Solid State Communications,
Vol. 8, pp. 665—668, 1970.
Printed in Great Britain
OPTICAL SPECTRA OF EVEN-ELECTRON RARE EARTH IONS IN THE ORTHOFERRITES* A.P. Malozemoff and R.L. %Vhite Department of Materials Science, Stanford University, Stanford, California
(Received 13 February 1970 b~’H. Suhi)
Two nearly degenerate even-electron singlet states have a spin Hamiltonian which differs from the usual Kramers’ case. Such a Hamiltonian describes optical Zeeman spectra on TmFeO ~ and HoFeO 3 by showing two differently oriented, unidirectional, ground state g-factors of 8.1 along the z-axis for thulium and of 14.9 at 31°from the y-axis in the xy-plane for holmium.
THE OPTICAL Zeeman spectra and magnetic properties of ions in crystals often differ strongly depending on whether the ion has an odd or an even number of aelectrons. In symmetry the odd-electron case, an ion in site of low has its
or in other words if several states are nearly ‘accidentally degenerate.’ In the particular case of two singlet states close1together and has shown far from other state, Griffith field can be that theirany behavior in a magnetic
energy levels broken up into Kramers’ degenerate pairs which split apart linearly in an applied magnetic field. As is well-known, the behavior of these energy levels can be described in terms of a spin Hamiltonian
be described by a spin Hamiltonian which has spin one-half but which otherwise has a very different form from the Hamiltonian for the oddelectron case: I~HZ~ZSZ + DS~, S = 1/2
This Haniiltonian has a R-factor along only one direction, and the g-factor in orthogonal directions i~strictly zero. Furthermore, this direction of the only R-factor is not set by the site symmetry alone but also depends on the transformation properties of the two singlet states which are being described. And finally there is a crystal—field term linear in spin but with a spin direction orthogonal to that of the operator describing the Zeeman term, although as Griffith has emphasized, the specific directions of these two spin operators have no physical significance. Nne of these characteristics are true of the
where ~ is the Bohr magneton, S the fictitious spin angular momentum operator with spin onehalf, H the magnetic field, and g a symmetric tensor, By contrast, in the even electron case, the ion in a site of low symmetry has its energy levels broken up into singlet states which are in general unaffected by a magnetic field in first order. Strong Zeeman effects can occur only if several of these singlet states happen to lie close together relative to the Zeeman energy, *
odd-electron spin Hamiltonian, and the Zeeman splittings predicted for the two cases are quite
This work was supported in part by the United
States Army Research Office, Durham, in part by the National Science Foundation, and in part by the Advanced Research Projects Agency through the Center for Materials Research at Stanford University.
different. Several trivalent even-electron rare earth ions in the orthorhombically distorted perovskite
RARE EARTH IONS IN THE ORTHOFERRITES
structure have shown precisely the conditions for a spin Hamiltonian theory, namely ground singlet states lying close together but two far from the nearest excited state.2 We have applied group theoretical methods to the analysis of these cases and have found the surprising result that there are several different possible g-factor directions for this single perovskite crystal structure. Further, we have taken data on trivalent holmium and thulium in the orthoferrites which supports all aspects of the theory and brings out the contrast with the odd-electron ions, Griffith ‘ has shown that the direction of the only g-factor is along the vector g defined by g = 2, where a> and b> are the two isolated lowest lying crystal—field states of the system, and L and S are the usual total orbital and spin angular momentum operators. In discussing how this g-direction is determined, Griffith overemphasizes the role of the state space spanned by a> and b> and underemphasizes the role of site symmetry. In a simple application of group theory to the case of rare earth ions in the orthoferrites, we will now show that g is determined both by the site symmetry and by the transformation properties of the states. The ions have site symmetry C
1,~, that is, a single mirror plane perpendicular to the z-direction of transform the crystal, ~ and sotothe crystal— field states must according one of
the xy-plane given by the vector (gm, g 21, 0), 2) ~/2 and g~,= (g÷ g It is good to bear in mind that except for their orthogonality, the particular choice of directions for the two spin operator components S~and S~,is arbitrary and was made simply to conserve the symmetry of the g-tensor H.g.S. In addition we stress again the unusual result that there are in this case two different spin Hamiltonians to describe ions in the same crystal structure, the difference stemming from the transformation properties of the singlet states. Before comparing these expressions to the actual data on the orthoferrites, it is necessary to make several other steps. There are four rare earth ions per unit cell and one can use group theory to relate the spin Hamiltonians for these different sites among themselves. In addition one must account for the dipole and exchange interactions with the other magnetic ions. Our data were taken above the ordering temperature of the rare earth lattices (6.5°K for HoFeO~)but well below the ordering temperature of the iron lattices (Ca. 700~K),5 that is, in the temperature range 8—30°K. In this range, one may ignore rare earth interactions and can treat the interactions with iron sublattices as a constant molecular field whose direction is known 6 Afrom full neutron account of diffraction andwill group theory. these effects be presented elsewhere,7 and
the two representations of C
1~.Using thefind group 4 we theoretical matrix element theorem, that if the two singlet states belong to the same representation (Case 1), then the vectorg points along the z-direction, and the spin Hamiltonian can have the form =
using results, the spinexpressions Hamiltoniansfordescribed above the yield the following the Zeeman splittings E(H) of the ground levels for the two even-electron cases and also for the comparable odd-electron case: Even-electron case I:
where v is any direction in the .vv-plane of the crystal.
On the other hand, if the two singlet states belong to opposite representations (Case II), then
the vector g points in the xv-plane, and the Spin Hamiltonian can have the form =
DS~+ f3H~g~.S~DS~+ /~(H7g,
where v is in this case a particular direction in
[D2 + (~+ g~f3H~±g3H)2]h/2
2H~ + (~ + g~f3H 2~f3 1± (~~ ±g~f3H~+ g~j3H~,,)2]~/2
Here H~,H~,and H~are the components of the applied magnetic field referred to the
RARE EARTH IONS IN THE ORTHOFERRITES
Table 1. Parameters for the fit of the even-electron spin Hamiltonians to optical data on the two ground singlet states of trivalent holmium and thulium in the orthoferrites. Parameters as defined intext D (in cm’) A (in cm~)
this work ~3 ~6.5, ~7.2
<0.25 <0.25 8.1 ±0.5
7.7 ±0.4 t 12.8 ±0.4t <0.25
7.2 ±0.3 7.0 ±0.6 13.4 ±1.4 3.2 ±0.4
*At 20°K. These values are derived using our theory from data taken along the [110}-axes by Schuchert et a!? and agrees within experimental error with the direct experimental results of the next column. orthorhombic axes of the crystal, the g’s are the various g-factors, the A’s describe the effect of molecular fields, D is the crystal—field splitting in the even-electron cases, and the ±refers to two different kinds of rare earth sites. The g’s A’s and D’s can be viewed simply as parameters to be identified in experiment. It is interesting to note that from this point of view, the oddelectron expression is general enough to fit the even-electron data as a special case; nevertheless the even-electron expressions are distinctly different because they require many fewer parameters to fit the data. To check these predictions, we have studied a previously unreported sharp-line optical Zeeman spectrum of TmFeO 3, and have also extended 2 on the optical Zeeman work of Schuchert et a!. HoFeO 3. These spectra arise from transitions within the 4f shell of the trivalent rare earth ion which in both cases has an even number of electrons. The data were taken in the temperature range 8—30°K, where both crystals have the magnetic configuration, 6 in terms of which we have done our molecular field analysis. We have found that the data on the ground state of HoFeO3 fits Case II as described above. The spectrum of TmFeO is complicated by large numbers of satellite lines, but the strong lines of the spectrum indicate two ground levels separated by 17.4 cm~ which split in a magnetic field according to Case I. Parameters for these fits are listed in Table 1 and lead tofor unidirectional 8.1 along the z-axis thulium, andg-factors of 14.9 atof31°from the y-axis in the xy-plane for holmium,
Schuchert et a!.2 have previously interpreted their data on holmium in terms of an odd-electron expression, which, as we have shown, happens to be general enough to account for the evenelectron splittings also. However, we have been able to take their data in the xy-plane and fit it within experimental accuracy to the simpler even-electron expressions of Case II (cf. Table 1). Although their data shows a splitting along the z-direction, our data on several different samples in fields up to 45koe shows no measurable splitting, as predicted by the theory. In interpreting our data, we have checked that demagnetizing effects and perturbations from higher levels were negligible; we also confirmed the quality of our crystals by X-ray measurements, which agreed to 0.02 perconclude cent withthen thethat standard results of Treves. ~ We at least as far as our measurements are concerned, trivalent holmium and thulium support all aspects of the even-electron theory for the orthoferrites as developed above. The concept of unidirectional g-factors thus provides a simple model for visualizing the magnetic properties of even-electron ions in the perovskites; that is, we can expect the ions of Case Ito show splittings only along the z-direction, and those of Case II only in the xy-plane. Data on other crystals of the perovskite family supports this contrast of even and oddelectron ions. YbA1O , isomorphic to the 8orthoand ferrites, has been studied by Hiifner al. but shows strictly no splitting along the et z-axis significant splittings in the xy-plane, indicating
RARE EARTH IONS IN THE ORTHOFERRITES
that the even-electron ion terbium in the ortho-
aluminate belongs to Case II. By contrast, Zeeman data on DyFeO and on ErFeO~ has
express their gratitude to Prof. R.M. White for continuous advice in this work, to R.S. Feigelson for the orthoferrite crystals, and to G.W. Martin
shown that for odd-electron cases all the g-parameters of the odd-electron theory are
The authors wish to
for X-ray analysis of the crystals.
needed to fit the data.
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SCHlJCHERT H., HUFNER S. and FAULHABER R., Z. Phys. 220, 280 (1969).
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TINKHAM M., Group Theory and Quantum Mechanics, p. 80, McGraw—Hill, New York (1964).
TREVES D., J. app!. Phys. 36, 1033 (1965).
WHITE R.L., J. app!. Phys. 40, 1061 (1969).
MALOZEMOFF A.P., Thesis, Stanford University.
HUFNER S., HOLMES L., VARSANYI F. and VAN UITERTL.G., Phys. Rev. 171, 507 (1968). SCHUCHERT H., HUFNER S. and FAULHABER R., Z. Phys. 220, 273 (1969).
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L’hamiltonien de spin pour deux singlets presque d~g~nér~s diffêre de celui pour un doublet de Kramers. Les niveaux fondamentaux de TmFeO 3 et HoFeO3, ~tudiés par spectroscopie optique, obéissent un tel hamiltonien; ils montrent une valeur non-nulle de g, dans une direction seulement, de 8,1 dans la direction z pour le thulium et de 14,9 31c de Ia direction y dans le plan xy pour le holmium.