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Terahertz spectroscopy of crystal-field transitions in magnetoelectric TmAl3(BO3)4

Dynamic magnetic properties of magnetoelectric TmAl3(BO3)4 borate have been investigated by terahertz spectroscopy. Crystal field (CF) transitions within the ground multiplet 3H6 of Tm3+ ions are observed and they are identified as magnetic-dipole transitions from the ground singlet A1 to the next excited doublet E of Tm3+ ions. Unexpected fine structure of the transitions is detected at low temperatures. The new modes are assigned to local distortions of the sites with D3 symmetry by Bi3+ impurities, which resulted in the splitting of A1 ->E transition. Two types of locally distorted sites are identified and investigated. The main contribution to the static magnetic susceptibility is shown to be determined by the matrix elements of the observed magnetic transitions. We demonstrate that even in case of local distortions the symmetry of the undistorted crystal is recovered for magnetic and for quadratic magnetoelectric susceptibilities.

The magnetic and magnetoelectric properties of ferroborates RFe 3 (BO 3 ) 4 are rather complex because of the antiferromagnetic ordering of the Fe spins below T N ~ 30-35 K and their strong coupling with rare-earth moments [8,9,[14][15][16]. Besides spontaneous magnetic phase transitions from easy-plane to easy axis states (R=Gd, Ho) [17,18], they exhibit significant spontaneous magnetic field-induced electric polarization as well as colossal (/ ~ 300%) changes of the dielectric constants below T N in ferroborates with R = Sm, Ho [19,20]. As has been shown recently [21], strong magnetoelectric excitation at microwave frequencies (electromagnon) representing one of the antiferromagnetic resonance mode in Fe-subsystem is responsible for this magnetodielectric effects.
In spite of the significant role of the antiferro-magnetically ordered Fe-subsystem in magnetic and magnetoelectric phenomena of ferroborates, it is the rare-earth subsystem which is responsible for the magnetoelectric coupling, since in YFe 3 (BO 3 ) 4 ferroborate with non-magnetic Y 3+ ions the magnetoelectric properties are very weak [1]. Moreover, the Fe-subsystem in borates has been shown as not essential to establish a large electric polarization in external magnetic fields.
Thus, the largest bilinear (quadratic) magnetoelectric effect was observed recently in alumoborates TmAl 3 (BO 3 ) 4 [4], HoAl 3 (BO 3 ) 4 [3] and TbAl 3 (BO 3 ) 4 [22]. The absence of Fe ions and of R-Fe exchange interactions makes RAl 3 (BO 3 ) 4 compounds to an appropriate model system to study the rare-earth contribution to magnetoelectric properties. The quantum theory of magnetoelectricity in rare-earth borates was developed in Ref. [23]. This theory is based on a single ion mechanism due to electric dipole moment of rare-earth ion 4f shells and due to ion displacement induced by exchange magnetic fields. This approach has been applied to describe temperature and field dependences of polarization in Nd, Sm and Eu ferroborates [23] as well as in alumoborates with R = Tm, Ho, Tb [22,24]. An important role of low-energy levels, i.e. the ground state of the R-ions in a crystal field of D 3 local symmetry has been recognized.
Usually, crystal field (CF) states are determined from the analysis of optical transitions from the ground to exited multiplets [10,11]. Direct determination of the actual CF levels responsible for the magnetic and magnetoelectric properties requires terahertz (or sub-terahertz) spectroscopic experiments. In addition, magnetic and magnetoelectric properties in real borates crystals may be influenced by local distortions of R-sites due to defects (see for example Ref. [25,26]).
In this work we present the results of terahertz experiments of the low-energy crystal-field states of Tm 3+ ions in thulium alumo-borate which are responsible for magnetic and magnetoelectric behavior in this material. Magnetic susceptibility of TmAl 3 (BO 3 ) 4 reveal strong anisotropy at low temperatures with a significant increase in the basal ab-plane (easy directions) compared to the c-axis [6,24,27]. The observed simultaneous increase of field-induced electric polarization in TmAl 3 (BO 3 ) 4 [4] indicates the prevailing influence of the low-energy crystal-field Tm 3+ states not only on magnetic but also on magnetoelectric properties of the crystal.
Electronic structure of Tm 3+ in lightly doped (Y 1-x Tm x )Al 3 (BO 3 ) 4 crystals has been studied recently by optical absorption and emission spectroscopy [25]. The CF states of the non-Kramers Tm 3+ ions occupying the positions of D 3 symmetry form the singlets A 1,2 and the doublets E. It was found that the ground CF state of the Tm 3+ ions is a singlet A 1 , the next excited state is a doublet (E) at ~29 cm -1 , and the remaining crystal-field states lie above 100 cm -1 . Similar structure of the crystal-field states of the ground multiplet 3 H 6 of Tm 3+ was also observed in pure TmAl 3 (BO 3 ) 4 by optical spectroscopy [27]. Therefore, direct A 1 E electronic transitions in the ground multiplet are in the frequency range of the terahertz spectroscopy. The actual energy structure of the low-lying CF Tm 3+ states in TmAl 3 (BO 3 ) 4 was not studied yet. Here we show that the energy levels are affected by local distortions of Tm 3+ sites in a real crystal and discuss theirs influence on magnetic and quadratic magnetoelectric properties.

Experiment
Most results in this work have been obtained using quasi-optical terahertz BWO-spectroscopy [28] in transmittance geometry. The beam path of the terahertz spectrometer is shown in the lower panel of Fig. 1. This spectrometer utilizes linearly polarized monochromatic radiation in the frequency range from 8 cm -1 to 37 cm -1 which is provided by backward-wave-oscillators (BWO's).
BWO's are tunable sources of the electromagnetic radiation based on the periodic modulation of the electron beam in a high-vacuum tube. The variation of the output frequency is realized by changing the accelerating voltage in the range of few kilovolts. The output power of the BWO's is typically 1-100 mW. Opto-acoustic Golay cell or He-cooled bolometer are used as detectors of the radiation.
Frequency dependent transmission spectra have been be analyzed using the Fresnel optical formulas for the transmittance T = |t| 2 of a plane parallel sample Here r is the reflection amplitude at the air-sample interface, t 1 is the "pure" transmission amplitude,  and  are the complex dielectric permittivity and magnetic permeability of the sample, respectively, d is the sample thickness, and  is the radiation wavelength. Equation (1) can be applied for anisotropic crystals as well, provided that the incident radiation is linearly polarized along the principal optical axes.
Equation (1) contains two unknown material constants,  and , and thus cannot be inverted in general case. Fortunately, in present material system the dielectric contribution  is only weakly frequency dependent and the magnetic permeability  is limited to a narrow frequency range which can be approximated by a sum of Lorentzians: where ν k , ν k and  k are resonant frequency, linewidth and the contributions of the k-th mode to the magnetic permeability .
Direct fitting of the spectra for samples with different thickness using Eqs. (1,2) allows to separate electric and magnetic contributions without additional measurements. In case of transparent samples Fabry-Pérot oscillation on the sample surfaces are visible (see, e.g. Fig. 3 below). The periodicity and amplitude if these oscillations allows to obtain the dielectric permittivity of the sample outside the magnetic modes via Eq. (1).
Static magnetic susceptibility has been measured using commercial Physical Property Measurement System. Single crystals of TmAl 3 (BO 3 ) 4 were grown from flux as described in Ref. [29].
Oriented samples were prepared in the form of plane-parallel bc-plane and ab-plane oriented plates with several thicknesses from 0.18 up to 1.7 mm and with lateral dimensions of about 10x10 mm 2 . Transmission experiments on samples with different thickness are necessary to increase the dynamic range of the spectrometer in case of strong absorption. To obtain the mode parameters, the spectra of samples with different thickness have been fitted self-consistently. Figure 2 shows the static magnetic susceptibility of TmAl 3 (BO 3 ) 4 along the a-axis and along the caxis. The susceptibility along the a-axis agrees well with Ref. [4,24,27] both qualitatively and quantitatively. We note that the c-axis susceptibility in Ref. [4] is roughly by factor of two larger than that shown in Fig. 2, and that measured in Refs. [24,27], which might be due to small fraction from the a-axis signal. The same effect probably explains small upturn of  c at the lowest temperatures in our data (inset to Fig. 2).

Results and discussion
Typical transmittance spectra of TmAl 3 (BO 3 The contribution to the magnetic permeability of the two components of the A 1 E transitions (i.e. A 1 E + , A 1 E -) can be calculated as [32]: Here µ x = µ y are the matrix elements of the A 1 E transition for ac magnetic field along x(a) or y(b) axes, respectively; E 0 is the ground state energy, E 1 is the energy of the doublet; N is the concentration of the R-ions, and Z is the partition function.
Local distortions of the crystal field of the Tb 3+ ion are illustrated in Fig. 6. Based on this scheme we attribute the observed structure of the magnetic transitions to local distortions of the crystal field that splits the doublet E into two singlet components E ± . We identify four resonance modes (Figs. 4, 5) as two pairs of transitions A 1  E k ± corresponding to two kinds of distorted R-sites (k=1, 2, left panel of Fig. 6). Strongly split components ν 2 ± can be assigned to the Tm 3+ ions being nearest to Bi impurities (right panel of Fig. 6) which arise during the crystal growth and which occupy the R-sites [26,33,34]. The weakly split central components ν 1 ± can be attributed to remaining Tm 3+ ions which are farther away from Bi-impurities and which are subjected to weaker CF distortions. As no further splitting of the spectra is observed, this scheme reasonably explains the sequence of the magnetic transitions in TmAl 3 (BO 4 ) 3 . Similar Tm 3+ splitting due to lowsymmetry components of the crystal field induced by random lattice strains was observed by optical spectroscopy and analyzed theoretically in Ref. [35]. The line shape in this case was predicted to be asymmetrical due to distribution of distortions. However, in a reasonable approximation two central components can still be described as Lorentzians, Eq. (2), since the mode splitting does not allow to specify the line shape.
In spite of local distortions, the excitation conditions of the observed resonances appear to be similar to the D 3 symmetry state. This result is surprising on the first glance. However, it can be explained by averaging of the magnetic response over several equivalent Tm 3+ sites, which results in the recovery of the initial excitation conditions for undistorted CF states. In contrast to the undistorted sites, the CF transitions at the distorted positions occur between the singlet states AE k ± , where E k ± are the components of initial doublets (k = 1, 2, see Fig. 6). Thus, the tensor of the complex magnetic permeability in a given frequency range contains four resonance terms at frequencies ν k = ν 1± , ν 2± corresponding to the transitions AE 1 ± and AE 2 ± at sites #1 and #2, respectively. Here where x, y, z axes coincide with the crystallographic a, b, c axes for one of the three equivalent trigonal a-axis directions. We omit the transitions indexes (±) and the site numbers (#1, #2) for simplicity. Due to local D 3 symmetry of Bi impurities, other Tm positions shown in Fig. 7 can be obtained starting from the positon Nr. 1 by rotations corresponding to this symmetry. Six symmetry operations, which are necessary in this case, may be described in a unified manner by following transformation matrices: where α 1,2,3,4,5,6 = 0, 120°, -120°, 0, 120°, -120° are possible rotation angles and η 1,2,3 =1, η 4,5,6 =-1, respectively.
By adding the contributions from all possible six positions, the average tensor of the magnetic contributions for six equivalent sites #2 from the first coordinate sphere is obtained as: From Eq. (6) we see that the off-diagonal components of general magnetic susceptibility are cancelled out by symmetry and the resulting tensor corresponds to that of D 3 symmetry. Therefore, due to averaging over the possible sites the excitation conditions of the magnetic transitions are unaffected by the distortions. In addition, we note that for random distortions closely similar averaging can be performed using continuous distribution.
Using the procedure analogous to that of Eq. (3) the magnetic susceptibility perpendicular to the c-axis can be easily obtained. Within the present assumptions the final expression is given by: are effective matrix elements of the magnetic dipolar transitions averaged over six equivalent Tm 3+ sites , E 0 and E k ± are the CF states of the ground singlet and the two components of the split doublet, respectively, c k is the concentration of the rare-earth ions of distorted sites of k-type. Equation (7)  Within a good approximation, the splitting of the magnetic modes does not influence the static magnetic susceptibility which is simply given by the sum of all observed contributions. The calculated total contribution of four A-E k ± transitions to the static susceptibility along the a-axis is given by dashed line in Fig. 2 and is provides good description of the experimental data. In present experiments no dynamic magnetic contribution along the c-axis has been observed within the sensitivity of our spectrometer. This agrees with the weakness of the static susceptibility along the c-axis,  c << a , as shown in Fig. 2. The remaining value of  c can be explained by quadratic terms in  z as a function of crystal-field perturbations, while the observed splitting of the doublet states E is linear with respect to these perturbations.
The splitting of the crystal-field states of Tm 3+ ions was confirmed by experiments in external magnetic fields. Representative spectra in external magnetic fields parallel to the c-axis are shown in Fig. 8. External magnetic field additionally splits the E i ± singlets as: Here the position indexes are omitted for clarity. Total macroscopic polarization is obtained by averaging over all positions and taking into account the interrelation of local axes as determined by Eq. (5). After the averaging procedure, Eq. (10) has the form of Eq. (9) with effective magnetoelectric susceptibilities given by: , respectively. Eq. (9) reproduces the quadratic behavior of the electric polarization along the a-axis in external magnetic field H ‖ a as observed in Ref. [4]. In experiments of Ref. [4] similar field dependence has been detected for P ‖ c clearly contradicting the predictions by Eq. (9). Further investigations are necessary to resolve this problem.

Conclusions
Crystal field (CF) transitions of the ground multiplet 3  Due to site averaging, even in the presence of distortions the symmetry of the magnetic response as well as quadratic magnetoelectric susceptibility correspond to that of an undistorted crystal. We believe that the recovery of macroscopic symmetry of physical properties in crystals with random local distortions could also be observed in other systems.