Studies of semiconducting pyrite and marcasite compounds using many-body perturbation theory in the GW approximation

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Mitglied der Helmholtz-Gemeinschaft

Studies of semiconducting pyrite and marcasite compounds using

many-body perturbation theory in the GW approximation

2nd April 2014, DPG Spring-meeting 2014 in Dresden

Timo Schena, Gustav Bihlmayer, Christoph Friedrich and Stefan Blügel

Peter Grünberg Institut & Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, Germany

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Motivation

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FeS₂ pyrite: a promising photovoltaic material ?

large optical absorption

I. J. Ferrer et al. Solid State Communications 74 (1990)

abundance well-suited band gap

FeS₂: 0.95eV [1]

Shockley-Queisser-Limit

Source: Wikipedia

[1]: A. Ennaoui et al. (1993)

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Motivation

2

abundance well-suited band gap

•  size of fundamental band gap

•  defects, surfaces, device geometry open-circuit voltage

 200mV too small!

Source: Wikipedia FeS₂: 0.95eV [1]

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Motivation

2

abundance well-suited band gap ??

•  size of fundamental band gap

•  defects, surfaces, device geometry open-circuit voltage

 200mV too small!

Source: Wikipedia

FeS₂: 0.95eV [1]

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Methods

^{DFT, G}0W₀ and QSGW

DFT:

whole zoo of xc-functionals GW approximation:

⌃ = GW non-local and energy-dependent

✓ ~

2mr² + v_ext + v_H

◆

· ⁱ(r) + Z

dr⁰⌃(r,r⁰,"^QP_i ) · ⁱ(r⁰) = "^QP_i _i(r)

✓ ~

2mr² + v_ext +v_H + v_xc

◆

· ⁱ(r) = "_{i i}(r)

mean field ^G⁰^W⁰ ⌃(r,r⁰,"^QP_i )

“Hermitianize”

QSGW selfconsistent cycle:

•  fast

•  unreliable band gap prediction

•  slow

•  (usually) systematically improved band gaps

•  in many cases a too large band gap as compared to experiment

•  start-point independent Kotani et al. Phys. Rev. B 76, 165106 (2007)

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FeS

₂

Pyrite

Lattice Structure

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Fe S

a

d_{S S} = p

3a · (1 2u)

•  simple cubic containing 12 atoms

•  S-S dimer distance has crucial influence on electronic structure

•  almost octahedral coordination of S around Fe (t_2g- and e_g-states split)

•  24 symmetries, inversion included

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FeS

₂

Pyrite

Electronic Structure within PBE and G₀W₀

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: S 3p : Fe 3d

: PBE : G₀W₀

•  fundamental band gap: Fe 3d – S 3p transition

•  G₀W₀ reduces band gap size (convergence not simple) more details: Schena et al. Phys. Rev. B 88, 235203 (2013)

PBE

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FeS

₂

Pyrite

Optical Absorption

more details: Schena et al. Phys. Rev. B 88, 235203 (2013) 6

•  optical absorption dominated by “Fe 3d – Fe 3d” transitions

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: PBE

: G₀W₀ 4x4x4 Optical Absorption

more details: Schena et al. Phys. Rev. B 88, 235203 (2013)

FeS

₂

Pyrite

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•  G₀W₀: minor differences regarding the optical band gap

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Similarities Differences

•  octahedral symmetries

•  S-dimers (crucial for electronic structure)

•  orthorhombic (6 atoms in unit cell)

•  edge-shared octahedrons a=4.44Å b=5.42Å c=3.39Å

u=0.200 v=0.378

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FeS

₂

Marcasite

Lattice Structure

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: S 3p : Fe 3d

: PBE : G₀W₀

FeS

₂

Marcasite

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Electronic Structure within PBE and G₀W₀

more details: Schena et al. Phys. Rev. B 88, 235203 (2013)

•  more promising band gap as compared to FeS₂ pyrite PBE

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Pyrites/Marcasites

Choice of Compounds

9 webelements.com

Pyrites

Marcasites

Pyrite FeS₂ RuS₂ OsS₂ ZnS₂ V[in Å³] 159.0 176.6 207.5 211.1

u 0.385 0.382 0.392 0.401

Marc. FeS₂ FeSe₂ FeTe₂ V[in Å³] 81.6 99.6 127.8 u 0.200 0.213 0.224 v 0.378 0.369 0.362

Structural parameters from ICSD database

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FeS2 RuS2 OsS2 ZnS2 FeS2 FeSe2 FeTe₂ 0.0

0.5 1.0 1.5 2.0 2.5

Egap(eV)

exp PBE G₀W₀

Pyrites [1-4] Marcasites [5,6]

Pyrites/Marcasites

Band Gaps: PBE, G₀W₀ and experiment

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•  different orbital character at the

band edges defining the fundamental band gap

•  large screening effects

Agreement suffers in case of …

Ref. for exp. band gaps:

[1]: Ennaoui et al. (1993) [2]: Huang et al. (1988)

[3]: Jaegermann et al. (1988) [4]: Bullet (1982)

[5]: Jagadeesh et al. (1980) [6]: Harada (1998)

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•  GW-correction mainly consists of two parts:

influence on gap : ì î

Pyrites/Marcasites

The role of the screening

11 11

⌃_GW = ⌃_ex + ⌃_corr

Pyrite FeS₂ RuS₂ OsS₂

E_gap(DFT) 0.62 0.12 0.67

E_gap(GW) 0.25 0.14 0.63

: S 3p : M 3d

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Si C MgO NaCl Ar GaAs InSb -100

-50 0 50 100

(E

th gap

Eexp gap

)/E

exp gap

(in%)

PBE G₀W₀ QSGW

~280%

metallic

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What is about QSGW?

Simple Compounds:

Pyrites:

Ref. for exp. band gaps:

Si : 1.25 eV , Ortega et al. (1993) C : 7.3 eV , Hellwege et al. (1982) MgO : 7.7 eV , Adachi et al. (1999) NaCl : 8.5 eV , Poole et al. (1975) GaAs: 1.52 eV , Hellwege et al. (1982) InSb : 0.24 eV , Ortega et al. (1993)

PBE G₀W₀ QSGW

FeS₂(P) 0.62 0.25 ~0.9

ZnS₂ 1.89 2.94 ~3.7

transition _!

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Take-Home Messages

•  new band gap measurements needed for the pyrite and marcasite compounds (be aware of low-intensity conduction states)

•  G₀W₀@PBE calculations for systems with large screening can cause a reduction of the band gap size

•  QSGW calculations usually lead to overestimated gap sizes,

however they might be important for pyrite and marcasite compounds

•  in general: more investigations on transition metal compounds exhibiting a peculiar p-d transition should be conducted

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Take-Home Messages

Acknowledgements

•  new band gap measurements needed for the pyrite and marcasite compounds (be aware of low-intensity conduction states)

•  G₀W₀@PBE calculations for systems with large screening can cause a reduction of the band gap size

•  QSGW calculations usually lead to overestimated gap sizes,

however they might be important for pyrite and marcasite compounds

•  in general: more investigations on transition metal compounds exhibiting a peculiar p-d transition should be conducted

•  Martin Schlipf, Markus Betzinger, Irene Aguilera, and Gregor Michalicek for fruitful discussions

•  Support by BMBF under project number 03SF0402A (NADNuM) and the Jülich Supercomputing Center (JSC) for computation time

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Backup-Slides

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Pyrites: FeS₂Overview

Backup-Slides

•  crystal field splitting in Fe 3d t_2g- and e_g-states

•  bonding and anti-bonding ssσ-, ppσ- and ppπ-states from S-S dimers

•  significant hybridization between Fe 3d- and S 3p-orbitals ssσ

ssσ*

ppσ, ppπ, ppπ*

ppσ*

t_2g e_g

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Pyrites: RuS₂

: S 3p : Ru 3d

0.12 eV

Backup-Slides

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: S 3p : Os 3d

0.75 eV

Pyrites: OsS₂

Backup-Slides

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Pyrites: ZnS₂

: S 3p : Zn 3d

1.40 eV

Backup-Slides

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Marcasites: FeSe₂

: Se 3p : Fe 3d

0.28 eV

Backup-Slides

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Marcasites: FeTe₂

: Te 3p : Fe 3d

Backup-Slides

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Structural dependence

•  position of p-band with respect to d-states

delicately depending on Wyckoff parameter u

•  Structural relaxation has a strong influence on band gap size

u #

E_gap #

splitting ppσ, ppσ* _#

d_{S S} "

I I I

d_{S S} = p

3a·(1 2u)

Backup-Slides

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Orbital-dependent optical absorption:

electrical dipole

matrixelement orbital contributions µ, ν of states i, f

Orbital-

decomposed contributions multiplied by factors to obtain full optical

absorption

solving a system of linear

equations.

Optical Absorption

A

^µ!⌫

(!) = ⇡!

✏

₀

ncV

X

k

X

i,f

|h f | d ˆ | i i|

²

· (! ("

_f

"

_i

)) · ⇢

^⌫_f

· ⇢

^µ_i

Backup-Slides

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The role of the screening – Effect on HSE06 calculations

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Backup-Slides

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The role of the screening – Effect on HSE06 calculations

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G₀W₀ HSE FeS₂(P) 0.25 2.23 RuS₂ 0.40 1.33 OsS₂ 0.63 1.96 ZnS₂ 2.94 2.99 FeS₂(M) 1.44 2.69 FeSe₂ 1.18 2.52 FeTe₂ 1.17 1.60

! transition (in eV):

•  25% HF-exchange too much, 1/ε better