73546 - Solid State Physical Chemistry M

Learning outcomes

The aim of the course is to give tools for the quantum-mechanical study and interpretation of the electron transport in devices made up by molecules or molecular layers or nanostructures between metallic contacts, with a special emphasis on the current flow when a voltage is applied across the device. The course proposes theoretical models to interpret the electric conductance on the atomic scale and to describe phenomena taking place when the dimensions of the system are progressively increased. The course can be considered an introduction to the recent rapidly growing research field named Molecular Electronics.

Course contents

Prerequisites: Basic knowledges of analysis, linear algebra,    differential and integral calculus, classical physics and general chemistry given by the courses of the triennium. It is also useful to have basic knowledges of the Hamiltonian formalism and  Maxwell electromagnetism.

Suggestion: it is useful to integrate this Course with that of  Chimica Fisica dei Materiali Solidi that represents a complement of the Course of Solid State Physical Chemistry.

Program:

• BASIC QUANTUM MECHANICS: the Stern-Gerlach experiment - Analogies with experiments made with polarized light - Development of a mathematical formalism for the interpretation of the SG experiment: Ket, Bra and Operators;  Ket and Bra complementary spaces; scalar products;  physical observables as Hermitian operators, physical states of the system as ket and bra vectors of complementary spaces with scalar product - The secular problem: matrix representation of Hermitian operators, eigenvalues, eigenkets, transition probabilities and their physical interpretation -  Compatible and non-compatible observables, and the uncertainty principle - Symmetry and quantum theory of angular momenta.                                
• WAVE QUANTUM MECHANICS: position and momentum observables, generalization of the theory for operators with continuous spectrum - Representations of the Hamiltonian on the basis of the position and momentum operators;  examples and applications to prototype systems.                                                                   
• TIME-INDEPENDENT and TIME-DEPENDENT SCHROEDINGER EQUATIONS: properties of atomic and molecular Hamiltonians and of their eigenfuntions. Time evolution of quantum states.

• METHODS OF QUANTUM CALCULATION: variational methods: Hartree-Fock, Configuration Interaction, Density Functional Theory - Perturbative Theory
  

• MOLECULAR QUANTUM MECHANICS: Born-Oppenheimer approximation - Molecular Orbital Theory - Valence Bond Theory - Calculation of excited states and energies of atoms and molecules - Calculation of roto-vibrational energy levels and states of molecules.

• ELEMENTS OF SOLID STATE PHYSICS:  Geometrical description of crystals: simple lattices and composite lattices, Bravais lattices; primitive and unitary cells, Wigner-Seitz cells. Reciprocal lattices: definitions and basic properties; planes and directions in Bravais lattices; Brillouin zones. Translational simmetry in quantum-mechanics: Bloch theorem and electronic wavefunctions, bands and density of states; cyclic boundary conditions.
  

• ELECTRONIC TRANSPORT IN NANODEVICES:   Modelling of a nanoscale transistor - An atomistic view of the electrical resistence - Energy levels diagram - Flow of electrons and rate equation - Current in one-level channel - The quantum of conductance. The quantum theory of the electronic trasport in nanodevices will be test using a specific code for modelling of the electronic transport in nanodevices.

Readings/Bibliography

The use of notes taken during the lections will be crucial, together with lecture notes and other material provided by the teacher. For further investigations, the following books are recommended:           

• J.J. Sakurai, Modern Quantum Mechanics, ed. Wiley.
• P.Atkins, R.Friedman Molecular Quantum Mechanics, ed. Oxford University Press.
• G.Grosso and G.Pastori-Parravicini, Solid State physics, ed. Academic Press.
• S.Datta, Quantum Transport. Atom to Transistors, ed. Cambridge.

Teaching methods

The course is organized in frontal lectures, where basic concepts, fundamental principles and mathematical/computational techniques of basic and molecular quantum mechanics are presented and discussed together with simple models of nanotransistors. After the theoretical explanation of each topic, lections will be devoted to the solution of exercises and specific problems involving prototype atomic, molecular and solid state systems, and electronic transport in simple nanoelectronic devices. This procedure aims to aid the student in acquiring the ability to convert a physical problem into a theoretical/computational procedure able to give quantitative results.

Assessment methods

The final test is an oral examination, that verifies the achievement of the following teaching targets:
- knowledge of basic concepts, mathematical structure and main computational methods of quantum mechanics and of its application to molecular systems;
- knowledge of elements of the quantum transport theory;
- ability of using quantum mechanics to formulate and solve problems relative to the calculation of structural and electronic properties of atoms, molecules;
- ability of modeling the electronic transport in simple nanoelectronic devices.
The oral exam ends with the request of solving a simple problem on one of the topics of the course. The solution of the problem does not require explicit numerical calculations.

Teaching tools

Lectures notes and other didactic material are made available in electronic format. Attending to the Chimica Fisica Dei Materiali Solidi course, that is complementary to this course, is strongly suggested.

Office hours

See the website of Renato Colle