Artificial Control of Dot Sites for Ga Droplet Arrays on CaF ₂ Films by Surface Steps and Electron Beam Modifications

Abstract

Spontaneous loss of charge by charged black holes by means of pair-creation of charged Dirac particles is considered. We provide three examples of exact calculations for the spontaneous discharge process for 4D charged black holes by considering the process on three special non-rotating de Sitter black hole backgrounds, which allow to bring back the problem to a Kaluza-Klein reduction. Both the zeta-function approach and the transmission coefficient approach are taken into account. A comparison between the two methods is also provided, as well as a comparison with WKB results. In the case of non-zero temperature of the geometric background, we also discuss thermal effects on the discharge process.ArXiv ePrint: 0906.1520 JHEP08(2009)028 determines a negative energy potential which at classical level must be discarded. However, at quantum level, negative energy states must be included, and a quantum interpretation to this couple of potentials can be given. The positive energy potential determines the allowed positive energy states, whereas the negative energy potential determines the allowed negative energy states. The usual separation of these states occurring in absence of external fields is not ensured a priori, and there can be regions where an overlap of positive and negative states for the particle is allowed, i.e. the Klein paradox takes place. In these level-crossing regions, by means of tunneling between negative and positive states, pair production of charged particles can take place with a rate determined by the transmission probability for the particle to cross the forbidden region between the two potentials, and can be computed e.g. in the WKB approximation.We improved this semiclassical picture in the case of anti de Sitter Reissner-Nordström black holes showing that the potentials have a direct interpretation at the quantum level without referring to the classical H-J equation [11]. Then, for the class of de Sitter Reissner-Nordström black holes we found that level-crossing is always present, due to the peculiar occurrence of both a black hole event horizon and a cosmological event horizon [12], and we also considered a particular limit case, when the black hole horizon radius r + equates the cosmological horizon radius r c : the Nariai black hole [13][14][15]. The aforementioned class of solutions contains further limit cases, corresponding to the extremal cases r − = r + = r c , which are called ultracold solutions of type I and II [13,15]. A careful WKB analysis was also performed for the Nariai case and the ultracold ones.Herein, we develop our analysis of the pair-creation process associated with the black hole electrostatic field, and fully exploit the fact that the aforementioned special backgrounds allow an exact calculation of the vacuum instability. As a consequence, we can provide for the first time, to our knowledge, exact results for the instability of 4D charged black holes. We point out that our backgrounds are of a special character: in all the cases the geometry involved is the one o...