Substitutional carbon doping of the honeycomb-like boron nitride (BN) lattices in two-dimensional (nanosheets) and one-dimensional,(nanoribbons and nanotubes) nanostructures was achieved via in situ electron beam irradiation in an energy-filtering 300 kV high-resolution transmission electron microscope using a C atoms feedstock intentionally introduced into the microscope. The C substitutions for B and N atoms in the honeycomb lattices were demonstrated through electron energy loss spectroscopy, spatially resolved energy-filtered elemental mapping, and in situ electrical measurements. The preferential doping was found to occur at the sites more vulnerable to electron beam irradiation. This transformed BN nanostructures from electrical Insulators to conductors. It was shown that B and N atoms in a BN nanotube could be nearly completely replaced with C atoms via electron-beam-induced doping. The doping mechanism was proposed to rely on the knockout ejections-of B and N atoms and subsequent healing of vacancies with supplying C atoms.
In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.
In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.