Kindly tell me in detail about the possible outcomes of the "Anomalous Hall Effect" measurement.
A simple answer would go like this. The ordinary Hall effect in semiconductors appears in response to an applied magnetic field. The drifting charge carriers experience a Lorentz force in the presence of an applied field and thus accumulate on one side of the sample to create a Hall-voltage. It is now almost intuitive that a sample that is magnetized even in the absence of a magnetic field will create a Hall-voltage as well. This Hall-effect which appears in the absence of an applied magnetic field is the anomalous Hall effect. Obviously one can learn about the magnetism/spin-polarization of the sample through such measurements. It is tempting to say that the anomalous Hall effect originates from the internal field that accompanies the magnetization and thus is basically similar to the ordinary Hall effect. However, this is not true. The actual microscopic mechanism is different, depends on spin orbit coupling and spin dependent scattering.
Thank you dear Christian. So basically the internal magnetic field of magnetization plays role and create 'Hall Voltage'. So in case of a 'ferromagnetic semiconducting' material what can we learn about the magnetism or spin polarization of the carriers??
Regarding dilute magnetic semiconductors (DMS), you can perform temperature-dependent magnetotransport measurements and hysteresis curves (magnetic moment and magnetoresistance), in order to determine the kind of "Hall system" you might have, and whether the system forms long-range magnetic order (ferromagnetism, Curie temperature) or not (superparamagnetism, blocking temperature). At least.
In addition, you can perform X-Ray Magnetic Dichroism (XMCD) in order to determine the orbital and spin magnetic moments of the sample as function of, say, temperature or concentration of the magnetic atoms. The temperature dependence of magnetization (or of the orbital and spin moments) tells you if the system behaves as a spin-glass, spin-ice, Heisenberg ferromagnet, Ising model, or an intinerant RKKY system, a Kondo system, etc... You can perform spin-resolved photoemission spectroscopy, map the Fermi surface and identify the electronic states near Fermi level which are responsible for the magnetic behavior, determine their spin-polarization and correlate them with the behavior of the orbital and spin moments or to the magnetotransport behavior. There are many things you can do. Of course, the underlying mechanisms of what is observed should be elucidated with the help of theoretical models of these results.
If the electrons are spin-polarized along z and the current j flows along x, there will be a voltage in the y direction. An obvious reason is the normal Hall effect produced by the magnetic field of the spin polarized electrons. However there is an additional (anomalous) contribution which is directly proportional to spin polarization, but is NOT caused by the associated magnetic field. This additional contribution is due to spin-orbit interaction, however the exact mechanisms are still a subject of debate
The anomalous voltage can be presented as V_an = alpha*j*P, where P is spin polarization and alpha is a coefficient proportional to spin-orbit interaction. If alpha were known in advance, one could deduce spin polarization. This is not normally the case. Rather, one deduces P from magnetization, or otherwise (e.g. optical measurements in DMS) and thus determines alpha.
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