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The gate leakage current of metal-oxide-semiconductors (MOSs) composed of hafnium oxide (HfO2) exhibits temperature dependence, which is usually attributed to the standard Poole-Frenkel (P-F) transport model. However, the reported magnitudes of the trap barrier height vary significantly. This paper explores the fundamental challenges associated with applying the P-F model to describe transport in HfO2/ SiO2 bilayers in n/p MOS field-effect transistors composed of 3- and 5-nm HfO2 on 1.1-nm SiO2 dielectric stacks. The extracted P-F trap barrier height is shown to be dependent on several variables including the following: the temperature range, method of calculating the electric field, electric-field range considered, and HfO2 thickness. P-F conduction provides a consistent description of the gate leakage current only within a limited range of the current values while failing to explain the temperature dependence of the 3-nm HfO2 stacks for gate voltages of less than 1 V, leaving other possible temperature-dependent mechanisms to be explored.