Quasi-Exact Solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lam\'e potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure

Hydrogen behaviour in amorphous Si/Ge nano-structures after annealing

The H behaviour in a-Si, a-Ge, a-SiGe is still debated, also thanks to their employment in photovoltaic solar cells whose performance depends on dangling bonds passivation by H doping. a-SiGe can be obtained by depositing alternating nano-layers of a-Si and a-Ge and intermixing the 2 atoms by annealing. Here results on H behaviour upon annealing of nano-structures made of 50 couples of very thin (3 nm each) alternating layers of a-Si and a-Ge are given. The superlattice nano-structures were deposited by sputtering. Hydrogen was added at flow rates of 0.4 to 6 ml/min. ERDA of a-Si and a-Ge single layers showed that for flows ≥1.5ml/min the incorporated H saturates at 16 at% and 7 at% in Si and Ge, respectively. IR optical absorbance showed that H is mostly incorporated as Si and Ge monohydrides. Annealing was done at 673 K for times between 1 and 10 h. The H behaviour in nano-structures as a function of annealing and H content was followed by IR optical absorbance, AFM and ERDA. With increasing annealing temperature/time the surface morphology degrades with formation of bumps and craters whose size and density increase with increasing H content. Upon annealing the signals of Ge-H and Si-H complexes disappear in the IR spectra indicating that H is released to the lattice. This supports the conclusion that it is the released H that produces bumps and craters when the bumps blow up because of the high internal pressure of H. ERDA of a-Si and a-Ge single layers, showing a faster H release in a-Ge than in a-Si, suggests that in the superlattice nano-structures H is first released from the a-Ge layers upon annealing. This agrees with literature reporting on the lower binding energy of Ge-H with respect to Si-H. It also shows that H is unstable against annealing

Silicide formation reactions in a Si Co multilayered samples

Solid state reactions between amorphous Si and crystalline Co have been investigated by synchrotron radiation at Bessy Berlin, Germany . The multilayered samples with 10 periods of a Si 15 nm Co 15 nm layers were produced by magnetron sputtering and isothermally heat treated at temperatures between 523 and 593 K. From the time evolution of the XRD spectra first the growth rate of the CoSi phase as well as the decay rate of the Co layer we determined at 523 and 543 K . The kinetics were described by a power law; tk, and for the growth of CoSi k 0.65 while for the loss of the Co the k 0.77 was obtained, respectively. At higher temperatures at 573 and 593 K the formation and growth of the Co2Si layer, at the expense of the Co and already existing CoSi layers, was observed with exponents of about 1 for all the above kinetics. These results, together with the results of resistance kinetics measurements, in similar multilayered as well as bi layered samples at similar temperatures, providing similar exponents will be presented. Possibility of the interface reaction control and or the effect of the diffusion asymmetry which was recently published for the interpretation of solid state reactions with non parabolic kinetics on the nanoscale will be discusse