classic model ofspin transportmodel of spindown/spinup bandsmore chapters on this topic:IntroductionBasic Transport equationsSpin and charge currentsSpin drainNonmagnetic metalsFerromagnetic metalsSemiconductors (Basic)Threshold spin currentSpin gain/dampingSpin RelaxationSpin Hall/ Inverse Spin Hall effectseeinteractionclassic model ofspin transportmodel of spindown/spinup bandsmore chapters on this topic:IntroductionBasic Transport equationsSpin and charge currentsSpin drainNonmagnetic metalsFerromagnetic metalsSemiconductors (Basic)Threshold spin currentSpin gain/dampingSpin RelaxationSpin Hall/ Inverse Spin Hall effectseeinteraction 
The Rate of the Spin Relaxation
Spin and Charge Transport. Classical model of the spinup/spindown band.It is important!!!! All data on this page are calculated based on the model of the spinup/spindown bands. The model of the spinup/spindown bands ignores the fact that the spin is often rotated after spinindependent scatterings (See here). Therefore, some predictions based on this model may be incorrect and differ from a experimental observation.For the modified model, which includes all abovementioned facts, click here or hereThe same content can be foundin V. Zayets Phys. Rev. B 86, 174415 (2012) (clich here to download pdf);or http://arxiv.org/abs/1205.1278 Abstract:Utilizing simple approximations I will show that the spin relaxation is linearly proportion to spin chemical potential .
In the case when the energy is conserved during a spinflip scattering, the rate of spin relaxation is calculated as where are the probability for transition from spin up band to spin down band and transition from spin down band to spin up band, respectively are the density of states for spinup to spindown bands; F(E) is FermiDirac distribution; is Fermi energy in case when there is no charge accumulation; is the change of the Fermi energy due to a charge and/or spin accumulation. In the case when there is no spin accumulation, there is no spin relaxation , That leads to . In the case when the charge and spin accumulations are small Eqn. (1) is simplified to Simplifying Eqn. (3), we obtain where Eqn (4) is Since the constant A does not depend on the spin accumulation or charge accumulation, the spin relaxation rate is linearly proportional to the spin chemical potential. The expression (6) states that if the condition (2) is satisfied, the spin relaxation is linearly proportional to the spin chemical potential and the spin diffusion length does not depend on spin or charge accumulations. In the case when condition (2) is not satisfied, the A will be not a constant, but a function of spin and charge accumulations Since the spin accumulation is proportional to the spin chemical potential and the charge accumulation is proportional to the second derivative of charge chemical potential the Eqn. (7) can be rewritten as The physical meaning of Eqn. (7) is the following. At sufficiently large spin or charge accumulation, the spin diffusion length depends on the magnitude of the spin and charge accumulations. This should be considered in cases when the spin current close to the threshold spin current and in a charge accumulation region in the vicinity of the tunnel barrier.

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