Quantum transport in graphene Hall bars: Effects of vacancy disorder

Using the tight-binding model, we investigate the influence of vacancy disorder on electrical transport in graphene Hall bars in the presence of quantizing magnetic fields. Disorder, induced by a random distribution of monovacancies, breaks the graphene sublattice symmetry and creates states localized on the vacancies. These states are observable in the bend resistance, as well as in the total DOS. Their energy is proportional to the square root of the magnetic field, while their localization length is proportional to the cyclotron radius. At the energies of these localized states, the electron current flows around the monovacancies and, as we show, it can follow unexpected paths depending on the particular arrangement of vacancies. We study how these localized states change with the vacancy concentration, and what are the effects of including the next-nearest-neighbor hopping term. Our results are also compared with the situation when double vacancies are present in the system. Double vacancies also induce localized states, but their energy and magnetic field dependencies are different. Their localization energy scales linearly with the magnetic field, and their localization length appears not to depend on the field strength.

Figure 1

Graphene Hall bar system (left) and three studied disorder types (right). Widths of the horizontal and vertical leads are set to $w_h=49.71$ nm and $w_v=49.94$ nm, respectively, while lengths of the horizontal and vertical arms ($l_v$ and $l_h$) are equal, and set to 50 nm. Before disorder is introduced, all edges are considered to be clean, meaning that there are no dangling bonds on them.

Figure 2

Modified vector potential ${\vec{A}}^{\prime }(x,y)$. Arrows show the direction of ${\vec{A}}^{\prime }(x,y)$, and their color represents its intensity. System shape is marked by the gray area.

Figure 3

Average Hall (${\overline{R}}_H={\overline{R}}_{13,42}$; red curves) and bend (${\overline{R}}_B={\overline{R}}_{12,43}$; blue curves) resistances for various types and concentrations of vacancy disorder. Disorder type and concentration are shown in every subplot, in the lower-left corner. Results for the bend resistance depend strongly on the vacancy concentration; therefore we scale ${\overline{R}}_B$ by multiplying it with a scaling coefficient, in order to present all results in the same range. A scaling coefficient is presented in every subplot, above the vacancy concentration. Thin gray vertical lines mark the energy of Landau levels given by Eq. (15) for $B=20$ T, while black vertical lines mark the position of the new peaks at $E=\pm 33.9$ meV. The temperature is equal to 16 K.

Figure 4

Averaged DOS (black curves; arbitrary units) and distributions of eigenenergies in a closed system (orange histograms in the background). Red lines mark the positions of the two new peaks at $E=\pm 33.9$ meV, while dotted lines mark the energies of Landau levels. Similarly to the resistances in Fig. 3, for every vacancy type and concentration, DOS is first smoothed and then averaged over $N=10$ different vacancy samples. In the case of the eigenenergies, results for positive energies for $N=10$ distributions are summed without smoothing or averaging, and mirrored around $E=0$ axis. We used the same sets of vacancy distributions as in Fig. 3. Magnetic field is $B=20$ T, and $T=16$ K.

Figure 5

(a) DOS for a single DV distribution ($n=0.01\%, B=14$ T). (b) LDOS for a single SVA distribution ($n=0.005\%, B=20$ T) at the vacancy localization energy ($E=33.9$ meV). (d) LDOS for a single DV distribution ($n=0.01\%, B=14$ T) at the localization energy [$E=1.45$ meV; marked with red vertical line in (a)]. (c) and (e) Zoom around particular vacancies in (b) and (d). Vacancies are marked with green circles and green dots in the center. Temperature is set to $T=0$ K.

Figure 6

(a)–(d) LDOS for SV disorder type, for different vacancy concentrations: 0.001%, 0.005%, 0.01%, and 0.02%, and at different energies: 33.9 meV, 33.9 meV, 45 meV, and 52 meV, respectively. Vacancy positions are marked with green circles. (e)–(h) DOS for SV disorder type, for the corresponding concentrations. Energies at which we calculated LDOS in (a)–(d) are marked with red vertical lines in (e)–(h). Magnetic field is $B=20$ T, and $T=0$ K.

Figure 7

DOS for different values of magnetic field and different disorder distributions: (a) $n=0.001\%$ SV distribution, (b) $n=0.005\%$ SV distribution, (c) single divacancy located at the center of the system, (d) $n=0.01\%$ DV distribution. The two distributions of monovacancies are the same as those used in Figs. 6 and 6, respectively. In all four cases $T=0$ K. The green arrows mark ($E,B$) points at which we study LDOS in Figs. 8, 9, and 10.

Figure 8

The evolution of LDOS around a single SV vacancy for $(E,B)$ values lying on the parabola in Fig. 7 (marked by the green arrows). The magnetic field strengths are 5 T, 10 T, 15 T, and 20 T, and the corresponding energies and cyclotron radii are (a) $E=16$ meV, $R_c=36.7\text{Å}$; (b) $E=23.4$ meV, $R_c=26.8\text{Å}$; (c) $E=29$ meV, $R_c=22.1\text{Å}$; and (d) $E=33.9$ meV, $R_c=19.4\text{Å}$. The two circles with radii $R_c$ and $2R_c$ in each inset are centered at the vacancy site. Insets in the lower-right corners show the total DOS in a 6 meV energy range around the localization energy.

Figure 9

Same as Fig. 8 but now showing normalized LDOS, where LDOS in each subplot is divided by a maximum LDOS value for that subplot. The lower-right insets (showing the total DOS around the energy of a localized state) are also scaled, so that the DOS peak maximum is equal to 1.

Figure 10

Same as Fig. 9 but now showing normalized LDOS and DOS for a single divacancy located at the center of the system. The $(E,B)$ points at which we calculated LDOS and DOS are marked with green arrows in Fig. 7.

Figure 11

Decomposition of the bend resistance $R_B=\left(T_{41}{T}_{32}-T_{42}{T}_{31}\right)/D$, where $D=({\alpha }_{11}{\alpha }_{22}-{\alpha }_{12}{\alpha }_{21})S$. Results for two SV distributions ($n=0.01\%$) in (a) and (b) are decomposed in (c) and (d), respectively. Most of the terms in (c) and (d) are vertically displaced, with dashed lines marking the corresponding positions of the zero axes. Magnetic field is $B=20$ T, and $T=0$ K.

Figure 12

(a) and (b) Electron transmission functions from the first lead, for two specific disorder distributions [the same two distributions as used in Figs. 11 and 11, respectively]. The transmission functions are vertically displaced by $\mathrm{\Delta }T=2$ for clarity. (c)–(f) Current densities at energies marked in (a) and (b) by vertical red lines. Magnetic field is $B=20$ T, and temperature is $T=0$ K.

Figure 13

Smoothed and averaged bend resistances (${\overline{R}}_B$; red curves), and total DOS (blue curves) for increasing value of the next-nearest-neighbor hopping energy $t^{\prime }$. All results are obtained for the SV disorder type ($n=0.01\%$), for $N=10$ different disorder distributions. Magnetic field is $B=20$ T, and $T=16$ K. Green, dashed lines are linear fits of the ${\overline{R}}_B$ peak energy versus the NNN hopping energy $t^{\prime }$. Results for $t^{\prime }\ne 0$ are displaced horizontally by $\mathrm{\Delta }E=-3|t^{\prime }|$ in order to align the Landau levels. The $R_B$ peaks at the zeroth Landau level are cut off above 0.09 $h/\left(2e^2\right)$ for clarity.