1. Introduction

Parametric formulations to describe the dielectric function or complex refractive index of specific materials have long been used. Just to cite a few examples, the optical constants of amorphous semiconductors have been considered in the framework of the parametric model of Forouhi and Bloomer [1–3], who recently extended the formulation to consider metals [4]; the contribution of conduction electrons to the optical properties of metals has been repeatedly considered by means of the Drude model [5,6]; and it has been shown that the effective dielectric function of two- and three-dimensional metal-insulator composite materials can be described by the Lorentz model [7–9]. Drude-Lorentz type models have been applied when describing the dielectric function of materials whose optical properties have contributions from conduction and bound electrons [10], as well as those characterized by contribution from conduction electrons and holes, besides that of bound electrons as in transition metals [11].

Rhodium is a soft silver-white transition metal with a face-centered cubic structure at standard conditions of temperature and pressure [12]. It is used in the manufacture of thermocouples in platinum-rhodium alloys and as a coating in jewelry, for making laboratory vessels and in surgical tools, as a catalytic agent, in electroplating [13], and in plasmonic applications [14–16]. The temperature dependence of the Rh magnetic susceptibility was measured a long time ago [17], with first Hall coefficient measurement carried out in 1957 [18]. The study of its optical properties initiated with measurement of the reflectivity spectrum, for wavelengths between 0.5 and 9.2 μm, which shows an increase from 77% in the yellow visible region to 92% at 2.5 μm, and beyond and that it gradually increases to about 94% at 9 μm [19]. Additional measurements were carried out in 1939 [20]. The electron energy loss spectrum of Rh was reported by Lynch and Swan [21], and a set of optical constants, in the energy range 6.2 to 41.3 eV was obtained from reflectivity measurements for opaque electron beam evaporated Rh films at low and room temperatures [22]. Low temperature absorptivity measurements were carried out in the spectral range 0.20 to 4.4 eV, reflectivity measurements at room temperature through 3.0 to 40.0 eV were carried out to obtain the optical constants of opaque polished polycrystalline Rh films [23,24]. From reflectance measurements at various angles of incidence, the optical constants of thin Rh films were obtained by Windt et al. [25]. Effective masses of both conduction electrons and holes have been determined from measurements based on the De Haas–van Alphen effect [26], and band structure calculations have also been reported for Rh [27,28].

We have recently devised an extended version of the DL model to describe the spectral variation of the DF for the transition metals palladium and iridium [11,29], taking into account the presence of electrons and holes as charge carriers, and by writing the volume plasma frequencies of bound electrons involved in Lorentz susceptibility, in terms of the volume plasma frequency of the collective oscillation of the electrons, the oscillation strengths and the number of conduction electrons per atom. In this article, we report the application of the same formalism to model the DF of Rh through a large spectral range: from 0.2 to 39 eV. A summary of the model, and of the simulated annealing approach used to optimize the set of parameters involved in the DF model, are given in Section 2. Results of the optimization approach are given in Section 3 together with the report of the set of derived quantities. The plasmonic behavior is reported in Section 4, with evaluation from sum rules of the number of electrons and holes participating as charge carriers and in inter-band transitions given in Section 5.

2. Dielectric function model and parameter optimization

As other transition metals, the optical, charge-transport, thermal, plasmonic, and magnetic properties of Rh are influenced by itinerant electrons and holes with additional contributions from bound electrons [30]. According to reported band structure diagrams [27,28], there are hole sheets- (in the X-W and W-L directions) and pockets- (around the X- and L-symmetry points) states somewhat above the Fermi level E_F. The Fermi level is spectrally located just before the last sharp peak of the d-band, displayed over a background of density of states values provided by s-states [31]. The density of states at the Fermi level, ρ(E_F), has been calculated to be between 0.32 and 0.38 states/eV atom, depending on the specific quantum mechanics-based method used [28]. When the contribution of conduction electrons and holes is accounted for in the Drude term of the DF (ε=ε₁+iε₂), and incorporating the polarization and absorption due to bound electrons, an extended version of the Drude-Lorentz model has been devised [11,29,32,33]:

The formulation of the DF given by Eq. (1) has the following advantages in contrast with the traditional form of applying DL models: (1) the energy dependence of Drude term is self-consistent with the physical behavior required for Im(ε) when the angular frequency tends to zero. It opens the possibility to obtain the static conductivity, and corresponding electrical resistivity, once the optimization has been carried out; (2) for the first time, contribution to itinerant holes is accounted for in the Drude term of the DF of Rh, making it possible to use the same approach to describe the dielectric function of semiconductors; (3) all parameters involved in the formulation of DF have known physical interpretation consistent with the role played in Eq. (1); (4) the number of parameters depends on the specific material and on the spectral range being considered. The number of parameters is not chosen under the criterion that it is the minimum number of parameters required to minimize the objective function described later. This number is chosen from previous information available in the literature about conduction properties (electrons and/or holes), spectrophotometric measurements, energy loss spectra, and band structure calculations; (5) the present formulation satisfies Kramers-Kronig (KK) dispersion relations because the Drude and Lorentz susceptibilities do [35]; and (6) a large set of derived parameters can be evaluated and some of them can be compared with measured or calculated values to gain confidence in the physical modeling of the DF for a particular material. These six features make the present formulation robust enough to be applied even in the case of large spectral ranges, and materials with significant spectral structure in the DF, as is the case of Rh. The proposed method can be applied to other materials like semiconductors and dielectrics, in bulk or thin film form. In the case of dielectrics, the Drude susceptibility term must be suppressed. When considering thin films, other derived quantities related with surface resistivity and grain boundaries can be obtained. The range of applicability is extensive, both in terms of the type of materials to be considered, as well as the range of energies that can be considered.

Within the context of the optimization method described below, the following three substitutions are made in Eq. (1): γ_oh=ηγ_oe, Ω_ph²=βχΩ_pe², and ε_hf=1/f_o with β=m_e/m_h, χ=n_h/n_e=z_h/z_e, and f_o with values between zero and unity. The counterparts of n_e and z_e are n_h and z_h for holes, respectively, with n_e$\ne$n_h because Rh is a non-compensated metal [18]. A frequency-dependent relaxation time has been assumed to improve the fitting of infrared optical properties of some metals by the Drude model [36]. Similar dependences on angular frequency for both electrons and holes are assumed: 1/τ_e=1/τ_oe+Λ_eω² and 1/τ_h=1/τ_oh+Λ_hω², respectively. This ω²-dependence of 1/τ_e has been phenomenologically attributed to the scattering of the charge carriers at grain boundaries [37]. Under this assumption, it would be expected that Λ_e and Λ_h show low values when the samples have been annealed. Theoretical formulations attribute this ω²-dependence to the electron-electron interactions between charge carriers [10,38]. The corresponding energies associated with these scattering frequencies are: γ_e=γ_oe[1+(Γ_eΩ)²] and γ_h=γ_oh[1+(Γ_hΩ)²], where Γ_e and Γ_h are two additional parameters to be optimized: Γ_k=(2πΛ_k/hγ_ok)^1/2 with the subscript letter k = e for electrons, and k = h for holes. We call these two parameters renormalized time between grain boundary scattering events. These expressions for γ_e and γ_h substitute γ_oe and γ_oh respectively in Eq. (1). The parameters to be optimized in the Drude susceptibility term are Ω_pe, γ_oe, β, χ, η, Γ_e, and Γ_h. They determine the contribution of the conduction electrons and holes to the dielectric function of the metal. Moreover, the background polarization due to inter-band transitions beyond the spectral range considered is accounted for the inverse of the optimized value of f_o. With the definitions given through this paragraph, the set of 8 Drude DF parameters to be optimized is f_o, Ω_pe, β, $\chi $, γ_oe, η, Γ_e, and Γ_h.

The contribution of inter-band excitations to the dielectric function is approximated by the classical Lorentz summation term in Eq. (1) in which these excitations are modeled by resonance oscillators, each one associated with a j^th population of electrons bounded to the metal ions whose resonance frequency (energy) is ω_j (Ω_j), i.e. ε_L=ε_L,1+ε_L,2+…+ε_L,K with ε_L,j(Ω)=f_jΩ_pe²/{z_e[(Ω_j²-Ω²)-iΩγ_j]}. The resonance energies (Ω_j), Lorentzian widths (γ_j), and oscillator strengths (f_j) appear in this term. In a quantum mechanical picture, the resonance frequencies correspond to electronic transitions between occupied and vacant states close to critical points in the band structure or involving parallel bands. The N_p=9 + 3 K parameters in the DL model of the dielectric function are determined by minimizing the following merit function:

Due to its nonlinear form, heuristic optimization algorithms are used due to their ability to evade local minima and to approach the global minimum. In this work, an Acceptance-Probability-Controlled Simulated Annealing (APCSA) approach is used [39–41]. The subindex J in Eq. (2) stands for the J^th cycle temperature. We have implemented our own APCSA code, with some improvements as described in [11,29,32,33]. The merit function value is analogous to energy in the context of simulated annealing (SA) methods. Variations giving decreased values of the merit function compared to its previous value are accepted, and those giving increased values can be either accepted or rejected; the choice is made through application of the Metropolis algorithm [42,43]. The acceptance of variations giving positive changes to the merit function, allows these optimization schemes to avoid the neighborhood of local minima, and to move towards solutions in the vicinity of the global minimum. As the global minimum is approached, the effective temperature of the system decreases according to the specific scheme that characterizes the SA method being used. Temperature in the context of the present approach is a ‘measure’ of the dispersion of the accepted merit function (energy) values corresponding to random variations of the parameters being optimized.

Figure 1 depicts the spectral variation of the optical constants of Rh, i.e., the refractive index and extinction coefficient. The known values of the DF involved in the evaluation of the merit function F_J through the optimization procedure were obtained from the optical constants [24]: ${\bar{\varepsilon }_1}$=n²-k² and ${\bar{\varepsilon }_2}$=2nk [44]. These optical constants were obtained from measured spectra of absorptivity (0.2-4.4 eV) and reflectivity (3.0-40 eV) through KK analysis, from a large (bulk) polycrystalline sample subjected to annealing in vacuum conditions [23,24]. Other materials, in bulk or thin film form, can be considered once their optical constants or dielectric functions are obtained from ellipsometry or spectrophotometric techniques. For Rh, the values of ${\bar{\varepsilon }_1}$ (${\bar{\varepsilon }_2}$) are between -1328 (649) at the lowest infrared energy of 0.20 eV and 0.50 (0.38) at the largest vacuum ultraviolet energy of 39.0 eV. The number of degrees of freedom, i.e., the number of terms that can change in the minimization of F_J is 2κ minus the number of quantities that are optimized, including F_J itself. This definition of the merit function allows us to retrieve both ε₁(ω) and ε₂(ω) simultaneously with minimized error throughout the whole spectral range being considered.

 

Fig. 1. Spectral variation of the optical constants of Rh [refractive index n(λ) and extinction coefficient k(λ)] taken from [24].

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To be able to compare values with those of previous reports of Drude parameters, if available in the literature, the added Drude dielectric susceptibilities corresponding to the second and third terms in the right side of Eq. (1) [χ_Drude=-Ω_pe²/(Ω(Ω+iγ_e)-Ω_ph²/(Ω(Ω+iγ_h)] can be expressed approximately as χ_eff=-Ω_o²/[Ω(Ω+iγ_eff)]. The effective values of γ_eff are obtained from the relation: γ_eff=Ω$\cdot$Im(χ_eff)/│Re(χ_eff)│. As expected, this effective scattering frequency shows a dependence on energy that can be modeled by γ_eff=γ_o(1+(Γ_oΩ)²), with regression coefficients close to unity. The fitting of γ_eff allows us to calculate both γ_o (τ_o= h/2πγ_o) and Γ_o. For each spectral point, with α=γ_eff/Ω, the value of the effective volume plasma energy is obtained from the relation Ω_o=[(-Re(χ_eff)+Im(χ_eff))Ω²(1+α²)/(1+α)]^1/2, showing very low spectral dispersion. Its average value is evaluated from the set of Ω_o-values. The γ_o, Γ_o, and Ω_o parameters can be considered as effective Drude quantities.

Table 1 contains a glossary of those parameters involved in the DL model and definition of the merit function. The explicit parameters are those that appears explicitly in the definition of the merit function or in it through the expression of the DF given in Eq. (2). Those related with the explicit parameters are denoted as implicit ones. Once the optimization is carried out, a set of derived quantities can be evaluated as shown in Table 2, demonstrating in full extend the fundamental role of the dielectric function. We are assuming that holes contribute to paramagnetic and diamagnetic susceptibilities in the same manner that conduction electrons do [45].

Table 1. Glossary of the physical parameters explicitly or implicitly involved in the definition of the merit function, and in the Drude Lorentz model of the dielectric function.

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Table 2. Physical quantities that can be evaluated once the optimization of the DL parameters is carried out.

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In the two-band model, the Hall coefficient listed can be written as R_H=-(1-χq²)/[ez_eN(1+χq)²] with q=β/η [47]. Both electrons and holes contribute to the intrinsic conductivity in fractions given by P_e=1/[1+χq] and P_h=1-P_e, respectively. Besides the spectral dependence of ε₂(ω), the absorption of light is considered through the dynamic conductivity, σ(ω)=ε_oωε₂(ω), and through the bulk and surface energy loss functions. In the Gaussian system, the dynamic conductivity is given by σ_G(ω)= ωε₂(ω)/4π [σ(ω) = 4πε_oσ_G(ω) with σ in S/m and σ_G in 1/s].Within this formulation, the static conductivity is given by σ_o=ε_oωε₂(ω) when ω tends to zero, with no inter-band transitions contribution in this limit. From Eq. (1), the limit is given by n_ee²τ_oe/m_e+n_he²τ_oh/m_h which is the known formula for the static conductivity.

2.1 Some derived quantities reported in the literature for Rh

Some of the quantities included in Table 2 have been reported in the literature from measurements or calculations: the electrical resistivity at RT, ρ_o=4.78 μΩ cm, for millimeter-sized samples annealed at 1300 °C [48,49], with the corresponding value for the static conductivity σ_o=21.3 × 10⁴ S/cm. The experimental value of γ_exp=4.65 mJ/mol K² have been reported for the total heat capacity coefficient [50], which implicitly includes electron-phonon and electron exchange interactions. The Hall coefficient R_H=5.05 × 10⁻¹¹ m³/A s [51], the total magnetic susceptibility χ_m=0.990 × 10⁻⁶ cm³/g [52], and the effective volume plasma energy Ω_o=8.8 eV [53] have also been reported. Values for the average effective mass of the conduction electrons and holes have been estimated to be close to 1.39 and 1.19 respectively [26], which give an experimental value for β close to 1.17. The RT Lorenz number of Rh has been determined to be L_N=2.41 × 10⁻⁸ WΩ/K², close to the theoretical value L_o=π²k_B²/3e²=2.45 × 10⁻⁸ WΩ/K² [54]. The thermal conductivity has been measured at RT in κ_th=151 W/m K [54]. The resonance energies involved in the Lorentz term of the DF (with K=11) can be optimized from initial values estimated from reported band structure calculations and from optical analysis based on spectrophotometric measurement: Ω₁=0.40, Ω₂=1.40, Ω₃=3.05, Ω₄=4.70, and Ω₅=5.30 eV [28]; Ω₆=11.6, Ω₇=14.6, Ω₈=19.3, Ω₉=24,0, and Ω₁₀=30.5 eV [23]; and Ω₁₁=48.0 eV [25]. The presence of impurities could have its effect in most of the derived parameters: charge carrier and thermal conductivities, Hall coefficient, heat capacity coefficient, magnetic susceptibilities. This is mainly due to scattering of the charge carriers by impurities, which affect the intrinsic values of the mean free paths, relaxation times, Fermi velocities, etc. Those optimized parameters involved in the Lorentz term would be less affected by the presence of impurities at low level concentrations.

3. Simulated annealing optimization and numerical results

The optimization process has two steps: a first optimization entirely based on the APCSA approach is carried out to obtain a solution close to the global minimum. Then, a set of solutions are obtained by making random variations around the first solution just mentioned. From this set of solutions, average values and standard deviations are obtained for the optimized parameters and the derived ones. Simulated annealing methods approximate the global minimum whatever is the initial approximation (seed) made of the parameters being optimized. Of course, the further (closer) the seed is from the position of the global minimum, the longer (shorter) will be the computational time required to approach that minimum.

3.1. Solution from an APCSA optimization

Figure 2 shows the result of a first fitting obtained from application of the APCSA method. The global minimum is approached by means of this solution. This optimization stage is very expensive in terms of computing time. The quality of the fitting is remarkable despite the large spectral range considered and the large range of variation of both real and imaginary parts of the DF, with the merit function reaching the small value F_J=3.872 × 10⁻³ with J=389 cycles of temperature. This is the value of the merit function when reporting the optimized parameters with three decimal places. The value is basically the same when increasing the number of decimal places. For example, when using six decimals, F_J=3.848 × 10⁻³. The F_J-value increases about one order of magnitude when the optimized parameters are reported with two decimal places. We assume that three is the number of significant decimal places in the values of the optimized parameters to be reported.

 

Fig. 2. A fitting of: (a) real and (b) imaginary parts of the Rh dielectric function from an APCSA optimization (blue solid lines) involving J=389 temperature cycles. Red dots are experimental values. The merit function reached the value F_J=3.872 × 10⁻³ after J=389 cycles of temperature.

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Figure 3 shows the behavior of five of the Drude parameters and z_e as the number of temperature cycles increases from 1 to J=389. Figure 4 displays the convergent behavior of some of the derived quantities corresponding to this first optimization. The converged values are indicated in the corresponding figure captions. Each of these figures is part of an extensive display panel that shows the convergence for each of the parameters being optimized, as well as some of the derived physical parameters. Another display panel, with two figures, shows us the behavior of the real and imaginary parts of the dielectric function that is being evaluated at the end of each temperature cycle, and it is compared with the experimental values that enter in the definition of the merit function. The optimization was carried out until the solidification condition was satisfied: the lowest F_J-value for each temperature cycle is compared with the lowest values for the three preceding consecutive temperatures [39]. The simulated annealing process is finished when the corresponding three relative differences are lower than the specified numerical tolerance η_ο, i.e., when│(F_J-F_J-k)/F_J│<η_ο for k=1,2,3. We have used η_ο=10⁻⁴ with κ=208 as the number of spectral DF values.

 

Fig. 3. Variation of Drude parameters being optimized through the temperature cycles (between 1 and 389), as well as the number of conduction electrons per atom. The convergent values are: (a) Ω_pe=5.878 eV, (b) γ_oe=0.245 eV, (c) β=m_e/m_h=1.071, (d) χ=z_h/z_e=1.198, (e) η=γ_oh/γ_oe=0.092, and (f) z_e=0.662.

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Fig. 4. Variation of physical quantities obtained from some of the optimized parameters (as indicated in Table 2) through the temperature cycles (between 1 and J=389, in logarithmic scale). The convergent values are: (a) static electrical resistivity ρ_o=3.53 μΩ cm, (b) Hall coefficient R_H=9.73 × 10⁻¹¹ m³/As, (c) average electron effective mass m_e=1.92 m, (d) average hole effective mass m_h=1.79 m, (e) total heat capacity coefficient γ_hcc=1.85 mJ/K²mol, and (f) logarithmic value reached by the merit function log(F_J)=-2.41.

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3.2 Set of solutions from SPGM assisted APCSA optimizations

Given the numerical stability of the solution when considering three as the number of significant decimal figures of the optimized parameters, we apply the APCSA method with a starting point close to that obtained from the first solution, by making random variations up to ±0.005 in the values of the optimized parameters obtained from the first optimization stage, to generate 101 solutions and using mobile boundaries for the parameters being optimized. The approach of this first solution is schematically represented by the blue broken line in Fig. 5 finishing in the blue point, while those randomly generated are shown by the red points. The required convergence to obtain each solution has been accelerated by applying a Spectral Projected Gradient Method (SPGM) [55,56], once finished each temperature cycle of the APCSA process characterized by an acceptance ratio lower than 10%. Figure 6 shows results of one of these solutions, obtained from the SPGM assisted APCSA method. When running the program for a solution, with the acceptance ratio being lower than 10%, that obtained by the APCSA method at each temperature cycle is taken as the starting point in the SPGM whose solution is taken for APCSA as initial point in its next temperature cycle, and so on. Each final solution is obtained when the merit function is around F_J=3.872 × 10⁻³. Average values and standard deviations are calculated from the set of 101 solutions, with the results reported in Table 3 for the optimized parameters, and in Table 4 for the derived ones.

 

Fig. 5. Schematic representation of the global minimum of the merit function. The first F_J-APCSA solution is given by the blue point approached by the broken blue line, and the set of 101 solutions is represented by the red points. From them, the average value of the optimized merit function and its standard deviation, σ_F, are obtained, as well as those corresponding to the optimized and derived parameters.

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Fig. 6. Behavior of the merit function F_J in terms of the number of temperature cycles of the APCSA application followed by using a SPGM method to improve the APCSA result.

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Table 3. Average values and standard deviations of SPGM assisted APCSA optimized Drude-Lorentz parameters used to calculate the dielectric function of a polycrystalline annealed bulk sample of Rh [23,24]. Parameters are given in eV, except the dimensionless f_o and the oscillator strength f_j-values. Values of Γ_e and Γ_h are given in 1/eV. The average value of the merit function and its standard deviation is F=(3.873 ± 0.009)x10⁻³. The parameters in bold are Drude parameters.

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The effective mass of conduction electrons is slightly larger than that of conduction holes (β=1.074). The number of conduction holes per atom is about 20% larger than that of electrons (χ=1.201). The scattering frequency of holes is about one order of magnitude lower than that of electrons (η=0.091). The electrical conduction in transition metals involves itinerant electrons doing quantum transitions from s- states bellow the Fermi level to d-states somewhat above the Fermi level, and holes making transitions between d-states above the Fermi level. Due to the more spatially localized character of d-states as compared to s-states, the probability of intra-band transition between d-states (P_dd) of neighboring atoms is lower than that between s- to d-states (P_sd). It is expected to have γ_oh<γ_oe for this reason, and consequently η<1. According to Fermi’s golden rule, the scattering frequencies are also proportional to the corresponding densities of states at the Fermi level of electrons and holes. Both densities of states, ρ(E_e) and ρ(E_h) are almost equal, 0.396 and 0.392 states/eV atom, respectively. Thus, the low value of η is an indication of the significant difference in the degree of spatial overlapping between s-d and d-d states. Due to the itinerant character of the conduction electron and holes participating in these intra-band transitions, it is expected to have an inverse proportionality of the transition rates and scattering frequencies with mean free path of the corresponding charge carriers. The probability that electrons or holes are in the vicinity of a unit cell, to participate in an inter-band transition, is proportional to z_kN_a a/L_k with k = e or h, L_k being the intrinsic mean free path, a is the lattice constant, and with N_a atoms per conventional unit cell. The larger the mobility of a charge carrier, the lower its localization and the probability of participating in an inter-band transition. With the obtained mean free paths of electrons and holes (L_e=1.80 nm and L_h=22.4 nm, respectively), z_e and z_h, and the corresponding carriers mobilities, the renormalized densities of states D_e=z_eN_aa${\cdot}$ρ(E_e)/(L_eμ_e) and D_h=z_hN_aa${\cdot}$ρ(E_h)/(L_hμ_h) can be calculated. The ratio between them is D_h/D_e=0.008. This means that the low value of η is due to the lower overlapping of the state wave functions involved in the d-d transitions, as well as the reduced renormalized density of states for the conduction holes. We have modeled the DF of iridium following the same approach [29]. In this case, a=0.384 nm, ρ(E_e) = 0.27 states/eV atom, ρ(E_h) = 0.09 states/eV atom, L_e=4.35 nm, and L_h=10.8 nm, μ_e=7.45 cm²/Vs, μ_h=20.2 cm²/Vs, with η=0.987. Consequently, D_h/D_e=0.038. For these two transition metals, we see a correlation between the renormalized density of states and the η-value (the ratio between scattering frequencies of holes and electrons): [D_h/D_e]_Ir/[D_h/D_e]_Rh∼5 and η_Ir/η_Rh∼11.

The total dynamic conductivity (σ=σ_L+σ_D) is displayed in Fig. 7, with decoupled Lorentz (σ_L) and Drude (σ_D) contributions. Extrapolation to the zero-frequency limit gives the static conductivity: σ(ω$\to$0)=σ_D(ω$\to$0)=σ_o=28.3 × 10⁴ S/cm, with no contribution from bound electrons (σ_L(ω$\to$0) = 0). The corresponding resistivity is ρ_o=3.54 μΩ cm, which is slightly lower than the 4.78 μΩ cm experimentally measured. The static conductivity is determined to be about 7% by conduction electrons and in about 93% by holes. The experimental values displayed in the figure were calculated by Weaver et al. from the imaginary part of the DF of Rh depicted in Fig. 2(b). Following an analogous procedure, Pierce and Spice obtained the DF of Rh for a 150 μm thick film deposited by electron beam evaporation [53]. Their spectrum of the dynamic conductivity displays the same spectral features obtained by Weaver et al. with peaks at 1.1, 2.5, and 5.6 eV. By decoupling the Drude and Lorentz contributions of the dynamic conductivity, one can see the contribution of the low energy peak correlated with Ω₁=0.24 eV.

 

Fig. 7. Dynamic conductivity of Rh obtained from the imaginary part of the DF modelled with the APCSA-optimized parameters. Dots display experimental data obtained by Weaver et al. [23]. Lorentz and Drude contributions are denoted by σ_L and σ_D respectively, with the total conductivity given by σ=σ_L+σ_D.

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Tripathi et al. have compared the σ-values of Weaver with those obtained from their band structure calculations [28], including the decoupling of inter- and intra-band contributions. Their calculations display more structure around the resonance energies just aforementioned. The spectral positions of these peak are correlated with the resonance energies Ω₂=1.18, Ω₃=2.58, and Ω₅=5.79 eV, respectively. The peak close to Ω₄=3.0 eV is not resolved probably due to superposition with the Ω₃-peak.

3.3 Electrical and thermal conductivities

The electrical resistivity obtained from the APCSA optimization is lower than that measured for Rh. This is an implicit result of using low temperature absorptivity measurements in the spectral range 0.2– 4.4 eV, and room temperature reflectivity ones from 3.0 to 40.0 eV to obtain the DF of Rh [23]. This introduces an overestimation of the electrical conductivity at room temperature: σ_o=σ_e+σ_h=28.3 × 10⁴ S/cm with σ_e=1.87 × 10⁴ S/cm and σ_h=14.2σ_e when optimizing through the whole spectral range. The electrical conductivity is significantly dominated by holes, with a similar behavior obtained for the thermal conductivity. This overestimation of σ_o can be shown in the following way: from the reflectivity (R) measurements reported by Weaver et.al. at short wavelengths (see Fig. 2 in [23]), for energies between 0.2 and 0.4 eV, and from application of the Hagen-Rubens relation [R=1-2(2ωε_o/σ)^1/2 when σ/ωε_o much larger than unity], one can estimate by extrapolation the static conductivity. We obtain σ_o=46.6 × 10⁴ S/cm (ρ_o=2.15 μΩ cm), with σ/ωε_o between 43 and 181 in the mentioned spectral range. The RT resistivity of the sample used by Weaver et al. is smaller than the expected one.

The contributions to thermal conductivity by each charge carrier type are obtained from: κ_e=n_ev_e²γ_hcc,eTτ_oe/3N_A=8.9 W/m K and κ_h=n_hv_h²γ_hcc,hTτ_oh/3N_A=151 W/m K with T=295 K [12]. Avogadro’s number has been introduced to obtain the thermal conductivity with the units specified when the heat capacity coefficient is given in J/K² mol. The total conductivity is κ_th=κ_e+κ_h=160 W/m K. The Lorenz numbers are: L_N,e=κ_e/σ_eT=1.62 × 10⁻⁸ WΩ/K² and L_N,h=κ_h/σ_hT=1.94 × 10⁻⁸ WΩ/K², with its average value given by L_N=κ_th/σ_oT= (σ_eL_N,e+σ_hL_N,h)/σ_o=1.92 × 10⁻⁸ WΩ/K². This value is significantly lower than the measured one due to the overestimated value of the electrical conductivity.

3.4 Magnetic susceptibility

Total magnetic susceptibility obtained once the optimization was carried out is χ_m=2.81 × 10⁻⁶ cm³/g. The molar susceptibility can be determined from the total mass susceptibility: χ_M=Mχ_m=2.89 × 10⁻⁴ cm³/mol with M=102.9055 g/mol as the atomic weight of Rh. From Curie’s law for paramagnetic materials [57], and the relation between diamagnetic and paramagnetic susceptibilities (see the thirteenth line in Table 2), the number of effective Bohr magnetons is given by p=(3k_B/μ_oN_av)^½(χ_MT)^½/μ_B with N_av as Avogadro’s number [12]. The average magnetic moment per atom is μ_avg=pμ_B. The evaluation for Rh gives p=0.23 at RT. Calculations of p for Rh_n clusters of n-atoms from quantum mechanics formalisms have yielded to dispersive results. The value p=0.23 is close to p=0.28 obtained by Villaseñor-González et al. for Rh₄₃ atoms in a fcc structure using self-consistent tight-binding calculations [58]. Values between 0.80 and 0.08 have been reported by Cox et al. for Rh_n atomic clusters with n between 9 and 55 [59,60]. The largest values are displayed for small clusters and decreasing dispersive values were obtained as the size of the cluster increases, with p±Δp=0.26 ± 0.02. When the exchange correlation effect is included, χ_M,ec=χ_M/[1-ρ(E_F)I_F] where I_F is the exchange correlation integral and ρ(E_F)I_F is the Stoner parameter [61,62]. The evaluations give χ_M,ec=4.91 × 10⁻⁴ cm³/mol and p=0.30 when using reported values for ρ(E_F) = 1.33 states/eV atom and I_F=0.309 eV [63], with the enhancement factor being [1-ρ(E_F)I_F]^-1=1.698, and the recalculated value of χ_m=4.77 × 10⁻⁶ cm³/g. With the values for ρ(E_F) and I_F reported by Janak [64],1.32 states/eV atom and 0.327 eV, respectively, χ_M,ec=5.09 × 10⁻⁴ cm³/mol and p=0.32.

The electrical conduction is due to itinerant electrons and holes charge carriers. This means that these charge carriers partially behave like free charge carriers or like localized ones. The presence of this partial localization or spatial confinement of them would decrease the magnetic susceptibility, as shown by Cahaya who extended the known relations of Pauli and Landau susceptibilities by incorporating two facts in his derivation [65]: (a) by considering an infinite quantum well, he assumed null values of the charge carrier wave functions at the boundaries of the medium instead of periodic boundary conditions, and (b) an artificial local confinement is introduced through a spatially periodic magnetic field whose spatial period λ_q is tuned through the wave number q=2π/λ_q. This field introduces a local confinement of the charge carriers in those planes perpendicular to the direction of the magnetic field. He obtains the same expressions for Pauli and Landau magnetic susceptibilities, multiplied by corresponding factors given by

To apply Cahaya’s formalism linked with the initially calculated magnetic Pauli and Landau susceptibility components for both electrons and holes, we write the corresponding mean free paths in terms of the spatial period of the magnetic field: L_k=N_P/L,_kλ_q,P/L,k and x_k becomes x_k=πN_P/L,_k/L_kk_F,k with k_F,k=(3π²n_k)^1/3 [k_F,e=11.25 nm, k_F,h=11.95 nm]. The fraction of each contribution to the total susceptibility (F_P,k and F_L,k with k = e and h) is calculated from the results of the optimization, after applying the enhancement due to exchange correlations. By assuming that the same fractions can be associated with the reported experimental value, the four contributions giving the measured susceptibility are evaluated. For each component, the equation │F_P/L,k${\cdot}$χ_exp - Z_P/L,k${\cdot}$χ_k│=δ, with δ close to or lower than 10⁻¹⁰, has been solved for χ₁=χ_e,ec, χ₂=χ_h,ec, χ₃=χ_e’_,ec, and χ₄=χ_h’_,ec. Table 5 shows the results. The lower value of λ_q,L when compared with λ_q,P suggests that the charge carriers contributing to diamagnetism are the electrons which are involved in s-p transitions. The s-states and p-states of neighboring atoms are less separated than states d-d. Holes contribute more significantly to the paramagnetic susceptibility.

Table 4. Set of average values and standard deviations of derived quantities for the Rh bulk sample whose DF is depicted in Fig. 2 with the optimized Drude parameters indicated in Table 3. The parameters in bold are effective Drude parameters.

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Table 5. Diminution factors for both conduction electrons and holes contributing to the Pauli and Landau magnetic susceptibilities. The Rh lattice constant is a=0.380 nm.

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3.5 Heat capacity coefficient

The reported value of the charge-carrier heat capacity coefficient is γ_exp=4.65 mJ/K² mol [50]. This measurement includes the contributions from electron-phonon coupling (EPC), and exchange correlation (EC) between charge carriers. The value obtained from the optimization process is significantly lower, γ_hcc=1.86 mJ/K² mol, about 2/5 of the reported one, which means that the contributions from the EPC and EC are large in rhodium. Regarding the total heat capacity coefficient γ, it is given by the relation γ=π²k_B²ρ(E_F)(1+λ)/3 with contributions from charge carriers and electron-phonon interactions. By doing γ=γ_exp=γ_hcc(1+λ) with λ as the coupling strength of the charge carriers both to the lattice phonons and to other charge carriers [69], one obtains λ=(γ_exp-γ_hcc)/γ_hcc=1.50. These evaluations do not account for the effect of EC between electrons at the Fermi surface nor EPC in the value of γ_hcc obtained from the optimization. The electronic specific heat is given by C=γT, with T as the absolute temperature and γ the specific heat coefficient, which can be also written as γ=(πk_B)²nm/($\hbar$k_F)² where k_F=(3π²n)^1/3 is the Fermi momentum, n is the number density of charge carriers, and m the effective mass of each one. If the previous equation for γ is applied to an ideal electron gas, with no interactions between charge carriers or between them and the phonons, γ_k$\to$γ_o,k=(πk_B)²n_km/($\hbar$k_F,k)² with k = e or h. In the context of the present optimization approach, γ_k=(πk_B)²n_km_k/($\hbar$k_F,k)². Consequently, γ_k/γ_o,k=m_k/m. This relation is valid in the limit of T$\to$0 where the Fermi momentum is not changed by many-body effects according to the Luttinger theorem [70,71]. The optimized values of the relative effective masses of electrons and holes become enhancement (λ_k>0) or diminution (λ_k<0) factors [m_k/m${\equiv}$1+λ_k] that must be applied to the first estimations of the corresponding specific heat coefficients. The corrected values of the specific heat coefficients are γ_hcc,e$\to$(1+λ_e)${\cdot}$γ_hcc,e=1.79 mJ/K² mol and γ_hcc,h$\to$(1+λ_h)${\cdot}$γ_hcc,h=1.66 mJ/K² mol. The recalculated total heat capacity coefficient is γ_hcc=3.45 mJ/K² mol. The evaluation of Papaconstantopoulos, based on tight-binding calculations, gives 3.24 mJ/K² mol [72]. Calculated values between 3.12 and 3.61 mJ/K² mol have been reported [73]. The corrected value of the electron-phonon coupling parameter is λ=0.35. This value is close to λ=0.34 reported for iridium [74]. We are not aware of previous reports of λ-value for Rh.

4. Bulk and surface energy loss functions

The absorption of energy by a material from that being transported by the propagating electromagnetic wave is determined by the energy loss function (ELF) [75]. The bulk-ELF can be written in the form

A similar approach is followed to carry out the decomposition of the surface-ELF [77]:

Figure 8(a) displays the results for the bulk-ELF values, and Fig. 8(b) corresponds to the surface-ELF spectra. Is not easy to distinguish surface from volume plasmon excitations, or their nature (due to bound electrons or charge carriers) in experimental energy loss spectra [21,78]. The decoupling of the total ELF in charge carriers and inter-band contributions simplify this task. This happens for Rh with the largest peaks displayed for energies lower than 10 eV which arises from contributions by charge carriers and bound electrons. The Drude bulk ELF displays small peaks at 0.8 and 2.3 eV, with the main peak at 8.6 eV, associated with the light absorption by the dominant volume plasmon excitation. A broader peak is shown at 33.1 eV. Two shoulders are spectrally located at 5.4 and 28.7 eV. The contribution to the total ELF from bound electrons or inter-band transitions shows a dominant peak at 32.7 eV with another one at 8.8 eV. Regarding the surface ELF, the dominant surface plasmon excitation of charge carriers is displayed at 7.8 eV, superimposed on a surface excitation of bound electrons displayed at 7.9 eV. Other broader surface excitation of bound electrons is seen at 24.1 eV.

 

Fig. 8. Total bulk (a) and surface (b) energy loss functions (ELFs) for the Rh sample whose modelled DF is displayed in Fig. 2. The ELFs of the charge-carriers (Drude) and bound electrons (Lorentz) are also displayed. In figure (a) A_o=A_o^(B) and B_o=B_o^(B), and in figure (b) A_o=A_o^(S) and B_o=B_o^(S) according to Eqs. (5) and (6), respectively.

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5. Effective numbers of charge carriers and bound electrons

Given the bulk-ELF, the effective total number of bulk electrons and holes per atom participating as charge carriers and bound electron per atom involved in inter-band transitions, in the spectral range considered, can be evaluated from the partial sum rule [79]

The evaluations give $Z_{eff}^{(cc)}$=0.013 and $Z_{eff}^{(it)}$=13.653. $Z_{eff}^{(cc)}$ is significantly lower than z_e+z_h which is an indication that electron-hole recombination in Rh plays a meaningful role. The ELF associated with the bulk Lorentz contribution to the DF can also be decomposed to display the relevant absorption peaks of each oscillator and the presence of collective oscillations of bound electrons. The starting points is the equation

In this way $L_L^{(B)}(\Omega ) = \sum\limits_{j = 1}^K {L_{L,j}^{(B)}(\Omega )}$. From the decomposition of the Lorentz ELF, the effective number of electrons per atom participating in each inter-band transition can be evaluated:

The following results are obtained: Z_eff⁽¹⁾=4.207 × 10⁻⁴, Z_eff⁽²⁾=3.229 × 10⁻¹, Z_eff⁽³⁾=4.097 × 10⁻³, Z_eff⁽⁴⁾=1.080 × 10⁻², Z_eff⁽⁵⁾=1.176 × 10⁻¹, Z_eff⁽⁶⁾=1.694 × 10⁻¹, Z_eff⁽⁷⁾=7.653 × 10⁻¹, Z_eff⁽⁸⁾=4.879 × 10⁻¹, Z_eff⁽⁹⁾=9.359 × 10°, Z_eff⁽¹⁰⁾=9.484 × 10⁻¹, and Z_eff⁽¹¹⁾=1.467 × 10°. The sum of these eleven Z_eff^(j)-values gives 13.653.

6. Summary

The dielectric function of Rhodium has been modeled through an extensive energy range going from the infrared (0.2 eV) to the vacuum ultraviolet (39.0 eV). A Drude-Lorentz model has been applied with its parameters being optimized through an acceptance-probability-controlled simulated annealing method. The dielectric function model incorporates in a self-consistent way the contributions of electrons and holes to the Drude term of the dielectric function. Each one of the optimized parameters is characterized by its physical meaning, and is compared, when possible, with reported values found in the literature: high frequency dielectric constant, volume plasma frequencies, electron scattering frequency, average number of itinerant conduction electrons contributed by each atom, resonance frequencies, resonance widths, and oscillator strengths. An extensive set of derived physical quantities is evaluated from some of the optimized parameters. They cover optical, magnetic, charge transport, and plasmonic properties of this transition metal: electron and hole effective masses, number of holes contributed by each atom, hole scattering frequency, relaxation times of electron and holes, static and dynamic conductivities, electrical resistance, Hall coefficient, heat capacity coefficients and electron-phonon interaction coefficients, mobilities of electron and holes, paramagnetic and diamagnetic susceptibilities of conduction electron and holes, effective number of Bohr magnetons, thermal conductivity, Fermi energies of charge carriers (electrons and holes) and corresponding densities of states, Fermi velocities and intrinsic mean free paths. Additionally, the bulk and surface energy loss function obtained from the modeled dielectric function has been decoupled in terms of the charge carriers and inter-band transition contributions, making possible the visualization of bulk and surface plasmonic behavior of Rh. In future work, it would be desirable to compare the performance of APCSA with other global optimization algorithms such as genetic algorithm or particle swarm optimization. Other interesting options, related to the effect of the uncertainty of experimental spectral points, are approaches like Bayesian optimization, which can be used to evaluate the robustness of the solution.