A divot will come naturally if you are hitting down on the ball. The key is to bottom out your swing AFTER you hit the ball.
If you are picking it clean, with your short irons, you need to change your swing a bit. I could say to experiment with ball position, but you aren't taking any divot, so your swing bottoms out above the ground. Not optimal.
With the 6 iron and higher I like to "sweep it". With the shorter irons though, you need to "hit down" on the ball. Try to "squish" the ball into the ground with the middle of your clubface.
When you are sweeping with the short irons, too many bad things can happen. Because your swing path through the ball is so horizontal, when you do take grass, it's fat. When you time it perfectly, it's a nice shot. When you come in a little high, it's a skull. Over time you decide the fat shots don't go anywhere, so it's always good, a bit thin, or a dead skull.
Your swing needs to bottom out past the ball. It's easiest to do this with your left wrist suppinated.
Here's a surprise

, read 5 Lessons by Ben Hogan. This is the first year I am taking a divot, and my scores are dropping. Not a coincidence I am guessing.