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Non-magnetic metals (beta=0)

Spin and Charge Transport. Classical model of the spin-up/spin-down band.

It is important!!!! All data on this page are calculated based on the model of the spin-up/spin-down bands. The model of the spin-up/spin-down bands ignores the fact that the spin is often rotated after spin-independent 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 above-mentioned facts, click here or here

The same content can be foundin V. Zayets Phys. Rev. B 86, 174415 (2012) (clich here to download pdf);or




A non-magnetic metal is defined here as a conductive material, the conductivity of which is independent on spin polarization (beta=0). The case of the spin-independent conductivity is the simplest case to analyze the spin and charge transport. In this case the spin and charge currents flow independently and a spin accumulation does not cause a charge accumulation and vise versa. The spin current is diffusive and the charge current is drifted. In the case of non-magnetic metals the spin/charge transport equations converge to the Valet-Fert spin diffusion equation.

Fig.1 Properties of drift charge current and diffusion spin current in non-magnetic metals..


Which materials may have spin-independent conductivity?

Even in the case of non-magnetic materials, in case when there is a spin accumulation in the material, the material's conductivity usually becomes spin-dependent. The condition, that conductivity of a material is spin-independent, implies that the conductivity should be also independent of a charge accumulation. It is a rather rare case. It is only the case when the density of states in a non-magnetic metal is a constant with respect of to energy near the Fermi level. In the case of small charge and spin accumulations, the conductivity of most of non-magnetic metals can be considered as spin-independent.

Transport Equations

General spin and charge transport equations (which were derived here)

is simplified in case of beta=0 as

The eqn. (3) describes a charge transport. Eqn. (4) is the Valet-Fert equation (see the original papar here), which describes spin diffusion. The independent equations for charge and spin mean that neither a spin current nor a spin accumulation affects the charge accumulation and charge current. Neither a charge current nor a charge accumulation affects the spin accumulation and spin current.

The spin and charge currents can be calculated as

Drift of Charge

In case when an external electrical field is applied to a conductive material, the electrons as charged particles are drifted along the direction of the electrical field.

A solution of Eqn (3) describing a drift charge current , which is drifted along an applied electrical field , is


Diffusion of Spin

The spin diffusion is a flow of spin from region of large spin accumulation into regions of smaller spin accumulation. In the case of non-magnetic metals the spin diffusion is described by the Valet-Fert spin-diffusion equation (4)

The Valet-Fert spin-diffusion equation (4) has a general solution

where s is an unit vector directed toward the diffusion direction of the spin current. Therefore, along the diffusion direction the decrease in e-times over spin diffusion length .

For example, in the case when the spin diffuses along the x-direction, the spin chemical potential and the spin current are described as

Also, spin may diffuse in the opposite direction

It is important to notice that the spin currents that flows in forward and backward direction will not interact. Therefore, the total spin chemical potential is a sum of spin chemical potentials for currents flowing in the forward and backward directions.

It is also important to notice that for a fixed spin current, the spin chemical potential will be larger in materials with smaller conductivity and longer spin diffusion length.