Electronic transport across linear defects in graphene

We investigate the low-energy electronic transport across grain boundaries in graphene ribbons and infinite flakes employing two distinct techniques. Using the recursive Green's-function method, we compute the electronic transmittance across different types of grain boundaries in graphene ribbons and flakes. We use the charge and current density spatial distributions to enhance our understanding of their electronic transport properties. We find that electronic transport depends both on the grain boundaries' microscopic details and on their orientation. In addition, we employ the transfer-matrix formalism to analytically study the electronic transport across a class of zigzag grain boundaries with periodicity 3. We find that these grain boundaries give rise to intervalley scattering.

Figure 1

Schematic representation of the different linear defect structures considered as grain boundaries. On the top is the 5757 structure. In the middle we show the 585 linear defect. At bottom is the bilayer graphene of length $L$, formed by the overlap of two monolayer regions. The shadow areas represent the left and right semi-infinite contacts.

Figure 2

(a) Resistivity across the grain-boundary structures 5757 and 585, compared to the experimental results of Tsen et al. [12]. Inset: Transmission function. Here we used an armchair ribbon with $g_s=2, g_v=1, N=56$ atoms along the width and making further correspondence to $W=1\mu \mathrm{m}$. (b) Spatial distribution of charge densities and current densities over the GB for different energies: $E={10}^{-7}$ eV, $E=7\times {10}^{-4}$ eV, and $E=14\times {10}^{-4}$ eV. The charge densities are schematically represented here for a narrower ribbon. The current densities are evaluated at different sites. The color of the arrow represents the magnitude of the electric current between any two neighboring sites, which are linearly normalized to the maximum value.

Figure 3

Schematic representation of a linear defect 585 at different orientation angles (a) $0^{\circ }$, (b) ${30}^{\circ }$, (c) ${60}^{\circ }$, and (d) ${90}^{\circ }$. (e) Transmission probability for each angle. The ribbon has $N=56$ sites in the width.

Figure 4

Spatial distribution of charge densities and current densities over the linear defect 585 with a orientation angle of $\theta ={30}^{\circ }$, corresponding to the energies indicated by the numbered arrows in Fig. 3: $E=-0.21$ eV, $E=1\times {10}^{-7}$ eV, and $E=0.21$ eV.

Figure 5

(a) Schematic representation of the overlap between two graphene monolayers, forming a bilayer region of length $L$. (b) Transmission throughout a bilayer region of length $L$ corresponding to 80 atoms superposed. Both $AA$ and $AB$ stackings are considered for the overlap (bilayer region). For comparison is shown the transmission through a pristine monolayer and pristine bilayers $AA$ and $AB$ of same width. (c) Same as in (b), showing now the transmission throughout a bilayer region of length $L$ corresponding to 320 atoms superposed. (d) Band structures for pristine monolayer and pristine bilayers $AA$ and $AB$ of same width.

Figure 6

Charge and current density distributions for the nanostructure composed of two graphene (monolayer) ribbons that partially overlapped ($AB$ stacked in the overlap/bilayer region). The two panels stand for two different energies: (a) $E=0.001$ eV, which corresponds to a minimum of transmission (indicated by arrow 1 in Fig. 5). (b) $E=0.1$ eV, which corresponds to a maximum of transmission (indicated by arrow 2 in Fig. 5).

Figure 7

Charge and current density distributions for the nanostructure composed of two graphene (monolayer) ribbons that partially overlapped ($AA$ stacked in the overlap/bilayer region). The two panels stand for two different energies: (a) $E=0.11$ eV, which corresponds to a low transmission (indicated by the arrow 3 in Fig. 5). (b) $E=0.2$ eV, which corresponds to a maximum of transmission (indicated by the arrow 4 in Fig. 5).

Figure 8

Scheme of two grain boundaries proposed in the context of ab initio works both on graphene and on boron nitride. (a) The 7557 defect line [33] and (b) the $t7t5$ defect line [32]. In these schemes we highlight in blue the region of the grain boundary.

Figure 9

(a) FBZ of pristine graphene (whose direct lattice vectors can be chosen to be ${\mathbf{u}}_1=a(1,0)$ and ${\mathbf{u}}_2=a(-1,\sqrt{3})/2$). (b) FBZ of graphene whose unit cell has thrice the size of that of pristine graphene in the zigzag direction (direction of ${\mathbf{u}}_1$). In this case, the direct lattice vectors can be chosen to be $3{\mathbf{u}}_1$ and ${\mathbf{u}}_2$ (as done in Figs. 15 and 16). (c) Spectrum of pristine graphene projected along the $k_x$ direction (parallel to the GB). (d) Comparison between the $k_x$-projected spectrum of (unfolded) pristine graphene (light green) and the same spectrum after a triple folding (red): in the latter, the two valleys (and the $\mathrm{\Gamma }$ point) are mapped into $k_{x}a=0$.

Figure 10

Transmittance in terms of the incidence angle (incoming electron chosen to be on the valley ${\mathbf{K}}_{+}$ Dirac point) for the 7557 grain boundary. Panels (a)–(d) correspond to scattering processes occurring at energies of, respectively, $0.01t, 0.1t, 0.3t$, and $0.5t$. The hopping parameters (see Fig. 15) were set at $\xi =0.98, \gamma =0.94$, and $\beta =0.1$. The dark blue curve stands for the transmittance preserving the valley degree of freedom (electron from the ${\mathbf{K}}_{+}$ below the GB scattering scatters to the same valley above the GB, i.e. $T_{++}={\left|{\tau }_{++}\right|}^2$). The dashed light blue curve stands for the intervalley transmittance (electron at the ${\mathbf{K}}_{+}$ valley below the GB scattering into the valley ${\mathbf{K}}_{-}$ above the GB, i.e. $T_{-+}=(v_{-}/v_{+}){\left|{\tau }_{-+}\right|}^2$; $v_{\pm }$ stands for the velocity of the mode $l\pm$).

Figure 11

Transmittance in terms of the incidence angle (incoming electron chosen to be on the valley ${\mathbf{K}}_{+}$ Dirac point) for the $t7t5$ grain boundary. Panels (a)–(d) correspond to scattering processes occurring at energies of, respectively, $0.01t, 0.1t, 0.3t$, and $0.5t$. The hopping parameters (see Fig. 16) were set at ${\xi }_1=1.06, {\xi }_2=0.95, {\xi }_3=0.83, {\xi }_4=0.80, {\xi }_5=1.30, {\xi }_6=1.05, {\gamma }_1=1.23, {\gamma }_2=1.20, {\gamma }_3=1.18$, and ${\gamma }_1=1.36$. The dark blue curve stands for the transmittance preserving the valley degree of freedom (electron from the ${\mathbf{K}}_{+}$ below the GB scattering scatters to the same valley above the GB, i.e., $T_{++}={\left|{\tau }_{++}\right|}^2$). The dashed light blue curve stands for the intervalley transmittance (electron at the ${\mathbf{K}}_{+}$ valley below the GB scattering into the valley ${\mathbf{K}}_{-}$ above the GB, i.e., $T_{-+}=(v_{-}/v_{+}){\left|{\tau }_{-+}\right|}^2$; $v_{\pm }$ stands for the velocity of the mode $l\pm$).

Figure 12

Cut of the energy spectrum around the two valleys (${\mathbf{K}}_{+}$ in red and ${\mathbf{K}}_{-}$ in blue) at energies $0.5t$ (left) and $0.3t$ (right). The trigonal wrapping of the spectrum in each valley is clearly seen, as well as that the spectra around the two valleys are connected by $\mathbf{k}⟶-\mathbf{k}$, i.e., $E_{{\mathbf{K}}_{\pm }}\left(\mathbf{k}\right)=E_{{\mathbf{K}}_{\mp }}(-\mathbf{k})$. The critical angle ${\theta }_c$ for the intervalley scattering is defined by the smallest angle of incidence ${\theta }_c^{\left(i\right)}$ for which it is possible to satisfy linear momentum conservation along the $x$ direction for a scattering between the valley ${\mathbf{K}}_{+}$ and the valley ${\mathbf{K}}_{-}$. That is, $k_x^{+}\left({\theta }_c^{\left(i\right)}\right)=k_x^{-}\left(0\right)$, where $k_x^{+}\left({\theta }_c^{\left(i\right)}\right)$ [$k_x^{-}\left(0\right)$] stands for the $k_x$ momentum in ${\mathbf{K}}_{+}$ [${\mathbf{K}}_{-}$] valley for an angle of incidence ${\theta }_c^{\left(i\right)}$ (see red arrows) [transmission zero (see blue arrows)] at the energy $\epsilon$. From the figure one can see that since the trigonal wrapping decreases with decreasing energy, then the critical angle decreases for lower energy.

Figure 13

Variation of the transmission probabilities $T_{++}$ and $T_{-+}$ (of the 7557 grain boundary) upon modifications of the original hoppings at the grain boundary ($\xi =0.98, \gamma =0.94$, and $\beta =0.1$) at an energy $E=0.01t$ [compare with Fig. 10]. The green full and dashed curves in (a) stand for the $T_{++}$ and $T_{-+}$ of a 7557 grain boundary with all the original hoppings divided by 2, i.e., $\xi /2, \gamma /2$, and $\beta /2$, while the red full/orange dashed curves were obtained with all the original hoppings divided by 10, i.e., $\xi /10, \gamma /10$, and $\beta /10$. Panel (b) stands for $T_{++}$ and $T_{-+}$ with all the hoppings set to $t$ (green curves) and to $2t$ (red/orange curves). The transmission curves shown in panel (c) were obtained by keeping all the hoppings fixed except $\xi$ that was multiplied by $0.8$ (green curves) and by $1.2$ (red and orange curves). Similarly, the transmission curves shown in panel (d) were obtained by keeping all the hoppings fixed except $\gamma$, that was multiplied by $0.8$ (green curves) and by $1.2$ (red and orange curves).

Figure 14

Variation of the transmission probabilities $T_{++}$ and $T_{-+}$ (of the $t7t5$ grain boundary) upon modifications of the original hoppings at the grain boundary (${\xi }_1=1.06t, {\xi }_2=0.95t, {\xi }_3=0.83t, {\xi }_4=0.80t, {\xi }_5=1.30t, {\xi }_6=1.05t, {\gamma }_1=1.23t, {\gamma }_2=1.20t, {\gamma }_3=1.18t$, and ${\gamma }_1=1.36t$) at an energy $E=0.01t$ [compare with Fig. 11]. The green full and dashed curves in (a) stand for the $T_{++}$ and $T_{-+}$ of a $t7t5$ grain boundary with all the original hoppings divided by 2, i.e., ${\xi }_i/2$ and ${\gamma }_i/2$, while the red full/orange dashed curves were obtained with all the original hoppings divided by 10, i.e., ${\xi }_i/10$ and ${\gamma }_i/10$. Panel (b) stands for $T_{++}$ and $T_{-+}$ with all the hoppings set to $t$ (green curves) and to $2t$ (red/orange curves). The transmission curves shown in (c) were obtained by keeping all the hoppings fixed except ${\xi }_1, {\xi }_2, {\xi }_3, {\xi }_5$, and ${\xi }_6$ that were multiplied by $0.8$ (green curves) and by $1.2$ (red and orange curves). Similarly, the transmission curves shown in (d) were obtained by keeping all the hoppings fixed except ${\gamma }_1, {\gamma }_2, {\gamma }_3$, and ${\gamma }_4$, that were multiplied by $0.8$ (green curves) and by $1.2$ (red and orange curves).

Figure 15

Crystalline structure of the 7557 grain boundary [33]. The region of the defect line is highlighted in blue.

Figure 16

Crystalline structure of the $t7t5$ defect line [32]. The region of the defect line is highlighted in blue.