Context. Recent studies on stellar evolution have shown that the properties of compact objects strongly depend on the metallicity of the environment in which they were formed. Aims. Using some very simple assumptions on the metallicity of the stellar populations, we explore how this property affects the unresolved gravitational-wave background from extragalactic compact binaries. Methods. We obtained a suit of models using population synthesis code, estimated the gravitational-wave background they produce, and discuss its detectability with second- (advanced LIGO, advanced Virgo) and third- (Einstein Telescope) generation detectors. Results. Our results show that the background is dominated by binary black holes for all considered models in the frequency range of terrestrial detectors, and that it could be detected in most cases by advanced LIGO/Virgo, and with Einstein Telescope with a very high signal-to-noise ratio. The observed peak in a gravitational-wave spectrum depends on the metallicity of the stellar population.