The critical resistivity of an antiferromagnetic metal

Physica 96B (1979) 54 - 70 © North-Holland Publishing Company

THE CRITICAL RESISTIVITY OF AN ANTIFERROMAGNETIC METAL I. BALBERG and A. MAMAN The Racah Institute of Physics, The Hebrew University of Jerusalem, 91000 Israel Received 27 January 1978 Revised 30 May 1978

The critical resistivity is analysed in detail for the first time and is found to yield a critical amplitude ratio as well as critical exponents. Thus, critical resistivity is shown to be a helpful tool for the determination of critical parameters. In the present analysis, the published data of Rao and coworkers on the resistivity of dysprosium are used. The results indicate that the critical resistivity of this material has a magnetic energy-like temperature dependence, both above and below the critical point, Tc, Since antiferromagnetic metals belong to the only class of materials for which this universal dependence of the resistivity was questioned, the present results, as well as theoretical considerations given here, establish this universality. The specific heat critical parameters obtained here, a = a' = -0.04 -+0.005 and IA/A'I = 1.44 -+0.05, are the only critical parameters obtained thus far for dysprosium, which were shown to be temperature independent over two orders of magnitude of the reduced temperature t (3 X 10 - 4 ~
1. Introduction

systematic method of analysis and computational difficulties. This paper has a three-fold purpose: To establish the universality o f the above behavior by showing that both theoretical considerations and proper analysis of available data indicate that the critical resistivity o f antiferromagnetic metals is proportional to the magnetic energy [ 1]. To show the usefulness o f the critical resistivity for determination of critical parameters, and to extract the specific heat critical parameters o f dysprosium. For this purpose we apply a specific heatlike analysis [14] to the available data on the resistance o f dysprosium [ 12]. We further suggest some improvements and modifications that can be helpful for the analysis of specific heat [14], critical resistivity and critical behavior o f optical absorption [1 ]. At first, in section 2, we shall show that close enough to Tc, i.e., where the squared magnetization (S)2 can be treated by perturbation theory, the above conclusion is indeed expected to prevail in antiferromagnetic metals. In order to appreciate the need for presenting a systematic method of analysis we review in section 3 the analyses used in the past and their main disadvan-

By now the critical resistivity o f systems that undergo a second order phase transition is fairly well understood [ 1, 2]. The conclusion o f the corresponding theories is, that the temperature dependence of the critical resistivity will be the same as that o f the magnetic energy in all possible physical systems [ 1, 2]. While this conclusion on a universal behavior is widely accepted theoretically and experimentally for ferromagnetic metals and magnetic semiconductors, it is not well established regarding antiferromagnetic metals. There were some disagreements between theoretical predictions [ 1 - 4 ] concerning the behavior o f the latter materials below the critical temperature T c. The experimental data [ 5 - 1 3 ] on the magnetic metals studied were shown to indicate only qualitative or, at best, semiqualitative [ 1 0 - 1 3 ] agreement with the above conclusion. Furthermore, the analyses of the available data have not enabled thus far a reliable determination o f critical parameters from these data. The main obstacles to achieve a quantitative comparison between theory and experiment were the uncertainty regarding the theoretical result, the lack o f a 54

I. Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal tages. The present method of analysis is demonstrated in section 4, using the experimental data reported by Rao et al. [12] for dysprosium. We have chosen their results for several reasons. As far as we know, these are the most detailed high precision resistance data available. The analysis used by Rao et al. [12] for their data is, however, unsatisfactory and does not use the more interesting part (the close vicinity of Tc) of their high quality data. Another consideration for the use of those data is that for dysprosium different critical exponents were concluded by various authors. The exponents that were obtained thus far seem to depend on the method of analysis used. Moreover, some o f the exponents were interpreted [ 12] as being in disagreement with the conclusion of energy-like behavior. The fact that the qualitative behavior of all rare-earth antiferromagnets is the same, but different analyses have yielded different critical exponents for the various materials seems also to be contradictory to the conclusion of universality. Further, such unsatisfactory analyses have led to misinterpretation of the critical fluctuations associated with the critical behavior [13]. Last, but not least, is the fact that the critical parameters of the helical antiferromagnetic metals, Tb, Dy and Ho were calculated recently by Bak and Mukamel [15] who showed that the critical behavior of these materials corresponds to that of an anisotropic n = 4-component vector model. The present analysis is also a response to their call for an experimental test of their predictions. In our analysis we first examine the analysis of Rao et al. Then we analyze their data in the close vicinity of T c, finding the temperature range over which the results are temperature independent. Finally some suggestions for analysis o f critical data are presented. In section 5 we discuss the physical significance of the results for antiferromagnetic metals in general and for the rare earth metals in particular.

55

The discussion of antiferromagnets is more refined, and in particular contradictory predictions for the resistivity below the critical temperature, To, appear in the literature [1 4]. The same applies to nonmagnetic transitions (such as antiferroelectric and order-disorder) which occur at a finite wave vector Q and change the lattice periodicity. In ref. l it was argued that the leading terms in the close vicinity o f T c should be energy-like, i.e., proportional to It[ 1 --c~ where t is the reduced temperature ( T To)IT c and a is the specific heat critical index. More recently, Binder, Meissner and Mais [3] have also derived !he temperature dependence of the resistivity for antiferromagnets below T c and found that the leading term was II2 = Itl 2t3, where (SQ) is the sublattice magnetization and/3 is the critical index of the spontaneous magnetization. Since the discussion of the relevant point in ref. 1 was very brief and since the question of the universal behavior of the resistivity was raised we would like to discuss this point in more detail. In addition to the determination of the critical behavior of the resistivity of antiferromagnetic metals, we comment on the basic assumptions adopted thus far in the available theories [1 4]. The case of antiferromagnetic metals has more subtle points than the cases of other magnetic and electronic structures [1, 2]. Some of these points will be mentioned. Since our interest in the present work is confined to the functional temperature dependence of the critical resistivity, we do not elaborate on the qualitative features of this dependence farther away from T c. The latter features (such as increase or decrease of the resistivity when Tincreases through To) are known to depend on the details of the electronic and magnetic structures. The interaction assumed in all theories dealing with the problem is given by the Hamiltonian [1 4] : ~'int = GO ~, 6(r - R ) s r • SR(f),

(2.1)

R

2. Theory For ferromagnets it is by now generally accepted [ 1 - 3 ] that the electrical resistivity should display energy-like critical behavior, both above and below the Curie point. This reflects the fact that the resistivity is determined by the large-momentum-transfer scattering and therefore by the corresponding asymptotic large momentum correlation functions.

where G O is a constant. One assumes an electron, located at the point r, which has a spin st, and a localized spin S R at the lattice site R and the time L The other assumption shared by all existing theories is that one can treat the scattering, due to the above interaction, in the Born approximation. Hence the cross section for a scattering event is proportional to IMk,k,I 2 where Mk, k, is the matrix element ofg~in tTo determine the matrix element one considers a spin-

L Balbergand A. Maman/Criticalresistivity ofan antiferromagneticmetal

56

up electron with a wave-vector k and a Bloch wavefunction eik" ruk(r)l~). This electron is scattered to a state with a wave-vector k' and the energy exchange in this scattering event is/leo. The calculation can be generalized if the states It) and I+) are assumed to have the same probability. Dealing with antiferromagnetic ordering (assuming that SR (t) is, say, a spin-up ion) we note that the spin at the point R' is SR,(t) eiQ. (R -R') where Q is the magnetic-reciprocallattice vector (the point of instability) and SR,(I ) is the "ferromagnetic spin" at R'. To be able to cope with the problem one must make two simplifying assumptions [1, 2, 4] : (i) the fluctuations are associated with the nearest point of instability, and (ii) there is no interaction between fluctuations belonging to different points of instability. These amount to the evaluation of the cross-section (and the mean-free-time) for one point of instability. For the system described above one can easily show that:

where N is the number of lattice sites and q = Iq[. We can write then [1, 3] (IMk, k, [2) =

G2(rlq_Ql(Co)+ m2(0)6(q

- Q)6(w)), (2.5)

where m(0) is the amplitude of
r'q,= f r'q,(¢o)d¢o

(2.6)

IMk, k,I 2 = Cl R,R'

where G 1 = C1G21uk,(Ro)[2 luk(R0)l 2, C 1 is a constant, R 0 is an arbitrary lattice site and q = k' - k. To calculate the a priori probability of scattering event with momentum transfer fiq and energy transfer "K~owe have to sum up over all the states of the spin system with their corresponding Boltzmann factor, i.e., take the thermodynamical average of (2.2). This procedure yields

which corresponds to the quasielastic approximation is known in the above limit. In particular, it is well known that it has an energy-like temperature dependence [1 ]. The probability of the scattering process is obtained by multiplying (IMk, k' 12)by the occupancy of the k state and the availability of the k' state. Using the condition of detailed balance one can show that the resultant.probability is proportional to ([Mk, k, 12) ×/~hw(e~ ~to - 1) -1 where ~= 1/kBT. With this result the solution of the Baltzmann equation for the meanfree-time r is given by [1, 2] :

(IMk,k,I 2) = G 2(Slq _Qi, o • S_lq _Qi ' _to),

l / z : f f d3kd3k'G3gP(k',k)

X f(SR(O)'SR,(f))eiQ'(R-R')eitot'dt,

(2.2)

__ot~

(2.3)

where G 2 = C2G 1, C 2 is a constant and ( ) denotes the thermodynamical average. Eq. (2.3) is usually expressed in terms of the spinspin correlation function, that for an antiferromagnet, with a point of instability Q, can be defined by: r'qAF(~o) = Fiq_Ol(w ) = [1/(2zrN)]

x f

-). ( s R , ( O

-

)>

-*~ R , R '

X e i(q-Q)" (R'-R) eitof dr,

(2.4)

F.S. oo

X f (IMk,k,12)~w(e ~ t o - 1)-ldw,

(2.7)

--era

where the integration is over the Fermi surface and all possible energy transfers. Here G 3 = C3G2, C3 is a constant and the function q~k, k') is determined by the Fermi surface [1,2] (see below). In the available treatments of the problem at hand, it was assumed that/~iw ,< 1 and thus the quasielastic approximation was adopted. This approximation deserves justification in the case of critical resistivity because for this prob-

I. Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal lem.the characteristic frequency of the fluctuations 6% = kBTe/~t and one is interested in T ~ T c. Recently, Helman and Balberg [1] have shown that while Pq,(¢o) is not known one can prove that the temperature derivative of the integral over co in (2.7) has the same temperature dependence as d(Fq, + mZ(O) 6(q' - Q))/ dTin the vicinity of T c. This work indicates then an energy-like behavior for both elastic and inelastic processes. In fact, this is tree not only for the present ~but also for all other transport mean-free-times. It has to be stressed that for metals the contribution of the inelastic processes to 1/r may be larger than that of the elastic processes, but for our purpose, i.e., the determination of the temperature dependence of r, one may proceed by using the quasielastic approximation. Hence:

1/r c~ ~ ~ d3kd3k'G3cb(k ', k)(Fiq-QI

+

57

where the function f(Q + q') includes both Fermi surface and interaction effects [I, 4]. For an antiferromagnet where Q is on the zone boundary and there is no interaction between different points of instability, the integration (2.9) is limited to [q'l < IQI. Hence one can develop (2.9) around the point of instability and obtain [1] : 1/r o: f(Q) f d3 q ' [Fq, + m2(O) (5(q')] + V f ( Q ) .fd3q'Fq,q' + . . . .

(2.10)

In the simple case of a spherical Fermi surface

f(q' + Q) = Iq' + QI and thus f(Q) = Q. The second term on the r.h.s, ofeq. (2.10) can be shown [1] to be:

m2(0)6(q - Q ) ) " -(3rr/2)

F.S.

dq t Fq,q t 3 .

(2.11)

(2.8) We should note that the possible anisotropy of the scattering cross-section in the case of antiferromagnets is unaccounted for by (2.5) and that in the available treatments of the problem at hand, the scattering cross-section was assumed to be isotropic. On the other hand, the effect of anisotropy of the Fermi surface on the mean-free-time was discussed for both ferromagnets and antiferromagnets [ 1, 2]. The latter anisotropy is treated by considering an explicit expression for (I,(k, k'). There is no reason why the function qb(k, k') cannot be generalized to include both Fermi surface and cross-section dependences on k and k'. Hence, in principle, the procedure used for dealing with an anisotropic Fermi surface can be adopted for this more general case. It has been shown that the temperature dependence of 1/T in the close vicinity of T c is not affected by the anisotropy of the Fermi surface even in extreme cases [ 1]. One may then reasonably assume that close enough to T c anisotropies in the electronic and/or magnetic system will not change the critical temperature dependence of 1/r. Again, this conclusion is sufficient for our purpose. Following the above considerations and by making the transformation q ~ q' + Q one can conclude that "close enough" to T c one may write [1,2, 4] :

1/7" (x fd3q'f(Q + q')(rq, + m2(O) 6(q')),

(2.9)

By the spin sum rule one immediately realizes that the first term on the r.h.s, o f e q . (2.10) is a constant independent of temperature. In the magnetic case this is exact because this integral is proportional to S(S + 1) where S is the magnitude of the ionic spin [ 1]. In a more general situation there may be corrections but these only involve the asymptotic correlation functions, and, at most, diverge like Itl 1 - ~ . As discussed in detail in ref. 1, integrals such as (2.11) cannot have stronger divergences than It[ 1 - a . One thus concludes that the strong It[ 2# behavior is cancelled out in all systems. Qualitatively, eq. (2.11) indicates that i / r will decrease as the temperature increases through T c. We would like to emphasize that eq. (2.10) and the resulting qualitative behavior as well as the Itl 1- ,~ dependence below Tc, are only valid for sufficiently small Itl (see below). The range of Itl for which this is valid depends on the actual parameters of the individual system. Thus a crossover to a different type of behavior with increasing It l may be observed. The above results do not hold for very low temperatures where the critical fluctuations vanish. For these temperatures we can apply the qualitative argument that, finally, for very low temperatures 1/r must decrease with decreasing temperature [ 1 ]. In the above discussion we assumed that f(q' + Q) in eq. (2.9) is temperature independent since it was

I. Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal

58

derived by using the Bloch functions of the magnetically unperturbed electronic system. Below Tc both the wave functions and the band structure will be affected by the periodic m(0) eiQ" (R -R') potential. In this case f(Q) will be a function of m(0) (and thus of temperature) and may yield [4, 8] a different critical temperature dependence of 1/r than that expected for" a temperature independent f(Q). Hence, the present discussion has to be restricted to the close vicinity of Tc where the magnetic periodic potential can be treated as a perturbation, and to lowest order one can assume that f(Q) is independent of m(0). In fact this is the point which was overlooked in ref. 3 and has caused the Itl 2~ dependence of the critical mean-free-time. In that paper it was assumed that the interaction Hamiltonian is hin t = G O ~ 6(r - R ) s r "(S R - (SR)).

(2.12)

R

If this hin t is used, one must add to the unperturbed, non-magnetic, Hamiltonian of the system a term of the form GO~,RS(r - R ) ( S R ) . Hence, the wave function, and thus G 2 in eq. (2.5) must be dependent on II. One may retain the contribution of (SR) in both hin t and G 2 or apply a perturbation approach, as we do here, and consider hin t and G 2 in the I
mentally the predictions given above. The answer is borne out by both the qualitative and quantitative features of the experimental results. It is found [1-3] that the general features of the temperature dependence can be explained by the results (2.10) and (2.1 1) without invoking the effect of superzone splitting. Further, the exact temperature dependence in the vicinity of Tc is in accord with the theory for 1/r and is not in accord with the expected effect of superzone splitting. While this will be discussed in section 5 we should stress here that the "close enough" vicinity of Tc is accessible for experimental studies. The predicted [ I ] critical temperature dependence of the resistivity of antiferromagnetic metals is summarized in fig. 1. The resistivity increases when T c is approached from above. It is expected to exhibit a transition from the mean-field, - t I/2, behavior to the critical, energy-like - t 1 - a , behavior around the Ginzburg reduced temperature [ 16] t G. The resistivity continues to rise as It jl - a below Tc until finally it decreases at low temperatures. The dashed curve signals the region where the qualitative behavior is understood [1] but the exact behavior may depend on changes in the electronic structure of the material [1, 2]. Indeed, the qualitative features of fig. 1 were observed in many antiferromagnetic metals [5, 6, 8]. A quantitative examination of this resistivity anomaly, is, of course, concerned with the exact temperature dependence (e.g., the value of a) in the Itl 1 - a regions. The energy-like behavior of the resistivity can be argued by considering the physical nature of the interaction between charge carriers and critical fluctuations

P

1 I

•:~

t½

/I

1

tc

r~p~tuxe Fig. 1. The predicted temperature dependence of the resistivity of an antiferromagnetic metal in the vicinity of Tc. t o corresponds to the Ginzburg temperature at the vicinity of which a transition from mean-field to critical behavior is expected.

I. Balbergand A. Maman/Criticalresistivity ofan antiferromagnetic metal in antiferromagnetic metals. As is well known, the dominant contribution to the resistivity in an elastic scattering process is that of large-angle-scattering of the charge carriers. Hence, large-momentum-transfers between the carriers and the scattering system will be the preferred contribution to the resistivity. In the case of magnetic materials this transfer implies an increase of the resistivity with the increase of the large-q spin fluctuations. In antiferromagnets the large-q fluctuations grow when the temperature decreases through T c since the antiferromagnetic order is associated with the large-wave-vector Q. In an antiferromagnetic metal where Q ~ kF, this growth yields an increase in the resistance upon temperature decrease through Tc, since the large-momentum-transfer is consistent with the conservation of energy and momentum. As in the case of ferromagnetic metals [ 1] a behavior dominated by small-q momentum transfers can take place only for Itl > 0.1. Thus, in the close vicinity of T c a large-q dominated behavior is to be expected. Within this vicinity a transition from the critical x-ltl1 - a to the mean-field-Y-ttll/2 behavior is expected both above and below T c, when T departs from T c. The agreement of the results deduced from the experimental data (see section 5) with these expectations enables one to conclude that band structure effects [4, 8] and long range order behavior [13] are not important in the temperature region considered here.

3. Previous analyses of critical resistivity In the early works [ 5 - 9 ] the resistivity was measured as a function of temperature. The "experimental" derivative dp/dT was calculated and then presented as a function of Itl on a log-log scale. T c was assumed to be either the temperature Te where dp/dThad its extremum value [9, 13] or as the temperature that has yielded the expected power law behavior [8]:

dp/dltl = c(1 - X)ltl -x + b

(3.1)

over a large range of log Itl. In this expression X is the critical exponent and b and c are constants. Sometimes the value of b was determined from estimates of the background resistivity [8, 9] and sometimes it was

59

eliminated by considering [5, 17] the linear dependence of log (d2p/dT 2) on log It L The slope was then assumed to yield - X - 1. There are many faults in this approach that become apparent when the nature of the experimental data is considered. Since one deals with discrete experimental points, one can at best claim that the true T c lies in the interval Te + AT ~> T /> T e - AT, where AT is the temperature interval between the adjacent experimental points in the vicinity of Te. In fact the situation may be even worse since the derivative is not necessarily a smooth function in the vicinity of Te and local peaks in its vicinity may increase the uncertainty in T c to a few AT on each side of Te. In principle, the latter problem can be overcome by using a large number of data points in the set used for calculating the derivative [8, 17], but this in turn causes "smearing" of dp/dT in the vicinity of T c. Moreover, as was demonstrated recently [18] this yields a dependence of the critical exponents on the number of points in the set. Hence, regardless of the differentiation method used, the results can be meaningful only for Itl >> 2xT/Tc. The third disadvantage in this method has to do with the fact that measurements thus far were limited to the range 10 -1/> Itl ~> 10 -4 and one expects a fit of a single X over at least two orders of magnitude of Itl to be meaningful [16]. Since the results at the small Itl end are more sensitive to the choice of Tc, the power-law behavior extracted by this method was usually the X corresponding to large Itl end [8]. When a single X could not account for the data, two X's were assumed [5, 6]. Their values were "graphically" determined and thus were quite subjective. The uncertainty in the results obtained by these early analyses is appreciated if the early works on Ni and Dy are examined. In 1970, the values X = 0.1 -+0.1 and X' = - 0 . 3 -+ 0.1 were reported [17] for Ni. This was in excellent agreement with the values known then for thecritical indices of the specific heat, but different from the theoretical expectations of X = X'. To date, we know quite certainly [19] that the specific heat measurements yield X = X' = - 0.12 +- 0.02. In 1973 before the unjustified approximation in the theory of Suezaki and Mori [4] was realized [1, 2] the analysis of the data on dysprosium was interpreted [8] to be in accord with their value of X = 0.3. It was apparent already then that the same data (with a different choice of Tc) can yield a nega-

60

I. Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal

tire value for X, but the 0.3 value was chosen. These examples show how flexible were the results obtained by the early analyses and why, to date, we can recognize them only as qualitative estimates of the correct behavior. As with the specific heat data analyses [20] the above qualitative approach was replaced by a linear [10, 11] dr a linear-like [12, 13] least squares method of analysis. Fits were attempted between the experimental data and the expression given by eq. (3.1) or its integral: p(lt{) = clt{ 1 - x + bltl + a.

(3.2)

Hence, a five parameter fit (with the parameters a, b, c, X and T c for T ;> Tc and the same but primed parameters for T < Tc) was required. In order to utilize the linear methods which were available, this fit was usually reduced to a three parameter fit. While these methods of analysis are more reliable than the early qualitative methods they have required some dubious assumptions in order to overcome the non-linear nature of the functions (3.1) or (3.2). Since the disadvantages of using such linear least squares methods for non-linear functions have been discussed already in detail [21,22] we will list here only the principle drawbacks that are pertinent to the present study: (a) Tc has to be pre-selected not on the basis of least square fits [10, 13] ; (b) the computed errors in the parameters were either determined subjectively [12] or they were non-realistically small [22] (the error in a parameter was determined when all other parameters were "frozen"); (c) the computation is rather elaborate when some parameters are not pre-selected since numerous five parameter sets have to be examined [12]. The last drawback is not merely a technicality but appears to be the main reason to the fact that no critical amplitudes [i.e., c in eqs. (3.1) and (3.2)] and no reliable critical exponents were derived from critical resistivity. This is also the reason why the dependence of the parameters on the various temperature intervals has not been examined, and why the results of the analysis have not been correlated with explicit specified constraints imposed upon the parameters. The determination of the amplitude ratio is important for the physical conclusions since reports [10, 13] of X = X', without giving the value o f c / c ' are not enough to confirm scaling [16, 19]. Another odd

effect of linear analysis is that the functions fitted to the data [12, 17] were found to be discontinuous at T c while the data exhibits continuity (see next section). Moreover, the "crossover" from a behavior characterized by a given X to a behavior characterized by a different X [5, 13] has not been analyzed and a criterion for such a "crossover" has not been given. All the above difficulties can be overcome nowadays since non-linear least-squares computer programs are available. The data can be analyzed in detail and in a systematic manner. This was realized first by Kornblit and Ahlers [14] who used the Marquardt maximum-neighborhood method [21, 23] to analyze specific heat data. Such a non-linear method has not been used before for analysis of critical resistivity data and thus no reliable critical parameters were extracted from the data. In this paper we use this non-linear method [23, 24] to extract more information on the critical behavior, from the temperature dependence of the resistivity.

4. Analysis The experimental data of Rao et al. [12] on dysprosium (T c ~ 180 K) exhibit the well-known Nshaped resistivity anomaly shown in fig. 1. Since in their analysis they could not find a single power-law behavior to describe the entire data they have considered the data in the temperature intervals 60 t> T - Tc/> 2.5 K and 15 t> T e - T~> 2.5 K for which temperature-independent exponents were found. While these temperature ranges are wide on the T scale they are quite narrow on the log [tl scale, and their correspondence to the critical regime (where scaling arguments [16] apply) is in question. This is especially so since the long-range RKKY interaction in the rare earth metals [25] implies a narrow (Itl < 10 - 2 ) critical regime (see section 5). Hence in re-analyzing their data we have stressed the analysis of the data points in the I T - Tel < 2.5 K intervals. Following this we also analyzed the "crossover" between the two temperature ranges. Finally, all the analyses given thus far for both specific heat and critical resistivity had the implicit assumption that there is no statistical error in the experimental temperature readings. Hence the analyses were least-squares-ordinate-fittings rather than best curve-fittings. We shall suggest here a curve

L Balbergand A. Maman/Criticalresistivity of an antiferromagnetic metal fitting that we believe to be a better way for parameter determination. The present analysis of the experimental data of the c-axis resistance of dysprosium is based on a nonlinear-least-squares-fit between the data and an energylike expression of the resistance R. Hence we try to fit the temperature dependence of the resistance in the close vicinity of T c with the function [1,20] : R ( T ) = (h/~)[Itll-~/(1 - ~ ) - I t l ]

+Bltl + C

over these temperature intervals. Following the work of Rao et al. we have chosen their T c (= 180.33 K). The results obtained over the temperature range: 241.454 >~ T>~ 182.515 K were: 0.403 ....9.477 34.943 118.214

~
(4.3)

(4.1)

both above and below T c. Here c~ is the critical exponent of the specific heat, A is the "critical amplitude of the resistivity", and B and C are constants. There are then ten parameters to be fitted here, five for the data above T c and five (that will be denoted here by primes) for T < T c. The computations are done here by using Marquardt maximum-neighborhood method [23, 24] that was applied already for specific heat data analysis [14, 22]. The method is base~t on minimizing the sum N di)= ~ [ZXR(Ti)] 2 i=l

61

(4.2)

by a gradient search [21]. Here ZXR(Ti)= R(Ti) - Rexp(Ti), where R(Ti) is the value of the function (4.1) at the point T i and Rexp(Ti) is the experimental value at the temperature Ti. The parameters of R are those which yield the minimum of (4.2), and their uncertainty at this minimum will be given here by their nonlinear limits of confidence [24]. These limits are measures of the uncertainty of a parameter when all other parameters are allowed to vary [ 2 1 , 2 3 ] . The least-root-mean-squared deviation o (dPmin/N)l/2 will be used to characterize the quality of the fit of the data with the function R given by (4.1). =

4.1. Analysis for I T - Tcl> 2.5K In the original analysis [12] of the data these were the temperature intervals studied. Since they are relatively remote from Tc, it is expected that different computation methods will yield similar results. To confirm this and to find the parameters a, b and c [see eq. (3.2)] that have not been reported by Rao et al. [12] we have used the present computation method

where a is dimensionless and A, B and C have here the dimensions of m~2. Over the temperature range 178.549 >/T~> 164.765 K the results were: -0.3292 527.925 -919.44 119.374

~
-0.3288 528.185 918.38 119.401

~< - 0 . 3 2 8 5 K 528.446 ~<--918.33 ~< 119.428.

(4.4)

We have given here in addition to the computed parameters their limits of confidence. For simplicity we will present from now on the values of the parameters and the deviations of the limits of confidence from these values. As expected, our results for a = X and a' = X' are in excellent agreement with those of Rao et al. [12]. There is thus no doubt that the values obtained here:

a-C=l18.215+-O.OO2, b-~B

A/a=58.41+O.O1,

c - A / [ c f f l - a)] = - 3 9 . 3 3 - 0.01,a' = 119.40+-0.03, b' = 687.5 -+ 0.5 and c' -- - 1 2 0 8 . 9 -+ 0.4 are the values that should yield the best fit in the analysis of Rao et al. [12]. These numbers already indicate the problems raised by limiting the analysis to the above temperature intervals. First, a differs from a' much beyond the experimental error implying a discontinuity in the R(T) function at T c (or a mismatch of 0.3 K in the values of T c and T~), in contrast with the physical expectations and the continuity of the data. Second, the derivative dR/dTdiverges as T c is approached from above, but it exhibits a cusplike behavior when T c is approached from below. And third, IA/c~l and A'/a' are of the order of B and B' respectively, indicating that the dominant contributions to the resistance in the temperature intervals considered are of noncritical nature.

1. Balbergand A. Maman/Criticalresistivity ofan antiferromagnetic metal

62

4.2. Analysis for I T -

Tel <

2.5 K

127.80

The data of Rao et al. [12] in these intervals has not been analyzed before. We have thus carried out the analysis in these intervals as a function o f the interval ends using the method described above. The parameters of the best fit in the interval 182.515 I> T 1> 178.079 K were:

128.70 -r 125.60

E _J

i,i

124.so-

123.q0-

z 122.30-

T c = 180.366 -+ 0.013 C = 118.885 -+0.04 a = - 0 . 0 4 1 2 + 0.002 A =-231-+1 B = 890---4

T'c = 180.357-+ 0.03 C'= 118.885 -+0.04 c~' = -0.0351-+ 0.005 A'= 160-+2 B' = - 3 4 4 - + 5 . (4.5)

Since there are only few experimental points between 178.079 K and the value of T c obtained (11 points) we have repeated the analysis by expanding the temperature interval below T c. The results were exactly the same (withinthe uncertainty limits) for the 15 points between 175.194 K and this value of T c. In fig. 2 we show a computer generated plot of the data for T > T c as well as the function (4.1) with the parameters given by eq. (4.5). A similar plot is shown in fig. 3 for the data below Tc. The results given in eq. (4.5) demonstrate some noteworthy features: 118.90 118.55-

-r- 118.20ED Jl17.85-

117.50~ 117.15-

t--co ~

I16.80~

Q:2 116 45

116.101 180.2

,

,

,

,

,

181.4

,

,

182.6

TEMPERRTURE K Fig. 2. c-axis resistance of Dy in the close vicinity of T c for T > T c. The data points are taken from ref. 12 and the curve presents our best fit of the data to eq. (4.1).

I.-. 03

121,20LI-I

120.I0119.00 175.6

178.0 TEMPERATURE

180.q

K

F~. 3. c-axis resistance of Dy in the close vicinity of Tc for T < Te. The data points are taken #om reL 12 and the curve presents our best fit of the data to eq. (4.1). (a) T c = T'c and C = C' within the confidence limits, i.e., the best fit yields a continuity between the two functions described by the parameters (4.5). The values of Tc, Tc, C and C' are between the values of the two experimental points (at 180.312 K and 180.369 K) which are the closest experimental points to the point T e = 180.340 K where dRexp/dT obtains its minimum value. This is in accord with the expectations mentioned in section 3 and it indicates that the data is of good quality even at the closest vicinity o f T c. (b) The derivatives of the functions obtained, - A / a + B and A'/a' - B' are equal within the uncertainty limits. This is in accord with expectations from the behavior of specific heat [14, 22], (c) The scaling expectation a = a' is fulfilled within the confidence limits and the values of a, a' and IA/A'I are reasonable [15, 26] (see section 5). To get an idea of the quality of the fit we show, in fig. 4, the temperature dependence of the deviations z~(Ti) of the data from the fitted curves shown in figs. 2 and 3. It is clearly seen that the deviations are o f the order o f 0.1%, which is at least as good as the fit found in specific heat data analysis [ 14, 22]. For a more quantitative characterization of the fit one can use the root-mean-square deviations o and o'. In the present case, the fit shown in fig. 4 for T > Tc (triangles) corresponds to o = 3.50 × 10 -2 and the fit shown for T < T c (circles) corresponds to o' = 3.99 × 10 -2. At this point we should

L Balberg and A. Maman/Critieal resistivity o fan antiferromagnetic metal

63

180.38F

CZ2

<~ %

m

A

q

.36~

ii

A A

n . . . . . . .

@

AA ~

A A r ~

m

m

-q

A

34~

mm

A 6 A

Ag . mmm

~_u

.32i

180.30L

n

0

1

2

3

I

-5

t

-q

I

I

t

3

4

5

6

102 ]tmaxm

mn

8

. io

t

-2

LOG IT/Tc-lq Fig. 4. The deviation of the data points from the fitted curves, shown in fig. 2 (triangles) and in fig. 3 (squares), as a function of log ItJ. mention that an addition of an analytic term of the form [14, 22] E[tl 2 to eq. (4.1) did not change neither the parameters obtained nor the o's. This can be gathered from the fact that zXR does not change significantly with decreasing It I. So far we have considered the temperature regions, T - Tc < 2 . 5 K a n d T c - T < 5 K . A priori there is nothing special about these intervals except that they were left out in the analysis of Rao et al. [ 12]. It is then important to find what are the temperature intervals for which the results (4.5) are the best description o f the data, since this will give an evaluation for the reliability and extension [ 14, 22] o f the results. We have thus repeated the same computations for different temperature intervals in the close vicinity o f T c. This, with the restriction that the a's obtained do not exceed significantly the above values (o, o' 5 X 10-2). As a first stage we have used the data points between 180.369 K (denoted here Train) and a higher temperature Tmax, while for the other side of T c we have used the data points between 180.312 K (denoted here T;nin) and some lower temperature Tmax. The parameters obtained in the corresponding intervals are presented in figs. 5 - 8 as a function of the parameter Itmaxl = (Tma x - Tc)/T c or [tmax[ = ( T c - Tmax)/T c. The error bars of the parameter represent the nonlinear limits o f confidence. The first property to be checked is the continuity o f the function described by (4.1) at T c. For this purpose the Itmaxl dependence of the parameters T c, T c, C and C' was examined. The results for T c and T c are shown in fig. 5. It is seen that, within the uncertainty limits of these parameters,

Fig. 5. The Itmaxl dependence of the values of Tc (circles) and T'c (dots) obtained from the best fit of the data to eq. (4.1). Tmin and Tmin are the fixed ends of the temperature intervals. The error bars represent the non-linear limits of con fidence. T c = T c over the temperature intervals studied, except for the vicinity Oftma x ~ 2 X 10 -2. This may be associated with an experimental error in the temperature readings there (see section 4.3). The values of C and C' were found to be a constant (C-- C' = 118.885) over the Itmaxl range shown in fig. 5, except for the points around tma x ~ 2 X 10 -2 where C ~ 119.110, again a little beyond the statistical uncertainty. However, on the whole the T c = T c and the C = C' results establish the continuity o f the two branches of the function (4.1) at T c. The other general criterion is the continuity of the function's derivative [14, 22] at Tc, i.e., that - A / ~ + B = A'lct' - B'. In fig. 6 we show the dependence of the latter quantities on Itmaxl. The results indicate clearly the existence of this continuity over the entire Itmaxl range. (Note that this is equivalent to the continuity of the specific heat.) While the properties mentioned above are in accord with general physical expectations there are specific theoretical predictions [15, 26] for the values o f ~ = a'. The results for ct and a ' in the above Itmaxl 6000~

---

-

I

t

o !flI

.~

4000

3oooL 0

I I 1

i I

• TT c

~

, 2

3

5

102 Itmaxl Fig. 6. The Itmaxl dependence of the values of A/a B (circles) and -A'/a: + B' (dots) obtained from the best fit.

1. Balberg and A. Maman/Critical resistivity of art antiferromagnetic metal

64

ranges are shown in fig. 7. It is apparent that while there is a difference between a and a', this difference is smaller than the corresponding limits of confidence. In fig. 8 we show the values of [A[ and A ' as a function of [tmax[. The size of the points in fig. 8 is larger than the uncertainty o f the parameters and thus a ratio of [A/A'I = 1.45 is well established. Both the values of a and a' and [A/A'[ are not the exact values predicted by the theory [15] but as will be discussed in section 5 they are certainly in agreement with the trend predicted by the theory. It should be stressed that in the above discussion, while we show the results for T > T c up to ]tmax[ ~ 2.5 X 10 -2 the same values are obtained up to [tmax] ~ 0.1 except that o increases beyond the above selected value of 5 X 10 -2 as Itmax] increases beyond 2.5 X 10 -2. As a second stage we have analyzed the data using experimental points between 182.515 K (denoted Tmax) and some lower temperature Tmin > 180.369 K for T > T c, and the data points between 175.794 K (denoted Tmax) and some higher temperature T ~ n < 180.312 K. Here the results are presented as a function of the parameter Itminl = (Tmin - Tc)/T c or [train[ : ( T c - T ~ i n ) / T ' c. In fig. 9 we show the values of T c

~ .34 o Tc

~- .32 i 180.30 0

oT~ I

10

I

30 2O 104 ]tmin]

40

50

Fig. 9. The Itminl dependence of the values of Tc (circles) and T'c (dots) obtained from the best fit. Tmax and Tmax are the fixed ends of the temperature intervals. and T c' as a function of Itmin I. Again, as in fig. 5, the continuity of R(T) is apparent and again the same values are obtained for both T c and T c. The continuity of the derivative as a function o f [train[ is also similar to that obtained for the [tmax[ dependence and so are the values of a, a', A and A'. Trying other sub-intervals between 174.795 and 184.860 K,that include at least 10 points on each side of Tc,have yielded the same results. We shall not elaborate on this since the data is available [12] and the results can be checked for every desired sub-interval o f the intervals discussed here.

10t I

-0.02

oct

"~ -0.03 -0.04 -0.05 -0.06i

i

I

3

2

I

4

5

102 It.~xl Fig. 7. The Itmaxl dependence of the values of c~(circles) and a' (dots) obtained from the best fit.

o

oO

o

oo

o

o A/160.35 • A'/160.35

o

1.o °.8o

Tmax= 182.515K T~aax= 175.794 K

4.3. The IT - Tcl < 2.5 K region re-analyzed

-0.01,

~1.4 1"6I -~ 1.2

180.38.

1

l

I

[

I

I

2

3

4

5

IO2 ItmaxI Fig. 8. The Itmaxl dependence of the values of A (circles) and A' (dots) obtained from the best fit. There are no error bars shown since they are smaller than the size of the points.

In the above computations the least-squares fit was carried out by finding the minimum of qb given by eq. (4.2). The method used here, as well as the method used in specific-heat-data analysis [14], represents in fact a least-squares fit of the ordinates. In other words, while a statistical scatter is implied for the values o f the resistance measured, it is implicitly assumed that there is no statistical scatter in the temperature readings. Furthermore, the ordinate deviations A R ( T i ) are expected to give an exaggerated larger weight to the points close to T c since the changes of the resistance there are relatively large. To check whether these drawbacks are important in the present analysis and whether the results mentioned above depend on the computational method we have considered a different deviation than AR(Ti) for the least-squares fit. In view of the above it seems that it will be more reliable to take the deviation of the data points from the fitted curve than to take the ordinate deviation. Thus instead of using [AR(Ti)] 2 in eq. (4.2) we should use (ARif) 2 = (Rexp(Ti) - R ( T f ) ) 2 + ( T i - T/-)2 where (R(Tf), T£)

L Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal is the point on the fitted curve which is the closest to (Rexp(Ti) , Ti). Assuming that the temperature intervals between subsequent data points AT i are small enough, (2~Rf() 2 can be approximated by (Rex_(Ti) - R(Ti)) 2 cos20 where 0 arctan Hence the best curve fitting will be obtained by the minimization of the function:

[(dg/dr)rif.

=

N

[Rexp(Ti ) _ R(Ti)] 2/[1 + (dR/dT)2Ti ] . (4.6)

,I,=

i=1 The outweighing of the points close to T c by considering xI, instead of q5 is apparent here since dR/dT increases strongly as T c is approached. Since the present approach (4.6) seems to be a more reliable one we have repeated all the computations mentioned in section 4.2, by using xI,. The results o f this work were all, within the limits of confidence, equal to those of eq. (4.5), and the values of o were very close to those obtained by using qs. The differences were only in the fine details of the Itmax[ and Itminl dependences of the parameters. To appreciate these fine details we show in fig. 10 the T c and T'c dependence on Itmaxl as computed by using qs. Comparison of fig. 10 and fig. 5 indicates, for example, that on the whole the same values of the parameters are obtained. However, while in fig. 5 the largest deviations from the mean value of T c were obtained at tmax ~ 2 × 10 -2, in fig. 10 the largest deviation appears around tmax ~ 1.2 × 10 -2. In view of (4.6) this difference is believed to indicate that around tmax ~ 2 × 10 -2 there was a relatively large experimental error in the temperature reading. On the other hand the errors in most of the data points (Rexp(Ti), Ti) seem to result from the uncertainties in the resist-

°38f36

-¥

OTc

.34i .32 180.30 0

I

2

3 102 Itmaxl

Fig. 10. The Itmaxl dependence of the values of Tc (circles) and the values of T'c (dots) obtained from the best-curve-fit. While the temperature intervals are the same as in fig. 5, the fit is determined by eq. (4.6) rather than eq. (4.2).

65

ance readings. We should point out that in experiments where the data are not as precise as those used here the results obtained when q5 is replaced by qs, may differ substantially from each other [27].

4. 4. The "transition" region The exponents obtained in sections 4.1 and 4.2 are quite in agreement with those of Meaden et al. [5] on terbium and those of Singh and Geldart [13] on holmium. In all cases a transition from a temperature dependence with an exponent ?t = 0.45 -+ 0.1 to a dependence with a negative X was observed when T c was approached from above. Since this "crossover" appears to be a general property of the rare earth metals and since it may be misinterpreted [13] (see section 5) we would like to consider it on a more quantitative basis. So far only a breaking point between the two behaviors was given but its determination [5, 13] has not been discussed. From the present work we do know that the result (4.5) is the best fit in the close vicinity of T c, while the result (4.3) is the best fit in the farther temperature region. The above analysis indicates also, that one can examine the transition between the two results in terms of the quality of the fit of each of these results in the closer and farther temperature regions. To illustrate this point we show in fig. 11 the deviation AR(Ti) as a function of temperaturein the vicinity of T c for the two results. It is seen that when the result (4.5) is used (triangles) the deviations are relatively small throughout the region and are evenly distributed around AR = 0. On the other hand, the use of (4.3) yields AR(Ti) (circles) that increase when T c is approached and that is always positive. Over the range N I(AR(Ti)) 2./N] 1/2 of (4.5), that will shown, Z -= [Y'i= be denoted here ~5, is about 3 X 10 -2 for any Tmax in the temperature interval shown, while Y~3 [Y~ of (4.3)] is an order of magnitude larger (4 X 10 -1) for these intervals. Extending Tmax, we found that such a large value of Z is obtained for the results (4.5) at Tmax = 184.860 K. Hence, if we define the transition region as the temperature interval T 1 < T < T 2 for which Z 3 ( T 1 ) > ]~3(T;> T1) , 1~5(T2)> Y.5(T< T2) and 2;3(T1) = Y~5(T2) = 10o we get that in the present case this interval is between 182.515 and 184.860 K. Of course, this choice of 10o = Y~3= ~;5 = 4 × 10 -1 is somewhat arbitrary but a Y~which is an order of mag-

66

L Balberg and A. Maman/Critical resistivity o f an antiferromagnetic metal 87 .S

"°

o

75.0 62.5

50.0

e i

CE:

o<~ 37,5

'"

CD

25 .O o "oo

12.5

la

ijl

•

!

!

AAI,b

0.0

"

""

g :?.o..

"

"l

-12.5

'

'

i

i

181 .q

180.2 T

i

J

182.6

[K]

Fig. 11. Deviation of the data from the function (4.1) for T > Tc. This, when the parameters that correspond to the laxge-t range [eq. (4.3)] are used (circles), and when the parameters that correspond to the small-t range [eq. (4.5)] are used (triangles). nitude larger than o o f the best fits [eq. (4.5) below 182.515 K and eq. (4.3) above 200 K] seems a reasonable criterion. In the present case the situation is even better defined since the transition is easily identified due to sharp rise in the 22s (~3 for T < T 2 and ]~5 for T > T1). Hence a different choice of the above ~3 = 2;5 values will alter the interval T 1 ~
To find the origin of this "crossover" we have reanalyzed the data for T < T c with variable temperature intervals. We found that for every temperature subinterval in the range 172.228 ~< T~< 180.312 K the results within the uncertainty limits were those given in (4.5). For every temperature interval for which Tmax > 150 K but T~i.n = 180.312 K the results were also (within the uncertainty limits) those of eq. (4.5) (though o increased with decreasing T'max). On the other hand, for each temperature interval below Tc,that did not include the 178.549 ~< T ~< 180.312 K range, the results were those of eq. (4.4). All these facts yield a consistent clear picture if we recall that the peak of the resistivity of dysprosium [12] (see fig. 1) is around 167 K. The effect of the many experimental points below 167 K is to bring about a large negative X [eq. (4.4)]. On the other hand, the experimental points between the peak and T c do yield the result (4.5). Thus the value X = - 0 . 3 is associated with the "bending" of the resistivity around 167 K. Since this behavior is not a critical phenomenon (it occurs far from Tc) and may result either from band structure effect [4, 8] or from the monotonicity of the magnetic energy [1 ], the value a' - 0 . 3 is an artifact of the Rexp(Ti) bending. It has nothing to do with the "real" critical behavior. Moreover the results in all temperature sub-interval that correspond to 5 × 10 -2 ~> Itl/> 2.5 X 10 --4 are an excellent evidence that only the ~' = - 0 . 0 3 5 1 value and the other results of (4.5) are meaningful parameters for this material (and probably for holmium and terbium). To compare the fits of the results (4.4) and (4.5) with the data, we show in fig. 12 the ternI 0.0 t

e

0.0

,

" • A

¢

e

¢

:

• e

C~-lO .0 C3

o

-20.0 -30.0 -q0.0 I73.0

i

e

e

177.0 T

e

A

J

181.0

[K]

Fig. 12. Deviation of the data from the function (4.1), for T < T c. This, when the parameters (4.4) are used (circles) and when the parameters (4.5) axe used (triangles).

L Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal perature dependence of the corresponding AR(Ti). As in fig. 11 we see that the results (4.4) (circles) deviate from the AR = 0 line as T approaches Tc while the results (4.5) (triangles) are evenly distributed around this line.

5. Discussion

In the present work we have presented the first systematic analysis of a critical resistivity. This analysis has yielded in addition to the values of the critical exponents, the value of the critical amplitude ratio. We have shown that improper analysis cannot yield the latter value on the one hand, and may yield subjective (see section 2) or wrong values (such as a' = - 0 . 3 for dysprosium) for the exponents on the other hand. In previous works [5-13] no attention was paid to some general physical requirements that have to be fulfilled by the results such as the continuity of the resistivity and its derivative at T c. Here we overcame these shortcomings by conducting a non-linear least-squares-fit analysis that has yielded, in addition to meaningful values for the parameters a, a' and [A/A'[, the fulfillment of these physical expectations. The high quality of the data o f R a o et al. [12] has enabled the determination of these critical quantities without imposition of any constraints. Such constraints were necessary, for example, in specific heat [14, 22] or optical absorption [18] data analyses. The main purpose of this paper was to establish by both theoretical considerations and data analysis that the critical resistivity of antiferromagnetic metals has a magnetic energy-like temperature dependence. This is in contradiction to some previous theoretical suggestions [3, 4] and previous conclusions based on unsatisfactory or incomplete analyses of experimental data [8, 12, 13]. The above purpose was indeed achieved since the resistivity of dysprosium at the close vicinity of Tc, i.e., [t[ < 3 × 10 -2 was shown to h'ave a magnetic energy-like behavior both above and below T c. Recalling that t3 in this material is 0.335 there is no way to interpret the present (and previous [ 12, 13 ]) results with negative a's in terms of a it[ 2a or [t[&like temperature dependence, in this it[ range. Since hand structure effects were predicted to yield such a dependence [4, 8] we can conclude that the results obtained from the experimental data are

67

not consistent with the manifestation of these effects. Hence our perturbation approximation (see section 2) is valid in this Itl range. The fact that similar exponents were derived [5, 13] for Tb and Ho indicates that in these materials as well as in the present material the critical resistivity reflects the effect of critical fluctuations on the mean free time of the carriers rather than the fhictuation's effect on the number of carriers. In the present case the conclusion on a magnetic energy-like behavior is supported by the proximity of the value of [A/A'[ to the theoretical predictions (see below). The transition to the t 1/2 mean field behavior with increasing t (that we found as in previous works [5, 13], to be quite abrupt) is also in accord with the effect of the large-q spin fluctuations on the mean free time of the carriers. As was shown in detail [ 1 ] this effect yields a p oc tl/2 dependence. If small-q spin fluctuations would have become dominant, as suggested by Singh and Geldart [13], then a t In t type dependence should have been observed [1 ]. Further, as expected on physical grounds and as was shown in ref. 1, tire transition from a behavior dominated by large-q fluctuations to a behavior dominated by small-q fluctuations will be on a much broader log It1 range than the transition observed here. We should also note that in an antiferromagnetic metal where Q ~ k F such a transition will occur [1] for [ti > 0.1, i.e., outside the critical regime [16]. We can thus conclude that in the entire temperature range where the resistivity decreases with temperature (205 >/T~> 167 K, in the present material, see figs. 1 - 3 ) the perturbation approximation used in section 2 holds. This is quite a wide region (t < 0.1), and thus, one can safely apply the perturbation approximation when critical behavior is considered [1,28]. This approximation, which is supported by the present findings, enables the calculation of critical resistance [ 1] and magnetoresistance [28] in antiferromagnets. The physical mechanism described in section 2 applies then to the It[ < 0.1 temperature region. Farther away from Tc, band structure effects and other scattering mechanisms may be important, but this is beyond the present interest, namely, the critical behavior. As was pointed out in the introduction, this paper is also meant to be a response to the call for experimentally determined critical parameters [15, 29]. The values of such parameters are of special impor-

68

L Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal

tance since "the accuracy of the e-expansion (e being 4 - d where d is the dimensionality of the system) is probably not very great when e = 1, and the actual values of these new exponents should be sought by experiment" [30]. Hence, we should compare the critical parameters obtained here with the theoretical predictions [15, 29]. Before making such a comparison one should examine the accuracy of both experimental and theoretical values. Let us review then the available experimentally determined critical parameters for Dy and the accuracy of the theoretical predictions, a = a' = - 0 . 1 7 . By M6ssbauer spectroscopy [31] an average value of/3, which was determined over the range 2 × 10 -3 ~< Itl-<< 3 X 10 -1 , was found to be 0.335. The early data of specific heat measurements has shown [32, 33] a clear transition to logarithmic dependence as Itl decreases below 3 X 10 -2. (While in Tb a transition to a sublogarithmic dependence was found [34] ). This is in accord with the present t G value for the transition from mean field to critical behavior. An indirect and insensitive method to determine a from magnetic susceptibility measurements [35] has yielded a = 0.03 -+ 0.02, which is again in accord with a logarithmic-like dependence of the specific heat. The more recent specific heat [36] results were improperly analyzed yielding a transition from an a = a' = - 0 . 0 2 -+ 0.01 behavior to an cz = a' = +0.18 + 0.08 behavior with decreasing Itl. Since the shortcomings of such analysis were discussed in detail already [22], we will only mention here that the latter values are an artifact of the analysis. The interpretation of these results in terms of a crossover to a dipolar dominated behavior (which is expected to be irrelevant to antiferromagnets [30, 37] ) is thus unfounded. As was discussed in detail in section 4, the values a = 0.4 and a' = - 0 . 3 , obtained in the previous analysis [ 12] of the data used here, do not describe the behavior in the critical regime [16]. The critical parameters obtained in the present work, a = a' = - 0 . 0 4 -+ 0.005 and IA/A'I --- 1.44 + 0.05, seem to be the most accurate and reliable critical parameters obtained thus far for dysprosium since they do not vary over the interval 3 X 10 --4 < Itl < 3 × 10 -2 and its subintervals. The present parameters are independent of the computational details and yield the physical expectations of continuity of the resistivity as well as the continuity o f its temperature derivative at T c. All this, without any constraints on the parameters.

The theory of Bak and Mukamel [15, 29] shows that the critical behavior of Dy-like materials can be described by a model of n = 4 components orderparameter and a Hamiltonian that has a tetragonal symmetry. The critical behavior was studied in the exact renormalization group to second order in e. The exponents found for the stable tetragonal fixed point are equal to those of the n -- 4 isotropic fixed point. Generally, it is known however, that critical exponents may be significantly different [30, 38], when they are developed to higher order in e. In the present case they may vary from those of the n = 4 isotropic fixed point [29], and even the fixed point which determines the critical behavior, may be different from the predicted tetragonal point [29]. Additional inaccuracy is added if one tries to deduce numerical values for the parameters by substituting e = 1 in the e-expansion. For example: in the present case, the theoretical predictions to order e 2 and with e = 1 are [29], a = a' = - 0 . 1 7 and v = 0.70 (u being the correlation length exponent). If we use the scaling relation a = 2 - ud with the above value of v we get a = a' = - 0 . 1 0 . Since in this case there are no series calculations to compare with, we may take the discrepancy between the above theoretical a = a' values as an estimate o f the order of uncertainty in the numerical values predicted by the theory. It is not impossible then to attribute the large discrepancy between the above theoretically calculated values, - 0 . 1 7 < a = c~' < - 0 . 1 0 , and the present experimental values, a = a' = - 0 . 0 4 , to the uncertainty in the former values. On the other hand, since the values predicted by the theory are reasonable (see below) one cannot be content with such an explanation for the discrepancy. Below, we consider other possible explanations. While the above uncertainty applies to the present anisotropic case, the predicted values are close to what one would expect for the n = 4 isotropic case. This is since the values a = a ' = - 0 . 1 7 are smaller than the values obtained by series expansion for the isotropic n = 3 model [19, 26] (a = a ' = - 0 . 1 4 ) and since this is expected from the trend o f decreasing ct's with increasing n [39]. In this context it is also worth considering the critical parameter IA/A'I for which we do not as yet have a prediction for the anisotropic n = 4 model but we do have the prediction [26] :

IA/A'I = 2~(1 + e)(n/4) + ¢ ( e 2)

(5.1)

for the generalized classical Heisenberg ferromagnet.

L Balberg and A. Maman/Critical resistivity ofan antiferromagnetic metal If we use this relation with a = a' = - 0 . 1 7 , e = 1 and n = 4 we obtain iA/A'[ = 1.75. Again, while one cannot be certain about this numerical value, the above trend-argument indicates that this is a reasonable estimate since with n = 1,2, and 3 one gets IA/A'[ = 0.50, 0.99 and 1.36 respectively [26]. As the theoretical values appear to be reasonable for the n = 4 isotropic case, one should consider experimental reasons for the above discrepancy. In particular, material problems such as the smearing of the transition that may occur in real crystals [12]. This does not, however, appear to be the present case because the smearing of T c is very unlikely to yield equal exponents and reasonable values for [A/A'[. More important, such smearing should make the exponent more negative [22, 27] and thus would have "improved" the agreement between experiment and theory. In fact, if there is any smearing it will imply that the true experimental exponents are less negative than - 0 . 0 4 . The above discussed discrepancy may also be attributed to the fact that the form [eq. (4.1)] used for the analysis is too simple and that the proper correction terms have not been included [22]. Let us examine this possibility. We know that the critical behavior of complicated systems is described by a series of crossovers as the critical point is approached [30]. On symmetry grounds [15] and under the assumption that the anisotropic exchange is an irrelevant variable in the corresponding e-expansion [29], the following sequence of crossovers are feasible for Dy-like systems: Gaussian ~ n = 4 isotropic ~ n = 4 tetragonal, or Gaussian ~ n = 2(xY) isotropic ~ n = 4 tetragonal (the first case corresponds to u >> w = 2v, and the second case to w = 2v >> u in the Hamiltonian of ref. 29). Within the framework of this model one may argue that the experimentally analyzed temperature region (3 X 10 4 ~< It[ ~< 3 X 10 -2) is a crossover region to the tetragonal behavior. Thus the addition of the proper correction-to-scaling terms [37] may improve the agreement between theory and experiment [22]. Mukamel [29] has calculated the correction exponents col = P)ti for the tetragonal fixed point and found the values 601 = co2 = 0.12 and co3 = 0.32. Hence we have tried to find the parameters of the best fit of the experimental data (in the I T - Tc[ < 2.5 intervals), to the functions [22, 37] :

69

[A/a(1 - a)l Itl 1 -~(1 + Dltl wi) + (B - A/a)ltl + C. (5.2) The only variation in comparison with the function (4.1) is the introduction of the correction terms (1 + Dlt]~°i). We found that these corrections have neither changed the values of the ten parameters given in eq. (4.5) (e.g., we find here that a = - 0 . 0 4 3 7 + 0.002 for coI = co2, and a = - 0 . 3 9 8 -+ 0.02 for co3) nor the values of o and o' ( ~ 4 X 10-2). This leads to the conclusion that the deviation of the critical parameters,in the present work,from the theoretically calculated ones, is not due to the experimental data or its analysis. In view of the reliability of the theoretical values and the presently obtained values, one would like to find an explanation to the above discrepancy that will be compatible with both. Recalling that the anisotropy energy is relatively large in Dy [25] (i.e., w = 2u > u in terms of ref. 29) it is suggested that the second sequence of the above crossovers takes place, and that the tetragonal behavior will be observed only for relatively small Itrs. if this is the case, then the n = 2 behavior [26] (a = a' = - 0 . 0 2 , [A/A'[ = 0.99) may be observed in the presently investigated temperature range. This is quite an attractive explanation since these X Y predicted values are quite close to the present findings a = a' = - 0 . 0 4 and [A/A'I = 1.44. In the above discussion we did not consider the RKKY oscillatory nature of the exchange. While this is justified in the case of ferromagnets (where this interaction is an irrelevant variable in the e-expansion) it is not justified a priori in the present case. Here the period of the planar spiral spin arrangement [25] 1/Q is very close to 1/k F and the interaction may be relevant [40]. This problem has not been investigated thus far, and it is not impossible that such an investigation will yield theoretical values which are in better agreement'with the experimental values obtained here. In conclusion, the present analysis establishes the magnetic energy-like behavior of the resistivity in antiferromagnetic metals. It shows that in the rareearth antiferromagnets a transition from a mean field behavior to a critical behavior takes place at it[ 3 × 10 -2. The results in the critical regime yield the most reliable critical parameters for the present

70

I. Balberg and A. Maman/Critical resistivity of a n antiferromagnetic metal

material and indicate the usefulness o f critical resistivity m e a s u r e m e n t s for the d e t e r m i n a t i o n o f these parameters. The discrepancy between the values o f the critical parameters which were obtained here and the available theoretical values m a y be explained by the relative importance o f the anisotropy energy and the R K K Y exchange in the critical behavior o f dysprosium'.

Acknowledgements The authors are indebted to S. Alexander and D. Mukamel for m a n y stimulating discussions and to A. A h a r o n y for his helpful c o m m e n t s on the manuscript.

References [1] S. Alexander, J. S. Helman and I. Balberg, Phys. Rev. B13 (1976) 304 have reviewed the recent developments in the field and discussed the critical resistivity of antiferromagnets in the quasielastic approximation. The physical nature of the corresponding interaction and references for the transport problem can be found in I. Balberg, Physica 91B (1977) 71. A recent justification for the use of the quasielastic approximation for the close vicinity of T c was given by J. S. Helman and I. Balberg, Solid State Commun. 27 (1978) 41. [2] D. J. W. Geldart and T. G. Richard, Phys. Rev. B12 (1975) 5175; B15 (1977) 1502 have discussed the effect of anisotropic Fermi surface on the critical resistivity of ferromagnets and antiferromagalets. [3] K. Binder, G. Meissner and H. Mais, Phys. Rev. B13 (1976) 4890. [4] Y. Suezaki and H. Mori, Prog. Theor. Phys. 41 (1969) 1177. [5] G. T. Meaden, N. H. Sze and J. R. Johnston in Dynamical Aspects of Critical Phenomena, J. I. Budnik and M. P. Kawatra, eds. (Gordon and Breach, New York, 1972) p. 315. [6] C. Akiba and T. Mitsui, J. Phys. Soc. Japan 32 (1972) 644. [7] H. Taub, S. J. WiUiamson, W. A. Reed and F. S. L. Hsu, Solid State Commun. 15 (1974) 185. [8] R. A. Craven and R. D. Parks, Phys. Rev. Letters 31 (1973) 383. [9] I. Balberg, S. Alexander and J. S. Helman, Phys. Rev. Letters 33 (1974) 836. [10] L. W. Shacklette, Phys. Rev. B9 (1974) 3789. [11] D. S. Simons and M. B. Salamon, Phys. Rev. B10 (1974) 4680.

[12] K. V. Rao, O. Rapp, Ch. Johanneson, D. J. W. Geldart and T. G. Richard, J. Phys. C. Solid State Phys. 8 (1975) 2135. [13] R. L. Singh and D. J. W. Geldart, Solid State Commun. 20 (1976) 501. [14] A. Kornblit and G. Ahlers, Phys. Rev. B8 (1973) 5163. [15] P. Bak and D. Mukamel, Phys. Rev. B13 (1976) 5086. [ 16] S. K. Ma, Modern Theory of Critical Phenomena (W. A. Benjamin, New York, 1976). [17] F. C. Zumsteg and R. D. Parks, Phys. Rev. Letters 24 (1970) 520. [18] I. Balberg and A. Maman, Phys. Rev. B16 (1977) 4535. I. Balberg, A. Manan and S. Alexander, Solid State Commun. 28 (1977) 701. [19] G. Ahlers and A. Kornblit, Phys. Rev. B12 (1975) 1938. [20] F. L. Lederman, M. B. Salamon and L. W. Shacklette, Phys. Rev. B9 (1974) 2981. [21] P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969). Chaps. 9-11. [22] A. Kornblit and G. Ahlers, Phys. Rev. B l l (1975) 2678. [23] D. J. Marquardt, J. Soc. Indust. and Appl. Math. 11 (1963) 43. [24] D. J. Marquardt, Share (IBM) DPE NLIN 3094 (IBM Report, Unpublished, 1964). [25] B. R. Cooper, Solid State Physics 21 (1968) 393. [26] E. Br6zin, J. C. Le Guillou and J. Zinn-Justin, Phys. Lett. A47 (1974) 285 and in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976) Vol. VI. p. 125. [27] Such cases were encountered in the analysis of published data on the critical resistivity of some magnetic alloys (I. Balberg, unpublished). [28] I. Balberg and J. S. Helman, Phys. Rev. B18 (1978) 303. [29] D. Mukamel, Phys. Rev. Letters 34 (1975) 481. [30] A. Aharony, Phys. Rev. B8 (1973) 4270 and in Phase Transitions and Critical Phenomena (Academic Press, New York, 1976) Vol. VI. p. 357. [31] E. Loh, C. L. Chien and J. C. Walker, Phys. Letters A49 (1974) 357. [32] M. Griffel, R. C. Skochdopole and F. H. Spedding, J. Chem. Phys. 25 (1956) 75. [33] E. B. Amitin, Yu. A. Kovolevskaya, F. S. Rahkmenkulov and J. E. Paukov, J. Soy. Phys. Solid State 12 (1970) 599. [34] L. D. Jennings, R. M. Stanton and F. H. Spedding, J. Chem. Phys. 27 (1957) 909. [35] L A. Boyarskii, F. M. Zemerov and A. I. Romanenko, J. Sov. Phys. Solid State 17 (1975) 669. [36] F. L. Lederman and M. B. Salamon, Solid State Commun. 15 (1974) 1373. [37] A. D. Bruce and A. Aharony, Phys. Rev. B10 (1974) 2078. [38] A. Aharony, Phys. Rev. B10 (1974) 3006. [39] M. E. Fisher, Rev. Mod. Phys. 46 (1974) 597. [40] D. Mukamel and A. Aharony, Private Communication.