Physically unclonable function (PUF) is a lightweight encryption technique that generates random digital keys (responses) using intrinsic process variations of devices, which is a promising solution for Internet of Things (IoT) security due to its compatibility with constrained resources. Recent attempts to adopt nonvolatile memory (NVM) into PUFs have enhanced stability through a write-back technique that maintains consistent responses from the enrollment phase even under wide environmental variations by storing the response in the NVM device. However, the stability of the previous NVM PUFs is limited by the low on/off ratio of the NVMs. In addition, the circuit required to implement the write-back technique poses challenges of increased area and energy consumption. Considering the hardware limitations and power constraints of IoT devices, this paper proposes a ferroelectric field-effect transistor (FeFET) PUF as a suitable security solution. The high on/off ratio of FeFET and the proposed auto write-back technique that does not require additional circuitry realize the stability improvement (a bit error rate of <0.0001%) under wide environmental variations without incurring area and energy overheads. The negligible off current of FeFET prevents static power consumption, which leads to the lowest energy consumption of 6.70e-15 J during the response generation of the FeFET PUF. In addition, the compact PUF cell composed of two FeFETs achieves a high density of 87.37 F2.