公募研究：「磁性金属におけるハイブリッド密度汎関数理論の新展開」

ハイブリッド密度汎関数と結晶中電子の局在性Jul. 8, 2013
合田 義弘
Y. Gohda
Department of Physics
The University of Tokyo

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In collaboration with:
S. Tsuneyuki

Semi-local functionals in DFT
have been successful.
generalized gradient approximation (GGA)based on QMC results for homogeneous electron gas (HEG)

accurate for most of itinerant electrons in normal metals
and narrow-gap semiconductors

However, ...
fails in describing localized electrons
self-interaction error
Jul. 8, 2013

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Y. Gohda

DFT+U method
Density of states [arb. unit]up

Gd-doped GaN

Ndf 4fFej2 3d

Nd2Fe14B
μs=24.9μB/f.u.

Ndf 5d0
Ndf 5dFej2 3ddown−10Gohda	  &	  Oshiyama,	  PRB	  78,	  161201(R)	  (2008).

−8−6−4−202

Energy relative to εF [eV]Ndf 4f468

useful for localized (large U/t) electrons
Itineracy described by U=0fails for intermediately localized electronse.g. O 2p states: occupancy dependent on projection radius in DFT+U
Jul. 8, 2013

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Y. Gohda

Perdew’s ladder(DMC)(DFT+DMFT)TDDFT, GW
Hybrid DFT
MGGA, DFT+U
GGA
LDA
M. Chagall 1973

a convenient & routinely applicable scheme?
unified descriptions of itinerant and localizedelectrons?
Jul. 8, 2013

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Y. GohdaHybrid-DFT
2.78 eV (exp)2.57 eV (HSE)
1.60 eV (GGA)

zincblende GaP

Optical conductivity [arb. units]

direct gap (Γ)

0HSE
PBE

14 January 2013

Volume 102 Number 2

Applied Physics
Letters

apl.aip.org

012

3

4

5

Frequency [eV]678Gohda	  &	  Tsuneyuki,	  APL	  102,	  023901	  (2013).

reasonably accurate with moderate computationalcost, at least, for semiconductors/insulators
Jul. 8, 2013

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Y. Gohda

Koopmans’ theorem in HF
It	  is	  saGsﬁed	  thatif	  all	  the	  atomic	  posiGons	  and	  all	  the	  relevant	  orbitals	  are	  kept	  ﬁxed.
In	  reality,	  orbitals	  relax	  upon	  electron	  addiGon.
Hartree-Fock (HF)
no electron correlation
self-interaction free
Jul. 8, 2013

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Y. Gohda

Generalized Koopmans condition
Zunger	  &	  Freeman,	  PRB	  16,	  2901	  (1977).

Perdew	  et	  al.,	  PRL	  49,	  1691	  (1982).

(with	  derivaGve	  disconGnuiGes)
integraGon	  of	  Janak’s	  theorem
Lany	  &	  Zunger,	  PRB	  80,	  085202	  (2009).

Exact condition to be fulfilled under fixed atomic positions
Π: self-interaction energy due to addition of N+1st electron
without electronic relaxation
Σ: energy due to electronic relaxation
HF: Π=0, Σ<0;Jul. 8, 2013

DF(LDA/GGA): generally Π+Σ>0
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Y. Gohda

Functionals satisfying GKC
single Fe in AlN

10E(N+1)−E(N)6α=0.12
Energy relative to εi(N)[eV]

α=0.08Energy relative to εi(N)[eV]8420εj(N+1)

εj(N)

εi(N)

εi(N-1)−2

E(N)−E(N−1)−4−600.05

0.1

0.15

0.2

HF fraction α in HSE

single Ni in AlN

8E(N+1)−E(N)64εj(N+1)2εj(N)εi(N)

0εi(N-1)−20.25HF−4E(N)−E(N−1)
00.05

0.1

0.15

0.2

HF fraction α in HSE

0.25

HF

optimization of mixing parameter α in Hybrid-DFT
conventional HSE06: α=0.25
Jul. 8, 2013

α(Fe in AlN)=0.08 < α(Ni in AlN)=0.12
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Y. Gohdabcc Fe
determination of α by constraint hybrid-DFT: α=0.02
Density of states [arb. units]c.f. fcc Ni : α=00−10−50Energy relative to εF [eV]510μ(α=0.02)=2.27μB; μ(α=0)=2.20μB ; μ(exp.)=2.2μB
Jul. 8, 2013

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Y. Gohda

locality
benzene A

benzene B

exact condition: EA(N+1) + EB(N) = EA(N+0.5) + EB(N+0.5)
DF(GGA): 2·E(N+0.5) < E(N+1) + E(N)overdelocalized

Hartree-Fock: 2·E(N+0.5) > E(N+1) + E(N)overlocalized

opGmized	  α:	  measure	  of	  locality
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Y. Gohda

states near εFNiegO

locality at Ni siteAlNNiO: k(-1/4, 1/4, 0) 17th band, up+down, unocc.

Jul. 8, 2013

AlNiO: 195th band, up+down

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Y. Gohda

summary
hybrid-DFT scheme to describe bothitinerant & localized electrons
utilizing the generalized Koopmans condition
suitable for relatively large scale calculations

localization of 3d electrons described bymixing parameter αoxides > point defects in AlN > metalsJul. 8, 201312 /15

Y. Gohda