Screening of Coulomb Impurities in Graphene

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, $Z|e|/r$, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter $Z\alpha$, where $\alpha$ is the "fine structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge $Z_{eff}$, determined by the localized induced charge. We find that an impurity with bare charge Z=1 remains subcritical, $Z_{eff} \alpha < 1/2$, for any $\alpha$, while impurities with $Z=2,3$ and higher can become supercritical at certain values of $\alpha$.Comment: 4 pages, 2 figure

The Induced Charge Generated By The Potential Well In Graphene

The induced charge density, $\rho_{ind}(\bm r)$, generated in graphene by the potential well of the finite radius $R$ is considered. The result for $\rho_{ind}(\bm r)$ is derived for large distances $r\gg R$. We also obtained the induced charges outside of the radius $r\gg R$ and inside of this radius for subcritical and supercritical regimes. The consideration is based on the convenient representation of the induced charge density via the Green's function of electron in the field.Comment: 12 pages, 2 figures, version published in Phys.Rev.

High-energy e+e−e^+e^- photoproduction cross section close to the end of spectrum

We consider the cross section of electron-positron pair production by a high-energy photon in a strong Coulomb field close to the end of electron or positron spectrum. We show that the cross section essentially differs from the result obtained in the Born approximation as well as form the result which takes into account the Coulomb corrections under assumption that both electron and positron are ultrarelativistic. The cross section of bremsstrahlung in a strong Coulomb field by a high-energy electron is also obtained in the region where the final electron is not ultrarelativistic.Comment: 20 pages, 4 figure

New quantum-mechanical phenomenon in a model of electron-electron interaction in graphene

A quantum mechanical model of two interacting electrons in graphene is considered. We concentrate on the case of zero total momentum of the pair. We show that the dynamics of the system is very unusual. Both stationary and time-dependent problems are considered. It is shown that the complete set of the wave functions with definite energy includes the new functions, previously overlooked. The time evolution of the wave packet, corresponding to the scattering problem setup, leads to the appearance of the localized state at large time. The asymptotics of this state is found analytically. We obtain the lower bound of the life time of this state, which is connected with the breakdown of the continuous model on the lattice scale. The estimate of this bound gives one a hope to observe the localized states in the experiment.Comment: 10 pages, 2 figure

Relativistic Coulomb Green's function in dd-dimensions

Using the operator method, the Green's functions of the Dirac and Klein-Gordon equations in the Coulomb potential $-Z\alpha/r$ are derived for the arbitrary space dimensionality $d$. Nonrelativistic and quasiclassical asymptotics of these Green's functions are considered in detail.Comment: 9 page

Corrections to the energy levels of a spin-zero particle bound in a strong field

Formulas for the corrections to the energy levels and wave functions of a spin-zero particle bound in a strong field are derived. General case of the sum of a Lorentz-scalar potential and zero component of a Lorentz-vector potential is considered. The forms of the corrections differ essentially from those for spin-1/2 particles. As an example of application of our results, we evaluated the electric polarizability of a ground state of a spin-zero particle bound in a strong Coulomb field.Comment: 7 pages, 1 figur