline film (Sample SC), (b) template-stripped polycrystalline film (Sample TS), and (c) rough polycrystalline film (Sample Ro).
Second, smooth polycrystalline films (Sample TS) were obtained by thermal evaporation of Ag followed by template stripping [24, 32]. Ag (99.99%, Kurt J. Lesker) was deposited at 0.2 Å/s and 2 x 10−6 Torr via a Kurt J. Lesker Nano 36 onto clean Si wafers (sonicated in acetone, isopropanol, methanol, and water, each for 10 min). A glass counter substrate was then attached to the as-deposited Ag surface via thermal epoxy (EpoTek 377), which was cured at 130 °C for 1 h. The glass and metal film were then peeled off the silicon to reveal a Ag surface with 0.7-nm RMS roughness [Fig. 1(b)]. (We note that ε can be further improved in polycrystalline films if lower pressures and faster rates are used [33].)
Lastly, rough polycrystalline films (Sample Ro) were prepared by sputtering Ag at roomtemperature at 160 nm/min onto clean Si wafers using 470 W, 10 mTorr Ar pressure, and a target distance of 13 cm. A glass counter substrate was then attached to the as-deposited surface with optical epoxy (Norland NOA63) followed by template stripping [24, 32]. The measured roughness of the exposed Ag surface varied from sample to sample between 2 and 5 nm RMS depending on the region [see Fig. 1(c) for one example]. The particular sample used for dielectric-function measurements had an RMS roughness of 4.5 nm, determined from AFM images from two regions.
2.2 Ellipsometry measurements
To quantify the optical quality of the films we determined their complex dielectric functions via ellipsometry. The magnitude of εR indicates how much energy the metal can store, and εIm is related to loss. Plasmonic materials with a large negative εR and a small positive εIm are typically sought.
The experiments used freshly prepared, pristine films (2 days old). Each sample type was first tested in a liquid-N2 bath to confirm the absence of delamination upon cooling. Identical fresh films were then placed into a spectroscopic ellipsometer (J. A. Woollam, VUV-VASE) equipped with a cryostat. Data were collected under high vacuum (4 x 10−9 Torr) at temperatures of 298, 200, 100, 50, and 5 K at an incidence angle of 70°. By fitting to a two-layer vacuum-metal model (WVASE32 software) we extracted values for εR and εIm. We note that these represent effective dielectric functions that include both bulk and surface effects.
3. Results and discussion
3.1 Dielectric functions
Figures 2(a) and 2(b) plot εR and εIm versus wavelength, respectively, for Sample SC as afunction of temperature. With decreasing T, εR becomes slightly less negative while εIm decreases more significantly. The latter indicates that losses decrease at lower T, consistent with a reduction in scattering (confirmed by our analysis below). Figures 2(c) and 2(d) plot εR and εIm at 298 and 5 K for all three types of samples. (Digital files containing the actual ε values are available from the authors). The same trends seen in the single-crystalline film appear for the polycrystalline samples. For εIm, Sample Ro exhibits higher values overall. In such films, temperature-independent surface scattering remains significant even at low T. Samples SC and TS, which are smoother, reach lower values, with Sample SC being the best, as expected due to the lack of grains.
Fig. 2 The real (a) and imaginary (b) parts of the dielectric function for a single-crystalline Ag film (Sample SC) at various temperatures. The real (c) and imaginary (d) parts of the dielectric functions for: (i) Sample SC, (ii) a template-stripped smooth polycrystalline film (Sample TS), and (iii) a rough film (Sample Ro) at 298 (RT) and 5 K. The data were extracted from ellipsometry measurements.
3.2 Drude parameters
The Drude model permits a simple analysis of the temperature dependence. Because it assumes a free-electron gas moving against a fixed background of positive ions, it cannot account for interband electronic transitions, which become important for Ag in the blue region of the visible spectrum. For longer wavelengths, the complex dielectric function can be written as
where ε∞ is the dielectric constant at infinite frequency, ω is the angular frequency, ωp is the plasma frequency, and ωc is the damping rate (or collision frequency). ωc includes contributions from electron-electron scattering, ωee [22], and electron-phonon scattering, ωep [21], defined as ωee is a frequency and temperature-dependent term with Γ as the Fermi-surface average for the scattering probability, Δ the fractional Umklapp scattering coefficient, EF the Fermi energy (5.5 eV for Ag), ℏ Planck’s constant divided by 2π, and kB the Boltzmann constant. ωep is a frequency-independent term that depends on a constant, ω0, and the Debye temperature, θD (215 K for Ag). Because kBT is much smaller than ω, electron-electron scattering is very weakly temperature dependent [Eq. (2)]. Meanwhile, electron-phonon scattering is strongly temperature dependent [Eq. (3)]. Therefore, reduced electron-phonon scattering should be the origin of the decreased εIm at low T. [We note that, because Eqs. (1) to (3) neglect interband transitions as well as surface roughness and grain boundaries, they can only strictly be applied at long wavelengths to smooth single-crystalline films. However, below we use them to fit the optical data for all three of our sample types.]With this model, we extracted the Drude parameters for our three samples at each temperature. Table 1 summarizes the results. We also list the percent error (or normalized standard deviation) for each data set. The normalized deviations between the calculated and measured εR and εIm are determined at each wavelength. The RMS deviation for a set of all such values is then reported. This method was used to weight the real and imaginary deviations equally. While the extracted errors for all three sample types are small, they are slightly larger for the single-crystalline sample for unknown reasons.
Table 1. Drude Parameters Extracted from Fits of the Measured Dielectric Functions
Table 2 summarizes a second set of fits, exploiting a one-oscillator Drude-Lorentz model,
While even better agreement (<0.5% error) could be obtained with this model, which contains three additional parameters (s1, ω1, and γ1), it provides less insight into the physics of the temperature dependence. Thus, below we focus on the Drude model.Table 2. Drude-Lorentz Parameters Extracted from Fits of the Measured Dielectric Functions
As seen in Table 1, the electron-phonon scattering rate decreases monotonically with T for all films, as expected. Because the product ΓΔ is the fitting parameter that primarily compensates for the effect of surface scattering in the film, ΓΔ increases with grain density and surface roughness. Also, while ΓΔ was fit without constraints at each temperature, it varies only ~10% from the average for each film. This agrees with Eq. (2), which assumes these values are temperature independent. Lastly, note that the measured ΓΔ for Sample Ro is ~0.3, consistent with a previous estimate of ~0.4 [19].
So far we have not considered thermal expansion in the Drude model. The plasma frequency should be temperature dependent through the linear thermal expansion coefficient, γ, as
with T0 as the reference temperature. According to this expression, the plasma frequency should then increase with decreasing T. However, assuming 1.5 x 10−5 K−1 for γ [34], we expect only a + 0.7% change in ωp in going from 298 to 5 K. Experimentally, we find the opposite trend. The extracted plasma frequencies (Table 1) decrease by several percent upon cooling. In practice, the fits use ωp and ε∞ to correct for deviations from the Drude model, including the influence of interband transitions. When we instead constrain ωp according to Eq. (5), much worse fits are obtained.3.3 Plasmon propagation lengths
As shown above, cooling reduces εIm and thus plasmonic losses. Because losses limit the propagation distance of the SPPs, this distance should increase at low T. Using our measured dielectric functions, we calculated the expected SPP propagation lengths, LSPP, at the vacuum-Ag interface [1, 2] (see Fig. 3). Figure 4(a) shows the expected enhancement in LSPP at various temperatures (compared to 298 K) for Sample SC. Figure 4(b) compares the expected enhancement at 5 K for our three sample types. We see a clear dependence on the crystallinity and roughness of the Ag. Compared to SPP propagation lengths at room temperature, Sample SC at 5 K should show enhancements from 35 to 55% between 500 and 800 nm. Sample TS should show enhancements from 23 to 40% in this range. While the roughness of the single-crystalline and template-stripped films is similar, grain boundaries in the latter contribute to additional temperature-independent losses. Sample Ro had a much higher roughness and smaller grains. Consequently, it should exhibit smaller enhancements from 10 to 22% between 500 and 800 nm.
Fig. 3 Expected propagation lengths at various temperatures for a rough film (blue), template-stripped polycrystalline film (red), and single-crystalline film (black). These values are calculated from the measured dielectric functions using a two-layer vacuum-Ag model.
Fig. 4 (a) Expected enhancements at various temperatures (relative to 298 K) in the surface-plasmon-polariton (SPP) propagation length, LSPP, for Sample SC calculated from the measured dielectric functions. (b) The expected enhancements in LSPP at 5 K (relative to 298 K) for Samples SC, TS, and Ro. (c) Direct measurements of LSPP from single-crystalline Ag films at 298 (RT) and 25 K. 200-nm-wide slits and grooves were milled into four identical films using a focused ion beam. Illumination on the bottom of the film launches SPPs along the vacuum-Ag interface from the slit towards the groove. Scattered light from the groove is collected using a spectrometer. LSPP is extracted by fitting data from various slit-groove separations, d, to an exponential decay. Error bars represent standard deviations for the four samples. (d) Measured enhancements in LSPP determined from the data in (c).
To test these predictions, we also measured LSPP directly using the slit-groove technique [24, 28]. Pairs of parallel slits and grooves were etched into single-crystalline films with a dual-beam focused-ion beam (FIB) operating at 30 kV and 33 pA (FEI, Helios). All slits and grooves were 200 nm wide and 20 µm long; the slits were etched through the entire film andthe grooves were 50 to 60 nm deep. Each slit-groove pair was separated by a distance d, which we varied from 10 to 40 µm. When the films were illuminated from the backside with a 250-W tungsten-halogen lamp, SPPs were generated at the slits. They then propagated towards the groove. Spectra were obtained by collecting the scattered light at 298 and 25 K. Propagation lengths at each wavelength were determined by fitting an exponentially decaying function to the spectra for a range of d.
Figure 4(c) shows the measured LSPP values averaged from four different single-crystalline Ag films. As in previous slit-groove experiments on such samples [28, 29], our measured values are less than those predicted from our ε data (e.g. 12.5 versus 60 µm for 630 nm at RT, see Fig. 3). However, we note that unlike in Ref [28], we assumed that the SPPs are launched from the slits only in the normal direction. If we included an angular distribution of the launched SPPs, our measured LSPP would increase by up to 150%. Lower LSPP values could also be caused by film contamination (during the FIB process or general oxidation/sulfidation). Our sample preparation required more than a week for AFM, sample shipment, and FIB etching before optical measurements could be performed. However, despite these issues, the measured enhancements in LSPP [Fig. 4(d)] are in reasonable agreement with the expected values [Fig. 4(b)], especially at wavelengths longer than 580 nm where enhancements between 40 and 60% are observed.
4. Conclusion
In summary, we report experimental values for the dielectric functions of Ag films at cryogenic temperatures. The measurements were fit to Drude and Drude-Lorentz models and the parameters tabulated. The results are consistent with reduced electron-phonon scattering upon cooling. We show that this can lead to a 40 to 60% increase in the surface plasmon polariton propagation length for red wavelengths at low temperature. While our focus here has been on plasmonic applications, the dielectric functions and model parameters reported are useful for many low-temperature experiments and calculations on planar or structured Ag.
Acknowledgments
We acknowledge funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement Nr. 339905 (QuaDoPS Advanced Grant) and from GACR through grant number 15-13778S. S.-H.O. acknowledges support from the U.S. National Science Foundation (NSF CAREER Award) and J.H.P. from a Korea Institute of Science and Technology internal project.
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