Free Electron Gas Model Research Articles

Development of a new analytical solution for electronic heat capacity for higher electron temperatures

A new analytical solution has been found for the electronic heat capacity of typical s-band metals. It is valid for high electron temperatures of the order of tens of kK, commonly reached in fs pulsed laser ablation. We have used the density of states (DOS) from the free-electron gas model (FEG) and a special type of function called polylogarithms. From the exact analytical solution, a third order Taylor expansion which works well up to 50kK could be derived. A new term was added to the DOS for d-band metals, which allowed us to find a new semi-analytical solution which predicts accurately the electronic heat capacity of Cu.

Carbon nanotube electron blackbody and its radiation spectra

An optical blackbody is an ideal absorber for all incident optical radiation, and the theoretical study of its radiation spectra paved the way for quantum mechanics (Planck's law). Herein, we propose the concept of an electron blackbody, which is a perfect electron absorber as well as an electron emitter with standard energy spectra at different temperatures. Vertically aligned carbon nanotube arrays are electron blackbodies with an electron absorption coefficient of 0.95 for incident energy ranging from 1 keV to 20 keV and standard electron emission spectra that fit well with the free electron gas model. Such a concept might also be generalized to blackbodies for extreme ultraviolet, X-ray, and γ-ray photons as well as neutrons, protons, and other elementary particles.

Anomalous electronic energy losses in protons passing through Gd thin films

We report a theoretical analysis of recent data on the unexpectedly high electronic energy loss protons in a polycrystalline gadolinium target. These results led the authors of this data to conclude that the free electron model used to analyse these results fails. In this work we provide a quantitative explanation of the experimental results, using an approach based on density functional theory within the framework of the free electron gas model. We performed semi-classical trajectory simulations (SCTS) and employ the local-density-approximation model (LDA), using an inhomogeneous electron density distribution and the polycrystalline character of Gd samples. The good agreement obtained, delineates the need of considering a realistic description of a particle trajectory and the effective electron density ‘seen’ along it, whose description remains within a FEG model.

Magic Numbers in Octahedral Ligated Metal-Chalcogenide Superatoms.

The attainment of the superatomic state offers a unifying framework for the periodic classification of atomic clusters. Metallic clusters attain the superatomic state via the confined nearly free electron gas model that leads to groupings of quantum states marked by radial and angular momentum quantum numbers. We examine ligated octahedral metal-chalcogenide clusters where the nearly free electron gas model is invalid; however, the high symmetry can also lead to the bunching of electronic states. For octahedral TM6E8L6 clusters (TM = transition metal; E = chalcogen; L = ligand), the electronic shells are filled for valence electron counts of 96, 100, and 114 electrons. These magic electron counts are marked by large highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gaps, high ionization energies, and low electron affinity─all classic signatures of the superatomic state. We also find that clusters with electron counts differing from the magic counts show periodic patterns reminiscent of those observed in the periodic table of elements.

Evaluation of exchange-correlation effects on the heat-shielding performance of carrier electrons in LaB6 using momentum-transfer resolved electron energy-loss spectroscopy

Exchange-correlation (XC) effects in carrier electrons have a significant influence on the dielectric properties and electric characteristics of a material. In this study, momentum-transfer (q) resolved electron energy-loss spectroscopy was conducted to experimentally evaluate the XC effects of carrier electrons in LaB6 bulk crystals, whose nanoparticles have been used for near-infrared-light shielding filters. By measuring q dependence of plasmon energy due to carrier electrons and evaluating the deviation from the free electron gas model in random phase approximation, the dielectric correction factor due to the XC effects, i.e., the local field correction G(q), for the carrier electrons in LaB6 were experimentally derived. This experimental result confirmed that the XC effects are non-negligible for the carrier electrons in LaB6. Because the XC effects influence dipole surface plasmon energy of LaB6 nanoparticles, the evaluation of the XC effects is important for a precise understanding of the optical properties of LaB6 nanoparticles.

The free electron model and the electronic energy losses of protons at low velocities interacting with polycrystalline tantalum

In this letter, we report experimental data and theoretical work on the electronic energy loss and energy loss straggling of protons transmitted through self-supported thin films of tantalum in a polycrystalline tetragonal phase . Low-energy protons with energies below 10 keV and Ta films with nominal thickness of 6, 9 and 12 nm are used. The aim of this work is to understand the unexpected values of the Ta stopping power for low-energy proton backscattering reported recently, which are far from the prediction of the standard free electron gas model and semi-empirical approaches. This had led the authors to conclude the failure of the free electron model. Our transmission measurements confirm these experimental results. In this work, a qualitative discussion and quantitative explanation of our experimental results is given, using an approach based on the density functional theory within the framework of the free electron gas (FEG) model. We performed semiclassical deterministic trajectory simulations and employ the local density approximation model, using an inhomogeneous electron density distribution and the polycrystalline character of Ta samples.

Thermodynamic restrictions on the heat capacity of a fermion gas

Using the thermodynamic restrictions imposed by Carnot theorem over an Otto cycle that operates at maximum work regime and applying them to obtain the heat capacities of electron gases stemming from different metals which provide enough electrons to behave as fermion gases with CV=αT−βT3, the coefficients α and β are calculated from a free electron gas model. These values agree with the experimental data for metals with high electronegativities; that is, only these metals with high electronegativities provide a fermion gas that is in accordance with the thermodynamic restrictions presented in this work.

Electron–phonon coupling factor and electron heat capacity of 6H-SiC

ABSTRACT For high-energy single ion events, the relationship between electronic energy loss mechanism and atomic processes is generally described by the inelastic thermal spike model. However, the parameters required for the model are not accurately known and are often estimated from a free electron gas model. To ensure this model is more reliable and predictive, a more accurate calculation for the parameters should be taken. In this paper, the temperature dependence of the electronic heat capacity and electron–phonon coupling factor (e-ph) has been calculated, using density functional theory (DFT). The calculation results demonstrate that the effect of electron thermal excitation on the thermodynamic parameters is sensitive to the bandgap, and these parameters vary immensely with the electron temperature. Besides, to verify the accuracy of obtained parameters, both the swift heavy ions (SHI) irradiation in crystalline 6H-SiC and amorphous 6H-SiC are simulated, using the inelastic thermal spike model. And the results are more consistent with the experiment conclusion, for the obtained parameters by DFT. Hence, the calculated temperature-dependent parameters are more suitable for the inelastic thermal spike model.

Sc3Ir4Si13+x and Sc4Ir7Ge6 – the perovskite-related crystal structures

Abstract The crystal structure of Sc3Ir4Si13+x (x = 0.22) [space group P m 3 ‾ n $Pm\bar{3}n$ , a = 8.4651(1) Å] is found to be a new disordered variant of the primitive cubic Yb3Rh4Sn13 Remeika prototype. The silicide is stable in the narrow temperature range of 1283–1397 °C and reveals metallic properties. The crystal structure of Sc4Ir7Ge6 [U4Re7Si6 type, space group I m 3 ‾ m $Im\bar{3}m$ , a = 8.1397(8) Å] is refined for the first time. The electronic band structure calculations reveal that the properties of this germanide can be explained based on the free electron gas model. Both compounds reveal close structural relationships to the simple perovskite structure.

Vacancy-induced enhancement of electron–phonon coupling in cubic silicon carbide and its relationship to the two-temperature model

The electron–phonon coupling factor was calculated for both pristine and vacancy-rich 3C-SiC. Ab initio calculations were performed within the framework of the density functional perturbation theory. Wannier functions were used to interpolate eigenvalues into denser grids through the electron–phonon using Wannier code. The coupling factor was determined through calculations of the electron self-energy, electron–phonon relaxation time, and electronic specific heat. These parameters were extrapolated to high temperatures using a hybrid model which mixes band calculations for electrons below an energy cutoff with the free electron gas model for electrons above the energy cutoff. The electron relaxation times, specific heats, electron drift mobilities, and electron–phonon coupling factors were calculated as a function of electron temperature. Si and C vacancies were found to have a profound effect on electron–phonon coupling for all temperatures, while electronic specific heat capacity was found to be most affected at cryogenic temperatures. The electron drift mobility was calculated at different temperatures using the scattering time. Calculated mobilities were validated with Hall mobility measurements reported in the literature. The importance of structural defects on the electron–phonon coupling is discussed in the context of the two-temperature model, a model that has been widely used to understand aspects of the interaction of solids with pulsed laser irradiation and swift heavy ion irradiation.

Electrochemical Impedance Imaging on Conductive Surfaces

Electrochemical impedance spectroscopy (EIS) is a powerful tool to measure and quantify the system impedance. However, EIS only provides an average result from the entire electrode surface. Here, we demonstrated a reflection impedance microscope (RIM) that allows us to image and quantify the localized impedance on conductive surfaces. The RIM is based on the sensitive dependence between the materials' optical properties, such as permittivity, and their local surface charge densities. The localized charge density variations introduced by the impedance measurements will lead to optical reflectivity changes on electrode surfaces. Our experiments demonstrated that reflectivity modulations are linearly proportional to the surface charge density on the electrode and the measurements show good agreement with the simple free electron gas model. The localized impedance distribution was successfully extracted from the reflectivity measurements together with the Randles equivalent circuit model. In addition, RIM is used to quantify the impedance on different conductive surfaces, such as indium tin oxide, gold film, and stainless steel electrodes. A polydimethylsiloxane-patterned electrode surface was used to demonstrate the impedance imaging capability of RIM. In the end, a single-cell impedance imaging was obtained by RIM.

Ultraslow electron-phonon scattering and polaron formation in magnetite

The comprehensive analysis of AC electrical conductivity in magnetite was performed in order to find relations between the formation of polarons, phonons and conduction by a virtual free electron gas. The analysis performed here for the first time shows experimental data for the behavior of electrons for magnetite with the scattering time shifted to the GHz region. According to our study, the DC electrical conductivity can be described by the virtual free electron gas model, and high frequency conductivity can be described by the combination of the Drude model for disordered materials and Jonscher's universal power law. The observed peak at the imaginary part of AC conductivity was related to the scattering time of the electron-phonon coupling. This interaction between electrons and thermally formed phonons results in the formation of large polarons, and these are responsible for high frequency conductivity in magnetite.

Plasmonic Edge, Centre and Breathing Modes in Two-Dimensional Disks

Recently, plasmonic edge and breathing modes are obtained by electron energy loss spectroscopy in aluminum nanotriangles and silver nanodisks. In this article, based on the free-electron gas model and the linear response theory, we have derived the energy absorption spectrum and charge distributions under applied external fields to study the plasmons sustained by two-dimensional disks. Plasmonic edge, center and breathing modes are further observed by the charge distributions. In addition, with the help of energy absorption spectrum, the evolutionary characteristics of plasmon modes with the system parameters such as size, atomic separation and electron filling are further confirmed, and the effect of different applied external fields on plasmon excitations is displayed.

Anomalous Thermodiffusion of Electrons in Graphene

We reveal a dramatic departure of electron thermodiffusion in solids relative to the commonly accepted picture of the ideal free-electron gas model. In particular, we show that the interaction with the lattice and impurities, combined with a strong material dependence of the electron dispersion relation, leads to counterintuitive diffusion behavior, which we identify by comparing a two-dimensional electron gas (2DEG) and single-layer graphene. When subject to a temperature gradient ∇T, thermodiffusion of massless Dirac fermions in graphene exhibits an anomalous behavior with electrons moving along ∇T and accumulating in hot regions, in contrast to normal electron diffusion in a 2DEG with parabolic dispersion, where net motion against ∇T is observed, accompanied by electron depletion in hot regions. These findings bear fundamental importance for the understanding of the spatial electron dynamics in emerging materials, establishing close relations with other branches of physics dealing with electron systems under nonuniform temperature conditions.

Stopping and straggling of 60–250-keV backscattered protons on nanometric Pt films

The stopping power and straggling of backscattered protons on nanometric Pt films were measured at low to medium energies (60--250 keV) by using the medium-energy ion scattering technique. The stopping power results are in good agreement with the most recent measurements by Primetzhofer Phys. Rev. B 86, 094102 (2012) and are well described by the free electron gas model at low projectile energies. Nevertheless, the straggling results are strongly underestimated by well-established formulas up to a factor of two. Alternatively, we propose a model for the energy-loss straggling that takes into account the inhomogeneous electron-gas response, based on the electron-loss function of the material, along with bunching effects. This approach yields remarkable agreement with the experimental data, indicating that the observed enhancement in energy-loss straggling is due to bunching effects in an inhomogeneous electron system. Nonlinear effects are of minor importance for the energy-loss straggling.

Anomalous neutralization characteristics in Na+ neutralization on Al(111) surfaces

The jellium model of free electron gas and its extended version have been widely used to understand the neutralization of alkali-metal ions on metal surfaces. We report an unexpected deviation from its prediction that we observed in the neutralization of $\mathrm{N}{\mathrm{a}}^{+}$ ions scattering from an Al(111) surface. We find that the neutralization probability decreases monotonically with increasing ion velocity for the specular scattering condition, which is consistent with the well-known parallel velocity effect. However, the neutralization probability exhibits an unexpected bell shape with the variation of outgoing angle for a given incident energy. Calculations based on the jellium model using the rate equation and including the dynamic parallel velocity effect are presented. Their results agree with the velocity dependence of the neutral fraction, but completely fail in reproducing the angle dependence. This anomalous angle dependence could be due to the appearance of inelastic processes, corresponding to inner $2p$ electron promotion in hard encounters with Al atoms for large incidence angles, when the interatomic distances become small. This can lead to the formation of autoionizing Na states that result in the formation of extra $\mathrm{N}{\mathrm{a}}^{+}$ ions, not accounted for in the jellium model.

On the determination of stopping cross-sections in ion scattering in solids and deviations from standard models

When atomic particles traverse solids they suffer energy losses due to elastic scattering from nuclei and excitation of electron–hole pairs. These are referred to as nuclear and electronic stoppings respectively. In this paper we discuss methods of determining energy losses and ‘stopping’ cross-sections in ion transmission through thin films, as well as in large angle and grazing backscattering from surfaces. This is done on the basis of deterministic simulations of ion scattering by following ion trajectories as they pass through the solid and sample regions of different electron densities depending on the distance from atomic nuclei. The ab initio calculated electron densities in the crystal are used to determine the stopping power, as predicted by the free electron gas model, and including a threshold value for d electron excitation. We discuss some aspects that are not included in standard descriptions based on the use of free electron models and averaged effective electron densities. In this context, we point out the possibility of inelastic processes involving inner-shell excitations, and briefly summarise main findings.

Effect of resonant coherent excitation on the electronic stopping of slow channeled ions

We calculated the nonlinear screening and the electronic stopping of energetic helium ions in a III-V compound semiconductor at low speeds below stopping maximum under channeling conditions. For the range of velocity considered, it is found the energy loss of the intruding ions show pronounced dependence on the electronic screening from the induced charge. Different with the prevalent scenario described by free-electron gas model that the induced charge distribution lags behind the projectile forming the so-called wake potential. We found through real-time time-dependent density functional theory that the induced charge by the channeling ion keeps oscillating back and front when channeling through the zinc-blende crystalline GaN, which is interpreted as a consequence of temporally oscillating Coulomb field arising from the periodical atomic arrangement along ion trajectory. When one of the frequencies coincides with the transition energy of the ionic charge, it would be resonantly excited by the oscillating field, which further gives rise to a depletion in electronic screening by bound electrons to the ion, resulting in odd enhanced stopping trend at narrow velocity windows.

Morphological evolution of Ag nanoclusters grown on hydrogen-terminated si(111)-(1 × 1) surface: Appearance of quantum size effect at room-temperature

We have observed the early stages of Ag nanocluster evolution on a hydrogen-terminated Si(111)-(1 × 1) surface at room temperature (RT) by using scanning tunneling microscopy and low energy electron diffraction, and examined changes in number density, height, base diameter, volume and shapes of the nanocluster with a deposition amount from 0.2 to 8 ML. It is found that Ag nanoclusters evolve by repeating the stages of nucleation/growth and coalescence, and change the growth morphology and the growth mode after the coalescence. Transition from initially formed dome-like three dimensional clusters to flat-top two dimensional (2D) islands evidently shows that quantum size effect emerges during the growth process of Ag nanoclusters even at RT. The characteristic stability of 8 atomic layer height in the 2D islands is well interpreted based on a free-electron gas model for free-standing nanofilms.

Symmetries and local conservation laws of variational schemes for the surface plasmon polaritons

The relation between symmetries and local conservation laws, known as Noether's theorem, plays an important role in modern theoretical physics. As a discrete analog of the differentiable physical system, a good numerical scheme should admit the discrete local conservation laws and inherent mathematical structures. A class of variational schemes constructed for the hydrodynamic-electrodynamic model of lossless free-electron gas in a quasi-neutral background shows good properties in secular simulations of surface plasmon polaritons [Q. Chen et al., Phys. Rev. E 99, 023313 (2019)]. We show the discrete local conservation laws admitted by these schemes. Based on the gauge symmetry of the discrete action functional, a discrete charge conservation law is realized locally, which is consistent with the discrete Euler-Lagrange equations obtained from the variational schemes. Based on the discrete Euler-Lagrange equations, discrete local momentum and energy conservation laws are derived directly, which are rigorous in theory. The preservation of the discrete local conservation laws and Lagrangian symplectic structure ensure that the numerical scheme is correct in physics.