Zeeman Interaction Terms Research Articles

Magnetoelectric polarizability and optical activity: Spin and frequency dependence

We extend a microscopic theory of polarization and magnetization to include the spin degree of freedom of the electrons, introducing a general spin-orbit coupling and Zeeman interaction term in the Hamiltonian. At finite frequencies and including spin, the magnetoelectric polarizability tensor is replaced by two separate tensors, one that relates the polarization P to the magnetic field B and a separate tensor that relates the magnetization M to the electric field E. When combined with other relevant response tensors a third-rank tensor that relates the induced current density to gradients in the electric field can be introduced; it is gauge invariant, in a form suitable for numerical calculations, and describes optical activity---including spin effects---even in materials that may lack time-reversal symmetry.

Spin blockade in hole quantum dots: Tuning exchange electrically and probing Zeeman interactions

Spin-orbit coupling is key to all-electrical control of quantum-dot spin qubits, and is frequently stronger for holes than for electrons. Here we investigate Pauli spin blockade for two heavy holes in a gated double quantum dot in an in-plane magnetic field. The interplay of the complex Zeeman and spin-orbit couplings causes a blockade leakage current anisotropic in the field direction. The period of the anisotropic leakage is critically dependent on the relative magnitude of Zeeman interaction terms linear and cubic in the magnetic field. The current and singlet-triplet exchange splitting can be effectively adjusted by an appropriate choice of field direction, providing a simple control variable for quantum information processing and a way of tailoring magnetic interactions in hole spin qubits.

Competition between external and synthetic magnetic fields on a spin-1 ultracold Bose gas

We study a spin-1 ultracold Bose gas in the presence of both external and synthetic magnetic fields via mean-field approach corresponding to both signs of the spin-dependent interaction potential. The synthetic field is included through the hopping frequencies via Peierls coupling and the external field through the Zeeman interaction term to explore the influence of one over the other. The antiferromagnetic case shows that at low values of external field strengths, the spin singlet state is stable and the system is more likely in Mott insulating (MI) phase as the synthetic field is enhanced. But at large external fields, due to a competition between the hopping and the Zeeman interaction terms, the former tends to stabilize the MI phase compared to the superfluid (SF) phase, while the latter tries to destabilize the MI phase by suppressing the formation of singlet pairs. In the ferromagnetic case, insensitivity of the phase properties to the external magnetic field leads to stabilization of the MI phase with increasing synthetic magnetic field. Further the magnetization shows that there are distinct signatures of transverse polar and longitudinal ferro SF phases for both antiferromagnetic and ferromagnetic interactions, respectively.

Comments on critical electric and magnetic fields from holography

Abstract We discuss some aspects of critical electric and magnetic fields in a field theory with holographic dual description. We extend the analysis of [1], which finds a critical electric field at which the Schwinger pair production barrier drops to zero, to the case of magnetic fields. We first find that, unlike ordinary weakly coupled theories, the magnetic field is not subject to any perturbative instability originating from the presence of a tachyonic ground state in the W-boson spectrum. This follows from the large value of the ’t Hooft coupling λ, which prevents the Zeeman interaction term to overcome the particle mass at high B. Consequently, we study the next possible B-field instability, i.e. monopole pair production, which is the S-dual version of the Schwinger effect. Also in this case a critical magnetic field is expected when the tunneling barrier drops to zero. These Schwinger-type criticalities are the holographic duals, in the bulk, to the fields E or B reaching the tension of F1 or D1 strings respectively. We then discuss how this effect is modified when electric and magnetic fields are present simultaneously and dyonic states in the spectrum can be pair produced by a generic E − B background. Finally, we analyze finite temperature effects on Schwinger criticalities, i.e. in the AdS-Schwarzshild black hole background.

Theoretical Explanation of the Optical and Electron Paramagnetic Resonance Spectral Data for Trivalent Ytterbium Ions in β-Lead Fluoride

ABSTRACT Six crystal-field energy levels and three electron paramagnetic resonance (EPR) data [g factor and hyperfine structure constants A(171Yb3+) and A(173Yb3+)] for trivalent ytterbium (Yb3+) ions in the cubic β-lead fluoride (β-PbF2) crystal are calculated by using a diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are added to the Hamiltonian in the conventional diagonalization method, and so the optical and EPR data can be calculated in a unified way. The calculated results are in reasonable agreement with the experimental values, and the signs of constants A(171Yb3+) and A(173Yb3+) are determined. The results are discussed.

Theoretical studies of the spin-Hamiltonian parameters and defect structure for the tetragonal Gd3+ center in cubic c-RbZnF3 crystal

A diagonalization (of the energy matrix for which the Hamiltonian contains the free ion, crystal-field interaction and Zeeman interaction terms) method was applied to calculate the spin-Hamiltonian parameters (g factors, g // and , and five zero-field splittings, ) for the tetragonal Gd3+ center in cubic c-RbZnF3 perovskite. The calculated results are in reasonable agreement with experimental values. From the calculation, the defect model that the tetragonal Gd3+ center is attributed to Gd3+ occupying the 12-fold coordinated Rb+ site associated with a nearest Rb+ vacancy, VRb, along C 4 axis owing to charge compensation is confirmed and the defect structural data of this Gd3+ impurity center in c-RbZnF3 are acquired. The results are discussed.

Spin-Hamiltonian parameters and defect structures for the trigonal Yb3+ centers in wurtzite-type ZnS and CdS crystals

A complete diagonalization (of the energy matrix, where the corresponding Hamiltonian consists of free-ion, crystal-field, Zeeman and hyperfine interaction terms) method is applied to calculate the spin-Hamiltonian parameters (g factors g||, g⊥ and hyperfine structure constants A∥, A⊥) for the trigonal Yb3+ centers in wurtzite-type ZnS and CdS crystals. The calculated results are in reasonable agreement with the experimental values. In the calculations, the crystal-field parameters are obtained from the superposition model, which enables correlation of the spin-Hamiltonian parameters with the defect structure of the studied impurity center. The defect structures of Yb3+ centers (characterized by the local atom-position parameters uloc, which differ from the corresponding parameter in the host crystal) in both crystals are therefore estimated from the calculations. The results are discussed.

Studies of the spin-Hamiltonian parameters and defect structures for Gd 3+ ions in zircon-structure silicates MSiO 4 (M = Zr, Hf, Th)

The spin-Hamiltonian parameters ( g factors g ∥, g ⊥ and zero-field splittings b 2 0 , b 4 0 , b 4 4 , b 6 0 , b 6 4 ) for 4f 7 ion Gd 3+ at the tetragonal M 4+ site of zircon-structure silicates MSiO 4 (M = Zr, Hf, Th) are calculated from a diagonalization (of energy matrix) method. The Hamiltonian concerning this energy matrix contains the free-ion, crystal-field interaction and Zeeman interaction terms and the 56 × 56 energy matrix is constructed by considering the ground multiplet 8 S 7/2 and the excited multiplets 6 L 7/2 ( L = P, D, F, G, H, I). The defect structures of Gd 3+ centers in the three MSiO 4 crystals are yielded from the calculation. The results are discussed.

Anderson Localization Triggered by Spin Disorder—With an Application to Eu x Ca1−x B6

The phenomenon of Anderson localization is studied for a class of one-particle Schr\"odinger operators with random Zeeman interactions. These operators arise as follows: Static spins are placed randomly on the sites of a simple cubic lattice according to a site percolation process with density x and coupled to one another ferromagnetically. Scattering of an electron in a conduction band at these spins is described by a random Zeeman interaction term that originates from indirect exchange. It is shown rigorously that, for positive values of x below the percolation threshold, the spectrum of the one-electron Schr\"odinger operator near the band edges is dense pure-point, and the corresponding eigenfunctions are exponentially localized. Localization near the band edges persists in a weak external magnetic field, H, but disappears gradually, as H is increased. Our results lead us to predict the phenomenon of colossal (negative) magnetoresistance and the existence of a Mott transition, as H and/or x are increased. Our analysis is motivated directly by experimental results concerning the magnetic alloy Eu_x Ca_1-x B_6.

A Theoretical Study on the Spin-Hamiltonian Parameters for Samarium(III) Ion in Potassium Yttrium Tungstate Crystal

The nine spin-Hamiltonian (SH) parameters (g-factors gi and hyperfine structure constants 147Ai and 149Ai for 147Sm3+ and 149Sm3+ isotopes, where i = x, y, z) for the Samarium(III) ion in monoclinic potassium yttrium tungstate [KY(WO4)2] crystal are calculated within the rhombic symmetry approximation from a diagonalization of energy matrix method. Differing from the conventional diagonalization method used in the calculation of crystal-field levels, in the present method, we attach the Zeeman (or magnetic) and hyperfine interaction terms to the conventional Hamiltonian and construct the 66×66 energy matrix for 4f5 ions in rhombic crystal field and under an external magnetic field by considering all the ground-term multiplets 4HJ. The calculated results are in reasonable agreement with the experimental values

Unified calculations of the EPR g factors and zero-field splitting for the tetragonal Gd 3+ center in Pr 1.98Gd 0.02CuO 4

A diagonalization (of energy matrix) method is applied to study the spin-Hamiltonian parameters ( g factors g //, g ⊥ and zero-field splittings b m n ) of the tetragonal Gd 3+ center in Pr 1.98Gd 0.02CuO 4 compound. The Hamiltonian concerning this energy matrix contains the free-ion, crystal-field interaction and magnetic (or Zeeman) interaction terms and so these spin-Hamiltonian parameters can be calculated together. The calculated results are in good agreement with the experimental values. The local structural data of Gd 3+ impurity center in Pr 1.98Gd 0.02CuO 4 are also obtained from the calculation. The results are discussed.

Defect model and spin-Hamiltonian parameters for the tetragonal Nd 3+ center in the tetragonal phase of SrTiO 3 crystal

The spin-Hamiltonian parameters ( g factors g ∥, g ⊥ and hyperfine structure constants 143A ∥, 143A ⊥, 145A ∥ and 145A ⊥) of the tetragonal Nd 3+ center in the low-temperature ( T≈4.2 K) tetragonal phase of SrTiO 3 are calculated from a diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are attached to the conventional Hamiltonian and a 52×52 energy matrix concerning the ground term 4 H J ( J=9/2, 11/2, 13/2, 15/2) is constructed. The Nd 3+ center is attributed to Nd 3+ occupying the 12-fold coordinated Sr 2+ site in SrTiO 3. Differing from the defect model assumed in the previous paper that the tetragonal distortion of this Nd 3+ center is due to the association of one interstitial oxygen ion at a nearest neighborhood of Nd 3+ and the Nd 3+ displacement Δ z along C 4 axis, we suggest that it is due to the distortion of SrTiO 3 lattice in the tetragonal phase. The calculated g factors g ∥ and g ⊥ show good agreement with the experimental values, suggesting that our defect model of Nd 3+ center in SrTiO 3 is reasonable. The experimental hyperfine structure constants were not reported and so our calculated results remain to be checked by EPR experiment.

Spin-Hamiltonian parameters and defect structure for the rhombic Dy 3+ center in AgCl crystal

Based on the defect model that the rhombic Dy 3+ center in AgCl crystal is formed by substitutional Dy 3+ ion associated with two nearest Ag + vacancies ( V Ag) along the 〈1 1 0〉 and 〈 1 ¯ 1 ¯ 0 〉 axes owing to charge compensation, the spin-Hamiltonian parameters ( g factors g i and hyperfine structure constants 161 A i and 163 A i , where i = x, y, z) of this rhombic Dy 3+ center are calculated from a diagonalization (of energy matrix) method. In the method, the Zeeman (or magnetic) and hyperfine interaction terms are attached to the classical Hamiltonian used in the calculation of crystal-field energy levels and a 66 × 66 energy matrix concerning this Hamiltonian is constructed by taking all the ground-term multiplets 6 H J ( J = 15/2, 13/2, 11/2, 9/2, 7/2, 5/2) into account. The calculated results ( g factors g i and average | A ¯ ( 161 D y 3 + ) | and | A ¯ ( 163 D y 3 + ) | ) are in reasonable agreement with the experimental values. From the calculations, the above defect model of rhombic Dy 3+ center is confirmed, the defect structure of this Dy 3+ center (characterized by the displacement of Cl − ligand caused by V Ag) is obtained and the components of hyperfine structure constants A i ( 161Dy 3+) and A i ( 163Dy 3+) are predicted. The results are discussed.

A study of the spin-Hamiltonian parameters for the trigonal Sm 3+ center in La 2Mg 3(NO 3) 12·24H 2O crystal

This paper reports a theoretical calculation of spin-Hamiltonian parameters ( g factors g //, g ⊥ and hyperfine structure constants 147 A //, 147 A ⊥, 149 A //, 149 A ⊥) for 147Sm 3+ and 149Sm 3+ isotopes in the trigonal La 3+ site of La 2Mg 3(NO 3) 12·24H 2O crystal from a diagonalization (of energy matrix) method. In the method, the Hamiltonian concerning the energy matrix includes the Zeeman and hyperfine interaction terms and so there is no perturbation calculation in it. The crystal-field parameters in the energy matrix are calculated using the superposition model, in which the structural data of 12-fold coordinated site rather than those of the incorrect 6-fold coordinated site given in the previous paper are applied. The calculated results are in reasonable agreement with the experimental results. The results are discussed.

Investigations of the spin-Hamiltonian parameters for Yb3+ in the tetragonal phase of SrTiO3 crystal

The spin-Hamiltonian parameters (g factors, , , and hyperfine structure constants, , ) for Yb3+ at the 12-fold coordinated Sr2+ site of tetragonal SrTiO3 crystal at T ≈ 2 and 50 K were calculated using the completed diagonalization (of the energy matrix) method (CDM). Differing from the conventional CDM, the Hamiltonian concerning the energy matrix in this CDM contains the Zeeman and hyperfine interaction terms and so the spin-Hamiltonian parameters can be calculated directly from CDM. The crystal parameters used in the energy matrix are calculated from the superposition model, in which the tetragonal distortions and hence the order parameters φ at both temperatures are applied. From the calculations, the spin-Hamiltonian parameters at both temperatures are explained reasonably and the signs of hyperfine structure constants for the 171Yb3+ and 173Yb3+ isotopes in SrTiO3 are suggested. The results are discussed.

Spin‐Hamiltonian parameters for Dy3+ ion at the trigonal 12‐fold coordinated La3+ site of La2Mg3(NO3)12·24H2O

AbstractThe spin‐Hamiltonian parameters (g factors g‖, g⊥ and hyperfine structure constants 161A‖, 161A⊥, 163A‖, 163A⊥) for 161Dy3+ and 161Dy3+ isotopes in the trigonal 12‐fold coordinated La3+ site of La2Mg3(NO3)12·24H2O crystal are calculated from a diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are added to the conventional Hamiltonian used in the studies of crystal‐field energy levels, and a 66×66 energy matrix concerning the ground multiplet 6H15/2 and the first to fifth excited multiplets 6H13/2, 6H11/2, 6H9/2, 6H7/2 and 6H5/2 are applied. The calculated results are discussed. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A theoretical investigation of the spin-Hamiltonian parameters for Dy 3+ ion in CaWO 4 crystal

The spin-Hamiltonian parameters, g factors g ∥ and g ⊥ and hyperfine structure constants 161 A ∥, 161 A ⊥, 163 A ∥ and 163 A ⊥, for Dy 3+ ion at the tetragonal (S 4) Ca 2+ site of CaWO 4 crystal are calculated from a diagonalization (of energy matrix) method. In the method, the Zeeman (or magnetic) and hyperfine interaction terms are included in the Hamiltonian, and the 66×66 energy matrix (note: the crystal-field energy levels can also be obtained by diagonalizing the matrix) corresponding to the ground-term states 6 H J (where J=15/2, 13/2, 11/2, 9/2, 7/2 and 5/2) for 4f 9 ion in tetragonal crystal field and under an external magnetic field is established. In the matrix, the free-ion parameters are taken as the mean values obtained for Dy 3+ ions in many crystals and the crystal-field parameters B kq are those obtained from the optical spectra of CaWO 4:Dy 3+. The calculated results are close to the experimental values.

Studies of the EPR parameters and defect models for three Er 3+ impurity centers in ThO 2 crystals

The electron paramagnetic resonance (EPR) parameters ( g factors and hyperfine structure constants of the ground state, and g factors of the first excited state) of three (one cubic and two trigonal) Er 3+ centers in fluoride-type ThO 2 crystal are studied from a diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are added to the classical Hamiltonian, and a 52 × 52 energy matrix concerning the ground multiplet 4I 15/2 and the first to third excited multiplets 4I 13/2, 4I 11/2 and 4I 9/2 is established for a 4f 11 ion in trigonal crystal field and under an external magnetic field. From the studies, the EPR parameters for three Er 3+ centers in ThO 2 are reasonably explained, the defect models for the two trigonal Er 3+ centers suggested in the previous paper are confirmed and the defect structures of the two trigonal centers are obtained. The results are discussed.

Ferromagnetism in Cu-doped ZnSe semiconducting quantum dots

The projection of integrating optical, magnetic and electronic functionalities into a single material have aggravated passionate attention in mounting wide band gap diluted magnetic semiconductor (DMS) in the midst of room temperature ferromagnetism. We report the evidence of ferromagnetism in Cu-doped ZnSe quantum dots (QDs) below room temperature, grown from a single source precursor by lyothermal method with the sizes of approximately 3.2–5.14 nm. QDs mainly exhibit paramagnetic behavior between 80 and 300 K, with a weak ferromagnetic/anti-ferromagnetic exchange at lower temperature as observed by superconducting quantum interference device (SQUID) magnetometer. From the Curie–Weiss behavior of the susceptibility, Curie temperature (T c) of Cu-doped ZnSe sample has been evaluated. From EPR, we obtain the Lande-g factor in the Zeeman interaction term as 2.060. Photoluminescence and EPR measurements support and confirm the view that Cu2+ substitutes for Zn2+ in Cu-doped ZnSe quantum dots.

Theoretical calculations of the spin-Hamiltonian parameters for the tetragonal Dy3+ center in HfSiO4 crystal

The spin-Hamiltonian parameters (g factors g(parallel to), g(perpendicular to) and hyperfine structure constants (161)A(parallel to), (161)A(perpendicular to), (163)A(parallel to), (163)A(perpendicular to)) for Dy(3+) at the tetragonal H(f)(4+) site of H(f)SiO(4) crystal are calculated from the diagonalization (of energy matrix) method. In the method, we add the Zeeman (or magnetic) and hyperfine interaction terms to the conventional Hamiltonian used in the calculations of crystal-field energy levels. The 66 X 66 energy matrix concerning the new Hamiltonian is established by considering the ground multiplet (6)H(15/2) and the first to fifth excited multiplets (6)H(J) (where J=13/2, 11/2,9/2, 7/2 and 5/2), and the crystal-field parameters in the energy matrix are calculated from the superposition model. From the calculations, the spin-Hamiltonian parameters of H(f)SiO(4):Dy(3+) are explained reasonably and the signs of hyperfine structure constants for (161)Dy(3+) and (161)Dy(3+) isotopes in H(f)SiO(4) are suggested. The results are discussed. (C) 2010 Elsevier B.V. All rights reserved.