Trigonal Anisotropy Research Articles

Magnetic field induced ab -plane rotation of the Eu magnetic moments in trigonal EuMg2Bi2 and EuMg2Sb2 single crystals below their Néel temperatures

The thermodynamic and electronic-transport properties of trigonal ${\mathrm{EuMg}}_{2}{\mathrm{Bi}}_{2}$ in $ab$-plane magnetic fields ${\mathbf{H}}_{x}$ and the A-type antiferromagnetic structure have recently been reported. The Eu magnetic moments with spin $S=7/2$ remain locked in the $ab$ plane up to and above the $ab$-plane critical field ${\mathbf{H}}_{x}^{\mathrm{c}}=27.5$ kOe at which the Eu moments become parallel to ${\mathbf{H}}_{x}$. Here additional measurements at low fields are reported that reveal a new spin-reorientation transition at a field ${H}_{c1}\ensuremath{\approx}465$ Oe where the Eu moments remain in the $ab$ plane but become perpendicular to ${H}_{x}$. At higher fields, the moments cant toward the field resulting in $M\ensuremath{\propto}{H}_{x}$ up to ${\mathbf{H}}_{x}^{\mathrm{c}}$. Similar results are reported from measurements of the magnetic properties of ${\mathrm{EuMg}}_{2}{\mathrm{Sb}}_{2}$ single crystals, where ${H}_{\mathrm{c}1}\ensuremath{\approx}220$ Oe is found. Theory is formulated that models the low-field magnetic behavior of both materials, and the associated anisotropies are calculated. The $ab$-plane trigonal anisotropy in ${\mathrm{EuMg}}_{2}{\mathrm{Sb}}_{2}$ is found to be significantly smaller than in ${\mathrm{EuMg}}_{2}{\mathrm{Bi}}_{2}$.

Thermodynamics of the Nonrelativistic Free-Electron Fermi Gas in One, Two, and Three Dimensions from the Degenerate to the Nondegenerate Temperature Regime

The thermodynamic and electronic-transport properties of trigonal EuMg2Bi2 in ab-plane magnetic fields Hx and the A-type antiferromagnetic structure have recently been reported. At a temperature of 1.8 K < TN, the Eu magnetic moments with spin S = 7/2 remain locked in the ab plane up to and above the ab-plane critical field Hxc = 27.5 kOe at which the Eu moments become parallel to Hx. Here additional measurements at low fields are reported that reveal a new spin-reorientation transition at a field Hc1 = 465 Oe where the Eu moments remain in the ab plane but become perpendicular to Hx. At higher fields, the moments cant towards the field resulting in M proportional to Hx up to Hxc. Similar results are reported from measurements of the magnetic properties of EuMg2Sb2 single crystals, where Hc1 = 220 Oe is found. Theory is formulated that models the low-field magnetic behavior of both materials, and the associated anisotropies are calculated. The ab-plane trigonal anisotropy in EuMg2Sb2 is found to be significantly smaller than in EuMg2Bi2.

Origin of field-induced discontinuous phase transitions in Nd2Fe17

Magnetic properties of a trigonal ferromagnet $\mathrm{N}{\mathrm{d}}_{2}\mathrm{F}{\mathrm{e}}_{17}$ have been studied on single crystals in steady (14 T) and pulsed (32 T) magnetic fields. The easy-magnetization direction lies close to the [120] axis, deviating from the basal plane by $2.{9}^{\ensuremath{\circ}}$ (at $T=5\phantom{\rule{0.16em}{0ex}}\mathrm{K}$). Of particular interest is the low-temperature magnetization process along the high-symmetry axis [001], which is the hard direction. This process is discontinuous and involves two first-order phase transitions (FOMPs). One of them (at 20 T) is a symmetry FOMP similar to that observed in $\mathrm{S}{\mathrm{m}}_{2}\mathrm{F}{\mathrm{e}}_{17}$. The second transition (at 10.4 T) is unusual: as the magnetization turns abruptly toward the applied field, it also changes its azimuthal orientation (the angle $\ensuremath{\varphi}$) by ${60}^{\ensuremath{\circ}}$. Both transitions can be reasonably accounted for by the presence of a significant sixth-order trigonal anisotropy term.

Magnetic anisotropy ofSm2Fe17single crystals

The previously accepted notion that the spontaneous magnetization of Sm2Fe17 lies in the basal plane of the crystal is true only approximately, and then only around room temperature. At low temperatures the magnetization, whose orientation is not fixed by the symmetry, is found to deviate from the basal plane by as much as 10 degrees. The threefold symmetry axis is a hard direction; to magnetize the crystal in this direction, a magnetic field of about 9 T is required. The hard-axis magnetization arrives at saturation discontinuously, by way of a first-order phase transition. The behavior is a general one for trigonal ferromagnets where K-1 < 0 and the leading trigonal anisotropy constant is nonzero, K-2' not equal 0. Although of universal occurrence, the first-order transition is only visible at low temperatures, where it is accompanied by a magnetization anomaly of sufficient size.

Origin and spectroscopic determination of trigonal anisotropy in a heteronuclear single-molecule magnet

W-band ({\nu} ca. 94 GHz) electron paramagnetic resonance (EPR) spectroscopy was used for a single-crystal study of a star-shaped Fe3Cr single-molecule magnet (SMM) with crystallographically imposed trigonal symmetry. The high resolution and sensitivity accessible with W-band EPR allowed us to determine accurately the axial zero-field splitting terms for the ground (S =6) and first two excited states (S =5 and S =4). Furthermore, spectra recorded by applying the magnetic field perpendicular to the trigonal axis showed a pi/6 angular modulation. This behavior is a signature of the presence of trigonal transverse magnetic anisotropy terms whose values had not been spectroscopically determined in any SMM prior to this work. Such in-plane anisotropy could only be justified by dropping the so-called 'giant spin approach' and by considering a complete multispin approach. From a detailed analysis of experimental data with the two models, it emerged that the observed trigonal anisotropy directly reflects the structural features of the cluster, i.e., the relative orientation of single-ion anisotropy tensors and the angular modulation of single-ion anisotropy components in the hard plane of the cluster. Finally, since high-order transverse anisotropy is pivotal in determining the spin dynamics in the quantum tunneling regime, we have compared the angular dependence of the tunnel splitting predicted by the two models upon application of a transverse field (Berry-phase interference).

Enhanced magnetoelastic coupling in hexagonal multiferroic HoMnO3

We have investigated the magnetoelastic coupling in HoMnO${}_{3}$ by measuring the temperature and magnetic field dependence of two elastic moduli ${C}_{11}$ and ${C}_{33}$ with an ultrasonic velocity technique. In comparison with similar data in YMnO${}_{3}$, we have identified an enhanced elastic softening, predominantly on ${C}_{11}$, that scales with the spin susceptibility over a wide temperature range. Spin fluctuations induced by Ho-Mn interactions are responsible for this softening: when the softening is quenched by an in-plane magnetic field with a concomitant stabilization of the Mn order, it is rather enhanced for a field oriented along the $c$ axis. The H-T phase diagrams deduced from elastic anomalies appearing at the different phase transitions are consistent with previous microwave data. The origin of the Ho-Mn coupling is discussed in terms of a fourth-order trigonal anisotropy term in the spin Hamiltonian.

Nonlocal Landau theory of the magnetic phase diagram of highly frustrated magnetoelectricCuFeO2

A nonlocal Landau-type free-energy functional of the spin density is developed to model the large variety of magnetic states, which occur in the magnetic-field\char21{}temperature phase diagram of magnetoelectric ${\text{CuFeO}}_{2}$. Competition among long-range quadratic exchange, biquadratic antisymmetric exchange, and trigonal anisotropy terms, consistent with the high-temperature rhombohedral $R\overline{3}m$ crystal symmetry, are shown to all play important roles in stabilizing the unusual combination of commensurate and incommensurate spin structures in this highly frustrated triangular antiferromagnet. It is argued that strong magnetoelastic coupling is largely responsible for the nonlocal nature of the free energy. A key feature of the analysis is that an electric polarization is induced by a canting of the noncollinear incommensurate spin structure. Application of the model to ordered spin states in the triangular antiferromagnets ${\text{MnBr}}_{2}$ and ${\text{NaFeO}}_{2}$ is also discussed.

Trigonal Anisotropy in Graphite and Carbon Nanotubes

We discuss here how the trigonal warping effect of the electronic structure is relevant to optical processes in graphite and carbon nanotubes. The electron-photon, electron-phonon, and elastic scattering matrix elements have a common factor of the coefficients of Bloch wave funtions of the A and B atoms in the graphite unit cell. Because of the three fold symmetry around the Fermi energy point (the K or K′ point), the matrix elements show a trigonal anisotropy which can be observed in both resonance Raman and photoluminescence spectroscopy. This anisotropy is essential for understanding the chirality dependence of the Raman intensity and the optical response of single wall carbon nanotubes.

Threefold Symmetric Magnetic Two-Ion Coupling in hcp Rare-Earth Metals

The heavy rare earths crystallize in the hcp structure. Most of the mag- netic couplings between two ions in these metals are independent of the two different orientations of the hexagonal layers. However, trigonal anisotropy terms may occur, reflecting that the c-axis is only a threefold axis. In the presence of a trigonal coupling the symmetry is reduced, and the double-zone representation in the c-direction ceases to be valid. The strong interaction between the transverse optical phonons and the acoustic spin waves prop- agating in the c-direction of Tb detected more than twenty years ago, was the first example of a trigonal coupling found in these systems. A few years ago a careful neutron-diffraction study of the c-axis modulated magnetic structures in Er showed the presence of higher harmonics at positions along the c-axis translated by odd multiple of 2π/c. This indicates distortions of the structures due to trigonal couplings, and the same characteristic phe- nomenon has now also been observed in Ho. Additionally, mean-field calcu- lations show that a trigonal coupling in Ho is required, in order to explain the increase in the commensurable effects observed for the 8 and 10 layered periodic structures, when a field is applied along the c-axis.

Neutron scattering investigation of the mixed magnetic system Fe1-xMnxCl2

Neutron scattering has been used to study both sublattice magnetisation and magnetic excitations in the Fe1-xMnxCl2 mixed magnetic system for low Mn concentrations (x<or=0.25). First the temperature dependence of magnetic Bragg reflections exhibits an anomalous behaviour at low temperature which is imputed to the magnetisation of Mn2+ in the exchange field with Fe2+ ions. On the other hand, by inelastic neutron scattering the authors observe an overall decrease of the dispersion curves of the host excitations with Mn concentration. A comparison of the inelastic results with predictions of (i) a cluster model, and (ii) a mean crystal model suggests that, in addition to the effect of mixing on exchange, a strong concentration dependence of the trigonal anisotropy must be taken in account.

Theory of stripe-domain nucleation in garnet films

Micromagnetics predicts that domain nucleation in an in-plane field in mechanically rigid uniaxial films with easy axis normal to the faces should take the form of stripes parallel to the field. This prediction does not agree with observation in some real films. It has been shown previously that the magnetoelastic interaction in deformable films may cause the stripes to be perpendicular to the field when the films are thick enough. Quantitative assessment of this effect is difficult because the magnitudes of the magnetoelastic constants are not accurately known, and because high-quality epitaxial films are at least partly constrained from deforming. In this paper we discuss another interaction which can cause the stripes to be perpendicular to the field in a [111]-oriented cubic film, such as bubble-device garnet. The interaction energy is a dipolar energy brought into play by the effectively trigonal anisotropy. It vanishes when H is along [112̄] and is maximum when H is along [1̄10]. Quantitative evaluation shows that the interaction can account for a large part but not all of the behavior observed in recent experiments.

Trigonal Magnetocrystalline Anisotropy in Hexagonal Oxides

Torque measurements of the magnetocrystalline anisotropy between 77\ifmmode^\circ\else\textdegree\fi{}K and 300\ifmmode^\circ\else\textdegree\fi{}K are reported for single crystals of two different ferrimagnetic oxides having structures related to that of the mineral magneto-plumbite. The compounds, with chemical compositions ${\mathrm{Co}}_{2}$${\mathrm{Ba}}_{2}$${\mathrm{Fe}}_{12}$${\mathrm{O}}_{22}$ and ${\mathrm{Co}}_{2}$${\mathrm{Ba}}_{3}$${\mathrm{Fe}}_{24}$${\mathrm{O}}_{41}$, are known as ${\mathrm{Co}}_{2}Y$ and ${\mathrm{Co}}_{2}Z$, respectively. Both compounds, after suitable magnetic field cooling treatment, display trigonal anisotropy in the basal plane at 77\ifmmode^\circ\else\textdegree\fi{}K. The field cooling is shown to have the effect of placing the magnetization into one of two energetically equivalent orientations, each of which leads to a trigonal term of different sign. Examination of the crystal structures shows that in the case of ${\mathrm{Co}}_{2}Y$ an additional term ${{K}_{2}}^{\ensuremath{'}}{sin}^{3}\ensuremath{\theta}\ifmmode\times\else\texttimes\fi{}cos\ensuremath{\theta}cos3\ensuremath{\phi}$ (where $\ensuremath{\theta}$ and $\ensuremath{\phi}$ are the polar coordinates of the magnetization) should be added to the generally accepted phenomenological hexagonal anisotropy expression. The anisotropy constant ${{K}_{2}}^{\ensuremath{'}}$ is evaluated as 6\ifmmode\times\else\texttimes\fi{}${10}^{5}$ erg/${\mathrm{cm}}^{3}$ at 117\ifmmode^\circ\else\textdegree\fi{}K. In the case of ${\mathrm{Co}}_{2}Z$ this trigonal anisotropy term is not consistent with the crystal symmetry, although it is appropriate for structural subgroups of the unit cell. Its trigonal anisotropy is explained in terms of a "puckered" magnetization pattern whereby the sign of the $c$-axis component of magnetization is different for adjacent subgroups. The existence of this puckered pattern implies that the exchange coupling across the boundaries between subgroups is relatively weak. A new rotational hysteresis effect in ${\mathrm{Co}}_{2}Y$ is described and explained phenomenologically. An atomic theory assigning the origin of the trigonal anisotropy of both compounds to the cobalt ions is presented.