Impurity-induced magnetic moments on the graphene-lattice Hubbard model: An inhomogeneous cluster dynamical mean-field theory study

Defect-induced magnetic moments are at the center of the research effort on spintronic applications of graphene. Here, we study the problem of a nonmagnetic impurity in graphene with a new theoretical method, inhomogeneous cluster dynamical mean-field theory (I-CDMFT), which takes into account interaction-induced short-range correlations while allowing long-range inhomogeneities. The system is described by a Hubbard model on the honeycomb lattice. The impurity is modeled by a local potential. For a large enough potential, interactions induce local antiferromagnetic correlations around the impurity and a net total spin $\frac{1}{2}$ appears, in agreement with Lieb's theorem. Bound states caused by the impurity are visible in the local density of states (LDOS) and have their energies shifted by interactions in a spin-dependent way, leading to the antiferromagnetic correlations. Our results take into account dynamical correlations; nevertheless they qualitatively agree with previous mean-field and density functional theory (DFT) studies. Moreover, they provide a relation between impurity potential and on-site repulsion $U$ that could in principle be used to determine experimentally the value of $U$.

Figure 1

The graphene lattice with an impurity at site $i=0$. The two-site basis, with sublattice $a$ (blue) and $b$ (red), as well as the lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, are defined at the bottom-right corner.

Figure 2

Evolution of the LDOS at (a) the impurity site, (b) at a site next to the impurity, (c) at a second neighbor site on the sublattice $a$, and (d) at a third neighbor site on sublattice $b$ as a function of impurity potential ${\epsilon }_0$. Note that the $y$-axis scale is broken and has a larger scale on the upper panel of (a) to fit in the very large peaks.

Figure 3

Cluster-bath system used in this work. The numbered blue circles are the cluster sites per se. The different bath-cluster hybridization terms are indicated by colored links.

Figure 4

The supercluster used in this work. The bath sites of each of the 19 six-site clusters are hidden for clarity. The superlattice basis vectors are indicated with the green arrows. Intercluster links are the brown dotted lines.

Figure 5

Density of states for the noninteracting case (blue, exact) and $U=2$ (red, CDMFT) in the absence of impurity. A Lorenzian broadening $\eta =0.05$ was used.

Figure 6

Net spin of each site in the I-CDMFT solution for $U=2$ and ${\epsilon }_0=11$, represented by colored circles (blue is down, red is up). The area of each circle is proportional to $\langle S_i\rangle$, which ranges from $0.00315$ to $0.0771$.

Figure 7

Antiferromagnetic order parameter per site $\langle \widehat{M}\rangle$ of the supercluster (top panel) and total spin $\langle \widehat{S}\rangle$ (bottom panel), as a function of impurity potential ${\epsilon }_0$. The inset shows the relation between $U$ and the threshold values ${\epsilon }_{0,c}$ (full line) and ${\epsilon }_{0,v}$ (dashed line). At $\epsilon \ge {\epsilon }_{0,c}$ the net spin takes a finite value and then, at $\epsilon \ge {\epsilon }_{0,v}$, it reaches $1/2$ where it saturates.

Figure 8

LDOS for sublattice $a$ (left panels) and for sublattice $b$ (right panels) for sites in the vicinity of the impurity. The impurity is on one of the sites of sublattice a. The distance from the impurity is color coded: the insets define the colors associated with each position around the impurity (black square). The LDOS for sites closer to the impurity is darker. [(a) and (d)] LDOS calculated from the $T$-matrix formalism presented in section 3 with ${\epsilon }_c=11$. [(b) and (e)] LDOS calculated with I-CDMFT, but without interactions ($U=0$, ${\epsilon }_c=11$). [(c) and (f)] LDOS calculated with I-CDMFT and interactions ($U=2$, ${\epsilon }_c=11$) for both up and down spin as indicated on the figure. A Lorenzian broadening $\eta =0.05$ is used and the sign of the LDOS is reversed for spin down in the lower panels.

Figure 9

Difference between the density of states for spin-up and spin-down for sublattices $a$ (blue) and $b$ (red). Each curve is the sum of the LDOS over all sites of a given sublattice within the supercluster (57 sites for each sublattice). The shading represents the sum over frequencies up to the Fermi level and highlights the contribution of each sublattice to the spin polarization.