Su-Huai Wei

Bejing Computational Science Research Center

Recent Development of APW-based Methods and Band Structure of Semiconductors

Outline

• Brief history of APW-based DFT methods (APW, LAPW, FLAPW, etc.) for electronic structure calculations

• Issues related to the standard APW and LAPW methods

• Some recent developments (LAPW+LO, APW+LO, etc.)

• Second variation method and effects of P1/2 orbital on SO coupling

Thanks for David Singh for providing me some of the slides

Density Functional Theory (DFT)

Hohenberg-Kohn theorem

The total energy of an interacting inhomogeneous electron gas in the presence of an external potential Vext(r ) is a functional of the density ρ

H(ρ) = T(ρ) + Veff(ρ)

][)()( ρρ FrdrrVE ext += ∫

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

Walter Kohn, Nobel Prize 1998 Chemistry

Kohn Sham equations

][||)()(

21)(][ ρρρρρ xcexto Erdrd

rrrrrdrVTE +′

−′′

++= ∫∫

Total energy

Ekineticnon interacting

Ene Ecoulomb Eee Exc exchange-correlation

1-electron equation (Kohn Sham)

)()())(())(()(21 2 rrrVrVrV iiixcCext

Φ=Φ+++∇− ερρ

∑≤

Φ=FEi

irε

ρ 2||)(

vary ρ

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)

12∇² + V(r) + Vxc(r) ϕi

k = ϕikεi

k

relativisticsemi-relativisticnon relativistic

Full potential : FP“Muffin-tin” MTpseudopotential (PP)

Local density approximation (LDA)Generalized gradient approximation (GGA)Beyond the LDA: e.g. LDA+U

Spin polarizednon spin polarized

non periodic (cluster)periodic

plane waves : PWaugmented plane waves : APWlinearized “APWs” (LAPW) atomic oribtals. e.g. Gaussians

Basis functions

Treatment of spin

Representationof solid

Form ofpotential

exchange and correlation potentialRelativistic treatment

of the electrons

Kohn-Sham equations

How to solve the Kohn Sham equations

APW based schemes• APW (J.C.Slater 1937)

– Non-linear eigenvalue problem– Computationally very demanding

• LAPW (O.K.Andersen 1975)– Generalized eigenvalue problem– Full-potential

• LAPW+LO (D. J. Singh 1991)– Local orbitals (for semi-core states)

• APW+LO (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000)– Better convergence by combination of most efficient

schemes– Basis for large inhomogeneous system

The Augmented Planewave (APW) MethodJ.C. Slater, Phys. Rev. 51, 846 (1937); T. L. Loucks, APW method, (1967).

ul(r,El)Ylm(r)

ei(G+k)⋅r Space is divide into two regions:• MT spheres centered at atomic sites

• Interstitial

“Basis” consists of planewaves in the interstitial region and atomic partial waves in the spheres.

ϕk(r) =Σ cGΩ-1/2 ei(G+k)⋅r r∈Interstitial

G Σ aklm ul(r,El) Ylm(r) r∈MT Sphere

lm

• ul(r) are the radial solutions of Schrődinger’s equation at the energy of interest. alm are determined by matching the basis functions at the MT boundary.

The standard APW method requires use of an energy dependent secular equation and is not practical for complex systems

Atomic partial waves

are energy dependent basis functions. One had tonumerically search for the energy, for which the det(H-ES) vanishes.

∑m

mKm rYErua

)(),(

HY=ESY

H HamiltonianS overlap matrix

The Augmented Planewave (APW) Method

• Key points:• The ul are orthogonal to “core” states if the core states have zero

wavefunction at the MT sphere.

• Since the basis functions are indexed by G one may think that there is a connection with the planewave pseudopotential formalisms. Indeed, one can show that with a unitary transformation:

<φ|H|φ>x = ε< φ|φ>x

So, APW is like a very soft non-norm-conserving pseudopotential. However, it is highly non-transferable:

ul(r,E) cannot be used at another energy (because u is very energy dependent - ∂u/∂E is usually very large).

HPS SPS

There is a trade-off between transferability and softness (nothing is free). The story of linearization and local orbitals is related to this.

The Augmented Planewave (APW) Method

O.K. Andersen, Phys. Rev. B 12, 3060 (1975). D. D. Koelling and G. O. Arbman, J. Phys. F (1975).

Key Ideas:• The problem with the APW method is the energy dependence of the

secular equation which is a result of the energy dependence of the augmenting function.

• Solution: Add variational freedom: particularly ů(r) = ∂u(r)/∂E.

• Where El is the pivot energy, alm and blm are determined by matching the value and first derivative of the basis functions at the sphere boundary.

ϕk(r) =Σ cGΩ-1/2 ei(G+k)⋅r r∈ Int.

G Σ [aklm ul(r,El)+bk

lm ů(r,El) ]Ylm(r) r∈ MTlm

The Linearized Augmented Planewave (LAPW) Method

Linearization of energy dependence of the radial function

)ˆ()],()(),()([ rYrEukBrEukA mnm

mnmkn

∑ +=Φ

rnKkie

).( + Atomic sphere

PWs

PW

O.K.Andersen,Phys.Rev. B 12, 3060 (1975)

Join PWs in value and slope

LAPW

LAPWs

Features of the LAPW basis:1. Basis is flexible enough to use a single diagonalization (energy errors

are now O((E-El)4).

2. The transferability makes the use of full-potential easier.

3. The additional matching conditions to connect both u and ů to the planewaves means that for a given level of convergence, more planewaves are needed.

The full potential, all electron nature combined with the flexible basis made the FLAPW method the state-of-the-art for calculating electronic structures, especially for transition elements and their compounds – Many groups developed codes since then.

The Linearized Augmented Planewave (LAPW) Method

The LAPW method

Development of the (F)LAPW codes:1. LAPW codes with MT approximation are developed in late 1970’s.(Koelling et al., 1975); (Jepsen et al., 1978); (Krakauer et al., 1979)

2. FLAPW was developed for slab calculations.(E. Wimmer, H. Krakauer, M. Weinert, and A. Freeman PRB24, 864 (1981))

3. General bulk FLAPW code was developed.(Wei and Krakauer, 1984); (Jansen and Freeman, 1984); (Blaha and Schwarz, 1990)

Other improvement and modifications:Force (Yu, Singh, and Krakauer, 1991)

FLAPW+LO (Singh, 1991)

Linear Response (Yu, Wang, Krakauer, 1994)

APW+LO (Sjostedt, Nordstrom, and Singh, 2000)

Stress and Pressure (Thonhauser, Ambrosch-Draxl, and Singh, 2002)

Electric Field Gradient (EFG) calculation for Rutile TiO2 as a function of the Ti plinearization energy

P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992).

Complication in the LAPW method

What went wrong?

• Many elements (e.g., Ti) have extended core states that have non-zero wavefunction on the sphere boundary so the u and ů is not orthogonal to them

• The LAPW method requires non-overlapping spheres, there are serious limits to how large RMT can be, especially in oxides, nitrides, carbides, and H containing system

Electronic Structure

E

Ti- 3p

O 2pHybridized w.Ti 4p, Ti 3d

Ti 3d / O 2pEF

Complication in the LAPW method

Problems with semi-core states

Complication in the LAPW method

1. Using two windows, one for the semicore state and one for the valence state.

The states calculated in the two windows are not exactly orthogonal to each other

2. Adding more variation freedom (SLAPW).

Requires extra matching condition that leads to high planewave cut-off

3. Using LOCAL ORBITALS

Solutions

ϕ(r) = Σ cG Ω-1/2 ei(G+k)⋅r

G Σ [almul(r)+blmů(r)+clmül(r) ]Ylm(r)lm

D. Singh, Phys. Rev. B 43, 6388 (1991).

ϕ(r) = Σ cG Ω-1/2 ei(G+k)⋅r

G Σ [almul(r, E1)+blmů(r,E1)]Ylm(r)lm

+Σ clm(a’lmul(r,E1)+b’lmůl(r,E1)+u(2)l(r,E2)) Ylm(r)

lm

The LAPW+LO MethodLAPW+LO basis is:

The variational coefficients are: (1) cG and (2) clm

The non-variational coefficients are alm, blm, a’lm, & b’lm• alm and blm are determined by matching the value and

derivative on the sphere boundary to the planewaves as usual.• a’lm and b’lm are determined by matching the value and

derivative on the sphere boundary to zero. Thus this part (a’lmul(r)+b’lmůl(r)+u(2)

l(r))Ylm(r) is formally a local orbital.

ϕ(r) = Σ cG Ω-1/2 ei(G+k)⋅r

G Σ [almul(r, E1)+blmů(r,E1)]Ylm(r)lm

+Σ clm(a’lmul(r,E1)+b’lmůl(r,E1)+u(2)l(r,E2))Ylm(r)

lm

Key Points:

1. We are trading a large number of extra planewave coefficients for some clm.

2. LAPW+LO converges like LAPW. The LO adds a few basis functions (e.g., 3 per atom for p states). Can also use LO to relax linearization errors, e.g. for a narrow d or f band

3. Two energy parameters, one for u and ů and the other for u(2). Choose one at the semicore position and the other at the valence

<G|G>

RKmax

La

D. Singh, Phys. Rev. B 43, 6388 (1991).

Cubic APW

QAPW

The LAPW+LO Method

To lower RKMAX even further, one can try to go back to use the APW and add few local orbitals:

• Structures with small atoms and large empty spaces.

• Structures with some “hard” atoms embedded in a matrix of “soft” atoms: e.g. Mn impurities in Si.

• Now only the value on the boundary matches. This means that there are extra APW-like kinetic energy terms in the Hamiltonian and forces.

• APW+LO is equivalent to LAPW not LAPW+LO. It is not suitable for handling semicore states.

E. Sjostedt, L. Nordstrom and D.J. Singh, Solid State Commun. 114, 15 (2000).

ϕ(r) = Σ cG Ω-1/2 ei(G+k)⋅r

G Σ [almul(r, E1)]Ylm(r)lm

+Σ clm(a’lmul(r,E1)+u(2)l(r,E2)) Ylm(r)

lm

The APW+LO Method

Convergence of the APW+LO Method

Ce

G. Madsen, P. Blaha, K. Schwarz, E. Sjostedt and L. Nordstrom PRB 64, 195134 (2001).

x100

Remarks on the APW-based Methods

• Several augmentation schemes are developed to make the APW-based method more efficient and accurate.

• There is no requirement that all atoms or angular momenta be augmented in the same way. APW+LO could be used for those atoms for which a high Gmax would otherwise be needed, whereas LAPW could be used for all others.

• Applications:• LAPW+LO (to treat semi-core states and reduce linearization error).

• APW+LO (to lower GMAX in the LAPW method).

• APW+2LO (APW+LO with semi-core treatment).

• p1/2 local orbitals (for spin-orbit).

• …

Second Variation Method

HR = HSR + HSO

HSR ΨSR = εSR ΨSR

ΗR ΨR = HSR ΨR+ HSO ΨR

ΨR = Σn Cn ΨSRn

In standard FLAPW code, ∆SO is calculated using the Second Variation Methodwhere the spin-dependent p½ and p3/2 orbitals are replaced by the pl=1 orbital.

Because the p½ orbital has non-zero value at the nuclear site, but pl=1 orbital has zero value at r=0, so pl=1 orbital does not represent well the p½ orbital near r=0.

What is the effect of p½ orbital on the calculated ∆SO?

Arsenic 4 p Bismuth 6 p

r (a.u.) r (a.u.)

Effects of the p½ local orbitals on the spin-orbit splitting of semiconductors

Spin

-orb

it sp

littin

g [m

eV]

Effects of the p½ orbitals

6 p anions ~250 meV

5 p anions ~ 40 meV

4 p anions ~ 10 meV

p½ corrections: