  We have investigated the superconducting ground state of the newly discovered
superconductor ThFeAsN with a tetragonal layered crystal structure using
resistivity, magnetization, heat capacity and transverse-field (TF) muon-spin
rotation ($\mu$SR) measurements. Our resistivity and magnetization measurements
reveal an onset of bulk superconductivity with $T_{\bf c}\sim$ 30 K. The heat
capacity results show a very small anomaly, $\Delta$C$_{ele}$$\sim$0.214
(J/mol-K) at $T_{\bf c}\sim$ 30 K and exhibits exponential behavior below
$T_{\bf c}$, which fits better to two superconducting gaps rather than a single
gap. Further a nonlinear magnetic field dependence of the electronic specific
heat coefficient $\gamma$(H) has been found in the low temperature limit, which
indicates that the smaller energy gap is nodal. Our analysis of the TF-$\mu$SR
results shows that the temperature dependence of the superfluid density is
better described by a two-gap model either isotropic $s$+$s$-wave or
$s$+$d$-wave than a single gap isotropic $s$-wave model for the superconducting
gap, consistent with other Fe-based superconductors. The combine $\gamma$(H)
and TF-$\mu$SR results confirm $s$+$d$-wave model for the gap structure of
ThFeAsN. The observation of two gaps in ThFeAsN suggests multiband nature of
the superconductivity possibly arising from the d-bands of Fe ions.
Furthermore, from our TF-$\mu$SR study we have estimated the magnetic
penetration depth, in the polycrystalline sample, of
$\lambda_{\mathrm{L}}$$(0)$ = 375 nm, superconducting carrier density $n_s =
4.6 \times 10^{27}~ $m$^{-3}$, and carrier's effective-mass $m^*$ =
2.205\textit{m}$_{e}$. We will compare the results of our present study with
those reported for the Fe-pnictide family of superconductors.
