In bulk
semiconductorband 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
Band diagram of the
band-bending at the interface of (a) a low
work function metal and n-type
semiconductor, (b) a low work function metal and a p-type semi conductor, (c) a high work function metal and an n-type semi conductor, (d) a high work function metal and a p-type semi conductor. (Figure adapted from H. Luth's Solid Surfaces, Interfaces, and Thin Films, p. 384.[1])
As mentioned above, when a
metal is deposited onto a
semiconductor, even when the metal film 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.[2]
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,[3] and
J. Tersoff,[4][5] 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
, where is the spin orbit splitting of at the point.
is the indirect conduction band minimum.
Metal–semiconductor contact point barrier height
Band diagram of the contact point potential barrier at the interface of a metal and semiconductor. Shown are , the energy of the barrier, and , the maximum band bending in the semiconductor. (Figure adapted from H. Luth's Solid Surfaces, Interfaces, and Thin Films, p. 408 (see Refs.)
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 [6] and later revised [7] to be:
where and 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:
where is a parameter adjustable for the specific metal, dependent mostly on its electronegativity, . Tersoff showed that the experimentally measured fits his theoretical model for
Au in contact with 10 common semiconductors, including
Si,
Ge,
GaP, and
GaAs.
is dependent on the charge densities of the both materials
density of surface states
work function of metal
sum of dipole contributions considering dipole corrections to the jellium model
semiconductor gap
Ef – Ev in semiconductor
Thus can be calculated by theoretically deriving or experimentally measuring each parameter. Garcia-Moliner and Flores also discuss two limits
(The
Bardeen Limit), where the high density of interface states pins the Fermi level at that of the semiconductor regardless of .
(The
Schottky Limit) where varies with strongly with the characteristics of the metal, including the particular lattice structure as accounted for in .
Applications
When a bias voltage is applied across the interface of an n-type semiconductor and a metal, the Fermi level in the semiconductor is shifted with respect to the metal's and the band bending decreases. In effect, the capacitance across the depletion layer in the semiconductor is bias voltage dependent and goes as . This makes the metal/semiconductor junction useful in
varactor devices used frequently in electronics.
References
^H. Luth, Solid Surfaces, Interfaces, and Films, Springer-Verlag Berlin Heidelberg, New York, NY, 2001.
^L. Pauling, The Nature of the Chemical Bond. Cornell University Press, Ithaca, 1960.
^Hannay, N. Bruce; Smyth, Charles P. (1946). "The Dipole Moment of Hydrogen Fluoride and the Ionic Character of Bonds". Journal of the American Chemical Society. 68 (2). American Chemical Society (ACS): 171–173.
doi:
10.1021/ja01206a003.
ISSN0002-7863.
^Garcia-Moliner, Federico and Flores, Fernando, Introduction to the theory of solid surfaces, Cambridge University Press, Cambridge, London, 1979.