From quantum confinement to quantum Hall effect in graphene nanostructures

We study the evolution of the two-terminal conductance plateaus with a magnetic field for armchair graphene nanoribbons (GNRs) and graphene nanoconstrictions (GNCs). For GNRs, the conductance plateaus of $\frac{2e^2}{h}$ at zero magnetic field evolve smoothly to the quantum Hall regime, where the plateaus in conductance at even multiples of $\frac{2e^2}{h}$ disappear. It is shown that the relation between the energy and magnetic field does not follow the same behavior as in “bulk” graphene, reflecting the different electronic structure of a GNR. For the nanoconstrictions we show that the conductance plateaus do not have the same sharp behavior in zero magnetic field as in a GNR, which reflects the presence of backscattering in such structures. Our results show good agreement with recent experiments on high-quality graphene nanoconstrictions. The behavior with the magnetic field for a GNC shows some resemblance to the one for a GNR but now depends also on the length of the constriction. By analyzing the evolution of the conductance plateaus in the presence of the magnetic field we can obtain the width of the structures studied and show that this is a powerful experimental technique in the study of the electronic and structural properties of narrow structures.

Figure 1

Three graphene nanostructures considered in this work. (a) An armchair graphene nanoribbon. (b) A nanoconstriction with zero length (bowtie). (c) A graphene nanoconstriction with finite length $L$. All three structures in the figure have the same number of dimer lines, $N_{\text{DL}}=9$.

Figure 2

Color plot of the conductance of a graphene nanoribbon ($N_{\text{DL}}=101$) as a function of the Fermi energy and $\sqrt{B}$. The lines present in the plot are isovalues of the conductance. The numbers in the graph show the values of conductance in units of $G_0=\frac{2e^2}{h}$.

Figure 3

Conductance vs energy for graphene nanoconstrictions with different lengths at zero magnetic field, with a constant width of $W=12.3$ nm. The curves for different values of $L$ are shifted by $\frac{2e^2}{h}$ for a better visualization.

Figure 4

Conductance vs energy for (a) $B=0$ T and (b) $\sqrt{B}=10.5\sqrt{\text{T}}$ for a graphene nanoribbon, in solid black lines, and a graphene nanoconstriction with zero and finite length, in dotted red and dashed blue lines, respectively.

Figure 5

Color plot of the conductance of a graphene nanoconstriction ($N_{\text{DL}}=101$) with zero length as a function of the Fermi energy and $\sqrt{B}$. The lines present in the plot are isovalues of the conductance. The numbers in the graph show the values of conductance in units of $G_0=\frac{2e^2}{h}$.

Figure 6

Position of the end of the plateaus $G=\frac{2e^2}{h}$ (circles) and $\frac{6e^2}{h}$ (stars) in $E$ and $\sqrt{B}$ for a graphene nanoribbon, in gray, and a graphene nanoconstriction with zero length, in black. For comparison, the expected values for an infinite graphene plane for $n=$ 1 and 2 are plotted as a dashed and dotted line, respectively. Inset: obtained magnetic field for the transition to the quantum Hall regime vs occupation (Landau-level) number.