Plasmon Spectroscopy for Subnanometric Copper Particles: Dielectric Function and Core–Shell Sizing

In the last years, there has been a growing interest in the study of transition metal nanoparticles (Nps) due to their potential applications in several fields of science and technology. In particular, their optical properties are governed by the characteristics of the dielectric function of the metal, its size and environment. This work analyses the separated contribution of free and bound electrons on the optical properties of copper Nps. Usually, the contribution of free electrons to the dielectric function is corrected for particle size through the modification of the damping constant, which is changed as usual introducing a term inversely proportional to the particle’s radius to account for the extra collisions with the boundary when the size approaches the electronic mean free path limit (about 10 nm). For bound electron contribution, the interband transitions from the d-band to the conduction band are considered together with the fact that the electronic density of states in the conduction band must be made size-dependent to account for the larger spacing between electronic energy levels as the particle decreases in size below 2 nm. Taking into account these specific modifications of free and bound electron contributions to the dielectric function, it was possible to fit the bulk complex dielectric function, and consequently, determine optical parameters and band energy values such as the coefficient for bound electron contribution Qbulk = 2 × 1024, gap energy Eg = 1.95 eV, Fermi energy EF = 2.15 eV, and damping constant for bound electrons γb = 1.15 × 1014 Hz. With both size-dependent contributions to the dielectric function, extinction spectra of copper Nps in the subnanometer radius range can be calculated using Mie’s theory and its behaviour with size can be analysed. These studies are applied to fit experimental extinction spectra of very small spherical core–shell Cu–Cu2O Nps generated by ultrafast laser ablation of a solid target in water. Theoretical calculations for subnanometric core radius are in excellent agreement with experimental results obtained from core–shell colloidal Nps. From the fitting, it is possible determining core radius and shell thickness of the Nps, showing that optical extinction spectroscopy is a good complementary technique to standard high-resolution electron microscopy for sizing spherical nanometric-subnanometric Nps.


Introduction
The study of copper metal nanoparticles (Nps) is an area of active research due to their wide potential applications in chemistry, catalysis, material science and nanofluidics [1][2][3][4].The high conductivity, large extinction cross section, photosensitivity and low cost of copper makes it a promising material for the development of miniaturised devices that can integrate electronic, photonic and chemical features for use in biological nanosensors [5][6][7][8][9].It can also be mentioned its capability of insertion in host polymer matrices for nonlinear optical applications [10][11][12].
The spectral characteristics of plasmons of metal Nps may be used to characterise the size of noble metals Nps in the range 10-100 nm radius [13].Recently, we have shown a method for sizing gold and silver Nps below 10 nm by fitting extinction spectra with Mie's theory together with a conveniently modified bulk refractive index [14][15][16][17][18].The modification considers that the damping constant in Drude's model is increased due to additional collisions of free electrons with the boundary of the particle.This fact produces a size dependence of the dielectric function and consequently of the refractive index.For noble metals, transitions of bound electrons to conduction band levels contribute appreciably to the dielectric function.For silver and gold, Pinchuk et al. [19] analysed the influence of interband electronic transitions on the frequency, amplitude and bandwidth of the surface plasmon resonance in small metal clusters in the Rayleigh approximation.However, for the case of copper Nps, the influence of free and bound electrons on the dielectric function and its dependence with size has not been fully analysed.
In this work, the size effect over the dielectric function will be studied further in order to analyse its influence on the extinction spectra of copper Nps.Both free and bound electron contributions will be considered, and the values of the different parameters involved in the model will be determined.Extinction spectra analysis will be applied to sizing subnanometric core-shell copper-copper oxide Nps fabricated by femtosecond laser ablation in liquid media.

Theoretical Framework
In general, the complex dielectric function for bulk metals can be decomposed into two terms, a free-electron term and an interband, or bound-electron term.Since the dielectric function is additive, it can be written as, For bound electrons, the complex dielectric function arising due to transitions from the copper d-band to the conduction sp-band, can be calculated using the expression given by Inouye et al. [20]: where ħω g is the gap energy (E g ) of copper, F (x, T) is the Fermi energy distribution function of conduction electron of energy ħ x at the temperature T with Fermi energy E F ; γ b represents the damping constant in the interband transition and Q bulk is a proportionality factor.
For free electrons, the complex dielectric function can be written as: where ω p is the bulk plasma frequency and γ free is the bulk damping constant in the Drude model.Its values for bulk copper are taken from Shalaev [21].Figure 1a, b shows the real and imaginary components of bound (Eq.2) and free (Eq.3) electron contributions calculated separately, while its sum (full line) is the best fit to experimental bulk values derived from complex refractive index taken from Johnson and Christy [22] (represented with hollow circles).The simultaneous fit of the real and imaginary part of both contributions yields the optimum values of Q bulk , E g , E F and γ b which are included in Table 1 together with other used parameters.Parameters used and determined are given in Table 1 It is worth noting from Fig. 1b that the free electron contribution for the imaginary part is important for wavelengths larger than 600 nm while the bound electron contribution dominates for wavelengths shorter than 600 nm.This particular behaviour gives rise to the characteristic aspect of the experimental dielectric curve (hollow circles), and it influences the behaviour of the copper optical extinction spectrum in the short wavelength range.
When the particle's size is smaller than the mean free path of the conduction electrons in the bulk, the damping constant for the free-electron contribution (related to the inverse of the collision time) is increased by the collisions of the electrons with the particle boundary.So, it must be modified by adding a term to the bulk damping constant, which depends inversely on the particles radius [23]: where v F is the electron velocity at the Fermi surface and R is the radius of the particle.Fermi velocity for copper was taken from [24] to be v F 015.7×10 14 nm/s.C is a constant related to electron scattering processes within the particles whose value ranges between 0.5 and 1.2, as derived from first principles calculations [25].In our case, a C value of 0.8 was used.Figure 2a, b shows the behaviour of the real and the imaginary part of the dielectric function with wavelength, for different Np radii, considering only the free electron contribution using Eqs. 3 and 4.
It can be observed in Fig. 2a that the real part of the dielectric function considering only the free electron contribution, is very sensitive to size for R<2 nm.In the range 2-10 nm is much less sensitive and it is almost coincident with the bulk curve for R010 nm.On the other hand, the imaginary part plotted in Fig. 2b shows a similar limit behaviour for R>15 nm, but there seems to be also a limiting behaviour for R00.6 nm.
For the case of the dielectric function related to bound electrons contribution, the size dependence may be accounted for following the idea introduced by Logunov et al. [26] assuming that the electronic density of states is different for Nps of different size.These authors conclude that small particles have larger spacing between electronic states, so the density of states will be smaller for very small Nps.This fact can be taken into account [14] by changing the proportionally factor Q bulk in the contribution of bound electrons from its value accepted for the bulk to Þ , where R is the radius of the particle and R 0 is the scale factor.So, the size-dependent bound electron contribution to the dielectric function can be written as: Figure 3 shows the real (a) and imaginary (b) parts of the bound electron dielectric function considering this size  dependence for different radii values, according to the Eq. 5.
It can be observed in Fig. 3a that the curves are very sensitive to small changes of the radii under 1 nm, but the size correction is negligible for R≥2 nm, in agreement with Fig. 1a.In Fig. 1b, the imaginary part shows a similar limiting behaviour with size, being also negligible for R≥ 2 nm.The results shown in Figs. 2 and 3 related to the separated free and bound electron contributions to the complex dielectric function and its dependence with size will influence the features of the extinction spectra of nanometric and subnanometric particles.

Experimental Procedure
Metal Nps were fabricated by ultrafast pulse laser ablation in liquids.The target sample of copper used to carry out these experiments was a circular disk of 1 mm thick of high purity grade immersed in water.Laser ablation was performed using a Ti/sapphire chirped pulsed amplification (CPA) system from Spectra Physics, emitting pulses of 100 fs width at 1 kHz repetition rate centered at 800 nm wavelength.The maximum output energy was 1 mJ per pulse.A 5-cm focal length was used to focus the laser beam on the target disk surface.The energy per pulse used in this experiment was 500 μJ.Under these experimental conditions, a colloidal solution of core-shell Cu-Cu 2 O Nps is generated.Other authors [27] also report on the generation of oxide-coated copper Nps fabricated by nanosphere lithography.In our case, the fabrication process was done during 20 min, after which the water solution shows a typical greenish color, which is attributed to the presence of a large number of Nps in the solution.Optical extinction spectroscopy (OES) was conducted by means of a Shimadzu spectrophotometer from 300 to 1,000 nm.The sample preparation for this method precludes the development of agglomerates since no ligand compounds were added to the solutions.Measurements were performed on diluted and sonicated Nps solutions.

Results
Since the size of the copper Nps considered in this paper is very small compared with the incident wavelength, the response to optical extinction can be described using the electrostatic approximation.In this approach, the expression for the extinction cross section is where α is the polarisability, k 0 ¼ 2p n m l is the wavenumber in the medium surrounding the particle, n m is the refractive index of the medium and λ is the wavelength of the incident light in vacuum.For spherical Nps with core-shell structure, the polarisability can be written as [26]: À Á 3 is the ratio between inner and outer radius volumes, R0r core is the metal central core (copper), R′0r (core+coating) is the outer radius (copper core+copper oxide coat thickness), ε 1 0ε 1 (λ, R), ε 2 0ε 2 (λ) and ε m 0ε m (λ) are the dielectric functions of the core, coating (shell) and surrounding medium, respectively.Another parameter related with the extinction cross section is the extinction coefficient defined as Q ext ¼ C ext p R 02 .The complex dielectric function corresponding to the Cu 2 O shell thickness is plotted in Fig. 4 according to the data given by Palik [28].

(a)
Wavelength [  Figure 5a shows the extinction coefficient for a coreshell Cu-Cu 2 O Np for a subnanometric metal radius R0 0.7 nm covered by a thin layer of copper oxide (R′−R050 % R), calculated with and without bound electron size correction.It can be observed that the peak position is near 650 nm for both spectra, but differences are more noticeable for wavelengths lower than 650 nm, where the influence of imaginary part of bound electrons is more important.The correction of bound electron smooths the contrast I max À I min ð ÞI max = in the mentioned range, while for larger wavelengths the spectra are coincident.
Figure 5b shows the extinction coefficient for a subnanometric copper bare core, which is a special case of coreshell Np where R′0R.Here, the differences between the spectra are more evident for wavelengths shorter than 600 nm, and the plasmon resonance is blue shifted from 650 nm (with oxide shell) to a small shoulder at 600 nm without shell.
We may apply the above calculations to fit the experimental extinction spectra corresponding to the core-shell Nps obtained previously by laser ablation.Figure 6 shows the best fit of the experimental extinction spectrum corresponding to a colloidal solution of Cu-Cu 2 O Nps fabricated by fs laser ablation with pulse energy of 500 μJ.This fit was attained considering a linear combination of two types of core-shell Nps: one with 0.9 nm core radius and shell thickness of 40 % R and the other of the same core radius and a 150 % R shell thickness.The coefficients of this combination, 0.45 and 0.55 respectively, give the optimum relative abundance of the species.
It is interesting to notice that neither of the single species can fit the whole spectrum by itself.This fact can be seen in Fig. 7 where the normalised spectrum of each species is represented together with the experimental spectrum for comparison.It can be seen that the 40 % shell thickness curve correctly fits the plasmon peak position, but there is a serious disagreement for wavelengths shorter than 550 nm.The 150 % shell thickness curve has its peak position shifted to the IR in about 60 nm with respect to the experimental one, a shift that can be readily measurable with a commercial spectrophotometer and does not fit correctly the spectrum in the short wavelength range.
To analyse the sensitivity of the fitting procedure, Fig. 8 shows the behaviour of the theoretical extinction spectra compared with the experimental one for different values of the core radii and shell thickness.Figure 8a shows the spectra corresponding to the same shell thickness distribution as in Fig. 6 but different core radii.It can be observed that although the core radii are only 0.1 nm above and below the optimum value (0.9 nm), a small but observable difference with respect to the experimental spectrum can be seen.
Figure 8b shows the spectra corresponding to the same optimum core radius but different shell thickness distribution.The difference in the spectra is more noticeable below plasmon peak, where the influence of the bound electron is more important The 40 % R thickness remains fixed while the larger thickness component is changed to 120 and 180 % R from the optimum value of 150 % R. The calculated spectra show definite differences below 500 nm.So, the experimental spectrum can be fitted by both distributions for wavelengths larger than 500 nm but neither can fit below 500 nm.All these results support the fact that OES is a very simple, inexpensive and sensitive technique to size nanometric-subnanometric bare core or core-shell Nps colloids.This size range is complicated to reach for conventional HR TEM and AFM microscopies.Besides, the measurement statistics for OES is very large with typical values ranging from 10 12 to 10 14 particles, depending on particle size and sample concentration.

Conclusions
In this work, we have studied separately the behaviour of the free and bound electron contribution to the wavelengthdependent dielectric function of copper Nps with size.
The free electron contribution was modified as usual including a term inversely proportional to the particle radius in the expression of the bulk damping constant.With this modification, both real and imaginary parts of the free electron dielectric function show a noticeable dependence with size for Nps from 10 nm to less than 1 nm.This dependence shows a limiting behaviour to bulk for a radius R≈10 nm.
The bound electron contribution of transitions from the dband to the conduction band was modelled using the expression given by Inouye et al. [20] and analysed in two steps.Firstly, the parameters involved in the expression for the bulk bound electron dielectric function, such as Q bulk , E g , γ b and E F , were adjusted to fit simultaneously the real and imaginary experimental values for bulk copper complex dielectric function taken from Johnson and Christy [22].Secondly, for subnanometric Nps, a size dependence factor that accounts for the larger energy level spacing, was included in the former expression for the bound electron dielectric function.This correction is important from subnanometric size to a radius of about 2 nm, above which the correction is negligible.
When these expressions for the dielectric function were used in the calculation of the polarisability of metal Nps Fig. 7 Solid line corresponds to experimental spectral extinction of Cu-Cu 2 O Nps fabricated by laser ablation with 500 μJ energy.Squares with full line is calculated extinction spectrum corresponding to R0 0.9 nm and R′−R040 % R. Circles with dashed line is calculated extinction spectrum corresponding to R00.9 nm and R′−R0150 % R within Mie's theory, it was possible to show the influence of the bound electron size correction on the shape of the extinction spectrum of subnanometric Nps.For core-shell Cu-Cu 2 O Nps, the differences are more noticeable for wavelengths between 400 and 650 nm, where the influence of imaginary part of bound electrons is more important.For the case of bare core Cu Nps, the differences between the spectra with and without size correction are more evident for wavelengths shorter than 600 nm, where the influence of the bound electrons is more important.As an application of these theoretical studies, we have successfully fitted the experimental optical extinction spectra of core-shell Cu-Cu 2 O Nps fabricated by ultrashort pulse laser ablation of solid target in water.The optimum fitting using Mie's theory yielded a dominant core radius R00.9 nm with a shell thickness distribution of 40 and 150 % R. We also show that the method of spectral fitting is extremely sensitive to small variations in core radius or shell thickness.Calculated spectra for radii 0.1 nm above and below the optimum, yield curves that depart from the experimental spectrum right in the limit of experimental uncertainty.The same occurs for a 30 % R difference in shell thickness.
In this way, it is shown that OES can be applied as a complementary method to advanced microscopy techniques for sizing spherical bare core and core-shell metal Nps in the nanometer-subnanometer size range.OES has also the advantage of a very good measurement statistics provided by the large number of particles in the path of the spectrophotometer beam across the sample cell.Besides, it avoids coalescence effects since the measurement is made directly on the colloidal suspension.

Fig. 1
Fig.1Free and bound electron contributions to the complex dielectric function of bulk copper, calculated with Eqs. 1, 2 and 3. (a) Real component ε′ and (b) imaginary component ε″.Theoretical calculation is compared with experimental values from Johnson and Christy[22].Parameters used and determined are given in Table1

Fig. 2
Fig. 2 Dependence of the free electron contribution to the dielectric function with wavelength for different sizes: real (a) and imaginary (b) components

Fig. 3
Fig. 3 Dependence of the bound electron contribution to the dielectric function with wavelength for different sizes: real (a) and imaginary (b) components

Fig. 4 Fig. 5
Fig. 4 Real and imaginary parts of the complex dielectric function of Cu 2 O

Fig. 8
Fig. 8 Experimental and calculated extinction spectra for different core size and shell thickness distributions: a different core radii for optimum shell thickness distribution and b different shell thicknesses distribution for optimum core radii

Table 1
Optical parameters for bulk copper used and determined in this work