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Proximity induced pseudogap in mesoscopic superconductor/normal-metal bilayers

Recent scanning tunneling microscopy measurements of the proximity effect in Au/La$_{2-x}$Sr$_{x}$CuO$_{4}$ and La$_{1.55}$Sr$_{0.45}$CuO$_{4}$/La$_{2-x}$Sr$_{x}$CuO$_{4}$ bilayers showed a proximity-induced pseudogap [Yuli et al., Phys. Rev. Lett. {\bf 103}, 197003 (2009)]. We describe the proximity effect in mesoscopic superconductor/normal-metal bilayers by using the Bogoliubov-de Gennes equations for a tight-binding Hamiltonian with competing antiferromagnetic and d-wave superconductivity orders . The temperature dependent local density of states is calculated as a function of the distance from the interface. Bound state due to both d-wave and spin density wave gaps are formed in the normal metal for energies less than the respective gaps. If there is a mismatch between the Fermi velocities in the two layers we observe that these states will shift in energy when spin density wave order is present, thus inducing a minigap at finite energy. We conclude that the STM measurement in the proximity structures is able to distinguish between the two scenarios proposed for the pseudogap (competing or precursor to superconductivity).