Richardsonov model za superprevodna kovinska zrna ; Richardson's model in superconducting metallic grains

V delu predstavimo enoelektronske tranzistorje in kovinska zrna ter opišemo transport elektronov, ki zaporedoma tunelirajo skozi takšne naprave. Izpeljemo osnovne enačbe mikroskopske teorije superprevodnosti BCS ter predstavimo rezultate eksperimentov Ralpha, Blacka in Tinkhama z aluminijastim superprevodnim kovinskim zrnom. Uvedemo Richardsonov model za opis dogajanja v superprevodnih kovinskih zrnih in ga rešimo analitično, kar se prevede na reševanje sistema nelinearnih enačb. Te enačbe rešimo numerično in primerjamo rezultate z variacijsko rešitvijo. Ugotovimo, da se v limiti velikega kosa snovi rezultata obeh pristopov ujemata, za pravilen opis prehoda med superprevodnim in paramagnetnim stanjem pa je potrebno uporabiti analitično rešitev. Problem rešimo tudi z metodo renormalizacijske grupe gostotne matrike (DMRG), ki nam omogoča, da izračunamo nekatere zanimive korelacijske funkcije. Primerjava analitične in numerične rešitve z metodo DMRG potrdi veliko natančnost te metode. ; We present single-electron transistors, metallic grains and describe the sequential tunnelling of electrons through such devices. We derive the main equations of the BCS microscopic theory of superconductivity and present the results of the experiments of Ralph, Black and Tinkham done with alluminium superconducting metallic grain. We introduce the Richardson's model to describe the behaviour in superconducting metallic grains and solve it analitically, which amounts to solving a system of nonlinear equations. This is done numerically and the results are compared to the variational solution. We find that in the bulk limit both methods agree, however, to correctly describe the transition between the superconducting and normal states the analitical solution should be used. The problem is also solved with the use of density matrix renormalization group (DMRG) algorithm which allows us to calculate some interesting correlation functions. Comparing the analytical and DMRG results confirms the accuracy of this algorithm.