Influence of dynamic scattering on the critical transport properties of ferromagnetic metals

0038-1098/78/0701-0041 $02.00

Solid State Communications, Vol. 27, pp. 41 s.s. © Pergamon Press Ltd. 1978. Printed in Great Britain

INFLUENCE OF DYNAMIC SCATTERING ON THE CRITICAL TRANSPORT PROPERTIES OF FERROMAGNETIC METALS* J.S. Helmant and I. Balberg The Racah Institute of Physics, The Hebrew University, Jerusalem 91000, Israel

(Received 17 February 1978 by W'Low) Using the general properties of the dynamic spin-spin correlation function it is shown that the critical behavior of electrical transport coefficients will be the same as that of the magnetic energy. Hence, the temperature and magnetic field behavior of these coefficients, in the vicinity of the critical temperature, To, do not depend on whether the scattering processes are elastic or inelastic.

RECENT theoretical and experimental studies have indicated that the critical electrical resistivity of magnetic materials has the temperature [ 1-3] and magnetic field [4] dependences of the magnetic energy. Hence, in the vicinity of a critical temperature, Tc, one should expect that the relation:

dp/dT o: Cp

(1)

will be maintained. Here p is the resistivity associated with the interaction of charge carriers and critical spin fluctuations, dp/dT is its derivative with respect to the temperature and Cp io the specific heat. Most theories available thus far for p were based on the quasielastic [1-4] (or static) approximation. This amounts to the assumption that the cross section for the above interaction is independent of the possible energy transfer in the corresponding scattering processes. The justification of the above approximation was already questioned in the pioneer paper of Fisher and Langer [1]. They pointed out that while small momentum transfers may be considered as an elastic scattering process [5], this is not valid a priori for large momentum transfers which are relevant in the present case of carrier scattering by critical spin fluctuations [ I-3]. The question was raised again in relation to the critical behavior of the resistivity [6] and in relation to the critical behavior of the thermopower [7]. * This work was done within the framework of the Joint Program of Scientific Collaboration between the Israel National Council for Research and Development and the Mexican National Council for Science and Technology. One of us (J.S.H.)would like to thank Research Corporation for partially supporting this work. t Permanent address: Centro de lnvestigaci6n del Instituto Politdcnico Nacional, Ap. Postal 14.740, M~xico, D.F. Mexico.

Recently, Geldart [8] calculated the first correction to the static approximation under the initial assumption that inelastic effects are small relative to the quasielastic approximation, and concluded that dynamical scattering does not modify relation (1). This calculation involves a development in co/ka To, where co is the energy transfer during the scattering process and T c is the critical temperature. In the case of critical scattering, however, it is not obvious that this development is rapidly convergent because co/kB Tc may be of order one for large momentum transfers. In this paper we reach the same conclusion without any a priori assumption with regard to the magnitude of the inelastic effects, and avoiding the series development in co/kB Tc. Based on general properties of the dynamic spin correlation function, we show that the "dynamical" calculation leads also to equation (1). The present discussion, which is restricted to ferromagnetic metals, can be generalized for the other magnetoelectronic systems using the method of [3]. Such a generalization shows that the conclusion reached here for ferromagnetic metals applies to all other systems. Generally, the mean free time T associated with a transport coefficient in a metal is given by an integral of the form [9]: 2kF 2kF r-' = al f dqf~(q)J~(q)+aa ; dqfa(q)Ja(q) (2) 0

0

where a~ and aa are constants, k r is the effective radius of the Fermi surface andfn(q) and Jn(q) are functions of the momentum transfer q. In particular, J.(q)

-- ~

s(q, c o ) ~ c o ) " ( ~ - 1) -~ dco

(3)

where S(q, co) is the "structure factor" that depends on 41

42

TRANSPORT PROPERTIES OF FERROMAGNETIC METALS

the momentum transfer q and the energy transfer o , and [3 = 1/k s T. For the resistivity [9] as = 0 while for the thermal conductivity [9] and the thermopower [7] as¢O. The quasielastic approximation, which is given by

J, = ; S ( q , o ) d o ,

(4)

is obviously valid when/~o c ,~ 1, where o c is the characteristic energy of the system [7, 9]. In the present case of scattering by critical spin fluctuations [7] this approximation is questionable since one is interested in temperatures which are close to Te while oc ~ k B Tc. For the critical behavior discussed here the "structure factor" S(q, o ) to be considered is the dynamic spin-spin correlation function [7]:

Fq(o) = eaF ~ e-&qqlS,~lm)12a(o + el - e~).

(5)

l,r r l

Here: F = -- (1//3) In ~ e -&l

(6)

Vol. 27, No. 1

where t is the reduced temperature (T-- Tc)/Tc and a is the critical exponent of the specific heat. This is in fact the origin of the behavior described by equation (1). Below we use the term "diverging" behavior to characterize a temperature dependence of the type given in equation (9). Such a temperature dependence can be associated with a true divergence (a positive as for the Ising case) or with a cusp-like behavior (a negative but small, as for the X Y or Heisenberg cases). The method to be used in dealing with the problem at hand is to demonstrate that the little we do know about Fq(o) is enough to show that dJ1/dT and dJ3/dT are proportional to dFq [dT. Hence, while the magnitude of r -~ may be different in the "static" or "dynamic" cases, its temperature derivative will have the same divergence as Cp. We will be able then to validate the results obtained in the static approximation for the critical behavior of the transport properties. From the definition of F q ( o ) given in equation (5) it follows that Fq (09) is a real and positive function. In an isotropic and homogeneous system Fq (w) satisfies the detailed balance condition [5, 9]:

I

is the free energy of the magnetic system, ez is an energy eigenvalue of the system taken with respect to the ground state eo = 0, and q[Sq Ira) is the matrix element of

sq = ~ s~ ei*RJ

(7)

r~(--o)

= e-~

rq(o).

(10)

For a given functionf(w) one can then easily show that the condition (10) yields the relation:

; f ( o ) ( e ~w -- 1) -1Fo ( o ) d o = _oo

where S~ is the spin operator at the position R~. The sums in equations (5) and (6) are performed over the complete set of eigenstates of the magnetic system, and the sum in equation (7) is over all lattice sites. The problem in evaluating J~ and Ja in the present case lies in the lack of information concerning the explicit dependence of Pa ( o ) on q, on o or on T, and that only "little information is available concerning the behavior of P a ( o ) for large momentum transfers" [7]. On the other hand, the static correlation function [5], Fq

= ; Fq(o) do

(8)

;

i f ( o ) - - f ( - - ¢o)](e ~6° -- 1) -1 F a ( o ) do.

(11)

o

In particular it follows from equations (8) and (11) that for the f u n c t i o n g ( o ) = [2 sinh (flo)] [e# w - 1] -1 the relation

r~ = f g(o)r~ (o)

(12)

do

0

is obtained. Thus d r a / d r = j [ag(m)/a r ] r , ( o ) d o 0

is known explicitly for small q and in the asymptotic large-q limit [ 1, 3, 4]. Since it is known that in the first limit the quasielastic approximation is justified [ 1, 5, 7 ], and since the critical behavior is determined by the large momentum transfers, for all magnetoelectronic systems [3], we shall consider here only the asymptotic largo-q limits of F q ( o ) and F a . In particular, one recalls that close enough to Tc, F a has a magnetic energy-like behavior [1, 3] so that: dFq/dT c~ Cp ~ Itl -a

(9)

+ fg(o)[aro(o)/0r]

do.

(13)

0

Since g ( o ) and ag(o)[aT are continuous and bounded functions of o for o > 0 and since F a ( o ) is positive the mean value theorem of the integral calculus allows us to conclude that: Fa = g(og) f rq(w) d o , 0

(14)

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TRANSPORT PROPERTIES OF FERROMAGNETIC METALS

The leading term of dJn/dT (the only possible term that diverges as T-+ Te) is:

and that the first integral in (13) is:

[ag(co)/3Tlto= ~og,; Pq (co) dco

(15) 2 J~co)n[e~W --

O

where cog and cog, are some intermediate values of co. As is apparent from (14) the integral f o Pq(co) dco is Finite and is non-diverging as T ~ T¢. In fact, in the vicinity of Tc,g(cog) and [ag(co)/aT] ,~=w~, are practically constants implying that the integral in (t,0 and therefore the expression (15) have the temperature (and magnetic field) dependence of F u . It is worth mentioning that here as well as for the subsequent application of the mean value theorem, one can get better estimates of the intermediate values of 60 by considering the fact that they should be in the interval 0 ~ co <~toe. In order to evaluate the second integral in (13) we have to consider i~Pq(co)/~T. Using the definition (5) one readily finds that:

aG(co)/ar

= Ka(co)-- q3/T)E(r)Pq(co),

(16)

where E(T) = F - T(DF/3T) is the energy of the system and: Ka(co) = (l/T) eaF ~ ~eI ea~l l, r a

x IqlSq Im)12~(co + e z - era).

(17)

The second integral in (13) can then be written [see equations (14) and (16)] as:

;g(co)Kq (co) dco -- (]3/T)E(T)Pq/g(wg).

(18)

0

In view of the above discussion, the second term of(18) is proportional to E2(T) and thus, it is non-divergent as T-+ Tc. Hence, the only possible diverging term in equation (13) is the first integral of (18). Noting that Kq (co) is positive one can apply again the mean value theorem and conclude that the leading term of dFq/dT in the vicinity of Tc is

g(~) fG(co) dco 0

where c3 is an intermediate value of this integration interval. Considering equation (9) and (13) it follows then that .f~* K a (co) dco has a specific heat-like divergence. Turning now to the transport integrals Jn [equation (3)] and using theproperty (11) one can express them in the form: Jn = 2

f(/3co)nteat°- l]-'r,(w)dw. 0

43

(19)

l]-~Kq(co)dco.

(20)

0

Since ~co)'(e a°: - 1) -1 is a continuous and bounded function for co/> 0, we may apply the mean value theorem, and noting that ~co,)n [eaton --I ] -1 is practically constant in the vicinity of Tc (for any intermediate 60, in the integration interval), this yields the conclusion that the leading divergence of dJn[dT is the same as that off~* Ka(co) dco. Hence, dJn/dThas the temperature dependence of dl"q/dT and thus of Cp. It follows then, that for all transport properties that are described by equation (2), withS(q, 60) = Pq(co), the relation:

d(r-l)/dr ~ G

(21)

prevails. We can conclude now that the critical behavior of the transport properties will be the same as that of the magnetic energy. The existing experimental data are in accord with the prediction of equation (21). For the critical resistivity (which is the most studied transport property) all the qualitative findings indicate that equation (1) is satisfied [3]. The semi-quantitative analysis of the critical resistivity of iron [10] and gadolinium [11] indicate that this is indeed the ease. A recent quantitative analysis [12] of the published data on iron [10] shows that the critical parameters obtained (a =a' = - 0 . 2 + O.02,A/A' = 1.48 -+0.09) are in good agreement with theoretical predictions for a Heisenberg ferromagnet [13] (a = ~,' = -- 0.14 +-0.06, A/A' = 1.36 -+0.06). It is, however, more interesting to test the generality of the present predictions by measurements of other transport porperties, such as the thermopower, S, that involve also the integral Ja. A very recent study [14] of the critical transport properties o f a Crl_xAlx (x = 0.06) alloy has shown that dS/dTcc dp/dTboth above and below Tc. This is then a verification of the present prediction that the thermopower should also have the critical behavior of the magnetic energy. In summary, the use of the general properties of the dynamic spin correlation function yields the conclusion that the critical behavior of all electrical transport properties is the same as that of the magnetfc energy.

Acknowledgements - The authors are grateful to Prof. Amit for helpful discussion and to Prof. Sousa for sending his experimental results prior to publication.

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TRANSPORT PROPERTIES OF FERROMAGNETIC METALS

Vol. 27, No. 1

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NOTE ADDED IN PROOF In the thermal resistivity and thermopower coefficients there is a term with f3(q) ccq at the origin. Therefore, it is not obvious that the results derived here for asymptotally large q apply. However, a more detailed analysis of the problem by J.S. Helman, S. Alexander and I. Balberg (to be published elsewhere) shows that equation (21) holds also in those cases.