Analysis of some physical properties of cerium compounds in the Anderson model

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Analysis of some physical properties of cerium compounds in the Anderson model

P. Lethuillier, C. Lacroix-Lyon-Caen

To cite this version:

P. Lethuillier, C. Lacroix-Lyon-Caen. Analysis of some physical properties of cerium com- pounds in the Anderson model. Journal de Physique, 1978, 39 (10), pp.1105-1108.

�10.1051/jphys:0197800390100110500�. �jpa-00208850�

ANALYSIS OF SOME PHYSICAL PROPERTIES

OF CERIUM COMPOUNDS IN THE ANDERSON MODEL

P. LETHUILLIER and C. LACROIX-LYON-CAEN

Laboratoire Louis-Néel, C.N.R.S., 166X, ³⁸⁰⁴² Grenoble-Cédex, ^France (Reçu ^{le 26 mai} 1978, accepté ^{le 4} juillet ¹⁹⁷⁸⁾

Résumé.

Utilisant la théorie de Lacroix-Lyon-Caen ^{et al.} qui ^ont calculé la susceptibilité magnétique ^d’un système ^Kondo de cérium dans le modèle d’Anderson

^tenant compte ^{des effets} du champ cristallin,

déterminé la valeur du paramètre d’échange 0393n(EF) ^{et la} température

de Kondo TK ^de quelques composés de cérium à partir ^{de mesures de} susceptibilité. ^On peut remarquer

que | 0393| n(EF) augmente quand ^le coefficient de chaleur spécifique électronique 03B3 décroît et que le produit 03B3.0393n(EF) varie peu. L’analyse des courbes de résistivité de

composés (nous

notamment mesuré la résistivité électrique ^de CePb3) ^{montre la} validité des calculs effectués par Cornut et Coqblin

l’Hamiltonien d’Anderson lorsque T~ TK.

Abstract.

Using ^the theory ^of Lacroix-Lyon-Caen et ^al., who have calculated the magnetic susceptibility ^of

cerium Kondo system in the Anderson model, including ^the crystal ^field effects,

we

have determined the value of the exchange parameter 0393n(EF) ^{and the Kondo} temperature TK ^of

some

cerium compounds ^from susceptibility measurements. We observe that | 0393| n(EF) ^increases

when the electronic specific heat coefficient _03B3 decreases and that the product 03B3.0393n(EF) ^varies only slightly. ^The analysis ^{of the} resistivity

^{of these} compounds (in particular,

^have measured the electrical resistivity of CePb3) ^{shows the} validity of the calculations performed by Cornut and Coqblin

with the Anderson Hamiltonian for T ~ TK.

Classification

Physics ^Anstracts

72.15Q201375.20H201375.30C

1. Introduction.

Recently, Lacroix-Lyon-Caen

et al [1] (hereafter ^called paper I) have calculated the

magnetic susceptibility ^{of a} cerium Kondo system ⁱⁿ the Anderson model including ^the crystal field, ^for temperatures T > TK. ^In particular, ^this theory explains correctly the observed reduced susceptibility

of the dilute alloys La1 _xCexAl2. ^{In order to fit} precisely ^the experimental results for concentrated

compounds, it is useful to know the sign and the order of magnitude ^{of the RKKY} type interactions : we

discuss this evaluation in section 2. In section 3, ^we apply ^the theory developed ⁱⁿ paper 1 ^{to some} cerium

compounds ^{so as to} determine the parameter rn(EF) (where ^{r is the exchange integral and n(EF) is the} density ^{of states} of the conduction band at the Fermi level for one spin direction) and the Kondo tempe-

rature. Then, ^we compare, in section 4, Tn(EF) ^with

the electronic specific heat coefficient _y. Finally, ^we

discuss the resistivity ^{curves of some of these com-} pounds in section 5 and the temperature of the abso- lute thermopower maximum in section 6.

2. Discussion of the paramagnetic ^Curie tempe-

rature.

In cerium Kondo compounds, large negative

values of the paramagnetic ^Curie temperature 0p ^are

generally measured whereas the contribution 0 PRKKY

of the RKKY type interactions to 0p ^{is weak or even} positive. However, ^we observe that the sign ^{and the}

order of magnitude ^of 0 PRKKY in ^{these compounds}

may be determined by ^the application of de Gennes’

law (which predicts that, ^{in a} series of rare earth

compounds, ^the paramagnetic ^Curie temperatures ^are proportional ^to ( g - 1)2 J(J + 1 )) ^{with the} gadoli-

nium compounds ^as reference. For instance, ⁱⁿ CeIn3,

we have measured 0p

^{48 K whereas}

Buschow et al. [2] have found in GdIn3 : Hp = - 85 K

and the application of de Gennes’ law leads in CeIn3

to opcal

^{1 K. In CeAl2, Swift et al. [3] have}

determined 0p

^{33 K and Maple [4]} has measured

Op = - 40 ^K in the dilute alloys La1 _xCexAl2 (x - 0.02), ^{where the RKKY} interactions are negli- gible. ^Thus °PRKKY

^{+ 7 K, whereas} OPcal

^{+ 2 K.}

As these RKKY type interactions are weak, ^{we shall} apply ^the theory ^of paper I, valid for isolated impuri- ties, ^to concentrated cerium systems.

3. Détermination ofrn(EF) ^and TK.

^{We shall now}

determine the parameter rn(EF) and the Kondo tem-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390100110500

1106

perature TK by ^the analysis ^of susceptibility ^results

for some cerium compounds. ^{We shall use} equations (13), (32) ^and (34) of paper ¹ valid with a crystal ^field.

In the high temperature region (d /kB T « 1), ^the

formula (13) ^of paper I, ^which gives ^the magnetic susceptibility ^without magnetic (RKKY) interactions between the cerium ions, becomes :

where

Eo being ^the position of the 4f shell relative to the Fermi _energy, Vk ^the mixing parameter of the Ander-

son Hamiltonian, ^C

0.805 the Curie constant of the free tripositive ^cerium ion, ^J

5/2 ^{and D a}

cut-off parameter. ^{In this} high temperature region,

the equation (32) ^of paper I, giving ^{the Kondo tem-}

perature is reduced to :

result similar to the expression ^of Coqblin ^and

Schrieffer [5].

3. 1 CePb 3

Lethuillier et al. [6] have fitted the

susceptibility ^{curve of this} compound, using ^the

classical expression ^{of the} magnetic susceptibility (formula ^I.1 in reference [6]) ^{with a crystal field} splitting between the doublet and the quartet A

EQ - ^ED

^{66 K and a} molecular field coeffi- cient n

Hp/C ^{= 18 e.m.u. However, the study of}

the magnetic susceptibility ^{of the} compound

La0.SCe0.5Pb3 ^{between 4 and 300 K} has made it

possible ^{to determine} precisely ^a molecular field coefficient n

6 e.m.u. for CePb3. ^{Thus, in this} compound, ^the antiferromagnetic interaction with the conduction band reduces the magnetic suscepti- bility. Using ^{the formula} (34) ^of paper I, ^{it is} possible

to obtain a good agreement between the experimental

and calculated susceptibilities (Fig. 1) with the follow-

ing ^{set of} parameters ^D

⁷⁰⁰ K, rn(EF) = - 0.07,

n = 6 e.m.u. and d = EQ - ED = ^{100 K.} Using ^the

formula (32) ^of paper I, ^{we find a} very low Kondo temperature : TK

^{2 x 10-3 K.}

3.2 CeSn3. ^-This compound ^{becomes non} magne- tic under 150 K [7]. ^The variation of its lattice _para- meter between 77 and 300 K [8] shows that it keeps

a valence close to 3. We may thus think that CeSn3

exhibits a Kondo effect. The crystal field level scheme of this compound ^{is not known but, from} figure ⁷

in reference [6], ^we may extrapolate

FIG. 1.

Reciprocal magnetic susceptibility ^of CePb3

temperature. ^The continuous

has been calculated with

equation (34) of paper I, using ^the following parameters : ^D

⁷⁰⁰ K,

The paramagnetic ^Curie temperature ^is large [9] : Hp

^{203 K so that} 0 PRKKY ^is negligible. Using ^the

formula (1), ^we obtain, ^{from the} high temperature susceptibility ^{values :} rn(EF) = - 0.42 ^{with D}

^{700 K.}

From formula (2), ^{we deduce a Kondo} temperature TK = ^{120 K which is in} good agreement ^{with the}

susceptibility ^curve (Fig. 2). ^The large magnitude ^of TK justifies ^{the use} of the formula (2) instead of the

equation (32) ^of paper I, which takes into account the crystal field effect.

FIG. 2.

Reciprocal magnetic susceptibility ^of CeAl2 (taken ^from

reference 4), CeSn3 (taken from reference [10]), CeIn3 ^and CePb3.

3. 3 CeBe 13.

^This compound ^has magnetic pro-

perties ^{similar to} CeSn3 [10] and it is possible ^{to fit}

its high temperature susceptibility ^{curve with the} parameters ^D

^{700 K and} rn(EF) = - 0.22, using

the formula (1). From formula (2), ^{we deduce a}

Kondo temperature TK

^{60 K.}

In the paper I, ^{we have determined} with D

700 K and

for La, -,,Ce,,A’2 compounds (x = 0.02). ^These para- meters explain correctly the variation of the magnetic susceptibility of CeAl2 ^{with the} temperature, ^measured by Maple [4], if we take into account the 0 PR.... ^value.

Thus, ^the concentrated system CeAl2 exhibits the

same behaviour as the dilute alloys Lal-xCexAI2.

4. Comparison ^{with the} spécifie heat results.

We have reported, ^{in table} I, the values of the electronic

specific heat coefficients _y of CePb3, CeSn3, CeBel3, CeAl2 determined by Cooper et ^al. [10]. ^We give ^also

the value of _y for CeIn3 determined from the data of Van Diepen et ^al. [11] ^{between 12 and 20 K because}

this compound ^orders antiferromagnetically ^{at 10.5 K.}

For comparison, ^we have noted the electronic specific

heat coefficients of LaX3 (X

Sn, Pb, In) compounds

measured by Bucher et al. [12]. The values of _y mea-

sured in the cerium compounds ^are notably larger

than the values determined in the lanthanum com-

pounds ^{and we} may remark (Table I) that, ^when y

decreases, 11’ 1 n(EF) increases and that the product y.Tn(EF) ^varies only slightly.

TABLE 1

Electronic specific ^heat coefficient y, exchange para- meter rn(EF) ^and product y.Fn(EF) of ^{some cerium} compounds.

5. Discussion of the resistivity ^curves.

^{We have}

measured the electrical resistivity ^of CePb3 ^between

4 and 300 K by ^{an A.C. four} probe technique. ^The

relative precision ^{of the} measurements is 10-3. Unfor-

tunately, ^the sample contained free lead and became

superconducting under 7.3 K. The resistivity ^increases

very rapidly up ^to 25 K owing ^to the increase in the

population of the excited quadruplet. ^{Between 30 and}

50 K, the increase of the resistivity ^is only ¹ % (Fig. 3)

because the logarithmical ^Kondo resistivity term,

decreasing ^{with the} temperature, nearly compensates for the increase of the spin ^{disorder and} of the lattice

resistivity ^terms.

FIG. 3.

Electrical resistivity of CePb3

temperature.

In the _paper I, ^{we have} compared ^the resistivity

curves of CeIn3 ^and CeAl2. ^{These two} compounds

have similar crystal ^field level schemes and we remark- ed that CeIn3 ^{has no} resistivity ^{minimum at low}

temperature [13] contrary ^to CeAl2 [4] ^{because it has}

a higher ^Kondo temperature. Thus, ^the theory ^of

Cornut and Coqblin [14] (who have calculated the

resistivity ^{of a} cerium Kondo system in the Anderson model including ^the crystal field) ^which predicts ^a resistivity ^{minimum at low} temperature, ^{is not valid} for CeIn3 ^{when the} temperature ^{is close to} TK.

Similarly, CeSn3 ^{has no} resistivity minimum under 300 K [15] because its Kondo temperature ^is very

high.

6. Température ^{of the absolute} thermopower ^maxi-

mum of some cerium compounds.

^{In the} theory ^of

Peschel and Fulde [16], ^{valid for} magnetic impurities

in a host metal, ^the magnetic ^{ion is} considered as a

two levels system, separated by ^an energy 4. This

theory predicts ^a thermopower ^{maximum at about}

1108

J/3. ^In satisfactory agreement with this theory,

Bucher et al. [17] ^have observed such a maximum in

PrSn3, PrCuS, PrPb3, TmSb and TmCd at about d/2,

where A is the total crystal ^field splitting.

In CePb3 [10] ^and CeIn3 [18], thermopower ^maxima

have been observed respectively ^{at 40 K and 75} K, whereas the crystal ^field splitting ^{is A}

^{100 K} (or

66 K in the classical susceptibility fit) ⁱⁿ CePb3 ^and

d

150 K in Celn3 [11], confirming ^the previous

results. We have seen that CePb3 ^{has a} very low Kondo temperature ^{while for} CeIn3, ^we have calcu- lated TK

^{1.7 K} (paper I). ^{In both cases, the Kondo} temperature ^is notably smaller than the crystal ^field splitting ^{and the} general result observed by ^Bucher

et al. remains valid. However, ^{in the case of} CeSn3

and CeBel3 which have high TK values, the thermo- power maxima occur respectively ^{at 200 K and 100 K,}

which correspond approximately ^to their Kondo tem-

peratures.

Recently Bhattacharjee ^and Coqblin [19] ^{have cal-}

culated the thermopower ^{of cerium} compounds ^{in the}

Anderson model. In the case of a two-level system,

separated by ^an energy d, ^a thermopower ^maximum

is expected ^between 4 /6 ^and A/3.

7. Conclusion.

The comparison of the calculated and experimental magnetic susceptibilities ^{of cerium}

compounds ^{allows one to determine} FN(EF) ^{and the}

Kondo temperature ; this leads to a better understand-

ing ^{of the} properties ^{of these} compounds. ^In particular,

the large negative values of the paramagnetic ^Curie

temperature appear ^{as a} measurement of the reduction of the magnetic susceptibility by ^the antiferromagnetic

interaction with the conduction band. Our qualitative analysis ^{of the} resistivity ^{curves of some cerium com-} pounds ^{shows the} validity ^{of the} theory ^{of Cornut}

and Coqblin ^for T > TK. ^Thus, the Anderson model

explains correctly ^a large ^{number of} experimental

results. Finally, ^we may remark that our susceptibility

fits lead to a crystal field level scheme larger ^{than the}

exact value. For instance, ⁱⁿ CePb3, ^our calculation leads to 4

100 K whereas from the resistivity ^mea-

surements, ^one _may ^deduce L1

60 ± ^{15 K. This} arises from the fact that our perturbation calculation is limited to fourth order in Vk.

Acknowledgments.

^{We thank Dr. B.} Coqblin ^for helpful discussions.

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