Filling minimality of Finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu's isosystolic inequality to Finsler metrics, both for Holmes-Thompson and Busemann definitions of Finsler area.Comment: 16 pages, v2: improved introduction and formattin

Searching for Hyperbolicity

This is an expository paper, based on by a talk given at the AWM Research Symposium 2017. It is intended as a gentle introduction to geometric group theory with a focus on the notion of hyperbolicity, a theme that has inspired the field from its inception to current-day research

Electronic properties of Si/Si1–x–yGexCy heterojunctions

We have used admittance spectroscopy and deep-level transient spectroscopy to characterize electronic properties of Si/Si1–x–yGexCy heterostructures. Band offsets measured by admittance spectroscopy for compressively strained Si/Si1–x–yGexCy heterojunctions indicate that incorporation of C into Si1–x–yGexCy lowers both the valence- and conduction-band edges compared to those in Si1–xGex by an average of 107 ± 6 meV/% C and 75 ± 6 meV/% C, respectively. Combining these measurements indicates that the band alignment is type I for the compositions we have studied, and that these results are consistent with previously reported results on the energy band gap of Si1–x–yGexCy and with measurements of conduction band offsets in Si/Si1–yCy heterojunctions. Several electron traps were observed using deep-level transient spectroscopy on two n-type heterostructures. Despite the presence of a significant amount of nonsubstitutional C (0.29–1.6 at. %), none of the peaks appear attributable to previously reported interstitial C levels. Possible sources for these levels are discussed

Deep-level transient spectroscopy of Si/Si1–x–yGexCy heterostructures

Deep-level transient spectroscopy was used to measure the activation energies of deep levels in n-type Si/Si1–x–yGexCy heterostructures grown by solid-source molecular-beam epitaxy. Four deep levels have been observed at various activation energies ranging from 231 to 405 meV below the conduction band. The largest deep-level concentration observed was in the deepest level and was found to be approximately 2 × 10^15 cm^–3. Although a large amount of nonsubstitutional C was present in the alloy layers (1–2 at. %), no deep levels were observed at any energy levels that, to the best of our knowledge, have been previously attributed to interstitial C

Band offsets in Si/Si1–x–yGexCy heterojunctions measured by admittance spectroscopy

We have used admittance spectroscopy to measure conduction-band and valence-band offsets in Si/Si1–xGex and Si/Si1–x–yGexCy heterostructures grown by solid-source molecular-beam epitaxy. Valence-band offsets measured for Si/Si1–xGex heterojunctions were in excellent agreement with previously reported values. Incorporation of C into Si1–x–yGexCy lowers the valence- and conduction-band-edge energies compared to those in Si1–xGex with the same Ge concentration. Comparison of our measured band offsets with previously reported measurements of energy band gaps in Si1–x–yGexCy and Si1–yCy alloy layers indicate that the band alignment is Type I for the compositions we have studied and that our measured band offsets are in quantitative agreement with these previously reported results

Measurement of band offsets in Si/Si1–xGex and Si/Si1–x–yGexCy heterojunctions

Realization of group IV heterostructure devices requires the accurate measurement of the energy band offsets in Si/Si1–xGex and Si/Si1–x–yGexCy heterojunctions. Using admittance spectroscopy, we have measured valence-band offsets in Si/Si1–xGex heterostructures and conduction-band and valence-band offsets in Si/Si1–x–yGexCy heterostructures grown by solid-source molecular-beam epitaxy. Measured Si/Si1–xGex valence-band offsets were in excellent agreement with previously reported values. For Si/Si1–x–yGexCy our measurements yielded a conduction-band offset of 100 ± 11 meV for a n-type Si/Si0.82Ge0.169C0.011 heterojunction and valence-band offsets of 118 ± 12 meV for a p-type Si/Si0.79Ge0.206C0.004 heterojunction and 223 ± 20 meV for a p-type Si/Si0.595Ge0.394C0.011 heterojunction. Comparison of our measured band offsets with previously reported measurements of energy band gaps in Si1–x–yGexCy and Si1–yCy alloy layers indicates that the band alignment is type I for the compositions we have studied and that our measured band offsets are in quantitative agreement with these previously reported results

Strain relaxation kinetics in Si1–xGex/Si heterostructures

Strain relaxation in Si1–xGex/Si superlattices and alloy films is studied as a function of ex situ anneal treatment with the use of x-ray diffraction and Raman spectroscopy. Samples are grown by molecular-beam epitaxy at an unusually low temperature (≈365 °C). This results in metastably strained alloy and superlattice films significantly in excess of critical thicknesses previously reported for such structures. Significant strain relaxation is observed upon anneal at temperatures as low as 390 °C. After a 700 °C, 2 h anneal, superlattices are observed to relax less fully (~43% of coherent strain) than corresponding alloys (~84% of coherent strain). Also, the strain relaxation kinetics of a Si1–xGex alloy layer is studied quantitatively. Alloy strain relaxation is approximately described by a single, thermally activated, first order kinetic process having activation energy Ea=2.0 eV. The relevance of our results to the microscopic mechanisms responsible for strain relaxation in lattice-mismatched semiconductor heterostructures is discussed

A compactness theorem for complete Ricci shrinkers

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.Comment: 28 pages, final version, to appear in GAF

More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons

A recent precise formulation of the hoop conjecture in four spacetime dimensions is that the Birkhoff invariant $\beta$ (the least maximal length of any sweepout or foliation by circles) of an apparent horizon of energy $E$ and area $A$ should satisfy $\beta \le 4 \pi E$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies $\ell^2 \le \pi A$. We have tested these conjectures on the horizons of all four-charged rotating black hole solutions of ungauged supergravity theories and find that they always hold. They continue to hold in the the presence of a negative cosmological constant, and for multi-charged rotating solutions in gauged supergravity. Surprisingly, they also hold for the Ernst-Wild static black holes immersed in a magnetic field, which are asymptotic to the Melvin solution. In five spacetime dimensions we define $\beta$ as the least maximal area of all sweepouts of the horizon by two-dimensional tori, and find in all cases examined that $\beta(g) \le \frac{16 \pi}{3} E$, which we conjecture holds quiet generally for apparent horizons. In even spacetime dimensions $D=2N+2$, we find that for sweepouts by the product $S^1 \times S^{D-4}$, $\beta$ is bounded from above by a certain dimension-dependent multiple of the energy $E$. We also find that $\ell^{D-2}$ is bounded from above by a certain dimension-dependent multiple of the horizon area $A$. Finally, we show that $\ell^{D-3}$ is bounded from above by a certain dimension-dependent multiple of the energy, for all Kerr-AdS black holes.Comment: 25 page