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Graphene, discovered in 2004, is a two-dimensional array of carbon atoms arranged in a honeycomb lattice. The calculation of the energy spectrum in graphene reveals a linear dispersion relation around the corners of the Brillouin zone (K and K´ points):
 * $$E(\vec{q})\approx \pm v_F \hbar |\vec{q}|,\quad \vec{q}=\vec{k}-\vec{K}\; \text{or}\; \vec{q}=\vec{k} - \vec{K^\prime}, \qquad(1)$$

where vF represents the Fermi velocity, given by vF ≈ 106 m/s. This linear dispersion relation along with the two sublattices of graphene allows the Hamiltonian to be written as the relativistic Dirac equation
 * $$-i \hbar v_F \vec{\sigma} \cdot \nabla \psi (\vec{r}) = E \psi (\vec{r}), \qquad (2)$$

where and σ = (σx, σy) are the Pauli matrices. The spinor character, called “pseudospin”, is not based on real spin but results from the fact that there are two atoms in the unit cell of graphene (i.e. graphene lattice is made up of two interpenetrating triangular sublattices A and B as shown in Fig. 1). Carriers in graphene are said to behave as massless Dirac fermions. The term “massless” is used because the energy of a relativistic particle is given by, where m is the rest mass, p is the momentum, and c is the velocity (i.e. vF for the case of graphene). Since graphene has a linear dispersion relation of E = pc, then the rest mass of electrons in graphene must be zero. It is noted that although the rest mass is zero, the cyclotron mass is not zero and corresponds to mc = EF/vF2. The presence of massless Dirac fermions in graphene along with the pseudospin-related effects allow for the realization of unique quantum relativistic phenomena.

Conventional quantum Hall effect in a 2D electron gas
In 1980, Klitzing et al. found that when the electrons of a sample are well confined in two dimensions (e.g. semiconductor heterostructures) and in the condition of a strong magnetic field and low temperature, the measured Hall resistance ρH (i.e., ρxy) or Hall conductivity σxy is quantized
 * $$\rho_{xy}=\frac{U_y}{I_x}=\frac{h}{ie^2}=\frac{25812.8}{i}\Omega,\quad i=1,2,3... \qquad (3)$$

This effect is called integer quantum Hall effect (QHE). Interestingly, the values of the Hall resistivity are independent on the materials chosen in the measurements.

In fact, the quantum Hall effect is the characteristic of two-dimensional electron gases in strong magnetic fields. By solving the Schrödinger equation, two-dimensional confinement and strong magnetic fields will give rise to the emergence of discrete Landau levels
 * $$E_N=\left(N+\frac{1}{2}\right) \hbar\omega_c, \qquad (4)$$

where ωc is the cyclotron frequency, ωc = eB/mc. The degeneracy factor (nB) of each Landau level is nB = eB/ch. Additionally, in the presence of disorder in real materials Landau levels get broadened and localized states appear between different Landau levels, as shown in Fig. 2(a). When i Landau levels are just fully occupied, the carrier density satisfies n=inB. Then, if we start to increase the carrier density gradually (i.e., changing the gate voltage in experiments), extra electrons will first occupy the localized states between two successive Landau Levels, and they cannot participate in the conduction of currents. Thus, the Hall conductivity does not change [i.e., the existence of plateaus in Fig. 3(a)] until the carrier density is high enough so that extra electrons begin to occupy the (i+1)-th Landau Level and contribute to the conduction. Then, the Hall conductivity begins to change again and stops when this Landau level is fully occupied, as shown in Fig. 2(a).

The relation between (ρxy, ρxx) and (σxy, σxx) is given by
 * $$\sigma_{xy}=-\frac{\rho_{xy}}{\rho_{xx}^2+\rho_{xy}^2},\quad \sigma_{xx}=\frac{\rho_{xx}}{\rho_{xx}^2+\rho_{xy}^2}, \qquad (5)$$

where ρxx and σxx are the longitudinal resistivity and conductivity, respectively. By considering the spin degeneracy and possible sublattice degeneracy, we add a factor g to the conductivities and finally obtain their values in the quantized Hall regime (i.e., the presence of plateaus)
 * $$\sigma_{xy}=\mp i \frac{ge^2}{h},\quad \sigma_{xx}=0,\quad i=0,1,2... \qquad (6)$$

If we define a filling factor of Landau levels, ν = n/gnB = nhc/geB, as the ratio of the carrier density n to the total degeneracy of each Landau level gnB, then a quantized Hall conductivity σxy is always expected as long as the filling factor is around an integer, as shown in Fig. 3(a). Besides changing carrier density n, therefore, we can also change magnetic field B to make the filling factor be an integer and realize QHE.

QHE of Single-Layer Graphene
It was predicted that differing from the QHE in other two-dimensional electron system, Eq. (6), single-layer graphene should exhibit an anomalous QHE with half-integer filling factors, 9,10 i.e., the Hall conductivity is given by
 * $$\sigma_{xy}=\mp \left(i+\frac{1}{2}\right) \frac{ge^2}{h},\quad i=0,1,2..., \qquad (7)$$

where g=2×2 for graphene due to spin and sublattice degeneracy. To explain and understand this anomalous QHE, it is necessary to recognize the unique electronic structure of graphene whose quasi-particle excitation is governed by the massless Dirac equation and the presence of relativistic Landau levels. Since graphene consists of two nonequivalent sublattice A and B, there are two nonequivalent corners K and K´ (named Dirac point) in the first Brillouin zone (BZ). Close to the K or K´ point, the dispersion relation is linear, Eq. (1). It is interesting to notice that the Fermi velocity does not depend on the energy or momentum and the linear dispersion relation will make low energy electron or hole excitations have zero effective mass. DiVincenzo and Mele 11 and Semenoff 12 noticed that even though vF is much smaller than c, such massless excitations are still governed by a two-dimensional Dirac equation of relativistic quantum mechanics, and its form is given by Eq. (2). We reiterate that ψ is like a two-component spinor whose spin degree of freedom indicates the sublattices and is called the “pseudospin” to distinguish it from the real electron spin.13 The massless Dirac equation for the momentum around K’ has the same form as in Eq. (2) except for replacing (σx, σy) by (σx, -σy).2 When a uniform magnetic field B is applied perpendicular to the graphene plane, the operator -iħ∇ in Eq. (2) should be replaced by -iħ∇+eA/c. If we take the Landau gauge: A = B(-y, 0) and write the wave function as the form of ψ(x, y) = exp(ikx)ϕ(y), then the Dirac equation will read .	(8) Fig. 2: Different types of Landau quantization. Density of states of Landau levels for (a) conventional 2D electron system, (b) massless Dirac fermions in single-layer graphene, and (c) massive Dirac fermions in bilayer graphene. Red and blue colors are for electrons and holes, respectively.14 © 2007 Nature Publishing Group. After defining2 (9)	    	(10) then Eq. (8) becomes (11) Since the canonical communication relation: [, ] = 1 is satisfied, the eigenvectors can be constructed from the eigenfunctions of a harmonic oscillator, φN. The obtained eigenvectors and energy eigenvalues are given by2,3 (12) These discrete energy eigenvalues are the relativistic Landau levels, where plus and minus sign correspond to electron and hole excitations, respectively. At the opposite Dirac point, K´, we can obtain the same spectrum and hence each Landau level is doubly degenerate. An important difference from the non-relativistic Landau levels is the existence of a zero energy state (N=0), as shown in Fig. 2(b). It belongs to both electron and hole excitations, and its degeneracy is twice smaller than any other Landau levels with |N| > 0 whatever for electrons or holes.10,15 For a single-layer graphene, therefore, when the i-th Landau level is just fully occupied, the relationship between the carrier density n and the degeneracy factor of Landau levels is n = (i+1/2)gnB, which will lead to the half-integer filling factor ν = i+1/2 in the Hall conductivity of Eq. (7). A schematic illustration of the Hall conductivity as a function of the carrier density (i.e., filling factor) is shown in Fig. 3(c). Compared with Eq. (6) and Fig. 3(a), it is obvious that the Hall conductivity has a shift of ½ in the unit of ge2/h. In 2005, the anomalous QHE of single-layer graphene was successfully observed in experiments.5,16 The measured Hall conductivity and longitudinal resistivity as a function of the carrier density are shown in Fig. 3(d). The results are in good agreement with the theoretical prediction in Fig. 3(c), and quantized plateaus still survive even at room temperature.17

C.	QHE of Bilayer Graphene For bilayer graphene, the nearest-neighbor tight-binding method predicts a gapless state with parabolic bands touching at the K and K’ points, instead of conical bands, and further analysis18 shows that quasi-particles in bilayer graphene can be described by using the effective Hamiltonian (13) Fig. 3: Comparison of different types of QHE. Schematic illustrations of the Hall conductivity as a function of filling factor for (a) the conventional 2D electron systems, (c) single-layer graphene, and (e) bilayer graphene. In (a),(c),(e), orange and blue peaks represent the density of state of Landau levels. (b) Recovered conventional QHE (green) measured in bilayer graphene after chemical doping. The curve of undoped bilayer graphene (red) is also shown for comparison. (d) The measured Hall conductivity and longitudinal resistivity versus the carrier density in single-layer graphene at T = 4 K and B = 14 T. (f) The measured Hall conductivity versus the carrier density in bilayer graphene at T = 4 K and B = 12 T and 20 T. (a),(c),(e),(f) are adapted from Ref. 19. © 2006 Nature Publishing Group. (b),(d) are adapted from Ref. 14. © 2007 Nature Publishing Group. where (px, py)=p=-iħ∇+eA/c. This Hamiltonian combines the off-diagonal structure, similar to the Dirac equation, but with Schrodinger-like terms p2/2m. The exact solution of this effective Hamiltonian indicates that the resulting quasi-particles are chiral, similar to massless Dirac fermions, but have a finite mass m≈0.054m0.18 The Landau levels of such “massive Dirac fermions” are given by18 , 	 (14) The zero energy state has two degenerate levels corresponding to N=0 and 1, and hence its density of states is twice than that of monolayer, which is clearly shown in Fig. 2(c). Figure 3(e) shows the theoretically predicted QHE of bilayer graphene. Due to the doubled density of states (2gnB) at the zero-energy Landau level, the quantized Hall conductivity change is two times of ge2/h when the Fermi level crosses the zero energy (neutrality point) by changing the carrier density (i.e., filling factor). For other Landau levels, however, the heights of the Hall conductivity steps are still ge2/h because their densities of states remain gnB. Thus, bilayer graphene has integer QHE , 	 (15) However, the zero-energy plateau that exists in conventional integer QHE is missing here. Experimental observations successfully demonstrated such an anomalous feature in bilayer graphene in 2006,19 as shown in Fig. 3(f). It is interesting that the conventional QHE with all the plateaus present can be recovered in chemical doped bilayer graphene by the electric field effect. Chemical doping can shift the neutrality point to high gate voltage Vg. So when the neutrality point is achieved, the applied electric field (i.e., gate voltage) is high enough to break the symmetry between two graphene layers and hence induce a semiconducting gap.20 This induced gap eliminates the additional degeneracy of the zero-energy Landau level and leads to the uninterrupted QHE sequence by splitting the double step into two.18 The experimental realization of such a recovery was done in 2007,21 and the measured Hall conductivity is shown by green curves in Fig. 3(b).

II. Klein Tunneling in Graphene A.	Klein Tunneling in Single-Layer Graphene Fig. 4: Tunnelling through a potential barrier in graphene. (a) Spectrum of quasiparticles in single-layer graphene emphasizing the crossing energy bands associated with sublattices A and B. The pseudospin is denoted by σ and is fixed along the respective branches of the spectrum. (b) Potential barrier of height V0 and width D. The three diagrams in (a) show the positions of the Fermienergy E across such a barrier. (c) Spectrum for quasiparticles in bilayer graphene. Despite its parabolicity, the spectrum originates from the intersection of energy bands formed by equivalent sublattices.22 © 2006 Nature Publishing Group. Oskar Klein in 1929 predicted that if the relativistic Dirac equation is applied to electrons incident on a potential barrier, then the electron tunneling will not undergo exponential dampening as the Schrödinger equation would suggest for non-relativistic quantum mechanics, but will achieve near-unity transmission as the potential barrier becomes of the order of the electron rest mass, mc2.23 This is counterintuitive since we would expect the transmission probability to approach zero, not one, as the barrier height increases towards infinity. This relativistic effect is due to the fact that positrons will be attracted to the potential barrier such that positrons can reside inside the barrier and match the energy of the electrons outside the barrier.22 Matching the wavefunctions of the electrons and positrons results in the high transmission probability through the barrier.24 According to quantum electrodynamics, states at positive and negative energies are linked and described by the same spinor wavefunction, which is a property of the Dirac equation known as charge-conjugation symmetry.22 For a long time this effect had not been observed experimentally because of the large energy barrier (~mc2) and electric field (>1016 V/cm) required.25 Two important aspects of graphene allow it to demonstrate the realization of Klein’s prediction. First, the linear dispersion relation, Eq. (1), around the K and K´ points means that carriers in graphene resemble massless Dirac fermions and are described by the Dirac equation. Second, graphene’s quasiparticles must be described by two-component wavefunctions in order to define the contributions from both sublattices. As previously mentioned, this sublattice association is analogous to the spin index in quantum electrodynamics and is thus often referred to as pseudospin. The conical energy spectrum at the Dirac points contains branches corresponding to each sublattice. Therefore, electrons and holes on the same branch may have opposite energies and propagate in opposing directions, but their pseudospin points in the same direction [parallel (antiparallel) to the electron (hole) momentum].22 In typical semiconductor physics, electrons and holes are governed by separate Schrödinger equations, but in graphene the crystal structure containing two sublattices means that electrons and holes are coupled akin to the charge-conjugation symmetry in relativistic quantum mechanics. Furthermore, we can now introduce the quantity known as chirality, which is the formal projection of  on the direction of motion , and is positive (negative) for electrons (holes). Chirality essentially represents the fact that k electrons and –k hole states are intricately connected because they originate from the same sublattice.14 The Klein tunneling experiment that can be setup in graphene is explained as follows. Assume a square potential barrier (Fig. 4) such that: .	(16) For an electron travelling in the positive x-direction with angle ϕ with respect to the x-axis then the wavefunction components are:22 (17) where kF is the Fermi wavevector, kx = kFcos ϕ and ky = kFsin ϕ are the wavevector components outside the barrier,,  is the refraction angle, s = sgn E, and s´ = sgn(E – V0). The wavefunction must be continuous so the reflection coefficient r is given by:22 (18) If the barrier is very large then we assume θ = 0 so the reflection probability R = |r|2 simplifies to	 ,	(19) and the transmission probability T = 1- R simplifies to22 .	(20) From the equation for the transmission probability we notice two important effects. First, the transmission probability is unity for ϕ = 0. Hence, the barrier is transparent and Klein tunneling occurs for all electrons of normal incidence. Secondly, a resonant pattern is apparent since T = 1 when qxD = πn where n is an integer. For a p¬-n junction in graphene, as shown in Fig. 5, the transmission probability is given by26 ,	(21) Fig. 5: Angular dependence of quasiparticle transmission through a graphene n-p junction.26 © 2006 The American Physical Society. where we assume the junction is smooth and θ is not too close to π/2 (note that θ here is the angle with respect to the x-axis). Indirect evidence of Klein tunneling in graphene was provided by comparing conductance measurements of graphene p-n junctions with theoretically predicted values.27 Direct evidence was provided by studying the conductance oscillations measured as a function of magnetic field and charge density for a graphene device with a narrow (~20 nm) gate used for creating a locally gated region (i.e., potential barrier).28 An expected phase shift at a specified magnetic field that corresponded to normally incident carriers signified perfect transmission through a potential barrier, and thus, direct experimental evidence of Klein tunneling.

B.	Veselago Lens in Graphene The fact that carriers in graphene behave as massless Dirac fermions provides the potential for fabrication of electronic lenses. For optics, different materials and interfaces are used in order to manipulate light. In typical semiconductors the most common interface is the p¬-n junction, but these junctions generally are not appropriate for manipulating an electron beam, as optics are for a beam of light. However, because of the physics associated with Klein tunneling as discussed above, p¬-n junctions are potentially transparent to electrons in graphene and may be used as lenses for electrons.29 The analogous Snell’s law for an electron travelling from a n-type region to a p-type region in graphene is given by29 (22) where c and v denote the conduction band and valence band (i.e. n-type region and p-type region) respectively. Since n is negative, the junction will focus an incoming divergent electron beam similar to the case for light refracted by metamaterials with refractive indexes of -1.30 Therefore, an n-p¬-n junction in graphene could be used as a Veselago lens (Fig. 6). By varying the charge-density in the top-gated p¬-type region the focal point changes to a cusp and can be shifted. Furthermore, a prism shaped top-gate can be used to form a beam splitter as shown in Fig. 6.

Fig. 6: (a) Veselago lens for electrons formed by graphene n-p-n junction. (b) and (c) Prism-shaped focusing beam splitter in the ballistic n-p-n junction in graphene.29 © 2007 American Association for the Advancement of Science. References: 1	K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306 (5296), 666 (2004). 2	A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Reviews of Modern Physics 81 (1), 109 (2009). 3	J. W. McClure, Physical Review 104 (3), 666 (1956). 4	P. Avouris, Nano Letters, DOI: 10.1021/nl102824h (2010). 5	K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438 (7065), 197 (2005). 6	K. v. Klitzing, G. Dorda, and M. Pepper, Physical Review Letters 45 (6), 494 (1980). 7	K. v. Klitzing, Reviews of Modern Physics 58 (3), 519 (1986). 8	R. B. Laughlin, Physical Review B 23 (10), 5632 (1981). 9	Y. S. Zheng and T. Ando, Physical Review B 65 (24) (2002). 10	V. P. Gusynin and S. G. Sharapov, Physical Review Letters 95 (14) (2005). 11	D. P. Divincenzo and E. J. Mele, Physical Review B 29 (4), 1685 (1984). 12	G. W. Semenoff, Physical Review Letters 53 (26), 2449 (1984). 13	C. W. J. Beenakker, Reviews of Modern Physics 80 (4), 1337 (2008). 14	A. K. Geim and K. S. Novoselov, Nature Materials 6 (3), 183 (2007). 15	M. I. Katsnelson, Materials Today 10 (1-2), 20 (2007). 16	Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature 438 (7065), 201 (2005). 17	K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, Science 315 (5817), 1379 (2007). 18	E. McCann and V. I. Fal'ko, Physical Review Letters 96 (8) (2006). 19	K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal'ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, Nature Physics 2 (3), 177 (2006). 20	E. McCann, Physical Review B 74 (16) (2006). 21	E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, Jmbl Dos Santos, J. Nilsson, F. Guinea, A. K. Geim, and A. H. C. Neto, Physical Review Letters 99 (2007). 22	M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature Physics 2 (9), 620 (2006). 23	 O. Klein, Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamikvon Dirac. Z. Phys. 53, 157–165 (1929). 24	P. Krekora, Q. Su, and R. Grobe, Physical Review Letters 92 (4), 040406 (2004). 25	G. Soff, T. Beier, M. Greiner, H. Persson, and G. Plunien, Advances in Quantum Chemistry, Vol 30 30, 125 (1998). 26	V. V. Cheianov and V. I. Fal'ko, Physical Review B 74 (4) (2006). 27	N. Stander, B. Huard, and D. Goldhaber-Gordon, Physical Review Letters 102 (2), 026807 (2009). 28	A. F. Young and P. Kim, Nature Physics 5 (3), 222 (2009). 29	V. V. Cheianov, V. Fal'ko, and B. L. Altshuler, Science 315 (5816), 1252 (2007). 30	J. B. Pendry, Physical Review Letters 85 (18), 3966 (2000).