Contribution to the theory of Kondo resistivity in magnetic compounds

J. Phvs. Chem. Solids

Pergamon Press 1970. Vol. 3 I, pp. 2245-2250.

CONTRIBUTION RESISTIVITY

Printed in Great Britain.

TO THE THEORY OF KONDO IN MAGNETIC COMPOUNDS

F. E. MARANZANA Philips Research Laboratories, N.V. Philips’ Gloeilampenfabrieken,

Eindhoven, Netherlands

(Received 15 October 1969; in revisedfarm 8 December 1969) Abstract-The theory of Kondo exchange scattering scattering system consists of a periodic spin structure. 1. PRODUCTION

pages contain a variation of Kondo’s original calculation[ l] of the resistivity of dilute magnetic alloys. The present calculation applies rather to systems in which the magnetic ions make up a large percentage of the total number of atoms present in the system. Moreover the magnetic ions are arranged periodically so that the results to be obtained are not to be applied to alloys containing large concentrations of magnetic impurities. One should rather compare the present results to the experimental evidence available on the intermetallic compound CeAl, [2]. The experimental situation in the case of CeAl, is briefly summarized as follows: the resistivity vs. temperature curve has a maximum approximately at the ordering temperature (as determined by magnetic susceptibility measurements), then it decreases for increasing temperatures. it has a clear minimum, and it increases further. The minimum has been thought to be indicative of the presence of the Kondo effect [2]. Further, the partial substitution of Th for Ce influences the resistivity curve: with increasing Th percentages the maximum and the minimum become vaguer, and they finally disappear. For this Th concentration the resistivity versus temperature curve rises rapidly up to the transition temperature, it shows a plateau. and then starts rising again. The effects of the Th substjtution have been THE FOLLOWING

has been extended

to the case in which the

interpreted[2] by suggesting that the density of states of the system’s conduction band is diminished by the addition of Th. The results obtained on systems containing Th in place of Ce have been put forward as further support of the presence of Kondo anomalies, in view of the relevance of the density of states in the formulae applying to the Kondo effect in dilute alloys. The formula to be derived at present reproduces rather nicely the experimental evidence; this is not really surmising, if one recalls that the essential features to be obtained (a density-of-states dependent minimum in the resistivity vs. temperature curve), are already present in the dilute alloy calculation. It is on the other hand pleasant that a Kondolike formula can be constructed for a system containing many magnetic ions, so that the possibility of the presence of the Kondo effect in a concentrated system gains further credibility. There exist in fact theoretical studies on concentrated magnetic alloys [3] which predict that when the concentration of the magnetic ions is larger approximately than 1 per cent and there exists no long-range magnetic order, the minimum disappears. The point of the present calculation is that this is not true when the magnetic ion system consists of a periodic spin lattice, in which long-range order exists. Abrikosov[4] has considered the case of a concentrated ahoy in which long-range order exists. The results of this author apply to a modei which differs essentially from the

2246

F. E. MARANZANA

present one. Abrikosov places in fact the magnetic ions on a number of randomly selected lattice sites. The present calculation is imitated step by step from Kondo’s [ 11. It will consequently be described very briefly. 2. DESCRIPTION OF THE MODEL AND OF THE CALCULATION N magnetic ions are arranged on a simple cubic lattice in a crystal of volume V. The spins of the ions interact with one another through a Heisenberg interaction. The Heisenberg interaction is taken into account in the following by means of a motecuiar field: each spin is supposed to move in the presence of this temperature dependent field, and it moves otherwise independently of the other spins. Conventional molecular field theory is adopted, so that. for instance, the average length of each spin is determined by the usual implicit equation containing on the right-hand side a Brillouin function whose argument is again a function of the average length of the spin. The conduction of the system is accounted for by a Fermi sea of electrons residing in a band extending from -El to Ez with a constant density of states rz(EF). The Fermi level is taken as the zero of energy so that EF = 0. The interaction of the ‘conduction system’ and the ‘magnetic system’ amounts to the usual s-f interaction: &p’

=

_

$

C

on the basis of the fact that they probably contribute terms of the same order of magnitude and temperature dependence as the terms retained, and these can therefore be considered representative of the genera1 behaviour. (Notice that the procedure followed does not exclude, in second order, processes of the type k? + k”$ --+ k’? and the typical Kondolike processes). The transition probability terms proportional to J2 and J3 are given by the formulae: W,(n’,a)

-~fi(E*--E~,)/(a’j~In)/”C-2)

(3) The ket Jcr) represents a state which is labelled by the k vectors and the spins CTof al1 conand by the magnetic duction electrons, quantum numbers of all magnetic ions. Substitution off 1) into (2) gives:

eWL”R,,

kk'n

xexp

[i(k--k’)(R,-Z&l]. (4)

(Strictly speaking, one should derive the RKKY part of the Heisenberg interaction from the s-finteraction.) The calculation runs along the following lines: the transition probability up to third order in J is calculated. Only elastic scattering is considered, so that processes of the type kf + k’$ are neglected,

The transition probability term proportional to J3 is different for the two orientations of the spin of the electron being scattered from k to k’. Moreover for either spin direction four different intermediate states are possible involving the scattering of either one or two electrons in typical Kondo-like processes. Substitution of (1 f into (3) gives:

KONDO

RESISTIVITY

IN MAGNETIC

These factors are averaged the assumption that no internal the magnetic impurities are so ‘do not see one another’. In consideration the process of ceeds as follows: in the case of

wmmmn X

exp [i((k-

+

(k’-K’)R,qE k

-+ -5

(

xexp

k’)&+

(K’- k)Rm

2 mlm,exp{i[(k-k’)R1+

3 C m,[S(S+l)-m,(m,-+l)] > lm,.V [i{(k-k’)R,+

In

(z

+c.c. + T

( ;ci” )c --

by Kondo on field is present: dilute that they the case under averaging pro-

(k’-k)R,]}

(8)

for example,

(P-k)&}]

1

2247

COMPOUNDS

(5)

mm,m,

mEmnexp

{i[(k-k’)R,

+ykr-k)R.]l)

= z (w%)

xexp{i[(k-k’)R,+

(k’-k)R,]}

xexp{i[(k-k’)R,+

(k’-k)R,]}

h?ln,W’

xexp X

[i{(k-_‘)R1+(K’-_k)R,+(k’-_‘)R.)]

fib

-Ek + &v + i’f/

1

+c.c.

+

ml[S(S+

2

= N(m2) +(m)” X {-N++erms

l)-m,(m,Tl)]

containinga(k-k’)}

lm,k”

Xexp [i((k-k’)&+ X

s

k”

-Ek+Ept+iq

= N( (m - ( m) ) “) + terms containing 8.

(k’-k)R,}]

+c.c.

1.

The upper sign applies to the scattering of an eIectron with spin up, the lower sign applies to the scattering of an electron with spin down. The transition probability contains factors ot the type C mlmrm, exp (i[ (k- k’)R!

Im

+ (k”-k)R,+ z m[StS+

1) -m,(m,-

Xexp{i[(k-k’)R,+

(k’-k”)R,]}

(6)

I)1 (k’-k)R,]}

(7)

where mz is the azimuthal quantum number of the I-th spin, (S (S + 1)) 1/Zis the length of the spins.

In this calculation the periodicity of the magnetic system plays an essential role, because it allows one to split the term of equation (4) into a k-independent term and a contribution of the electron’s self-energy which is of no interest for the scattering. It is interesting to consider the effects of two types of deviation from periodicity, namely the case of a concentrated alloy in which the magnetic ions are placed at random in the matrix, and the case of a periodic spin structure in which some of the magnetic ions are replaced by non-magnetic ones. In the case of the concentrated alloy (treated in ref. 141 by a different technique) the positions R, of the magnetic ions are distributed at random. This has as a consequence that z exp [AS,] is equal to a delta function plus other terms dependent upon k. The form of these terms depends upon the distribution

2248

F. E. MARANZANA

of the magnetic ions and it is extremely difficult to compute in practice. In the case of the replacement of a small fraction of magnetic ions by non-magnetic ones, the results obtained above are not significantly modified. yiV terms appear besides N( (m - (m) ) “) but they do not contain N, and can be neglected as long as y < 1. When the above described averages are performed on !Vl and W, one obtains:

= $S(Ek-

Ekr)

The relaxation time is obtained in the conventional way by summing over k’: A spin independent relaxation time is obtained by summing the relaxation times proper to eiectrons with spin up and down. This relaxation time is introduced in the well-known formula for the conductivity and the result for the resistivity is finally obtained:

p

=

37r l -5-fie’jqvF2

“Jz((m-(m))2)

where vF is the Fermi velocity.

3. DISCUSSION

Inspection of equation (9) shows the following: the factor outside the square bracket corresponds to usual spin-disorder scattering in the elastic approximation. Its temperature dependence is such that the resistivity increases rapidly up to the transition temperature and thenceforth remains constant. In the case under consideration this factor is corrected by the square bracket. This bracket contains a temperature-independent correction and the Kondo logarithmic term. The former correction is temperature independent owing to the rough way in which the sums over k and k’ have been performed to obtain the relaxation time and the conductivity. This correction is anyway some four times smaller than the second one, because E, is presumably of the same order of magnitude as E2 while kT is approximately IO4 times smaller. The sign of the former correction depends upon the relative magnitude of E, and &. The latter correction is positive if J is negative and it represents the essential part of the correction to spin-disorder scattering. Below the transition temperature the temperature dependence of this correction is overridden by the T dependence of spin-disorder resistivity, but it appears on the contrary above the transition temperature; in this temperature region the total resistivity then decreases with increasing temperature. The maximum at the transition temperature shown by the experimental resistivity is thus reproduced. The minimum above the transition temperature is obtained, as usual, by adding to this resistivity an increasing term due to phonon scattering. It is also evident from equation (‘-3 that the density of states n(EF) plays a decisive role in producing the extrema in the resistivity curve. The extrema are washed out if the density of states decreases. Then normal spindisorder scattering remains. In essence equation (9) describes the folfow ing state of affairs: below the transition temperature an internal field acts on each magnetic ion’s spin and it prevents it from nipping. If

2249

KONDO RESISTIVITY IN MAGNETIC COMPOUNDS spin flip is forbidden the Kondo scattering processes do not take place and the In T term is quenched. At the transition temperature however the internal field drops to zero. Then above this temperature each magnetic ion is free to flip its spin under the influence of the s-f interaction and this gives rise to the In T term. Figure 1 shows a plot of resistivity vs. temperature as derived numerically from equation (9). The calculation of ( (WI- ( m) ) “) has been performed by assuming that the magnetic ions have total angular momentum equal to 4, like cerium ions. Each ion has six energy levels labelled by the value of the azimuthal quantum number m. The levels are degenerate in the absence of fields. In the present case c Y

e 3n’l ---

v 52

2h$Ni++

4 t

however the ions are under the influence of a cubic crystal field and the internal molecular field proportional to zgl’. where z is the number of nearest neighbouring magnetic ions, g is the Land& factor, and r is the Heisenberg coupling constant. If the internal field is absent, as in the paramagnetic region, the crystal field splits the sextet into a doublet and a quartet (5). It has been assumed for the calculation that the doublet is the ground state, the quartet being separated from the doublet by a distance of 80°K. This assumption seems to be contradicted by preliminary experimental indications (6) that suggest that the quartet is the ground state. However, the reversal of the sign of the crystal field would not change the general behaviour of spin-disorder resistivity. Also, these considerations on the internal structure of the magnetic ions have minor relevance with respect to the influence of the Kondo terms. The presence of the internal field splits the doublet and the quartet further. The magnetic transition temperature is such (= 10°K) that the internal field can be looked upon as a small correction to the crystal field. Within this model the relevant quantity ((m-(m))“>

(IO)

= (m2>-(m>2

is obtained through the equation (ma) = Spur (m,“X) where

1,2

(11)

H is the sum of the internal field plus crystal field hamiltonions m, is the z component of the spin operator S of one of the magnetic ions. The following parameters have been used in the numerical calculations: In IE1/E21 = 1; E, = 5 eV; zgl’ = 7°K; the curves of Fig. 1 give the behaviour of the resistivity vs. temperature. The variable on the ordinate axis is p divided by the numerical factors appearing on the right-hand side of equation (9). The different curves have been obtained by attributing to the parameter 2Jn ( EF) /N the values reported in the figure. zgl’(m)

Fig. 1. Resistivity vs. temperature curve obtained from equation (9) in the text. Each curve is labelled by the correspondingvalue of the parameter2J, (4) /N.

n=

2250

F. E. MARANZANA

A comparison of Fig. 1 with the experimental curves (2) shows that the agreement is quite satisfactory. In this respect however a word of caution is in order. The theory assumes that the density of states can be changed without modifying the magnetic ion system. In practice the substitution of thorium for cerium breaks up the translational symmetry of the magnetic ion system, making the application of the theory doubtful at least at large Th concentrations. Also, the fact that the number of Ce ions decreases changes the transition temperature experimentally. The theory does not take this effect into account. Finally of course one should be suspicious of a calculation based on perturbation theory and the molecular field theory.

author is grateful to Dr. H. J. van Daal and Dr. K. H. J. Buschow for a number of fruitful discussions.

Acknowledgements-The

REFERENCES KONDO J., Prog. theor. Phys. 3237 (1964). i: BUSCHOW K. H. J. and VAN DAAL H. J.. Phys. Rev. f&t. 23,408 (1969). 3. HARRISON R. J. and KLEIN M. W., Phys. Rev. 154. 540 (1967). 4. ABRIKOSOV A. A., Physics 2,61 (1965). 5. LEA K. R.. LEASK M. J. M.. and WOLF W. P.. J. 6. Phys. Chem. Solids23, 1381 (1962). WALLACE W. E., CRAIG R. S., THOMPSON A.. DEENADAS C., DIXON M., AOYAGI M., and MARZOUK M., Proc. du Collogue International sur les Elements des Terres Rares. Paris-Grenoble 5-10 May (1969); to be published. See however HILL R. W. and MACHADO da SILVA J. M., Phys. Lett. 3OA, 13 (1969).