Here's the trailer in question:
What is this compelling song?
Released in 2002 for his album American IV: The Man Comes Around.
Originally written and performed by Nine Inch Nails.
Director James Mangold did an interview with Empire magazine, where he explained his choice of music as "standing out":
Obviously I have a connection and a fondness for Johnny Cash, and his tone and his message and his music. But the real driver in all these decisions is trying to separate ourselves, in an accurate way, from the other superhero movies.
We think we’re going to deliver something a little different and we want to make sure we’re selling audiences on the difference. Sometimes even when a movie’s a little different, the studio’s trying to market the movie just like all the others. [Cash’s] music, in a way, separates us from the standard, bombastic, brooding orchestral, swish-bang, doors opening and slamming, explosions kind of methodology of some of these movies.
Same website also provides some parallels between the lyrics and Logan himself.