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156 CHAPTER 10. FERMI LIQUID THEORY renormalised f-band pinned to Fermi level (0) k F E F k F Hybridisation between broad s, p, d-band and f-band causes large Fermi surface broad band from s, p, d orbitals k Figure 10.10: Schematic band structure for heavy fermion materials moments at high temperatures, as if the formation of band states could be ignored. On the other hand, the Curie form of the susceptibility does not extend all the way to zero temperature, but rather it crosses over the to a constant value in the low temperature limit, just as the Sommerfeld coefficient of the heat capacity did. This suggests that at low temperatures, a local moment picture of these materials is not appropriate, and we must instead consider them as heavy Fermi liquids. It is difficult to reconcile these two ways of thinking about the same material. 10.4.3 Renormalised band picture for heavy fermion systems A qualitative understanding of the origin of the heavy fermion state can be obtained by considering the hybridisation between the bands associated with the more extended s,p, and d-orbitals on the atoms and the bands which arise from the very tightly localised atomic f-orbitals. Because a partially filled f-orbital will always lie inside filled s, p and even d orbitals with a higher major quantum number, there is negligible hybridisation between f-orbitals on neighbouring atoms – they are just too far apart. This results in a very flat band from the atomic f-states. If we consider single-electron states naively, then we find that the f-band formed from the atomic f-orbitals is well below the chemical potential, and should therefore be completely full. In such a scheme there would be no local moments at high temperature and no heavy fermion behaviour at low temperature. The scheme fails, because it ignores the strong Coulomb repulsion between electrons sharing the same f-state. Instead, once a single electron has occupied an f-orbital, the energy cost for a second electron hopping onto the same orbital is very high, almost prohibitive. We can modify our single particle picture to ensure that the f-orbitals have an average occupancy of one, by renormalising
10.4. FERMI LIQUIDS AT THE LIMIT: HEAVY FERMIONS 157 Figure 10.11: Fermi surface sheets of heavy quasiparticles, detected by quantum oscillation measurements in UPt 3 . (or shifting) the energy of the f-states close to the chemical potential, near E F . This is the renormalised band picture, shown in Fig. 10.10. Similar to what happened in the band structure of copper in the Lent term problem, the renormalised, narrow f-band and the broad band arising from atomic s, p, and d orbitals hybridise, producing an anticrossing very close to the Fermi level. This causes the new dispersion E(k) to cross E F at a much reduced slope compared to the broad s, p, d-band. As the slope dE/dk gives the Fermi velocity v F and is inversely proportional to the effective mass, this scheme can explain the enhanced effective masses observed in materials in which partially occupied f-orbitals are present. Moreover, note that k F is larger for the hybridised system than it would be without the hybridisation. In fact, the volume of the Fermi surface in heavy fermion systems is large enough to contain not only the electrons on s, p, and d-bands but also the f-electrons.
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