In solid-state physics, metal-induced gap states are electron states that exist near the surface of a semiconductor due to the presence of a metal on the surface. They have energies that fall within the semiconductor's bandgap thus are forbidden in the bulk of the semiconductor. In bulk semiconductor band structure calculations, it is assumed that the crystal lattice (which features a periodic potential due to the atomic structure) of the material is infinite. When the finite size of a crystal is taken into account, the wavefunctions of electrons are altered and states that are forbidden within the bulk semiconductor gap are allowed at the surface. Similarly, when a metal is deposited onto a semiconductor (by thermal evaporation, for example), the wavefunction of an electron in the semiconductor must match that of an electron in the metal at the interface. Since the Fermi levels of the two materials must match at the interface, there exists gap states that decay deeper into the semiconductor.
Band-bending at the metal-semiconductor interface
As mentioned above, when a metal is deposited onto a semiconductor, even when the metal film is as small as a single atomic layer, the Fermi levels of the metal and semiconductor must match. This pins the Fermi level in the semiconductor to a position in the bulk gap. Shown to the right is a diagram of band-bending interfaces between two different metals (high and low work functions) and two different semiconductors (n-type and p-type). Volker Heine was one of the first to estimate the length of the tail end of metal electron states extending into the semiconductor's energy gap. He calculated the variation in surface state energy by matching wavefunctions of a free-electron metal to gapped states in an undoped semiconductor, showing that in most cases the position of the surface state energy is quite stable regardless of the metal used.
Branching point It is somewhat crude to suggest that the metal-induced gap states (MIGS) are tail ends of metal states that leak into the semiconductor. Since the mid-gap states do exist within some depth of the semiconductor, they must be a mixture (a Fourier series) of valence and conduction band states from the bulk. The resulting positions of these states, as calculated by C. Tejedor, F. Flores and E. Louis, and J. Tersoff, must be closer to either the valence- or conduction- band thus acting as acceptor or donor dopants, respectively. The point that divides these two types of MIGS is called the branching point, E_B. Tersoff argued
E B = 1 2 [ E V ¯ + E C ¯ ] {\displaystyle E_{B}={\frac {1}{2}}[{\bar {E_{V}}}+{\bar {E_{C}}}]}
E V ¯ = E V − 1 3 Δ s o {\displaystyle {\bar {E_{V}}}=E_{V}-{\frac {1}{3}}\Delta _{so}} , where Δ s o {\displaystyle \Delta _{so}} is the spin orbit splitting of E V {\displaystyle E_{V}} at the Γ {\displaystyle \Gamma } point.
E C ¯ {\displaystyle {\bar {E_{C}}}} is the indirect conduction band minimum.
Metal–semiconductor contact point barrier height
In order for the Fermi levels to match at the interface, there must be charge transfer between the metal and semiconductor. The amount of charge transfer was formulated by Linus Pauling and later revised to be:
δ q = 0.16 e V | X M − X S C | + 0.035 e V 2 | X M − X S C | 2 {\displaystyle \delta q={\frac {0.16}{eV}}|X_{M}-X_{SC}|+{\frac {0.035}{eV^{2}}}|X_{M}-X_{SC}|^{2}}
where X M {\displaystyle X_{M}} and X S C {\displaystyle X_{SC}} are the electronegativities of the metal and semiconductor, respectively. The charge transfer produces a dipole at the interface and thus a potential barrier called the Schottky barrier height. In the same derivation of the branching point mentioned above, Tersoff derives the barrier height to be:
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