Theory of large-scale electronic structure calculation and nanoscale mechanical property in fracture behavior of silicon
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Theory of large-scale electronic structure calculation and nanoscale mechanical property in fracture behavior of silicon Takeo Hoshi1,2 , Ryu Takayama3,1 a , Yusuke Iguchi1 and Takeo Fujiwara1,2 1
Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan. Core Research for Evolutional Science and Technology (CREST-JST), Japan Science and Technology Agency, Honcho, Kawaguchi-shi, Saitama, Japan. 3 Research and Development for Applying Advanced Computational Science and Technology (ACT-JST), Japan Science and Technology Agency. 2
ABSTRACT Several theories and program codes were developed for large-scale atomistic simulations with fully quantum mechanical description of electron systems. The fundamental concepts are generalized Wannier state and Krylov subspace. Test calculations were carried out with upto 106 atoms using a standard workstation. How electronic state is described in large-scale calculation was demonstrated on Si(001)-(2 × 1) surface. As a practical application, cleavage fracture of silicon was simulated with 10-nm-scale samples for investigating its nanoscale mechanical behavior. Discussions are focused on the unstable (001) cleavage mode and the stable (experimentally observed) (111)-(2 × 1) cleavage mode. As well as elementary surface reconstruction, step formation and bending in cleavage path were observed. These results were compared with experiments, such as scanning tunneling microscope (STM). INTRODUCTION Large-scale atomistic simulation with quantum mechanical freedom of electrons requires manipulation of a large Hamiltonian matrix. In order to calculate physical quantities of a system, we should obtain either one-electron eigenstates or the one-body density matrix of the system. The calculation of eigenstate is usually reduced to matrix diagonalization procedure and this procedure results in severe computational cost for a large-scale system. Any physical quantity X can be evaluated by means of the one-body density matrix ρ as ˆ = Tr[ˆ ˆ = hXi ρX]
Z Z
drdr 0 ρ(r, r 0 )X(r 0 , r).
(1)
Even though the density matrix ρ(r, r 0 ) is of long-range, only the short-range behavior of the ˆ is a short-range operator. The energy and forces density matrix is necessary in case that X acting on an individual atom are really this case and the locality of the Hamiltonian has this advantage in large-scale calculation. [1] Therefore, the essential methodology for large-scale electronic structure calculation is how to obtain the density matrix ρ without calculating eigenstate.
a
Present address:Canon Inc., Analysis technology center, Morinosato-Wakamiya, Atsugi-shi, Kanagawa, Japan
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THEORY We have developed a set of theories and program codes for large-scale electronic structure calculation with obtaining density matrix, instead of eigenstate. [2, 3, 4, 5, 6, 7, 8, 9] These theories are founded by generalized Wannier state [2, 3, 4] or Krylov subspace. [6, 8] A bench mark is shown in Fig. 1(a), in which one of our methods, called the ‘perturbativeWannier-state m
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