In this article, based on a recent theorem by Lieb er al, we shall prove two theorems on Ihe momentum distribution functions of the half-filled Hubbard model on a ddimensional simple cubic lattice in a mathematically rigorous way. More precisely, we shall first show that the half-filled positive-U and negative4 Hubbard models have the same momentum distribution functions n (4) and n+(q). Then, we will show that n.(q) are symmetric functions about the value E = 3. Fmally. we shall briefly discuss some possible applications of these theorems to the further numerical investigations on the ground state of the Hubbard model at half-filling.

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The growth of III-V materials on Si has been pursued for two decades to facilitate the monolithic integration of light emitters with existing Si device technology. While room-temperature GaAs/AlGaAs lasers and even vertical cavity lasers have been demonstrated on Si (100) the device characteristics were only marginal due to micro-cracks and high dislocation density in the GaAs buffer. In our approach to develop a technology for mismatched epitaxy on Si, we depart from the thick metamorphic buffer.Instead, we have used AlSb QDs followed by an AlSb/GaSb superlattice buffer. The strain relief is achieved by a combination of 90¡ misfit dislocations at the AlSb/Si interface combined with surface undulations rather than threading dislocations as in the case of GaAs grown metamorphically on Si. The AlSb based buffer is free from threading dislocations. As we grow the AlSb quantum dots, they initially coalesce and then proceed to grow planar. This planarity however changes to an undulating (100) plane in order to relieve residual strain. These undulations enable the AlSb epi to increase its surface area and hence accommodate the 13% mismatch without causing threading dislocations in the material. Misfit dislocations in the material that run parallel to the substrate facilitate the undulations. We are presently working to perfect this growth technique to allow the misfit disclocation grid to fully relieve strain thereby eliminating material undulations. The material is characterized using HR-TEM, AFM, HR-XRD, RHEED and PL. We also demonstrate filamentary photopumped lasing from an InGaSb quantum well based laser structure grown on Si (100) using an AlSb quantum dot nucleation layer. . The growth of devices on the material requires a superlattice on the AlSb layer. The superlattice consists of AlSb and GaSb layers and helps return the undulating surface to a (100) plane. The resulting surface resembles step-flow growth. InGaSb PL structures grown on the superlattice have shown intensities comparable to those grown directly on GaSb substrates. The PL is fairly uniform across a 2 inch wafer, indicating uniformity of the material. Current driven devices are presently being developed. Initial data shows small leakage current at the interface, however, we believe this can be fully eliminated by optimized growth. AFM images of AlSb on Si showing defect-free nucleation to coalescence and planar growth. 0.0001 I 0.01 I P 0.03 I P 0.25 I P 0.4 I P 0.63 I P Photopumped lasing spectrum from an InGaSb/ GaSb QW structure. TEM images of the AlSb/Si interface. 674 0-7803-9217-5/05/$20.00©2005 IEEE WAA5 17:15 -17:30

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No abstract