Theory of the magnetic-field-induced insulator in neutral graphene sheets

Recent experiments have demonstrated that neutral graphene sheets have an insulating ground state in the presence of an external magnetic field. We report on a $\pi$-band tight-binding-model Hartree-Fock calculation which examines the competition between distinct candidate insulating ground states. We conclude that for graphene sheets on substrates the ground state is most likely a field-induced spin-density wave and that a charge-density-wave state is possible for suspended samples. Neither of these density-wave states support gapless edge excitations.

Figure 1

(Color online) Upper panel: Schematic representation of valley polarized CDW, SDW and F spin-polarized broken-symmetry solutions that can be obtained in a self-consistent mean-field calculation of graphene under a perpendicular magnetic field at half filling. Each arrow represents the filling of one $n=0$ Landau level of a given spin and valley. Lower panel: From left to right $E^{\text{CDW}}-E^{\text{SDW}}$, $E^{\text{CDW}}-E^{\text{F}}$, and $E^{\text{SDW}}-E^{\text{F}}$ total-energy differences per electron in eV as a function of the on-site repulsion $U$ and ${\epsilon }_r$ obtained from a data mesh of $9\times 10$ points, calculated neglecting the Zeeman term and for a magnetic field of 792 T corresponding to 1/100 of a flux quantum per honeycomb hexagon. For smaller values of $U$ the CDW solutions are energetically favored whereas for larger values of $U$ the SDW solutions are favored in a wide range of ${\epsilon }_r$. The F solutions are never the lowest in energy. The zero energy red contour lines indicate degeneracy between two different solutions.

Figure 2

(Color online) One-dimensional representation of the dispersionless band structure of a graphene sheet under a strong magnetic field $B=440\text{T}$ represented in the momentum coordinate $k$ parallel to the narrower direction of the unit cell. The band gaps follow the $B^{1/2}$ scaling law expected from the continuum model. The red color is used to represent up spin while blue is used for down spin. Left Panel: Band structure for CDW, SDW, and F solutions obtained for $U=5\text{eV}$ and ${\epsilon }_r=4$. For these interaction parameters the SDW state has the lowest energy and the largest gap at the Fermi level. Right Panel: Band structures for $U=5\text{eV}$ and ${\epsilon }_r=2$. When the on-site repulsion is sufficiently weak the energetically favored solution corresponds to the CDW state and this solution then has the largest gap.

Figure 3

(Color online) Total energy per site differences $\Delta E_{tot}$, separated into kinetic energy $\Delta E_{kin}$, electrostatic energy $\Delta E_{DI}$, and exchange energy $\Delta E_{XI}$ contributions as a function of magnetic length ${\ell }_B$ in lattice constant $a=2.46\text{Å}$ units. The total-energy differences were fitted to a $C_{3/2}{B}^{3/2}+C_2{B}^2$ curve. The fitting parameters are listed in Table . Left panel: Energy differences between F and SDW solutions $\Delta E^{\text{F}/\text{SDW}}=E^{\text{F}}-E^{\text{SDW}}$. These results were obtained with interaction parameter values $U=5\text{eV}$ and ${\epsilon }_r=4$ for which SDW is the lowest energy configuration. The more negative values of exchange energy in the SDW state compensates the kinetic-energy penalty related to the inhomogeneous accumulation of the electron wave functions at alternating lattice sites. The electrostatic energy differences are zero thanks to the uniform electron density for both solutions. Right panel: Same as the previous figure but for $\Delta E^{\text{F}/\text{CDW}}=E^{\text{F}}-E^{\text{CDW}}$. The interaction parameters in this case are $U=5\text{eV}$ and ${\epsilon }_r=2$ for which CDW is the lowest-energy configuration. When the on-site repulsion $U$ is small enough that the electrostatic energy penalty for the inhomogeneous charge distribution is small, exchange is the main contribution driving the CDW instability. However, in the case illustrated here $U$ is so small that the electrostatic part of the Hamiltonian does play an important role in favoring the CDW state. The energy contributions follow a magnetic-field decay law that deviates more from $B^{3/2}$ than in the previous case because the on-site interaction $U$ plays an essential role.

Figure 4

(Color online) Tilt-angle $\theta$ dependence of the critical magnetic field required to induce a transition to the F state. The black solid curve and dashed blue curve represent the critical magnetic fields starting from the SDW and CDW states, respectively. In the CDW curve we observe a larger deviation from a simple $\text{cos}\left(\theta \right)$ law due to a stronger influence of lattice scale physics described by the $C_2$ coefficient in Eq. (3).

Figure 5

(Color online) Representation of the unit-cell choices for armchair and zigzag edge-terminated graphene nanoribbons.

Figure 6

(Color online) Band-structure and spin-density distributions in armchair ribbons calculated for $U=4\text{eV}$ and ${\epsilon }_r=4$ for a magnetic field corresponding to ${\ell }_B=4.24a$ $\left(B=606\text{T}\right)$ using a $100\times 2$ lattices in the unit cell and 80 $k$-point sampling in the Brillouin zone. The red color is used to represent up spin while blue is used for down spin. Upper row: Armchair ribbon band structures for the F, SDW, and CDW solutions. Only the F solution is metallic while the other two configurations are insulating. For this choice of parameters SDW is the energetically favored state but the energies of the other states are similar and their band gaps also have a similar size. Lower row: Spin-$\uparrow$ and spin-$\downarrow$ density distributions across one of the zigzag rows in the unit cell of an armchair ribbon. The influence of Landau-level mixing on the three solutions is apparent in the differences between the three sets of density distributions. In the case of SDW solutions Landau-level mixing enhances the local spin density while in the case of the CDW solution the magnitude of the charge-density oscillation is reduced by Landau-level mixing. The dashed horizontal lines represent the occupation in absence of Landau-level mixing.

Figure 7

(Color online) Quantum-hall edge domain-wall wave functions and energy bands corresponding to an F solution calculated for a zigzag graphene nanoribbon with 20 carbon atom pairs in the unit cell $\left(W=42.6\text{Å}\right)$ under a magnetic field of $B=5660\text{T}$. Left panel: In the upper figure we represent the edge-state wave functions for the occupied and unoccupied $\uparrow$-spin and $\downarrow$-spin states when hybridization is not included. The lower figure represents the coefficients of the hybridized domain-wall state. All wave functions are plotted at the Bloch wave vector indicated by arrows on the right panel. Right panel: Quantum-hall edge-state bands with collinear-spin states represented by black and red symbols and the noncollinear-spin edge states formed by optimizing the hybridization of up- and down-spin orbitals represented by the blue symbols. We observe a clear single-particle energy gap related to the energy gained by forming the noncollinear domain-wall structure.