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Electron paths and double-slit interference in the scanning gate microscopy

We analyze electron paths in a solid-state double-slit interferometer based on the two-dimensional electron gas and their mapping by the scanning gate microscopy (SGM). A device with a quantum point source contact of a split exit and a drain contact used for electron detection is considered. We study the SGM maps of source-drain conductance ($G$) as functions of the probe position and find that for a narrow drain the classical electron paths are clearly resolved but without any trace of the double-slit interference. The latter is present in the SGM maps of backscattering ($R$) probability only. The double-slit interference is found in the $G$ maps for a wider drain contact but at the expense of the loss of information on the electron trajectories. Stability of $G$ and $R$ maps versus the geometry parameters of the scattering device is also discussed. We discuss the interplay of the Young interference and interference effects between various electron paths introduced by the tip and the electron detector.