Threshold Voltage, VT
The threshold voltage is defined as the gate bias that must be applied in order to induce sufficient number of charge carriers in the channel for conduction. The polarity of induced charge is opposite to that of the free carriers in the bulk substrate, and the concentration of the induced charge must be at least the free carrier concentration in the bulk. This is the so-called inversion condition.
In finding the threshold voltage VT, it is easier to start from a flat semiconductor band. However, since the real-world MOS samples have a bent band in the semiconductor due mainly to the difference in workfunction between metal and semiconductor and the positive oxide charge present in oxide, it is necessary to know how much bias should be applied to achieve the flat band condition for the semiconductor. Then starting from this flat band condition, we can calculate how much additional bias is needed to obtain the inversion condition.
If the work function is ,then the Fermi levels of the bulk metal and the bulk semiconductor lie at different position with respect to their common reference point, namely the vacuum level. In thermal equilibrium the Fermi level in the MOS structure must be constant, meaning that the Fermi level of bulk metal must be moved until it coincides with the Fermi level of the bulk semiconductor, and this induces the band bending in semiconductor.
The processing-dependent oxide charge is always positive and lies near the oxide-semiconductor interface. This positive oxide charge will therefore induce a negative charge in semiconductor, causing a downward band bending in semiconductor.
With the above two effects combined, the following applets show you the equilibrium band diagram for MOS, allows you to calculate the bias that you need to apply to gate for flat band condition in the semiconductor, and allows you to change the bias and observe the band diagram.