Double detonations in sub-Chandrasekhar mass carbon-oxygen white dwarfs with helium shell are a potential explosion mechanism for a Type Ia supernova (SNe Ia). It comprises a shell detonation and subsequent core detonation. The focus of our study is on the effect of the progenitor metallicity on the nucleosynthetic yields. For this, we compute and analyse a set of eleven different models with varying core and shell masses at four different metallicities each. This results in a total of 44 models at metallicities between 0.01$Z_odot$ and 3$Z_odot$. Our models show a strong impact of the metallicity in the high density regime. The presence of $^{22}$Ne causes a neutron-excess which shifts the production from $^{56}$Ni to stable isotopes such as $^{54}$Fe and $^{58}$Ni in the $alpha$-rich freeze-out regime. The isotopes of the metallicity implementation further serve as seed nuclei for additional reactions in the shell detonation. Most significantly, the production of $^{55}$Mn increases with metallicity confirming the results of previous work. A comparison of elemental ratios relative to iron shows a relatively good match to solar values for some models. Super-solar values are reached for Mn at 3$Z_odot$ and solar values in some models at $Z_odot$. This indicates that the required contribution of SNe Ia originating from Chandrasekhar mass WDs can be lower than estimated in orevious work to reach solar values of [Mn/Fe] at [Fe/H]$=0$. Our galactic chemical evolution models suggest that SNe Ia from sub-Chandrasekhar mass white dwarfs, along with core-collapse supernovae, could account for more than 80% of the solar Mn abundance. Using metallicity-dependent SN Ia yields helps to reproduce the upward trend of [Mn/Fe] as a function of metallicity for the solar neighborhood. These chemical evolution predictions, however, depend on the massive star yields adopted in the calculations.
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