woptic: optical conductivity and adaptive k-integrationP. Wissgott1 E. Assmann1 J. Kuneš2 A. Toschi1 P. Blaha3 K. Held1

1Inst. of Solid State Physics, TU Vienna 2Inst. of Physics, Czech Academy of Sciences 3Inst. of Materials Chemistry, TU Vienna

Brioullin-zone integration

I application of Bloch theoremI indispensable for numerical study

of periodic solidsI traditionally, uniform samplingI in some cases, small k -space regions

contribute disproportionately, e.g.:I transportI low-energy spectral propertiesI here: optical conductivity σ

; adaptive k -mesh refinementω

[eV

]

σk(ω

)[1

06Ω−

1eV−

1]

W L Γ X W K

−2

0

2

ω[e

V]

0.8

1

1.2

1.4

1.6

1.8

0.2

0.1

0

fcc-Al: k -resolved contributions to σ , and bandstructure

Adaptive tetrahedral k-mesh

I k -space is divided into tetrahedra,which are successively selectivelyrefined

I in 3d, refinement is not unique,sub-tetrahedra are not congruent

I Kuhn triangulation: 2 classes oftetrahedra

Γ

W

L

KX

adaptive k -mesh for fcc-Al after six refinements

woptic: optical conductivity of interacting electrons

I part of Wien2Wannier package

I works with Wien2k, Wannier90,and any self-energy

I two main parts:I woptic_main computes σ

I refine_tetra manages k -mesh

I Vertex corrections are neglected(in DMFT, they vanish)

I for non-interacting electrons (c: conduction, v : valence):

σαβ(Ω)∼ ∑c,v ,k

δ (εc(k )−εv(k )−Ω)Ω ∇

αvc(k )∇

β

cv(k )

I for interacting electrons (; spectral density Anm (k ,ω)):

σαβ(Ω)∼ ∑k ,ω

f (ω)−f (ω+Ω)Ω tr

∇

αA(k ,ω +Ω)∇βA(k ,ω)

f = Fermi function, ∇αnm = 〈ψnk |∂β |ψmk〉

I The trace is expensive, adaptive integration worthwhile

fcc-Al: convergence and comparison to Wien2k

experimentwien2k optic, 22776 k-pointswoptic - ℓ = 1, 145 k-pointswoptic - ℓ = 10, 2705 k-pointswoptic - ℓ = 17, 13497 k-points

σ[1

03Ω−

1cm

−1]

ω [eV]

00 1 21.50.5

5

10

15

I contributions to σ stronglypeaked in k

I match Wien2k results,but much cheaper

I precise shape depends onbroadening schemeI Wien2k: Lorentzian (0.1 eV)I woptic: Σ ←−iδ

I `: # refinement iterations

SrVO3 (correlated metal, including DMFT self-energy)

experiment

woptic, non.-int.,ℓ= 1, 165 k-points

woptic, non.-int. ℓ = 5, 6545 k-points

woptic, int.. ℓ = 1, 165 k-points

woptic, int.. ℓ = 5, 6506 k-points

σ[1

03Ω−

1cm

−1]

ω [eV]0

5

5

10

10 15

ω[e

V]

σk(ω

)[1

03Ω−

1eV−

1]

Γ M R Γ X M

−5

0

5

ω[e

V]

0

4

8

12 8

4

0

Tetrahedral Hygiene

I enforce regularity: at most one “hanging node”per edge, to avoid unstable convergence

I and shape stability: no highly-distortedtetrahedra, to avoid large errors

I no k -point is wastedI k -points re-used in subsequent iterations,

lapw1 and optic only called for new onesI take advantage of symmetries

hanging nodes (red) in 2d;for regularity, refine center triangle

I tetrahedron T selected for refinement ifintegration error estimate εT ≥ Θ max

T ′εT ′ (Θ = "harshness")

Practical considerations

I inputs:I momentum matrix elements ∇α

nm (k )(from Wien2k’s optic)

I Hamiltonian H(k ) (from Wannier90)I self-energy Σ (k ,ω)

(e.g. from DMFT)

I random-phase problem:I ψk has random phase in Wien2k

(independent diagonalizations)I may affect σ in interacting caseI to be fixed by interpolating ∇(k )

modes of operation:I full momentum-matrix ∇α

nm (k )I Peierls approximation: ∇ 7→ ∂H

∂kI 2 non-interacting modes

References

woptic P.W., E.A. J.K., A.T., P.B., K.H. in preparationWien2Wannier J.K. et al., Comput. Phys. Commun. 181, 1888 (2010)W2k optic Ambrosch-Draxl and Sofo, Comput. Phys. Commun. 175, 1 (2006)Wannier90 A.A. Mostofi et al., Comput. Phys. Commun. 178, 685 (2008)

http://wannier.org

tetrahedral refinement Ong, SIAM J. Sci. Comput. 15, 1134 (1994)Al exp. Ehrenreich et al., Phys. Rev. 132, 1918 (1963)SVO exp. Makino et al., PRB 58, 4384 (1998)

Take-home messages

I woptic calculates the optical conductivityI from a many-body calculationI using an adaptive k -mesh

I the two are separated, refine_tetra could be used for other quantitiesI distributed as part of Wien2Wannier:http://www.wien2k.at/reg_user/unsupported/wien2wannier/

I “random-phase problem” is being worked on