Operator theory plays an important role in theoretical physics. A brief introduction to operator theory will be given first. The notion of homogeneous operators will be introduced, their interesting properties will be discussed. Some explicit simple examples will be presented.

Considerable efforts have been made to obtain high quality semiconductor heterostructures for electronic and optical applications. In the 3–5m mid wavelength infrared range (MWIR), an alternative to theHgCdTe dominating material technologies can be the InAs/GaSb superlattices(SL) and InSb quantum dots. In many cases, defects at the interfacesaffect the performance of the films in electronic and optoelectronicapplications. Thus, controlling the defect microstructure is of criticalimportance in these materials systems to realize their potential. In alarge lattice-mismatched system, it is almost impossible to suppress thegeneration of misfit dislocations until the epilayer grows to a usablethickness for device applications. In this talk we present GaSb/InAssupperlattice (SL) structures grown on GaSb (practically no latticemismatch) and GaAs (a large lattice mismatch) substrates and InSb QDstructure grown grown on GaSb substrate. Because of its extremely highspatial resolution transmission electron microscopy (TEM) is a naturaltechnique to apply to the nanometer-scale characterization of thesesemiconductor heterostructures. In particular, cross-sectionaltransmission electron microscopy is a powerful tool for investigatingnanometer-scale interface properties of semiconductor materials anddevices.

Atomic diffusion in amorphous and nanocrystalline alloys has been a subject of great interest, as it governs the changes in the structure of these alloys. Depth profiling using radioactive tracer or SIMS have been the most extensively used techniques for such studies. However, typical depth resolution of these techniques is a few nm, and this limits the minimum diffusion length that can be measured. It may be noted that the thermal stability of amorphous and nanocrystalline alloys is generally not very high, and therefore, diffusion annealing have to be done at relatively low temperatures (typically 400K-700K). As a result, the diffusion lengths achievable within a reasonable annealing time can be as small as a nanometer. Also, both amorphous and nanocrystalline alloys exhibit structural relaxation at still lower temperatures and a possible study of the effects of structural relaxation on atomic diffusivity would involve measurement of still smaller diffusion lengths. We have used nuclear resonance reflectivity of x-rays and neutron reflectivity from isotopic multilayers for precise measurement of self-diffusion in chemically homogeneous systems. The alternate layers have the same chemical composition and differ only in the isotopic abundance of one of the species (e.g., 57Fe). If the energy of the incident radiation is tuned to the nuclear resonance energy of 57Fe, large scattering contrast develops between layers containing natural Fe and 57Fe due to strong nuclear resonance scattering from 57Fe nuclei. This results in a Bragg peak in the reflectivity corresponding to the bilayer periodicity of the multilayer. Similar Bragg peak is observed in neutron reflectivity due to different scattering lengths for 57Fe and 56Fe. Height of this Bragg peak can be monitored to get information about the interdiffusion of 57Fe isotope across the interfaces. Results will be presented on self-diffusion of Fe in Fe-N and Fe-Zr alloys. In nanocrystalline Fe60N40, variation in diffusivity due to structural relaxation at temperatures as low as 393K could be observed. In Fe85Zr15 alloy films, neutron reflectivity measurements show that Fe diffusivity in amorphous and nanocrystalline states is very similar. It is suggested that in nanocrystalline phase atomic diffusion occurs mainly via grain-boundary regions which have structure similar to that in the parent amorphous phase. Effects of external stress on diffusivity will also be presented.

We consider a model for a single molecule with a large frozen spin sandwiched in between two BCS superconductors at equilibrium, and show that this system has a $pi$ junction behavior at low temperature. The $pi$ shift can be reversed by varying the other parameters of the system, e.g., temperature or the position of the quantum dot level, implying a controllable $pi$ junction with novel application as a Josephson current switch. In contrast to previous works the importance of the contribution from the continuum of states above the superconducting gap is brought out. The free energy for certain configuration of parameters shows a bistable nature, which is a necessary pre-condition for achievement of a qubit.