MagnetoTransport
Literature, University of Naples "Federico II", Department of Physics, 2022-03-28
N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall Effect, Rev. Mod. Phys. 82, 1539 (2010).
The Hall effect was frequently called the queen of solid-state transport experiments.
The AHE problem involves concepts based on topology and geometry.
Normal Hall Effect:
In 1879 Edwin H. Hall, conductor
In 1881 Edwin H. Hall, ferromagnetic iron
Anomalous Hall effect:
Berry phase (Berry, 1984).
earlier dubbed “anomalous velocity” by Luttinger, arose naturally in the first microscopic theory of the AHE by Karplus and Luttinger (1954).
KL showed that when an external electric field is applied to a solid, electrons acquire an additional contribution to their group velocity.
experiments of Pugh (1930) and Pugh and Lippert (1932) stablished that an empirical relation
Hall resistivity $\rho_{xy} = R_0H_z + R_sM_z$
The second term represents the Halleffect contribution due to the spontaneous magnetization.
Kondo considering skew scattering from spin excitations (Kondo, 1962), it may be seen that $\rho_{xy}$ also varies as $\rho^2$ at finite T.
Smit argued that the main source of the AHE currents was asymmetric (skew) scattering from impurities caused by the spin-orbit interaction (SOI) (Smit, 1955, 1958.)
Three mechanisms
Since the 1980s, the quantum Hall effect in two dimensional electron systems in semiconductor heterostructures has become a major field of research in physics( Prange and Girvin, 1987).
Both the integer (Thouless et al., 1982) and fractional quantum Hall effects can be explained in terms of the topological properties of the electronic wave functions.
Some concepts I should understand.
Berry-phase
On the theoretical front, the adoption of the Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents… The intrinsic AHE can be expressed in terms of the Berry-phase curvatures and it is therefore an intrinsic quantum-mechanical property of a perfect crystal.
Readers who are not familiar with the Berry-phase concepts may find it useful to consult the elementary review by Ong and Lee (2006) and the popular commentary by MacDonald and Niu (2004).
metallic dilute magnetic semiconductors
the case of a material containing magnetic impurities e.g., Mn embedded in a nonmagnetic host such as Cu (the dilute Kondo system).
Skew scattering
An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets.
Kubo and Keldysh formalisms
In addition, more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms are reviewed, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime.
2.