Trace Routing for a 0.5mm Pitch BGA
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Dog-bone routing is preferred as it is lower cost and less risk of soldering issues. It also provides space for 0201 decoupling caps to be placed directly opposite the BGA pins. You should be able to use a 6 mil via drill and route the signals on the inner layers using ~3.3 mil traces. The trace can increase width or join a diff pair as soon as it exists the via array. Via in pad can be used but should be a last resort.
So it should look something like this, these number can be tweaked slightly (but do not go less than 3/3 space/trace, also try not to go thicker than a 0.62 board if you use a 6 mil drill):