Magnetic Order Parameter Research Articles

Gonihedric (and Fuki-Nuke) order

A 3D Ising model with a purely plaquette, 4-spin interaction displays a planar flip symmetry intermediate between a global and a gauge symmetry and as a consequence has a highly degenerate low temperature phase and no standard magnetic order parameter. This plaquette Hamiltonian is a particular case of a family of 3D gonihedric Ising models defined by Savvidy and Wegner. An anisotropic variant of the purely plaquette gonihedric model, originally discussed as the ‘Fuki-Nuke’ model by Suzuki, is non-trivially equivalent to a stack of 2D Ising models, each of which can magnetize independently at the phase transition point. Consideration of this anisotropic model suggests that a suitable order parameter in the isotropic case may also be constructed using a form of planar magnetization, in which nearest neighbour correlators 〈σiσj〉 summed over planes replace a sum over spin values 〈σi〉. We conduct Monte-Carlo simulations to investigate this and related candidate order parameters in a toy model of gonihedric ground states, the Fuki-Nuke model and the isotropic plaquette gonihedric model itself.

Spin gating electrical current

The level of the chemical potential is a fundamental parameter of the electronic structure of a physical system, which consequently plays an important role in defining the properties of active electrical devices. We directly measure the chemical potential shift in the relativistic band structure of the ferromagnetic semiconductor (Ga,Mn)As, controlled by changes in its magnetic order parameter. Our device comprises a non-magnetic aluminum single electron channel capacitively coupled to the (Ga,Mn)As gate electrode. The chemical potential shifts of the gate are directly read out from the shifts in the Coulomb blockade oscillations of the single electron transistor. The experiments introduce a concept of spin gating electrical current. In our spin transistor spin manipulation is completely removed from the electrical current carrying channel.

Weak ferromagnetism and internal magnetoelectric effect in LiFeP2O7

The magnetic, thermodynamic, and pyroelectric properties of LiFeP$_2$O$_7$ single crystals are investigated with emphasis on the magnetoelectric interaction of the electrical polarization with the magnetic order parameter. The magnetic order below T$_N\simeq$ 27 K is found to be a canted antiferromagnet with a weak ferromagnetic component along the $b-$axis. A sharp peak of the pyroelectric current at T$_N$ proves the strong internal magnetoelectric interaction resulting in a sizable polarization decrease at the onset of magnetic order. The magnetoelectric effect in external magnetic fields combines a linear and a quadratic field dependence below T$_N$. Thermal expansion data show a large uniaxial magnetoelastic response and prove the existence of strong spin lattice coupling. LiFeP$_2$O$_7$ is a polar compound with a strong interaction of the magnetic order parameter with the electric polarization and the lattice.

Role of thermal vibrations in magnetic phase transitions

This article addresses the contradiction that exists in studies on magnetic phase transitions between theoretical models that ignore the role of thermal vibrations and analysis of neutron diffraction data that always incorporates them. Ignoring thermal vibrations in both theoretical models and analysis of diffraction data leads to the latter giving different magnetic-order parameters for different reciprocal lattice lines. This leads to a unique consequence, the assumption to neglect a physical phenomenon turns a single-valued experimental observable into a multiple-valued one where all values are equally valid. This assumption is clearly unacceptable and must be rejected. Diffraction data constrain all theoretical models to incorporate thermal vibrations and represent the exchange interaction as temperature dependent, Jij (T), instead of the current practice, Jij (0).

1H and 195Pt NMR Study of the Parallel Two-Chain Compound Per2[Pt(mnt)2

1H and 195Pt NMR are used to probe the spin ½ anion chain in the quasi-one-dimensional conductor Per2[Pt(mnt)2], which exhibits nearly simultaneous charge density wave (CDW) and spin-Peierls (SP) transitions at low temperatures (Tc ~ 8 K). Below Tc the [Pt(mnt)2] chain forms a spin-singlet state that is evident in 1H NMR spectra and spin relaxation (1/T1) rates; however minority unpaired Pt spins may remain in the SP ground state. With increasing magnetic field, the SP and CDW order parameters decrease in unison, indicating they are coupled up to a critical field Bc ~ 20 T. Above Bc, the spin singlet evolves into a spin-polarized configuration. The 195Pt NMR signals vanish as either Tc or Bc are approached from within the SP ground state, suggesting the hyperfine field of the Pt nucleus is significantly stronger than at the proton sites. Simulations yield a consistent picture of the angular, temperature, and magnetic field-dependent spectral features.

Multiferroic and Magnetoelectric Oxides: The Emerging Scenario.

Multiferroics were considered to be rare because magnetism and ferroelectricity require entirely different criteria for the materials. Several multiferroic oxides have, however, been discovered in the past few years by virtue of novel operating mechanisms, the most effective one being ferroelectricity driven by magnetism itself. Many such oxides where the magnetic and electric order parameters interact also exhibit magnetoelectric or magnetodielectric properties. In this Perspective, properties of manganites, ferrites, and other monophasic multiferroic oxides with spin-induced electric polarization are described. Multiferroic properties arising from charge ordering are examined. The present status of BiMnO3, which is an unusual example of a ferromagnetic-ferroelectric, is presented. Recent findings suggest that it is likely that many more multiferroic and magnetoelectric oxide materials exhibiting magnetically induced ferroelectricity will be found in the future.

Magnetoelastic coupling and multiferroic ferroelastic/magnetic phase transitions in the perovskite KMnF3

The magnetic, elastic, and anelastic behavior of single-crystal KMnF${}_{3}$ have been investigated by superconducting quantum interference device (SQUID) magnetometry and resonant ultrasound spectroscopy (RUS) through the sequence of phase transitions: phase I, $Pm$$\overline{3}$$m$ $\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}({T}_{\mathrm{c}1}$ $=$ 185 K) \ensuremath{\rightarrow} phase II, $I$4/mcm $\ensuremath{\rightarrow}({T}_{\mathrm{c}2}$ $=\phantom{\rule{0.16em}{0ex}}{T}_{\mathrm{N}}$ $=$ 87 K) \ensuremath{\rightarrow} phase III, antiferromagnetic, Cmcm \ensuremath{\rightarrow} (${T}_{\mathrm{c}3}$ $=$ 82 K) \ensuremath{\rightarrow} phase IV, canted ferromagnet, Pnma. It is concluded that observed changes in the elastic properties can be explained simply in terms of strain/order parameter coupling for the octahedral tilting transitions. There appears to be no evidence in the present data or in data from the literature for coupling between the magnetic order parameter and shear strains. Any coupling between the magnetic and structural transitions is therefore weak, probably occurring only biquadratically through a small common volume strain. The combined data show unambiguously that, for the crystal used, the N\'eel point and the structural transition at 87 K are coincident. In other crystals, with slightly different stoichiometries and defect contents, this need not be the case, however, and the overlap of transition temperatures in KMnF${}_{3}$ is essentially accidental. Strong acoustic dissipation at \ensuremath{\sim}0.1--1 MHz in the stability field of phase II is attributed to the local mobility of transformation twin walls under externally applied stress. A Debye-like loss peak near 130 K is attributed to pinning of at least some twin walls by defects, but relatively high levels of acoustic dissipation below this freezing temperature imply that some of the twin walls remain mobile due to weak pinning or the absence of any pinning. Acoustic losses continue in the stability field of phase III (Cmcm) but diminish substantially in the stability field of phase IV (Pnma), implying quite different twin mobilities in the different structure types. Overlap of the structural and magnetic instabilities in KMnF${}_{3}$ opens up possibilities for manipulation of ferroelastic twinning by application of a magnetic field and for creation of materials in which the ferroelastic twin walls have quite different magnetic properties from the matrix in which they lie.

Resonant soft X-ray scattering studies of multiferroic YMn2O5

We performed soft X-ray resonant scattering at the MnL 2,3- and OK edges of YMn2O5. While the resonant intensity at the Mn L 2,3 edges reflects the magnetic order parameter, the resonant scattering at the O K edge is found to be directly related to the macroscopic ferroelectric polarization. The latter observation reveals the important role of the spin-dependent Mn-O hybridization for the multiferroicity of YMn2O5. We present details about how to obtain correct energy dependent lineshapes and discuss the origin of the resonant intensity at the O K edge.

Thermodynamics of f.c.c.-Ni–Fe Alloys in a Static Applied Magnetic Field

Within the scope of the self-consistent field and mean (‘‘molecular’’) self-consistent field approximations, applying the static concentration wave method, the thermodynamics of f.c.c.-Ni–Fe alloys undergoing the static applied magnetic field effects is studied in detail. Under such conditions, the analytical corrections to expressions for the configuration-dependent part of free energy of macroscopically ferromagneticL12-Ni3Fe-type orL10-NiFe-type ordering phases are taken into account. The obtained results for thermodynamically equilibrium states are compared with the refined phase diagram for f.c.c.-Ni–Fe alloys calculated recently without taking into account the applied magnetic field effects. Considering the specific character of microscopic structure of the magnetic and atomic orders in f.c.c.-Ni–Fe alloys, the changes of shape (and in arrangement) of order-disorder phase-transformation curves (Kurnakov points) are thoroughly analysed. A special attention is addressed to the investigation of the concentration, temperature, and magnetic-field induction-dependent atomic and magnetic long-range order parameters, especially, near their critical points. As revealed unambiguously, influence of a static applied magnetic field promotes the elevation of Kurnakov points for all the atomically ordering phases that is in an overall agreement with reliable experimental data. On the base of revealed phenomenon, the magneto external field analog-to-digital converter of the monochromatic radiations (X-rays or thermal neutrons) is hypothesized as a claim.

The frustrated Heisenberg antiferromagnet on the honeycomb lattice: J1–J2 model

We study the ground-state phase diagram of the frustrated spin- antiferromagnet with J2 = xJ1 > 0 (J1 > 0) on the honeycomb lattice, using the coupled-cluster method. We present results for the ground-state energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. We find a paramagnetic PVBC phase for xc1 < x < xc2, where xc1 ≈ 0.207 ± 0.003 and xc2 ≈ 0.385 ± 0.010. The transition at xc1 to the Néel phase seems to be a continuous deconfined transition (although we cannot exclude a very narrow intermediate phase in the range 0.21 ≲ x ≲ 0.24), while that at xc2 is of first-order type to another quasiclassical antiferromagnetic phase that occurs in the classical version of the model only at the isolated and highly degenerate critical point . The spiral phases that are present classically for all values are absent for all x ≲ 1.

Landau theory for the phase diagram of multiferroic Mn1−x(Fe,Zn,Mg)xWO4

We present a theoretical analysis of the temperature-magnetic field-concentration phase diagram of the multiferroic Mn$_{1-x}$M$_{x}$WO$_{4}$ (M=Fe, Zn, Mg), which exhibits three ordered phases, with collinear and non-collinear incommensurate and with a commensurate magnetic order. The middle phase is also ferroelectric. The analysis uses a semi-phenomenological Landau theory, based on a Heisenberg Hamiltonian with a single-ion anisotropy. With a small number of adjustable parameters, the Landau theory gives an excellent fit to all three transition lines, as well as the magnetic and the ferroelectric order parameters. The fit of the magnetic and ferroelectric order parameters is further improved by including the effect of fluctuations near the transitions. We demonstrate the highly frustrated nature of these materials and suggest a simple explanation for the dramatic effects of doping with different magnetic ions at the Mn sites. The model enables an examination of different sets of exchange couplings that were proposed by a number of groups. Small discrepancies are probably a consequence of small errors in the experimental magnetic parameters. In addition, using the Ginzburg criterion we estimate the temperature range in which fluctuations of the order parameters become important.

Phase diagram of Ba1−xKxFe2As2

We report the results of a systematic investigation of the phase diagram of the iron-based superconductor Ba${}_{1\ensuremath{-}x}$K${}_{x}$Fe${}_{2}$As${}_{2}$ from $x$ $=$ 0 to $x$ $=$ 1.0 using high-resolution neutron and x-ray diffraction and magnetization measurements. The polycrystalline samples were prepared with an estimated compositional variation of $\ensuremath{\Delta}x$ $\ensuremath{\lesssim}$ 0.01, allowing a more precise estimate of the phase boundaries than reported so far. At room temperature, Ba${}_{1\ensuremath{-}x}$K${}_{x}$Fe${}_{2}$As${}_{2}$ crystallizes in a tetragonal structure with the space group symmetry of $I$4/mmm, but at low doping, the samples undergo a coincident first-order structural and magnetic phase transition to an orthorhombic (O) structure with the space group Fmmm and a striped antiferromagnet (AF) with the space group ${F}_{\mathrm{c}}mm$\ensuremath{'}$m$\ensuremath{'}. The transition temperature falls from a maximum of 139 K in the undoped compound to 0 K at $x$ $=$ 0.252, with a critical exponent as a function of doping of 0.25(2) and 0.12(1) for the structural and magnetic order parameters, respectively. The onset of superconductivity occurs at a critical concentration of $x$ $=$ 0.130(3), and the superconducting transition temperature grows linearly with $x$ until it crosses the AF/O phase boundary. Below this concentration, there is microscopic phase coexistence of the AF/O and superconducting order parameters, although a slight suppression of the AF/O order is evidence that the phases are competing. At higher doping, superconductivity has a maximum ${T}_{\mathrm{c}}$ of 38 K at $x$ $=$ 0.4 that falls to 3 K at $x$ $=$ 1.0. We discuss reasons for the suppression of the spin density wave order and the electron--hole asymmetry in the phase diagram.

Neutron scattering study of underdoped Ba1−xKxFe2As2(x=0.09and0.17) self-flux-grown single crystals and the universality of the tricritical point

We present a combination of elastic neutron scattering measurements in zero and 14.5 T and magnetization measurements in zero and 14 T on under-doped superconducting Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$ x=0.17, and the same measurements in zero field on a non-superconducting crystal with x=0.09. The data suggest that the under-doped materials may not be electronic phase separated but rather have slightly inhomogeneous potassium doping. The temperature dependence of the magnetic order parameter (OP) below the transition of the sample with x=0.09 is more gradual than that for the case of the un-doped BaFe$_{2}$As$_{2}$, suggesting that this doping may be in the vicinity of a tricritical point. We advance therefore the hypothesis that the tricritical point is a common feature of all superconducting 122s. For the x=0.17 sample, while T$_{c}$ is suppressed from $\approx$17 K to $\approx$8 K by a magnetic field of 14 T, the intensity of the magnetic Bragg peaks (1 0 3) at 1.2 K is enhanced by 10$%$ showing competition of superconductivity (SC) and antiferromagnetism (AFM). The intensity of the magnetic Bragg peaks (1 0 3) in the (T$_{c}$, T$_{N}$) temperature interval remain practically unchanged in 14.5 T within a 10$%$ statistical error. The present results are discussed in the context of the existing literature.

A linear-quadratic order parameter coupling model for magnetoelastic phase transitions in Fe1−xO and MnO

The combined structural and antiferromagnetic phase transition in wüstite, Fe1−xO, at the Néel temperature, TN ∼ 195 K, is treated in terms of coupling between two macroscopic order parameters related to separate discrete instabilities. These each couple with a rhombohedral shear strain to give an indirect mechanism of linear (structural)–quadratic (magnetic) coupling between them. Based on patterns of lattice parameter and magnetic ordering data from the literature, it appears that the overall behaviour matches the general pattern of linear-quadratic coupling for a system with two rather similar instability temperatures. At low pressures, the magnetic instability occurs at a higher temperature than the structural instability but the coupling results in a single phase transition dominated by the influence of the magnetic order parameter. For Fe0.99O the magnetic order parameter, m, varies with temperature as ∼m8 ∝ (TN − T), while for Fe0.94O the temperature dependence is of ∼m4 or ∼m2. It is proposed that, with increasing pressure, there is a crossover of instability temperatures such that the structural instability occurs first at pressures above ∼13 GPa. This would be expected to give rise to a first-order phase transition, as appears to occur, but magnetic ordering would still occur simultaneously if the coupling is sufficiently strong. Symmetry analysis and comparison with the behaviour of MnO shows that there are a number of different possible magnetically ordered structures which could be stabilized by pressure or non-hydrostatic stress to give topologically rich phase diagrams.

Magnetic properties of spin-diluted iron pnictides fromμSR and NMR in LaFe1−xRuxAsO

The effect of isoelectronic substitutions on the microscopic properties of LaFe1-xRuxAsO, for 0< x< 0.8, has been investigated by means of muSR and 139La NMR. It was found that Ru substitution causes a progressive reduction of the N\`eel temperature (T_N) and of the magnetic order parameter without leading to the onset of superconductivity. The temperature dependence of 139La nuclear spin-lattice relaxation rate 1/T_1 can be suitably described within a two-band model. One band giving rise to the spin density wave ground-state, while the other one is characterized by weakly correlated electrons. Fe for Ru substitution yields to a progressive decrease of the density of states at the Fermi level close to the one derived from band structure calculations. The reduction of T_N with doping follows the predictions of the J_1-J_2 model on a square lattice, which appears to be an effective framework to describe the magnetic properties of the spin density wave ground-state.

Effect of Uniaxial Strain on the Structural and Magnetic Phase Transitions inBaFe2As2

We report neutron scattering experiments probing the influence of uniaxial strain on both the magnetic and structural order parameters in the parent iron pnictide compound, BaFe2As2. Our data show that modest strain fields along the in-plane orthorhombic b axis can affect significant changes in phase behavior simultaneous to the removal of structural twinning effects. As a result, we demonstrate in BaFe2As2 samples detwinned via uniaxial strain that the in-plane C4 symmetry is broken by both the structural lattice distortion and long-range spin ordering at temperatures far above the nominal (strain-free) phase transition temperatures. Surprising changes in the magnetic order parameter of this system under relatively small strain fields also suggest the inherent presence of magnetic domains fluctuating above the strain-free ordering temperature in this material.

Ground-state phases of the frustrated spin-12J1–J2–J3Heisenberg ferromagnet (J1&lt;0) on the honeycomb lattice withJ3=J2&gt;0

We study the ground-state (gs) properties of the frustrated spin-1/2 $J_{1}$--$J_{2}$--$J_{3}$ Heisenberg model on a honeycomb lattice with ferromagnetic (FM) nearest-neighbor ($J_{1}=-1$) exchange and frustrating antiferromagnetic (AFM) next-nearest-neighbor ($J_{2}>0$) and next-next-nearest-neighbor ($J_{3}>0$) exchanges, for the case $J_{3}=J_{2}$. We use the coupled cluster method in high orders of approximation, complemented by the exact diagonalization of a lattice with 32 sites, and calculate the gs energy, magnetic order parameter, and spin-spin correlation functions. We find a quantum phase transition between regions characterized by FM order and a form of AFM ("striped") collinear order at $J^{c}_{2} \approx 0.1095 \pm 0.0005$. We compare results for the FM case (with $J_{1}=-1$) to previous results for the corresponding AFM case (with $J_{1}=+1$). While the magnetic order parameters behave similarly for the FM and the AFM models for large values of the frustration parameter $J_{2}$, there are considerable differences between them for $J_{2}/|J_{1}| \lesssim 0.6$. For example, the quasiclassical collinear magnetic long-range order for the AFM model (with $J_{1}=+1$) breaks down at $J^{c_{2}}_{2} \approx 0.60$, whereas the "equivalent" point for the FM model (with $J_{1}=-1$) occurs at $J^{c}_{2} \approx 0.11$. Unlike in the AFM model (with $J_{1}=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical AFM phases (with N\'eel and striped order) for $J^{c_{1}}_{2} < J_{2} < J^{c_{2}}_{2}$, with $J^{c_{1}}_{2} \approx 0.47$, we find no clear indications in the FM model for an intermediate magnetically disordered phase between the phases exhibiting FM and striped order. Instead, the evidence points strongly to a direct first-order transition between the two ordered phases of the FM model.

High-order Ho multipoles in HoB2C2 observed with soft resonant x-ray diffraction

Soft resonant x-ray Bragg diffraction (SRXD) at the Ho M4,5 edges has been used to study Ho 4f multipoles in the combined magnetic and orbitally ordered phase of HoB2C2. A full description of the energy dependence for both σ and π incident x-rays at two different azimuthal angles, as well as the ratio Iσ/Iπ as a function of azimuthal angle for a selection of energies, allows a determination of the higher order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The Ho 4f multipole moments have been estimated, indicating a dominant hexadecapole (rank 4) order with an almost negligible influence from either the dipole or the octupole magnetic terms. The analysis incorporates both the intra-atomic magnetic and quadrupolar interactions between the 3d core and 4f valence shells as well as the interference of contributions to the scattering that behave differently under time reversal. Comparison of SRXD, neutron diffraction and non-resonant x-ray diffraction shows that the magnetic and quadrupolar order parameters are distinct. The component of the magnetic order exhibits a Brillouin type increase below the orbital ordering temperature TQ, while the quadrupolar order increases more sharply. We conclude that the quadrupolar interaction is strong, but quadrupolar order only occurs when the magnetic order gives rise to a quasi-doublet ground state, which results in a lock-in of the orbitals at TQ.

Frustrated multilayer ferromagnet-antiferromagnet structures: Beyond the scope of the exchange approximation (a review)

The frustrations of exchange interaction between ferromagnetic and antiferromagnetic layers, which arise at the uncompensated interface between the layers due to the interface roughness, have been described. The distribution of magnetic order parameters in the vicinity of the interface between the layers has been investigated, and the “layer thickness-roughness” magnetic phase diagram has been obtained in the case of the two-layer ferromagnet-antiferromagnet system and the ferromagnet-antiferromagnet-ferromagnet spin-valve system. An analysis has been performed taking into account the single-ion anisotropy energy, i.e., beyond the scope of the exchange approximation. It has been demonstrated that the number of easy axes in the layer plane, in many respects, determines the existence of an exchange shift of the hysteresis loop of the ferromagnet due to its interaction with the antiferromagnetic substrate.

Variations in elastic and anelastic properties of Co3O4 due to magnetic and spin-state transitions

Elastic anomalies and acoustic dissipation associated with spin-state transitions of Co3+ in Co3O4 have been investigated using resonant ultrasound spectroscopy (RUS) at high frequencies (0.1-1.5 MHz) between 6 and 1107 K, and dynamic mechanical analysis (DMA) at low frequencies (0.1-50 Hz) between 127 and 775 K. Above ~800 K, the shear modulus decreases with increasing temperature and the acoustic dissipation increases. The amount of softening scales linearly with an empirical order parameter for low spin → low spin + high spin states derived from analysis of changes in unit-cell volume with temperature. The mechanism for anelastic losses is not understood but must be due to relaxation of strains coupled either to changes in spin state or to changes of cation configuration between tetrahedral and octahedral sites occurring by electron exchange. The pattern of shear modulus softening, in proportion to the spin order parameter, is likely to be general for cubic structures containing cations, which undergo spin-state transitions, such as Fe2+ in (Mg,Fe)O. Below ~30 K, RUS data show a large softening of the shear modulus and a steep increase in acoustic dissipation due to the magnetic transition from paramagnetic to antiferromagnetic. This could be understood in terms of the influence of a symmetry breaking strain coupled to the magnetic order parameter such that cubic lattice geometry is not maintained. Possible magnetic space groups for the antiferromagnetic structure are considered through the use of the group theory.