RKKY interaction in graphene bubbles

By means of the Lanczos method, a numerically efficient theoretical approach, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in a graphene bubble. We find that the RKKY interaction in a graphene bubble can be larger or smaller than the corresponding result in pristine graphene by a few orders of magnitude, depending on the sublattice attribution and pseudomagnetic field strength where two magnetic impurities are positioned, which is due to the sublattice polarization of the low-energy electronic states in a strong pseudomagnetic field. If two magnetic impurities are both in the bubble region, the $R^{-3}$ decay rate of the RKKY interaction found in pristine graphene breaks down. But it recovers when one magnetic impurity is far away from the bubble center no matter where another impurity is located. When the Fermi level deviates from the Dirac point by carrier doping, the antiferromagnetic RKKY interaction between two magnetic impurities located at the opposite sublattices can be inverted to be ferromagnetic by altering the bubble height properly.

Figure 1

(a) The side view and (b) top view of a Gaussian graphene bubble with the center at the origin of Cartesian coordinates. In (b) ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the primitive lattice vectors of pristine graphene. The large dashed circle denotes the contour of the half height of the bubble ($r=\sigma$). The spatially nonuniform pseudomagnetic field in the bubble has threefold rotational symmetry. The six small solid circles specify the maximal pseudomagnetic field regions, with $+$ and $-$ signs denoting the pseudomagnetic field in the $\pm z$ directions, respectively. Points $A, A^{\prime }, A^{\prime \prime }, B$, and $B^{\prime }$ denote the typical positions of the magnetic field to calculate the RKKY interactions.

Figure 2

The RKKY interactions $J_{AA}\left(\mathit{R}\right)$ along (a) the armchair direction and (c) the zigzag direction and $J_{AB}\left(\mathit{R}\right)$ along (b) the armchair direction and (d) the zigzag direction in pristine graphene as functions of $R=|{\mathit{r}}_{lj\alpha }-{\mathit{r}}_{l^{\prime }{j}^{\prime }{\alpha }^{\prime }}|$. The Lanczos method and eigenfunction expansion method yield consistent results, and the insets are the results in main panel replotted in a larger $R$ range in logarithmic scale in order to show clearly the $R^{-3}$ decay law in the large-$R$ limit.

Figure 3

The RKKY interactions (a) $J_{AA}\left(\mathit{R}\right)$ and $J_{BB}\left(\mathit{R}\right)$ and (b) $J_{AB}\left(\mathit{R}\right)$ along the armchair direction ($y$ axis) in a graphene bubble as functions of $R=|{\mathit{r}}_{lj\alpha }-{\mathit{r}}_{l^{\prime }{j}^{\prime }{\alpha }^{\prime }}|$. In these results one magnetic impurity is fixed at some typical lattice points specified in the legend. The corresponding result of pristine graphene is also plotted for comparison. The inset replots the results in the main panel in a larger $R$ range in logarithmic scale in order to show clearly the recovery of the $R^{-3}$ decay law in large-$R$ limit. The LDOS spectra at representative lattice points: (c) $(11,-11,A)$ (dotted line) and $(-11,11,A)$ (solid line) and (d) $(0,0,A)$ (solid line) and $(0,0,B)$ (dashed line). In addition, the LDOS of pristine graphene (dotted line) is plotted in (d) for comparison.

Figure 4

The LDOS spectra in a graphene bubble at representative lattice points along a zigzag line (a) at $(1,1,A)$ (solid line) and $(1,1,B)$ (dashed line) and (b) at $(20,20,A)$ (solid line) and $(20,20,B)$ (dashed line). The RKKY interactions (c) $J_{AA}\left(\mathit{R}\right)$ and (d) $J_{AB}\left(\mathit{R}\right)$ along a zigzag line in a graphene bubble as functions of $R=|{\mathit{r}}_{lj\alpha }-{\mathit{r}}_{l^{\prime }{j}^{\prime }{\alpha }^{\prime }}|$. The pinned positions of one magnetic impurity for these results are specified in the legend. The inset replots the results in the main panel in a larger $R$ range in logarithm scale in order to show clearly the recovery of oscillation with a period of three lattice points in the large-$R$ limit.

Figure 5

The RKKY interactions for the magnetic impurities fixed at the typical lattice points $A^{\prime }, A^{\prime \prime }$, and $B^{\prime }$ as labeled in Fig. 1. (a) $|J_{\alpha {\alpha }^{\prime }}\left(R\right)|$ versus $h$ with $\sigma =5\text{nm}$; (b) $|J_{\alpha {\alpha }^{\prime }}\left(R\right)|$ versus $\sigma$ with $h=3.5\text{nm}$. Note that in (b) the crystal coordinates of magnetic impurities vary with $\sigma$ as $h$ is fixed. In the cases of (a) and (b) the Fermi level is fixed at $E_F=0$, but in the case of (c) $E_f=0.04t$ to account for the carrier doping.