# MESOSCOPIC AND TOPOLOGICAL MATERIALS

**Academic Year 2022/2023**- Teacher:

**FRANCESCO MARIA DIMITRI PELLEGRINO**

## Expected Learning Outcomes

The course aims to provide the student with the fundamentals of the physics of mesoscopic systems with reference to experiments and theory, focusing on graphene and topological systems.

** Knowledge and understanding. **Critical understanding of the most advanced developments of Modern Physics, both theoretical and experimental, and their interrelations, also across different subjects. Adequate knowledge of advanced mathematical and numerical tools, currently used in both basic and applied research. Remarkable acquaintance with the scientific method, understanding of nature, and of research in Physics.

**Applying knowledge and understanding. **Ability to identify the essential elements in a phenomenon, in terms of orders of magnitude and approximation level, and being able to perform the required approximations. Ability to use an analogy to apply known solutions to new problems (problem-solving).

**Making judgments.*** *Ability to convey own interpretations of physical phenomena, when discussing within a research team. Developing one's own sense of responsibility, through the choice of optional courses and the final project.

**Communication skills. **Ability to discuss advanced physical concepts, both in Italian and in English.

**Learning skills.*** *Ability to acquire adequate tools for the continuous update of one's knowledge. Ability to access to specialized literature both in the specific field of one's expertise and in closely related fields. Ability to exploit databases and bibliographical and scientific resources to extract information and suggestions to better frame and develop one's study and research activity. Ability to acquire, through individual study, knowledge in new scientific fields.

## Course Structure

Frontal lectures.^{*}

^{*}Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

## Required Prerequisites

Fundamental knowledge:

Quantum Mechanics: Harmonic oscillator, angular momentum and Pauli theory of spin, approximate methods;

Structure of Matter: Quantum Statistics.

Important knowledge:

Solid State Physics: Free electrons, electrons in a crystal, phonons.

Useful knowledge:

Quantum Mechanics: Scattering Theory;

Statistical Mechanics: Kinetic Theory.

## Detailed Course Content

**Semiclassical theory**: Semiclassical Boltzmann equation, Relaxation time approximation, Elastic scattering, Diffusive limit, Inelastic scattering, Thermoelectric effects.**
Scattering approach to quantum transport**: Scattering region, leads, and reservoirs, Scattering matrix, Conductance from scattering, Resonant tunneling, Introduction to the localization.**
Fluctuations and correlations**: Definition and main characteristics of noise, Scattering approach to noise, Boltzmann-Langevin approach, Introduction to the effect of noise on quantum dynamics.**
Single-electron effects**: Charging energy, Tunnel Hamiltonian and tunneling rates, Master equation, Cotunnelling.**
Graphene**: Electron structure in monolayer graphene, electrical doping, Landau levels in monolayer graphene.**
Topological materials one and two dimensions**: SSH model, Berry phase and polarization, Chern number, Current operator and particle pumping, Chern insulators (QWZ model), 2-dimensional time-reversal invariant topological insulators, Electrical conduction of edge states.

## Textbook Information

[1] T. T. Heikkilä, The Physics of Nanoelectronics: Transport and Fluctuation Phenomena at Low Temperatures, Oxford Master Series in Physics (2013).

[2] M. I. Katsnelson, Graphene: Carbon in Two Dimensions, Cambridge University Press (2009).

[3] J.K. Asbóth, L. Oroszlány, A. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions, Springer (2016).

[4] S. M. Girvin, K. Yang, Modern Condensed Matter Physics, Cambridge University Press (2019).

## Course Planning

Subjects | Text References | |

1 | Semiclassical theory (5h) | [1] Chapt. 2 |

2 | Scattering approach to quantum transport (8h) | [1] Chapt. 3 |

3 | Fluctuations and correlations (6h) | [1] Chapt. 6 |

4 | Single electron effects (4h) | [1] Chapt. 7 |

5 | SSH model (3h) | [3] Chapt. 1 |

6 | Berry phase, polarization, and Chern number (3h) | [3] Chapt. 2-3 |

7 | Current Operator and Particle Pumping (2h) | [3] Chapt. 4-5 |

8 | Chern insulators (2h) | [3] Chapt. 6 |

9 | Time-Reversal Symmetric Two-Dimensional Topological Insulators (2h) | [3] Chapt. 8 |

10 | Electrical Conduction of Edge States (2h) | [3] Chapt. 10 |

11 | Graphene (5h) | [2] Chapt. 1-2 |

## Learning Assessment Procedures

Learning verification is an oral exam that consists of a discussion of three (3) distinct topics of the course contents, of which the first is chosen by the student (presented with attention to formal details).*

For the evaluation, the following aspects are taken into account: the relevance of the answers, the level of analysis, the ability to connect different topics of the course, the ability to report examples, the correct use of formal tools, language properties, and clarity of presentation.

**Learning assessment may also be carried out online **if the conditions require it.*

## Examples of frequently asked questions and / or exercises

The questions below are not a complete list but they are just some examples

"What are quantities to compare to distinguish the transport regimes and what are the respective transport equations?", "What are the features of the Landau levels in monolayer graphene?", "How does one define the topological properties of the SSH model?".