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FURIO CORA’

MODELLING FUNCTIONAL MATERIALS(BEYOND LOCAL DFT)

University College LondonDept of Chemistry f.cora@ucl.ac.uk

PERSONAL INTEREST

SOLIDS WITH TRANSITION METAL IONS

-VERY RESPONSIVE TO ENVIRONMENT (CATALYSIS, SENSORS)

-EXTREME PROPERTIES (MICROELECTRONICS)-CHALLENGE TO THEORY

ZEOLITES, ALPOS DENSE SOLIDS

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IN THE BEGINNING …

( ) ( ) ( ) ( )1 21 1KStot 1 22 21 12

ˆ v ,E EN

xcσi σi σi ext α β

σ α,β i

ρ r ρ rn r ρ r dr dr dr ρ ρ

r

σ

ϕ ϕ−= =

= ∆ + + + ∑ ∑ ∫ ∫r r

r r r r rr

Kinetic Kinetic EnergyEnergy

Coulomb energyCoulomb energyNuclear (external) Nuclear (external) potentialpotential

ExchangeExchange--correlation correlation energyenergy

BUT !ONE ELECTRON SHOULD NOT INTERACT WITH ITSELF

REQUIRES

ElectronElectron--electron interactionelectron interaction

( ) ( ) [ ]1 21 1 2212

0E xcρ r ρ r

dr dr ρr

+ =∫r r

r rr

NOT SATISFIED IN DFT WHEN ρ ISTHE TOTAL EL. DENSITY: SELF-INTERACTION

ρρρρ = TOTAL EL. DENSITY

ELECTRONIC STRUCTURE STUDY OF SOLIDS

FURIO CORA’

ELECTRONIC STRUCTURE STUDY OF SOLIDS

FURIO CORA’

Mn

O

H

P

O

O O

Ni

f)

LOCALISED d ELECTRONS

( ) ( ) [ ]1 21 1 2212

0E xcρ r ρ r

dr dr ρr

+ =∫r r

r rr

SELF-INTERACTION TERMscales as 1/r12CRITICAL FOR LOCALISED STATES

STRONGLY CORRELATED SYSTEMS

but the problem is the EXCHANGE

MnO O

O

O

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SOLUTION STRATEGIES:

1) INTRODUCE SELF-INTERACTION IN DFT-LDA+U-SIC-LDAself-interaction is different for each electron;the SI correction must be orbital-dependent(traditional DFT only uses the TOTAL electronic density)

2) GO BEYOND DFT-Quantum Monte Carlo (QMC)-Wavefunction techniques (post-HF: MP, CI, CP)-Dynamical Mean Field Theory (DMFT)

1b) THE OLDEST QM THEORY (HARTREE-FOCK) IS INFERIOR TO DFT, BUT IS SELF-INTERACTION FREE

< I J | K L > - < I K | J L > = 0 if I=J=K=L COMBINE HF AND DFT DESCRIPTIONS:

HYBRID FUNCTIONALS

ELECTRONIC STRUCTURE STUDY OF TM PEROVSKITES

FURIO CORA’UCLUCLUCLUCL

HYBRID FUNCTIONALS

One-electron equations can be cast in the same structure

-HF hHF= t + VN(R) + j + XHFαXHF+(1-α)VX

-DFT hKS= t + VN(R) + j + VXC

Tempting to mix the two treatments, BUT! this has nofoundation in quantum mechanics, and needs to be treatedas empirical.

HYBRID FUNCTIONALS ARE GOOD IF AND ONLY IFWE ACHIEVE SYSTEMATIC IMPROVEMENTS

ELECTRONIC STRUCTURE STUDY OF TM PEROVSKITES

FURIO CORA’UCLUCLUCLUCL

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1 PARAMETER ONLY – AMOUNT OF HF EXCHANGEPRIMARY GOAL: REMOVE SELF-INTERACTION

(1 )hyb GGA GGAxc x x x x cE a E a E E= + − +

EVALUATE EFFECT OF ax (0

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DISPLACEMENT OF CHARGES DURINGFERROELECTRIC DISTORTION

M

A CUBIC PHASE2e- in O-2p AO

FE PHASE2e- in MO BOND

e-

e-

+ -

THE POLARISATION HASNUCLEAR AND ELECTRONIC CONTRIBUTIONS

ACCURACY OFHYBRID FUNCTIONALSRELATIVE IONIC SIZES, DISTORTION, POLARISATION

POLARISATION

EXPT

BEST VALUE

DISTORTION

EXPT

BEST VALUE

Ferroelectric BaTiO3

EXPT

RELATIVE IONIC SIZES

LDA,GGABEST VALUE

EXACT EXCHANGE

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MAGNETIC PEROVSKITES: KMnF3

MAPPING ONTO AN ISING SPIN HAMILTONIAN

0 0 , ,

12M ij z i z ji j

E E E E J S S≠

= + = + ∑Jse

Jd

HYBRID DFT IN THE SOLID STATE

FURIO CORA’UCLUCLUCLUCL

MAGNETIC PEROVSKITES

KMnF3

EXPT

26AFM FM

se z

E E E

J S

∆ = −

=

Jse

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UCLUCLUCLUCLUCLUCLUCLUCLBand Structures of FeO

0%HF 35%HF20%HF

BLYP functional gives a metallic solutionBLYP functional gives a metallic solution

R X M Γ F

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

Ha

rtre

e

R X M Γ F

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

Hart

ree

R X M Γ F

Ef

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

Ha

rtre

e

B3LYP transition between Fe3d/O2sp -Fe3dB3LYP transition between Fe3d/O2sp -Fe3d

H35B65LYP transition between Fe3d/O2sp-Fe4s

H35B65LYP transition between Fe3d/O2sp-Fe4s

Experimentally weak absorption of 0.5 eV between Fe3d/O2sp-Fe4sand stronger transition of 2.4 eV between Fe3d/O2sp-Fe3dBagus et al. PRL (1977)

Experimentally weak absorption of 0.5 eV between Fe3d/O2sp-Fe4sand stronger transition of 2.4 eV between Fe3d/O2sp-Fe3dBagus et al. PRL (1977) Alfredsson et al. PRB (2004)

TERNARY SOLIDS, ABOnEQUILIBRIUM DISTORTION

PEROVSKITES: IDEAL RATIO OF IONIC SIZESTol = (RA+RO)/(RB+RO) = √√√√2O

A

B

RATIO OF AO/BO EQUIL.BOND DISTANCES

VARIES WITH % HF EXCHANGE

DIFFERENT FUNCTIONALS GIVEDIFFERENT IONIC SIZES

COVALENCE CAN DELOCALISEELECTRONS AND REDUCE

SELF-INTERACTIONMAKES CATIONS WITH COVALENT

BONDS (B) BIGGERCaTiO3, BaTiO3, CaSO4, AlPO4 etc

EXPT

LDA,GGA

EXACT EXCHANGE

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INITIAL CONCLUSIONS- Hybrid functionals improve accuracy of DFT for solids- Best hybridisation depends on property, and isdifferent from molecular values (more localised states)

HYBRID DFT IN THE SOLID STATE

FURIO CORA’UCLUCLUCLUCL

Equil. Geom, Lattice param. 60-80(structural distortions) >601 el. properties: P, Jse 40-60Transition pressures >502nd deriv: phonons, Elast. prop 20

Property α (%)

Ref. for initial systematic workF. Corà et al, ‘The performance of hybrid density functionals in solid state chemistry’,Sructure and Bonding vol 113, 2004, 171-232.

CaCu3Ti4O12 (CCTO)

Giant dielectric constant (104 – 105) subject of debate.

Barrier layer effect?

Require semi-conducting grains/domains separated by insulating surfaces.

Mark Michel

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Bulk electronic properties:Magnetic coupling and

Spin localisation

Depend on %HF exchangenot on local DFT flavour

Bulk Structuredepends on both choice of functional and

amount of HF exchange.

Best agreement:20-60% HF exchange With PBE and BLYP functionals.

SURFACE PROPERTIES OF CCTO (Ca0.25Cu0.75TiO3)

(001) SURFACE STRUCTURE

2) ONLY ONE OF THE THREESURFACE Cu2+ IONS IS STILL INSQUARE PLANAR COORDINATION

TWO Cu2+ IONS ARE NOT INSTABLE COORDINATION1) IN LARGE WINDOW3) IN SMALL WINDOW

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3

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DISPROPORTIONATION1) Cu1+ IN LARGE WINDOW3) Cu3+ IN SMALL WINDOW

CALCULATED ∆E2Cu2+ � Cu1+ + Cu3+ = 0.52 eV

HYBRID DFT IN THE SOLID STATE

UCLUCLUCLUCL

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SURFACE PROPERTIES OF CCTO (Ca0.25Cu0.75TiO3)

TOTAL AND PROJECTED DENSITY OF STATES

Cu DISPROPORTIONATION

GENERATES

SURFACE TRAP STATES

Cu1+ TRAPS HOLES ���� Cu2+

Cu3+ TRAPS e- ���� Cu2+

NO MOBILE CARRIER

AT THE SURFACE

HYBRID DFT IN THE SOLID STATE

UCLUCLUCLUCL

20% HF: surface disproportionationinto Cu1+ and Cu3+.

60% HF: Charge transfer from surface O2- to Cu2+, yielding Cu1+/O1- ions.

Redox chemistry creates trap states which trap mobile charge carriers.

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3

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(001) SURFACE – CHANGE %HF

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INTERNAL BARRIER LAYER MODEL FOR CCTO

Furio Cora, Amy Poole, Andrew Wills

Prussian Blue, FeIII4(FeII(CN)6)3(H2O)6

Unit cell of Prussian Blue with iron sites

indicated

FeIII

FeII

FeIII

• Cubic Structure:

– FeIII - NC - FeII - CN - FeIII

– Lattice parameter = 10.2Å

– Space Group = Pm3m

• Colour due to charge transfer between FeIII and FeII cations

• Historically determined:– 0.008 µB ± 0.028*

*Ref: P. Day, F. Herrn, A. Ludi et al; Helvetica Chemica Acta,63, 148 (1980)

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• FeIII is paramagnetic– S=5/2

• FeII, C and N are diamagnetic– S=0 FeII ;

– But measured magnetisation (0.008±0.028µB)

• Ferromagnetic order below TC = 5.7K

Bonding in Prussian Blue

ILL-D20 Experimental output

Polarised neutron diffraction and Experimental Spin Density Maps

Spin density map {1 0 0} plane

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III

III

II

IIIII

Theoretical spin density map

{2 0 0} plane

Theoretical spin density map

{1 0 0} plane

Experimental spin density map

{1 0 0} plane

Experimental and Theoretical spin density maps

{1 0 0} plane

Spin Density Maps of Prussian Blue

Analysis of the Spin Density

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II IIIII

III

II

• Spin Density

– FeIII• Expt = 3.5 µB• Model = 4.3 µB

– FeII• Expt = 0.40 µB• Model = 0.36 µB

spin density map {2 0 0} plane

Spin Density Map

PB Model with no vacancies (only FeIII(CN)6): spin on FeII ~ 0.2 µBModel with vacancies (trans-FeIII(CN)4(H2O)2): spin on FeII ~ 0.36 µB

• Spin Density on Fe(II) depends on local environment. Range from 0.07 to 1.2 electrons (40%HF)

• Coupled to structural distortion

Spin Density Maps of Prussian Blue –Disordered vacancies

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Average value of spin density on FeII increases on increasing disorder in structure

• Spin Density FeII

• Expt = 0.40 µB• Model = 0.072 µB ordered vacancies

0.073/1.112 disordered (1)0.157/0.179 disordered (2)

Highest when there is FeIII(CN)5(H2O)Oxygen of the H2O is π donor; this

reduces the dimension in which spin can be transferred; more charge is transferred to the FeII opposite H2O

Spin Density Maps of Prussian Blue –Disordered vacancies

CONCLUSIONS

-Hybrid functionals improve accuracy of DFT for solids- Systematic trends, but- Best hybridisation depends on property, and isdifferent from molecular values (more localised states)

-Defects, surfaces and situations that break local symmetryare more challenging. Underconstrained solutions difficult to reproduce reliably with DFT; importance ofaccurate reference from experiment or modelling (post-HF)

-In general, beware of anything unusual coming from a DFTstudy!

HYBRID DFT IN THE SOLID STATE

FURIO CORA’UCLUCLUCLUCL