Magnetoresistance of a two-dimensional electron gas in a random magnetic field.

We report magnetoresistance measurements on a two-dimensional electron gas (2DEG) made from a high mobility GaAs/AlGaAs heterostructure, where the externally applied magnetic field was expelled from regions of the semiconductor by means of superconducting lead grains randomly distributed on the surface of the sample. A theoretical explanation in excellent agreement with the experiment is given within the framework of the semiclassical Boltzmann equation. PACS numbers: 72.20.M, 73.50.J, 73.50.-h, 73.40.-c Typeset using REVTEX 1 The response of a 2DEG to a spatially inhomogenous magnetic field is a subject of considerable interest both theoretically and experimentally. One of the experimental techniques proposed is to deposit a pattern of small magnets on the semiconductor containing the 2DEG. This technique, however, only gives rise to a very weak modulation. Another method is to grow the 2DEG on a substrate with a modulated thickness. The applied magnetic field experienced by the electrons in the plane of the curved 2DEG will vary with the thickness modulation. The feasibility of this method is however limited by the technological difficulties of the MBE regrowth techniques required. So far, the most simple technique, originally proposed by Rammer and Shelankov for studying weak localisation effects in inhomogenous magnetic fields, employs a type II superconducting gate on top of the heterostructure containing the 2DEG. For a type II superconducting gate an applied magnetic field will create the so-called mixed state in the superconductor above the lower critical field Bc1. In this state the magnetic field penetrates the film as flux tubes. Each flux tube will contain an integral number of (superconductivity) flux quanta Φ0 = h/2e. For a perfect type II superconductor the mixed state is accomplished by the formation of a two-dimensional hexagonal lattice of vortices. In a real superconductor inhomogeneities will tend to pin the vortices, so a random distribution of flux tubes is more likely to occur rather than the regular lattice. The magnetoresistance of the type II superconductor gated samples have been investigated experimentally in various limits of 2DEG properties. Bending et al. and Geim have studied the weak localisation effects predicted by Rammer and Shelankov for a low mobility GaAs/AlGaAs heterostructure with a Pb gate and a thin Bi film evaporated on a Nb/Mo substrate respectively. Kruithof et al. have studied the mechanisms of voltage induction in the 2DEG of a Si MOSFET caused by flux flow in a Nb/Mo superconducting gate. The above experiments have probed the diffusive properties of the 2DEG’s. In the ballistic regime, where the electronic mean free path is much longer than the vortex diameter, a series of experiments were performed by Geim et al.8–10, and the results interpreted by treating the vortices as scatterers. The effect of single vortices have also been studied both experimentally and theoretically.11–13 2 In this letter we demonstrate a new and very simple technique for creating a strong magnetic field modulation. We also propose a semiclassical model based on the Boltzmann equation to explain the measured magnetoresistance caused by the inhomogeneous magnetic field. Our model gives excellent agreement with the experiment, and in addition is applicable to the results of Geim et al. The inhomogeneous magnetic field was achieved by means of small lead grains randomly distributed on the surface of a high mobility GaAs/AlGaAs heterostructure. The lead grains (approximated as spheres) used had a size (average diameter) distribution as shown in Fig. 1. For these grain sizes Pb is a type I superconductor. Below the critical field given by