Analytical treatment of synchronization of spin-torque oscillators by microwave magnetic fields

An analytical approach is presented for the study of synchronization effects in spin-transfer-driven nanomagnets subjected to radio-frequency magnetic fields. The conditions are derived and discussed under which the current-induced magnetization precession is synchronized by the radio-frequency field. Exact analytical results are obtained for the case when the problem exhibits uniaxial symmetry around the axis perpendicular to the device plane. It is demonstrated that the magnetization dynamics under nonzero current and nonzero rf field is identical in structure to that under zero current. On this basis, analytical predictions are obtained for: the existence of phase-locking between current-induced magnetization precession and rf field oscillations; the frequency pulling effect in proximity of phase locking; the occurrence of hysteresis effects in phase-locking as a function of the spin-polarized current. The proposed approach is valid for arbitrary rf field amplitude and current intensity.     

The generating functions formalism for the analysis of spin response to the periodic trains of RF pulses. Echo sequences with arbitrary refocusing angles and resonance offsets

The generating functions (GF) formalism was applied for calculation of spin density matrix evolution under the influence of periodic trains of RF pulses. It was shown that in a general case, closed expression for the generating function can be found that allows in many cases to derive analytical expressions for the generating function of spin density matrix (magnetization, coherences). This approach was shown to be particularly efficient for the analysis of multi-echo sequences, where one has to average over various frequency isochromats. The explicit analytical expressions for the generating function for echo amplitudes in a Carr–Purcell–Meiboom–Gill (CPMG) echo sequence, a multiecho sequence with incremental phase of refocusing pulse, a gradient echo sequence including transient period were obtained for an arbitrary flip angle and an arbitrary resonance offset. Comparison of the theory and the spin-echo experiments was done, demonstrating a good agreement.     

Cross-correlations between low-γ nuclei in solids via a common dipolar bath

Correlation of chemical shifts of low-γ nuclei (such as 15N) is an important method for assignment of resonances in uniformly-labeled biological solids. Under static experimental conditions, an efficient mixing of low-γ nuclear spin magnetization can be achieved by a thermal contact to the common reservoir of dipole-dipole interactions in order to create 15N-15N, 13C-13C, or 15N-13C cross-peaks in a 2D correlation spectrum. A thermodynamic approach can be used to understand the mechanism of magnetization mixing in various 2D correlation pulse sequences. This mechanism is suppressed under magic-angle spinning, when mixing via direct cross-polarization with protons becomes more efficient. Experimental results are presented for single-crystalline and powder samples of 15N-labeled N-acetyl-L-15N-valyl-L-15N-leucine (NAVL). In addition to the thermodynamic analysis of mixing pulse sequences, two different new mixing sequences utilizing adiabatic pulses are also experimentally demonstrated.     

Theoretical study of a magnetoresistive magnetic field transmitter with a ring sensor

Theoretical results are presented concerning the performance of thin-film anisotropic magnetoresistive magnetic field sensors with a single- and multilayered ring configuration. A mathematical model is proposed that allows analytical calculation of magnetization fields, is suitable for sensor optimization, and requires a reasonable amount of computing time. A system of equations is derived for the magnetization angle of rotation. On this basis, the response, sensitivity, and angular characteristics of particular sensors are optimized.     

Analytical Approach for Piecewise Linear Coupled Map Lattices

A simple construction is presented which generalizes piecewise linear one-dimensional Markov maps to an arbitrary number of dimensions. The corresponding coupled map lattice, known as a simplicial mapping in the mathematical literature, allows for an analytical investigation. In particular, the spin Hamiltonian which is generated by the symbolic dynamics is accessible. As an example, a formal relation between a globally coupled system and an Ising mean-field model is established. The phase transition in the limit of infinite system size is analyzed and analytical results are compared with numerical simulations.     

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.     

An E.P.R. study of the anion radical of diplieadane

A study of radiation damping e ects in inhomogeneously broadened systems leads to an analytical theorem that relates the time integral area of the nuclear magnetic resonance signal to the tipping angle of the magnetization caused by the circuit generated reaction field. The theorem is applied to predict relative signal areas in examples of pulse sequences. In a few selected cases these predictions are also checked by experiment.     

Nonlinear magnetization dynamics under circularly polarized field

Exact analytical results are presented for the nonlinear large motion of the magnetization vector in a body with uniaxial symmetry subject to a circularly polarized field. The absence of chaos, the existence of pure time-harmonic magnetization modes with no generation of higher-order harmonics, and the existence of quasiperiodic magnetization modes with spontaneous breaking of the rotational symmetry are proven. Application to ferromagnetic resonance and connection with the Stoner-Wohlfarth model are discussed.     

Exact and quasiclassical Green's functions of two-dimensional electron gas with Rashba–Dresselhaus spin–orbit interaction in parallel magnetic field

New exact and asymptotical results for the one particle Green's function of 2D electrons with combined Rashba–Dresselhaus spin–orbit interaction in the presence of in-plane uniform magnetic field are presented. A special case that allows an exact analytical solution is also highlighted. To demonstrate the advantages of our approach we apply the obtained Green's function to calculation of electron density and magnetization.     

Non-Gaussian scattering by a random phase screen

A theoretical investigation of non-Gaussian scattering by a smoothly varying deep random phase screen is presented. New analytical results, valid for arbitrary illuminated area, are derived for the contrast of the intensity pattern in the Fraunhofer region and the effect of two scale sizes in the screen is calculated.     

Quadrupole relaxation enhancement--application to molecular crystals

A general theory of field dependent spin-lattice relaxation for nuclei of the spin quantum number 1/2 (1H, 19F, 13C) caused by dipole-dipole interactions with neighboring quadrupolar nuclei (nuclei possessing a quadrupolar moment) is presented. The theory is valid for arbitrary motional conditions and should be treated as a quadrupolar counterpart of the paramagnetic relaxation enhancement theory. When the energy level splitting of the dipolar spin (I=1/2) matches one of the transition frequencies of the quadrupolar nuclei one can observe a local enhancement of the dipolar spin relaxation (referred to as "quadrupolar peaks"). To see such effects the dynamics modulating the spin interactions has to be relatively slow. This brings the system beyond the validity range of perturbation approaches and requires the stochastic Liouville equation to be applied. The presented theory describes the quadrupolar relaxation enhancement (QRE) for an arbitrary spin quantum number of the quadrupolar nuclei and includes the asymmetry of the quadrupolar coupling. It has been applied to interpret the shape of magnetization curves (amplitude of 1H magnetization versus magnetic field) for the molecular crystal [C3N2H5]6[Bi4Br18] ([C3N2H5]-imidazolium). The magnetization curves show several dips (local minima) attributed to 1H-14N quadrupolar relaxation enhancement effects. In addition, as a limiting case a perturbation approach to QRE has been presented and its validity conditions have been discussed.     

Modulated phases and the mean-field theory of magnetism with competing interactions

The mean-field theory of an Ising magnet with infinitely weak, infinitely long-range potentials of arbitrary sign is presented in terms of a variational principle for the magnetization. Previous studies of the theory have revealed paramagnetic, ferromagnetic, and modulated phases. For a particular choice of potential, which is an obvious continuous version of the between-plane ANNNI model interaction, exact solutions of the stationary condition implied by the variational principle are obtained. This leads us to formulate a trial magnetization to well describe the modulated phase in general. To illustrate the utility of the trial magnetization, both analytic and numerical calculations are performed, which determine the wavenumber in certain portions of the modulated phase for the above-mentioned potential.     

Energy eigenvalues from an analytical transfer matrix method

A detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems. The derivation is strict and complete. The energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method. The anharmonic oscillator potential and the rational potential are two important examples. Checked by numerical techniques, the results for the two potentials by the present method are proven to be exact and reliable.     

RF Gradient BIRD/TANGO Sequence to Eliminate Uncoupled Magnetization

A modification of the BIRD and TANGO sequences is presented which employs radiofrequency field gradients to eliminate the net magnetization from uncoupled spins, while completely preserving coupled magnetization. The standard BIRD and TANGO sequences cause selective nutation of protons directly bound to a coupling partner, while returning uncoupled magnetization to +z. These sequences lend themselves naturally to modification using RF gradients, which require no increase in pulse-sequence complexity while providing substantial suppression of uncoupled resonances and elimination of typical antiphase and multiple-quantum error terms that arise from improperly set pulse lengths or delays. In the RF-gradient BIRD/TANGO sequence, the uncoupled magnetization is dephased in a plane orthogonal to the RF axis, while the desired signal components are refocused, effectively in a rotary echo. The sequence has applications to solvent suppression and selective isotopomer excitation. It is demonstrated for selective excitation of the satellites in a sample of chloroform, yielding suppression of the uncoupled magnetization by a factor of approximately 800.     

Random-walk technique for simulating NMR measurements and 2D NMR maps of porous media with relaxing and permeable boundaries

We revisit random-walk methods to simulate the NMR response of fluids in porous media. Simulations reproduce the effects of diffusion within external inhomogeneous background magnetic fields, imperfect and finite-duration B(1) pulses, T(1)/T(2) contrasts, and relaxing or permeable boundaries. The simulation approach consolidates existing NMR numerical methods used in biology and engineering into a single formulation that expands on the magnetic-dipole equivalent of spin packets. When fluids exhibit low T(1)/T(2) contrasts and when CPMG pulse sequences are used to acquire NMR measurements, we verify that classical NMR numerical models that neglect T(1) effects accurately reproduce surface magnetization decays of saturated granular porous media regardless of the diffusion/relaxation regime. Currently, analytical expressions exist only for the case of arbitrary pore shapes within the fast-diffusion limit. However, when fluids include several components or when magnetic fields are strongly inhomogeneous, we show that simulations results obtained using the complete set of Bloch's equations differ substantially from those of classical NMR models. In addition, our random-walk formulation accurately reproduces magnetization echoes stemming from coherent-pathway calculations. We show that the random-walk approach is especially suited to generate parametric multi-dimensional T(1)/T(2)/D NMR maps to improve the characterization of pore structures and saturating fluids.     

A general calculation and sensitivity analysis of reflector scanning systems using skew-ray tracing

A generic analytical method based on a skew-ray tracing approach is proposed for controlling the parameters of a reflector scanning system (RSS) comprising an arbitrary number of reflectors in such a way that the output light ray correctly traces a user-defined curve in 3-D space. In addition, a technique is presented for evaluating the sensitivity of the RSS output ray and scanning speed to changes in the RSS parameters. The feasibility of the proposed methodology is demonstrated using an RSS comprising a single reflector for illustration purposes. The results confirm that the proposed approach provides an effective means of analyzing and designing RSS systems comprising an arbitrary number of reflectors.     

Ground-state energy density of a one-dimensional fermion system with repulsive δ-function interaction: A pseudo-linear response treatment

An analytical treatment for the ground-state energy density of the system of one-dimensional fermions interacting via repulsive δ-function potential is presented. The treatment rests on a direct approximation to the pair distribution function and coupling constant integration. The analytical results, obtained for arbitrary coupling strength values, are in good agreement with those given by the numerical solution of an exact integral equation.     

Nonequilibrium spin transport through a diluted magnetic semiconductor quantum dot system with noncollinear magnetization

The spin-dependent transport through a diluted magnetic semiconductor quantum dot (QD) which is coupled via magnetic tunnel junctions to two ferromagnetic leads is studied theoretically. A noncollinear system is considered, where the QD is magnetized at an arbitrary angle with respect to the leads’ magnetization. The tunneling current is calculated in the coherent regime via the Keldysh nonequilibrium Green’s function (NEGF) formalism, incorporating the electron–electron interaction in the QD. We provide the first analytical solution for the Green’s function of the noncollinear DMS quantum dot system, solved via the equation of motion method under Hartree–Fock approximation. The transport characteristics (charge and spin currents, and tunnel magnetoresistance (TMR)) are evaluated for different voltage regimes. The interplay between spin-dependent tunneling and single-charge effects results in three distinct voltage regimes in the spin and charge current characteristics. The voltage range in which the QD is singly occupied corresponds to the maximum spin current and greatest sensitivity of the spin current to the QD magnetization orientation. The QD device also shows transport features suitable for sensor applications, i.e., a large charge current coupled with a high TMR ratio.     

Forced vibrations of oscillators with a purely nonlinear power-form restoring force

Harmonically excited oscillators with non-negative real-power geometric nonlinearities and no linear term in the restoring force are considered. Perturbation approaches are developed for the cases of weak and strong nonlinearity. Frequency-amplitude equations are derived for an arbitrary value of the non-negative real power of the restoring force as well as analytical expressions for the steady-state response at the frequency of excitation. It is shown that the system response is of a softening type for the powers lower than unity and of a hardening type for the powers higher than unity. Frequency-response curves of the antisymmetric (constant force) oscillator, the restoring force of which has a zero power, are also discussed. Comparisons with numerical results are presented for confirmation of the analytical results obtained.     

Eine mikromagnetische Ableitung der „Grundregeln“ für magnetische Bereichsstrukturen in massiven Kristallen

A result of experimental investigations of magnetic domain pattern was, simply said, that the domains are always aranged in such a way, that magnetic stray-fields are avoided everywhere, and only easy directions are occupied by the magnetization. This is true for domains in the bulk of the specimen as well as for the surface. This result — which may be described by two basic rules for magnetic domains — will now be derived theoretically. The idea is to examine the stability of an arbitrary volume with respect to a certain variation of the magnetization; the variation leads two what may be called an incomplete Bloch-wall. It results from the calculation, that in a ferromagnetic crystal every volume element, which is homogenously magnetized and large as compared to the thickness of a Bloch-wall, can be looked at as more or less independently variable. The basic rules follow from this argument. The range of validity, the significance and possible applications of our rules are discussed.