84535 - SYMMETRIES, ELECTRONS AND PHONONS

Conoscenze e abilità da conseguire

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.

Contenuti

1. Atomic structure

1.1. Introduction to symmetry

• Introduction to symmetry in crystallography. Isometric trasformations. Direct and opposite congruence. Symmetry operations and symmetry elements. Matrix description.
• Giacovazzo: 1.1, 1.2, 1.A
• Lecture “Introduction to Symmetry in Condensed-Matter” available on virtuale.unibo.it.

• Elements of group theory for crystallography. Group axioms. Abelian groups. Order of a group, multiplication tables. Cyclic groups. Subgroups. Classes and conjugate elements. Molecular symmetry. Point groups. Equivalent symmetry elements. Crystallographic point groups, nomenclature. Classes of molecular symmetry operations. Group representations. Matrix representation. Reducible and irreducible representations. The Great Orthogonality Theorem and its consequences. Properties of irreducible representations. Character tables. Basis functions. Schrödinger equation and symmetries. Wave functions as bases for irreducible representations. Representation of Abelian groups: cyclic groups, translation group (Bloch’s theorem), rotation group. Degenerate case, relation with the dimension of irreducible representations. Application: splitting of d electrons in an octahedral field.
• Cotton: 1, 2, 3, 4, 5.1, 9.1, 9.2, 9.3
• Tinkham 2, 3 and 4
• Lecture “Elements of Group Theory in Condensed Matter Physics” available on virtuale.unibo.it.

1.2. Crystal structures in 2D.

2D lattices. Primitive vectors. The five 2D lattices. Crystal structures: lattices with a basis. Example: the graphene lattice. Unit cells: primitive, conventional, Wigner&Seitz. Space group. Symmorphic and non symmorphic space groups. The 17 2 dimensional space groups (plane groups). The asymmetric unit. Wyckoff symbols and positions.
• Marder 1.2, 1.3
• Lecture “2D crystals” available on virtuale.unibo.it.

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.

• Marder 2
• Ashcroft & Mermin 4 (and 7).
• Lecture “3D crystals” available on virtuale.unibo.it.

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.
• Marder 3.2, 3.3 e 3.4 (part)
• Ashcroft & Mermin 5 and 6.
• Lecture “Scattering – Reciprocal Lattice” available on virtuale.unibo.it.

1.5. Beyond crystals; quasicrystals
Alloys, amorphous solids, liquid crystals. Quasicrystals: solids with long range order without symmetry by translation. Penrose tilings. Fibonacci sequence as 1D quasicrystal; its construction as a 1D projection of a 2D square lattice. Fourier transform of a Fibonacci sequence and “diffractions peaks”. Icosahedral packing. Natural quasicrystals.

• Marder 5.10 (part)
• Vainshtein 5.1
• Lecture “Quasicrystals” available on virtuale.unibo.it.

 

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.
• Ashcroft & Mermin 2
• Marder: from 6.1 to 6.5
• Kittel 6
• Grosso & Pastori Parravicini 3
• Lecture “Fermi gas” available on virtuale.unibo.it.


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. Point symmetries and Bloch electrons; symmetry and degeneracy.
• Ashcroft & Mermin 8
• Marder 7.1, 7.2, 7.3
• Lecture “Independent electrons in Periodic Potential” available on virtuale.unibo.it.

• 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.
• Ashcroft & Mermin 9
• Marder from 8.1 to 8.3
• Cohen & Louie 3.3
• Lecture “Electrons in Weak Periodic Potential” available on virtuale.unibo.it.


• 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.
• Ashcroft & Mermin 10
• Marder 8.4
• Kittel 9
• Lecture “Tight binding” available on virtuale.unibo.it.

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.
• Kaxiras and Joannopoulos 5

• 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.
• Yu & Cardona 8.1
• Marder 23.6.1 and 23.6.2

• Lecture “Band structure - photoemission” available on virtuale.unibo.it.

2.4. Beyond the indepedent electron approximation
Hartree and Hartree-Fock equations. Koopman’s theorem. Hartree – Fock for the free electron gas. Screening; Thomas – Fermi approach to screening.

• Ashcroft & Mermin 17
• Marder: 9.1 and 9.2
• Grosso & Pastori Parravicini 4.7
• Cohen and Louie 6.1 – 6.3
• Lecture “Beyond independent electrons” available on virtuale.unibo.it.

Cohesion, vibrations and phonons

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.
• Marder 11
• Ashcroft & Mermin 19 and 20
• Kittel 3 (first part)
• Lecture “Classification and cohesion in solids” available on virtuale.unibo.it.

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.
• Ashcroft & Mermin 21
• Lecture “Static Lattice Failures” available on virtuale.unibo.it.

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.
• Grosso & Pastori Parravicini 8.1
• Rigamonti & Carretta 10
• Lecture “Lattice Vibrations – introduction and classical” available on virtuale.unibo.it.

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. The Born force constant model. Elasticity. Strain, dilation, stress, elastic constants. Elasticity in cubic crystals. Elastic waves in cubic crystals. Elasticity of isotropic solids.
• Marder 13
• Ashcroft & Mermin 22
• Lecture “Lattice Vibrations – introduction and classical” available on virtuale.unibo.it.


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.
• Marder 13.3
• Ashcroft & Mermin 23
• Lecture “Phonons” available on virtuale.unibo.it.

Inelastic phonon scattering.

Inelastic scattering of neutrons, light and x-rays as probes of vibrational structure. Energy and wavelength of thermal neutrons. Conservation of energy and momentum in inelastic neutron scattering, measurement of phonon dispersion relations. Raman and Brillouin scattering. Inelastic x-ray scattering.
Neutron scattering cross section: differential cross section and double differential cross section. The scattering length and the Fermi pseudopotential. Cross section for scattering by an ensemble of nuclei. Coherent and incoherent scattering. Scattering by a crystal: elastic scattering, the phonon expansion and coherent one phonon scattering.

• Marder 13.4
• Ashcroft & Mermin 24
• Squires 1, 2 and 3.1 – 3.7
• Lecture “Inelastic phonon scattering” available on virtuale.unibo.it.

Testi/Bibliografia

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).

Metodi didattici

Lezioni frontali con presentazioni.

Modalità di verifica e valutazione dell'apprendimento

Esame orale della durata di circa 30 minuti. L'obiettivo dell'esame è la valutazione della conoscenza del significato fisico degli argomenti illustrati nel corso. Di norma, verranno discussi due argomenti. L'iscrizione alla lista di esame avviene tramite almaesami in gruppi di 3 studenti ogni ora e mezza, 12 studenti al giorno.

Strumenti a supporto della didattica

Presentazioni utilizzate a lezione, disponibili on-line.

Orario di ricevimento

Consulta il sito web di Federico Boscherini