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**(1) Ohm's Law (Electric Field Only) - The law of V = IR (V - voltage, I - current, and R - resistance) was discovered by Georg Ohm in 1827 - a pre-Hall discovery in early 19th Century (see "History of Ohm's Law"). The modern derivation :**

The electron gas model considers the electrons as gas molecules bumping each other but the average velocity is zero. A net velocity v_{d}is developed when an electric field E = V/L is applied to the conducting material. The current I is defined as I = nev_{d}A, and the current density J = I/A, where n is the number density of the electron gas, A is the cross-section and L the length of the conductor (Figure 13-08u). The following derivation is to develop a relationship between I and V, the resistance R is just a proportional constant depending on the property of the conductor.

#### Figure 13-08u Ohm`s Law in Electricity [view large image]

**In super-conductive state, the electrons move coherently as a whole; thus = and R = 0 so that the current I can run forever without applying any voltage.**

**The cgs units for I - C(Coulomb)/sec = ampere, R - erg-sec/C**^{2}= ohm, V - erg/C, J - amp/cm^{2}, E - erg/C-cm,_{0}- 1/ohm-cm = 1/sec.-
**(2) Hall Effect (Classical) - As shown in Figure 13-08v, applying a magnetic field B perpendicular to a conducting slab (in the +z direction and with current I in the +x direction) would create a force in the -y direction. Eventually, the electrons accumulated at one side develop an electric field E**_{y}perpendicular to the original E_{x}. The usefulness of such effect is to enable the measurement of the electron density, which was unknown 137 years ago. **(3) Hall Effect (2-D Classical) - In two dimensional thin film at low temperature with strong magnetic field, the z component of the drift velocity v = v**_{d}can be neglected so that the Lorentz force formula can be written down in terms of v and relaxation time as (vector is denoted by boldface letter, the dimension of number density n is #/cm^{2}instead of #/cm^{3}in the 3-D case) :

**(4) Hall Effect (2-D Quantum) - The quantum treatment is closely related to the classical version. The additional feature is the discrete energy levels leading to the quantization of the conductivity. Although the formulation (both classical and quantum) is for one electron, as all the electrons move along almost collisionless, it is applicable to every one of them. Following the conditions for the current density J and the electric field E deduced in the classical case above, the interaction terms are eEx and A = (0,Bx,0) so that the Hamiltonian :**

#### Figure 13-08w Landau Distribution [view large image]

**Actually, the = eB/m defined above is the cyclotron frequency related to the circular motion of electron confined by the magnetic field B, while = (K/m)**^{1/2}is for the harmonic osciallation where K is the force constant of the spring. Similarly,_{}= (/eB)^{1/2}is the cyclotron radius, now the angular momentum L = m_{}= instead of mvr in the classical definition. Therefore, the orbital angular momentum in this case is quantized with an integer quantum number 1.

**The energy level in this case is referred to as Landau levels. Due to impurity etc., the density of states will evolve from sharp Landau****levels to a broader spectrum of levels as shown in Figure 13-08w. Since the spin and k**_{y}do not appear in the Landau levels, it is degenerate as shown in the same image. The different electron orbitals for different Landau levels are displayed at the bottom as well. Experimental measurement gives_{xy}= 1/25812.807557 ohm^{-1}, which allows precise determination of some physical constants (see "Quantum Hall Effect Applications"). Figure 13-08x1 shows the step-wise increment of the resistivity/conductivity as function#### Figure 13-08x1 QH Conductivity, Experiment [view large image]

**of the magnetic field B and the number density n (negative number represents hole with +|e| ). The**_{xx}is also present as it is not exactly equal to zero in the real world. The quantum steps can be interpreted as a topological phenomenon, for which changes to the system's properties could not occur smoothly.

**The Chern class in topology is used to classify geometric objects into different categories. As shown in Figure 12-08x2,b, it is defined as the surface integral in a patch of the object. The concept is now applied to the low temperature physics, in which it is discovered that the magnetic property in k-space can be identified to the topological formula (with the Chern number replaced by C, see Figure 13-08x2,a). Thus, in the bulk of the topological material C = 0, while it is C = 1, 2, 3, ... in different energy levels at the edge; the values of the conductivity/resistivity would vary in steep steps corresponding to the change in Chern number (in discrete steps). Ultimately, it is the variation of the wave function (from whatever cause in k-space) that induces the phenomena in the various kinds of Hall effects. The reason behind**#### Figure 13-08x2 Topological Interpretation

_{}**the appearance of the Chern number becomes apparent if the surface integral is transformed to a line integral (of a closed loop around the surface in k-space) by the Stokes' Theorem :****There is a similar derivation for "Magnetic Flux Quantization" in superconducting loop. It is noticed that while the magnetic flux quanta**_{0}= h/e ~ 4x10^{-15}joules/amp, the Hall resistivity changes in step of h/e^{2}~ 2.5x10^{4}joules/amp-C ~ 4x10^{-15}joules/amp since there are 6x10^{18}electrons in 1 C (Coulomb). Both cases have their origin in current loop, the extra "e" power in Hall resistivity comes from a collection of electrons instead of from a single one.

**In Figure 13-08w, the Landau levels have a four-fold degeneracy with two for spin up and down and two more for particular values****of k**_{y}at the valley between the valence and conduction bands making a total of 2x2 = 4 (see Figure 13-08y). In addition, if there is a boundary in the y-direction for example, then the degeneracy for k_{y}is 2(/_{0}), where is the magnetic flux equals to BxS (S is the surface area) and_{0}= h/e = 4x10^{-15}Web is the fundamental quantum flux constant. Thus, the degeneracy normally is a huge number more than 10^{15}, so that all the free electrons in the system sit in only a few Landau levels. It is under such condition where the quantum Hall effect arises.#### Figure 13-08y Landau Level Degeneracy [view large image]

**See "Spin-Valley Coupling", and "Landau Quantization".**

**(5) Hall Effect (2-D Quantum Spin) - It turns out that there is another aspect of the Quantum Hall Effect with the spinning electrons, which are forced to move along the edge to travel in one way (for both spin up and down in separate path, see Figure 13-08z1,b). This effect bends the Landau levels at the edge upward and splits each one in two. These are edge states as represented by the line width in each Landau level in Figure 13-08w.**#### Figure 13-08z1 Hall Effect 2-D Quantum Spin [view large image]

**It can be visualized that the electron charge (current) gives rise to one facet of the Hall Effect, while its spin (current) presents a different view, and thus ushers in the "spintronic" era.**

**A theoretical model was proposed in 2004 to account for the possible existence of the Quantum Spin Hall (QSH) effect (Figure 13-08z1,b). The material is a very thin two dimensional inslulor such as the two layers (bilayer) graphene. Because the different geometric shape at the edge, the state of the electrons in there is influenced by the spin-orbit coupling, i.e., between the electron spin and the orbital motion, which directs the spin up electrons to move in one layer, while the spin down electrons run in opposite direction in the other layer without the assistance of external magnetic field (see Figure 13-08z2,e in which the red and blue represent different electron spins, while the atoms in the two layers are labeled with open and filled circles). Figure 13-08z2,d shows the contact between the valence and conduction bands. The electrons can move freely through the contact and make the top bands conductive. However, it is doubtful that the spin-orbit interaction in the bilayer graphene is strong enough to realize the QSH effect. A new material in the form of Mercury Telluride (HgTe) sandwiched between layer of Cadmium Telluride (CdTe) is proposed in 2006 to do the trick. Note that the strength of spin-orbit coupling is proportional to atomic number, thus the suitable material would involve heavy elements such as mercury (Hg) with atomic number 80 - very close to the limit where the element would become radioactive. The process is summarized in Figure 13-08z2,a,b,c and explained briefly below.**

- a. For the thickness of the HgTe d < 6.5 nm, the conduction band E1 is normally over the valence band H1 and separated by a gap. As the thickness increase (implying stronger S-L coupling) to d > 6.5 nm, the placement is inverted (Figure 13-08z2,a).
- b. When d > 6.5 nm, edge states allow traveling of electrons appear in the gap turning the edge into conductor (Figure 13-08z2,b).
- c. The graph in the left of Fgiure 13-08z2,c shows very large resistance, while the one in the right for the inverted bands depicts much lower resistance with a peak at h/2e
^{2}.

#### Figure 13-08z2 QSH Effect

[view large image]**Such QSH effect is observed one year after the theoretical prediction. Cheaper compounds based around bismuth are now available even in 3-D. Its prospects in applications such as quantum computing etc. has opened up a new class of sought-after material, e.g., the topological insulator.**-
**(6) Hall Effect (Anomalous) - The Anomalous Hall Effect (AHE) was observed in 1881 by the same Edwin Hall. It is similar to the classical Hall effect except that it does not require the application of external magnetic field (Figure 13-08z4,a). It occurs in ferromagnetic material with magnetization M, which behaves as a magnetic field in k-space (the modern material is topological insulator). An anomalous velocity is imparted to the spinning electrons curving to the left or right according to the direction of the spin. The depreciation of the number of spin up electrons is a reflection of de-population when the corresponding band is lifted up by ferro-magnetization (see Figure 13-08z4,a, which also shows the similarity of Classical Hall Effect / Anomalous Hall Effect and attributes the AHE to spin-orbit coupling). The more sophisticated quantum version in the following reveals more details with a minimum of mathematical formulas.**#### Figure 13-08z4 QAH Effect

[view large image]

**As shown earlier in QHE, by virtue of the Stokes' Theorem and the the definition of Chern number in low temperature physics, the Berry phase is just such number. It explains the steep transition of the conductivity in QAHE with the fact that the Chern number could change only in discrete step (from = 0 in the bulk to = 1 at the edge in this case).**

**Figure 13-08z4,c shows the relationship between conductivity**_{xy}and applied magnetic field B(T). The top one with various gate voltages V_{g}, and with various temperatures in the next two rows at V_{g}= 0.2V and -7.0V respectively. The red (blue) arrows and curves represent the process of increasing (decreasing) magnetic field. It illustrates the conductivity increases (decrease) in steep step near B(T) = 0 for all cases. The graphs also shows that the effect dissolves at high gate voltage and high temperature. Actually, most of the Hall effects have to be run at low temperature, because thermal agitation will destroy the rather minuscule interaction. The newly observed effect may have potential applications in future electronic devices.

## Figure 13-08v Hall Effect, Classical |
On the other hand with known electron density, it can be turned into a magnetometer to measure the magnetic field from the reading of V_{H} (Figure 13-08v,b). |

Figure 13-08z3 illustrates the origin of band structure in terms of the dependence of the energy E on the linear momentum k. The dashed line represents the completely free electron gas within the material. In reality, the electron does interact with the atomic nuclei producing the discrete segment of curves with a gap in between. It is spatially corresponding to the site of the nuclei (see reciprocal separation of distance "n/a" at the k axis). The level of zero energy is often shifted up resulting in negative energy as shown in Figure 13-08z2,b. In addition, the mirror image in k are also displaced in units of 2n(/a) to produce valley and peak instead of disjointed segments.
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## Figure 13-08z3 Band Structure [view large image] |
See the definitions of various bands in Figure 13-08z2,d, and "Topological Insulator" for explicit derivation of 2-D spin-orbit coupling etc. |

**(7) Spin-Orbit Coupling in Topological Insulator (2-D, 3-D), and Topological Superconductor -**

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
^{2}k^{2}/2m (denote by k^{2} 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 k_{x}i k_{y} terms acting like magnetic field, see 2-D Hall Effect) : |

## Figure 13-08z6 3D Topological Insulator |

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). |

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.

- 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).
- 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 e
^{ikx}and the Bloch Wave u_{n,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 u_{n,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. - 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.
- 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.

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 E_{F} can be considered as
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## 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 E_{F} depending on the type of energy band). |

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## 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. |

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