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11Short cite this: Magn. Reson. Solids 21, 19601 (2019)
doi: 10.26907/mrsej-19601
EPR spectra and magnetization of XY-type rare-earth ions in pyrochlores Y2Ti2O7:RE3+ (RE=Yb, Er)
R.G. Batulin, M.A. Cherosov, I.F. Gilmutdinov, A.G. Kiiamov, V.V. Klekovkina*, B.Z. Malkin, A.A. Rodionov, R.V. Yusupov
Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia *E-mail: Vera.Klekovkina@kpfu.ru
(Received November 24, 2019; revised December 8, 2019; accepted December 9, 2019; published December 10, 2019)
The results of studies of Y2Ti2O7 single crystals doped with Er3+ and Yb3+ 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 Er3+ or Yb3+ 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 Er3+ and Yb3+ 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 4f13 configuration of Yb3+ ions with the charge transfer states.
PACS: 71.70.Ch, 75.10.Dg, 76.30.Kg, 71.70.Ej.
Keywords: rare-earth pyrochlores, electron paramagnetic resonance, magnetization, magnetic susceptibility, crystal field, g-factors, hyperfine interaction.
1. Introduction
Extensive experimental and theoretical studies of magnetic structures and excitations in the so called XY pyrochlores containing rare-earth (RE) Er3+ and Yb3+ ions with the easy-plane magnetic anisotropy have been carried out over the last twenty years ( [1-3] and references therein). A diversity of magnetic phases in the concentrated materials Yb2M2O7 and Er2M2O7 (M=Ge4+, Ti4+, Pt4+, Sn4+) was revealed at low temperatures (noncoplanar or coplanar antifer-romagnetic order [4-7], antiferromagnetic phase of Palmer-Chalker [8,9], splayed ferromagnetic order [10-14]).
RE ions in pyrochlore (space group Fd3m) crystal lattice occupy 16d Wyckoff positions with the local trigonal symmetry D3d. Four magnetically non-equivalent RE sublattices form a cornersharing tetrahedral network, the basis vectors of RE3+ ions in the unit cell in the crystallographic system of coordinates with the origin at the center of a corresponding tetrahedron are as follows: ri = (1,1,1)/8, r2 = (-1,-1,1)/8, r3 = (-1,1, —1)/8, r4 = (1, -1, —1)/8 in 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 Er3+ and Yb3+ ions in the trigonal CF transform accordingly to the irreducible representations r4 or r56 of the D3d point symmetry group. Note that magnetic properties of these states are substantially different, in particular, the so called dipole-octupole doublets r56 are split only by the magnetic field parallel to the trigonal symmetry axis (the transversal g-factor g± = 0) while both the longitudinal (gy) and
the transversal g-factors of the r4 doublets are non-zero.
The CF splittings of the electronic 4fN - multiplets and spectroscopic g-factors of Er3+ (N = 11) and Yb3+ (N = 13) ions in dilute and concentrated pyrochlores were extensively studied earlier. The g-factors of the ground state of Yb3+ ions were determined from 170Yb Mossbauer absorption measurements in Y2Ti2O7:Yb3+ (1 at.%) sample enriched with the 170Yb isotope in Ref. [15], but g-factors of Er3+ ions were estimated only from calculations based on CF parameters which had been used for simulations of dc-magnetic susceptibility of Er2Ti2O7 [27]. Some CF energies of Yb3+ ions in Y2Ti2O7:Yb3+ and Yb2Ti2O7 were measured by means of optical spectroscopy [17]. CF excitations corresponding to transitions between sublevels of the ground multiplets of Yb3+ and Er3+ in Yb2Ti2O7 and Er2M2O7 (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 basis for construction of theoretical models of interactions between RE ions and a crystal lattice and for understanding magnetic properties of concentrated RE compounds. A number of CF parameter sets for Er3+ and Yb3+ in the pyrochlores with different chemical compositions were proposed in 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 Y2Ti2O7 single crystals doped with Er3+ and Yb3+ 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 present work is to obtain additional information about electronic structures of pyrochlores containing XY-type RE ions (Er3+ or Yb3+).
We present low-temperature magnetic field dependences of magnetization and temperature dependences of the bulk dc-susceptibility of Y2Ti2O7:Yb3+ and Y2Ti2O7:Er3+ (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.
2. 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 0-9T 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 Y2Ti2O7:Yb3+ and Y2Ti2O7:Er3+ single crystals with the content of 0.5at.% 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 diluted samples.
For weak magnetic fields, B < 1T, the magnetization M(B,T) is practically isotropic, as
Figure 1. Measured (symbols) and calculated (solid curves) field dependences of the magnetization of Y2Ti2O7:Er3+ (a) and Y2Ti2O7:Yb3+ (b) single crystal for magnetic fields B directed along the crystallographic axes at temperature T=2K.
one may expect in case of global cubic symmetry of the studied systems. The temperature dependences of the static magnetic susceptibility x(T) = M(B,T)/B of Y2Ti2O7:Yb3+ and Y2Ti2O7:Er3+ single crystals measured in the magnetic fields B=0.4T and 0.3 T, respectively, are presented in Figure 2.
The measured susceptibility curves can be satisfactorily approximated by the Curie law, x(T) = C/T with the Curie constant C = 2cNa§2J(J + l)^B/3kB at elevated temperatures and C = 2cNA(gj^ + 2g"2_)^B/3fcB at low temperatures (here c is the concentration of RE ions, na is the Avogadro number, ^b is the Bohr magneton, kB is the Boltzmann constant, g is the Lande factor of the ground multiplet with the total angular moment J, gy 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 l). Of particular interest for justification of the CF parameter sets is the difference in the relative shifts of magnetization curves of Er3+ and Yb3+ ions in magnetic fields along the tetragonal [001] axis, in this case the observed magnetization in strong magnetic fields has maximum values for the Yb3+ ions but minimum values for the Er3+ ions. The gaps of ~ 600 cm-1 and ~ 50 cm-1 between the ground r4 doublet and the first excited state of Yb3+ and Er3+ 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 Y2Ti2O7:Er3+ than in Y2Ti2O7:Yb3+ single crystals. The EPR spectra were measured with the commercial Bruker ESP300 X-band spectrometer in static magnetic fields up to 550 mT. The registered spectra of the Y2Ti2O7:Yb3+ and Y2Ti2O7:Er3+ (0.5 at.%) single crystals with the magnetic fields directed along the crystallographic axes are shown in Figures 3 and 4, respectively. Using the measured resonant magnetic fields of even isotopes, we found the corresponding effective g-factors.
The spectra measured with the magnetic field B directed along the tetragonal [001] axis con-
T (K)
Figure 2. Measured (symbols) and calculated (solid curves) temperature dependences of the renormal-ized dc-susceptibilities x[Y2-2cRE2cTi2O7]/c (RE=Er, Yb, c=0.005).
tain one intense line corresponding to the g-factor g[001] = y^gy /3 + 2g2±/3 of four magnetically equivalent paramagnetic centers (here and below gy and g± are g-factors in 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 RE3+ 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 a-sites (the magnetic field is declined from the local trigonal axis by the angle of ~ 35.3 degrees) and two ¿S-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 g^] = gy,
gfm] = y/gy/9 + 8gy/9, g^g = ^/2g|/3 + g2 /3, g^g = g± are observed. The sum of 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 (167Er with the nuclear spin I = 7/2 and natural abundance of 22.9%, 171Yb (14.3%) and 173Yb (16.2%) with I =1/2 and I = 5/2, respectively). The measured values of the g-factors, gy and g^, and the hyperfine structure parameters for odd isotopes are presented in Table 2. We note that the g-factors of Er3+ ions in the concentrated Er-pyrochlope Er2Ti2O7 gy = 2.6 from Ref. [27] (polarized neutron diffraction measurements) and g^ = 2.7 [28] (low-temperature EPR transmission-type spectroscopy) are rather far from our values.
YTi O :Yb3+
2 2 7
R.G. Batulin, M.A. Cherosov, I.F. Gilmutdinovet al. -Experiment -Simulation
B||[001]
B ||[111]
B ||[110]
-1-1-1-1-1- H—1—I—"-//n-1-r~
150 200 250 120 180 360 390
200
300
5(mT)
Figure 3. EPR spectra of the Y2Ti2O7 :Yb3+ (0.5 at.%) single-crystal for three orientations of the mag-
3. Discussion
In order to simulate the EPR spectra, magnetization and susceptibility data, we considered the following Hamiltonian of the Er3+ ion
Here Hfi is the free ion standard parameterized Hamiltonian [29] that operates in the total space of 364 states of the electronic 4fn configuration. HCF is the CF energy of 4f electrons, HZ is the electronic Zeeman energy, and Hhf corresponds to the hyperfine interaction. The free-ion Hamiltonian is written as follows
Hfi = Z ^ ^ ■ li + aL(L + 1) + ^ (fkfk + Pkpk + Tktk + Mkmk)
i k (2)
+ PG(G) + 7G(Rt).
Values of the parameters in (2) determined from simulations of the optical spectra are the following (in units of cm-1): F2 = 96748, F4 = 67943, F6 = 55532, Z = 2368, a = 18, P = -570, y = 1631, P2 = 653, P4 = 326, P6 = 65, M0 = 3.6, M2 = 0.56Mo, M4 = 0.31Mo, T2 = 451, T3 = 61, T4 = 100, T6 = -245, T7 = 305, T8 = 160 (conventional notations).
The Hamiltonian of the Yb3+ ion is considered in the framework of the model derived in Ref. [24] where we account for the hybridization of the 4f states of a hole at the Yb3+ ion with the charge transfer states of the complex containing the Yb3+ ion and two its nearest neighbor oxygen ions (namely, with the states of the holes at the 2p oxygen shells due to transfer of an electron from O2- ions into the ytterbium 4f13 shell).
netic field B (v = 9.41553 GHz, T = 15K). Experimental data and the results of simulations are shown by solid black and red lines, respectively. Linewidths of the spectral components of the simulated spectra had the values of a = 0.105 GHz for B || [001], 0.12 GHz for B || [110], and ai = 0.09GHz and a2 = a3 = a4 = 0.12GHz for B || [111].
H = Hfi + HCF + HZ + HHF .
(1)
YTi O :Er
2 2 7
3+
• Experiment
• Simulation
s
J
.. .. OVV*"
BII [111]
I ■ .//■■■
70 140 250 300
B || [110]
T
T
T
100 150 200
5(mT)
Figure 4. EPR spectra of the Y2Ti2O7:Er3+ (0.5 at.%) single-crystal for three orientations of the magnetic field B (v = 9.41553 GHz, T = 20K). Experimental data and the results of simulations are shown by solid black and red lines, respectively. Linewidths of the spectral components of the simulated spectra had the values of a = 0.18 GHz for B || [001] and B || [110], and o-i = 0.15 GHz and a2 = a3 = a4 = 0.225 GHz for B || [111].
The local systems of coordinates defined in [30] for RE ions at sites with the radius-vectors rn are used in calculations: the Zn axes are parallel to rn and the Xn axes are in the planes containing rn 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,
hcf = ^ (B0O0 + B0O0 + B43O43 + B0O0 + B63O63 + b606),
(3)
determined by six non-zero CF parameters Bp (here Op 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 dc-susceptibility and the magnetization data are given in Table 1. The operator corresponding to the Zeeman energy has the form of Hz = — ^ ■ B, where M = —№(kL + 2S) is the magnetic moment of a RE ion, L and S are orbital and spin moments, respectively, and k is the orbital reduction factor (k = 0.99 for the Er3+ ion and 0.98 for the Yb3+ ion).
Table 1. CF parameters Bk (cm 1) for the impurity RE3+ ions in Y2Ti2O7:Er3+ and Y2Ti2O7:Yb3+.
pk 20 40 43 60 63 66
Er3+ 239.8 311.8 —2305.2 45.7 666.6 753.2
Yb3+ 264.8 270.8 —2155.2 44.9 636.6 683.2
The Hamiltonian of hyperfine interactions Hhf contains magnetic Hhfm and quadrupole Hhfq contributions:
Hhfm = №Yn^(r-3)4f E {2I ■ l + O0 (3szIz - s ■ I) + 3O| (sa/a - SyIy)
4f / y {2I 1 + O 2 (3Sz1 z s I) + 3O2 (sx^x sy1 y) xIy + syIx) +6O2 (szIx + SxIz) + 6O2 (syIz + szIy)
(4)
Hhfq = i7Te2Q-1T {(1 - Y~) E qL (3Iz2 - I (I + 1))
- (1 - Rq) <r-3>4f E [O0 (3Iz2 -1 (I +1)) + 3O2 (Ix2 - Iy2) + 3O-2 (IxIy + Iy Ix) +6O2 (Iz Ix + IxIz) + 6O-1 (Iy Iz + Iz Iy)]} .
(5)
Here yn is the nuclear gyromagnetic ratio (yn/2^ = -1.18, 6.9 and -1.98MHz/T for 167Er, 171Yb and 173Yb [31] respectively), (r-3)4f is the expectation value of the r-3 operator over a 4f-electron radial wave function equal to 11.07 and 12.5 at. units for Er3+ and Yb3+ [32], respectively, Q is the quadrupole moment of the nucleus (Q =5.88 and 1.35 in units of 10-28 m2 for 167Er and 173Yb [31], respectively), y^> = -80 and Rq = 0.1 are Sternheimer antishielding and shielding factors [33], h is the Planck constant. The first term in Eq. (5) comes from the crystal lattice contribution to the electric field gradient at the nucleus and contains the sum over the host lattice ions with charges qL and radius vectors rL relative to the considered RE ion.
The CF energies of RE ions were found by numerical diagonalization of the operator H0 = HFI + HCF. The matrices of the operators HZ and HHF were computed in the space of eigenfunctions of H0, and the subsequent calculations were carried out with the Hamiltonian H = E0 + HZ + HHF (here E0 is the diagonal matrix with elements equal to eigenvalues of H0).
The average values of magnetic moments of RE ions versus the magnetic field B and temperature T were computed in the local systems of coordinates accordingly to the definition
= Tr[Mexp(-(E0 - M ■ B)AbT)] WB'T Tr [exp(-(E0 - M ■ B)/kBT)] ( )
The obtained field dependences of the magnetization
4
4B"
M(B,T) = ^jB^E B -<Mn)B,T, (7)
n=1
where the sum is taken over four nonequivalent RE sites in the unit cell, are compared with the experimental data in Figure 2. The magnetic susceptibility tensor in the local system of coordinates is diagonal and has only two different components = XXX = Xyy, X|| = Xzz. The computed temperature dependences of the isotropic bulk magnetic susceptibilities X(T) = [X||(T) + 2x±(T)]/3 ofY2Ti2O7:Er3+ and Y2Ti2O7:Yb3+ are compared with the results of measurements in Figure 2.
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 Hs 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)
HS = $||№BzSz + (BxSx + BySy) + A||izSz + A± (IxSx + IySy) , (8)
Table 2. The spin-Hamiltonian parameters for impurity Yb3+ and Er3+ ions in Y2Ti2O7 (the signs of the hyperfine constants are obtained from theoretical analysis).
171 Yb 171Yb 173 Yb 173 Yb 167Er 167Er
Exper. Theory Exper. Theory Exper. Theory
gy 1.787 1.864 1.787 1.864 2.29 2.305
g± 4.216 4.181 4.216 4.181 6.76 6.811
Ay (MHz) 1235 1353 -353.8 -388 -277.4 -252
A± (MHz) 3064 3050 -881.4 -875 -810.9 -735
where the effective spin S = 1/2, gy and g± are components of the g-tensor, Ay and A± are the magnetic hyperfine constants. Exactly these spin-Hamiltonian parameters were determined from the analysis of the measured EPR spectra (see Table 2, columns "Exper.").
The magnetic hyperfine interaction (4) can be written as follows: Hhfm = a ■ I, here the components of the vector a
ax = №YN^(r-3)4f V [2lx - (O0 - 3O2)sx + 3O-2Sy + 6O^sz], (9)
ay = №YN^(r-3)4f V [2ly - (O0 + 3O2)sy + 3O-2Sx + 6O2-1Sz], (10)
az = 2^B7N^(r-3)4f V [lz + O0Sz + 3Olsx + 3O221Sy] (11)
operate in the space of electronic wave functions. The components of the g-tensor and the hyperfine constants in the spin-Hamiltonian are calculated using the expressions
gy =2 <+| kLz + 2Sz |+>, g± = 2 (+| kLx + 2Sx |-), (12)
AM =2 <+| az |+>, A± = 2 <+| ax |->, (13)
where |+> and |-> are the eigenfunctions of the Hamiltonian H0 corresponding to the ground CF doublet of a Kramers ion. The equations (12), (13) pave a bridge between the spin-Hamiltonian and crystal-field approaches. The calculated parameters of the spin-Hamiltonian (Table 2, columns "Theory") agree with the experimental data. We note that the obtained values of g-factors for Yb3+ ions are close to those communicated in Ref. [15] (gy=1.79, g^=4.27) for 170Yb3+ ions in the sample of Y2Ti2O7:Yb3+ (1 at.%). The estimated shifts of the hyperfine sublevels of the ground doublets of the 167Er3+ and 173Yb3+ ions induced by the quadrupole hyperfine interaction (5) which are less than 250 MHz, were not revealed in the measured spectra.
The relative integral intensities Wj of the resonant magnetic dipole transitions i ^ j between the eigenstates of the spin-Hamiltonian (8) with energies Ej, Ej and wave functions |i> and |j> induced by the microwave magnetic field B1 directed along the unit vector e perpendicular to the constant field B can be written as
I 12
Wjj = |(i|(g±Sxex + g±Syey + gySzez)|j>| . (14)
The spectral distribution of absorption intensities for the fixed microwave frequency v in the magnetic field B is approximated by the sum of Gaussians with the varied linewidth a
I (B) - V Wjj(pj - pj)-^exp[-(Ajj - 2nhv)2/2a2], (15)
^r V2na
jj
here Aji = Ej (B) - Ei(B), pi(B) 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 Figures 3 and 4, the simulated spectral envelopes dI(B)/dB reproduce well the registered EPR signals at different orientations of the magnetic fields.
4. Summary
We carried out detailed experimental and theoretical studies of static and dynamic magnetic properties of impurity Er3+ and Yb3+ ions in Y2Ti2O7 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 pyrochlores Y2Ti2O7:Yb3+ and Y2Ti2O7:Er3+ containing small concentrations of 0.5 at.% of RE ions. The shapes of EPR lines including the fine hyperfine structures due to odd 171Yb, 173Yb and 167Er isotopes were successfully modeled and the corresponding linewidths 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 Er2Ti2O7 and Yb2Ti2O7.
Acknowledgments
Financial support of the Russian Science Foundation under Grant N. 19-12-00244 is acknowledged.
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