Nuclear magnetic relaxation rates of unconventional superconductivity in doped topological insulators

We study the temperature dependence of nuclear magnetic relaxation (NMR) rates to detect a sign of topological superconductivity in doped topological insulators, such as $M$($=$Cu,Nb,Sr)$_{x}$Bi$_{2}$Se$_{3}$ and Sn$_{1-x}$In$_{x}$Te. The Hebel-Slichter coherence effect below a critical temperature Tc depends on the superconducting states predicted by a minimal model of doped topological insulators. In a nodal anisotropic topological state similar to the ABM-phase in $^{3}$He, the NMR rate has a conventional $s$-wave like coherence peak below Tc. In contrast, in a fully-gapped isotropic topological superconducting state, this rate below Tc exhibits an anti-peak profile. Moreover, in a two-fold in-plane anisotropic topological superconducting state, there is no coherence effect, which is similar to that in a chiral $p$-wave state. Thus, we reveal that the NMR rates shed light on unconventional superconductivity in doped topological insulators.