Negative hopping magnetoresistance of two-dimensional electron gas in a smooth random potential.

We show that the tunnel coupling between semiclassical states localized in different minima of a smooth random potential increases when magnetic field is applied. This increase originates from the difference in gauge factors which electron wave functions belonging to different electron “lakes” acquire in the presence of the field. In the common case of a narrow barrier between two lakes, the characteristic magnetic field is determined by the area of the lakes, and thus may be quite small. The effect of the field on coupling constants leads to a negative magnetoresistance in low-temperature conduction. PACS numbers: 73.50.Jt, 71.55.Jv Deep in the insulating regime, the low-temperature conductivity of a two-dimensional electron gas is dominated by phonon-assisted electron hops between localized states. The conventional picture [1] of hopping implies that localized states are formed by individual impurities. The lower the temperature, the larger the separation of the localized states between which a typical hop takes place. In this picture, the magnetic field affects hopping transport in two ways: by shrinking the wave functions of the localized states, and by changing the contributions of different tunneling paths to the amplitude of a hop. The latter means that in the course of tunneling between initial and final states an electron can “visit” a sequence of virtual states localized on neighboring impurities, so that the amplitude of a hop represents a sum of partial amplitudes. In a magnetic field, each path acquires a phase factor. These factors suppress the destructive interference of different paths, thus causing the decrease [2,3] of the resistance in a weak magnetic field. It was shown in [2] that this decrease comes from specific hops for which tunneling amplitudes are close to zero in the absence of a field. Although the portion of these hops is small, the rise of conductivity with magnetic field is related to the increase of the tunneling probabilities for these particular hops, since they are most sensitive to the field. A different realization of the insulating regime emerges if the random potential is smooth. In the latter case, the electron gas breaks up into separate lakes, each lake accommodating many electrons. Within a certain temperature range the electron transport is provided by tunneling of electrons through saddle points of barriers separating the adjacent lakes. This picture is different from the standard picture of variable range hopping [1], so that the straightforward application of the theory of magnetoresistance [2,3] is impossible. In the present paper, we study magnetoresistance of electron gas in this regime. It is important to note that the states active in transport (i.e., with energies close to the Fermi level) correspond to high-number levels of size quantization in the minima of the random potential. As a result, the amplitude of a hop is determined not only by the transmission coefficient at the saddle point, but also depends significantly on the overlap between the wave functions on the both sides of the barrier. This overlap is normally small because of the oscillatory behavior of the wave functions. If the applied magnetic field is weak enough, its prime role is to affect this overlap. We argue that the overlap increases with magnetic field, i.e., the coupling between the states, localized in adjacent minima of the potential, becomes stronger. In other words, unlike the conventional [2,3] model, magnetoresistance of each elementary link appears to be negative thus leading to the net negative magnetoresistance of the system. Consider two minima separated by a barrier. In the vicinity of a saddle point the barrier potential has the form