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Hall Effect (Classical and Quantum)

Ohm's Law, Hall Effect, Hall Effect (2-D), QHE, QSHE, QAHE, Topological Insulator

The Hall effect was discovered in 1879. It emerges by the combination of electric and magnetic fields and leads to interesting observation at that time. Followings are steps to trace its progression from ancient to modern. Those who are not mathematically inclined can just browse through the comments at the beginning and end of each step. Sorry about the inconvenience. The difference between classical and quantum Hall effects is in the origin of the phenomena. While the classical cases rely on applying a longitudinal current to deflect electrons toward the transversal direction, the quantum effects in step-wise change of conductivity / resistivity depend on the intrinsic property of the material.
2-D Topological Insulator Normally, the electron is not really free within the crystal lattices, its wave function is in the form of Bloch Wave (see QAHE). However, the spin-orbit coupling could create edge states to link the two separated bands together making them in contact at a point where the electron behaves like a free particle (see Figures 13-08z2 and 13-08z3). The eigen-energy at that point is 2k2/2m (denote by k2 henceforth to simply the formulas). It has four-fold degeneracy - 2 from spin up and down ((0), (1)); another 2 from the the directions of the momentum k. By shifting the origin of the E-k space to

Figure 13-08z5 2-D Topological Insulator [view large image]

the contacting point (Figure 13-08z5,a), the effect of spin near this point can be written in term of the spin states and k dependent coefficients (in k-space and matrix notation with the kxi ky terms acting like magnetic field, see 2-D Hall Effect) :

There is a similar derivation for 3-D topological insulator (see "Berry Curvature") by adding the kz component into the formulation. The net effect is the replacement of edge states by the surface states in which the spinning electrons can wander on the surface instead of running around in an one dimensional loop at the edge (Figure 13-08z6). The k-space magnetic field is non-vanishing and is in the form :
3D Topological Insulator

Figure 13-08z6 3D Topological Insulator [view large image]

The effect of 3-D topological insulator has been observed with the BiSb alloys in 2009 (see "Experiments of a Topological Insulator"). Meanwhile see more claims on monopole discovery in "Synthetic Magnetic Monopole in Superfluid".

Majorana Quasi-particle If the topological insulator is placed in close contact with a superconductor, the constituent in the surface state becomes Majorana quasiparticle. It is a bound state of a hole and an electorn resulting in a quasiparticle which is its own anti-quasiparticle and often mis-labeled as Majorana fermion which is a fundamental particle like Majorana neutrino (Figure 13-08z7,a,b). Anyway, this kind of material is known as topological superconductor. In term of the E-k space, it is the product of merging a band in the bulk (the tl) into a band in the superconductor. Different energy states appear as shown in the right

Figure 13-08z7 3D Topological Superconductor

side of Figure 13-08z7,c. This novel material is anticipated to have important quantum computing applications via its properties of non-abelian statistics and non-local interaction (entanglement).
A 2-D topological superconductor in a hetero-structure sample constituting of a topological insulator Bi2Se3 film and a s-wave superconductor NbSe2 has been realized in 2011 (Figure 13-08z7,d). BTW the s-wave is referred to the opposite spins of the electron-hole pair ; it is something like the s, p, d orbitals in atomic configurations.

See a review article in "Search for Majorana Fermions in Superconductors".

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