Large-Scale Electronic Structure Calculations with Real-Space grid
Density-Functional Theory code
Jun-Ichi Iwata 111

Graduate School of Enginnering, The University of Tokyo, Japan

Introduction

First-principles quantum-mechanical electronic structure
calculations based on the density-functional theory (DFT) is an
important ingredient of material sciences. The system sizes
handled in usual DFT calculations are limited within a thousandof atoms due to the computational complexity of fully quantummechanical approaches.
Real-space DFT (RSDFT) code, which has been developedin our group, is a promising application to overcome the sizedifficulty in first-principles quantum mechanical calculations.
RSDFT is developed based on the real-space grid formulation ofDFT [1] instead of conventional reciprocal-space formulation, inwhich frequent use of fast Fourier transform (FFT) is inevitable,
so that it’s suitable for massively-parallel computers such as K
computer [2]. Recently, we have performed electronic structure
calculations for silicon nanowires with 10,000 to 1000,000
atoms by using RSDFT on the K computer with very highperformance, and we got Gordon-Bell prize in 2011 with thiswork.
In this paper, I review the first-principles electronic structure
calculations in DFT and our program code RSDFT. In addition,
I give an introduction on the technological importance of ourtarget system silicon nanowire (SiNW), and discuss the resultsof the calculations for that system.
2First-principles
quantummechanicalcalculations based on the density functional
theoryThe foundation of DFT has been given by Hohenberg and Kohn
in 1964 [3], and subsequently, a formulation which is suitablefor practical calculations has been given by Kohn and Sham in1965 [4]. In the Kohn-Sham formulation of DFT, total energy ofthe system and the electron density are obtained by a set offunctions, called the Kohn-Sham orbitals ! n (r) , as follows [1]:Coulomb interaction between electrons and ions [5]. E XC iscalled the exchange-correlation energy which arise fromquantum mechanical effects. The exact functional form of theexchange-correlation energy is unknown, so that simple
approximate forms, such as local-density approximation [6], areutilized for practical calculations.
Minimization of the total-energy functional with respect tothe Kohn-Sham orbitals, with ortho-normalization conditions
among the orbitals, leads the following equation:ĥKS! n (r) = ! n" n (r)(3)

1! (r#) ! E XC [ " ]ĥKS = ! " 2 + $ d r#+2
r ! r# !" (r#)

(4)

N ION
kk+vION (r) + % vPP(r) $ d r#vPP
(r#)
k=1

This is called the Kohn-Sham equation [4] which is the basic
equation of first-principles electronic structure calculations. It isworse noting that Kohn-Sham equation is not a usual eigenvalue
problem but nonlinear one, because the integro-differentialoperator Eq. (4) depends on the eigenvector itself through theelectron density Eq. (2). Therefore, we have to solve the KohnSham equation (nonlinear eigenvalue problem) self consistently.
3Real-space grid DFT code

In the real-space grid calculations, we discretize the Kohn-Shamequation in 3-D spatial grid as shown in Fig. 1, and represent thedifferential operator in a higher-order finite-difference
approximation. Then the Kohn-Sham equation is recast into a
conventional linear-algebraic eigenvalue problem.

N1% dr!n (r)&2!n (r)n=1 2

E !"{! }#$ = '+1! (r)! (r()% dr % d r( r ) r( + EXC [ ! ]2(1)+ % dr! (r)vION (r)N N IONkk+' ' % dr! n (r)vPP(r) % d r(vPP(r()! n (r()
n=1 k=1

whereN! (r) = ! ! n (r)

2(2)

n=1

is the electron density, N and NION is the number of electronsa represent theand ions in the system, respectively, vION and vPPFig. 1Real-space grid representation of the unit cell ofcrystalline silicon. The eight sub cells are allocated on each
CPU (or node) for parallel computations.

Read initial atomic configuration
Construct initial density
and initial potentials
Generate initial orbitals
Construct ionic pseudo potentials

Update orbitals
using Conjugate-Gradient (CG)
Perform Gram-Schmidt ortho-normalization (GS)Update density

Update Hartree andexchange-correlation potentials
Perform subspace diagonalization (SD)Has SCF converged ?

NoYes
endFig. 2
Flow chart of self-consistent procedure of RSDFTcalculations.
We divide the spatial region into several sub regions, andallocate the work on each region to each CPU (or node). In thefinite-difference calculations, we need to exchange boundary
values between next-nearest nodes, and we use MPI_ISEND andMPI_IRECV routines for this purpose. For global summations(integrations), we use MPI_ALLREDUCE. In addition to thegrid parallelization, we also parallelize in the Kohn-Shamorbitals. In each node, we can also parallelize tasks in thread byusing OpenMP.
The flow chart of the RSDFT calculations is shown in Fig. 2.
The overall algorithm is the subspace-iteration method, which isa standard way to solve large-sparse eigenvalue problems. As
mentioned previously, our problem is nonlinear eigenvalue
problem, so that we have additional density and potentialfunction update processes in the flow chart to achieve the selfconsistent solution.
The computational cost of the RSDFT is proportional to the
cube of NION, the number of ions (atoms) in the system.
However, the most of computations in O(NION3) routines (e.g.
Gram-Schmidt) can be performed with level 3 BLAS, so that thecomputational time is not so much increased as the system size
increases.
4Results of electronic structure calculaiotns forsilicon nanowires

SiNW Field-Effect Transistors (FETs) are expected to beboosters in post-scaling semiconductor technology [7]. Clearscaling of short-channel effects vs. NW sizes with the fixed gate
lengths has been observed [8]. Reducing the dimensions ofSiNWs has been shown to improve the short-channel control.
Furthermore, SiNW FETs with channel dimensions of 5.0 nm ×
6.3 nm have been fabricated [8]. A recent experiment showedthat the optimal dimension of the cross section of the SiNW
FET is around 10 nm [9]. At such small scales, quantum
confinement becomes prominent, and therefore reliable quantum
mechanical calculations are highly desired to investigate thecharacteristics of SiNW devices.

Fig. 3
Dependence of electron states on the morphology of
the cross sectional shapes of the silicon nanowires.
We have studied the electron states of SiNWs with several
cross-sectional shapes and channel lengths. The size of thesystems is ranging from 10,000 atoms to 100,000 atoms. Thesenumbers of atoms are the same as those of real device sizes ofSiNWs used for surrounding-gate type FETs.
In Fig. 3, we show the calculated electron density of states of
SiNW with circular and elliptic cross sectional shape. Clearly,
the electron states depend on the cross-sectional shape ofSiNWs due to quantum confinement effect.
From the first-principles calculations, we can calculate thecurrent and current-voltage characteristics of SiNW FETs.
These are helpful information for the development of realdevices. We believe that the RSDFT code opens a way to study
SiNW FETs from first-principles of quantum mechanics in real
device sizes.
5Summary

We have developed RSDFT code to perform large-scale firstprinciples quantum mechanical calculations on massively
parallel computers. RSDFT efficiently works on the K computer,and it makes calculations for 10,000-atom to 100,000-atomsystems possible. We got Gordon-Bell prize in 2011 with thiswork.
We perform first-principles calculations for SiNWs in realdevice sizes, and confirm the dependence of electron states on
the cross-sectional shape of SiNW due to quantum confinement
effect. The results show that RSDFT opens a way to study
SiNW FETs from first-principles.
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Iwata, J., Takahashi, D., Oshiyama, A., Boku, T., Shiraishi, K.,Okada, S., Yabana, K., “A massively-parallel electronic-structure
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Journal of Computational Physics 229, 2339 (2010).
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