Theory and Modelisation of Electronic Structures of materials

Mebarek Alouani (IPCMS)

Due to recent progress in high performance computing, ab initio electronic structure methods can now describe and predict the electronic properties of materials with high accuracy. These methods of calculating the electronic structure of materials are therefore widely used in physics, chemistry and biology. The main objective of this course is to teach students these modern methods, which are widely used for the computation of the electronic properties of materials.

In this course we will first learn that chemical bonds and magnetism are non classical and that their origins derive from the symmetry of the wave functions of identical particles. This notion of symmetry permits the understanding of the Heisenberg Hamiltonian and Hund’s rules. The physics of the identical particles and the use of the mean-field approximation give a qualitative description of the electronic structure of atoms, molecules and solids, by solving numerically the Schrödinger equation. We will becomes also familiar with the effects of the octahedral and tetrahedral crystal field on the magnetism of transition metal oxides, e.g., super-exchange and double-exchange lead to antiferromagnetic and ferromagnetic coupling, respectively. We will also learns how a paramagnetic system acquires a spontaneous magnetization due to the competition between the exchange field and the kinetic energy. This first part of the course will end with the resolution of the Schrödinger equation of a solid in the almost free-electron approximation and will be illustrated by computing the aluminum band structure.

The density functional theory will be the subject of the second part of this course. We will first integrate the notion of Hartree-Fock approximation of an electron gas in an external potential, which will facilitate the understanding of the electronic structure of atoms, molecules and solids. The introduction of the density functional theory of Hohenberg and Kohn (DFT) [1] and the notion of effective potential for solving the Kohn-Sham equations [2] will thus be facilitated. The DFT will then be generalized to spin-polarized fermions. Other functionals will be also introduced, in particular the generalized gradient approximation [3], the so-called DFT + U functionals, where the Hubbard U interaction is added to correlated sites [4], and so-called hybrid functionalities. In addition to this chapter on DFT, we will be introduced to the pseudo- potential method[7] and all-electron methods, such as the projected plane wave (PAW) method [8], and the augmented plane wave (LAPW)method [9]. Spin-orbit coupling (SOC) and its importance in magnetism will be illustrated by some applications for non-collinear magnetism, spin waves, Curie temperature, magnetic anisotropy [5], and magnetic circular X-ray dichroism [6]. The course will finish by introducing the empirical tight-binding method[10] and its illustration for some typical examples, such as the calculation of the electronic structure of chains of atoms with or without Peierls distortion, graphene, and cubic materials.

Program :

I. Identical particles

  1. Origin of the chemical bond and magnetic coupling
  2. Schödinger equation and Dirac equation
  3. Axiom of spin-statistics: Fermions and Bosons
  4. Heisenberg model
  5. Hund’s rules
  6. Numerical solution of the Schrödinger equation for atoms
  7. Electronic structures of atoms, molecules and solids.
  8. Electronegativity of Pauling/Mulliken and charge transfer in molecules and solids

II. Theory of the electronic structure

  1. The periodic potential
  2. Bloch’s theorem
  3. Born von Karman conditions
  4. Bloch functions in plane waves
  5. Calculation of the density of electronic states: tetrahedron method.
  6. Solution of the Schrödinger equation for weak potential
  7. Eigenvalues and eigenvectors
  8. Calculation of the electronic structure of aluminum

III. Crystal field and Stroner model

  1. Octaedric and tetrahedral crystalline fields
  2. Coulomb energy / crystalline field
  3. Jahn-Teller Distortions
  4. Colors of minerals.
  5. Super Exchange and Double Exchange
  6. Magnetic orders in molecules and solids
  7. Spontaneous magnetization in paramagnetic metals

IV. Hartree and Hartree-Fock methods

  1. Adiabatic approximation
  2. Hartree and Hartree- Fock (HF) approximation for the many-body problem
  3. The variational principle
  4. The exchange hole
  5. HF symmetry and spin polarization
  6. Koopmans theorem
  7. Practical implementation of Hartree-Fock for molecules

V. Density Functional Theory

  1. The theorems of Hohenberg and Kohn
  2. The Kohn-Sham method
  3. The variational principle
  4. The exchange hole and correlation
  5. The polarization of spins
  6. Functional of Exchange and Correlation: Local density approximation
  7. Functional Exchange and Correlation: RPA and GGA approximations
  8. Beyond semi-local approximations: hybrid functionals , LDA + U
  9. Examples and Results
  10. Spin-Orbit Coupling: Magnetic Anisotropy and Magnetic Dichroism of X-rays

VI. Tight binding method

  1. Why the tight-binding method
  2. Combinations of atomic orbitals
  3. Different orbitals for each type of material: metal / semiconductor
  4. The Slater-Koster integrals and parameters of the tight-binding method
  5. The symmetry of the Slater-Koster matrix
  6. Hybridization and basic construction by symmetry
  7. Calculation of a molecular chain and a periodic chain
  8. Effect of the approximation beyond first neighboors
  9. 2D and 3D cases
  10. Origin of the band gap and its adjustment in semiconductors and insulators
  11. Peierls distortion in periodic chains
  12. s-p ybridization : silicon case
  13. Example of graphene band structure
  14. General Formulation: multi-atom material
  15. How to write a tight-binding program

VII. Pseudo-potential method and all-electron methods

  1. Core electrons and valence electrons
  2. Difficulties of plane wave approach to solve Kohn-Sham equations
  3. The all electron LAPW and PAW methods
  4. Herring’s idea, exploited by Phillips and Kleinman
  5. Norm conserving pseudopotentials of Hamann, Schlüter, and Chiang
  6. Non-Local Pseudopotentials of Kleinman Bylander
  7. Vanderbilt’s soft pseudopotentials

Objectives in terms of skills: This course (theory of electronic structure) allows the student to become familiar with the modern methods for computing the electronic structure of materials. Its objectives in terms of skills are to help the students understand the foundations of the density functional theory and the different implementations of Kohn-Sham equations, such as the pseudo-potential and all-electron methods (LAPW, PAW). This course introduces the different approximations to the exchange-correlation potential and prepares the student to apply these modern methods to different problems in condensed matter, chemistry, and biology.

Perequisites: To take full advantage of this course, the student must master the basics in quantum mechanics and solid-state physics.

Bibliography

[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
[2] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965); R.G. Parr and W. Yang, Density- Functional Theory of Atoms and Molecules, Oxford University Press, 1989.
[3] J. P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, 16533 (1996).
[4] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991).
[5] I. Galanakis, M. Alouani, and H. Dreyssé, Phys. Rev. B 62, 6475 (2000).
[6] V. Kanchana, G. Vaitheeswaran, M. Alouani, and A. Delin, Phys. Rev. B 75(R), 220404 (2007). [7] Electronic Structure: Basic Theory and Practical Methods, by R. M. Martin, Cambridge University Press, 2004.
[8] P. E. Bloechl, Phys. Rev. B 50, 17953 (1994).
[9] Planewaves, Pseudopotentials, and the LAPW method, by David Singh and Lars Nordstrom, Springer.

[10] Simplified LCAO Method for the Periodic Potential Problem, J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).