Is every Doctor of the Church a Saint? What are the requirements to be made a Doctor of the Church? This is a question directed at Roman Catholics, or anyone familiar with the history of the Roman Catholic church.

Every Doctor of the Church is a saint.  I was curious if this is perhaps a requirement to be recognized as a Doctor, or if it is merely a "coincidence" (i.e. unintentional by the church) that all Doctors also happen be saints?

I've also noticed that some Doctors of the Church are also Early Church Fathers.  The opposite is not true though - not all Fathers of the church are necessarily Doctors (see Origen, for example).

What exactly are the requirements to be appointed a Doctor of the Church?
Why the titled is conferred


  Certain ecclesiastical writers have received this title on account of
  the great advantage the whole Church has derived from their doctrine.


Who confers it


  To these great names others have subsequently been added. The
  requisite conditions are enumerated as three: eminens doctrina,
  insignis vitae sanctitas, Ecclesiae declaratio (i.e. eminent learning,
  a high degree of sanctity, and proclamation by the Church). Benedict
  XIV explains the third as a declaration by the supreme pontiff or by a
  general council. But though general councils have acclaimed the
  writings of certain Doctors, no council has actually conferred the
  title of Doctor of the Church. In practice the procedure consists in
  extending to the universal church the use of the Office and Mass of a
  saint in which the title of doctor is applied to him. The decree is
  issued by the Congregation of Sacred Rites and approved by the pope,
  after a careful examination, if necessary, of the saint's writings. It
  is not in any way an ex cathedra decision, nor does it even amount to
  a declaration that no error is to be found in the teaching of the
  Doctor. It is, indeed, well known that the very greatest of them are
  not wholly immune from error. No martyr has ever been included in the
  list, since the Office and the Mass are for Confessors. Hence, as
  Benedict XIV points out, St. Ignatius, St. Irenæus, and St. Cyprian
  are not called Doctors of the Church. - Source: Doctors of the Church | New Advent