Solid StateCommunications,Vol.17,pp. 1241—1247,1975. PergamonPress. Printedin GreatBritain

DYNAMICAL SUSCEPTIBILITYOF INTERMEDIATE VALENCE SYSTEMS

C.A. Balseiro* andA. Làpez

CentroAtómico Bariloche,ComisiónNacionalde EnergiaAtómica,Instituto de FisicaDr. JoseA. Balseiro,

UniversidadNacionalde cuyo,Bariloche,Argentina(Received27May 1975 by S. Lundqvist)

The transversesusceptibilityof magneticsystemsdescribedby theAndersonmodeliscalculatedevaluatingtheselfenergyfor thetwo particleGreenfunction.Theresultsare appliedto semiconductorsandmetals.In thefirstcasea shift in the resonancefrequencyis obtainedaswell asa continuumof excitationscorrespondingto thepromotionof electronsfrom thelocal-izedstateto the conductionbandstates.In themetallic casea Korringarelationfor therelaxationtimein themagneticlimit is obtained.Therealandimaginarypartsof the selfenergyare given asfunctionsof the distancefrom thelocalizedlevel to the Fermilevel.

1. INTRODUCTION AND METHOD OF 4ffl and4f~ ionic like configurations,of thebandCALCULATION like conductionelectronstatesandhybridization

matrix elementsbetweenthesetwo kind of states.5’6THE MIXTURE of two configurationsm the groundstateof anumberof rare-earth(RE) andtransitionmetal compoundshasbeenconjectured.’3In thecase In thisnote we studythedynamicalspin suscepti-of the REcompoundsthesesubstancesarecharacter- biity for asmallgapmagneticsemiconductorandforizedby asmall energygapbetweenthe

4ffl and4fn_l a metalwithin thecontextof theAndersonmodel7

5d’ configurations.Theexperimentalconsequences in the limit of infinite correlation.of this admixtureare discussedin reference4 andthey include intermediatelattice parameters,saturating Obviously thephysicalpictureimpliedby thislow temperaturestaticsusceptibilitiesandanomalous modelis over-simplifiedin that it doesnot correctlyspecificheats.Recentexperimentsin X-ray photo- describethe complexintra atomicinteractionsthatemissionspectroscopy2haveclearly showntheco- determinethemagneticpropertiesof the4ffl ions.existenceof two configurationsin themetallicphases Furthermoretheoneelectronhybridizationtermisof severalRE chalcogenidesandchalcogenbased probablyastrongsimplification of the morerealisticcompounds.Thereareindicationsthat in the semi- Hamiltonianwhich would follow from a first principlesconductingphasesthis configurationadmixtureis calculation.Nonetheless,we feelthat theimportantalsopresent.2 featuresof the dynamicalsusceptibilityof real systems

do appearalreadyin this simplecalculation.The theoreticalsituationis notcompletelyclari-

fiedbut thereare reasonsto believethat a complete We usetheHamiltonian for the systemin amag-treatmentof thesesystemsshouldincludeasbasic neticfield H:ingredientsa correctdescriptionof thehighly correlated .~,, — +

— f~ +unane+~ekackacko

* Fellowof ConsejoNacionalde Investigaciones ~- + ~.

+ E ~V~c~& + i.c.Cientificasy Teciucas,Argentma. ko

1241

1242 DYNAMICAL SUSCEPTIBIUTYOF INTERMEDIATE VALENCE SYSTEMS Vol. 17,No. 10

wheref~(f0) is a creation(annthilatioñ)operatorfor a ([v, r]~>— 2ir ((Ø~~)>a localizedelectronof energyE0 = — abLfH (a = ±); ~(w) = . (4)u is the Coulombrepulsionbetweentwo localizedelec-trons,n0 =ff0 is thenumberoperator;c~0(ck~)is a In thestrongcorrelationlimit (u ~+00) we havecreation(annihilation)operatorfor an electronof en- ([~~r]) = — ~ V~(~~f~>— ~ V,~(c~f~>.(5)ergy ~ = — ap8H. Vk is thehybridizationmatrix k

element. To calculatethe meanvaluesin equation(5) weusethe equationsof motion for ((ck~,f~>)and((ft , cZt>),

Wehavecalculatedthe dynamicaltransverseim- which leadusto the oneparticleGreenfunctionpurity susceptibilityx(w) which isgiven in termsof ((f0 , f~)),with a = C, l~.For this function we usethetheretardedanticommutatorGreenfunctionG(w) = form givenby Hewson.

9((ft~f~,f~ft>)= ((ak ~J))

1 °~ / w’ \ Im G(w’ + iO) It would appearmorelogical to usehere the onex(w) = ; J dw’ tanh ~ ~ — s.,’ + jO particleGreenfunctionof reference8(b), butsincewe

haveno evidencethat this latterform gives a betterwhereIm standsfor imaginarypart. descriptionof themagneticimpurity,we preferthe

simpler form of reference9.If oneneglectsinteractionsbetweenspinslocal-

izedat differentsitesthe totaldynamicalsusceptibility On theotherhandwe have:for a concentratedsystemis givenby cx(w)wherec is = ~ VkVk’((f~ckt,f~ckt))theconcentrationof magneticions. kk’

We follow the equationof motionmethodin the + ~ V~VI~’((cxtft c~tft>>kk’form put forwardby GötzeandWolfle.8 We assumeG(w) tobe of theform — ~ VkV,~((f~ck~,c~ft>)

ha(Se)

G(w) = — (1) — ~ V,~’V~’ ((c~tf4,f~ca’t)>. (6)

2irw—w1—~(w) ha’

whereS~is thez-componentof the impurity spin, TheGreenfunctionsin thefirst line of (6) give a con-(...)isa statisticalaverageandWf =

2J1fH is theZee- tribution of fourth order whichwe neglect.To evaluate

mansplitting. The equationof motionfor G(w) can the retardedGreenfunctionsin the secondline webe written in theform usea decouplingof thecorrelationfunctionswhich

leadsto thefollowing expressionof theretardedtwo1 ~ + 1 ~ a]+) particleGreenfunction in termsof theoneparticleG(w) = — ____ ____

27r w — — Wf ~2ir ones:

— ((~+~))1 (2) ((f~ca~,Ca’~ft))~=

W —

with+ , + 2 ~° ftw

1)+f(~2)—2f(w1)f(w2)= [a~,H’] = aH —H a — J d~1f dw2w + w1 — w2 +10

whereH’ is thehybridizationHaniiltonian. (7)x Irn((ft,fj

1))L,+Io Im((ck~,c~’~))L,+IoFrom equation(1) we obtainthe relation

G(w) = 1 (SZ> + 1 ~ (~)G(~). (3) A similar expressioncanbe obtainedfor the otherGreenfunction.For ((C~

0,c~’0>)we takethe unperturbedex-WWf (~)Wfpression.Thefinal form for ~ (w) turnsoutto be

In orderto recognize~ (~)to secondorderin V wereplaceG(w) in ther.h.s.of(3) by G°(w),the unper-turbedGreenfunction.This allows us thento identify

Vol. 17, No. 10 DYNAMICAL SUSCEPTIBIUTYOFINTERMEDIATE VALENCE SYSTEMS 1243

(1 —ni>~(w) = <~2>( ~ VkI2[( E?)2+r2( )It )(f Im~I’+ (‘ E~\ 1

[2 2irkT\ _i~—~)J)— ~ ~ (I —ni) 2_____________ I f(~)( E~)J\1

a (eat—E~)2+F(w)[ t —E~)(f(eat)_Im~[_+~ 1 i~ )j

+ E V,~2(1 — n~>$ dw1 f(.~i) +f(e~)— 2f(w i)f(ea~) F(w)/ir

k W + C~— + 10 (w~—E~)2+ r2(~)

+ ~ lVaI2(1 nt> f dw i)+at)2~’~~t)k -~ 1 C~—~ + ~at +10 (w1 —E~)

2+ F2(w)) (8)

where Forthe electronparamagneticresonanceexperimentsI’(w) = Im ~ lV~2 oneis interestedin thepole of x(w)which lies close

a w — to w,~.Iti perturbationtheory isgiven (to secondorderin V) by

= E0 + Re ~ I v~12 +

k WEaWe find:

and~11(z)is the digammafunction.This formsallows us 2ftUto studydifferentcasesof interest. Re ~ ~(2)’(w) = ~ IV~l

2k (ek E

0)2(10)

2. SEMICONDUCTORSYSTEMS Im ~(2)~w) ~(2)” (w) = 0

2.1. Perturbationexpressionfor ~(w) in thepresence whereofa magneticfield = ~

Fora semiconductivematrix, in which theFermi Thusthepole liesatlevel CF lies betweenthe impurity level andthe bottomof theconductionband,perturbationtheorycanbe = + ~ I Va 2 2LiH 2IleffH (11)applied.Theperturbationform of ~(w) canbeobtained with k (ca E

0)2 =

from (8) by settingI’ = 0 andE~= E0. The ensuing ________

expressionfor ~(w) is Peff = + k (Ca —E0)2~

= ~> ( ~lVaI2 (1—nt>~ —E~

~IVal2 <‘flt>— [f(eat)—f(Et)Ja eat—Tt

+ ~ IV~l2 <~ ~ [f(eat)+f(Et)—2f(ea~)f(Et)]a w+Et—e~

4+iO

+ ~IV~l2 (1 —nt> [f(eat)+f(Et)—2f(eat)f(Et)] (9)

k w+eatE4+iO

1244 DYNAMICAL SUSCEPTIBIUTYOF INTERMEDIATE VALENCE SYSTEMS Vol. 17,No. 10

Thisis the sameshift inp asobtainsfrom a directcal-culationof the staticsusceptibility.10In this approxi-

E. = —300 ~K

2)0. -0000mationandforp~H<E

0 wehavean infinite relaxation3)0 00~

time.

2.2. Perturbationexpansionfor ~(o.,)atzero magneticfield

It is interestingto studyx(w) in a widerrangeof _____

frequenciesasprovidedfor instanceby theneutron _______________________________________0 500 5000 1500 2000

scatteringexperiments. ~0

In the absenceof amagneticfield the first twotermsof expression(9) vanish.If we takea densityofstatesfor theconductionbandof theform: (1) T = 3000

(2) Is 1000

N(e) = ~~_~jv~W2—(e— W)2 3)

,TW1)

we obtainin the low temperaturelimit

—n> 4 fE0+~c—W= ~ V

2 ~Re ~

or

~ — (~~— ~)]f(E0) 0 1000 1500(12)

(1—n> 2 ~3W= — ~ ~ V coth—~ (2) r200 ~‘<

(I(3 ) r 150 ~K

x {N(Eo)+w)[f(Eo)—f(Eo +w)1(13)

For ~> E0 I the imaginarypartof ~(w) is nonvanish-ing. The imaginarypartof the susceptibilityx”(w) is

forw>0 I/ p2’,~ { 0(lEoI w) 6(w)x (w) = — — — — 0(w — IE0I) 0 500 1000 ~500

2ir +q1 FIG. 1. Fouriertransformof the correlationfunction

x1w — ~(2)~(~,)]2 + E~(2)”(w)]2J tanh for impuritiesin a semiconductorasfunction of fre-quency.(a) T= 100°K.W = 10

40K,I’ = 300°K.(b)(14) E

0 = —400°K,W= l0~°K,F= 300°K.(c)Eo=

where0(x) is the step functionand 400°K,W= 1040K, T= 50°K.

n 4V2 2IE0( ~ E0 (21E01)1q = ~ (~[i + — ~ (15) 2.3. Non-perturbationexpansionfor ~(w) atzerofie~

Figure 1 givesaplot of theFourier transformof the In this sectionwe discussthe form (8) for ~(w)correlationfunction(at, 0>(w) whichgivesthe cross for thecaseof a semiconductor,whenthe Fermilevelsectionfor neutronscattering,for differentvaluesof liesbetweenthelocalizedstateandthebottom of thetheparametersE0, F, W andkT.The delta function at conductionband.Thedensityof statesof the localized

= 0 is not indicated. level is givenby

Vol. 17, No. 10 DYNAMICAL SUSCEPTIBILITYOF INTERMEDIATE VALENCE SYSTEMS 1245

~ (1—n)~(w—Eo) w<0

1 +a

PL(W) = (1—n> F(w) (1—n) i~(w) 0<w<2W (16)ir (w—E~)2+F2(w) 1 —(4V2/W2)}(w—E

1)2+i~2(w)

0 w>2W

where4V2

E0 =E0+-~-(E0—W+V’2IE0IW)

600

a = — W~~E~2— W2 + [1(]

4V2F E~’—WE

0—(4V2/W) 4V2 400

=1 _(

4V2/W2)E0w~E0)(I) (1(1= 300 •0

2(1= 100 )(

L1(w) = 4V2 ~/W2— (w — W)2. 200 (3(1= 10 ‘II

In the low temperaturelimit it follows that(3)

4V2 11500 2000

= —~-~~—>j’do.,1 pL(wl) 0 500 1000

~2w (J(wi + w — W\2 FIG. 2. Realandimaginarypartsof the selfenergyx [-j~- — Re w ) — 1 function for a metallic host, as functions of freq,uency

forE0 = — 500°K,F = 300°K.Thecurve for ~ (w)

— (w -~ — o~))~ (17) correspondsto T = 300°K.

V2

~ (w) = — ~ $ dwj {[f(w1) +f(w1 + w)

—2f(wj)f(wi +w)]N(w1 +w)(0)

+(w ~~~w)}pL(w1)

V2 (1(1 300~I(

(2) 1 100 ~I’= _ir~fdwi F(w,wl)pL(wl). 1000(3(1 301(

Analyzingtheseexpressionsonefinds

500= E~2’(wE”)[1 +a]’ +O[V4] (18)and -i irV2(1—n)“I,,~ ~w) = ~(2)”(wEfl)E1 +a] — ________

2Wt F(w, w

1) A(w1) do.,1 ~ -sooo -500X ~ 1 —(4V

2/W2)(w1—E1)

2+~2(w,)~0

FIG. 3. Imaginary part of the self-energy function for(19) a metallichost,asfunctionofE

0, for threedifferent

In ~‘(w) theterm of order V4 is irrelevantsince temperatures.

is alwaysgreaterthanthis term.Forthe imagi-

narypart,however,thesecondtermin (19)maybetheimportantonewhen I(2)”(w) vanishes. Forvery smallw (w ((kT= 1fl3) we havethen

1246 DYNAMICAL SUSCEPTIBILITYOF INTERMEDIATE VALENCE SYSTEMS Vol. 17,No. 10

4,rV2(1—n> 00

= 2 kT Jdwi —N(wl)pL(wl)00

4irV2(1 —n> 1 ~f _______________= (Sf> 1 — (4V2/ W2) kTj dw

1 ~—N(w1) ( —E~)2+ i~2(w

1) (20)

if kT<eF,wehave: <

=

(1) 4 ~V2~2 e~F [1 + exei(_x)(l + x)]

whereei(x) is theexponentialintegralandx = j3E1. Ifbesidesx~ l,this reduces to ~

(2) 1: 100K

1 /1 — \ (AIJ

2\211P’o2

= ~ ~~~I_r__II~LI e~F (21) (0) 1:

w s~\ w) ~E1) 2

Onemay assume4V2/W 250°Kand W 1040Kand

so this mechanismwould give a relaxationtimeof theorderof: 1

Tr = 4x lO~(E)2 e~FtT 2

For1E1 I = 400°KandT= 20°Kthis would give a line-

widthL~&H 4 G which is accesseibleto measurementin an EPR experiment.It shouldbenotedthat theT

2dependenceis a consequencefor themodified Lorent-

500 100 15% 2000

ziandensityof statespL(w) assumedfor the localized [~

oneparticlestate. FIG. 4. Fouriertransformof correlationfunction forimpuritiesin a metal,asfunctionof frequency.ForE

0= — l000°K,F = 200°K.The bumpnearE0corres-

3. METALLIC SYSTEMS pondingto thepromotionof electronsto the Fermilevel,tendsto disappearwhenthetemperaturerises

Forametallic systemwe are mterestedm a theo- and alsowhenE0 approachesthe Fermienergy.reticaldescnptionthatwould allow usto discussthebehaviourof themagneticion whenE0 crossestheFermi level. In this sectionwe shalltakeCF as thezeroof energy. ~2”(w) = _irkTy OV] = —irkT’y[N(0)J]

2

In the limit /lfH =°~kT andassuminga constantden- (sity of statesN(0)for theconductionbandwe get,from where‘y isof the orderone andJ2 = V4/(E~+ f~2)is

equation (8) 00 a modified Schrieffer—Wolff11 exchange parameter.

= ~ $dwi (_~~L~’~ .

(Sz) ao.)iJ Ourexpressionfor therelaxationtime coincidesr/ir with the Korringalike expressionobtainedby Gotze

X 2 2~ andWölfle8 for thes—dmodel.(w—Eo) +F

As it is knownin themetalliccase the conditionPfH~ In general,theexpressionfor ~“(w) canbewrittenkT~ 1’ canbe achievedandin this rangewe havefor asE

0 <0

Vol. 17, No. 10 DYNAMICAL SUSCEPTIBILITYOFINTERMEDIATE VALENCE SYSTEMS 1247

— —N 0 ~2 (1 — n> ~ as a function of E0, the distance from the localizedkw) — ( ) co a ~2) energylevel to the Fermilevel. In realsystemsthis

/ — parametermay be governedby pressure5’6orby chang-

x Im ‘I’ — + —f-- (1 —

1E0 w ing the host composition, as is probablythecasefor2 2irT F JJ Ce impurities in a La—Th matrix.

12Figure 4 gives the

spin—spin correlation function for the metallic case,for two different temperatures.

whereIm representsthe imaginarypart and‘1’(z) is thediagammafunction. In closing,we shouldmentionthat thedynamical

susceptibility of magnetic impurities in metals has beenTherealpart turnsoutto be: calculatedby otherauthors,but thesetheoriesdo not

1 (1 — cover thecaseswe haveconsideredheresincethey refer= — — N(0)V2 $ do.,

1 [I — 2f(w1)] either to thes—dmodel,13 or arebasedon theHartree—

iT Fockapproximationfor the AndersonHamiltonian,

F 1 - w1 + w and are thus inapplicable in the highcorrelationlimit.

14x(E)2+F2Re [w(~+~ 2irT )

Acknowledgements— The authorsare gratefultoB. Alascio and E. Martinez for manyusefuldis-

In Fig. 2 we give ~‘(w) andE”(w) asfunction of w for cussions. Interesting conversations with M.C.G.differentvaluesof theparameters.Figure 3 shows~“(0) Passeggiand S. Oseroffare also acknowledged.

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