Critical effects in the electrical resistivity of samarium

Journal of the Less-Common Metals, 41 (1975) 149 - 156 @ Elsevier Sequoia &A., Lausanne - Printed in the Netherlands

CRITICAL EFFECTS SAMARIUM

IN THE ELECTRICAL

149

RESISTIVITY

OF

G. KRITHIVAS Department

of Physics, Brandeis

University,

Waltham, Mass. 02154

(U.S.A.)

G. T. MEADEN Artetech

International,

Cockhill House,

Rowbridge

BA14

9BG (Gt. Britain)

(Received January 3, 1975)

Summary Logarithmic divergences are reported for the 14 K magnetic critical point of samarium in both electrical-resistivity and heat-capacity data. This agrees with the Fisher-Langer and Mannari critical scattering theories. It is also shown that application of de Gennes’ magnetic resistivity relation to the second transition point at 104 K yields reasonable values for the exchange constant and the effective mass of the conduction electrons.

Introduction The effect on the flow of conduction electrons of thermal fluctuations in magnetic-spin systems has been much studied in recent years. Particular attention has been devoted to the relationship between critical fluctuations and the electrical resistivity, following the discovery that the temperature derivative of the resistivity diverges as the temperature approaches the critical temperature, T,. This kind of behaviour has been investigated at ferromagnetic-paramagnetic transitions, as in nickel and iron [ 1,2] , and at antiferromagnetic-paramagnetic transitions, as in europium [ 3 - 51, terbium [4 - 61, and dysprosium [4, 7] . In the present paper, new results of highprecision, electrical-resistivity measurements on samarium of good purity at its two magnetic transition temperatures, 14 K and 104 K, are reported. Because the results are to be discussed in the light of critical scattering theory, a brief outline of the essential theory is appropriate here. In 1958 de Gennes and kiedel [8] showed that within the first Born approximation, the cross section for scattering of a conduction electron is proportional to the spin-spin correlation function. This function was evaluated in the Ornstein-Zemike form for all spin separations, R, at temperature, T. The resulting expression for &/dT may be set in the form dp/dT = (T,-T)/(IT,-TI)(A

log E + B)

(1)

150

where E = (T,--T)/T and A and B are either constants or weakly temperaturedependent terms. The purpose of the factor, (Z’,- T)/( IT,- TI), is to show that the sign of dp /dT reverses at the transition temperature, Te,causing an upward cusp in p. Although this behaviour resembles the well-known anomalous behaviour of europium, and of c-axis gadolinium, many ferromagnets behave quite differently, displaying instead a monotonic temperature dependence with a sharp decrease near the Curie point. When the problem was taken up by Fisher and Langer [9] and Mannari [lo], it was found that the d~a~~rnent originates from the use of the Ornstein-Zemike correlation function which does not correctly describe the microscopic behaviour of the spin system. Instead, it was found that above T, the principal contributions to the energy, U,,(T),and to the resistivity, pn,(Z’), are from short-range spin-spin correlations. Hence, the dominant singularities at 7’, in the magnetic specific heat, C,,(- dU,,/dT), and in necessarily resemble each other. The critical divergence is posi(dp IdT)nuct tive for T 2 T,,so there is no cusp in p sOur experimental-resistivity data for samarium aound 14 K, and the published heat capacity data of Lounasmaa and Sundstrom [ 111, are analysed in order to test for this kind of correspondence.

The samarium sample was initially in the form of a thin, rolled sheet, of thickness 0.27 mm. A specimen, 24 mm X 1.8 mm, was cut from it with the length parallel to the direction along which the sample had been rolled. A test run, down to liquid helium ~mperature, gave a resistivity at 4.2 K of 6.5 pohm-cm with a residual resistivity ratio of 15. The specimen was then annealed at 773 K for one hour. Because of the high vapour pressure of samarium at elevated temperatures, the quartz tube containing the specimen was filled with high-p~ity argon to about one third atmospheric pressure, in order to lessen sublimation effects. After this treatment the resistivity at 4.2 K was lowered to almost 1 &ohm-cm and the residual res~tivi~ ratio improved to 55. The resistivity data were obtained using the accurate measuring equipment described elsewhere [ 4,121. Data analysis and results The tern~~t~e variation of the electrical resistivity in the neighbourhood of the lower critical transition is shown in detail in Fig. 1. A singularity in the dp/dT US.T curve occurs at 13.9 K. For the purpose of the present analysis, this is taken to be the location of the Neel ~mperatu~, TN,insofar as the magnetism modifies the flow of the conduction electrons. The qualitative nature of this curve indicates the scattering effect of fluctuations among localised 4f spins on the conduction electrons. In analysing the experimental data, it is helpful to express dp/dT in the Form &/dT = (Cf~)(e-h-l) + D, (2)

151

where C and D are constants (or only weakly temperature-dependent) and X is the exponent characterising the critical-resistive behaviour. In this form, the equation reveals a power-law dependence when d2p/dT2 is plotted against E, whereas, for h = 0, the divergence is logarithmic. If the accuracy of the experimental data is insufficient to permit the computation and plotting of d2p/dT2, an alternative method due to Sze [4] may be employed. With p. as the residual resistivity, and aT the phonon resistivity, consider the total resistivity p(T) at temperature T, P(T) =~,,,a +P~T)

+ QT+

(3)

~0,

and its particular value at T,, PC!',)=~rnag+~dTc)

+aTc

(4)

+~.a.

Subtracting eqn. (4) from eqn. (3) and rearranging, P(T) --p(T,)= T-T,

~nucU)-~dT,)

IT--J

+(y

’

(5)

When (dp /dT)s,, varies logarithmically, a test is readily made by plotting a semilogarithmic graph between the experimentallydetermined factor (p(T) - p(T,))/(ITT,I) and IT- T,I. A straight line of constant slope would indicate a logarithmic dependence. Figure 2 shows such a logarithmic dependence for T > TN between TN + 0.6 and TN + 2.3, and is consistent with the theoretical proposal of Fisher and Langer [9] and Mannari [lo]. For T < TN, (dp/dT)o,, is usually dominated by the onset of long range order. It is seen from Fig. 2 that below TN the logarithmic dependence extends from TN - 0.6 to TN - 5. It may be noted that the nature of these plots assumes that the non-fluctuation-resistivity contributions to the total resistivity are either constant or linear in temperature. The fact that these assumptions are likely to be true is suggested by the logarithmic graphs which result in Fig. 2. However, at temperatures below TN - 5 the phonon term is known to be non-linear. It was mentioned that on theoretical considerations (dp/dT)s,,, and C- might behave similarly. In an attempt to verify this, we have analysed the carefully-tabulated, heat-capacity data of Lounasmaa and Sundstrom II111 in the neighbourhood of 14 K. The magnetic heatcapacity was separated from the total heat capacity by subtracting the electronic and lattice contributions in the following manner. From the very-low-temperature work of Lounasmaa [ 131 on samarium, the electronic contribution, C,, was taken to be 10.2 mJ/ mol deg. The lattice part was calculated using the Debye function by taking 164 K for the Debye temperature as indicated by Rosen’s elastic constant measurements [ 141. Figure 3 shows the resulting temperature dependence which we derived for the magnetic specific heat. The transition temperature appears to be 13.3 K if it is taken to be the location of the C,, peak; this is quite close to the temperature of the peak in the (dp/dT) us. T plot of Fig. 2. The next step is to find whether this magnetic heat capacity might vary logarithmically.

152

Fig. 1. Electrical resistivity of samarium below 18 K in pohm-cm showing the singularity in dp/dT (@ohm-cm K-‘) which is centred on 13.9 K.

IT-T,I

Fig. 2. Logarithmic plots of the critical resistivity above and below TN,

From Fig. 4, in which & is plotted against logIT- TNI,it appears that there is a possible logarithmic dependence up to 1.6 deg for T--TN and up to 5 deg for TN - I’, with breakdowns occurring for IT- TN15 0.5 deg. The upper transition point Despite several experimental ~ves~gations carried out over the years, the origin of the upper transition point of samarium at 104 K remained obscure until quite recently. The confusion for so long had doubtless been, at least partly, related to the variability of the often low purity of the samples. For example, Lock [ 151 discovered a kink in the magnetic susceptibility curve between 105 and 150 K on a solid specimen which also displayed a maximum at 14.8 K. In the susceptibility work of Graf [16] , however, there was no indication of a Neel temperature effect around 14 K, while the appearance of a rather pronounced kink at around 105 K was att~but~ to the presence of SmN. In addition to the above work on solid samples, Leipfinger [17], Jelinek et al. [18] and Schieber et al. [19] have reported

153

Fig. 3. Magnetic specific heat of samarium deduced from the published data of Lounasmaa et al. [ 91, showing a maximum at about 13.3 K. Fig. 4. Plot of the magnetic specific heat of samarium against logIT - T,I.

the susceptibility behaviour of powder samples. They all observed a lowtemperature peak at around 14.8 K but nothing in the region of lOO/llO K. On the other hand, the electrical resistivity data of Alstad et al. [ 201 and Arajs and Dunmyre [ 211, together with the susceptibility measurements of the latter, supported the occurrence of two transitions. Furthermore, the heat capacity data of Jennings et al. [22] showed maxima near 13 and 105 K. These authors ascertained that the excess heat-capacity contribution at 105 K corresponds to that expected for samarium if the transition had a magnetic origin. However, in the sample of Offer et al. [23] a Mossbauer examination below 77 K did not reveal an ordered spin state. Clarification was finally provided by the neutron diffraction work of Koehler and Moon [24] which was carried out on a large, single crystal enriched with the isotope ‘“Sm. These experiments provided the evidence that the upper transition is caused by magnetic moments ordering on hexagonal-type lattice sites, and the lower transition by moments ordering on cubic-type sites. In the following we examine the electrical resistivity of samarium between 4 and 300 K in order to make various deductions concerning the effects of these two magnetic transitions on the electron scattering. The general appearance of the resistivity curve (Fig. 5) is similar to that of Alstad et al. [20] and Arajs and Dunmyre [ 211. It clearly shows the occurrence of two prominent changes in electron scattering: one at about 14 K and the other at 104 K. Below 104 K the resistivity decreases much more rapidly with decreasing temperature than it does above this temperature. This suggests that above this transition, the magnetic moments caused by localised 4f spins disordered on the hexagonal-type sites lead to a constant spin-disorder resistivity (p mW). From the graph we find this to be 44 pohm-cm, which is close to the value quoted by Taylor [ 251 who probably

154

w

0

I

,

I

15

I,0

225

T

100

(IO

Fig. 5. Electrical resistivity of samarium from 4 to 300 K.

used the resistivity data of Al&ad et al. [20] (namely, higher than the figure of 39 Ccohm-cm as derived from and Dunmyre [21]. It is instructive to test our graphical value for the tivity, P magrusing the formulae of de Gennes [ 261 for tivity and paramagnetic Curie temperature. Firstly, the magnetic-moment disorder resistivity

3Nm*n Pmas = 2 e2EFh

46 pohm-cm). It is the results of Arajs spin-disorder the magnetic

resisresis-

is given by

(6)

G2(g-l)2j~+1)

(m* is the effective mass of the conduction electrons, e the electronic charge, N the number of atoms per unit volume, EF the Fermi energy, g the Land6 factor, j the total angular momentum, and G a coupling constant with dimensions of energy times volume). For temperatures well above the magnetic ordering temperature, p mog is constant. Accordingly, one may assume that for samarium above 104 K (i.e., in the paretic region) the slope of the linear part of the resistivity curve of Fig. 5 is caused solely by phonon scattering. In such a case, the experimental value of 44 pohm-cm for the spindisorder resistivity is just equal to p_ of eqn. (6). A direct test of this equation is not possible because G and m* are not ~dependen~y known. However, if the paramagnetic Curie temperature, fiP, is accessible to compu~tion or to experimental dedication, it can be combined with eqn. (6) in order to calculate G and m*. The indirect exchange interaction between the atomic spins, when employed in a molecular field model, leads to the following expression [ 26 ] for 0 D, kB p = - 3nZ2G2&1

V+‘(g-

1)2j(j + 1) nZ o F(ZK,R,,).

k is the Roltzmann factor, V the atomic volume, and P(X) = (X cosx is the oscillating Ruderman-Kittel function. For 2 = 3 (the number of duction electrons per atom) and for the h.c.p. structure [27], the sum -0.0065. Then, if m is the free electron mass, the result of combining (6) and (7) is

sinx).x-4 conis eqns.

155

(g-l12J(J

+ I)

Fig. 6. Dependence of the paramagnetic Curie temperature, eP, on the ionic

(g- 1)2j(i + 1). m*/m = 7.7 pmsglep

parameter

(8)

at-K! +!!I

9&n,, m* [ (g-1)2j(j+

% 1) I

’

(9)

In these equations pmagis expressed in pohm-cm, eP in Kelvins and G in eVA3. Strictly speaking, samarium has a rhombohedral lattice, but it is possible to reinterpret it in terms of a non-primitive hexagonal form with a c/u axial ratio of c/(4.%2). Because the sum appearing in eqn. (7) is not known for the rhombohedral lattice, the results in eqns. (8) and (9) for the h.c.p. lattice may be adopted as an approximation. Appropriate constants are taken by noting that the ground term of the Sm3’ ion is 6 H,,, , so that j = 5/2 and g = 2/7. Hence, the de Gennes ionic parameter (g- 1)2jg’ + 1) = 4.46. In order to obtain 8,, eqn. (7) is considered in its application to the rare-earth series as a uniform group of elements. Using Fig. 6, which provides the variation of the observed eP with the ionic parameter (g- 1)2j(j + l), the paramagnetic Curie temperature, 8,, is then estimated as 90 K for samarium. Substitution for &, and pmagin eqns. (8) and (9) finally yields 3.8 for m*/m and 2.6 eVA3 for G. These values of m*/m and G for samarium can be compared with the sequence 2.6, 2.8, 2.9, 2.5, 4.2, and 5.7 for m*/m and 3.1, 3.2,3.0, 2.9, 2.2 and 1.9 for G in the case of the six metals, gadolinium, terbium, dysprosium, holmium, erbium, and thulium, which succeed samarium in the rare-earth series. Hence, we conclude that despite the nature of the assumptions made, the calculated values of m*/m (= 3.8) and G(= 2.6 eVA3) for samarium compare satisfactorily with what might be expected on general grounds from the position of this element in the magnetic rare-earth series. It further supports, too, the proof by Koehler and Moon [24] that the upper critical point at 104 K is a magnetic- paramagnetic transition.

156

Acknowledgment We wish to thank Dr N. H. Sze for his help in various ways during the course of this investigation. References 1 P. P. Craig, W. I. Goldburg, T. A. Kitchens and J. I. Budnick, Phys. Rev. Letters, 19 (1967) 1334. 2 Ya. A. Kraftmaker and T. Yu. Romashina, Fiz. Tverd. Tela, 9 (1967) 1851; (English trans.) Sov. Phys.-Solid State, 9 (1967) 1459. 3 N. H. Sze and G. T. Meaden, J. Phys. F, 2 (1972) 693. 4 N. H. Sze, Ph. D. Thesis, DaIhousie University, Halifax, Canada, 1972. 5 G. T. Meaden, N. H. Sze and J. R. Johnston, in J. I. Budnick and M. P. Kawatra (eds.), Dynamical Aspects of Critical Phenomena, Gordon and Breach, New York, 1972, p. 315. 6 R. A. Craven and R. D. Parks, Phys. Rev. Letters, 31 (1973) 383. 7 K. V. Rao, 0. Rapp, T. G. Richard and D. J. W. Geldart, J. Phys. C, 6 (1973) L 231. 8 P. G. de Gennes and J. Friedel, J. Phys. Chem. Solids, 4 (1958) 71. 9 M. E. Fisher and J. S. Langer, Phys. Rev. Letters, 20 (1968) 665. 10 I. Manna& Phys. Letters, 26A (1968) 134. 11 0. V. Lounasmaa and L. J. Sundstrom, Phys. Rev., 158 (1968) 591. 12 G. T. Meaden and N. H. Sze, J. Low-Temp. Phys., 6 (1969) 567. 13 0. V. Lounasmaa, Phys. Rev., 133 (1964) A 219. 14 M. Rosen, Phys. Rev. Letters, 19 (1967) 696. 15 J. M. Lock, Proc. Roy. Sot. (London), B 70 (1957) 566. 16 P. Graf, Z. Angew. Chem., 11 (1961) 534. 17 H. Leipfinger, Z. Physik., 150 (1958) 415. 18 F. J. Jelinek, E. D. Hill and B. C. Gestem, J. Phys. Chem. Solids, 26 (1965) 1475. 19 M. Schieber, S. Foner, R. Delco and E. McNiff, J. Appl. Phys., 39 (1968) 885. 20 J. K. AIstad, R. V. Colvin, S. Legvold and F. H. Spedding, Phys. Rev., 121 (1961) 1637. 21 S. Arajs and G. R. Dunmyre, Z. Naturforsch., 21a (1966) 1856. 22 L. D. Jennings, E. D. Hill and F. H. Spedding, J. Chem. Phys., 31 (1959) 1240. 23 S. Offer, E. SegaI, I. Nowik, E. R. Bauminger, L. Grodzins, A. J. Freeman and M. Schieber, Phys. Rev., 137 (1965) A 627. 24 W. C. Koehler and R. M. Moon, Phys. Rev. Letters, 29 (1972) 1468. 25 K. N. R. Taylor, Contemp. Phys., 11 (1970) 423. 26 P. G. de Gennes, J. Phys. Radium, 23 (1962) 510. 27 Y. A. Rocher, Advan. Phys., 11 (1962) 32.