  To study the coupling system of space-time and Fermions, we need the explicit
form of the energy-momentum tensor of spinors. The energy-momentum tensor is
closely related to the tetrad frames which cannot be uniquely determined by the
metric. This flexibility increases difficulties to derive the exact expression
and easily leads to ambiguous results. In this paper, we give a detailed
derivation for the energy-momentum tensor of Weyl and Dirac spinors. From the
results we find that, besides the usual kinetic energy momentum term, there are
three kinds of other additional terms. One is the nonlinear self-interactive
potential, which acts like negative pressure. The other reflects the
interaction of momentum $p^\mu$ with tetrad. The third is the spin-gravity
coupling term which is a higher order infinitesimal in weak field, but may be
important in a neutron star. This term is also closely related with magnetic
field of a celestial body. These results are based on the decomposition of
usual spin connection into geometrical part and dynamical part, which not only
makes calculation simpler, but also highlights their different physical
meanings. In addition, we get a new tensor $S^{\mu\nu}_{ab}$ in calculation of
tetrad formalism, which plays an important role in the interaction of spinor
with gravity.
