84535 - Symmetries, Electrons and Phonons

Learning outcomes

At the end of the course the student will learn the basic notions regarding: symmetries of the atomic structure of molecules and crystals and their description using group theory; electronic states in crystals in the independent electron approximation (band theory) and lattice vibrations in classical and quantum approaches.

Course contents

1. Atomic structure

1.1. Introduction to symmetry

• Introduction to symmetry in crystallography and basic elements of group theory for crystallography.

1.2. Crystal structures in 2D.

2D lattices. Primitive vectors. The five 2D lattices. Crystal structures: lattices with a basis. Unit cells: primitive, conventional, Wigner&Seitz. Space group.

1.3. 3D crystal structures
Monoatomic lattices. Coordination number. FCC and BCC lattices: conventional, primitive and Wigner&Seitz unit cell. Hexagonal lattice and HCP structure. Stacking of spheres. Selected examples: diamond, NaCl, CaF2, zincblende, wurtzite, perovskite. The 14 Bravais lattices and the 7 crystal systems. The 32 crystallographic point groups and the 230 space groups.

1.4. Determination of crystal structure by scattering of waves and particles; the reciprocal lattice.
X-rays, electrons and neutrons. Energy – wavelength relations for x-rays, neutrons and electrons and practical sources for diffraction experiments. Approximations adopted: fixed atoms, elastic scattering, single scattering. Scattering from an atom in the origin and in an arbitrary position. Scattering from an ensemble of atoms. Atoms on a lattice, lattice sums. Laue conditions. The reciprocal lattice. Lattice planes and Miller indeces. Lattice directions. Relation between lattice planes and reciprocal lattice vectors. The 1st Brillouin zone; relation between the volume of primitive unit cells in real and reciprocal space. Diffraction from a general crystal structure, the geometrical structure factor. The Ewald sphere. Brief mention of various scattering geometries. Effect of lattice vibrations: the Debye Waller factor.

 

2. Electronic structure

2.1. The free electron gas
Basic Hamiltonian for condensed matter. The free electron gas: static lattice and independent electron approximations. Plane wave solution for the one – electron Schrödinger equation. Born – von Karman boundary conditions. Construction of the ground state. The Fermi sphere, surface, energy, wave vector and temperature; numerical estimates. Density of states. Specific heat. Comparison with experiments.

 

2.2. Non interacting electron in a periodic potential

• Consequences of symmetry on electron states General properties of electron states in a periodic potential. Ideal periodicity, existence and importance of lattice defects and impurities. Bloch’s theorem. Born - von Karman boundary conditions and Bloch wave vectors. Expansion of periodic functions in plane waves (in real and reciprocal space). Band index and crystal momentum. Band structure. First Brillouin zone. Extended, reduced and repeated schemes for band structure. The ground state, differences between metals and non metals. The density of states and van Hove singularities. Velocity of Bloch electrons.

• Independent electrons in a weak periodic potential. Calculation of the effect of a weak periodic potential using second order perturbation theory: non degenerate and degenerate case. Energy gap and Bragg planes; constant energy surfaces near Bragg planes. One and three dimensional band structure representation. The Fermi surface. Higher order Brillouin zones. The copper Fermi surface. Fermi surface for a metal with valence 4. Effect of an atomic base. Analogy with x-ray scattering.

• Tight binding. Tight binding Hamiltonian. Wannier functions. Linear combinations of neraly degenerate atomic orbitals. Eigenvalue equation. The case of an s band for an FCC structure. Bandwidth and overlap integrals. Orthogonality of Wannier unctions based on diferent lattice sites.

2.3. Complements on electronic structure

• Overview of band structures. Band structures, charge densities and density of states in selected cases. 3D crystals: nearly free electron metals (Al), selected FCC and BCC metals, ionic insulators and covalent solids, semiconductors (Si, Ge, GaAs, others). 2D crystals: graphene and BN, silicene and MoS2.
• Photoemission. The work function. Experimental scheme. Initial and final states in many body and one electron schemes. Angle integrated photoemission, density of states. Angle resolved photoemission, band structure determination; example: graphite.

 

2.4 Selected examples of band properties and electronic instabilities

• Charge density wave in metals, topological properties, superconductivity

 

3. Cohesion, vibrations and phonons

3.1 Classification and cohesion of solids

Classification of solids: molecular, ionic, covalent, metallic and hydrogen bonded. Cohesive energies. Atomic radii. Noble gasses and the Lennard Jones potential. Ionic crystals and the Madelung constant. Cohesion in covalent and hydrogen bonded solids.

 

3.2 Failures of the static lattice model.

Equilibrium properties: T^3dependence of the specific heat, existence of thermal expansion. Transport properties: finite electrical and thermal conductivity, deviations from the Weideman-Franz law, existence of superconductivity, thermal conductivity of electrical insulators, transmission of sound. Interaction with radiation: infrared reflectivity of ionic crystals, inelastic scattering of light, neutrons and x-rays. Reduced amplitude of elastic scattering x-ray peaks.

 

3.3 Adiabatic approximation and revision on molecular vibrations.

Separation of the Schrödinger equation in two equations describing electronic states and nuclear motion; the electronic energy eigenvalue as potential energy for the motion of the nuclei. Harmonic approximation. Quick revision on rotations and vibrations in diatomic molecules and inelastic light scattering as an experimental probe; normal modes of polyatomic molecules in classical and quantum approaches. Labelling of normal modes with Mulliken symbols from group theory.

 

3.4 Lattice vibrations in the harmonic approximation: classical limit.

One dimensional monoatomic chain, dispersion relation. 1D chain with basis, acoustic and optical branches of the dispersion relation. Lattice vibrations in 3 dimensions, periodic boundary conditions. The dynamical matrix. Vibrations in lattices with bases.

 

3.5 Lattice vibrations in the harmonic approximation: quantum limit.

Hamiltonian of the harmonic crystal in terms of creation and annihilation operators. Phonons. Bose – Einstein statistics. Density of states. Lattice specific heat. High temperature limit and the Dulong and Petit law. Low temperature limit, T3 dependence. Einstein and Debye models.

 

Readings/Bibliography

Main textbooks

• F. Albert Cotton, Chemical Applications of Group Theory, Third Edition, Wiley (1990)
• Michael P. Marder, Condensed Matter Physics, Second Edition, Wiley (2010)
• Neil W. Ashcroft and N. David Mermin, Solid State Physics, Saunders College Publishing (1976)
• Charles Kittel, Introduction to Solid State Physics, Eighth Edition, Wiley (2005)

Complementary textbooks

• General condensed matter
• Efthimios Kaxiras and John D. Joannopoulos, Quantum Theory of Materials, Cambridge University Press (2019)
• Marvin Cohen and Steven Louie, Fundamentals of Condensed Matter Physics, Cambridge University Press (2016)
• Attilio Rigamonti and Piero Carretta, Structure of Matter, Springer (2015)
• Gian Franco Bassani e Umberto Grassano Fisica dello Stato Solido, Bollati Boringhieri (2000), only in Italian.
• Feng Duan and Jin Guojon, Introduction to Condensed Matter Physics, Volume 1, World Scientific (2005).
• Giuseppe Grosso and Giuseppe Pastori Parravicini, Solid State Physics, Second Edition, Academic Press (2014)
• Peter Y. Yu and Manuel Cardona, Fundamentals of Semiconductors, Fourth Edition, Springer (2010)
• G.L. Squires, Introduction to the theory of thermal neutron scattering, Cambridge University Press, 1978
  • Crystallography
• Carmelo Giacovazzo (editor), Fundamentals of Crystallography, Third Edition, Oxford University Press (2011).
• Boris K. Vainshtein, Fundamentals of Crystals: Symmetry, and Methods of Structural Crystallography, Second Enlarged Edition, Springer (1994).
• Boris K. Vainshtein, Vladimir M. Fridkin and Vladimir L. Indenbom, Modern Crystallography II: Structure of Crystals, Springer (1979)
• Gerald Burns and Anthony M. Glazer, Space groups for solid state scientists, Academic Press (2013)
  • Group theory
• Michael Tinkham, Group Theory and Quantum Mechanics, Dover (2003).

Teaching methods

Frontal lectures using the blackboard for the demonstrations and a slide projector for viewgraphs and graphic renderings.

During the lecture the teachers will consider activate learning by interacting directly with students.

Assessment methods

Oral examination. The exam date can be arranged by contacting the teacher by email (s.sanna@unibo.it) at least 10 days in advance of the desired date.

Typically the student is asked two main questions selected by the teacher.

For each argument the student will be asked to develop a theory/calculations to obtain a physical law, by illustrating the main conceptual steps, including considerations and approssimations used, and to explain its physical meaning and its use to study some physical property of an ideal and/or real physical system, paying attention to the order of magnitude of the physical quantities at play. The typical duration of the exam is 30 minutes.

The purpose of the oral exam is to verify the student's knowledge and his/her ability to apply it and to make the necessary logical-deductive connections.

Graduation of the final grade: Preparation on a very limited number of topics covered in the course and analytical skills that emerge only with the help of the teacher, expression in overall correct language → 18-19;

Preparation on a limited number of topics covered in the course and autonomous analysis skills only on purely executive issues, expression in correct language → 20-24;

Knowledge of a large number of topics addressed in the course, ability to make independent choices of critical analysis, proper use of specific terminology → 25-29;

Substantially exhaustive preparation on the topics covered in the course, ability to make independent choices of critical analysis and connections, full mastery of the specific terminology and ability to argue and critical thinking → 30-30L

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en [http:] ). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

Teaching tools

Blackboard, overhead projection.

Office hours

See the website of Samuele Sanna