## Abstract

Optical, charge carriers transport, quantum mechanics, magnetic, thermal, and plasmonic properties of the transition metal rhodium are considered. An extended Drude-Lorentz (DL) model is applied to describe the dielectric function (DF) of rhodium in a spectral range going from the mid-infrared (12.4 μm) to the vacuum ultraviolet (32 nm). The Drude term of the DF includes, as optimization parameters, the inverse of the high frequency dielectric constant, the volume plasma frequency and scattering frequency of the electrons, the scattering frequency of holes relative to that of electrons, the ratio between the effective masses of electrons and holes, the number of holes per atom relative to that of electrons, and the renormalized times between grain boundary scattering events for electrons and holes. The Lorentz contribution to the DF includes the number of conduction electrons per atom, the oscillator strengths, the resonance energies, and the Lorentzian widths. Values of the parameters involved in the DF are optimized by an acceptance-probability-controlled simulated annealing method that minimizes spectral differences between the real and imaginary parts of the DF values obtained from the literature and those evaluated from the DL parametric formulation, accounting for the presence of electrons and holes as charge carriers. Once an optimized spectral description of the DF of rhodium is obtained, a large set of charge-transport, magnetic, thermal, plasmonic, and quantum mechanics derived quantities are evaluated: mobilities, relaxation times, Fermi velocities, effective masses, electrical and thermal conductivities, heat capacity coefficients, Hall coefficient, diamagnetic and paramagnetic susceptibilities, effective number of Bohr magnetons, Fermi energies and corresponding densities of states, energy loss functions, effective number of charge carriers participating in conduction, and effective number of electrons involved in inter-band transitions.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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=ε_{1}+*i*ε_{2}), 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]:

*h*ν=

*h*ω/2π with (ω) ν as the (angular) frequency of the incident light,

*h*is Planck´s constant, ε

_{hf}is the high frequency dielectric constant associated with the high energy inter-band transitions spectrally located beyond the energy range considered, Ω

_{pe}=

*h*ν

_{pe}(Ω

_{ph}=

*h*ν

_{ph}) with corresponding ω

_{pe}(ω

_{ph}) as the volume plasma angular frequency of the collective oscillation of electrons (holes), whose scattering frequency is γ

_{oe}(γ

_{oh}). Namely, ω

_{pe}=(n

_{e}

*e*

^{2}/ε

_{o}m

_{e})

^{1/2}where ε

_{o}is the free space permittivity,

*e*is the electron’s charge,

*n*

_{e}as the number density of conduction electrons, and

*m*

_{e}as their effective mass. The scattering frequencies, γ

_{oe}and γ

_{oh}for electrons and holes respectively, are related with corresponding relaxation times: τ

_{oe}=

*h*/2πγ

_{oe}and τ

_{oh}=

*h*/2πγ

_{oh}. The index j goes from 1 to

*K*, with

*K*equal to the number of oscillators considered. In the Lorentz term, resonance energies are denoted by Ω

_{j},

*f*

_{j}are the oscillator strengths, and γ

_{j}are the Lorentzian widths. Equation (1) specifies the DF-model in terms of 5 Drude parameters (ε

_{hf}, Ω

_{pe}, Ω

_{ph}, γ

_{oe}, and γ

_{oh}), with the Lorentz term involving 1 + 3 K parameters (z

_{e}, Ω

_{j},

*f*

_{j}, and γ

_{j}). The relative number density of conduction electrons, i.e., the number of conduction electrons per atom, is

*z*

_{e}=

*n*

_{e}/

*N*with

*N*as the number density of metal ions.

*N*can be calculated from the lattice constant

*a*and the number of atoms per conventional unit cell. For Rh, it corresponds to a face-centered cubic (fcc) structure with

*a*=0.380 nm at room temperature (RT) [34], with

*N*

_{a}=4 atoms per conventional unit cell [

*N*=

*N*

_{a}/

*V*

_{cc}with

*V*

_{cc}=

*a*

^{3}as the volume of the conventional unit cell]. It gives

*N*=7.29 × 10

^{22}atoms/cm

^{3}. It is not assumed,

*a priori*, that all valence electrons become conduction electrons nor is the effective mass of electrons assumed to be equal to their rest mass. These two assumptions, often found in textbooks and in reported modeling of the optical properties of metals, based on the Drude model, significantly obscure how accurate the model is. The

*a priori*adoption of these two assumptions is based in the fact that there is no simple methodology to obtain

*z*

_{e}and

*m*

_{e}from spectrophotometric measurements. In this sense, what is reported in this article fills such a gap. The optimized way of calculating

*z*

_{e}and

*m*

_{e}is what validates the results of all derived parameters, including Fermi velocities and corresponding densities of states.

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_{ph}^{2}=βχΩ_{pe}^{2}, 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

_{e}ω

^{2}and 1/τ

_{h}=1/τ

_{oh}+Λ

_{h}ω

^{2}, respectively. This ω

^{2}-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 ω

^{2}-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

_{e}Ω)

^{2}] and γ

_{h}=γ

_{oh}[1+(Γ

_{h}Ω)

^{2}], where Γ

*and Γ*

_{e}*are two additional parameters to be optimized: Γ*

_{h}_{k}=(2πΛ

_{k}/

*h*γ

_{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

*, and Γ*

_{e}*. 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*

_{h}*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*_{j}Ω_{pe}^{2}/{*z*_{e}[(Ω_{j}^{2}-Ω^{2})-*i*Ωγ_{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]: ${\bar{\varepsilon }_1}$=*n*^{2}-*k*^{2} and ${\bar{\varepsilon }_2}$=2*nk* [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 ε_{1}(ω) and ε_{2}(ω) simultaneously with minimized error throughout the whole spectral range being considered.

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_{pe}^{2}/(Ω(Ω+*i*γ_{e})-Ω_{ph}^{2}/(Ω(Ω+*i*γ_{h})] can be expressed approximately as χ_{eff}=-Ω_{o}^{2}/[Ω(Ω+*i*γ_{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_{o}Ω)^{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}))Ω^{2}(1+α^{2})/(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].

In the two-band model, the Hall coefficient listed can be written as *R*_{H}=-(1-χ*q*^{2})/[*ez*_{e}*N*(1+χ*q*)^{2}] with *q*=β/η [47]. Both electrons and holes contribute to the intrinsic conductivity in fractions given by *P*_{e}=1/[1+χ*q*] and *P*_{h}=1-*P*_{e}, respectively. Besides the spectral dependence of ε_{2}(ω), the absorption of light is considered through the dynamic conductivity, σ(ω)=ε_{o}ωε_{2}(ω), and through the bulk and surface energy loss functions. In the Gaussian system, the dynamic conductivity is given by σ_{G}(ω)= ωε_{2}(ω)/4π [σ(ω) = 4πε_{o}σ_{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_{o}ωε_{2}(ω) when ω tends to zero, with no inter-band transitions contribution in this limit. From Eq. (1), the limit is given by *n*_{e}*e*^{2}τ_{oe}/*m*_{e}+*n*_{h}*e*^{2}τ_{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 × 10^{4} S/cm. The experimental value of γ_{exp}=4.65 mJ/mol K^{2} 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} m^{3}/A s [51], the total magnetic susceptibility χ_{m}=0.990 × 10^{−6} cm^{3}/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/K

^{2}, close to the theoretical value

*L*

_{o}=π

^{2}k

_{B}

^{2}/3

*e*

^{2}=2.45 × 10

^{−8}WΩ/K

^{2}[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: Ω

_{1}=0.40, Ω

_{2}=1.40, Ω

_{3}=3.05, Ω

_{4}=4.70, and Ω

_{5}=5.30 eV [28]; Ω

_{6}=11.6, Ω

_{7}=14.6, Ω

_{8}=19.3, Ω

_{9}=24,0, and Ω

_{10}=30.5 eV [23]; and Ω

_{11}=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.

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.

#### 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.

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_{k}*N*_{a} *a*/*L _{k}* with

*k*=

*e*or

*h*,

*L*being the intrinsic mean free path,

_{k}*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

_{e}N

_{a}

*a*${\cdot}$ρ(E

_{e})/(L

_{e}μ

_{e}) and D

_{h}=z

_{h}N

_{a}

*a*${\cdot}$ρ(E

_{h})/(L

_{h}μ

_{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 cm

^{2}/Vs, μ

_{h}=20.2 cm

^{2}/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$0)=σ_{o}=28.3 × 10^{4} S/cm, with no contribution from bound electrons (σ_{L}(ω$\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 Ω_{1}=0.24 eV.

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 Ω_{2}=1.18, Ω_{3}=2.58, and Ω_{5}=5.79 eV, respectively. The peak close to Ω_{4}=3.0 eV is not resolved probably due to superposition with the Ω_{3}-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 × 10^{4} S/cm with σ_{e}=1.87 × 10^{4} 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 × 10^{4} 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*_{e}v_{e}^{2}γ_{hcc,e}*T*τ_{oe}/3*N*_{A}=8.9 W/m K and κ_{h}=*n*_{h}v_{h}^{2}γ_{hcc,h}*T*τ_{oh}/3*N*_{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/K^{2} 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_{e}*T*=1.62 × 10^{−8} WΩ/K^{2} and *L*_{N,h}=κ_{h}/σ_{h}*T*=1.94 × 10^{−8} WΩ/K^{2}, with its average value given by *L*_{N}=κ_{th}/σ_{o}*T*= (σ_{e}*L*_{N,e}+σ_{h}*L*_{N,h})/σ_{o}=1.92 × 10^{−8} WΩ/K^{2}. 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} cm^{3}/g. The molar susceptibility can be determined from the total mass susceptibility: χ_{M}=*M*χ_{m}=2.89 × 10^{−4} cm^{3}/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*=(3*k*_{B}/μ_{o}*N*_{av})^{½}(χ_{M}*T*)^{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 Rh_{43} 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*±Δ*p*=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} cm^{3}/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} cm^{3}/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} cm^{3}/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

*x*

_{k}=

*q*/2

*k*

_{F,k}, with

*k*

_{F,k}as the Fermi momentum of the

*k*

^{th}charge carrier type. These two parameters tend to unity when

*q*tends to zero, i.e., the known expressions of paramagnetic and diamagnetic susceptibilities are recovered when the local confinement disappears, and the charge carriers behave like free particles. Otherwise,

*Z*and

_{P}*Z*are lower than unity, i.e., they are diminution factors,

_{L}*Z*

_{P,k}being the Lindhard function [66]. Under the absence of an external magnetic field, other effects could contribute to a spatially periodic or quasi-periodic internal magnetic field: thermal fluctuations of atomic nuclei around their equilibrium positions [67], and small variations of the quasi-stationary electrostatic screening potential due to thermal fluctuations of the position of the center of mass of the electronic gas of charge carriers [68].

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},_{k}λ_{q,P/L,k} and *x*_{k} becomes *x*_{k}=π*N*_{P/L}*,*_{k}/*L*_{k}*k*_{F,k} with *k*_{F,k}=(3π^{2}*n*_{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 χ_{1}=χ_{e,ec}, χ_{2}=χ_{h,ec}, χ_{3}=χ_{e}’_{,ec}, and χ_{4}=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.

#### 3.5 Heat capacity coefficient

The reported value of the charge-carrier heat capacity coefficient is γ_{exp}=4.65 mJ/K^{2} 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/K^{2} 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=π^{2}k_{B}^{2}ρ(*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*=γ*T*, with *T* as the absolute temperature and γ the specific heat coefficient, which can be also written as γ=(πk_{B})^{2}*nm*/($\hbar$*k*_{F})^{2} where *k*_{F}=(3π^{2}*n*)^{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})^{2}*n _{k}m*/($\hbar$

*k*

_{F,k})

^{2}with

*k*=

*e*or

*h*. In the context of the present optimization approach, γ

_{k}=(πk

_{B})

^{2}

*n*

_{k}

*m*

_{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/K

^{2}mol and γ

_{hcc,h}$\to$(1+λ

_{h})${\cdot}$γ

_{hcc,h}=1.66 mJ/K

^{2}mol. The recalculated total heat capacity coefficient is γ

_{hcc}=3.45 mJ/K

^{2}mol. The evaluation of Papaconstantopoulos, based on tight-binding calculations, gives 3.24 mJ/K

^{2}mol [72]. Calculated values between 3.12 and 3.61 mJ/K

^{2}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

*ε*and

_{D}*ε*as the Drude and Lorentz dielectric functions [76]. In this way, $L_D^{(B)}(\Omega ) ={-} A_0^{(B)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} [1/{\varepsilon _D}(\Omega )]$ and $L_L^{(B)}(\Omega ) ={-} B_0^{(B)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} [1/{\varepsilon _L}(\Omega )]$ are the bulk-ELFs corresponding to the charge-carriers and bound electrons, respectively. The values of the $A_0^{(B)}$ and $B_0^{(B)}$ real coefficients are obtained at each spectral point by solving the Eq. (5a) together with

_{L}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.

## 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]

_{max}=39.0 eV in this case. The integration is carried out with Simpson’s rule and Richardson extrapolation [80]. The optimized parametric form of the DF is used to extrapolate ${L^{(B)}}(\Omega )$ from Ω

_{min}=0.2 eV to zero frequency. Once the evaluation is carried out,

*Z*=13.666. Similar sum rules hold for the charge-carriers and bound electrons per atom. They can be evaluated using the decomposition of the ELF to obtain the effective number of charge-carriers contributing to conduction [$Z_{eff}^{(cc)}$], as well as the effective number of bound electrons participating in inter-band transitions [$Z_{eff}^{(it)}$]:

_{eff}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

*K*-1, where the real coefficients $A_j^{(B)}$ and $B_j^{(B)}$ are obtained by solving Eq. (9) for their real and imaginary parts. From them, the ELFs of each one of the first j-1 oscillators are given by

*K*oscillator

^{th}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.

## Acknowledgments

The author thanks the support given by the Universidad de Costa Rica to carry out this work.

## Disclosures

The author declares no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are available in Ref. [24]. A pseudo-code of the APCSA method is available upon reasonable request.

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