EPR linewidth of local magnetic moments diluted in Ce intermediate-valence compounds

PHYSICAL REVIEW

B

VOLUME 46, NUMBER 2 1JULY 1992-II

EPR

linewidth

of

local

magnetic

moments

diluted

in

Ce

intermediate-valence

compounds

Pablo

A.

Venegas

Departamento de Ftsica, Faculdade de Ciencias,

RESP,

Campus Bauru, R.Engenheiro Luis Coube s/n, Bauru, Sgo Paulo, Brazi

Gaston

E.

Barberis

Instituto de Fisiea, Universidade Estadual de Campinas (UNICAMP), 13081Campinas, Sito Paulo, Brazil (Received 23 December 1991)

Anomalous thermal behavior on the EPR linewidths has been observed for Gd impurities diluted in

Ce„Al

„B„(A

=La,

Y,

B

=Ir,

Os,Rh,Pd) intermediate-valence compounds. In this work we show that

the exchange interaction between the local magnetic moments and the intermediate-valence host ions has an important contribution to the relaxation rates ofthe local moments. We calculated the relaxa-tion, using the Redfield formalism and the ideas contained in the interconfigurational fluctuation model

ofHirst. We show that the exchange interaction contribution has an exponential dependence on the ex-citation energy ofthe intermediate-valence ions.

I.

INTRODUCTION

Ce intermetallic compounds have been extensively studied in the literature and many

of

them show anomalies typical

of

intermediate-valence (IV) alloys.'

By doping Ce compounds with Gd + ions, their

ESR

spectra allow us to study locally the influence

of

the intermediate-valence Ce ions. In metals, the local mo-ments

ESR

linewidth,

hH,

isexpected

to

increase linearly with the temperature (Korringa ). In contrast, in

Ce„A,

„Pd3

(A=Ag,

Y)and Ce„La& „Oszthe Gd

ESR

spectra show a nonlinear increase

of

hH

in the tempera-ture range

4.2&

T&300

K.

At low temperatures the slope d(b,

H)ldT

is smaller than in the isostructural

non-IV compounds MPd3

(M=Sc,

Y,

La), whereas at high temperatures the resonance line is strongly broadened and the slope asymptotically approaches the value mea-sured in LaPd3. The temperature dependence

of

hH

be-comes nonlinear and cannot be explained only by invok-ing the usual Korringa mechanism.

EPR

studies in metallic hosts, which show interconfigurational fluctuations, suggest that the

in-direct exchange interaction between the local magnetic impurities and the Ce

4f

electrons have an appreciable contribution to the linewidth

of

the magnetic impurities.

The Ce interconfigurational fluctuations are transferred via the Ruderman-Kittel-Kasuya-Yoshida

(RKKY)

in-teraction to the Gd site. These fluctuations generate an effective alternating magnetic field, which relaxes the Gd

spins. A similar mechanism was also used for the inter-pretation

of

the NMR relaxation rates

of

noble metals with Kondo impurities, for perturbed angular

correla-tion measurements

of

' Cein (La&

„Y„)A12

alloys, ' and

for Gd

ESR

on concentrated Van Vleck paramagnets.

"

We use this mechanism to explain the anomalous

behav-ior

of

the linewidth

of

Gd diluted in the Ceintermediate valence compounds.

The Gd linewidth can be calculated with use

of

the ideas contained in the "interconfigurational fluctuation"

model

of

Hirst.'

'

This model has been used toexplain

various physical measurements and particularly magnetic susceptibility and Mossbauer studies in intermediate-valence compounds. In their approach the expectation

value

of

the measured physical quantity (isomer shift, magnetic moment,

etc.

)iscalculated by "averaging" over

all possible

4f"

configurations. The occupation probabil-ity

of

the n configuration is usually expressed as

P„(T)

~

exp(

E„/T„)

—

with

T„=T+b„.

"

'

Here

_E„

and b,

„are

the energy and width

of

the

4f"

configuration. The quantity measured thus depends ex-ponentially on the excitation energy,

E,„,

defined as the energy required to delocalize a single electron from the

4f"

configuration (i.

e.

,

E,

„=E„E„&).

Al—though this

theoretical approach is phenomenological, it successfully explains most

of

the Europium Mossbauer spectra and susceptibility studies in mixed-valence compounds. '

'

In this work we use this approach and the Redfield

for-malism' to calculate the contribution

of

an excited

configuration

of

the host Ce ions

to

the linewidth

of

the impurity spectra. We fit the experimental data and

com-pare with other theoretical approaches.

'

II.

THE MODEL

The spin Hamiltonian describing the Ce ions in the n

configuration can be written

H=H, f+H,

,

where the crystal-field Hamiltonian for Ceis

H,

t

=

g

[8404(

J'")+8606(J'")

] I

with

04(J'")

and

06(J'")

the cubic combinations

of

Stevens' equivalent operators

of

the fourth and sixth de-gree, respectively,

84

and

86

are the crystalline-field pa-rameters

of

the fourth and sixth degree, respectively, and

J'

'_{isthe total angular momentum}

_of

_the

1th Ce ion. The

Zeeman Hamiltonian for Ceis

PABLO A. VENEGAS AND GASTON

E.

BARBERIS 46

where

_pz

is the Bohr magneton, gz is the Lande factor, and H isthe external magnetic field.

The Gd Hamiltonian can be written as

H'

_=H,

_'f+H,

'+H„.

₍₄₎

III.

THELINEWIDTH

In our metallic host the impurity

ESR

linewidth

calcu-lations must consider the energy transfer between the im-purity spin

S,

the spin

of

the host rare-earth ions (Ce), the spins

of

the conduction electrons as well as the lattice, in the presence

of

static external magnetic field, and asmall alterating field. In the present work we shall assume that

the system is in the unbottlenecked regime,

i.

e.,the con-duction electrons and the host magnetic ions are in equi-librium with the lattice. In addition we shall assume that

the transverse susceptibility associated with the impurity

at the impurity resonance frequency, coo, is much larger

than those

of

the other spin systems at the same

frequen-cy. In this limit one can consider only the transverse magnetization,

M,

of

the impurities and neglect the

in-teraction

of

the

rf

field with the magnetic host ions and the conduction electrons. In other words, the magnetic host ions and the conduction electrons are "passive dissi-pative systems.

"

On the other hand, the static magnetic field splits the Gd resonance in

2S

=7

fine-structure lines, which are well resolved at low temperatures. In the present case, forsimplicity, we assume that the fine

struc-ture is collapsed and suppose the Gd with an effective spin

S

=

—,

'.

With these assumptions the impurity

linewidth can be expressed as

EH=a+bT+EIF

(6)

where a isthe residual linewidth, b isthe usual Korringa

contribution originating from the impurity-conduction-electron exchange interaction, and EIF is the impurity linewidth due to the exchange interaction with the Ce

host ions. The last contribution arises from the

RKKY

coupling between the Ceand Gd ions, which transfers the

Cefluctuations tothe Gd site.

The

ESR

relaxation

of

an impurity due to (stable)

ex-cited configurations

of

the host ions can be calculated us-ing the Bloch-Redfield-Wangsness formalism, ' in analo-gy with impurity relaxation in Van-Vleck compounds.

"

Then the relaxation rate can be written as

Here

_H,

'f

and

H,

' are defined in a similar form as

H,

f and

H„with

the obvious substitution

of

J'"

by the spin

of

the Gd,

S.

The Gd impurity interacts with the neighbor-ing host Ceions via exchange interaction:

Zp

H,

„=

—

(g

—

1)8

g

S

J'",

1=1

where

8

is the exchange intergal between the two ions and zo is the number

of

Ceneighbor ions

of

each

impuri-ty.

For

simplicity, however, we have considered only the first Ceneighbors. We calculate the relaxation rate

of

the impurity

if

it does not create a perturbation

of

the Ce

host Hamiltonian.

bH=

[()((gJ

—

1)] [k

(coo)+k,

(coo)+2k„(0)],

(7) 2A'

where co& is the impurity resonance frequency.

kqq(co)(q

=x,

y,z) isthe Fourier transform

of

the spectral functions defined as

k

=zoK

(co) with Kqq(co) defined as

Kqq(cu)=

f

(5J

(t)5J )e

dt

(10)

In the present case, the Ce interconfigurational fluctua-tion frequency, Ace, is much greater than the impurity resonance frequency, coo. Physically for our model it means that for a very short correlation time, the spectral density

of

the fluctuating field is

"white"

to the frequen-cies far above the resonance frequency. So that if we view the problem from a rotating frame (that is one where the magnetization is rotating at the Larmor fre-quency), the

x,

y, and z directions are equivalent and the effects

of

the external magnetic field in the fluctuation

spectra can be neglected. In this casewe have that

K,

„(a)0)

=—

K„„(0);

_K

_(coo)

—

=

_{K (0)}

and for cubic symmetry

K„„(0)

=

Ky

(0)

=K„(0)

.

(12)

With these assumptions the calculation

of

spectral functions (see

Ref.

11)leads to the following expression

forthe linewidth:

,

[~(gJ

—

1)]'zoF(o)

_y

_p.

I(«I

_J,

I«&

I',

(14) $2

where

_Ina)

and

E„are

the eigenvectors and eigenvalues

of

H,

F(0)

a frequency distribution function due to the finite width

of

the Ce

4f"

states, and

_p„

=

_exp(

_E„

_/T)/Z,

with

Z

—

=

_g„,

exp(

E„,

/T).

—

Ce ions ffuctuate between the

4f

and the

_4f

' configurations.

If

we assume the

4f

configuration as the ground state (with

J,

=0),

the contribution

of

the excited

4f

configuration to the impurity linewidth can be writ-ten as

—_{F. /T}

ex

IF where

2

=

(

2~/fi

)_[

8(

_g1

—

1)_]

zoF (0)

I(1

a

I

J,

I1

a

&I

0 0

k

(~)=

f

g

5J'"'(t)

g

5J'"'

e'

'dt.

qq ₂ q q

k=1 k=1

Here

_{( )}

denotes a thermal average and 5Jq")the ffuctua-tion in the _q component

of

the total angular momentum

of

Cedue to the interconfigurational fluctuations, defined as

5J(k)

—

J(k)

₍

J(k)

₎

q q q

In the absence

of

pair correlations,

i.

e., terms like

EPRLINE%'IDTH OFLOCAL MAGNETIC MOMENTS

DILUTED.

.

.

913

and

E,

_„

is the excitation energy required to delocalize a

single electron from the Ce

_4f

'configuration.

I800

IV. RESULTSAND DISCUSSIQN I400 Using the expression obtained in

Eq.

(15)we can

calcu-late the total linewidth

of

the Gd resonance, given by

Eq.

(7). Figures 1 and 2 plot the calculated linewidth as a function

of

the temperature for selected cases

of

Ce La) „Osz and

Ce„Y)

„Pd3

powered samples. The

residual linewidth, which is sample dependent, is not

con-sidered in the plots (see figure captions).

To

fit the data

we adjust the parameters b, A, and

E,„,

where b represents the

"apparent"

Korringa parameter. This is done so because we are not considering the exchange nar-rowing effect

of

the fine structure

of

Gd at low

T.

' As stated previously, here we assume an effective spin

S=

—,'

for the magnetic ion. Rigorously, the linewidth calcula-tions have to consider the contribution

of

the fine

struc-ture associated with the spin

S

=

—',

of Gd.

However, the exchange narrowing effect collapses the spectra in a sin-gle line, which can be treated by an

S

=

—,' effective spin.

This collapse, at low T,ismore evident forbig values

of

b and the resonance iswell described by this efFective spin.

For

the cases where b is small and the fine-structure

con-tribution, not considered here, is important we use an ap-parent Korringa parameter, associated to the

S

=

—,'

effective spin. An example

of

the last case isthe Gd di-luted in CePd3 resonance. Comparing our theoretical

re-sults with the experimental data we can see that in most

of

the cases we got an excellent fit. The same calculations have been made for Ce(Pd&

„Ag„)3

and Ce(Pd&

„Rh„)3

with similar results. According to our model, at low

T

the main contribution tothe linewidth isoriginated in the usual Korringa mechanism. At high temperature the population

of

the excited Ce

4f

configuration is

in-500

—IOOO CI IJJ

600

200

50 I00 150 200

TEMPERATURE (K)

250 300 350

FIG.

2. Same asFig.1forCe„Y& „Pd3.

creased, and the exponential contribution given by hi+ is the most important. The parameters used to fit the ex-perimental data are shown in Table

I.

Looking at the values

of

the fitting parameters the

effect

of

the Ce concentration seems clear. The increase in the excitation energy value when the Ce concentration

isreduced agree with the interconfigurational fluctuation model.' ' The model predicts an increase

of

the

excita-tion energy when we cross from the intermediate valence

to the magnetic regime. This is the case, for example, when we go from CeOsz to LaOsz. On the other hand, the A value depends on the strength

of

the Cefluctuation

spectra and we can expect an increase with the Ce con-centration. The small value

of

b obtained for high Ce

concentrations, at low T, agree with the prediction

of

the "hybridization hole" model for Gd diluted in the CePd3 powered sample and with that obtained by us ' for the monocrystalline spectra

of

the sample compound. This

result agrees also with that obtained by Hirst, for low T, but not with the result obtained in

Ref.

19,which pre-dicts a higher value for b. However, the thermal

behav-ior

of

the linewidth obtained by the latter authors agree,

at least qualitatively, with that obtained here.

300

X

I-C5

& 300

TABLE

I.

Obtained values ofthe parameters b, a, and

_E,

_„

that fitEq.(6)for diluted Gd. The last column shows the exper-imental value of

E,

„published in Ref. 23, tobetaken as a com-parison.

I00

b [G/IC] A [G]

E,

„(theor.

)

E,

„(expt.

)

50 I00 I50 200

TEMPERATURE (K)

250 350

FIG.

1. Temperature dependence of the linewidth for different values of the concentration

x

in Ce La& „Os2. The

full lines are the plot ofthe theoretical expression [Eq. (6)].The

fitting parameters can be found in Table I(a). The residual

linewidths are not realisitic, for the data was shifted in order to avoid superimposed curves.

1

0.95

0.9

0.85

0.8

1

0.91

0.8

1.27 1.31

3.28

45

5.18

0.54

0.69

0.76

(a) Ce„La, „Os2 9547

9580 7364 3112

94

672 678 849 1132 1681

(b) Ce„Y, „Pd3

1543 576

1170 609

642 517

914 PABLO A. VENEGAS AND GASTON

E.

BARBERIS 46

It

is important to observe that in contrast to the hy-bridization hole model, here we have supposed a constant density

of

states asin a normal metal. The nonlinear

con-tribution to the linewidth, according to the present mod-el, is originated in the exchange interaction

of

the mag-netic impurities with the Ce ions. Our calculations, re-garding the hybridization hole model, permit a quantita-tive description

of

the linewidth, but using a lesser num-ber

of

adjustable parameters. In the other hand, is neces-sary topoint out that for small values

of x,

where the AIF value is small in comparison with the Korringa

contribu-tion, the numerical simulation

of

AH leads to unexpected small values

of

E,

„.

However, as we can clearly seein the

Ce

La,

„Os2 case, b asymptotically approaches the

LaOs2 b value when

x

goes to

0.

Note that the values obtained for the excitation energy

are, within the experimental error, close to that founded by Sereni, Olcese, and Rizzuto. This agreement with experimental results supports this interpretation.

ACKNOWLEDGMENTS

The authors want to acknowledge the International Center

of

Theoretical Physics, Trieste, Italy, for the hos-pitality received there, where part

of

this work was per-formed. This work was partially supported by CNPQ

and

FAPESP,

Brazil.

'J.

M.Robinson, Phys. Rep. 51,1(1979).

J.

Korringa, Physica 16, 601 (1950).

H.Schaeffer and

B.

Elschner, Z.Phys. 53,109(1983).

4T. Gambke,

B.

Elschner,

R.

Kremer, and M.Schanz,

J.

Magn. Magn. Mater. 36,124(1983).

5C.

Y.

Huang and

K.

Sugawara,

J.

Low-Temp. Phys. 28, 229 (1977).

H.Alloul, Physica 86-88B,449(1977).

7D.

E.

MacLaughlin, O.Penna, and M.Lysak, Phys. Rev.B23, 1039(1981).

A. Narath, in Magnetism, edited by V. H. Suhl (Academic,

New York, 1973), p. 150.

B.

Giovaninni, P.Pincus, G.Gladstone, and A.

J.

Heeger,

J.

Phys. (Paris) Colloq. 321, C1-163 (1971).

M.Mekata, T.Uemura, and

T.

Kuriyama, Hyperfine Interact. 10,833(1981).

D. Davidov, V. Zevin,

R.

Levin, D. Shaltiel, and

K.

Ba-berschke, Phys. Rev.B15,2771(1977).

L.L.Hirst, Phys. Kondens. Mater. 11,255(1970).

~3L.L.Hirst, in Magnetism and Magnetic Materials (San Fran

cisco, 1974),Proceedings ofthe 20th Annual Conference on

Magnetism and Magnetic Materials, AIP Conf. Proc. No.24, edited byC. D.Graham, G.H.Lander, and

J. J.

Rhyne (AIP,

New York, 1975).

L.L.Hirst, in Valence Instabilities and Narrow Band

Phenom-ena, edited by

R.

D.Parks (Plenum, New York, 1977),p.3. 'sD.

K.

Wohlleben, in Proceedings ofthe International Confer

ence on Valence Fluctuations in Solids, edited by L.Falicov, W. Hanke, and M.

B.

Maple (North-Holland, Amsterdam, 1981).

'

_E.

_{R.Bauminger, D.Froindlich,}

_I.

_{Nowik, S.}

Ofer,

I.

Feiner,

and

I.

Mayer, Phys. Rev.Lett. 30,1053(1973).

B. C.Sales and D.Wohlleben, Phys. Rev. Lett. 35, 18(1975); 35,1240(1975).

' _{A.G.Redfield, Adv. Magn. Res. 1,1(1965); IBM}

_J.

_Res.Dev. 1, 19 (1957),see also C. P. Slichter, Principles ofMagnetic Resonance, edited by M. Cardona, P. Fulde, and H.

J.

Queisser, Springer Series in Solid State Sciences, 2nd ed.,Vol. 1(Springer, New York, 1978).

' _{M. M. Ochi and O. L. T.de Menezes, Solid State Commun.}

75,355(1990).

T.

Gambke and T.Elschner,

J.

Phys. (Paris) Colloq. 40, C5-331(1979).

'P.

A. Venegas and G.E.Barberis, Solid State Commun. 58, 447(1987).

T.

Gambke,

B.

Elschner, and L. L.Hirst, Phys. Rev. Lett. 40, 1290(1978).