Transport properties of rare earth-non-transition-metal compounds exhibiting long-range magnetic order

E. GRATZ

3. Transport properties of rare earth-non-transition-metal compounds exhibiting long-range magnetic order

3.1. Theoretical introduction

The purpose of section 3.1 is to give a detailed description of the transport properties of those RI compounds whose magnetic properties are due to the localized magnetic moments of the rare earth component only. The collective magnetic properties of such compounds are based on the polarization of the conduction band by the rare earth moments. The resultant exchange interaction is known as the Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction between sp- and f-electrons. Since the magnetic properties of these systems have a strong influence on the transport properties it is appropriate to begin the theoretical discussion with a description of the magnetic moments of free rare earth ions. This section continues with a description of the RKKY interaction and with theoretical predictions for the magnetic effect on the electrical resistivity in several temperature regions.

The influence of magnetic order on the thermopower and thermal conductivity is not as well understood theoretically as is the resistivity. Theoretical models have been presented in a small number of publications only and they are in reasonably good agreement with experimental data. A short description of some of these calculations will be given in section 3.1.4 below.

3.1.1. Rare earth magnetic moments

The free ions of the lanthanides are usually trivalent both in their atomic state and in the solid state. However, exceptions are often found for the elements Ce, Eu, and Yb in the solid state, since these elements tend to have either empty, half-full or full 4f shells.

The electronic structure of rare earth atoms can be described in terms of a core of filled shells equivalent to a xenon atom plus the following configuration

4f"5d°-16s 2 (n = 1, 2 . . . 14).

This configuration can be described as follows:

(i) a 4f shell which is gradually filled and may give rise to a magnetic moment,

136 E. GRATZ AND M.J. ZUCKERMANN

(ii)

(5s25p 6)

shells which are completely filled and screen the 4f shell, (iii) (5d°-16s 2) valence shells which form conduction bands in the solid state.

As noted by Liu (1961) a partially filled 4f shell has a magnetic moment of g~BJ. /xB is the Bohr magneton and J is the total angular moment. From the Russell-Saunders coupling scheme J = L + S or IL - S] depending upon whether the 4f shell is more than or less than half filled. The Land6 g-factor, g, is given by

g = 1 + J ( J + 1 ) - L ( L + 1)+ S ( S + 1)

2J(J + 1) (18)

A description of the exchange interaction between two rare earth magnetic moments is given in section 3.1.2.

3.1.2. Exchange interaction

Liu (1961) noted that the wave functions of the 4f electrons on different rare earth (R) atoms in the solid state do not usually overlap. This is because the radius of the 4f shell is almost 0.35 A and the wave functions are therefore zero on the Wigner-Seitz sphere. There can therefore be no direct exchange, and exchange interactions between different R-magnetic moments must be mediated by the conduction electrons. Liu points out that there are two possible interaction types. In the first type the 4f magnetic moment on the R-atom polarizes the sp conduction bands of the compound via a direct s-f exchange interaction given by

/')sf = - 2 ( g - 1) ~_~ ~(ri - Rj)s,. Jj, (19) where s~ and r~ are the spins and spatial coordinates of the conduction electrons, respectively and dj and Rj are those of the R-atoms, respectively. ~ is the exchange integral for this interaction and has been calculated by several authors (Liu, 1961; Kondo, 1962). Eq. (19) is important for the analysis of transport properties.

The polarization of the conduction band due to the Hamiltonian of eq. (19) oscillates spatially with a long-range damping factor. The polarized conduction electrons can now scatter magnetically with another R atom. The resultant exchange interaction is known as the RKKY interaction and accounts for most of the magnetic properties of RI compounds.

The Hamiltonian for the RKKY interaction is given by

Rex = --2(g -- 1) 2 ~. Jex(Raj)J~ "Jj, (20)

where the indirect exchange ~ex(R) is now given by

~ex(R) = m ~ [2kFR cos(2kFR)- sin(2kFR)]. (21)

4~r~lRI ~

kF is the Fermi wave vector and J¢ is the direct exchange interaction between sp- and f-electrons of eq. (19). It is important to note that ~¢x(R) of eq. (21) is an

TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 137

isotropic function of R since it was derived in the free electron approximation.

However, the position of the nodes and peaks of ~ex(R) will be shifted when crystal structure effects are included in the calculation.

As described by Liu (1961) the nature of the exchange interaction changes when the energy required to promote an electron from the conduction band into the 4f band is small. In this case the admixture interaction between conduction and 4f electrons must be taken into account. This interaction leads to'°an effective attractive sp-f exchange interaction of the Schrieffer-Wolff kind though the form of the Hamiltonian is still given by eq. (19). Such admixture interactions are applicable to Ce and Yb alloys (see Coqblin and Blandin, 1968) and lead to a Kondo effect.

3.1.3. Influence of localized magnetism on resistivity

In section 2.1.1 it was assumed that the temperature dependence of the non-magnetic RI compounds is determined by the residual resistivity p0 and the lattice resistivity Pph 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) + Pm,g(T). (22)

In fig. 13 the temperature dependence of the electrical resistivity of RI compounds without d-electron contribution is shown schematically. For the case of resistivity the contribution Pmag(T) will now be the subject of discussion. Omag(T)

Cexp

Fig. 13. Schematic temperature dependence of the electrical resistivity of magnetic RI compounds with negligible d-electron contribution.

138 E. GRATZ AND M.J. ZUCKERMANN is characterized by the following traits:

(i) a temperature independent behaviour for T > Tord, (ii) a pronounced kink at T = Tor~,

(iii) a strong decrease for T < Tord with decreasing temperature.

Tord is the transition temperature for magnetic ordering. A short discussion of the theoretical models describing such a behaviour will now be given.

3.1.3.1. P a r a m a g n e t i c r e g i m e (T > Tord)

Several authors (Kasuya, 1956; de Gennes and Friedel, 1958; Dekker, 1965) were able to solve the conduction electron scattering problem in the paramagnetic regime (T > Tord) by using the Hamiltonian of eq. (19). The magnitude Pmag(T > rord) ~- Pspd is then given by

3 7 r N m *

Pspd = ~ Io~12(g -- 1)2J(J + 1). (23)

Eq. (23) was derived by solving the Boltzmann equation in the relaxation time approximation. The conduction electrons were considered to be free particles with an effective mass m*. The transition probability was obtained in the first Born approximation. Furthermore, it was assumed that in the paramagnetic region the local magnetic moments are completely decoupled with each moment having (2J + 1) possible orientational states in a given direction z (neglecting the influence of the crystal field). For the other symbols used in eq. (23) see Dekker (1965). Eq. (23) shows that O~pd is proportional to the de Gennes factor (g - 1) 2

x Y(J + 1). pspa provides a large additive contribution to the total resistivity as shown in fig. 13. We assume that the linear increase of the total p ( T ) curve is caused by electron-phonon scattering (in accordance to the Bloch-Grtineisen law of eq. (2). An extrapolation procedure from high temperature to absolute zero then gives the magnitude of Ospd.

Eq. (23) can be written as

3 7 r N m * 2 L _ .

pspd(X) {x[~al (g,, - 1)zJ,,(Ja + 1)+ (1 -x)l~bl2(gb -- 1)2jb(Jb + 1) - x(1 - X)[~a(ga ^-^- 1)Ja -¢b(gb - 1)Jb]2}, (24) for a compound consisting of two different magnetic R elements where ~¢a and Jb are the (sp)-f direct exchange integrals for the alloy components and g,, gb are the respective Land6 g-factors. The expression in eq. (24) is derived under the assumption that the domains of ferromagnetically aligned spins have dimensions which are small compared to the conduction electron mean free path. When the system contains only one magnetic rare earth component as in (GdxYl-x), eq. (24) reduces to

3 ~ r N m * ,

mpa(x)

= ~ t g - 1)2l¢12jx(1 +

Jx).

(25)

Dekker (1965) proposed the following expression for po(x) 3 ~ r N m * , ,

po(X) = ~ x t ~ - x)[v~b + l¢12(g - 1)212] (26)

TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 139 for the case where the mean free path of the conduction electrons is much greater than the size of the magnetic domains for a system with a single magnetic component. Yah = Va - Vb is the difference in the Coulomb interaction of the conduction electrons with the two components of the binary system.

3.1.3.2. Influence of crystal field on spin-disorder resistivity

The expression for the spin-disorder resistivity has the following form in the presence of a crystal field:

3zrNm* ji2(g

pspd(Z) = ~ -- l) = Y~

I(m'si'ls" dlmj)12pJi~,.

(27)

ms and m'~ represent the spins of the conduction electrons in the initial and final states, i and i' are the initial and final crystal field levels of the rare earth ion, p~

is the Boltzmann probability that the rare earth ion is in a crystal field level with energy Ei and is given by

Pi = e x p ( - E i / k B T ) , (28)

e x p ( - Ei/kBT)

E~ depends on the extent of the crystal field splitting and f~, is given by 2

f~' = 1 + exp[(Ev- E~)/kBT]" (29)

Rao and Wallace (1970) applied eq. (27) to experimental data for CeA12 in order to explain the unusually broad "knee" in the p vs. T curve at about 70 K. They ascribed this behaviour to the influence of the crystal field on pspa given by eq.

(27). One of the important consequences of the influence of crystal fields on resistivity in the paramagnetic state is that Pspd now becomes temperature dependent.

3.1.3.3. K o n d o effect and K o n d o lattices

The occurrence of minima in the temperature dependence of the electrical resistivity of metallic systems has been known for a long time in the metallic systems consisting of elements such as Cu, Ag or Au as solvents with very small amounts (~<0.1 at%) of magnetic impurities such as Cr, Mn, Fe as solutes (Van den Berg, 1964). The theoretical explanation of this effect is due to Kondo (1964). Kondo showed that the spin-disorder resistivity is not a constant (as given by eq. (23)) but increases with decreasing temperature under certain circumstances. This result follows from third-order perturbation theory using the Hamiltonian given by eq. (19). In Kondo's model it is shown that the conduction electrons can be scattered from an initial state to a final state via intermediate states. The efficiency of this additional "channel" is large in the low-temperature range and gives rise to the observed increase of the spin-disorder resistivity for

< 0 (see below). The appearance of a minimum in the p vs. T curve follows from the combined effect of the increased spin-disorder resistivity and the decreasing lattice scattering resistivity on cooling the sample. The expression for

140 E. GRATZ AND M.J. ZUCKERMANN the Kondo resistivity is then given by

PKondo(T) = Pspd[1 + ctN(eF)p ln(T/TF)]. (30)

where a depends on the nature of the local moment, Tv is the temperature corresponding to the Fermi energy, ~ is the direct exchange integral of eq. (19) and N(eF) is the density of states at the Fermi energy.

Summing up it can be said that the Kondo anomaly should be observable if:

(a) localized magnetic moments exist in a metallic matrix,

(b) these local moments are decoupled and spin-flips can take place, (c) the sign of ~ is negative due to sp(d)-f admixture interactions.

Some of the Kondo systems investigated in this section are not in fact dilute magnetic alloys. Instead they are "Kondo lattices" in which the magnetic moments lie on a sublattice of the RI compound (e.g. CeA12). The R-atom in all these compounds is Ce for which the d-f admixture interaction mentioned in section 3.1.2 is dominant. Such RI compounds are therefore Kondo systems in which eq. (30) for the resistivity holds. However, for temperatures T < TK the magnetic moment is usually not totally compensated in a Kondo lattice since now there are too few conduction electrons to achieve full compensation on every magnetic lattice site.

Another feature of the Kondo lattice is that the same d-f admixture interaction which gives rise to the Kondo behaviour also generates an RKKY interaction between magnetic moments on different sites. The competition between the Kondo compensation and the magnetic ordering due to the RKKY interaction then determines the magnetic properties of the system. For example, CeA12 exhibits a Kondo-like minimum in the resistivity above the Ne61 temperature TN. Below TN the compound is an antiferromagnet. In contrast, CeA13 is non-magnetic because its f- and d-band are too broad to support magnetic moments.

3.1.3.4. Critical phenomena at T ~ Tc or TN

As was discussed in section 3.1.3 the resistivity curves for the RI compounds in question show pronounced kinks at Tord. This is a critical phenomenon which is due to the influence of magnetism on electrical resistivity as shown schematically in fig. 13. It is therefore useful at this point to review the theory of critical phenomena in magnetically ordered systems. In particular, the Curie point Tc of a ferromagnet and the Ne61 point TN Of an antiferromagnet are examples of critical points which are analogous to critical points of liquid-vapour systems.

Order-disorder phase transitions involving long-range order parameters occur at these magnetic critical points. For the ferromagnetic case the order parameter is the spontaneous magnetization Mo. At T = 0 the ferromagnetic spin systems of a pure crystal are completely ordered but with increasing temperature, M0 decreases continuously. The breakdown of long-range magnetic order is initially slow, because when most of the spins are well aligned, a large amount of energy is required to reverse a spin in the presence of the exchange field. However, as the temperature increases, a breakdown of long-range order occurs more rapidly until the long-range order vanishes at the Curie point. Nevertheless, critical spin fluctuations still maintain a limited order among the spins in small regions of the

TRANSPORT PROPERTIES OF RE INTERMETALLIC COMPOUNDS 141 lattice which contain comparatively few spins well above Tc or TN. When the temperature is sufficiently high (T >> Tc or TN) complete disorder can be assumed. However, close to the critical point the amplitude of spin fluctuations is small, but the fluctuating domains are quite extended in space. The total net spontaneous magnetization is always zero above Tc.

Many physical properties are influenced by critical fluctuations. These include both equilibrium properties such as the specific heat and the magnetic suscep- tibility and non-equilibrium phenomena such as the transport properties. Here we distinguish between two classes of magnetic materials. One is the class which can be described in terms of localized spins represented by the rare earth compounds. The other class comprises the weak ferromagnetic materials whose magnetic properties can satisfactorily be explained using the band model. Our experiments on RI compounds show that an anomaly in the resistivity occurs only for the first class of magnetic materials for which the magnetic moments are localized. In rare earth compounds containing A1, Ni, Cu, Ag, Au, etc. the kink in the p vs. T curve in the vicinity of Tc or TN is found to be extremely sharp, whereas practically no anomaly could be found at Tc for weak ferromagnetic substances like ZrZn2 (Ogawa, 1976). De Gennes and Friedel 0958) examined the effect of short-range order spin fluctuations on the resistivity of ferromagnetic metals with localized spins interacting via the exchange Hamiltonian of eq. (19). The calculation was carried out in the first Born approximation using the Ornstein-Zernicke method for evaluating the static spin-spin correlation function needed to derive

Pmag(T).

They found that

Prnag(T)

close to Tc is given by

with b > 0.

Fisher and Langer (1968) improved the analysis by introducing a modified correlation function which gave a theoretical expression for the divergence in dpmag/dT similar to that found for the specific heat in the temperature range just above Tc. Richard and Geldart (1973) extended this model and showed that the result of Fisher and Langer also holds for temperatures below To. In antifer- romagnetic materials dpmag/dT at TN is influenced by the existence of superzones (Elliott and Wedgwood, 1963). The anomalous behaviour of transport properties at the magnetic critical points have been studied in detail by Ausloos et al. (1980) and applied to TbZn. Zoric et al. (1973) show that the product of the thermopower S and the resistivity p can be written as follows in the critical regime:

pSJT = Apn + Bpc + CF(2kFT). (32)

Se is the electron diffusion thermopower, Pn is the normal resistivity and pc is the critical resistivity given by

2k F

kF 4 f F(k, T)k 3 dk. (33)

142 E. GRATZ AND M.J. ZUCKERMANN

F(k, T) is the spin-spin correlation function, kF is the Fermi momentum, T is the temperature, and A, B, C are constants. If Pn is assumed to be linear in T, the derivation of eq. (32) can be written

d(pSe/T)/dT = A' + B' dpc/dT + C' dF/dT. (34)

Zoric et al. (1973) then use eq. (34) in conjunction with "mental differentiation"

of the curve of F(k, T) vs. T of Fisher and Langer (1968) (see inset to fig. 54) to suggest the following qualitative differences between dp/dT and d(pSJT)/dT in the critical region:

(i) The anomaly in dp/dT should be more articulated than that in d(pSo/T)/dT.

(ii) The minimum in do/dT should be closer to Tc than d(p/r)/dT.

These results were used by Zoric et al. (1973) to analyse the critical behaviour of Se for GdNi2. This is discussed in section 3.2.2.4.

3.1.3.5. Intermediate temperature range ( 0 . 2 Z o r d ~ T ~ Tord)

Pmag is usually temperature dependent in this temperature range. Below Tc the localized spins begin to order so that the conduction electrons scattering sharply decrease causing the pronounced kink at the ordering temperature. As was first demonstrated by Kasuya (1956) one can use the molecular field approximation instead of the s-f exchange interaction given in eq. (19) in the temperature regime T ~ Tc or TN. This model neglects correlations between the spins and only the interaction of a spin with the average exchange field produced by the neighbouring spins is taken into account. The scattering of the conduction electrons by a particular local moment at R~ is now due to the difference between S and (S)r where (S)r is the average ionic spin. The Hamiltonian for the molecular field model is then given by

/~ = -o~s. (S - (S)r). (35)

The magnetic contribution Pmag to the resistivity can now be calculated using/~ of eq. (35) and the result is given by

Pmag(T) ⁼ Pspd [1 (S)2 ] (36)

S(S + 1)J'

where pspd is the spin-disorder resistivity (see above). A knowledge of (S)r as a function of temperature enables us to obtain the temperature dependence of Pmag for T ~< T~ or Ts.

3.1.3.6. Low-temperature regime (T ~ T¢ or TN)

In the lowest temperature regime the magnetic properties of the RI compounds discussed in this section can be described satisfactorily using the spin wave model. The low-energy collective excitations of a system of coupled 4f moments are known as spin waves. Therefore at low temperatures the conduction electrons can be scattered both by phonons and by spin waves. In most of the magnetic materials electron spin wave scattering gives a large contribution to the total resistivity. In particular, several authors (Kasuya, 1959; Mannari, 1959;

Volkenshtein et al., 1973) derived a T a dependence for the spin wave contribution to the total resistivity in ferromagnetic materials. Such calculations

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