Optical Constants and Optical Anisotropy of Ultrathin Gold Films

Continuous and homogeneous thin gold films are fabricated on Si–SiO2 substrates using an additional seed layer based on Cu oxide or oxidized Cu. The optical properties of these gold films with thicknesses from 3 to 50 nm are studied with the help of variable angle spectroscopic ellipsometry. The optical constants and the dielectric functions of thin Au films in the wavelength range of 240–1700 nm are extracted by fitting measured ellipsometric data using the Fresnel isotropic and anisotropic models. By applying the Drude approximation to the infrared range, values of the plasma frequency ωp and relaxation time τ are obtained as a function of film thickness. It is found that plasma frequency ωp of the gold films is mostly constant in the thickness range of 3–50 nm with an average being close to the bulk value ωp ≈ 8.45 eV. The relaxation time decreases dramatically from the bulk values of τ ≈ 14 fs to τ ≈ 2 fs for gold films of 3 nm in an agreement with the confinement effect. It is established that the thinnest gold films (≈3 nm) are described better by an anisotropic layer and the underlying reasons for such optical anisotropy are discussed.


Introduction
Nanotechnology has revolutionized material science by enabling fabrication of ultrathin films with thicknesses ranging from nanometers to micrometers. [1,2]Recently, discovery of 2D atomic materials [3] and van der Waals heterostructures [4] made possible to reduce thickness of fabricated films to an atomic level (<1 nm).Among the 2D materials, one can easily find dielectrics represented by hexagonal boron nitride, semimetals represented by graphene, and semiconductors represented by transitionalmetal dichalcogenides.At the same time, ultrathin layers of metals did not receive widespread attention despite they represent an interesting and important part of Lego-like van der Waals heterostructures.[7] It is worth noting that gold is the metal of choice for successful label-free optical biosensing. [8]The reduction of the thickness of gold films to a nanometer level can allow one to extend surface plasmon resonances to infrared wavelengths and provide a broad dynamically tunable optical response important for ultrasensitive biosensing and electro-optical modulation at telecom wavelengths. [5]Therefore, fabrication of ultrathin metal films (UTMFs) on various substrates and study of their optical properties is of substantial technological and scientific interest.
Several interesting effects are expected to happen in UTMFs.Density-functional theory study of UTFMs [9] predicts that the plasma frequency of electrons in UTMFs should drop for the thicknesses ≤3 nm.Additional scattering caused by the confinement effect leads to a modification of the electron collision time and could result in significant changes of the permittivity of thin metal films. [10]Furthermore, in metal films of small thickness, the presence of the boundaries breaks the symmetry of an electron motion which leads to intrinsic anisotropy of UTMFs. [9]TMFs could also acquire optical anisotropy through sample fabrication procedures [11] due to a columnar growth, the presence of noticeable surface roughness, etc.
There are several challenges in fabricating UTMFs that made an access to the metal films of thicknesses %5 nm problematic.First, it is difficult to keep thin fabricated layers of metals continuous and homogenous due to wetting properties of metals.Second, it will be beneficial to use conventional metal deposition methods for obtaining high-quality UTMFs on common substrate materials, which puts an additional constraint.Third, methods of mechanical exfoliation-very popular for fabrication of family of monolayer materials such as graphene [12] and transitional-metal dichalcogenides [4] -could not be used in this case due to the face-centered cubic (fcc) lattice of Au and difficulties of fabricating crystalline gold films at normal conditions.Recently, it was reported that atomically thin gold nano-sheets with thickness of only 0.47 nm (two atomic layers thick) can be synthesized via a one-step aqueous approach at 20 °C, using methyl orange as a confining agent. [13]However, this chemical method of producing bilayer Au does not demonstrate a high degree of structural perfection and leads to many defects.In addition, the sizes of the fabricated gold flakes were at the 100 nm level, [13] which is too small for advanced plasmonic applications. [14]Overall, conventional modern deposition techniques result in Au films of thickness well above 10 nm. [15]It was also reported that a monolayer of MoS 2 could be used as a prospective adhesion layer for deposition of continuous conductive gold films with thickness of only 3-4 nm. [16]This method has its own limitations connected to small sizes of exfoliated MoS 2 flakes, defects in larger MoS 2 flakes produced by chemical vapor deposition (CVD) technique that lead to island formation in grown Au films, and possible influence of a conductive MoS 2 sublayer on the properties of gold films and their morphology.Also, ultrahigh vacuum %10 À10 mbar and clean Si(111) (7 Â 7) surfaces could be used to grow ultrathin Au films. [17]This technique allows one to achieve continuous gold films up to 4 nm thicknesses.Unfortunately, these films are stable only in ultrahigh vacuum environment. [17]he tendency of metal atoms to form 3D islands during the growth on a solid substrate in the VolmerÀWeber growth mode [18,19] is the main problem in fabricating UTMFs using conventional deposition methods such as electron-beam evaporation, chemical vapor deposition, electroplating, and sputtering.This natural tendency towards island growth could be suppressed by the introduction of seed layers (typically Cu, Ge, Ni, etc.) on the substrate before deposition of ultrathin (<10 nm in thickness) high-quality gold films. [5]An addition of a seed layer provides an access to fabrication of UTMFs (%3 nm thick) such that their properties could deviate significantly from those of the bulk materials due to factors such as electron confinement, surface effects, and film-substrate interactions.Main difference in the physical properties of thick and ultrathin films is conditioned by the strongly confined electrons in sublayers.Confinement of electrons could even lead to discrete quantum-well states that dominate the dielectric and electronic properties [20,21] and could cause quantized electrical conductance. [22,23]ere, we report fabrication of continuous sub-10 nm gold films on Si-SiO 2 substrates with the introduction of a CuO [24] or room-temperature oxidized Cu seed layer. [5]Two methods of growing ultrathin gold films (UTGFs) on commercial Si-SiO 2 substrates were used: 1) fabrication of a 0.5 nm thick CuO layer directly deposited from the corresponding target, deposition of ultrathin Au films by thermal evaporation; 2) electron-beam deposition of a Cr layer of thickness 1 nm, then a 0.5-1 nm thick Cu layer, room-temperature exposure in ambient air of the Cu layer that allows a formation of an oxidized layer of Cu, and finally the growth of Au films with desired thickness in the same technological process.The developed procedures allowed us to fabricate homogeneous and continuous gold films with thickness as low as 3 nm.Then, we performed thorough morphological and optical characterizations of the fabricated UTGFs.Variable angle spectroscopic ellipsometry was used to extract complex refractive index and hence the dielectric functions of the ultrathin gold across a broad spectral range of 240-1700 nm.The extracted dielectric functions were fitted by the Drude model in the near-infrared region and plasma frequencies and the relaxation times were determined as a function of film thickness.We found that the relaxation time decreases considerably in thin films due to the confinement effect while the plasma frequency remained mostly constant and close to the bulk value of 8.45 eV down to the film thicknesses of %3 nm.We also found that the thinnest gold films (%3 nm thick) are described better by an optically anisotropic layer and discussed underlying reasons for such anisotropy.

Sample Preparation
UTGFs were prepared using two similar methods in two different laboratories: 1) in the Barcelona Institute of Science and Technology, Spain; these samples are labeled as B(Au-nm); 2) and in the University of Manchester, UK; these samples are labeled as M(Au-nm).In both methods, thin Au films were prepared by an electron-beam or thermal evaporation on commercial 0.55 mm thick Si(100) substrates covered by a %290 nm thick thermally grown SiO 2 layer.In the first method, the gold films deposition was performed by conventional thermal evaporation on the Si-SiO 2 substrates covered by CuO or Cu using sputtering technique.The preparation was performed in high vacuum at a base pressure of 1 Â 10 À7 Torr and the substrates were held at room temperature during evaporation.A very small rate of deposition less than %1 Å s À1 was chosen to increase accuracy of Au thickness using calibration techniques.Growth of the metal films was monitored by a calibrated quartz microbalance (CQM).In the second method, gold deposition was performed using electron-beam evaporation with the same fabrication procedure for all films with thickness ranging from 3 to 47 nm.To enhance the adhesion of Au to the substrate, an additional thin layer of Cr of thickness 1 nm was deposited on Si-SiO 2 substrates by electron-beam evaporation prior to gold deposition.Strong adhesion of gold to the substrate is very important for creating biosensors working in aggressive environments and also for fabrication of nanoplasmonic devices using electron-beam/optical lithography.To achieve continuous UTGFs with amorphous-like morphology with the help of the second method, higher deposition rates at the level of %2-3 Å s À1 during deposition of Cr layer were used.It is worth noting that rapidly evaporated films normally reproduce the surface smoothness of the substrate better.To obtain homogenous and continuous gold film, we introduced a seed layer between adhesive Cr and Au layers.A very thin 1 nm thick oxidized Cu film was chosen as a seed layer as proposed in ref. [5].To make the seed later, a Cu film was evaporated on the top of metallic Cr layer with elevated rates (%3.0Å s À1 ) and thicknesses of 0.5-1 nm without breaking vacuum between Cr and Cu depositions.Then, the substrate with two fabricated layers was moved from high vacuum into the air condition for 30 min to oxidize the Cu seed film.Finally, a metallic Au layer with desired small thickness was evaporated by electron-beam method in high vacuum after pumping the chamber.Our deposition rate was controlled at 1 Å s À1 at the base pressure of 1.0 Â 10 À6 Torr.Such procedure of preparing ultrathin gold film allows one to achieve better morphology of continuous film and hence better properties of the ultimate devices.Schematics of ultrathin gold film preparation are shown in Figure 1a.

Morphology of UTGFs
The Ultra Plus Carl ZEISS scanning electron microscope (SEM) was used for high-resolution imaging of fabricated thin gold films.SEM provides information about the samples' surface morphology.The unique in-lens SEM detector gives resolution of the order of 1.0 nm at 15 kV, dependent on the type of samples.Figure 1b shows an optical image of a 5.5 nm thick gold films on a Si-SiO 2 -Cr-CuO substrate, which confirms homogeneity of the sample M(Au-5.5 nm).SEM images in Figure 1c-e show surface morphology of Au films of different thicknesses on Si-SiO 2 -Cr-oxidized Cu substrates, the samples M(Au-3 nm), M(Au-4 nm), and M(Au-5 nm).The SEM images confirm continuity of the fabricated gold films and the absence of 3D islands of irregular shapes over a large area.The irregularities in the fabricated UTGFs are larger for the thinnest Au film, the sample M(Au-3 nm), Figure 1c.We suggest that the oxidized Cu seed layer promotes an increase in Au adatoms diffusion, which leads to formation of a large number of nucleation centers followed by the constitution of the fine grain structure of the film being grown in the Frank-van der Merwe mode. [18]The slightly different morphology of Au %3 nm thick films as compared to ref. [5]  can be explained by the use of Cr adhesion layer and slight differences in deposition parameters.Using an atomic-force Dimension 3100 microscope, we checked that the surfaces of all the films studied are atomically smooth with roughness R q < 0.3 nm, which is close to the roughness of commercial Si-SiO 2 substrates used in our work.Such roughness provides less than 5% error in extraction of gold optical constants in the long wavelength region as our modeling suggests.

Characterizations: Optical Properties of UTGFs
We characterized the optical properties of the thin gold films by recording ellipsometric spectra with subsequent extraction of the complex dielectric functions of the top film for various thicknesses ranging from 3 to 47 nm.The ellipsometry measurements were done with J. A. Woollam Co. M2000F focused beam ellipsometer that allowed us to perform variable angle spectroscopic measurements in the range of wavelengths 240-1700 nm and angles of incidence from 45°to 75°.The spot size on the sample was %40 Â 70 μm 2 at %60°-70°angles of incidence.The ellipsometry essentially measures sample reflection for two different polarizations of incident light.It yields two spectral parameters, Ψ and Δ, which are related to the amplitude and the phase of a complex reflectance ratio ρ, such that ρ = r p /r s = (tanΨ)exp(iΔ), where r p and r s are the amplitude reflection coefficients for pand s-polarized light. [25]Since ellipsometry measures the ratio of the reflection coefficients for p-polarized and s-polarized light, it provides much better accuracy as the amplitude noise of a light source is cancelled.In addition to ellipsometric parameters Ψ and Δ, the ellipsometer can measure separately the spectra of intensity reflections for pand s-polarized light, namely R p = |r p | 2 and R s = |r s | 2 , at various angles of incidence.The ellipsometric spectra allow accurate determination of the complex refractive index and the permittivity of the Au thin films.To retrieve the complex refractive index, N = n þ ik, from the measured data, we performed fitting of the measured spectra at different angles using the Fresnel's equations of a multilayer model consisting of a Si-SiO 2 substrate with either CuO-Au or Cr-oxidized Cu-Au layers on the top of a substrate using Woollam variable angle spectroscopic ellipsometry (WVASE) software of J. A. Woollam Company.The initial fitting was performed point by point for each wavelength.These data were then used for a normal fit where continuous spectral dependences of material constants are expected.The bare Si-SiO 2 substrates were also characterized by ellipsometry which allowed us to extract the thickness of SiO 2 layer used later in our models.The refractive indexes for Si and SiO 2 were in excellent agreement with Palik data. [26]The optical properties of the substrate covered by Cr-oxidized Cu layers were also measured and modeled using WVASE software.
To model the measured ellipsometric data, we first apply an isotropic model to the fabricated Au thin films with the aim to extract values of the optical constants n and k of the top gold layer.The Fresnel model we used in the calculations is shown in Figure 2a. Figure 2b shows the measured and the fitted spectra of the ellipsometric parameter Ψ for the sample B(Au-6 nm) as a function of angle of incidence.For the fit, the optical constants of Si, SiO 2 , and CuO were taken from the WVASE library.The thickness of the SiO 2 layers was found by fitting the ellipsometric spectra measured on the bare substrate.The optical constants of the top gold layer were fitted to achieve the best agreement with the measured spectroscopic parameters Ψ and Δ in the whole range of the wavelengths and angles of incidence.Ψ demonstrates prominent minima which are strongly dependent on the angle of incidence due to the interference in the structure.A very good agreement between the measured and modeled ellipsometric spectra for both Ψ and Δ was observed with the use of an isotropic model for a gold layer when UTGFs were thicker than 3 nm, see Figure 2b.The mean squared error (MSE)-a measure of the goodness of the model fit to the data in the WVASE software based on χ 2 analysis-was low (below 5 which we found to guarantee an agreement of experimental data with fitted data at the accuracy of experimental errors) and rose up to 7-10 as the film thickness decreased to a value just below 6 nm.The values of the gold film thickness corresponding to the readings of the CQM monitor during evaporation were used as the initial values during fitting of ellipsometric characteristics Ψ and Δ.More precise values of the thickness, d, refractive index, n, and extinction coefficient, k, of Au thin films were obtained after fitting the experimental results.The difference between fabricated thicknesses of Au film and extracted from the fitting of ellipsometric data was less than 0.25 nm. Figure 2c,d shows the SEM images of the structures with 3 and 6 nm gold film.From SEM images, we can conclude that a UTGF with thickness of 6 nm is homogeneous, a UTGF with thickness of 3 nm possesses some grain structure.We note that in the case of these films, a significant time-more than 1 year-passed from the deposition to characterization and morphological rearrangement due to Au mobility cannot be excluded. [27]Figure 2e,f plots the complex refractive index and dielectric functions of UTFGs for samples fabricated without additional adhesive Cr layer.We pay more attention to the dependence of the dielectric constants ε 1 and ε 2 because they are closely related to the electronic structure of the solids and are more directly comparable with theory. [28]e found that the real and imaginary parts of the complex dielectric function, ε 1 and ε 2 , of UTGFs depend on thickness of Au film.Figure 2 shows that values of ε 1 and ε 2 increase with a decrease of the gold film thickness.
The data for the samples featuring UTGFs with an adhesive Cr (%1 nm) layer used during fabrications-the Manchester samples-are shown in Figure 3. Figure 3a plots the measured ellipsometric spectra as well as the calculated fit of the spectra performed with the WVASE software.The Fresnel model used for the fit is shown in Figure 2a.As mentioned earlier, the optical constants of Si, SiO 2 , Cr, and CuO were taken from the WVASE library and the thickness of the SiO 2 layers was found by fitting the ellipsometric spectra measured on the bare substrate.The optical constants of the top gold layer were fitted to achieve the best agreement with the measured spectroscopic parameters Ψ and Δ in the whole range of the wavelengths and angles of incidence.The extracted values of optical constants of the top gold layer are shown in Figure 3b. Figure 3c,d shows the extracted dielectric functions of UTGFs of different thicknesses produced with the Cr sublayer.By comparing Figure 2 and 3, we see that the real and imaginary dielectric functions of UTGFs does not depend on the presence of the Cr sublayer.It is worth noting that the real part of dielectric permittivity ε 1 is negative in the whole spectrum range for all UTGFs investigated by us, Figure 2 and 3, indicating continuity of the fabricated Au films. [29]As in case of UTGFs produced without Cr sublayer, the real and imaginary parts of the complex dielectric function increase with a decrease of the gold film thickness, Figure 3c,d, which is connected with surface scattering as explained later.Finally, the inset of Figure 3d shows the measurements of Hall resistance as a function of magnetic field for a Hall cross made of a UTGF with thickness of 6 nm (B-Au-6 nm) at two different temperatures which we will use later to calculate the electron density.

Results and Discussion
Many optical properties of metals are mediated by their conducting electrons.The conducting electrons in a metal are delocalized over comparatively large distances, so that they can be treated, at least in the first approximation, as electron plasma confined by a fixed positively charged atomic ions.Properties of conducting electron plasma can be described well by the Drude model. [30]The Drude model considers a classical motion of free electrons governed by the two parameters: the plasma frequency, ω P , which is proportional to the density of electrons; and the electron collision time, τ, which is also known as the relaxation time (in parenthesis, we note that the Drude model provides a good description of conducting electrons in metals, however, it neglects band structure effects and also provides no physical insight into the underlying electron interactions [30,31] ).As we discussed in introduction, the plasma frequency of electrons in UTGFs should decrease when thickness decrease below ≤3 nm. [9]We did not reach these thicknesses in our experiments.However, the presence of boundaries separated by a small distance given by film thickness leads to an additional surface scattering, which modifies the electron collision time [18] due to the confinement effect.This implies that the observed tendencies of an increase of the dielectric constants ε 1 and ε 2 for thinner films, see Figure 2 and 3, could be directly connected to the renormalization of the relaxation time in UTGFs.
To check this hypothesis, we have fitted the extracted optical constants of UTGFs in the near-infrared part of the spectrum by the Drude expression as the Drude contribution becomes prevalent in the near-infrared optical response of the metal. [30]Figure 4 shows the behavior of the extracted plasma frequency and the relaxation time of conducting electrons in UTGFs as a function of gold film thickness.We found that the plasma frequency of electrons in UTGFs does not significantly change for the whole range of the studied thicknesses, see Figure 4a, and was close to the bulk values of 8.45 eV. [32]This is in agreement with theory developed in ref. [9] but in stark contrast with the results obtained on UTGFs fabricated on MoS 2 substrates [16] where it was found that the plasma frequency decreases when thickness falls below 6 nm. [33]This discrepancy is not surprising as the presence of a conductive MoS 2 substrate could significantly change the extracted values of plasma frequency due to the resistive coupling through the conductive sublayer. [34,35]It is worth noting that the Drude parameters of the films fabricated in Manchester and Barcelona are roughly the same despite slightly different methods of sample preparation, see Figure 4.
To confirm the fact that the plasma frequency of UTFGs does not change much in the studied thickness range, we evaluated electron density in UTFGs using an alternative method based on the Hall effect.To this end, we fabricated a Hall cross using deposited UTFGs, placed it in the magnetic field and measured the dependence of the Hall voltage on the applied magnetic field.Dependence of Hall resistance on the magnetic field for film B(Au-6 nm) is shown in the inset of Figure 3d.The relationship between the Hall resistance R xy and the applied magnetic field B is given by B = ÀndeR xy , where n is the carrier density, d is the thickness of the film, and e is the electron charge.This allows us to extract the electron density n and to calculate the plasma frequency as ω P = (ne 2 /(mε 0 )) 1/2 , where m is the mass of electron and ε 0 is the vacuum permittivity.From the inset of Figure 3d, we see that the surface electron density does not depend on the temperature and can be evaluated as n = 3•10 16 cm À2 , which is many orders of magnitude larger than the electron density in graphene. [3]From this concentration, we estimate the plasma frequency of UTFGs as 8.4 eV, which agrees well with the bulk values of the plasma frequency [32] and with the results obtained from the Drude fitting of the optical spectra, Figure 4a.Hence, the Hall measurements confirm the Drude parameters extracted from our optical data.The thickness dependence of the electron collision time, Figure 4b, is more complicated.In bulk metal, the electron free path is given by v F τ b , where v F is the Fermi speed (v F = 1.4 Â 10 6 m s À1 for bulk Au) [30] and τ b is the relaxation time (τ b % 14 fs for bulk Au). [32]When the thickness of the metal film d becomes small such that d < v F τ b , which is 17 nm for bulk Au, electrons will experience additional surface scattering on the film boundaries.For UTGFs with very small thicknesses, this surface scattering will become prevailing leading to the thickness dependent relation time τðdÞ % d=v F .For intermediate thicknesses, an empiric approximation is often used to calculate the electron relaxation time.This approximation assumes that the bulk scattering and surface scattering are statistically independent processes [18] and results in the following expression: This expression yields the correct thin-film limit τðdÞ % d=v F for d ( v F τ b as well as the correct thick-film limit τðdÞ % τ b for d ≫ v F τ b .We found that the expression (1) provides good description of the relaxation times of UTGFs extracted from the Drude model, see Figure 4b. Figure 4b also confirms the fact that the relaxation times of UTGFs with d < 10 nm indeed follow the dependence τðdÞ % d=v F due to the electron confinement effect.
An application of a more elaborate Fuchs' approach [36] to UTGFs is debatable due to anisotropy of the UTGFs films.Indeed, due to the thin-film symmetry of UTGFs, scattering of electrons moving in-plane of the film is different from scattering of electrons moving out of plane, which leads to intrinsic anisotropy of optical properties of UTGFs.The situation is somewhat similar to graphene and other 2D materials where the in-plane optical constants of an atomic material can be drastically different from the out-of-plane optical constants. [37]It is worth noting that a columnar growth and roughness of surfaces can also lead to significant anisotropy of UTGFs which is different from intrinsic anisotropy.The presence of anisotropy was confirmed in our modeling of optical spectra of UTGFs.As we mentioned earlier, the MSE for the fit was small (below 10) for all films with thicknesses above 3 nm, which indicates that these films can be described well by an isotropic model.At the same time, MSE becomes large for the isotropic fit of optical properties of UTGFs with thicknesses %3 nm and the fit obtained with isotropic model becomes nonideal, see Figure 5a.Hence, we have modeled UTGFs of the smallest thickness obtained using an anisotropic model for UTGFs and obtained an excellent fit shown in Figure 5b.From this fit, we extracted the behavior of in-plane optical constants (n || and k || ) and out-of-plane constants (n ⊥ and k ⊥ ) shown in Figure 5c,d, respectively.We see that the in-plane and out-of-plane constants demonstrate different spectral behavior: the in-plane constants correspond to continuous metal while the out-of-plane constants correspond to a localized plasmon resonance observed at wavelength of 900 nm.As only the film roughness could lead to the presence of out-of-plane localized plasmons, we conclude that the surface roughness of the thinnest films is the main contributor to the optical anisotropy.This also implies that fabricated UTGFs of thickness %3 nm are near the percolation threshold.This conclusion is in agreement with the results of ref. [17], where the percolation threshold for ultrathin gold films fabricated in ultrahigh vacuum was observed at thicknesses of 2-4 nm and the plasma frequency of ultrathin gold films extracted from the optical measurements showed complex behavior near the percolation.
Finally, we compare our results with those existing in the literature.There is a wide range of different values of the plasma frequencies and relaxation times quoted for ultrathin gold films.Table 1 summarizes these values.
We see a variety of different values of the Drude constants observed for gold films.This is not surprising as properties of thin metal films depends on methods of their fabrication which could lead to different roughness of films, different spatial sizes of this roughness, amorphous or crystalline structures of films and islands.In addition, the properties of a substrate used can also affect the properties of metal films under study due to proximity effects.As far as the thickness dependence of plasma frequency of our ultrathin gold films is concerned, we found that it is mostly constant in the studied range of thicknesses 3-50 nm.This is in agreement with the density functional calculations [9] and experiments performed on ultrathin TiN films [38] where the detectable changes were observed only at thicknesses ≤2 nm.It has to be noted that our experiments and TiN experiments were performed on films with extremely small roughness (<0.3 nm), which is probably the reason of such agreement.Also, our results are in broad agreement with those obtained in ref. [15].At the same time, our results are in disagreement with the conclusion [16,33] that the plasma frequency should decrease when the thickness of metal film becomes smaller than 15 nm.This conclusion was supported by some phenomenological theory. [39]However, our Hall effect measurements (shown in the inset of Figure 3d) confirm that the electron density does not change much even in the very thin films, which is in line with our optical data.Hall measurements provide unambiguously the electron concentration of metal films and do not dependent on a model used (in contrast to processing of optical data).Hence, we believe that the disagreement with the previous works [16,33] is connected to the properties of Au films deposited on MoS 2 , namely, their roughness and substrate conductivity.
Lastly, we note that roughness does not only affect electron properties of UTMFs but also affect extraction of optical constants of thin metal films using ellipsometry. [25]The effect of roughness on modeling of ellipsometry data will depend on film optical constants, thicknesses of the films and roughness parameters (out-of-plane roughness, spatial sizes of roughness in plane, etc.) as well as on properties of a substrate. [25]For example, Au dot arrays deposited on a conductive Cr substrate could have localized surface plasmon resonances completely suppressed due to the resistive coupling through the substrate and the overall structure would look like a continuous gold film with much smaller plasma frequency. [35]

Conclusion
In summary, an introduction of a seed layer opens an access to fabrication of continuous and homogeneous sub-10 nm ultrathin gold films over large area.We described two methods for making UTGFs on commercial Si-SiO 2 substrates covered by a CuO or oxidized Cu seed layer.The gold film morphology was analyzed using SEM with resolution of about 1 nm.Flat continuous gold films with thickness as low as 3 nm were produced with and without an adhesive 1 nm thick Cr layer.Using variable angle spectroscopic ellipsometry, we determine the complex optical constants and the dielectric functions of UTGFs in spectral range from 240 to 1700 nm by fitting ellipsometric parameters to the Fresnel isotropic and anisotropic models.Near-infrared constants of UTGFs are described well by the Drude model with the constant plasma-frequency-and thickness-dependent relaxation time conditioned by the electron confinement effects.The large density of electrons in UTGFs evaluated from the Hall measurements confirms the high value of the plasma frequency for small thicknesses.We found that UTGFs of the thickness of %3 nm have to be described by an anisotropic model with different optical constants for in-plane and out-of-plane electron motion.

Figure 1 .
Figure 1.Process of fabrication of ultrathin gold films and nanostructure characterization.a) A schematic illustration of the fabrication of continuous ultrathin gold films.b) An optical image of thin Au film with thickness of 5.5 nm.c-e) Scanning electron microscope (SEM) images of the surface morphologies of ultrathin gold films (UTGFs) fabricated on Si/SiO 2 /Cr/oxidized CuO substrate for different Au thicknesses: 3.3, 4, and 5 nm for the samples M(Au-nm).

Figure 2 .
Figure 2. Properties of UTGFs deposited on Si/SiO 2 /CuO substrates without an adhesive Cr (%1 nm) layer.a) Schematic of the structure substrate-ultrathin Au film (substrates [Si-SiO 2 -(Cr)-CuO]) and ellipsometric measurement.b) Spectral dependences of ellipsometric parameter Ψ observed at different angles for gold film B(Au-6 nm)-the measured (solid lines) and modeled (dash-dot) Ψ spectra.c,d) SEM images of UTGFs with thickness d = 3 and 6 nm, respectively.e) Extracted optical constants of UTGFs with d = 3 and 6 nm.f ) Evaluated complex dielectric functions of UTGFs with d = 3 and 6 nm.

Figure 3 .
Figure 3. Properties of UTGFs deposited on Si/SiO 2 /oxidized Cu substrates with an adhesive Cr (%1 nm) layer.a) Spectra of ellipsometric parameter Ψ measured at different angles for the sample M(Au-3.3nm).The measured data are shown by the solid lines and the fits are shown by the dash-dot lines.b) Extracted optical constants of UTGFs for thicknesses d = 3.3; 4.2; 5.1, and 10 nm.c,d) The measured real ε 1 and imaginary ε 2 parts of the dielectric functions of UTGFs for d = 3.3; 4.2; 5.1, and 10 nm.The inset shows the Hall measurements for the sample B(Au-6 nm) that yields the high-electron concentration of the film as 3.07 Â 10 16 cm À2 .

Figure 4 .
Figure 4.The Drude parameters of UTGFs as a function of film thickness.a) The plasma frequency, ω P .The dotted line shows the bulk value of ω P .b) The relaxation time, τ.The blue solid line shows the d/v F fit and the magenta dotted line gives a fit produced by Equation (1).Error bars were calculated by fitting the extracted long-wavelength spectral dependence of n and k of the samples to the Drude model.

Figure 5 .
Figure 5.Comparison of isotropic and anisotropic model for UTFG of thickness 3 nm-the sample B(Au-3 nm).a) A fit of ellipsometric parameters with an isotropic gold film resulting in mean squared error (MSE) of 9.3.b) A fit of ellipsometric parameters with an anisotropic gold film resulting in MSE of 1.4.c) Values of the in-plane and out-of-plane refractive indices extracted from the anisotropic model.d) Values of the in-plane and out-of-plane absorption coefficient extracted from the anisotropic model.

Table 1 .
The plasma frequency and relaxation time of gold films measured in different studies.