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.
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