Are anomalous Hall effect and spin Hall effect mutually exclusive?

Question

In many papers that cover an analysis of Hall effects, the spin Hall effect is often qualitatively described as being nearly the same as the anomalous Hall effect except for the fact that it doesn't cause a traverse voltage drop across the sample. It is said in these papers that this is because materials that exhibit spin Hall effects have equal populations of spin-up and spin-down electrons while materials that exhibit anomalous Hall effects do not.

However, this paper ¹ is implicitly based on the notion that materials can exhibit anomalous Hall and spin Hall effects concurrently. I have no chance of understanding this paper until I have a coherent qualitative understanding of what the actual difference is between anomalous and spin Hall effects, but I cannot find a clear explanation anywhere.

Could someone please help me understand the actual difference between anomalous Hall and spin Hall effects?

Reference:

1. Y. Zhang et al., "Strong, anisotropic anomalous Hall effect and spin Hall effect in chiral antiferromagnetic compounds Mn₃X (X = Ge, Sn, Ga, Ir, Rh and Pt)", Phys. Rev. B 95, 075128 (2017), arXiv:1610.04034.

Answer 1

Many papers said that AHE and SHE share the same mechanism. A simple description of this theory is that the charge carriers have different spin directions can have opposite transverse movements by some mechanism. Therefore, one can get SHE. In general, this happens to non-ferromagnetic materials and there is no applied magnetic field in SHE. Although there are charge carriers with opposite spin at two sides of the sample, the amount of these two kinds of carriers are equal. That's why one can not get a voltage difference between two sides of sample.

However, if the material is ferro-magnetic, then the amount of spin-up and spin-down are usually not equal, no matter the applied field is zero or not. In this case, if a SHE happens, an AHE also happens. Since it is much easier to measure charge instead of spin, we focus on its AHE and SHE is ignored. Also, personal opinion, people tend to connect SHE with no magnetic field situation(also non-magnetic materials). As for AHE, it is related to magnetic field and magnetic materials. Even when the applied field is zero, ferro-magnetic materials have an inner field. So maybe that's why the misunderstanding comes from that AHE and SHE are exclusive.

Finally, back to the paper you mentioned. It talks about a special situation, which is anti-ferromagnetism. Those antiferromagnetic materials have special spin textures. To be simple, you can just imagine that the amount of spin-up and spin-down are equal. So, when there is no applied field, a normal SHE happens. When there is an applied field, spin-up are more than spin-down, so you get AHE. BTW, actually this Mn3X has a complex spin texture, not just spin-up and spin-down, but it is easier for you to understand.

Answer 2

In my opinionit makes no sense to debate the mutual exclusiveness of SHE and AHE, as they are one effect. Namely, the transverse scattering of charge carriers based on their spin.

In general, with disordered materials, scattering is random leading only to transverse spin accumulation on the material surface. This is the so-called spin Hall effect.

As a result, if any spin orientation is dominant a greater absolute number of charge carriers will be scattered in the same direction creating an electrical potential difference. The anomalous Hall effect emerges.

There are also edge cases, whose classification as SHE/AHE I will not decide: A compensated ordered antiferromagnet, will create scattering maxima for each of its sublattices (e.g. scatter in two opposite directions for a simple collinear antiferromagnet). Because the antiferromagnet is compensated, the number of charge carriers in each direction is still equal, so no voltage develops. But the scattering is also not isotropic, so it is different from the first case too.