Theory of Interedge Superexchange in Zigzag Edge Magnetism

A graphene nanoribbon with zigzag edges has a gapped magnetic ground state with an antiferromagnetic interedge superexchange interaction. We present a theory based on asymptotic properties of the Dirac-model ribbon wave function which predicts $W^{-2}$ and $W^{-1}$ ribbon-width dependencies for the superexchange interaction strength and the charge gap, respectively. We find that, unlike the case of conventional atomic-scale superexchange, opposite spin orientations on opposite edges of the ribbon are favored by both kinetic and interaction energies.

Figure 1

Hubbard model mean-field calculations for ${\gamma }_0=2.6\mathrm{eV}$ and $U=2\mathrm{eV}$ for a zigzag ribbon with $N=20$. Left: Mean-field energy bands for the AFM state: $E^{\mathrm{AFM}}(k)=\pm [{\Delta }^{\mathrm{AFM}}(k{)}^2+t^2(k){]}^{1/2}$ for both spins with the low-energy states of one spin concentrated on one side of the ribbon and the low-energy states of the opposite spin concentrated on the opposite side. $t(k)$, the left-right hopping parameter, is quite insignificant for this ribbon width in the edge-state region. ${\Delta }^{\mathrm{AFM}}(k)$ is dominant in the edge-state region because of large local spin polarizations. Right: Mean-field energy bands for the FM state: $E_{\sigma }^{\mathrm{FM}}(k)=\sigma {\Delta }^{\mathrm{FM}}(k)\pm |t(k)|$. Note that ${\Delta }^{\mathrm{AFM}}$ and ${\Delta }^{\mathrm{FM}}$ are nearly identical. The bands are periodic with periodicity $2\pi /a$ and inversion symmetric.

Figure 2

Left: Dependence of ${\tilde{\Delta }}^{\mathrm{AFM}}$ and ${\gamma }_0\tilde{t}$ on the scaled coordinate $\tilde{q}=qW$ and the corresponding $N=60$ AFM state quasiparticle bands. Note that the self-consistently calculated ${\tilde{\Delta }}^{\mathrm{AFM}}$ approaches a well-defined function at large $N$. The positions $k_c$ and $k^{*}$ are, respectively, the values of $k$ at which ${\Delta }^{\mathrm{AFM}}=t$ and the band gap minimum occurs. Right: Scaling collapse of antiferromagnetic state self-consistent left-right polarization $\tilde{P}$ and symmetric-antisymmetric polarization $\tilde{T}$ represented in the scaled coordinate $\tilde{q}$. Note that $P^2+T^2=1$ by definition.

Figure 3

$k$-resolved contributions to the kinetic (${\tilde{\epsilon }}_T$) (upper panel) and exchange (lower panel) energies (${\tilde{\epsilon }}_X$) of the FM and AFM states and the corresponding $\mathrm{FM}-\mathrm{AFM}$ differences as a function of the scaled momentum coordinate $\tilde{q}=qW$. ${\epsilon }_T^{\mathrm{FM}}$ and ${\epsilon }_T^{\mathrm{AFM}}$ are the integrands in the kinetic energy expression Eq. (13), and ${\epsilon }_X^{\mathrm{FM}/\mathrm{AFM}}$ is the corresponding quantity for the exchange interaction energy. The discontinuities in the ferromagnetic case are due to the crossing between the majority-spin symmetric and minority-spin antisymmetric bands at $k_c$, indicated by a thin vertical line. Although the kinetic energies roughly double the interaction energies at most $k$ values, the exchange contribution to superexchange is much larger because of weaker cancellation between $|k|<k_c$ and $|k|>k_c$ regions.

Figure 4

The difference between the FM and AFM states in total energy ($\Delta E$), exchange energy ($\Delta E_X$), and kinetic energy ($\Delta T$) plotted on a linear scale (left) and a logarithmic scale (right). The total energy difference follows a $W^{-2}$ decay law at large $W$ and is dominated by exchange energy contribution. The kinetic energy contribution is substantially smaller and the asymptotic decay law develops only for sufficiently large ribbon width. The total energy $\Delta E$ was fitted with $2.7/(W^2+280)$, the exchange energy $\Delta E_X$ with $2.1/(W^2+100)$, and the fitting for the kinetic contribution was obtained from the difference between both resulting in $0.6/W^2$ (represented with $\times 20$ magnification), all terms given in $\mathrm{eV}{\AA }^{-2}$ units. 12 K $k$ points were sampled in the 1D Brillouin zone.