Separate Schottky barrier height from band bending

The metal work function $W_m$ is the energy between its Fermi level and vacuum. Semiconductor electron affinity $\chi$ is the energy between the conduction edge and vacuum. In the ideal Schottky–Mott picture, aligning vacuum levels before equilibration gives an electron barrier $\Phi_{Bn}=W_m-\chi$. These symbols denote energies, often tabulated in electronvolts. A corresponding hole barrier is the remainder of the semiconductor gap, provided the same band-edge picture applies and the interface has no additional gap-changing structure.

The electron barrier is not identical to the bulk-to-interface band bending. In neutral n-type bulk the conduction edge already lies above the Fermi level by $\Delta_b=E_c-E_F$. Only the difference between barrier energy and this bulk separation becomes the built-in electrostatic drop times q. The distinction matters even in an ideal contact: changing doping changes the bulk separation and depletion width without necessarily changing the ideal interface barrier set by the metal and semiconductor.

A depleted n-type semiconductor contains positively charged ionized donors near the contact, balanced by negative charge on the metal. Most of the electrostatic drop lies in the semiconductor because the metal screens over a short distance. For uniform complete ionization, integrating Poisson's equation gives a one-sided depletion width. The relevant voltage is semiconductor band bending, not the full electron barrier energy divided by q. An accumulation contact has different mobile charge and cannot use the depletion formula unchanged.

In a thermionic-emission regime, carriers with sufficient energy cross the barrier. The reverse saturation scale is exponentially sensitive to barrier energy, with a temperature-squared prefactor in the elementary three-dimensional model. A forward current expression also assumes a specified voltage drop across the contact and neglects complicating interface transport. The Richardson constant depends on effective mass and material assumptions; treating it as a universal numerical constant for every semiconductor or interface is unjustified. Barrier lowering and tunneling can further alter the apparent temperature dependence.

Interface states, chemical bonding, and dipoles can pin the barrier so that changing metal work function produces much less barrier change than Schottky–Mott alignment predicts. Pinning is a statement about the interface's electronic response, not a claim that the bulk semiconductor Fermi level cannot move with doping. The ideal formula remains a useful reference for identifying which physical effects must be added, but is not a guaranteed quantitative barrier prediction from two tabulated material numbers.

Heavy doping makes the depleted region narrow and can permit substantial tunneling through a barrier whose height remains nonzero. Such a contact may have a small, approximately linear resistance over the operating range and be called ohmic. Low barrier height is therefore not the only route to a useful contact. Conversely, a nominally favorable metal can still have large contact resistance because of contamination, an interfacial layer, or current crowding. Contact resistance, rectification, and ideal barrier height describe related but distinct properties.