Book contents
- Frontmatter
- Contents
- Preface
- Part I Phenomenologlcal theories of superconductivity
- Part II The microscopic theory of a uniform superconductor
- 25 The Cooper problem: pairing of two electrons above a filled Fermi sea
- 26 The BCS theory of the superconducting ground state
- 27 Elementary excitations: the Bogoliubov–Valatin transformation
- 28 Calculation of the thermodynamic properties using the Bogoliubov–Valatin method
- 29 Quasiparticle tunneling
- 30 Pair tunneling: the microscopic theory of the Josephson effects
- 31 Simplified discussion of pairing mechanisms
- 32 The effect of Coulomb repulsion on Tc
- 33 The two-band superconductor
- 34 Time-dependent perturbations
- 35 Nonequilibrium superconductivity
- Part III Nonuniform superconductivity
- Appendix A Identical particles and spin: the occupation number representation
- Appendix B Some calculations involving the BCS wavefunction
- Appendix C The gap as a perturbation through third order
- Superconducting transition temperature, thermodynamic critical field, Debye temperature and specific heat coefficient for the elements
- References
- Additional reading
- List of mathematical and physical symbols
- Index
27 - Elementary excitations: the Bogoliubov–Valatin transformation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Part I Phenomenologlcal theories of superconductivity
- Part II The microscopic theory of a uniform superconductor
- 25 The Cooper problem: pairing of two electrons above a filled Fermi sea
- 26 The BCS theory of the superconducting ground state
- 27 Elementary excitations: the Bogoliubov–Valatin transformation
- 28 Calculation of the thermodynamic properties using the Bogoliubov–Valatin method
- 29 Quasiparticle tunneling
- 30 Pair tunneling: the microscopic theory of the Josephson effects
- 31 Simplified discussion of pairing mechanisms
- 32 The effect of Coulomb repulsion on Tc
- 33 The two-band superconductor
- 34 Time-dependent perturbations
- 35 Nonequilibrium superconductivity
- Part III Nonuniform superconductivity
- Appendix A Identical particles and spin: the occupation number representation
- Appendix B Some calculations involving the BCS wavefunction
- Appendix C The gap as a perturbation through third order
- Superconducting transition temperature, thermodynamic critical field, Debye temperature and specific heat coefficient for the elements
- References
- Additional reading
- List of mathematical and physical symbols
- Index
Summary
We recall expressions (26.19) for the expectation value of the reduced Hamiltonian
Let us calculate the energy required to add an electron to the system in a state |k↓〉 assuming its companion state | –k↓〉 is empty. In order to do this we must: (i) account for the energy increase due to removing the amplitude of the pair associated with this wave vector k and (ii) add the energy of the lone electron introduced into the state | k〉. When we remove the bound pair from the ground state, according to (27.1) the energy of the system changes by an amount
(the factor 2 in the second term arises because the chosen pair state (k, – k) occurs twice since we have a double sum). Using Eq. (26.23) we may write the form (27.2) as
Adding to (27.3) the energy ξk of one (unbound) electron then yields the quasiparticle excitation energy,
where we used Eqs. (26.24) and (26.27). Thus the energy needed to add an electron in state k↓ is σk. If we calculate the energy required to remove an electron in a state – k↓ we also obtain σk. Note the minimum excitation energy is Ak; i.e., the excitation spectrum has an energy gap.
- Type
- Chapter
- Information
- Superconductivity , pp. 208 - 211Publisher: Cambridge University PressPrint publication year: 1999