E. GRATZ
2. Transport properties of non-magnetic rare earth intermetallic compounds 1. Theoretical introduction
2.1.1. Electrical resistivity
We begin by assuming that the resistivity of non-magnetic RI compounds obeys Matthiessen's rule. This rule states that the temperature dependence of the resistivity is given by
O(T)
= O0 + pph(r), (1)where p0 is the residual (defect) resistivity and 0ph(T) is the lattice resistivity due to the electron-phonon interaction. The effect of the electron-electron inter- action is neglected since RI compounds are usually not of sufficiently high purity (Ziman, 1960; Volkenshtein, 1973). In the compounds consisting of non-magnetic rare earth with noble metals, O0 was frequently found to be in the range of a few /xl) cm (see section 2.2). 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, especially if the system crystallizes in the MgCu2 structure (e.g. RAI:) or in the CeCu: structure (e.g. RCu2). The scattering of electrons by vacancies, dislocations, impurities, etc. is assumed to be elastic. As the tem- perature rises O0 remains constant, but Pph increases rapidly because of electron scattering by the quanfized 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, belonging to s-p conduction bands, are scattered by the electron- phonon and electron-impurity interaction into vacant energy states within these bands. The general result for
pph(T)
in this case is given by the Bloch-Grtineisen law (Ziman, 1972)ODIT
x 5 dx (2)
Oph(T)=gR(~D)'
I (l_e_X)(eX_l).0
The constant R includes the electron-phonon coupling constant, the atomic masses of the different types of atoms and a characteristic temperature 0D (Debye temperature) for the phonons. Using the limits of integration indicated,
122 E. GRATZ AND M.J. ZUCKERMANN
the standard integral reduces to an expression proportional to (OD/T) 4 at high temperatures and to a constant equal to 124.4 at low temperatures. Pph is therefore proportional to T 5 at low temperatures and proportional to T at high temperatures. The derivation of eq. (2) is based on four assumptions:
(i) The electrical field E and the current density are described by semiclassical theory. This implies that the E field "accelerates" electrons according to k = - e E / h and the current density is caused by the resulting excess of conduc- tion-band electrons with a velocity v = -(1/h)~Tke(k).
(ii) The scattering events are separated spatially by a sufficient number of wavelengths so that an electron "recovers" from a previous collision before experiencing the next one. For free electrons the criterion is [kF[/"> 1 (kF is the Fermi wave vector, l is the mean free path).
(iii) The effective mass approximation is applicable.
(iv) The phonons are in thermal equilibrium and can be described by the Debye model.
Resistivity data for non-magnetic RI compounds are discussed in conjunction with eq. (2) in section 2.2.
2.1.2. T h e r m o p o w e r
The thermopower for a simple metal can be written as follows:
S = Se + Sg, (3)
where Se and S~ 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 (Ziman, 1960)
S~(T) = (4)
31e[ L~r(e) oE J,=,F"
or(e) is the contribution to the electrical conductivity from the electron energy surface e(k) = constant. At high temperatures the total thermopower is usually a linear function of temperature 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 phonon-phonon interaction over the electron-phonon interaction at these temperatures (Bailyn, 1960). At lower temperatures the deviation of the total thermopower from linearity is usually ascribed to the phonon-drag term Sr However, Gu6nault (1971) was able to show that Sg is quenched at low temperatures by strong scattering between phonons and impurities. This is in agreement with the experimental data for metals such as Ag containing small amounts of Hg (Craig and Crisp, 1978), Ag-As (Song and Crisp, 1978) and Cu-Ga alloys (Crisp, 1978).
However, in some metals, the introduction of impurities results in an increase in the magnitude of the thermopower at low temperature. In particular this was found in the case of dilute aluminium (Huebener, 1968) and indium alloys (Duddenhoeffer and Bourassa, 1977).
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 123 By taking into account second-order effects in the electron-phonon inter- action, Nielsen and Taylor (1974) have suggested that the low-temperature deviation from linearity of the thermopower may be partially explained in terms of the diffusion component itself. The Nielsen-Taylor effect is sometimes referred to as "phony phonon drag".
2.1.3. T h e r m a l c o n d u c t i v i t y
The thermal conductivity h of metals has two components, an electronic component he and a lattice component he for which the phonons are the heat carriers. The electronic component he can be written as follows in terms of the thermal resistivity W of the electron system
he = 1/Wo. (5)
We is in general given by
We = w0 + wph, (6)
where W0 is the thermal resistivity due to scattering of electrons by imperfection and Wph is due to electron-phonon scattering: eq. (6) assumes the validity of Matthiessen's rule. Wo is related to the residual resistivity P0 by the Wiedemann- Franz law
Wo = A / T . (7)
A = po/Lo and Lo is given by
L 0 = T - - - = 2 . 4 4 × 1 0 - S W f I K -l.
Eq. (7) is only true if the relaxation times for W0 and O0 are equal. The following expression of Wph was obtained by Wilson (Ziman, 1960) for all temperatures using the variational principle and assuming a constant relaxation time
4A T 5 1
etdT
J. ( ? ) = f x° ex dx
e x - 1 " (8b)
0
0D is the Debye temperature and N is the number of ions per unit volume. At low temperatures eq. (8a) becomes
Wph(T) ~-- B T 2 ( T .~ 0D)- (9)
Eqs. (5)-(9) then give the following expression for he at low temperatures:
T
he = A + B T 3" (10)
124 E. GRATZ AND M.J. ZUCKERMANN In the limit where Wph ~ Wo, eq. (10) shows that he is given by
T Lo T. (11)
he - A P0
We now assume that the electron and the lattice contribution to h are additive, giving the following result for the total thermal conductivity:
h = ho + he. (12)
Then from eqs. (11) and (12)
he = h - LoT/po (Wph ~ W0). (13)
This expression is used in section 2.2.3 to obtain values for he from the experimental results for h at very low temperatures.
Another ansatz used in the literature (Wtosewicz et al., 1979) assumes that the Wiedemann-Franz law holds for the total electrical resistivity
he - LoT (14)
/90 + Pph(T)'
where 0ph is the resistivity due to electron-phonon scattering given by eq. (2).
Then he is given from eqs. (12) and (14) by
he = h L o T (15)
0o + Pph(T)"
Eq. (15) is also used in section 2.2:3 to analyse the experimental data.
2.1.4. M e c h a n i s m s f o r saturation of the resistivity at high temperatures
It is well known that the high-temperature resistivity of several intermetallic compounds and alloys saturates at high temperatures. The best-known example of a non-magnetic compound in which such saturation effects occur is Nb3Sn (Woodard and Cody, 1964). These phenomena have also been observed for enhanced paramagnetic compounds and alloys, such as YCo2 and CeRu2, and itinerant ferromagnetic metals well above the Curie temperature (see sections 4 and 2.2.1.1 below). The saturation phenomenon has been theoretically in- vestigated in a rigorous and complete way by Allen (1980). Allen shows that the Boltzmann equation for transport phenomena is only valid under the conditions given in section 2.1.1. It is important to note that the theoretical results presented in sections 2.1.1-2.1.3 are all derived from the Boltzmann equation.
Allen notes how sophisticated and successful the Boltzmann equation is in spite of these limitations.
Allen has analysed the following eight mechanisms for the saturation effect.
The first three lie within the formalism of the Boltzmann equation and the remaining do not.
(a) Anharmonicity: This word is used to convey the idea that the phonon frequencies to are temperature dependent. Normal anharmonic behaviour causes (to)2 to decrease with increasing temperature and therefore causes an increase in
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 125
p(T)
above linearity. However, (0)) 2 is known to increase abnormally for A15 compounds. Bader and Fradin (1976) have estimated that at least one third of the deviation from linearity associated with saturation can be explained by anhar- monicity.(b) Fermi smearing: This is due to the fact that the derivative of the Fermi function with respect to kinetic energy is not a delta function at non-zero temperatures. This clearly affects the resistivity significantly but Allen points out that this mechanism probably does not lead to a systematic tendency to saturation.
(c) Alteration of the band structure with temperature: Allen regards this as a small effect which is less significant than (a).
(d) Anderson localization: The Anderson criterion for localization of electrons is connected with the degree of disorder of the system. Allen remarks that it has been shown that the electrical conductivity vanishes at zero temperature when the degree of disorder is close to a critical value. He, however, indicates that there is no hint of an excess conductivity before localization occurs and that the situation is unclear at finite temperatures.
(e) Debye-Waller factors: Allen regards the effect of Debye-Waller factors on the resistivity as requiring more theoretical investigations. One objection is that the Debye-Waller factor should have the same effect on both weak and strong scattering metals and secondly, that multiphonon effects and Debye-Waller effects enter in the same order of perturbation theory in the resistivity. The degree of cancellation of the two effects is not yet known.
(f) Phonon drag: Allen discounts phonon drag as a saturation mechanism since it is only expected to be important for pure metals with weak phonon anharmonicity at low temperatures.
(g) Phonon ineffectiveness: Morton et al. (1978) and Cote and Meisel (1978) have both proposed that phonons of wave vector q become ineffective scatterers when q( < 1. Allen feels this mechanism is dubious and, if at all true, should be related to phonon drag.
(h) Non-classical conduction channels: Allen states that the generalized Boltzmann equation due to Chakraborty and himself (1978, 1979) can be used to solve the effect of temperature-dependent electron bands on the resistivity. He also points out that non-classical terms related to interband dipole transition and interband electron currents must be included. He further notes that the theory is complicated and that conclusions are difficult to make at this point. However, he feels that this is the most likely mechanism for explaining the saturation tendency.
We have given a summary of Allen's excellent review article. We refer the reader to the original papers for further details. Allen clearly points out that it is important to include interband effects in the formalism. This also follows from experiments since saturation effects in the resistivity at high temperatures seem to occur in metallic compounds and alloys with unfilled d or f bands. Allen and Chakraborty's theory may therefore lead to the understanding of saturation effects if s-d(f) scattering and electron-phonon scattering effects are treated on
126 E. G R A T Z A N D M.J. Z U C K E R M A N N
the same footing. Clearly Matthiessen's rule will no longer hold in this case, particularly if the electrons are strongly scattered by the phonons.
Another scattering mechanism which may lead to saturation of the resistivity in enhanced paramagnetic materials is the scattering of electrons by spin fluctuations. This point is discussed further in section 4.
Allen (1980) points out that the maximum resistivity for d-band compounds and alloys is of the order of 150 ixl~ cm. He suggests that this value is reasonable from a theoretical point of view since it is related to the lower limit of the mean free path, i.e. the interatomic distance. However, certain alloys, such as Y6(Fe, Mn):3 and Y(Fe, Co)z have resistivities which saturate above 220/xO cm (see section 4 for details). This may well be due to resonance scattering effects for the d-electrons in disordered pseudobinaries (Harris et al., 1978).
2.2. Experimental results ancl discussion 2.2.1. Electrical resistivity data
2.2.1.1. RAI2 (R = Y, La, Yb, Lu), CeTz (T = Co, Ru, Ir)
All the compounds under consideration crystallize in the cubic MgCu: struc- ture. The R-A1 compounds are typical examples of compounds with negligible d-electron effect on resistivity. Eq. (2) should then be applicable. A nearly linear temperature dependence was indeed found by Van Daal and Buschow (1970) in the high-temperature region for non-magnetic RA12 compounds (fig. 1).
Fig. 2 exhibits the temperature dependence of the resistivity for some Ce- transition metal compounds (Van Daal and Buschow, 1970). In all these com- pounds Ce is in its tetravalent state. These compounds are therefore all non- magnetic but the susceptibility is about one order of magnitude higher than in the RA12 compounds (Buschow, 1977, 1979). As mentioned in section 2.1.4 the influence of d-electrons in the enhanced paramagnetic compounds gives rise to
5 0
4 0
3 O
2 0 ¸
10-
~
\LoA/2/ / ~YbA/2
~
- Y A / 2tO0 Z~oo 3oo
Fig. 1. p vs. T c u r v e s of s o m e non- m a g n e t i c RA12 c o m p o u n d s (Van Daal and B u s c h o w , 1970).
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 127
80
Celr
GC
T[K]
0 100 200 300
Fig. 2. P vs. T curves of some CeT2 com- pounds (T = Co, Ru, Rh, Ir) (Van Daal and Buschow, 1970).
pronounced curvature in the p vs. T curve as shown in fig. 2. The residual resistivity p0 is subtracted in both fig. 1 and fig. 2.
2.2.1.2. Fit of the Bloch-Griineisen law to experimental data
The p vs. T curves of the non-magnetic R-compounds YA12, LuA12, YCu2, LuCu2, and LaNi measured by Nowotny and Gratz (1981) are shown in fig. 3.
The authors fitted eq. (2) to the experimental data and the results are given in table 1. Although this fit is within an error of about 1% the calculated Debye temperatures 0D are considerably smaller than those obtained from specific heat measurements (e.g. 0D = 473 K for YA12 and 384 K for LuAlz). The discrepancy can be understood in terms of the theory proposed by Kelly and McDonald (1952). These authors state that the Debye temperature 0o obtained from specific heat comes from low-temperature contributions, whereas 0D obtained from resistivity measurements represents an average over all temperatures.
2.2.1.3. La(Agl_xln~), Ce(Ag~_xlnx)
The presence of the 5d electrons at the Fermi level is important not only for magnetic coupling but also for the stability of the crystal structure (Ihrig and Methfessel, 1976a, b). La(Ag, In) and Ce(Ag, In) crystallize in the CsC1 structure and undergo a martensitic transition to a tetragonal phase at low temperatures.
This cubic to tetragonal transition is ascribed to the Jahn-Teller effect. The effect of such a transition on the resistivity in the La(Ag, In) pseudobinary system is shown in fig. 4 (Balster, 1972). The corresponding measurements in the
128 E. GRATZ AND M.J. ZUCKERMANN
z,O
3 0
20
10
[HQcm] L U ~
I I I
~ T [K]50 100 150 200 250
Fig. 3. p vs. T curves of YA12, LuA12, YCu2, LuCu2, and LaNi (Nowotny and Gratz, 1981).
Ce(Ag, In) system are given in fig. 5 (Ihrig, 1973). The hysteresis observable in the p vs. T curve is due to this structural transition. Fig. 6 gives a survey of the transition temperatures of such systems (Ihrig and Methfessel, 1976a). The transition temperature was obtained from the midpoint of the hysteresis loops.
The anomalies in the resistivity behaviour can therefore be understood by a dramatic change of the phonon spectrum at this transition.
TABLE 1
The Debye temperatures 0 D and mean square deviations obtained by a fit procedure of the Bloch-Griineisen law (eq.
(2)) to the experimental data.
Mean square Compound Crystal structure 0D(K) deviation (%) YA12 ~ MgCu2 structure 289 0.25
LuA12 J 269 0.19
169 1.17 YCu2 CeCu2 structure
LuCu2 177 0.90
LaNi CrB structure 166 1.28
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 129
30 lO
~[jPcr.l LaAgl-x In
X~O
601
=01, o ~ ~ ,
160 ~ = O. 5
140
180 =0.7
160 ~ t r a g o n c t l ! cubic
1 7 0 ~ ~ = 0 . 8
150 =0.9
130 , , , ~ , T[K~
0 100 200 300
Fig. 4. p vs. T curves of the La(Ag~ xlnx)
pseudobinary system (Balster, 1972).
2.2.1.4. YbAl3, YbAI2
The resistivity for these c o m p o u n d s is shown in fig. 7. Havinga et al. (1973) analysed both the resistivity data and the t h e r m o d y n a m i c data of these com- pounds in terms of a two-band system.
The lower level is a narrow r e s o n a n c e (virtual bound) state which represents non-magnetic Yb 2÷ ions. The upper state represents the magnetic Yb 3÷ ion with a total angular m o m e n t u m of J = 7/2. The model was fitted to the susceptibility of YbAI3 and the value of 270 K was f o u n d for the energy difference AE b e t w e e n the two levels. The value of the paramagnetic Curie t e m p e r a t u r e for the u p p e r level was given by 0 = 220 K. Havinga et al. (1973) then derived the following expression for the resistivity using the two-level model
p ( T ) = b(1 - c)[1 - exp(-aTZ/b)] + Cpspd , (16) where c is the c o n c e n t r a t i o n of the Yb ions in the 3+ state and is given b y
c = 8 exp(-AE/kBT)/[1 + 8 exp(-AE/kBT)]. (17)
a and b are p a r a m e t e r s related to the virtual bound state. Pspd is, as usual, the spin-disorder resistivity.
Eq. (16) was fitted to the resistivity of YbA13 after the lattice contribution had been subtracted. The results of the fits are shown in fig. 7. The authors r e m a r k that the value of h E obtained b y fitting the resistivity is given b y h E = 290 K and is in good agreement with that obtained f r o m the susceptibility fit. H o w e v e r ,
130 E. GRATZ AND M.J. ZUCKERMANN
GO 50"
4 0
7,: I
220- 210.
195|
0
Ce
Ag1_ In
x=O
I I
.,.~tetragonal cubic"
I i i I ~
,b
s w Fig. 5. p vs. T curves of the Ce(Ag>xlnx) pseudobinary system (Ihrig, 1973).the values of Pspd and 0 obtained from the fit are at least one order of magnitude larger than one would expect. This indicated to the authors that the two-level model was s o m e w h a t too simple for Yb systems.
The authors also fit the resistivity data of YbA12 using the same m e t h o d (see fig. 7). In this case the value of AE obtained from the resistivity was found to be 1770 K which was again in good agreement with the value of 1800 K obtained from susceptibility. H o w e v e r , the values of Pspd and 0 found from the fit to the resistivity and suceptibility data were again too high.
The authors concluded by stating that the T 2 term in the resistivity m a y well be fitted, that such a behaviour can be ascribed to the spin fluctuations (see section 4).
2.2.2. Thermopower data
2.2.2.1. RAI2 (R = Y, La, Lu), RCu2 (R = Y, Lu)
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 r m o p o w e r for several non-magnetic RI c o m p o u n d s is shown in fig. 8 (LaA12), fig. 9 (YAI2, LuA12) , and fig. 10 (YCu2, LuCu2).
Most of the S vs. T curves are characterized b y a nearly linear t e m p e r a t u r e
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 131
300
200
100 r M tKl
LaAg
/n x
0 0.5 tO Fig. 6. Concentration dependence of the tran-
CoAg x Coin sition temperatures, TM, in La(Ag, In) and
LOAg Iclln Ce(Ag, In) (Ihrig and Methfessel, 1976a).
dependence above 150K. This is in agreement with eq. (4) assuming that a In cr/aE 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 in section 2.1.2 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 of section 2.1.2.
Although the structure of the YA12, LaA12, and LuA12 compounds is the same and the rare earth ions are all in the 3+ state, the shape of the S vs. T curves is considerably different for different compounds. The significant difference be-
40
30 20
10
0 ~oo 200 3o0 4oo 5o0 600
Fig. 7. Resistivity data of YbA13 and YbAI2.
The solid lines give the experimentally obtained Ape w vs. T curves corrected for residual and electron-phonon scattering con- tributions. The dotted lines show the estimated contributions due to spin-disorder scattering by the Yb 3÷ magnetic moments described by the second term of eq. (16). The dashed lines show the estimated contribution connected with scattering processes on Yb 2÷ virtual bound states which is described by the first term of eq. (16) (Havinga et al., 1973).