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Physical properties of rhodium retrieved from modeling its dielectric function by a simulated annealing approach

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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 [13], 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 [79]. 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 [1416]. 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 EF. 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(EF), 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]:

$$\varepsilon = {\varepsilon _{hf}} - \frac{1}{\Omega }\left( {\frac{{\Omega _{pe}^2}}{{\Omega + i{\gamma_{oe}}}} + \frac{{\Omega _{ph}^2}}{{\Omega + i{\gamma_{oh}}}}} \right) + \frac{1}{{{z_e}}}\sum\limits_{j = 1}^K {\frac{{{f_j}\Omega _{pe}^2}}{{\Omega _j^2 - {\Omega ^2} - i\Omega {\gamma _j}}}}$$
where Ω=hν=hω/2π with (ω) ν as the (angular) frequency of the incident light, h is Planck´s constant, εhf is the high frequency dielectric constant associated with the high energy inter-band transitions spectrally located beyond the energy range considered, Ωpe=hνpeph=hνph) with corresponding ωpeph) as the volume plasma angular frequency of the collective oscillation of electrons (holes), whose scattering frequency is γoeoh). Namely, ωpe=(nee2ome)1/2 where εo is the free space permittivity, e is the electron’s charge, ne as the number density of conduction electrons, and me as their effective mass. The scattering frequencies, γoe and γoh for electrons and holes respectively, are related with corresponding relaxation times: τoe=h/2πγoe and τoh=h/2πγoh. 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 Ωj, fj are the oscillator strengths, and γj are the Lorentzian widths. Equation (1) specifies the DF-model in terms of 5 Drude parameters (εhf, Ωpe, Ωph, γoe, and γoh), with the Lorentz term involving 1 + 3 K parameters (ze, Ωj, fj, and γj). The relative number density of conduction electrons, i.e., the number of conduction electrons per atom, is ze=ne/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 Na=4 atoms per conventional unit cell [N = Na/Vcc with Vcc=a3 as the volume of the conventional unit cell]. It gives N=7.29 × 1022 atoms/cm3. 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 ze and me from spectrophotometric measurements. In this sense, what is reported in this article fills such a gap. The optimized way of calculating ze and me 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): γoh=ηγoe, Ωph2=βχΩpe2, and εhf=1/fo with β=me/mh, χ=nh/ne=zh/ze, and fo with values between zero and unity. The counterparts of ne and ze are nh and zh for holes, respectively, with ne$\ne$nh 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/τe=1/τoeeω2 and 1/τh=1/τohhω2, respectively. This ω2-dependence of 1/τe has been phenomenologically attributed to the scattering of the charge carriers at grain boundaries [37]. Under this assumption, it would be expected that Λe and Λh 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: γeoe[1+(ΓeΩ)2] and γhoh[1+(ΓhΩ)2], where Γe and Γh are two additional parameters to be optimized: Γk=(2πΛk/hγok)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 γe and γh substitute γoe and γoh respectively in Eq. (1). The parameters to be optimized in the Drude susceptibility term are Ωpe, γoe, β, χ, η, Γe, and Γh. 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 fo. With the definitions given through this paragraph, the set of 8 Drude DF parameters to be optimized is fo, Ωpe, β, $\chi $, γoe, η, Γe, and Γh.

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 jth population of electrons bounded to the metal ions whose resonance frequency (energy) is ωjj), i.e. εLL,1L,2+…+εL,K with εL,j(Ω)=fjΩpe2/{ze[(Ωj22)-iΩγj]}. The resonance energies (Ωj), Lorentzian widths (γj), and oscillator strengths (fj) 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 Np=9 + 3 K parameters in the DL model of the dielectric function are determined by minimizing the following merit function:

$${F_\textrm{J}} = \frac{1}{{2\kappa - {N_p} - 1}}\sum\limits_{i = 1}^\kappa {\left[ {{{\left( {\frac{{{\varepsilon_1}({\omega_i}) - {{\bar{\varepsilon }}_1}({\omega_i})}}{{{{\bar{\varepsilon }}_1}({\omega_i})}}} \right)}^2} + {{\left( {\frac{{{\varepsilon_2}({\omega_i}) - {{\bar{\varepsilon }}_2}({\omega_i})}}{{{{\bar{\varepsilon }}_2}({\omega_i})}}} \right)}^2}} \right]}$$
where κ is the number of known spectral DF values (${\bar{\varepsilon }_1} + i{\bar{\varepsilon }_2}$). In terms of the optimization parameters and photon energy, the DF is given by
$$\varepsilon = \frac{1}{{{f_o}}} - \frac{{\Omega _{pe}^2}}{\Omega }\left( {\frac{1}{{\Omega + i{\gamma_{oe}}[1 + {{({\Gamma _e}\Omega )}^2}]}} + \frac{{\beta \chi }}{{\Omega + i\eta {\gamma_{oe}}[1 + {{({\Gamma _h}\Omega )}^2}]}}} \right) + \frac{1}{{{z_e}}}\sum\limits_{j = 1}^K {\frac{{{f_j}\Omega _{pe}^2}}{{\Omega _j^2 - {\Omega ^2} - i\Omega {\gamma _j}}}}$$

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 [3941]. The subindex J in Eq. (2) stands for the Jth 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 FJ 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 FJ is 2κ minus the number of quantities that are optimized, including FJ 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.

 figure: Fig. 1.

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) [χDrude=-Ωpe2/(Ω(Ω+iγe)-Ωph2/(Ω(Ω+iγh)] can be expressed approximately as χeff=-Ωo2/[Ω(Ω+iγeff)]. The effective values of γeff are obtained from the relation: γeff$\cdot$Im(χeff)/│Re(χeff)│. As expected, this effective scattering frequency shows a dependence on energy that can be modeled by γeffo(1+(ΓoΩ)2), with regression coefficients close to unity. The fitting of γeff allows us to calculate both γoo= h/2πγo) and Γo. For each spectral point, with α=γeff/Ω, the value of the effective volume plasma energy is obtained from the relation Ωo=[(-Re(χeff)+Im(χeff))Ω2(1+α2)/(1+α)]1/2, showing very low spectral dispersion. Its average value is evaluated from the set of Ωo-values. The γo, Γo, and Ωo 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].

Tables Icon

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.

Tables Icon

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

In the two-band model, the Hall coefficient listed can be written as RH=-(1-χq2)/[ezeN(1+χq)2] with q=β/η [47]. Both electrons and holes contribute to the intrinsic conductivity in fractions given by Pe=1/[1+χq] and Ph=1-Pe, respectively. Besides the spectral dependence of ε2(ω), the absorption of light is considered through the dynamic conductivity, σ(ω)=εoωε2(ω), and through the bulk and surface energy loss functions. In the Gaussian system, the dynamic conductivity is given by σG(ω)= ωε2(ω)/4π [σ(ω) = 4πεoσG(ω) with σ in S/m and σG in 1/s].Within this formulation, the static conductivity is given by σooωε2(ω) when ω tends to zero, with no inter-band transitions contribution in this limit. From Eq. (1), the limit is given by nee2τoe/me+nhe2τoh/mh 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, ρo=4.78 μΩ cm, for millimeter-sized samples annealed at 1300 °C [48,49], with the corresponding value for the static conductivity σo=21.3 × 104 S/cm. The experimental value of γexp=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 RH=5.05 × 10−11 m3/A s [51], the total magnetic susceptibility χm=0.990 × 10−6 cm3/g [52], and the effective volume plasma energy Ωo=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 LN=2.41 × 10−8 WΩ/K2, close to the theoretical value Lo2kB2/3e2=2.45 × 10−8 WΩ/K2 [54]. The thermal conductivity has been measured at RT in κth=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 FJ=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, FJ=3.848 × 10−3. The FJ-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: Fig. 2.

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 FJ=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 ze 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 FJ-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 ηο, i.e., when│(FJ-FJ-k)/FJ│<ηο for k=1,2,3. We have used ηο=10−4 with κ=208 as the number of spectral DF values.

 figure: Fig. 3.

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) Ωpe=5.878 eV, (b) γoe=0.245 eV, (c) β=me/mh=1.071, (d) χ=zh/ze=1.198, (e) η=γohoe=0.092, and (f) ze=0.662.

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 figure: Fig. 4.

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 ρo=3.53 μΩ cm, (b) Hall coefficient RH=9.73 × 10−11 m3/As, (c) average electron effective mass me=1.92 m, (d) average hole effective mass mh=1.79 m, (e) total heat capacity coefficient γhcc=1.85 mJ/K2mol, and (f) logarithmic value reached by the merit function log(FJ)=-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 FJ=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.

 figure: Fig. 5.

Fig. 5. Schematic representation of the global minimum of the merit function. The first FJ-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, σF, are obtained, as well as those corresponding to the optimized and derived parameters.

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 figure: Fig. 6.

Fig. 6. Behavior of the merit function FJ 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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Tables Icon

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 fo and the oscillator strength fj-values. Values of Γe and Γh 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.

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 (Pdd) of neighboring atoms is lower than that between s- to d-states (Psd). It is expected to have γohoe 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(Ee) and ρ(Eh) 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 zkNa a/Lk with k = e or h, Lk being the intrinsic mean free path, a is the lattice constant, and with Na 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 (Le=1.80 nm and Lh=22.4 nm, respectively), ze and zh, and the corresponding carriers mobilities, the renormalized densities of states De=zeNaa${\cdot}$ρ(Ee)/(Leμe) and Dh=zhNaa${\cdot}$ρ(Eh)/(Lhμh) can be calculated. The ratio between them is Dh/De=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(Ee) = 0.27 states/eV atom, ρ(Eh) = 0.09 states/eV atom, Le=4.35 nm, and Lh=10.8 nm, μe=7.45 cm2/Vs, μh=20.2 cm2/Vs, with η=0.987. Consequently, Dh/De=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): [Dh/De]Ir/[Dh/De]Rh∼5 and ηIrRh∼11.

The total dynamic conductivity (σ=σLD) is displayed in Fig. 7, with decoupled Lorentz (σL) and Drude (σD) contributions. Extrapolation to the zero-frequency limit gives the static conductivity: σ(ω$\to$0)=σD$\to$0)=σo=28.3 × 104 S/cm, with no contribution from bound electrons (σL$\to$0) = 0). The corresponding resistivity is ρo=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.

 figure: Fig. 7.

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 σL and σD respectively, with the total conductivity given by σ=σLD.

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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: σoeh=28.3 × 104 S/cm with σe=1.87 × 104 S/cm and σh=14.2σe 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 σo 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(2ωεo/σ)1/2 when σ/ωεo much larger than unity], one can estimate by extrapolation the static conductivity. We obtain σo=46.6 × 104 S/cm (ρo=2.15 μΩ cm), with σ/ωεo 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: κe=neve2γhcc,eTτoe/3NA=8.9 W/m K and κh=nhvh2γhcc,hTτoh/3NA=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 κtheh=160 W/m K. The Lorenz numbers are: LN,eeeT=1.62 × 10−8 WΩ/K2 and LN,hhhT=1.94 × 10−8 WΩ/K2, with its average value given by LNthoT= (σeLN,ehLN,h)/σo=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 χm=2.81 × 10−6 cm3/g. The molar susceptibility can be determined from the total mass susceptibility: χM=Mχm=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=(3kBoNav)½MT)½B with Nav as Avogadro’s number [12]. The average magnetic moment per atom is μavg=B. The evaluation for Rh gives p=0.23 at RT. Calculations of p for Rhn 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 Rhn 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, χM,ecM/[1-ρ(EF)IF] where IF is the exchange correlation integral and ρ(EF)IF is the Stoner parameter [61,62]. The evaluations give χM,ec=4.91 × 10−4 cm3/mol and p=0.30 when using reported values for ρ(EF) = 1.33 states/eV atom and IF=0.309 eV [63], with the enhancement factor being [1-ρ(EF)IF]-1=1.698, and the recalculated value of χm=4.77 × 10−6 cm3/g. With the values for ρ(EF) and IF reported by Janak [64],1.32 states/eV atom and 0.327 eV, respectively, χM,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 λq is tuned through the wave number q=2π/λq. 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

$${{\rm Z}_{P,k}} = \frac{1}{2} + \frac{{1 - x_k^2}}{{4{x_k}}}\ln \left|{\frac{{{x_k} + 1}}{{{x_k} - 1}}} \right|,$$
$${{\rm Z}_{L,k}} = \frac{3}{{8x_k^2}}\left[ {1 + x_k^2 - \frac{{{{(1 - x_k^2)}^2}}}{{2{x_k}}}\ln \left|{\frac{{{x_k} + 1}}{{{x_k} - 1}}} \right|} \right] = \frac{3}{{4x_k^2}}[{1 - (1 - x_k^2){Z_{P,k}}} ],$$
where xk=q/2kF,k, with kF,k as the Fermi momentum of the kth 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, ZP and ZL are lower than unity, i.e., they are diminution factors, ZP,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: Lk=NP/L,kλq,P/L,k and xk becomes xkNP/L,k/LkkF,k with kF,k=(3π2nk)1/3 [kF,e=11.25 nm, kF,h=11.95 nm]. The fraction of each contribution to the total susceptibility (FP,k and FL,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 │FP/L,k${\cdot}$χexp - ZP/L,k${\cdot}$χk│=δ, with δ close to or lower than 10−10, has been solved for χ1e,ec, χ2h,ec, χ3e,ec, and χ4h,ec. Table 5 shows the results. The lower value of λq,L when compared with λq,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.

Tables Icon

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.

Tables Icon

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.

3.5 Heat capacity coefficient

The reported value of the charge-carrier heat capacity coefficient is γexp=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, γhcc=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=π2kB2ρ(EF)(1+λ)/3 with contributions from charge carriers and electron-phonon interactions. By doing γ=γexphcc(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=(γexphcc)/γhcc=1.50. These evaluations do not account for the effect of EC between electrons at the Fermi surface nor EPC in the value of γhcc obtained from the optimization. The electronic specific heat is given by CT, with T as the absolute temperature and γ the specific heat coefficient, which can be also written as γ=(πkB)2nm/($\hbar$kF)2 where kF=(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, γk$\to$γo,k=(πkB)2nkm/($\hbar$kF,k)2 with k = e or h. In the context of the present optimization approach, γk=(πkB)2nkmk/($\hbar$kF,k)2. Consequently, γko,k=mk/m. This relation is valid in the limit of T$\to$0 where the Fermi momentum is not changed by many-body effects according to the Luttinger theorem [70,71]. The optimized values of the relative effective masses of electrons and holes become enhancement (λk>0) or diminution (λk<0) factors [mk/m${\equiv}$1+λk] that must be applied to the first estimations of the corresponding specific heat coefficients. The corrected values of the specific heat coefficients are γhcc,e$\to$(1+λe)${\cdot}$γhcc,e=1.79 mJ/K2 mol and γhcc,h$\to$(1+λh)${\cdot}$γhcc,h=1.66 mJ/K2 mol. The recalculated total heat capacity coefficient is γhcc=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

$${L^{(B)}}(\Omega ) ={-} {\mathop{\rm Im}\nolimits} \left[ {\frac{1}{{\varepsilon (\Omega )}}} \right] ={-} A_0^{(B)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} \left[ {\frac{1}{{{\varepsilon_D}(\Omega )}}} \right] - B_0^{(B)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} \left[ {\frac{1}{{{\varepsilon_L}(\Omega )}}} \right],$$
with εD and εL 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
$${\mathop{\rm Re}\nolimits} \left[ {\displaystyle{1 \over {\varepsilon (\Omega )}}} \right] = A_0^{(B)} (\Omega )\cdot {\mathop{\rm Re}\nolimits} \left[ {\displaystyle{1 \over {\varepsilon _D(\Omega )}}} \right] + B_0^{(B)} (\Omega )\cdot {\mathop{\rm Re}\nolimits} \left[ {\displaystyle{1 \over {\varepsilon _L(\Omega )}}} \right].$$

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

$${L^{(S)}}(\Omega ) ={-} {\mathop{\rm Im}\nolimits} \left[ {\frac{1}{{\varepsilon (\Omega ) + 1}}} \right] ={-} A_0^{(S)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} \left[ {\frac{1}{{{\varepsilon_D}(\Omega ) + 1}}} \right] - B_0^{(S)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} \left[ {\frac{1}{{{\varepsilon_L}(\Omega ) + 1}}} \right]$$

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.

 figure: Fig. 8.

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) Ao=Ao(B) and Bo=Bo(B), and in figure (b) Ao=Ao(S) and Bo=Bo(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]

$${Z_{eff}} = \frac{2}{{\pi \textrm{ }\Omega _{pe}^2}}\int\limits_0^{{\Omega _{\max }}} {\Omega \cdot {\mathop{\rm Im}\nolimits} [ - 1/\varepsilon (\Omega )]d\Omega } = \frac{2}{{\pi \textrm{ }\Omega _{pe}^2}}\int\limits_0^{{\Omega _{\max }}} {\Omega \cdot {L^{(B)}}(\Omega )d\Omega }$$
with Ωmax=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 Ωmin=0.2 eV to zero frequency. Once the evaluation is carried out, Zeff=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 [$Z_{eff}^{(it)}$]:
$$Z_{eff}^{(cc)} = \frac{2}{{\pi \textrm{ }\Omega _{pe}^2}}\int\limits_0^{{\Omega _{\max }}} {\Omega \cdot L_D^{(B)}(\Omega )d\Omega },$$
$$Z_{eff}^{(it)} = \frac{2}{{\pi \textrm{ }\Omega _{pe}^2}}\int\limits_0^{{\Omega _{\max }}} {\Omega \cdot L_L^{(B)}(\Omega )d\Omega }.$$

The evaluations give $Z_{eff}^{(cc)}$=0.013 and $Z_{eff}^{(it)}$=13.653. $Z_{eff}^{(cc)}$ is significantly lower than ze+zh 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

$$\frac{1}{{{\varepsilon _L}(\Omega )}} = \frac{{A_j^{(B)}(\Omega )}}{{{\varepsilon _{L,j}}(\Omega )}} + \frac{{B_j^{(B)}(\Omega )}}{{\sum\limits_{i = j + 1}^K {{\varepsilon _{L,i}}(\Omega )} }}$$
with j=1,2, …, 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
$$L_{L,j}^{(B)}(\Omega ) ={-} \left[ {\prod\limits_{i = 0}^{j - 1} {B_i^{(B)}(\Omega )} } \right]A_j^{(B)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} [{1/{\varepsilon_{L,j}}(\Omega )} ]$$
and for the last Kth oscillator
$$L_{L,K}^{(B)}(\Omega ) ={-} \left[ {\prod\limits_{i = 0}^{K - 2} {B_i^{(B)}(\Omega )} } \right]B_{K - 1}^{(B)}(\Omega ) \cdot {\mathop{\rm Im}\nolimits} [{1/{\varepsilon_{L,K}}(\Omega )} ]$$

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:

$$Z_{eff}^{(j)} = \frac{2}{{\pi \textrm{ }\Omega _{pe}^2}}\int\limits_0^{{\Omega _{\max }}} {\Omega \cdot L_{L,j}^{(B)}(\Omega )d\Omega }$$

The following results are obtained: Zeff(1)=4.207 × 10−4, Zeff(2)=3.229 × 10−1, Zeff(3)=4.097 × 10−3, Zeff(4)=1.080 × 10−2, Zeff(5)=1.176 × 10−1, Zeff(6)=1.694 × 10−1, Zeff(7)=7.653 × 10−1, Zeff(8)=4.879 × 10−1, Zeff(9)=9.359 × 10°, Zeff(10)=9.484 × 10−1, and Zeff(11)=1.467 × 10°. The sum of these eleven Zeff(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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39. A. D. Rakic, J. M. Elazar, and A. B. Djurisic, “Acceptance-probability-controlled simulated annealing: A method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995). [CrossRef]  

40. A. B. Djurisic, A. D. Rakic, and J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997). [CrossRef]  

41. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef]  

42. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953). [CrossRef]  

43. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983). [CrossRef]  

44. F. Wooten, Optical Properties of Solids (Academic Press, 1972).

45. A. H. Morrish, The Physical Principles of Magnetism (IEEE Press, 2001).

46. E. G. Batyev, “Pauli paramagnetism and Landau diamagnetism,” Phys.-Usp. 52(12), 1245–1246 (2009). [CrossRef]  

47. J. M. Ziman, Electrons and Phonons (Clarendon Press, 1960)

48. G. T. Meaden, Electrical Resistance of Metals (Plenum Press, 1965).

49. G. K. White and S. B. Woods, “Thermal and electrical conductivity of rhodium, iridium, and platinum,” Can. J. Phys. 35(3), 248–257 (1957). [CrossRef]  

50. G. T. Furukawa, M. L. Reilly, and J. S. Gallagher, “Critical analysis of heat-capacity data and evaluation of thermodynamics properties of ruthenium, rhodium, iridium, and platinum from 0 to 300 K. A survey of the literature data on osmium,” J. Phys. Chem. Ref. Data 3(1), 163–209 (1974). [CrossRef]  

51. C. M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, 1972).

52. F. E. Hoare and J. C. Walling, “An absolute measurement of the susceptibility of tantalum and other metals,” Phys. Stat. Sol. (b) 64, 337–341 (1952). [CrossRef]  

53. D. T. Pierce and W. E. Spicer, “Optical properties of rhodium,” Phys. Stat. Sol. 60(2), 689–694 (1973). [CrossRef]  

54. G. S. Kumar and G. Prasad, “Experimental determinations of the Lorenz number,” J. Mat. Sci. 28(16), 4261–4272 (1993). [CrossRef]  

55. M. Raydan, “The Barzilai and Borwein gradient method for the large-scale unconstrained minimization problem,” SIAM J. Optim. 7(1), 26–33 (1997). [CrossRef]  

56. A. Ramírez-Porras and W. E. Vargas-Castro, “Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method,” Appl. Opt. 43(7), 1508–1514 (2004). [CrossRef]  

57. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1985).

58. P. Villaseñor-González, J. Dorantes-Dávila, H. Dreyssé, and G. M. Pastor, “Size and structural dependence of the magnetic properties of rhodium clusters,” Phys. Rev. B 55(22), 15084–15091 (1997). [CrossRef]  

59. A. J. Cox, J. G. Louderback, and L. A. Bloomfield, “Experimental observation of magnetism in rhodium clusters,” Phys. Rev. Lett. 71(6), 923–926 (1993). [CrossRef]  

60. A. J. Cox, J. G. Lounderback, S. E. Apsel, and L. A. Bloomfield, “Magnetism in 4d-transition metal clusters,” Phys. Rev. B 49(17), 12295–12298 (1994). [CrossRef]  

61. S. H. Vosko and J. P. Perdew, “Theory of the spin susceptibility of an inhomogenous electron gas via the density functional formalism,” Can. J. Phys. 53(14), 1385–1397 (1975). [CrossRef]  

62. P. B. Allen, “The electron-phonon coupling constant,” in Handbook of Superconductivity, C. P. Poole editor (Academic Press, 1999) pp.478-483.

63. M. M. Sigalas and D. A. Papaconstantopoulos, “Calculations of the total energy, electron-phonon interaction, and Stoner parameter for metals,” Phys. Rev. B 50(11), 7255–7261 (1994). [CrossRef]  

64. J. F. Janak, “Uniform susceptibilities of metallic elements,” Phys. Rev. B 16(1), 255–262 (1977). [CrossRef]  

65. A. B. Cahaya, “Paramagnetic and diamagnetic susceptibility of infinite quantum well,” Journal of Materials Science, Geophysics, Instrumentation and Theoretical Physics 3, 61–67 (2020). [CrossRef]  

66. J. Lindhard, “On the properties of a gas of charge particles,” Kgl. Danske Videnskab. Selskab Mat. Fys. Medd. 28, 1–57 (1954).

67. I. A. Dorofeev and E. A. Vinogradov, “Structure of electromagnetic fields near the surface of an ionic crystal,” . Synch. Investig 6(5), 796–804 (2012). [CrossRef]  

68. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).

69. G. R. Stewart, “Measurement of low-temperature specific heat,” Rev. Sci. Instrum. 54(1), 1–11 (1983). [CrossRef]  

70. J. M. Luttinger and J. C. Ward, “Ground-state energy of a many-fermion system II,” Phys. Rev. 118(5), 1417–1427 (1960). [CrossRef]  

71. J. J. Kas, T. D. Blanton, and J. J. Rehr, “Exchange-correlation contributions to thermodynamic properties of the homogenous electron gas from a cumulant Green’s function approach,” Phys. Rev. B 100(19), 195144 (2019). [CrossRef]  

72. D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids from Z=1 to Z=112 (Springer, 2015).

73. M. Yahaya and G. C. Fletcher, “The bandstructure of rhodium, its groundstate and the effects of change of volume,” J. Phys. F: Metal Phys. 9(7), 1295–1305 (1979). [CrossRef]  

74. W. L. McMillan, “Transition temperature of strong-coupled superconductors,” Phys. Rev. 167(2), 331–344 (1968). [CrossRef]  

75. M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press, 2003).

76. Y. Sun, H. Xu, B. Da, S. F. Mao, and Z. J. Ding, “Calculation of energy-loss function for 26 materials,” Chin. J. Chem. Phys. 29(6), 663–670 (2016). [CrossRef]  

77. O. Madelung, Introduction to Solid-State Theory (Springer, 1996).

78. M. M. Thiam, V. Nehasil, V. Matolín, and B. Gruzza, “The AES and EELS study of small rhodium clusters deposited onto alumina substrates,” Surf. Sci. 487(1-3), 231–242 (2001). [CrossRef]  

79. S. Tanuma, C. J. Powell, and D. R. Penn, “Inelastic mean free paths of low-energy electrons in solids,” Acta Phys. Pol. A 81(2), 169–186 (1992). [CrossRef]  

80. R. L. Burden, D. J. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015).

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  39. A. D. Rakic, J. M. Elazar, and A. B. Djurisic, “Acceptance-probability-controlled simulated annealing: A method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
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  42. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
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  43. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
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  45. A. H. Morrish, The Physical Principles of Magnetism (IEEE Press, 2001).
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  48. G. T. Meaden, Electrical Resistance of Metals (Plenum Press, 1965).
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    [Crossref]
  50. G. T. Furukawa, M. L. Reilly, and J. S. Gallagher, “Critical analysis of heat-capacity data and evaluation of thermodynamics properties of ruthenium, rhodium, iridium, and platinum from 0 to 300 K. A survey of the literature data on osmium,” J. Phys. Chem. Ref. Data 3(1), 163–209 (1974).
    [Crossref]
  51. C. M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, 1972).
  52. F. E. Hoare and J. C. Walling, “An absolute measurement of the susceptibility of tantalum and other metals,” Phys. Stat. Sol. (b) 64, 337–341 (1952).
    [Crossref]
  53. D. T. Pierce and W. E. Spicer, “Optical properties of rhodium,” Phys. Stat. Sol. 60(2), 689–694 (1973).
    [Crossref]
  54. G. S. Kumar and G. Prasad, “Experimental determinations of the Lorenz number,” J. Mat. Sci. 28(16), 4261–4272 (1993).
    [Crossref]
  55. M. Raydan, “The Barzilai and Borwein gradient method for the large-scale unconstrained minimization problem,” SIAM J. Optim. 7(1), 26–33 (1997).
    [Crossref]
  56. A. Ramírez-Porras and W. E. Vargas-Castro, “Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method,” Appl. Opt. 43(7), 1508–1514 (2004).
    [Crossref]
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    [Crossref]
  59. A. J. Cox, J. G. Louderback, and L. A. Bloomfield, “Experimental observation of magnetism in rhodium clusters,” Phys. Rev. Lett. 71(6), 923–926 (1993).
    [Crossref]
  60. A. J. Cox, J. G. Lounderback, S. E. Apsel, and L. A. Bloomfield, “Magnetism in 4d-transition metal clusters,” Phys. Rev. B 49(17), 12295–12298 (1994).
    [Crossref]
  61. S. H. Vosko and J. P. Perdew, “Theory of the spin susceptibility of an inhomogenous electron gas via the density functional formalism,” Can. J. Phys. 53(14), 1385–1397 (1975).
    [Crossref]
  62. P. B. Allen, “The electron-phonon coupling constant,” in Handbook of Superconductivity, C. P. Poole editor (Academic Press, 1999) pp.478-483.
  63. M. M. Sigalas and D. A. Papaconstantopoulos, “Calculations of the total energy, electron-phonon interaction, and Stoner parameter for metals,” Phys. Rev. B 50(11), 7255–7261 (1994).
    [Crossref]
  64. J. F. Janak, “Uniform susceptibilities of metallic elements,” Phys. Rev. B 16(1), 255–262 (1977).
    [Crossref]
  65. A. B. Cahaya, “Paramagnetic and diamagnetic susceptibility of infinite quantum well,” Journal of Materials Science, Geophysics, Instrumentation and Theoretical Physics 3, 61–67 (2020).
    [Crossref]
  66. J. Lindhard, “On the properties of a gas of charge particles,” Kgl. Danske Videnskab. Selskab Mat. Fys. Medd. 28, 1–57 (1954).
  67. I. A. Dorofeev and E. A. Vinogradov, “Structure of electromagnetic fields near the surface of an ionic crystal,” . Synch. Investig 6(5), 796–804 (2012).
    [Crossref]
  68. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).
  69. G. R. Stewart, “Measurement of low-temperature specific heat,” Rev. Sci. Instrum. 54(1), 1–11 (1983).
    [Crossref]
  70. J. M. Luttinger and J. C. Ward, “Ground-state energy of a many-fermion system II,” Phys. Rev. 118(5), 1417–1427 (1960).
    [Crossref]
  71. J. J. Kas, T. D. Blanton, and J. J. Rehr, “Exchange-correlation contributions to thermodynamic properties of the homogenous electron gas from a cumulant Green’s function approach,” Phys. Rev. B 100(19), 195144 (2019).
    [Crossref]
  72. D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids from Z=1 to Z=112 (Springer, 2015).
  73. M. Yahaya and G. C. Fletcher, “The bandstructure of rhodium, its groundstate and the effects of change of volume,” J. Phys. F: Metal Phys. 9(7), 1295–1305 (1979).
    [Crossref]
  74. W. L. McMillan, “Transition temperature of strong-coupled superconductors,” Phys. Rev. 167(2), 331–344 (1968).
    [Crossref]
  75. M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press, 2003).
  76. Y. Sun, H. Xu, B. Da, S. F. Mao, and Z. J. Ding, “Calculation of energy-loss function for 26 materials,” Chin. J. Chem. Phys. 29(6), 663–670 (2016).
    [Crossref]
  77. O. Madelung, Introduction to Solid-State Theory (Springer, 1996).
  78. M. M. Thiam, V. Nehasil, V. Matolín, and B. Gruzza, “The AES and EELS study of small rhodium clusters deposited onto alumina substrates,” Surf. Sci. 487(1-3), 231–242 (2001).
    [Crossref]
  79. S. Tanuma, C. J. Powell, and D. R. Penn, “Inelastic mean free paths of low-energy electrons in solids,” Acta Phys. Pol. A 81(2), 169–186 (1992).
    [Crossref]
  80. R. L. Burden, D. J. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015).

2021 (1)

A. R. Forouhi and I. Bloomer, “Optical dispersion equations for metals applicable to the Far-IR through EUV spectral range,” J. Phys. Commun. 5(2), 025002 (2021).
[Crossref]

2020 (1)

A. B. Cahaya, “Paramagnetic and diamagnetic susceptibility of infinite quantum well,” Journal of Materials Science, Geophysics, Instrumentation and Theoretical Physics 3, 61–67 (2020).
[Crossref]

2019 (3)

J. J. Kas, T. D. Blanton, and J. J. Rehr, “Exchange-correlation contributions to thermodynamic properties of the homogenous electron gas from a cumulant Green’s function approach,” Phys. Rev. B 100(19), 195144 (2019).
[Crossref]

A. R. Forouhi and I. Bloomer, “New dispersion equations for insulators and semiconductors valid throughout radio-waves to extreme ultraviolet spectral range,” J. Phys. Commun. 3(3), 035022 (2019).
[Crossref]

W. E. Vargas, N. Clark, F. Muñoz-Rojas, D. E. Azofeifa, and G. A. Niklasson, “Optical, charge transport and magnetic properties of palladium retrieved from photometric measurements: approaching the quantum mechanics background,” Phys. Scr. 94(5), 055101 (2019).
[Crossref]

2018 (1)

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 8010064 (2018).
[Crossref]

2017 (2)

2016 (4)

A. Ahmadivand, R. Sinha, S. Kaya, and N. Pala, “Rhodium plasmonic for deep-ultraviolet bio-chemical sensing,” Plasmonics 11(3), 839–849 (2016).
[Crossref]

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

L. V. Rodríguez-de Marcos and J. I. Larruquert, “Analytic optical-constant model derived from Tauc-Lorentz and urbach tail,” Opt. Express 24(25), 28561–28572 (2016).
[Crossref]

Y. Sun, H. Xu, B. Da, S. F. Mao, and Z. J. Ding, “Calculation of energy-loss function for 26 materials,” Chin. J. Chem. Phys. 29(6), 663–670 (2016).
[Crossref]

2012 (1)

I. A. Dorofeev and E. A. Vinogradov, “Structure of electromagnetic fields near the surface of an ionic crystal,” . Synch. Investig 6(5), 796–804 (2012).
[Crossref]

2009 (1)

E. G. Batyev, “Pauli paramagnetism and Landau diamagnetism,” Phys.-Usp. 52(12), 1245–1246 (2009).
[Crossref]

2008 (1)

W. E. Vargas, D. E. Azofeifa, N. Clark, and X. Márquez, “Collective response of silver islands on a dielectric substrate when normally illuminated with electromagnetic radiation,” J. Phys. D: Appl. Phys. 41(2), 025309 (2008).
[Crossref]

2004 (2)

2001 (3)

M. M. Thiam, V. Nehasil, V. Matolín, and B. Gruzza, “The AES and EELS study of small rhodium clusters deposited onto alumina substrates,” Surf. Sci. 487(1-3), 231–242 (2001).
[Crossref]

H. Y. Li, S. M. Zhou, J. Li, Y. L. Chen, S. Y. Wang, Z. C. Shen, L. Y. Chen, H. Liu, and X. X. Zhang, “Analysis of the Drude model in metallic films,” Appl. Opt. 40(34), 6307–6311 (2001).
[Crossref]

S. Czerczak, J. P. Gromiec, A. Palaszewska-Tkaez, and A. Swidwinska-Gajewska, “Nickel, ruthenium, rhodium, palladium, osmium, and platinum,” Patty’s Toxicology 1, 653–768 (2001).
[Crossref]

1998 (1)

1997 (4)

A. B. Djurisic, A. D. Rakic, and J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[Crossref]

P. Villaseñor-González, J. Dorantes-Dávila, H. Dreyssé, and G. M. Pastor, “Size and structural dependence of the magnetic properties of rhodium clusters,” Phys. Rev. B 55(22), 15084–15091 (1997).
[Crossref]

M. Raydan, “The Barzilai and Borwein gradient method for the large-scale unconstrained minimization problem,” SIAM J. Optim. 7(1), 26–33 (1997).
[Crossref]

J. W. Arblaster, “Crystallographic properties of rhodium,” Platinum Metals Rev. 41, 184–189 (1997).

1995 (1)

A. D. Rakic, J. M. Elazar, and A. B. Djurisic, “Acceptance-probability-controlled simulated annealing: A method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
[Crossref]

1994 (3)

C. B. Pedroso and M. B. Santos, “The Lorentz model applied to composite materials,” J. Phys.: Condens. Matter 6(12), 2395–2402 (1994).
[Crossref]

A. J. Cox, J. G. Lounderback, S. E. Apsel, and L. A. Bloomfield, “Magnetism in 4d-transition metal clusters,” Phys. Rev. B 49(17), 12295–12298 (1994).
[Crossref]

M. M. Sigalas and D. A. Papaconstantopoulos, “Calculations of the total energy, electron-phonon interaction, and Stoner parameter for metals,” Phys. Rev. B 50(11), 7255–7261 (1994).
[Crossref]

1993 (2)

A. J. Cox, J. G. Louderback, and L. A. Bloomfield, “Experimental observation of magnetism in rhodium clusters,” Phys. Rev. Lett. 71(6), 923–926 (1993).
[Crossref]

G. S. Kumar and G. Prasad, “Experimental determinations of the Lorenz number,” J. Mat. Sci. 28(16), 4261–4272 (1993).
[Crossref]

1992 (1)

S. Tanuma, C. J. Powell, and D. R. Penn, “Inelastic mean free paths of low-energy electrons in solids,” Acta Phys. Pol. A 81(2), 169–186 (1992).
[Crossref]

1990 (1)

M. I. Markovic and A. D. Rakic, “Determination of optical properties of aluminium including electron reradiation in the Lorentz-Drude model,” Opt. Laser Technol. 22(6), 394–398 (1990).
[Crossref]

1988 (2)

1986 (1)

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34(10), 7018–7026 (1986).
[Crossref]

1983 (2)

G. R. Stewart, “Measurement of low-temperature specific heat,” Rev. Sci. Instrum. 54(1), 1–11 (1983).
[Crossref]

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

1979 (1)

M. Yahaya and G. C. Fletcher, “The bandstructure of rhodium, its groundstate and the effects of change of volume,” J. Phys. F: Metal Phys. 9(7), 1295–1305 (1979).
[Crossref]

1977 (3)

J. F. Janak, “Uniform susceptibilities of metallic elements,” Phys. Rev. B 16(1), 255–262 (1977).
[Crossref]

K. Carrander, M. Dronjak, and S. P. Hörnfeldt, “De Hass-van Alphen effect in rhodium,” J. Phys. Chem. Solids 38(3), 289–296 (1977).
[Crossref]

J. H. Weaver, C. G. Olson, and D. W. Lynch, “Optical investigation of the electronic structure of bulk Rh and Ir,” Phys. Rev. B 15(8), 4115–4118 (1977).
[Crossref]

1975 (1)

S. H. Vosko and J. P. Perdew, “Theory of the spin susceptibility of an inhomogenous electron gas via the density functional formalism,” Can. J. Phys. 53(14), 1385–1397 (1975).
[Crossref]

1974 (2)

G. T. Furukawa, M. L. Reilly, and J. S. Gallagher, “Critical analysis of heat-capacity data and evaluation of thermodynamics properties of ruthenium, rhodium, iridium, and platinum from 0 to 300 K. A survey of the literature data on osmium,” J. Phys. Chem. Ref. Data 3(1), 163–209 (1974).
[Crossref]

S. R. Nagel and S. E. Schnatterly, “Frequency dependence of the Drude relaxation time in metal films,” Phys. Rev. B 9(4), 1299–1303 (1974).
[Crossref]

1973 (2)

N. E. Christensen, “The band structure of rhodium and its relation to photoemission experiments,” Phys. Stat. Sol. (b) 55(1), 117–127 (1973).
[Crossref]

D. T. Pierce and W. E. Spicer, “Optical properties of rhodium,” Phys. Stat. Sol. 60(2), 689–694 (1973).
[Crossref]

1971 (1)

1970 (2)

O. K. Andersen, “Electronic structure of the fcc transition metals Ir, Rh, Pt, and Pd,” Phys. Rev. B 2(4), 883–906 (1970).
[Crossref]

M. L. Theye, “Investigation of the optical properties of Au by means of thin semitransparent films,” Phys. Rev. B 2(8), 3060–3078 (1970).
[Crossref]

1968 (2)

M. J. Lynch and J. B. Swan, “The characteristic loss spectra of the second and third series of transition metals,” Aust. J. Phys. 21(6), 811–816 (1968).
[Crossref]

W. L. McMillan, “Transition temperature of strong-coupled superconductors,” Phys. Rev. 167(2), 331–344 (1968).
[Crossref]

1960 (1)

J. M. Luttinger and J. C. Ward, “Ground-state energy of a many-fermion system II,” Phys. Rev. 118(5), 1417–1427 (1960).
[Crossref]

1959 (1)

R. N. Gurzhi, “Mutual electron correlation in metal optics,” Soviet Physics JETP 35, 673–675 (1959).

1957 (2)

B. R. Coles and J. C. Taylor, “Sign reversal of the Hall effect in rhodium,” J. Phys. Chem. Solids 1(4), 270–274 (1957).
[Crossref]

G. K. White and S. B. Woods, “Thermal and electrical conductivity of rhodium, iridium, and platinum,” Can. J. Phys. 35(3), 248–257 (1957).
[Crossref]

1954 (1)

J. Lindhard, “On the properties of a gas of charge particles,” Kgl. Danske Videnskab. Selskab Mat. Fys. Medd. 28, 1–57 (1954).

1953 (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

1952 (2)

F. E. Hoare and J. C. Walling, “An absolute measurement of the susceptibility of tantalum and other metals,” Phys. Stat. Sol. (b) 64, 337–341 (1952).
[Crossref]

F. E. Hoare and J. C. Matthews, “The magnetic susceptibilities of platinum, rhodium and palladium from 20 to 290 K,” Proc. Royal Soc. A 212, 137–148 (1952).
[Crossref]

1939 (1)

W. W. Coblentz and R. Stair, “Note on the spectral reflectivity of rhodium,” Journal of Research of the National Bureau of Standards 22(1), 93–95 (1939).
[Crossref]

1936 (1)

N. F. Mott, “The electrical conductivity of transition metals,” Proc. Roy. Soc. A 153, 699–717 (1936).
[Crossref]

1911 (1)

W. W. Coblentz, “The reflecting power of various metals,” Bulletin of the Bureau of Standards 7(2), 197–225 (1911).
[Crossref]

1900 (1)

P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306(3), 566–613 (1900).
[Crossref]

Ahmadivand, A.

A. Ahmadivand, R. Sinha, S. Kaya, and N. Pala, “Rhodium plasmonic for deep-ultraviolet bio-chemical sensing,” Plasmonics 11(3), 839–849 (2016).
[Crossref]

Allen, P. B.

P. B. Allen, “The electron-phonon coupling constant,” in Handbook of Superconductivity, C. P. Poole editor (Academic Press, 1999) pp.478-483.

Andersen, O. K.

O. K. Andersen, “Electronic structure of the fcc transition metals Ir, Rh, Pt, and Pd,” Phys. Rev. B 2(4), 883–906 (1970).
[Crossref]

Apsel, S. E.

A. J. Cox, J. G. Lounderback, S. E. Apsel, and L. A. Bloomfield, “Magnetism in 4d-transition metal clusters,” Phys. Rev. B 49(17), 12295–12298 (1994).
[Crossref]

Arblaster, J. W.

J. W. Arblaster, “Crystallographic properties of rhodium,” Platinum Metals Rev. 41, 184–189 (1997).

Arendt, P.

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).

Avendano, E.

W. E. Vargas, F. Muñoz-Rojas, E. Avendano, V. Quirós, and M. Hernández-Jiménez, “Optical, charge transport, magnetic, and quantum mechanical properties of bulk iridium,” (in manuscript form to be submitted to OSA Continuum).

Azofeifa, D. E.

W. E. Vargas, N. Clark, F. Muñoz-Rojas, D. E. Azofeifa, and G. A. Niklasson, “Optical, charge transport and magnetic properties of palladium retrieved from photometric measurements: approaching the quantum mechanics background,” Phys. Scr. 94(5), 055101 (2019).
[Crossref]

W. E. Vargas, D. E. Azofeifa, N. Clark, and X. Márquez, “Collective response of silver islands on a dielectric substrate when normally illuminated with electromagnetic radiation,” J. Phys. D: Appl. Phys. 41(2), 025309 (2008).
[Crossref]

Barreda, Á. I.

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Batyev, E. G.

E. G. Batyev, “Pauli paramagnetism and Landau diamagnetism,” Phys.-Usp. 52(12), 1245–1246 (2009).
[Crossref]

Blanton, T. D.

J. J. Kas, T. D. Blanton, and J. J. Rehr, “Exchange-correlation contributions to thermodynamic properties of the homogenous electron gas from a cumulant Green’s function approach,” Phys. Rev. B 100(19), 195144 (2019).
[Crossref]

Bloomer, I.

A. R. Forouhi and I. Bloomer, “Optical dispersion equations for metals applicable to the Far-IR through EUV spectral range,” J. Phys. Commun. 5(2), 025002 (2021).
[Crossref]

A. R. Forouhi and I. Bloomer, “New dispersion equations for insulators and semiconductors valid throughout radio-waves to extreme ultraviolet spectral range,” J. Phys. Commun. 3(3), 035022 (2019).
[Crossref]

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34(10), 7018–7026 (1986).
[Crossref]

Bloomfield, L. A.

A. J. Cox, J. G. Lounderback, S. E. Apsel, and L. A. Bloomfield, “Magnetism in 4d-transition metal clusters,” Phys. Rev. B 49(17), 12295–12298 (1994).
[Crossref]

A. J. Cox, J. G. Louderback, and L. A. Bloomfield, “Experimental observation of magnetism in rhodium clusters,” Phys. Rev. Lett. 71(6), 923–926 (1993).
[Crossref]

Brener, N. E.

G. S. Tripathi, N. E. Brener, and J. Callaway, “Electronic structure of rhodium,” Phys. Rev. B 38(15), 10454–10462 (1988).
[Crossref]

Burden, A. M.

R. L. Burden, D. J. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015).

Burden, R. L.

R. L. Burden, D. J. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015).

Cahaya, A. B.

A. B. Cahaya, “Paramagnetic and diamagnetic susceptibility of infinite quantum well,” Journal of Materials Science, Geophysics, Instrumentation and Theoretical Physics 3, 61–67 (2020).
[Crossref]

Callaway, J.

G. S. Tripathi, N. E. Brener, and J. Callaway, “Electronic structure of rhodium,” Phys. Rev. B 38(15), 10454–10462 (1988).
[Crossref]

Carrander, K.

K. Carrander, M. Dronjak, and S. P. Hörnfeldt, “De Hass-van Alphen effect in rhodium,” J. Phys. Chem. Solids 38(3), 289–296 (1977).
[Crossref]

Cash, W. C.

Chen, L. Y.

Chen, Y. L.

Christensen, N. E.

N. E. Christensen, “The band structure of rhodium and its relation to photoemission experiments,” Phys. Stat. Sol. (b) 55(1), 117–127 (1973).
[Crossref]

Clark, N.

W. E. Vargas, N. Clark, F. Muñoz-Rojas, D. E. Azofeifa, and G. A. Niklasson, “Optical, charge transport and magnetic properties of palladium retrieved from photometric measurements: approaching the quantum mechanics background,” Phys. Scr. 94(5), 055101 (2019).
[Crossref]

W. E. Vargas, D. E. Azofeifa, N. Clark, and X. Márquez, “Collective response of silver islands on a dielectric substrate when normally illuminated with electromagnetic radiation,” J. Phys. D: Appl. Phys. 41(2), 025309 (2008).
[Crossref]

Coblentz, W. W.

W. W. Coblentz and R. Stair, “Note on the spectral reflectivity of rhodium,” Journal of Research of the National Bureau of Standards 22(1), 93–95 (1939).
[Crossref]

W. W. Coblentz, “The reflecting power of various metals,” Bulletin of the Bureau of Standards 7(2), 197–225 (1911).
[Crossref]

Coles, B. R.

B. R. Coles and J. C. Taylor, “Sign reversal of the Hall effect in rhodium,” J. Phys. Chem. Solids 1(4), 270–274 (1957).
[Crossref]

Cox, A. J.

A. J. Cox, J. G. Lounderback, S. E. Apsel, and L. A. Bloomfield, “Magnetism in 4d-transition metal clusters,” Phys. Rev. B 49(17), 12295–12298 (1994).
[Crossref]

A. J. Cox, J. G. Louderback, and L. A. Bloomfield, “Experimental observation of magnetism in rhodium clusters,” Phys. Rev. Lett. 71(6), 923–926 (1993).
[Crossref]

Cox, J. T.

Czerczak, S.

S. Czerczak, J. P. Gromiec, A. Palaszewska-Tkaez, and A. Swidwinska-Gajewska, “Nickel, ruthenium, rhodium, palladium, osmium, and platinum,” Patty’s Toxicology 1, 653–768 (2001).
[Crossref]

Da, B.

Y. Sun, H. Xu, B. Da, S. F. Mao, and Z. J. Ding, “Calculation of energy-loss function for 26 materials,” Chin. J. Chem. Phys. 29(6), 663–670 (2016).
[Crossref]

de la Osa, R. Alcaraz

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 8010064 (2018).
[Crossref]

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Ding, Z. J.

Y. Sun, H. Xu, B. Da, S. F. Mao, and Z. J. Ding, “Calculation of energy-loss function for 26 materials,” Chin. J. Chem. Phys. 29(6), 663–670 (2016).
[Crossref]

Djurisic, A. B.

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).
[Crossref]

A. B. Djurisic, A. D. Rakic, and J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[Crossref]

A. D. Rakic, J. M. Elazar, and A. B. Djurisic, “Acceptance-probability-controlled simulated annealing: A method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
[Crossref]

Dorantes-Dávila, J.

P. Villaseñor-González, J. Dorantes-Dávila, H. Dreyssé, and G. M. Pastor, “Size and structural dependence of the magnetic properties of rhodium clusters,” Phys. Rev. B 55(22), 15084–15091 (1997).
[Crossref]

Dorofeev, I. A.

I. A. Dorofeev and E. A. Vinogradov, “Structure of electromagnetic fields near the surface of an ionic crystal,” . Synch. Investig 6(5), 796–804 (2012).
[Crossref]

Dressel, M.

M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press, 2003).

Dreyssé, H.

P. Villaseñor-González, J. Dorantes-Dávila, H. Dreyssé, and G. M. Pastor, “Size and structural dependence of the magnetic properties of rhodium clusters,” Phys. Rev. B 55(22), 15084–15091 (1997).
[Crossref]

Dronjak, M.

K. Carrander, M. Dronjak, and S. P. Hörnfeldt, “De Hass-van Alphen effect in rhodium,” J. Phys. Chem. Solids 38(3), 289–296 (1977).
[Crossref]

Drude, P.

P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306(3), 566–613 (1900).
[Crossref]

Elazar, J. M.

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).
[Crossref]

A. B. Djurisic, A. D. Rakic, and J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[Crossref]

A. D. Rakic, J. M. Elazar, and A. B. Djurisic, “Acceptance-probability-controlled simulated annealing: A method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
[Crossref]

Everitt, H. O.

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Faires, D. J.

R. L. Burden, D. J. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015).

Finkelstein, G.

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Fisher, R. F.

Fletcher, G. C.

M. Yahaya and G. C. Fletcher, “The bandstructure of rhodium, its groundstate and the effects of change of volume,” J. Phys. F: Metal Phys. 9(7), 1295–1305 (1979).
[Crossref]

Forouhi, A. R.

A. R. Forouhi and I. Bloomer, “Optical dispersion equations for metals applicable to the Far-IR through EUV spectral range,” J. Phys. Commun. 5(2), 025002 (2021).
[Crossref]

A. R. Forouhi and I. Bloomer, “New dispersion equations for insulators and semiconductors valid throughout radio-waves to extreme ultraviolet spectral range,” J. Phys. Commun. 3(3), 035022 (2019).
[Crossref]

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34(10), 7018–7026 (1986).
[Crossref]

Furukawa, G. T.

G. T. Furukawa, M. L. Reilly, and J. S. Gallagher, “Critical analysis of heat-capacity data and evaluation of thermodynamics properties of ruthenium, rhodium, iridium, and platinum from 0 to 300 K. A survey of the literature data on osmium,” J. Phys. Chem. Ref. Data 3(1), 163–209 (1974).
[Crossref]

Gallagher, J. S.

G. T. Furukawa, M. L. Reilly, and J. S. Gallagher, “Critical analysis of heat-capacity data and evaluation of thermodynamics properties of ruthenium, rhodium, iridium, and platinum from 0 to 300 K. A survey of the literature data on osmium,” J. Phys. Chem. Ref. Data 3(1), 163–209 (1974).
[Crossref]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

González, F.

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 8010064 (2018).
[Crossref]

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Gromiec, J. P.

S. Czerczak, J. P. Gromiec, A. Palaszewska-Tkaez, and A. Swidwinska-Gajewska, “Nickel, ruthenium, rhodium, palladium, osmium, and platinum,” Patty’s Toxicology 1, 653–768 (2001).
[Crossref]

Grüner, G.

M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press, 2003).

Gruzza, B.

M. M. Thiam, V. Nehasil, V. Matolín, and B. Gruzza, “The AES and EELS study of small rhodium clusters deposited onto alumina substrates,” Surf. Sci. 487(1-3), 231–242 (2001).
[Crossref]

Gurzhi, R. N.

R. N. Gurzhi, “Mutual electron correlation in metal optics,” Soviet Physics JETP 35, 673–675 (1959).

Gutiérrez, Y.

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 8010064 (2018).
[Crossref]

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Hass, G.

Hernández-Jiménez, M.

W. E. Vargas, F. Muñoz-Rojas, E. Avendano, V. Quirós, and M. Hernández-Jiménez, “Optical, charge transport, magnetic, and quantum mechanical properties of bulk iridium,” (in manuscript form to be submitted to OSA Continuum).

Hoare, F. E.

F. E. Hoare and J. C. Walling, “An absolute measurement of the susceptibility of tantalum and other metals,” Phys. Stat. Sol. (b) 64, 337–341 (1952).
[Crossref]

F. E. Hoare and J. C. Matthews, “The magnetic susceptibilities of platinum, rhodium and palladium from 20 to 290 K,” Proc. Royal Soc. A 212, 137–148 (1952).
[Crossref]

Hörnfeldt, S. P.

K. Carrander, M. Dronjak, and S. P. Hörnfeldt, “De Hass-van Alphen effect in rhodium,” J. Phys. Chem. Solids 38(3), 289–296 (1977).
[Crossref]

Hunter, W. R.

Hurd, C. M.

C. M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, 1972).

Janak, J. F.

J. F. Janak, “Uniform susceptibilities of metallic elements,” Phys. Rev. B 16(1), 255–262 (1977).
[Crossref]

Kas, J. J.

J. J. Kas, T. D. Blanton, and J. J. Rehr, “Exchange-correlation contributions to thermodynamic properties of the homogenous electron gas from a cumulant Green’s function approach,” Phys. Rev. B 100(19), 195144 (2019).
[Crossref]

Kaya, S.

A. Ahmadivand, R. Sinha, S. Kaya, and N. Pala, “Rhodium plasmonic for deep-ultraviolet bio-chemical sensing,” Plasmonics 11(3), 839–849 (2016).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Kumar, G. S.

G. S. Kumar and G. Prasad, “Experimental determinations of the Lorenz number,” J. Mat. Sci. 28(16), 4261–4272 (1993).
[Crossref]

Larruquert, J. I.

Li, H. Y.

Li, J.

Li, P.

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Lindhard, J.

J. Lindhard, “On the properties of a gas of charge particles,” Kgl. Danske Videnskab. Selskab Mat. Fys. Medd. 28, 1–57 (1954).

Liu, H.

Liu, J.

X. Zhang, Y. Gutiérrez, P. Li, Á. I. Barreda, A. M. Watson, R. Alcaraz de la Osa, G. Finkelstein, F. González, D. Ortiz, J. M. Saiz, J. M. Sanz, H. O. Everitt, J. Liu, and F. Moreno, “Plasmonics in the UV range with rhodium nanocubes,” Proc. SPIE 9884, 98841E–1 (2016).
[Crossref]

Lorentz, H. A.

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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.

24. J. H. Weaver, “Optical properties of metals,” in Handbook of Chemistry and Physics, 73rd ed., (CRC Press, 1993), Chapter 12, pp. 111–126.

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Figures (8)

Fig. 1.
Fig. 1. Spectral variation of the optical constants of Rh [refractive index n(λ) and extinction coefficient k(λ)] taken from [24].
Fig. 2.
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 FJ=3.872 × 10−3 after J=389 cycles of temperature.
Fig. 3.
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) Ωpe=5.878 eV, (b) γoe=0.245 eV, (c) β=me/mh=1.071, (d) χ=zh/ze=1.198, (e) η=γohoe=0.092, and (f) ze=0.662.
Fig. 4.
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 ρo=3.53 μΩ cm, (b) Hall coefficient RH=9.73 × 10−11 m3/As, (c) average electron effective mass me=1.92 m, (d) average hole effective mass mh=1.79 m, (e) total heat capacity coefficient γhcc=1.85 mJ/K2mol, and (f) logarithmic value reached by the merit function log(FJ)=-2.41.
Fig. 5.
Fig. 5. Schematic representation of the global minimum of the merit function. The first FJ-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, σF, are obtained, as well as those corresponding to the optimized and derived parameters.
Fig. 6.
Fig. 6. Behavior of the merit function FJ in terms of the number of temperature cycles of the APCSA application followed by using a SPGM method to improve the APCSA result.
Fig. 7.
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 σL and σD respectively, with the total conductivity given by σ=σLD.
Fig. 8.
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) Ao=Ao(B) and Bo=Bo(B), and in figure (b) Ao=Ao(S) and Bo=Bo(S) according to Eqs. (5) and (6), respectively.

Tables (5)

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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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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 fo and the oscillator strength fj-values. Values of Γe and Γh 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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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.

Equations (15)

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ε = ε h f 1 Ω ( Ω p e 2 Ω + i γ o e + Ω p h 2 Ω + i γ o h ) + 1 z e j = 1 K f j Ω p e 2 Ω j 2 Ω 2 i Ω γ j
F J = 1 2 κ N p 1 i = 1 κ [ ( ε 1 ( ω i ) ε ¯ 1 ( ω i ) ε ¯ 1 ( ω i ) ) 2 + ( ε 2 ( ω i ) ε ¯ 2 ( ω i ) ε ¯ 2 ( ω i ) ) 2 ]
ε = 1 f o Ω p e 2 Ω ( 1 Ω + i γ o e [ 1 + ( Γ e Ω ) 2 ] + β χ Ω + i η γ o e [ 1 + ( Γ h Ω ) 2 ] ) + 1 z e j = 1 K f j Ω p e 2 Ω j 2 Ω 2 i Ω γ j
Z P , k = 1 2 + 1 x k 2 4 x k ln | x k + 1 x k 1 | ,
Z L , k = 3 8 x k 2 [ 1 + x k 2 ( 1 x k 2 ) 2 2 x k ln | x k + 1 x k 1 | ] = 3 4 x k 2 [ 1 ( 1 x k 2 ) Z P , k ] ,
L ( B ) ( Ω ) = Im [ 1 ε ( Ω ) ] = A 0 ( B ) ( Ω ) Im [ 1 ε D ( Ω ) ] B 0 ( B ) ( Ω ) Im [ 1 ε L ( Ω ) ] ,
Re [ 1 ε ( Ω ) ] = A 0 ( B ) ( Ω ) Re [ 1 ε D ( Ω ) ] + B 0 ( B ) ( Ω ) Re [ 1 ε L ( Ω ) ] .
L ( S ) ( Ω ) = Im [ 1 ε ( Ω ) + 1 ] = A 0 ( S ) ( Ω ) Im [ 1 ε D ( Ω ) + 1 ] B 0 ( S ) ( Ω ) Im [ 1 ε L ( Ω ) + 1 ]
Z e f f = 2 π   Ω p e 2 0 Ω max Ω Im [ 1 / ε ( Ω ) ] d Ω = 2 π   Ω p e 2 0 Ω max Ω L ( B ) ( Ω ) d Ω
Z e f f ( c c ) = 2 π   Ω p e 2 0 Ω max Ω L D ( B ) ( Ω ) d Ω ,
Z e f f ( i t ) = 2 π   Ω p e 2 0 Ω max Ω L L ( B ) ( Ω ) d Ω .
1 ε L ( Ω ) = A j ( B ) ( Ω ) ε L , j ( Ω ) + B j ( B ) ( Ω ) i = j + 1 K ε L , i ( Ω )
L L , j ( B ) ( Ω ) = [ i = 0 j 1 B i ( B ) ( Ω ) ] A j ( B ) ( Ω ) Im [ 1 / ε L , j ( Ω ) ]
L L , K ( B ) ( Ω ) = [ i = 0 K 2 B i ( B ) ( Ω ) ] B K 1 ( B ) ( Ω ) Im [ 1 / ε L , K ( Ω ) ]
Z e f f ( j ) = 2 π   Ω p e 2 0 Ω max Ω L L , j ( B ) ( Ω ) d Ω
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