Description and contextualization of the subject

Este curso aborda temas avanzados de física de la materia condensada, y se divide en dos partes dedicadas respectivamente a fenómenos cuánticos en semiconductores y a técnicas computacionales. En la primera parte, se exploran el transporte cuántico en estructuras bidimensionales, incluido el efecto Hall cuántico, y materiales novedosos como quantum dots y quantum wires. Los conceptos clave incluyen el papel del desorden, la estructura de bandas y los efectos relativistas, junto con la dinámica de espín y los mecanismos de control esenciales para el diseño de puertas cuánticas. La segunda parte cubre técnicas computacionales, especialmente la Teoría del Funcional de la Densidad (DFT), sus principios y aplicaciones. Se incluyen las ecuaciones de Kohn-Sham, los funcionales de intercambio y correlación, y los métodos de implementación utilizando software como QUANTUM ESPRESSO y SIESTA. También se introducen las funciones de Wannier y aplicaciones en la interpolación de la estructura de bandas y en cálculos de la fase de Berry.

Learning outcomes of the subject

Knowledge or content:

RCO1. Demonstrate the ability to explain the fundamental principles of the quantum world, both at a basic and technical level.

RCO2. Have a basic knowledge of the relevant literature in quantum mechanics and be capable of effectively reading and understanding research articles.

RCO3. Be able to initiate the development of original ideas and applications within the context of quantum physics research.

RCO4. Possess the capacity for independent research, synthesis, and be able to present in a clear and structured way complex issues related to the various areas of quantum mechanics addressed in this Master¿s program.

RCO5. Under supervision, demonstrate the ability to write and defend original work that meets the quality standards required for publication in high-impact indexed journals.

RCO6. Be able to identify opportunities for innovation and technology transfer in the field of quantum science and technology.

RCO9. Know the basic literature and demonstrate the ability to solve standard problems in the field of Quantum Statistical Physics.

RCO11. Know the basic literature and demonstrate the ability to solve standard problems in the field of Condensed Matter Physics.

RCO12. Know the basic literature and demonstrate the ability to solve standard problems in the field of Quantum Technologies.



Competencies:

RC1. Possess and understand knowledge that provides a basis or opportunity for developing and/or applying original ideas, often in a research context.

RC2. Apply acquired knowledge and problem-solving skills in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their field of study.

RC3. Demonstrate the ability to integrate knowledge and address the complexity of formulating judgments based on incomplete or limited information, including reflection on social and ethical responsibilities linked to the application of their knowledge and judgments.

RC4. Communicate conclusions, as well as the underlying knowledge and rationale, clearly and unambiguously to both specialized and non-specialized audiences.

RC5. Possess learning skills that enable continued study in a largely self-directed or autonomous manner.



Abilities or skills:



RHE1. Demonstrate proficiency in using tools for bibliographic resource searches.

RHE2. Exhibit critical capacity to read research articles and incorporate their findings into one¿s own work.

RHE3. Write and present original work in one of the official languages and in English.

RHE4. Communicate scientific concepts and results clearly and effectively to both specialized and non-specialized audiences, through presentations and publications.

RHE5. Demonstrate the ability for autonomous learning and staying current with scientific and technological advances.





RHT1. Understand and apply the fundamental principles of quantum mechanics to analyze and solve problems in basic research in quantum science.

RHT2. Understand and apply the fundamental principles of quantum mechanics to analyze and solve problems in quantum technology.

RHT3. Effectively integrate into a fundamental or applied research project involving quantum aspects, and solve problems in multidisciplinary environments.

RHT4. Evaluate and select appropriate tools and techniques for research in fundamental physics.

RHT5. Evaluar y seleccionar las herramientas y técnicas adecuadas para la obtención de aplicaciones tecnológicas basadas en la física cuántica.

Temary

Quantum phenomena in semiconductors: transport and control

Two-dimensional structures and modern nanotechnologies. Quantum Hall effect. Quantum standards. Novel single and double-layer two-dimensional materials.

Quantum wires: conductance and conductivity. Quantum transport. Critical role of disorder patterns. ¿¿

Quantum dots: zero-dimensional quantum systems. Semiconductor spin-based qubits.

Band structure of solids. Relativistic effects in semiconductors. Spin-orbit coupling, spin relaxation, and quantum spin transport. ¿¿

Spin dynamics and control by time-dependent electric fields: application to design of quantum gates. ¿



Computational Techniques in Condensed Matter

Introduction to Density Functional Theory (DFT). Historical background and importance of DFT. Basic principles and the Hohenberg-Kohn theorems. The Kohn-Sham equations. Exchange-correlation functionals: Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA).

Computational Techniques and Implementation of DFT Theories. Plane Wave Expansion. Concept of cutoff and its relation to spatial resolution. Pseudopotentials. Linear Combination of Atomic Orbitals (LCAO). Introduction to the QUANTUM ESPRESSO and SIESTA codes.

Linear Response and the Sternheimer Equation. Phonon calculations using linear response theory Supercell method vs. linear response.

Relativistic Calculations and Non-Collinear Spin Magnetism. Spin matrix formalism within DFT.

Introduction to Wannier Functions. Definition and basic properties of Wannier functions. Maximally localized Wannier functions (MLWFs). Methods to construct Wannier functions: Marzari-Vanderbilt method.

Applications of Wannier Functions. Band structure interpolation. Calculating Berry phases and Berry curvatures. Wannier functions in tight-binding models.

Bibliography

Basic bibliography

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

D. Sholl, J. A. Steckel, Density Functional Theory: A Practical Introduction, Wiley, 2009.

In-depth bibliography

Journals

J.M. Soler et al., The SIESTA Method for Ab Initio Order-N Materials Simulation, J. Phys. Condens. Matter 14, 2745 (2002).

S. Baroni, S. de Gironcoli, A. Dal Corso, Phonons and Related Crystal Properties from Density-Functional Perturbation Theory,

Rev. Mod. Phys. 73, 515 (2001).

N. Marzari and D. Vanderbilt, Wannier Functions and Their Applications, Rev. Mod. Phys. 84, 1419 (2012).