self-interaction correction (sic) scheme and f-electron...

Computational Science & Engineering DepartmentCSEDaresbury Laboratory

SelfSelf--Interaction Correction (SIC) Scheme Interaction Correction (SIC) Scheme and fand f--Electron MaterialsElectron Materials

Z. (Dzidka) SzotekZ. (Dzidka) Szotek

Daresbury Laboratory, UKDaresbury Laboratory, UK


Lanthanides = Rare EarthsLanthanides = Rare Earths


OutlineOutline

§§ Introduction: some issues in fIntroduction: some issues in f--systemssystems

§§ SICSIC--LSD method: advantages and disadvantagesLSD method: advantages and disadvantages

Ø Full implementation (T=0K)•• Applications: valence and valence transitionsApplications: valence and valence transitions

Ø Local SIC (Multiple Scattering Theory implementation)•• Finite temperature phenomena: Ce phase diagramFinite temperature phenomena: Ce phase diagram•• Finite Temperature Magnetism of the Heavy Rare EarthsFinite Temperature Magnetism of the Heavy Rare Earths

§§ ConclusionsConclusions


Electrons in SolidsElectrons in Solids

Bloch pictureBloch picture

Mean FieldMean Field: : Delocalised StatesDelocalised States(band formation)(band formation)

☺☺ Chemical and structuralChemical and structuralLL Correlation Correlation

HeitlerHeitler--LondonLondon

Atomic SolutionsAtomic Solutions: : Localized StatesLocalized States(multiplets)(multiplets)

LL Chemical and structuralChemical and structural☺☺ CorrelationCorrelation

δbandδloc

ε0ε0


Materials Specific Calculations for Materials Specific Calculations for ff--Electron SystemsElectron Systems

Traditional Approaches:Traditional Approaches:– f-core– f-band

Compare with experimental resultsCompare with experimental results

Depending on system, all variations of fDepending on system, all variations of f--core and fcore and f--band could band could be found be found


0.5 1.0 1.5

0.4

0.8

5f in U

4f in gd

Rad

ial d

ensi

ty

Radius (Angstrom)

Lanthanides and ActinidesLanthanides and Actinides

Gd

(Borrowed from M.S.S. Brooks)(Borrowed from M.S.S. Brooks)


Poor screening of nuclear charge by 4f electrons

LANTHANIDESLANTHANIDES

Similarity in properties, with gradual changes occurring across the lanthanide series Ø a size effect from the Lanthanide Contraction

4f n electrons are contracted into the core and unable to participate in bonding


Rare Earth Chalcogenides & PnictidesRare Earth Chalcogenides & Pnictides

Note different behaviour in the lattice constants Note different behaviour in the lattice constants between the chalcogenides and the pnictidesbetween the chalcogenides and the pnictides


DFTDFT

?e-

Veff

? ({xi})

?

Exact mapping of aExact mapping of amanymany--body problem body problem onto a oneonto a one--electronelectron

problem in an problem in an effective potentialeffective potential

e-

TT

(r))n(en(r)rdnE hxc

LSDxc ∫= 3][][][][][ nEnUVnTnE xcexts +++=

ns

LSDxcexts

LSD

nT

nnnnnn

nEnUVnTnE

αα

αψ

ψψα

∆−=

=+=

+++=

∑↓↑↓↑

}{min][

),(;

][][][][

)()()()();()()]([)( rrrrVrerrVrH xcHexteffeffLSD rrrrrrrr υυυψψψ ++==+∆−≡

LSDLSD

DFT: Local Spin Density (LSD)DFT: Local Spin Density (LSD)

Source of unphysicalSource of unphysicalselfself--interaction!interaction!

KS:KS:


Localized Electrons and SICLocalized Electrons and SIC

p d/f s p s

• LDALDA introduces unphysical interaction, of an electron with itself:• SelfSelf--interactioninteraction:

ü vanishes for extended states ü important for atomic-like states.

Self-interaction corrected (SIC)-LDA: E SIC − LSD = E LSD − δ αSIC

α

occ .

∑δδαα

SICSIC = self-interaction energy for localized state α= gain in energy associated with localization

vvxcxc[n]: L[n]: Localocal DDensityensity AApproximationpproximation (LDA)(LDA)• Exc: homogeneous electron gas• Assumes delocalized electrons

Gain in kinetic energy (WW) ⇔ On-site Coulomb interaction (UU)

W >> UW >> ULDA worksLDA works

U >> WU >> WLDA failsLDA fails

band pictureband picture localized picturelocalized picture

p s

localized/delocalized picturelocalized/delocalized picture• SICSIC--LDA LDA combines band picture and localized picture

• localized state → gain in SIC energy• delocalized state → gain in band formation energy

U H (r) = dr'n(r ')

|r − r'|∫

(Perdew & Zunger, 1981)


Global energy minimum determines ground state configurationGlobal energy minimum determines ground state configuration

NNvalencevalence = Z = Z -- NNcorecore--NNSIC(loc)SIC(loc)

Self-interaction free total energy functional (Perdew & Zunger, 1981):

? Orbital dependent potential differentiates between localized Orbital dependent potential differentiates between localized and delocalised electronsand delocalised electrons

?? Gain in band formation Gain in band formation vs.vs. gain in localization (SIC energy)gain in localization (SIC energy)?? Study of various localization/delocalisation configurationsStudy of various localization/delocalisation configurations

( )criteriononlocalizatiVV

VHH

SICSIC

SICLSD

0

;)(

=−

==+= ∑ββαα

αββαββ

αβαααα

ψψ

δψψψλψψ

{ })0,();,(];[][][

][][][}][{

↑↓↑↓↑ =+==+=

+++−∆−= ∑αααααα

αααααα

δ

δψψψ

nnandnnnnnnnEnUn

nEnUVnE

LSDxc

SIC

LSDxcext

SICSIC

SelfSelf--InteractionInteraction--CorrectedCorrected--LSDLSD

Repeated Transformations between Bloch Repeated Transformations between Bloch and Wannier representationsand Wannier representations

(Eq. 10)(Eq. 10)

W.M. Temmerman et al., in Electronic Density Functional Theory: Recent Progress and New Directions (eds. Dobson, Vignale, Das) (Plenum N.Y. 1998).


Splitting of the d/fSplitting of the d/f--Electron ManifoldElectron Manifold

/d /d


{ } nLSDxc

SIC nEnUnK ∑ −−∆−=α

ααααψψψ

α

][][min][}{

][][][~

nEnEnE LSDSICxc

LSDSIC −∆+=

{ } ][][][}][{ nEnUVnE LSDxcext

SICSIC +++−∆−= ∑α

ααααα δψψψ

SIC in KohnSIC in Kohn--Sham RepresentationSham Representation

][][][][~

nEnUVnKnE LSDxcext

SICSIC +++=

][][][][~

nEnUVnTnE SICxcexts

SIC +++=

][][][][ nKnTnEnE SICs

LSDxc

SICxc +−=

])[][(][][][][~

nEnEnEnUVnTnE LSDxc

SICxc

LSDxcexts

SIC −++++=

][][][ nEnEnE LSDxc

SICxc

LSDSICxc −=∆ −

KSKS--SIC Functional:SIC Functional:

Change inChange in the exchangethe exchange--correlationcorrelationfunctional makes the differencefunctional makes the difference

A. Svane, PRBA. Svane, PRB5151, , 7924 (1995).7924 (1995).


Unified HamiltonianUnified Hamiltonian

___________________________________________________________________________________________________________________________________________________________

HH00 = H= HLSDLSD;

________________________________________________________________________________


Applications: Rare EarthsApplications: Rare Earths


P. Strange, A. Svane, W.M. Temmerman,

Z. Szotek and H. Winter, Nature 399, 756 (1999).

Rare Earths & CompoundsRare Earths & Compounds

Trivalent

Divalent

Total Energy DifferencesTotal Energy Differences

Number of Itinerant fNumber of Itinerant f--ElectronsElectronsCalculated vs. Experimental Lattice ParametersCalculated vs. Experimental Lattice Parameters

ValenceValence is determinedby the number of

localized f-electrons, but valence transitionsvalence transitions

are driven bythe number of

itinerant f-electrons

NNvalencyvalency = Z = Z -- NNcorecore--NNSIC(loc)SIC(loc)

+ - RE sulphides? - RE metals

? - RE sulphides

+ - experiment

+ - experiment? - RE metals

EII

-E

III


Structural and Localisation Transitions in CePStructural and Localisation Transitions in CeP

A. Svane et al., Solid State Commun. A. Svane et al., Solid State Commun. 102102, 473 (1997); A. Svane et al., J. Phys.: Condens. Matter , 473 (1997); A. Svane et al., J. Phys.: Condens. Matter 1010, 5309 (1998);, 5309 (1998);A. Svane et al., Phys. Rev. B A. Svane et al., Phys. Rev. B 5959, 7888 (1999)., 7888 (1999).


W. M. Temmerman et al., Phase Transitions 80:4, 415W. M. Temmerman et al., Phase Transitions 80:4, 415--443 (2007).443 (2007).

Phase Transitions in Ce Pnictides & ChalcogenidesPhase Transitions in Ce Pnictides & Chalcogenides


Ytterbium CompoundsYtterbium Compounds

Yb2+

Yb3+

Pt ~ 75 kbar (SIC-LSD)Pt ~ 100 kbar (Experiment)

Nval ~ 2.3 (SIC-LSD)Nval ~ 2.4 (Experiment)

SIC: W.M. Temmerman et al., Phys. Rev. Lett. 83, 3900 (1999). Combined approach: B. Johansson’s group (A. Delin et al., PRL

79, 4637 (1997).)

Total f-count difference vs. valence energy difference

DivalentDivalent

TrivalentTrivalent


Samarium CompoundsSamarium Compounds

DivalentDivalent

CalibratedCalibrated

? - Experiment

TrivalentTrivalent

DivalentDivalent

?E

TrivalentTrivalent

A. Svane et al., Phys. Rev. B 71, 045119 (2005); A. Svane et al., Phys. Stat. sol. (b) 241, 3185 (2004).

? E=EIII - EII


Europium CompoundsEuropium Compounds

M. Horne et al., J. Phys.: Condens. Matter 16, 5061 (2004).

TrivalentTrivalent

DivalentDivalent

All DivalentAll Divalent

? E=EIII - EII ? E=EIII - EII


Applications: ActinidesApplications: Actinides


Rare EarthsRare Earths: P. Strange, A. Svane, W.M. Temmerman, Z.Szotek and H. Winter, : P. Strange, A. Svane, W.M. Temmerman, Z.Szotek and H. Winter, Nature 399 Nature 399 (1999) 756.(1999) 756.

ActinidesActinides: L. Petit, A. Svane, W.M. Temmerman and Z. Szotek, Solid State : L. Petit, A. Svane, W.M. Temmerman and Z. Szotek, Solid State Communications Communications 116 (2000) 379.116 (2000) 379.

Trivalency vs. Divalency in fTrivalency vs. Divalency in f--SystemsSystems

Divalent

Trivalent


1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6

1 . 4

1 . 6

1 . 8

2 . 0

2 . 2

2 . 4

3 d 4 d 5 d

Wig

ner-

Sei

tz R

adiu

s (A

ngst

rom

)

E l e m e n t

Eu Yb

Es

WignerWigner--Seitz RadiiSeitz Radii

(Borrowed from M.S. S. Brooks)(Borrowed from M.S. S. Brooks)


Light ActinidesLight Actinides

A. Svane et al., PRB (submitted); condA. Svane et al., PRB (submitted); cond--mat/0610146v2.mat/0610146v2.


Vexp = 168 (au)3

VLS = 218 (au)3 (+30%)

L. Petit et al., Molecular Physics Reports L. Petit et al., Molecular Physics Reports 3838, 20, 20--29 (2003).29 (2003).

100100 150150 200200 250250 300300-- 0.140.14

-- 0.100.10

-- 0.060.06

-- 0.020.02

V (a.u.)V (a.u.)33

E

E ––

EE0 0 (e

V)

(eV

)


Pu and Am in jjPu and Am in jj--RepresentationRepresentation


3+

4+

2+


ActinidesActinides



PuOPuO22±±xx

Until recently, PuOUntil recently, PuO22 was accepted to be the stable Pu oxidewas accepted to be the stable Pu oxide andanda compound of choice for long time storage of Pu a compound of choice for long time storage of Pu

2000: Discovery of PuO2000: Discovery of PuO2+x2+x by Haschke et al.by Haschke et al.

Oxidation reaction in the presence of water, at temperatures in Oxidation reaction in the presence of water, at temperatures in the the range 25range 25°° to 350to 350°° C : PuOC : PuO22+xH+xH22OO——>PuO>PuO2+x2+x+xH+xH22

Existence of PuOExistence of PuO2+x 2+x remains controversialremains controversial

SICSIC--LSDLSD: Theoretical study of the changes in electronic structure of : Theoretical study of the changes in electronic structure of PuOPuO2 2 under oxidation under oxidation


PuOPuO22±±xx from SICfrom SIC--LSDLSD

PuO2+x (x=0.25)

PuO2+x

PuO2 Pu4+

PuPu4+4+ in in PuOPuO22, , PuPu5+5+ in PuOin PuO2.252.25, ,

and and PuPu3+3+ in in PuOPuO1.751.75

At T=0K PuOAt T=0K PuO22is stable and is stable and

it is an insulatorit is an insulator

While for While for dd--Pu Pu dynamical valence and spin dynamical valence and spin

fluctuations have to befluctuations have to beConsidered, Considered, PuOPuO22±±xx seems well seems well

described with SICdescribed with SIC--LSD, .LSD, .

L. Petit et al., Science 301, 498 (2003).L. Petit et al., Molecular Physics Reports 38, 20

(2003).

J.M. Haschke et al., Science 287, 285 (2000).


U (5fU (5f3 3 6d6d1 1 7s7s22))Np (5fNp (5f5 5 6d6d0 0 7s7s22))Pu (5fPu (5f6 6 6d6d0 0 7s7s22))Am (5fAm (5f776d6d0 0 7s7s22))Cm (5fCm (5f7 7 6d6d1 1 7s7s22))

Actinide Pnictides and ChalcogenidesActinide Pnictides and Chalcogenides

W. M. Temmerman et al., Phase Transitions 80:4, 415W. M. Temmerman et al., Phase Transitions 80:4, 415--443 (2007).443 (2007).

(6)(6)(7)(7)

(8)(8)(9)(9)(10)(10)


Local SIC (LLocal SIC (L--SIC) Approach and ApplicationsSIC) Approach and Applications


Full approach: DMFT or DCPAFull approach: DMFT or DCPA

Simplified approach:Simplified approach:– starting with the static limitstatic limit within multiple scattering theory (multiple scattering theory (KKRKKR))– self-interaction correction (SICSIC): identify important configurationsidentify important configurations– use CPACPA--DLMDLM (pseudo-alloys) to invoke valence and spin fluctuations–– NLNL--CPACPA to allow for SRO, and DLM technology for calculating TSRO, and DLM technology for calculating Tcc

–– dynamics dynamics (possibly two level system-like approach/R-matrix/ensemble averaging)

LSDLSD SICSIC--LSDLSD

ββ

αβαααα ψεψψ ∑=+= )( SICLDA VHHεψψ =LSDH

Valence, orbital, and charge orderValence, orbital, and charge order

HUTSEPOTHUTSEPOT

AbAb--initio Method for Strongly initio Method for Strongly Correlated Electron SystemsCorrelated Electron Systems

• itinerant electrons • good description for weakly

correlated metals

• electrons are localised

• good description for insulatorsIntermediate range: • partial localisation • fluctuations• DMFT


SelfSelf--Interaction Correction Interaction Correction in Multiplein Multiple--Scattering TheoryScattering Theory

scattering center

Phase shiftsPhase shifts ?? llWigner delay timeWigner delay timeis a measure of localisationis a measure of localisation

• Delocalised states:– fast electrons, broad bandwidth, weak scatterer, short Wigner delay time– basically no self-interaction

• Localised states:– narrow bands, sharp resonance, long Wigner delay time– larger self-interaction

0

W

( )l

E E

d EdE

ητ

=

=

single-site scattering

EEErZErZrn iiLL

iL

E

E

iLlm d)(),(),()( F

B

1σσσπσ τℑ−= ∫


Coherent Potential Approximation (CPA):Coherent Potential Approximation (CPA):Effective medium: on average no extra scattering by impuritiesEffective medium: on average no extra scattering by impurities

Generalized alloy picture:Generalized alloy picture:Chemical disorderChemical disorder (e.g. CuNi)(e.g. CuNi)Moment fluctuationsMoment fluctuations ((Hubbard IIIHubbard III) )

– static spin fluctuations– disordered local moments (DLM)

Valence fluctuationsValence fluctuations ((Hubbard IIIHubbard III) ) – static fluctuations between valence configurations– mixed valence: YbS

LL--SIC: SICSIC: SIC--KKRKKR--CPA+DLMCPA+DLM

cA + cB =

So far:So far: fluctuations around LSDfluctuations around LSD,, now:now: fluctuations around SICfluctuations around SIC--LSDLSD

A B C ccAA + c+ cBB =1=1


LL--SIC: Finite TemperaturesSIC: Finite Temperatures

For each temperature, volume and concentrations cFor each temperature, volume and concentrations cii

Identify different local (spin/valence) configurations lying cloIdentify different local (spin/valence) configurations lying close in energyse in energy

[ ]

∑∑∫

−=

−=

−−+−=

++=

i

ivibiBvib

iiiBmix

Bel

vibmixel

SckS

cckS

ffffndkS

SSSS

ln

))(1ln())(1()(ln)()( εεεεεε ββββ

0}]{,,[~

}{}{ min

=∂∂

≡ cc

ii

i

cVTFc

withV,


SICSIC--KKRKKR--CPACPAUse alloy analogy:Use alloy analogy:

Localised (SIC) and delocalised states Localised (SIC) and delocalised states (LSD):(LSD):

– static valence fluctuations– a -? Ce phase transition

Orientations of local moments:Orientations of local moments:– static spin fluctuations– disordered local moments (DLM)

Use Coherent Potential Approximation Use Coherent Potential Approximation (CPA)(CPA)

– Effective medium: on average no extra scattering by impurities

M. Lüders et al., Phys. Rev. B71, 205109 (2005).

?? phase:phase:

–Paramagnetic (local moments)–Large volume–Localized f-electron

aa phase:phase:

–Non-magnetic–Small volume–Delocalized f-electron

1515--17% volume collapse17% volume collapsecritical pointcritical point

–Smooth crossover at higher temperatures


Free energy: F E ST= −

LL--SIC:Ce at Finite TemperaturesSIC:Ce at Finite Temperatures

Gibbs free energy

Free energy

),,(),),,,((),,( cTppVcTcTpVFcTpG +=V(a.u.)

p[kb

ar]

200 K400 K

600 K800 K

1000 K1200 K

1400 K1500 K

1600 K

LSICLSIC--KKRKKR--CPACPA with DLMwith DLM

Use alloy analogy (Hubbard III) Use alloy analogy (Hubbard III) with localised (SIC) and delocalised with localised (SIC) and delocalised (LSD) states.(LSD) states.

vibmixel SSSS ++=


LL--SIC: Ce Phase DiagramSIC: Ce Phase DiagramMinimize Gibbs free energy

),,(),),,,((),,( cTppVcTcTpVFcTpG +=

for each (p,T ) with respect to cc, the concentration of the ? -phase

experiment

theory

T [K]

T T ∆∆SS

∆∆EEtt

-- p p ∆∆VV

Mixing entropyMixing entropy

Magnetic entropyMagnetic entropy

Electronic entropyElectronic entropy

Lüders et al., Phys. Rev. B 71, 205109, (2005)

B. Amadon et al., PRL 96, 066402 (2006).


Finite Temperature Magnetism in Heavy Rare Finite Temperature Magnetism in Heavy Rare EarthsEarths


Disordered Local Moments for GdDisordered Local Moments for Gd

Combining SICCombining SIC--LSD with the disordered local moment (DLM) picture LSD with the disordered local moment (DLM) picture of magnetism:of magnetism:

• Using GdGd as a prototype for late rare earths (REs)

•• SICSIC--LSDLSD corrects the wrong magnetic structure of GdGd calculated by LSD.

•• DLMDLM: calculation of the magnetic susceptibility in the paramagnetic phase: peak in χ(q) provides the magnetic ordering vector and transition temperature.


Gd: Magnetic Order in Heavy Rare EarthsGd: Magnetic Order in Heavy Rare Earths

DLMDLM: calculation of the magnetic susceptibilityin the paramagnetic phase - peak in χχ(q)(q) provides the magnetic ordering vector.

Onset of Magnetic OrderOnset of Magnetic OrderBelow TTCC: paramagnetic state is unstable:an infinitesimal magnetic field will induce a finite magnetization. Divergence of the paramagnetic susceptibility indicates Curie (Neel) temperature.

LSD

SIC-LSD

Interaction between moments


Investigate magnetic order as a function of lattice parameters (Investigate magnetic order as a function of lattice parameters (Wigner Seitz radius Wigner Seitz radius and c/a ratio).and c/a ratio).

Magnetic Order in Heavy Rare EarthsMagnetic Order in Heavy Rare Earths

c/a =1.54c/a =1.54


Fermi Surface NestingFermi Surface Nesting


Phase Diagram for Heavy Rare EarthsPhase Diagram for Heavy Rare Earths

I. Hughes et al., Nature 446, 650 (2007).A.V. Andrianov (private communication)A.V. Andrianov (private communication)

q=(0,0,0.13) – present workq= (0,0,0.11) - experiment


Summary for GdSummary for Gd

We verify the importance of the c/a, but discover that the We verify the importance of the c/a, but discover that the lanthanide

contraction plays a separate, completely distinct role in determining plays a separate, completely distinct role in determining

the magnetic properties of the heavy RE elements.the magnetic properties of the heavy RE elements.


ConclusionsConclusions

SICSIC--LSD provides improved treatment of localized statesLSD provides improved treatment of localized states

Competition between localization energy and band formation energCompetition between localization energy and band formation energy leads to a useful definition y leads to a useful definition of valence even for metallic systemsof valence even for metallic systems

Description of change in valence as a function of pressure and cDescription of change in valence as a function of pressure and chemical compositionhemical composition

Phase transitions of fPhase transitions of f--electron systems at finite temperatureselectron systems at finite temperatures

Localization energy and band formation energy are treated on equLocalization energy and band formation energy are treated on equal footingal footing


CollaboratorsCollaborators

M. LM. Lüüdersders and W. M. Temmerman (and W. M. Temmerman (DaresburyDaresbury))A.A. Svane et al. (Svane et al. (AarhusAarhus))L. Petit (L. Petit (ORNL & AarhusORNL & Aarhus))

J.B. Staunton and I. Hughes (J.B. Staunton and I. Hughes (WarwickWarwick))B. L. GyB. L. Gyöörffy (rffy (BristolBristol))

D. KD. Köödderitzsch (dderitzsch (MunichMunich) ) M. DM. Dääne, A. Ernst and ne, A. Ernst and W. Hergert (W. Hergert (HalleHalle))

P. Strange et al. (P. Strange et al. (KeeleKeele//KentKent))H. Winter (H. Winter (KarlsruheKarlsruhe))J. Poulter (J. Poulter (BangkokBangkok))

self-interaction correction (sic) scheme and f-electron...

Documents