Is my dentist responsible for my crown that broke off only after 6 months?

You can sue anybody for anything, but I doubt you would win.  Firstly, why did you need the root canal in the first place?  Likely the tooth was either compromised by an impact injury or have enough decay that the pulp was infected, which means there was not really much natural tooth left.  Why did the crown break off -- what were you doing at the time.  Virgin teeth break off sometimes.  Since you worked in a dental office, you should understand that there is no guarantee in dentistry or medicine that any particular treatment will work. Having a crown on a severely weakened tooth  is standard of care.  Sorry that things did not work out, but as they say, "Stuff happens."