<figure><img src="image_1.png" /></figure>

HAL
open science

# A magnetic tight-binding model: surface properties of transition metals and cobalt nanoparticules

Jacques Réné Eone

► To cite this version:

Jacques Réné Eone. A magnetic tight-binding model: surface properties of transition metals and cobalt nanoparticules. 2022. hal-01949869v3

HAL Id: hal-01949869

https://hal.science/hal-01949869v3

Preprint submitted on 22 Jan 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
# A magnetic tight-binding model: surface properties of transition metals and cobalt nanoparticles

Jacques René Eone

► To cite this version:

Jacques René Eone. A magnetic tight-binding model: surface properties of transition metals and cobalt nanoparticles. 2025. hal-01949869v4

HAL Id: hal-01949869

https://hal.science/hal-01949869v4

Preprint submitted on 2 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A magnetic tight-binding model : surface properties of transition metals and cobalt nanoparticles

Jacques R. Eone II

Department of Physics, University of Strasbourg, Strasbourg, France

(Dated: January 20, 2019)

The magnetic and surface properties of some transition metals have been investigated through the tight-binding approximation including Coulomb correlations. These surface properties are derived from a charge neutrality rule restricted to the d-band leading to a charge distribution including sp surface states in agreement with a Linear Muffin-Tin Orbital (LMTO) calculation. This new approach describes the local magnetism, surface energies and work functions without recourse to the total energy. Our investigation focuses on fcc cobalt, bcc iron, fcc nickel and fcc platinum surfaces with an exploration of fcc cobalt nanoparticles.

## I. INTRODUCTION

The electronic structure of a transition metal is described as a set of more or less delocalized electrons dominated by a d-band containing more localized electrons than the sp-band made up of nearly-free electrons. These two bands are correlated and the action of the electrons occupying the sp-band broadens the d-bandwidth and makes the d-electrons less localized [1]. These electrons give metallic and cohesive properties that can be easily obtained numerically from the Kohn-Sham equation [2] through a density functional theory (DFT) calculation. Although efficient, a DFT calculation is limited by the number of atoms in the studied system. Thus, semi-empirical methods such as the tight-binding approximation, if they rigorously integrate the rules governing the behavior of electrons, are more adequate to study these systems. The tight-binding approximation is often limited to the d-band neglecting the s- and p-states, which unfortunately leads to incorrect energies [1]. The impact of s- and p-electrons on the d-band is therefore a very important rule to describe a transition metal. The rules governing the electrons of a transition metal are even more important at the surface. In fact, at the surface of a transition metal, one can define an obvious rule of charge neutrality. Some calculations show that the total charge is conserved at the surface of transition metals and transition metal alloys per atomic site, per orbital and per chemical species [3-5]. In this work, this charge neutrality rule is restricted to the d-electrons which are more localized than the s- and p-electrons and have a more important role on bond formation and cohesion. By applying this charge neutrality rule on the d-band, the delocalized sp-band containing the s- and p-electrons gives at the Fermi level free sp surface states for a layer beyond the surface (S + 1) which represents the vacuum. The charge neutrality rule leads to a self-consistency treatment to find surface properties like surface energies and work functions using empirical laws but also to deduce the surface magnetism. The surface magnetism is derived from the Stoner model and is obtained by shifting the non-magnetic local density of states (LDOS). This is done by considering that the Coulomb correlations of

the d-band are conserved at the surface. In this work we extend this method to find the magnetic properties of fcc cobalt nanoparticles.

## II. METHODOLOGY

In the tight-binding approximation, the atomic potential $H^{at}$ is perturbed by a weak perturbation $\Delta U(r)$ due to the interaction with neighboring atoms. The atomic energy of the d-band in the atomic basis $\psi_{\lambda}$ can be described as shifted by an integral $\alpha$.

$$ \epsilon_d = \underbrace{\int \psi_{\lambda}^{*}(r) H^{at} \psi_{\lambda}(r) d^3 r}_{\text{local}} + \underbrace{\int \psi_{\lambda}^{*}(r) \Delta U(r) \psi_{\lambda}(r) d^3 r}_{\alpha: \text{perturbation}} \quad (1) $$

The strength of this perturbation integral depends directly on the overlap between two $\lambda$ orbitals from one atom to his neighbors. At the surface, the coordination number is lower and the atomic potential is perturbed differently than in the bulk by the presence of the neighbors. The impact of this new potential at the surface can be described by a simple shift of the atomic energies by a quantity $\alpha$ respecting a rule of a charge neutrality. This process is a self-consistency procedure correcting the electronic structure at the surface before studying the magnetic proprieties. The local magnetism is derived from the local Hubbard Hamiltonian.

$$ H = -t \sum_{i,j,\sigma} (c_{i\sigma}^{\dagger} c_{j\sigma} + \text{h.c.}) + U_d \sum_{\lambda} n_{\lambda\uparrow} n_{\lambda\downarrow} \quad (2) $$

Where we consider $n_0 = 5$ d orbitals noted $\lambda$, $t$ the hopping integral and $U_d$ the effective Coulomb repulsion in one orbital $\lambda$. If the studied d-band is derived from a basis where the effects of the s and p-states are taken into account, $U_d$ therefore contains all the correlations and the d-bandwidth is broader.
# A magnetic tight-binding model: surface properties of transition metals and cobalt nanoparticles

Jacques R. Eone II

Department of Physics, University of Strasbourg, Strasbourg, France

(Dated: 2019)

The magnetic and surface properties of some transition metals have been investigated within the tight-binding approximation, including Coulomb correlations. These surface properties are calculated after applying a charge neutrality rule that is restricted to the d-band. This formalism gives a charge distribution containing delocalized sp-states in agreement with a linear muffin-tin orbital calculation. It enables the description of local magnetism, surface energies, and work functions without recourse to the total energy. The present investigation is focused on the study of fcc cobalt, bcc iron, fcc nickel, and fcc platinum surfaces, as well as an exploration of fcc cobalt nanoparticles.

## I. INTRODUCTION

The electronic structure of a transition metal is characterized by delocalized electrons, with a predominance of electrons in a d-band over those in the sp-band. The d-band is distinguished by its greater localization of electrons compared to the sp-band, which consists of nearly free electrons. A correlation has been observed between the two bands, and the action of the electrons occupying the sp-band has been shown to broaden the d-bandwidth and reduce the localization of the d-electrons [1]. The resultant metallic ground state can be readily described using density functional theory (DFT) [2]. While DFT calculations are known to be efficient, their application is constrained by the number of atoms in the system under study. Therefore, semi-empirical methods, such as the tight-binding approximation, are more appropriate for studying these systems, provided that they rigorously integrate the rules governing the behavior of electrons. The tight-binding approximation is frequently restricted to the d-band, resulting in the neglect of the s- and p-states. This approximation yields energies that are not accurate [1]. It is imperative to consider the impact of sp-band on the d-band when describing a transition metal. The rules that govern the electrons of a transition metal are especially important at the surface. Indeed, a clear principle of charge neutrality can be applied at the surface of a transition metal. According to some authors, the total charge is conserved at the surface of transition metals and transition metal alloys per atomic site, per orbital and per chemical species [3–5]. In this study, the charge neutrality rule is restricted to the d-band because the d-electrons play a more significant role in the formation of bonds and the cohesion. By applying this charge neutrality rule to the d-band, the delocalized sp-band generates free sp-states for a layer beyond the surface (S + 1), which represents the vacuum. The charge neutrality rule leads to a self-consistency approach useful for the calculation of surface properties like surface energies and work functions using empirical laws, as well as deducing surface magnetism. The surface magnetism is derived from the Stoner model and is obtained by shifting the nonmagnetic local density of states (LDOS). This

is achieved by the assumption that the Coulomb correlations of the d-band are conserved at the surface. This formalism is generalized to obtain the magnetic properties of fcc cobalt nanoparticles.

## II. METHODOLOGY

In the tight-binding approximation, the atomic potential $H^{at}$ is perturbed by a weak perturbation $\Delta U(r)$ due to the interaction with neighboring atoms. The atomic energy of the d-band in the atomic basis $\psi_{\lambda}$ can be described as shifted by a quantity $\alpha$:

$$ \epsilon_d = \underbrace{\int \psi_{\lambda}^{*}(r) H^{at} \psi_{\lambda}(r) d^3 r}_{\text{atomic}} + \underbrace{\int \psi_{\lambda}^{*}(r) \Delta U(r) \psi_{\lambda}(r) d^3 r}_{\alpha : \text{perturbation}} \quad (1) $$

The strength of this perturbation $\Delta U(r)$ depends on the overlap between d orbitals from one atom and its neighboring atoms. The coordination number is reduced at the surface, and the atomic potential is subject to different perturbations from those in the bulk due to the presence of neighboring atoms. The impact of this novel potential at the surface can be delineated by a shift in atomic energies by a quantity $\alpha$, in accordance with a charge neutrality rule. This process is a self-consistency procedure that corrects the electronic structure at the surface prior to the study of the magnetic properties. The local magnetism is derived from the local Hubbard Hamiltonian:

$$ H_i = -t \sum_{j,\sigma} (c_{i\sigma}^{\dagger} c_{j\sigma} + \text{h.c.}) + U_d \sum_{\lambda} n_{\lambda\uparrow} n_{\lambda\downarrow} \quad (2) $$

where $\lambda$ represents a d orbital, $t$ denotes the hopping integral and $U_d$ is the effective onsite Coulomb repulsion. In the context of a basis that incorporates the effects of the s and p-states, the d-bandwidth is broadened and all Coulomb correlations are encompassed by $U_d$.
A. Stoner local magnetism

The spin magnetic moment $\mu$ and the total number of electrons in the d-band $n_d$ can be written by the charge fluctuation [6] :

$$ \mu = n_0 \langle n_{\uparrow} - n_{\downarrow} \rangle \quad \text{and} \quad n_d = n_0 \langle n_{\uparrow} + n_{\downarrow} \rangle \quad (3) $$

The average population per spin is given by :

$$ \langle n_{\uparrow} \rangle = \frac{1}{2n_0}(n_d - \mu) \quad \text{and} \quad \langle n_{\downarrow} \rangle = \frac{1}{2n_0}(n_d + \mu) $$

The second term of Eq. (2) can be decomposed in the mean field approximation [6] :

$$ \begin{aligned} U_d \sum_{\lambda} n_{\lambda\uparrow} n_{\lambda\downarrow} &\approx U_d \sum_{\lambda} n_{\lambda\uparrow} \langle n_{\downarrow} \rangle + n_{\lambda\downarrow} \langle n_{\uparrow} \rangle - \langle n_{\uparrow} \rangle \langle n_{\downarrow} \rangle \\ &= U_d \sum_{k, \sigma} n_{k\sigma} \langle n_{-\sigma} \rangle - n_0 U_d \langle n_{\uparrow} \rangle \langle n_{\downarrow} \rangle \\ &= \frac{U_d}{2n_0} \sum_{k\sigma} (n_d - \sigma\mu) c_{k\sigma}^{\dagger} c_{k\sigma} \\ &\quad - n_0 U_d \frac{1}{4n_0^2} (n_d - \mu)(n + \mu) \\ &= \frac{U_d}{n_0} \sum_{k\sigma} \left(\frac{n_d}{2} - \frac{\sigma}{2\mu}\right) c_{k\sigma}^{\dagger} c_{k\sigma} - \frac{U_d}{n_0} \left(\frac{n_d^2}{4} - \frac{\mu^2}{4}\right) \end{aligned} $$

The Hubbard Hamiltonian of Eq. (2) becomes :

$$ H = \sum_{k\sigma} \left( \epsilon_k + \frac{n_d U_d}{2n_0} - \frac{\sigma}{2} \frac{U_d \mu}{n_0} \right) c_{k\sigma}^{\dagger} c_{k\sigma} - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \quad (4) $$

The band structure $\epsilon_{k\sigma} = \epsilon_k + \frac{n_d U_d}{2n_0} - \frac{\sigma}{2} \frac{U_d \mu}{n_0}$ is then dependent on the spin $\sigma$ and the bands are then shifted by an exchange splitting $\Delta\epsilon$. We deduce the Stoner relation :

$$ \Delta\epsilon = \frac{U_d\mu}{n_0} = I\mu \quad \text{and} \quad \mu = \frac{n_0}{U_d}\Delta\epsilon \quad (5) $$

Where $I$ is the Stoner parameter. $I$ and $U$ are self-consistency parameters used to obtain a correct magnetic moment. We can derive the total band energy $E_b$ of a magnetic system by making the summation in Eq. (4) depending on the spin :

$$ E_b = \left\{ \begin{array}{ll} \sum_{k\uparrow} \epsilon_k^{\uparrow} + \frac{n_d U_d}{2n_0} - \frac{1}{2} \frac{U_d \mu}{n_0} N_{\uparrow} & - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ \sum_{k\downarrow} \epsilon_k^{\downarrow} + \frac{n_d U_d}{2n_0} + \frac{1}{2} \frac{U_d \mu}{n_0} N_{\downarrow} & - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \end{array} \right. \quad (6) $$

By making this summation of the bands spin up and spin down containing respectively $N_{\uparrow}$ et $N_{\downarrow}$ electrons, we obtain :

$$ E_b = \left\{ \begin{array}{ll} \epsilon_{band}^{\uparrow} + \frac{n_d U_d}{2n_0} N_{\uparrow} - \frac{1}{2} \frac{U_d \mu}{n_0} N_{\uparrow} & - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ \epsilon_{band}^{\downarrow} + \frac{n_d U_d}{2n_0} N_{\downarrow} + \frac{1}{2} \frac{U_d \mu}{n_0} N_{\downarrow} & - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \end{array} \right. \quad (7) $$

Or linearly :

$$ \begin{aligned} E_b &= \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} + \frac{n_d U_d}{2n_0} (N_{\uparrow} + N_{\downarrow}) \\ &\quad - \frac{1}{2} \frac{U_d \mu}{n_0} (N_{\uparrow} - N_{\downarrow}) - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ &= \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} + \frac{n_d U_d}{2n_0} n_d - \frac{1}{2} \frac{U_d \mu}{n_0} \mu - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ E_b &= \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} + \frac{1}{4n_0} U_d n_d^2 - \frac{1}{4n_0} U_d \mu^2 \end{aligned} $$

The variation of the energy when we make the transition from a non-magnetic state to a magnetic state is then given by :

$$ \Delta E^{mag} = E_b^{mag} - E_b^{non.mag} \quad (8) $$

$$ = \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} - \epsilon_{band}^{non.mag} - \frac{1}{4n_0} U_d \mu^2 \quad (9) $$

$$ \Delta E^{mag} = \Delta E_{coh} - \frac{1}{4n_0} U_d \mu^2 \quad (10) $$

This transition energy $\Delta E^{mag}$ is negative for all ferromagnetic materials.

B. Surface effects : a self-consistency treatment

At the surface, a lower coordination decreases the bandwidth. The potential felt and the charge at the Fermi level are different from that in the bulk. The relaxation allows to decrease the interatomic distances and to increase the overlap between orbitals and thus the bandwidth. This effect requiring a total energy can be included in a simple correction involving the shift of atomic energies so that we obtain a conservation of the charge and the bandwidth. The electrons of the d-band participating in the cohesion are more affected by this charge neutrality. Assuming that only the d-electrons undergo an atomic energy shift $\delta\epsilon_d$ and define the surface Fermi level at the surface (S), we obtain for the delocalized sp-band, free states beyond the Fermi level ($S+1$). This extra charge mainly from the p-band is no longer included in the calculation of the properties of the studied surface. This treatment gives to the charge distribution from the Slater-Koster tight-binding approach similar features as a LMTO calculation [7]. The sp-band has therefore an important effect on the surface properties and should not be neglected. Calculations even using a total energy neglecting the $s$ and $p$ states lead to incorrect surface energies [8, 9]. The surface energy $\gamma$ results from an empirical law as the difference of the band energies after the charge neutrality procedure.

$$ \gamma = \frac{1}{3} \left[ \sum_{\lambda} \left( \int_{-\infty}^{E_f} \mathrm{En}(E, \delta\epsilon_{\lambda}) dE - N_e(\lambda) \delta\epsilon_{\lambda} \right) - E_{band}^{bulk} \right] \quad (11) $$
A. Local magnetism in the Stoner model

The spin magnetic moment $\mu$ and the total number of electrons in the d-band $n_d$ can be expressed in terms of charge fluctuations [6]:

$$ \mu = n_0 \langle n_{\uparrow} - n_{\downarrow} \rangle \quad \text{and} \quad n_d = n_0 \langle n_{\uparrow} + n_{\downarrow} \rangle, \quad (3) $$

where $n_0=5$ d orbitals. The average population per spin is approximate by:

$$ \langle n_{\uparrow} \rangle = \frac{1}{2n_0}(n_d - \mu) \quad \text{and} \quad \langle n_{\downarrow} \rangle = \frac{1}{2n_0}(n_d + \mu) $$

The second term of Eq. (2) can be decomposed in the mean-field approximation [6]:

$$ \begin{aligned} U_d \sum_{\lambda} n_{\lambda\uparrow} n_{\lambda\downarrow} &\approx U_d \sum_{\lambda} n_{\lambda\uparrow} \langle n_{\downarrow} \rangle + n_{\lambda\downarrow} \langle n_{\uparrow} \rangle - \langle n_{\uparrow} \rangle \langle n_{\downarrow} \rangle \\ &= U_d \sum_{k,\sigma} n_{k\sigma} \langle n_{-\sigma} \rangle - n_0 U_d \langle n_{\uparrow} \rangle \langle n_{\downarrow} \rangle \\ &= \frac{U_d}{2n_0} \sum_{k\sigma} (n_d - \sigma\mu) c_{k\sigma}^{\dagger} c_{k\sigma} \\ &\quad - n_0 U_d \frac{1}{4n_0^2} (n_d - \mu)(n + \mu) \\ &= \frac{U_d}{n_0} \sum_{k\sigma} \left( \frac{n_d}{2} - \frac{\sigma}{2\mu} \right) c_{k\sigma}^{\dagger} c_{k\sigma} - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \end{aligned} $$

The local Hubbard Hamiltonian of Eq. (2) becomes:

$$ H_i = \sum_{k\sigma} \left( \epsilon_k + \frac{n_d U_d}{2n_0} - \frac{\sigma}{2} \frac{U_d \mu}{n_0} \right) c_{k\sigma}^{\dagger} c_{k\sigma} - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \quad (4) $$

The band structure $\epsilon_{k\sigma} = \epsilon_k + \frac{n_d U_d}{2n_0} - \frac{\sigma}{2} \frac{U_d \mu}{n_0}$ is then dependent on the spin $\sigma$ and the bands are then shifted by an exchange splitting $\Delta\epsilon$. The Stoner relation is deduced as follows:

$$ \Delta\epsilon = \frac{U_d\mu}{n_0} = I\mu \quad \text{and} \quad \mu = \frac{n_0}{U_d}\Delta\epsilon, \quad (5) $$

where $I$ is the Stoner parameter. $I$ and $U$ are self-consistency parameters that are employed to obtain a correct spin magnetic moment. The band energy $E_b$ of a magnetic system can be derived through the summation in Eq. (4) depending on the spin:

$$ E_b = \left\{ \begin{array}{ll} \sum_{k\uparrow} \epsilon_k^{\uparrow} + \frac{n_d U_d}{2n_0} - \frac{1}{2} \frac{U_d \mu}{n_0} & - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ \sum_{k\downarrow} \epsilon_k^{\downarrow} + \frac{n_d U_d}{2n_0} + \frac{1}{2} \frac{U_d \mu}{n_0} & \end{array} \right. \quad (6) $$

The summation of the bands spin up and spin down containing respectively $N_{\uparrow}$ et $N_{\downarrow}$ electrons gives:

$$ E_b = \left\{ \begin{array}{ll} \epsilon_{band}^{\uparrow} + \frac{n_d U_d}{2n_0} N_{\uparrow} - \frac{1}{2} \frac{U_d \mu}{n_0} N_{\uparrow} & - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ \epsilon_{band}^{\downarrow} + \frac{n_d U_d}{2n_0} N_{\downarrow} + \frac{1}{2} \frac{U_d \mu}{n_0} N_{\downarrow} & \end{array} \right. \quad (7) $$

Or linearly:

$$ \begin{aligned} E_b &= \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} + \frac{n_d U_d}{2n_0} (N_{\uparrow} + N_{\downarrow}) \\ &\quad - \frac{1}{2} \frac{U_d \mu}{n_0} (N_{\uparrow} - N_{\downarrow}) - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ &= \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} + \frac{n_d U_d}{2n_0} n_d - \frac{1}{2} \frac{U_d \mu}{n_0} \mu - \frac{U_d}{n_0} \left( \frac{n_d^2}{4} - \frac{\mu^2}{4} \right) \\ E_b &= \epsilon_{band}^{\uparrow} + \epsilon_{band}^{\downarrow} + \frac{1}{4n_0} U_d n_d^2 - \frac{1}{4n_0} U_d \mu^2 \end{aligned} $$

The variation in energy that occurs during the transition from a non-magnetic state to a magnetic state can be expressed as follows:

$$ \Delta E^{\text{mag.}} = E_b^{\text{mag.}} - E_b^{\text{nonmag.}} \quad (8) $$

$$ = \epsilon_{\text{band}}^{\uparrow} + \epsilon_{\text{band}}^{\downarrow} - \epsilon_{\text{band}}^{\text{nonmag.}} - \frac{1}{4n_0} U_d \mu^2 \quad (9) $$

$$ \Delta E^{\text{mag}} = \Delta E_b - \frac{1}{4n_0} U_d \mu^2 \quad (10) $$

The energy change $\Delta E^{mag}$ is negative for all ferromagnetic materials.

B. Surface effects: a self-consistency treatment

At the surface, a lower coordination number results in a decrease in bandwidth. The potential and the charge at the Fermi level are distinct from those in the bulk. The relaxation makes the distance between atoms smaller and increases the overlap between orbitals. This increases the bandwidth. Although this effect requires a total energy, it can be incorporated into a rudimentary correction. This correction involves shifting the atomic energies so that the charge and bandwidth are conserved. The electrons in the d-band, which are more involved in the cohesion, are used to achieve charge neutrality. Assuming that only the d-electrons undergo an atomic energy shift $\delta\epsilon_d$, and defining the surface Fermi level at the surface (S), delocalized sp-states that lie beyond the Fermi level (S + 1) are obtained. This additional electronic charge, which is primarily from the p-band, is no longer included in the calculation of the properties of the studied surface. This procedure provides a charge distribution that is similar to the one obtained from a Linear Muffin-Tin Orbital (LMTO) calculation [7]. Therefore, it can be concluded that the sp-band exerts a significant influence on the surface properties and must not be disregarded. It is evident that calculations, even when employing a total energy approach that disregards the s and p states, result in erroneous surface energies [8, 9]. The surface energy $\gamma$ is derived from an empirical law, which is defined as the difference between the band energies subsequent to the charge neutrality procedure:
Eq. (11) is the mean value of the contribution of all the bands $\lambda = s, p, d$. We assume that $\delta\epsilon_s = 0$ and $\delta\epsilon_p = 0$. $n(E, \delta\epsilon_\lambda)$ is the shifted local density of states (LDOS) at the surface respecting the charge neutrality, $N_e(\lambda)$ the number of electrons in the band $\lambda$ and $E_{\text{band}}^{\text{bulk}}$ the band energy of the bulk. This expression contains the contribution $N_e(\lambda)\delta\epsilon_{i\lambda}$ which takes into account the energy for shifting the atomic levels of the band $\lambda$ by a quantity $\delta\epsilon_\lambda$.

The surface magnetism comes from the Stoner model applied on the LDOS. From the non-magnetic local density of states (LDOS), we create two LDOS spin up and spin down and we shift these LDOS by several values of the exchange splitting $\Delta\epsilon$ [10]. At the surface, we define the work function by [11]:

$$W = E_{\text{vacuum}} - E_f \qquad (12)$$

Where $E_{\text{vacuum}}$ is the energy to extract an electron from the surface to the vacuum without an additional kinetic energy. This vacuum energy depends on the surface properties and is derived from the mean value of the band energies after the self-consistency charge neutrality.

$$E_{\text{vacuum}} = \frac{1}{3} \left[ \sum_{\lambda} \frac{1}{N_e(\lambda)} \left( \int_{-\infty}^{E_f} E n_{\lambda}(E, \delta\epsilon_{\lambda}) dE \right) \right] - 3\gamma \qquad (13)$$

We add to this expression the magnetic contribution $\Delta E^{\text{mag}}$ for the bulk and the surface for a magnetic work function. Eq (13) is also an empirical law giving a qualitative description of the work function.

### III. RESULTS

Our calculations are based on the Slater-Koster hopping parameters and atomic energies to build the hopping integral and the tight-binding Hamiltonian. These hopping parameters $ss\sigma, sp\sigma, sd\sigma, pp\sigma, pp\pi, pd\sigma, pd\pi, dd\sigma, dd\pi, dd\delta$ along with the atomic energies $\epsilon_s, \epsilon_p$ and $\epsilon_d$ are obtained by fitting the tight-binding band structure with the one obtained with a Density Functional Theory (DFT) calculation using an all-electron and full-potential basis. The fit (Figs. 1 and 2) and our tight-binding hamiltonian is restricted to the first neighbors. This approximation is enough to have a good accuracy especially for a fcc crystal structure. However for bcc iron, the parameters are taken from Ref. [12].

#### A. Results in the bulk

The magnetic properties of the bulk of the ferromagnetic elements through the tight-binding approximation has already been studied. The magnetic moment results from the shift of the non-magnetic LDOS while keeping the charge in the d-band $\mu = n_{d\uparrow} - n_{d\downarrow} =$

<figure><img src="image_2.png" /><figcaption>FIG. 1. Band fitting of non magnetic fcc Co.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 2. Band fitting of fcc Pt.</figcaption></figure>

$n_d(E - \frac{\Delta\epsilon}{2}) - n_d(E + \frac{\Delta\epsilon}{2})$. The curve representing this magnetic moment is intercepted by another curve of the magnetic moment from the Stoner relation in Eq. (5) giving the value of $U_d$ in the bulk (Fig. 3) corresponding to a coherent magnetic moment. For a d-band having the effects of the s- and p-states, $U_d$ containing the correlations is about 4.98 eV for fcc iron, 5.93 eV for fcc cobalt and 6.80 eV for fcc nickel. These values are summarized in Table (I). The obtained values (Table I) with our new hopping parameters are slightly different from those from a previous work [1].

<figure><img src="image_4.png" /><figcaption>FIG. 3. Calculation of $U_d$ for fcc Co.</figcaption></figure>
$$ \gamma = \frac{1}{3} \left[ \sum_{\lambda} \left( \int_{-\infty}^{E_f} En(E, \delta\epsilon_{\lambda})dE - N_e(\lambda)\delta\epsilon_{\lambda} \right) - E_b^{\text{bulk}} \right] \quad (11) $$

This equation is the mean value of the contribution from all the bands $\lambda = s, p, d$. It is posited that $\delta\epsilon_s = 0$ and $\delta\epsilon_p = 0$. $n(E, \delta\epsilon_{\lambda})$ is the shifted local density of states at the surface after the application of the charge neutrality procedure, $N_e(\lambda)$ is the number of electrons in the band $\lambda$ and $E_b^{\text{bulk}}$ is the band energy of the bulk. This expression contains a double counting contribution $N_e(\lambda)\delta\epsilon_{i\lambda}$, which includes the energy for shifting the atomic levels of the band $\lambda$ by a quantity $\delta\epsilon_{\lambda}$.

The surface magnetism is derived from the Stoner model, which is applied to the LDOS. The non-magnetic LDOS is the initial point of this process, from which two LDOS spin up and spin down are created. These LDOS are then shifted by multiple values of the exchange splitting $\Delta\epsilon$ [10]. The work function is defined at the surface as follows [11]:

$$ W = E_{\text{vacuum}} - E_F, \quad (12) $$

where $E_{\text{vacuum}}$ denotes the energy required to extract an electron from the surface to the vacuum, without accounting for any additional kinetic energy. This vacuum energy depends on the surface properties and is derived from the mean value of the band energies subsequent to the self-consistency charge neutrality:

$$ E_{\text{vacuum}} = \frac{1}{3} \left[ \sum_{\lambda} \frac{1}{N_e(\lambda)} \left( \int_{-\infty}^{E_F} En_{\lambda}(E, \delta\epsilon_{\lambda})dE \right) \right] - 3\gamma \quad (13) $$

In order to calculate the magnetic work function, it is necessary to add the quantity $\Delta E^{\text{mag}}$ for the bulk and the surface. Eq (13) is also empirical, providing an approximate description of the work function.

### III. RESULTS

The calculations were performed using the Slater-Koster hopping parameters and atomic energies to construct the hopping integral and the tight-binding Hamiltonian. These hopping parameters – $ss\sigma$, $sp\sigma$, $sd\sigma$, $pp\sigma$, $pp\pi$, $pd\sigma$, $pd\pi$, $dd\sigma$, $dd\pi$, $dd\delta$ – along with the atomic energies – $\epsilon_s$, $\epsilon_p$ and $\epsilon_d$ – are obtained by fitting the tight-binding band structure with the band structure obtained from a DFT calculation. The fit (Figs. 1 and 2) and the tight-binding hamiltonian is restricted to the first-neighbor approximation. This approximation is adequate for achieving satisfactory accuracy, particularly in the context of fcc crystal structures. Nevertheless, the hopping parameters for bcc iron are obtained from Ref. [12].

<figure><img src="image_2.png" /><figcaption>FIG. 1. Nonmagnetic fcc Co band structures obtained using the tight-binding approximation and DFT after fitting.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 2. fcc Pt band structures obtained using the tight-binding approximation and DFT after fitting.</figcaption></figure>

#### A. Results for bulk materials

The magnetic properties of ferromagnetic elements were studied using tight-binding approximation in another work [1]. The spin magnetic moment results from the shift of nonmagnetic LDOS while keeping the electronic charge in the d-band $\mu = n_{d\uparrow} - n_{d\downarrow} = n_d(E - \frac{\Delta\epsilon}{2}) - n_d(E + \frac{\Delta\epsilon}{2})$. The spin magnetic moment curve is crossed by another spin magnetic moment curve from the Stoner relation in Eq. (5). This gives the value of $U_d$ in the bulk (Fig. 3). This value should correspond to the experimental spin magnetic moment. For a d-band including the effects of the sp-band, $U_d$ is about 4.98 eV for fcc iron, 5.93 eV for fcc cobalt and 6.80 eV for fcc nickel. These values are summarized in Table I. The results of this study, employing new hopping parameters, exhibit slight discrepancies from those reported in a previous study [1].

<table><caption>TABLE I. $U_d$, $\Delta\epsilon$ and $\Delta E_{\text{bulk}}^{\text{mag}}$ for ferromagnetic metals.</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th></tr></thead><tbody><tr><td>$U_d$ [eV]</td><td>4.98</td><td>5.93</td><td>6.80</td></tr><tr><td>$\Delta\epsilon$ [eV]</td><td>2.20</td><td>1.91</td><td>0.83</td></tr><tr><td>$\mu$ [$\mu_B$]</td><td>2.22</td><td>1.61</td><td>0.61</td></tr><tr><td>$\Delta E_{\text{bulk}}^{\text{mag}}$ [eV]</td><td>-0.28</td><td>-0.21</td><td>-0.03</td></tr></tbody></table>
<table><caption>TABLE I. Values of $U_d$, $\Delta\epsilon$ and $\Delta E_{bulk}^{mag}$ in the bulk of the ferromagnetic elements.</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th></tr></thead><tbody><tr><td>$U_d$ [ev]</td><td>4.98</td><td>5.93</td><td>6.80</td></tr><tr><td>$\Delta\epsilon$ [ev]</td><td>2.20</td><td>1.91</td><td>0.83</td></tr><tr><td>$\mu$ [$\mu_B$/atom]</td><td>2.22</td><td>1.61</td><td>0.61</td></tr><tr><td>$\Delta E_{bulk}^{mag}$ [eV]</td><td>-0.28</td><td>-0.21</td><td>-0.03</td></tr></tbody></table>

## B. Results at the surface

### 1. Non-magnetic surface

By applying a self-consistent loop on the *d*-band in order to obtain a charge neutrality at the surface (S), the *d* atomic energies are then shifted by $\delta\epsilon_d$ depending on the crystallography direction (Table II). This surface self-consistency *d* charge neutrality as stated previously creates at the Fermi level free charges in the *sp*-band.

<table><caption>TABLE II. Shift of the *d* atomic energy to obtain the *d* charge neutrality in Co, Ni, Fe and Pt compared to the bulk.</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th><th>Pt</th></tr></thead><tbody><tr><td>$\delta\epsilon_d(111)$ [ev]</td><td>-</td><td>0.34</td><td>0.33</td><td>0.58</td></tr><tr><td>$\delta\epsilon_d(110)$ [ev]</td><td>0.08</td><td>-</td><td>-</td><td>-</td></tr><tr><td>$\delta\epsilon_d(100)$ [ev]</td><td>0.30</td><td>0.42</td><td>0.43</td><td>0.81</td></tr></tbody></table>

We can notice that the value of $\delta\epsilon_d$ at the surface depends on the crystallographic direction. We obtain almost similar values of $\delta\epsilon_d$ for fcc Co and fcc Ni for the two considered crystallographic directions. Concerning the charge distribution, Table (III) gives the charge at the surface (S) and the layer (S + 1) for fcc Co in the crystallographic directions: (100) and (111). The sum of these *sp* surface states is in agreement with a LMTO calculation for fcc Co [7]. We obtain almost the same contribution to the *sp* surface states at the layer (S + 1) for fcc Ni and fcc Pt but not enough to generalize that the population of this layer is constant for a crystalline structure. However, for bcc Fe, the total contribution to (S + 1)(100) and (S + 1)(110) is respectively 0.47 and 0.30 electrons which is larger than in the fcc structure. In Table (IV), the non-magnetic surface energies are calculated by using Eq. (11).

<table><caption>TABLE III. Population per orbital at the surface (S) and in the layer (S+1) for a non-magnetic fcc Co.</caption><thead><tr><th></th><th>s</th><th>p</th><th>d</th><th>Total</th></tr></thead><tbody><tr><td>$N_e(100)(S)$</td><td>0.48</td><td>0.28</td><td>7.87</td><td>8.61</td></tr><tr><td>$N_e(100)(S + 1)$</td><td>0.11</td><td>0.26</td><td>0.01</td><td>0.38</td></tr><tr><td>$N_e(111)(S)$</td><td>0.54</td><td>0.35</td><td>7.86</td><td>8.75</td></tr><tr><td>$N_e(111)(S + 1)$</td><td>0.05</td><td>0.19</td><td>0.02</td><td>0.26</td></tr></tbody></table>

<table><caption>TABLE IV. Non-magnetic surface energies of fcc Co, fcc Ni, bcc Fe and fcc Pt</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th><th>Pt</th></tr></thead><tbody><tr><td>$\gamma(111)$ [ev]</td><td>-</td><td>0.88</td><td>0.71</td><td>1.02</td></tr><tr><td>$\gamma(110)$ [ev]</td><td>1.29</td><td>-</td><td>-</td><td>-</td></tr><tr><td>$\gamma(100)$ [ev]</td><td>0.88</td><td>1.19</td><td>0.97</td><td>1.43</td></tr></tbody></table>

The value of the surface energy $\gamma_{Fe}(100)$ is overestimated compared to the experimental value of about 0.87 eV [13]. The surface energy of Pt(100) is also overestimated: this surface is actually reconstructed [14]. The value given by our calculation is a non-reconstructed surface energy. The reconstruction of the surface Pt(100) gives a hexagonal structure [14] which has a smaller surface energy than the non-reconstructed surface. Nevertheless, the surface energy of Pt(111) is consistent with the experimental value of 1.03 eV [13] and a DFT calculation [15].

### 2. Surface magnetism

We assume that the charge neutrality in the *d*-band leads to the conservation of the bandwidth and thus the conservation of the Coulomb parameter $U_d$. This postulate is very important to find the magnetic moment at the surface as well as the variation of energy $\Delta E^{mag}$. Using the value of $U_d$ obtained in the bulk of fcc Co, we have 1.77 $\mu_B$ and 1.86 $\mu_B$ as magnetic moment (Figs. 3 and 4) respectively for the surfaces Co(111) and Co(100) which are in agreement with another calculations [7, 16]. By applying the same procedure with the other ferromagnetic elements, we obtain the values in Table (V). In Table (V), the magnetic moment at the surface is underestimated in the case of bcc Fe and overestimated in the case of fcc Ni compared to a DFT calculation. These discrepancies may be due to the fact that the charge neutrality does not describe correctly the surface properties of these materials and there is likely a small amount of charge transfers between the orbitals.

We can deduce the magnetic surface energy by adding to the Eq (11) the variation of the band energy due to the magnetism $\delta\Delta E^{mag} = \Delta E^{mag}(100/111/110) - \Delta E_{bulk}^{mag}$.

<table><caption>TABLE V. Magnetic moment and $\Delta E^{mag}$ at the surface of Co, Ni and Fe</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th></tr></thead><tbody><tr><td>$\mu(111)$ [$\mu_B$]</td><td>-</td><td>1.77</td><td>0.70</td></tr><tr><td>$\mu(110)$ [$\mu_B$]</td><td>2.54</td><td>-</td><td>-</td></tr><tr><td>$\mu(100)$ [$\mu_B$]</td><td>2.65</td><td>1.86</td><td>0.83</td></tr><tr><td>$\Delta E^{mag}(111)$ [ev]</td><td>-</td><td>-0.39</td><td>-0.04</td></tr><tr><td>$\Delta E^{mag}(110)$ [ev]</td><td>-0.52</td><td>-</td><td>-</td></tr><tr><td>$\Delta E^{mag}(100)$ [ev]</td><td>-0.61</td><td>-0.45</td><td>-0.07</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>FIG. 3. Calculation of $U_d$ for fcc Co.</figcaption></figure>

## B. Results for transition metal surfaces

### 1. Non-magnetic surface

The application of a self-consistent procedure to the *d*-band is necessary to achieve charge neutrality at the surface (S). This results in a shift of the *d* atomic energies by a quantity $\delta\epsilon_d$. This shift is dependent on the crystallographic direction (Table II). As previously stated, the surface self-consistency *d* charge neutrality leads to the formation of free electronic charges in the *sp*-band at the Fermi level.

<table><caption>TABLE II. Energy shift of *d* atomic levels for achieving charge neutrality in Co, Ni, Fe and Pt.</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th><th>Pt</th></tr></thead><tbody><tr><td>δε<sub>d</sub>(111) [eV]</td><td>-</td><td>0.34</td><td>0.33</td><td>0.58</td></tr><tr><td>δε<sub>d</sub>(110) [eV]</td><td>0.08</td><td>-</td><td>-</td><td>-</td></tr><tr><td>δε<sub>d</sub>(100) [eV]</td><td>0.30</td><td>0.42</td><td>0.43</td><td>0.81</td></tr></tbody></table>

The obtained $\delta\epsilon_d$ value at the surface depends on the crystallographic direction. A comparison of the two crystallographic directions reveals that the $\delta\epsilon_d$ values for both fcc Co and fcc Ni are nearly equivalent. With regard to the distribution of charge, the Table III presents the charge at the surface (S) and the layer (*S* + 1) for fcc Co in the crystallographic directions (100) and (111). The total number of *sp* states in the (*S* + 1) layer is consistent with an LMTO calculation for fcc Co [7]. The same contribution to the *sp* states at the layer (*S* + 1) is obtained for both fcc Ni and fcc Pt. However, this is insufficient to conclude that the population of this layer is constant for a crystalline structure. However, for bcc Fe, the total contribution to (*S* + 1)(100) and (*S* + 1)(110) is 0.47 and 0.30 electrons respectively. A magnitude larger than in the fcc structure. In Table IV, the nonmagnetic surface energies are calculated using Eq. (11).

The value of the surface energy $\gamma_{Fe}(100)$ is found to be greater than the experimental value of approximately

<table><caption>TABLE III. Electronic population per orbital at the surface (S) and in the layer (S+1) for a nonmagnetic fcc Co.</caption><thead><tr><th></th><th>s</th><th>p</th><th>d</th><th>Total</th></tr></thead><tbody><tr><td>N<sub>e</sub>(100)(S)</td><td>0.48</td><td>0.28</td><td>7.87</td><td>8.61</td></tr><tr><td>N<sub>e</sub>(100)(S + 1)</td><td>0.11</td><td>0.26</td><td>0.01</td><td>0.38</td></tr><tr><td>N<sub>e</sub>(111)(S)</td><td>0.54</td><td>0.35</td><td>7.86</td><td>8.75</td></tr><tr><td>N<sub>e</sub>(111)(S + 1)</td><td>0.05</td><td>0.19</td><td>0.02</td><td>0.26</td></tr></tbody></table>

<table><caption>TABLE IV. Nonmagnetic surface energies for fcc Co, fcc Ni, bcc Fe and fcc Pt.</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th><th>Pt</th></tr></thead><tbody><tr><td>γ(111) [eV]</td><td>-</td><td>0.88</td><td>0.71</td><td>1.02</td></tr><tr><td>γ(110) [eV]</td><td>1.29</td><td>-</td><td>-</td><td>-</td></tr><tr><td>γ(100) [eV]</td><td>0.88</td><td>1.19</td><td>0.97</td><td>1.43</td></tr></tbody></table>

0.87 eV [13]. Additionally, the Pt(100) surface energy is overestimated. A surface reconstruction occurs at this particular surface [14]. The calculated values are obtained using a non-reconstructed surface. The reconstruction of the Pt(100) surface results in the formation of a hexagonal structure, which exhibits a reduced surface energy relative to the non-reconstructed surface. Notwithstanding, the Pt(111) surface energy is consistent with the experimental value of 1.03 eV [13], as well as a DFT calculation [15].

### 2. Surface magnetism

It is hypothesized that charge neutrality in the *d*-band results in the conservation of the bandwidth and, consequently, the conservation of the Coulomb parameter $U_d$. This parameter is used to determine the spin magnetic moment at the surface and the variation of energy $\Delta E^{mag}$. The $U_d$ value obtained in the bulk of fcc Co was utilized to calculate the spin magnetic moment of Co(111) and Co(100) surfaces. The resulting values of 1.77 $\mu_B$ and 1.86 $\mu_B$ (Figs. 3 and 4) are consistent with the results reported in another calculation [7, 16]. Subsequent application of the identical procedure to the remaining ferromagnetic elements yielded the values presented in Table V. As illustrated in Table V, the spin magnetic moment at the surface is underestimated in the case of bcc Fe and overestimated in the case of fcc Ni compared to a DFT calculation. The observed discrepancies may be attributable to the charge neutrality used to describe the surface properties of these metals.

The magnetic surface energy can be obtained by adding to Eq. (11) the variation of the energy due to magnetism: $\delta\Delta E^{mag} = \Delta E^{mag}(100/111/110) - \Delta E^{mag}_{bulk}$.

The magnetic surface energies are: $\gamma_{Fe}^{mag}(100) = 1.18$ eV, $\gamma_{Fe}^{mag}(110) = 0.80$ eV for bcc Fe, which are in close agreement with the experimental value of 0.89 eV
<figure><img src="image_5.png" /><figcaption>FIG. 4. Magnetic moment at the surface Co(100).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. Magnetic moment at the surface Co(111)</figcaption></figure>

The magnetic surface energies are then :
$\gamma_{Fe}^{mag}(100) = 1.18$ eV, $\gamma_{Fe}^{mag}(110) = 0.80$ eV in fcc
Fe which is not far from the experimental value :
0.89 eV [13, 17]. For fcc Co, $\gamma_{Co}^{mag}(100) = 0.94$ eV,
$\gamma_{Co}^{mag}(111) = 0.83$ eV in consistent with the experimen-
tal value : 0.87 eV [13, 17] and finally the magnetic
surface energies of fcc Ni are $\gamma_{Ni}^{mag}(100) = 0.94$ eV and
$\gamma_{Ni}^{mag}(111) = 0.70$ eV in agreement to the experimental
value of about 0.79 eV [18] and another calculation of
about 0.679 eV [16].

The work functions are calculated by applying Eq.
(12). The calculated work functions in Table (VI) are
close to the values obtained in another calculation [7]
and are in agreement with the experimental values. Our
method studying magnetic properties from bulk to sur-
faces can be extended to the study of nanoparticles. In
this work, we limit our investigation to fcc Co nanopar-
ticles (cuboctahedrons) but the method can be applied
on any magnetic nanoparticle.

C. Nanoparticles

The proprieties of a nanoparticle are calculated by
making the assumption that all the atomic sites with the
same coordination in a first neighbor approximation have
the same proprieties. In this approximation, we consider

classes of atomic sites. In a cuboctahedron there are five
classes of sites : The bulk (coordination number : 12),
the edges (coordination number : 7), the vertexes (co-
ordination number : 5), the faces (100) (coordination
number : 8) and the faces (111) (coordination number :
9). We consider fcc Co cuboctahedrons with a size going
from 55 atoms to 1415 atoms. The selfconsistency surface
charge neutrality procedure is the same : we fix a gen-
eral Fermi level almost defined by the bulk and we shift
the *d* atomic energies of each class at the surface until
at that Fermi level, the charge in the *d*-band is the same
than in the bulk. The magnetic properties are calculated
by shifting for each class the non-magnetic LDOS with
different values of the exchange splitting (so we have five
curves defined by $\mu = N_{\uparrow} - N_{\downarrow}$), these curves are in-
tercepted by the magnetic moment defined in Eq. (5)
conserving the Coulomb correlations $U = 5.93$ eV (Fig.
7). This procedure gives a magnetic moment depend-
ing on the coordination in Fig (6) and summarize in the
Tables (VII) and (VIII).

<figure><img src="image_7.png" /><figcaption>FIG. 6. Magnetic moment depending on the coordination number and the size for fcc Co cuboctahedrons calculated using $U_d = 5.93$ eV.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 7. Calculation of the magnetic moment of fcc Co nanoparticle.</figcaption></figure>

In the magnetic nanoparticles, there is an oscillation
of the magnetic moment depending of the size [21]. This
size effect can appear in our simple model. This oscil-
lation can also be observed in the variation of the work
function depending of the size of the particle. However,
the surface energy is dependent on the size without a
significant oscillation.
<figure><img src="image_5.png" /><figcaption>FIG. 4. Spin magnetic moment at the surface Co(100).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. Spin magnetic moment at the surface Co(111)</figcaption></figure>

<table><caption>TABLE V. Spin magnetic moment and $\Delta E^{mag}$ at the surface of Co, Ni and Fe</caption><thead><tr><th></th><th>Fe</th><th>Co</th><th>Ni</th></tr></thead><tbody><tr><td>μ(111) [μ<sub>B</sub>]</td><td>-</td><td>1.77</td><td>0.70</td></tr><tr><td>μ(110) [μ<sub>B</sub>]</td><td>2.54</td><td>-</td><td>-</td></tr><tr><td>μ(100) [μ<sub>B</sub>]</td><td>2.65</td><td>1.86</td><td>0.83</td></tr><tr><td>ΔE<sup>mag</sup>(111) [eV]</td><td>-</td><td>-0.39</td><td>-0.04</td></tr><tr><td>ΔE<sup>mag</sup>(110) [eV]</td><td>-0.52</td><td>-</td><td>-</td></tr><tr><td>ΔE<sup>mag</sup>(100) [eV]</td><td>-0.61</td><td>-0.45</td><td>-0.07</td></tr></tbody></table>

[13, 17]. For fcc Co, the magnetic surface energies are calculated to be $\gamma_{Co}^{mag}(100) = 0.94$ eV, $\gamma_{Co}^{mag}(111) = 0.83$ eV, which are in agreement with the experimental value of 0.87 eV [13, 17]. Similarly, the magnetic surface energies for fcc Ni are $\gamma_{Ni}^{mag}(100) = 0.94$ eV and $\gamma_{Ni}^{mag}(111) = 0.70$ eV, which are in agreement with the experimental value of about 0.79 eV [18] and another calculation of 0.679 eV [16].

The calculation of work functions is achieved through the application of Eq. (12). As demonstrated in Table VI, the calculated work functions are in close to the values obtained in another calculation [7]. Furthermore, these values align with the experimental values. The methodology employed in the study of magnetic properties from the bulk to surfaces can be extended to the study of nanoparticles. In this study, the focus is on

fcc Co nanoparticles (cuboctahedrons). However, the methodology can be extended to encompass a broader range of magnetic nanoparticles.

### C. Nanoparticles

The properties of a nanoparticle are determined through the assumption that all atomic sites with equivalent coordination in a first-neighbor approximation possess analogous properties. In this approximation, the nanoparticle has different classes of atomic sites. A cuboctahedron is composed of five distinct classes of sites. The bulk (coordination number: 12), the edges (coordination number: 7), the vertices (coordination number: 5), the facet (100) (coordination number: 8), and the facet (111) (coordination number: 9). In this work, fcc Co cuboctahedrons are studied with dimensions ranging from 55 atoms to 1,415 atoms. The self-consistency surface charge neutrality procedure remains the same. A general Fermi level is established, which is predominantly defined by the bulk. The d atomic energies of each class at the surface are then shifted until the charge in the d-band at that Fermi level is equivalent to the charge in the bulk. The magnetic properties are calculated by shifting the nonmagnetic LDOS for each class with different values of the exchange splitting (so that five curves are defined by $\mu = N_{\uparrow} - N_{\downarrow}$). These curves are interpreted by the spin magnetic moment defined in Eq. (5) with the same Coulomb parameter $U = 5.93$ eV (Fig. 7). The procedure yields a spin magnetic moment depending on the coordination depicted in Figure (6). A comprehensive overview of the results is provided in Tables VII and VIII.

<figure><img src="image_7.png" /><figcaption>FIG. 6. Spin magnetic moment depending on the coordination number and the size for fcc Co cuboctahedrons calculated using $U_d = 5.93$ eV.</figcaption></figure>

Depending on their size, magnetic nanoparticles exhibit oscillatory behavior of the spin magnetic moment [21]. This size effect can occur in this elementary model. This oscillation can also be observed in the variation of the work function depending on the size of the particle. Nonetheless, the surface energy is dependent on the nanoparticle's size without a significant oscillation.
<table><caption>TABLE VI. Work functions [eV] for the Fe, Co and the Nickel</caption><thead><tr><th></th><th>Fe (100)</th><th>Fe (110)</th><th>Co (100)</th><th>Co (111)</th><th>Ni (100)</th><th>Ni (111)</th><th>Pt(111)</th></tr></thead><tbody><tr><th>W (Non mag.)</th><td>6.02</td><td>4.93</td><td>6.40</td><td>5.48</td><td>5.52</td><td>4.79</td><td>6.24</td></tr><tr><th>W (Ferro)</th><td>5.70</td><td>4.69</td><td>6.16</td><td>5.30</td><td>5.49</td><td>4.78</td><td>-</td></tr><tr><th>W (Expt)</th><td></td><td>4.17 [19]</td><td></td><td>5.00 [20]</td><td></td><td>5.15 [20]</td><td>5.65 [20]</td></tr></tbody></table>

<table><caption>TABLE VII. Magnetic moment, work function, surface tension for a nanoparticle of 1415 atoms</caption><thead><tr><th></th><th>Bulk</th><th>Vertexes</th><th>Edges</th><th>(100)</th><th>(111)</th></tr></thead><tbody><tr><th>μ [μ<sub>B</sub>]</th><td>1.60</td><td>1.93</td><td>1.88</td><td>1.86</td><td>1.76</td></tr><tr><th>W [eV]</th><td>-</td><td>7.99</td><td>6.59</td><td>6.09</td><td>5.30</td></tr><tr><th>γ [eV]</th><td>-</td><td>1.90</td><td>1.31</td><td>1.09</td><td>0.81</td></tr></tbody></table>

<table><caption>TABLE VIII. Magnetic moment, work function, surface tension for a nanoparticle of 309 atoms</caption><thead><tr><th></th><th>Bulk</th><th>Vertex</th><th>Edges</th><th>(100)</th><th>(111)</th></tr></thead><tbody><tr><th>μ [μ<sub>B</sub>]</th><td>1.56</td><td>1.89</td><td>1.85</td><td>1.81</td><td>1.74</td></tr><tr><th>W [eV]</th><td>-</td><td>7.64</td><td>6.43</td><td>5.89</td><td>5.05</td></tr><tr><th>γ [eV]</th><td>-</td><td>1.79</td><td>1.26</td><td>1.02</td><td>0.72</td></tr></tbody></table>

## IV. CONCLUSION

Nowadays, we find nanoparticles in several fields and the methods to study their properties are crucial. Unfortunately, the most efficient approach using an *ab initio* calculation is limited to about hundreds of atoms. In this work, we have introduced a tight-binding approximation which encompasses the correlations and which allows to determine the magnetic and surface proprieties by just applying a rule of charge neutrality. This method shows its efficiency by computing values close to a density functional theory (DFT) calculation and experimental results. This approach can be applied to obtain some important properties of large size magnetic or non-magnetic nanoparticles without requiring the total energy. But the method can be extended to the calculation of the total energy to study more effectively certain phenomena like the relaxation and the reconstruction. The calculation of the electronic structure in this work being done in the real space, we can apply the model to non crystalline materials or structures with defects and distortions.

[1] J. R. Eone II, (2018), arXiv:1809.06096 [cond-mat.mtrl-sci].

[2] W. Kohn and L. J. Sham, Phys. Rev. **140**, A1133 (1965).

[3] L. Zosiak, C. Goyhenex, R. Kozubski, and G. Tréglia, Journal of Physics: Condensed Matter **27**, 455503 (2015).

[4] A. Jaafar, C. Goyhenex, and G. Tréglia, Journal of Physics: Condensed Matter **22**, 505503 (2010).

[5] M. Sansa, A. Dhouib, F. Ribeiro, B. Legrand, G. Tréglia, and C. Goyhenex, Modelling and Simulation in Materials Science and Engineering **25**, 084004 (2017).

[6] Y. Takahashi, *Spin Fluctuation Theory of Itinerant Electron Magnetism* (Springer, 2013) p. 12.

[7] M. Aldén, S. Mirbt, H. L. Skriver, N. M. Rosengaard, and B. Johansson, Phys. Rev. B **46**, 6303 (1992).

[8] C. Massobrio, H. Bulou, and C. Goyhenex, *Atomic-Scale Modeling of Nanosystems and Nanostructured Materials* (Springer, 2010) p. 127.

[9] C. Mottet, G. Tréglia, and B. Legrand, Surface Science **352-354**, 675 (1996), proceedings of the 15th European Conference on Surface Science.

[10] C. Goyhenex, G. Tréglia, and B. Legrand, Surface Science **646**, 261 (2016).

[11] N. D. Lang and W. Kohn, Phys. Rev. B **3**, 1215 (1971).

[12] D. A. Papaconstantopoulos, *Handbook of the Band Structure of Elemental Solids From Z = 1 To Z = 112* (Springer, 2015).

[13] W. Tyson and W. Miller, Surface Science **62**, 267 (1977).

[14] G. Ritz, M. Schmid, P. Varga, A. Borg, and M. Rønning, Phys. Rev. B **56**, 10518 (1997).

[15] L. Vitos, A. Ruban, H. Skriver, and J. Kollar, Surface Science **411**, 186 (1998).

[16] P. Parida, B. Ganguli, and A. Mookerjee, (2014), arXiv:1410.3185 [cond-mat.mtrl-sci].

[17] F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K. Niessen, *Cohesion in Metals* (North-Holland, 1988).

[18] E. A. Clark, R. Yeske, and H. K. Birnbaum, *Metallurgical Transactions A **11***, 1903 (1980).

[19] R. E. Simon, Phys. Rev. **116**, 613 (1959).

[20] H. B. Michaelson, Journal of Applied Physics **48**, 4729 (1977).

[21] J. L. Rodríguez-López, F. Aguilera-Granja, K. Michaelian, and A. Vega, Phys. Rev. B **67**, 174413 (2003).
<table><caption>TABLE VI. Work functions in eV for Fe, Co and Ni.</caption><thead><tr><th></th><th>Fe (100)</th><th>Fe (110)</th><th>Co (100)</th><th>Co (111)</th><th>Ni (100)</th><th>Ni (111)</th><th>Pt(111)</th></tr></thead><tbody><tr><th>W (Nonmag.)</th><td>6.02</td><td>4.93</td><td>6.40</td><td>5.48</td><td>5.52</td><td>4.79</td><td>6.24</td></tr><tr><th>W (Ferro)</th><td>5.70</td><td>4.69</td><td>6.16</td><td>5.30</td><td>5.49</td><td>4.78</td><td>-</td></tr><tr><th>W (Expt)</th><td></td><td>4.17 [19]</td><td></td><td>5.00 [20]</td><td></td><td>5.15 [20]</td><td>5.65 [20]</td></tr></tbody></table>

<table><caption>TABLE VII. Spin magnetic moment, work function, surface energy for a fcc Co nanoparticle of 1415 atoms.</caption><thead><tr><th></th><th>Bulk</th><th>Vertices</th><th>Edges</th><th>(100)</th><th>(111)</th></tr></thead><tbody><tr><th>μ [μ<sub>B</sub>]</th><td>1.60</td><td>1.93</td><td>1.88</td><td>1.86</td><td>1.76</td></tr><tr><th>W [eV]</th><td>-</td><td>7.99</td><td>6.59</td><td>6.09</td><td>5.30</td></tr><tr><th>γ [eV]</th><td>-</td><td>1.90</td><td>1.31</td><td>1.09</td><td>0.81</td></tr></tbody></table>

<table><caption>TABLE VIII. Spin magnetic moment, work function, surface energy for a fcc Co nanoparticle of 309 atoms.</caption><thead><tr><th></th><th>Bulk</th><th>Vertices</th><th>Edges</th><th>(100)</th><th>(111)</th></tr></thead><tbody><tr><th>μ [μ<sub>B</sub>]</th><td>1.56</td><td>1.89</td><td>1.85</td><td>1.81</td><td>1.74</td></tr><tr><th>W [eV]</th><td>-</td><td>7.64</td><td>6.43</td><td>5.89</td><td>5.05</td></tr><tr><th>γ [eV]</th><td>-</td><td>1.79</td><td>1.26</td><td>1.02</td><td>0.72</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>FIG. 7. Calculation of the spin magnetic moment for a fcc Co nanoparticle.</figcaption></figure>

IV. CONCLUSION

In the prevailing context, the presence of nanoparticles in diverse fields has become pervasive. Consequently, the development of methodologies for the study of their properties is imperative. It is unfortunate that the most efficient approaches using *ab initio* calculations are limited to approximately hundreds of atoms. In this work, a tight-binding approach encompassing the correlations and enabling the determination of magnetic and surface properties through the application of a charge neutrality rule was introduced. The efficacy of the method is demonstrated by its capacity to produce values that are close to those obtained from density functional theory (DFT) calculations and experimental results. This approach facilitates the calculation of the fundamental properties of magnetic and non-magnetic nanoparticles on a large scale, obviating the necessity of computing the total energy. However, the method can be extended to include the total energy, thereby enabling a more effective study of certain phenomena, such as relaxation and reconstruction. In this study, the calculation of the electronic structure was conducted in real space, thereby enabling the application of the model to non-crystalline materials or structures that exhibit defects and distortions.

[1] J. R. Eone II, (2018), arXiv:1809.06096 [cond-mat.mtrl-sci].

[2] W. Kohn and L. J. Sham, Phys. Rev. **140**, A1133 (1965).

[3] L. Zosiak, C. Goyhenex, R. Kozubski, and G. Tréglia, Journal of Physics: Condensed Matter **27**, 455503 (2015).

[4] A. Jaafar, C. Goyhenex, and G. Tréglia, Journal of Physics: Condensed Matter **22**, 505503 (2010).

[5] M. Sansa, A. Dhouib, F. Ribeiro, B. Legrand, G. Tréglia, and C. Goyhenex, Modelling and Simulation in Materials Science and Engineering **25**, 084004 (2017).

[6] Y. Takahashi, *Spin Fluctuation Theory of Itinerant Electron Magnetism* (Springer, 2013) p. 12.

[7] M. Aldén, S. Mirbt, H. L. Skriver, N. M. Rosengaard, and B. Johansson, Phys. Rev. B **46**, 6303 (1992).

[8] C. Massobrio, H. Bulou, and C. Goyhenex, *Atomic-Scale Modeling of Nanosystems and Nanostructured Materials* (Springer, 2010) p. 127.

[9] C. Mottet, G. Tréglia, and B. Legrand, Surface Science **352-354**, 675 (1996), proceedings of the 15th European Conference on Surface Science.

[10] C. Goyhenex, G. Tréglia, and B. Legrand, Surface Science **646**, 261 (2016).

[11] N. D. Lang and W. Kohn, Phys. Rev. B **3**, 1215 (1971).

[12] D. A. Papaconstantopoulos, *Handbook of the Band Structure of Elemental Solids From Z = 1 To Z = 112* (Springer, 2015).

[13] W. Tyson and W. Miller, Surface Science **62**, 267 (1977).
[14] G. Ritz, M. Schmid, P. Varga, A. Borg, and M. Rønning, Phys. Rev. B **56**, 10518 (1997).

[15] L. Vitos, A. Ruban, H. Skriver, and J. Kollar, Surface Science **411**, 186 (1998).

[16] P. Parida, B. Ganguli, and A. Mookerjee, (2014), arXiv:1410.3185 [cond-mat.mtrl-sci].

[17] F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K. Niessen, *Cohesion in Metals*

(North-Holland, 1988).

[18] E. A. Clark, R. Yeske, and H. K. Birnbaum, Metallurgical Transactions A **11**, 1903 (1980).

[19] R. E. Simon, Phys. Rev. **116**, 613 (1959).

[20] H. B. Michaelson, Journal of Applied Physics **48**, 4729 (1977).

[21] J. L. Rodríguez-López, F. Aguilera-Granja, K. Michaelian, and A. Vega, Phys. Rev. B **67**, 174413 (2003).