Theoretical Materials Science

Syllabus

A) Quantum Mechanics

Background

Waves and Particles, Hilbert space, Hermitian operators and observables, operator algebra, the postulates of Quantum mechanics.

The basic theorems

Eigenvalues of position, energy and momentum, Ehrenfest theorem, Uncertainty principle.

Harmonic Oscillator:

Algebraic method, eigenstates and eigenvalues, applications.

Angular momentum

Ladder operators, eigenvalues and eigenstates for rotations and for spin, addition of angular momenta, spin-orbit coupling, singlet-triplet states.

Many-body systems:

Spin-statistics theorem, Slater Determinants, Pauli principle, exchange and correlation effects, Density-Functional Theory.

B) Quantum Theory of Solids

From atoms (ions – external electrons) to solids

Equilibrium structures at minimum energy, Coulomb potential energy and quantum kinetic energy, Heisenberg’s uncertainty principle and minimum kinetic energy. Atoms size and energy, formation of molecules and solids, estimate of values for basic properties of elemental solids using fundamental principles and dimensional analysis.

Electron motion, the problem of electrical resistivity and the basic approximations in solid state physics

Drude formula. Atomic structure and bonding and the properties of solids, crystal lattices. Adiabatic (Born-Oppenheimer), independent particle, harmonic approximation. Periodicity, Bloch theorem, origin of energy bands and gaps. One-dimensional “crystals”, chain of classical coupled harmonic oscillators and phonon bands.

Linear combination of atomic orbitals

Diatomic molecule. One-dimensional tight-binding model, bands and gaps for more than one orbital per atom, diatomic unit cell. Analogy with wave propagation in one-dimensional “crystals” of coupled harmonic oscillators and phonons.

Semiclassical theory of conduction in metals

Electrons in a conduction band, free electron model, Fermi energy, total (kinetic) energy, density of states, effective mass, DC conductivity, materials response to EM field, oscillators model for conductivity and dielectric function, properties and uses of dielectric function.

Conduction in semiconductors

Electrons and holes, carrier effective mass, intrinsic semiconductor conductivity, carrier mobility and concentration, temperature dependence, chemical potential and Fermi energy. Impurities, donors and acceptors, Fermi level in extrinsic semiconductors, temperature dependence, carriers lifetime.

Learning Outcomes

The course is an introduction to the relationship between the atomic/electronic structure of solid materials and their macroscopic properties as well as the properties that render them invaluable in modern technology. The first part of the course is basic quantum mechanics and the second part is an introduction to the basic principles of solid-state physics. The course covers topics such as the postulates of quantum mechanics and implications, the relation between atomic configuration and electronic structure (electronic energy states, bands and gaps), how this determines conductors, semiconductors and insulators, the interaction of materials with the electromagnetic field. The learning goals that should have been achieved by the end of the course are:

1. Students learn the basics of quantum theory of matter.
2. Students understand the role of quantum theory in the stability of solids as well as their mechanical and electronic properties.
3. Students should be able to explore the interaction of materials with electromagnetic fields.
4. Students become familiar with the most important aspects of the electronic properties of materials so that they can understand the design and operation of electronic devices.

The course according to the European Qualifications Framework for Lifelong Learning belongs to level 7.

Recommended Bibliography

• Eugen Merzbacher, Quantum Mechanics, 3rd Edition, John Wiley & Sons (1998).
• Stefanos Trachanas, An Introduction to Quantum Physics: A First Course for Physicists, Chemists, Materials Scientists, and Engineers, 1st Edition, Wiley-VCH (2017).
• Charles Kittel, Kittel’s Introduction to Solid State Physics, Global Edition, 8th Edition, Wiley (2018).
• E.N. Economou, The Physics of Solids Essentials and Beyond, 1st Edition, Springer (2010).
• W.D. Callister, Jr., D.G. Rethwisch, Materials Science and Engineering: An Introduction, 10th Edition, Wiley (2018).
• I. Harald, L. Hans, Solid-State Physics: An Introduction to Principles of Materials Science, 4th Edition, Springer (2009).