System-size convergence of point defect properties: The case of the silicon vacancy

We present a comprehensive study of the vacancy in bulk silicon in all its charge states from 2+ to 2-, using a supercell approach within plane-wave density-functional theory, and systematically quantify the various contributions to the well-known finite size errors associated with calculating formation energies and stable charge state transition levels of isolated defects with periodic boundary conditions. Furthermore, we find that transition levels converge faster with respect to supercell size when only the Gamma-point is sampled in the Brillouin zone, as opposed to a dense k-point sampling. This arises from the fact that defect level at the Gamma-point quickly converges to a fixed value which correctly describes the bonding at the defect centre. Our calculated transition levels with 1000-atom supercells and Gamma-point only sampling are in good agreement with available experimental results. We also demonstrate two simple and accurate approaches for calculating the valence band offsets that are required for computing formation energies of charged defects, one based on a potential averaging scheme and the other using maximally-localized Wannier functions (MLWFs). Finally, we show that MLWFs provide a clear description of the nature of the electronic bonding at the defect centre that verifies the canonical Watkins model.Comment: 10 pages, 6 figure

Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a $(1-\epsilon)$-approximation for the number of perfect matchings of a $\lambda$-dense bipartite graph, using $O(n^{\frac{1-2\lambda}{8\lambda}+\epsilon^{-2}})$ samples. With size $n$ on each side and for $\frac{1}{2}>\lambda>0$, a $\lambda$-dense bipartite graph has all degrees greater than $(\lambda+\frac{1}{2})n$. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes

Single-particle-sensitive imaging of freely propagating ultracold atoms

We present a novel imaging system for ultracold quantum gases in expansion. After release from a confining potential, atoms fall through a sheet of resonant excitation laser light and the emitted fluorescence photons are imaged onto an amplified CCD camera using a high numerical aperture optical system. The imaging system reaches an extraordinary dynamic range, not attainable with conventional absorption imaging. We demonstrate single-atom detection for dilute atomic clouds with high efficiency where at the same time dense Bose-Einstein condensates can be imaged without saturation or distortion. The spatial resolution can reach the sampling limit as given by the 8 \mu m pixel size in object space. Pulsed operation of the detector allows for slice images, a first step toward a 3D tomography of the measured object. The scheme can easily be implemented for any atomic species and all optical components are situated outside the vacuum system. As a first application we perform thermometry on rubidium Bose-Einstein condensates created on an atom chip.Comment: 24 pages, 10 figures. v2: as publishe