Negative static permittivity and violation of Kramers-Kronig relations in quasi-two-dimensional crystals

We investigate the wave-vector and frequency-dependent screening of the electric field in atomically thin (quasi-two-dimensional) crystals. For graphene and hexagonal boron nitride we find that, above a critical wave-vector $q_c$, the static permittivity $\varepsilon(q \! > \!q_c,\omega \! = \!0)$ becomes negative and the Kramers-Kronig relations do not hold for $\varepsilon(q \! > \! q_c,\omega)$. Thus, in quasi-two-dimensional crystals, we reveal the physical confirmation of a proposition put forward decades ago (Kirzhnits, 1976), allowing for the breakdown of Kramers-Kronig relations and for the negative static permittivity. In the vicinity of the critical wave-vector, we find a giant growth of the permittivity. Our results, obtained in the {\it ab initio} calculations using both the random-phase approximation and the adiabatic time-dependent local-density approximation, and further confirmed with a simple slab model, allow us to argue that the above properties, being exceptional in the three-dimensional case, are common to quasi-two-dimensional systems.Comment: 9 pages, 13 figure

Electronic excitations in quasi-2D crystals: What theoretical quantities are relevant to experiment?

The ab initio theory of electronic excitations in atomically thin [quasi-two-dimensional (Q2D)] crystals presents extra challenges in comparison to both the bulk and purely 2D cases. We argue that the conventionally used energy-loss function $-$Im $1/\epsilon({\bf q},\omega)$ (where $\epsilon$, ${\bf q}$, and $\omega$ are the dielectric function, the momentum, and the energy transfer, respectively) is not, generally speaking, the suitable quantity for the interpretation of the electron-energy loss spectroscopy (EELS) in the Q2D case, and we construct different functions pertinent to the EELS experiments on Q2D crystals. Secondly, we emphasize the importance and develop a convenient procedure of the elimination of the spurious inter-layer interaction inherent to the use of the 3D super-cell method for the calculation of excitations in Q2D crystals. Thirdly, we resolve the existing controversy in the interpretation of the so-called $\pi$ and $\pi+\sigma$ excitations in monolayer graphene by demonstrating that both dispersive collective excitations (plasmons) and non-dispersive single-particle (inter-band) transitions fall in the same energy ranges, where they strongly influence each other.Comment: 19 pages, 6 figure

A comparison theorem for nonsmooth nonlinear operators

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity $f$ is $L^p$ function with $p > 1$. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.Comment: 12 page

Oblique derivative problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge

We consider an oblique derivative problem in a wedge for nondivergence parabolic equations with discontinuous in $t$ coefficients. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces.Comment: 26 page