Objective aims
Acquisition of the fundamentals of solid state structure
Knowledge and understanding
By attending the course students will:
know the basics of electronic structure of the solid state
know the simplest models to approach the quantitative description of the behavior of electrons in solids
know the basic mathematical tools used for the study of the electronic structure of solids
Applying knowledge and understanding
A student who has profitably attended the course will be able to:
explain the simplest properties of metals and crystalline solids in terms of their electronic structure
interpret the basic aspects of the band structure obtained from computations or experiments
Making judgements
The contents of the course are of mathematical-quantitative character, which force the student to develop the ability to understand the presented material in a thorough way, but autonomously and consciously.
Communication skills
The material presented in the classroom requires the introduction and use of a specific and rigorous language: mastering this language is one of the targets that students will be required to pursue.
Learning skills
The course is an elementary approach to the structure of the solid state; nonetheless, the concepts introduced in the classroom will form a solid cultural base allowing the students to develop more deeply specific arguments related to experimental and/or theoretical-computational problems arising in this field.

The course does not have any formal prerequisite. However, given the mathematical nature of the arguments that will be discussed, it will be useful for the students if they already have a basic knowledge of analysis which includes derivatives, integrals, functions of single/multiple variables, simple differential equations as well as elementary concepts of quantum mechanics.

Brief introduction to waves Periodic functions in one dimension Sinusoidal functions The complex exponential Stationary and traveling waves Waves in 2 and 3 dimensions Plane waves Introduction to Fourier series Heuristic derivation of the Fourier series Exponential form of the Fourier series Elementary properties of the Fourier series Fourier series in 2 and 3 dimensions The reciprocal lattice and its fundamental properties The free electron model for metals Fundamental assumptions of the model The solution of the Schrodinger equation The boundary conditions of Born and von Karman k space The allowed wave vectors and the dispersion relation Fermi-Dirac distribution The density of states Fundamentals of crystallography The structure of crystals Bravais lattices Primitive, unit, conventional and Wigner-Seitz cell Lattices with a basis Supercells Connection between direct and reciprocal lattice Miller indexes Diffraction Bragg and von Laue conditions Ewald sphere Electronic states in the presence of a periodic potential The Schrodinger equation with a periodic potential The Bloch theorem The central equation The empty lattice model and the band structure The nearly free electrons model and the band gap The physical origin of the band ga The tight-binding model Conductors, semiconductors and insulators

N.W.Ashcroft, N.D.Mermin, Solid State Physics, 1976, Harcourt Brace College Publishers
C.Kittel, Introduction To Solid State Physics, 1996, John Wiley & Sons, New York, 7th ed.
H.Ibach, H.Luth, Solid-State Physics: An Introduction to Principles of Materials Science , 2003, Springer, Berlin, 3rd ed.
A.P. French, Vibrations and Waves, M.I.T. Introductory Physics
R.G. Mortimer, Physical Chemistry, 2000, Academic Press, S.Diego, 2nd ed.

Periodic functions, Fourier series and waves
Periodic functions in one dimension
Sinusoidal functions
The complex exponential
Heuristic derivation of the Fourier series
Equivalent forms of the Fourier series
Exponential form of the Fourier series
Fourier series in two and three dimensions
The reciprocal lattice and its fundamental properties
Elementary properties of the Fourier series
Stationary and traveling waves
Waves in two and three dimensions
Plane waves
Complex exponential form of sinusoidal waves
The description of the structure of the crystalline solids
Bravais lattices
Primitive cell
Supercells
Wigner-Seitz primitive cell
Classification of the Bravais lattice
More about the reciprocal lattice
Brillouin zones
Lattice planes and reciprocal vectors
Miller indices
Brief introduction to diffraction theory
The diffraction model of Laue
The Laue condition
Diffraction and Brillouin zones
The Fermi gas model
Basic assumptions of the model
The solution of the Schrodinger equation
Periodic boundary conditions of Born e von Karman
The k space
The allowed wave vectors and the dispersion relation
The Fermi-Dirac distribution
The density of states
The electronic states in the presence of a periodic potential
The Schrodinger equation with a periodic potential
The Bloch theorem
The central equation
Another proof of the Bloch theorem
Periodicity in k space
The band structure
The “empty lattice” model
The band structure in the empty lattice model
The band structure representation in two or three dimensions
The Fermi surface
Density of states and band structure
The “nearly free electrons” model and the band gap
A mathematical explanation of the band gap
A physical explanation of the band gap
Band gap and diffraction
Band gaps in two or three dimensions
2D or 3D Fermi surface in the nearly free electrons model
The “tight binding” model
An “exaggerated” crystal
Electrons in a real crystal and the basis of the tight binding model
The tight binding Hamiltonian
A general expression for the energy
The application of the tight binding approximations
The bands in the tight binding model
Conductors, semiconductors and insulators

The course consists of classroom lectures, during which all the presented theoretical aspects are derived and discussed on the blackboard. Lecture notes together with further teaching material are made available to the students from the web pages of the course

For any question you might have, please contact the lecturer by email

Individual exam. There will be at least 3 questions on the topics presented in classroom. Students are given the possibility to answer the questions orally or in writing. Students will have to demonstrate a good understanding of the fundamental concepts developed in class and to have understood the connections between them. The correct use of terminology and language proficiency will also be evaluated. The final mark will be out of thirty and the passing grade is greater than or equal to 18/30.
The evaluation grid adopted is as follows:
- Excellent (30 - 30 cum laude): excellent knowledge of the topics, excellent proficiency of the technical language, excellent analytical ability.
- Very good (27 - 29): good knowledge of the topics, notable fluency in technical language, good analytical ability.
- Good (24-26): good knowledge of the main topics, fair command of technical language; the student shows adequate analytical ability.
- Satisfactory (21-23): the student does not show full mastery of the main topics, despite possessing the fundamental knowledge; language skills are satisfactory and the analytical ability is sufficient.
- Sufficient (18-20): minimal knowledge of the main topics and technical language, limited analytical ability.
- Insufficient: the student does not have acceptable knowledge of the contents of the different topics of the program.