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
Band gap (ev) W Theoretical band gaps of AGNRs Armchair GNRs (AGNRs) 1 2 3 4 5 N a -2 N a -1 N a 6 5 4 3 2 1 0 E g of the isolated AGNRs N a = 7 LDA GW N a L. Yang et al., PRL 99, 186801 (2007). LDA (GGA) underestimates the band gaps of the AGNRs. 4
E g underestimation problem in LDA/GGA DFT: Only a ground state theory E g Exp. = E N 1 E N {E N E N + 1 } Self-interaction error LDA/GGA electron sees extra Coulomb repulsion by own charge. Energy levels of the occupied orbitals are destabilized. E g Ionization energy: I = ε N+1 N ε N N KS + δe xc δρ = E g DFT Electron affinity: A M=N+0 M=N 0 Derivative discontinuity 5
Beyond LDA/GGA SIC GW TDDFT Exact Exchange LDA+DMFT Hybrid functionals Quantum Monte-Carlo LDA+U Meta-GGA ρ, ρ (within GGA) + 2 ρ + t Success in improvement of E g for a wide variety of materials in a low computational cost comparable to GGA TPSS, RTPSS, M06L, mbj (TB09), Kinetic energy density 6
MBJ POTENTIAL 7
Details (1) Becke-Johnson (BJ) exchange potential [No empirical parameter] Becke and Johnson, J. Chem. Phys. 124, 221101 (2006). v x BJ r = v x BR r + 1 π 5 12 Becke-Roussel (BR) exchange potential v x BR r = 1 b r 2t r ρ r 1 e x r 1 r x r e x 2 x Nonlinear equation involving ρ, ρ, 2 ρ, and t b = x3 e x 8πρ 1 3 Kinetic energy density: t r = 1 2 Proposed to model the Coulomb potential created by the exchange hole Reproduce Slater s averaged exchange potential v x Sla. (Negative) N i=1 ψ i r ψ i (r) To correct the difference between v x Sla. and v x Exact (Positive) E g improvement was moderate 8
Details (2) Modified BJ (mbj) exchange potential [With empirical parameter] Tran and Blaha, Phys. Rev. Lett. 102, 226401 (2009). v x mbj r = cv x BR r + (3c 2) 1 π 5 12 2t r ρ r c = A + B g g = 1 1 V cell 2 cell ρ r ρ r + ρ r ρ r d 3 r Average of g = ρ ρ A = 0.012, B = 1.023 bohr 1 2 Fitting with the experimental E g of many different systems c = 1 Original v x BJ c > 1 Less negative potential in low ρ regions E g increases monotonically with respect to c. v x LDA potential is approximately recovered for any c for a constant ρ. Reproduce the derivative discontinuity of the v x Exact 9
Details (3) Correlation effects can be taken into account by adding v c 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. 10
EXAMPLES 11
Theoretical vs experimental Eg mbj+lda Standard LDA By WIEN2K The mbj potential yields E g in good agreement with experiments. PRL 102, 226401 (2009). 12
Nonmagnetic TM oxides and sulfides (1) The quality of the improvement of E g (ev) is not constant for all cases. << = << < The present determination of c < may not be general <<< enough. D. Koller et al., PRB 83, 195134 (2011). 13
Nonmagnetic TM oxides and sulfides (2) SrTiO 3 D. Koller et al., PRB 83, 195134 (2011). v mbj xc vpbe xc vbr x t/ρ -term d-e g d-t 2g Ti O More attractive The mbj potential around Ti is quite aspherical mainly due to the More repulsive t/ρ -term. SrTiO 3 CBM (Ti-3d-t 2g ) E g = 1.88 ev PBE 2.70 ev mbj Cu 2 O CBM (Cu-3d-e g ) E g = 0.53 ev PBE 0.82 ev mbj VBM (O-2p) VBM (Cu-3d-e g ) 14 Limit of the orbital-independent potential
Ferromagnetic Metals Overestimation of the magnetic moments (μ B ) < > Fe Strong increase of the exchange splitting Good error cancellation LDA(PBE): v x LDA v c LDA < mbj mbj: v x vlda c Not Good D. Koller et al., PRB 83, 195134 (2011). 15
(Anti-)Ferromagnetic insulators NiO PBE mbj Ni-4s Ni-3d-e g Improved using smaller or larger c Ni-3d-t 2g O-2p D. Koller et al., PRB 83, 195134 (2011). 16
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 Limits Merits & Limits of mbj Increase of the energy of unoccupied states (larger E g ) t/ρ term Moderate improvement of the E g 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 17
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. Summary 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 18