Magnetic phases in periodically rippled graphene

We study the effects that ripples induce on the electrical and magnetic properties of graphene. The variation of the interatomic distance created by the ripples translates in a modulation of the hopping parameter between carbon atoms. A tight-binding Hamiltonian including a Hubbard interaction term is solved self-consistently for ripples with different amplitudes and periods. We find that, for values of the Hubbard interaction $U$ above a critical value $U_C$, the system displays a superposition of local ferromagnetic and antiferromagnetic ordered states. Nonetheless, the global ferromagnetic order parameter is zero. The $U_C$ depends only on the product of the period and hopping amplitude modulation. When the Hubbard interaction is close to the critical value of the antiferromagnetic transition in pristine graphene, the antiferromagnetic order parameter becomes much larger than the ferromagnetic one, being the ground state similar to that of flat graphene.

Figure 1

(a) Pseudo-Landau levels appearing in rippled graphene. In the absence of interactions, $U=0$, there are eight zero-energy degenerated pseudo-Landau levels corresponding to states located in regions with positive, $B_{\mathrm{ef}}^{+}$, or negative, $B_{\mathrm{ef}}^{-}$ effective magnetic fields, with spin projection $\uparrow$ or $\downarrow$ and in Dirac points $\mathbf{K}$ or ${\mathbf{K}}^{\prime }$. In regions with $B_{\mathrm{ef}}^{+(-)}$, the wave functions have only amplitude in sublattice $A\left(B\right)$. A moderate interaction $U$ opens an exchange energy gap, favoring a local FM order in regions with $B_{\mathrm{ef}}^{+}$ and opposite polarized FM order in regions with $B_{\mathrm{ef}}^{-}$. In (b) we show schematically the real space magnetic order for moderate $U$. For larger values of the Hubbard interaction (c), rippled graphene gets a Néel order with a very weak FM modulation.

Figure 2

Critical Hubbard interaction for magnetic instabilities in rippled graphene, as function of the dimensionless parameter ${\overline{\delta }}_t=\frac{{\delta }_t}{\hslash v_{F}G}$. Dots with different colors correspond to different values of the period $L$. In the previous expression $G=2\pi /L$. In the inset we plot $U_C$ as function of ${\delta }_t$ and for different periods $L$.

Figure 3

(a) Graphene crystal structure, where $\mathbf{a}$ and $\mathbf{b}$ indicate the lattice vectors. $\mathbf{\delta }$ is the vector connecting the two atoms, A and B, in the unit cell. (b) Schematic view of a graphene ripple of period $2L$ and height $h_0$.

Figure 4

(a) Band structure near a Dirac point for graphene in the presence of a sinusoidal modulation of the hoping with period $L=246a$ and ${\delta }_t$ running from 0 to $0.1t$. (b) Same as (a) for $L=346a$ and ${\delta }_t=0.2t$. Continuous lines correspond to tight-binding results, whereas dashed lines are the dispersion obtained using Eq. (10). Inset in (b) shows, as function of the dimensionless parameter ${\overline{\delta }}_t$, the renormalized Fermi velocity at Dirac points for $L=178a$ (triangles) and $L=115a$ (squares) and values of ${\delta }_t$ ranging from zero to ${\delta }_t=0.2t$. The dashed line corresponds to expression Eq. (12).

Figure 5

Square of the wave functions obtained from the tight-binding calculation for wave vectors near the ${\mathbf{K}}^{\prime }$ [(a)–(b)] and $\mathbf{K}$ [(c)–(d)] points. The momentum $k_y$ is measured with respect to the Dirac points. For each $k_y$ we consider the two states, 1 and 2, closest to zero energy. Panels (a) and (c) correspond to the amplitude on sublattice $A$, whereas panels (b) and (d) correspond to amplitude on sublattice $B$. The results are obtained in the tight-binding approximation with a sinusoidal modulation of the hopping of period $L=346a$ and amplitude ${\delta }_t=0.2t$.

Figure 6

Ferromagnetic (top) and antiferromagnetic (bottom) local order parameter for a ripple with ${\delta }_t=0.1t_0, L=260a$ and $U=0.5t_0, t_0, 1.5t_0, 2t_0, 2.2t_0, 2.3t_0$, and $2.5t_0$.

Figure 7

Band structure near the Dirac point for graphene in the presence of a sinusoidal modulation of the hoping with period $L=246a, {\delta }_t=0.1t_0$ and different values of the Hubbard coupling. The bands are spin degenerated.