

We will establish the foundation and the methodology for electronic structure calculations in the following two aspects; 


Detail
(1) Large Scale Electronic Structure Calculations
Establishing the quantum mechanical molecular dynamics simulator and the technology of process simulator for nanoscale 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 tennanometre 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 CarParrinello method, owing to its heavy computational cost.
We will establish the quantum mechanical molecular dynamics simulation method
and the technology of process simulator for nanoscale systems of semiconductors and metals with
from ten thousands to ten millions atoms.



Method:
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
10^{4}10^{7} atoms.
Calculation is carried out by using the onebody density matrix or the Green's function,
instead of oneelectron eigenstates, which drastically reduces the computational cost.
The onebody density matrix ρ and a physical quantity <X> are defined
from occupied oneelectron eigenstates {ƒÓ_{k} },
as
One can find that, if the matrix X(r', r ) is of short range,
the offdiagonal longrange component of the density matrix does not contribute to the physical quantity <X>,
which is important for the practical success of largescale calculations.
We have developed two solvers, the Krylovsubspace method and the generalizedWannierstate method,
for largescale matrix computational scheme.
Moreover, we have developed two methods as the solver of the Krylovsubspace method,
Krylovsubspace diagonalization (KRSD) method
and the shifted conjugateorthogonal conjugategradient (shiftedCOCG) method.
Two methodologies, linear algebraic calculation and eigenvalue problem, are generally available
for solving largescale electronic structure calculation.
We adopt Krylovsubspace diagonalization (KRSD) method and Lanczos method as solvers of eigenvalue problem
and shiftedCOCG method as a solver of linear algebraic calculation.
The shiftedCOCG 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 roundoff 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 SlaterKoster (tightbinding) 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
KRSD: Krylovsubspace method with subspace diagonalization procedure
WSVR: Wannierstate method with variational procedure
WSPT: Wannierstate method with perturbative procedure



Research results:
(1) Development of LargeScale matrix computational scheme
Krylov subspace method and shiftedCOCG 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.15191524 (2004)]
A linear algebraic method named the shifted conjugateorthogonal conjugategradient (shiftedCOCG) method is introduced for largescale 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 larfescale 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 1078710802 (2006)]
The 10nmscale structure formed in silicon cleavage is studied by the quantum mechanical calculations of largescale 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 stepformation 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 10nmscale 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 oneelectron band theory and manyelectron theory.

Strongly correlated electron systems are systems which have strong Coulomb interactions comparable to electronhopping 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, metalinsulator transition, Colossal MagnetoResistance (CMR), Hightemperature 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 oneelectron 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 oneelectron band theory and manyelectron theory and apply it to Strongy correlated electron systems.
Method:

LDA+DMFT method
LDA+DMFT method is the combination method with firstprinciples electronic strucutre theory and the Dynamical Mean Feald Theory (DMFT).
LDA+DMFT method includes onsite 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 selfenergy, 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 onsite dynamical correlation is included and the interatomic 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 selfenergy is determined with the interpolation scheme
between the high frequency limit and the strong interaction limit (the atomic limit)
by using the second order selfenergy.
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 oneelectron band theory and manyelectron 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, multiatom (compound),
spinpolarized and strongly hybridized cases between s, p and dbands.


Figure 4: The selfconsistent loop of LDA+DMFT with IPT method

GW Approximation
GW approximation (GWA) is the first term approximation of the manybody perturbation series for onebody Green's function G
and the selfenergy is replaced by the lowest order term of the expansion as
Σ(1,2)=iG(1,2)W(1,2),
where W is a dynamically screened interaction and W in GWA is treated with the Random Phase Approximation (RPA)
(See Figure 5).
Moreover, onsite 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 unitcell.
The program code of GWA proposed in or Lab is applicable to the cases with more than twenty atoms inside a unitcell.


Figure 5: Feynman Diagrams for GW approximation 
Research results:
Electronic structure and quasiparticle of antiferromagnetic LaMnO_{3} in GW appriximation 

Figure 6: Partial DOS for antiferromagnetic LaMnO_{3}
Figure 7: The imaginary part of the Green's function of antiferromagnetic LaMnO_{3} 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 Atype 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 insulatortometal transition with antiferromagnetictoferromagnetic transition with photocarrier injection is attributed to the characteristic properties of excited electron states in Atype antiferromagnetic perovskite systems. The onsite dd Coulomb interaction is strongly screened at the low energy region by mobile eg electrons. (See Figure 6 and 7)


Electronic structure of ferromagnetic bccFe, fccNi and antiferromagnetic NiO in the LDA+DMFT method
[O,Miura and T. Fujiwara, in preparation]
We applied LDA+DMFT with IPT method to ferromagnetic bccFe and fccNi
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 chargetransfer insulator type and the band gap is 4.3 eV.
This comes from the onsite Coulomb interaction of nickel 3d bands,
which causes the drastic change of hybridization between Ni3d and O2p 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 fullLDA 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 La_{2x}Sr_{x}NiO_{4}
[S. Yamamoto, T. Fujiwara and Y. Hatsugai, Phys. Rev. B 76, 165114 (2007)]
Electronic structure of stripe ordered La_{2x}Sr_{x}NiO_{4} 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 manybody Hamiltonian derived from LDA calculations and including onsite and intersite 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. Intersite 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 shiftedCOCG method to manyelectron problem
ShiftedCOCG method was applied to calculate the spectrum of
La_{2x}Sr_{x}NiO_{4} (x=1/2).
The sucess of application of shiftedCOCG method implies that shiftedCOCG method is a powerful tool applicable to
manyelectron problem as well as largescale electronic structure calculations.


Figure 9: The intersite Coulomb interaction V dependence of excitation spectrum in
La_{2x}Sr_{x}NiO_{4}(x=1/2)



