We consider quantum point contacts (QPCs) defined within disorderedtwo-dimensional electron gases as studied by scanning gate microscopy. Weevaluate the conductance maps in the Landauer approach and wave functionpicture of electron transport for samples with both low and high electronmobility at finite temperatures. We discuss the spatial distribution of theimpurities in the context of the branched electron flow. We reproduce thesurprising temperature stability of the experimental interference fringes farfrom the QPC. Next, we discuss -- previously undescribed -- funnel-shapedfeatures that accompany splitting of the branches visible in previousexperiments. Finally, we study elliptical interference fringes formed by aninterplay of scattering by the point-like impurities and by the scanning probe.We discuss the details of the elliptical features as functions of the tipvoltage and the temperature, showing that the first interference fringe is veryrobust against the thermal widening of the Fermi level. We present a simpleanalytical model that allows for extraction of the impurity positions and theelectron gas depletion radius induced by the negatively charged tip of theatomic force microscope, and apply this model on experimental scanning gateimages showing such elliptical fringes.
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