Abstract
Multi-point probability measures along with the dielectric function of Dirac Fermions in mono-layer graphene containing particle-particle and white-noise (out-plane) disorder interactions on an equal footing in the Thomas-Fermi-Dirac approximation is investigated. By calculating the one-body carrier density probability measure of the graphene sheet, we show that the density fluctuation (ζ−1) is related to the disorder strength (ni), the interaction parameter (rs) and the average density (bar{{boldsymbol{n}}}) via the relation {{boldsymbol{zeta }}}^{-{bf{1}}}{boldsymbol{propto }}{{boldsymbol{r}}}_{{boldsymbol{s}}}{{boldsymbol{n}}}_{{boldsymbol{i}}}^{{bf{2}}}{bar{{boldsymbol{n}}}}^{-{bf{1}}} for which bar{{boldsymbol{n}}}to {bf{0}} leads to strong density inhomogeneities, i.e. electron-hole puddles (EHPs), in agreement with the previous works. The general equation governing the two-body distribution probability is obtained and analyzed. We present the analytical solution for some limits which is used for calculating density-density response function. We show that the resulting function shows power-law behaviors in terms of ζ with fractional exponents which are reported. The disorder-averaged polarization operator is shown to be a decreasing function of momentum like ordinary 2D parabolic band systems. It is seen that a disorder-driven momentum qch emerges in the system which controls the behaviors of the screened potential. We show that in small densities an instability occurs in which imaginary part of the dielectric function becomes negative and the screened potential changes sign. Corresponding to this instability, some oscillations in charge density along with a screening-anti-screening transition are observed. These effects become dominant in very low densities, strong disorders and strong interactions, the state in which EHPs appear. The total charge probability measure is another quantity which has been investigated in this paper. The resulting equation is analytically solved for large carrier densities, which admits the calculation of arbitrary-point correlation function.
Highlights
Since the discovery of graphene as a hybrid between metal and insulator, this material with relativistic energy spectrum of zero-gap Dirac Fermions[1,2] attracts attention due to its unique electronic properties and prospective applications in nanoelectronics[3,4,5,6]
For the undoped graphene in one loop approximation of the polarization function, it is shown that the static dielectric function ε(q) is a constant[6,39] and no collective modes are allowed within the PRA approximation
In addition to the screening length found by others, there is a characteristic length scale for the screening, namely qch which is related to the disorder strength and inter-particle dimensionless interaction factor to interactions via qch ≡ be defined in the text π 42 and dnirs[2] in which ni d is the substrate is the disorder strength, rs is distance
Summary
It is worth noting that in obtaining the above differential Fn2 a task which is true only for non-zero finite Fns. The original charge equation has electron-hole symmetry for the case μ = 0 which should result to an electron-hole symmetric form of Pn. Our approximation (considering Gn as a constant) violated this symmetry. In the case positive ns for μ = 0 Both of these functions have decreasing ζ′ → ∞, Pn approaches continuously to Pnlarge ζ′ which admits behavior in terms of larger carrier densities that is consistent with larger mean densities 〈n〉. To facilitate the procedure we restrict the calculations to positive densities (so that sgn(n) |n| = n), having in mind that the same calculations should be done for negative densities (for which a minus sign is necessary) In this case ζ become very large and we have: dP dr α[∂n(
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