Modeling of nanostructures and materials

Multi-scale methods allowing for modeling of nanostructures and novel materials on atomistic to macroscopic length scales will be discussed during this lecture. Students will get acquainted with ab initio methods in the framework of the Kohn-Sham realization of the density functional theory, ab initio and classical molecular dynamics, semi-empirical methods such as tight-binding, the concept of coarse graining, valence force field approaches, Monte Carlo methods, and some exemplary continuum theories (such as the theory of elasticity). Particular attention will be focused on various techniques to solve Kohn-Sham equations, which employ various expansion bases (such as plane-waves, localized atomic orbitals, real space integration). Further, the state-of-the-art numerical codes will be discussed. The students will practically use these codes to perform computations of electronic structure and resulting properties for the whole range of nanosystems and materials

Following topics will be discussed during the series of lectures:

1. Physics on Different Length- and Timescales

- Electronic/Atomic Scale

- Atomistic/Microscopic

- Microscopic/Mesoscopic

- Mesoscopic/Macroscopic

2. Computer Simulations and Computational Materials Science

- What is Computational Material Science on Multiscales ?

- What is a Model? – Scientific Method

- Hierarchical Modeling Concepts above the Atomic Scale

3. Computational Methods on Electronic/Atomistic Scale

(a) Ab-initio methods

- Hamiltonian for condensed matter systems

- The adiabatic and Born-Oppenheimer Approximation

(b) Density Functional Theory – Basic concepts

- Kohn-Sham realization of the Density Functional Theory

- Derivation of the Kohn-Sham equations

- Approximations to the exchange-correlation functionals

- Methods of solving the Kohn-Sham equations

- Concept of pseudopotentials and plane wave method

- Linear combination of atomic orbitals

- Linearized Augmented Plane Waves method (LAPW)

- Linearized Muffin-Tin Orbitals method (LMTO)

- Concept of multiple-scattering, Green's function, random systems

- Force calculations; The Hellmann-Feynman Theorem

(c) – Car-Parrinello Molecular Dynamics

(d) – Survey of numerical codes for solving K-S equations

(e) Theory of excitations

- GW method for energies of one-particle excitations

- Time dependent DFT

(f) Semi-empirical Methods

- Tight-Binding Method

- Semi-empirical pseudopotential method

4. Computational Methods on Atomistic/Microscopic Scale

(a) Fundamentals of Statistical Physics and Thermodynamics

- Statistical ensambles

- Virtual ensembles

- Entropy and temperature

(b) Classical Interatomic and Intermolecular Potentials

- Charged systems, Ewald summation

- Van der Waals Potential

- Covalent Bonds

- Embedded Atom Potentials

- Pair Potentials

- Valence Force Field Models

(c) Classical Molecular Dynamics Simulations

- Numerical Ingredients of MD Simulations

- Integrating the Equations of Motion

- Periodic Boundary Conditions

- Making Measurements

(d) Monte Carlo Method

- Basic concepts

- Markov chains

- Metropolis Algorithm

5. Computational Methods on Mesoscopic/Macroscopic Scale

(a) Physical Theories for Macroscopic Phenomena

- The Continuum Hypothesis

- Theory of elasticity as an example of continuum theory

- Bridging Scale Applications: Crack Propagation in a Brittle Specimen

(b) Gizburg-Landau/Cahn-Hiliard Field Theoretic Mesoscale Simulation Method

6. Perspectives in Multiscale Materials Modeling

K. Ohno, K. Esfarjani, Y. Kawazoe, "Computational Materials Science - From Ab Initio to Monte Carlo Methods", Springer-Verlag, Berlin, 1999.

Robert G. Parr and Weitao Yang, "Density-Functional Theory of Atoms and Molecules", Oxford University Press, New York, 1989.

R.M Dreizler and E.K.U Gross, "Density Functional Theory - An Approach to the Quantum Many-Body Problem", Springer-Verlag, Berlin, 1990.

Richard M. Martin, "Electronic Structure - basic theory and practical methods", Cambridge University Press, Cambridge, 2004.

W. E. Pickett, Pseudopotential Methods in Condensed Matter Applications, Computer Physics Reports 9, 115-198 (1989).

Feng Duan & Jin Guojun, “Introduction to Condensed Matter Physics”, World Scientific, New Jersey, 2005.

E. Engel & R. M. Dreizler, “Density Functional Theory, An Advanced Course”, Springer-Verlag, Berlin, 2011.

K. Varga and J. A. Driscoll, “Computational Nanoscience, Applications for Molecules, Clusters, and Solids”, Cambridge University Press, Cambridge, 2011.

D. Marx & J. Hutter, “Ab initio molecular dynamics: basic theory and advanced methods”, Cambridge University Press, Cambridge, 2009.

A. Gonis, “Theoretical Materials Science, Tracing the Electronic Origins of Materials Behavior, Materials Research Society, 2000.

D. Raabe, “Computational Materials Science”, Wiley & Sons, 1992.

J.M Haile, “Molecular Dynamics Simulations”, Wiley, 1992.

C. Massobrio, H. Bulou, Ch. Goyhenex (eds.), “Atomic-Scale Modeling of Nanosystems and Nanostructured Materials”, Lecture Notes in Physics 795, Springer, Berlin Heidelberg, 2010.

Dierk Raabe, “Computational Materials Science: The Simulation of Materials, Microstructures and Properties”, Wiley-VCH Verlag GmbH, 1998.

T. Saito, “Computational Materials Design”, Springer-Verlag, Berlin, 1999.

After the lecture, the students will get familiar with knowledge of the basic methods for modeling of nanostructures and materials on atomistic, mesoscopic, and macroscopic length.

Practical exercises will learn the students how to perform modeling of nanostructures and materials employing state-of-the-art numerical codes on high performance computing (HPC) environment.

There are three components of the final note:

(i) multiple-choice test from the lecture's material

(ii) points for the work during the computer exercises

(iii) small modeling project at the end of the course