Band bending

When the electrochemical potential of the free charge carriers around an interface of a semiconductor is dissimilar, charge carriers are transferred between the two materials until an equilibrium state is reached whereby the potential difference vanishes.

[1] The band bending concept was first developed in 1938 when Mott, Davidov and Schottky all published theories of the rectifying effect of metal-semiconductor contacts.

Devices such as the diode, the transistor, the photocell and many more play crucial roles in technology.

In this section metal-semiconductor contact, surface state, applied bias and adsorption induced band bending are discussed.

Figure 1 shows the ideal band diagram (i.e. the band diagram at zero temperature without any impurities, defects or contaminants) of a metal with an n-type semiconductor before (top) and after contact (bottom).

When the metal and semiconductor are brought in contact, charge carriers (i.e. free electrons and holes) will transfer between the two materials as a result of the work function difference

Under equilibrium the work function difference vanishes and the Fermi levels align across the interface.

Due to the low concentration of free charge carriers in the semiconductor, the electric field cannot be effectively screened (unlike in the metal where

, electrons are shared from the metal to the semiconductor, resulting in an electric field that points in the opposite direction.

One can envision the direction of bending by considering the electrostatic energy experienced by an electron as it moves across the interface.

An electron moving from the semiconductor to the metal therefore experiences a growing repulsion as it approaches the interface.

The result is that the semiconductor energy band bends downwards towards the metal surface.

The unpaired electrons in the dangling bonds of the surface atoms interact with each other to form an electronic state with a narrow energy band, located somewhere within the band gap of the bulk material.

For simplicity, the surface state band is assumed to be half-filled with its Fermi level located at the mid-gap energy of the bulk.

In disequilibrium, the Fermi energy is thus lower or higher than that of the surface states for p- and n-doping, respectively.

Due to the energy difference, electrons will flow from the bulk to the surface or vice versa until the Fermi levels become aligned at equilibrium.

Therefore, the Fermi energy of the semiconductor is almost independent of the bulk dopant concentration and is instead determined by the surface states.

As the molecule approaches the surface, an unfilled molecular orbital of the acceptor interacts with the semiconductor and shifts downwards in energy.

It follows from the general uncertainty principle that the molecular orbital broadens its energy as can be seen in the bottom of figure 3.

An electric field is set up and upwards band bending near the semiconductor surface occurs.

[6] Thus the potential difference between the bands is either increased or decreased depending on the type of bias that is applied.

It also approximates a sudden drop in charge carrier concentration in the depletion region.

term compensates for the existence of free charge carriers near the junction from the bulk region.

The p-n diode is a device that allows current to flow in only one direction as long as the applied voltage is below a certain threshold.

When a forward bias is applied to the p-n junction of the diode the band gap in the depletion region is narrowed.

The applied voltage introduces more charge carriers as well, which are able to diffuse across the depletion region.

Under a reverse bias this is hardly possible because the band gap is widened instead of narrowed, thus no current can flow.

When the transistor is in its so called 'off state' there is no voltage applied on the gate and the first p-n junction is reversed bias.

The electric field in the depletion region separates the electrons and holes generating a current when the two sides of the p-n diode are connected.

[10] Different spectroscopy methods make use of or can measure band bending:

Figure 1: Energy band diagrams of the surface contact between metals and n-type semiconductors. , the vacuum energy; , the maximum energy of the valence band; , minimum energy of the conduction band; , the metal work function; , the semiconductor work function; , the electron affinity of the semiconductor.
Figure 2: Energy band diagrams under influence of surface-induced band bending. CB, the conduction band; VB, the valence band; , the Fermi energy; , the vacuum energy.
Figure 3: Influence of the adsorption of an acceptor molecule (A) on the surface of an n-type semiconductor. [ 1 ]