Transport properties of rare earth intermetallic compounds (electrical resistivity, thermopower and thermal conductivity)

Journal of Magnetism and Magnetic Materials 29 (1982) 181-191 North-Holland Publishing Company

181

INVITED REVIEW

T R A N S P O R T P R O P E R T I E S OF RARE E A R T H INTERMETALLIC C O M P O U N D S (ELECTRICAL RESISTIVITY, T H E R M O P O W E R A N D T H E R M A L C O N D U C T I V I T Y ) *'** E. G R A T Z Inst. f Experimental Physics, Technical University Vienna, 1040 Wien, Austria

and M.J. Z U C K E R M A N N Dept. of Physics, McGill University, Montreal, Quebec, Canada

The temperature dependence of a number of characteristic transport data of magnetic and non-magnetic rare earth intermetallic compounds will be shown. These data are used to discuss the magnetic scattering contribution to electrical resistivity O, thermpower S and thermal conductivity X. The estimation of the influence of magnetic scattering is obtained from experimental data by comparing p(T), S(T) and h(T) of magnetic and non-magnetic isostructural compounds.

1. Introduction The rare earth intermetallic ( R I ) - c o m p o u n d s are good candidates for the study of the influence of the magnetic scattering on transport properties such as electrical resistivity p, thermopower S and thermal conductivity )~. This is because the nature of the magnetic order is quite well understood in R I c o m p o u n d s especially if the magnetism is only due to localized 4f m o m e n t s as is the case for the series REAl2, R E C u 2 , etc. ( R E = rare earth). Further it is possible to estimate the magnetic scattering contribution to the transport properties of R I - c o m p o u n d s b y comparing p, S and X of magnetic and non-magnetic isostructural RI comp o u n d s (e.g. YA12 and GdAI2). The assumption that the lattice contribution to the transport properties is the same in both the magnetic and the isostructural non-magnetic c o m p o u n d s allows us to estimate the contribution of magnetic scattering * This subject will be discussed in detail by the authors in vol. 5, ch. 42 in the Handbook on the Physics and Chemistry of Rare Earths, eds. K.A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam, 1982) p. 117. ** This paper is dedicated to Prof. S. Methfessel on the occasion of his 60th birthday.

to electrical resistivity, thermopower and thermal conductivity. The aim of the paper is to present characteristic transport data and to show how the magnetic scattering contributes to O, S and X using the above mentioned method.

2. Electrical resistivity In fig. 1 the temperature dependence of electrical resistivity of some non-magnetic RI comp o u n d s with different structures is shown (YAI 2, LuA12 cubic MgCuE-structure; Y C u 2, L u C u 2 ort h o r h o m b i c CeCuz-structure; and LaNi orthor h o m b i c CBr-structure). Assuming that the resistivity of non-magnetic RI c o m p o u n d s obeys Matthiessen's rule, which states that p ( T ) is given by p ( T ) = Po + Pph(T),

(1)

where P0 is the residual resistivity caused by lattice defects etc. and Oph(T) is the resistivity due to the e l e c t r o n - p h o n o n interaction. It should be pointed out that in most of the cases it is extremely difficult to obtain reliable values for P0 because of the brittleness of the RI compounds. As the tempera-

0 3 0 4 - 8 8 5 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d

E. Gratz, M.J. Zuckermann / Transport properties of rare earth compounds

182

50

l;O

30

20

10

0

50

lO0

tSO

200

250

Fig. 1. P versus T c u r v e s of Y A l z , L u A l 2, Y C u 2 , L u C u 2 a n d LaNi.

ture rises P0 remains constant, but Pph increases rapidly because of electron scattering by the quantized lattice vibrations (phonons). It can be assumed that the influence of d-electrons on the physical properties of non-magnetic RI compounds with noble metals is negligible. The electrons responsible for the charge transport belong to s - p conduction bands and are scattered by the electron-phonon and electron-impurity interaction into vacant energy states within these bands. A general expression for Pph(T) is given by the Bloch-GrLineisen law [1]. In the high temperature limit Oph is proportional to T. This is found to be in agreement with the experimental data in RI compounds with negligible d-electron contributions. YCo 2 does not exhibit long range magnetic order it should be included in this considerations because it forms the non-magnetic basis for the magnetic R E C o 2 compounds which will be discussed later. The p versus T curve for Y C o 2 is shown in fig. 2. This shows that a pronounced curvature in the p versus T curve is seen near 100 K above which the resistivity tends to saturate. This curvature is not observed in the resistivity of Y non-transition metal compounds such as YA12 and YCu 2. The resistivity of YCo 2 was shown to follow a T z law below 20 K [2]. The T 2 law at low

temperature and the high temperature saturation of O for YCo z was interpreted in terms of the theory of spin fluctuations [3]. At this point it should be pointed out that also the magnetic R E C o 2 compounds which are isostructural to YCo 2 are characterized by such a saturation tendency at higher temperature. The compound Y C o 2 is an enhanced paramagnet. If the Co content of such compounds is sufficiently increased they become magnetically ordered. This is the case for the concentration range from YCo 3 to Y2C0~7. On this basis it was expected that Y4Co 3 would be a paramagnet and somewhat less enhanced than Y C o 2. However, Y4Co3 exhibits magnetic order below 6 K. In addition to the magnetic state a superconducting state was found below 2 K by ac-susceptibility measurements and confirmed by resistivity, specific heat and magnetization measurements [4-7]. In the case of non-magnetic RI compounds it was assumed that the temperature dependence of the non-magnetic RI compounds is determined by the residual resistivity P0 and the resistivity Oph due to electron-phonon interaction. In RI compounds showing magnetic order, an additional contribution to the resistivity must be taken into consideration. This contribution Pmag describes scattering processes of conduction electrons due to disorder in the arrangement of the magnetic moments. Assuming the validity of Matthiessen's rule it follows that the total temperature dependence of the resistivity in these compounds is given by p ( T ) = Po + Pph(T) + Pr~,g(T) •

(2)

t~crn J

150

I00

5C

o2

T[K3

50

/00

/50

200

250

Fig. 2. p versus T c u r v e s of Y C o 2 a n d Y4C03 .

183

E. Gratz, M.J, Zuckermann / Transport properties of rare earth compounds

In fig. 3 the temperature dependence of the electrical resistivity of RI compounds without d-electron contribution is shown schematically. For the case of resistivity the contribution P m a g ( T ) will now be the subject of discussion. P m a g ( T ) is characterized by the following traits: (i) a temperature independent behaviour for T >

,t~gcm~

jSmAi2

80

60

40

Tord;

(ii) a pronounced kink at T = Tord; and (iii) a strong decrease for T < Totd with decreasing temperature. To~d is the transition temperature for magnetic ordering. The temperature dependence of the resistivity of some heavy REAl 2 compounds (RE = Sm, Gd, Tb, Dy) is shown in fig. 4. Note the similarity of the p versus T curve to the behaviour predicted by the schematic picture in fig. 3. The dependence of P s p d ( ~ P m a g for T > T ~ ) on the Gennes factor (g--1)2×J(J+l) for the heavy REAl 2 compounds is shown in fig. 5. The agreement with the theoretical model, which states that Pspd is proportional to the de Gennes factor is seen to be satisfactory [8]. The Pspd values were obtained by the extrapolation procedure as shown in the schematic picture (see fig. 3). In the following we are presenting the p versus T curves of some examples of those RI-compounds which have a magnetic rare earth component and at least one second magnetic 3d transition metal component.

20 ¸

50

100

150

200

T[K] 300

250

Fig. 4 The p versus T curves of some magnetic REAl 2 compounds ( R E = Sm, Gd, Tb, Dy). The arrows indicate the Curie temperatures.

The magnetic and the transport properties of magnetic RE-non-transition metal compounds can be understood in terms of the R K K Y interaction between the localized moments which described the magnetic properties of these compounds. A more complex magnetic behaviour is expected for RI-compounds in which the second component is a 3d transition metal such as Mn, Fe or Co. The magnetic behaviour of the transition metal component is now based on the magnetic polarization of the electronic d-bands. Consequently also the nature of the transport properties becomes more complex in these compounds. What was mentioned above will now be demon-

? 60

,,

4(J 30. I

9,o,L

// ......

1o

Tb

" ,i 5

Fig. 3. Schematic temperature dependence of the electrical resistivity of magnetic Rl-compounds with negligible d-electron contribution.

tO

Gd

, i (g'l)2J(,, J ' l ) 15

Fig. 5. The dependence of Ps~ on the de Gennes factor (g - 1)2 × J ( J + 1) of some heavy REAl 2 compounds (O Van Daal et al. [13], • Gratz et al. [141).

E. Gratz, M.J. Zuckermann / Transport properties of rare earth compounds

184

strated on some RECo 2 compounds which are isostructural to REAl 2 compounds. In a large number of investigations concerning the RECo 2 compounds, it was shown that in DyCo 2 H o C o 2 and ErCo 2 the magnetic transition is a first order phase transition in contrast to the other R E C o 2 compounds where it is second order. In the case of a first order transition most of the physical parameters change abruptly at the transition temperature [9-12]. Fig. 6 shows the temperature dependence of the resistivity of the RECo 2 compounds in question. The discontinuities observed in the p versus T curve of the Er, Ho and Dy compound represent the influence of this first order transition on resistivity. Note that e.g. in ErCo 2 at 32 K the magnitude of the electrical resistivity increases by about 600% on heating up the sample by a few tenths of a degree. The magnitude of this j u m p in the p versus T curve decreases with increasing

[E~cm] 16C

120

4o:

I

o: J, ,

,oco

ordering temperature (ErCo2, Tc = 32 K; HoCo2, T~ = 78 K; DyCo 2, T~ = 135 K). This is due to the increasing effect of temperature [12]. The discontinuities at Tc are related to the sudden appearance of a high degree of order in rare earth moments immediately below T~ (in contrast to a second order transition where the alignment in the moments increases continuously below Tc). The extremely slow increase of the p versus T curve in the high temperature range experimentally found in all the RECo 2 compounds (YCo 2 included) is assumed to be due to the s - d scattering processes of conduction electrons into available d-states [12,

15]. Unusually interesting resistivity behaviour was found in the RE6(Fe X, Mn 1 x)23 compounds with the Th 6Mn 23 structure. Although the samples show sharp lines in the Debye-Scherrer photographs the resistivity behaviour measured for these types of R I - c o m p o u n d s is similar to that found in amorphous metallic compounds. The p versus T curves of two RE6(Fe, Mn)23 pseudobinary series are given in fig. 7 [Y6(Fe, Mn)23] and fig. 8 [Er6(Fe, Mn)23] [16]. The substitution of only a small fraction of Fe or Mn in the compounds RE6Fe23 or RE6Mn23 causes an extremely strong increase of the residual resistivity Po- In the middle concentration range the O versus T curves are dominated by a negative temperature gradient of resistivity irrespective of whether the RE-atom carries a local moment (Er-compounds) or not (Y-

20 40

250

2OO

150

100

~

x = 10

~/

~¢Fe,Mnl_~)2z

rtKJ

50'

ot 5

. . . . 50

100

150

200

250

Fig. 6. p versus T curves of ErC02, HoC02, DyCo 2 and TbCo 2 The arrows indicate the magnetic ordering temperature.

50

100

150

,200

250

Fig. 7. p versus T curves for various concentrations of the pseudobinary Y6(FexMnl x):3 system. Arrows indicate T~.

E. Gratz, M.J. Zuckermann / Transport properties of rare earth compounds

t~2cml

_

x.O.7 --x.0 8

T[K] o

5o

too

150

2O0

Fig. 8. p versus T curves for various concentrations of the pseudobinary Er6(FexMn 1-x)23 system. Arrows indicate Tc.

compounds). This behaviour is related to the complex crystallographic structure of these compounds [16]. In this case Mooij's rule may well apply. Mooij [17] proposed that if the mean free path of the conduction electrons is reduced to the size of the atomic distances, the P versus T curves become temperature independent. -We note that no anomalies were found at the Curie temperature in these highly resistive crystalline systems.

3. Thermopower The thermopower for a simple metal can be written as follows:

S(T) : Se(T ) + S~(T),

(3)

where Se and Sg are the thermopowers due to electron diffusion and phonon drag, respectively. The standard expression for the thermopower S¢ due to electron diffusion is given in the relaxation time approximation by ~r2k~T(

1 o(,)

0o(~)) a,

(4) .... '

a(() is the contribution to the electrical conductivity from the electron energy surface E ( k ) = constant. At high temperature the total thermopower is usually a linear function of temperature

185

for many metals as predicted by the theory of the electron diffusion thermopower, i.e. eq. (4). It is also in agreement with the phonon-drag theory which predicts that the phonon-drag contribution should be negligible at high temperatures due to the dominance of the p h o n o n - p h o n o n interaction over the electron-phonon interaction at these temperatures [18]. At lower temperatures the deviation of the total thermopower from linearity is usually ascribed to the phonon-drag term Sg. However, it could be shown that Sg is quenched at low temperatures by strong scattering between phonons and impurities [19]. This is in agreement with the experimental data for metals such as Ag containing small amounts of Hg [20]. However, in some metals, the introduction of impurities results in an increase in the magnitude of the thermopower at low temperature [21]. By taking into account second-order effects in the electron-phonon interaction, Nielsen and Taylor [22] have suggested that the low temperature deviation from linearity of the thermopower may be partially explained in terms of the diffusion component itself. The NielsenTaylor-effect is some times referred to as "phony phonon drag". The temperature dependence of thermopower for several non-magnetic RI-compounds is shown in fig. 9 (YA1 z, LuAlz) and fig. 10 (YCu 2 and LuCu2) [23]. Most of the S versus T curves are characterized by a nearly linear temperature dependence above 150 K. This is in agreement with eq. (4) assuming that [a In o ( c ) / ~ ] at the Fermi energy hardly changes with the temperature. In the low temperature range pronounced minima are observed in some of these compounds (YA12, YCu2). As discussed above there are two possible explanations for such anomalies. One possibility is that these minima are caused by phonon drag. The other mechanism is the Nielsen-Taylor effect. The same case as applied to resistivity, applies to thermpower where the data for YCo 2 and Y4Co3 should be discussed for the purposes of comparison. The magnitude of the thermopower of YCo 2 is three times larger than that of the isostructural compound YA12. Furthermore, the thermopower saturates above 200 K. This was not observed for YA12. We feel that this saturation of S(T) is characteristic of metal in which the d electrons

E. Gratz, M.J. Zuckermann / Transportproperties of rare earth compounds

186

"~'IS[tJV/K.1

S&V/KI

_

~

~

........

r~K:

~

2oo

250

o

~

~

~

~,

25o TfKJ

-2 -4

-6 k . . / ~ -8 -10~ -12 -14 Fig. 11. S versus T c u r v e s of Y C o 2 a n d Y4Co3 .

Fig. 9. S versus T c u r v e s of Y A l 2 a n d L u A I 2.

play a dominant role in the transport properties. In fact in ref. [24] an s - d scattering mechanism was used to show that the diffusion thermopower is a sensitive function of the density of states of the d-electrons and its derivatives at the Fermi level. The T 2 dependence of the low temperature resistivity of YCo 2 was analysed in terms of spin fluctuations [3], it is possible that the minimum in the thermopower of YCo 2 and 20 K may also be due to spin fluctuations though no analysis is as yet extant. The absolute thermopower of YCo 2 and YaCo3 is negative from 10 K to room temperature as can be seen in fig. 11 [25]. The magnitude of the thermopower of Y4Co 3 at high temperature is about 10 #tV/K which is comparable to the thermopower of YCo 2 in the same temperature range. For systems containing localized magnetic moments, the thermopower has not been theoretically investigated in such detail as the resistivity. An expression for the thermopower of ferromagnetic

st~v/K~

/o.>I// ycu2

~ " ,

~'~

/o.~'~/

o/°"

LuCu2

,o., . . . . . . . . . . . . . . . . . . . . . 5o ,oo ~5o 2oo 25o

Fig. 10. S versus T c u r v e s of Y C u 2 a n d L u C u 2.

rlK_l

materials with localized moments has been obtained from the Bloch equation in both the Born approximation and the spin wave approximation [26]. In the former case, a molecular field approximation was used to obtain the energy spectrum of the conduction electrons and the localized magnetic moments. The following result was obtained for the thermopower in ref. [26]:

s ( r ) = S e -~- Smag,

(5)

where Se is given by eq. (4) and Smagthe anomalous magnetic thermopower in the molecular field approximation, reads: Sn,ag

--

kB2Hc(1-exp(-tI°/kBT)) ec~ 1 7exp(-Ho/kBT

) '

(6)

~F is the Fermi energy, H 0 is the molecular field for the localized spin and H c is the molecular field for the conduction electrons. H c is given by: H c-- 2 ( g - - 1 ) ~ ( j : ) ,

(7)

where ~ is the direct exchange integral of the ( s p ) - f interaction and ( Z ) denotes the mean value of the z-component of the total angular momentum of the RE-ions. The expression for the anomalous magnetic thermopower in the spin wave approximation is quite complicated and will not be reproduced here. The magnon drag is analogous to phonon drag and was first analysed theoretically in ref. [27]. The magnons therefore create an additional thermopower by driving the electrons along their thermal gradient. It was shown that the expression for

E. Gratz, M.J. Zuckermann / Transport properties of rare earth compounds

the thermopower S m due to magnon drag is similar in form to the phonon drag thermopower Sg [27]. The experimental data for the thermopower of magnetic RI-compounds suggest that magnon drag effects may indeed influence the thermopower quite strongly. For example, a pronounced peak is found in GdA12 at 80 K whereas no such anomaly occurs in the thermopower of the isostructural YA12. This is discussed further below. To obtain the contribution of spin-disorder scattering to the thermopower in the paramagnetic temperature range the Nordheim-Gorter rule was used in the following manner [28]. The Nordheim-Gorter rule is given for non-magnetic RIcompounds by: s : a m = l°° So--~ PPh aph ,

p

(8)

P

and for magnetic RI-compounds by: PO- S0 + Pph Pspd sm = --~ "-~ Sph "~-7 S~pd,

(9)

Snm and Sm denote the diffusion thermopower of a non-magnetic RI-compound and a magnetic RIcompound, respectively, So, Sph and Sspa are the contributions to the thermopower due to impurity scattering, phonon scattering, and spin-disorder scattering, respectively, p, P0, Pph and Pspd represent the total resistivity, the residual resistivity, the phonon resistivity and the spin disorder resistivity, respectively. The first two terms in eq. (9) are replaced by Senm of eq. (8). This is equivalent to the assumption that to the first approximation the impurity and the phonon contribution of corresponding isostructural magnetic and non-magnetic compounds are equal. These considerations lead to the following expression for the spin disorder contribution for the thermopower Sspd ~- (Sem -- Senm)P/Pspd"

(10)

187

crystal structure. Gd was chosen as the magnetic component of the compound because it is in an s-state, and the effect of magnetic anisotropy on the thermopower is therefore avoided. The results are given in fig. 12 which shows that Sspd has a linear temperature dependence well above T~ for each of these compounds [28]. The S versus T curves of some REAl 2 (RE = Sm, Gd, Tb, Dy) are given in fig. 13 [23]. All four order ferromagnetically at low enough temperature. The arrows in fig. 13 indicate the magnetic ordering temperature Tc as given in the literature [29]. The following behaviour occurs: (a) A kink in the vicinity of T~ which indicates that magnetic scattering strongly influences the S(T)-behaviour. (b) An extremely pronounced non-linear behaviour is found for all these compounds below the ordering temperature. As mentioned above, this anomalous behaviour can be explained by the magnetic scattering of conduction electrons on spin waves [26]. (c) The temperature dependence of the thermopower in the paramagnetic regime is smooth and nearly linear with temperature. An application of the analysis as given by eq. (10) to these four

Sspd[P V/K2

GdAI2 A--

Z, 2 0 -2 -~;

T[KJ ,t

!

I

180

200 •

•

220 •

240

260 GdN i

•

-6 -8

The thermopower of the following RI-compounds were analysed using eq. (10): YA12 and GdAI 2, YCu z and GdCu z and LaNi and GdNi. The above selection of RI-compounds allows a comparison of the thermopower of a non-magnetic compound and that of a magnetic compound with the same

-tO

•

•

•

i ~ l _

~

GdCu2

Fig. 12. Temperature dependence of Sspd for GdA12, GdCu 2 and GdNi.

E. Gratz, M.J. Zuckermann / Transport properties of rare earth compounds

188

S[IJV/K]

S273[~ V/K]

...~-"

/ ....

~.~SmA~

~-~ %~

...~/

o

6

,,F

I

\

/ \

°A12 .

"

--o

~

"6

o

4

2'

2

O"

"0

"2'

2 n

a

~DYAI 2

"'4

6

Ce

i

La

4

rf

a

Fig. 13. S versus T curves of some REAl 2 compounds (RE = Sm, Gd, Tb, Dy). The arrows indicate T~ as given in the literature [29].

~

4"

-4

~'~'~"~"..o~

8

oJ

8

i°

i

Nd

Pr

a /

Srn Gd Oy i Eri t Yb i , t , i i , Prn Eu Tb Ho Tm Lu

Fig. 15. The magnitude of the thermopower at T = 2 7 3 K ($273) for the RECu 2 series together with the spin quantum number (S), orbital quantum number ( L ) and the total quantum number ( J ) across the rare earths.

~ SttuV/KJ

c o m p o u n d s shows a linear temperature dependence of Sspa [30]. The S versus T curves of some R E C u 2 ( R E -Sm, Gd, Tb, Dy) are shown in fig. 14 [23]. The arrows indicate the Ne61 temperature TN as obtained from initial susceptibility measurements. S m C u 2 is paramagnetic in the whole temperature range. The slope of the S versus T curves change in a systematic m a n n e r in going from SmCu 2 to D y C u z. Fig. 15 shows the r o o m temperature thermopower S273 for all the R E C u 2 c o m p o u n d s across the rare earth series. It can be seen that the dependence of $273 for R E C u z c o m p o u n d s on the

21 0 -2 -4 -6

~

TCK] GO

100

140

180

220

260

ErC~

-8 4 2 0 -2

100

140

-4

4 2

/ HoC~ 60

-4 S[uV/KI

]]I

et

oycu~

~1

, ,%,~&"°

1,~ ~ 0

~

:t

-6

t

.... ~

.OyCu2

~oo '

1o- - TbCu2

/~'~

~p~" o- "~

2o9~ ~ " 250 J

~

'

'

~.~

_ (

~

Fig. 14. S versus T curves of some R E C % compounds ( R E = Sm, Gd, Tb, Dy). The arrows indicate the Ne61 temperature T s as obtained by ac susceptibility measurements.

01

\

Fig. 16. S versus T curves of some heavy REC% compounds ( R E = T b , Dy, Ho, Er). The arrows indicate the magnetic ordering temperatures.

E. Gratz, M.J. Zuckermann / Transportproperties of rare earth compounds

RE-ato~nic number has a similar shape to that of the quantum numbers J and L of the 4f multiplet of the RE-ions. This implies a magnetic correlation, though no theoretical explanation is extant. The effect of a first order magnetic phase transition on the thermopower of RECo 2 compounds was studied in ref. [12]. The results are shown in fig. 16. The first order phase transition causes a discontinuity in the thermopower. The magnitude of the discontinuity increases with decreasing T¢, i.e. as the rare earth changes from Tb to Er. This is consistent with the resistivity data given in fig. 6. The curvature in the thermopower found for RECo 2 compounds above T¢ is nearly the same for all the compounds and has the same character as the thermopower of Y C o 2 above 70 K (see fig. l 1). The magnitude of S found in this temperature range is about 12-15 /~V/K. In ref. [12] it was postulated that the curvature is caused by an s-d scattering mechanism which cannot occur in compounds with a low density of d-states at the Fermi level. Note the similarity of the S versus T curve of Y C o 2 t o that of E r C o 2 above T~.

189

A = po/Lo and L 0 is given by L°-

,/7,2 k 2 - - 2 . 4 4 × 10 -8 W f ~ K -2.

3

e2

The general result for Wph(T ) is given by the Wilson formula [1]. At low temperatures the Wilson formula becomes Wph(r ) ~--BT 2

(T<
(14)

(O D denotes the Debye temperature).

Xe at low temperatures is given by:

Xo~-- T / ( A + BT3).

(15)

In the limit where Wph << W0 eq. (15) shows that Xe is given by:

X~ ~-- T/A = ( Lo/Po)T.

(16)

We now assume that the electrons and the lattice contribution to ~ are additive, giving the following result for the total thermal conductivity: ?~----Xe+X ,.

(17)

It follows from eq. (16) that

4. Thermal conductivity

X, ~ - X - ( L o T / O o )

The thermal conductivity ~, of metals has two components, an electronic component X¢ and a lattice component ?~l for which the phonons are the heat carriers. The electronic component h e can be written as follows in terms of the thermal resistivity W of the electron system h e = 1/W~.

Another ansatz used in ref. [31] assumes that the Wiedemann-Franz law holds for the total electrical resistivity

X~= LoT/[Po + Pph(r)],

(19)

where Pph is the resistivity due to electron-phonon scattering. ?~1 is than given by: At = X

(12)

where W0 is the thermal resistivity due to scattering of electrons by imperfection and Wph is due to electron- phonon scattering: eq. (12) assumes the validity of Matthiessen's rule. W0 is related to the residual resistivity P0 by the Wiedemann-Franz law:

Wo = A / T .

(18)

(11)

We is in general given by We-~ Wo-~- Wph ,

(Wph << Wo).

(13)

LoT Po + Pph(T) "

(20)

Figs. 17 and 18 show the temperature dependence of the thermal conductivity )~ for YA12 and LuA12 respectively [32]. The shape of the )~ versus T curves is considerably different from those of a pure metal such as A1. For pure and well annealed A1 samples an extremely pronounced peak in the low temperature range near 0.10 D is observable. As discussed above, the thermal conductivity of a metal consists of two contributions, namely an electronic part )% and a lattice part )~l- )~l is usually

E. Gratz, M.J. Zuckermann

190

/ Transport properties of rare earth compounds

VAl~

A [mWIcmK]

~, [mW/cmK3

200

250

,° > /

150 1.50 100 106 50 50"

T[K)

5b

,bo

,;o

2bo

2~o

Joo

5'o

~5o

2oo

250

300

T[K]

Fig. 17. X versus T curve of YAI 2 (for the meaning of ~ ] , Xu, Ae2, AI2 see the text).

assumed to give a small contribution to the total thermal conductivity for pure metals [1]. One of the problems in interpreting the thermal conductivity is to find a way of decomposing 2~ into }k e and X1- Two methods have been discussed above which will now be applied to experimental data. The results for YA12 and LuA12 are shown in figs. 17 and 18, respectively. It can be seen that the use of eq. (18) can only be made up to about 40 K (see Xel in figs. 17 and 18) since the total resistivity p can no longer be described by the residual P0 alone. Also eq. (18) is meaningless when X < L o T / p o. This implies that it is reasonable to use

~ rmwlc,~Kj

,b

Fig. 19. A versus T curves of SmAI 2, GdAI 2 and TbA12 (arrows indicate To).

the second method contained in eq. (20). The resulting curves for the temperature dependence of the electronic part ~'e2 and the lattice part X~2 obtained by the second method are also shown in figs. 17 and 18 for both compounds. The ?%2(T) curves should now obey Wilson's formula. The temperature dependence of the thermal conductivity X, of some magnetic REAl 2 compounds (RE -Sm, Gd, Tb) is given in fig. 19. The ordering temperatures are indicated by arrows in the versus T curves. At present there is no detailed

TcfK ]

W2sofcmK /W]

L u AI2

2oo

/ I

Iso I; ,oo I x\\

150

,o

/.1 / /~

200

~ #.~

x\\A",\

41

100 /'~12

9

"/ / \1 \ xx'x

I--Y,' 50

...

,~,~.~-~ ~ ' ~ 0

.

~

Ce 5'0

J 100

150

i 200

Lo 2;0

300

T~K]

Fig. 18. A versus T curve of LuAI 2 (for the meaning of hel, All, he2, Al2 see the text).

Nd Pr

5m Pm

.

.

.

Gd Eu

.

.

Oy Tb

"9

Er Ho

Yb Tm

Lu

Fig. 20. The thermal resistivity at T : 2 5 0 K (W250) is shown together with the Curie temperature Tc for the REAl 2 series across the rare earths.

E. Gratz, M.J. Zuckermann / Transport properties of rare earth compounds u n d e r s t a n d i n g o f the t e m p e r a t u r e d e p e n d e n c e of t h e t h e r m a l c o n d u c t i v i t y of t h e s e R E A l 2 c o m pounds. However, we believe that a systematic i n v e s t i g a t i o n o f t h e ~,(T) a c r o s s t h e r a r e e a r t h s c a n g i v e us a n i n d i c a t i o n o f the i n f l u e n c e o f local m a g n e t i c m o m e n t s o n the t h e r m a l c o n d u c t i v i t y . F o r e x a m p l e the a n a l y s i s of the t h e r m a l r e s i s t i v i t y at 250 K (Wzso) a c r o s s the R E A 1 2 - s e r i e s s h o w s t h a t t h e r e exists a n i m p o r t a n t c o n t r i b u t i o n to the t h e r m a l r e s i s t i v i t y f r o m s - f s c a t t e r i n g o f the c o n d u c t i o n e l e c t r o n s [30]; this is s h o w n in fig. 20 w h i c h also s h o w s t h e m a g n i t u d e o f Tc o f the REAl 2 compounds.

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191

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