Exponential GLM


lets assume I have some data where my response follows an exponential distriubtion (also inlcuding zeros), what is a proper way of modelling it via rstanarm?

Unfortunately, the family objects used for stan_glm() doesn’t contain Exponential() (is there an obvious reason why not?).
Or is it possible to use Gamma() instead but with a fixed shape parameter to 1 and still allowing also response values of zero?