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Effects of Impurity Vertex Correction on NMR Coherence Peak in S-Wave Superconductors

We study the effects of non-magnetic impurity vertex correction on nuclear spin-lattice relaxation rate $1/{T_1} $ of conventional s-wave superconductors within the Eliashberg formalism. We obtain, with a self-consistent $t-$matrix treatment of impurity scatterings, the expressions for impurity vertex function and nuclear spin-lattice relaxation rate. The $1/T_1$ is evaluated with a simple approximation on angular average, and found to agree in clean limit with the previous result that $1/(T_1 T)$ remains unrenormalized under the impurity vertex correction. As dirtiness is increased, on the other hand, the coherence peak in $1/(T_1 T)$ is found to increase due to the impurity vertex correction. This result is discussed in connection with the experimental observations on conventional superconductors.