2024 Faculty Courses School of Materials and Chemical Technology Department of Materials Science and Engineering Graduate major in Materials Science and Engineering

Quantum theory of metals

Academic unit or major: Graduate major in Materials Science and Engineering
Instructor(s): Kan Nakatsuji / Yoshihiro Gohda
Class Format: Lecture (Face-to-face)
Media-enhanced courses: -
Day of week/Period (Classrooms): 3-4 Mon / 3-4 Thu
Class: -
Course Code: MAT.M430
Number of credits: 200
Course offered: 2024
Offered quarter: 3Q
Syllabus updated: Mar 14, 2025
Language: English

Syllabus

Course overview and goals

This lecture covers fundamentals in physics of metals, where electron theory as the basics of materials science is mainly discussed. The course teaches the electronic band theory from the basics which describes the electronic states of solids. The "nearly free-electron model" and the "tight-binding approximation" will be introduced as the simplest and most valuable models in the band theory. First-principles electron theory is discussed as one of the most important topics in advanced solid state physics. First-principles electron theory is cutting-edge non-empirical electron theory applied for numerical simulations. Hartree-Fock approximation is discussed, where the exchange effect coming from the Fermi-Dirac statistics is taken into account. Then, wave-function theory and density functional theory is introduced. These topics will help to understand quantum theory of metals.

Course description and aims

By completing this course, students will be able to:
1) Understand that the electronic states govern the material properties microscopically.
2) Understand the free-electron metallic states as the simplest itinerant electron system.
3) Understand that the electron states of solid crystals become Bloch states.
4) Understand that many-body effects among electrons reduce the Coulomb-repulsion energy.
5) Understand the basics of first-principles electron theory to describe electronic states non-empirically.

Keywords

crystal structure, reciprocal lattice, Brillouin zone, electronic band structure, nearly free-electron model, tight-binding approximation, phonons, first-principles electronic-structure theory, density functional theory, exchange interactions, metallic magnetism

Competencies

Specialist skills
Intercultural skills
Communication skills
Critical thinking skills
Practical and/or problem-solving skills

Class flow

At the beginning of the class (not each class), solutions to exercise problems assigned during the previous class are reviewed.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	Fermi-Dirac statistics and free electron gas	Explain how the energy of free-electron gas is described in Fermi-Dirac statistics.
Class 2	Space symmetry and crystal structures	Explain the Bravais lattice and typical lattice structures.
Class 3	Reciprocal lattice and Brillouin zone	Calculate the reciprocal lattice vectors of some typical lattices and draw the Brillouin zone.
Class 4	Bloch's theorem	Understand that the wave functions of valence band electrons in crystals obey Bloch's theorem.
Class 5	Nearly free electron model	Explain nearly free electron (NFE) model and the origin of the band gap.
Class 6	Fermi surface	Understand the relationship between the shape of the Fermi surface and crystal and electronic structures.
Class 7	Tight-binding model	Understand how the tight-binding approximation is applied to describe the electronic structure of materials.
Class 8	Phase equilibria	Understand the free energy and phase equilibria.
Class 9	Bose-Einstein statistics and quantum harmonic oscillators	Understand Bose-Einstein statistics and quantum harmonic oscillators.
Class 10	Phonons	Understand phonons by statistical physics.
Class 11	Force constants and electronic states	Understand relationship between force constants and electronic states.
Class 12	Effective spin-spin interactions	Understand exchange interactions.
Class 13	Density functional theory	Understand density functional theory.
Class 14	Ferromagnetism in metals	Understand first-principles electron theory of ferromagnetism in metals

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

Lecture notes are distributed if necessary.

Reference books, course materials, etc.

Kittel: Introduction to Solid State Physics,
R.M. Martin, Electronic structure,
Oshiyama et al.: Computations and Materials　(in Japanese),
A.A. Abrikosov: Fundamentals of the Theory of Metals.

Evaluation methods and criteria

Learning achievement is evaluated by reports or a final exam.

Related courses

MAT.M409 ： Thermodynamics for Phase Equilibria
MAT.A203 ： Quantum Mechanics of Materials
MAT.M202 ： Statistical Mechanics(M)
MAT.M206 ： Electronic Structure and Physical Properties of Metals
MAT.C414 ： Introduction to Solid State Science

Prerequisites

It is desirable that the students have learned basics of quantum mechanics.
Not available for those who are credited with MAT.M407-01 Advanced Solid State Physics a.