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| First introduced by M. Pollak,<ref>M. Pollak. Discuss. Faraday Soc. 50 (1970), p. 13
| | The Internet has become an extremely crowded place. Couple of years back the number of websites about the Web wasn"t as big because it is today. Today the number of sites online is in millions. In this example it is very hard for almost any website to stand-out of the group. No website today can endure without proper promotional efforts to create your website a well visited one. To make sure a regular movement of traffic towards your site you have to get it search-engine enhanced. Search Engine Optimization is vital nowadays. Learn further on our favorite partner portfolio - Click here: [http://pom.chem.uw.edu.pl/dokuwiki/doku.php?id=Enhance_Search_Engine_Rankings_Using_Online_Monopoly_Techniques_41652 enhance_search_engine_rankings_using_online_monopoly_techniques_41652 [POM Do]. No site may survive today without seo. Seo makes it possible to in getting the internet site discovered. Your site rating can be enhanced and this can be your key to online success. My co-worker found out about [http://about.me/recommendedreadingboard look at this] by searching the San Francisco Post. <br><br> |
| </ref> the '''Coulomb gap''' is a soft gap in the Single-Particle [[Density of states|Density of States]] (DOS) of a system of interacting localized electrons.
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| Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap.
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| == Theory ==
| | The biggest benefit of search engine optimization is the fact that your site ranking improves through search engine optimization. [http://www.ww117.com/get-search-engine-optimisation-seo-done-in-a-day-2/ Open In A New Browser] includes extra information concerning how to deal with this view. He or she gets the effect in few seconds, when someone enters a in the search bar of a search engine. The websites that look towards the top of this effect page get more odds of getting more readers. The visits of individuals on your own site are really important. Those internet sites, which have more visitors, are good gross earners. The earning of a website is quite much dependent on how many readers a website gets. If a site gets more visitors, the probability of increase in the amount of business are more. If this really is not the case, you"ll not manage to make sales. Getting more internet traffic through search engine marketing not just helps you in increasing your sales, it also helps in making money through lending promotional support to other websites. If you spot the links of other websites in your website, you may charge them with this service. Ergo in this way, search engine marketing makes it possible to to make money online. <br><br>Seo is vital. You can become search engine improved by publishing your site to the search engines. You need to change the content of your site too. Also, you can place the links of one"s site on some places on the web; this will help you in getting more guests. Seo will help you to make more income on line in this manner..<br><br>If you have any concerns pertaining to wherever and how to use [http://ugliestpage7555.pen.io child health], you can call us at the website. |
| At zero temperature, a classical treatment of a system gives an upper bound for the DOS near the [[Fermi]]-energy, first suggested by Efros and Shklovskii.<ref name="EfrosShklovskii1">A L Efros and B I Shklovskii , J. Phys. C8, L49 (1975)</ref> The argument is as follows:
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| Let us look at the [[ground state]] configuration of the system. Defining <math> E_i </math> as the energy of an [[electron]] at site <math> i </math>, due to the disorder and the Coulomb interaction with all other electrons (we define this both for occupied and unoccupied sites), it is easy to see that the energy needed to move an electron from an occupied site <math> i </math> to an unoccupied site <math> j </math> is given by the expression:
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| :<math>\Delta E=E_j-E_i-e^2/r_{ij} </math>.
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| The subtraction of the last term accounts for the fact that <math> E_j </math> contains a term due to the interaction with the electron present at site <math> i </math>, but after moving the electron this term should not be considered. It is easy to see from this that there exists an energy <math> E_f </math> such that all sites with energies above it are empty, and below it are full (this is the Fermi energy, but since we are dealing with a system with interactions it is not obvious a-priori that it is still well-defined).
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| Assume we have a finite single-particle DOS at the Fermi energy, <math> g(E_f) </math>. For every possible transfer of an electron from an occupied site i to an unoccupied site j, the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e., <math>\Delta E>=0 </math>.
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| Assuming we have a large system, let us consider all the sites with energies in the interval <math> [E_f-\epsilon, E_f+\epsilon]. </math> The number of these, by assumption, is <math> N= 2 \epsilon g(E_f). </math> As explained, <math> N/2</math> of these would be occupied, and the others unoccupied. Of all pairs of occupied and unoccupied sites, let us choose the one where the two are closest to each other. If we assume the sites are randomly distributed in space, we find that the distance between these two sites is of order:
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| <math> R \sim (N/V)^{-1/d} </math>, where d is the dimension of space.
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| Plugging the expression for <math> N </math> into the previous equation, we obtain the inequality:
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| <math> E_j-E_i-C e^2 (\epsilon g(E_f)/V)^{1/d} >0 </math> where <math> C</math> is a coefficient of order unity. Since <math> E_j-E_i <2\epsilon </math>, this inequality will necessarily be violated for small enough <math> \epsilon </math>. Hence, assuming a finite DOS at <math> E_f </math> led to a contradiction. Repeating the above calculation under the assumption that the DOS near <math> E_f </math> is proportional to <math> (E-E_f)^\alpha </math> shows that <math> \alpha>=d-1 </math>. This is an upper bound for the Coulomb gap. Efros <ref>A. L. Efros, J. Phys. C: Solid State Phys 9, 2021 (1976)</ref> considered single electron excitations, and obtained an integro-differential equation for the DOS, showing the Coulomb gap in fact follows the above equation (i.e., the upper bound is a tight bound).
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| Other treatments of the problem include a mean-field numerical approach,<ref>M. Grunewald, B. Pohlmann, L. Schweitzer, and D.Wurtz,J. Phys. C: Solid State Phys., 15, L1153 (1982)</ref> as well as more recent treatments such as,<ref>M. Muller and S. Pankov, Phys. Rev. B. 75, 144201 (2007)</ref> also verifying the upper bound suggested above is a tight bound. Many Monte-Carlo simulations were also performed,<ref>J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Lett. 49, 758 - 761 (1982)
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| </ref><ref>A. Mobius, M. Richter, and B. Drittler, Phys. Rev. B 45, 11568 (1992)</ref> some of them in disagreement with the result quoted above. Few works deal with the quantum aspect of the problem.<ref>G. Vignale, Phys. Rev. B 36, 8192(1987)
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| </ref>
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| == Experimental observations ==
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| Direct experimental confirmation of the gap has been done via tunneling experiments, which probed the single-particle DOS in two and three dimensions,.<ref>J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266 (1995)</ref><ref>V. Y. Butko, J. F. Ditusa, and P. W. Adams, Phys. Rev. Lett. 84, 1543 (2000).</ref> The experiments clearly showed a linear gap in two-dimensions, and a parabolic gap in three-dimensions.
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| Another experimental consequence of the Coulomb gap is found in the conductivity of samples in the localized regime.
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| The existence of a gap in the spectrum of excitations would result in a lowered conductivity than that predicted by [[Variable range hopping|Mott Variable range hopping]]. If one uses the analytical expression of the Single-Particle DOS in the Mott derivation, a universal <math> e^{-1/T^{1/2}} </math> is obtained, for any dimension.<ref>B. Shklovskii and A. Efros, Electronic properties of doped
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| semiconductors (Springer-Verlag, Berlin, 1984).</ref> The observation of this is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the Coulomb gap. The transition from Mott to so-called Efros-Shklovskii Variable Range Hopping has been observed experimentally for various systems.<ref>A. Y. Rogatchev and U. Mizutani, Phys. Rev. B. 61, 15550
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| (2000).</ref> Nevertheless, no rigorous derivation of the Efros-Shklovskii conductivity formula has been put forth, and in some experiments <math> e^{-1/T^{\alpha}} </math> behavior is observed, for alpha which fits neither the Mott nor the Efros-Shklovskii theories.
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| ==See also==
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| * [[Coulomb's Law]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| {{DEFAULTSORT:Coulomb Gap}}
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| [[Category:Condensed matter physics]]
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| [[Category:Statistical mechanics]]
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| [[Category:Concepts in physics]]
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