Semimetal-antiferromagnetic insulator transition in graphene induced by biaxial strain

We report first-principles calculations on the antiferromagnetic spin ordering in graphene under biaxial strain. Using hybrid functional calculations, we found that the semimetallic graphene sheets undergo a transition to antiferromagnetic insulators at a biaxial strain of 7.7$\%$ and that the band gap rapidly increases after the onset of this transition before reaching 0.9 eV at a biaxial strain of 12$\%$. We examined the competition of the antiferromagnetic spin ordering with two-dimensional Peierls distortions upon biaxial strain, and found that the preceding antiferromagnetic insulator phase impedes the Peierls insulator phase. The antiferromagnetic insulator phase is destabilized upon carrier filling but robust up to moderate carrier densities. This work indicates that biaxially strained graphene represents a noble system where the electron-electron and electron-lattice interactions compete with each other in a simple but nontrivial way.

Figure 1

Band gap opening in graphene by charge, spin, and bond ordering. (a) The charge and spin population of pristine graphene; each C $2p_z$ orbital has a single electron without spin asymmetry. (b) The charge-ordering-induced sublattice symmetry breaking of hexagonal BN sheets. (c) The spin-ordering-induced sublattice symmetry breaking of the antiferromagnetic insulator phase. (d) The bond ordering of the Peierls insulator phase (the inverse Kekulé distortion),[15] where the bond lengths of intracell and intercell bonds with respect to the Wigner-Seitz cell (red hexagon) differ. The inverse Kekulé distortion produces intervalley mixing potentials.

Figure 2

Semimetal to antiferromagnetic insulator transition in biaxially strained graphene. (a) The energy gain by antiferromagnetic ordering, $\Delta E_{\mathrm{AF}}$, and the staggered magnetic moment as a function of equibiaxial strain, $\varepsilon$. Inset, the spin density plot at $\varepsilon =10\%$. (b) The electronic band structure at $\varepsilon =10\%$. In this figure, every band is doubly degenerate, which corresponds to spin-up and spin-down components. (c) The wave functions of the valence band maximum and the conduction band minimum for each spin component. The radius and color of a circle at each atomic site reflect the wave function coefficients of the C $2p_z$ orbitals with the phase angle of 0, $2\pi /3$, and $4\pi /3$, which are represented by red, green, and blue, respectively. (d) The band gap as a function of $\varepsilon$.

Figure 3

Hopping integral as a function of $\varepsilon$. The hopping integral between the nearest-neighbor sites, $t$, is estimated from the band gap at the $M$ point, which amounts to $2t$ in the nearest-neighbor tight-binding model.[4] For semimetallic graphene, $t$ decreases almost linearly as $\varepsilon$ increases [i.e., $t=t_0(1-\alpha \varepsilon )$ with $t_0\approx 2.5$ eV and $\alpha \approx 1.5$].

Figure 4

Antiferromagnetic (AF) versus Peierls insulator phases in biaxially strained graphene. (a) The energy gain of the antiferromagnetic and Peierls insulator phases over the semimetal phase. (b) The band gap of each phase; both of the energy gain and the band gap exhibit a crossover at $\varepsilon =15.3\%$. (c) The energy gain of the Peierls insulator phase as a function of the relative intracell bond lengths; blue lines were obtained by spin-unpolarized calculations, and the red lines were obtained by spin-polarized calculations with antiferromagnetic spin ordering. The decrease and increase in the intracell bond lengths correspond to the inverse Kekulé (A) and Kekulé (B) distortions (upper panels), respectively. (d) The staggered magnetic moment of the antiferromagnetic insulator phase as a function of the lattice distortion.

Figure 5

Stress as a function of biaxial strain $\varepsilon$. Whereas the antiferromagnetic (AF) spin ordering only slightly changes the mechanical properties, the Peierls ordering (the inverse Kekulé distortion) decreases the mechanical instability point to $\varepsilon \approx 14.9$$\%$.

Figure 6

Filling-control metal-insulator transition of the antiferromagnetic insulator phase. (a) The band gap of the antiferromagnetic insulator phase as a function of carrier filling for $\varepsilon =$ 8–16$\%$. (b) The metal-insulator phase diagram in the plane of carrier density and biaxial strain, which was obtained from (a). Lattice distortions are not considered here.