Magnetism in strained graphene dots

We study the magnetization of square and hexagonal graphene dots. It is shown that two classes of hexagonal dots have a second-order phase transition at a critical Hubbard energy $U$, whose value is similar to the one in bulk graphene, albeit the dots do not have a density of states proportional to the absolute value of the energy, relatively to the Dirac point. Furthermore, we show that a particular class of hexagonal dots, having zigzag edges, does not exhibit zero-energy edge states. We also study the effect of uniaxial strain on the evolution of the magnetization of square dots and find that the overall effect is an enhancement of magnetization with strain. The enhancement can be as large as 100% for strain on the order of 20%. Additionally, stress induces a spatial displacement of the magnetization over the dot, moving it from the zigzag to the armchair edges.

Figure 1

(Color online) Illustration of the honeycomb lattice with the $A$ and $B$ sublattices, the lattice vectors ${\mathit{\delta }}_i$ $\left(i=1,2,3\right)$, the primitive vectors $\mathit{a}$ and $\mathit{b}$, and the hoppings $t_i\left(i=1,2,3\right)$ used in Sec. 4. The abscissas are along the zigzag edge (horizontally in the figure), and $a_0$ is the equilibrium carbon-carbon distance.

Figure 2

(Color online) Types of hexagonal and square dots studied in this work: (a) hexagon with $D_6$ symmetry and no zigzag edges (termed HEX2 in the figures below); (b) hexagon with $D_6$ symmetry and zigzag edges (termed HEX1 in the figures below); (c) hexagon with $D_3$ symmetry with zigzag edges; (d) an antidot with the external and internal boundaries made of $D_6$ symmetry with no zigzag edges; (e) square where the vertices are of zigzag type (termed SQR1); and (f) square where the vertices are of armchair type (termed SQR2). The size of the figures is characterized by the number $L$ of carbon atom horizontal lines (zigzag type of lines), for example, in panel (a) one has $L=14$ and in panels (e) and (f) one has in both cases $L=10$.

Figure 3

(Color online) Illustration of the local magnetization for dots of the same type illustrated in Fig. 2, but with a much larger number of atoms, thus avoiding finite-size effects. Upright triangles refer to positive magnetization and the down ones to negative values. The open triangles refer to the points where the magnetization has it maximum value. In panel (a) we have a HEX2 hexagon with $U=2.3$; for (b) and (c) hexagons $U=2$. In panel (d) we show an antidot, where the external boundary is from a EHX2 hexagon and the internal one is from a HEX1 hexagon. Panels (e) and (f) are of type SQR1 and SQR2, respectively.

Figure 4

(Color online) Variation in the magnetization as function of $U$ for squares and hexagons of different types. The vertical dashed lines refer to the values of $U$ used in Refs. 38, 53 (see text in Sec. 2 for a discussion about these choices).

Figure 5

(Color online) Variation in quantity $\Delta \left(N\right)$ with the number of atoms for square and hexagonal dots. From this figure alone we can understand that the spectrum of hexagons of types HEX1 and HEX2 behaves exactly in the same way, not showing the development of zero-energy edge states, a consequence of the $D_6$ symmetry alone.

Figure 6

(Color online) Variation in $\Delta E$ with the Coulomb interaction $U/t$ for hexagons of the type HEX1 and squares of the type SQR1.

Figure 7

(Color online) Representation of the effect of stress on the length of the nearest-neighbors carbon atoms. On the left we depict the $\mathcal{Z}\mathcal{Z}$ case and on the right the $\mathcal{A}\mathcal{C}$ one. For the $\mathcal{Z}\mathcal{Z}$ case, the stress $\sigma$ induces an increase in the hopping associated with the vertical bond, due to the Poisson effect. The hoppings associated with bonds 1 and 3 are reduced, and the system tends to dimerize at large deformations. For tension along $\mathcal{A}\mathcal{C}$, $\sigma$ is oriented along bond 2. In this case all hoppings are reduced, but the one associated with bond 2 decreases more than the other two, leading to a set of quasi-one-dimensional chains. The quantitative change in the hoppings upon stress was studied quantitatively using ab initio methods in Ref. 61.

Figure 8

(Color online) Variation in the magnetization as function of the Hubbard interaction $U$ for different strain values $\epsilon$. The top panels refer to stress along the $\mathcal{Z}\mathcal{Z}$ edge and the other to stress along the $\mathcal{A}\mathcal{C}$ edge. The Poisson ratio used was that of a personal electronic transfer substrate $\left(\nu \simeq 0.3\right)$, considered in study of the Raman redshift of the $G$ and $2D$ peaks of graphene (Ref. 58). The vertical dashed lines refer to the values of $U$ used in Refs. 38, 53 (see text in Sec. 2 for a discussion about these choices).

Figure 9

Variation in the critical value, $U_c$ with the amount of strain for the cases $\mathcal{Z}\mathcal{Z}$ (top left) and $\mathcal{A}\mathcal{C}$ (top right). In both cases we used HEX2-type hexagons with $L=8$. In the bottom row we have the dependence of the maximum value of magnetization, $m_{\text{max}}$, on the amount of strain, using $U/t=2$, and for the cases $\mathcal{Z}\mathcal{Z}$ (down left) and $\mathcal{A}\mathcal{C}$ (down right). In both cases we used SQR2-type squares with $L=10$.

Figure 10

(Color online) The top row shows the variation in the nearest-neighbor hoppings $t_{1,2,3}$ and the average hopping, $⟨t⟩$ under uniaxial strain with strain applied along the $\mathcal{Z}\mathcal{Z}$ (left) and $\mathcal{A}\mathcal{C}$ directions (right). The bottom left panel consists of the variation in $⟨t⟩$ with strain for different orientations of the uniaxial deformation. In the last panel on the bottom right we present $U_c/t_0$ and $U_c/⟨t⟩$ for the two representative directions.

Figure 11

(Color online) Spatial variation in the magnetization as function of strain. (a) $\epsilon =0$; (b) and (c) corresponds to $\epsilon =0.14,0.22$ with stress along the armchair edge; and (d), (e), and (f) corresponds to $\epsilon =0.10,0.14,0.22$ with stress along the zigzag edge. In both cases there is an increase in the magnetization along the armchair edge as $\epsilon$ increases. The upright triangles represent positive magnetization and the down ones represent negative values.