Surface Inversion and Gate Voltage:
Surface inversion occurs in a MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) when the gate voltage is equal to the threshold voltage of the device. This phenomenon is essential for the operation of MOSFETs and plays a crucial role in controlling the flow of current through the device.

Explanation:

Threshold Voltage:
The threshold voltage (Vth) is the minimum voltage required on the gate of a MOSFET to establish a conducting channel between the source and drain terminals. Below this threshold voltage, the device remains in the off state, and no significant current flows.

Surface Inversion:
When the gate voltage equals the threshold voltage, the surface of the semiconductor material beneath the gate electrode undergoes a transition. This transition is known as surface inversion, where an electron-rich layer is formed at the surface of the semiconductor. This layer acts as a conducting channel, allowing current to flow between the source and drain terminals.

Gate Voltage and Surface Inversion:
- When the gate voltage is less than the threshold voltage, the MOSFET remains in the off state, and no inversion layer is formed.
- When the gate voltage is equal to the threshold voltage, surface inversion occurs, and the device enters the on state with a conducting channel.
- If the gate voltage exceeds the threshold voltage, the device remains in the on state, but the channel may experience further enhancements in conductivity.
In conclusion, surface inversion in a MOSFET occurs when the gate voltage reaches the threshold voltage, leading to the formation of a conducting channel at the semiconductor surface. Understanding this relationship is crucial for designing and analyzing MOSFET circuits in various applications.