Critical behaviour of the thermal conductivity near the Curie point of gadolinium

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Critical behaviour of the thermal conductivity near the Curie point of gadolinium

J.B. Sousa, R.P. Pinto, M.M. Amado, M.F. Pinheiro, J.M. Moreira, M.E.

Braga

To cite this version:

J.B. Sousa, R.P. Pinto, M.M. Amado, M.F. Pinheiro, J.M. Moreira, et al.. Critical behaviour of

the thermal conductivity near the Curie point of gadolinium. Journal de Physique, 1980, 41 (6),

pp.573-578. �10.1051/jphys:01980004106057300�. �jpa-00209281�

Critical behaviour of the thermal conductivity

near the Curie point ^of gadolinium (*)

J. B. Sousa, ^R. P. Pinto, ^{M. M.} Amado, ^{M. F.} Pinheiro, ^{J. M.} Moreira and M. E. Braga

Centro de Fisica da Universidade do Porto, Porto, Portugal

(Reçu ^{le 23 août} 1979, ^{révisé le} 24 janvier, accepté ^{le 29} février 1980)

Résumé.

Une nouvelle technique ^{de mesures} relatives de la conductivité thermique (K resolution 1 : 104) ^au voisinage ^des points critiques ^{a été} appliquée ^{au Gd. Les résultats sont discutés en relation avec} les contributions :

électronique, ^des phonons ^et de la conduction bipolaire. ^Des résultats très precis ^sur la résistivité (p) ^{et sa dérivée} d03C1/dT dans le Gd sont utilisés pour analyser ^la dépendance de la conductivité thermique ^{en fonction de la tem-} pérature, K(T).

Abstract.

High resolution data (1 : 104) ^is presented ^on the behaviour of the thermal conductivity (K) ^{of Gd}

near the Curie point. The results are analysed ^{in terms of} electronic, lattice and bipolar conduction. Previous information on the electrical resistivity (p) ^{and the} temperature derivative dp/dT is also used to understand the temperature dependence ^{of K.} Detailed reference is made on the experimental method used.

Classification Physics ^Abstracts

72.15E

1. Introduction.

Previous work on the thermal

conductivity (K) ^of polycrystalline Gd failed to reveal _any anomaly ^{at the Curie} temperature 7c [1, 2].

However, well below Tc ^some anomalous behaviour

was found, ^{but the} experimental ^{results were incon-}

sistent. Whereas in ref. [1] ^a minimum is observed in K around 230 K and a sharp ^{increase occurs in} dK/dT ^{near 270 K in ref.} [2] both features _appear

practically together, ^at around 270 K.

For monocrystalline samples, ^{the c-axis thermal} conductivity ^{exhibits a minimum near 270 K, but no} anomaly ^{was found at} the Curie temperature [3].

For the a-direction, ^a change ^{in the} slope ^{of the}

thermal conductivity (dK/dT; ^from negative ^to posi-

tive values) ^was observed, ^{but at a} temperature (T*) slightly above the Curie point, ^T*

T, ^{+ 1 K} [3].

In spite ^{of such} results, ^the anomaly in the elec- trical resistivity (p) ^occurs consistently ^{at the same} temperature (Tc), ^{both for} poly- ^and monocrystalline samples. Therefore, K-anomalies observed well below

Tc ^are probably associated with secondary effects,

not directly ^{linked to the} ferro-paramagnetic ^transi-

tion in Gd.

(*) ^Work financially supported by ^INIC (Portugal) ^{and a NATO}

Research Grant (no. 1481).

From this previous ^{work [1-3] we can} simply

conclude that the K-anomaly ^{near the Curie} point

is very weak, ^{and so} far detected only along ^{the a-} axis, where it is linked to a slight ^{increase in the} slope dKIdT ^{as the} sample ^{enters the} paramagnetic ^state.

The apparent discrepancy ^{T* =1=} Tc ^{remains an} unexplained ^feature.

In view of this state of affairs, ^{we started a detailed} study ^{of the thermal} conductivity ^{of Gd near the}

Curie point. ^{The recent} improvement ^{of an} experi-

mental method ^to detect minute changes ^{in K near}

a transition point [4, 5] ^{enabled us to observe a small}

but well defined anomaly ^{in K,} precisely ^located

at 7e (not T*), ^{and in} good agreement with the elec- trical resistivity ^data (p, dpIdT) ^{obtained in the same} sample. Besides the observed change ^{in the} slope dK/dT, ^more complex features appear in the imme- diate vicinity ^{of Tc, where a} slight depression ^{is observ-}

ed in K. This suggests the existence of extra scattering

for the heat carriers very ^near Tc. ^{On the other} hand,

our results enabled (for ^{the first} time, ^we believe)

the experimental observation of a Curie point ^K- anomaly ⁱⁿ polycrystalline ^Gd.

The results are discussed in terms of electronic, phonon ^and bipolar contributions to K in Gd. Physi-

cal arguments ^are also advanced to understand _pre- vious K-anomalies well below Tc, ^{and the} _apparent discrepancy between T* and Tc ^values.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004106057300

574

2. Experimental ^{method. q In a} thermal conducti-

vity measurement (K = f.Q.ð.T-1), apart from ^a form factor f, ^{we must} accurately ^{measure the small} temperature difference ð.T and the heat flow inside

the sample (Q ).

The accurate measurement of Q ^is very difficult, because it involves ^the estimation of all heat losses from the sample; ^this explains the usual difficulty

in achieving ^{even 0.1} % accuracy in standard mea- surements. However, significant progress ^can be made in the case of critical phenomena, ^{since the} important changes ^{occur here over a} very restricted range of temperature, within which the non critical

K-contributions hardly change. Therefore, ^{if we are} able to keep Q ^{and heat losses constant over such} reduced _range we can write :

i.e. it becomes unnecessary ^to measure Q (simply ^a constant) ^{in order to} obtain relevant information for critical phenomena ^studies (changes ⁱⁿ K).

According ^{to (1),} high K-resolution requires high

resolution in AT. For a typical ^{value AT}

^{0.5 K,}

and thermocouple thermometry ^{with a} sensitivity

S ‘ 10 AV.K-’, ^{we must measure a small} voltage

ð.V

S.AT ~ 5 gV. ^To achieve K-resolutions of a

few parts ⁱⁿ 104, ^{as it is} usually required ⁱⁿ phase

transition studies, ^nanovolt resolution becomes impe-

rative. In addition, ^the proper study of the critical

region usually requires ^a period ^of several hours.

Thus it becomes also essential to ensure an intrinsic

stability ^{of the} measuring equipment ^{down to the nV}

level over a fairly long period ^{of time.}

By using very stable digital nanovoltmeters (Teke-

lec 925; 4 1/2 digits ; 1 ^nV resolution), ^a tight ^control

on the room temperature, ^and by taking exceptional precautions ^{in the} design ^and setting up of the measur-

ing equipment, sample assembly ^and experimental

chamber (shielding, ^thermal anchoring, 5 1 oil-bath to increase thermal stability; reduced heat losses),

we succeeded in obtaining ^the required ^resolution

and stability ^to perform ^the measurements. For

example, ^{with the} sample ^{heater on} (AT rr ^0.5 K),

the emf. e corresponding ^{to AT could be} kept ^vir- tually ^{constant (to within I} nV) ^over very long periods

of time :

In order to check the reproducibility, ^we intentionally produced ^transient perturbations ^{of the} temperature of the oil-bath. As shown in figure 1, ^{the emf. 8} always

returns to the same initial value (to ^{within 1} nV),

even for E-departures ^from equilibrium ^as high ^as

200 nV.

With such high degree of intrinsic stability, ^resolu-

tion and reproducibility, ^{we could} safely ^{avoid the}

check on zeros between consecutive points. ^{In the}

Fig. ^1.

Stability of the thermal emf. (E) in the differential ther-

mocouple (AT). A-B : bath in thermal equilibrium; e ^{stable to}

within 1 nV over periods ^{of 1 h.} B-C : transient perturbation ^{of the}

bath produces ^a sharp departure ^{of 8 from} equilibrium (150 nV).

However, the system ^recovers stability. (C-D) ^{and E reaches} practi- cally ^{the same} initial value (to ^{within 2} nV). Repetition ^{of the}

transient perturbations gives practically ^{the same results} (E-F- G-H).

standard technique, ^{this means} switching-off ^the sample ^heater (Q

0) ^and waiting ^{until a new} equi-

librium is reached (zero readings). Unfortunately,

this very process alters the temperature distribution in the experimental ^chamber (and ^heat losses) beyond

an acceptable limit under our present ^aims of accu;

racy (1 : 104). ^{In our} method, ^we keep ⁱⁿ turn Q

constant all _{the way} through the transition region, using ^{a d.c.} power supply ( I : 104 stability) perma-

nently ^{switched on to the} sample ^heater.

The temperature difference AT ^0.5 K) ^{is measur-} ed directly ^{on a} differential Cu-constant thermo-

couple ^{with its} junctions ⁱⁿ good ^{thermal contact with}

the sample, by ^a special method which uses metallic solder but avoids electrical contacts. The mean tem-

perature ^{T of the} sample is measured with a Cu- constantan thermocouple; ^absolute precision ^{in T}

is of the order of 0.03 ^K (1 : ^{104 in relative terms,}

for Tc & ³⁰⁰ K).

The reproducibility ^{of K-data in different} experi-

mental runs is of the order of a few parts ⁱⁿ 10’, ^but by scaling ^the corresponding K-values to K at a cer-

tain temperature, e.g. K(T,), ^the experimental ^curves

can be reproduced ^{to within a few} parts ⁱⁿ 104. Further details have been published ^elsewhere [4], ^{and a}

more extended version will be given ^{in due course} [6].

3. Thermal conductivity ^{of Gd} ( T = yj.

^Ther-

mal conductivity data in the vicinity of the Curie temperature ^of gadolinium (T/Tc - ^I ;5 10- 2) ^is presented ^{for a} fairly pure sample ^{with a} resistivity

ratio p(300)Jp(4.2 K) = ^{86 and} p(4.2)

^1.47 pH. ^cm.

The available sample ^{was a} polycrystal with dimen- sions appropriate ^{to this type of} measurements

(I

^1.5 cm, 0 - 4 mm). ^The temperature ^{rate used}

to span the transition region ^was slightly ^below

0.005 K . min. -1, ^{and the} vacuum was better than 10-6 torr.

Figure ² shows K in the transition region, ^with

data taken at 0.08 K intervals, giving ^{a total of}

Fig. ^2. Temperature dependence ^of the thermal conductivity (K) of Gd in the vicinity ^{of the critical} point T,

^{292.92 K.}

50 points ^{in a} temperature range where K changes only by ^0.7 %. ^{The Curie} temperature, ^{as determined} from accurate dp/dT measurements (Fig. ^{3), is}

Fig. ^3.

^{Lower curve :} critical behaviour of (1/ Pc) (dp/dT) compared ^with (K - Klin) ; ^{see text for} Klin. ^{Inset :} (K - Ke) ^{vs. T.}

marked by ^{an arrow at} Tc

^{292.92 K.} The absolute value of K in our polycrystalline sample ^{at the critical} point (0.124 W.cm-’.K-’) compares favourably

with the available data for polycrystalline ^Gd (0.140 W. cm - 1 . K - 1; ^ref. [1]) ^{and mono-} crystalline samples (Ka

^{0.103 W . cm-1. K-1,} K,

^0.108 W . cm-’ . K-1; ^ref. [3]). ^{In this}

context the data given ⁱⁿ [2] appear rather low

(K = 0.07 W. cm-’. K- 1).

An important ^{feature in the} results of figure ^{2 is the}

clear increase in the slope dK/dT ^as gadolinium ^enters

the paramagnetic ^state. As shown in section 4, ^this is mainly ^{due to a} mean-free-path ^{effect on the elec-}

tron contribution ^{to the thermal} conductivity.

Besides the change ⁱⁿ slope, ^our K-results indicate

a more complex ^behaviour very ^near Tc. ^{A better} insight ^can be obtained from (K - Klin), ^where Klin is the extrapolation of the linear part ^{of K in the}

ferromagnetic phase; this is shown in figure ^{3. It}

becomes clear that the thermal conductivity ^is depres-

ed in the transition region, possibly ^{due to extra} scattering ^caused by ^{the critical} fluctuations. This

seems consubstantiated by the behaviour of the tem-

perature derivative of the electrical resistivity, ^{as also} given ⁱⁿ figure ^{3 : the} sharp peak ⁱⁿ dp/dT ^at Tc

^{292.92 K indeed} coincides with the minimum in (K - Klin).

A relevant result from our investigation ^{is that it}

appears ^not legitimate ^to identify ^{the Curie} point

with the intersept ^{of the linear} extrapolations ^{of K}

from the ferro- and paramagnetic phases ; ^{in our case}

one would get ^T*

Tc ^{+ 0.9 K. This} may explain why ^T*

T, ^{+ 1 K in} previous ^{results for} gadoli-

nium [3], ^{where the} scarcity ^of data-points naturally prevented ^the subtle but relevant deviations from

linearity ^near Tc ^{to be} experimentally ^observed.

In an attempt ^to pin the relevant contributions to K, let us estimate the order of magnitude of the electro- nic contribution (Ke), ^{as derived from the electrical} resistivity ^data (p) ^{and the use of the Wiedemann-}

Franz law :

with Lo

^{2.45 x 10-8 W.Q.K-l. For rare earth} metals, ^this procedure ^is justified ^{both at low} tempe-

rature [7] ^{and at} high temperatures (T > 0 z ¹⁵⁰ K;

see ref. [3]). ^In absolute terms, ^{at T}

300 K we have p

126 J.I!l.cm, ^{so that} Ke ^{= 0.06} W . cm -1. K -1.

Since K = 0.12 W .cm - 1 . K -1 experimentally, ^we

conclude that (extra) important K-contributions do exist in Gd, ^{a fact to} keep ^{in mind when} interpreting

the critical behaviour of K near Tc.

4. Discussion.

4.1 CHANGE OF SLOPE (dK/dT)

NEAR Tc.

^{Let us} consider first what would be

expected from the electronic contribution only. ^Since

K, ^is proportional ^to the electronic specific ^heat

576

(Ce ^oc T) ^{and mean free} path ^{(/.) we can write imme-} diately :

Below the Curie point, A ^increases rapidly with decreas-

ing T, ^{due to the} increasing ^{order in the} system

(d2/dT 0); ^{in the} paramagnetic phase ^{one has}

instead d£/dT = ⁰ (complete disorder). An increase in the slope then follows from _eq. (3) ^{when the} sample

enters the paramagnetic state, in qualitative agree- ment with the results of figure ^{2. In addition, since À}

above Tc ^is practically ^constant (spin-disorder ^domi- nance) ^and C, ^oc T, ^{one should} expect ^a quasi-linear increase of K with T in the paramagnetic state, which appears consistent ^{with the} experimental ^results.

Further information can be gained ^{if we extract}

from K(T) ^the electronic contribution Ke(T). ^As expected, ^the difference in the slopes above and below

Tc is reduced. However, ^a significant ^{mismatch still} persists, ^{as can be seen from the curve} (K ; Ke)

in the inset of figure 3, ^{where the} positive slope ^above

the Curie point ^{(3.5 x 10-4} W.cm-1.K-2) ^is _clearly bigger ^{than below} T,. We should notice that a positive slope has also been found by ^{Nellis and} Legvold [3] in

the (K- Ke) ^curve for Gd above Tc, precisely ^{with the}

same order of magnitude ( N ^10-4 W . cm-1. K- 2).

These results suggest ^{an intrinsic feature in} Gd, possibly associated with extra contributions to K which appear enhanced above Tc.

4.2 DIFFERENT K-CONTRIBUTIONS.

An impor-

tant contribution to K _comes, of _course, from pho-

nons (Kg), ^but unfortunately ^the corresponding tempe-

rature dependence (dKgldT ^{0) cannot explain the}

observed positive slope ⁱⁿ (K - Ke).

An interesting ^effect which increases with ^T has been invoked ^{for rare earth} metals, ^{due to the} bipolar

electronic contribution (Kb) ^to the thermal conduc-

tivity [8, 9]. ^In principle ^{this appears} when several electronic bands overlap ^{near the} Fermi level (e.g.

5d and 6s bands in Gd). The idea is that whereas for conduction in one band only, ^the boundary ^condition imposed in K-measurements (zero electrical current) permits ^{no net} particle ^{flux, when carriers of both} signs ^are present, equal particle ^{currents in the two}

bands are allowed. There is still no net transfer of

charge (electron-hole) ^{but the} pairs ^do carry ^an energy equal ^{to the sum of the two} transport energies

relative to the Fermi level [8], ^and this increases with the thermal excitation KT. Of _course, the Kb ^contri-

bution is expected ^{to be small at low} temperatures.

We can now write for the total thermal conductivity :

In doing ^{so, we are} ignoring possible contributions from _magnons, an assumption ^{which seems} justified

for temperatures rather close and above T, [10, 11].

It would be of interest to estimate the relative impor-

tance of Kg ^{and Kb in Gd. As pointed out by Kle-}

mens [12], ^{the accurate thermal} conductivity separa- tion is difficult at high temperatures. ^{Therefore we} will be satisfied with an approximate ^value of Kg ^itself,

for which more reliable information exists.

For such order of magnitude ^{estimate, we consider}

the lattice conduction ^not much different _among all

heavy ^{rare earths. We then} choose elements which do not exhibit complications ^{due to} magnetic ^transi-

tions neither (anomalous) positive slopes ⁱⁿ (K - Ke)

at high temperatures. The elements Yb and Lu are

in such favourable conditions ; indeed, K-data for both elements are remarkably ^{close over} the range of temperatures where the phonon contribution dominates (T ^{100 K ;} [13]).

Since we are interested in Kg ^at high temperatures,

we use available K-data for Lu in the range 200- 300 K (b-axis ; [14]), ^{assuming it to be rather close to} Kg.

The following temperature dependence ^is then obtain- ed :

withA

6.34 W.cm-1 and B

0.02 W.cm-1.K-’.

We interpret ^{the T-1 term as due to} phonon-

electron scattering, ^{and B due to} phonon-phonon scattering; ^this gives Kg = 0.04 W . cm -1. K -1 at T

300 K, ^{and from} (2) ^and (4), Ke

0.06, Kb = 0.02 W . cm -1. K -1. In absolute terms, ^we then conclude that an appreciable part ^{of the K-} contribution in Gd near Tc ^{is due to} phonons. ^How-

ever, in terms of slope ^we expect ^from (5),

for T

300 K. From our experimental ^value

it necessarily follows that a significant positive slope

is expected ^{for the extra} contribution in (3), ^{of the} order dK/dT = ^{+ 3.2 x} 10-4 W .cm-1 K-2.

4. 3 DEPRESSION OF K NEAR T,.

^{We must be}

aware that the localized depression ^{of K near} Tc ^is

not eliminated by subtraction of Ke ^from K, ^{as the} inset of figure ³ clearly ^{shows. In} view of the smallness of the effect, ^{it could be} argued ^{that minute} differences in absolute temperature (or reproducibility) ^{in the} independent p(T) ^and K(T) measurements could

produce ^a Ke(T)-curve slightly mismatched with K,

so as to cause an apparent minimum in the difference K - Ke. ^Such reasoning appears inconsistent in the present ^{case, since} Ke(T) ^{is a} monotonically increasing

function of T near T,(K,,

LT/p; dp/dT positive,

see Fig. 3). Therefore it cannot produce by ^{itself a}

minimum in the difference ^{K -} Ke ; ^{we believe that}

the slight ^{and localized} depression ^{of K near} T,

has a true physical origin.

A similar depression ^{in K has} been observed in the

ferromagnetic compound ^TbZn [15], and in Fe

near T, [16]; ^{it also} appears ^to be present ^{in Ni} [17].

Since these metals have an appreciable ^electrical resistivity, ^{the role of} phonons ^is here enhanced.

Near T, ^the large spatial inhomogeneities ^{due to}

critical fluctuations should affect, ⁱⁿ principle, ^the propagation ^{of the} long wavelength phonons. ^A depression ^{in the} phonon conductivity is in fact predict-

ed near the Curie point [18-20]. ^As explicitly ^shown by Papoular [20], ^such effects become only ^relevant

in the immediate vicinity ^of Tc, ^an observation which links well with the localized character of the depres-

sion experimentally ^{observed in K.}

The reason why ^the K-depression appears rather small in the metals referred above probably ^derives

from the fact long wavelength phonons give ^{a rela-} tively ^small contribution to the energy transport (then K) ^at high temperatures, and also because the

coupling between lattice (phonons) ^and spins ^is usually ^{rather weak in metals. For} insulating ^mate-

rials however, ^{a severe} reduction in the phonon pro-

pagation ^is readily ^{observed near} magnetic phase

transitions [21, 25], particularly ^{in the case of low} Tc materials where long wavelength phonon ^{heat trans-} port ^dominates [23-25].

4.4 K-ANOMALIES BELOW Tc.

^{On the} light ^{of the}

considerations presented above, ^{it would be} appro-

priate to comment on the puzzling differences observ- ed in the behaviour of K for Gd at temperatures ^well below Tc [1-3].

In particular ^one would like to understand why ^the

c-axis dK/d T derivative changes sign ^{at - 270 K}

whereas along the a-axis such change ^occurs slightly

above Tc ; ^{in both cases, the} ferromagnetic ^transition

as given by dp/dT - ^data precisely ^{occurs at the}

same temperature, Tc

^{293 K.}

We can understand these features ⁱⁿ terms of the

peculiar behaviour of dKe/dT, ^with Ke given by ^the

Wiedemann-Franz law (see ^ref. [26]) :

According ^{to our recent c-axis} resistivity ^measure-

ments on single crystals ^{of Gd} (Fig. ^4); (see ^also

ref. [27]), ^{at T = 200 K we have}

so that negative ^{values are} expected ^for dKe/dT, ⁱⁿ

agreement with K-data [1, 3]. However, ^{for T = 220K} the p-derivative ^{shows a} dramatic decrease which is particularly enhanced around 270 K, ^where

According ^{to (6) this} favours the onset of positive (dKIdT), ^at temperatures considerably ^{lower than} T,.

Along ^the a-axis, (1/p).(dp/dT)a ^exhibits apprecia-

ble values _up to the Curie point, ^{so that} dK/dT keeps

the negative sign. However, just ^above Tc, dp/dT

has a sharp ^{break and} rapidly ^{reaches a} sufficiently

low value ^to produce ^{a reversal in the} sign of dK/dT.

Such an interpretation ^is, again, consistent with the

experimental ^facts.

Fig. ^4.

Temperature dependence ^of dp/dT ⁱⁿ monocrystalline

Gd samples, along ^{a- and} c-directions. The anomaly ^{at T = 225 K} corresponds ^{to the} spin reorientation transition in Gd.

The case of polycrystals ^{is more} complex, depend-.

ing ^{on the} particular sample investigated. ^However,

one can safely claim that (1/p) (dp/dT) ⁱⁿ polycrystal-

line samples [28] ^{should be} depressed ^with regard ^to

the corresponding ^a-axis value, ^{due to the} sharp

reduction in (1/p) (dpldT)c ^{which occurs at} temperatu-

res still well below Tc. Therefore, the inversion in the

sign ^of dK/dT ^is likely ^{to occur} definitely ^below Tc,

as has been observed in all polycrystalline ^Gd samples

so far investigated [1-2].

Acknowledgments.

^{The authors are} deeply ^indebt-

ed to Prof. C. S. Furtado for the loan of the Gd sam-

ple, ^{to Prof. J. M.} Araujo for valuable comments

and help, ^{and to} Eng. ^{J. Bessa for} technical aid.

578

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