EPR spectra and magnetization of XY-type rare-earth ions in pyrochlores Y$_2$Ti$_2$O$_7$:RE$^{3+}$ (RE=Yb, Er)

The results of studies of Y$_{2}$Ti$_2$O$_7$ single crystals doped with Er$^{3+}$ and Yb$^{3+}$ ions by means of electron paramagnetic resonance (EPR) and dc-magnetometry are reported. EPR signals of the trigonal centers with the characteristic hyperfine structure of Er$^{3+}$ or Yb$^{3+}$ ions were observed. Field dependences of magnetization of single crystals for magnetic fields directed along the crystallographic axes and temperature dependences of magnetic susceptibilities were measured. Spin Hamiltonian parameters ($g$-factors and parameters of the hyperfine interaction) for Er$^{3+}$ and Yb$^{3+}$ ions were obtained from analysis of experimental data. The registered EPR spectra and magnetization curves were successfully reproduced by simulations in framework of the crystal-field approach, in particular, with an account for hybridization of ground 4f$^{13}$ configuration of Yb$^{3+}$ ions with the charge transfer states.

RE ions in the crystal lattice of the pyrochlore (space group Fd3m ) occupy 16d Wyckoff positions with the local trigonal symmetry D 3d . Four magnetically non-equivalent RE sublattices form a corner-sharing tetrahedral network, the basis vectors of RE 3+ ions in the unit cell in the crystallographic system of coordinates with the origin at the center of a corresponding tetrahedron are as follows: 1 (1,1,1) / 8  r , 2 ( 1, 1,1) / 8    r , 3 ( 1,1, 1) / 8    r , 4 (1, 1, 1) / 8    r in the units of the lattice constant a . The first coordination shell of RE ions contains eight oxygen ions which form strongly distorted cubic polyhedron with the two nearest neighbor oxygen ions at one of the four crystallographic [111] axes. Specific magnetic properties of different RE pyrochlores are determined, first of all, by the single RE ion magnetic characteristics and the energy spectrum in the trigonal crystal field (CF).
The wave functions of the ground Kramers doublet of Er 3+ or Yb 3+ ions in the trigonal CF transform accordingly to the irreducible representations 4  or 56  of the D 3d point symmetry group. Note that magnetic properties of these states are substantially different, in particular, the so called dipole-octupole doublets 56  are split only by the magnetic field parallel to the trigonal symmetry axis (the transversal gfactor 0 g   ) while both the longitudinal ( g  ) and the transversal g-factors of the 4  doublets are non-zero.
The CF splittings of the electronic 4f N -multiplets and spectroscopic g-factors of Er 3+ (N=11) and Yb 3+ (N=13) ions in dilute and concentrated pyrochlores were extensively studied earlier. The g-factors of the ground state of Yb 3+ ions were determined from 170 Yb Mössbauer absorption measurements in Y 2 Ti 2 O 7 :Yb 3+ (1 at.%) sample enriched with the 170 Yb isotope in Ref. 15, but g-factors of Er 3+ ions were estimated only from calculations based on the CF parameters which had been used for simulations of the dc-magnetic susceptibility ofEr 2 Ti 2 O 7 [16]. Some CF energies of Yb 3+ ions in Y 2 Ti 2 O 7 :Yb 3+ and Yb 2 Ti 2 O 7 were measured by means of optical spectroscopy [17]. CF excitations corresponding to transitions between sublevels of the ground multiplets of Yb 3+ and Er 3+ in Yb 2 Ti 2 O 7 and Er 2 M 2 O 7 (M=Ge, Ti, Sn, Pt, Ru) were studied by means of inelastic neutron scattering in Refs. [18,19] and [20,21], respectively.
The CF approach serves as a basis for construction of theoretical models of interactions between RE ions and a crystal lattice and for understanding of magnetic properties of concentrated RE compounds. A number of CF parameter sets for Er 3+ and Yb 3+ in the pyrochlores with different chemical compositions were proposed in the literature (see [1,22,23] and references therein). However, the comment written 18 years ago [15], namely, "All are inappropriate as each corresponds to a crystal level scheme and to the ground state wave functions which are not compatible with the experimental data", is actual nowadays as well. Determination of the physically consistent CF parameters describing both the measured g-factors and CF energies remains a topical problem.
In recent work [24], stimulated by the increasing interest to the unconventional magnetic properties of geometrically frustrated RE pyrochlores, we presented the measured electron paramagnetic resonance (EPR) and site-selective emission and excitation optical spectra of Y 2 Ti 2 O 7 single crystals doped with Er 3+ or Yb 3+ ions (0.5 at.%). The CF parameters were calculated in the framework of the semi-phenomenological exchange charge model and then corrected to fit the experimental data. The main goal of the present work is to obtain additional information about electronic structures ofpyrochlores containing XY-type RE ions (Er 3+ or Yb 3+ ). We present low-temperature magnetic field dependences of the magnetization and temperature dependences of the bulk dc-susceptibility of Y 2 Ti 2 O 7 :Yb 3+ and Y 2 Ti 2 O 7 :Er 3+ (0.5 at. %) single crystals. The shape and the hyperfine structure of EPR signals presented in [24] and magnetization curves are analyzed by making use of the corresponding sets of CF parameters.

Experimental details and results
The measurements of the magnetization M(B,T)parallel to the external magnetic field B with magnetic fields in the range of 0−9 T applied along the crystallographic axes [100], [111] and [110] were performed using a vibrating sample magnetometer option of the PPMS-9 system (Quantum Design) on Y 2 Ti 2 O 7 :Yb 3+ and Y 2 Ti 2 O 7 :Er 3+ single crystals with the content of 0.5 at.% of impurity ions grown by the optical floating-zone method [25]. The details of the single crystal growths were described in [24]. The pyrochlore structure of the samples was confirmed by X-ray diffraction measurements. The obtained magnetic field dependences of the magnetization at the temperature T of 2 K are shown in Figure 1. The experimental data in Figure 1 are corrected by accounting for diamagnetic contributions [26] into the measured magnetization which are substantial in case of strongly dilute samples.
For weak magnetic fields, B< 1 T, the magnetization M(B,T) is practically isotropic, as one may expect in the case of global cubic symmetry of the studied systems. The temperature dependences of the static magnetic susceptibility ( ) of Y 2 Ti 2 O 7 :Yb 3+ and Y 2 Ti 2 O 7 :Er 3+ single crystals measured in the magnetic fields B=0.4 T and 0.3 T, respectively, are presented in Figure 2. The measured susceptibility curves can be satisfactorily approximated by the Curie law, ( )  is the Bohr magneton, k B is the Boltzmann constant, g is the Lande factor of the ground multiplet with the total angular moment J, g || and g  are the g-factors of the ground Kramers doublet determined from the EPR spectra (see below)).As the magnetic field increases, a noticeable dependence of the magnetization on the field direction appears due to mixing of wave functions of the ground and excited CF levels by the Zeeman interaction (see Figure 1). Of particular interest for justification of the CF parameter sets is the difference in the relative shifts of magnetization curves of Er 3+ and Yb 3+ ions in magnetic fields along the tetragonal [001] axis, in this case the observed magnetization in strong magnetic fields has maximum values for the Yb 3+ ions but minimum values for the Er 3+ ions.The gaps of ~600 cm -1 and ~50 cm -1 between the ground  4 doublet and the first excited state ofYb 3+ and Er 3+ ions [24], respectively, differ by more than an order of magnitude, and, in agreement with the nonlinear mechanism of the single-ion magnetic anisotropy in cubic paramagnetic centers, it is remarkably stronger in Y 2 of the four magnetically equivalent paramagnetic centers (here and below g || and g  are g-factors in the local coordinate frames with the Z-axis along the trigonal symmetry axis of the corresponding center). If the external magnetic field is parallel to a trigonal [111] axis, there are two magnetically non-equivalent centers of RE 3+ ions: one site whose local anisotropy axis is parallel to the static magnetic field B and three sites have their local anisotropy axis at ~109.5 degrees from the vector B. In the magnetic field applied along the two-fold rhombic [110] axis, there are also two magnetically non-equivalent centers: two so-called  -sites (the magnetic field is declined from the local trigonal axis by the angle of ~35.3 degrees) and two  -sites (the field is perpendicular to the local trigonal axis). Therefore in cases of static magnetic field directed along trigonal or rhombic axes, two EPR lines with different intensities corresponding to the g-factors (1) [111] [110] g g   are observed. The sum of the squared g-factors over four ion sites in the unit cell for each direction of the magnetic field that determines the linear magnetic susceptibility is a constant. Along with the most intensive signals from even isotopes, the spectra contain weaker components of the hyperfine structure originating from odd isotopes ( 167 Er with the nuclear spin I=7/2 and natural abundance of 22.9 %, 171 Yb (14.3 %) and 173 Yb (16.2 %) with I=1/2 and I=5/2, respectively).The measured values of the g-factors, g || and g  , and the hyperfine structure parameters for odd isotopes are presented in Table 1. We note that the g-factors of Er 3+ ions in the concentrated Erpyrochlore Er 2 Ti 2 O 7 g || =2.6 from Ref. 27 (polarized neutron diffraction measurements) and g  =7.6 from Ref. 28 (low-temperature EPR transmission-type spectroscopy) are rather far from our values.

Discussion
In order to simulate the EPR spectra, magnetization and susceptibility data, we considered the following Hamiltonian of a rare-earth ion, The local systems of coordinates defined in [30] for RE ions at sites with the radius-vectors r n are used in calculations: the Z n axes are parallel to r n , and the X n axes are in the planes containing r n and the selected crystallographic tetragonal axis coinciding with the z-axis of the global coordinate frame.The CF Hamiltonian written in the local system of coordinates, is determined by six non-zero CF parameters k p B (here k p O are linear combinations of the single-electron spherical tensor operators [22]), the sum is taken over 4f electrons with radius vectors r, orbital and spin moments l and s, respectively. The values of the CF parameters used to simulate the EPR spectra, the dcsusceptibility and the magnetization data are given in The obtained field dependences of the magnetization where the sum is taken over four nonequivalent RE sites in the unit cell, are compared with the experimental data in Figure 1. The EPR spectra corresponding to magnetic dipole transitions between sublevels of the well isolated ground doublet split by the external magnetic field can be described by the spin-Hamiltonian H S which represents the projection of the total Hamiltonian H of an ion on the 2-dimensional ground state manifold. In the local system of coordinates, the spin-Hamiltonian is written as follows (we neglect nuclear Zeeman and quadrupole energies) where the effective spin S=1/2, g  and g  are components of the g-tensor, A  and A  are the magnetic hyperfine constants. Exactly these spin-Hamiltonian parameters were determined from the analysis of the measured EPR spectra (see Table 1, columns "Exper.").The magnetic hyperfine interaction (3) (14) here is the population of the state i, and the sum over i and j is taken over all electron-nuclear sublevels of the electronic doublet in case of odd isotopes. As shown in Figures3 and 4, the simulated spectral envelopes ( ) / dI B dB reproduce well the registered EPR signals at different orientations of the magnetic fields.

Summary
We carried out detailed experimental and theoretical studies of static and dynamic magnetic properties of impurity Er 3+ and Yb 3+ ions in Y 2 Ti 2 O 7 single crystals. The crystal field approach was used to interpret the measured magnetic field and temperature dependences of the magnetization and the characteristics of the EPR spectra. The sets of the CF parameters determined earlier from the analysis of the optical spectra [24] allowed us to reproduce successfully the values of resonance fields and relative intensities of EPR signals for different orientations of the applied field and the magnetic anisotropy induced by strong fields in cubic pyrochloresY 2 Ti 2 O 7 :Yb 3+ and Y 2 Ti 2 O 7 :Er 3+ containing small concentrations of 0.5 at.% of RE ions. The shapes of EPR lines including the fine hyperfine structures due to odd 171 Yb, 173 Yb and 167 Er isotopes were successfully modeled and the corresponding line widths and parameters of hyperfine interactions were determined from the fitting procedure.
The obtained new information on single-ion spectral and magnetic properties of diluted RE pyrochlores can be used to revise parameters of the anisotropic exchange interactions in Er 2 Ti 2 O 7 and Yb 2 Ti 2 O 7 and to advance investigations of the intriguing magnetic properties of concentrated pyrochlores.