# How does spin Hall effect happen?

In the spin Hall effect, electrons with different spins accumulate in opposite directions without any external magnetic field. Broadly, the reason behind this is the spin-orbit interaction. But, the question is how? How does the spin-orbit interaction leads to this. A physical explanation will suffice.

My usual disclaimer, you will get a better answer than this and I am writing it to learn about the effect myself, so please bear that in mind :)

My answer will be based around the similarities and differences between the two "varieties" of Hall effect. It is also based, in part on Wikipedia Spin Effect and the relevant links from that page.

The Spin Hall Effect results in the aggration of spin (the spin directions at one edge of a current carrying conducting material being opposite to the other edge). In most cases the current carrying conductor will be a wire, upon the surface of which the spins will wind around the surface. Image Source: Wikipedia Spin Hall Effect

The spin Hall effect is analogous to the lines of the magnetic field produced by the current, with the important distinction that the value of the spin polarization is far greater than the (almost ignorable ) equilibrium spin polarization in this magnetic field. The degree of polarisation on the boundary is (not surprisingly) proportional to the current carried, and reversing the current "flips" the spins.

A further analogy can be made to the "classical" normal Hall effect, where charges of opposite signs accumulate, because of the Lorentz effect in a magnetic field.

A list of the difference between the accumulation of charge and that of spin;

1. The spin polarization is found in layers, usually of the order of $1 µm$, and is limited by spin relaxation. The width of the spin layer is dependent on the spin diffusion length, as listed above. (In contrast to the normal Hall effect, which is associated with the much smaller Debye screening length).

2. The spin Hall effect (SHE) does not require the presence of a magnetic field, in fact a $\overrightarrow B$ field orthogonal to the spin polarisation direction will remove the effect.

Coupling of spin and charge currents

Although you ask for a physical (intuitive ) picture, please forgive a little math for reasons of brevity. The coupling of the spin and charge currents can be linked as follows.

Introduce the charge and spin flow densities, $q^{(0)}$ and $q_ {ij}^{(0)}$, (the $i$ index indicates the flow direction, while the $j$ indicates which component of the spin is flowing) which would exist in the absence of spin-orbit interaction: where;

$µ$ is the mobility coefficient

$D$ is the diffusion coefficient, connected by the Einstein relation,

$n$ is the electron concentration

$E$ is the electric field,

$P$ is the vector of spin polarization density

The upper equation above is the usual electron drift-diffusion formula.

By contrast, the second equation describes polarized electron spin current which may actually exist even in the absence of spin-orbit interaction, since spins are transported by electron flow. The small possible dependence of mobility on spin polarization is discounted in this example. The above equations will need to be modified for influences such as a temperature gradient.

Spin-orbit interaction couples the two currents and gives corrections to the values of the primary currents $q^{(0)}$ and $q_{ij}^ {(0)}$. For an isotropic material with inversion symmetry, we have:

Referring to the above equation, where $q_{i}$ and $q_{ij}$ are the corrected currents,

$ε_{ijk}$ is the unit antisymmetric tensor (whose non-zero components are $ε_{xyz} = ε_{zxy} = ε_{yxz} = – ε_{yxz} = – ε_{zyx} = – ε_{xzy} = 1$)

$γ$ is the small dimensionless parameter which is proportional to the strength of the spin-orbit interaction.

The difference in signs in the above equations can be attributed to the disimilar properties of charge and spin currents w.r.t time inversion. So, for example, a $q_{xy}$ spin current will induces a charge flow in the $z$ direction $(q_z)$, and inversely; a flow of charge in the $z$ direction will induce the spin currents $q_{xy}$ and $q_{yx}$.

As a classical analogy, this can be thought of in terms of the Magnus Effect, a spinning soccer ball will "stray" from its normal straight path in a direction dependent on it's sense of rotation.