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Comparison of the Richardson and BCS models of superconductivity based on calculations of ground state energies

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Authors
Kuusela, Petri
Date
2017
Discipline
Teoreettinen fysiikkaTheoretical Physics

 
Suprajohtavuuden Richardsonin ja BCS mallien vertailu perustilojen laskettujen energioiden perusteella. Suprajohtavuus on edelleen aktiivinen tutkimuksen alue, koska sen tarjoamista mahdollisuuksista huolimatta sitä ei vielä ymmärretä kokonaisvaltaisesti. Tässä pro gradu -tutkielmassa käyn läpi perusteet kahdelle suprajohtavuuden mallille, Richardsonin ja BCS malleille. BCS teoria on ensimmäinen ja käytetyin mikroskooppinen suprajohtavuuden teoria. Richardsonin malli on harvemmin käytetty malli, josta saadaan redusoidun BCS Hamiltonin operaattorin tarkat ominaistilat. Lasken molempien mallien perustilojen energiat sekä redusoidulle että täydelle BCS Hamiltonin operaattorille. Sama lasketaan myös yleiselle tarkan määrän vain pariutuneita elektroneja sisältävälle tilalle. Tälle eri Hamiltonin operaattorien energioiden erotus riippuu ainoastaan hiukkasmäärästä, ja siten ei ole merkitystä, kumpaa Hamiltonin operaattoria sellaisten tilojen energioiden vertailuissa käytetään. Energioiden vertailuista huomaan, että tasavälisten energiatilojen systeemillä ja kolmiulotteisella vapaiden elektronien systeemillä Richardsonin ja BCS mallien perustilojen energiat ovat lähellä toisiaan, Richardsonin mallin antaessa matalamman perustilan energian. Tästä päättelen, että BCS malli kuvaa hyvin näitä systeemejä. Kaksitilasysteemillä energioiden erotus on huomattava, Richardsonin mallin antaessa merkittävästi matalamman perustilan energian. Tämä on merkki siitä, että BCS malli ei kuvaa tällaista systeemiä hyvin. ...
 
Superconductivity remains an active area of research because there still is no comprehensive understanding of the phenomenon despite all the possibilities it offers. In this thesis I go through the basics of two models for superconductivity, the BCS model and the Richardson model. The BCS theory is the first and most used successful microscopic theory of superconductivity. Richardson model is a less used model which gives the exact eigenstates of the reduced BCS Hamiltonian. I calculate the ground state energies for both the reduced and the full BCS Hamiltonian for both models. This is done for a general paired state with fixed number of electrons as well. For this state I find that the difference between the full and reduced Hamiltonian energies depends only on the number of electrons, and thus conclude that it does not matter for comparison between such states which Hamiltonian is used. I find that in a system with equally spaced energies and in the free-electron three-dimensional system the ground state energies are very close to each other, with the Richardson model ground state energy being lower. From this I infer that the BCS model is a good description of these systems. The ground state energies of a two-level system however differs considerably, with the Richardson model ground state energy being significantly lower. This is an indication that the BCS model is not suitable for describing this system. ...
 
Keywords
Richardsonin malli superconductivity Richardson model ground state energy suprajohtavuus
URI

http://urn.fi/URN:NBN:fi:jyu-201711134240

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