A printable postcript version of this document can be found
here.
Condensed Matter Physics on the
Computer -
A Laboratory Style Graduate Course
Michael Schreiber and Thomas Vojta
Institut für Physik, Technische Universität Chemnitz,
D-09107 Chemnitz, Germany
1. Teaching Computational Physics
2. Design of the course
3. Physics and computational topics
4. Example problems
Teaching Computational Physics, Trest, 31 August 2000
Teaching computational physics:
What do we want to achieve?
Teach students to solve a physical problem using a computer simulation
This includes:
- analyzing the physical problem
- reformulating the problem in a way suitable for a simulation
- choosing an efficient numerical algorithm
- writing the computer code
- running the computer simulation
- analyzing and interpreting the data obtained
A laboratory style computational condensed
matter physics course
- taught several times at the universities of Dortmund, Mainz and Chemnitz
- aimed at beginning graduate students (3rd and 4th year students in Germany)
Structure of the course:
- consists of separate condensed matter physics problems to be solved using computer simulations
- each course unit comprises a lecture (one hour) and tutored computer work (two hours)
- the lecture covers basic physical ideas and algorithmical aspects
- computer work: students develop programs, run simulations and analyze and visualize the data
What it is not ...
Course concentrates on computational physics aspects
It is therefore
- not a replacement for a conventional course in condensed matter physics; instead, it is best taught in parallel to or
right after such a course
- not a course in numerical mathematics; algorithmical questions are discussed only to the extend necessary to solve
the problem; usually a rather simple method is used
- not a programming course; some programming experience is expected, most students have already used a
programming language at this stage
Implementational details:
- students develop programs from scratch
- no code is provided except for very few subroutines, e.g. for eigenvalue problems
- FORTRAN 90/95 is the language supported in the course
- students are allowed to use a different programming language (usually C, C++ or Pascal)
- visualization of the results via GNUPlot
Summary of topics
Algorithmic preliminaries
- series expansions (power series, Fourier series)
- random numbers (generation, distributions)
Random walks and growth models
- polymer statistics (simple walks and self-avoiding walks)
- percolation, diffusion limited aggregation (self-similarity, fractals)
Phonons
- dispersion of linear chain and square lattice
- phonon focusing
- molecular dynamics simulations
Electronic states
- band structure (dispersion relation, density of states)
- electronic hopping transport (nearest neighbor and variable range hopping)
- Anderson localization (localization length, metal-insulator transition)
Thermodynamics and related problems
- Ising model (equilibrium properties, phase transition)
- cellular automata for chemical reactions
- simulated annealing, traveling salesman
Summary of computational techniques
- integration of ordinary differential equations (e.g., Newtons equations)
- eigenvalue problems (e.g. stationary Schrödinger equation)
- transfer matrix methods
- Stochastic methods:
- random walks (simple sampling Monte Carlo)
- Metropolis algorithm (importance sampling Monte Carlo)
- dynamical Monte Carlo
- stochastic optimization
Conclusions
- computational physics combines a unique set of techniques from theoretical physics, experimental physics,
numerical mathematics, and computer science
- teaching computational physics, i.e. teaching how to solve a physics problem using computer simulations, therefore
requires specific types of courses
- we have successfully used a laboratory style course to teach computational condensed matter physics
course consists of a set of separate physics problems the students solve by
- analyzing the physical problem
- reformulating the problem in a way suitable for a simulation
- choosing an efficient numerical algorithm
- writing the computer code (from scratch)
- running the computer simulation
- analyzing and interpreting the data obtained
selection of problems is made to
- cover a wide variety of computational techniques
- provide connections to state-of-the-art research topics
Condensed Matter Physics on the Computer
-
Example problems
Michael Schreiber and Thomas Vojta
Institut für Physik, Technische Universität Chemnitz,
D-09107 Chemnitz, Germany
- Hopping conductivity in semiconductors
- Damage spreading in kinetic Ising models
- Anderson localization of disordered electrons
Hopping conductivity and damage spreading:
extensions and generalizations of basic equilibrium Monte Carlo (Metropolis) algorithms as taught in the Ising model
problem
Hopping conductivity: dynamics close to equilibrium
Damage spreading: dynamics far from equilibrium
Anderson localization of disordered electrons:
transfer matrix methods, finite size scaling M. Schreiber (Saturday)
Example: Hopping conductivity in semiconductors
Prerequisites: Problems on random numbers and on the Monte Carlo simulation of the Ising model
Goal: Investigation of the electrical transport in weakly doped semiconductors at low temperatures
Physics background:
- at low temperatures weakly doped semiconductors are insulators
- electrons can move by thermally (phonon) assisted hopping between localized impurity states
- without external electric field electrons can hop in all directions with equal probability no resulting current
- with electric field hopping probability depends on field net current results
Specific tasks:
- determine dependence of current on external voltage
- calculate conductivity
- study its temperature dependence
A simple model for the hopping transport problem
- linear, square or cubic lattice of (impurity) sites i with one electron state per site
- potential vi at site i is random, drawn from an interval [-W,W], to model the random environment of an impurity
- hopping is possible downwards in energy by emission of a photon or upwards in energy by absorption of a photon
- hopping rate Wij between two sites depends on distance rij=|rj-ri|
and on potential difference Vij = vj- vi- E(rj-ri) (E is the external electric field)
(a = localization length of impurity state, T = temperature)
- hopping process with these rates can be simulated by Monte Carlo procedure
Schedule for the computer work
First day:
- develop a Monte-Carlo program for thermally assisted hopping in a one-dimensional impurity band
- study diffusion of electrons without external electric field
Second day:
- determine the dependence of the current on the electric field, check Ohm's law, calculate the conductivity
- determine the temperature dependence of the conductivity
- discuss the various regimes (nearest neighbor hopping and variable range hopping
Optional:
- try other approaches to calculate conductivity (Kubo formula)
Connection to state-of-the-art research
Course problem: hopping transport of non-interacting electrons
well understood in general
However: real electrons interact via Coulomb potential
hopping transport of interacting electrons not fully understood today
Questions: What are the hopping entities, single electrons, pairs, clusters?
How do they couple to the phonons?
What is the temperature dependence of the resulting conductivity?