Welcome to the Wefugees Community and thank you for reaching out to us.
As your wife is an EU citizen, the EU/freedom of movement law will be relevant for your mobility (including employment and social aspects) within the EU. Unfortunately, I am not an expert in this field. However, EU citizens are entitled to receive social benefits in Germany as long as they are employees. Students on the other hand, whose study program might qualify them to receive student benefits (Bafög), are not entitled to receive the unemployment/social money (Arbeitslosengeld 2/SGBII) from the so-called "Jobcenter". But as far as I know, being enrolled in a PhD does not qualify a person to receive the Bafög (student money), it is considered as employment. If her salary is not sufficient to cover her monthly demand (this needs to be individually calculated), she may apply for the social money additionally. If you are holding the permanent residence permit EU, you also should not face problems in living and working in Germany. If I am not mistaken, you are entitled to get social benefits as spouse of an EU citizen anyway (as long as the respective EU citizen is working or at least seriously looking for a job).
I will also link two of our experts here - I am sure they can give more profund information. Dear @mbeon-Christine and @mbeon-Ruth, were my explanations correct and/or is there anything you would like to add?
In any case, please don't hesitate to get back to us with any further questions.
All the best,