 Solution

Higher donor concentration moves the semiconductor Fermi level toward the conduction band, reducing the semiconductor work function ${\Phi }_s=\chi +(E_c-E_F)$ when these quantities are expressed as energies. Equilibration with the metal aligns electrochemical potentials and creates band bending.

For an ideal n-type Schottky interface the barrier energy ${\Phi }_{Bn}={\Phi }_m-\chi$ is not automatically reduced merely by raising doping. The decisive change is the narrower depletion region:Large $N_D$ allows electrons to tunnel through the thin barrier, reducing the contact resistance and promoting an approximately Ohmic contact. Contact annealing can further modify the local interface and doping. This allows low-resistance source/drain injection while preserving a more rectifying gate contact; real interface-state pinning can modify the ideal work-function picture.