On Tractable Exponential Sums

We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}}$, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N

Fermi-level position at a semiconductor-metal interface

We have investigated the phenomenon of Fermi-level pinning by charged defects at the semiconductor-metal interface. Two limiting cases were investigated. In the first case we modeled an infinitely thick metallic coverage. In the second case we modeled a submonolayer coverage by using a free semiconductor surface containing defects. In both cases we assumed that most of the defect-induced interface states are localized inside the semiconductor, not more than a few angstroms away from the metal. Under these conditions we have estimated the difference in Fermi-level position between n- and p-type semiconductors to be less than 0.05 eV in the case of a thick metallic coverage. This difference was shown to be the maximum possible one, and it occurs only when there is no pinning. When there is pinning, this difference is even smaller. No such upper bound on the difference in Fermi-level position exists in the case of submonolayer coverage. We have also found that the defect density required to pin the Fermi level is ∼10^14 cm^-2 in the case of a thick metallic coverage, but only ∼10^12 cm^-2 in the case of a submonolayer coverage

Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.Comment: to appear in SIAM Journal on Discrete Mathematic

Rare-earth doped glasses and light managing in solar cells

Glasses doped with rare earth elements possess unique photoluminescence properties. They find application in several devices, such as lasers, optical amplifiers, and sensors. More recently, rare-earth doped glass thin films have been the subject of investigation for the development of frequency-converting layers able to increase the efficiency of silicon solar cells. Another approach to the improvement of the performance of a solar cell is based on the capture of a larger flux of light by the detector, which can be obtained by surface texture, plasmonics, or waveguide structures. Here, the recent advances in this area will be briefly reviewed

The Quark Model and bb Baryons

The recent observation at the Tevatron of $\Sigma_b^{\pm}$ ($uub$ and $ddb$) baryons within 2 MeV of the predicted $\Sigma_b - \Lambda_b$ splitting and of $\Xi_b^-$ $(dsb)$ baryons at the Tevatron within a few MeV of predictions has provided strong confirmation for a theoretical approach based on modeling the color hyperfine interaction. The prediction of $M(\Xi^-_b) = 5790$ to 5800 MeV is reviewed and similar methods used to predict the masses of the excited states $\Xi_b^\prime$ and $\Xi_b^*$. The main source of uncertainty is the method used to estimate the mass difference $m_b - m_c$ from known hadrons. We verify that corrections due to the details of the interquark potential and to $\Xi_b$--$\Xi_b^\prime$ mixing are small. For S-wave $qqb$ states we predict $M(\Omega_b) = 6052.1 \pm 5.6$ MeV, $M(\Omega^*_b) = 6082.8 \pm 5.6$ MeV, and $M(\Xi_b^0) = 5786.7 \pm 3.0$ MeV. For states with one unit of orbital angular momentum between the $b$ quark and the two light quarks we predict $M(\Lambda_{b[1/2]}) = 5929 \pm 2$ MeV, $M(\Lambda_{b[3/2]}) = 5940 \pm 2$ MeV, $M(\Xi_{b[1/2]}) = 6106 \pm 4$ MeV, and $M(\Xi_{b[3/2]}) = 6115 \pm 4$ MeV. Results are compared with those of other recent approaches.Comment: 20 pages, 1 figure, to be published in Annals of Physics. Eq. (58) correcte

Model-independent analysis for determining mass splittings of heavy baryons

We study the hyperfine mass differences of heavy hadrons in the heavy quark effect theory (HQET). The effects of one-gluon exchange interaction are considered for the heavy mesons and baryons. Base on the known experimental data, we predict the masses of some heavy baryons in a model-independent way.Comment: 14 pages, 1 figur