Vacancy effects on electronic and transport properties of graphene nanoribbons

We analytically study vacancy effects on electronic and transport properties of graphene nanoribbons and nanodots using Green's function approach. For semiconducting systems, the presence of a vacancy induces a zero-energy midgap state. The spatial pattern of the wave functions critically depends on the atomistic edge structures and can be used as an unambiguous probe of the edge structure. For metallic systems, the midgap vacancy state does not exist. In these systems, the vacancy mainly works as a source of electronic scattering and modifies electronic transmission. We derive that the electronic transmission coefficient can be written as $\mathcal{T}={cos}^2\left(\alpha \right)$, where $\alpha$ denotes the phase angle of the on-site Green's function at the vacancy site of the ideal systems. At small energies, $\mathcal{T}$ exhibits distinctly different functional form depending on edge structures.

Figure 1

Schematic of graphene lattice. In the longitudinal ($x$) direction, the lattice is viewed as a repetition of a supercell defined by the shaded rectangle. In the transverse ($y$) direction, it is a collection of zigzag chains. Each lattice site is coordinated as $\left(m,n,\nu \right)$, where $m$ refers to the supercells, $n$ to the zigzag chains, and $\nu =A,B$ the sublattices. The $x$ coordinates are chosen in such a way that $x_{\nu }(m,n)=m$ if the site $(m,n,\nu )$ is to right of $(m,n,\overline{\nu })$. Otherwise, $x_{\nu }(m,n)=m-\frac{1}{2}$. Schematic of graphene nanoribbons: (b) armchair nanoribbons (ARs) and (c) zigzag nanoribbons (ZRs). Crossed sites represent the boundaries of structures where the wave function should vanish. $a_T$ denotes the length of the AR unit cell.

Figure 2

Energy spectrum for armchair graphene ribbons with $M=7$. (a) Energy dispersion obtained by Eq. (14). Note the region of parameter $p$ is $0\le p\le 2\pi$. (b) Band structure of AR (see Fig. 2 in Ref. [33]) is obtained by folding the shaded region in (a) into the central region. $k_y$ is the crystal momentum along the $y$ direction in ARs.

Figure 3

(a) Band structures for ZR with $N=6$. (b) Numerical solution of Eq. (16) is plotted in $k-p$ space, where the curves beyond $p=\pi$ represent the imaginary part of $N$-th root, i.e., $p_N=\pi -i\xi$, which gives edge states drawn in red lines in (a).

Figure 4

Energy dependence of $G_0^{AR}\left(\varepsilon \right)=F^{AR}\left(\varepsilon \right)-i\pi {\rho }_0^{AR}\left(\varepsilon \right)$. (a) ${\rho }_0^{AR}\left(\varepsilon \right)$ and (b) $F^{AR}\left(\varepsilon \right)$. $G_0^{AR}$ critically depends on whether $x_0=x_A(m_0,0)$, sits on the nodal line of ${\Psi }_{r_0,ps}^{AR}$ [Eq. (12)] or not. When the site sits on this line ($x_0=4.5$), ${\rho }_0^{AR}$ vanishes at $\varepsilon =0$. Otherwise ($x_0=5.0$) it stays constant. $\eta =0.005$ has been chosen.

Figure 5

Energy dependence of $G_0^{ZR}\left(\varepsilon \right)=F^{ZR}\left(\varepsilon \right)-i\pi {\rho }_0^{ZR}\left(\varepsilon \right)$. (a) ${\rho }_0^{ZR}\left(\varepsilon \right)$ and (b) $F^{ZR}\left(\varepsilon \right)$. Here, $N=20$. $G_0^{ZR}$ critically depends on whether the site $(0,n_0,A)$ is closer to the edge of A-sublattice or not. If so ($n_0=5$), ${\rho }_0^{ZR}$ diverges at $\varepsilon =0$. Otherwise ($n_0=15$), it vanishes there. $\eta =0.005$ has been chosen.

Figure 6

Low-energy behavior of $G_0^{ZR}\left(\varepsilon \right)=F^{ZR}\left(\varepsilon \right)-i\pi {\rho }_0^{ZR}\left(\varepsilon \right)$ for ZR with width $N=20$. $n_0$ indicates the position of the vacancy. (a) Enlargement of Fig. 5, showing that the LDOS for ZR behaves as ${\left|\varepsilon \right|}^{{\nu }_{\rho }}$ close to $\varepsilon =0$. ${\nu }_{\rho }\approx \frac{2n_0-N}{N-1}$. (b) Log-log plot of ${\rho }_0^{ZR}\left(\varepsilon \right)$ and (c) that of $F^{ZR}\left(\varepsilon \right)$. Red circles indicate numerical results. Solid lines in (b) represent analytical form of ${\rho }_0^{ZR}\sim {\varepsilon }^{{\nu }_{\rho }}$ while those in (c) stand for that of $F^{ZR}\sim {\varepsilon }^{{\nu }_F}$. See that $F^{ZR}$ follows a similar power law, but with slightly different exponent ${\nu }_F$. (d) Vacancy position ($n_0$) dependence of ${\nu }_F$ and ${\nu }_{\rho }$, obtained by Eq. (28). ${\nu }_F$ changes abruptly at $n_0=N/2$.

Figure 7

Spatial profile of ${\psi }_0^{\tau }$ for (a) sAR, (b) BZR, and (c) QD. The color bar is shared between (a) and (b). The vacancy is indicated by a ‘+’ in each panel. ${\psi }_0^{\tau }$ is normalizable in each case, but it has different asymptotic behaviors (see text).

Figure 8

(a) Setup in calculating the electronic transmission $\mathcal{T}$ of ZR with a vacancy at $(0,n_0,A)$. An electron wave incidents from the distant right and transmits to the distant left. Energy dependence of $\mathcal{T}\left(\varepsilon \right)$ in ZR for vacancy (b) far from or (c) closer to the A-sublattice edge. In the former case ($n_0=14,19$), $\mathcal{T}\left(\varepsilon \right)$ is a constant except for a dip near zero energy. In the latter case ($n_0=3,6$), an extra dip is found at finite energy. The dips happen due to the vanishing of $F^{ZR}\left(\varepsilon \right)$ at those energies, implying the occurrence of quasibound states. (d) Energy dependence of $\mathcal{T}$ of mAR with a vacancy at $(m_0,0,A)$ for various values of $x_0=x_A(m_0,0)$. As seen here, there is perfect transmission at all energies if the vacancy sits on the nodal lines of ${\Psi }_{r_0,ps}^{AR}$, e.g., $x_0=4.5$. Otherwise ($x_0=5,9.5$) the vacancy blocks the transmission.

Figure 9

(a) Schematic of bearded zigzag ribbons (BZRs). (b) Band structure of a BZR with $N=6$. (c) Energy dependence of $G_0^{BZR}=F+i{\rho }_0$ at site $(0,n_0,{\nu }_0)$. Depending on whether ${\nu }_0=A$ or not, the LDOS shows either a peak or a zero. (a) Position of poles of $T^{BZR}\left(\varepsilon \right)$: ${\varepsilon }_{BZR}=\sqrt{\mathcal{I}(n_0,N)/\mathcal{I}(n_0,A,N)}$. For ${\varepsilon }_{BZR}<{\Delta }_{BZR}$ these poles represent midgap bound states (shaded region). Otherwise, they represent resonant states. Here ${\Delta }_{BZR}=sin\left(\frac{\pi }{N+1}\right)$. (b) Numerical evaluation of $\mathcal{I}(n,N)$ and $\mathcal{I}(n,A,N)$. The latter is almost a constant.

Figure 10

Numerical results for (a) ${\mathcal{I}}_{l,1}$ and ${\mathcal{I}}_{l,2}$ defined in Eq. (A13) (with $l=1,10$) and (b) their averages ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ [Eq. (A15)]. (c) Ratio of ${\mathcal{I}}_1$ to ${\mathcal{I}}_2$. Note that this ratio relies on whether $x$ is an integer or a half integer. (d) Green's function $G_{BZR}^{AB}(m,n,0,50;0)$, which shows peaks at the position of the vacancy $(0,50,B)$ for $x_{AB}=x_A(m,n)-x_B(0,50)$ being integers or at the neighboring site if $x_{AB}$ is a half integer.

Figure 11

(a) Schematic of a rectangular QD with size $M\times N$, with wave functions vanishing on crossed sites. (b) Eigenstates of QD.