Jacques Friedel and the physics of metals and alloys

HAL Id: hal-01896085

https://hal.archives-ouvertes.fr/hal-01896085

Submitted on 16 Oct 2018

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Jacques Villain, M. Lavagna, Patrick Bruno.

Jacques Friedel and the physics of metals

and alloys.

Comptes Rendus Physique, Centre Mersenne, 2016, 17 (3-4), pp.276 - 290.

Contents lists available atScienceDirect

Comptes

Rendus

Physique

www.sciencedirect.com

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

Jacques

Friedel

and

the

physics

of

metals

and

alloys

Jacques

Friedel

et

la

théorie

des

métaux

et

alliages

Jacques Villain

a

,

∗

,

Mireille Lavagna

b

,

c

,

Patrick Bruno

d a_{Theorygroup,InstitutLaue-Langevin,38054 Grenoblecedex 9,France}

b_{UniversitéGrenobleAlpes,INAC–SPSMS,38000 Grenoble,France} c_{CEA,INAC–SPSMS,38000 Grenoble,France}

d_{Theorygroup,EuropeanSynchrotronRadiationFacility,38054 Grenoblecedex 9,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline22December2015 Keywords: Electronsinmetals Screening Friedeloscillations Kohnanomaly Kondoeffect

Tightbindingapproximation Mots-clés :

Electronsdanslesmétaux Effetd’écran

OscillationsdeFriedel AnomaliesdeKohn EffetKondo

Approximationdesliaisonsfortes

Thisis an introduction to the theoretical physics of metals for studentsand physicists fromotherspecialities.CertainsimpleconsequencesoftheFermistatisticsinpuremetals arefirstaddressed,namelythePeierlsdistortion,Kohnanomaliesandthe Labbé–Friedel distortion.Thenthephysicsofdilutealloysisdiscussed.Theanalogywithnuclearcollisions wasafruitfulstartingpoint,whichsuggestedoneshouldanalyzetheeffectsofimpurities intermsofascatteringproblemwiththeintroductionofphaseshifts.Startingfromthese concepts,Friedelderived atheoryoftheresistivityofalloys,and acelebratedsumrule relatingthe phaseshifts attheFermilevel tothe number ofelectrons intheimpurity, whichturnedouttoplayaprominentrole laterinthecontextofcorrelated impurities, as for instance in the Kondo effect. Friedel oscillations are also an important result, relatedtoincommensuratemagneticstructures. Itisshown howtheycan bederivedin various ways: fromcollision theory, perturbation theory, self-consistent approximations andGreen’sfunctionmethods.Whilecollisiontheorydoesnotpermittotakethecrystal structure into account, which is responsible for electronic bands, those effects can be includedinotherdescriptions,usingforinstancethetightbindingapproximation.

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

r

é

s

u

m

é

Cet article est une introduction à la théorie électronique des métaux. Il s’adresse aux étudiants et aux physiciens non spécialistes. On commence par décrire certaines conséquences simples de la statistique de Fermi–Dirac dans les métaux purs, comme la distorsion de Peierls, les anomalies de Kohn et la distorsion de Labbé–Friedel. On discuteensuitelaphysiquedesalliagesdilués.L’analogieavecleproblème descollisions nucléairesfutunpointdedépartfructueux,quiamenaàconsidérerl’effetdesimpuretés comme un problème de diffusion, dans lequel apparaissent les déphasages de l’onde électronique diffusée. Friedel élabora ainsi une théorie de la résistivité des alliages, et établit une règle de somme qui relie les déphasages au niveau de Fermi à la charge de l’impureté. Cette règle de somme joua plus tard un rôle essentiel dans le cas d’électrons fortement corrélés, notamment dans l’effet Kondo. Une autre découverte importantefutcelledesoscillationsdeFriedel,responsablesparexempledelaformation

*

Correspondingauthor.

E-mailaddress:jvillain@infonie.fr(J. Villain). http://dx.doi.org/10.1016/j.crhy.2015.12.010

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

1. Introduction

The language of physicistsevolves. In thisissue ofthe C. R.Physique devoted to Jacques Friedel, articles by physicists ofvariousgenerationsandvarioussub-specialitiescoexist,andtheirlanguageisdifferent. Thismakescommunicationwith non-specialistsandstudentsdifficult.Theaimofthepresentintroductionistobridgethegapbetweenexpertsand newcom-ers,toallowstudentstoreadoldtextbooks,whichmaybeveryusefulbecausetheywerewrittenbythosewhodiscovered the phenomena andoftenunderstand thembetter because they rememberthe difficultiesthey met,the mistakeswhich shouldbe avoided,theapproximationsthat failedandthose thatsucceeded. Wetryto presenta syntheticviewofa few essentialconceptsofmetalphysics.

2. Simplemechanismsinpuremetals

Thephysicsofmetals,whereelectronsarenotlocalized,butitinerant,isadifficultmatter.Inthemiddleofthetwentieth century,afewphysiciststriedtodefinebasicconceptsthatcanbringabetterunderstanding.Amongthem,JacquesFriedel’s partwasparticularlyimportant.

Thetaskwasfacilitatedbypreviousdiscoveries,forinstancetheJahn–Tellerdistortion,discoveredin1937[1],whichwill nowbebrieflyrecalled[2].Assumeahighlysymmetricsystem(acomplexoranimpurityinacrystal)whereanelectronic state is (for instancetwice) degenerate andoccupiedby asingle electron.The Jahn–Teller theorem statesthat this high symmetrystate isunstableanddistorts.Thereasonisthat aweakdistortion(oranysmallperturbation)always splitsthe degeneratelevelsintoonewhichisathigherenergyandonewhichisatlowerenergy(Fig. 1a).Theelectrongoesintothe loweststate,andthustheenergyisloweredbythedistortion.TheJahn–Tellerdistortionoccurswheneveranelectronicstate

hasan-folddegeneracyandisoccupiedby lessthann electrons.Anexample(Fig. 1b) isgivenby anoctahedralcomplex

inwhichthecentralionhasasinglevalenceelectrononap-shell.Theelectronhastochoosebetween3porbitalsthatare directedalongthethreeaxesoftheoctahedron.Ifitchoosesforinstancethez axis,thedistancea betweentheligandson thisaxisisdifferentfromthedistanceb betweentheligandsonaxes y andz.

TheJahn–Tellerdistortionoccursinsmallcomplexesorininsulatorscontaining impurities,butsimilar effectsoccurin metalsbecauseelectronicstatesjustaboveandjustbelowtheFermisurfacearenearly degenerate.AnexampleisthePeierls

distortion [3–5].It isan instability ofa one-dimensional conductorformed byelectrons interacting withregularlyspaced

ions.Theionslowertheelectronicenergyiftheymodulatetheir distancewithaperiod

π

/

kF,wherekFistheFermiwave vectoroftheelectrongas.Thereasonisthefollowing:thedistortionmixeselectronicstatesofwavevectorsk andk

−

2kF. Ifk isveryslightlylargerthankF,thenk

−

2kF isvery slightlylarger than

−

kFandthecorresponding energies



(

k

)

and



(

k

−

2kF

)

arealmost equal.The distortionlowers thelower energyandincreasesthe higherlevel,butsince thereisno electrononthislevel,thetotalelectronicenergyislowered.

The Peierls distortion (Fig. 2) is a property of one-dimensional conductors (which make them insulating or semi-conducting!).Butsimilareffectsoccurincertainthree-dimensionalmetals.Ithasevenbeensuggested[5]thatthecrystal structureofmanycovalent materialsmaybeconsideredasresultingfromaPeierlsdistortionofafictitiousmetal.In sub-section2.2,adistortionparticulartocertainthree-dimensionalcrystalstructureswillbeaddressed.

Fig. 1. TheJahn–Tellereffect.a)Energyofanelectronasafunctionofthedistortiona/b ofthesurrounding.Iftherearetwo(ormore)degeneratestatesin thehighlysymmetricstate,aslightdistortionincreasesoneoftheenergiesandlowerstheotherone(s).Ifthereisasingleelectron,itsenergyistherefore lowered,b)Exampleofanioninanoctahedralenvironmentwithapartlyoccupiedpshell.

Fig. 2. ThePeierlsdistortionofaone-dimensionalconductor.Alatticemodulationofperiod2kFlowerstheenergyoftheelectronsbelowtheFermilevel

andincreasestheenergyabovetheFermilevel.Thetotalelectronicenergyisthereforelowered.

Fig. 3. If q joinstwopointsoftheFermisurfacewherethetangentplanesareparallel,thephononfrequencyωph(q)hasaKohnanomalyforthisvalue

of q.‘Nesting’occurswhenlargepartsoftheFermisurfaceareveryclosetothetangentplanes.

2.1. Kohnanomaly

Thephononfrequency

ω

ph

(

q

)

ofametalmayhaveanomaliesforcertainvaluesofthewavevector q,whicharerelated

totheFermisurface[6].SuchanomaliesarecalledKohnanomalies.

Tointroducethesubject,we choosethecaseofaveryanisotropicconductor,whichisnearlyone-dimensional,butnot sufficiently toundergothePeierls transition.Withappropriate, butnotessentialsimplifying assumptions,thepositions xi

oftheions(assumedidenticalwithmassm)obeythelinearizedequationsofmotion

m d2xi

/

dt2

=



j

Ai j

(

xj

−

xi

)

(1)

sothattheFouriertransformxqobeys

m d2xq

/

dt2

=

A

(

q

)

xq (2)

Thequantity A

(

q

)

=



_jAi jexp

[

iq

·(

Rj

−

Ri

)

]

thusappearstobetherestoringforceoftheoscillator.Thephononfrequency

ω

ph

(

q

)

isreadilydeducedfrom(2),namely

ω

_ph2

(

q

)

=

A

(

q

)/

m (3)

When the Peierls transitionisapproached(for instanceby loweringthetemperature), thedistortedphase approaches stability andthereforethe restoringforce A

(

q

)

approaches0 forqx

=

2kF,where x isthe directionof highconductivity. Thereforethephononfrequency

ω

ph

(

q

)

approaches0too.Asafunctionof q,ithasaminimumforqx

=

2kF.1 Such anoma-liesofquasi-one-dimensionalconductors havebeenextensivelyinvestigatedinOrsay [4].Moregenerally,WalterKohn[6]

proposed tousethephonon spectrum(whichcan beobtainedfrominelasticneutronscattering)to obtain“imagesofthe Fermi surface”. Henoted that,if q joins two points ofthe Fermisurface wherethe tangentplanes areparallel, then the gradient of

ω

ph

(

q

)

is infinite(Fig. 3). However, such anomalies are generallyvery difficult to observe. In practice,Kohn anomaliesoftendesignateanyanomalyofthephonon frequencythatisduetoconductionelectrons.Itcanbeforinstance a minimum,or adiscontinuity oftheslope (Fig. 4). Kohnanomalies canbe observed iftheFermi surface hasan appro-priate shapeallowing forthepropertyofnesting.Thismeansthat,iftheFermisurface istranslatedby acertainvector Q, large portions ofthetranslatedsurface almost coincidewithlarge portionsofthe original Fermisurface.Palladium isan example[7].

Inasimpletheoreticaltreatment[4],theeffectofconductionelectronsisconsideredasaperturbationoftheion–ion in-teraction.Theresultingcorrectiontoformula(3)isanadditivetermproportionaltotheLindhardfunction(seeformula(18)

below), whichgivesthe responseof an electrongasto a perturbation.Adifficulty arisesfromthe factthat the Lindhard

Fig. 4. Typical phonon dispersion curves which are usually considered as possible Kohn anomalies in metals.

Fig. 5. Labbé–Friedeldistortion.a)Inthecubicstructure,thereexistthreefamiliesofconductingchainsinorthogonaldirectionsthathavethesamedensity ofelectronicstateswithasingularityjustabovetheFermilevel.b)Inthepresenceofatetragonaldistortion,thesingularitiesoccuratdifferentlocations forthethreechains,sothatthestatescorrespondingtothesingularityinthedensityofstatesofoneortwochainscanbecompletelyoccupied.

function depends on the frequency

ω

of the perturbation. The appropriate value of

ω

would be the phonon frequency

ω

ph

(

q

),

whoseself-consistentdetermination isnot possibleanalytically.This problemcan oftenbesolved by replacing

ω

by0in(18).Thisistheadiabaticapproximation,oftenjustifiedbythefactthatphononfrequenciesaregenerallysmallwith

respecttoelectronicfrequencies. 2.2. TheLabbé–Friedeldistortion

A transition from a cubic structure to a tetragonal one is observed at low temperature in intermetallic materials of formula V3X (X

=

Si, Ga, Ge, Sn, etc.) or Nb3Sn. Labbé and Friedel [8]explained this distortion asa kindof collective Jahn–Tellereffect.VorNbatomsformconductingchainsthathavethreeperpendiculardirections.Ifinterchainconduction isneglected,thedensityofstates

ν

(



)

divergesatthetopofthebandandatthebottom.Indeedtheenergyofanelectron ofwavevector k,countedfromthebottomoftheconductionband,is



(

k

)

= ¯

hk2

_/

_{m, wherem istheeffectivemass.Thus}

k

(



)

is proportional to

√



whereas

ν

(



),

whichfor a one-dimensional system is proportional to dk/d



, is proportional to 1/

√



.In thecompounds addressedhere(called A15by metallurgists), the Fermilevel iscloseto the singularity. Ifa distortionoccurs(Fig. 5) thethree familiesofchains havetheir singularitiesatdifferentlocations, andoneor twoofthe chainsmayhavethewholesingularitybelowtheFermilevel,whichimpliesa largeenergygain.Theelectronic energyof theother chainsorchainisincreased,butthisenergylossissmallerthantheenergygain.The detailedcalculation [8]is more complicatedthan forthetrue Jahn–Teller effect,butthe analogyis clear:the two energylevels of theJahn–Teller effectarejustreplacedbytwosingularities.

The Peierls distortion andthe Labbé–Friedeldistortion are properties ofcertain pure metals that arise froma simple mechanism. Similarly, simpleconcepts havebeen proposedfor dilutealloys, orofimpurities inmetals.A pioneerof this conceptualizationwasMott[9],butconsiderableadvanceshavebeenmadebyFriedelandhiscollaborators,asrecalledby Georges[10].

Inthenextsections,certainpropertiesofdilutealloys,orofimpuritiesinmetals,willbesummarized.

3. Impuritiesinmetalsanddilutemetallicalloys

Dilutealloyswereforphysiciststhesourceofvariousproblems,especiallythethreefollowingones.

i) The electrical resistivity does not vanish atlow temperature, in contrastwith that of pure metals. At temperature T

=

0,thereisaresidualresistivityproportionaltotheimpurityconcentration.Whatisthisresistivity?

ii)There isoftenanelectriccharge localizedontheimpurity.Thischarge isscreenedby conductionelectrons,so that the electricfield vanishes farfrom theimpurity. Howis the screeningcharge distributed? What is the resultingelectric potential?

iii)Theremayalsobeamagneticmomentlocalizedontheimpurity.Thismomentis,sotospeak,screenedbythespins oftheconductionelectrons,sothatthegroundstateisasinglet,i.e.anon-magneticstate.ThisistheKondoeffect. 3.1. Impuritiesinmetals:ascatteringphenomenon

The electricalresistivityofalloysatlowtemperatureresultsfromthescatteringofconductionelectronsby impurities. Scattering of free particles is a well-known problem ofquantum mechanics [11], andFriedel hadthe idea to apply the standardformalismtoalloys.

Letthisstandard formalismbe recalled. Thesimplestcaseiswhenthescatteringpotential hassphericalsymmetry. In theabsenceofscattering,theelectronicwavefunctionwouldbe2

ψ

_k0

(

r

)

=

exp

(

ik

·

r

)

(4)

IfascatteringpotentialwithsphericalsymmetryaroundtheoriginOisintroduced,thewavefunctioncanbeexpanded ina seriesofLegendrepolynomials P

(cos

θ ),

where

θ

istheanglebetweenvectors k and r.Thescatteringpotential will

beassumedtovanishbeyondacertaindistancea.Thenitcanbeshown[11]thatthewavefunctionhasthefollowingform forr

>

a,

ψ

k

(

r

)

=

∞



=0

(

2



+

1

)

iexp

(

i

δ



)

[

j

(

kr

)

cos

δ



−

n

(

kr

)

sin

δ



]

P

(

cos

θ )

(5)

where j

(

u

)

and n

(

u

)

are respectively spherical Bessel andNeumann functions (actuallyjust polynomialsin 1/u, sin u

andcos u).Thequantities

δ



(

k

)

dependonthescatteringpotential, theyarerealandarecalledphaseshifts.Expression(5)

should bevalidevenwhenthescatteringpotentialvanishes, andthereforealsoappliestotheplanewave (4).Inthatcase thephaseshiftsvanish:

ψ

_k0

(

r

)

=

exp

(

ik

·

r

)

=

∞



=0

(

2



+

1

)

ij

(

kr

)

P

(

cos

θ )

(6)

Formula(5)canalsobewrittenas

ψ

k

(

r

)

=

exp

(

ik

·

r

)

+ (

1

/

r

)

fk

(θ )

exp

(

ikr

)

(7)

with

fk

(θ )

= (

1

/

k

)

∞



=0

(

2



+

1

)

sin

δ

P

(

cos

θ )

exp

(

i

δ



)

(8)

Ifka

<<

1,thephaseshiftsoforder



>

0 arenegligible,andthescatteredwavereducestoitsterm



=

0,sothatithas sphericalsymmetrybecauseP0

(

u

)

=

1.

Phaseshiftscanalsobedefinedintwo-dimensionalconductors.Inone-dimensionalconductors,thereisasinglephase shift

δ

0.

3.2. Resistivityofadilutealloy

Using formula(8),Friedelandhis collaboratorswere able tocalculate theresidualresistivity ofa dilute alloy,i.e.the resistivityattemperatureT

=

0.Theyderivedtheformula[12,13]

ρ

=

4

π

e 2_c Nm



kF







sin2

(δ

−1

− δ



)

(9)

where c istheimpurityconcentration,



istheatomicvolume,e the electroniccharge, m theelectronic massandN the numberofelectronsperunitvolume.Phaseshiftswerecalculatedrepresentingeachimpurity(i.e.eachatomoftheminority element)byapotentialdependingonaparameter

κ

:forinstancescreenedCoulombpotentialdecayingasexp(

−

κ

r

)/

r.Then the phaseshifts shouldbe calculatedasfunctionsof

κ

.Finally

κ

shouldbe determined.This determinationcanbe done usingFriedel’ssumrule,whichwillbediscussedinthenextsubsection.

Usingformula(9),FagetdeCastelnauandFriedel[12] obtainedagoodagreementwithexperimentalresultsforZn,Ga, GeandAsimpuritiesincopper(Fig. 6).

Fig. 6. ResidualresistivityofvariousimpuritiesincopperaccordingtothetheoryofFagetdeCastelnauandFriedel

[12]

(thick,continuouscurve)compared withexperimentalpointsandaprevioustheoryofMottandJones

[9]

(dashedline).Theunitisthemicroohm·cmper%.

3.3. Friedel’ssumrule

As aneffect oftheelectron-impurity potential, acertain (average) number

n of conductionelectrons are usually at-tractedtotheimpuritysite.Thisnumbermaybevery closetoanintegerbut, sincethemediumisametalwithitinerant electrons,itisusuallynotaninteger.

Friedel’ssumrule[14]gives

n asafunctionofthephaseshifts,namely

n

=

2

π





(

2



+

1

)δ



(

kF

)

(10)

The powerofthisrule liesinthe factthat, mostofthephase shifts oftenvanish.Anextremecaseis whenonly one phase shift(usually

δ

0) isdifferentfrom0.Then Friedel’ssumrulecontains nosummation! Thisisthe casein Nozières’ treatmentoftheKondoeffect[15](seesection9belowandreference[10]).

4. Friedeloscillations:thethree-dimensionalcase

4.1. Screening

The Coulomb potential islong ranged. However, in a metal orin an electrolyte, two distant charges do not exert an appreciableforce uponeachother becauseother chargesscreentheCoulomb force.The questionis:howisthe screening chargedistributed? Ifthischargeiscarriedbyaclassical fluid,thescreenedelectricpotential andthescreeningchargeat distancer froma chargedimpurity are foundto decreaseexponentially withr, asexp(

−

κ

r

)/

r.This resultisan element oftheDebye–Hückeltheory ofelectrolytes[16].Theargumentisasfollows:theelectricpotential V

(

r

)

andthescreening charge

ρ

(

r

)

atdistancer ofthescreenedchargearerelatedbythePoissonequation

∇

2_V

₍

_r

₎

_{= −}

_ρ

₍

_r

_)/

_

0.Ontheotherhand, theprobabilityofhavingachargeatpoint r isproportionaltoexp

[−β

V

(

r

)

]

accordingtoBoltzmann’sstatistics(β

=

1/kBT ,

kB

=

Boltzmann’s constant,T

=

temperature).At longdistances,V issmall,sothat

ρ

(

r

)

isjustproportional to

−

V

(

r

)

and thePoissonequationreducesto

∇

2

ρ

(

r

)

=

κ

2

_ρ

₍

_r

_),

_where

_κ

2 _{isproportionalto1/T .Thesolutionis}

_ρ

₍

_r

₎

_∼

_exp(

₋

_κ

_r

_)/

_r.

Todescribe thescreening ofa chargedimpurity inmetals,the firstattempt was toapply an approximation proposed by Thomas[17] andby Fermi[18] in1927. Inthe Thomas–Fermitheory,thescreened potential

δ

V

(

r

)

andthescreening charge

δ

ρ

(

r

)

are postulated to be proportional, as they are in electrolytes.As a result, the decay is againexponential,

ρ

(

r

)

∼

exp(

−

κ

r

)/

r,but,incontrastwithelectrolytes,

κ

isindependentoftemperature.3

However, theassumption ofalocal proportionalityrelation

δ

ρ

(

r

)

∼ δ

V

(

r

)

is acceptableonly forlocalizedcharges,not for itinerantelectrons that propagate nearly freely withan almost constant velocity. This was realized in the middle of the last century,whenFriedeland hiscoworkers showedthat thedecayis oscillatory.The validity oftheThomas–Fermi

3 _{Thescreeninglength1}

/κisobtainedbywritingthattheFermilevelFisthesameeverywhere.Thus,ifthereisalocalexcesspotentialδV that

shiftstheelectronicenergybyeδV ,theelectronsofenergyhigherthanFshouldgoaway.TheirnumberperunitvolumeiseδV D(F)where D(E)

istheelectronicdensityofstates(numberofelectronic statesperunitvolumeand energyunit).Forfreeelectrons, D(E)= (2m3_E)1/2_/(2π2_¯h3_).Thus δρ(r)=e2_D(E

Fig. 7. A typical screened Coulomb potential showing Friedel oscillations (full curve) compared with the Thomas–Fermi approximation (dotted curve).

approximationwasthendiscussedmoresystematicallybyHohenbergandKohn[19],whoshowedthatitcorrectlydescribes long-wavelengthchargevariationsonly.

Friedeloscillationscanbederivedbyvariousmethods.WeshallfirstgiveFriedel’soriginalargument,orrathera simpli-fiedversioninwhichallphaseshiftsareassumedtovanishexcept

δ

0.Thegeneraltreatmentjustinvolvesmorecumbersome equations.

4.2. Friedeloscillationsandphaseshifts

InFriedel’sfirstpapers,whosehistoricalimportanceisrecalledbyDaniel[20],thecrystalstructurewasneglected.Then, intheabsenceofimpurity,theelectronic statesareplanewavescharacterizedbytheirwavevector k.Inthepresenceofa (single) impurity,electronicwavefunctions

ψ

k

(

r

)

havetheform(5),butthey canstillbecharacterizedbythewavevector

k oftheincidentwave.The electronicdensityisthesumofthecontributions

|ψ

k

(

r

)

|

2 ofthevariouswavefunctions.

Sub-tractingtheelectronicdensity

|ψ

_k0

(

r

)

|

2 intheabsenceofimpurity,oneobtains theexcesselectronicdensitycorresponding tothewavevector k,namely

k

(

r

)

= |ψ

k

(

r

)

|

2

− |ψ

k0

(

r

)

|

2

= [

j0

(

kr

)

cos

δ

0

−

n0

(

kr

)

sin

δ

0

]

2

−

j20

(

kr

)

or

k

(

r

)

= [

n20

(

kr

)

−

j20

(

kr

)

]

sin2

δ

0

−

2 j0

(

kr

)

n0

(

kr

)

cos

δ

0sin

δ

0 Rememberingthat j0

(

u

)

=

sin u/u andn0

(

u

)

= −

cos u/u,oneobtains

k

(

r

)

= (

kr

)

−2

[

cos

(

2kr

)

sin

δ

0

+

sin

(

2kr

)

cos

δ

0

]

sin

δ

0 (11) Itisofinteresttodefinethelocaldensityofstatesg

(

r

,



),

definedfornon-interactingelectronsby

g

(

r

,



)

=



k

|ψ

k

(

r

)

|

2

δ(



−



k

)

(12)

where

ψ

k

(

r

)

is thenormalized wave function ofan electronwith wave vector k and



k isits energy. In theabsence of

impurity, g

(

r

,



)

is independent of r. Animpurity atthe originmodifies g

(

r

,



)

by an amount

g

(

r

).

Assuming that



k

dependsonlyon

|

k

|

,relation(11)yields

g

(

r

)

= (

A

/

r2

)

sin

[

2k

(



)

r

+

ϕ

]

(13)

where2k(



)

istheinversefunctionof



(

k

),

A isaconstantand

ϕ

= δ

0.Fig. 7showstypicalFriedeloscillationscompared withtheThomas–Fermiapproximation.

ThelocaldensityofstatesattheFermilevelcanbemeasuredbyascanningtunnelmicroscope[21,22](STM).Thecase of the (111)surface of Cu is particularlyspectacular because ithas an electronic surface state whose Fermi“surface” is nearlyacircle,sothatthetwo-dimensionalversionof(13)(with1/r insteadof1/r2)holds.ThecomplicatedFermisurface ofbulkcopperplays norole.Itisworthsayingafew wordsabouttheremarkableversatilityoftheSTM,whichisableto observethesurfaceatoms,butalso(byadjustingthevoltage)toforgetthemandfocusonelectronicstates.

Theexcesselectronic density

ν

(

r

)

canbeobtainedbyintegrating(13)overtheenergyaftermultiplyingbytheFermi function(i.e.theoccupationprobabilityofeachstate).Atzerotemperature,thismeansthat(13)shouldbeintegratedover k

<

kF.Friedeloscillationsareagainobtained,butthedependenceinr isdifferent,namely1/r3 insteadof1/r2 [23].Indeed, forlarger,(13)isthederivativewithrespecttokFof

ν

(

r

)

= −

B

bulk, andthe oscillating interaction influences the positionof theimpurities. This effecthas beenobserved by scanning tunnelmicroscopy[25].

Ifthe impurity is a magneticone, the interaction isbetweenthe magnetic moments ofthe impurities.The magnetic momentscanbe nuclearspins[26] orelectronic ones[27,28].TheresultingRudermann–Kittel–Kasuya–Yosida interactions (RKKY)areoscillatingfunctionsofthedistance,alternativelyferro- andantiferromagnetic,andtheyareresponsibleforthe spinglassstructureofcertainalloys[29].

Actuallythemagneticmomentsdonotneedtobe thoseofimpurities.Oscillatinginteractionsalsoexistinpuremetals andareresponsibleforhelicalormodulatedmagneticstructuresofCrandofcertainrareearths.4

6. Amorerealisticapproach:thetight-bindingapproximation

Thesuccessofthecalculationsdescribedinsections3and4israthersurprising,sincethetrueelectronicwavefunctions in the pure metal are very differentfrom theassumed plane wave exp(ik

·

r

).

The true wave functionsare Bloch waves, uk

(

r

)

exp(ik

·

r

),

whereuk

(

r

)

isa periodicfunction,butintheneighborhood ofeach atomatpoint R,theBlochfunctionis

notverydifferentfromawavefunction

ϕ

(

r

−

R

)

ofthefreeatom.Forinstance,insodium,thewavefunctionsofconduction electronslocallylooklike2sorbitals.Thus,theelectronscanoftenbeassumedtobetightlyboundtotheneighboringatom, sothatBlochwavefunctionsofthepuremetalhavetheform

ψ

k

(

r

)

=

N−1/2



R

ϕ

(

r

−

R

)

exp

(

ik

·

R

)

where N is the numberof lattice sites.The lattice is assumedto contains a single atom per unit cell. The factor N−1/2 ensures normalization.Assumingasingle, non-degenerateorbital

ϕ

persite, ignoringthespin andneglecting interactions betweenelectrons,theHamiltoniancanbewrittenas

H

=



RR

tRRc+_RcR (15)

where c+_R andcR are electron creationanddestruction operators on theatom R. Theconstants tRR arecalledresonance

integrals.Indeedtheirevaluationrequirestheintegrationofanexpressionthatdependsonthewavefunctions

ϕ

(

r

−

R

)

and

ϕ

(

r

−

R

)

andontheelectron–ioninteraction.

Adifficulty arisesfromthefact that theatomicwave functions

ϕ

(

r

−

R

)

and

ϕ

(

r

−

R

)

attwoneighboringsites R,R

havenoreasontobeorthogonal.Theoristssometimesignorethisproblemandassumeorthogonality.

Formula(15)holdsinapuremetal.Inanalloy,thedifferentenergiesofelectronsondifferentsitesshouldbetakeninto accountandthetight-binding5Hamiltonianreads

H

=



RR tRRc+_RcR

+



R



Rc+_RcR (16)

Moreover,interactionsbetweenelectronsmaybetakenintoaccount.ThepopularHubbardmodelintroducesarepulsive (Coulomb) interaction between two electrons on the same site only. Since there is a single orbital per atom, the two electrons necessarilyhaveoppositespins. TheHubbardmodelisthe simplestmodelofitinerantmagnetism. Ifonewants toanalyzetheeffectofasinglemagneticimpurity(e.g.,theKondoeffect),itissufficienttotakeintoaccounttheCoulomb interactionontheimpuritysite.ThisistheAndersonmodel,addressedbyGeorges[10] andinsection9ofthispaper.

Thus,theelectronicstatesofinterestaregeneratedbytheactionofoperatorscRandc+_R onthegroundstate

|

0



.These

operators destroyorcreatea conductionelectronatsiteR.TheirFouriertransformsck andc+_k are alsouseful.Thedirect

spaceisnolongerconsideredcontinuousasintheprevioussections,butdiscrete.

In manycases, theconduction band consists ofseveral orbitalsthat should be labeled by an additionalindex

α

.For instance,in a transitionmetal asFe, the conductionbands are the4s and3d bands. Andof course, each orbital can be occupiedbytwoelectronswithoppositespins.

Applicationsofthetight-bindingapproximationtobulkmetalsandtheirsurfacesarereviewedinreferences[5,10,30,31].

4 _{Rareearthsandchromiumaresomewhatdifferent.Inrareearth,magnetismisduetolocalized,4felectrons,andtheoscillatinginteractionresultsfrom}

theactionofconductionelectrons.Inchromium,themagnetismitselfresultsfromconductionelectrons.

7. Perturbativeandself-consistenttreatments

As seenabove, theoriginal derivationofFriedeloscillations ignoredthe consequencesofthecrystalstructure andthe propertiesofBlochwaves.Apossiblealternativemethodisjustperturbationtheory.Ifaweakpotential Vq

(

ω

)

exp

[

i(q

·

r

−

ω

t

)

]

isappliedtothegasofBlochelectrons(whoseinteractionsareneglected),theperturbationoftheelectronicdensityis

ν

q

(

ω

)

exp

[

i(q

·

r

−

ω

t

)

]

where

νq

(

ω

)

=

e

χ

(

q

,

ω

)

Vq

(

ω

)

(17)

wheretheso-calledLindhardfunctionhasbeenintroduced,

χ

(

q

,

ω

)

=

1 N





k f0

(



k+q

)

−

f0

(



k

)



k+q

−



k

− ¯

h

ω

−

i

η

(18)

where



is thevolumeoftheunit celland f0

(



)

=

1/

[

1

+

exp(



−



F

)/

kBT

]

istheFermifunction attemperatureT . The quantity

η

isreal,positive andverysmall.Itappearsbecausetheperturbationisassumedtobe switchedonfromatime t

= −∞

ataveryslowrate

η

,correspondingtoanimaginaryfrequency.Itavoidsanydivergenceof(18)forrealfrequencies. Inthecaseofinteresthere,

ω

=

0 andtheimpuritypotential hasmanyFouriercomponents Vq,theeffectofwhichis

additive.

Thebandstructure,andthereforethecrystalstructure,aretakenintoaccountthroughthefunction



k.Theexplicit

calcu-lationcanbedonenumericallyforeachparticularmaterial.Weonlynotethat,inathree-dimensionalmetal,formula(14), andthereforeFriedeloscillations,canbederived [32]attemperatureT

=

0 from(17)and(18)if



k isassumedtodepend

only on

|

k

|

.If



k hasaweak angulardependence(asitreallyhas),thebehavioroftheelectron densityatlongdistances

fromthe impurity maybe modified,butFriedeloscillations persist,aswillbe argued insubsection8.4.Ofcourse,ifthe energy



kandthepotentialVkareknown,theresponse(17)canbecalculatednumerically.

In firstorderperturbationtheory,formulae(17)and(18)arevalidforanymetal,includingone- and two-dimensional ones. Foraone-dimensionalmetal,

χ

(

q

,

0)divergesasln(1/

|

q

−

2kF

|)

near2kF.Thiscorresponds toFriedeloscillationsof

theform

(1/

r

)

cos(2kFr

+

ϕ

)

inthechargedensity.

7.1. Self-consistenttreatment

Actually equations(17)and(18)arenot limitedto thelinearresponse approximationandmaybe regardedasa self-consistent approximate solution,provided Vq is not the applied potential Vq0,buttakes screening intoaccount. In three

dimensions,thisimplies[33]that

Vq

(

ω

)

=



1

−

4

π

e 2 q2

χ

(

q

,

ω

)



−1 V_q0

(

ω

)

(19) 8. Green’sfunctions

In thecurrentliterature, itisusual toexpressphysicalquantities intermsofGreen’sfunctions.Theyarederived from correlationfunctionssuchas



c+_R

(

t

)

cR



=



exp(itH)c+_Rexp(

−

itH)cR



,where

H

istheHamiltonian.WeshallseethatGreen’s functionsprovideamoregeneralwaytoobtainFriedeloscillations.Thetemperaturewillbeassumedtobe0.

In cases of interest here, electrons donot interact. One can introduce the one-electron states andthe corresponding creationanddestructionoperatorsc+_{α and}cα.ThentheHamiltoniancanbewrittenas

H

=



α

¯

h

ω

αc+αcα (20)

ThelocaloperatorscR canbewrittenaslinearcombinationsoftheeigenmodeoperatorscα :

cR

=



α

ψ

α

(

R

)

cα (21)

Thecorrelationfunctions



c+_R

(

t

)

cR



arethereforelinearcombinationsofthefunctions

0

|

c+_α

(

t

)

c_α

|

0

 =

0

|

c+_αc_α

|

0



exp

(

i

ω

αt

)

(22)

wherethe temperaturehasbeentakenequalto0.It isconvenienttomultiplythisexpressionby theHeavisidefunction6

θ (

t

),

sothattheFouriertransformreads

whicharelinearcombinationsof

ˆ

Gα

(

t

)

=

i

θ (

t

)



c+_α

(

t

)

cα



−

i

θ (

−

t

)



cαc+α

(

t

)



(25)

whoseFouriertransformGα

(

ω

)

canbeobtainedattemperatureT

=

0 byadditionof(23)andananalogousterm:

G_α

(

ω

)

=

∞



−∞

ˆ

G_α

(

t

)

exp

(

−

i

ω

t

)

dt

=

1

ω

−

ω

α (26)

Itfollowsfrom(26)and(21)that

GR R

(

ω

)

=



α

ψ

_α∗

(

R

)ψ

α

(

R

)

1

ω

−

ω

α (27)

whichcanbewritteninaconciseformas

GR R

(

ω

)

=

0

|

c+R 1

ω

−

H

/

h

¯

cR

|

0

 +

0

|

cR 1

ω

−

H

/

h

¯

c + R

|

0



(28)

8.1. Green’sfunctionsandlocaldensityofstates

Thelocaldensityofstates(12)canbeobtainedfrom(27)ifthefrequencyhasacomplexvalue

ω

−

i

η

.Wedefine

G−_{R R}

(

ω

)

=

GR R

(

ω

−

i

η

)

=



α

ψ

_α∗

(

R

)ψ

α

(

R

)

1

ω

−

ω

α

−

i

η

=



α

ψ

_α∗

(

R

)ψ

α

(

R

)

ω

−

ω

α

+

i

η

(

ω

−

ω

α

)

2

+

η

2 (29)

AfunctionG+_{R R}

(

ω

)

canbedefinedinthesameway,replacing

−

i

η

by

+

i

η

Inthelimit

η

→

0(29)implies

ImG−_{R R}

(

ω

)

=

2

π



α

ψ

_α∗

(

R

)ψ

α

(

R

)δ(

ω

−

ω

α

)

(30) Thisformulaisofparticularinterestfor R

=

R.Then,itdefinesthelocaldensityofstates gR

(

ω

),

alreadyintroducedby (12),whichisaspecialformof(30)whentheeigenfunctionsoftheHamiltoniancanbecharacterizedbyawavevector k. Thus gR

(

ω

)

=

2

π

ImG − R,R

(

ω

)

(31)

8.2. Green’sfunctionsinapurecrystal

Relation(31)showsthatGreen’sfunctionshaveaphysicalsignificance.Buthowcanthey becalculated?Howcanthey beusedtoderiveFriedeloscillationsaroundanimpurity?Thefirstthingtodoistocalculatetheminthepuremetal.Then theeigenstates

α

are Blochfunctionscharacterized bytheir wavevector k. TheGreen’s functionG0_{R R}

(

ω

)

=

G0

(

r

,

ω

)

only

dependsonr

=

R

−

R,and(29)reads(omittingthetilde)

G−₀

(

R

−

R

,

ω

)

= (

1

/

N

)



k

1

ω

−

ω

k

−

i

η

exp

[

ik

(

R

−

R

)

]

(32)

where N is thenumberof atoms(or unit cells, becausewe assume one atom perunit cell). The factor 1/N arises from thenormalizationconditionofthewave functions,andensuresthat theGreen’sfunctionhasafinitevalue foraninfinite sample, since the summationis over N vectors. Assuming spherical symmetry ofthe function

ω

k, one obtains inthree

dimensions G−₀

(

r

,

ω

)

=

2

π

a 3 r



_{sin kr}

ω

−

ωk

−

i

η

k dk

wherea3isthevolumeoftheunitcell.Atlongdistancer,thesineoscillatessorapidlythattheonlyimportantcontribution comesfromk veryclosetok

(

ω

),

theinversefunctionof

ω

(

k

).

Itfollowsthat

G−₀

(

r

,

ω

)



2

π

2_k

₍

_ω

₎

α

r

[

cos k

(

ω

)

r

+

i sin k

(

ω

)

r

]

(33)

where

α

=

d

ω

/dk.

8.3. Green’sfunctionsandscattering

AlinkbetweenGreen’sfunctionsandthescatteringproblemmaybeobtainedasfollows.Foranincidentwave



_k0

(

R

)

=

exp(ik

·

R

)

andascatteringpotential VR,theoutgoingwaveis[34]:



k

(

R

)

= 

k0

(

R

)

−

1 4

π



R G+₀

(

R

−

R

,

ω

)

VR



k

(

R

)

(34)

Thisformulaholdsinthetightbindingapproximationwithoneorbitalpersite.Amoregeneralformulahasbeengiven forinstancebyBlandin[34].

Anotherproperty ofGreen’sfunctionsisthat theydeterminethe wavefunction atalllattice sites R when particlesof energyh

¯

ω

areconstantlycreatedatR

=

0.IndeedtheSchrödingerequationwrites



_R

H

R R

(

R

)

− ¯

h

ω

(

R

)

=

α

δ

0R

(0),

where

α

isaconstant.Inversionofthisrelationyields,using(28),7

(

R

)

=

G0

(

R

,

ω

)(

0

)

(35)

Equation (34) can be solved by iteration. At first order,



k

(

R

)

can be replaced by



_k0

(

R

).

From the iteration series

emergestheT-matrixTR R

(

ω

)

definedbytherelation

G

(

R

,

R

,

ω

)

=

G0

(

R

−

R

,

ω

)

+



R

G0

(

R

−

R

,

ω

)

TR,R

(

ω

)

G0

(

R

−

R

,

ω

)

(36)

ThisformulayieldstheGreen’sfunctioninthepresenceofanimpurity.TheT-matrixisusuallyshortranged,sothat,at longdistance,(13)canbederivedfrom(36),(33)and(31).

8.4. Long-distancebehavioroftheoscillationsinarealcrystal(withanon-sphericalFermisurface)

AlthoughtheGreenfunctions,thechargedensity,andthelocaldensityofstatescanbeinprinciplecalculatedfrom(36), explicit formulaeas(13)haveonlybeenderived assuming that theelectronenergyh

¯

ω

k dependsonlyon

|

k

|

,whichis a

completelyunphysicalassumption.TheGreen’sfunctionsofareal,perfectcrystalhavebeendiscussedbyBlandin[34],on the basis ofan apparently unpublished thesisof Laura Roth [35]. However, adetailedcalculation was published by Roth et al.afewyearslater[36].

In thesimplestcases,thelong-distancebehavior isdescribedina purecrystalbya formulaanalogousto(33),i.e.the Green’sfunctiondecaysascos(K r

−

ϕ

)/

r inanygivendirection,butK dependsonthedirectionof r,andofcoursealsoon

ω

. Moreprecisely, K is thevalue ofk atthe pointswherethe normaltothesurface

ω

(

k

)

=

ω

is parallelto r.Ofcourse thereareatleasttwosuchpoints, k and

−

k.Buttheremaybemorepointsifthesurface

ω

(

k

)

=

ω

isnotconvex.Inthat casetheGreen’sfunctiondecaysasalinearcombinationoffunctionsoftheformcos(K r

−

ϕ

)/

r,withdifferentvaluesof K foragivendirectionr

/

r.Fig. 8showsacasewithtwovectorsK1 andK2.

In practice,the long-distancebehavior isnotextremelyimportant,sincethemain physicaleffects resultfromthefirst oscillations. However, it is oftenconvenient to write analytic formulaeas (13)and(14),and itis good toevaluate their validity.Forinstance,(13)isoftencorrect,exceptthatk

(



)

shouldbereplacedbyadifferentvaluek

(



,

r

/

r

)

ineachdirection. Knowing theGreen’sfunction ofthepure crystal,one candeducethelocaldensityofstatesaround animpurity from

(31)and(36).Anintegrationthengivesthechargedensity.DetailscanbefoundinarticlesbyGautier[37] andbyRothet al.[36].SinceseveralpropagationvectorsK1,K2...maybepresent,dependingonthedirectionR

/

R,Friedeloscillationsmay bemuchlessregularthanimpliedby(14),andobservedbySTMinCu(111)andingraphene[38].

9. Kondoeffect:phaseshiftsandquasiparticleFermiliquidtheory

Followingtheideas andtheoriesdevelopedbyJacquesFriedelandcollaboratorstodescribelocalimpuritiesinmetallic alloyswiththeintroductionoftheconceptofvirtualboundstatesandphaseshiftsexperiencedbythewavefunctionofthe conductionelectronwhenscatteredofftheimpurity,P.W.AndersonintroducedthefollowingmodelHamiltonian[39],HAnd

HAnd

=

Hd

+

Hc

+

V (37)

Fig. 8. Ifthesurfaceω(k)=ωisnotconvex,itmaybenormalto r intwopointsk1andk2(ormore).Inthecaseshownbythefigure,theGreen’sfunction G0(r,ω)decaysatlongdistancer as[A1cos(k1·r− φ1)+A2cos(k2·r− φ2)]/r,whereA1,φ1,A2,φ2dependonωandr/r.

with Hd

=



σ

ε

0d†σdσ

+

Un↑n_↓ Hc

=



k,σ

εk

σck†σckσ V

=



k,σ

(

tσc_k†_σdσ

+

h

.

c

.)

where Hd describesthelocalizedelectrons inthe impuritysitewithspin S

=

1/2 interacting amongthemselvesthrough the repulsiveCoulomb interaction, U ; Hc describesthe conductionelectrons with spin s

=

1/2 in the metallic host, and

V the hybridizationbetweenthe localizedandconduction electrons.nσ

=

d†σ dσ is the numberoflocalized electrons for

spin

σ

,

ε

0theirenergy(inthepresenceofamagneticfield B,

ε

0shouldbereplacedby

ε

σ

=

ε

0

+

σ

g

μ

BB

/2 resulting

from Zeemansplitting),andtσ thehybridizationmatrixelementbetween

|

k

σ



and

|

σ



states,assumedtobek-independent.At equilibrium, theenergies oftheelectrons inthemetal aredistributedaccordingto theFermi–Dirac distributionfunction: nF

(

ε

kσ

)

= [

exp

[(

ε

kσ

−

μ

)/

kBT

)

]

+

1

]

−1,where

μ

isthechemicalpotentialinthemetal(

μ

istakenastheoriginofenergy inthefollowing).

Initsoriginalformorconsidering itsfurtherextensions,thismodelprovidesthebasistodescribecorrelatedmetalsor superconductorswiththefundamentalissueabouttheexistenceofaMottmetal–insulatortransitioninducedbycorrelations

[40],aswell asheavy-fermion systems[41] inwhichinsteadofhavingoneisolated impurity,onehasaperiodic arrayof “impurities”(anomalousrare-earthoractinideatoms),withthepossibleonsetofquantumphasetransitions8_andnon-Fermi liquidbehavior observed inthevicinity ofa quantum criticalpoint.This modelhasexperienced a resurgence ofinterest theselasttwodecades,sinceitisbelievedtobetheappropriatemodeltodescribethequantumdot[42–45],inwhichthe centralregionofthedotisassimilatedtotheimpurity,andthetwoleads–leftandright–constitutethetwopartsofthe metallicreservoir. Interestingly,thequantumdotcan bedriveninnon-equilibriumconditionswhenforinstanceadcbias voltageV appliedbetweenthetwoleadssetsadifferencebetweenthechemicalpotentialsofeachlead.

IntheabsenceofCoulombinteraction(U

=

0),theHamiltonianissolvableandleadstotheformationofFriedel’svirtual boundstatewiththedensityofstatesoflocalizedelectronsofspin

σ

, Adσ

(

ω

)

givenby

Adσ

(

ω

)

=

1

π



σ

(

ω

−

ε

0

)

2

+ 

σ2 (38)

where



σ

=

π



k

|

tσ

|

2

δ(

ε

0

−



kσ

)

(equalto

π

|

tσ

|

2

ρ

0σ inthelimitwhentheconductionelectronbandisofinfinitewidth, leading to a constant densityof states forspin

σ

givenby

ρ

0σ ). The expression (38)corresponds to a Lorentzian peak centeredattheenergy

ε

0 oftheimpuritylevel,ofwidth



σ resultingfromthevirtualexcursionortransferofthelocalized electron into the conduction metallic host given by Fermi’s golden rule. This peak is precisely the virtual bound state introducedbyJacquesFriedel.Itiseasytocheckthatthephaseshiftat

ω

=

0,

δ(0)

experiencedbythewavefunctionofthe conductionelectronwhenscatteredofftheimpurity,isdirectlyrelatedtothenumberofelectronsofspin

σ

intheimpurity site,ndσ

=

0

−∞Adσ

(

ω

)

d

ω

accordingto

δ

σ

(

0

)

=

π

ndσ (39)

corresponding to Friedel’s sumruleasoutlined in Section 3.3. Thisresultreflects the completescreening ofthe positive chargeoftheimpuritybythenegativechargeoftheconductionelectronssurroundingtheimpuritysite.

Theoppositelimit(V

=

0)correspondstotheatomiclimit.Thetotalnumberofelectrons,nd intheimpuritysitevaries withtheenergyoftheimpurity level,

ε

0 fromnd

=

2 (doubly-occupiedsite)when

ε

0

= −

U ,tond

=

0 (emptysite)when

ε

0

=

0.Animportantcaseistheparticle-holesymmetriclimitcorresponding to

ε

0

= −

U

/2;

Inthislimit,thetotaldensity ofstatesoflocalizedelectronshastwopeaks,locatedat

ω

=

ε

0 and

ω

=

ε

0

+

U ,symmetrictoeachother withrespectto

ω

=

0,leadingtond

=

1 (singly-occupiedsite).

ThequestionthenistounderstandtheeffectoftheCoulombinteractions,U onthisvirtualboundstatepicture. Remark-ably,inthepresenceofinteractions,thephaseshiftattheFermilevel,

δ(0),

isrelatedtothenumberofelectronsofspin

σ

intheimpuritysite,ndσ ,byexactlythesameabove-mentionedFriedel’ssumruleintheabsenceofCoulombinteraction– seeEq.(39).Theproofwas givenbyLangreth[46]makinguseofgeneralWardidentities.Friedel’ssumruleiscompletely generalwhatevertheoccupancyoftheimpuritylevelandthestrengthoftheinteractionsare.

AnimportantstepfurtherwastakenbySchriefferandWolff[47],whoshowedthatintheso-calledlocalmomentlimit

|

ε

0

|

 

and

|

ε

0

+

U

|

 

(nd



1), the Anderson model maps at small energy into the Kondo model defined by the

Hamiltonian, HK(withinascatteringpotentialterm,whichisomitted)

HK

=

Hd

+

1 2J



S

(

0

)

·



k,k,σ,σ c†_kσ

σ



σ σckσ (40) where J

= |

tσ

|

2

1 εd

−

1 εd+U



,



S

(0)

=

1₂



_σ,σd†σ

σ



σ σdσ isthespinoftheimpurity,and

σ



arethePaulimatrices. J is pos-itive inthe localmoment regime consideredandcorresponds to antiferromagneticcoupling betweenthetwo spins. This modelgivesrisetotheKondoeffectatlowtemperature,correspondingtothedynamicalscreeningoftheimpurityspinby theconductionelectronspin.J.Kondo[48]developedaperturbationtheoryin J forthismodel.Heobtainedalogarithmic increase intheresistivitywithdecreasingT, explainingthe experimentalobservations.However thecalculationsalsolead tounphysicaldivergenceoftheresistivityatlowtemperature,meaningthattheperturbationtheoryin J ceasestobevalid

when T goeslower belowtheKondotemperature,9 thecharacteristiclow-energyscaleoftheKondomodel.BelowTK,the

perturbation theory in J breaks down andthe system crossesover to anotherregime whereone has to use alternative non-perturbativemethods.What happensbelow TK wasfirstforeseenby P.W.Andersonby devisingaperturbative renor-malizationgroupmethodthathenamedasPoorMan’sScaling[50],whichconsistsinperturbativelyeliminatingexcitations to the edges of thenoninteracting band. This methodindicated that, astemperature isdecreased, the effectivecoupling betweenthespinsoftheimpurityandoftheconductionbandelectron, Jeff,increasesto

+∞

,leadingtotheformationofa singletstatebetweenthespinoftheimpurityandthespinoftheconductionelectronsintheimpuritysite.Thisargument waspursuedinamorerigorouswaybyK.G.Wilson[49]usingtheNumericalRenormalizationGroup(NRG)method.Wilson was abletoshowthat whentheenergyscalegoestozero,thedistributionofeigenstates becomessimilar towhatwould prevailif J weregoing to

+∞

.Therefore,evenifthebare J issmall, thereisasmoothcrossoverfromtheweakcoupling tothestrongcouplingregime, J

= +∞

asT goesbelowTK.TherealbreakthroughmadebyNRGcalculationsistodescribe thewholecrossoverfromtheweakcouplingregimetothestrongcouplingregimewhenT orB decreases.

SoonafterWilson’spaper,itwas realized[15]by PhilippeNozièresthatthe“lowtemperature-endofthestory”canbe describedasalocalFermiliquid.LetusremindthattheconceptofFermiliquidtheoryintroducedbyLandauin1957–1959 isbasedonthe assumptionthatthesystemexhibitsan adiabaticcontinuitywhentheinteractionsare turnedon adiabat-ically, andthen thereisa one-to-one correspondencebetweenthesingle-particle excitationsofthegasofnoninteracting particles (i.e. the bare electron)andthe single-particle excitations ofthe gas ofinteracting particles atsufficiently short timescales(definingthe“quasiparticle”).Thegasofinteractingparticlesisthendescribedasasystemofweakly-interacting “quasiparticles”. Nozières formulated a local Fermi-liquid theory for the quasiparticles and expressed it in terms ofthe phase shiftacquired bythe quasiparticlewhen scatteredoffthe Kondo singlet,enlargingFriedel’s phaseshift conceptto thecaseofcorrelatedimpurity.Usuallythephaseshiftdependsonlyonthekineticenergy

ε

ofthequasiparticle,buthere inthespiritoftheLandautheoryofFermiliquid,Nozièresconsideredtheadditionaldependenceofthephaseshiftonthe distributionfunction

δ

nσ

(

ε



)

oftheotherquasiparticleswithwhichthequasiparticleinteracts:

δ

σ

(

ε

)

= δ

0

(

ε

)

+



εσ

ϕ

σ,σ

(

ε

,

ε



)δ

nσ

(

ε



)

(41)

Since in the strongcoupling regime (low T and B), only thestates inthe vicinity of theFermi surface matter, one can expandthequantities

δ

0

(

ε

)

and

ϕ

σ σ

(

ε

,

ε



)

aroundtheFermienergy

δ

0

(

ε

)

= δ

0

+

αε

+ β

ε

2

+ . . .

(42)

ϕ

σ,σ

(

ε

,

ε



)

=

ϕ

σ,σ

+ ψ

σ,σ

(

ε

+

ε



)

+ . . .

(43)

9 _{InKondo’swork,T}

Kisidentifiedasthetemperatureatwhich,intheperturbationtheoryin J ,thethird-ordertermofthetransmissionequalsthe

viewedasphenomenologicalFermi-liquid parameters.Theycorrespondrespectivelytoenergy-dependentelasticscattering andlocalinteractionbetweenthequasiparticlesresultingfromthepolarizationoftheKondosingletcomplex.Theycanbe relatedtothe zero-temperature impurityspin susceptibility,

χ

s,theinverseof whichdefinestheKondo temperature, TK, thecharacteristiclow-energy scaleoftheKondo model.Theirexactvalues canbe extractedfromtheresultsobtainedby WilsonusingNumericalRenormalized Groupmethod.Indeed,byresortingsimplephysicalargumentsasthefactthat the KondosingularityistiedtotheFermienergy,itcanbeshownthat

α

and

ϕ

areconnectedthroughtherelation

α

−

2

ρ

0

ϕ

a

=

0 (45)

UsingthisquasiparticleFermiliquidtheory,Nozièreswasabletoderivethelow-T and-B physicalpropertiesofthemodel. HerecoverstheanomalousWilsonratioR (thedimensionlessratioofthespinsusceptibilitytothelinearcoefficientofthe specific heat) witha value R

=

2, and predicts a quadratic dependenceof the resistivity in T and B, characteristic ofa Fermi-liquidbehavior,withthedeterminationofthecoefficientsoftheT2 _andB2_terms.10

Independently,YosidaandYamada[52]developedadiagrammaticFermi-liquidtheorybasedonperturbationcalculations in successiveorders in U .These calculationsshow that the groundstate of the Andersonmodel isa Fermi liquidin all parameterregimeswhatevertheoccupancyoftheimpuritylevelandthestrengthoftheinteractionsare.Theircalculations, firstperformedatthesecondorderinU ,reproducethemainresultsobtainedbythequasiparticleFermiliquidtheory.By usingWardidentities,theywereabletoshowthattheresultsholdevenuptoinfiniteorderinU .Thistypeoftheoryisat theoriginofrenormalizedperturbationtheory[53,41]asreviewedinGeorges’contribution[10].

BothofthesequasiparticleanddiagrammaticFermi-liquidtheoriesprovedtobeextremelyuseful.Theycouldbeapplied tovariousextensionsoftheKondoorAndersonmodel.Theyhavethegreatadvantagetobeapplicabletoout-of-equilibrium situations, as for instance when a finite bias voltage is applied between the left and right leads of a quantum dot as mentionedatthebeginningofthissection.ThisisquiteremarkablegiventhatthealternativemethodstotreattheKondo modelorAndersonmodelatintermediateorstrongU –basedoneitherNumericalGroupRenormalization,orBetheAnsatz, orconformalfieldtheory–arenoteasilytransposabletothesituationofnon-equilibrium.

ThissectionhighlightstheimportanceofFriedel’sideasforthestudyofcorrelatedimpuritiesasdescribedbytheKondo model or the Anderson impurity model. It turns out that Friedel’s sum rule about the phase shift plays a key role in the description ofthese relatedscatteringproblems, triggering the development ofFermi-liquid theoriesto describe the low-temperatureregimeofthesemodels.Thisfieldhasexperiencedaresurgenceofinterestduringtheselasttwodecades withtherevivaloftheKondoeffectinthecontextofmesoscopicphysics.

10. Conclusion

TheexampleofFriedeloscillationsshowshowaphenomenoncanbederivedfromvariousmethods. Oscillationsappearinthreedifferentproperties:

i)theelectronicdensity(orcharge density)aroundanimpurity,givenindimension3byformula(14).Indimension2, thefactor1/r3 isreplacedby1/r2 andindimension1,itisreplacedby1/r.Grapheneisanexceptionalcase[21,22];

ii)thelocaldensityofchargearoundanimpurity,givenindimension3byformula(13).Indimension2,thefactor1/r2 isreplacedby 1/r.Thisquantity isparticularlyimportantattheFermienergy,sinceit isrelatedtomeasurablequantities suchasthetunnelmicrograph;

iii)theone-electronGreenfunctionofthepurecrystal,givenindimension3by(33).

Themethods usedbyJacquesFriedelwere notsopreciseasmoderntechniques.Hewas attheoriginofmanyfruitful ideas, butrarelytried toexpresstheminan elaborated form.He didencouragehiscollaboratorstodoso, andmuch has beendoneinthisdirectionbyAndréBlandin.AgreatmeritofJacquesFriedelistohavebeenmoreconcernedbythefuture ofhismanycollaboratorsthanbyhisown.

References

[1]H.Jahn,E.Teller,Stabilityofpolyatomicmoleculesindegenerateelectronicstates.I.Orbitaldegeneracy,Proc.R.Soc.Lond.A161(1937)220. [2]I.B.Bersuker,TheJahn–TellerEffect,CambridgeUniversityPress,2006.

10 _{Inaveryinterestingrecentpaper,Moraetal.}

_[51]

_{succeedsinextendingthequasiparticleFermiliquidtheorytoanyregimeoftheAndersonmodel}

withtheintroductionof4(insteadof2)Fermi-liquidparametersexpressedasafunctionofthespinandchargesusceptibilities(andderivatives)which theyagaindeterminefromtheBetheAnsatzandWilson’sNumericalRenormalizationGroupmethods.