Charged particle transport in antidot lattices in the presence of magnetic and electric fields: Langevin approach

Magnetotransport experiments on antidot lattices show a rich variety of physical phenomena. Depending on the value of the particle mean-free path $l$ in relation to the period $\lambda$ of regular scatterers, two very different regimes can be distinguished: the strongly diffusive $\left(l⪡\lambda \right)$ and the weakly diffusive $\left(l\gtrsim \lambda \right)$. We study particle transport in two-dimensional periodic landscapes based on a classical Langevin equation. The model covers both regimes and exhibits many of the features found experimentally at low magnetic fields. The most interesting observation is the presence of anomalous peaks in the magnetoresistance as a function of the magnetic field in the weakly diffusive regime. The roles of finite temperatures and of the electric field are also discussed.

Figure 1

Upper panel: a finite portion of the two-dimensional periodic potential [Eq. (3)] for parameters $a=5$, $b=0.8$. Lower panel: potential profiles $V\left(x,0\right)$ for fixed $a=10$ (left) and fixed $b=0.8$ (right). $V_0=1$ in both panels.

Figure 2

Resistivity ratio $\rho$ vs the resistivity ratio when $V_0=0$ $\left({\rho }_0=B/\tilde{\gamma }\right)$ for $\tilde{E}=1.75$, $\tilde{\gamma }=5$, $T=0\text{K}$, and potential parameters: ${\tilde{V}}_0=0.175$, $a=5$, $b=0.7$ (solid line), and $b=0.9$ (dotted line).

Figure 3

Hall resistance ${\rho }_{xy}$ and magnetoresistance ${\rho }_{xx}$ vs magnetic field $B$ for the same parameters as in Fig. 2.

Figure 4

(Color online) Trajectories for $B=1.2$, 2.9, 4.1, and 6.5 tesla, with increasing angles corresponding to increasing values of $B$. Results are shown for two different potential parameters, $b=0.7$ (solid lines) and $b=0.9$ (dotted lines). The dashed lines indicate the particle flow direction in the absence of antidots.

Figure 5

Magnetoresistance ${\rho }_{xx}$ (upper panel) and resistivity ratio $\rho$ (lower panel) vs magnetic field extending to high $B$ values for different temperatures. $T=0\text{K}$ (solid line), $T=2\text{K}$ (dashed line), and $T=20\text{K}$ (dotted line). The potential parameters are $\tilde{E}={\tilde{V}}_0=1.75$, $a=5$, and $b=0.7$.

Figure 6

Comparison of deflection angles for $b=0.7$ and $b=0.9$, calculated in terms of $\beta \equiv \rho \left(b=0.9\right)-\rho \left(b=0.7\right)$. The dotted lines correspond to the magnetic-field values of the trajectories in Fig. 4. Values of other parameters are as in Fig. 2.

Figure 7

(Color online) Typical trajectories at $B=0.35\text{tesla}$ for different values of the friction parameters indicated in the panels. In each case we include two trajectories in the presence of obstacles, as well as the trajectory in the absence of obstacles. In all panels the remaining parameters are $b=0.7$, $a=5$, $\tilde{E}=0.02$, ${\tilde{V}}_0=2$, and $T=0\text{K}$.

Figure 8

(Color online) Left panel: single trajectories for two different $B$ values. Right panel: Fragments of single trajectories for different $B$ values. (a) $\left(B=0.5\right)$ and (b) $\left(B=0.8\right)$ are trapped trajectories, (c) $\left(B=0.5\right)$ and (d) $\left(B=0.4\right)$ pinned orbits (encircling one and four antidots respectively), and (e) $\left(B=0.6\right)$ and (f) $\left(B=1.0\right)$ are runaway paths. In both panels the $\tilde{\gamma }=0.001$ and the remaining parameters are as in Fig. 7.

Figure 9

Magnetoresistance ${\rho }_{xx}$ and Hall resistance ${\rho }_{xy}$ vs magnetic field $B$ for $\tilde{\gamma }=0.001$, $\tilde{E}=0.02$, ${\tilde{V}}_0=2$, $T=0\text{K}$, $a=5$, $b=0.7$, and three different system lengths: $L=10$ (solid line), $L=20$ (dashed line), and $L=50$ (dotted line).

Figure 10

Resistivity ${\rho }_{xx}$ vs magnetic field $B$ for different temperatures: $T=0\text{K}$ (solid line), $T=10\text{K}$ (dashed line), $T=30\text{K}$ (dotted line), and $T=100\text{K}$. System size is $L=50$ and parameters are $\tilde{\gamma }=0.001$, $\tilde{E}=0.02$, ${\tilde{V}}_0=2$, $a=5$, and $b=0.7$.

Figure 11

Magnetoresistance in small (top) and large (bottom) systems for different $\tilde{E}$ values: 0.02 (solid line), 0.07 (dashed line), and 0.1 (dotted line). $T=0\text{K}$; system parameters are the same as in previous figure.

Figure 12

Magnetoresistance ${\rho }_{xx}$ vs $\tilde{E}$ for three different $B$ values: 0.35 (solid line), 0.85 (dashed line), and 1.5 tesla (dotted line). The lines simply connect the computed values. System size is $L=50$ and parameters are: $\tilde{\gamma }=0.001$, ${\tilde{V}}_0=2$, $T=0\text{K}$, $a=5$, and $b=0.7$.