# Research Interests

My research interest is the development of a theoretical understanding of the properties of disordered systems with emphasis on electron and photon localization, photonic crystals, random lasers, left-handed materials, random magnetic systems (spin glasses and random fields), effects of disorder on nonlinear problems and amorphous semiconductors. The theoretical techniques used to attack these problems are analytical (Coherent Potential Approximation and Mean Field Theory) and numerical (Monte Carlo and molecular dynamics simulations and finite scaling analysis). The theoretical models developed are often quite sophisticated to accurately reflect the complexity of real materials and processes and as a consequence, utilization of large scale computers has become routine. Considerable effort is also spent in the investigation of analytical approaches.

Please Select a Topic:

- Electron and Light Localization
- Random Magnetic Systems
- Photonic Band Gaps
- Random Lasers
- Left-Handed Materials
- Wave Propagation in Complex Media
- Amorphous Semiconductors

### 1. Electron and Light Localization

Over the last five years we have developed a theory for the behavior of the electronic and transport properties near the band edges of disordered materials. An analogy between the localization problem with that of a bound state in a potential well was developed. The potential-well analogy (PWA) theory permits explicit calculations of different properties such as localization lengths, conductivities, mobility edges, etc., from quantities that can be obtained from mean field theories such a Coherent Potential Approximation (CPA). We have demonstrated that the PWA coupled with the CPA is capable in producing results not only in qualitative but in quantitative agreement with an independent, very reliable numerical approach. The advantage of the PWA-CPA approach over numerical approaches is analytical and gives results, even for complicated quantities, almost immediately. The PWA-CPA approach applied by our group to electronic localization, phonon localization, and recently to light localization with tremendous success. We currently use these techniques to understand the fundamental mechanisms responsible for the localization of light and other classical waves in random systems. We hope that these studies will enable us to establish the correct theory in describing localization of light and will give the range of parameters most promising to search for a mobility edge in the classical wave localization problem.

### 2. Random Magnetic Systems

Random magnetic fields are predicted to have drastic effects on the phases and phase transitions that occur in magnetic materials. The transition behavior is much more elaborate and cannot be discussed using equilibrium statistical mechanics. Entirely new experimental and theoretical issues arise connected with both the accessibility and uniqueness of the ground state. A simple realization of the random field Ising model (RFIM) is provided by a dilute anti-ferromagnet (DAF) in a uniform magnetic field. We have completed a systematic study of the site-dependent mean field theory of the RFIM and DAF in a field. We have recently started studies on the investigation in the non-quilibrium time-dependent structure factor S(q,t) of the RFIM as it is slowly cooled from a high-temperature paramagnetic phase to low enough temperatures. It has been heuristically suggested without any firm evidence that the loss of long range order changes the ‘-function Bragg peak to a Lorentzian square lineshape. This can be used as a signature of the presence of disorder in experimental systems. These results are expected to be tested experimentally on surface overlayers using a precise LEED optics facility that Tringides has under construction.

### 3. Photonic Band Gaps

There is a joint theoretical and experimental program which is intended to design, fabricate and characterize a new class of composite materials which possess forbidden ranges of frequencies, in which electromagnetic waves cannot propagate in any direction. These materials are called "photonic crystals" and the forbidden frequencies are called "photonic gaps", and they can be regarded as photonic analogues of electronic semiconductors with electronic semiconductors with electronic gaps. This class of material will exhibit many interesting physical properties, and will find important practical applications in lasers, mirrors, resonators, filters, and quantum optical devices. The theoretical effort will be directed at designing periodic dielectric structures that gives the optimal frequency gap for various applications with special emphasis on the fabricability of these structures, especially in the sub-micron length scale where these materials will find applications in optical instruments. The main purpose of the experimental effort is to fabricate the structures designed by theory in the micron and submicron length scales, using micro-fabrication patterning and etching techniques. The structural and optical properties of these micro-structures will be characterized and studied using optical techniques. The effect of disorder, defects and structural imperfections on the propagation of electromagnetic waves through these photonic crystals will also be studied both theoretically and experimentally.

### 4. Random Lasers

Disordered systems that both scatter and amplify light (the so-called random lasers) has been a fascinating subject to study. Over the last five years, there have been substantial theoretical and experimental efforts to unravel the mechanism that gives rise to this amazing behavior. A model to simulate the phenomenon of random lasing is presented. It couples Maxwell's equations with the rate equations of electronic population in a disordered system. Finite difference time domain methods are used to obtain the field pattern and the spectra of localized lasing modes inside the system. A critical pumping rate exists for the appearance of the lasing peaks. The dependence of this critical rate on the length of the system and the strength of disorders is obtained. The number of lasing modes increases with the pumping rate and the length of the system. There is a lasing mode repulsion. This property leads to a saturation of the number of modes for a given size system and a relation between the localization length and average mode length. Similar behavior is expected to be seen in photonic crystals, too.

### 5. Left-Handed Materials

The recent demonstration of a Left-Handed Material (LHM) by the UCSD group of Dr. Schultz, underscores the relevance of utilizing structured materials to create electromagnetic responses not available in naturally occurring materials. This LHM made use of an array of conducting, nonmagnetic elements to achieve a negative effective permeability m, and an array of conducting wires to achieve a negative effective permittivity e, the simultaneous combination of which has never before been observed in any previously known material. Since the product of (-m)(-e) is a positive number, Maxwell’s equations permit propagation, and the LHM thus offers exciting new opportunities for controlling and modifying the scattering of electromagnetic radiation. As Veselago discussed many years ago, LH materials will display unique “reversed” electromagnetic properties. This suggests that in LHM Snell’s law is reversed, the group velocity (i.e. energy velocity) and the phase velocity (wave vector direction) must be opposite. The Doppler and Cherenkov effects will also be reversed in a LHM. An approaching source will appear to radiate at a lower frequency. Although these counteractive properties follow directly from Maxwell’s equations, which still hold in these unusal materials, they have yet to be demonstrated experimentally. We therefore will design new geometries and construction possibilities and to collaborate with experimental groups (UCSD, Boeing, UCLA and Bilkent Univ.) to achieve novel structures, with the ultimate goal of developing LH metamaterials with unique electromagnetic functionality. We will use our transfer matrix methods and FDTD codes in these studies.

### 6. Wave Propagation in Complex Media

The information will be added at a later date.

### 7. Amorphous Semiconductors

Recently existing progress has been achieved by the theory group at MRC in modeling amorphous semiconductors. This work was initiated by Rana Biswas with the computer-generation of large realistic models of amorphous Si (a-Si) using molecular dynamics simulations of quenched molten Si. Models with 500 2000 Si atoms, incorporating periodic boundary conditions, are easily made. The models had distributions of bond lengths and bond angles that agreed well with experiment. A novel feature of the computer-generated models is the presence of both dangling bonds as well as five coordinated atoms. This led to a dramatic localization of low-frequency model consisting primarily of vibrations of undercoordinated atoms as well as high-frequency modes. Our new thrust has been to develop a tight-binding electronic structure program, that can describe well the electronic densities of states and optical absorption of amorphous silicon. We will also investigate with molecular dynamics simulations how the recombination energy of photoexcited carriers leads to metastable changes (Staebler-Wronski effect), particularly the breaking of weak Si-Si bonds. The understanding of the fundamental mechanisms responsible for the Staebler-Wronski effect in Si will be of considerable benefit to the photovoltaic industry.