1. Introduction

Silver (Ag) has been the preferred material for plasmonic applications at optical frequencies due to its significantly lower losses than other noble metals. Nevertheless, loss due to grain boundaries and surface roughness still presents a serious challenge. For example, the propagation length of surface plasmon polaritons (SPPs) is limited to a few microns in thermally deposited polycrystalline Ag films in the visible frequency range [1]. Such limitations have been greatly reduced in template striped films with atomically smooth surfaces [2] and single crystalline Ag films that further remove the grain boundaries [1]. A number of recent studies have demonstrated the growth of high-quality single crystalline films using either molecular beam epitaxy (MBE) [3–5] or wet-chemistry synthetic methods [5–9]. Careful characterization measurements have demonstrated the superior properties of these single crystalline films with SPP propagation length exceeding 100 µm in the visible frequency range [4,5,9].

Knowledge of the dielectric function of Ag in the long wavelength range plays an important role to understand and predict the optical properties of infrared (IR) plasmonic devices. However, the results reported in previous measurements show significant discrepancies due to different sample preparation procedure and most of these studies only cover a limited energy range [1,4,10–20]. In this paper, we report the optical constants of Ag in the long wavelength range (1.24 – 7 µm) by performing spectroscopic ellipsometry (SE) measurements on a MBE- grown, atomically smooth, single crystalline Ag film. We confirm that the extracted optical dielectric constants satisfy the Kramers-Kronig (K-K) relation. We extend the previous measurements [4] of optical constants of single crystalline Ag film in the visible frequency range to the IR frequency range of 0.18 – 1 eV. These measurements require a sample with an extremely flat surface, large lateral dimensions (∼ cm), and thickness (> 300 nm). Such samples have only become available recently following the development of a rapid MBE growth method [21]. Our measurements have demonstrated that the loss is significantly lower in the IR frequency range than the widely quoted Palik’s optical constants [10]. Using the measured optical constants, we compare other optical properties including SPPs propagation length and Q-factors with that of other films. These results will inform researchers working with Ag as an optical material in the long wavelength range.

2. Experiments and results

Atomically smooth epitaxial silver film. Our previous studies of epitaxial Ag and Al thin films, grown by MBE in an ultra-high-vacuum (UHV) chamber, are obtained via a time-consuming two-step method [3–5,22], making it impractical to produce optically thick Ag film for characterization and applications in the IR frequency range. In order to overcome this challenge, a new MBE growth method with a rapid deposition rate 3 nm/min was introduced [21]. A single-crystalline, atomically smooth, and optically thick (≥ 300 nm) epitaxial Ag film was grown on heavily doped Si(111)-7 × 7 substrate (see Appendix A for sample growth detail). The high-quality surface of epitaxial Ag film with oxide capping layer was confirmed by atomic force microscopy (AFM) shown in Fig. 1(a), showing a root mean square (rms) surface roughness of 0.44 nm. Furthermore, the crystallinity of the epitaxial Ag film was characterized by X-ray diffraction (XRD) analysis and only one diffraction peak at 38.2° is observed in the 2θ scan, originating from the Ag(111) crystal plane, as shown in Fig. 1(b). The full width at half-maximum (FWHM) of the Ag(111) peak is about 0.8°, demonstrating the high quality of crystallinity of the thick Ag film.

 

Fig. 1. (a) AFM image of the epitaxially grown 300 nm Ag film (about 2 nm Al₂O₃ capping layer). (b) XRD 2$\theta $ pattern of the epitaxially grown Ag films with different thickness (100, 150, and 300 nm). The Ag(111) peak of 300 nm Ag film shows a FWHM of ∼ 0.8°. (c) Layered structure of our thick Ag film sample with a self-oxidized cap.

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SE measurement in the long wavelength range. Standard reflection measurements with K-K analysis have been used to determine the complex dielectric function of opaque materials. However, uncertainty can exist in the extracted dielectric constants [23]. In contrast, SE simultaneously measures both the amplitude ratio and phase difference of polarized light, $\rho = {\raise0.7ex\hbox{${{r_p}}$} \!\mathord{\left/ {\vphantom {{{r_p}} {{r_s}}}} \right.}\!\lower0.7ex\hbox{${{r_s}}$}} = \tan \psi {e^{ - i{\Delta}}}$, which provides direct access to the real and imaginary parts of the dielectric function without having to rely on a K-K analysis procedure [24]. In our SE measurements, two variable-angle spectroscopic ellipsometers (VASE and IR-VASE, J. A. Woollam) were used in the spectral range of 0.50 – 4.2 eV and 0.18 – 0.62 eV, respectively. All SE measurements were taken at three angles of incidence of 60°, 67.5°, and 75°, and analyzed simultaneously using WVASE software (J. A. Woollam). No incident angle dependence in the effective dielectric function was observed, verifying the expected isotropic optical properties of Ag. In order to extract the complex dielectric function, $\varepsilon = {\varepsilon _1} + i{\varepsilon _2}$, we fit the entire data set with a combination of the Drude model and a parametric oscillator model [25] (see Appendix C for details). In the energy region of 0.18 – 1 eV that we focus on, the Drude model is applied. These models ensure that the extracted optical constants satisfy the K-K consistency requirement over the entire spectral range (see Fig. 6. in Appendix E). Moreover, an iterative fitting method is applied based on the layered structure shown in Fig. 1(c). The multilayer structure consists of a 2 nm Al₂O₃ capping layer, a 300 nm epitaxial Ag film, and a Si substrate. We start the fitting from a set of initial dielectric constants for each layer using independently measured values for the Si substrate and a reference data file included in the WVASE software for the 2 nm Al₂O₃ cap layer. We do not expect that either the values for Si or Al₂O₃ have appreciable influence on the extracted optical constants of Ag. The 300 nm Ag film is optically thick, which was confirmed while analyzing the ellipsometric data, therefore the Si substrate has no influence. Thus, the extracted optical constants are due to the epitaxial Ag.

Complex dielectric function of epitaxial silver film. Negative real part ($- {\varepsilon _1}$) and positive imaginary part (${\varepsilon _2}$) of the optical dielectric constants of the epitaxial Ag in the spectral range of 0.18 – 1 eV are plotted in Fig. 2(a) and 2(b) (red). The results from Palik’s Handbook of optical constants [10] (green), Johnson and Christy (JC) [11] (gray), Yang et al. [16] (blue), and McPeak et al. [17] (purple) are shown for comparison. This comparison demonstrates the variations among the existing literature values in this relatively narrow spectral range. The insets in Fig. 2(a) and 2(b) zoom in the region difficult to distinguish between different sets of measurements. The residues in Fig. 2(a) and 2(b) are the difference between the calculated effective dielectric function and the experimentally measured effective dielectric function. All residues in this region are centered around zero, suggesting that the experimental SE data is fitted well with the chosen model. In the narrow range of 0.5 – 0.6 eV, one can observe noise that is higher than the other spectral region. We suggest that these higher uncertainties are associated with the residual instrumental errors since we included two data sets taken by two different ellipsometers in this spectral region. The two data sets are simultaneously fitted to extend the spectral range. Large uncertainty of ${\varepsilon _2}$ exists in JC’s data [11] (∼ 40%) as indicated by the gray shaded curves in the energy range of 0.64 – 1 eV. This uncertainty likely resulted from the poor detector efficiency in the long wavelength range in those previous measurements. The dielectric constants in the range of 1 – 4.2 eV are shown in Appendix D because they are not the focus of our study.

 

Fig. 2. (a) Negative real part and (b) imaginary part of dielectric function of epitaxial Ag film from 0.18 to 1 eV (red). Data from Ref. [10,11,16,17] are shown together with gray shaded curve representing the uncertainties in the JC data for comparison. The fitting residues for the epitaxial Ag film are plotted below (a,b). See Data File 1 for optical constants.

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Fig. 3. Plot (a)-(c) show the calculated surface plasmon propagation lengths (L_SPP) and the quality factors for localized surface plasmon resonances (Q_LSPR) and surface plasmon polaritons (Q_SPP) in the infrared spectral ranges from this work and literatures [10,16].

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3. Discussion

Optical properties of epitaxial silver and Drude model. Optical properties of metals are determined by both free and bound electrons. The free electron contribution to the dielectric function is significantly increased with decreasing frequency, where the optical properties are dominated by intraband transitions within the conduction band. In this low energy region, the complex dielectric function of a metal, described by the Drude model with the approximation of a non-interacting conduction electron gas, is given by

In order to demonstrate the potential performance improvement of IR-plasmonic structures based the epitaxial Ag, we calculate the SPPs propagation lengths for SPPs and quality factors [31] for localized surface plasmon resonances (LSPR) using the measured optical constants, as shown in Fig. 3 (details on SPPs propagationg lengths and quality factors in Appendix B). Table 1 shows these values at a few selected wavelengths and compares them with previous reports. Our measurements show a slight improvement in the broad spectral range from visible to IR range in comparison with those data from Yang (template-stripped film). For example, a factor of 17 – 20% improvements are shown in ${L_{SPP}}$, ${Q_{LSPR}}$, and ${Q_{SPP}}$ at $\lambda $ = 7 $\mu m$ (see Appendix F for optical performances). The improvement over Palik’s work (polycrystalline film) is more notable. we show that our epitaxial film exhibits 158% improvement in ${L_{SPP}}$, a 66% improvement in ${Q_{LSPR}}$, and a 155% improvement in ${Q_{SPP}}$ at $\lambda $ = 7 $\mu m$, in comparison to Palik’s data. These comparisons highlight the advantages of the epitaxial films in different plasmonic applications.

Table 1. The calculated surface plasmon propagation lengths (L_SPP) are shown along with the quality factors for localized surface plasmon resonances (Q_LSPR) and surface plasmon polaritons (Q_SPP) based on the dielectric constants from this work (red) and literatures^{a, b} at ultraviolet (370 nm), visible (650 nm), near-infrared (1000 nm), telecommunication (1550 nm), and infrared (7 µm) wavelengths.

View Table

4. Conclusion

In summary, we report optical dielectric constants of epitaxially-grown, optically-thick single crystalline Ag film from 0.18 to 1 eV by SE. As expected, intrinsic loss of our thick Ag film is larger than that of the thin Ag film grown by a more elaborate two-step method [4] but smaller than that found in thermally deposited polycrystalline films even after template stripping is applied to remove surface roughness. The dielectric constants reported here represent the intrinsic optical properties of bulk Ag in the long wavelength range. We expect that these measured data will guide the simulations for IR-plasmonic applications based on single crystalline Ag.

Appendix

A. Epitaxial silver film preparation

We recently developed a two-step growth method by MBE under ultra-high vacuum (below 5 × 10⁻¹⁰ Torr) [21]. We combine a room-temperature, high-rate deposition, and a high-temperature-annealing procedure to prepare optically thick epitaxial Ag film with a greatly improved growth efficiency. In this way, only a few hours of one growth cycle is required to ensure the high quality of an optically thick, atomically smooth, single-crystalline Ag film. First, Ag is evaporated onto the Si (111) substrate at room temperature with a deposition rate of about 3 nm/min, which is about 30 times faster than a previous method to grow epitaxial thin Ag films [4]. Then, the Ag film is transferred to the molybdenum heater equipped with tungsten filaments in the same MBE chamber for in-situ annealing. The annealing temperature is maintained at 500 °C for half an hour. Since exposing bare Ag film to the ambient can result in its rapid oxidization and contamination, in-situ capping of AlO_x is applied to prevent such deteriorations. 7 ML (∼1.6 nm) of epitaxial Al layer is first grown on the Ag film and exposure to high purity oxygen under 1.5 ×10⁻⁶ Torr pressure for 10 minutes follows. As a result, a high-quality AlO_x capping layer on the Ag film is formed to protect the surface.

B. SPP propagation length and quality factors

The dispersion relationship for SPPs propagating along the interface between a metal and a dielectric is given by [26]

(2)$${k_{SPPs}} = \frac{\omega }{c}\sqrt {\frac{{{\varepsilon _m}{\varepsilon _d}}}{{{\varepsilon _m} + {\varepsilon _d}}}} $$

where $\omega $ is the angular frequency, c is the speed of light, ${\varepsilon _m}$ and ${\varepsilon _d}$ are the permittivity of the metal and dielectric, respectively. The SPP propagating length, ${L_{SPPs}}$, is determined from the imaginary part of the SPP wavevectors, ${k_{SPPs}}$,

(3)$${L_{SPPs}} = \; \frac{1}{2k_{SPPs}^{\prime\prime}} = {\lambda _0}\frac{{{{({\varepsilon_m^{\prime}})}^2}}}{2\pi \varepsilon _m^{\prime\prime}}{\left( {\frac{{\varepsilon_m^{\prime} + {\varepsilon_d}}}{{\varepsilon_m^{\prime}{\varepsilon_d}}}} \right)^{\frac{3}{2}}}$$

where $\varepsilon _m^{\prime}$ and $\varepsilon _m^{\prime\prime}$ are the real and imaginary part of ${\varepsilon _m}$, respectively. When metal exhibits low loss, satisfying the condition $|{\varepsilon_m^{\prime}} |\gg |{{\varepsilon_d}} |$, ${L_{SPPs}}$ can be approximated as ${L_{SPPs}} \approx {\lambda _0}\frac{{{{({\varepsilon_m^{\prime}} )}^2}}}{2\pi \varepsilon _m^{\prime\prime}}$.

To evaluate the loss of a noble metal in plasmonic applications, dimensionless quality factors are defined for both localized surface plasmon resonances (LSPRs) and surface plasmon polaritons (SPPs) as [31]

(4)$$\begin{array}{c} {Q_{LSPR}} = - \varepsilon^{\prime} / \varepsilon^{\prime\prime}, \\ Q_{SPP} = \varepsilon^{{\prime}{2}}/\varepsilon ^{\prime\prime}. \end{array}$$

Higher quality factors are often correlated with sharper resonances and the stronger local field enhance in plasmonic applications. We plot these quantities to evaluate our Ag film quality.

C. Parametric optical constant model

The parametric model (PM) developed by Johs et. al. has been used to fit optical dielectric constants over a broad spectral region with improved accuracy [25]. In this work, Psemi-M0 model, a generalized PM provided in WVASE32 software, is used to describe the interband transition around 4 eV. A representative, asymmetric function for ${\varepsilon _2}$ is plotted in Fig. 4 and the parameters in this model are defined as follows:

• A, E₀, B = Amplitude, center-energy position, and broadening.
• WR = Endpoint position relative to the center-energy position (E₀). It changes the spectral width on the right side of the central energy.
• PR = Horizontal position of the right control point relative to the center-energy position (E₀) and the endpoint.
• AR = Relative magnitude of the right control point (compared to the A)
• O2R = Coefficient for the 2^nd order terms in the polynomials on the right side of the oscillator. It provides additional flexibility to the function.

 

Fig. 4. Representative functions of the Psemi-M0 oscillators are plotted to show how the parameters change the functions in panels (a-d).

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D. Optical dielectric function of epitaxial silver film from 1 to 4.2 eV

In the energy range from 1 to 3.8 eV, Palik’s optical constants show the highest ${\varepsilon _2}$ values among the literature values represented in Fig. 5. For JC’s data, large uncertainties in the optical constants (i.e., up to 40%) exist. We note that ${\varepsilon _2}$ values in Wu’s optical constants are lower than other measurements because the highest quality films were measured. The comparison between our current work and those from Yang and McPeak’s results will be discussed further in Appendix D.

 

Fig. 5. (a) Negative real part and (b) imaginary part of dielectric function of epitaxial 300 nm Ag from 1 to 4.2 eV (red). Data from Palik’s Handbook of optical constants [10] (green), JC [11] (gray), Wu et al. [4] (orange), Yang et al. [16] (blue), and McPeak et al. [17] (purple) are shown for comparison. The gray shaded curve represents the uncertainties in the JC data and the fitting residues for the epitaxial Ag film are plotted below (a,b) for our measurements. The fitting yields higher errors in the interband transition region. The inset of (a) shows $-{\varepsilon _1}$ near 3.8 eV in linear scale, where the transition from negative to positive values concurs at the interband transition.

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E. Kramers-Kronig consistency

 

Fig. 6. Real (blue) and imaginary (green) part of optical dielectric function of epitaxial Ag film from ellipsometry measurement are shown together with a Kramers-Kronig (K-K) consistent fit (red). The inset is plotted in linear scale from 3.6 to 4.2 eV to emphasize K-K consistency near the interband transition.

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F. Optical performance of epitaxial silver film

In a broad spectral range from 0.18 to 3.6 eV, the calculated L_SPP and Q-factors in Fig. 7 demonstrate better optical performances of plasmonic structures based on our epitaxial Ag than those based on the optical constants reported by Palik and Yang. In the range of 1.24 to 3 eV, Wu’s measurement shows the best optical performances due to the highest quality of film. McPeak’s measurements suggest better optical performances than those from Yang’s even though both studies were performed on template-stripped Ag film. It could be due to better vacuum condition (∼ 10⁻⁸ mbar) in McPeak’s work compared to Yang’s work (∼ 10⁻⁶ mbar) during film deposition processes, or different growth rates that result in different grains. In this higher energy range, our results are comparable to those from McPeak within error bars.

 

Fig. 7. Calculated SPP lengths (L_SPP) and the quality factors for localized surface plasmon resonances (Q_LSPR) and surface plasmon polaritons (Q_SPP) in the (a) infrared and (b) visible spectral ranges, respectively.

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Funding

National Science Foundation (DMR-1808042); Welch Foundation (F-1662, F-1802); Ministry of Science and Technology, Taiwan (105-2112-M-007-011-MY3, 105-2633-M-007-003); Air Force Office of Scientific Research (FA9550-15RYCOR162, FA9950-15RYCOR159).

Acknowledgments

The collaboration between National Tsing-Hua University and The University of Texas at Austin is facilitated by the Global Networking Talent 3.0 Program, Ministry of Education in Taiwan. We would like to thank Tom Tiwald at J. A. Woollam for help in analysis of ellipsometry measurements.

Disclosures

The authors declare no competing interests.

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