Abstract
Violation of time reversal and spatial inversion symmetries has profound consequences for elementary particles and cosmology. Spontaneous breaking of these symmetries at phase transitions gives rise to unconventional physical phenomena in condensed matter systems, such as ferroelectricity induced by magnetic spirals, electromagnons, nonreciprocal propagation of light and spin waves, and the linear magnetoelectric (ME) effect—the electric polarization proportional to the applied magnetic field and the magnetization induced by the electric field. Here, we report the experimental study of the holmiumdoped langasite, Ho_{x}La_{3−x}Ga_{5}SiO_{14}, showing a puzzling combination of linear and highly nonlinear ME responses in the disordered paramagnetic state: its electric polarization grows linearly with the magnetic field but oscillates many times upon rotation of the magnetic field vector. We propose a simple phenomenological Hamiltonian describing this unusual behavior and derive it microscopically using the coupling of magnetic multipoles of the rareearth ions to the electric field.
Introduction
Novel materials and physical effects are essential for the continuing progress in nanoelectronics^{1}. The coupling between electric and magnetic dipoles in magnetoelectric (ME) and multiferroic materials enables electric control of magnetism that can significantly reduce power consumption of magnetic memory and data processing devices^{2,3,4,5,6}.
The linear ME effect^{7,8,9} occurs in centrosymmetric crystals, such as Cr_{2}O_{3}^{10} and LiCoPO_{4}^{11}, in which both time reversal and inversion symmetries are spontaneously broken by an antiferromagnetic spin ordering inducing toroidal, monopole, and other unconventional multipole moments^{12}. This effect allows for control of antiferromagnetic domains and gives rise to unidirectional light transmission at THz frequencies associated with electromagnons^{13,14,15}.
However, the linear ME effect in centrosymmetric crystals requires a perfect match between magnetic and lattice structures, which narrows the pool of ME materials substantially. Symmetry requirements are less stringent for magnets with noncentrosymmetric crystal lattices, such as boracites^{16}, copper metaborates^{17}, and the skyrmion material, Cu_{2}OSeO_{3}^{18}, in which diverse magnetic orders exhibit ME behavior.
A bilinear ME effect—electric polarization proportional to the second power of the applied magnetic field—does not even require timereversal symmetry breaking and can occur in the paramagnetic state of noncentrosymmetric magnets through a variety of miscroscopic mechanisms^{19,20}. This higherorder effect is by no means weak: the magnetically induced electric polarization in the recently studied holmium hexaborate is much higher than the polarization of linear MEs at comparable magnetic fields^{21}.
The Hodoped langasite studied in this paper does not seem to fit into this classification of ME effects. Langasite, La_{3}Ga_{5}SiO_{14}, and related compounds have been synthesized as early as the 1980s^{22,23} and actively studied since then^{24} for their interesting nonlinear optical, elastic and piezoelectric^{25,26} properties and applications in acousto and electrooptics^{27,28}. The noncentrosymmetric crystallographic space group P321 of langasites allows for optical rotation and piezoelectricity, but not for spontaneous electric polarization due to the presence of orthogonal threefold and twofold symmetry axes. The study of ME properties was focused on Fesubstituted langasites showing noncollinear periodically modulated orders of Fe spins^{29}. The interplay between magnetic frustration, structural and spin chiralities gives rise to a rich variety of multiferroic behaviors^{30,31}.
The Hodoped langasite does not order down to lowest temperatures. Yet, its magnetically induced electric polarization oscillates 6 times when the direction of the magnetic field rotates through 360^{∘} in the abplane, which is indicative of a highly nonlinear ME response proportional to (at least) the sixth power of the magnetic field. On the other hand, the magnitude of the induced polarization grows linearly with the applied field, which seems to be at odds with the absence of any magnetic ordering in this material. This ME behavior is very different from that of classical MEs, such as Cr_{2}O_{3}. We show that it can be understood on the basis of langasite symmetry and a hierarchy of energy scales in the spectrum of the magnetic Ho ion.
Results
Experiment
The samples used in this work are diluted rareearth langasites, Ho_{x}La_{3−x}Ga_{5}SiO_{14}, with x = 0.043 ± 0.005. The crystal structure of R_{3}Ga_{5}SiO_{14} langasites^{32,33} is shown in Fig. 1a. The rareearth sites, R, have a rather low C_{2} symmetry with the rotation axis along the aaxis of the crystal, allowing for electric dipole moments on Rsites, which in the case of magnetic Ho ions can depend on an applied magnetic field.
The fielddependence of magnetization (Fig. 1b) shows saturation at fields ~1 T at low temperatures, which suggests that the magnetism of Ho^{3+} (^{5}I_{8}) ions in langasites is dominated by the two lowestenergy levels split by the crystal field from other states of the J = 8 multiplet. The level splitting in this socalled nonKramers doublet, Δ, is estimated to be 3.1 K. As discussed below, based on symmetry arguments and magnetization fits, the Isingtype magnetic moments of Hoions are oriented perpendicular to the local symmetry axis, i.e., in the X_{i}Y_{i} plane, forming an angle γ ≈ 30^{∘} with the crystallographic caxis (Fig. 1b). γ is the only free magnetic anisotropy parameter in our model that reproduces well the field dependence of the magnetization along the b^{*} and caxes (see Fig. 1a) but underestimates the magnetization for H∥a, which is indicative of deviations of the magnetic moments from the X_{i}Y_{i}plane (see discussion below).
Figure 2 shows the angular dependence of the electric polarization along the caxis, P_{c}, for the magnetic field vector, H, rotated in the ac and abplanes. The complex angular dependence is indicative of a highorder ME effect resulting from the C_{3} symmetry of langasite. Although the expansion of the magnetically induced electric dipole of the Hoion in powers of H begins with terms ∝ H^{2}, the bilinear ME effect is cancelled in the sum over the dipole moments of the Ho dopants in the three La sublattices.
The lowestorder phenomenological expression for P_{c} allowed by C_{3} and C_{2} symmetries is
where we use the Cartesian coordinates with \(\hat{{\bf{x}}}=\hat{{\bf{a}}}\), \(\hat{{\bf{y}}}={\hat{{\bf{b}}}}^{* }\), \(\hat{{\bf{z}}}=\hat{{\bf{c}}}\) and a_{4}(T) is a material constant. Fourthorder ME effect in crystals with trigonal crystal symmetry was discussed earlier^{34,35,36}. Equation (1) implies that \({P}_{z}\propto {\sin }^{2}\theta \sin 2\theta \cos 3\varphi\), where θ and φ are, respectively, the polar and azimuthal angles describing the magnetic field direction. This angular dependence is in good agreement with the results of experimental measurements at high temperatures shown in Fig. 2a, whereas at low temperatures the experimental θdependence becomes more complex, indicating substantial higherorder contributions to the ME effect.
The fourthorder ME effect described by Eq. (1) gives zero electric polarization for a magnetic field in the basal ab plane. However, the experimentally measured electric polarization shown in Fig. 2b is of the same order as that for outofplane fields. The electric polarization shows a periodic dependence on the azimuthal angle, φ, with the period of 60^{∘} apparently resulting from a sixthorder ME effect. The lowestorder expression for the electric polarization induced by an inplane magnetic field is, indeed, of sixth order in H:
The sixthorder ME response has not been discussed earlier. The righthand side of Eq. (2), found using symmetry arguments^{34,35,36}, is the lowestorder polynomial of H_{x} and H_{y} that transforms as P_{z}. The sawtooth shape of the angular dependence of the polarization at low temperatures (Fig. 2b) indicates contributions of ME effects of yet higher orders.
Equations (1) and (2) imply that the electric polarization is proportional to the 4th or 6th power of the applied magnetic field, depending on the orientation of H. Instead, at low temperatures we obseve a nearly linear dependence of P_{z} for any field direction (see Fig. 3). Importantly, phenomenological Eqs. (1) and (2) only hold for weak magnetic fields, ∣μ_{0}H∣ ≪ max(Δ, k_{B}T), μ_{0} being the saturated magnetic moment of a Ho^{3+} ion (these equations are obtained in Methods by expansion in powers of H). At low temperatures, their validity is limited to fields ≲ 1 T (see Fig. 1b). The saturation of the Ho magnetic moment at higher magnetic fields gives rise to higherorder harmonics in the observed angular dependence of the electric polarization.
Counterintuitively, the onset of a more complex θ and φdependence correlates with the crossover to a linear dependence of P_{z} on the magnitude of the magnetic field (Fig. 3). In what follows we discuss the theory behind this unusual ME effect.
Theory
In the absence of inversion symmetry, the rareearth ions interact with the electric field, E, through an effective dipole moment operator, \(\hat{{\bf{d}}}\), which in the subspace of the groundstate multiplet can be expressed in terms of the quadrupole and higherorder magnetic multipole moments of the ions^{37}. The spectrum of the lowenergy states depends then on both electric and magnetic fields, which gives rise to a ME response of the rareearth ions. The description of the ME effect in Holangasite can be simplified by projecting the Hamiltonian describing the ME behavior of the groundstate multiplet on the subspace of the two states, \(\leftA\right\rangle\) and \(\leftB\right\rangle\), forming the nonKramers doublet (see “Methods”).
Alternatively, the Hamiltonian describing the ME coupling of the nonKramers doublet can be directly deduced from symmetry considerations. Within the unit cell of Holangasite, there are three equivalent rareearth positions with local C_{2} symmetry axes along the \({\hat{Z}}_{i}\) (i = 1, 2, 3) directions related by 120^{∘}rotations around the crystallographic c axis. At site 1, \({\hat{Z}}_{1}\parallel a\), \({\hat{Y}}_{1}\parallel c\) (along the C_{3}symmetry axis) and \({\hat{X}}_{1}=[{\hat{Y}}_{1}\times {\hat{Z}}_{1}]\). In the local coordinate frame, the ME Hamiltonian is,
(the index i = 1, 2, 3 labeling the rareearth ion is omitted for simplicity). Equation (3) is a general symmetryallowed expression provided one of the states forming the doublet is even and another is odd under \({C}_{2}={2}_{{Z}_{i}}\), in which case the magnetic moment of the doublet lies in the X_{i}Y_{i} plane and thus can have a nonzero component along the crystallographic c axis. If both states would transform in the same way under \({2}_{{Z}_{i}}\), the magnetic moment would be parallel to the Z_{i} axis and the magnetic moments of all Ho ions would lie in the ab plane, in disagreement with the experiment.
The Ycomponent of the polarization (P∥c) measured in our experiment, results from the third term in Eq. (3). For E_{X} = E_{Z} = 0 and E_{Y} = E_{c}, the total effective Hamiltonian in the subspace spanned by the \(\leftA\right\rangle\) and \(\leftB\right\rangle\) states is:
where (σ_{x}, σ_{y}, σ_{z}) are Pauli matrices, Δ is the crystalfield splitting, μ = (μ_{X}, μ_{Y}, 0) and d_{Y} are, respectively, the (transitional) magnetic and electric dipole moments of the Ho^{3+} ion. The form of \({\hat{H}}_{{\rm{e}}ff}\) is constrained by 2_{Z} and timereversal symmetries of the Ho ions. In addition, 3_{c} symmetry implies that the (real) coefficients, d_{Y}, μ_{X}, μ_{Y} and g_{YZ} are the same for all Hosublattices.
The free energy of the doublet, f_{i} (i = 1, 2, 3), can now be easily calculated and the magnetically induced electric polarization is given by
where n_{Ho} is the density of Ho ions and M_{i} is related to the average magnetic moment of the ith Ho ion by 〈μ_{i}〉 = (μ_{X}, μ_{Y}, 0)M_{i} and equals
where +ϵ_{i} and −ϵ_{i} are energies of the two states forming the doublet, \(({\boldsymbol{\mu }}\cdot {{\bf{H}}}_{i})={\mu }_{X}{H}_{{X}_{i}}+{\mu }_{Y}{H}_{{Y}_{i}}\) and β = 1/k_{B}T. In zero applied electric field, \({\epsilon }_{i}=\sqrt{{({\boldsymbol{\mu }}\cdot {{\bf{H}}}_{i})}^{2}+{(\Delta /2)}^{2}}\) (for details see the Methods section and refs. ^{38,39,40,41}). The resulting magnetic fielddependence of the electric polarization (dashed lines in Figs. 2 and 3) is in reasonable agreement with experimental data.
The largest discrepancy between theory and experiment is found in the angular dependence of the electric polarization shown in Fig. 2. The main reason for this deviation is the intrinsic disorder in the langasite crystal structure (Fig. 1). The four positions in the local surrounding of rareearth ions are randomly occupied by two Si and two Ga ions^{24,32} resulting in six inequivalent Ho ions in each sublattice. The concomitant lattice distortions affect magnetic anisotropy and ME interactions of the ions. Importantly, our simplified model based on average symmetry of rareearth ions captures the basic physics of the ME response of Holangasite.
Discussion
Figure 3 demonstrates that the electric polarization is a linear function of the applied field strength in the broad range of temperatures and external magnetic fields. The region of linear effect agrees well with the regime of saturated magnetic moments shown in Fig. 1b. This can be understood by noting that Eq. (5) can be obtained from the ME coupling, \({g}_{YZ}{E}_{c}{H}_{{Z}_{i}}{M}_{i}\), on the ith Ho sublattice. At low temperatures, βϵ_{i} ≫ 1, M_{i} grows with the applied magnetic field and approaches unity at rather weak fields, ∣μ_{0}H∣ ~ Δ, above which the magnetic moment saturates and the ME coupling becomes linear.
Importantly, the ME coupling in Eq. (3) originates from the admixture of higherenergy excited states from the J = 8 multiplet to \(\leftA\right\rangle\) and \(\leftB\right\rangle\) in the presence of electric and magnetic fields (see Methods). The linear ME effect is a consequence of a relatively large energy separation, W, of the nonKramers doublet from the higherenergy states: it is observed for Δ ≪ ∣μ_{0}H∣ ≪ W, when the dipole magnetic moment of the doublet is saturated while the admixture of higherenergy states remains small. Rearearth ions with Kramers doublets can also show linear ME response, which makes the whole series of the rareearth dopands potentially interesting for studies of this unusual phenomenon.
There is an additional ME coupling that results from the matrix elements of the electric and magnetic dipole operators between the \(\leftA\right\rangle\) and \(\leftB\right\rangle\) states in Eq. (4) and does not involve highenergy excited states. This coupling, however, is proportional to second and higher powers of the electric field and does not contribute to the magnetically induced electric polarization measured in zero applied electric field, which can be traced back to the fact that the electric dipole is even and magnetic dipole is odd under time reversal.
Next we explain how the linear dependence of P_{c} on the magnetic field strength can be compatible with the complex dependence on the direction of the field (see “Methods” for more details). The sixfold periodicity of the polarization with the azimuthal angle is related to the presence of three rareearth sublattices. Each sublattice separately gives rise to oscillations with the period of 180^{∘} (see Eq. (17)), since the Isinglike Ho magnetic moment changes sign as the magnetic field rotates in the abplane. Similar twofold periodicity is observed in the ME Co_{4}Nb_{2}O_{9} with Heisenberg spins, where the Néel vector describing an antiferromagnetic ordering of Co spins rotates together with the magnetic field vector^{42}. In addition, the ME effect in Holangasite is a sum of the responses of the three kinds of Ho positions with magnetic moments rotated through 120^{∘} around the c axis, which gives rise to oscillations with the period of 60^{∘}.
Holangasite can show the inverse effect—an electrically induced magnetization ∝ E_{c}—even though the ME coupling is an even function of H. The sensitivity to the magnetic field sign, important for applications of ME materials, is recovered under a bias magnetic field. An additional, electrically induced magnetization changes sign under the electric field reversal. If the bias field is strong enough to saturate Ho magnetic moments, the inverse ME effect is linear (see “Methods” for more details).
We have shown that the magnetic and ME properties of Holangasite may be understood taking into account the two lowest crystalfield levels of Ho^{3+} and the interplay of the local and global symmetries. Our theory reconciles a complex oscillating dependence of the polarization on the magnetic field direction with a linear dependence on its magnitude. The electric polarization is enabled by the absence of inversion symmetry at the local Ho positions. In weak magnetic fields, when the Zeeman energy is small compared to the energy splitting in the nonKramers doublet, the polarization is described by Eqs. (1), (2) and is highly nonlinear in H. It becomes linear in the strongfield limit, when the induced magnetic moment of Ho ions saturates, but it remains a complicated function of the magnetic field direction.
Importantly, the doping of noncentrosymmetric materials by magnetic rareearth ions at lowsymmetry positions provides a route to a linear ME response that does not require special magnetic orders and crystal lattices, as it occurs already in the paramagnetic state. Oscillatory dependence of magnetic and ME responses on the direction of the applied magnetic field is governed by crystal symmetry and is a generic property of such systems.
Methods
Experimental
Single crystals of doped holmium langasite Ho_{x}La_{3−x}Ga_{5}SiO_{14} were grown by the Czochralski technique. The crystals were proven to be righthanded single domain^{24,32}. The handedness of langasites is likely a result of the growth procedure^{24}. The orientation and crystallinity of the samples have been controlled using Laueanalysis. The actual concentration of holmium has been obtained using the total reflection Xray fluorescence spectroscopy. The sample showed homogeneous polarization rotation of about 5^{∘} for the green light, which agrees with literature data^{43}.
For the electric polarization experiments a planeparallel plate with thickness d = 2.13 mm and area A = 106 mm^{2} has been cut with the flat surface perpendicular to the caxis (ccut). Silver paste electrodes were applied to two surfaces of the sample parallel to the abplane. The polarization was measured using an electrometer adapted to a Physical Property Measuring System, with magnetic fields of up to 14 T and temperatures down to 2 K. By changing the orientation of the sample in the cryostat, the direction of the magnetic field relative to the crystal axes was adjusted. The magnetization was measured using a commercial Vibrating Sample Magnetometer with magnetic fields up to 6 T. In these experiments, smaller samples from the same batch were used.
Microscopic theory of ME effect in Holangasite
The symmetry of the rareearth environment in langasites is described by the point group C_{2}. This cyclic group has two onedimensional representations with respect to the rotation through an angle π around the twofold symmetry axis (local Z_{i}axis): the symmetric, A, and antisymmetric, B, representations. Ho^{3+} with an even number of felectrons is a nonKramers ion. The crystal field with C_{2} symmetry completely removes the degeneracy of the ground ^{5}I_{8} multiplet that splits into 2J + 1 = 17 singlets with the wave functions transforming according to either A or B irreducible representations of the C_{2} group.
In the absence of inversion, the rareearth ions interact with the electric field, E, through an effective dipole moment operator, \(\hat{{\bf{d}}}\), which in the space of the Ho^{3+} groundstate multiplet can be expressed in terms of the quadrupole and higherorder magnetic multipole moments of the ions^{37}:
where \({\hat{Q}}_{\alpha \beta }=\frac{1}{2}[{\hat{J}}_{\alpha }{\hat{J}}_{\beta }+{\hat{J}}_{\beta }{\hat{J}}_{\alpha }\frac{2}{3}J(J+1){\delta }_{\alpha \beta }]\) is the quadrupole moment operator, \(\hat{{\bf{J}}}=({\hat{J}}_{X},{\hat{J}}_{Y},{\hat{J}}_{Z})\) being the total angular momentum operator of the Ho^{3+} lowestenergy multiplet, a, b and c are phenomenological constants and the higherorder multipoles are omitted. Equation (7) is written in the local coordinate frame, (X_{i}, Y_{i}, Z_{i}), and the index i = 1, 2, 3 labeling the rareearth ion is omitted for simplicity. The phases of the wave functions forming the groundstate multiplet can be chosen in such a way that the matrix elements of the electric dipole (even under timereversal) are real, whereas the matrix elements of the magnetic dipole (odd under timereversal) are all imaginary.
The ground state of the Ho^{3+} ion in the crystalfield potential, V_{cf}, is a quasidoublet (two closeby singlets with the gap Δ ~ 2 cm^{−1}) which are separated by the energy W ~ 30–50 cm^{−1} from the excited crystalfield states. The wave functions, \(\leftA\right\rangle\) and \(\leftB\right\rangle\), of the quasidoublet can belong either to the same or to different representations. As discussed in the main text, the ccomponent of the magnetization measured in the experiment is nonzero only in the latter case.
The response of Ho^{3+} ions to the applied electric and magnetic fields is described by \(\hat{V}=\hat{{\bf{d}}}\cdot {\bf{E}}+{\mu }_{{\rm{B}}}{g}_{{\rm{L}}}\hat{{\bf{J}}}\cdot {\bf{H}}\), where the first term describes the coupling of the effective dipole operator Eq. (7) to the electric field and the second term is the Zeeman coupling, μ_{B} and g_{L} = 5/4 being, respectively, the Bohr magneton and the Lande factor. To describe the ME response of the Holangasite, we project the total Hamiltonian, \(\hat{H}=\hat{V}+{\hat{V}}_{{\rm{c}}f}\), on the quasidoublet subspace:
where i, j = 1, 2 label the states of the nonKramers doublet, \(\leftA\right\rangle\) and \(\leftB\right\rangle\), and the effect of the higherenergy states from the ^{5}I_{8} multiplet (k = 3, 4, …, 17) is treated in the second order of perturbation theory assuming \( \langle k \hat{V} i\rangle  \ll W\), and \({W}_{k}={E}_{k}\frac{({E}_{1}\,+\,{E}_{2})}{2}\). We note that this approach can also be used to analyze the rareearth contribution to magnetoelastic and optical properties.
The matrix elements of \(\hat{V}\) in the quasidoublet subspace are constrained by C_{2} symmetry (cf. Eq. (4)):
where I is the 2 × 2 identity matrix, \({\bf{d}}\cdot {{\bf{E}}}_{i}=\langle A \hat{{\bf{d}}}\cdot {{\bf{E}}}_{i} B\rangle ={d}_{X}{E}_{{X}_{i}}+{d}_{Y}{E}_{{Y}_{i}}\) and \({\boldsymbol{\mu }}\cdot {{\bf{H}}}_{i}=i{\mu }_{{\rm{B}}}{g}_{J}\langle A \hat{{\bf{J}}}\cdot {{\bf{H}}}_{i} B\rangle ={\mu }_{X}{H}_{{X}_{i}}+{\mu }_{Y}{H}_{{Y}_{i}}\). Due to the 3_{c} symmetry of the langasite crystal, the real coefficients d_{0}, d_{X}, μ_{X}, etc. are the same for all Ho sites, i.e., are independent of i. The phases of the wavefunctions \(\leftA\right\rangle\) and \(\leftB\right\rangle\) are chosen such that the matrix elements of the timeeven electric dipole operator involve real matrices, I and σ_{x}, whereas the timeodd magnetic dipole operator is proportional to σ_{y} with imaginary matrix elements.
In the last term of Eq. (8) we only leave the offdiagonal parts proportional to the product of the matrix elements of \(\hat{{\bf{d}}}\cdot {\bf{E}}\) and \({\mu }_{{\rm{B}}}{g}_{{\rm{L}}}\hat{{\bf{J}}}\cdot {\bf{H}}\) giving rise to the ME coupling,
(cf. Eq. (3)). Here, the coefficients g_{XY}, g_{YX}, etc. are real numbers. This interaction is invariant under C_{2} and timereversal symmetries, and since it involves both the electric and magnetic dipole operators, the matrix elements of \({\hat{H}}_{{\rm{m}}e}\) in the quasidoublet subspace are imaginary.
Adding all contributions to the effective twolevel Hamiltonian and assuming \({E}_{{X}_{i}}={E}_{{Z}_{i}}=0\) and \({E}_{{Y}_{i}}={E}_{c}\) (i = 1, 2, 3), we obtain Eq. (4). The energies of the states forming the nonKramers doublet at the ith Ho^{3+} site, obtained by diagonalization of the Hamiltonian Equation (4), equal ±ϵ_{i}, where
The free energy is then given by
where n_{Ho} is the density of Ho ions and k_{B} is the Boltzmann constant. The average magnetic moment measured at zero applied electric field is
where \(\beta =\frac{1}{{k}_{{\rm{B}}}T}\), μ_{0} is the magnetic moment of the nonKramers doublet and m_{i} is a unit vector defined by
The relation between the unit vectors in the hexagonal, Cartesian and the three local frames (see Fig. 1a) is:
On all three sites \({\hat{Y}}_{i}=(0,0,1)\) in the Cartesian coordinates, \({\hat{Z}}_{2,3}=(\frac{1}{2},\pm\! \frac{\sqrt{3}}{2},0)\) and \({\hat{X}}_{2,3}=(\!\mp\! \frac{\sqrt{3}}{2},\frac{1}{2},0)\).
The ccomponent of the electric polarization measured in our experiment is given by
Note that since the electric polarization is measured at zero applied electric field, d_{Y} that enters into the expression for the energy levels Eq. (11), drops out from the expression for P_{c}, in which \({\epsilon }_{i}=\sqrt{{({\boldsymbol{\mu }}\cdot {{\bf{H}}}_{i})}^{2}+{(\Delta /2)}^{2}}\). The energy splitting in the nonKramers doublet by both electric and magnetic fields (see Eqs. (9) and (11)) does not result in the ME coupling linear in electric field, because the electric and magnetic dipoles transform with opposite signs under time reversal. The observed ME effect originates solely from the admixture of higherenergy states in applied electric and magnetic fields (the last term in Eq. ((8)). In addition, Ho ions carry a “builtin” electric dipole moment along the local Z_{i} axis in the ab plane, which results from the first term in Eq. (9) and is independent of the magnetic field. These electric dipoles on three Ho sites are rotated through 120^{∘} around the c axis with respect to each other and give no net contribution to the electric polarization.
In the limit of strong fields and low temperatures, ∣μ ⋅ H_{i}∣ ≫ Δ, k_{B}T, when the magnetic moment saturates, the electric polarization,
shows a complex sawtooth dependence on the direction of the magnetic field in the ab plane (due to the signfunctions) and, at the same time, a simple linear dependence on the strength of the magnetic field, in agreement with our experimental observations.
In the opposite limit of weak fields and high temperatures, k_{B}T ≫ ∣μ_{0}H∣, Δ,
where H is the magnitude of the magnetic field and \(\gamma =\arccos ({\mu }_{Y}/{\mu }_{0})\), which agrees with Eq. (1) obtained by symmetry and gives a_{4}(T) ∝ T^{−3}. At zero temperature, \({({k}_{{\rm{B}}}T)}^{3}\) in Eq. (18) is replaced by Δ^{3}/12.
For magnetic field applied in the ab plane, the electric polarization at high temperatures,
is proportional to H^{6} and oscillates six times as φ varies by 2π, in agreement with Eq. (2) (a_{6} ∝ T^{−5}). At zero temperature, \({({k}_{{\rm{B}}}T)}^{5}\to {\Delta }^{5}/90\).
The coefficients a_{4} and a_{6} in Eqs. (1, 2) can be estimated from the lowfield slopes of experimental data shown in Fig. 3. As demonstrated in the insets to Fig. 3, they indeed follow the expected a_{4} ∝ T^{−3} and a_{6} ∝ T^{−5} behavior.
The ME coupling ∝ E_{c}, which leads to Eq. (16) for the magnetically induced polarization,
also gives rise to the electrically induced magnetization, \(\delta {\bf{M}}=\frac{\partial {f}_{{\rm{m}}e}}{\partial {\bf{H}}}\).
Figure 4a shows the magnetization at E_{c} = 0 (Eq. (13)) plotted vs the angle φ describing the orientation of the magnetic field in the ab plane. In general, M is not parallel to H. In the saturation regime, it depends on the sign of the magnetic field projection on the easy axes of Ho ions, but not on the magnetic field strength. Figure 4b shows the electrically induced magnetization, δM, divided by E_{c}. M and \({\left(d{\bf{M}}/d{E}_{c}\right)}_{{E}_{c} = 0}\) have opposite symmetries with respect to φ → − φ: M_{x} is a symmetric function of φ and M_{y}, M_{z} are antisymmetric, whereas for dM/dE_{c} it is the other way around.
Interestingly, the inverse ME response is amplified when the magnetic field is normal to the easy axis of one of the Hosublattices. At zero temperature and for μ_{0}H ≫ Δ, δM shows sharp peaks at (m_{i} ⋅ H) = 0,
Data availability
The data that support the findings of this study are available from the authors on reasonable request, see author contributions for specific data sets.
Code availability
The code of the model used to produce the fits will be provided by A.A.M. or M.M. upon request.
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Acknowledgements
We thank Peter Kregsamer and Christina Streli for help in determining the actual Hocontent in our samples. This work was supported by the Russian Science Foundation (161210531) and by the Austrian Science Funds (W 1243, I 2816N27, P 27098N27).
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A.P., A.A.M., and M.M. supervised the project; Th.K. and A.S. designed the experiment; B.V.M. grew the samples; L.W., L.B., A.P., E.C., D.S., and V.Y.I. characterized the samples using various techniques; L.W., V.Y.I., and A.M.K. conducted the experiments and analyzed the data under the supervision of A.P. and A.A.M.; N.V.K., A.K.Z., A.I.P., A.A.M., and M.M. developed the theory. A.P., L.W., A.A.M., and M.M. contributed to the writing of the manuscript.
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Weymann, L., Bergen, L., Kain, T. et al. Unusual magnetoelectric effect in paramagnetic rareearth langasite. npj Quantum Mater. 5, 61 (2020). https://doi.org/10.1038/s41535020002639
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DOI: https://doi.org/10.1038/s41535020002639
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