Lattice theory of pseudospin ferromagnetism in bilayer graphene: Competing interaction-induced quantum Hall states

In mean-field theory, bilayer graphene’s massive Dirac fermion model has a family of broken-inversion-symmetry ground states with charge gaps and flavor-dependent spontaneous interlayer charge transfers. We use a lattice Hartree-Fock model to explore the lattice scale physics of graphene bilayers, which has a strong influence on ordering energy scales and on the competition between distinct ordered states. We find that inversion symmetry is still broken in the lattice model and estimate that the transferred areal densities are $~$${10}^{-5}$ electrons per carbon atom, that the associated energy gaps are $~$${10}^{-2}\hspace{0.5em}\mathrm{eV}$, that the ordering condensation energies are $~$${10}^{-7}$ eV per carbon atom, and that the differences in energy between competing ordered states is $~$${10}^{-9}$ eV per carbon atom. We find that states with a quantized valley Hall effect are lowest in energy, but that the coupling of an external magnetic field to spontaneous orbital moments favors the broken-time-reversal-symmetry states that have quantized anomalous Hall effects. Our theory predicts nonmonotonic behavior of the band gap at neutrality on the potential difference between layers, in qualitative agreement with recent experiments.

Figure 1

Left panel: Band structure of graphene represented in the primitive zone defined by the reciprocal lattice vectors ${\mathrm{b⃗}}_1$ and ${\mathrm{b⃗}}_2$ of the honeycomb lattice. The equilateral triangle represents the region in the primitive cell we need to sample in order to fully describe a system in which the symmetry between $K$ and $K^{\prime }$ valleys is broken. The smaller triangles enclose inequivalent regions in $k$ space associated with $K$ and $K^{\prime }$ valleys. Right panel: An example of coarse sampling of $k$ points in the primitive cell supplemented by a finer grid in hexagons generated around those coarse $k$ points near the $K$ and $K^{\prime }$ valleys. The illustrated example consists of $32\times 32$ coarse points and a $16$-fold enhancement of density resulting in an effective sampling of $512\times 512$ points in the primitive cell near the valleys.

Figure 2

Left panel: Noninteracting biased tight-binding model (dotted line) and unbiased Hartree-Fock (solid line) band structures for bilayer graphene for $k$ moving away from the $\Gamma$ point toward the Dirac point $K$. The broken-symmetry-state bands are compared with noninteracting electron bands with inversion symmetry explicitly broken by an externally applied electric field $\mathcal{E}=0.1$ V/nm. The interacting system bands exhibit enhanced velocities and a band gap due to spontaneously broken inversion symmetry. These results were obtained for a model with hopping parameters ${\gamma }_0=-3.12\hspace{0.5em}\mathrm{eV}$, ${\gamma }_1=-0.377\hspace{0.5em}\mathrm{eV}$, ${\gamma }_3={\gamma }_4=0$, and dielectric screening parameter ${\varepsilon }_r=4$. Right panel: On-site $k$-dependent exchange potentials of the spontaneously broken-symmetry state on the four bilayer sublattices. The on-site potential is larger in magnitude on the low-energy sites that do not have an opposite-layer neighbor, and lower on average in the layer with the larger density.

Figure 3

Left panel: Berry curvatures associated with the two valence bands for a Hartree-Fock (HF) solution, and for noninteracting bilayer graphene in the presence of an interlayer bias. We use a thicker line to represent the curvature of the low-energy band and a thinner line for the band farther from the Fermi level. The small remote band contribution has been magnified $\times 20$. An interlayer electric field of $\mathcal{E}=0.1$ V/nm added to the noninteracting electron model gives a band gap comparable to the one which emerges from our mean-field calculations with ${\varepsilon }_r=4$. Right panel: Orbital magnetization contributions from the two valence bands in the vicinity of a valley point. We compare results for the self-consistent broken-symmetry states with those for noninteracting electrons with an external interlayer bias.

Figure 4

Temperature dependences of mean-field-theory charge transfer per valley spin and the associated band gap. Top panel: Total charge per unit area (${10}^{11}$ cm${}^{-2}$) transferred from one layer to another $\Delta n_l$ per each valley-spin polarized component as a function of temperature. We represent the dependence for different values of the relative dielectric constant ${\varepsilon }_r$. Middle panel: Mean-field-theory band gap as a function of temperature. Bottom panel: Ratio between the band gap and transferred charge density.

Figure 5

Bias dependence of the total energies of the system for different valley-spin polarization configurations for a system with ${\varepsilon }_r=4$. The system undergoes transitions from the AF to Fi and then to the F state as a function of external field, each one displaying clearly different quantum Hall conductivity properties. The origin of energy has been arbitrarily shifted for presentation convenience.

Figure 6

Upper panel: Each branch represents the bias dependence of the valley-resolved layer-polarization density associated with one valley-spin component. These results were obtained for a system with ${\varepsilon }_r=4$. The multiplication factors represent the number of times each branch needs to be repeated to give the total charge density to complete the contributions from the four components. Lower panel: Band gap as a function of bias for F, Fi, and AF states for a system with ${\varepsilon }_r=4$. We observe a competition in the band-gap opening between exchange and Hartree contributions which can lead to a reduction of the band gap as the bias is increased.

Figure 7

External bias dependence of layer-resolved charge density, band gaps, and sublattice charge asymmetry. We have considered the F configuration where all four valley-spin components are polarized toward the same layer. Top panel: Charge accumulation dependence as a function of bias in presence of Hartree and Fock terms and charge accumulation dependence as a function of bias in presence of Hartree screening only. The evolution of $\Delta n_l$ as a function of bias has a slightly larger slope in the Hartree-only approximation than the Hartree-Fock case. Middle panel: Band gaps obtained for Hartree-Fock and Hartree approximations using different values of ${\varepsilon }_r$ as a function of external electric field $\mathcal{E}$. We notice a substantial enhancement of the band gap when the exchange term is included. Bottom panel: Measure of deviation from neutrality in the charge differences for each one of the sublattices in a given layer normalized by the total sloshed charge. The asymmetric distribution of the charge between low-energy sublattice $A$ and high-energy sublattice $B$ of a given graphene layer is enhanced in the presence of electron exchange. This asymmetry becomes smaller as the external bias becomes stronger.