Probing decoupled edge states in a zigzag phosphorene nanoribbon via RKKY exchange interaction

Phosphorene is an anisotropic puckered two-dimensional hexagonal lattice of phosphorus atoms. The edge modes in a zigzag phosphorene nanoribbon are quasiflat in nature and fully isolated from the bulk states, which are unique in comparison to the other hexagonal lattices like graphene, silicene, etc. We theoretically investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between two magnetic impurities placed on the nanoribbon and extract the signatures of the flat edge states via the behavior of it. Due to the complete separation of the edge states from the bulk, we can isolate the edge mode contribution via the RKKY interaction from that of the bulk by tuning the external gate potential when both the impurities are placed at the same edge. The bulk-induced RKKY interaction exhibits very smooth oscillation with the distance between the two impurities, whereas for edge modes it fluctuates very rapidly. We also explore the effect of tensile strain both in absence and presence of gate voltage and reveal that the RKKY interaction strength can be boosted under suitable doping, when both the impurities are within the bulk.

Figure 1

(a) Schematic view of the phosphorene lattice structure is presented. Two different colors, blue and red, are used to denote the atoms belonging to two different planes. (b) A projected view of the lattice in $x\text{−}y$ plane is demonstrated. The five nonequivalent hopping parameters associated with the lattice are marked by $t_1$, $t_2$, $t_3$, $t_4$, and $t_5$. $a$ and $b$ are the two lattice constants along the $x$ and $y$ directions, respectively.

Figure 2

Schematic of ZPNR with the atoms of two different zigzag chains denoted by two different colors, blue (dark gray) and pink (light gray). The magnetic impurities are denoted by vertical green (light gray) arrow sign. The line numbers along the $x$ and $y$ direction are denoted by the index $n$ and $m$, respectively.

Figure 3

Energy band dispersion of ZPNR for three different values of gate voltage (in units of $t_1$) is shown. The width of the ribbon is taken as $N=10$. The position of the Fermi level is denoted by the dashed line. Here, ${\epsilon }_z=0$.

Figure 4

The features of energy band dispersion of ZPNR for three different strengths of tensile strain is illustrated when $U=0$. The width of the ribbon is considered to be the same as mentioned in Fig. 3. Two different positions of the Fermi level are denoted by a dashed line.

Figure 5

Energy band dispersion of ZPNR is illustrated for three different strengths of tensile strain in presence of a fixed nonzero gate voltage $U=3t_1$. The width of the ribbon is the same as mentioned in Fig. 3. The position of the Fermi level is marked by the dashed line.

Figure 6

The behavior of RKKY exchange interaction (in units of $C={(J_c\hslash /2)}^2$) between two magnetic impurities is demonstrated as a function of the distance (in units of the lattice parameter $b$) between them when (a) both the impurities are at the same edge, (b) both are within the bulk of ZPNR, and (c) one at the edge, while the other one is located in the interior of the ZPNR. The size of the undoped ribbon is $10\times 100$. For each case, three different gate voltages, $U=0$, 1.5, and 3 (in units of $t_1$) are considered. Here, we choose the strain ${\epsilon }_z=0$.

Figure 7

The behavior of LDOS is shown as a function of energy for (a) edge states ($n=1$) and (b) bulk states ($n=5$). Here, we have considered $m={26}^{th}$ unitcell.

Figure 8

The feature of RKKY interaction strength (in units of $C$) is illustrated with respect to the gate voltage for three different spatial distributions (in units of $b$) of two magnetic impurities.

Figure 9

The behavior of RKKY exchange interaction (in units of $C={(J_c\hslash /2)}^2$) between two magnetic impurities is illustrated as a function of the distance (in units of lattice parameter $b$) between them when (a) both the impurities are at the same edge, (b) both are within the bulk, and (c) one at the edge while the other one within the bulk. The size of the undoped ($E_F=0$) ZPNR is $10\times 100$. For each case, three different values of strain, ${\epsilon }_z=0$, $10\%$ and $20\%$ (in units of $t_1$) are taken into account in absence of any gate voltage ($U=0$).

Figure 10

The behavior of RKKY interaction strength (in units of $C$) between two impurities is illustrated as a function of distance (in units of $b$) between them when both the impurities are inside the bulk. The ZPNR is doped at $E_F=1.6t_1$ and $U=0$. We choose the same size of the ribbon as mentioned in Fig. 9.

Figure 11

The feature of RKKY exchange interaction (in units of $C$) between two magnetic impurities, located inside the bulk of a ZPNR, is demonstrated as a function of the distance (in units of $b$) between them. We choose $U=3t_1$, $E_F=0$ and the same size of the ribbon as mentioned in Fig. 9.

Figure 12

The behavior of RKKY interaction strength (in units of $C$) is illustrated as a function of degree of strain for three different spatial distributions (in units of $b$) of two impurities. We consider finite doping $E_F=-2$ eV and gate voltage $U=3$ eV.