Actually, both deaths are physical.
28 Marvel not at this: for the hour is coming, in the which all that are in the graves shall hear his voice, 29 And shall come forth; they that have done good, unto the resurrection of life; and they that have ndone evil, unto the resurrection of damnation
The Holy Bible: King James Version, Electronic Edition of the 1900 Authorized Version. (Bellingham, WA: Logos Research Systems, Inc., 2009), Jn 5:28–29.
The evil are resurrected just like the righteous. The difference is the righteous go into eternal life. The evil go to the second death. That is their resurrection to damnation or their second death. They will literally die a second time.
There is nothing in all of Scripture that speaks of a "spiritual death". At least not pertaining to man. One could say that the demons being destroyed is a form of spiritual death. However, that has nothing to do with humans. Humans aren't spirits so they can't have spiritual death.