I am searching for a Doctoral Advisor in Germany. Is the official term Doktorvater/Doktormutter? It sound a bit strange to write a Professor asking if they could be my "Doktormutter". Which terms are used in German universities?
Here's the definition from Duden. The bold part highlights the part that clearly states that it is used while writing the dissertation.
Doktorvater/Doktormutter: Universitätsprofessor(in), der/die dem Doktoranden das Thema für die Doktorarbeit gibt [und ihn bei der Abfassung seiner Dissertation berät]
As already pointed out in all the other answers, Betreuer or the verb betreuen are the preferred terms.
The important thing to note is that a doctoral advisor can be called both Betreuer and Doktorvater. Once you finished your dissertation, your advisor is naturally not your advisor any longer. You would then refer to him as "früherer/damaliger Betreuer".
The term Doktorvater, however, is still valid. It could be a little confusing if you write a second dissertation, though... ;)
For more information about Doktorvater, please see Wikipedia.
And finally here's a quote from zeit.de that once again clearly shows that Doktorvater is used while still writing the dissertation.
Laut einer im Dezember 2012 erschienenen Studie des Hochschul-Informations-Systems ist jeder fünfte Doktorand mit seiner Betreuung unzufrieden. Doch viele trauen sich nicht, die Probleme offen anzusprechen, schließlich sind sie weiterhin vom Wohlwollen des Doktorvaters abhängig.
The term Doktorvater is indeed used, although it may be perceived as old-fashioned by some. Whether the term Doktormutter is used as widely and in the same way would be a different question.
However, the word Doktorvater is not what you are looking for, because it describes your thesis advisor after you have received your doctorate degree.
As the others have already noted, the word Betreuer(in) in mostly used. Of course it lacks the thesis as in thesis advisor, so some context is needed. One could argue whether the betreuen refers to the candidate or his or her work, I think the word is used in both ways.
Yet another question is whether you should write to the professor in German at all or just stick to English. The best choice will depend on the field and the person, so it is hard to judge.
I find Doktorvater and Doktormutter really cute. I am sure the professor would smile if you asked him/her like this. The way how we call it in German is Betreuer / Betreuerin, regardless of what type of work (Betreuer der Diplomarbeit, Praktikumsbetreuer etc.) and the verb is betreuen. So you could ask your professor:
Ich möchte Sie hiermit herzlich bitten, meine Doktorarbeit/Dissertation zu betreuen.
Just to summarize, Doktorvater/Doktormutter is an unofficial old-fashioned term, intended for more informal use (kind of a "cute" way to call your Doctoral Advisor). Students prefer to say Betreuer/Betreuerin. I heard that Doktormutter is even less used, probably because of the small number of female professors, which made the term unfamiliar to most students.
Doktorvater (I have never heard the term Doktormutter). It is a superficially endearing term and conjures up warm images of the 'family'. One then thinks of one's relationship to one's own family. The warmness in some cases is sustainable, but in others it is most definitely not. One then talks, as I have done, to students writing a Ph.D. at a German university about their 'doctoral dads' and the warmness can quickly turn cold to frosty. In my 15 years' experience of teaching at a German university, I can say that students prefer Betreuer/in. But Americans seem to like the older term. Probably the cuddly sentimentality of it.