Formal logic is logic as concerned with the pattern of valid inference which makes any proof a proof regardless of subject matter. For example, the subject of formal logic of the first operation of the mind (i.e. simple apprehension) is the term (i.e. "A sign out of which a simple proposition is constructed"), but formal logic does not investigate the intension of the term, simply its relation to other terms. Formal logic doesn't ask what X means in the statement, "All X are Y". This is logic as we understand it today, a study of the formal correctness and consistency of our reasoning.
Material logic is logic as concerned with proofs within a specific subject matter, so yes it has to do with soundness because the truth of premises are examined. Material logic is focused on the truth in knowing the intension of X in "All X are Y." For example, the subject of material logic of the first operation of the mind is the disposition of the universal (signified by the term), so material logic asks what are the conditions necessary for the universal.
W.D. Ross introduces the beginning of Aristotle's Posterior Analytics with this distinction between formal and material logic and shows the need for material logic:
Syllogistic inference involves, no doubt, some scientific knowledge,
viz. the knowledge that premisses of a certain form entail
a conclusion of a certain form. But while formal logic aims simply
at knowing the conditions of such entailment, a logic that aims
at being a theory of scientific knowledge must do more than this;
for the sciences themselves aim at knowing not only relations
between propositions but also relations between things, and if the
conclusions of inference are to give us such knowledge as this,
they must fulfil further conditions than that of following from
certain premisses. To this material logic, as we might call it in
opposition to formal logic, Aristotle now turns (Aristotle's Prior and Posterior Analytics, 51).
See John of Poinsot, Tractatus de Signis, ed. John Deely, p. 24/10-13. Also, John Deely, Four Ages of Understanding, 601.