The advantage of trenching to your barn is that you can think of other options in the trench.
Example, you can add a security line from the house to barn so if someone breaks in you get a signal in the house. OR, a video cable to add a camera, how about a light switch at the house so you can turn on the outside lights of the barn from the house.
The options are endless when trenching and using conduit rather that just dropping direct burial cable in the dirt.
Take you time, think out many options and always go bigger cable than needed for either future or to keep the cable cooler.
Copper in my opinion is the only way to go.
I ran a 600 ft cable and added a video cable conduit, and a phone cable conduit all at the same time. Years later I was glad I did because I was able to easily add video monitoring because of break ins, and the phone conduit was big enough to add a 25 pair cable for phones, internet, and securoty.
It was cheaper doing it with an open trench than digging another trench while trying to miss the original conduit later on.