We have combined structural and optical characterizations to investigate the tensile-strained state and the band gap engineering of Ge layers grown on Si(001) using molecular beam epitaxy. The tensile strain is generated in the Ge layers due to a difference of thermal expansion coefficients between Ge and Si. The Ge growth on Si(001) was proceeded using a two-step growth process: a low-temperature step to produce relaxed buffer layers, followed by a high-temperature step to generate the tensile strain in the Ge layers. For the low-temperature step, we have evidenced the existence of a substrate temperature window from 260 to \(300\circ\)C in which the well-known Stranski-Krastanov Ge/Si growth mode transition from two-dimensional to three-dimensional growth can be completely suppressed. We show that the value of the tensile strain in the Ge layers lineally increases with increasing the growth temperature and reaches a saturation value of \(\sim 0.24\)% in the temperature range of \(700-770\circ\)C. Post-grown cyclic thermal annealing has allowed to increase the tensile strain up to 0.30%, which is the highest value ever reported to date. Finally, photoluminescence measurements reveal both an enhancement of the Ge direct band gap emission and a reduction of its energy due to the presence of tensile strain in the layers.