A spintronic theory is proposed for a magnetic tunnel junction with a single-crystal barrier. This theory is founded on optical theory and the Patterson function approach and can adequately account for the influences of the barrier periodicity and lattice distortion. Based on the proposed theory, we investigated the temperature characteristics of the tunnel magneto-Seebeck effect in MgO-based magnetic tunnel junctions. In low-temperature approximation, the temperature only influences the Seebeck coefficients through the Fermi distribution function. However, in the proposed theory, the temperature can modify the potential parameter of the barrier through lattice distortion and further alter the Seebeck coefficients. As the tunneling electrons are scattered by the single-crystal barrier, the corresponding coherence leads to the oscillation of the Seebeck coefficients. The results can theoretically interpret the non-monotonic temperature characteristics of the Seebeck coefficient in the parallel configuration and TMS in MgO-based magnetic tunnel junctions. Furthermore, the physical mechanism of the non-monotonic temperature characteristics is elucidated. In addition, we investigated the influences of the parameters of the lattice distortion, such as strain, defect concentration, and recovery temperature, on the temperature characteristics of the tunnel magneto-Seebeck effect. It is found that the amplitudes of the non-monotonic variations of the Seebeck coefficient in the parallel configuration can be modulated by the defect concentration and recovery temperature.