We use the transfer matrix method to study the quantum tunnelling through an indirect-band-gap double-barrier like the GaAs/A1As/GaAs/A1As/GaAs heterostructures along the[001]axis,which is described by the tight-binding model.The X-valley quasi-bound state gives rise to the Fano resonance different from the direct double-barrier transition in a resonance-tunnelling diode.The quantitative calculations demonstrate that a relatively high spin-polarization of the transmission probability can be achieved as compared with the single-barrier tunnelling case.Moreover the extension to the multi-barrier device is provided and leads to an important observation that the spin polarization increases with the number of barriers.