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

2

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), …

6

Kinetic energy density

ＭBJ POTENTIAL

7

Details (1) Becke-Johnson (BJ) exchange potential [No empirical parameter]

8

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]

9

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

12

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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