Cryogenic atomic layer etching (ALE) represents a promising technique for achieving subnanoscale material removal in semiconductor processes, owing to its unique self-limiting surface-adsorbing reactions. This paper presents a cryogenic ALE method for SiN, utilizing surface modification with a hydrogen fluoride (HF) dose and an Ar etch step for removing the modification layer. The surface reactions and etching mechanism were examined using in situ monitoring techniques, including spectroscopic ellipsometry and attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR). Our observations reveal a self-limiting etching behavior for SiN and a reduction in the etch amount per cycle (EPC) with a decreasing substrate temperature. During the HF dose step, in situ ATR-FTIR spectra indicate the formation of a surface-adsorbed layer containing ammonium fluorosilicate (AFS) on the SiN surface. Subsequently, during the Ar plasma etching step, both the AFS layer and the surface-adsorbed species were removed. At lower substrate temperatures, the stability of the AFS layer and surface-absorbed species increased, resulting in a reduction in EPC. Through the control of Ar ion energy and substrate temperature, the manipulation of EPC ranging from several nanometers to a few angstroms in atomic layer etching is achieved, offering potential utility in nanoscale device applications utilizing silicon nitride.