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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 the bulk 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. At the edge, by shifting the origin of the E-k space to the contacting point (Figure 13-08z5,a), the effect

Figure 13-08z5 Topological Insulator, 2D

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.

Figure 13-08z7 3D Topological Superconductor [view large image]

Different energy states appear as shown in the right 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".

The coming of age for topological insulators prompted the quest for more of them as there are only a few hundreds such substances out of about 200,000 compounds in material databases. Finally in 2017, a method is developed to search for these precious materials. It unifies the concepts of lattice (real) space (from the perspective of chemistry) and band structure in k-space (from the point of view in physics) to come up with a way to check out the characteristic of the object. The original paper by the title of "Topological Quantum Chemistry" is highly technical. The following is an attempt to summarize the essentials by elaborating on one of the illustrations in the article.

Prior to run the test for topological insulator, it compiles all of the possible ways energy bands in a solid can be connected throughout the Brillouin zone to obtain all realizable band structures in all non-magnetic space groups.
    Energy Bands, Types The example of graphene below is a semimetal in which both the bulk and edge has overlap between the conduction and valence bands. Whereas for topological insulator these bands in the bulk are disconnected, only the edge (or surface in the case of 3-D) has overlapping bands and is robust against perturbations. Nevertheless, the methodology is also applicable to discover the characteristics of metals and semimetals as well as topological insulators (see Figure 13-08z8, in which the Fermi Energy EF can be considered as

    Figure 13-08z8 Energy Bands, Types [view large image]

    the highest filled energy level at absolute temperature T = 0. For finite temperature, there could be some diffusion of electrons into energy level higher than EF depending on the type of energy band).
    BTW, semiconductors are the basic components for all kinds of devices in the electronic age. There are three different types, namely, the intrinsic, p and n (see Transistor for the bending of the valence and conduction bands in such application). Now back to the main subject :

    Tow Views of TOpo-Inso
  1. Unification of the Real and Reciprocal Spaces - From the orbitals in the real lattice space, a tool has been developed to translate the spatial symmetries to the high-symmetry points in the Brillouin zone of the reciprocal space (such as the in the middle, K at the corners, and M the middle of the hexagonal side, see Figure 13-08z9,1).
  2. kp Theory - The quantum states of an electron in the periodic potential V is determined by the Schrodinger Equation as shown in Figure 13-08z9,2. The wave function can be separated into a cyclic part eikx and the Bloch Wave un,k(x), where k denotes the reciprocal vector in the k-space, a.k.a., reciprocal / momentum space. By shifting the origin of the k-space to the high-symmetry points (thus, k = 0), the solution is just another periodic function un,0(x) where n refers to each individual band. There is no analytic solution for k 0, a perturbation method (kp theory) can be used to obtain approximation for small k such as the dispersion energy shown in Figure 13-08z9,2.
  3. Graph Theory - The features around each of the high-symmetry points are patched together by considering group constraints and graph theory mapping to estimate the global band structure.
  4. Wannier Functions - The Wannier function is essentially the Fourier transform of the Bloch function from k-space to points near the lattice site. Based on the connections as shown in Figure 13-08z9,3, the Wannier functions (yellow spheres) can be established at the site of the lattice.
  5. Figure 13-08z9 Topo-Insu, Two Views United [view large image]

    Topological semimetal or insulator can be deciphered according to whether the Wannier functions of the two energy bands are connected at the site or not as shown in Figure 13-08z9,4.
The methodology has been applied to all 230 crystal symmetry groups. The possible band structures, which are topologically non-trivial, are classified. It reveals some promising candidate materials such as Ph2O and other examples (see original article).

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