Emissivity measurements on noble metals: validation of a free-electron description including the anomalous skin effect

Eur. Phys. J. Plus (2024) 139:1043 https://doi.org/10.1140/epjp/s13360-024-05819-3 Regular Article Emissivity measurements on noble metals: validation of a free-electron description including the anomalous skin effect Iñigo González de Arrieta1,a, Telmo Echániz2,b, Albert Adibekyan3,c, Christian Monte3,d, Jörg Hollandt3,e, Gabriel A. López1,f 1Physics Department, University of the Basque Country UPV/EHU, 48940 Leioa, Spain 2Applied Mathematics, University of the Basque Country UPV/EHU, 48013 Bilbao, Spain 3Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany Received: 23 July 2024 / Accepted: 8 November 2024 © The Author(s) 2024 Abstract The optical properties of noble metals (Cu, Ag, and Au) have been studied in the mid-infrared using emissivity measurements at 473 K. Their optical responses can be described using classical free-electron expressions in the Mott–Zener approximation with an additional term to account for the extra emission due to the anomalous skin effect, with no free parameters. Clear differences between the specular and diffuse surface scattering regimes can be observed. These results constitute metrological-quality validations of the thermal radiative properties of these important metals. 1 Introduction The optical and thermal radiative properties of metals are determined by several electron interactions that can take complicated forms, depending on the metal and the spectral range [1,2]. In particular, theoretical descriptions of the role of surface scattering have been traditionally complicated, especially for high-conductivity noble metals. The optical response of free-electron metals is often calculated by means of semiclassical transport equations, such as the Boltzmann equation, by assuming a local relationship between the electric field and the current [1,3]. This is often a good approximation, as the electric field is exponentially damped when penetrating a metal surface, a phenomenon described by the skin depth δ(ω)c/√2πωσ(ω). The picture of a frequency-dependent local conductivity σ(ω) breaks down for good conductors, when the electron mean free path lvFτ(with vFthe Fermi velocity and τthe relaxation time) becomes close to the value of the skin depth [4]. In this case, the carriers experience varying values of the electric field between scattering events, which leads to the current at each point being a function of the field at previous locations. The notion of optical constants thus becomes ill-defined in this spectral range [3]. Integral formulations of the transport equations are often the way to achieve numerical results to this complex problem [5–12]. The anomalous skin effect is particularly relevant for understanding the thermal radiative properties of noble metals, for which even minor contributions to their already low emissivities lead to significant variations. Thus, a quantitative description of the emissivity of noble metals and the quantification of the contribution from the anomalous skin effect are relevant questions. The difficulty of performing reliable emissivity measurements in these materials is evident from the low signal emitted by these highly reflecting metals. Therefore, most literature data is reported as total hemispherical emissivity values, measured using calorimetric methods [13–16]. However, for analytical purposes, spectrally resolved data in the infrared range are the most useful for direct verification of theoretical predictions. Unfortunately, the existing literature on these measurements for noble metals reports scattered results and contradictory interpretations [2,3,11,17–24]. In this work, we provide new data on the emissivity of noble metals and use them to validate this free-electron framework, particularly in connection to the role played by the anomalous skin effect. This effect is expected to be relevant not only for infrared optics, but also for its connections to the general behavior of these metals and their applications [11,25–28]. ae-mail: [email protected] (corresponding author) be-mail: [email protected] ce-mail: [email protected] de-mail: [email protected] ee-mail: [email protected] fe-mail: [email protected] 0123456789().: V,-vol 123 1043 Page 2 of 9 Eur. Phys. J. Plus (2024) 139:1043 2 Theoretical background In principle, the mid-infrared response of noble metals can be described in rather basic terms within the free-electron framework, because of the simplicity of their Fermi surfaces. Their good electrical conductivities and associated large relaxation times (even at moderately high temperatures) allow a further simplification in this frequency range: the Mott–Zener approximation ωτ > 1[2, 29]. In this approximation, the bulk emissivity of the metal is described simply as 2/ω pτ,whereωpis the plasma frequency [2]. Then, the influence of the anomalous skin effect can be modeled as an additional absorption by electron-surface scattering, which can be described by means of an empirical electronic specularity parameter, p. Therefore, to a first approximation, the emissivity in this high-frequency region can be calculated as follows [2]: ε2 ωpτ+(1−p)3vF 4c.(1) The calculation method is based on the Mott–Zener approximation described above, with no additional absorption mechanisms being considered. Thus, the emissivity of each metal is completely determined by intrinsic free-electron properties, which can be determined by non-optical means, as well as the phenomenological electronic specularity parameter p. This method is not applicable to metals with complex Fermi surfaces. In the case of simple metals under conditions which violate the Mott–Zener condition, more complex expressions can be used to retrieve the emissivity and the influence of the anomalous skin effect [2,21]. However, additional steps have been introduced to generalize the emissivity calculation procedure. In the free-electron model, only two parameters, the electron density and relaxation time, are required for a complete description of the optical response of the metal. Therefore, a generalization of Eq. (1) that depends only on directly measurable non-optical parameters is given by the following relations [2]: ε5.263 ·10−6(rs/a0)3/2Tr(T), (2) r(T)2411 (rs/a0)3 ρ∗ μ(T) T+p1995 (rs/a0)5/2T,(3) where r(T) is a generalization of the relaxation frequency that includes both the bulk and surface scattering terms, a0is the Bohr radius, rsis the free-electron radius of each metal, and ρ∗ μis the effective electrical resistivity in μ·cm. The latter parameter must be introduced to take into account the difference between the parameters defined at DC and those in the infrared, and is given by the following expression: ρ∗(T)ρ(T)me m∗1/2τDC τIR ,(4) wheremestandsforthebareelectronmassandm∗fortheeffectiveopticalmass.Therelaxation-timequotientarisesfromanisotropic scattering at the Fermi surface and can be determined from numerical integrals [2,13,18]. Typically, the two quotients compensate each other to give a number on the order of 1 (correction factors in Table 1). Therefore, the emissivity of these metals is directly computed from available data, with the necessary parameters given in Table 1. It must be noted that this approach is only valid for temperatures higher than the Debye temperature, which is the case for this study. Free-electron radius data have been taken from Ref. [30], while the temperature-dependent electrical resistivity data have been interpolated from the recommended values given in the critical review by Matula [31]. The only free parameter left in this approach is the electronic specularity factor p, which can take values from 0 to 1. This parameter accounts for the nature of electron reflection at the sample surface: 0 corresponds to diffuse scattering and 1 to perfectly specular reflection. The combined ratio of the parameters in Eq. (4) has been determined based on optical data and Fermi surface integration [13,18]. Similar calculations have been made in recent times based on ab initio methods, including a larger number of metals with more complex Fermi surfaces [32]. It must be noted that most integral calculations of the anomalous skin effect do not take into account such corrections [5, 21]. Because the infrared relaxation value tends to the DC value at sufficiently long wavelengths [22], it is unclear how this effect should be incorporated into a full spectral representation of the emissivity in the region of validity of the anomalous skin effect. It would be necessary to incorporate a frequency-dependent relaxation time, instead of a high-frequency limit. In any case, available studies point to the potential crossover point taking place at wavenumbers lower than 400 cm−1[19,22]. Thus, further studies in the far-infrared would be necessary to clarify this point, but the present technique, based on thermal emission, is very difficult to Table 1 List of free-electron parameters used in the calculations. Electrical resistivity values in μ cm 10−8mare obtained by linear interpolation of recommended values Material rs/a0[30]ρμ(473 K) [31] Correction factor [13,18] Ag 3.02 2.70 1.24 Cu 2.67 2.90 1.09 Au 3.01 3.74 1.43 123 Eur. Phys. J. Plus (2024) 139:1043 Page 3 of 9 1043 Table 2 Roughness characterization of the samples. Rastands for the average roughness of the profile, Rqfor the root-mean-square roughness, Rzfor the mean height of the profile peaks, and RSm for the average distance between profile elements Sample Ra(μm) Rq(μm) Rz(μm) RSm (μm) Cu 0.062 0.087 0.548 285 Ag 0.048 0.062 0.347 186 Bulk Au 0.036 0.046 0.286 219 Au mirror 0.01∗––– *data reported by the manufacturer perform at low frequencies for poorly emitting materials. Studies in the far-infrared and submillimeter range could be performed using non-optical techniques, such as impedance measurements in resonant cavities [33]. 3 Materials and methods High-purity Cu and Ag sputtering targets from Goodfellow®were used (purities: 99.99 mass%). Additionally, two gold samples of >99% purity, in the form of a bulk disk and a protected yttria-coated mirror film, were also studied. Some of these samples have been used as reference materials for testing new emissometers and in metrological intercomparisons [34]. The copper and silver samples were mechanically polished, whereas the gold ones were measured as received. The surface roughnesses were determined using standard profilometry, with the resulting parameters being reported in Table 2. The only exception is the gold mirror, whose surface is not directly accessible, due to the Y2O3coating. Instead, the average roughness value provided by the manufacturer is given. Infrared emissivity measurements were carried out in the Reduced-Background Calibration Facility (RBCF2) of the PhysikalischTechnische Bundesanstalt (PTB) [34,35]. This facility is a recently upgraded version of the previous RBCF and is now located on an ISO 5 cleanroom. The RBCF2 system connects a source chamber to a detector chamber through a beamline cooled by liquid nitrogen. The source chamber contains multiple radiation sources that can be precisely positioned along the beamline’s optical axis. A temperature-controlledspherewithin the source chamber holds a sampleforemissivity measurements. Thedetectorchamberfeatures infrared detectors and an ellipsoidal mirror that focuses radiation from the source chamber onto a Fourier transform spectrometer via the cooled beamline. This setup allows for vertical and horizontal scanning of both sources and detectors, enabling the measurement of their homogeneity. Further information can be found in Ref. [35] and references therein. Emissivity measurements are performed using a direct radiometric method, by comparing the radiance emitted by the sample to those of two reference blackbodies of high metrological quality. The sample is mounted on a rotating heating plate, located inside a thermally controlled spherical enclosure painted with a black paint of known emissivity. Distinctive features of this system include a high vacuum throughout (10−6hPa), the use of a liquid-nitrogen-cooled beamline to connect the detector and sample chambers, and a liquid-nitrogen-cooled blackbody as a low-temperature reference. The high-temperature blackbody is kept at a temperature close to the measured sample. The measurements were performed in vacuum in the mid-infrared range with a DLaTGS detector. The samples were heated by conduction, with Apiezon®thermal grease ensuring good thermal contact to the heating plate. The sample surface temperature was obtained from a Monte Carlo simulation of a heat transfer model, based on readings by resistance thermometers [36]. The dominant sources of uncertainty using this method are signal repeatability at high wavenumbers and sample temperature at low wavenumbers. In the case of highly reflecting materials, however, their exceedingly small emission signal motivates the use of an enclosure heated to 80 ◦C, in order to increase the total radiation coming out of the sample surface to detectable limits [37]. In this case, the correction for reflected enclosure wall radiation dominates the uncertainty budget at low wavenumbers and leads to correlated uncertainties throughout the spectral range, which affect the statistical properties of the results. 4 Results and discussion The normal spectral emissivities of the four samples are shown in Figs. 1and 2, where all samples feature very low and spectrally flat emissivities. The uncertainty decreases with wavenumber, as the weight of reflected signals from other infrared sources decreases in intensity. Measuring the spectral emissivity of such low-emitting materials with a standard uncertainty below 0.03 is a very challenging task, one of the main motivations behind the development of the RBCF2 instrument [38]. The theoretical values for the emissivities of the three metals under study, calculated using Eq. (2), are also shown in Figs. 1 and 2. The agreement between these theoretical relations and the experimental results is excellent, which proves the reliability of these expressions and allows discussion on the role of the surface scattering. On the one hand, both copper and silver could be reproduced accurately by the p0 approximation, with the experimental uncertainties being sufficiently low to rule out the p1 123 1043 Page 4 of 9 Eur. Phys. J. Plus (2024) 139:1043 Fig. 1 Normal spectral emissivities of copper and silver at 473 K, compared to the predictions of Eq. (2) for extreme values of p. Shaded regions correspond to standard uncertainties (k1). Note that the copper measurements have been performed at a higher resolution Fig. 2 Normal spectral emissivities of two gold samples (bulk and mirror film) at 473 K, compared to the predictions of Eq. (2) for extreme values of p. Shaded regions correspond to standard uncertainties (k1) possibility. Given that these samples were polished by the same method, it is not surprising that they both feature similar surface parameters. In contrast, the picture of the gold samples was more complex: The solid sample could be accurately reproduced by the p1 approximation, but the film had a higher near-normal emissivity, more consistent with the p0 approximation. It must be noted that these differences in specularity refer only to scattering at the electronic scale, as all samples have very low roughnesses compared to mid-infrared wavelengths (as shown in Table 2). It is well known that the emissivities of thin films can be higher than those of their bulk counterparts, even for well-prepared samples [17,18,39]. It is worth noting that one of the first experimental studies of the anomalous skin effect in silver films observed an important reduction in their electrical conductivity, which increased the emissivity [17]. We should also take into account the differences in temperature dependence of noble metal films when compared to their bulk counterparts [40]. These differences can be traced back to structural and morphological properties (i.e., grain growth and surface roughness, among others), not only differences in electrical properties. Nevertheless, the biggest effects are predicted for nanometric films or grain sizes, which are the most susceptible to structural degradation upon heating, so only thick samples were used in this study. Furthermore, the chosen temperature (473 K) is close to 0.3Tmfor all metals, where Tmis the melting point. This temperature is typical for annealing procedures, but is not high enough to induce irreversible changes. Additional studies at oblique angles reveal no additional features in any of the bulk spectra (Figs. 3,4,and5). In the case of the coated gold film (not shown), spectral peaks corresponding to the vibrational modes of the yttria coating are apparent at high angles. In contrast with other directional measurements [20], there is no spectral feature contradicting the Mott–Zener picture of a spectrally flat emissivity for the other three samples. These measurements will be used below to calculate the total hemispherical emissivities. An important result of this work is that there is no straightforward relation between the roughness parameters shown in Table 2and their surface specularities from an electronic scattering viewpoint. This points to the complexity of defining a specularity parameter and quantifying its relation to measurable surface properties. Because it accounts for electron scattering, its characteristic length 123 Eur. Phys. J. Plus (2024) 139:1043 Page 5 of 9 1043 Fig. 3 Directional spectral emissivity of the copper sample at 473 K Fig. 4 Directional emissivity of the silver sample at 473 K Fig. 5 Directional emissivity of the bulk gold sample at 473 K scale is not the light wavelength, but the de Broglie wavelength of electrons in these materials (3 Å for copper and 5 Å for silver and gold) [18]. Based on diffraction arguments, it seems that the roughness needs to be almost atomically low in order to have an electronic specularity different than 0. Interestingly, there is experimental evidence that somewhat rougher surfaces can apparently still have important specularities [17,20], a point which is supported by some theoretical results [41]. Some authors have also claimed 123 1043 Page 6 of 9 Eur. Phys. J. Plus (2024) 139:1043 Fig. 6 Total hemispherical emissivity of copper [14,15], compared to the literature data and theoretical predictions (4/3 of the values predicted by Eq. (2)). The error bar corresponds to the standard uncertainty (k1) Fig. 7 Total hemispherical emissivity of silver, compared to the literature data [14,16,24]and theoretical predictions (4/3 of the values predicted by Eq. (2)). Error bars correspond to standard uncertainties (k1) that partial effective specularity is allowed for any sample surface [9]. Along this line of argument, it is notable that the bulk gold sample does reproduce the p1 expression, which might be related to the absence of a polishing process prior to measurement, even if its roughness is similar to or higher than those of the other samples. In any case, clarification of this point has still not been achieved, nor is it expected to be further developed without more sophisticated surface science studies. The point of this work is to provide precise data on the basic thermal radiative properties of noble metals and validate a simplified but accurate theoretical picture. Nevertheless, the influence of this specularity on the infrared and visible properties of noble metals is of great importance, given their low bulk emissivities, but it is also crucial when discussing the low-to-high-frequency transition regime [9,10]. At lower frequencies, both surface scattering regimes differ by only a 9/8 absorption factor [9]. Finally, a comparison to the literature data is shown for completeness. Total hemispherical data, obtained by integration of the spectral directional results, are compared to reference values. The integration procedure to retrieve this property from angle-resolved data is based on a fitting procedure to Fresnel’s relations [34,36]. Figure 6shows the result for copper, together with the literature data and theoretical predictions (for good conductors, εH4/3·ε)[2,13]. Total hemispherical measurements for copper were reviewed in Ref. [13] and found to be reproducible with free-electron theories accounting for surface scattering and effective resistivities. Data from Ramanathan and Yen [14] are described in that reference and shown to be experimentally reproducible within the bounds set by the theory. Additional up-to-date low-temperature data on copper films [15] are shown, which point toward an intermediate value for the electronic surface specularity. The data point reported in this work is consistent with the literature values and the theoretical results for a perfectly diffuse surface at the electronic scale (p0). Similar agreements are found for silver and gold (Figs. 7and 8), although some of the literature datasets report values higher than those presented in this work. Despite evidence of the quality of the measurements, unambiguous comparisons to the theoretical predictions are complicated by the relatively large standard deviations, which are unavoidable in this kind of measurements. This complication is intrinsic to an emission measurement method when used to characterize highly reflecting materials. However, because the theory predicts 123 Eur. Phys. J. Plus (2024) 139:1043 Page 7 of 9 1043 Fig. 8 Total hemispherical emissivity of gold, compared to the literature data [15,24]and theoretical predictions (4/3 of the values predicted by Eq. (2)). Error bars correspond to standard uncertainties (k1) Fig. 9 Results of the fitting of the normal spectral data to a constant function, compared to the theoretical bounds for each metal (Eq. (2)). Data points correspond to the averages of the fitted values obtained assuming either perfectly correlated (r1) or uncorrelated (r0) spectral points. Error bars represent the standard deviations of the corresponding rectangular probability distributions spectrally constant values for materials which obey the Mott–Zener approximation, a solution can be formulated by performing a statistical comparison of the normal values to a constant function f(x)f0. The fitting was made by minimizing the following objective function: χ2rTσ−1r,(5) where ry−f(x) is the vector containing the residual at each point, and σis the variance–covariance matrix. For the particular case of uncorrelated spectral data, the off-diagonal terms become 0. The use of Eq. (5) is motivated by the covariant nature of these measurements. As mentioned above, uncertainties corresponding to the sample temperature and the correction of radiation reflected by the sample introduce correlations between spectral data points, especially at low wavenumbers [37]. These correlations are a priori unknown, but they significantly affect the fitting result. Thus, following the principle of maximum entropy, we have performed two fits for each metal, considering extreme values of the correlation coefficient: rij 0or1;i j,∀i,j. The variance–covariance matrices were constructed in the usual manner from the experimental uncertainties and these assumed values of the correlation coefficient. Negative correlation was not considered, because there was no physical mechanism in the measurement model that could provide such behavior. The fittings were made using the curve_fit method of SciPy 0.19, which allows specifying an input covariance matrix. The standard uncertainty due to the fitting process itself was negligible, but differences between the results obtained assuming no correlation or perfect correlation between wavenumbers were significant. The results are shown in Fig. 9, where the average of the two fits is represented for each metal, and the standard deviation corresponds to that of a rectangular distribution between both extremes: u(a+−a−)/√12. This approach assumes that all non-negative values of the correlation coefficient are equally probable, and thus, it produces the best estimate of the true value of the underlying constant emissivity assumed by our model. As expected from inspection of the spectral curves, three of the fitted emissivities are consistent with the values predicted by Eq. (2)(p1 for bulk gold, p0 for silver, and an intermediate value for copper). The emissivity of the gold mirror is slightly higher than the maximum value, although by less than one standard deviation. A possible explanation of this deviation is the presence 123 1043 Page 8 of 9 Eur. Phys. J. Plus (2024) 139:1043 of the protective yttria film, which can act as an anti-reflective layer. Nevertheless, all results are compatible with the theoretical predictions with only one free parameter. In summary, this test serves as a robust statistical validation of the hypothesis of the article, taking all the knowledge about the measurement uncertainty into account. 5 Conclusions To sum up, this work reports on good-quality emissivity data that can be accurately reproduced using simple expressions from free-electron theory in the Mott–Zener approximation and parameters determined by non-optical methods, with only the electronic specularity factor pas a variable parameter. The need to account for differences between DC and optical properties of these metals is highlighted. Tentative differences are observed between a bulk sample of gold (which suggests better agreement to p1), and silver and copper (which are closer to the predicted values for p0). Moreover, differences between two gold samples (bulk vs film) can be accounted for by different values of the specularity parameter. Despite the large uncertainties at low wavenumbers, the results reproduce the theoretical relations accurately and lend validity to its foundations. Further data at lower wavenumbers would be interesting to validate the model in a region where the anomalous skin effect theory predicts a departure from the results of this work. Acknowledgements The authors acknowledge financial support from the Basque Government. I. González de Arrieta also thanks the PhysikalischTechnische Bundesanstalt (PTB) for inviting him to their laboratory. Author contributions Iñigo González de Arrieta, Telmo Echániz, and Gabriel A. López contributed to the conceptualization and design of the study. Material preparation, data collection, and analysis were performed by Iñigo González de Arrieta and Albert Adibekyan, under the supervision of Christian Monte and Jörg Hollandt. The first draft of the manuscript was written by Iñigo González de Arrieta, and all authors commented on the previous versions of the manuscript. All authors read and approved the final manuscript. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was supported financially by the Basque Government (Eusko Jaurlaritza, Grant Number IT-1714-22). Data availability statement Datacontainedinthisarticle areavailable inZenodo(https://doi.org/10.5281/zenodo.12801555).Themanuscript has associated data in a data repository Code availability The calculations reported can be reproduced from the information available in the manuscript. Declarations Conflict of interest The authors have no relevant financial or non-financial interests to disclose. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. F. Wooten, Optical properties of solids (Academic Press, Cambridge, 1972) 2. A. Sievers, Thermal radiation from metal surfaces. J. 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