naturally when we consider phonon-mediated pairing. This kind of superconduc- tivity typically occurs in simple metals, where the conduction electrons can move nearly freely through the lattice. In the case of strongly correlated electron systems however, where the electrons may be more strongly localized, or have a largely en- hanced effective mass, the retardation of the electron-phonon interaction is no longer sufficient to overcome the Coulomb repulsion between the electrons. Nevertheless
1
The superconducting coherence volume is defined asξaξbξcwhereξa,ξbandξcare the coherence
lengths along the crystallographic axes. These correspond to the spatial extent of the Cooper pair along the respective axis.
2The interaction potential is of the formV
kk0 = Ω−1RV(r)ei(k’−k).rdr, where Ω is the normal-
superconductivity does occur in some strongly correlated materials, for example in the case of the heavy fermion superconductors or high-Tc cuprates. In these cases
the superconductivity still results from the pairing up of electrons, as the flux quan- tisation is shown to correspond to charge carriers of charge 2e [59], however the pairing mechanism is different.
Before moving on to discuss alternative possible pairing mechanisms, let us first consider the form of the Cooper pair wavefuction:
Ψ(r1, s1;r2, s2) =f(r1−r2)χ(s1, s2) (3.2.1)
which we have here written as a product of the pair’s orbital and spin wavefunctions,
f andχrespectively [60]. As a Cooper pair is made up of two indistinguishable spin- half particles, its wavefunction must be antisymmetric under particle exchange, i.e. Ψ(r1, s1;r2, s2) =−Ψ(r2, s2;r1, s1). The spin part can either have total spin S = 0
(singlet) or S = 1 (triplet). The orbital part can be written in terms of spherical harmonicsYlm, where its parity is given by (−1)l. The namess-, p-,d-, f-wave are given to the states with angular momentum l= 0,1,2,3, as per convention3.
The requirement that the overall Cooper pair wavefunction pick up a minus sign under particle exchange therefore leads to the following possible combinations of spin and spatial parts:
S= 0 =⇒ χ(s1, s2) = 1 √ 2(| ↑↓i − | ↓↑i), l= 0,2,4, ... S= 1 =⇒ χ(s1, s2) = | ↑↑i 1 √ 2(| ↑↓i+| ↓↑i), l= 1,3,5, ... | ↓↓i}
We call all pairing states with l > 0 unconventional. In general, angular momen- tum states with l > 0 are favored when the pairing mechanism is other than the electron-phonon interaction, as for such states there is a vanishing probability of the electrons encountering each other at the origin (asf(r)α rl, r →0), which reduces the Coulomb repulsion felt by the electrons. The spatial overlap of the electrons in thes-wave state is only possible because of the retarded nature of phonon-mediated pairing. A key difference between conventional and unconventional superconductiv- ity is that whereas the phase of the gap remains constant around the Fermi surface
3
In the presence of a crystal lattice,S and l are no longer good quantum numbers. Therefore technically the “lattice-free”s-,p-,d-,f-wave nomenclature is no longer accurate. However in the case of the materials studied in this thesis, spin-orbit coupling and lattice effects are weak enough that this treatment should provide a good approximation.
in the conventional case, it depends onkin the unconventional case such that:
X
k
∆(k) = 0, (3.2.2)
where the sum is around the Fermi surface.
∆(k), which is directly related to the Cooper pair wavefunction, is often referred to as the superconducting order parameter, in reference to Ginzburg Landau theory (which is discussed further in section 3.5.4). Indeed, it is a quantity which grows continuously from zero at a second order phase transition, in which all of the in- formation relating to the symmetries and spin configuration of the superconducting state are encoded.
Pairing in unconventional superconductors does not generally arise from inter- actions with the crystal lattice, but instead from electron-electron interactions. In their 1965 paper, Kohn and Luttinger showed that Cooper pairing could occur based solely on the repulsive Coulomb interaction [61], and that no weakly interacting elec- tronic system would remain normal down to absolute zero temperature. However this model leads to extremely low transition temperatures. Other types of pair- ing interactions, mainly based on spin-fluctuation exchanges, were later predicted. In these models the spins, which can either be localized or those of itinerant elec- trons, form a polarizable medium that can provide an attractive interaction between electrons, in a manner analogous to the polarizable crystal lattice for the electron- phonon interaction. These types of interactions are expected to occur in materials close to a magnetic instability. For example, in a 1986 paper which looks at the three-dimensional Hubbard model, it is shown that spin-fluctuation exchange occur- ring near a spin-density wave instability leads tod-wave pairing of electrons [62]. In general however there is no over-arching model which describes pairing in uncon- ventional superconductors. Instead these are often studied on a case by case basis, and proving what the pairing mechanism is in each material is a very active field of research.