An efficient magnetic tight-binding method for transition metals and alloys

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An efficient magnetic tight-binding method for

transition metals and alloys

Cyrille Barreteau, Daniel Spanjaard, Marie-Catherine Desjonquères

To cite this version:

Cyrille Barreteau, Daniel Spanjaard, Marie-Catherine Desjonquères. An efficient magnetic

tight-binding method for transition metals and alloys. Comptes Rendus Physique, Centre Mersenne, 2016,

17 (3-4), pp.406 - 429. �10.1016/j.crhy.2015.12.014�. �cea-01384597�

Contents lists available atScienceDirect

Comptes

Rendus

Physique

www.sciencedirect.com

Condensed matter physics in the 21st century: The legacy of Jacques Friedel

An

efficient

magnetic

tight-binding

method

for

transition

metals

and

alloys

Un

modèle

de

liaisons

fortes

magnétique

pour

les

métaux

de

transition

et

leurs

alliages

Cyrille Barreteau

a

,

b

,

∗

,

Daniel Spanjaard

c

,

Marie-Catherine Desjonquères

a a_{SPEC,CEA,CNRS,UniversitéParis-Saclay,CEASaclay,91191Gif-sur-Yvette,France}

b_{DTUNANOTECH,TechnicalUniversityofDenmark,ØrstedsPlads344,DK-2800Kgs.Lyngby,Denmark} c_{Laboratoiredephysiquedessolides,UniversitéParis-Sud,bâtiment510,91405 Orsaycedex,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline21December2015

Keywords: Tight-binding Magnetism StonerModel Spin–orbitcoupling Magneto-crystallineanisotropy Hartree–Fock Mots-clés : Liaisonsfortes Magnétisme ModèledeStoner Couplagespin–orbite Anisotropiemagnéto-cristalline Hartree–Fock

Anefficientparameterizedself-consistenttight-bindingmodelfortransitionmetalsusing s, panddvalence atomicorbitalsasabasissetispresented.Theparametersofour tight-bindingmodelforpureelementsaredeterminedfromafittobulkab-initio calculations. A very simple procedure that does not necessitate any further fitting is proposed to deal with systems made of several chemical elements. This model is extended to spin(and orbital) polarized materials by adding Stoner-likeand spin–orbit interactions. Collinearandnon-collinearmagnetism aswellas spin-spiralsare considered.Finallythe electron–electron intra-atomic interactions are taken into account in the Hartree–Fock approximation.Thisleadsto anorbital dependence oftheseinteractions, whichis ofa greatimportanceforlow-dimensionalsystemsandforaquantitativedescriptionoforbital polarizationandmagneto-crystallineanisotropy.Severalexamplesarediscussed.

©2015TheAuthors.PublishedbyElsevierMassonSASonbehalfofAcadémiedes sciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Nousprésentons unmodèle de liaisons fortes paramétré et auto-cohérentutilisant une base d’orbitales atomiquess, p, et d pour décrire lesélectrons de valence desmétaux detransition. Lesparamètres du modèle sont déterminés àpartir d’un ajustement non linéairesurdesrésultatsdecalculsabinitio d’élémentspursenvolume.Notreprocédurene nécessiteaucunparamètreniajustementsupplémentairepourl’étendreauxsystèmesavec plusieurs atomesde natures chimiques différentes.Nous avons généralisé notre modèle auxmatériauxprésentantunepolarisationdespinetorbitaleàl’aidedetermesdeStoner etde couplage spin–orbite. Nous traitons aussi bien lemagnétisme colinéaire que non colinéaireainsiquelesspiralesdespin.Enfinnousmontronscommentprendreencompte l’interactionélectron–électron intra-atomiquedansl’approximationdeHartree–Fock.Cela introduitunedépendanceorbitale desinteractionsquipeuts’avérerimportantedansles systèmes de basse dimensionalité et pour décrire correctement l’anisotropie

magnéto-*

Correspondingauthor.

E-mailaddresses:cyrille.barreteau@cea.fr(C. Barreteau),daniel.spanjaard@u-psud.fr(D. Spanjaard).

http://dx.doi.org/10.1016/j.crhy.2015.12.014

1631-0705/©2015TheAuthors.PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

cristalline et lapolarisation orbitale. Nous illustronsnotre propos àl’aide de plusieurs exemples.

©2015TheAuthors.PublishedbyElsevierMassonSASonbehalfofAcadémiedes sciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

EventhoughDensityFunctionalTheoryhasbecomeanextremelyefficientmethodwidelyusedinmanyareasofphysics, chemistry, andmaterialscience, thetight-binding (TB)descriptionofthe electronic structureremains very popular,since it provides a physically transparent interpretation in terms oforbital hybridization and bond formation. In addition, its moderatecomputationalcostpermitstohandleratherlargeandcomplexsystems,anditsstraightforwardimplementation allowsmanygeneralizationsandapplications.Inaddition,inrecentyears,withtheincreasinginterestinelectronictransport andtheexplosionofstudiesingraphenenanostructures,therehasbeenarenewalofinterestforTBcalculations.

HistoricallytheTBmethodwasintroducedbySlaterandKoster[1].Itwasoriginallythoughtasasemi-empiricalmodel todescribetheelectronicstructureofsolidswithareasonablysmallnumberofparametersthatcanprovidereliable semi-quantitativeresultswhentheseparametersaredeterminedfromafitonfirst-principlescalculations.JacquesFriedelinthe 1960swasoneofthepioneersintheapplicationofTBtothephysicsoftransitionmetals[2,3].Thesemodelswere essen-tiallybasedonaphysicaldescriptionofthebandstructure,butnorealargumentsaboutthetotalenergywere developed. Qualitative explanations ofthe trends inthe variation ofthe totalenergy whensome parameters are varied (numberof d electrons, concentration) were proposed, but only based on the band contribution to the total energy. Jacques Friedel was particularlytalentedindevelopingsimplemodelswitha simplifiedschematicdescription oftheelectronic densityof states(suchastherectangularbandmodel[4])thatcouldneverthelessdescribesurprisinglywellmanyphysicalproperties ofmaterials.Lateron,physicistsstartedtoaddaphenomenologicalrepulsivepair-potential tothebandenergy[5].It was howevernot veryclearwhatwas“hidden”behindthisphenomenologicalterm.Overtheyears,TBmethodshaveacquired amoresolidfundamentalbasis.Inparticular,withtheworkofHarrisandFoulkes[6,7],itwasshownhowatight-binding formalismcanbederivedfromDensityFunctionalTheory.Nowadays,thereexistmanyelectronic structurecodesbasedon variousversionsofTBwithdifferentdegreesofapproximations[8–10].

Veryearly,TBincombinationwithaStoner-likemodel[11] wasrecognizedasan adequatetooltodescribemagnetism intransitionmetals,andJacquesFriedelwas indeedveryactiveinthisfield[12,13].Indeed,magnetisminacrystalis inti-matelyrelatedtoitsbandstructure.TBhasbeenappliedtoalargevarietyofmagneticsystemsinvariouscrystallographic structures,dimensionalities(frombulktoclusters),orderedalloysorpresentingsomekindofdisorder[14].Itisnotthegoal ofthispapertoprovideanexhaustivepresentationofthiswealthofresearchinmagnetism.Wewillratherconcentrateon thepresentationofaTBmodelthatwehavedevelopedovertheyearsandthatisabletodescribeaccuratelyandefficiently awiderangeofmagneticphenomenaandmaterials.Itisan empiricalTBmethodwithparametersfittedonab-initio data.

Wewillshowhow,withalimitednumberofsimpleandwellcontrolledapproximations,wehavebeenabletogeneralize ourmodeltoalloysandincludenon-collinearmagnetism,spin-spiralsaswellasspin–orbitcoupling.

The paperis organized asfollows:we will presentthe generalconcepts ofthe tight-binding descriptionof electronic structureanditsimplementationinans,panddatomicorbitalbasissetfornon-magneticmaterials(Section2).Wewillpay particularattentiontodescribeproperlyfeaturesthatareoftennotdiscussedthoroughlyinpublications:non-orthogonality of theTB basis set, self-consistent treatment, proper definitionof local quantities,etc. Then, in Section 3, we will show how, usinga simpleStonermodel,spin-polarizationcan be included,firstfor collinearmagnetism(Section 3.1), then for non-collinearconfigurations(Section3.2).Spin–orbitcouplingandmagneto-crystallineanisotropywillalsobediscussedin detail.Section3willbeendedbyadiscussionofmoreelaboratedHartree–FocklikeHamiltoniansthatcanplayanimportant roleinlow-dimensionaloranisotropicsystems.FinallywewilldrawconclusionsinSection4.

2. Thetight-bindingmethod

Thetight-bindingapproximationisakindofcounterpartofthefree-electronjelliummodelofsolids.Indeed,whileplane wavesarethenaturalfunctionstodescribethedelocalizationofelectronsinajellium,theBlochfunctionsareexpandedover asetofatomic-likefunctionsintheTBapproach.Therefore,theTBmodelwillinprincipleapplybetter tosystemswhere the valenceelectrons originate fromwell-localized atomic functionsthat donot overlap too much betweenneighboring atoms. However, its domain ofapplicability has proved tobe larger than expected. Fora review of thedifferent aspects ofthetight-binding method,the readercan refertothefollowingbooksorreview articles[15–20].The specificityofour approachconsistsinthederivationofasystematicproceduretodeterminetheparametersofourTBmodel,whichisable todescribetheelectronic,magneticandenergeticpropertiesoftransitionmetalsandtheiralloyswithanaccuracycloseto first-principlecalculations.

2.1. Generalities

2.1.1. Theelectronpotentialinacrystal

Agood firstapproximationforthepotential seenbyan electroninacrystal(orassembly ofatoms)istowrite itasa sumofisolatedneutralatomic-likepotentials,

V

(

r

)

=



n

Vat

(

r

−

Rn

)

(1)

wherethesummationrunsoverthelatticevectorsRn ofthecrystal. Vat

(

r

)

isanatomic-likepotentialofradialsymmetry. Note that this decomposition is not unique and moreover the potential is not necessarily a truly atomic potential, but could be “modified” to take into account the confinement felt by the electrons in the crystal. In addition, since we are only interestedindescribing the valence electrons, thispotential israther a“pseudo” potential validina givenrange of energyfordescribingtheinteractionofvalenceelectrons withtheions.Duetothelatticeperiodicity,thecrystalpotential isperiodic.Itcanalsobegeneralizedtothecaseofseveralatomsperunitcell.Wewillthenwrite:

V

(

r

)

=



i,n

V_iat

(

r

−

Rn

−

τ

i

)

(2)

The summationover i is running over the Nat atoms in theunit cell.To avoidlengthy notations,we will usually write thepotential inthefollowingcondensedmannerbymakinguseofanoperatorformalism V

ˆ

=



_i,nV

ˆ

at

i,n.The one-electron Hamiltonianoperatoristhereforewritten:

ˆ

H

= ˆ

T

+



i,n

ˆ

V_iat_,n (3)

ˆ

T beingthekineticenergyoperator.

2.1.2. Linearcombinationofatomicorbitals

Once thepotential ofthe crystalhasbeen writtenasa sumof atomic-likepotentials,we consider thecorresponding atomicwavefunctions

φ

_iat_λ,solutionsoftheSchrödingerequationforatom i:

( ˆ

T

+ ˆ

V_iat

)φ

at_iλ

=

ε

at_iλ

φ

at_iλ (4)

Asmentionedpreviously,thereissomeflexibilityinthewaypotential V

ˆ

at

i isdefined.Inparticular,forpracticalreasons, it iswishedthatthespatial extensionoftheassociatedatomicwave functionsisofasmallerrangethanthat ofthetrue atomicwavefunctions.Inaddition,sincethevalencewavefunctionsusually includemorethanonetype ofatomicorbital, theindex

λ

(

=

1

. . .

Norb_{)refers tothesymmetryoftheangularpartoftherealorbitalss, p}

x,py,pz,dxy,dyz,dxz,dx2_−y2,

d_z2,etc.,

φ

at_iλ

(

r

)

=

R

iλ

(

r

)¯λ(

ˆ

r

)

(5)

where

R

iλ

(

r

)

and

¯λ(ˆ

r

)

arerespectivelytheradialandangularpartsoftheatomicorbital(ˆr beingtheangularcoordinates of thevector r with respectto some givenx, y, z axes). Inthe following,forconvenience, we will ratherusethe Dirac notations

|

in

λ



todenotetheketassociatedwiththeatomicorbital

φ

_iat_λ

(

r

−

Rn

−

τ

i

)

= 

r

|

in

λ



ofatomi inthecelln.

Thecentralapproximationofthetight-bindingmodelreliesontheassumptionthattherestrictedHilbertspacespanned by atomic-like orbitals is sufficient to describe the wave functions solutions of the Schrödinger equation (at least in a restrictedenergyrange)inthecrystal.Thewavefunctionsarethereforewrittenascombinationsofatomic-likeorbitals:

| =



inλ

Cinλ

|

in

λ



(6)

Notethateventhoughtheatomicorbitalsformanapproximatebasisoftherestricted“valence”space,thisbasishasno reasontobeorthonormal.Indeed,thescalarproductoftwoatomicwavefunctionslocatedondifferentatomicsitesdefines theso-calledoverlapintegrals:

Sinλ,jmμ

= in

λ

|

jm

μ



(7)

Forconvenience,theoverlapintegralshaveoftenbeenneglectedtobuildupsimplifiedmodelswhereitisassumedthat the non-orthogonality can be takenintoaccount through a renormalizationofthe other tight-binding parameters. These models canbeveryefficientandquiteprecisewhenoneisinterestedmoreintheenergeticsratherthanintheelectronic structure, since energeticsis mainlygoverned by delectrons, forwhich orthogonalityisa good approximation.However, thisapproximationisrathercrudeifonewantsagooddescriptionofthebandstructure,whichneedstheintroductionofs andpelectronsinthebasissetsincetheyaredelocalized.

2.1.3. One-,two-,andthree-centerintegrals

Letusnow write thevarious matrixelements of theHamiltonian inthisatomicbasis set.Making use ofEq.(4),the diagonaloron-sitetermscanbewritten:

ε

inλ

= 

in

λ

| ˆ

H

|

in

μ

 =

ε

atiλ

δ

λ,μ

+ 

in

λ

|



jm=in

ˆ

Vat_jm

|

in

μ



(8)

The second termofpure electrostatic character isthe so-calledcrystal-fieldintegral

α

iλμ. The additionalcontribution

α

iλobtainedfor

λ

=

μ

onlyshiftsthevalueoftheatomiclevel.Offdiagonaltermwillbeignoredinthefollowing.

Inthesamespirit, theoff-diagonalmatrixelementsofthe Hamiltonianobtainedfor

(

in

)

= (

jm

)

canbe splitintotwo typesofterms:

in

λ

| ˆ

H|jm

μ

 =

ε

at_jμSinλ,jmμ

+ in

λ

| ˆ

Vinat

|

jm

μ

 + in

λ

|



kp=in kp=jm

ˆ

V_kpat

|

jm

μ



(9)

whereatomofindexi

=

0 istakenastheorigin.

The firsttwo terms[21] involvetwo-centerintegrals (thecenterof thepotential isthesame asoneof thetwo wave functions) andare the onlyones that willbe retained since their contribution ismuch larger than thelast three-center integral term. In the following, the off-diagonal elements of the Hamiltonian (involving two-center integrals only) will be denoted

β

inλ,jmμ. These are the so-calledhoppingintegrals, whichare crucial ingredients in thetight-binding model

since theymeasure theability ofanelectron to“jump” fromone atomto theother anddecayrapidlywiththe distance betweenneighboringsites.Therefore,ausualapproximationconsistsincutting-offtheinteractionaboveacertainthreshold radius Rc.This isa very importantpoint since the rathershort-rangedextension of thehoppingintegrals makes theTB Hamiltonianmatrixsparse andallowsthe useofspecific efficientalgorithms.In addition,duetothe sphericalsymmetry oftheatomicpotentials,thehoppingintegralsbetweentwositesconnectedbyavector withanarbitraryorientationcan bewrittenasalinearcombinationofasetofhoppingintegrals

β

γ

=

ss

σ

,sp

σ

,sd

σ

,pp

σ

, pp

π

, pd

σ

,pd

π

,dd

σ

,dd

π

,dd

δ

definedinthecasewheretheorientationisalongthez-axis.ThissetofSlater–Kosterintegralsarethemainingredientsin theTBmodel,andthe

β

γ

(

R

)

aredecayingfunctionswithdistance.

2.1.4. BlochstatesandHamiltonianmatrixinthereciprocalspace

Inaperiodiccrystal,theHamiltoniancanbediagonalizedusingtheBlochstateswrittenintheform:

|

iλ

(

k

)

 =

1

√

N



n eik·(Rn+τi)

|

_in

_λ



₍₁₀₎

wherek is avector lying inthe reciprocalspaceandN the numberofprimitiveunit cells in thecrystal(where we use theBorn–vonKarman periodicboundaryconditions).Theeigenfunctions

|

(α)

₍

_k

₎

_

_{oftheTBHamiltonianarewritten asa} combinationofBlochstates:

|

(α)

₍

_k

₎

_{ =}



iλ

C(_iλα)

(

k

)

|

iλ

(

k

)



(11)

TheC(_inα_λ)andC_i(_λα)

(

k

)

expansioncoefficientsarethusrelatedbytheexpression:

C_in(α_λ)

=

√

1 NC

(α) iλ

(

k

)

e

ik·(Rn+τi) ₍₁₂₎

Eacheigenstate

α

isnowlabelledbyabandindex

α

(runningfrom1toNat_Norb_{)andawavevectork.TheHamiltonian} matrixofsize

(

Nat_Norb

_×

_Nat_Norb

₎

_{canbewrittenas:}

Hiλ,jμ

(

k

)

= 

iλ

(

k

)

|H

|

jμ

(

k

)

 =



m

eik·(Rm+τj−τi)

i0

_λ

|H

|

_jm

_μ



₍₁₃₎ Asimilarexpressionisderivedfortheoverlapmatrix

Siλ,jμ

(

k

)

= 

iλ

(

k

)

|

jμ

(

k

)

 =



m

eik·(Rm+τj−τi)



_i0

_λ

|

_jm

_μ



₍₁₄₎

2.1.5. SolvingtheTBSchrödingerequation

Inaperiodiccrystal,writingtheSchrödingerequationwithawavefunctiongivenbyEq.(11)leadstoamatrixequation oftheform:

H

(

k

)

Cα

(

k

)

=

ε

α

(

k

)

S

(

k

)

Cα

(

k

)

(15) H

(

k

)

and S

(

k

)

are the k-dependentHamiltonian andoverlap matrices while C α

(

k

)

is the vector built fromthe coeffi-cients C α_iλ

(

k

)

. Due to the presence of the definite and positive S

(

k

)

matrix, Eq. (15) is called a generalized eigenvalue problemthatneedstobesolvedforeachk vectorintheirreducibleBrillouinzone.

Theeigenvalues

ε

α

(

k

)

formtheso-calledband-structureofthecrystalwhenthek vectorisspanningtheBrillouinzone.

Thisgeneralizedeigenvalueequation simplifiesintoastandardonewhentheoverlapintegralsareneglected.The pres-enceoftheoverlapmatrixslightlycomplicatestheformalism,especiallywhenoneneedstodefine localquantitiesaswill beseeninthefollowingsections.

2.2. ThelocalquantitiesintheTBmethod

2.2.1. Quantummechanicsinanon-orthogonalbasisset

Whenusinganon-orthogonalbasisinquantummechanics,manyformulaehavetobemodified.Thisisthecaseofthe so-called closure relation.For the sake of simplicity (and generality), let us drop the i

λ

andn indexes and consider a non-orthogonal basisset

|

a



,anditscorrespondingoverlapmatrix Sa,b

= 

a

|

b



.Itisveryconvenienttointroducethedual basis

|˜

a



definedasfollows[22,23]:

|˜

a

 =



b

(

S−1

)

a,b

|

b



(16)

Bydefinition,

˜

a

|

b



= 

a

|˜

b



= δ

a,b,anditispossibletodefineageneralizedclosurerelation:



a

|˜

a



a

| =



a

|

a

˜

a

| = ˆ

Id (17)

where

ˆ

Idistheidentityoperator.Itisthenstraightforwardtoprovethatthetraceofanoperator A takes

ˆ

theform

Tr

( ˆ

A

)

=



a

˜

a| ˆA|a =



a

a| ˆ

A|˜a (18)

Averyusefuloperatorinquantummechanicsisthedensityoperator,whichisdefinedinacondensedwayas

ρ

ˆ

=

f

( ˆ

H

)

where f istheFermifunction.

ρ

ˆ

takesamoretransparentexpressionintheeigenfunctionbasisset

|

α



oftheHamiltonian.

ˆ

ρ

=



α

|

α



f

(

ε

α

)



α

|

(19)

fα

=

f

(

ε

α

)

beingtheoccupationfactoroftheeigenstate ofeigenvalue

ε

α .Inthecaseofaperiodic system,parameter

α

denotesthecombinationofthediscreteindex

α

previouslyintroducedandofthecontinuousk-vector.Thereforeusingthe previousrelationofthetrace,itcomesoutthatthetraceofthedensityoperatorthatdefinesthetotalnumberofelectrons inthesystemcanbewrittenas:

Ne

=

Tr

(

ρ

ˆ

)

=

Tr

(

ρ

S

)

(20)

Theelementsofthedensitymatrix

ρ

aregivenby:

ρ

a,b

=



α

C_aαC_bα fα

= c

+bca



(21)

The C α_a

= ˜

a

|

α



are the expansion coefficients of the eigenfunction

|

α



in the basis set

|

a



andc+_b, ca the creation and annihilationoperators intheatomicorbitals

|

b



and

|

a



,respectively.Finally,anaturalwayofdefining achargeprojected onorbital

|

a



istotaketherealpartofthematrixelementNa

=

Re

(

ρ

S

)

a,a,whichcanalsobewritteninamoresymmetric way as 1₂

(

˜

a

| ˆ

ρ

|

a



+ 

a

| ˆ

ρ

|˜

a

)

. This compact expression is in fact nothing else than the usual Mulliken charge of a given orbital a: Na

=

Re





α C_aα



C_aαfα



,



Cα_b

=



b Sa,bCbα (22)

Obviously,withthisdefinition,thesummationover alltheorbitals

|

a



allows usto recoverthetotalcharge, i.e., Ne

₌



ana.IftheoverlapintegralsareignoredintheexpressionofthelocalchargegivenbyEq.(22)(i.e. byreplacing



C αa by C αa), thentheconservationofthechargeisnolongerpreservedandwecallthecorrespondingquantitythe“net”charge.These relations can readily be transcribed todefine thelocal charge of a givenorbital symmetry Niλ in the context ofthe TB formalism.

2.2.2. Densityofstates

The densityofstatesis afundamental quantity that provides acrucial informationaboutthelevel distributionofthe Hamiltonianspectrum.Thetotaldensityofstatesisdefinedas:

n

(

E

)

=



α

Itsimply “counts”thenumberof statesbetweenE and E

+

dE divided bydE. Bydefinition,an integrationup tothe Fermilevel givesthe totalnumber ofelectrons inthe system. Ina similar way,asdone forthe Mulliken charge,a local densityofstatesprojectedonagivenorbitalcanbedefined:

na

(

E

)

=

Re





α C_aα



Cα_a

δ(

E

−

Eα

)



(24)

ThelocalchargeisthenrecoveredbyanintegrationweightedbytheFermifunction:

Na

=



na

(

E

)

f

(

E

)

dE (25)

2.2.3. Bandenergy

Aswillbeseeninthefollowing,thebandenergythatcanbeobtained,atzerotemperature,bysummingalleigenvalues belowthe Fermilevel is an essential componentofthe total energyofthe system. At non-zerotemperature, it is easily generalizedandcanbeformulatedinvariousways:

Eband

=

Tr

(

ρ

ˆ

H

ˆ

)

=



α

ε

αfα

=



En

(

E

)

f

(

E

)

dE (26)

Inasimilarmanner,asforthechargeortheprojectedDOS,onecandefinealocalcontributiontothebandenergy:

E_aband

=



Ena

(

E

)

f

(

E

)

dE (27)

In some cases, it is more convenient to work with the grand-potential



, which is defined at zero temperature as

Eband

₋

_Ne_E F:



=

EF



(

E

−

EF

)

n

(

E

)

dE (28)

whichcanbegeneralizedatfinitetemperatureT [16].Thelocalcontributionfromsitea isobviouslyobtainedinthesame wayasthelocalbandenergy.

2.3. Anspd tight-bindingHamiltonianfortransitionmetalsandalloys

In the following, we will present the TB model that we have developed over the years and applied to many dif-ferent systems. The non-magnetic part of this tight-binding Hamiltonian is similar to that of Mehl and Papaconstan-topoulos. We use a minimal non orthogonal basis set containing s, p, and d orbitals centered at each site i

,

|

i

,

λ



(

λ

=

s

,

px

,

py

,

pz

,

dxy

,

dyz

,

dzx

,

dx2_−y2

,

d_z2). Let usfirst consider the onsitematrix elements inthe case of a system with

asinglechemicalelement.InasimilarwaytoMehlandPapaconstantopoulos[8],weassumethatateach interatomic dis-tancethereferenceenergyischoseninsuchamannerthatthetotalenergyEtot isobtainedbysumminguptheoccupied oneelectronenergylevels:

Etot

=



α

ε

αfα (29)

Note that in this condition, the total energy can be calculated without introducing an empirical repulsive potential ascurrentlydone in pure d band calculations.Indeed, therepulsive contribution tothe total energyisaccounted forby the onsiteterms, which,consequently, must depend on the localenvironment (number of neighborsandbond lengths). FollowingRef.[8],wewrite:



iλ

=

aλ

+

bλ

ρ

_i1/3

+

cλ

ρ

_i2/3

+

dλ

ρ

_i4/3

+

eλ

ρ

2i (30) with:

ρ

i

=



j=i exp

[−

2Ri j

]

Fc

(

Ri j

)

(31)

whereFc

(

Ri j

)

isacut-offfunctionand



isaparameter.

Letusconsidernowthehoppingandoverlapintegrals.Inordertoreproducemorecloselytheir variationwithdistance onalargescale,wehavefoundusefultousealawsomewhatmorecomplicatedthanasimpleexponential,asdonein[8]:

β

γ

(

R

)

= (

pγ

+

fγR

+

gγR2

)

exp

[−

h2γR

]

Fc

(

R

)

(32) where

γ

indicatesthetypeofinteraction(e.g.,ss

σ

,sp

σ

,etc.)andFc

(

R

)

isacut-offfunction:

Fc

(

R

)

=

1

1

+

exp

[(R

−

R0

)/

l]

(33)

Inpractice,thisfunctionfixesthenumberofneighborsthataretakenintoaccount.InthecalculationswetookR0 equal to14Bohrandl equalto0.5 Bohr,theinteractions arestrictlycut-offfordistancesabove Rc

=

R0

+

5l

=

16

.

5 Bohr.Letus recallthat inthisparameterization,theband energyisequaltothetotalenergy.Limitingourselvestos,p,andd orbitals, we haveintroduced 16parameters intheexpressionoftheon-sitetermsandfourforeachexpressionofthetenhopping and ten overlap integrals. Thus, to completely specify our model, we need to determine at most96 parameters. These parameterscanbeobtainedbyafittoab-initio calculationsofthenon-magneticbandstructureandtotalenergyforseveral distancesandsimplecrystallographicstructures.

If we considernow a metallic alloy madeoftwo chemical elements A and B,the followingprocedure hasbeen car-ried out withsuccess.Afitforboth chemicalelements isperformedseparately,butwiththe samevalue of



.Then the intra-atomictermsofagivenatominthesystemwillonlydependonthenatureoftheconsideredatombythecoefficients

aλ

,

bλ

,

cλ

,

dλandeλ.Thevaluesofthehoppingandoverlapintegrals

β

γA−A (

β

γB−B) betweentwoidentical A (B)atomsare

the sameasinthe pure element,whilefora mixedcoupleofneighbors A

−

B,

β

_γA−B is obtainedbyusing thecommon approximationthatconsistsintakingthearithmeticaverageofthecorrespondinghomonuclearquantities:

β

_γA−B

=

1

2

(β

A−A

γ

+ β

γB−B

)

(34)

and thereforedoesnot necessitate additionalparameters. Finally,we apply a local charge neutrality proceduresuch that the localMulliken chargeofagivenatom i isequaltothenumberofvalenceelectrons ofthespecieoccupyingsitei.Its practicalimplementationwillbeexplainedinthenextsection.

Theadvantageofourapproachisthatitdoesnotrequireanyfurtherfittingtoab-initio dataforthebinarysystems.This procedureworksverywellfortheelectronicandmagneticpropertiesofalloys;however,ifoneisinterestedintheenergetics of alloys(mixingenergies etc.), itissometimes necessaryto slightlyrescalethe value ofthehetero-nuclearhoppingand overlap integralswithrespect tothe arithmetic average.Thisis thecasefor theFeCr system, whichnecessitates a small increase in theFe–Cr hoppingandoverlapparameters toreproduce accurately quantitiessuch asthemixingor interface energies[24].

2.4. Self-consistentcorrections

2.4.1. Localchargeneutralityandthenotionofpenalization

Whendealingwithinhomogeneoussystemsusingthistypeofsemi-empiricalTBHamiltonian,oneisusuallyfacedwith theproblemofchargetransfer.Indeed,ifsucha modelisappliedforexampletothecaseofasurface,oraclusterwhere atomshaveverydifferentgeometricalenvironments,unphysicallylargechargetransfersareobtained.Inmetallicsystemsin whichstrongscreeningeffectspreventtheaccumulationofcharges,itisimportanttoavoidsucheffects.Astraightforward manner to solve thisproblemisto use aso-called penalization technique that consistsinadding a positiveterm tothe total energy,which iszeroifthe charge neutralityis fulfilled,andverylarge otherwise.Letusstart fromthe expression ofthetotal energyofthesystemwrittenintermsoftheexpansion (real)coefficientsC α_inλ ontheatomicbasis,whichfor simplicityisassumedtobeorthogonal(thegeneralizationtothecaseofanon-orthogonalbasissetispresentedattheend ofthissection).ThetotalenergyofasystemdescribedbyanHamiltonianH0beforepenalizationiswrittenas:

Etot,0

=



αocc

inλ,jmμ

C_inα_λH0_inλ,jmμCα_jm_μ (35)

A minimization ofthisfunction withrespect to C α_in_λ underthe normalizationconstraint



_inλ

(

C α_in_λ

)

2

=

1 leads to the usual eigenvalueSchrödingerequation H0C α

=

ε

0

αC α 

.Letusnowadd apenalizationtermofquadraticformtothetotal energy: Epen

=



in Ui 2

(

Nin

−

N 0 i

)

2 (36)

where Ui isalarge positivequantity andN_i0 thecharge that onewants toimposeon sitei.Minimizingthetotalenergy

Etot,0 towhichthepenalizationtermisaddedleadstoa similareigenvalueequationwherethediagonalmatrixelements oftheHamiltonianhavebeenmodified:

Hinλ,jmμ

=

Hin0λ,jmμ

+

Ui

(

Nin

−

N0_i

)δ

inλ,jmμ (37)

Inaperiodiccrystal, Nin

=

Ni

,

∀

n,andtheapplicationoftheBlochtheoremleadstothecorrespondingmodified Hamil-tonianink space:

Thisequation canbe generalizedtothe caseofanon-orthogonal basis set.Providedthat thecharge isdefinedinthe Mullikenfashion,themodifiedHamiltonianreads:

Hiλ,jμ

(

k

)

=

Hi0λ,jμ

(

k

)

+

1

2

(

Ui

(

Ni

−

N

0

i

)

+

Uj

(

Nj

−

N0j

))

Siλ,jμ

(

k

)

(39) 2.4.2. Self-consistencyinthetight-bindingmethod

From Eq. (37) it is clear that this equation must be solved self-consistently, since the Hamiltoniannow dependson thelocalcharges.A straightforwardprocedurewouldbe tostart froman initial“input”charge distribution,andoncethe corresponding Hamiltonianisdiagonalized,the “output”charges areused asnewinputsandthe processis iterateduntil theinputandoutput chargesdifferbylessthanagiventhresholdnumber.However,itiswellknownthatsuchamethod generallyfailstoconverge.Thesimplestwaytogetridofthisproblemistointroduceonlyaportionoftheoutputcharge byalinearmixingstrategy.Unfortunately,thisisveryuneffectiveandsomemorerefinedmixingtechniquesarenecessary, such asBroyden mixing [25], based on a kindof Newton method that greatly improves the convergence.Nevertheless, convergenceinhighlyanisotropicsystemscanstillbechallenging,especiallywhenmagnetismisintroduced.

2.4.3. Double-countingcorrections

Oncetheself-consistencyhasbeenachieved,totalenergycanbeexpressedintermsoftheC α_iλ

(

k

)

coefficients;however, itisusuallymorepracticaltousethebandenergyofthemodifiedHamiltonian,whichcanbewrittenas:

Eband

=



αkocc

ε

α

(

k

)

=



αkocc Cα_iλ

(

k

)

H0_iλ,jμ

(

k

)

Cα_jμ

(

k

)

+



i Ui

(

Ni

−

Ni0

)

Ni (40)

Thereforeitcomesoutthatthetotal(penalized)energyisnotsolelygivenbythebandenergy,butshouldbecorrected byaso-calleddoublecountingterm.Thentotalenergyreads:

Etot

=



αkocc

ε

α

(

k

)

−



i Ui 2

(

N 2 i

− (

Ni0

)

2

)

(41)

Thisexpressionalsoholdsforanon-orthogonalTBmodelaslongasthechargesaredefinedinMulliken’smanner.

2.4.4. ForceTheorem

Inquantummechanics,theabsoluteenergiesofasystemarenotmeaningful,sincetheydependontheenergyreference. Therefore,mostofthetime,whatisreallyneededisthedifferenceoftotalenergiesbetweencases(orsystems)thatinvolve changesintheinteractingpotential.Whenthemodificationofthepotentialislarge,aself-consistentcalculationisnecessary tocomparethetotalenergiesoftwosystems.However,therearemanysituationswherethesechangesarerelativelysmall, andinthesecasesaveryusefulapproximation,commonlycalledtheForceTheorem(FT),canbeapplied.FTisalsoknown asAndersen force theorem,since it was firstintroduced by Andersen [26] (see alsoa detaileddiscussion inRef. [27] by V.Heine).AmagneticversionofthistheoremispresentedinRef. [28]whereit isappliedtocalculateeffectiveexchange interactionsinmetallicferromagnets.Wewillnowbrieflyrecallitsprinciples,butamoredetailedderivationcanbefound in Ref. [29] in the context of a tight-binding scheme. Let us consider a perturbative external potential

δ

Vext, which in a tight-binding scheme is often a perturbation of the on-site levels of the Hamiltonian, but could be any other small perturbation.Due to self-consistenteffects, theexternal potential slightlymodifiesthe localcharges, whichconsequently producesanadditionalinducedpotential

δ

Vind.However,itcanbeeasilyshownthatthevariationoftotalenergybrought by thisinduced perturbative potential isexactly compensated(tofirstorder) by its correspondingdouble countingterm. Therefore,thechangeintotalenergyisequaltothechangeofbandenergyinducedbytheexternalpotentialonly:



Etot

≈ δ





αocc fα

ε

α

(42)

calculatedbyignoringtheself-consistentcorrections.Thismeansthattheeigenvaluesoftheperturbedsystemareobtained afterasinglediagonalizationofthe“perturbed”TBHamiltonian(includingtheexternalpotentialonlyandnottheinduced potential).The Force Theorem,besides saving much computationaltime, is also oftenmore precise than thebrute force approach,whichconsistsinperformingfullself-consistentcalculations,thereforerequiringanextremeprecision.

Letusalso add that thefirst-order variation ofthe free energy F at fixed numberofelectrons is equal thevariation ofthegrand-potential



atfixedchemical potential[16].Therefore theForceTheorem canbeequivalently appliedtothe grand-potential.

3. Magnetisminthetight-bindingmethod

3.1. Collinearmagnetism

Untilnow,thespinoftheelectronhasbeenignoredandthetwospinorbitalsweredegeneratedsincenospin-dependent potential waspresenttoremovethedegeneracybetween“up”and“down” spins.Oneofthesimplest, yetamazingly

effi-cient,waytointroducemagnetisminaTBHamiltonianisbasedonamodelinitiallyproposedbyStoner[11],whichwillbe described below.Thewavefunctionsaswellasthepseudo-atomicorbitalsformingtheTBbasis

|

in

λ

σ



nowhavetwospin components,and

σ

= ±

1 forupanddownspins,respectively.ThespinmagnetizationexpressedinunitofBohrmagnetons isthendefinedasthedifferencebetweenthenumberofmajority(up)andminority(down)electrons,M

=

N_↑

−

N_↓.

3.1.1. Stonermodel

Toremovethedegeneracybetweenthetwospins,itisnecessarytointroduceapotentialthatgeneratesabandsplitting between“up”and“down”spins.IntheTBmodel,anaturalwayofcreatingthisspinsplittingistoconsideralocal(on-site) potential that shiftsdown (up) the majority(minority) spin bands.Moreover, since thispotential intrinsicallly originates from electron–electroninteractions, exchange splittingshould depend itselfon the magnetization ofthesystem andvice versa.The spin-dependentStonerpotential V

ˆ

Stoner

σ σ takesthesimpleforminanorthogonalTBbasis withonlyonetype of

orbitals(dforthetransitionmetals):

ˆ

V_{σ σ}Stoner

= −

1 2



inλ

|

in

λ

σ

(

σ

IiMi

δ

σ σ

)



in

λ

σ



|

(43)

where Ii are the Stoner parameters, and Mi the spin magnetization of atom i (i.e. the difference between the number of electrons withup and down spins summed over all the orbitals ofsame character). The Stonerparameter is usually expressed inenergyunits,whilethe magnetizationis inBohrmagnetons. Ii onlydependsonthe chemicalnature ofthe atomsittingatsitei.Thislocaltermproducesashift



exch_i

=

IiMi,calledexchangesplitting.

An alternative (but equivalent) manner to introduce the Stoner model is to start from the total energy of the non-magnetic system Etot,0 _{to whicha negative componentis added,}

₋

1

4



iIiM2i.Then following thesame procedureasfor charge penalization,itcomesoutthattheoriginalnon-magneticHamiltonianshouldbecorrectedby alocalpotentialthat hasexactlytheformoftheStonerpotential.Then,takingintoaccountthedoublecountingterms,thetotalenergyofthe magneticsystemcanbewritten:

Etot

=



ασ,occ

ε

α,σ

+



i Ii 4M 2 i (44)

where

ε

α,σ aretheeigenvaluesofthespin-dependentHamiltonian. 3.1.2. Stonercriterion

Let usconsiderthe caseofa systemwhereall atomsare equivalent (typicallya crystalwithone atom per unit cell). TheeigenvaluesoftheHamiltoniancanbesplitintoupanddownspinenergies,andthedensityofstatesofspin

σ

,nσ

(

E

)

is simplyobtainedby arigidshiftofthedensityofstatesper spin ofthenon-magnetic systemn0

(

E

)

(without exchange interaction):

nσ

(

E

)

=

n0

(

E

+

σ

2I M

)

(45)

From thisuniquepropertyofthe densityofstates,acriterion fortheexistence(or not)ofa magneticsolutioncanbe derived. Indeed,theequation defining theFermilevel EF fromthe totalnumberofelectrons N and theone defining the totalspinmagnetization M aresufficienttocompletelydefinethesystem(i.e. determinethemagnetization)aslongasthe originalnon-magneticdensityofstatesisknown:

N

=

EF



n0

(

E

+

I 2M

)

dE

+

EF



n0

(

E

−

I 2M

)

dE (46a) M

=

EF



n0

(

E

+

I 2M

)

dE

−

EF



n0

(

E

−

I 2M

)

dE (46b)

Eq.(46a)definestheFermilevelprovidedthatthetotalnumberofelectronsisfixed.Ituniquelydefines EF

(

M

)

. Substi-tuting EF

(

M

)

forEFinEq.(46b)leadstoanequationofthetype M

=

F

(

M

)

where:

F

(

M

)

=

E



F(M)

(

n0

(

E

+

I 2M

)

−

n0

(

E

−

I 2M

))

dE (47)

F

(

M

)

isanoddfunctionthatnecessarilysaturatestoamaximumvalue.Anon-zerosolutionforM existsonlywhenthe slopeof F

(

M

)

atM

=

0 islargerthanonethatisequivalentto F

(

0

)

≥

1.Itistheneasilyverifiedthatthisisequivalent to theinequality:

In0

(

EF

)

≥

1 (48)

Thisis thefamous Stonercriterion [11], whichcanbe derivedin manydifferentmanners. Forexample,itcanalsobe obtainedbyminimizingtotalenergy.

Table 1

StonerparameterineVforvariouselementsobtainedfroma pro-cedureoffinetuningofthem(aalat)curveforagiven

crystallo-graphicstructureasexplainedinthetext. Inthe caseofFewe usedthebccstructure,andforCo,NiandPtthefccstructure.For Pttheequilibriumstructureisnon-magneticbutthiselement un-dergoesaferromagnetictransitionatagivenexpandedlattice.In thecaseofCrweusedthebccantiferromagneticphaseandfitted theatomicmagnetizationprojectedononetheatomofthelattice.

Element Cr Fe Co Ni Pt

Id(eV) 0.82 0.88–0.95 1.10 1.05 0.6

3.1.3. ThemagneticspdTBmodel

Wehavebeenabletoincludemagnetismandlocalchargeneutralitycorrectionswithinanon-orthogonalspdTBmodel andourHamiltonianmatrixtakesthefollowingexpression:

H

=

HTB

+

VLCN

+

VStoner (49)

where HTB isthe non-magnetic(diagonal in spin)Hamiltonianmatrix described inSection 2.3, VLCN isthe localcharge neutralitycorrectionwhosematrixelementsaregivenby:

V_inLCN_λσ,jmμσ

=

1

2

(

Ui

(

Ni

−

N

0

i

)

+

Uj

(

Nj

−

N0j

))

Sinλ,jmμ

δ

σ σ (50)

ThiscontributiondoesnotdependonthespineithersincethelocalchargesNi,Nj includeasummationoverbothspin components.VStoner _{istheStonerpotential:}

V_inStoner_λσ,jmμσ

= −

σ

2

(

Ii,λMi,d

)δ

inλ,jmμ

δ

σ σ (51)

HerewehaveconsideredStonerparametersthatdependontheatomicnatureoftheatomcenteredatsitei,butalsoon theorbitalcharacter(s,p,ord).Mi,d isthespinmagnetizationofatomi summedovertheorbitalsofdcharacteronly.The spinpolarizationsofsandpelectronshavebeenneglectedsincetheyareverysmall(Eq. (2)of[30]).Letusalsomention that forreasons that willbe explained inSection 3.5presenting theHartree–Fockinteraction in theTB model,thelocal magneticmomentisthe“net”magneticmoment(i.e.,withoutincludingtheoverlapterms).Inpractice,sincetheoverlaps ofdorbitalsarerathersmall,thisdoesnotsignificantlymodifytheresultscomparedtoacasewheretheMullikencharges areused.

Atthispoint,itisimportanttonotethattheStonerpotentialisdiagonalinspin space,i.e. therearenocouplingterms involvingbothspins.Inpractice,onehastodiagonalizeseparatelytwoHamiltonians,whichleadstotwosetsofeigenvalues correspondingtothe“up”

ε

α,↑

(

k

)

and“down”

ε

α,↓

(

k

)

spins.Thisprocedureshouldobviouslybeperformedself-consistently since, asin the caseof the local charge neutrality, the potential dependson the local magnetizations.Therefore similar mixingtechniquestothoseexplainedaboveareusedtoreachconvergence.

3.1.4. DeterminationoftheStonerparameter

Intransitionmetals,magnetizationisverydominantlybornebydorbitals(whichexplainstheformofourTBpotential whereonly Mi,d istakenintoaccount)andtheStonerparameterofd orbitals Id mainlydeterminesthemagnetization of the materialandwe haveset Is

=

Ip

=

Id

/

10.As seen fromtheStonercriterion, the densityofstatesatthe Fermilevel plays acrucialrole intheonsetofmagnetisminamaterial.Inaddition,since thewidthoftheDOSgets narrowerwhen theinteratomicdistanceincreases, thisnecessarilyproduceshigherDOS,sothat normalizationispreserved.Consequently, a uniformbondstretchingisalmostalwaysaccompaniedbyanincreaseofthemagnetization.Evennon-magneticmaterials willbecome magneticforalarge-enough latticespacing, sincemostisolatedatomsare magnetic.Thisargumentcanalso be applied tothe understanding ofthe generaltrend ofthe variationof magnetization withthe averagecoordination of a system. Indeed,similarlyto theargument putforwardto explain theeffectof bondstretching, theaverage DOS width isdecreasingwhenthe averagecoordinationdecreases.Thisbasically explainswhylow-coordinatedsystemstendtohave largermagnetizations.

Theonsetofmagnetismwithlatticestretchingisusually ratherabruptandthisexplainswhywehavechosen totune the Stonerparameterby trying to reproducethisvariation asprecisely aspossible.Inpractice, themagneticmoment M

iscomputedasafunctionofthelatticeparameteraalatofagivencrystallographicstructureforseveralvaluesofId.These curvesarecomparedwithab-initiodataandthisverysimpleprocedureallowsaprecisedeterminationofId withinarather smallenergyrange.AfewvaluesarepresentedinTable 1.ForthecaseofFe,we givetwo values0.88and0.95 eV,since wefoundthat,eventhough Id

=

0

.

88 eV leadstospinmomentsinbetteragreementwithab-initio datathephasestability isbetterreproducedwithId

=

0

.

95 eV asdiscussedinthenextsection.

Fig. 1. Total energyperatomasafunctionoftheatomicvolumeforbulkFeinthebodycenteredcubic(bcc),facecenteredcubic(fcc),andhexagonal closedpack(hcp)structure.Foreachstructure,wehaveconsideredtheferromagnetic(FM)andthenon-magnetic(NM)configurations.

3.1.5. Totalenergyandmagneticphasestability

Magnetismbeingextremelysensitivetothelocalenvironmentofatoms,inparticularthebond-lengthsandthe coordi-nationnumber,itisnotsurprisingthatitplaysacrucialroleonthephasestabilityofmagneticmaterials.Asanillustrative exampleletusmentionthecaseofsmallmagneticclustersaswellasbulkironsincetheyareparticularlyinstructive.

Cluster physics isvery rich andcomplex dueto thelarge numberof possiblegeometrical atomicconfigurations (that growsexponentiallywiththenumberofatomsinthecluster).Thiscomplexityincreaseswhenmagnetismisenteringinto playsince,besidesthemanymetastablegeometricalconfigurationstobeconsidered,therealsooftenexistsseveralmagnetic solutions[31,32].Inadditionthebond-lengthsoftherelaxedstructuresareusuallydependingonthemagneticsolution.As a generaltrenditis foundthat solutionswithhighermagnetic momentsfavorlargerinteratomic spacings,showingonce againthatmagnetismandstructureareintimatelyentwined.

The intrication between magnetism and atomic structure is also very important in bulk materials. One of the most emblematic example isthe caseof iron. Indeed,ifFe were not magnetic its crystallographicstructure should have been hexagonalclosedpackedasfoundbystandardnonspin-polarizedDensityFunctionalcalculations. Magnetismstrongly sta-bilizesthebodycenteredcubicstructureinironandfavorsitwithrespecttomorecompactstructures.Thisisillustratedin

Fig. 1 wherewepresenttheresultofourTBtotalenergycalculationsforbulkFeasafunctionoftheatomicvolume.Note thatforthesetotalenergycalculationswehavetakenId

=

0

.

95 eV,sincethisvaluegivesabetterdescriptionofthephase stability while themagnetization isslightlyoverestimated. Thereasonforthissmallinconsistencyis duetothefact that theenergydifferencebetweenfccandbccisoverestimated(comparedtoab-initio results)inthenon-magneticphase.Asa consequence,thestabilizationofbccrequiresalargerStonerparametertogainmoremagneticenergy.

3.2. Non-collinearmagnetism

Intheprevioussection,wehaveexplainedhowmagnetismcanbeintroducedinourTBschemeandweusedtheterms of “up”and“down” spins, whichimplicitly supposed that thespin quantization axiswas taken along themagnetization direction that willbe denoted as z in thefollowing. However the z axishasno reasonto be relatedto any crystallo-graphic directionofthe crystal,sincespin andorbitalvariablesare belongingtodifferentspaces.Itwillbe seenhowthe spin–orbitcouplingpotentialisconnectingthesetwospaces.Beforediscussingthispoint,wewillfirstexpressthemagnetic TBHamiltonianinfixedspincoordinateaxes,whichismoreappropriatetodealwithnon-collinearmagneticconfigurations, sinceinsuchsystemsthespinquantizationaxesarechangingfromonesitetotheother.

Whenstudyingnon-collinearmagnetism,itisnecessarytouseaspin-orbitalfunctionwithtwo-components

|

↑

₍

_k

₎

_

|

↓

₍

_k

₎

_



wherebothcomponentsareusuallynon-zero,incontrastwiththecollinearcasewherethewavefunctionscanbeseparated into “up”and“down” spin wavefunctionscorresponding tothetwo “independent”spin channels. Theoperators nowact simultaneouslyonbothcomponentsofthespin-orbitals,suchasthePaulimatrices:

ˆ

σ

x

=

0 1 1 0



; ˆ

σ

y

=

0

−i

i 0



; ˆ

σ

z

=

1 0 0

−

1



3.2.1. Localandglobalspincoordinateaxes

We willfirst introduceglobalxyz and localxyz spin axes. Bydefinition,thelocal axesare chosen such that z is pointinginthedirectionofthelocalmagnetization.Startingfromtheglobalxyz axesofthecrystal,anewframexyzis obtainedbyarotationofanangle

φ

around z,finallyafurtherrotationofanangle

θ

around ygivesthelocalspinframe

xyz.

Thespinone-halfrotationmatrix[33]permitstochangecoordinatefromlocaltoglobal(andvice-versa)spinframe:

U

(θ, φ)

=



e−iφ2_cosθ 2

−

e−i φ 2_sinθ 2 eiφ2 _sinθ 2 ei φ 2 _cosθ 2



=

e−iφ2σˆz_e−iθ2σˆy ₍₅₂₎

The componentsoftheeigenvectors

|

↑

loc inthe globalframeare givenbythe followingmatrixequation U

(θ,

φ)



1 0

so that:

| ↑

loc

=

e−i φ 2 _cos

θ

2

| ↑ +

e iφ_2sin

θ

2

| ↓

(53)

Asimilarreasoninggives:

| ↓

loc

= −

e−i φ 2 _sin

θ

2

| ↑ +

e iφ_2cos

θ

2

| ↓

(54)

where

|

↑

loc and

|

↓

loc arethe spinstatesthatdiagonalizethe Paulimatrix

σ

z (localframe), while

|

↑

and

|

↓

arethe spineigenstatesdiagonalizing

σ

z (globalframe).

3.2.2. TheStonermodelintheglobalaxes

By construction the Stonerpotential matrixis diagonal in spin when expressed in the localspin axes andtakes the form:



V_inStoner, loc_λ,jmμ

= −

1

2Iiλ

(

Mi,d

σ

ˆ

z

)δ

inλ,jmμ (55)

Itisstraightforwardtoshow,usingthetransformation



VStoner, glob

₌

_U

_(θ,

_φ)

_VStoner, loc_U−1

_(θ,

_φ)

_{thattheStonerpotential} expressedintheglobalaxesisgivenby:



V_inStoner, glob_λ,jmμ

= −

1

2Iiλ

(

Mi,d

.

σ

ˆ

)δ

inλ,jmμ (56)

where Mi,d

=

Mi,d

(

sin

θ

icos

φ

i

,

sin

θ

isin

φ

i

,

cos

θ

i

)

isthemagnetizationvector inx

,

y

,

z coordinates,and

σ

ˆ

= ( ˆ

σ

x

,

σ

ˆ

y

,

σ

ˆ

z

)

is thevectoroperator builtfromthePaulimatrices.Thisisa ratherintuitive expressionwhosesimpleformcomesfromthe factthatthePaulimatrixvectoralsotransformslikearegularthree-dimensionalspacevectorunderaspacerotationmatrix. Notethatthe Stonerpotential istheonlyone thatisspin dependent,since HTB_{and V}LCN _{arepurely diagonalinspin} space andindependent of the spin, while VStoner is diagonal in real spaceand non-diagonal in spin-space.The density matrixcanaswellnolongerbesplitinto“up”and“down”spincontributions.Itslocalcomponentsarenowwritten:



ρ

iλ

=



ρ

_i↑↑_λ

ρ

_i↑↓_λ

ρ

_i↓↑_λ

ρ

_i↓↓_λ



(57) where

ρ

_iσ σ_λ 

=

Re





α Cα_{iλ σ}



Cα_iλσ



(58)

The localchargesandmagnetic momentsare then expressedasan appropriate trace overthe product ofdensityand Paulimatrices: Niλ

=

Tr

(

ρ



iλ

)

=

ρ

_i↑↑_λ

+

ρ

_i↓↓_λ (59a) M_ix_λ

=

Tr

(

ρ



iλ

σ

ˆ

x

)

=

ρ

i↑↓λ

+

ρ

↓↑ iλ (59b) M_iy_λ

=

Tr

(

ρ



iλ

σ

ˆ

y

)

=

i

(

ρ

_i↑↓_λ

−

ρ

_i↓↑_λ

)

(59c) M_iz_λ

=

Tr

(

ρ



iλ

σ

ˆ

z

)

=

ρ

i↑↑λ

−

ρ

↓↓ iλ (59d)

Finallyan importantpointhastobementionedbeforeclosingthissection:intheabsenceofspin–orbitcoupling,even foranon-collinearconfiguration,aglobalrotationofthespinmagnetizationdoesnotaffectthetotalenergyofthesystem. Thesystemissaidtobeinvariantbyrotation.

Fig. 2. Left: NoncollinearNéelmagneticstructureofatriangularchromiumlatticeobtainedfromaTBcalculation.Right:Totalenergyperatomofthe non-magnetic(NM),ferromagnetic(FM),row-wiseantiferromagneticorder(AF)andnon-collinearNéelphaseasafunctionofthefirst-neighbordistanceof thefreestandingCrtriangularlattice.

3.2.3. Anexampleofnon-collinearmagneticconfiguration

Non-collinear configurations usually show up in two cases:in very inhomogeneoussystems, orin frustrated systems. Thefirstsituationtypicallyoccursinsmallclusters.Eveninstronglyferromagneticmaterialssuchasiron,onecanobserve departuresfromcollinearitywhichhoweverremainsmall[34].Thesecondsituationgenerallyleads tomoreobvious non-collinearconfigurations.Toillustratethissituation,letusconsideraverysimplesystemwherestrongantagonistinteractions arepresent:thetriangularlatticeofchromium.

Chromiumisamaterialwithaclearantiferromagnetic(AF)first-neighborinteractions.Inbulkbcc chromium,the mag-neticgroundstateisantiferromagnetic[35].WhenCratomsarelocatedonatriangularlattice,aperfectAFordercannotbe established,sinceitisnotpossiblethatallfirst-neighborpairshavemagneticmomentspointingintooppositedirectionsin atriangularnetwork.WehavecarriedoutaseriesofmagneticTBcalculationsontheCrtriangularlatticeforlattice param-eters rangingfrom2.1 Åto3 Å.The magneticgroundstateisactually noncollinear,forminga well-knownNéelstructure asthe oneshowninFig. 2resulting fromour non-collinearTBcalculation.Arow-wisecollinear anti-ferromagneticorder (see Fig. 1bofreference[36])doesexistinthissystem,butatahigherenergy.Inaddition,abovea givenlatticespacing, a ferromagneticordercan alsodevelop,butits energyiseven higherthanthe AFone.Thisis inperfectagreementwith

ab-initio results[36].

3.2.4. GeneralizedBlochtheoremandspinspirals

Somesystemshaveanothertypeofnon-collinearmagneticgroundstatescalledspinspirals.Inthesemagneticstructures, the magnetization isrotatedbya constantangle froma unit cellto thenext one,thisanglebeingdefinedbysome spin spiralvectorq.Sincethereisnospin–orbitcoupling,aglobalrotationofthespincanalwaysbringthespinspiralstructure to thecasewhereq istheaxisofrotation.Thespiralisthen fullycharacterized byitsaxisdirection,theazimuthal angle thatevolveslinearlyalongtherotationaxisandcanbewrittenatagivenatom

φ

= φ

0

+

q

·

R,andtheconstantpolarangle

θ

= θ

0.Forconvenience,thedirectionoftheq vectoristakenasthez axisandthespinspirallooksliketheonepresented inFig. 3foramonatomiclinearchainwithasingle atomperunitcell.Themagnetization vectorcanbewritten: M

(

R

)

=

M

(

cos

(φ

0

+

q

·

R

)

sin

θ

0

,

sin

(φ

0

+

q

·

R

)

sin

θ

0

,

cos

θ

0

)

. The z axishas noreasonto berelatedto anyspecific crystallographic axis,buttoavoidlengthynotations,wewillkeep thesamenotationxyz to denotethecrystallographicaxes andthespin spiralaxes.Thefollowingdemonstrationwillbe doneintheglobalspinframeworkandwe cancheckthatthefinalresult doesnot dependonthechosen axes.Abruteforcestrategyto describethistypeofsystemswouldconsist inconsidering a large super-cell containing an entireperiodof the spiral. However, thiswouldbe verytime consumingandinefficient since smallq vectors would necessitate extremelylong super-cells. In addition, withthis approach the spiralshould be commensurate withthe lattice.Fortunately itis possibleto usetheso-calledgeneralizedBloch theorem [37].Considering that a lattice translation of R accompanied with a rotation q

·

R leaves the system invariant, a generalized Bloch wave functionmustverifytherelation:

U

(

0

,

q

·

R

)





_k↑_,q

(

r

+

R

)



_k↑_,q

(

r

+

R

)



=

eik·R





_k↑_,q

(

r

)



_k↓_,q

(

r

)



(60)

sothatitcanbewrittenas

|

σ

₍

_k

₎

_{ =}



n

eik·Rn_U−1

₍

₀

_,

_q

·

_R