"A piece of tape" is a pretty iffy way to control the depth of a hole, especially when you want to drill multiple (six!) holes that all come out the same - tape move easily, and really easily as it warms up. There is a product called a "drill stop" that's a metal ring that locks onto the bit with a set screw, they're usually sold in sets of several different sizes. Or - the discount way - you can cut some plastic tubing, like BiC pens or such, to various lengths. It doesn't have to be the perfect length, as you can slide the drill bit in and out a ways before you tighten the chuck.