The Geography of Non-formal Manifolds

We show that there exist non-formal compact oriented manifolds of dimension $n$ and with first Betti number $b_1=b\geq 0$ if and only if $n\geq 3$ and $b\geq 2$, or $n\geq (7-2b)$ and $0\leq b\leq 2$. Moreover, we present explicit examples for each one of these cases.Comment: 8 pages, one reference update

Optical band gap and the Burstein–Moss effect in iodine doped PbTe using diffuse reflectance infrared Fourier transform spectroscopy

Optical absorption edge measurements are performed on I doped PbTe using diffuse reflectance infrared Fourier transform spectroscopy. The Burstein–Moss shift, an increase in the absorption edge (optical band gap) with increasing doping level, is explored. The optical gap increases on the order of 0.1 eV for doping levels ranging from 3 × 10^(18) to 2 × 10^(20) cm^(−3), relevant doping levels for good thermoelectric materials. Chemical potential is estimated from transport measurements—specifically, Hall effect and Seebeck coefficient—using a single band Kane model. In heavily doped semiconductors, it is well-known that the band gap shrinks with increasing doping level. This effect, known as band gap renormalization, is fit here using an n^(1/3) scaling law which reflects an electron–electron exchange interaction. The renormalization effect in these samples is shown to be more than 0.1 eV, on the same order of magnitude as the band gap itself. Existing models do not explain such large relative changes in band gap and are not entirely self-consistent. An improved theory for the renormalization in narrow gap semiconductors is required