The covariant derivatives and energy momentum tensor of spinors
  To understand the coupling behavior of the spinor with spacetime, the explicit form of the energy-momentum tensor of the spinor in curved spacetime is important. This problem seems to be overlooked for a long time. In this paper we derive the explicit form of energy momentum tensors and display some equivalent but simple forms of the covariant derivatives for both the Weyl spinor and the Dirac bispinor in curved spacetime. 
Ying-Qiu Gu, Bijan Saha