Electronic transport in graphene with particle-hole-asymmetric disorder

We study the conductivity of graphene with a smooth but particle-hole-asymmetric disorder potential. Using perturbation theory for the weak-disorder regime and numerical calculations, we investigate how the particle-hole asymmetry shifts the position of the minimal conductivity away from the Dirac point $\varepsilon =0$. We find that the conductivity minimum is shifted in opposite directions for weak and strong disorder. For large disorder strengths, the conductivity minimum appears close to the doping level for which electron and hole doped regions (“puddles”) are equal in size.

Figure 1

Schematic picture of the setup considered for the conductance calculation: A short and wide $(W\gg L)$ graphene strip is connected to two highly doped leads. The leads are modeled by graphene strips with potential $u\to \infty$. A voltage $V$ is applied to the contacts and we are interested in the resulting current $I=GV$.

Figure 2

Diagrams that contribute to the conductance correction. Crosses represent energy contributions and dashed lines correspond to disorder averages. Diagram (a) yields the second-order energy correction to the graphene sample's conductance. Diagram (b) is needed to determine the second-order correction in the disorder potential. The two points for which the disorder average is performed are only separated by a small distance of order of the disorder correlation length $\xi$. Finally, (c) generates the conductance correction of a disorder potential's particle-hole asymmetry. Here, three points which are close to each other have to be considered for the third disorder-averaged moment. The energy acts at ${\mathbf{r}}^{\prime }$ which does not have to be close to the other coordinates. The resulting conductance contribution is presented in Eq. (21).

Figure 3

Parabolic fit of the numerically obtained conductivity data. The parameters are $\alpha =0.16/2\pi , L/\xi \sqrt{2}=40$, and $W/L=5$.

Figure 4

System-size dependence of the energy value ${\varepsilon }_{\min }$ at which the conductivity minimum is found. Upper panel: disorder strength $\alpha =0.04/2\pi$. Lower panel: $\alpha =16/2\pi$. The aspect ratio $W/L=5$ in both panels. The disorder average was performed for 500 disorder realizations at $L/\xi \sqrt{2}=40$ and for $200\left(25\right)$ realizations at $L/\xi \sqrt{2}=60\left(100\right)$.

Figure 5

Location ${\varepsilon }_{\min }$ of the conductivity minimum for weak and intermediate particle-hole-asymmetric disorder. The data points are the result of numerical calculations with the disorder potential chosen according to the model (26). The result (23) of the weak-disorder perturbation theory is indicated with a black line. The inset gives a log-log plot showing the agreement between the numerical results and the perturbation theory.

Figure 6

Location ${\varepsilon }_{\min }$ of the conductivity minimum for the full range of disorder strengths. The data points are the results of our numerical calculations, the lines represent the naive estimates ${\varepsilon }_{\mathrm{size}}$ (solid, black) and ${\varepsilon }_{\mathrm{charge}}$ (dashed, green). For large disorder strengths, the numerical data show a good agreement between ${\varepsilon }_{\min }$ and the energy value corresponding to equal sizes of electron and hole puddles inside the sample $\left({\varepsilon }_{\mathrm{size}}\right)$.