We investigate the Fock-Darwin states of the massless chiral fermions confined in a graphitic parabolic quantum dot. In light of Klein tunneling, we analyze the condition for confinement of the Dirac fermions in a cylindrically symmetric potential. New features of the energy levels of the Dirac electrons as compared to the conventional electronic systems are discussed. We also evaluate the dipole-allowed transitions in the energy levels of the dots. We propose that in the high magnetic field limit, the band parameters can be accurately determined from the dipole-allowed transitions.
Abstract. We show how the problem of nontermination proving can be reduced to a question of underapproximation search guided by a safety prover. This reduction leads to new nontermination proving implementation strategies based on existing tools for safety proving. Our preliminary implementation beats existing tools. Furthermore, our approach leads to easy support for programs with unbounded nondeterminism.
A phenomenological model defined in a two dimensional lattice is employed to investigate the d-wave superconductivity and the competing antiferromagnetic order in cuprate superconductors. Near the optimally doped regime, we show that it is possible to induce the spin density wave (SDW) and the accompanying charge density wave (CDW) orders with stripe modulations by applying a magnetic field. The periods of the magnetic field induced SDW and CDW are 8a and 4a, respectively. The spatial profiles of the local density of states (LDOS) at various bias energies have also been numerically studied. Near and beyond the energies of the vortex core states, we found that the LDOS may display the CDW stripe-like modulation throughout the whole magnetic unit cell. For energies closer to the zero bias, the stripes appear to be rather localized to the vortex. The intensity of the integrated spectrum of the LDOS shows that the strength of the stripe modulation is decaying away from the vortex core. This feature is in good agreement with STM experiments. The case for the magnetic-field induced SDW/CDW with 4-fold symmetry has also been studied. [3] seems to confirm the coexistence of the static charge modulation and the superconductivity on Bi 2 Sr 2 CaCu 2 O 8 (BSCCO) under a magnetic field. The authors reported that a four unit cell checkerboard pattern is localized in a small region around the vortex, and its intensity is exponentially decaying away from the vortex core. All these experiments indicate that the AF fluctuations could be pinned by the vortex cores and they may form a static SDW/charge-density-wave (CDW) like modulations in certain samples of cuprate superconductors. Theoretically, a number of works proposed that the observed checkerboard patterns could be explained by the SDW order with the two-dimensional (2D) [4,5,6] or stripe [7,8,9] modulations. Both the 2D-and stripe-SDW orders induced by the magnetic field are well known to have the accompanying CDW modulations. In particularly, the checkerboard pattern has been attributed to the superposition of stripe modulations of the CDW [7,10] oriented along x-and y-directions. In all these studies, the STM spectra obtained from the experiment have been directly interpreted in terms of the CDW order induced by the magnetic field. This is because that the conventional wisdom leads us to believe that the symmetry of the CDW order should be reflected in the STM spectra. On the other hand, the CDW order represents the charge density configuration in the ground state while the STM spectrum or the spatial distribution of the local density of states (LDOS) is determined by the behavior of the low-energy excitations in the system. It is therefore extremely interesting to know the difference between the CDW order and the spatial profile of the LDOS. This issue has not yet been addressed in the literature, and thus is the main purpose of the present paper.The phonemenological t − t ′ − U − V model defined in a 2D lattice and the Bogoliubov-de Gennes' equations will be used to n...
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