An efficient magnetic tight-binding method for transition metals and alloys
Cyrille Barreteau a, b, ⁎
 , Daniel Spanjaard c , Marie-Catherine Desjonquères a
a SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France
b DTU NANOTECH, Technical University of Denmark, Ørsteds Plads 344, DK-2800 Kgs. Lyngby, Denmark
c Laboratoire de physique des solides, Université Paris-Sud, bâtiment 510, 91405 Orsay cedex, France
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Abstract
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An
efficient parameterized self-consistent tight-binding model for
transition metals using s, p and d valence atomic orbitals as a basis
set is presented. The parameters of our tight-binding model for pure
elements are determined from a fit to bulk ab-initio
calculations. A very simple procedure that does not necessitate any
further fitting is proposed to deal with systems made of several
chemical elements. This model is extended to spin (and orbital)
polarized materials by adding Stoner-like and spin–orbit interactions.
Collinear and non-collinear magnetism as well as spin-spirals are
considered. Finally the electron–electron intra-atomic interactions are
taken into account in the Hartree–Fock approximation. This leads to an
orbital dependence of these interactions, which is of a great importance
for low-dimensional systems and for a quantitative description of
orbital polarization and magneto-crystalline anisotropy. Several
examples are discussed.
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Résumé
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Nous
présentons un modèle de liaisons fortes paramétré et auto-cohérent
utilisant une base d'orbitales atomiques s, p, et d pour décrire les
électrons de valence des métaux de transition. Les paramètres du modèle
sont déterminés à partir d'un ajustement non linéaire sur des résultats
de calculs ab initio d'éléments purs en volume. Notre procédure
ne nécessite aucun paramètre ni ajustement supplémentaire pour l'étendre
aux systèmes avec plusieurs atomes de natures chimiques différentes.
Nous avons généralisé notre modèle aux matériaux présentant une
polarisation de spin et orbitale à l'aide de termes de Stoner et de
couplage spin–orbite. Nous traitons aussi bien le magnétisme colinéaire
que non colinéaire ainsi que les spirales de spin. Enfin nous montrons
comment prendre en compte l'interaction électron–électron intra-atomique
dans l'approximation de Hartree–Fock. Cela introduit une dépendance
orbitale des interactions qui peut s'avérer importante dans les systèmes
de basse dimensionalité et pour décrire correctement l'anisotropie
magnéto-cristalline et la polarisation orbitale. Nous illustrons notre
propos à l'aide de plusieurs exemples.
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Introduction
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Even
though Density Functional Theory has become an extremely efficient
method widely used in many areas of physics, chemistry, and material
science, the tight-binding (TB) description of the electronic structure
remains very popular, since it provides a physically transparent
interpretation in terms of orbital hybridization and bond formation. In
addition, its moderate computational cost permits to handle rather large
and complex systems, and its straightforward implementation allows many
generalizations and applications. In addition, in recent years, with
the increasing interest in electronic transport and the explosion of
studies in graphene nanostructures, there has been a renewal of interest
for TB calculations.
Historically the TB method was introduced by Slater and Koster [[1]].
It was originally thought as a semi-empirical model to describe the
electronic structure of solids with a reasonably small number of
parameters that can provide reliable semi-quantitative results when
these parameters are determined from a fit on first-principles
calculations. Jacques Friedel in the 1960s was one of the pioneers in
the application of TB to the physics of transition metals [[2], [3]].
These models were essentially based on a physical description of the
band structure, but no real arguments about the total energy were
developed. Qualitative explanations of the trends in the variation of
the total energy when some parameters are varied (number of d electrons,
concentration) were proposed, but only based on the band contribution
to the total energy. Jacques Friedel was particularly talented in
developing simple models with a simplified schematic description of the
electronic density of states (such as the rectangular band model [[4]])
that could nevertheless describe surprisingly well many physical
properties of materials. Later on, physicists started to add a
phenomenological repulsive pair-potential to the band energy [[5]].
It was however not very clear what was “hidden” behind this
phenomenological term. Over the years, TB methods have acquired a more
solid fundamental basis. In particular, with the work of Harris and
Foulkes [[6], [7]],
it was shown how a tight-binding formalism can be derived from Density
Functional Theory. Nowadays, there exist many electronic structure codes
based on various versions of TB with different degrees of
approximations [[8], [9], [10]].
Very early, TB in combination with a Stoner-like model [[11]]
was recognized as an adequate tool to describe magnetism in transition
metals, and Jacques Friedel was indeed very active in this field [[12], [13]].
Indeed, magnetism in a crystal is intimately related to its band
structure. TB has been applied to a large variety of magnetic systems in
various crystallographic structures, dimensionalities (from bulk to
clusters), ordered alloys or presenting some kind of disorder [[14]].
It is not the goal of this paper to provide an exhaustive presentation
of this wealth of research in magnetism. We will rather concentrate on
the presentation of a TB model that we have developed over the years and
that is able to describe accurately and efficiently a wide range of
magnetic phenomena and materials. It is an empirical TB method with
parameters fitted on ab-initio data. We will show how, with a
limited number of simple and well controlled approximations, we have
been able to generalize our model to alloys and include non-collinear
magnetism, spin-spirals as well as spin–orbit coupling.
The
paper is organized as follows: we will present the general concepts of
the tight-binding description of electronic structure and its
implementation in an s, p and d atomic orbital basis set for
non-magnetic materials (Section 2).
We will pay particular attention to describe properly features that are
often not discussed thoroughly in publications: non-orthogonality of
the TB basis set, self-consistent treatment, proper definition of local
quantities, etc. Then, in Section 3, we will show how, using a simple Stoner model, spin-polarization can be included, first for collinear magnetism (Section 3.1), then for non-collinear configurations (Section 3.2). Spin–orbit coupling and magneto-crystalline anisotropy will also be discussed in detail. Section 3
will be ended by a discussion of more elaborated Hartree–Fock like
Hamiltonians that can play an important role in low-dimensional or
anisotropic systems. Finally we will draw conclusions in Section 4.
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