Nonlinear Magnetic Susceptibility Research Articles

Nonlinear Transverse Magnetic Susceptibility under Electric Toroidal Dipole Ordering

An electric toroidal dipole (ETD) moment is one of the fundamental dipole moments as well as electric and magnetic ones. Although it directly couples to neither an electric nor magnetic field due to its spatial inversion and time-reversal parities, its ordered state leads to unconventional transverse responses of the conjugate physical quantities. We here theoretically investigate nonlinear transverse magnetic susceptibility under the ETD ordering. By performing a self-consistent mean-field calculation for a five $d$-orbital model under a tetragonal crystalline electric field and using the nonlinear Kubo formula, we show that a third-order transverse magnetic susceptibility corresponding to a uniform magnetization perpendicular to the external magnetic field becomes nonzero once the ETD moment is ordered under tetragonal crystalline electric field. Moreover, we find that spin-orbital entanglement and a low-lying first excited crystal-field level are important for realizing large transverse responses.

Two-dimensional coherent spectrum of interacting spinons from matrix product states

We compute numerically the second- and third-order nonlinear magnetic susceptibilities of an Ising ladder model in the context of two-dimensional coherent spectroscopy by using the infinite time-evolving block decimation method. The Ising ladder model couples a quantum Ising chain to a bath of polarized spins, thereby effecting the decay of spinon excitations. We show that its third-order susceptibility contains a robust spinon echo signal in the weak-coupling regime, which appears in the two-dimensional coherent spectrum as a highly anisotropic peak on the frequency plane. The spinon echo peak reveals the dynamical properties of the spinons. In particular, the spectral peak corresponding to the high-energy spinons, which couple to the bath, is suppressed with increasing coupling, whereas those corresponding to low-energy spinons do not show any significant changes.

Nonlinearity and harmonics of the magnetic susceptibility of a Co-=SUP=-2+-=/SUP=- single-ion magnet in the paramagnetic region above the magnetic ordering temperature

In single-ion complexes based on Co2+ ions, the second and third harmonics of the magnetic susceptibility were found at temperatures of 2-4 K, which exceed the Neel temperature. The maxima of the second and third harmonics of the magnetic susceptibility are observed at a frequency of ~1 Hz, at which the maximum of the first harmonic is observed in a field of 3.2 kOe. An analysis of the dependences of the second and third harmonics of the magnetic susceptibility on the field and temperature showed that the nonlinearity arises as a result of the formation of a spin glass state at temperatures slightly higher than the Neel temperature. In this state, there is no long-range spin order, but there are spin clusters in the spin glass state. The spin-glass state in a compound with a Co2+ ion with a high magnetic anisotropy is unusual in that the exchange interaction is much smaller than the single-ion anisotropy energy. Keywords: molecular magnets, nonlinear magnetic susceptibility, nanostructures, spin dynamics, spin glass.

Effect of Er2O3 and ErF3 on the optical properties of sodium oxyfluoroborate glasses

Glasses based on B2O3–Na2O–NaF doped with Er2O3 and ErF3 were prepared by the melt quench technique. Density was measured and molar volume was calculated. The density and molar volume increased with the increase of Er2O3 and ErF3 content. Optical absorption spectra were measured to study the optical properties and their dependence on the Er2O3 and ErF3. The refractive index is estimated and correlated with the wavelength, Er2O3, ErF3, density, and O/F ratio. Nonlinear refractive index, optical and magnetic susceptibilities, Abbe number, molar refractivity, electronic polarizability, and energy gap were also calculated and correlated with Er2O3 and ErF3 contents.

Universal excitonic superexchange in spin-orbit-coupled Mott insulators

We point out the universal presence of the excitonic superexchange in spin-orbit-coupled Mott insulators. It is observed that the restriction to the lowest spin-orbit-entangled “J” states may sometimes be insufficient to characterize the microscopic physics, and the virtual excitonic processes via the upper “J” states provide an important correction to the superexchange. We illustrate this excitonic superexchange from a two-dimensional 5d iridate Sr2IrO4 and explain its physical consequences such as the orbital-like coupling to the external magnetic flux and the nonlinear magnetic susceptibility. The universal presence of the excitonic superexchange in other spin-orbit-coupled Mott insulators such as 3d Co-based Kitaev magnets and even f electron rare-earth magnets is further discussed.

Temperature assisted size dependent synthesis and magnetic properties of rare-earth chromium oxide nanoparticles

We report the synthesis of homogeneous single-phase nanoparticles of RCrO4 and RCrO3 (R = Sm, Gd, Dy, and Er) compounds using a modified sol–gel hydrothermal method followed by heat treatment. Annealing of the as-prepared powder sample at ambient pressure enables chromium to remain in the metastable Cr5+ (4s03d1) state by crystallizing into a zircon-type tetragonal RCrO4 (S.G. I41/amd, D4h19 symmetry) structure at 773 K, while it chooses to form orthorhombic RCrO3 (S.G. Pbnm, D4h19 symmetry) structure with a stable Cr3+ (4s03d3) state at 973 K. The analysis of DC magnetization (M) versus temperature (T) in an external magnetic field, H = 100 Oe for polycrystalline RCrO4 indicates the magnetic transition temperatures, TC ∼ 20 K. The derivative of zero-field cooled magnetic susceptibility with respect to T, dχZFCdT, versus T, indicates the presence of competing magnetic interactions due to the ferromagnetic (FM) and antiferromagnetic (AFM) ordering of Cr5+ and R3+ in RCrO4 respectively near magnetic transition. This feature specifies magnetic frustration that leads to metamagnetism in RCrO4. The fit of non-linear magnetic susceptibility, χ=MH, to χ=χ0+CT-θ, with a non-zero χ0 determined from high-T data gives effective magnetic moment, μeff, which is in accordance with the theoretical value. The analysis of M vs. T indicates that RCrO3 undergoes a paramagnetic (PM) to canted-antiferromagnetic (CAFM) transition of Cr sublattices at Néel temperature (TNCr) because of antisymmetric Dzyaloshinskii-Moriya (DM) interaction due to Cr–O–Cr superexchange with a possible antiferromagnetic (AFM) ordering of the rare-earth at low temperature (TNR < 20 K). The linear fit of χ-1 vs. T with a zero χ0 shows Curie-Weiss law is more appropriate at T>TC resulting expected μeff values for RCrO3.

Superconducting and chiral-glass to insulator transition in phase-glass models in a magnetic field

We investigate the quantum phase transitions of a phase-glass model in a magnetic field with frustration parameter f = 1/2, describing the effects random negative Josephson-junction couplings in two-dimensional superconductors. The critical behavior is obtained by a scaling analysis of path-integral Monte Carlo simulations at zero temperature, including corrections to finite-size scaling. A single superconducting and chiral-glass to insulator transition occurs above a disorder threshold with divergent nonlinear magnetic susceptibility, unaffected by the additional magnetic-field frustration. The relevance of this transition for nanohole superconducting thin films doped with magnetic impurities is discussed.

Engineering of the XY Magnetic Layered System with Adeninium Cations: Monocrystalline Angle-Resolved Studies of Nonlinear Magnetic Susceptibility.

An original example of modular crystal engineering involving molecular magnetic {CuII[WV(CN)8]}− bilayers and adeninium cations (AdeH+) toward the new layered molecular magnet (AdeH){CuII[WV(CN)8]}·2H2O (1) is presented. 1 crystallizes within the monoclinic C2 space group (a = 41.3174(12), b = 7.0727(3), c = 7.3180(2) Å, β = 93.119(3)°, and V = 2135 Å3). The bilayer topology is based on a stereochemical matching between the square pyramidal shape of CuII moiety and the bicapped trigonal prismatic shape of [WV(μ-CN)5(CN)3], and the separation between bilayers is significantly increased (by ∼50%; from ca. 9.5 to ca. 14.5 Å) compared to several former analogues in this family. This was achieved via a unique combination of (i) a 1D ribbonlike hydrogen bond system {AdeH+···H2O···AdeH+···}∞ exploiting planar water-assisted Hoogsteen···Sugar synthons with (ii) parallel 1D π–π stacks {AdeH+···AdeH+}∞. In-plane 2D XY magnetism is characterized by a Tc close to 33 K, Hc,in-plane = 60 Oe, and Hc,out-of-plane = 750 Oe, high values of in-plane γ critical exponents (γb = 2.34(6) for H||b and γc = 2.16(5) for H||c), and a Berezinskii–Kosterlitz–Thouless (BKT) topological phase transition, deduced from crystal-orientation-dependent scaling analysis. The obtained values of in-plane ν critical exponents, νb = 0.48(5) for H||b and νc = 0.49(3) for H||c, confirm the BKT transition (νBKT = 0.5). Full-range angle-resolved monocrystalline magnetic measurements supported by dedicated calculations indicated the occurrence of nonlinear susceptibility performance within the easy plane in a magnetically ordered state. We refer the occurrence of this phenomenon to spontaneous resolution in the C2 space group, a tandem not observed in studies on previous analogues and rarely reported in the field of molecular materials. The above magneto-supramolecular strategy may provide a novel means for the design of 2D molecular magnetic networks and help to uncover the inherent phenomena.

Spin-glass dynamics in the presence of a magnetic field: exploration of microscopic properties

The synergy between experiment, theory, and simulations enables a microscopic analysis of spin-glass dynamics in a magnetic field in the vicinity of and below the spin-glass transition temperature T g. The spin-glass correlation length, ξ(t, t w; T), is analysed both in experiments and in simulations in terms of the waiting time t w after the spin glass has been cooled down to a stabilised measuring temperature T < T g and of the time t after the magnetic field is changed. This correlation length is extracted experimentally for a CuMn 6 at. % single crystal, as well as for simulations on the Janus II special-purpose supercomputer, the latter with time and length scales comparable to experiment. The non-linear magnetic susceptibility is reported from experiment and simulations, using ξ(t, t w; T) as the scaling variable. Previous experiments are reanalysed, and disagreements about the nature of the Zeeman energy are resolved. The growth of the spin-glass magnetisation in zero-field magnetisation experiments, M ZFC(t, t w; T), is measured from simulations, verifying the scaling relationships in the dynamical or non-equilibrium regime. Our preliminary search for the de Almeida–Thouless line in D = 3 is discussed.

Geometrical surface pinning in the nonlinear AC susceptibility of HTS tapes

The analytic expression for the nonlinear magnetic susceptibility of a thin disk derived from Bean’s model with a uniform J C for predicts a peak-peak ratio of 7.1 (5.4) for the real (imaginary) components of the third harmonic χ 3 susceptibility. Our measurements show a peak-peak ratio closer to 1.1 in the real component and a noise limited lower estimate of ≈ 7 in the imaginary component. The anomalous third harmonic can be explained by the presence of a geometrical surface barrier for flux entry and exit which we have included as a small region of enhanced current density, J C,G, close to the surface of the disk. The geometrical surface current density is predicted to be approximately equal to J C,G ≈ 2B C1/μ 0 d. In 2G HTS tapes with d ≈ 1μm and B C1 ≈ 5mT this is J C,G ≈ 8GAm−2. We have measured both J C,B and J C,G using the nonlinear susceptibility of a Superpower tape without artificial pinning in fields orthogonal to the tape up to 35T. We find a geometrical surface current of J C,G ≈ 10GAm−2 and an average critical current density J C which is in good agreement with transport measurements performed on material from the same reel of tape.

Stochastic magnetic resonance of absorption effects of surface morphology in thin film magnetics

The effect of surface roughness in thin film magnetics on magnetic susceptibility and Dynamic Magnetic Hysteresis (DMH) loops are investigated using Brown’s continuous diffusion of magnetic spins model. The magnetic relaxation time of the magnetic thin films is determined by regressing the two-mode approximation for the magnetic susceptibility onto experimental data obtained for Ni45Fe55. There is a moderate linear correlation between the magnetic relaxation time and the surface roughness to the thickness (SRT) ratio. The area of the DMH loops and the non-linear dynamic magnetic susceptibility strongly depend on the magnetic relaxation time and the SRT of the films. The thin films exhibit a strong linear relationship between the coercivity and the SRT ratio, along with a strong logarithmic dependence of the dynamic magnetic susceptibility on the SRT ratio.

A millimeter-wave Bell Test using a ferrite parametric amplifier and a homodyne interferometer

A combined ferrite parametric amplifier and millimeter-wave homodyne interferometer are proposed as an ambient temperature Bell Test. It is shown that the non-linear magnetic susceptibility of the yttrium iron garnet (YIG) ferrite, on account of its narrow line-width Larmor precessional resonance, make it an ideal material for the creation of entangled photons. These can be measured using a homodyne interferometer, as the much larger number of thermally generated photons associated with ambient temperature emission can be screened out. The proposed architecture may enable YIG quantum technology-based sensors to be developed, mimicking in the millimeter-wave band the large number of quantum optical experiments in the near-infrared and visible regions which had been made possible by use of the nonlinear beta barium borate ferroelectric, an analogue of YIG. It is illustrated here how the YIG parametric amplifier can reproduce quantum optical Type I and Type II wave interactions, which can be used to create entangled photons in the millimeter-wave band. It is estimated that when half a cubic centimeter of YIG crystal is placed in a magnetic field of a few Tesla and pumped with 5 Watts of millimeter-wave radiation, approximately 0.5 × 1012 entangled millimeter-wave photon pairs per second are generated by the spin-wave interaction. This means an integration time of only a few tens of seconds is needed for a successful Bell Test. A successful demonstration of this will lead to novel architectures of entanglement-based quantum technology room temperature sensors, re-envisioning YIG as a modern quantum material.

Magnetic susceptibility and phase transitions in LiNiPO4

Detailed studies of specific heat, magnetization, and magnetic torque of a single crystal of ${\mathrm{LiNiPO}}_{4}$ olivine are presented. Olivines attract attention as promising for application as cathodes in Li-ion batteries and exhibiting a unique set of properties. ${\mathrm{LiNiPO}}_{4}$ is the unique olivine, in which antiferromagnetic order develops in two steps, i.e., at 21.8 K, the second-order transition to an incommensurate phase and then, at 20.9 K, the first-order transition to a commensurate phase appears. Specific heat studies, supplemented by the ``slope analysis procedure,'' revealed a splitting of the specific heat anomaly accompanying the first-order transition, which suggests that actually, these are two coupled transitions, one of which can be the ferroelectric one. The specific heat was measured as a function of temperature for a series of fixed external magnetic field $B$ values. As the result, analytical equations describing the phase transition lines in the $T\text{\ensuremath{-}}B$ plane were determined and evolution of a shape of the specific heat anomalies accompanying the phase transitions, observed in the powder sample under influence of $B$, was modeled. Angular dependence of magnetic torque and of magnetization for $B$ rotating within the $a\text{\ensuremath{-}}c$ and $b\text{\ensuremath{-}}c$ crystalline planes was measured for several fixed temperature and $B$ values. Based on these results, we found a new effect, that we called ``off-diagonal nonlinear magnetic susceptibility,'' i.e., we found that for each main crystallographic axis $(a,b$, and $c)$, an additional component of magnetic susceptibility, proportional to the square of the perpendicular to this axis component of $B$, exists. A phenomenological model of this effect, describing the experimental results correctly, was proposed.

Evidence of long-range ferromagnetic order and spin frustration effects in the double perovskite La2CoMnO6

We present a comprehensive study on the magnetic structure, dynamics, and phase evolution in the single-phase double perovskite $La_2CoMnO_6$. The mixed valence state due to oxygen deficiency is verified by X-ray photoelectron spectroscopy, and confirms a double ferromagnetic transition observed in DC magnetization. Neutron diffraction reveals that the magnetic structure is dominated by long-range ferromagnetic ordering, which is further corroborated by a critical exponents analysis of the paramagnetic to ferromagnetic phase transition. An analysis of the magnetization dynamics by means of linear and nonlinear ac magnetic susceptibilities marks the presence of two distinct cluster glass-like states that emerge at low temperatures. The isothermal entropy change as a function of temperature and magnetic field (H) is exploited to investigate the mechanism of stabilization of the magnetic phases across the H-T phase diagram. In the regime of the phase diagram where thermal energy is sufficiently low, regions of competing interactions due to local disorder become stabilized and display glass-like dynamics. The freezing mechanism of clusters is illustrated using a unique probe of transverse susceptibility that isolates the effects of the local anisotropy of the spin clusters. The results are summarized in a new H-T phase diagram of $La_2CoMnO_6$ revealed for the first time from these data.

Three-Dimensional Time Domain Simulation of the Quantum Magnetic Susceptibility

A way of using the Finite Difference Time Domain method is described to simulate the magnetic susceptibility of a quantum toroid. This simulation is based on the direct implementation of the time-dependent Schrodinger equation in three dimensions. First, the ground state eigenenergy and eigenstate are found. Next, the expectation value of the quantum magnetic dipole operator is calculated as a function of the applied magnetic field strength with a static magnetic field, and the results are compared with classical results. Then the magnetic dipole moment is calculated with a time-oscillating magnetic field applied. These expectation values are used to calculate the linear and nonlinear magnetic susceptibility of a torus, both without a grating and with a grating to increase irregularities in the shape, by repeating the calculations at various frequencies. The results are consistent with the expected results. This method can be used to calculate the quantum magnetic susceptibility of any structure in order to search for structures with better nonlinear properties.

Measurements of the NMR Knight shift tensor and nonlinear magnetization in URu2Si2

URu$_2$Si$_2$ exhibits an anomalous peak in the nonlinear magnetic susceptibility at the hidden order transition. In order to investigate this anomaly, we conducted direct magnetization measurements and investigated the detailed angular dependence of the $^{29}$Si nuclear magnetic resonance Knight shift tensor. We find that the nonlinear magnetization is smaller than previously reported, and the analogous nonlinear Knight shift tensor is below the detection limit. Our results suggest that the magnitude of the anomalous peak is sample dependent.

Demagnetization harmonic effects on the magnetization of granular systems on a macroscopic scale: the superconducting case

A model has been developed to determine the effective ac magnetic response of magnetic systems, taking into account the demagnetization effects arising from the sample geometry which determine the out-of-phase components of the applied fundamental frequency and higher harmonic components. Indeed, demagnetization fields and their intermodulation can significantly affect the ac magnetic response. This approach provides a system of self-consistent linear equations relating the magnetic response to the external magnetic field by means of nonlinear magnetic susceptibility. The model is extended to the magnetic response of granular systems in terms of the contributions of the individual grains and of the whole sample in the presence of demagnetization effects of the whole sample and of the grains on a macroscopic scale. In particular, our model is applied to a granular superconducting system. The comparison between the performed numerical simulations and the experimental data shows that the demagnetization fields of the single grains and of the whole sample, and their intermodulation, are relevant if magnetic measurements are used to extract detailed information about the analyzed material.

Isotropic-Heisenberg to isotropic-dipolar crossover and finite-size scaling in Cr75−xFe25+x (x = 0, 5) thin films

‘Zero-field’ linear ac magnetic susceptibility, χ1(T), of the Cr75−xFe25+x (x=0, 5) thin films with thickness, t, ranging from 980 to 10nm has been measured at temperatures close to Tc, the temperature at which the ferromagnetic-paramagnetic phase transition occurs. An elaborate analysis of χ1(T⩾Tc) for the films with t⩾40nm yields the temperature dependence of the effective critical exponent for susceptibility, γeff(T), that is characteristic of the three-dimensional (3D) isotropic Heisenberg-to-3D isotropic dipolar crossover. In the asymptotic critical region (ACR), these thin-film samples behave as a 3D isotropic dipolar (ID) ferromagnet. As the film thickness reduces from t≃ 980nm to 40nm, ACR narrows down while the temperature, Tdip, at which a dip in γeff(T) occurs and the temperature, TIH∗, that marks the onset of the 3D isotropic Heisenberg (IH) behavior, shift to lower temperatures. For a given t, the width of ACR as well as the characteristic temperatures Tdip and TIH∗ increase with decreasing (increasing) Fe (Cr) concentration. Consistent with these observations, the ratios involving nonlinear ac magnetic susceptibilities obey the generalized magnetic equation of state with 3D ID critical exponents and the value of Tc same as that determined from χ1(T). A quantitative comparison between theory and experiment highlights certain limitations of the existing theories. The films with t≲20nm do not exhibit 3D IH-to-3D ID crossover. Instead, the critical behavior of Cr70Fe30 thin films with t=21nm and t=11nm is that of a 3D IH and 3D Ising ferromagnet, respectively. By contrast, a 3D Ising (spin glass) critical behavior is observed in the Cr75Fe25 thin film with t=19nm (t=12nm). Curie temperature, Tc, decreases with film thickness in accordance with the finite-size scaling.

Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields

The interplay between quantum fluctuations and disorder is investigated in a spin-glass model, in the presence of a uniform transverse field $\Gamma$, and a longitudinal random field following a Gaussian distribution with width $\Delta$. The model is studied through the replica formalism. This study is motivated by experimental investigations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility $\chi_3$, which have led to a considerable experimental and theoretical debate. We analyzed two situations, namely, $\Delta$ and $\Gamma$ considered as independent, as well as these two quantities related as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature $T_f$; moreover, $T_f$ decreases by increasing $\Gamma$ towards a quantum critical point at zero temperature. The situation where $\Delta$ and $\Gamma$ are related appears to reproduce better the experimental observations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility. In this later case, by increasing $\Gamma$, $\chi_3$ becomes progressively rounded, presenting a maximum at a temperature $T^*$ ($T^*>T_f$). Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature $T^*$, one for $T_f<T<T^*$, and another one for $T>T^*$. It is argued that the conventional paramagnetic state corresponds to $T>T^*$, whereas the temperature region $T_f<T<T^*$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.

Effect of Magnetic Fluctuations on Spin Current

Spin Hall effect (SHE) and its inverse enable the interconversion between charge current and spin current. It is widely recognized that the SHE occurs in a nonmagnetic material with strong spin–orbit interaction. However, it can be generated even in magnetic materials such as ferromagnets, antiferromagnets, and those mixtures, i.e., spin glass. Here we review the SHEs in two typical magnetic systems. One is the SHE in a weak ferromagnetic metal in the vicinity of the Curie temperature where a nonlinear magnetic susceptibility can be detected. The other is the SHE in a spin glass metal where fluctuations at the magnetic impurity sites can be measured electrically in a very sensitive way. We argue that the spin current could be utilized as a sensor to detect a small magnetic fluctuation.