Billiard model of a ballistic multiprobe conductor

Resistance measurements in a ballistic narrow channel in a two-dimensional electron gas show a complex, nonmonotonic dependence on a weak perpendicular magnetic field B. Phenomena which have drawn particular attention are the "quenching of the Hall effect"" (a suppression of the Hall resistance around zero field), the "negative Hall resistance," the "last Hall plateau"~* (reminiscent of quantum Hall plateaus, but occurring at much lower B), "bend resistances" (associated with current passing around the corner at a junction), and "magnetically reduced backscattering" (a decrease of the longitudinal resistance in weak magnetic fields). The theoretical effort in this field" has focused on models of quantum-mechanical propagation and scattering, äs in an electron waveguide, Quantum-mechanical phase coherence is certainly necessary for some of the fine structure which appears experimentally only at the lowest (mK) temperatures, but the phenomena listed above have a relatively weak temperature dependence —suggesting a different origin. In this Letter we demonstrate that a model based on classical junction scattering, äs in an electron billiard, exhibits all these phenomena, which can thus be classified äs classical magneto-size effects in a degenerate electron gas. Our investigation builds on two recent papers:'" To explain the nonadditivity of the contact resistance of two opposite constrictions, we first pointed out that a flared (hornlike) constriction collimates the beam of injected electrons, äs a result of the adiabatic invariance of the product of width and transverse momentum. Baranger and Stone have proposed (on the basis of a quantummechanical calculation of the low-field Hall resistance) that this collimation causes the quenching of the Hall effect in a (realistic) cross geometry with rounded corners, by suppressing the coupling of the currentcarrying channel to the side probes used to measure the Hall voltage. We summarize our main results. Our calculations of the low-field Hall resistance RH show a quenched äs well äs a negative RH, depending on the geometry and consistent with the experiments of Ford et al. in which different geometries were compared. We find that a strong suppression of the coupling to the side probes is not necessary for a drastic reduction of RH below its 2D value—a relatively weak collimation of the injected beam to a cone of 90° angular opening being sufficient. At higher fields a strikingly broad and flat Hall plateau appears—although the model contains no quantization. Its origin is the guiding-center drift along the curved channels walls at the junction. This classical effect enhances RH to the contact resistance of the lead, which is approximately independent of B over a wide field ränge—hence the plateau. Geometrical resonances cause oscillations on the Hall plateau, resembling the oscillations in the experiments.' Magnetic guiding reduces backscattering, thereby suppressing the longitudinal resistance RL and the bend resistance RBAs in the experiments" we find an "overshoot" in RB from a negative to a positive value before it drops to zero, due to destruction of collimation before guiding becomes effective. We consider the geometry of a long channel with two intersecting side channels (Fig. l, right inset). An elec-