modified becke-johnson (mbj) exchange potential -...
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Modified Becke-Johnson (mBJ) exchange potential
Hideyuki Jippo Fujitsu Laboratories LTD.
2015.12.21-22 OpenMX developer’s meeting @ Kobe
Overview: mBJ potential
The semilocal exchange potential adding corrections of the screening effect to the local approximation of the exact exchange potential
Band gaps with an accuracy similar to hybrid functionals or GW methods for a wide range of different materials
Low computational cost comparable to standard DFT
Very fascinating potential for large-scale calculations
• Transport calculations for the semiconducting channel devices
1
Outline
Background
Details of mBJ exchange potential
Prospects towards implementation in OpenMX
Some examples
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BACKGROUND
3
Theoretical band gaps of AGNRs
4
L. Yang et al., PRL 99, 186801 (2007).
Na = 7
Eg of the isolated AGNRs
Na
LDA (GGA) underestimates the band gaps of the AGNRs.
Ban
d g
ap (
eV)
0
1
2
3
4
5
6 LDA GW Armchair GNRs (AGNRs)
・・・
・・・
・・・
・・・
1 2 3 4 5
Na
Na-1 Na-2
W
DFT: Only a ground state theory
Self-interaction error
LDA/GGA electron sees extra Coulomb repulsion by own charge.
Energy levels of the occupied orbitals are destabilized.
⇒ 𝐸𝑔 ↓
𝐸𝑔 underestimation problem in LDA/GGA
5
Electron affinity: 𝐴 Ionization energy: 𝐼
= 휀𝑁+1 𝑁 − 휀𝑁 𝑁 KS +𝛿𝐸𝑥𝑐
𝛿𝜌 𝑀=𝑁−0
𝑀=𝑁+0
𝐸𝑔DFT Derivative discontinuity
𝐸𝑔Exp.
= 𝐸 𝑁 − 1 − 𝐸 𝑁 − {𝐸 𝑁 − 𝐸 𝑁 + 1 }
=
Beyond LDA/GGA SIC
GW
TDDFT
Exact Exchange
LDA+DMFT
Hybrid functionals
Quantum Monte-Carlo
LDA+U
Meta-GGA
𝜌, 𝛻𝜌 (within GGA) + 𝛻2𝜌 + 𝑡
Success in improvement of 𝐸𝑔 for a wide variety of materials in a low computational cost comparable to GGA • TPSS, RTPSS, M06L, mBJ (TB09), …
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Kinetic energy density
MBJ POTENTIAL
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Details (1) Becke-Johnson (BJ) exchange potential [No empirical parameter]
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Becke and Johnson, J. Chem. Phys. 124, 221101 (2006).
𝑣𝑥BJ
𝐫 = 𝑣𝑥BR 𝐫 +
1
𝜋
5
12
2𝑡 𝐫
𝜌 𝐫
Becke-Roussel (BR) exchange potential
𝑣𝑥BR 𝐫 = −
1
𝑏 𝐫1 − 𝑒−𝑥 𝐫 −
1
2𝑥 𝐫 𝑒−𝑥 𝐫
• 𝑥 ← Nonlinear equation involving 𝜌, 𝛻𝜌, 𝛻2𝜌, and 𝑡
• 𝑏 =𝑥3𝑒−𝑥
8𝜋𝜌
1
3
Proposed to model the Coulomb potential created by the exchange hole
Reproduce Slater’s averaged exchange potential 𝑣𝑥Sla. (Negative)
※Kinetic energy density:
𝑡 𝐫 =1
2 𝛻𝜓𝑖
∗ 𝐫 ∙ 𝛻𝜓𝑖(𝐫)
𝑁
𝑖=1
To correct the difference between
𝑣𝑥Sla. and 𝑣𝑥
Exact (Positive)
𝐸𝑔 improvement was moderate…
Details (2) Modified BJ (mBJ) exchange potential [With empirical parameter]
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Tran and Blaha, Phys. Rev. Lett. 102, 226401 (2009).
𝑣𝑥mBJ
𝐫 = 𝑐𝑣𝑥BR 𝐫 + (3𝑐 − 2)
1
𝜋
5
12
2𝑡 𝐫
𝜌 𝐫
… Average of 𝑔 =𝛻𝜌
𝜌
Fitting with the experimental 𝐸𝑔 of many different systems
𝑐 = 𝐴 + 𝐵 𝑔
𝑔 =1
𝑉cell
1
2
𝛻𝜌↑ 𝐫
𝜌↑ 𝐫+
𝛻𝜌↓ 𝐫
𝜌↓ 𝐫𝑑3𝑟
cell
𝐴 = −0.012, 𝐵 = 1.023 bohr12
• 𝑐 = 1 ⇒ Original 𝑣𝑥BJ
• 𝑐 > 1 ⇒ Less negative potential in low 𝜌 regions
𝐸𝑔 increases monotonically with respect to 𝑐. 𝑣𝑥
LDA potential is approximately recovered for any 𝑐 for a constant 𝜌. Reproduce the derivative discontinuity of the 𝑣𝑥
Exact
Details (3) Correlation effects can be taken into account by adding 𝑣𝑐
LDA.
Cannot be used for the force calculation since the potential is not the derivative of an energy functional.
After the geometry optimization using LDA or GGA, only the band structure calculation should be performed using the mBJ potential.
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EXAMPLES
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Theoretical vs experimental Eg
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PRL 102, 226401 (2009).
mBJ+LDA
Standard LDA
By WIEN2K
The mBJ potential yields Eg in good agreement with experiments.
Nonmagnetic TM oxides and sulfides (1) The quality of the improvement of 𝐸𝑔 (eV) is not constant
for all cases. The present determination of 𝑐 may not be general enough.
13 D. Koller et al., PRB 83, 195134 (2011).
= <<
<<< <
< <<
◎
○
×
Nonmagnetic TM oxides and sulfides (2)
14
SrTiO3
𝑣𝑥𝑐mBJ
− 𝑣𝑥𝑐PBE 𝑣𝑥
BR 𝑡/𝜌 -term
Ti
O
The mBJ potential around Ti is quite aspherical mainly due to the 𝑡/𝜌 -term.
d-eg d-t2g
More attractive
More repulsive
PBE mBJ CBM (Ti-3d-t2g)
VBM (O-2p)
SrTiO3
PBE mBJ CBM (Cu-3d-eg)
Cu2O
VBM (Cu-3d-eg)
Eg = 0.53 eV 0.82 eV Eg = 1.88 eV 2.70 eV
Limit of the orbital-independent potential
D. Koller et al., PRB 83, 195134 (2011).
Ferromagnetic Metals Overestimation of the magnetic moments (μB)
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< >
Fe Strong increase of the exchange splitting
LDA(PBE): 𝑣𝑥LDA 𝑣𝑐
LDA
Good error cancellation
mBJ: 𝑣𝑥mBJ
𝑣𝑐LDA
<
Not Good…
D. Koller et al., PRB 83, 195134 (2011).
(Anti-)Ferromagnetic insulators
16
NiO Ni-4s
Ni-3d-eg
Ni-3d-t2g
O-2p
Improved using smaller or larger 𝑐
PBE mBJ
D. Koller et al., PRB 83, 195134 (2011).
Merits
Highly accurate energy band gaps, magnetic moments, and electron densities in most semiconductors and insulators
Computationally cheap and similar quantitative predictive power as much more sophisticated and expensive methods (Hybrid, GW, …)
More positive potential in low 𝜌 regions due to the screening 𝑡/𝜌 term
⇒ Increase of the energy of unoccupied states (larger Eg)
Limits
Moderate improvement of the 𝐸𝑔 and the magnetic moment for some semiconductors and insulators
Some space for improving the determination of 𝑐
Overestimation of the magnetic moment for the ferromagnetic metals
Less effective for materials whose densities of the VBM and CBM extend in similar spatial regions, ascribed to using the orbital-independent potentials
Merits & Limits of mBJ
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The mBJ exchange potential:
Semilocal approximation to an exact-exchange potential and a screening term
Just a XC-potential, not a XC-energy functional
Forces cannot be calculated.
Band gaps with an accuracy similar to hybrid functionals or GW methods for a wide range of different materials even though failure in some cases
Low computational cost comparable to LDA/GGA
Current: Implementation in OpenMX
Summary
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