Symmetry-protected coherent transport for diluted vacancies and adatoms in graphene

We study the effects of a low concentration of adatoms or single vacancies in the linear–response transport properties of otherwise clean graphene. These impurities were treated as localized orbitals, and for each type two cases with distinct coupling symmetries were studied. For adatoms, we considered top- and hollow-site adsorbates (TOP and HS). For vacancies, we studied impurity formation by soft bond reconstruction (REC), as well as the more symmetric case of charge accumulation in unreconstructed vacancies (VAC). Our results indicate that the transport is determined by usual impurity scattering when the graphene-impurity coupling does not possess $C_{3v}$ symmetry (TOP and REC). In contrast, VAC impurities decouple from the electronic states at the Dirac points, and yield no contribution to the resistivity for a sample in charge neutrality. Furthermore, the inversion-symmetry-conserving HS impurities also decouple from entire sets of momenta throughout the Brillouin zone, and do not contribute to the resistivity within a broad range of parameters. These behaviors are protected by $C_{3v}$ and inversion symmetry, respectively, and persist for more general impurity models.

Figure 1

Single impurities in a graphene lattice: (a) Top-site adsorbate, (b) hollow-site $s$-level adsorbate, (c) symmetric vacancy, and (d) reconstructed vacancy. Lattice sites in red and blue belong to sublattices $A$ and $B$, respectively.

Figure 2

$|{\mathrm{\Theta }}_I{\left(\mathbf{k}\right)|}^2$ for (a) vacancies $\left(I=V\right)$ and (b) reconstructed vacancies $\left(I=R\right)$; (c) hollow-site adatom $\left(I=H\right)$ upper band, and (d) hollow-site adatom lower band. The equipotential contours show topographic details at low energies, and the hexagons indicate the boundaries of the first Brillouin zone. The couplings vanish at $K$ and $K^{\prime }$ for all cases except $I=R$, and only $|{\mathrm{\Theta }}_H^{\pm }\left(\mathbf{k}\right)|$ display line nodes at specific angles about these points.

Figure 3

The spectral densities of TOP, REC, VAC, and HS impurities for different local energies ${\varepsilon }_d$, using ${\mathrm{\Gamma }}_0/D=0.05$. The amplitude at $\varepsilon =0$ vanishes as a power law for the $C_{3v}$ impurities VAC and HS, in stark contrast to the non-$C_{3v}$ impurities REC and TOP, whose spectral densities are singular at zero energy.

Figure 4

Resistivity as a function of temperature for TOP and REC $\left(n=0\right)$, and VAC and HS impurities $\left(n=0.95\right)$. Results are shown in charge neutrality $\left(\mu =0\right)$ for impurity local energies (a) ${\varepsilon }_d=-5{\mathrm{\Gamma }}_0$, (b) ${\varepsilon }_d=0$, and (c) ${\varepsilon }_d=5{\mathrm{\Gamma }}_0$, with ${\mathrm{\Gamma }}_0=0.05D\sim 0.5$ eV.

Figure 5

Low-temperature resistivity as a function of chemical potential. (a), (b), and (c) Correspond to a graphene sample with only (non-$C_{3v}$-symmetric) TOP and REC impurities. (d) through (i) Correspond to a mixture with a fraction $n$ of $C_{3v}$-symmetric and $1-n$ of non-$C_{3v}$-symmetric impurities. Results are shown for impurity local energies ${\varepsilon }_d=-5{\mathrm{\Gamma }}_0,0,5{\mathrm{\Gamma }}_0$ (left to right) at a temperature $T={10}^{-4}D\sim 10$ K. We have used ${\mathrm{\Gamma }}_0=0.05D$, which gives an impurity-graphene coupling $V\approx 800$ meV.

Figure 6

Resistivity of adatoms and vacancies as a function of the fraction $n$ of $C_{3v}$ impurities (HS and VAC) for fixed chemical potential $\mu =-5{\mathrm{\Gamma }}_0$. (a) For vacancies, the case of $\mu ={\varepsilon }_d=-5{\mathrm{\Gamma }}_0$ corresponds to the resistivity maximum of Fig. 5, which is independent of $n$. For adatoms this is also the maximum peak, but its amplitude goes to zero as ${(1-n)}^{1/2}$ as $n$ goes to one. (b) In the case of a resonant impurity $\left({\varepsilon }_d=0\right)$ the resistivity for adatoms also vanishes as ${(1-n)}^{1/2}$, whereas for vacancies it remains finite (away from charge neutrality) up to $n=1$.

Figure 7

The current-current correlation function is represented by a polarization diagram and an interaction vertex $\mathrm{\Gamma }$. Lines represent full graphene Green's functions in the presence of the impurities, and point vertices at the top and bottom represent the current operators.