Electron-solid and electron-liquid phases in graphene

We investigate the competition between electron-solid and quantum-liquid phases in graphene, which arise in partially filled Landau levels. The differences in the wave function describing the electrons in the presence of a perpendicular magnetic field in graphene with respect to the conventional semiconductors, such as GaAs, can be captured in a form factor which carries the Landau-level index. This leads to a quantitative difference in the electron-solid and -liquid energies. For the lowest Landau level, there is no difference in the wave function of relativistic and nonrelativistic systems. We compute the cohesive energy of the solid phase analytically using a Hartree-Fock Hamiltonian. The liquid energies are computed analytically as well as numerically, using exact diagonalization. We find that the liquid phase dominates in the n=1 Landau level, whereas the Wigner crystal and electron-bubble phases become more prominent in the n=2 and 3 Landau level.