Effect of Disorder on Magnetotransport in Semiconductor Artificial Graphene

Magnetotransport in mesoscopic samples with semiconductor artificial graphene has been simulated within the Landauer–Büttiker formalism. Model four-terminal systems in a high-mobility two-dimensional electron gas have a square shape with a side of 3–5 μm, which is filled with a short-period (120 nm) weakly disordered triangular lattice of antidots at the modulation amplitude of the electrostatic potential comparable with the Fermi energy. It has been found that the Hall resistance ${{R}_{{xy}}}(B)$ in the magnetic field range of B = 10–50 mT has a hole plateau ${{R}_{{xy}}} = - {{R}_{0}}$, where R₀ = h/2e² = 12.9 kΩ, at carrier densities in the lattice below the Dirac point n < n_1D and an electron plateau ${{R}_{{xy}}} = {{R}_{0}}$ at n > n_1D. Enhanced disorder destroys the plateaus, but a carrier type (electrons or holes) holds. Long-range disorder at low magnetic fields suppresses quantized resistance plateaus much more efficiently than short-range disorder.

Significant progress has been achieved in recent years in the formation of semiconductor lattices with a period of 70–130 nm in a high-mobility two-dimensional electron gas. Technological imperfections that can destroy the energy spectrum of lattices were minimized. Particular attention is paid to the development and test of devices with the triangular lattice of antidots [1], which allow one to fabricate and study semiconductor artificial graphene. These devices with an undoped GaAs/AlGaAs heterostructure are equipped with two gates separated by an insulator [2]. The bottom gate is perforated with a lattice of holes, is placed on the surface of the semiconductor, and pulls electrons from lateral contacts to the GaAs working layer. The top gate, which is separated from the bottom one by a thin dielectric layer, forms cylinders that penetrate into the holes and create depletion domains, i.e., finite-height barriers (antidots) in the two-dimensional electron gas under the holes. As a result, a smooth sinusoidal potential with a hexagonal symmetry appears in the plane of the two-dimensional electron gas. Differences in the diameters, shape, and positions of the holes in the bottom gate and of the cylinders of the top gate introduce long-range disorder in the lattice potential [2]. The typical fluctuations of such disorder cover several periods of the lattice.

Using the tight binding method, the authors of [3] predicted the behavior of the Hall conductivity ${{\sigma }_{{xy}}}(E)$ for energies in the band gaps of Landau minibands for some atomic lattices, including hexagonal and square. An infinite lattice in a perpendicular magnetic field was considered, and the ratio of the magnetic flux through a unit cell ϕ to the flux quantum ${{\phi }_{0}} = h{\text{/}}e$ was chosen to be a rational fraction $\phi {\text{/}}{{\phi }_{0}} = p{\text{/}}q$. An anomalous behavior of the Hall conductivity in the hexagonal lattice near the Dirac point σ_xy = $\pm (2N + 1)2{{e}^{2}}{\text{/}}h$, where $N = 0,1,2, \ldots$ (degeneracy in spin is taken into account), was predicted in [4] before the experimental observation of the quantum anomalous Hall effect in graphene. The authors of [3] showed that the quantum anomalous Hall effect exists for a class of lattices up to certain energies. These energies are related to Van Hove singularities and are located on both sides of the Dirac point. It was also revealed that the conductivity undergoes a giant jump upon transition through Van Hove singularities with a change in the sign of the conductivity. Thus, it was shown in [3] that the Hall conductivity is positive below the first Van Hove singularity and is negative above the second Van Hove singularity and varies in energy in these regions at a step of $2{{e}^{2}}{\text{/}}h$ according to the conventional quantum Hall effect. An example demonstrating the dependence ${{\sigma }_{{xy}}}(E)$ was given for a “low” magnetic flux $\phi = {{\phi }_{0}}{\text{/}}31$; the corresponding magnetic field strength for graphene is B = 2538 T. The magnetic field corresponding to the flux $\phi {\text{/}}{{\phi }_{0}} = 1{\text{/}}31$ for artificial graphene with a period of 120 nm is as low as B = 0.011 T (the ratio of magnetic fields is proportional to the square of the ratio of lattice constants). The results obtained in [3] cannot be directly transferred to semiconductor artificial graphene because this is an absolutely different electron system with a smooth effective potential and significant di-sorder.

We previously simulated the triangular lattice of finite antidots (a period of 100 nm, disorder is absent) and found that the quantum anomalous Hall effect appears near Dirac points in the perpendicular magnetic field of several milliteslas [5]; current below and above the Dirac point is carried by holes and electrons, respectively. This result primarily holds at short-range disorder, which makes it possible to suppress interference oscillations associated with the size of the sample [6]. It remains unclear whether long-range disorder will destroy the quantization of the Hall resistance R_xy and whether hole and electron conductivities can remain near the Dirac points at suppressed values R_xy ≪ R₀ = h/2e² = 12.9 kΩ.

In this work, we compare the effect of short- and long-range disorder on quantum transport through a hexagonal lattice. We calculate the Hall resistance R_xy for a device in the four-terminal measurement scheme at a fixed modulation of the periodic potential and at a fixed level of disorder. We show that short-range disorder with reasonable amplitudes does not destroy the conventional quantum Hall effect, whereas long-range disorder can significantly reduce the Hall resistance but does not change its sign.

The calculations are performed with the potential specified by the formula [7] $U({\mathbf{r}}) = {{V}_{0}}\sum \cos ({{{\mathbf{g}}}_{{\mathbf{i}}}} \cdot {\mathbf{r}})$, where ${{V}_{0}}$ is the amplitude of potential modulation and g_i are the reciprocal lattice vectors given by the expressions

$${\mathbf{g}_{1}} = {{g}_{0}}(1,1{\text{/}}\sqrt 3 ),$$

$${\mathbf{g}_{2}} = {{g}_{0}}(0,2{\text{/}}\sqrt 3 ),$$

$${\mathbf{g}_{3}} = {\mathbf{g}_{1}} - {\mathbf{g}_{2}} = {{g}_{0}}(1, - 1{\text{/}}\sqrt 3 ).$$

Here, ${{g}_{0}} = 2\pi {\text{/}}L$, where L is the period of the lattice. The band structures of the perfect hexagonal lattice for different ${{V}_{0}}$ values were discussed in [2]. It was shown that the band structure depends only on the dimensionless amplitude of potential modulation ${{w}_{0}} = 0.5{{V}_{0}}{\text{/}}{{E}_{0}}$, where ${{E}_{0}} = \frac{{8{{\pi }^{2}}}}{9}\frac{{{{\hbar }^{2}}}}{{m{\text{*}}{{L}^{2}}}}$. For the effective mass in GaAs $m\text{*} = 0.067{{m}_{e}}$ and the period of the lattice L = 120 nm, ${{E}_{0}} = 0.693$ meV.

Figure 1 shows low minibands calculated for the lattice with the total amplitude of the periodic potential from the minimum to the maximum $9{{w}_{0}}{{E}_{0}} \approx$ 1.56 meV and the modulation ${{w}_{0}} = 0.25$. This modulation is quite low at which the third miniband overlaps with the second miniband. Nevertheless, miniband features are already clearly manifested in the Hall resistance. Two low minibands cross at the Dirac point with the energy E_1D = 0.49 meV. The first and second Van Hove singularities correspond to the energies E_VH1 = 0.32 meV and E_VH2 = 0.57 meV at which the curvature of the energy dispersion changes sign. The concentration of particles at the first Dirac point corresponds to the filling of the first miniband with two electrons: ${{n}_{{1{\text{D}}}}} = 2{\text{/}}(\sqrt 3 {{L}^{2}}{\text{/}}2)$ = 1.6 × 10¹⁰ cm^–2. With increasing modulation, the third band rises and the band gap appears between the second and third bands at ${{w}_{0}} \geqslant 0.75$.

Fig. 1.

(Color online) Low minibands for the perfect lattice with the period L = 120 nm and the modulation ${{w}_{0}} = 0.25$. The VH1 and VH2 arrows indicate the energy positions of the first and second Van Hove singularities and the 1D arrow marks the first Dirac point.

Disorder was specified in the calculation by the function V_d(r), which was added to the periodic potential $U({\mathbf{r}})$. This procedure allows one to independently vary the amplitudes of modulation and disorder. Since the potential was numerically determined on the discrete square lattice with a step of ${{h}_{x}} = {{h}_{y}} = 8$ nm, disorder is most simply introduced as a random addition in each ($i,j$) site of the computational grid. The implementation of disorder V_d was specified by a sequence of random numbers ${{\delta }_{{i,j}}}$ drawn from the uniform di-stribution in the range from –0.5 to 0.5: ${{V}_{d}}({{{\mathbf{r}}}_{{i,j}}}) = {{\delta }_{{i,j}}}{{V}_{r}}$, where ${{V}_{r}}$ is the amplitude of disorder. This local (i.e., short-range) disorder induces scattering even in high magnetic fields of 1–2 T, at which the magnetic length remains larger than the step of the grid. The other type of disorder is more realistic. It is specified according to our experience of electrostatic calculations of three-dimensional structures with the triangular lattice [2]. According to these calculations with allowance for the spread of diameters of holes in the bottom gate, the largest deviations of the potential from the ideal shape occur in the region of antidots, whereas variations of the potential between antidots in channels filled with electrons are much smaller because of self-screening. Consequently, it is reasonable to specify disorder of this type in the form of the sum of Gaussians ${{V}_{r}}{{\Sigma }_{i}}{{\delta }_{i}}\exp ( - {{({\mathbf{r}} - {{{\mathbf{r}}}_{i}})}^{2}}{\text{/}}{{\sigma }^{2}})$, where ${{\delta }_{i}}$ are the random numbers in the range from –0.5 to 0.5, ${{{\mathbf{r}}}_{i}}$ specify the nodes of the triangular lattice, V_r is the common scaling factor, and σ = 45 nm is the half-width chosen close to the distance from the gate to the plane of the two-dimensional electron gas in the simulated structure. Figure 2 shows 1 × 1 μm fragments of the map of the potential of the lattice with the modulation ${{w}_{0}} = 0.25$ for two types of disorder with the same amplitude ${{V}_{r}} = 2$ meV, where humps of the potential (antidots) are shown in yellow–red and channels between them are indicated in blue. It is seen that, when disorder is specified by Gaussians, “ridges” and “valleys,” which cover a group of periods, appear in the potential, whereas the potential on neighboring periods at site disorder does not vary on average.

Fig. 2.

(Color online) Maps of the two-dimensional potential of the lattice with the period L = 120 nm and the modulation ${{w}_{0}} = 0.25$ at (top panel) long-range and (bottom map) local disorder with the amplitude ${{V}_{r}} = 2$ meV. The color potential scale in millielectronvolts is given on the right. The isoline (black circles on the top panel) corresponds to the energy of the first Dirac point.

The problem of scattering of electrons in the model square sample is solved with the KWANT package [8], which is designed to calculate multiterminal quantum transport. Four-terminal resistances are reconstructed by the Büttiker formulas from calculated transmission coefficients between contacts in the system [9]. In the simulation, input channels from above and below horizontally approach lateral sides of the square on which the lattice is specified. These channels are marked by gray vertical stripes on the right and left sides of the sample (see Figs. 3 and 4). The widths of all four channels in the reported calculations were equal to 560 nm. The potential in the channels was constant and equal to the minimum of the potential of the lattice in the absence of disorder. Figures 3 and 4 present the current density of particles with a given energy E that are injected in the lattice through the bottom left channel, are scattered by the lattice, and are emitted through the two top channels and the bottom left one. The energy E at zero temperature means the Fermi level. The trajectories of electrons and holes, which are two types of charge carriers in the lattice, are separated in the magnetic field. The Lorentz force in a magnetic field of B = 30 mT pushes particles with the energy $E = 0.5$ meV in Fig. 3 to the bottom edge of the lattice (electron conductivity). Fairly strong local disorder with the amplitude ${{V}_{r}} = 3$ meV does not destroy the edge state, which means that the Hall resistance is quantized. Figure 4 shows the current for the energy $E = 0.8$ meV in a magnetic field of B = 30 mT. Charge carriers in the absence of disorder (top panel) are holes. Particles move upward along the left wall and bypass clockwise the center of the lattice. Long-range disorder (bottom panel) destroys the edge state and mixes particles over the entire lattice; in this case, the resistance is negative and small in magnitude (hole conductivity dominates).

Fig. 3.

(Color online) Current density distribution $J(x,y)$ for the $3.6$ μm lattice at ${{w}_{0}} = 0.25$, $E = 0.5$ meV, $B = 0.03$ T, and local disorder with the amplitude ${{V}_{r}} =$ 3 meV. Gray vertical strips mark the places of connection of four horizontal channels. Particles enter the sample through the bottom left channel, are scattered by the lattice, and can leave through the top left, top right, and bottom right channels.

Fig. 4.

(Color online) Current density distribution $J(x,y)$ for 3.6 μm lattice at ${{w}_{0}} = 0.25$, $E = 0.8$ meV, and $B = 0.03$ T (top panel) in the absence of disorder ${{V}_{r}} = 0$ and (bottom panel) in the presence of long-range disorder with the amplitude ${{V}_{r}} = 2$ meV. Sizes on the axes are given in microns.

The density of states and the Hall resistance R_xy were calculated as follows. From the energy dependence of the density of states DoS(E) in the absence of disorder at given ${{w}_{0}}$ values and $B = 0$, we determined the position of the Dirac point (dip between the linear slope and rise in DoS$(E)$) and the positions of Van Hove singularities (points of maxima of DoS$(E)$). Then, magnetic field calculations of the resistance and DoS$(B)$ were performed for different levels and types of disorder at energies below and above the Dirac point. We note that all energy and magnetic field dependences were smoothed because of the presence of strong interference oscillations. Averaging corresponded to an effective temperature of 0.05 K. Model systems were squares with a side of 3.6–4.8 μm (30–40 periods of the lattice).

The electrostatically induced modulation of the periodic potential of the lattice in the experiment can decrease strongly under the gradual electron population of lower minibands because of self-screening. However, the behavior of the Hall resistance near the first Dirac point is similar at different modulations ${{w}_{0}} = 0.25{-} 2$ because of a similar dispersion relation for two lower subbands [2]. Differences in the behavior of ${{R}_{{xy}}}$ appear at higher densities n > n_1D and in high magnetic fields [6]. The calculations reported below concern the lattice with ${{w}_{0}} = 0.25$.

Figure 5 shows the energy dependences of the Hall resistance ${{R}_{{xy}}}(E)$ and the density of states DoS$(E)$ for ${{w}_{0}} = 0.25$ at different amplitudes ${{V}_{r}}$ of long-range disorder in a magnetic field of B = 25 mT. In the absence of disorder, ${{R}_{{xy}}} = \pm {{R}_{0}}$ resistance plateaus separate the N = 0, –1, and +1 Landau levels ${{E}_{0}} = 0.46$ meV, ${{E}_{{ - 1}}} = 0.34$ meV, and ${{E}_{{ + 1}}} = 0.6$ meV. Disorder and the magnetic field slightly shift in the energy positions of the Dirac point and Van Hove singularities compared to the values obtained from the dispersion relation presented in Fig. 1. The corresponding energies E_1D, E_VH1, and E_VH2 at low magnetic fields can be attributed to the points where the Hall resistance changes sign [3]. The Hall resistance again becomes negative in the interval from $E = 0.6$ meV to about $E = 1.2$ meV, but the magnitude of the resistance in this interval is much lower than that near the Dirac point. The ${{R}_{{xy}}} = \pm {{R}_{0}}$ plateaus disappear at ${{V}_{r}} > 1$ meV. The resistance near the Dirac point at ${{V}_{r}} = 2$ meV is much lower than ${{R}_{0}}$. The resistance in the range where the ${{R}_{{xy}}} = - {{R}_{0}}$ plateau existed in the absence of disorder decreased by several times, but it still corresponded to the hole conductivity. The ${{R}_{{xy}}} = {{R}_{0}}$ electron plateau above the Dirac point is transformed into a hump with a maximum of about ${{R}_{0}}{\text{/}}3$. Additional oscillations of ${{R}_{{xy}}}(E)$ occur at ${{V}_{r}} = 3$ meV and the Dirac point is smeared. The resistance at ${{V}_{r}} > 3$ meV has a small magnitude and predominantly positive sign. Thus, two regions E_VH1 < E < E_1D and E_VH2 < E < 1.2 meV of negative Hall resistances exist at ${{V}_{r}} \leqslant 2$ meV. The Landau levels in the density of states DoS$(E)$ at ${{V}_{r}} = 0$ are manifested as peaks. Three peaks corresponding to the N = 0, +1, and –1 Landau levels are wider than the conventional Landau levels because they are twice more degenerate (Fig. 5b). The reason is that the Fermi surface near the Dirac point is doubly connected (i.e., is split into two valleys). The magnetic flux through the area of the unit cell of the lattice at B = 25 mT is $\phi = 0.075{{\phi }_{0}}$. The corresponding magnetic field strength in natural graphene is B = 5925 T, which is unachievable. The N = +1 and –1 Landau levels at $\phi \approx 0.1{{\phi }_{0}}$ overlap with Van Hove singularities in agreement with the simulation of natural graphene (see Fig. 5с in [3]). Thus, the regime of high magnetic fields occurs at B = 25 mT and, correspondingly, Landau levels with higher N values are absent. At significant disorder, the energy regions with the hole and electron conductivity overlap; for this reason, the peaks of DoS$(E)$ are washed out rapidly and are hardly observed already at ${{V}_{r}} = 2$ meV. The peaks of DoS$(E)$ below and above Van Hove singularities can be interpreted as Landau levels coming from the bottom of the first miniband (electron conductivity) and from the top of the second miniband (hole conductivity).

Fig. 5.

(Color online) Energy dependences of the (a) Hall resistance ${{R}_{{xy}}}$ and (b) density of states for ${{w}_{0}} = 0.25$, B = 25 mT, and long-range disorder. The numbers 0, +1, and –1 in panel (b) mark the peaks of the N = 0, +1, and –1 Landau levels, respectively. The position of the Dirac point coincides with the N = 0 peak, whereas Van Hove singularities at B = 25 mT coincide with the N = +1 and ‒1 peaks.

Figure 6 shows the magnetic field dependence of the Hall resistance ${{R}_{{xy}}}(B)$ for ${{w}_{0}} = 0.25$ and $E =$ 0.4 meV at long-range disorder with different amplitudes. It is seen that long-range disorder very strongly affects ${{R}_{{xy}}}(B)$. The negative slope of ${{R}_{{xy}}}(B)$ at $B = 0$ indicates the hole conductivity. The resistance becomes positive at magnetic fields higher than 50–55 mT and then, oscillating, increases to ${{R}_{{xy}}} = {{R}_{0}}$. Deep minima, which are washed out by disorder, exist between the R = R₀, R₀/2, R₀/3, and R₀/4 plateaus.

Fig. 6.

(Color online) Magnetic field dependence of the Hall resistance ${{R}_{{xy}}}(B)$ for ${{w}_{0}} = 0.25$ and $E = 0.4$ meV (below E_1D) at long-range disorder with different amplitudes.

Figure 7 shows the magnetic field dependence of the Hall resistance ${{R}_{{xy}}}(B)$ for different sizes of the lattice and different types of disorder. All plateaus in the case of short-range disorder are very pronounced, whereas long-range disorder destroys the plateau at $B < 0.05$ T but holds the type of carriers: the resistance becomes negative in the region where the ${{R}_{{xy}}}(B) = - {{R}_{0}}$ plateau existed. Resistance curves for different implementations of a certain type of disorder have a qualitatively similar behavior.

Fig. 7.

(Color online) Magnetic field dependence of the Hall resistance ${{R}_{{xy}}}(B)$ at ${{w}_{0}} = 0.25$, $E = 0.4$ meV, and ${{V}_{r}} = 1.5$ meV for lattices specified on squares with a side of 3.6 and 4.8 μm at different implementations and types of disorder. The red solid and dashed blue lines refer to long-range disorder (disorder at antidots), whereas the green solid line corresponds to short-range disorder (disorder at sites).

Figure 8 shows the Hall resistance curves ${{R}_{{xy}}}(B)$ with (blue dotted lines) positive and (red solid lines) negative slopes at zero for various energies of electrons. It is seen that the slope of the ${{R}_{{xy}}}(B)$ curves at zero changes sign four times: at Van Hove singularities, at the Dirac point, and at an energy above $E = 1.3$ meV, where minibands overlap and the electron conductivity dominates. The second region of the hole conductivity is wider than the first one: it extends from 0 almost to B = 100 mT. We note that the carrier density in the sample can no longer be determined from the slope of the ${{R}_{{xy}}}(B)$ curve at zero.

Fig. 8.

(Color online) Hall resistance for ${{w}_{0}} = 0.25$ in the presence of long-range disorder with the amplitude ${{V}_{r}} =$ 2 meV at fixed energies indicated next to the curves. The ${{R}_{{xy}}}(B)$ curves are shifted vertically with a step of 0.5.

To summarize, we have shown that the Hall resistance changes sign under the variation of the energy (density) of electrons in the lattice or the magnetic field strength. Such an alternation of types of carriers is due to the miniband spectrum of the lattice. It has been found that short-range disorder in low magnetic fields hardly destroys the edge states of the lattice, whereas long-range disorder mixes particles over the entire system and suppresses the Hall resistance compared to quantized values in the absence of disorder.