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 effectseeinteraction 
Influence of the Coulomb electronelectron interaction on spin and charge transport
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:
The Coulomb electronelectron interaction (eeinteraction) may significantly influence the spin and charge transport in materials. Due to this interaction the material conductivity, the spin life time and the spin selectivity beta may depend of charge accumulation and spin accumulation. The most significant influence of the eeinteraction on transport properties may be for spin current, which is close to the threshold spin current. Also, due to the eeinteraction the spindependent conductivity of material becomes a matrix with nonzero diagonal elements. That modifies transport properties.Influence of charge and spin accumulations of materials parametersDue to the eeinteraction the material parameters should depend on magnitude of the charge accumulation. Indeed, the dependence of the material conductivity, the spin life time and the spin selectivity beta on a charge accumulation is perhaps the most significant influence of the ee interaction on the transport properties. Also, it could be expected that the spin life time and the conductivity spin selectivity beta on a charge accumulation also dependent on spin accumulation. It should be noted that in the case of nondegenerated semiconductors even without considering ee interaction, the conductivity is linearly proportional to charge accumulation and the spin selectivity beta is proportional to the spin accumulation (See here) (For example, in ntype semiconductors the conductivity is linearly proportional to the number of electrons in conduction band). Considering that the eeinteraction is weak and that the spin accumulation is proportional to and the charge accumulation is proportional to and taking into account the time inversion symmetry, the material conductivity, the spin life time and the conductivity spin selectivity beta can be described as where are small parameters describing the eeinteraction. It should be noticed that only in the materials, in which the time inversion symmetry is broken (there is a nonzero spontaneous magnetization) (for example, in ferromagnetic metals). Case of nonzero offdiagonal elements of conductivity matrixThere is another effect of the eeinteraction on the spin/charge transport, which was described in Ref.1
Without eeinteraction, the Ohm's low (Eqn.4 here) reads Eqns. (2a) mean that the flows of the spinup and spindown electrons are independent without any mutual influence. The Coulomb electronelectron interaction leads to interaction between electrons of opposite spins and more complicated expression for the Ohm's low where are symmetric and antisymmetric offdiagonal elements of spindependent conductivity, respectively. The transport equationsEven in the case when the conductivity is described by the matrix (3), it is rather easy to find the transport equations. Below we are following the procedure, which is described here. Introducing new variables Eqns. (2) are simplified to Since , utilizing Eqn. (5), we obtain Defining Eqns. (8) is simplified to and spin and charge transport equations will be The delta beta and the delta sigma are small values. In contrast, the spin selectivity beta is large in the case of the ferromagnetic metal)( near 50 %70% ) and in the case of semiconductors (9099 % near the threshold). Therefore, the influence of the delta beta and the delta sigma on spin transport may be significant only in the case of nonmagnetic metals, which will I will describe below. Nonmagnetic metalIn this case beta=0 and Eqns. (11) are simplified to In case when the conductivity matrix (3) is symmetric ( ) (This case was studied in Ref. [1])), the eeinteraction causes difference of the effective conductivity (or effective diffusion coefficient) for spin diffusion current and for the charge drift current (The spin drag effect). The result is the same as obtained in Ref. [1]. It should be noticed that in the case of ferromagnetic metals and semiconductors this effect will be overshadowed by largermagnitude effect of spindiffusionlength reduction due to the charge accumulation along spindiffusion (See here) The case of is interesting, because in this case the transport equations become similar to the cases of the ferromagnetic metals and semiconductors and the interesting effects for the spin/charge transport may occur. At first, it should be clarified whether it is possible that in material . Below I will check the time inversion symmetry of Eqn. (12) in order to find the basic properties of the delta beta. Obviously, the transport equations (12) should not change under the timereversal operator transformation. Noticing that under the time reversal transformation In order for transport Eqns. (12) to be invariants under the time reversal transformation, it should be That means that delta beta is spindependent. The delta beta can be spindependent only in two cases. The first possible case is when delta beta is proportional to spin accumulation . This case is similar to the case of the semiconductors where beta is proportional to spin accumulation . Another possible origin of could be the spinorbit interaction. In both cases the spin/charge transport will be similar to the spin/charge transport in the semiconductor. Since the value of delta beta is rather small, perhaps it is difficult to achieve condition for the threshold spin current. It should be noticed that in the case of the effective diffusion length will not be reduced as in the case of the ferromagnetic metals, but it will be elongated.

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