  Dirac electrons in solids show characteristic physical properties due to
their linear dispersion relation and two-band nature. Although the transport
phenomena of Dirac electrons in a normal state have intensively been studied,
the transport phenomena in a superconducting state have not been fully
understood. In particular, it is not clear whether Dirac electrons in a
superconducting state show Meissner effect (ME), since a diamagnetic term of a
current operator is absent as a result of the linear dispersion. We investigate
the ME of three dimensional massive Dirac electrons in a superconducting state
on the basis of Kubo formula, and clarify that Meissner kernel becomes finite
by use of the inter-band contribution. This mechanism of the ME for Dirac
electrons is completely different from that for the electrons in usual metals.
Our result shows that the Meissner kernel remains finite even when the
superconducting gap vanishes. This is an unavoidable problem in the Dirac
electron system as reported in the previous works. Thus, we use a prescription
in which we subtract the normal state contribution. In order to justify this
prescription, we develop a specific model where the Meissner kernel is obtained
by the prescription. We also derive the result for the electron gas by taking
the non-relativistic limit of Dirac Hamiltonian, and clarify that the
diamagnetic term of the Meissner kernel can be regarded as the inter-band
contribution between electrons and positrons in terms of the Dirac model.
