The layered trigonal antiferromagnets VX2 (X = Cl, Br, I) have the
crystal structure of CdI2 with space group PmL,
where the halide anions form hexagonal close-packed layers and the cations occupy
half the octahedral holes. Recently, VX2 have attracted extensive
interest due to the properties of quantum fluctuations (Bondarenko et al.,
1996; Rastelli and Tassi, 1996; Watabe et al., 1995), critical behaviours
(Kawamura, 1988) and Raman scattering (Guntherodt et al., 1979; Bauhofer
et al., 1980). In addition, Electron Paramagnetic Resonance (EPR) experiments
have been carried out on these systems and the anisotropic g factors g//
and g⊥ were measured at 300 K (Yamada et al., 1984).
In order to interpret these EPR experimental results, inspiring investigations have been made on VCl2 and VBr2 from the perturbation formulas of the g factors for a 3d3 ion in trigonal symmetry based on the cluster approach (Du and Li, 1994). In these formulas, not only the Spin-orbit (SO) coupling coefficient of the V2+ ion, but also that (as well as the p orbitals) of the ligands is taken into account (Du and Li, 1994). The theoretical results indeed show improvement compared with those based on only the SO coupling coefficient of the central ion. However, the trigonal crystal-field parameters in their calculations were taken as the adjustable parameters, without correlating with structural data of the studied systems. In addition, the experimental g factors of VI2 have not been explained uniformly. Since the S.O. coupling coefficient of the ligand I¯ is much larger than that of the V2+, the s-orbitals of the ligand may be important and lead to some contributions to the g factors. To investigate the g factors of VX2 to a better extent, in this study, the s-orbitals of the ligand are introduced to the single-electron wave functions of the 3d3 octahedral clusters. Then the previous theoretical formulas are also modified and applied to the studied systems.
Theory and Calculations
In VX2, V2+ locates on the octahedral site with slight
trigonal (D3d) distortion (Wyckoff, 1951). For a V2+(3d3)
ion in trigonal symmetry, its ground 4A2 state would be
split into two Kramers doublets due to the combination effect of the SO coupling
and the trigonal crystal-field interactions. Thus, the perturbation formulas
of the g-shifts Δgi (=gi - gs, where gs
= 2.0023 is the spin only value and I = // and ⊥) based on the cluster
approach can be expressed as (Du and Li, 1994):
Here the energy denominators Ei (I = 1 ~ 5 ) stand for the separations between the excited states 4T2, 2T2a, 2T2b, 4T1a and 4T1b and the ground state 4A2 in terms of the cubic field parameter Dq and the Racah parameters B and C for the studied 3d3 clusters (Du and Li, 1994). V and V are the trigonal field parameters. ζ and ζ' are the SO coupling coefficients, while k and k' are the orbital reduction factors. In the cluster approach of the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992), the contributions of the s-orbitals of ligands were usually neglected for simplicity. Unlikely, the above contributions are considered here. Thus, the total single electron wave function including the ligand s-orbital contributions may be written as
where φγ (γ = e and t denote the irreducible representations of the Oh group) are the d orbitals of the central metal ion. χpγ and χs stand for the p- and s- orbitals of the ligands. Nγ and λγ (or λs) are, respectively, the normalization factors and the orbital mixing coefficients. From the semiempirical method similar to those in the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992), we have the approximate relationships
and the normalization conditions
Here Sdpγ (and Sds) are the group overlap integrals. Usually, the mixing coefficients increase with increasing the group overlap integrals and one can approximately adopt the proportional relationship between the mixing coefficients and the related group overlap integrals, i.e., λe/Sdpe ≈ λs/Sds within the same irreducible representation eg. In general, the covalency factors ft and fe can be determined from the ratio of the Racah parameters for the 3d3 ion in a crystal to those in free state, i.e., ft ≈ fe ≈ C/C0.
According to the cluster approach containing the ligand s-orbital contributions, the SO coupling coefficients ζ, ζ' and the orbital reduction factors k, k' for the 3d3 octahedral clusters may be expressed as
where ζd0 and ζp0 are
the SO coupling coefficients of the free 3d3 and the ligand ions,
respectively. A denotes the integral, where
R is the metal-ligand distance of the studied systems. Obviously, when taking
Sds = λs = 0 and A = 0, the above formulas are reduced
to those in the absence of the ligand s-orbital contributions (Du and Li, 1994;
Du and Rudowicz, 1992).
From the superposition model (Newman and Ng, 1989), the trigonal field parameters V and V for the studied systems are written as follows:
where β is the angle between the metal-ligand bond and the C3 axis. The magnitude and nature (elongation or compression along the C3 axis) of trigonal distortion can be characterized by the value and sign of the angular difference δβ (= β- β0, where β0 ≈ 54.74° is the bonding angle in cubic symmetry). }(R) and }4 (R) are the intrinsic parameters with the reference bonding length R. For 3dn ions in octahedra, the relationships }4 (R) ≈(3/4)Dq and }2 (R)/}4 (R) ≈ 9 ~ 12 have been proved to be valid in many crystals (Newman and Ng, 1989; Yu et al., 1994; Edgar, 1976). Here we take }2(R) ≈ 9}4 (R). Therefore, the trigonal distortion (or local structure) is related to the low symmetrical parameters V and V and hence to the g factors (particularly the anisotropy g// - g⊥) of the studied systems.
For VX2 (Wyckoff, 1951), the metal-ligand bonding lengths R and the angles β between R and the C3 axis are collected in Table 1. From the distances R and the Slater-type SCF functions (Clementi and Raimondi, 1963), the group overlap integrals Sdpγ (as well as the integrals Sds and A) are obtained and shown in Table 1.
||The metal-ligand distances and angles, group overlap integrals,
the spectral parameters Dq, B and C (cm-1), Nγ
and λγ (and λs), SO coupling coefficients
(cm-1), orbital reduction factors and the trigonal field parameters
V and V (in cm-1) for VX2 (X = Cl, Br, I)
|| The gyromagnetic factors for VX2 at 300 K
|a Calculations based on neglecting of the contributions
from the s- orbitals of the ligands, i.e., similar to the treatments in
Du and Li (1994); Du and Rudowicz (1992). b Calculations based
on inclusion of the contributions from the s-orbitals of the ligands in
this work. c Yamada et al. (1984)
The spectral parameters Dq, B and C are acquired from the optical spectra for these systems (Erk and Hass, 1975) and collected in Table 1. By using Eq. 3 and 4 and the free-ion parameters B0 ≈ 766 cm-1 and C0 ≈ 2855 cm-1 (Griffith, 1964) for V2+, the covalency factor fγ and hence the molecular orbital coefficients Nγ and λγ can be calculated. From the free-ion values ζd0 ≈ 167 cm-1 (Griffith, 1964) for V2+ and ζp0 ≈ 587, 2460 and 5060 cm-1 for X = Cl, Br and I (McPerson et al., 1974), the parameters ζ, ζ, k and k' are obtained from Eq. 5 and also shown in Table 1.
Substituting the related parameters into Eq. 1, the theoretical g factors are obtained and compared with the observed values in Table 2. For comparisons, the calculated results by neglecting the contributions from the s-orbitals of the ligands (i.e., Sds = λs = 0 and A = 0, corresponding to the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992) ) are also given in Table 2.
From Table 2, one can find that the theoretical g factors including the ligand s-orbital contributions are in good agreement with the observed values. This means that the ligand s-orbital contributions seem not negligible in the analyses of the EPR g factors for VX2 systems, especially for the larger ligands Br and I.
For X = Cl, the magnitudes of the theoretical Δgi by including the ligand s-orbital contributions differ little (no more than 1%) from the results on neglecting the above contributions. So the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992) can be regarded as good approximations for this ligand. Nevertheless, for VBr2 and VI2 the calculated Δgi in the absence of the ligand s-orbital contributions are about 39% smaller and 28% larger than the experimental values, respectively. This means that the contributions from the SO coupling (which are much larger than that of V2+) of the ligands Br¯ and I¯ are somewhat overestimated, if only the contributions from the p-orbtials of the ligands are considered. In fact, inclusion of the s-orbitals of the ligands can modify the parameters Ne and λe, then change the magnitudes of k and ζ and finally lead to more reasonable Δgi. For X = Br, the significantly larger Δgi in magnitude by considering the ligand s-orbital contributions compared with those by neglecting the above contributions can be attributed to the noticeable (twice) increase in the positive ζ related to the very small value (about 13 cm-1) in the absence of the contributions. Therefore, the useful assumption that the contributions of the s orbitals of the ligands may be negligible for 3dn ions in octahedra (e.g., KNiF3) (Du and Li, 1994) seems not always valid for ligands having much larger SO coupling coefficient (e.g., I¯) and so the ligand s-orbital contributions should be considered in the studies of the g factors for VX2 here.
In the above calculations, the trigonal field parameters V and V are
determined from the structural data of the systems under study and the superposition
model, instead of taking as adjustable parameters. The calculated anisotropies
of the g factors are also comparable with experiment. The negative g//-g⊥
for VCl2 is in consistence with the fact that the ligand octahedron
is slightly compressed (i.e., δβ ≈ 0.22°>0). For VBr2
and VI2, the small calculated anisotropies agree largely with the
nearly isotropic g factors (the observed anisotropies are almost zero within
the experimental errors (Yamada et al., 1984). This point may be interpreted
as the small magnitudes of ζ (Table 1) due to the
larger ζp0 for both ligands (Eq. 1
and 5) and the very slightly elongated ligand octahedra (i.e.,
δβ ≈ -0.12° and -0.57°<0 for X = Br and I, respectively).
Thus, the trigonal field parameters obtained from the superposition model in
this work can also be regarded as reasonable.
The gyromagnetic factors for the layered antiferromagnets VX2 are theoretically studied by using the perturbation formulas of the g factors including the contributions from the s-orbitals of the ligands in this study. The above investigations seem to be useful to the experimentalists working on magnetic properties of these materials by mean of EPR technique.
This research was supported by the Youth Fundation of Science and Technology of UESTC under grant No. JX04022.