ecalj CMD2019 course menu
Sep.4th Wed 4 hours, Thus 6 hours , Fri 2 hours.
• Introduction
What we can do in ecalj?
• ecalj package
InstallMinimum usage
Band structure calculation. GW calculation.

• Theory
Band theory. LMTO+LAPW method, GW method• Usage, Read output and Practice

Give a glance to https://github.com/tkotani/ecalj/(This documents contains some theory --- I will change this documents more…)

The PMT method: a newlinearized method
PMT =APW + MTO
Muffin-tin

Takao Kotani (tottori-u)

Key point:1. Iteration cycle.
2. eigenfunctions are expanded withAPWs(augmented plane wave)
and MTOs(muffin-tin orbital)Independent particle picture and total energy
These can be obtained by the density functional theory(DFT) in LDA.
ElectronDensity 𝑛(𝒓)DFT in LDA

Fill up to Fermi energy

One-particle effective potential𝑉(𝒓)iterationcycle  2 V (r)  i (r)= i  i (r) 2mEigenvalues and Eigenfuncitons

 i , i (r)This iteration cycle until converged = total energy minimization

How to represent density and so on in computer?
How to solve it numerically?
Lists of the Full-potential methods•KKR•Pseudopotential method•PAW•LMTO•LAPWLinearized xxx methods(why do we call them ” linearized” method?
from the view of “exact ” APW method)

Finite basis set•Basis set (finite number of basis) We assume eigenfunctions are given as: p (r )    pj F j (r )j

F j : j  1, 2,...N V Fj ,2m
Overlap matrix Oij  Fi F j
Hamiltonian H ij  FijHO0 ijij 
Finite dimension problem(as the same as LCAO/Gaussian)iteration cycle
previous iteration

One-particle potential V(r)Basis set {Fi (r )}Hamiltonian H ij  Fi
 V Fj ,2m
Overlap matrix Oij  Fi Fj
Diagonalization (H   p O) p  0electron density n(r)Next iteration

How to choose the good basis?
• APW (augmented plane wave)• MTO ( muffin-tin orbital)
are the names of the basis functions.
Both of them are made by “augmentation”.

My conclusion:
To overcome shortcomings in APW basis and MTO basis, weshould use both of APW and MTO together. this means
the PMT method

But wait…
What is the APW and MTO? next page.

MT regions

Interstitial region

http://research.physics.illinois.edu/ElectronicStructure/598SCM-F04/lecture_notes/

Envelope function = PlaneWave(PW)ＡＰＷ
PW in MTMT region

MT region

3-components
PW ＋ Atomic-like part － Counter part:
PW within MT
0th
1st2nd•Good for Na(3s), high energy bands.
•Not so good for Cu(3d), O(2p)•SystematicLinearized APW

fcc Cuplane wave is not
so efficient for 3d
15Ry or more for~1mRy convergence

D.Singh et al PRB49,17424

ＭＴＯ

Atom-centered Hankel function, e.g,exp(  r )for S -channelr

Smooth Hakel function
MT region

MT region

3-components
Hankel ＋ Atomic-like part － counter part of Hankel
0th
1st2nd•Good for localized orbitals such as Cu(3d), O(2p)•Ｎot so good for extended states, surface.
•Not systematic

PMT =APW + MTO
T.K and M.van Schilfgaarde
Phys. Rev. B 81, 125117 (2010)

T.K， H. Kino, and H.Akai
Supercell calculations from H2 through Kr2.
Almost automatic setting of MTOs with APWs
(Energy cutoff 3~4Ry) .
J. Phys. Soc. Jpn. 82, 124714, (2013)
J. Phys. Soc. Jpn. 84, 034702 (2015)

Basis function
MT center at R. Radis R.
a basis F j (r) consists ofF0 j (r ),Envelope function
PW or smHankel

F1 j (r ),F2 j (r )Atomic-like partsfor r  R

Counter part
forr R

Augmentation parts
Cutoff: l  lmax ~ 4, Radial-part expansion

F j (r )  F0 j (r )  F1 j (r  R )  F2 j (r  R )Electron density n 𝒓 and potential 𝑉(𝒓) are expanded in a similar manner.

Local orbitals(lo): basis functions which are Non-zero only wihtin MTs.
e.g, see http://www.wien2k.at/lapw/index.html
This is also used together.

Nbasis=1081+36×2, 1973+36×２, 3025+36×２

Comparison with Gaussian

ＮＲ：non relativistic

ＧＴＯ：6-311+G(d,p)

Quasiparticle self-consistent GW
Takao kotani(tottori university)• Mean field theory.
What the eigenvalue means?
• GW, and QSGW method
Minimum and previous results.

Why ＱＳＧＷ？
We need good independent particle picture in order tocalculate linear responses (magnetic optical, transport…).
H0 (eigenvalue and eigenfunctions) are required.
＊ Band gap, Effective mass
＊Relative position of levels.
ＬＵＭＯ－ＨＯＭＯ, Fermi energy…

Metal

Fermi energy

Molecule
ＬＵＭＯ Molecule on
top of metal.
ＨＯＭＯ
18

Calculated(eV)BAND GAP
LDA, one-shot GW

QSGW(we see systematicover estimation).

Experimental Band gap (eV)

金属的なものＱＳＧＷでＯＫ．
ハイブリッド法では場合によりうまくいかない。

20

(12a.u.)3cell

LUMO
LUMO
HOMO
HOMO

計算の収束性は不明．．．

21ZnOBlack:QSGW
Red:LDA

3.87eV
0.71eV

Green:GLDAWLDA (Z=1,Offdiagonal included) 3.00eV
Blue：e-only self-consistency
3.64eV
Experiment(+correction) 3.60eV
22
Kotani et al PRB76,165106(2007)

d systems“No Pushing down 3d-band’’ in GW.
23

NiO

Black:QSGW Red:LDA Blue: e-only24Spin wave dispersion based on QSGW

25

srtio3

QSGW

S. A. Chambers et al, Surface Sci 554,81-89 (2004)26

Electron configuration
lift of degeneracy of 3d orbitals due tocrystal fieldegx2-y2∆𝐸 eg

3z2-r2[3d]Cu atomt2g

xycubic(regularoctahedron)yz, zxtetragonal(elongatedoctahedron)Around Ef, we have two eg bands.

Band structure:
La2CuO4(VWN)

Band width and ∆𝐸 eg
La2CuO4(QSGW)∆𝐸 eg

HgBa2CuO4(VWN)

HgBa2CuO4(QSGW)

28

Impact ionization rate(auger process)
Unoccupied bandOccupied band

Fermi’s Golden rult  transition rate. Sum up for all
final states. Matrix element of transition is＜1,3｜electron-electron interaction｜4,2＞（Energy conservation and Momentum conservation） 29

InAs impact ionization rate(=auger process= high energy electron lose energy with e-h pair)direct gap 0.38eV

QSGW

123eV

45

Present model: Sano and Yoshii JAP77 2020 (1995)

YH3(fcc structure 6x6x6)

GGA

L

QSGW

Γ

X

L

Γ

X

CuGaSe2 (eight atoms in cell, 2x2x2. ~3hours per iteration by single core)

1shotGW

T

Γ

QSGW

NT

Band gap GGA:0.4eV, GW(1shot noZ) 1.37 eV,
QSGW(1.79eV)
Exper. 1.63, 1.67, 1.73eV

ΓN

Band gap for CuAB2 （eV)
BA S

Se

Al3.62(3.49)

2.91(2.67)

Ga

2.83(2.43)

1.69(1.68)

In

1.22(u=0.22) ---1.48(u=0.23) (1.04)(1.53)

Experiments in (…)

CuInS2 band gap（eV)

0.22 0.23

uJ.Vidal, PRL 104, 056401 (2010)

3x3x3 calculation(need check)33

PdO(tetragonal , four atoms per cell 3x3x2 calculation , 1hour per iteration )

GGA

Γ

X

M

QSGW

Γ Z

RAZ

Γ

X

M

Γ ZRA

Z

Band gap about 0.7 eVLaNiO3M.J.Han, H.Kino, T.K (arXiv:1402.4915)

35

36

37

summary• Mean field theory and excitation energy
LDA, Hartree-Fock and QSGWJanak theorem, finite vs.infinite system• RPA total energy
QSGW method• Some Results
Band gap, GaAs and Na, Cu
NiO, Spin Wave, ZnO, LaMnO3,InAs (impact ionization), YH3, CuGaSe2,PdO• How QSGW works for atoms and molecules?
See F.Bruneval J.Chem.Phys 136,194107(2012)