High temperature paramagnon modifications in nearly magnetic metals

Solid State Communications, Vol. 11, pp. 683—687, 1972. Pergamon Press.

Printed in Great Britain

HIGH TEMPERATURE PARAMAGNON MODIFICATIONS IN NEARLY MAGNETIC METALS M.T. Béal-Monod Laboratoire de Physique des Solides,* Université de Paris-Sud, Centre d’Orsay, 91

—

Orsay, France

(Received 26 May 1972 by P.G. de Gennes)

The standard paramagnon structure in nearly magnetic metals is shown to be drastically modified at high temperatures. Qualitative consequences for the electron paramagnon temperature dependence of the electrical resistivity are considered. Behavior like that exhibited in aPu resistivity might be expected.

THE NOTION of enhanced spin fluctuations or ‘paramagnons’ in nearly magnetic metals is well known. 1 Their effects upon several properties have been extensively studied these last few years. In all these works, the paramagnons were defined and studied in the limit of vanishing ternperature. Although they are not really modes, they can be considered as elementary excitations in the sense that their existence is consistent with low temperature expansions, essentially in a range 0 ~ T << T~ 1,where T31 measures the characteristic temperature of the paramagnons of the order of 1/S times the Fermi temperature TF, S being the Stoner enhancement.

We believe, as will be shown in the following, that the very structure of the pararnagnon is considerably modified at high temperature and that this modification implies a drastic change in the electron—paramagnon resistivity. What happens where a finite mean free path is present? This has been partially answered by Fulde and Luther ~ who have studied the effect of impurities upon spin fluctuations. But this was still a low temperature treatment, and only the consequences for the specific heat were studied in their paper. We think that it would also be important to know how the transport properties would behave at high temperature in presence of a mean free path, either due to impurities or to phonons or

Then several questions can be considered:

any other renormalization. The phonon contribution indeed could be most important: for the resistivity, for instance, one can think of a for physical argument analogous to Rice’s one4

What happens at higher temperatures, say when T>> T~1(but 2still T << Tv)? Recently have generalized at Kaiser and Doniach T>> ~ the Mills and Lederer2 result for the resistivity due by electron—paramagnonscattering. They showed that the low temperature T2 resistivity switches to an increasing T contribution when

the Rondo problem; Rice pointed out that when the resistivity relaxation time is strongly energy dependent as in the Rondo problem, then Matthiessen’s rule breaks down, one cannot simply

T>> T~ 1. Their mathematical resistivity calculation is correct; however the main damping effect is probably missed in their treatment since they still used at high temperature the same low tern-

add phonon and impurity resistivities. In the present case, when the Debye temperature and the spin fluctuation temperature are of the same order of magnitude a similar remark may be~ernphasized

perature structure for the paramagnori propagator.

since the electron pararnagnon scattering depends very much on the involved energies. Therefore here too, phonon scattering and paramagnon

____________

*

Laboratoire associé au C.N.R.S. 683

684

HIGH TEMPERATURE PARAMAGNON MODIFICATIONS

Vol. 11, No. 5

scattering should not be separated but treated together.

A

Before calculating the high temperature electron—paramagnon resistivity, one must first of all study the high temperature structure of the paramagnon propagator with or without mean fee path present. A priori and according to the definition of elementary excitations, the paramagnon propagator should break down at high temperature. We will show in the following that that is actually what happens. Therefore one can reasonably expect that the magnitude of the scattering potential formed by pararnagnons will strongly decrease as will the electron—paramagnon resistivity,

x~

-~

FIG. 1. Renormalized vertex and particle—hole bubble taking into account a finite mean free path [formulas (4) and (7) of the text]. 1 -, ~‘c or I finite). One needs to calculate the electron—hole bubble modified by the renormalized vertex A which takes into account a finite mean free path (see Fig. 1). A is calculated by standard methods: ~

We just recall the usual interacting fermion gas Hamiltonian. 1

A (c~, ~

1

=

C~z

\

+

27rr

2

j —1

~=H~ +1n~n~

(1)

describing free electrons kinetic energy J{~plus the interacting part, I being the interaction, ‘~‘~ the electron spin densities spin up or down. This 1 to the paramagnon propagator leads within R.P.A. formula: x°(q, c~) x(q, ~ = 1-I ~ (q, ~) (2)

~

(4)

______________

We work with the temperature dependent Matsubara frequencies wn

÷i\

27TT(n

=

2) c~

~sgn c~n = W,~ -f-

with momentum q and frequency Cc. Within the usual low temperature, infinite mean free path theory, 1 the bare electron—hole bubble ~°(q, w) has a small q and c~j/V~qexpansion:

~

(q, w)

N 0

(

1

i7T

C,.)

.~. —

2

0(q2))

(3)

V~q

where V~is the Fermi velocity of the electrons, N0 the density of states at the Fermi level. In that case, for q = 0, w = 0, the static spin susceptibility ~(0,0) is strongly enhanced when IN0 is close to 1, the Stoner enhancement S being defined as (1—IN0 )_l

~(k

q)~

~(k) V~’

~—

.

~=

Vpqz

~

(5)

one assumes as usual that the paramagnon momenturn q is much smaller than the Fermi momentum kF of the electrons. The ~, c~_~are fermion frequencies; so as usual, the paramagnon frequency w~is a boson one. By analytical continuation the actual paramagnon frequency w is equal to ic~ (see Fig. 1). One easily finds K’

(&i,,.

—

w,~)

=

1

~L.__sgn

—

vF cIT

We want now to look more carefully at the behavior of x(q, oi) and first of all, of ~°(q, a) at finite temperature T ~Iat0.the Webehavior will work with a finite mean free path I = V~Tincluded in order to get a general formula which will

—

[1

—

9(w~.~

There

>,~.°is

V~q 1-sgn a~ (6)

)J arctg a~

.--

obtained by N

reduces to (3) for (T -~ 0, 1 -‘ ~.o), to Fulde and Luther’s result for (T 0, 1 finite), and to the formula we wish to present here for (T finite,

~°(q, w,~)= ljm A~

T

~

(~—i~) (~— ic~~

A (wa,

(

dz 0 ~

—

T wn --l

—

W7~-,~L)d~

.

q)

•42

-

2

~

A

(7)

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HIGH TEMPERATURE PARAMAGNON MODIFICATIONS

The constant A is introduced to prevent spurious divergences. The calculation is straight forward algebra. then finds, for q <
685

V

\

0 I Vp q 2 3 + T2 _______ + r’ (V~q) a)~+ Tt 3 (c~ T’)3

~ (q, w)

VF q

arctg x°(q, w~)=

(11)

N 0

~

—

..~±.)

V~.q

but T is low so that besides (10) (b) one 1/Tiis has thelarge, condition

T

V~q

1

VpqT

arctg

1 ÷—

a)

(12)

T

3

(8) Let us look at some asymptotic forms of the expression (8).

This was Then (11) the is simplified case studied andby oneFulde gets: and Luther

x

°(q

—

Small frequencies, low temperature, pure case: 1 — =

T

0 q. 0, ‘

—~—-

V

-~

0

(9)

1q /li-IT — 2 VFq)

(

N

)

1+i—n —~ 2 V~q

0

using the analytical continuation ~) = iw1~. One finds back the usual expression (3) leading to the standard low temperature paramagnon propagator (2)

2 0

1st case.

~°(q,w~j=N0

N

a)/.~)

~

T

(V~q) a)~+

(V~q)2 T

(13)

With a) = iw~,(13) leads back exactly to the result obtained by Fulde and Luther3 in their low temperature spin fluctuation problem in presence of impurities. (c) 1/T is small, T is large; now besides (10), one has the condition: (14) Then (11) with (14) gives

2 (V~q) 2 (V~q)2

xo~N0

2nd case.

3Ta))~

Large ‘effective’ frequency. We mean by ‘effective’ frequency the conbination (w~+ lPr): q -+ 0,

V

0

.

(10)

1

(15) At the limit 1/T= 0, (15) is reduced to o N

x

—-~

0

-a---)

(vF

q\2

N0 I Vp q\2

(16)

--~-~-)

The condition (10) may be verified in several ways, either because one of the two quantities w~and lIT are large compared to V~q,or because both are large. This leads to three sub cases:

(16) is a small quantity and real. Its analytic continuation is actually identical to the first term of the large frequency, zero temperature expansion of the Lindhard function. Comparing (16) to (3) one sees how different are the behaviors

(a) Both lIT and T are large; this means a high temperature regime and a short mean free path due itself either to many impurities present or to a large phonon contributicin, or any other renormalization factor. Then (8) with (10) gives

of x° at high and low temperatures. Therefore putting both expressions into (2), the structure of the paramagnon will be completely changed. As was physically expected, ~° and then the paramagnon itself ~ breaks down at high temperatures. This is of major importance then for the calculation of the resistivity.

686

HIGH TEMPERATURE PARAMAGNON MODIFICATIONS

Let us qualitatively examine some further consequences of these different cases. First of all the general formula (8) of course includes these different asymptotic cases: in principle (8) combined with (2) should give the long wavelength paramagnon behavior at any temperature with or without impurities. This would be useful to numerically compute any temperature dependent property of the enhanced material. Now in order to simplify things and restricting ourself to high temperatures, say T> T~1,(11) will typically reflect the behavior of >~.°. Speaking first of pure materials (no impurity present) where 1/T will be due to phonon contribution, then depending on the position of the Debye temperature °D with respect to the spin fluctuation temperature T31, then (13) or (15) will be more useful. According to both formulas ~ and then ~ (formula 2) will be small quantities, smaller and smaller for increasing temperatures. Since ~
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7 and Pu 7 for which the observed in aPu Fe they are enhanced general present feeling is5 that spin fluctuations systems; while for instance the last behavior is found in some cases quoted by Kaiser and Doniach2 as well as in metallic V 203 ~8

//

FIG. 2. Two possible temperature dependences

of the resistivity on the temperature respective magnitudes of the depending spin fluctuation and of the Debye temperature.

To conclude, we think that the paramagnon propagator is drastically modified when the ternperature increases: it breaks down at high ternperature following the general behavior of elementary excitations. Therefore the electron paramagnon resistivity will be strongly influenced by such a change at high temperature. Furthermore we point out that at high temperature the resistivities due to phonons and paramagnons cannot simply be added but must be treated as a whole. The resulting temperature dependence of the total resistivity is expected to be then strongly modified. It could, depending on the respective positions of the spin fluctuation temperature and of the Debye one, for instance exhibit a maximum at a temperature close to the spin fluctuation temperature followed by a decrease and then a pure phonon part increasing again as is observed in aPu.

Acknowledgements gratefultotothe Professor Friedel who drew—myI am attention high temperature resistivity problem. I enjoyed useful discussions with Pr. K. Maki and several comments from Drs. B. Coqblin and T.M. Rice. J.

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HIGH TEMPERATURE PARAMAGNON MODIFICATIONS

687

REFERENCES 1.

See for instance BERK N.F. and SCHRIEFFER J.R., Phys. Rev. Lert. 17, 433 (1966); DONIACH S. and ENGELSBERG S., Phys. Rev. Lett. 17, 750 (1966).

2.

KAISER A.B. and DONIACH S., mt. J. Magnetism, 1, 11(1970); MILLS D.L. and LEDERER P., i. Phys. Chem. Solids, 27, 1805 (1966); SCHINDLER A.I. and RICE M.J., Phys. Rev., 164, 759 (1967).

3.

FULDE P. and LUTHER A., Phys. Rev. 170, 570 (1968).

4.

RICE M.J., Phys. Rev. Lett. 23, 1108 (1969); RICE M.J. and BUNCE 0., Phys. Rev. B2, 3833 (1970).

5.

See for instance ABRIKOSOV A.A., GOR’KOV L.P. and DZIALOSHINSKI I.E., %!ethod of Quantum

6.

Field Theory in Statistical Physics. Prentice-Hall, Eriglewood Cliffs, N.Y. (1964). In this paper, for sake of simplicity, we have essentially considered that T’ was due to phonons or impurities. But, as was indicated at the beginning of the paper, at high temperature even in absence of impurities and if the phonon contribution can be neglected, all the Green’s functions will still contain a T~ factor which will then measure the damping time of the quasi particles; it contains in a phenomenological way the renormalization which has to be taken into account when the very notion of quasi particles breaks down as the temperature increases. In that case, T will depend itself on q and a)~ and should be calculated self consistently. It is not the object of the present paper to claim what will be the behavior of T’ for very large a)~ that is why the subsequent high temperature limit of the resistivity cannot be predicted exactly here. We just emphasize the fact that for increasing temperature and increasing T’, p should slow down.

7.

See for instance, MORTIMER M.J., Research Group report, Magnetic and Electronic Properties of Actinide metals. Harwell, England (1972); ARKO A.M. and BRODSKY MB., Proc. 4th mt. Conf. on Plutonium (1970), Vol. 17, p. 364, (edited by MINER W.N., Univ. Calif. Los Alamos, New Mexico, U.S.A.

8.

MCWHAN D.B. and RICE T.M., Phys. Rev. Lett. 22, 887 (1969).

On montre que la structure du paramagnon classique dans les m4taux presque magnétiques est fondamentalement modifide haute temperature. On en considère l’influence qualitative sur Ia dépendance en temperature de la résistivité. Des resistivites du type de celle du aPu pourraient étre ainsi expliquees.

a