Nanotechnology Engineering

Activity type

Chimica e fisica della materia

SSD

FIS/01

Year

1st year

Semester

2nd semester

CFU

6

Hours distribution

60 classroom hours

Lecturers

GIUSEPPE ZOLLO

Activity type

Chimica e fisica della materia

SSD

FIS/03

Year

1st year

Semester

1st semester

CFU

6

Hours distribution

60 classroom hours

Lecturers

PAOLO POSTORINO

The main purpose of the course is to transfer to the students the basic
knowledge concerning the multidisciplinary topics that found the
atomistic simulations. The course is focussed on the main aspects of
the classical models and the principal quantum models. Numerical
laboratories and exercises will help the students to develop the needed
technical skills.

Expected learning outcomes:

Knowledge and understanding (Dublin descriptor I)
At the end of the course the student will have the basic knowledge on the main atomistic simulations methods and techniques used to study , from an atomistic point of view, the nano-structures and systems of interest. They will then be able to understand the environment that surrounds it from the point of view of its microscopic and macroscopic structure. He will also be aware of the way the atomistic structure affects the materials properties and its relationship with other scientific disciplines and the need for continuous updating on the state of the art, due to the continuous progress of scientific knowledge and technology.

Applying knowledge and understanding (descriptor II)
At the end of the course of study the student will have developed the ability to understand the inner atomistic nature of some physical and chemical properties and their relation with the macrospoic properties of materials.

Making judgements (descriptor III)
At the end of the course the student will have to possess the tools to critically evaluate the limits of the different techniques and their potentiality.

Communication skills (descriptor IV)
At the end of the course the student must have acquired a good language property, especially with regards to a specific scientific terminology, so as to be able to clearly communicate their knowledge and conclusions to an audience composed from people with (or without) expertise in the field.

Learning skills (descriptor V)
At the end of the course the student must have developed a learning ability that will allow him to study and deepen the chemical aspects related to the field of nanotechnology in an autonomous way.

Knowledge and understanding (Dublin descriptor I)
At the end of the course the student will have the basic knowledge on the main atomistic simulations methods and techniques used to study , from an atomistic point of view, the nano-structures and systems of interest. They will then be able to understand the environment that surrounds it from the point of view of its microscopic and macroscopic structure. He will also be aware of the way the atomistic structure affects the materials properties and its relationship with other scientific disciplines and the need for continuous updating on the state of the art, due to the continuous progress of scientific knowledge and technology.

Applying knowledge and understanding (descriptor II)
At the end of the course of study the student will have developed the ability to understand the inner atomistic nature of some physical and chemical properties and their relation with the macrospoic properties of materials.

Making judgements (descriptor III)
At the end of the course the student will have to possess the tools to critically evaluate the limits of the different techniques and their potentiality.

Communication skills (descriptor IV)
At the end of the course the student must have acquired a good language property, especially with regards to a specific scientific terminology, so as to be able to clearly communicate their knowledge and conclusions to an audience composed from people with (or without) expertise in the field.

Learning skills (descriptor V)
At the end of the course the student must have developed a learning ability that will allow him to study and deepen the chemical aspects related to the field of nanotechnology in an autonomous way.

As a prerequisite it is mandatory that the students have attended the first module the course and have strong knowledge of physics and mathematics

Module: Principles of Quantum Mechanics with Elements of Structure of Matter

1) Lagrangian and Hamiltonian mechanics
N particles systems; generalized coordinates and conjugate momenta; phase space: micro-state and macro-state of a system; Lagrangian function and the Lagrangian equations of motion. The Hamiltonian function and the Hamiltonian equations of motion.

2) Introduction to statistical mechanics
Statistical ensamble; the Gibbs hypothesis; distribution function; the measure of a physical quantity; the Liouville theorem and its consequences.
Microcanonical ensamble: distribution and partition functions; entropy; therma, mechanical and chemical equilibrium; statistical macroscopic quantities: temperature, pressure, chemical potential. Intensive and extensive observables; the link with classical physics. Canonical ensamble: distribution and partition functions; energy probability density and density of states; the partition function of an ideal gas; the Helmoltz free energy; Maxwell and Maxwell-Boltzmann distributions; The equipartition theorem; Grand-Canonical Ensamble: Distribution and partition functions; the ideal gas in the granmd-canonical ensamble; quantum partition functions; the Bose-Einstein and the Fermi-Dirac distributions.

3) Monte-Carlo and statistical mechanics
Monte-Carlo integration; pseudo-random numbers. Random numbers generators, casuality tests; The Lehmer generator; importance sampling: inversion and rejection; Markov chains; the Metropolis-Monte-Carlo algorithm; technicalities: interaction potentials; periodic boundary conditions; truncation and shift etc.; orientational moves for polyatomic molecules; Metropolis Monte-Carlo in the micro-canonical and grand canonical ensambles.

4) Classical Molecular Dynamics
Equilibration and initialization; integrators: Verlet, Velocity Verlet, Gear Predictor –Corrector etc; the Verlet list; Lyapunov instability. Observables. Temperature, pressure and the Virial Theorem; energy conservation; structural observables: the Radial Distribution Function; Diffusion: the Green-Kubo formula; Molecular Dynamics in the canonical ensamble: the fluctuation of temperature; Thermostats: velocity rescaling; Andersen; the Nosè-Hoover thermostat. Barostats.

5) Potenziali classici
Fitting procedures; two bodies potentials; many-bodies potentials;Potentials for metals, semiconductors and soft-matter.

6) Introduction to quantum theory of many-body systems
The Schrödinger equations of many-bodies systems. frozen-core and Born-Oppenheimer approximations; the single electron approximation; the variational Rayleigh-Ritz principle; The Hartree equations; the Pauli exclusion principle; the Slater determinant and the Hartree-Fock equations; tight binding molecular dynamics; the tight binding method and the Bloch theorem; Linear Combination of Atomic Orbitals; parametrization and transferability; the Hamiltonian matrix and its diagonalization; the Hellmann-Feynman and the forces. The density functional theory. The Hohenberg-Kohn theorems and the Kohn-Sham equations;




Module: Atomistic Simulations
Introduction.
- Brief reminders of electromagnetism: Maxwell's equations, electromagnetic waves in vacuum and matter. Drude-Lorentz model.
- Crisis of classical physics: blackbody radiation, photoelectric effect, absorption spectra, hydrogen atom, Bohr model, tunneling effect.

Introduction to quantum mechanics.
- Wave-particle duality. Postulates of quantum mechanics. Physical observables and their measurement. Hamiltonian formalism. Principle of superposition. Time-dependent and time-independent Schrödinger equations. Heisenberg uncertainty principle.
- Conceptual applications: free particle, particle and potential barrier, particle in a box.
- Applications: harmonic oscillator, angular momentum, hydrogen atom. Spin operator.

Basic concepts of statistical mechanics:
- Boltzmann, Fermi-Dirac, and Bose-Einstein distributions.

Molecules.
- Typical energies, Born-Oppenheimer approximation. Harmonic and rigid rotor approximations.
- Morse potential.
- Hydrogen molecule and the method of atomic orbital approximation in the Linear Combination of Atomic Orbitals (LCAO).

Condensed matter: crystalline solids.
- direct lattice and reciprocal lattice. First Brillouine zone.
- One-electron model, Schroedinger equation in periodic potential, Bloch functions, band model, density of states.
- Conductors, insulators, semiconductors.
- Theory of conduction in metals & semiconductors.

Module: Principles of Quantum Mechanics with Elements of Structure of Matter
Class notes and textbook from the teacher.
a deeper understanding from:
1) first and second chapters: C. Kittel “Elementary Statistical Physics” John Wiley & Sons
2) third and fourth chapters Frenkel-Smit “Understanding Molecular Simulation”; Academic Press; Allen; Tildsley “Computer Simulation of Liquids”; Ed. Oxford Science;
3) fifth chapter: F: Ercolessi “A molecular dynamics primer”
WWW: http://www.fisica.uniud.it/~ercolessi/
4) sixth chapter: M.P. Marder “ Condensed Matter Physics” (Cap. 6; 7; 8; 9) John Wiley & Sons. R. M. Martin “Electronic Structure: Basic Theory and Practical Methods”, Cambridge University Press.



Module: Atomistic Simulations
Main Text: Quantum Mechanics (B. H. Bransden, C. J. Joachain)

Specific documents and lecture notes will be regularly added to the E-learning website.

Module: Principles of Quantum Mechanics with Elements of Structure of Matter
1) Per il primo e il secondo capitolo: C. Kittel “Elementary Statistical Physics” John Wiley & Sons
2) per il terzo e il quarto capitolo Frenkel-Smit “Understanding Molecular Simulation”; Academic Press; Allen; Tildsley “Computer Simulation of Liquids”; Ed. Oxford Science;
3) per il quinto capitolo: F: Ercolessi “A molecular dynamics primer”
WWW: http://www.fisica.uniud.it/~ercolessi/
4) per il sesto e il settimo capitolo: M.P. Marder “ Condensed Matter Physics” (Cap. 6; 7; 8; 9) John Wiley & Sons. R. M. Martin “Electronic Structure: Basic Theory and Practical Methods”, Cambridge University Press.



Module: Atomistic Simulations
Other texts for parts of the program indicated in class:
- Introduction to Modern Physics (G. Fowles)
- Optics Lecture Notes from the Department of Physics (P. Mataloni, A. Frova)
https://www.phys.uniroma1.it/biblioteca/web_disp/d6/dispense/Frova_Mataloni.pdf
- Quantum Mechanics for Applied Physics and Engineering (A.T. Fromhold Jr)
- Fundamental University Physics (Alonso – Finn)

Lectures are given in person. During the lessons, the instructor provides the basic knowledge needed to address atomistic-scale simulations, starting with the fundamentals of the statistical theory of materials at equilibrium and then addressing more technical issues related to the calculation of observable quantities under different macroscopic conditions. Although the course content is primarily theoretical, the instructor offers practical insights and examples to clarify the concepts illustrated. The final part of the course covers similar topics, but addressed in the field of many-body quantum theory.

Beside not being formally mandatory, attending the class is strongly recommended due to the intrinsic difficulty of the themes treated.

The examination consists in an oral interview and shall ensure that the following objectives are met:the students must demonstrate to have acquired the basic knowledge of the atomistic simulation theory and techniques.
The evaluation is based on:
verifying of the acquired knowledge (60%) ("Knowledge and understanding" and "Applying knowledge and understanding")
verifying of language property and clarity of presentation (20%) ("Communication skills")
verifying of the ability to apply the acquired knowledge in the field of nanotechnology (20%) ("Learning skills")

-) describe the main thermostats with particular emphasis on the Nosè-Hoover one
-) try to obtain the chemical potentia of an ideal gas in the grand-canonical ensamble
-) describe and detail the Metropoli MonteCarlo algoirithm needed to sample the phase space in various statistical ensables
-) descrime the main features and limist of the main integrators employed in molecula dynamics
-) describe and discuss the rationale and the main mathematical aspects that must be known to obtain the Hartree-Fock equations
-) detail and discuss the LCAO model for solid systems

Module: Principles of Quantum Mechanics with Elements of Structure of Matter

• 15 ore: introduzione alla meccanica statistica15 ore: metodi MonteCarlo e Metropolis15 ore: dinamica molecolare15 ore: introduzione alla teoria quantistica di sistemi a moti corpi
  • Books: Material provided by the teacher