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We study the transport properties of the quantum Hall bilayers systemslooking closely at the effect that disorder and electron-phonon interactionhave on the interlayer tunnelling current in the presence of an in-planemagnetic field $B_\parallel$. We find that it is important to take into accountthe effect of disorder and electron-phonon interactions in order to predict afinite current at a finite voltage when an in-plane magnetic field is present.We find a broadened resonant feature in the tunnelling current as a function ofbias voltage, in qualitative agreement with experiments. We also find thebroadening of this resonant feature due to electron-phonon coupling has anon-monotonic dependence on $B_\parallel$, related to the geometry of thedouble quantum well. We also compare this with the broadening effect due tospatial fluctuations of the tunnelling amplitude. We conclude that such staticdisorder provides only very weak broadening of the resonant feature in theexperimental range.

Effect of disorder and electron-phonon interaction on interlayer tunnelling current in quantum Hall bilayer