E. GRATZ AND M.J. ZUCKERMANN
4. Transport properties of rare earth-transition metal compounds exhibiting itinerant magnetism
4.1. Theoretical introduction
It was shown in section 3 that the transport properties of magnetic R-non- transition metal compounds can be understood in terms of a direct exchange interaction between conduction electrons and R localized moments. This inter- action leads to an indirect RKKY interaction between the localized moments which describes 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, in this section we summarize the theory of itinerant or band magnetism and its application to transport properties, We begin with the Stoner-Wohlfarth model and include a summary of recent works.
4.1.1. Theories for itinerant magnetism
The first theoretical description of itinerant ferro- and paramagnetism in transition metals is due to Stoner and Wohlfarth (Wohlfarth, 1976). These authors use Fermi statistics to describe the conduction electrons in the d-band.
They consider the case of a single conduction band in which the electrons interact via an intraatomic Coulomb interaction. This model is formally equivalent to the Hubbard model for correlations between electrons in a single conduction band.
Stoner and Wohlfarth were able to derive a self-consistent equation for the spontaneous magnetization M0 in the molecular field approximation (MFA) using the model described above. M0 is then written in terms of the average occupa- tion numbers (n~) for electrons of spin ~:
Mo = ~s{(n ~) - in + )}. (45)
The self-consistent equations for (n~) are given in the MFA by
(no) = f dE N(~)f(x~), (46)
0
x~ = (~ + I(n_~) - cr/XBH) cr = --- 1,
where I is an effective interaction energy between the electrons, N(e) is the density of states of conduction electrons of energy E, ~B is the Bohr magneton, H is the external magnetic field and f ( x ) is the Fermi function given by
f ( x ) = [exp(x - Ix)[kBT + 1] -1, (47)
tx is a chemical potential.
176 E. GRATZ AND M.J. ZUCKERMANN
From eqs. (45) and (46) it can be seen that the paramagnetic state is equivalent to taking (n 1)= (n ~). A ferromagnetic state occurs when (n 1) ¢ (n ~), i.e. when M0 ~ 0. The criterion for the occurrence of ferromagnetism is obtained by expanding eq. (46) in powers of M0 and is given by
1 ~< IN(EF). (48)
N(er) is the density of state at the Fermi surface and EF is the Fermi energy. Eq.
(48) is known as the Stoner criterion. Eq. (48) divides the range of values of IN(eF) into two regimes:
(i) Paramagnetic regime: IN(EF)< 1. In that case the kinetic energies of spin-up and spin-down electrons are equal and the static magnetic susceptibility X is given at zero temperature by
X = 2/x 2N(eF)/[ 1 -- IN(~F)]. (49)
1 --IN(eF) is known as the Stoner enhancement factor.
(ii) Ferromagnetic regime: IN(EF)> 1. In that case the spontaneous mag- netization M0 is non-zero. Eq. (46) shows that the kinetic energy of spin-up and spin-down conduction electrons are separated by a band gap Ab. In the absence of an external magnetic field Zab is given by
Ab "~" IMo/p,B (50)
and is known as the Stoner gap. Numerical methods are required to obtain values of Mo and Ab in general but analytical expressions exist for the Curie temperature predicted by the Stoner-Wohlfarth theory (T sw) and for the sus- ceptibility in the case of very weak ferromagnets. In particular T sw is given by
T sw~- A(ct - 1) 1/2 (Or = IN(eF)), (51)
where A depends on the band structure.
It should be noticed that the spontaneous magnetization Mo vanishes for temperatures T > T sw and therefore no magnetic moment, localized or other- wise is present in this temperature range.
In the case of weak ferromagnets (1 ~< IN@F)) Wohlfarth has shown that the Stoner model predicts a magnetic susceptibility for T > T sw which is non-Curie like and has the following form:
X oc T2 - (TSW)2. 1 (52)
One problem with the Stoner-Wohlfarth theory is that most systems to which it has been applied exhibit a Curie-Weiss law for T > To. The Stoner-Wohlfarth theory is a molecular field theory and therefore includes neither spin correlations nor dynamics, nor is the detailed band structure taken into account explicitly.
Several authors have recently turned their attention to this problem. Moriya and Takashashi (1978) used a renormalized theory of spin fluctuations (described in section 4.1.2) to obtain a Curie-Weiss law for ) above Tc for weak ferro- magnets. They also showed that the critical indices are different, for example
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 177
Tc z (a - 1 ) 3/4 from their theory. Also the ratio n ( T ) of the mean square of the local moment at T to that at T = 0 remains finite at Tc whereas it vanishes for the Stoner-Wohlfarth theory, i.e.
n (To) = 3/5 Moriya-Takahashi,
n ( T ) = 1 - (T[T¢) 2 Stoner-Wohlfarth. (53)
Liu (1976) used a quasispin model in conjunction with the coherent potential approximation to examine the properties of itinerant ferromagnets at high temperatures. He showed that local moments persist above T~ in the absence of long-range order. Capellmann (1979) and Korenmann and Prange (1979a, b) use equivalent descriptions of itinerant ferromagnetism which allow for local exchange splitting without long-range magnetic order. They find that
(i) The Curie temperature T ¢ ~ T sw, (the characteristic temperature of Stoner-Wohlfarth theory is given by eq. (51)).
(ii) Local magnetic moments persist above T¢ and vanish at T¢ . SW
Usami and Moriya (1980) and Moriya and Takahashi (1978b) applied an exten- sion of the theory of Moriya and Takahashi (1978a) to narrow band ferro- magnetism.
The most recent theory is that of Hasegawa (1980b). Hasegawa has been able to develop a single-site spin fluctuation theory by generalizing a local saddle- point theory proposed by himself (Hasegawa, 1980a). The theory was developed by combining the functional integral method with the alloy analogy (coherent potential approximation) and applying it to the Hubbard model. The author obtains a value for the Curie temperature of strong itinerant ferromagnetism within much less than T sw and predicts the existence of localized moments above To. The theory gives good semiquantitative agreement with experiment.
However, Hasegawa stresses that the explanation of the quantitative difference between the magnetization curves of Fe and Ni requires the inclusion of dynamical spin fluctuations in the formalism.
These recent theories have not been reviewed in detail (with the exception of Moriya's theory of section 4.1.2) since they have not yet been applied to transport properties. We do, however, suggest that such an application should be made. The effect of spin fluctuations and spin waves on the transport properties of itinerant systems is discussed separately in section 4.1.2.
4.1.2. Spin dynamic of itinerant magnetic systems and their effect on transport properties
We begin by describing the spin fluctuations for the paramagnetic regime (T > Tc or 1 > IN(EF)) in the random phase approximation (RPA) and related approximations. In the RPA spin fluctuations are described by the dynamic susceptibility
Xo(q, to) (54)
x(q, to) - 1 -/--Xo(q, ~o)'
where Xo(q, to) is the dynamic susceptibility of the conduction band, q is the
178 E. GRATZ AND M.3. ZUCKERMANN
wave vector and to is the frequency, x(q, to) of eq. (54) has poles for small q which represent critical damped excitations, i.e. whose "dispersion relation" is given by
tOq ~ i(1 - ot)l/2q, a = IN(eF). (55)
Eq. (55) indicates that the spin fluctuations are linear in q. Doniach (1967) showed that for small q, x(q, to) can be written
x ( q, to ) °~ -r-~-~q, to (56)
i.e. x(q, to) has a maximum at to = toq. When 1 - a/> 0, the maximum becomes a sharp low-lying peak for low enough q. Rice (1967) showed that, from his point of view, low-lying spin fluctuations could be considered as particles which obey Bose-Einstein statistics. These particles were called paramagnons and were used to describe the scattering of electrons by spin fluctuations, x(q, to) of eq. (54) reduces to X of eq. (49) in the limit of q --* 0 and oJ --. 0. The transverse component of the dynamic susceptibility has the same form as x(q, to) in eq. (54) for ferromagnetic metals (~ > 1) except that Xo(q, to) must be replaced by X-+(q, to) which now includes the splitting Ab of the conduction band. The poles of X-+(q, to) at small q now describe undamped spin waves which can be shown to have a quadratic dispersion relation. The longitudinal component of the dynamic susceptibility is more complicated due to the existence of a gap Ab between the two-particle (Stoner) excitations and the spin waves. Ab is given by eq. (50) at zero frequency. A continuum of two-particle excitations exists for to > 0. The spin waves spectrum merges with the two-particle continuum at a critical value of q and becomes progressively damped. The two-particle con- tinuum is a feature of magnetic metals and does not exist for magnetic insulators (Murata and Doniach, 1972).
Moriya and Kawabata (1973) made significant improvements to the RPA theory of spin excitations above by renormalizing the dynamic susceptibility.
Their theory is known as the self-consistent renormalization (SCR) theory. In the SCR theory X-+(q, to) is given by
x ;'+°(q' to) (57)
x;~+~(q' to) = 1 - Ix~+o(q, to) + x,~(q, to)"
x#~ now depends on the spontaneous magnetization M0 and ),~(q, to) is a renormalization function which must be determined self-consistently.
The main purpose of the SCR theory is to show that the spin fluctuations strongly affect the equilibrium properties of weak ferromagnets, in contrast to the RPA treatment (see section 4.1.1). The function h~(q, to) is taken to be independent of q and to in the long-wavelength approximation, i.e.
X~(q, ~o)--. x~(0, 0) = x~r. (58)
Also if the dependence of h ~ on I is assumed to be sufficiently weak, Moriya showed that the contribution of spin fluctuations to the free energy can be
T R A N S P O R T P R O P E R T I E S O F RE I N T E R M E T A L L I C C O M P O U N D S 179
written
AF = kBT ~, {ln[1 - Ix~t+o(q, ito~) + X~] + Ix~+o(q, iton)} (59)
qn
to,=2zmkBT, n = 0, ---1, -+2 . . .
Moriya then gives the following general relationship:
OAF(M, I)
aF0(M, T) 2 I M -+ - 2H, (60)
OM OM
H 1 _ l + ? t ~
M - XMx(0, 0) X~;(0, 0) I. (61)
X~(0, 0) is the static susceptibility. Eqs. (59) to (61) form a set of self-consistent equations from which XM~ can be calculated if X~a+o(q, to) is known. However, this function depends on the band structure and is usually unavailable in analytic form. Moriya (1979) points out that for weak ferromagnets an expansion of X~+o(q, to) to second order in (to~q) and q is sufficient.
The resulting analysis in the long-wavelength limit shows that the temperature dependence of )t is dominant and gives rise to the following results for weak ferromagnets:
(i) The Curie temperature Tc is given by Tc oc (a - 1) 3/4.
This is in contrast to the Stoner-Wohlfarth (SW) theory which gives sw 1)4/2
T~ oc (a - (see also section 4.1.1).
(ii) The static spin susceptibility X0 obeys a Curie-Weiss law for TF >> T -~ T~
whereas the SW theory gives the result of eq. (52).
(iii) The magnetization near T¢ is given by M oc (T 4/3 - T4/3) 1/2 whereas the SW theory gives the usual MF result, i.e. M oc (T 2 - T2) 1/2. Ueda and Moriya (1975) state that these results are in good agreement with the physical properties of weakly ferromagnetic metals ZrZn2 and Sc3In.
The SCR theory was then used to analyse the temperature dependence of the resistivity o(T) for weak ferromagnets. At low temperature o(T) can be written as follows
p(T) = Po + A T E. (62)
The coefficient A can be calculated using the ansatz that the (sp)-conduction electrons are responsible for carrying the electric current, whereas the d- electrons contribute mainly to the spin fluctuations (see Schindler and Rice, 1967;
Kaiser and Doniach, 1970). Schindler and Rice have shown that A can be written as follows for enhanced paramagnets:
A oc (1 - a) -1/2. (63)
This result has been confirmed by Ueda and Moriya (1975) using the SCR theory
180 E. GRATZ AND M.J. ZUCKERMANN
for both enhanced paramagnetic and weakly ferromagnetic metals far enough from To. They show that this temperature dependence is replaced by the following behaviour of p(T) with T in the neighbourhood of Tc
o ( T ) ~ T 513 (64)
which is in agreement with Mathon's earlier result (Mathon, 1968). This was confirmed by fits to resistivity data for ZrZnv
The analogous results for nearly and weakly antiferromagnetic metals are (Ueda, 1977)
p ( r ) = Po + A T 2, r "~ TN (65)
A ~ - I1-1/2,
p ( T ) o: T 3j2, T ~ TN.
Moriya notes that the resistivity p ( T ) as calculated using the SCR theory goes smoothly through To. No explicit reason is given, however.
The thermal resistivity W at low temperature is given by
W ~ A T . (66)
SCR theory shows that for weak ferromagnets
A oc (1 - or) 3/2. (67)
The above description of the SCR theory and its application to weak ferro- magnetic materials is taken from the review article by Moriya (1979). The reader is referred to the original articles by Ueda and Moriya (1975) for details.
The collective excitation spectrum of strong itinerant ferromagnets is dominated by the Stoner gap ,ab given by eq. (50) rather than the transverse spin fluctuations. As mentioned above the poles of X-+(q, to) are undamped spin waves with energies ~oq ~ zab in the long-wavelength limit.
The quadratic dispersion relation of the spin waves should therefore give rise to the usual resistivity for local moment ferromagnetism at low temperature, i.e.
p(T) = po + A T 2, (68)
where A is proportional to Ab.
The analogous result for antiferromagnets at low temperatures is
p ( T ) = Po + AT4. (69)
The critical region for strong itinerant ferromagnets has not yet been analysed.
This may be possible using Liu's theory (Liu, 1976) but it remains a difficult problem.
4.2. Experimental results and discussion 4.2.1. Resistivity data
4.2.1.1. YCo:, Y(Co1_~Fe)2:0.05 ~<x ~<0.15, Y(Irl-xFex)2: x = 0.2, 0.3
Although the compound YCo2 does not exhibit long-range magnetic order, it is included in this section because it forms the basis for the magnetic RCo2
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 181
YC
t o o l ~ ~
50
0
T[K]
i i i i t i i i t I i q i i I t , t , I i i , i l i i i i m
5O too 150 2OO 250
Fig. 59. p vs. T curves of YCo2 and Y4Co3 (Gratz et al., 1980a).
c o m p o u n d s . The c u r v e of 0 vs. T for Y C o 2 obtained b y G r a t z et al. (1980a) is shown in fig. 59. This shows that a p r o n o u n c e d curvature in the P vs. T c u r v e is seen near 100 K above which the resistivity tends to saturate. This curvature is not o b s e r v e d in the resistivity of Y-non-transition metal c o m p o u n d s such as YA12 and YCu2 (see section 2.2.1.2). The resistivity of YCo2 was shown to follow a T 2 law below 2 0 K b y Ikeda (1977a) (see fig. 60), in contrast to the low-
I00 50
18
5
I
0.5
0.1"
0.05
~, TiCo 2 @
o o
° YC°2 / 2
• ZrCo 2 / g
o NbCo 2 /oo °
jo oo
/ I, f
T 2
tl TCK] Fig. 60. l o g ( p - po) vs. log T plots of TiCo2, YCo2, ZrCo2, and NbCo2 (Ikeda,
1977a).
182 E. GRATZ AND M.J. Z U C K E R M A N N
temperature behaviour of TiCo2 and NbCo2 which is proportional to T 3 below 30 K. Ikeda (1977a) interpreted both the T: law at low temperatures and the high-temperature saturation of p for YCo2 in terms of the theory of spin fluctuations due to Ueda and Moriya (1975) described in section 4.1.2. He concluded that YCo2 is an enhanced paramagnet and that the resistivity is dominated by electron-paramagnon scattering. The T 3 behaviour of the resis- tivity of the other ACo2 compounds is also shown in fig. 60 and was interpreted in terms of s-d scattering by phonons. In a later article Ikeda (1977b) examines the resistivity of Y(Col-xFex)> He shows that the electron-spin fluctuation scattering is suppressed as the Fe concentration is increased as shown in figs. 61 and 62. He interprets this behaviour in terms of the polarization of Co atoms by neighbouring Fe atoms. Magnetic clusters then form and the Co atoms in the cluster can no longer contribute to electron-paramagnon scattering.
Van Dongen et al. (1980) measured the resistivity of Y(Col-xFex)2 for Fe concentrations in the range'0.02 ~< x ~< 0.15. The residual resistivity increases to 250/zIl cm at x = 0.1 and then decreases to about 100/xll cm at x = 0.15 (see fig.
63). This decrease is associated with the onset of magnetic order. The resistivity data are consistent with the occurrence of spin glass freezing for x <0.12 and with long-range ferromagnetic order for x >0.12. Small "kneelike" anomalies were seen in resistivity at the ordering temperatures as marked by the arrows in fig. 63. This implies a degree of localization of the magnetic moments on the Fe sites.
Van Dongen et al. (1980) suggest that the magnetic moments do not form on the Fe sites in Hf(Col_xFe~)2 below x = 0.2 since no spin-glass freezing occurs.
An ordered magnetic state appears in this system for x > 0.2. Van Dongen et al.
(1980) state that although in Y(Irl-xFe~)2 for x ~< 0.4 a non-zero hyperfine field is detected, no evidence is found for magnetic order from the susceptibility and magnetization measurements. They argue that the large contribution to the resistivity with a negative temperature coefficient in Y(Irl_xFex): for x < 0.4 (see
i
125 (Col_xFex)2 y
100 0 at % Fe
75 f 0.5
1,0
25
• 2 0
,
0 100 200 300
Fig. 61. The temperature variation of the resistivity due to magnetic ~cattering (pm) in Y(Col-xFex)2 (Ikeda, 1977b).
~ce~ml
200
o o 300 K
o o
100 ~ 4 . 2 K
at'/, Fe
o' i # 3
Fig. 62. The concentration depen- dence of the electrical resistivity of Y(Col-~Fe~)2 at 4.2K and 300K (Ikeda, 1977b).
2401 200 ¸ 200 2 5 0 170 200.
TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 183
1oo.
lOO
100 f 0
- . . _ _ _ ~ Y(lrT_xFex) 2
- x = 0 . 3 - - . . _ _ _ _
- 0 2
, . - ~ 0 8
i10 9 m '(CoT_xFeA2
r [g]
50 2bo
Fig. 63. p vs. T curves of Y(Col-xFexh and Y(Iq-xFe~)2 for different x. The arrows indicate the magnetic ordering temperature (Van Dongen et al., 1980).
fig. 63) is probably related to a short-range magnetic ordering mechanism. The M6ssbauer measurements indicate a polarization of the Co atoms in Y(Col-xFex)2 for x > xc, whereas in Y(Irl-xFex)2 no polarization is found on the Ir atoms. This means that more short-range magnetic ordering is expected for the latter system.
4.2.1.2. RCo2 ( R = Tb, Dy, Ho, Er), (HoxYl-x)Co2: O ~ x <~ 1.0
In a large number of investigations concerning the RCo2 compounds, it was shown that in DyCo2, HoCo2, and ErCo2 the magnetic transition is a first-order phase transition in contrast to the other RCo2 compounds where it is second order. In the case of a first-order transition most of the physical parameters such as lattice constants (Lee and Pourarian, 1976), magnetization (Lemaire, 1966), specific heat (Voiron et al., 1974), and electrical resistivity (Gratz et al., 1980b) change abruptly at the transition temperature. Fig. 64 shows the tem- perature dependence of the resistivity of the RCo2 compounds in question. The discontinuities observed in the p vs. T curve of the Er, Ho, Dy compounds represent the influence of this first-order transition on resistivity. Note that e.g.
in ErCo2 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. Precise measure- ments of the resistivity in the vicinity of Tc for ErCo2 show a hysteresis in temperature of about 0.8 K. The magnitude of this jump in the p vs. T curve
184 E. GRATZ AND M J . ZUCKERMANN
cS?cml
16~
120
40 . . . . . . . . . . . . . . . . . . . . . . . TIK___]
5 50 100 150 200 250
Fig. 64. Electrical resistivity as a func- tion of temperature for: ErCo2; HoCo2;
DyCo2; TbCo2. Arrows indicate Tc (Gratz et al., 1980b).
decreases with increasing ordering temperature (ErCo2: Tc = 32 K, HoCo2: Tc = 78 K, DyCo2: Tc = 135 K). This is due to the increasing effect of temperature (Gratz et al., 1980b). 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 T¢). Coupled with the ordering of the R-moments is a splitting of the Co 3d band and the appearance of an induced 3d moment of about 1/~B which enhances the ordering process. The extremely slow increase of the P vs. T curve in the high-temperature range experimentally found in all the RCo2 compounds (YCo2 included) is assumed to be due to the s-d scattering processes of conduction electrons into available d-states (Gratz et al., 1980b;
Steiner et al., 1977).
The magnetic properties of R C o 2 compounds have been analysed by Bloch et al. (1975) in the following way. The localized magnetic moments on the R sites were assumed to polarize the 3d band of the Co component and then to interact via this polarization. The polarization of the d-band is described by the Landau theory, the important parameter being the coefficient B of the term in the order