Senior Project (PHYS 491/492)

[Fall 2004]

Cellular Automata & Parallel Computing

The subject of Cellular Automata (CA) is one of the holy grails of scientific computing as well as statistical mechanics which was introduced in the late 1940's by John von Neumann and Stanislaw Ulam. Its practical importance was established in the late 1960's when John Horton Conway developed the Game of Life. CA's are discrete dynamical systems and are often described as a counterpart to partial differential equations, which have the capability to describe continuous dynamical systems. The meaning of discrete is, that space, time and properties of the automaton can have only a finite, countable number of states. The basic idea is not to try to describe a complex system from "above" - to describe it using difficult equations, but simulating this system by interaction of cells following easy rules. In other words: Not to describe a complex system with complex equations, but let the complexity emerge by interaction of simple individuals following simple rules.

An important asset of CA is its efficient and highly scalable performance in parallel computing platforms. In our case this will be our universities’ High Performance Computing Center. Therefore, I strongly encourage you to code in Fortran (F90).

Here is the task:

  • Read and learn the basics of CA and apply it to standard examples, such as Game of Life, the 2D Ising Model, and percolation transport
  • Learn the parallelization protocol, so-called MPI and run your parallel CA codes on our HPCC.

Previous Senior Projects

[Fall 2003]

Berry’s Phase

This is one of the most recent additions to our basic quantum mechanics knowledge by Michael Berry in 1983. By now, some graduate QM textbooks have already included this subject in their appendix, such as Sakurai (Modern QM, 2nd ed., pp. 464-480) and Böhm (QM, 3rd ed., pp. 571-658). Basically, a quantum system coupled to the rest of the Universe can be modeled by a Hamiltonian with a parametric dependence; for such systems the associated Berry’s phase is very useful and also unavoidable. The subject attracted my interest some years ago by an elegant formulation of the macroscopic polarization of solids using Berry’s phase techniques.

Here is the task:

  • Read and learn the basics of the Berry’s phase (a number of good reviews are available)
  • Study its applications like the Aharonov-Bohm effect and semiclassical electron dynamics in crystals

(Project done by Onur UMUCALILAR)

[Spring 2003]

Nonlinear Semiconductor Optics

The aim of this project is to develop familiarity (on the theoretical level) with nonlinear optics, a subject hardly touched upon in the undergraduate curricula. To limit the scope, mainly those pertaining to semiconductors will be intended, focusing on fundamental concepts. The final phase of the project will involve a representative computation of a nonlinear susceptibility of a III-V compound.

Here is the task:

  • Read and learn about the nonlinear optical susceptibility and its quantum mechanical theory; voluminous literature available*
  • Computation of a frequency-dependent second-order susceptibility, χ(2), of a semiconductor crystal.

(Project done by Gökhan ARIKAN)

[Fall 2002]

Electronic Dielectric Function of a Semiconductor

The aim of this computational project is to obtain the so-called longitudinal electric susceptibility of a semiconductor associated with electronic interband transitions. It is denoted by ε∞(q,ω) which corresponds to the electronic part of the frequency and wavevector dependent dielectric function. Having this function, you can readily extract several optical parameters, such as the index of refraction, absorption coefficient as well as the birefringence of certain crystal structures. What you need to fulfill this objective are the Bloch functions and energies of the valence and conduction bands of the semiconductor under investigation, say GaAs. Well, the empirical pseudopotential method (EPM) does just that. It  is a computationally-efficient band structure technique, based on the plane wave expansion of the non-core electronic states.

Here is the task:

  • Read about the EPM technique and do a survey on the processes that contribute to the dielectric polarizability of a semiconductor.
  • Then, write a code, essentially a number of subroutines, to calculate the  imaginary part of ε∞(q,ω).
  • Using the Kramers-Krönig relation obtain its real part, hence the index of refraction.
  • Compare your results with reality, i.e., experiments!

(Project done by Dündar YILMAZ)

[Spring 2002], [Fall 2001], [Fall 2000]

Monte Carlo Charge Transport Simulation

The push for even higher speed devices, scaled the typical gate lengths down to ~0.05 um. For such small dimensions, any reasonable electric field becomes a high field. As the electrons gain so much energy from the applied electric field, they become hot, in the sense that their energy exceeds by far the thermal energy. Analyzing the hot electron transport using pen and paper is a difficult task, some call it hopeless, as the governing equations (so-called Boltzmann transport equations) are of non-linear integro-differential nature. Analytical solutions to these equations often involve drastic approximations and miss out important physics. Not surprisingly the Monte Carlo (MC) technique has started to dominate this field over the last de cade. Some would find such a computational approach not so elegant; but if there is the possibility of new physics brought by complexity, then computer can become your best friend.

Here is the task:

  • Read about hot electron transport in a bulk III-V semiconductor such as GaAs and the associated quantum processes that scatter the electrons.
  • Write an MC code in F90 for multi-valley charge transport for a typical bulk III-V compound.

(Project done by Serdar Özdemir [S’02], Menderes IŞKIN [F’01], Engin DURGUN [F’00])

[Fall 2001]

Helical Quantum Wire

A quantum wire supports free motion along its axis, so that the axial component of the wave vector (kz) is a good quantum number and leads to a continuous energy spectrum. Quantum confinement in the plane perpendicular to wire axis traps the charge carriers, introducing a discrete spectrum; perpendicular wave vector is however, not a good quantum number, replaced by subband indices.

A helical quantum wire (HQW) is even more intriguing: as the axial translational symmetry is broken, kz bites the dust and is no longer a good quantum number. Good news is that, HQW has a special kind of symmetry, so-called glide symmetry (helix=translation+rotation), which enables us to introduce another quantum number!

HQW is not a mythological object, but exists naturally in organic (conducting) polymers; even DNA is in the form of a double helical structure.

Here is the task:

  • Analyze the energy spectrum of a HQW using his/her quantum mechanics knowledge (angular momentum, commutator algebra etc.).
  • Some numerical computation may be necessary as well.

 

(Project done by Muhammed YÖNAÇ)