there 2 kind of biceps, ones that peak easier (leaving a gap btween the head of the bicep and eblow) and ones made more for mass (peak seems to be shortened because the bicep is more stretched towards the elbow)
sounds like ur bicep is one of "mass" genetics. when ur arm is straight the bicep will tend to go more down to the elbow, this is normal for any bicep. keep training eventually the peak of your bicep will increase and you wont have the caved in look.
arnold likes to do concentration curls because they help increase your peak of the bicep. (according to him anyways, but ive never doubted arnold)
anyways without actually seeing your bicep i could be totally wrong and maybe ur arm is messed up, but if its both of your arms i really doubt there is anything wrong.
posting how long u been training for and a pic of ur bicep flexed and unflexed would help alot.