Scholarships for research stays abroad are offered by the German Academic Exchange Service (DAAD), postdocs may also apply at  Alexander von Humboldt Foundation.
Further more, the FAU Central Office of International Affairs has compiled a list of institutions which provide financial support for your stay abroad.
We recommend also to check our website with several scholarship databases for your own search as well as our website with general information about research stays abroad.
Further contacts, e.g. at your Faculty, can be found on (see section "Internationales")
If your doctoral project at FAU is funded already, do not forget to ask at your funding institution whether there are some special grants for a research stay abroad.