Dynamic RKKY interaction between magnetic moments in graphene nanoribbons

Graphene has been identified as a promising material with numerous applications, particularly in spintronics. In this paper we investigate the peculiar features of spin excitations of magnetic units deposited on graphene nanoribbons and how they can couple through a dynamical interaction mediated by spin currents. We examine in detail the spin lifetimes and identify a pattern caused by vanishing density of states sites in pristine ribbons with armchair borders. Impurities located on these sites become practically invisible to the interaction but can be made accessible by a gate voltage or doping. We also demonstrate that the coupling between impurities can be turned on or off using this characteristic, which may be used to control the transfer of information in transistorlike devices.

Figure 1

A schematic diagram of a graphene nanoribbon with $N$ atoms across its width. The lattice parameter is given by $a$. Impurities 1 and 2 are shown as numbered circles. The location of impurity $n$ is identified by the labels $D_{An}$ and $D_{Zn}$, which denote the unit cell depicted by the (blue) dashed rectangles and the carbon site inside this cell, respectively. The separation along the edge direction is given by $D_A=D_{A1}-D_{A2}$. Here, $D_{Z1}=2,D_{Z2}=3,D_A=5$, and $N=8$. The (red) arrows indicate lines with a vanishing local density of states.

Figure 2

Real and imaginary parts of ${\chi }_{1,1}^{+-}$ (in arbitrary units) as functions of the energy $E=\hslash \omega$ (in units of ${10}^{-3}t)$, where $\omega$ represents the frequency of the oscillatory field. For clarity, the bottom panel is plotted as a function of $E-E_R$, where $E_R=0.9639127436t$. The results are for a single impurity: (a) adsorbed to the edge of a nanoribbon $\left(N=5,D_Z=1\right)$; (b) adsorbed to a VLDOS site $\left(N=5,D_Z=3\right)$. $\mathrm{Re}{\chi }_{1,1}^{+-}$ (red dashed lines) and $\mathrm{Im}{\chi }_{1,1}^{+-}$ (black solid lines).

Figure 3

(a) Spectral density $S_1$ calculated as a function of energy $E=\hslash \omega$ (in units of ${10}^{-3}t)$ for a single impurity, and different values of the Fermi energy $E_F$. Each color corresponds to a specific value of $E_F$. The impurity is adsorbed to a site in the edge of a nanoribbon with $N=5$, along the line labeled by $D_Z=1$. (b) Local density of states $\left(\rho \right)$ at the pristine-ribbon site to which the impurity is adsorbed (black curve) as a function of the energy, and local density of Stoner modes $\left({\rho }_S\right)$ at the magnetic impurity site (brown curve) as a function of $E_F$. Color marks indicate the values of $E_F$ where the spectral densities depicted in panel (a) were calculated. The LDOSM ${\rho }_S\left(E_F\right)$ is represented by ${lim}_{\omega \to 0}\mathrm{Im}{\chi }_{11}^0\left(\omega \right)/\omega$.

Figure 4

Full width at half maximum $\mathrm{\Gamma }$ (black curve) and local density of Stoner modes ${\rho }_S$ (red curve) calculated as a function of the Fermi energy $E_F$ for a single impurity at the VLDOS site $D_Z=3$ in a nanoribbon of width $N=5$. Energies are expressed in units of the nearest-neighbor hopping integral $t$. The LDOSM ${\rho }_S$ is represented by ${lim}_{\omega \to 0}\mathrm{Im}{\chi }_{11}^0\left(\omega \right)/\omega$.

Figure 5

Full width at half maximum (FWHM) $\mathrm{\Gamma }$ of the spectral-lines peaks for impurities adsorbed at different distances $D_Z$ from the edge of a nanoribbon with width $N=29$. The FWHM reduces almost to zero for impurities adsorbed onto sites belonging to the VLDOS lines, indicating an extremely long-lived excitation in these cases. Energies are expressed in units of the nearest-neighbor hopping integral $t$.

Figure 6

Imaginary parts of the local ${\chi }_{11}^{+-}\left(\omega \right)$ (solid black line) and nonlocal ${\chi }_{21}^{+-}\left(\omega \right)$ (dashed red line) transverse spin susceptibilities as functions of the energy $E=\hslash \omega$ (in units of ${10}^{-3}t)$, for two impurities (1 and 2) adsorbed to non-VLDOS sites of a nanoribbon with $N=5$. Here, $D_{Z1}=D_{Z2}=1$, and $D_A=10$. The acoustic and optical modes are clearly visible, indicating that the magnetic impurities are ferromagnetically coupled, as expected.

Figure 7

Imaginary part of the local transverse spin susceptibility ${\chi }_{11}^{+-}\left(E\right)$ (solid black line) calculated as functions of the energy $E=\hslash \omega$ (in units of ${10}^{-3}t)$, for $D_{Z1}=D_{Z2}=1$. (a) $D_A=100$, (b) $D_A=300$, and (c) $D_A=500$. The red and green lines represent the acoustical and optical components of ${\chi }_{11}$, respectively.

Figure 8

Full width at half maximum $\mathrm{\Gamma }$ (in units of $t)$ of the acoustic mode as a function of the horizontal distance $D_A$ between the two impurities. (a) One of the two impurities is adsorbed to a non-VLDOS site along the line $D_Z=1$, and the other to VLDOS sites, along the line $D_Z=3$. The green dashed line represents the result for a single impurity adsorbed to a non-VLDOS site along the line $D_Z=1$. We notice that the results converge very rapidly to the single impurity value in this case. (b) Both impurities are adsorbed to non-VLDOS sites along the line $D_Z=1$. Here we observe that the convergence to the single impurity case takes place in an oscillatory way, and for very large separations only.

Figure 9

A comparison of the spectral lines calculated for magnetic impurities adsorbed to different lattice sites. (a) Spectral line for a single impurity adsorbed to a non-VLDOS site along $D_Z=1$ (black line), compared with the spectral line, computed on the same site, for a system with two impurities (red dots). Here the first impurity is adsorbed to the same non-VLDOS site $\left(D_{Z1}=1\right)$, while the second one lies atop a VLDOS site at $D_{Z2}=3$, and $D_A=10$. The nanoribbon width corresponds to $N=5$. (b) Spectral line for a system with two magnetic impurities adsorbed to non-VLDOS sites labeled by $D_{Z1}=D_{Z2}=1$, and $D_A=20$, in a nanoribbon with width $N=5$ (black line). The results are compared with the spectral line calculated for a system with three impurities. Two of them are adsorbed to the same non-VLDOS sites, while the third one sits in between, atop a VLDOS site labeled by $D_{Z3}=3,D_{A3}=8$ (red dots). Energies $E=\hslash \omega$ are in units of ${10}^{-3}t$.

Figure 10

Imaginary part of the local susceptibility as functions of the energy $E=\hslash \omega$ (in units of ${10}^{-3}t)$ for a system with two impurities (one non-VLDOS and one VLDOS) on the non-VLDOS site for different values of $E_F$. The impurities are located at $D_{Z1}=1,D_{Z2}=3$, with separation $D_A=14$ in a nanoribbon of width $N=5$