I have been using TEKS they are awesome! They require a number 3 Phillips bit. The below info is for the 2 3/4, but they have shorter ones you can use for 2x2's Good luck!
Here is the home depot link, not sure if it will work right, the model number is Model 21384 you might use the search function if it doesn’t
Teks self-tapping screws with wings provide secure wood-to-metal fastening with strong holding power, without the need to pre-drill! Engineered drill-point and flute remove material efficiently and properly size the hole for threads. Special wings ream a hole in the wood to prevent thread engagement during drilling. Ideal for fastening wood to thicker gauge (20-12 gauge) metal in applications such as wood headers and plywood to steel frames. Installs with a standard screwgun and a #3 Phillips bit.
No pre-drilling necessary
Screws tap their own threads
Secure fastening with strong holding power
Engineered drill point and flute (pilot section) removes material efficiently and sizes the hole for threads.
MFG Brand Name : Teks
MFG Model # : 21384
MFG Part # : 21384