= 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:
$$
W\simeq\sqrt{\frac{2\varepsilon_s V_{\rm bi}}{qN_D}}.
$$
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.
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