This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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The electronic properties of graphene

Annalen der Physik

For arbitrary values of n and l quantum numbers, we present a simple exact analytical solution of the D-dimensional (D ≥ 2) hyperradial Schrodinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact bound state energy eigenvalues (E
 nl
 ) are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions (ψ
 nl
 (r)) are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for a few typical LiH, CH, HCl, CO, NO, O2, N2 and I2 diatomic molecules for various values of n and l quantum numbers. Numerical tests using the energy calculations for the inter dimensional degeneracy (D = 2 − 4) for I2, LiH, HCl, O2, NO and CO are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.

Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules

For any arbitrary values of $n$ and $l$ quantum numbers, we present a simple exact analytical solution of the $D$-dimensional ($D\geq 2$) hyperradial Schr% \"{o}dinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact energy levels $(E_{nl})$ of all the bound-states are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions $% (\psi_{nl}(r))$ are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for the typical diatomic molecules $LiH,$ $CH,$ $HCl,$ $CO,$ $NO,$ $O_{2},$ $N_{2}$ and $I_{2}$ for various values of $n$ and $l$ quantum numbers. Numerical tests using the energy calculations for the interdimensional degeneracy ($D=2-4$) for $I_{2}, $ $LiH,$ $HCl,$ $O_{2},$ $NO$ and $CO$ are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.

Exact Quantization Rule to the Kratzer-Type Potentials: An Application to the Diatomic Molecules

The solutions of the Klein–Gordon equation with a Coulomb plus scalar potential in D dimensions are exactly obtained. The energy E(n,l,D) is analytically presented and the dependence of the energy E(n,l,D) on the dimension D is analyzed in some detail. The positive energy E(n,0,D) first decreases and then increases with increasing dimension D. The positive energy E(n,l D)(l≠0) increases with increasing dimension D. The dependences of the negative energies E(n,0,D) and E(n,l,D)(l≠0) on the dimension D are opposite to those of the corresponding positive energies E(n,0,D) and E(n,l,D)(l≠0). It is found that the energy E(n,0,D) is symmetric with respect to D=2 for D∈(0,4). It is also found that the energy E(n,l,D)(l≠0) is almost independent of the angular momentum quantum number l for large D and is completely independent of the angular momentum quantum number l if the Coulomb potential is equal to the scalar one. The energy E(n,l D) is almost overlapping for large D.

The klein–gordon equation with a coulomb plus scalar potential in d dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number ${n}_{j}$ of the bound states and the sum of the phase shifts ${\ensuremath{\eta}}_{j}(\ifmmode\pm\else\textpm\fi{}M)$ of the scattering states with the angular momentum $j$: ${\ensuremath{\eta}}_{j}(M)+{\ensuremath{\eta}}_{j}(\ensuremath{-}M)={\left({{(n}_{j}+1)\ensuremath{\pi},\mathrm{when}\mathrm{}\mathrm{a}\mathrm{}\mathrm{half}\mathrm{}\mathrm{bound}\mathrm{}\mathrm{state}\mathrm{}\mathrm{occurs}\mathrm{}\mathrm{at} E=M \mathrm{and} j=3/2 or \ensuremath{-}1/2}{{(n}_{j}+1)\ensuremath{\pi},\mathrm{when}\mathrm{}\mathrm{a}\mathrm{}\mathrm{half}\mathrm{}\mathrm{bound}\mathrm{}\mathrm{state}\mathrm{}\mathrm{occurs}\mathrm{}\mathrm{at} E=\ensuremath{-}M \mathrm{and} j=1/2 or \ensuremath{-}3/2}{{n}_{j}\ensuremath{\pi} ,\mathrm{the}\mathrm{}\mathrm{remaining}\mathrm{}\mathrm{cases}.}\right)$The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half-bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.

https://arxiv.org/pdf/quant-ph/9806006.pdf

Relativistic Levinson theorem in two dimensions

The irreducible bases of the icosahedral double groups I′ and I′h are explicitly presented in their respective group spaces. Applying these bases to the spin states |j, μ〉, we obtain a simple formula for combining the spin states into the symmetry-adapted bases which belong to a given row of given irreducible representations of I′ and I′h.

Correlations of spin states for icosahedral double group

The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established It is shown that $N_{m}\pi=\pi (n_{m}^{+}-n_{m}^{-})= [\delta_{m}(M)+\beta_{1}]-[\delta_{m}(-M)+\beta_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $\delta_{m}$ is named phase shifts The constants $\beta_{1}$ and $\beta_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$

Levinson's theorem for the Klein-Gordon equation in two dimensions

In the light of the Sturm-Liouville theorem, the Levinson theorem for the Schrodinger equation with both local and non-local cylindrically symmetric potentials is studied. It is proved that the two-dimensional Levinson theorem holds for the case with both local and non-local cylindrically symmetric cut-off potentials, which is not necessarily separable. In addition, the problems related to the positive-energy bound states and the physically redundant state are also discussed.

Xi Wen Hou

Papers

Relativistic Levinson theorem in two dimensions

Correlations of spin states for icosahedral double group

Levinson's theorem for the Klein-Gordon equation in two dimensions

Levinson's theorem for non-local interactions in two dimensions

Overtone spectra and intensities of tetrahedral molecules in boson-realization models