Touch two different materials and electrons flow from the one with the higher Fermi level until the Fermi levels are equal. The charge left behind is the contact potential.
Before contact the vacuum levels match and the Fermi levels differ. After contact the Fermi levels match and the vacuum levels differ. The amount is the same: φw,M2 − φw,M1.
Flow is fast at first and slows as the step builds, because the driving force is the remaining Fermi-level mismatch.
With silicon on one side the step appears mostly inside the silicon as band bending over a depletion region; between two metals it appears as a sharp step at the interface.
The physics behind it
Contact potential of M1 to M2: ψM1,M2 = φw,M2 − φw,M1 (eq. 1.4.1). For silicon φw = χ + Eg/2q + ΦF with χ = 4.05 V.
In a chain of materials the intermediate work functions cancel, and around any closed loop the steps sum to zero (eqs. 1.4.3 to 1.4.10).
The same physics sets ΦMS in the MOS structure of Chapter 2: the gate and the body are different materials connected through the external circuit.