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84 CHAPTER 4. HONEYCOMB LATTICE
superconducting alkali-metal GIC’s is that they are formed from constituents
which are not superconducting, yet upon intercalation they undergo normal-tosuperconducting
transitions. There exists a few materials which have honeycomb
lattice geometry and display superconductivity. One of these is the phonon mediated
BCS superconductor MgB 2 with the unusually high transition temperature
of 39 K [96]. Another example are the intercalated graphite compounds, e.g. the
recent discovery of superconductivity in C 6 Ca with transition temperature 11.5
K [97]. A detailed microscopic understanding of the superconducting mechanism
is not known yet for these compounds but interlayer states seem to play an important
role suggesting a more 3D behavior [98, 99]. There is also experimental
indication for intrinsic superconductivity in ropes of carbon nanotubes [100]. It is
worth noting that observation has been made of superconductivity in alkali-metal
doped C 60 [101] and this work has attracted a good deal of attention because of
the relatively high T c in K 3 C 60 .
The honeycomb lattice, like the square lattice, is bipartite and we would
expect that it would fit very well with the Néel state (one variety of each of the
spin lying respectively on each of the sublattice). However, in the square lattice,
the Néel magnetism is also associated with a perfect nesting of the fermi surface
(the nesting vector is the Q =(π, π) vector, which leads to a spin density wave
of vector Q). However, what is somewhat different in the honeycomb lattice
is that the Fermi surface at half-filling consists only of two points (see Fig.4.1).
Moreover, it is worth noting that the free electron density of states vanishes at the
Fermi surface. Therefore, the question regarding of the presence of magnetism at
half-filling remains an interesting one that we propose to address in this chapter.
This behavior of the free electron density of states is responsible for a Mott
metal-insulator transition in the half-filled Hubbard model on the honeycomb
lattice at a finite critical on-site repulsion of about U cr ≈ 3.6 t [102,103,104,105].
Therefore the system is a paramagnetic metal below U cr ,andaboveU cr an antiferromagnetic
insulator. In contrast the square lattice has a van Hove singularity in
the density of states at half–filling which leads to an antiferromagnetic insulator
already for an infinitesimal on–site repulsion.
As mentioned before, the t−J model is the effective model of the Hubbard
model in the limit of large on-site repulsion, and at half-filling it coincides with
the Heisenberg model. For both the square and the honeycomb lattices, the
Heisenberg model has an antiferromagnetic ground state. However the quantum
fluctuations are larger in the honeycomb lattice [106] due to the lower connectivity.
Upon doping, quite different properties are expected for the honeycomb
lattice as compared to the square lattice, since it has a van Hove singularity in
the free electron density of states at fillings 3/8 and 5/8. At these fillings the
system is expected to undergo spin density wave (SDW) instabilities, whereas in
the square lattice d-wave superconductivity can extend without disturbance up
to fillings of 1.4.
In that respect, the honeycomb lattice has more similarities with the hole