Intravalley multiple scattering of quasiparticles in graphene

We develop a theoretical description of intravalley scattering of quasiparticles in graphene from multiple short-range scatterers of size much greater than the carbon-carbon bond length. Our theory provides a method to rapidly calculate the Green’s function in graphene for arbitrary configurations of scatterers. We demonstrate that noncollinear multiple scattering trajectories generate pseudospin rotations that alter quasiparticle interference, resulting in significant modifications to the shape, intensity, and pattern of the interference fringes in the local density of states (LDOS). We illustrate these effects via theoretical calculations of the LDOS for a variety of scattering configurations in single layer graphene. A clear understanding of impurity scattering in graphene is a step toward exploiting graphene’s unique properties to build future devices.

Figure 1

(A) Honeycomb lattice structure of monolayer graphene consisting of two triangular lattices [denoted by $A$ (red) and $B$ (blue) carbon atoms] that are shifted relative to each other by $-b\mathrm{ŷ}$ where $b=1.42$ Å is the $C–C$ bond length. Each unit cell consists of an $A$ and $B$ lattice site [a typical unit cell is denoted by the dotted ellipse in Fig. 1a], with the position of the $j$th unit cell given by ${\mathrm{r⃗}}_j^A=m{\mathrm{a⃗}}_{+}+n{\mathrm{a⃗}}_{-}$ where $j\equiv [m,n]$ and ${\mathrm{a⃗}}_{\pm }=\pm \frac{\sqrt{3}b}{2}\mathrm{x̂}+\frac{3b}{2}\mathrm{ŷ}$ are the lattice vectors. (B) Scattering configuration in graphene for an incoming “wave” ${\Phi }_{\pm \mathrm{K⃗}}^{\mathit{in}}(\mathrm{r⃗},\mathrm{k⃗},\pm )=e^{i\mathrm{k⃗}·\mathrm{r⃗}}|{\theta }_{\mathrm{k⃗}}\rangle {}_{\pm \mathrm{K⃗}}$ incident upon a scatterer located at ${\mathrm{r⃗}}_n$.

Figure 2

The calculated relative change in the LDOS, $\frac{\delta \rho (\mathrm{r⃗},E)}{{\rho }_{\pm \mathrm{K⃗}}^{\text{clean}}(\mathrm{r⃗},E)}$, in the presence of (A) a single scatterer at ${\mathrm{r⃗}}_n$ and (B) an elliptical arrangement of $N=30$ identical scatterers (major and minor axes of 20 and 10 nm, respectively) in both graphene (solid, blue curve) and in a 2DEG or achiral system (dotted, red curve) as a function of (A) the distance from the single scatter or as a function of (B) the distance from the outermost scatterer along the semimajor axis of the ellipse. In both calculations, $a=10$ Å, $V_0=4$ eV, the scattering amplitudes were given by $s_l$ in Eq. (6), and $k=0.485\hspace{0.5em}{\text{nm}}^{-1}$ (corresponding to $E=0.32$ eV for graphene). In both (A) and (B), the first five partial waves ($l=0$ to $l=4$) were included in the calculations, which provides an accuracy of the numerical results to within $1\%$. (A) For the single scatterer case, the Friedel oscillations in the LDOS decrease as $(k{\rho }_n){}^{-2}$ for the graphene/chiral case (blue, solid curve), whereas they decrease as $(k{\rho }_n){}^{-1}$ in a 2DEG/achiral case (red, dotted curve). (B) Due to multiple scattering within the elliptical array of scatterers, the Friedel oscillations in the LDOS decrease at roughly the same rate as those found in a 2DEG and can be observed at distances up to 100 nm from the scatterers.

Figure 3

Theoretical calculations of $\frac{\delta \rho (\mathrm{r⃗},E)}{{\rho }_{\pm \mathrm{K⃗}}^{\text{clean}}(\mathrm{r⃗},E)}$ for a 200 Å by 400 Å rectangular array of $N=36$ identical scatterers [$V=4$ eV, $a=10$ Å] for wave vectors $k=0.318\hspace{0.5em}{\text{nm}}^{-1}$ [top, $E=0.21$ eV for graphene], $k=0.419\hspace{0.5em}{\text{nm}}^{-1}$ [middle, $E=0.2765$ eV for graphene], and $k=0.485\hspace{0.5em}{\text{nm}}^{-1}$ [bottom, $E=0.32$ eV for graphene] for both a 2DEG [right, Eq. (A8)] and graphene [left, Eq. (26)]. In all simulations, $\frac{\delta \rho (\mathrm{r⃗},E)}{{\rho }_{\pm \mathrm{K⃗}}^{\text{clean}}(\mathrm{r⃗},E)}$ was set to zero within 35 Å of each scatterer, $s_l$ was calculated using Eq. (6) in both graphene and the 2DEG, and $l_{max}=4$ was used in all calculations in order to ensure a relative accuracy of calculations to within one percent. Dramatic differences in the quasiparticle interference patterns between the graphene [left] and 2DEG [right] cases are observed due to the interplay between multiple scattering and the resulting pseudospin rotations of the scattering trajectories.

Figure 4

A comparison of the calculated $\frac{\delta \rho (\mathrm{r⃗},E)}{{\rho }_{\pm \mathrm{K⃗}}^{\text{clean}}(\mathrm{r⃗},E)}$ [Eq. (26)] using either the first five partial waves (left) or only the first or $s$-wave scattering ($l_{max}=0$, right) for [Fig. 4b] a 200 Å by 400 Å rectangular array of $N=36$ identical scatterers ($V=4.93$ eV, $a=10$ Å) and [Figs. 4a and 4c] an elliptical array (semimajor and semiminor axis of 200 and 100 Å, respectively) composed of $N=30$ identical scatterers [$a=10$ Å and either (A) $V=4.93$ eV or (C) $V=4$ eV]. In all cases, $\frac{\delta \rho (\mathrm{r⃗},E)}{{\rho }_{\pm \mathrm{K⃗}}^{\text{clean}}(\mathrm{r⃗},E)}$ was set to zero within 35 Å of the scatterers, and the choice of $l_{max}=4$ (left) was found to provide a relative accuracy of $1\%$. For (A) and (B), the energies were chosen such that the $p$-wave scattering amplitude, $\left|s_1\right|$, was larger than the $s$-wave scattering amplitude, $\left|s_0\right|$, which gave (A) $\left|s_0\right|=0.1429<\left|s_1\right|=0.9975$ ($k=0.328\hspace{0.5em}{\text{nm}}^{-1}$, $E=0.2163$ eV) and (B) $\left|s_0\right|=0.1237<\left|s_1\right|=0.9914$ ($k=0.303\hspace{0.5em}{\text{nm}}^{-1}$, $E=0.2$ eV). For the elliptical array in (C), however, even though $\left|s_0\right|=0.9319>\left|s_1\right|=0.0147$ ($k=0.424\hspace{0.5em}{\text{nm}}^{-1}$, $E=0.28$ eV), there are still slight differences in $\frac{\delta \rho (\mathrm{r⃗},E)}{{\rho }_{\pm \mathrm{K⃗}}^{\text{clean}}(\mathrm{r⃗},E)}$ near the array of scatterers between the calculations with $l_{max}=4$ (left) and $l_{max}=0$ (right).