Magnetic properties and magnetic order

2. SALIENT PHYSICAL PROPERTIES

2.2 Magnetic properties and magnetic order

The lanthanides are characterized by local magnetic moments coming from their highly localized 4f electron states. These moments polarize the conduction elec- trons which then mediate the long range magnetic interaction among them. The RKKY interaction is the simplest example of this mechanism. These long range magnetic interactions in lanthanide solids lead to the formation of a wide variety of magnetic structures, the periodicities of which are often incommensurate with the underlying crystal lattice. These are helical structures that have been studied in detail with neutron scattering (Sinha, 1978; Jensen and Mackintosh, 1991). In the later sections of this chapter, we shall elaborate on ourab initiostudy of the finite temperature magnetism of the heavy lanthanides and elucidate the role of the conduction electrons in establishing the complex helical structures in these sys- tems. This ab initio theory and calculations go beyond the ‘standard model’ of lanthanide magnetism (Jensen and Mackintosh, 1991).

The standard approach to describing the magnetism of lanthanides, and in particular their magnetic moments, is to assume the picture of electrons in an

Ce 0.85La 0.90 0.95 1.00 1.05

R3+ (Å)

Lu Yb Tm Er Ho Dy Tb Gd Eu Sm Pm Nd Pr

FIGURE 2 Ionic radii of trivalent lanthanides.

isolated atom. In the absence of spin-orbit coupling, the angular momenta of the electrons in an atom combine according to Russel-Saunders coupling to give a total orbital angular momentum^Land a total spin angular momentumSˆ, with the respective eigenvalues ofL Lð þ1Þh² and S Sð þ1Þh² (Gasiorowicz, 1974). In the lanthanides, however, spin-orbit coupling l L^

^Sˆ cannot be ignored (Strange, 1998) and therefore one has to introduce the total angular momentum^J¼L^ þS^. The wavefunctions describing the electrons then obey the eigenvalue relations:

^J²c_J;mj;L;S¼J Jð þ1Þh²c_J;mj;L;S

^J_zc_J;mj;L;S¼mjhc_J;mj;L;S

L^²c_J;mj;L;S¼L Lð þ1Þh²c_J;mj;L;S S^²c_J;mj;L;S¼S Sð þ1Þh²c_J;mj;L;S;

ð1Þ

where cJ,m_j,L,S are normalized atomic wavefunctions, which are assumed to be located on a single atom nucleus.

The energy associated with a magnetic field in the lanthanide is small in comparison to electronic energies and it is usual to treat it applying perturbation theory. The perturbing potential is simply the scalar product of the magnetic moment,m, and the magnetic field,^ B, experienced by the atom:

dV¼ ^m

^{B; ð2Þ}

where^mis given as

^

m¼ m_Bg_J^J: ð3Þ

Here,gJis the Lande´ g-factor:

g_J¼1 þ J Jð þ1Þ L Lð þ1Þ þS Sð þ1Þ

2J Jð þ1Þ : ð4Þ

If this model is correct, then it is only necessary to know the values of the quantum numbers J, L, andS to calculate the magnetic moment of the lantha- nides. These are determined by Hund’s rules:

(1) Swill be a maximum subject to the Pauli exclusion principle

(2) Lwill be a maximum subject to rule 1 and to the Pauli exclusion principle (3) If the shell is less than half-full, then the spin-orbit coupling coefficient lis

positive and J¼|LS| is the ground state. For a shell that is greater than or equal to half-fulllis negative and J¼LþSis the ground state.

InTable 1, we show the quantum numbers for the trivalent lanthanide ions, the Lande´ g-factor, and the high-temperature paramagnetic moment given as:

m_t¼ ffiffiffiffiffiffiffiffiffi h^m²i q

¼m_BgJ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J Jð þ1Þ

p ; ð5Þ

assuming that only the f-electrons contribute to the magnetic moment. If we know the energy levels of the lanthanide ions in a magnetic field, we can use standard statistical mechanics to calculate the susceptibility of the ions. For most of the ions,

the difference in energy between the first excited state and the ground state is much greater thank_BTat room temperature, wherek_Bis the Boltzmann constant, and essentially only the ground state is populated. This enables us to derive the Curie formula

w¼Nh^m²i

3kBT ð6Þ

for the susceptibility of a system ofNnon-interacting ions. More realistically, this formula should be replaced by the Curie-Weiss law, where theTin the denomi- nator inEq. (6)is replaced byTTc, whereT_cis the magnetic ordering tempera- ture. The Curie-Weiss formula was employed to determine the experimental magnetic moments, me, in Table 1. For Sm^3þ and Eu^2þ, the first excited level is withink_BTof the ground state and so is appreciably populated. To describe these two ions with numerical accuracy, it is necessary to sum over the allowed values of J and recall that each J contains 2J þ 1 states. The susceptibility then becomes considerably more complicated but does give a good description of Sm^3þand Eu^2þ.

The theoretical magnetic moments in Table 1 are for single trivalent ions assuming no inter-ionic interactions. However, the experiments are performed on metallic elements where each individual ion is embedded in a crystal and feels TABLE 1 Quantum numbers and total f-electron magnetic moments of the trivalent lanthanide ions.mtis the magnetic moment calculated fromEq. (5).meis the measured magnetic moment.

All magnetic moments are expressed in Bohr magnetons

S L J Ground state g_j mt mea

La 0.00 0.00 0.00 ¹S₀

Ce 0.50 3.00 2.50 ²F_5/2 6/7 2.54 2.4

Pr 1.00 5.00 4.00 ³H₄ 4/5 3.58 3.5

Nd 1.50 6.00 4.50 ⁴I_9/2 8/11 3.62 3.5

Pm 2.00 6.00 4.00 ⁵I₄ 3/5 2.68

Sm 2.50 5.00 2.50 ⁶H_5/2 2/7 0.85 1.5^b

Eu 3.00 3.00 0.00 ⁷F₀ 0.0 3.4^b,c

Gd 3.50 0.00 3.50 ⁸S_7/2 2 7.94 7.95

Tb 3.00 3.00 6.00 ⁷F₆ 3/2 9.72 9.5

Dy 2.50 5.00 7.50 ⁶H_15/2 4/3 10.65 10.6

Ho 2.00 6.00 8.00 ⁵I₈ 5/4 10.61 10.4

Er 1.50 6.00 7.50 ⁴I_15/2 6/5 9.58 9.5

Tm 1.00 5.00 6.00 ³H₆ 7/6 7.56 7.3

Yb 0.50 3.00 3.50 ²F7/2 8/7 4.54 4.5

Lu 0.00 0.00 0.00 ¹S₀

aAll experimental values taken fromKittel (1986).

bSee text.

c Eu usually exists in the divalent form.

the crystalline electric field, which arises from charges on neighbouring ions. The excellent agreement between the magnetic moment measured experimentally and that obtained fromEq. (5)implies that the 4f-electrons are so well shielded that the effect of the crystalline environment can be neglected and the agreement with Eq. (6)supports this view. With the agreement between theory and experiment displayed inTable 1, it might reasonably be assumed that the magnetic moments of the lanthanide metals are well understood. While this is probably true qualita- tively, it is certainly not true quantitatively. Questions about this model that may be raised are:

(1) In the solid state, the interaction with the sea of conduction electrons distorts the pure atomic picture, and multiplets other than the Hund’s rule ground state mix into the full wave function. In view of the success of the atomic approxima- tion, it is likely that this mixing of differentJquantum numbers is at the few percent level, nonetheless this is an assumption that should be tested.

(2) The eigenfunctions CJ,mj,L,S in Eq. (1) are generally quantum mechanically entangled, given as sums of Slater determinants constructed from atomic single-electron orbitals. The description of entangled quantum states in a solid state environment is extremely difficult. With the recent introduction of the DMFT (Section 3.8), one might have developed a scheme for this, however, more investigations are needed. Usually, one resorts to the indepen- dent particle approximation thus ignoring the ‘many-body’ nature of the wavefunction and the localized f-manifold is represented by a single ‘best choice’ Slater determinant. This is the approach taken in the various DFT schemes, including the SIC-LSD theory to be described in Section 3.5. How could such a picture arise from a more sophisticated and realistic description of the electronic structure of the lanthanides?

(3) The picture of lanthanide magnetism described above is for independent trivalent lanthanide ions. Thus, it does not explain cooperative magnetism, that is ordered magnetic structures which are the most common low- temperature ground states of lanthanide solids. Our view of a lanthanide crystal is of a regular array of such ions in a sea of conduction electrons to which each ion has donated three electrons. For cooperative magnetism to exist, those ions must communicate with one another somehow. It is generally accepted that this occurs through indirect exchange in the lanthanide metals, the simplest example of this being the RKKY interaction (Ruderman and Kittel, 1954; Kasuya, 1956; Yosida, 1957). However, for this to occur, the conduction sd-electrons themselves must be polarized. This conduction elec- tron polarization has been calculated using DFT many times and is found to be substantial. There have been many successes in descriptions of magnetic structures, some of which will be discussed in Section 7.2. However, in terms of the size of magnetic moments, the agreement between theory and experiment shown in Table 1 is considerably worsened (see discussion in Section 5). Why is this?

(4) If the atomic model were rigorously correct, a lanthanide element would have the same magnetic moment in every material and crystalline environment. Of course, this is not the case, and more advanced methods have to be employed to determine the effect of the crystalline and chemical environment on mag- netic moments with numerical precision. Examples of materials where the theory described above fails to give an accurate value for the lanthanide ion magnetic moment are ubiquitous. They include NdCo5(Alameda et al., 1982), fullerene encapsulated lanthanide ions (Mirone, 2005), and lanthanide pyro- chlores (Hassan et al., 2003).

In conclusion, it is clear that the standard model makes an excellent first approximation to the magnetic properties of lanthanide materials, but to under- stand lanthanide materials on a detailed individual basis, a more sophisticated approach is required.