Fujiwara Lab
We will establish the foundation and the methodology for electronic structure calculations in the following two aspects;
(1) Large Scale Electronic Structure Calculations
Establishing the quantum mechanical molecular dynamics simulator and the technology of process simulator for nano-scale systems of semiconductors and metals with from ten thousands to ten millions atoms.

Recent development of micro fabrication technique available for materials with less than100 nm encourages recent vigorous research on nanostructures, structures in nanometre and ten-nanometre scales.

Nanostructures has several states with different electronic structure (e.g. bulk region and surface region), coexisting and coflicnting with each other, and hense shows various and anormalous physical prooerties. Electronic structure theory plays a crucial role in understanding and controlling nanostructures. Dynamical simulation in these scales is, however, impractical for the conventional methodology, such as the Car-Parrinello method, owing to its heavy computational cost.

We will establish the quantum mechanical molecular dynamics simulation method and the technology of process simulator for nano-scale systems of semiconductors and metals with from ten thousands to ten millions atoms.

ELSES (Extra Large Scale Electronic Structure) project
In November, 2007, ELSES project was founded by our lab together with volunteer members of universities, institutes and companies, with the support from UCR-Workshops, Division of University Corporate Relations (DUCR), the University of Tokyo.
The aim of ELSES project is global standardization of nano-scale quantum material simulator from Japan.

For more information, please visit the main page of ELSES project: http://www.elses.jp

A set of theories and program codes have been developed in our group and a test calculation of Figure 1 shows that the computational cost is 'order N' or proportional to the system size (N) for the calculations with 104-107 atoms. Calculation is carried out by using the one-body density matrix or the Green's function, instead of one-electron eigenstates, which drastically reduces the computational cost.
The one-body density matrix ρ and a physical quantity <X> are defined from occupied one-electron eigenstates {k }, as

One can find that, if the matrix X(r', r ) is of short range, the off-diagonal long-range component of the density matrix does not contribute to the physical quantity <X>, which is important for the practical success of large-scale calculations.

We have developed two solvers, the Krylov-subspace method and the generalized-Wannier-state method, for large-scale matrix computational scheme. Moreover, we have developed two methods as the solver of the Krylov-subspace method, Krylov-subspace diagonalization (KR-SD) method and the shifted conjugate-orthogonal conjugate-gradient (shifted-COCG) method.

Two methodologies, linear algebraic calculation and eigenvalue problem, are generally available for solving large-scale electronic structure calculation. We adopt Krylov-subspace diagonalization (KR-SD) method and Lanczos method as solvers of eigenvalue problem and shifted-COCG method as a solver of linear algebraic calculation.

The shifted-COCG method gives an iterative solver algorithm of the Green's function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Green's function, and dynamically, at each step in dynamical simulations.

Practical calculations were carried out with Hamiltonians in the Slater-Koster (tight-binding) form. Each method proposed in our group is able to controll the accuracy and the computational cost by monitoring residuals for microscopic or basis freedoms.

Figure 1: The computational time of large scale electronic structure calculation as a function of the number of atoms (N).
Fcc Cu, liquid Carbon and bulk Si are calculated.

EIG: the conventional eigenstate calculation
KR-SD: Krylov-subspace method with subspace diagonalization procedure
WS-VR: Wannier-state method with variational procedure
WS-PT: Wannier-state method with perturbative procedure

Research results:
(1) Development of Large-Scale matrix computational scheme

Krylov subspace method and shifted-COCG method

Figure 2: Cleavage process of Silicon

Figure 3: Reformation of helical multishell Au nanowires

[R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, and T. Fujiwara, Phys. Rev. B 73, 165108, 9 (2006)]
[R. Takayama, T. Hoshi, T. Fujiwara, J. Phys. Soc. Jpn, vol. 73, No.6, pp.1519-1524 (2004)]

A linear algebraic method named the shifted conjugate-orthogonal conjugate-gradient (shifted-COCG) method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green's function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is applied to both a semiconductor and a metal.

(2) Application of the quantum mechanical molecular dynamics simulator to realitic larfe-scale systems

Cleavage process and surface reconstruction of Silicon.
[T.Hoshi, Y.Iguchi and T.Fujiwara, Phys. Rev. B 72, 075323 (2005)]
[T.Hoshi, R. Takayama, Y.Iguchi and T.Fujiwara, Physica B: Condensed Matter 376, pp. 975 (2006)]
[T. Hoshi and T. Fujiwara, J. Phys.: Condens. Matter 18 10787-10802 (2006)]

The 10-nm-scale structure formed in silicon cleavage is studied by the quantum mechanical calculations of large-scale electronic structure. The cleavage process was simulated and the results show not only the elementary process of the (experimentally observed) (111)-(2x1) surface reconstruction but also several step-formation processes. These processes are studied by analyzing electronic freedom and compared with scanning tunneling microscopy experiments. Several common aspects between cleavage and other phenomena are discussed from the viewpoints of nonequilibrium process and 10-nm-scale structure. (See Figure 2)

Helicity of Au nanowires
[Y.Iguchi, T.Hoshi and T.Fujiwara, Phys. Rev. Lett. 99, 125507 (2007)]

A model for the formation of helical multishell gold nanowires is proposed and is confirmed with quantum mechanical molecular dynamics simulations. The model can explain the magic number of the helical gold nanowires in the multishell structure. The elementary processes are governed by competition between energy loss and gain by s and d electrons together with the width of the d band. The possibility for the helical nanowires of platinum, silver, and copper is discussed. (See Figure 3)

(2) Beyond LDA:
Developping the novel method of the first principle electronic structure calculations with combination of one-electron band theory and many-electron theory.

Strongly correlated electron systems are systems which have strong Coulomb interactions comparable to electron-hopping integrals. Typical examples of a strongly correlated electron system are transition metal oxides. Much attention has been focused on strongly correlated electron systems, since these systems show anomalous physical properties such as various spin, charge and orbital order, metal-insulator transition, Colossal Magneto-Resistance (CMR), High-temperature superconductivity and so on. The valence orbitals in strongly correlated electron systems are partially filled and well localized 3d or 4f orbitals and hence play important roles. These various physical properties in strongly correlated electron systems have been extensively encouraged recent development of material and device designs.

The local density approximation (LDA) based on the density functional theory (DFT) is a quit successful for electronic structure calculation of many real materials with weakly correlated materials. However, the LDA is hardly applicable to strongly correlated electron systems, since the LDA one-electron potential is orbital independent and hence takes into account the Coulomb interaction as an averaged term. In particular, LDA overestimates the width of 3d bands and underestimates the band gap for strongly correlated electron systems.

To capture the physical phenomena on strongly correlated electron systems, more sophisticated method with dynamical electron correlation effects are needed.

We will establish the novel method of the first principle electronic structure calculations with combination of one-electron band theory and many-electron theory and apply it to Strongy correlated electron systems.


LDA+DMFT method

LDA+DMFT method is the combination method with first-principles electronic strucutre theory and the Dynamical Mean Feald Theory (DMFT). LDA+DMFT method includes on-site dynamical correlation when applyng to realistic aterials.

DMFT is the exact treatment for infinite dimensions. In infinite dimensions, the effects from neighbor sites are replaced with "mean field" and spatial fractuation is neglected, since the number of neighbor sites is infinite. In particular, the self-energy, corresponding to "mean field", is independet of the wave number and only dependent on frequency. In this condition, many electron systems in bulk is mapped onto single impurity atom embedded in effective medium, namely the single impurity problem. Mapped single impurity problem is solved accurately. Thus, the on-site dynamical correlation is included and the inter-atomic correlation is neglected In DMFT.

When calculating the local Green's function G(w) by using LDA Hamiltonian, realistic electronic strructure is included in the procedure of DMFT. This is called "LDA+DMFT" method. (See Figure 4.)

We adopt the Iterative Perturbation Theory (IPT) as a solver for mapped single impurity problem. Within the IPT, the self-energy is determined with the interpolation scheme between the high frequency limit and the strong interaction limit (the atomic limit) by using the second order self-energy.

Characteristic feature of LDA+DMFT method proposed in our Lab
Recently, various LDA+DMFT methods are proposed.
Charcteristic features of LDA+DMFT method proposed in our Lab are as follows:

(1) To adopt full LDA Hamiltonian without reducing its size
(2) To adopt IPT as a solver of mapped single impurity problem
(3) combination of one-electron band theory and many-electron theory by using the Ligand Field Theory (LFT)

Thus, the LDA+DMFT method proposed in our Lab is applicable to various realistic strongly correlated materials, both metallic and insulating, multi-atom (compound), spin-polarized and strongly hybridized cases between s, p and d-bands.

Figure 4: The self-consistent loop of LDA+DMFT with IPT method
GW Approximation

GW approximation (GWA) is the first term approximation of the many-body perturbation series for one-body Green's function G and the self-energy is replaced by the lowest order term of the expansion as


where W is a dynamically screened interaction and W in GWA is treated with the Random Phase Approximation (RPA) (See Figure 5).
Moreover, on-site Coulomb interaction U will be determined within the framework of GWA.

Characteristic feature of GW approximation proposed in our Lab
The computational task of GWA is much heavier than LSDA and hence conventinal GWA has been applied to the cases with less than a few atoms inside a unit-cell. The program code of GWA proposed in or Lab is applicable to the cases with more than twenty atoms inside a unit-cell.

Figure 5: Feynman Diagrams for GW approximation
Research results:
Electronic structure and quasiparticle of antiferromagnetic LaMnO3 in GW appriximation

Figure 6: Partial DOS for antiferromagnetic LaMnO3

Figure 7: The imaginary part of the Green's function of antiferromagnetic LaMnO3 at Γ point for the respective LSDA bands.

[Y. Nohara, A. Yamasaki, S. Kobayashi, T. Fujiwara, Phys. Rev. B 74, 064417 (2006)]

The electronic structure of A-type antiferromagnetic insulator LaMnO3 is investigated by the GW approximation. The band gap and spectrum are in a good agreement with experimental observation. The lifetime of electrons in conduction bands is much shorter than that of holes in valence bands. The insulator-to-metal transition with antiferromagnetic-to-ferromagnetic transition with photocarrier injection is attributed to the characteristic properties of excited electron states in A-type antiferromagnetic perovskite systems. The onsite d-d Coulomb interaction is strongly screened at the low energy region by mobile eg electrons. (See Figure 6 and 7)

Electronic structure of ferromagnetic bcc-Fe, fcc-Ni and antiferromagnetic NiO in the LDA+DMFT method
[O,Miura and T. Fujiwara, in preparation]

We applied LDA+DMFT with IPT method to ferromagnetic bcc-Fe and fcc-Ni and to antiferromagnetic NiO. In Fe and Ni, the width of occupied 3d bands becomes narrower than those in LDA and Ni 6 eV satellite is observed. In NiO, the resultant electronic structure is of charge-transfer insulator type and the band gap is 4.3 eV. This comes from the on-site Coulomb interaction of nickel 3d bands, which causes the drastic change of hybridization between Ni-3d and O-2p bands. These results are in good agreement with the experimental XPS.

The success of the implementation of present LDA+DMFT with IPT implies that present LDA+DMFT with IPT is suitably applicable to materials with much larger size and more complicated hybridization between more than two atoms by using full-LDA Hamiltonian without reducing its size. (See Figure 8)

Figure 8: Energy spectrum of antiferromagnetic NiO in the LDA+DMFT with IPT method.
Charge and spin stripe in La2-xSrxNiO4
[S. Yamamoto, T. Fujiwara and Y. Hatsugai, Phys. Rev. B 76, 165114 (2007)]

Electronic structure of stripe ordered La2-xSrxNiO4 is investigated. The system with x=1/3 is insulator, in LSDA+U, and shows charge and spin stripe, consistent with the experimental results. A highly correlated system of x=1/2 is studied by using exact diagonalization of the multiorbital many-body Hamiltonian derived from LDA calculations and including on-site and inter-site Coulomb interactions. The resultant ground state is an insulator with charge and spin stripe of the energy gap 0.9 eV, consistent with the observed one. Inter-site Coulomb interaction and anisotropy of hopping integrals, play an important role to form the charge and spin stripe order in a system of x=1/2. (See Figure 9)

Application of shifted-COCG method to many-electron problem

Shifted-COCG method was applied to calculate the spectrum of La
2-xSrxNiO4 (x=1/2). The sucess of application of shifted-COCG method implies that shifted-COCG method is a powerful tool applicable to many-electron problem as well as large-scale electronic structure calculations.

Figure 9: The inter-site Coulomb interaction V dependence of excitation spectrum in La2-xSrxNiO4(x=1/2)