From what I have heard about the difference between Shannon and Simpson index:

Shannon index puts more weights on richness (how many different species are there including the very rare one). Simpson put more attentions on dominant species (rare species don't make much difference)

From the TCR diversity point of view, which of these index make more sense? My personal opinion is Shannon index because that TCR-Seq only reflects a momentary TCR status of a dynamic process and those rare TCR may expand as long as they exist. I am not sue if this logic sounds right...