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Spin Hall effect and Inverse Spin Hall effect

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

##### For the modified model, which includes all above-mentioned facts, click here or here

Bellow is the "classical" description of the Spin Hall effect, which does not describe correctly all features of the effect.

The detailed description of the Spin Hall effect is here.

## Basic remarks Fig.1 Spin Hall effect. Charge current flows along the y-axis. Due to spin-orbit scattering, spin up electrons turn right and spin-down electrons turn left in the xy-plane

The spin-orbit interaction or spin-dependent scattering off impurities may cause a current flowing perpendicularly to the applied electrical field. The polarity of the current is different for spin up and spin-down electrons. The existence of this current is an origin of both the Spin Hall effect and the Inverse Spin Hall effect. We consider the case when in a material there is a preferential direction, for example, the z-axis is perpendicular to the normal of a quantum well. The electron flow deviates from a straight path in the xy-plane (See Fig.1). The spin-down electrons turn right and the spin-up electrons turn left with respect to the y-axis direction.

Then, the effect can be described by the spin-dependent conductivity tensor   where is effective conductivity.

The spin selectivity beta will be a tensor where is spin Hall conductivity.

The metal, in which spin selectivity is described by tensor (3), we will call a SH-metal.

In a SH-metal the charge/spin transport equations will be: Noticing that a skew-symmetric (antisymmetric) matrix M satisfies the Eqn. (4) are simplified to The spin/charge transport eqns. (6) for a SH-metal is exactly the same as equations for a non-magnetic metal and spin/charge transport in SH-metals is rather simple. The solutions of Eqns (6) are the same as the solution of the transport equations in the case of the non-magnetic metals. The first solution describes a drift current and the second solution describes a diffusion current. The same as in the case of the non-magnetic metals, in the SH-metals there is no interaction between the diffusion current and the spin current . Also, in the SH-metals there is no charge accumulation along spin diffusion as in the case of ferromagnetic metals or semiconductors.

## Drift current (Spin Hall effect) Fig.2 Spin Hall effect in metallic wire. Spin drift current converts into diffusion current at boundary

For example, if a drift current flowing along the x-axis , the spin and charge components of the drift current will be The drift spin current does not decay. There is no spin accumulation and there is no spin relaxation for flow of the spin drift current. For example, since there is no spin accumulation, the drift spin current can not be detected optically by the MO Kerr effect. However, in the case when the charge current flows inside the wire (See Fig.2) , the spin drift current is converted into the spin diffusion current at the boundary of the wire. The diffusion spin current decays from the wire boundaries towards the center of the wire. There is a spin accumulation along the flow of diffusion spin current and it can be detected by the MO Kerr effect.

## Diffusion spin current (Inverse Spin Hall effect) Fig.3 Inverse Spin Hall effect in metallic wire.

For example, if a diffusion spin current flowing along the x-axis , the spin and charge components of the diffusion current will be As was discussed here, the flow of a charge current requires both a charge source and a charge drain. Otherwise, charge accumulates and induces an electrical field in the opposite direction to the charge current. Therefore, the flow of charge current stops.

As is shown in Fig.3, the same is happening for the Inverse Hall effect in a metallic wire. A diffusion spin current flows along the wire. Due to the Inverse Hall effect, it generates a diffusion charge current flowing across the wire. Since this current does not have a drain, charge accumulates at the boarders of the wire. The accumulated charge induces an electrical field, which stops the charge current.

Steps of Figure 3.

Step 1. A diffusion spin current flows along the wire. Due to the Inverse Hall effect, it generates a diffusion charge current flowing across the wire.

Step 2. Charge accumulates at the boarders of the wire

Step 3. The accumulated charge induces an electrical field in the direction opposite to the flow of the diffusion charge current.

Step.4. Equilibrium. The drift charge current flowing along electrical field is of opposite sign and of the same amplitude as the diffusion charge current. Therefore, in total there is no charge current.

As is shown, a dc spin diffusion current does not induce a charge current due to the Inverse Hall effect. However, in case of the altering spin current, the charge accumulation is altering as well. In order to alter the charge accumulation the charge current is required. Therefore, the AC spin current induces the AC charge current.

It should be noted that in the case of semiconductors or ferromagnetic metals, there is a substantial charge accumulation along the diffusion spin current, which may exceed the charge accumulation due to the Inverse Spin Hall effect.

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