  We use time-of-flight (ToF) inelastic neutron scattering (INS) spectroscopy
to investigate the doping dependence of magnetic excitations across the phase
diagram of NaFe$_{1-x}$Co$_x$As with $x=0, 0.0175, 0.0215, 0.05,$ and $0.11$.
The effect of electron-doping by partially substituting Fe by Co is to form
resonances that couple with superconductivity, broaden and suppress low energy
($E\le 80$ meV) spin excitations compared with spin waves in undoped NaFeAs.
However, high energy ($E> 80$ meV) spin excitations are weakly Co-doping
dependent. Integration of the local spin dynamic susceptibility
$\chi^{\prime\prime}(\omega)$ of NaFe$_{1-x}$Co$_x$As reveals a total
fluctuating moment of 3.6 $\mu_B^2$/Fe and a small but systematic reduction
with electron doping. The presence of a large spin gap in the Co-overdoped
nonsuperconducting NaFe$_{0.89}$Co$_{0.11}$As suggests that Fermi surface
nesting is responsible for low-energy spin excitations. These results parallel
Ni-doping evolution of spin excitations in BaFe$_{2-x}$Ni$_x$As$_2$, confirming
the notion that low-energy spin excitations coupling with itinerant electrons
are important for superconductivity, while weakly doping dependent high-energy
spin excitations result from localized moments.
