Factorization of the evolution operator for a spin–orbit coupled system in an external magnetic field for an arbitrary L and

1976 ◽  
Vol 54 (23) ◽  
pp. 2295-2305
Author(s):  
R. Bogdanović ◽  
M. A. Whitehead ◽  
M. S. Gopinathan
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Evolution Operator ◽  
Coupled System ◽  
Projection Operators ◽  
Spin Orbit ◽  
Group Representations ◽  
Special Cases ◽  
The Matrix ◽  
Strong External Field

The evolution operator of the spin–orbit coupled system in an external magnetic field is factorized according to the operator equation:[Formula: see text]where HSO is the spin–orbit coupled Hamiltonian and HZ is the Zeeman Hamiltonian. The preceding equation is expressed in matrix form using the theory of the O(4) group representations. For the special case of [Formula: see text], an explicit form of the matrix C(t) is found. The operator corresponding to the matrix C(t) is found in a closed form using the projection operators constructed in the Appendix. Two special cases, corresponding to a weak and a strong external field, are considered, and possible applications of the results obtained are indicated.

SPIN-SPLITTING OF THE BOUND POLARON'S GROUND STATE BINDING ENERGY INDUCED BY THE RASHBA SPIN–ORBIT INTERACTION UNDER THE PERPENDICULAR MAGNETIC FIELD

2008 ◽  
Vol 22 (12) ◽  
pp. 1923-1932
Author(s):  
JIA LIU ◽  
ZI-YU CHEN
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Binding Energy ◽  
Ground State ◽  
Spin Splitting ◽  
Perpendicular Magnetic Field ◽  
Spin Orbit ◽  
Ground State Binding Energy ◽  
Bound Polaron ◽  
Rashba Spin

The influence of a perpendicular magnetic field on a bound polaron near the interface of a polar–polar semiconductor with Rashba effect has been investigated. The material is based on a GaAs / Al x Ga 1-x As heterojunction and the Al concentration varying from 0.2 ≤ x ≤ 0.4 is the critical value below which the Al x Ga 1-x As is a direct band gap semiconductor.The external magnetic field strongly altered the ground state binding energy of the polaron and the Rashba spin–orbit (SO) interaction originating from the inversion asymmetry in the heterostructure splitting of the ground state binding energy of the bound polaron. How the ground state binding energy will be with the change of the external magnetic field, the location of a single impurity and the electron area density have been shown in this paper, taking into account the SO coupling. The contribution of the phonons are also considered. It is found that the spin-splitting states of the bound polaron are more stable, and, in the condition of weak magnetic field, the Zeeman effect can be neglected.

The Normal Modes of Linearized Magnetohydrodynamics

1974 ◽  
Vol 52 (6) ◽  
pp. 509-515
Author(s):  
P. B. Corkum
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Normal Modes ◽  
Nonequilibrium Thermodynamic ◽  
Magnetohydrodynamic Equations ◽  
Minimal Set ◽  
Special Cases ◽  
Component Plasma ◽  
Phenomenological Coefficients ◽  
Central Purpose

The central purpose of this paper is to derive a general set of magnetohydrodynamic equations for a two component plasma in an external magnetic field and to find the eigenmodes of the linearized equations. The magnetohydrodynamic equations are derived from nonequilibrium thermodynamic principles. It is pointed out that a minimal set of phenomenological coefficients are found in this manner. The magnetohydrodynamic equations are linearized and then solved for the magnetohydrodynamic eigenmodes in the two special cases of the wave vector k parallel and perpendicular to the external magnetic field.

Influence of magnetic field and Rashba spin–orbit coupling on strong-coupling magnetopolarons in quantum disks

2014 ◽  
Vol 28 (27) ◽  
pp. 1450185
Author(s):  
Wei Xin ◽  
Chao Han ◽  
Eerdunchaolu
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Interaction Energy ◽  
Ground State ◽  
State Interaction ◽  
Orbit Coupling ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Phonon Interaction ◽  
Rashba Spin

On the basis of Lee–Low–Pines (LLP) unitary transformation, the influence of external magnetic field, Rashba spin–orbit coupling and quantum size effect on the ground-state interaction energy of strong-coupling magnetopolarons in quantum disks (QDs) is studied by using the Tokuda improved linear combine operator method. The results show that the ground-state interaction energy of magnetopolarons consists of four parts: the energy caused by the confinement potential of QDs, interaction energy between the electron and external magnetic field, electron and longitudinal-optical (LO) phonon interaction energy and additional term of Rashba effect originating from phonons. The electron–LO phonon interaction energy Ee- ph and additional term of Rashba effect are always negative; the absolute value |Ee- ph | increases with increasing transverse confinement strength ω0, cyclotron frequency of external magnetic field ωc and electron–LO phonon coupling strength α, but decreases with increasing the thickness of QDs L; the state properties of magnetopolarons are closely linked with the sign of the ground-state interaction energy of magnetopolarons E int and change of E int with ωc, ω0, α and L. In addition, the vibration frequency of magnetopolarons λ increases with increasing ωc, ω0 and α, but decreases with increasing L. For the ground state of magnetopolarons in QDs, the electron–LO phonon interaction plays a significant role, meanwhile, the influence of Rashba spin–orbit coupling effect cannot be ignored.

scholarly journals Switching of perpendicularly polarized nanomagnets with spin orbit torque without an external magnetic field by engineering a tilted anisotropy

2015 ◽  
Vol 112 (33) ◽  
pp. 10310-10315 ◽  
Author(s):  
Long You ◽  
OukJae Lee ◽  
Debanjan Bhowmik ◽  
Dominic Labanowski ◽  
Jeongmin Hong ◽  
...  
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Data Storage ◽  
Easy Axis ◽  
Film Plane ◽  
The Other ◽  
Spin Orbit ◽  
Applied Current ◽  
Density Data ◽  
Magnetic Easy Axis

Spin orbit torque (SOT) provides an efficient way to significantly reduce the current required for switching nanomagnets. However, SOT generated by an in-plane current cannot deterministically switch a perpendicularly polarized magnet due to symmetry reasons. On the other hand, perpendicularly polarized magnets are preferred over in-plane magnets for high-density data storage applications due to their significantly larger thermal stability in ultrascaled dimensions. Here, we show that it is possible to switch a perpendicularly polarized magnet by SOT without needing an external magnetic field. This is accomplished by engineering an anisotropy in the magnets such that the magnetic easy axis slightly tilts away from the direction, normal to the film plane. Such a tilted anisotropy breaks the symmetry of the problem and makes it possible to switch the magnet deterministically. Using a simple Ta/CoFeB/MgO/Ta heterostructure, we demonstrate reversible switching of the magnetization by reversing the polarity of the applied current. This demonstration presents a previously unidentified approach for controlling nanomagnets with SOT.

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