Deriving atmospheric parameters of M dwarfs from SPIRou spectra

M dwarfs are the most numerous stars in the solar vicinity. They constitute privileged targets in the search for planetary systems in the habitable zone and their characterization.
Several instruments were designed specifically to study M dwarfs, such as SPIRou, a high resolution spectro-polarimeter covering a large near-infrared range from 1 to 2.5 $\mu m$. Deriving the fundamental parameters of these stars is essential to the characterization of the companions, but remains a challenging task. In this talk, we discuss a method relying on the comparison to synthetic spectra to derive the effective temperature ($T_{\rm eff}$), the surface gravity ($\log{g}$) and metallicity ([M/H]) of M dwarfs from SPIRou observations. We demonstrate our ability to retrieve parameters in good agreement with the literature, with internal error bars of about 30 K in $T_{\rm eff}$, 0.05 dex in $\log{g}$ and 0.1 dex in [M/H] for a dozen of inactive M dwarfs with temperatures ranging from 3000 to 4000 K. Using both PHOENIX-ACES and MARCS synthetic spectra, we also show how the choice of model may lead to systematics of up to 50 K in $T_{\rm eff}$ and 0.4 dex in $\log{g}$ and [M/H].
We further extend our study to allow for the determination of alpha elements enhancement, [$\alpha$/Fe], discuss the impact of this parameter on the results, and extend our analysis to 40 SPIRou targets in order to provide the community with a reference database of accurate stellar parameters.