2012.03.17 東京大学本郷キャンパス工学部6号館2F63講義室
9:20-9:40

GW近似とその拡張による固体の電子状態計算

1,2佐久間怜・2菅原健人・2田中友理・3F. Aryasetiawan

1千葉大学大学院融合科学研究科
2千葉大学工学部ナノサイエンス学科
3Lund University

Contents• GW approximation including spin-orbit coupling• GW approximation for localized states

GW approximation
Σ = GW
Screened interaction

Green function

W = ε −1Vε = 1 − VP
P = GG
G = G0 + G0 ΣG• Tool for the quantitative description ofquasiparticle band structures• Starting point for (first-principles) many-bodytheory‣ GWΓ (Y. Takada)‣ GW+DMFT (S. Biermann et al.)Schilfgaarde et al. (2006)

Flow-chart of G0W0
Self-consistent DFT calculation
occ. unocc.

P(r, r '; ω ) = 2∑ ∑ ψ i* (r)ψ j (r)ψ *j (r ')ψ i (r ')
ij

⎡
⎤
11× ⎢
−⎥⎢⎣ ω − Δ ij + iη ω + Δ ij − iη ⎥⎦ε (r, r '; ω ) = δ (r − r ') − ∫ V (r, r '')P(r '', r '; ω )d 3r ''W (r, r '; ω ) = ∫ ε −1 (r, r ''; ω )V (r '', r ')d 3r ''+∞−1Σ(r, r '; ω ) =G0 (r, r '; ω + ω ')W (r, r '; ω ')dω '∫2π i −∞
EnQP = EnDFT + ψ n |Σ(EnQP ) − Vxc | ψ n

GW approximation including
spin-orbit coupling
Collaborators: C. Friedrich (FZ-Juelich)
S. Bluegel (FZ-Juelich)
T. Miyake (NRI-AIST)

Motivation
Intrinsic spin-Hall effect:(Murakami et al., Science 2003)

J-insulator: Sr2IrO4(Kim et al., PRL 2009)

Spin-dependent GW
Spin-dependent many-body Hamiltonian

Spin-dependent GW approximation
F. Aryasetiawan and S. Biermann,
Phys. Rev. Lett. 100, 116402(2008);
J. Phys. : Condens. Matter 21, 064232(2009)

Spin-dependent GWSpecial case: (one-particle) spin-orbit coupling + Coulomb interaction
0SOC
hαβ(r) = h 0 (r)δαβ + vαβ(r)1Vαβ ;γη (r, r ') =δαβδ γη
| r − r' |GW + SOC

Σαβ = GαβW

ε = 1 − VP
P = ∑ Gαβ Gβααβ−1

W =ε V• The self-energy and Green’s function now haveoff-diagonal matrix elements.

• The polarization P is calculated fromthe band dispersion including SO splitting

Hg Chalcogenides(HgX, X=S, Se, Te)

BER 2006

ð2Þ

For thickness d < dc (i.e., for a thin HgTe

Downloaded fHðkÞ ¼ eðkÞ þ d i ðkÞsi

Fig. 1. (A) Bulk energy
bands of HgTe and CdTe
near the G point. (B)The CdTe-HgTe-CdTequantum well in thenormal regime E1 > H1with d < dc and in the
inverted regime H1 >E1 with d > dc. In thisand other figures, G8/H1
symmetry is indicated inred and G6/E1 symmetry
Markus König,1 Steffen Wiedmann,1 Christoph Brüne,1 Andreas Roth,1 Hartmut Buhmann,1is indicated in blue.
Laurens W. Molenkamp,1* Xiao-Liang Qi,2 Shou-Cheng Zhang2

RESEARCH ARTICLES

Motivation

RESEARCH ARTICLES

Quantum Spin Hall Insulator State
QuantumSpin Hall Insulator Statein HgTe Quantum
Wells
in HgTe Quantum Wells

Markus König,1 Steffen Wiedmann,1 Christoph Brüne,1 Andreas Roth,1 Hartmut Buhmann,1Laurens W. Molenkamp,1* Xiao-Liang Qi,2 Shou-Cheng Zhang2

AUTHORS’Science
2006SUMMARYAUTHORS’ SUMMARYT

VOL 314SCIENCEScience 2007thhe discovery more than 25
Conductance
he discovery more than 25
Conductancestr
years ago of the quantum
channel withyears ago of the quantum
channel withcharge
up-spintu
Hall effect (1), in which the Hall effectup-spin charge(1), inwhich thecarrierselectrical” conth“Hall,” or “transverse electrical” concarriers“Hall,” or “transverseofductance of a material is quantized,ductance of a material is quantized,
came as a total surprise to the physics
tocame as a total surprise to the physics
This effect occurs invacommunity. This effect occurscommunity.
inlayered
metalsat
highmagnetic
layered metals at high magnetic
fields and results from the formaca
fields and results from the formation of conducting one-dimensionaloption of conducting one-dimensional
channels that develop at the edgessuchannels that develop at the edgesof the sample. Each of these edge
coof the sample. Each of these edge
channels, in which the current moves
only in one direction, exhibits a quantec
channels, in which the current moves
tized
conductancethatischaracterwww.sciencemag.org
saC
only in one direction, exhibits a quanisticofone-dimensionaltransport.
Thechcia
tized conductance that is characterConductance
Quantum
dnumber of edge channels in the samhich
istic of one-dimensional transport.ple
Theis directly related to the value ofchannel withwell
Quantum
down-spin
wenumber of edge channels in the samthe quantum Hall conductance. MoreSchematicof
thespin-polarizededge
channelsinaquantucharge carriersdeple is directly related to the valueof the charge wellover,carriers in these chaninsulator.
nels are very resistant to scattering.
gathe quantum Hall conductance. MoreSchematic of the spin-polarized edge channels in a quantum spin Hall
Not only can the quantum Hall effect be observed in macroscopic samples electronics states are filleusover, the charge carriers in these chaninsulator.

TInverted band structure(without SOC)

Hg 5d
conduction
band

Egap > 0“conduction”
band

Egap < 0Inverted band structure(with SOC)

Δ

Δ

“conduction” bands

Δ

LDA results
Eg

Δ

Δ

Eg

Eg

Eg

Δ

Δ

HgS

HgSe

HgTe

LDA

-0.66

-1.27

-1.20

Expt.

-0.15, -0.11-0.274-0.29, -0.30

LDA-0.12

+0.23

+0.78

+0.39,+0.38

+0.91

Expt.
.

GW results
Eg

ΔΔEg

Eg

Eg

ΔΔ

HgS

HgSe

HgTe

LDA-0.66

-1.27

-1.20

Expt.

-0.15, -0.11-0.274-0.29, -0.30

GW-0.02

-0.58

-0.60

LDA-0.12

+0.23

+0.78

+0.39,+0.38

+0.91

+0.32

+0.91

Expt.
.
GW-0.19

Energy [eV]

Energy [describeΓ8spin-orbi-1
perturbat-2
the spinthe self-e
-3QP
QP
DFT
QP
ΔEn ≡ Egn S − gESne =Teψ n |Σ(En ) − Vxc | ψ nof spatia
HHHgand cond
treated fu(b) Σx + Σc - Vxc
functions“conduction” bands
0.4
We ha
0.2
of mercu
0correction-0.2
structures-0.4
SO-splitted bands
due to st-0.6
are corre
tively, the gTe
SSgg
HHH
leading t
with prev
FIG. 3. (Color online) Self-energy correction for the states !6 ,improvem
and !8 : (a) only bare exchange and (b) full GW correction. The07

Self-energy correction

Possible application:
Searching new topological insulators

Eg

S. Chadov et al., Nature (2010)

Summary• SOC-included GW approximation is proposed• Application to Hg chalcogenides shows:‣ Improvement of the negative “band gap”
‣ Enhancement of spin-orbit splitting‣ State-dependent self-energy correction