Vacancy-induced magnetism in graphene and graphene ribbons

We address the electronic structure and magnetic properties of vacancies and voids both in graphene and graphene ribbons. By using a mean-field Hubbard model, we study the appearance of magnetic textures associated with removing a single atom (vacancy) and multiple adjacent atoms (voids) as well as the magnetic interactions between them. A simple set of rules, based on the Lieb theorem, link the atomic structure and the spatial arrangement of the defects to the emerging magnetic order. The total spin $S$ of a given defect depends on its sublattice imbalance, but some defects with $S=0$ can still have local magnetic moments. The sublattice imbalance also determines whether the defects interact ferromagnetically or antiferromagnetically with one another and the range of these magnetic interactions is studied in some simple cases. We find that in semiconducting armchair ribbons and two-dimensional graphene without global sublattice imbalance, there is a maximum defect density above which local magnetization disappears. Interestingly, the electronic properties of semiconducting graphene ribbons with uncoupled local moments are very similar to those of diluted magnetic semiconductors, presenting giant Zeeman splitting.

Figure 1

Examples of voids with different sublattice imbalance charges in the middle of a graphene ribbon. From left to right, the associated imbalance charges are $N_I=0,-1$ (vacancy), and 2.

Figure 2

(Color online) (a) Magnetic moments on lattice sites around a single vacancy. Inset: Probability density of the zero-energy state built with Gaussian functions on lattice sites. (b) Same as in (a) but for a triangular void with $N_I=2$. Inset: Same as in (b) but summing over the two zero-energy states.

Figure 3

(Color online) Density of states near the Dirac point for an armchair ribbon of $W=7a$ with two vacancies presenting imbalance charge of different signs and same modulus $\left(N_I=\pm 1\right)$. The solid lines correspond to the $B+A$ case (left lower inset) and the dashed lines correspond to the $A+B$ case (left upper inset). $A$ finite broadening has been added for visibility’s sake of the delta functions. The finite, but small energy splitting in the former case is not visible for this broadening. Right inset: Bonding-antibonding energy splitting as a function of the distance between vacancies for the two different spatial orderings.

Figure 4

(Color online) DOS projected on the vicinity of two triangular voids placed along the axis of a semiconducting ribbon for different relative distances. The imbalance charges are the same but differ in sign $\left(N_I=\pm 2\right)$.

Figure 5

(Color online) DOS for a single notch with imbalance charge $N_I=2$ (blue dashed line, right inset). The same notch with an additional notch nearby of charge $N_I=-1$ (black solid line, left inset).

Figure 6

(Color online) (a) The spin resolved DOS for a ribbon with $W=7a$ and $U=2\text{eV}$ with one vacancy. Spin $\uparrow$ $(\downarrow )$ is plotted as a positive (negative) number as a function of energy (we take the Fermi energy as zero). (b) Zoom of the spin-split midgap state. (c) Zoom of the conduction band minima. For clarity, we substitute the delta functions composing the DOS by Gaussian functions with a finite broadening.

Figure 7

(Color online) (a) Noninteracting $\left(U=0\right)$ energy gap (circles) and interacting $\left(U=2\text{eV}\right)$ midgap spin splitting (squares) as a function of the ribbon width $W$. (b) Midgap spin splitting as a function of $U$ for two ribbon widths $W=7a$ and $W=13a$. (c) $U=0$ inverse participation ratio for the midgap state. (d) Standard deviation of the magnetization ${\sigma }_1$, as defined in Eq. (9), as a function of $U$ for two ribbon widths.

Figure 8

(Color online) Spin-charge separation in single-atom vacancy. Left column: Neutral case. Right column: Charged case. Upper panels: Charge density $q_i-1$. Lower panels: $\left|⟨m_i⟩\right|/{\sum }_i\left|⟨m_i⟩\right|$. The local charge and local spin are zero everywhere for the neutral and charged cases, respectively. The spin texture of the neutral case is identical to the charge texture of the charged case.

Figure 9

(Color online) Emerging ferrimagnetic order in a rhombohedral void with imbalance charge $N_I=3-3=0$ situated in the middle of a ribbon with $W=10a$ for $U=2\text{eV}$. The largest magnetic moment per atom is $⟨m_i⟩=0.05$.

Figure 10

(Color online) Normalized standard deviation of the magnetization as a function of the distance between vacancies for four cases: ribbon with $W=7a$ and $U=2\text{eV}$, vacancies in the same sublattice (open circles); same ribbon but vacancies in different sublattices (full circles); ribbon with $W=13a$, vacancies in different sublattices (full squares); same as the previous case but with $U=4\text{eV}$ (stars).

Figure 11

(Color online) Left panel: Bottom of the conduction band spin splitting $\delta$ as a function of the vacancy distance and as a function of the linear vacancy density (inset) for a ribbon with $W=7a$ and $U=2\text{eV}$. Right panel: $\delta$ for the same ribbon for a fixed vacancy distance of $28a$ as a function of $U$.

Figure 12

(Color online) Linear size of the wave function of the vacancy $⟨R⟩$ of the net magnetization $M_d$ and of magnetization in the sublattice $A$, $M_d^A$, as function of the distance between vacancies. From top to bottom, the panels correspond to Hubbard constants, $U=0$, $U=1.5$, and $U=4.5\text{eV}$. The vacancies are located in site type $B$.

Figure 13

(Color online) (a) Midgap spin splitting as function of the distance between vacancies and (b) weight of the quasilocalized wave function on site $B$. The vacancies are located in the sublattice $B$ and $U=4.5\text{eV}$.

Figure 14

(Color online) Critical values for the occurrence of a magnetic texture in a system of vacancies on atoms of types $A$ and $B$ forming two square interpenetrated lattices. The inset indicates the variation of the standard deviation of the local magnetizations as a function of $U$ for a system with dimension of (26,60).