Spectral Approximation for Quasiperiodic Jacobi Operators

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into associated dynamical systems. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary to get detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-$K$ Jacobi operator in $O(K^2)$ operations, and use it to investigate the spectra of Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse potentials

DC-electric-field-induced and low-frequency electromodulation second-harmonic generation spectroscopy of Si(001)-SiO2_2 interfaces

The mechanism of DC-Electric-Field-Induced Second-Harmonic (EFISH) generation at weakly nonlinear buried Si(001)-SiO$_2$ interfaces is studied experimentally in planar Si(001)-SiO$_2$-Cr MOS structures by optical second-harmonic generation (SHG) spectroscopy with a tunable Ti:sapphire femtosecond laser. The spectral dependence of the EFISH contribution near the direct two-photon $E_1$ transition of silicon is extracted. A systematic phenomenological model of the EFISH phenomenon, including a detailed description of the space charge region (SCR) at the semiconductor-dielectric interface in accumulation, depletion, and inversion regimes, has been developed. The influence of surface quantization effects, interface states, charge traps in the oxide layer, doping concentration and oxide thickness on nonlocal screening of the DC-electric field and on breaking of inversion symmetry in the SCR is considered. The model describes EFISH generation in the SCR using a Green function formalism which takes into account all retardation and absorption effects of the fundamental and second harmonic (SH) waves, optical interference between field-dependent and field-independent contributions to the SH field and multiple reflection interference in the SiO$_2$ layer. Good agreement between the phenomenological model and our recent and new EFISH spectroscopic results is demonstrated. Finally, low-frequency electromodulated EFISH is demonstrated as a useful differential spectroscopic technique for studies of the Si-SiO$_2$ interface in silicon-based MOS structures.Comment: 31 pages, 14 figures, 1 table, figures are also available at http://kali.ilc.msu.su/articles/50/efish.ht