Tuning antiferromagnetism of vacancies with magnetic fields in graphene nanoflakes

Graphene nanoflakes are interesting because electrons are naturally confined in these quasi-zero-dimensional structures, whereas confinement in bulk graphene would require a band gap. Vacancies inside the graphene lattice lead to localized states and the spins of such localized states may be used for spintronics. We perform a tight-binding description of a nanoflake with two vacancies and include a perpendicular magnetic field via a Peierls phase. The tunnel coupling strength and from it the exchange coupling between the localized states can be obtained from the energy splitting between numerically calculated bonding and antibonding energy levels. This allows us to estimate the exchange coupling $J$, which governs the dynamics of coupled spins. We predict the possibility of switching in situ from $J>0$ to $J=0$ by tuning the magnetic field. In the former case, the ground state will be antiferromagnetic with Néel temperatures accessible by experiment.

Figure 1

Schematic of a hexagonal graphene nanoflake with zigzag (armchair) boundaries. Grey (and orange) dots—connected by solid (and dashed) lines—indicate the locations of carbon atoms. The zigzag (armchair) terminated flake is specified by the number of benzene rings (armchair sections) along each edge, $b \left(s\right)$, and the distance $d$ (in units of the atomic distance $a=1.42\AA$) of the vacancies, located at ${\mathit{r}}_{\mathrm{vac}}=(0,\pm y)$, from the Cartesian origin in the flake center. The sketched island has a $(b=4,d=2)\left[\right(s=2,d=2\left)\right]$ configuration. The vacancies (red dots) give rise to localized spin states (green shade) whose mutual dynamics is described by the exchange coupling $J$. A magnetic field $\mathit{B}\parallel {\mathit{e}}_z$ can be applied perpendicularly to the flake plane.

Figure 2

On-site probability densities for (a,b) an $\left(s=5,d=8\right)$ armchair terminated flake and (c,d) a $\left(b=10,d=10\right)$ zigzag terminated flake. For each, we plot the antibonding energy eigenstates $|1\rangle$ (a,c) as well as the localized vacancy states $|+y\rangle$ (b,d). Due to the restriction to nearest neighbor hopping, the on-site probability density of $|0\rangle$ is identical to the one of $|1\rangle$. The probability density of $|-y\rangle$ looks similar to the one of $|+y\rangle$, yet mirrored about the $x$-axis. There is a significant probability density at the edges for zigzag flakes but not for armchair flakes.

Figure 3

For $t^{\left(n\right)}=0 \left(n>1\right)$, we plot the exchange coupling $J$ for (a) armchair and (b) zigzag terminated flakes. In (c), we plot $J$ for zigzag terminated flakes and $t^{\left(n\right)}\ne 0 \left(n=1,2,3\right)$. The values of $J$ are listed in Tables 1, 2, and 3, respectively. Green circular (magenta square) markers correspond to $d=1+3n \left(d=2+3n\right)$, i.e., vacancy sites $(0,+y)$ with a nearest neighbor site at $(0,y+a)\left[\right(0,y-a\left)\right]$. Straight lines connect markers that belong to flakes of the same size, e.g., $s=3$ (a) or $b=15$ [(b) and (c)].

Figure 4

The exchange coupling $J$ (solid blue line and left axis) and the eigenenergies (dashed orange line and right axis) of the corresponding (anti-)bonding states are plotted against a magnetic field perpendicular to the plane of the flake. The flake parameters $(s,d)$ and $(b,d)$, respectively, the energy levels $E_0$ and $E_1$, as well as the magnetic field at which one flux quantum passes through the flake are indicated in the plots. The behavior of $J\left(B\right)$ is very specific and depends on flake size ($s$ or $b$), vacancy separation $\left(d\right)$, edge type, and $|t_{ij}^{\left(n\right)}|$ (see also Fig. 6). (a) and (b) show $J\left(B\right)$ for the configurations with numbers in boldface in Tables 1 and 2, respectively.

Figure 5

Results for a $(b=15,d=11)$ flake. (a) Two pairs of (anti-)bonding states satisfy the criteria (i)–(v). For the corresponding localized states, we plot the energy $\overline{E}$ (with respect to $E_0$) vs $\left|t\right|$. The hopping amplitude $\left|t\right|$ is maximal for the pair $\left\{\right|3\rangle ,|4\rangle \}$. (b)–(d) On-site probability densities of (b) the localized state $|+y\rangle$, (c) $|4\rangle$, and (d) $|3\rangle$. The on-site probability density of $|-y\rangle$ looks similar to the one of $|+y\rangle$, yet mirrored about the $x$ axis.

Figure 6

Similar to Fig. 2, but for the hopping amplitudes between neighbors up to third order. The exchange coupling $J$ (solid blue line) and the energy levels $E_m$ and $E_n$ of the corresponding (anti-)bonding states (dashed orange lines) are shown. Depending on the configuration of the zigzag terminated flake, $J$ can be tuned (a) only weaklyor (b) over an order of magnitude, or can be (c) switched off $\left(J=0\right)$ by tuning the spectrum into a degeneracy.