Band bending

Band bending refers to the local changes in the energy offset of a semiconductor's band structure near a junction, due to space charge effects. Because the common way to visualize the electron energy states and Fermi level in a material is to draw bands on an energy vs. distance plot (band diagram), band bending refers to bending observed in these diagrams and does not correspond to any physical (spatial) bending.

The primary principle underlying band bending inside a semiconductor is space charge: a local imbalance in charge neutrality. Poisson's equation gives a curvature to the bands wherever there is an imbalance in charge neutrality. Why is there charge imbalance? Although one expects a homogeneous material to be charge neutral everywhere (since it must be charge neutral on average) there is no such requirement for interfaces. Practically all types of interface develop a charge imbalance, though for different reasons:

• At the junction of two different types of the same semiconductor (e.g., p-n junction) the bands vary continuously since the dopants are sparsely distributed and only perturb the system.
• At the junction of two different semiconductors there is a sharp shift in band energies from one material to the other; the band alignment at the junction (e.g., the difference in conduction band energies) is fixed.
• At the junction of a semiconductor and metal, the bands of the semiconductor are pinned to the metal's Fermi level.
• At the junction of a conductor and vacuum, the vacuum level (from vacuum electrostatic potential) is set by the material's work function and Fermi level. This also (usually) applies for the junction of a conductor to an insulator.

Knowing how bands will bend when two different types of materials are brought in contact is key to understanding whether the junction will be rectifying (Schottky) or ohmic. The degree of band bending depends on the relative Fermi levels and carrier concentrations of the materials forming the junction. In the p-type semiconductor the band bends upward, while in n-type the band bends downward. Note that band bending is due neither to magnetic field nor temperature gradient. Rather, it only arises in conjunction with the force of the electric field.

See also

• Anderson's rule - approximate rule for band alignment of heterojunctions based on vacuum electron affinity.
• Schottky-Mott rule - approximate rule for band alignment of metal-semiconductor junctions based on vacuum electron affinity and work function.
• Field effect (semiconductor) - band bending induced by an electric field at the vacuum (or insulator) surface of a semiconductor.
• Thomas–Fermi screening - rudimentary theory of the band bending that occurs around a charged defect.
• Quantum capacitance - special case of band bending in field effect, for a material system containing a two-dimensional electron gas.

References

• James D. Livingston, Electronic Properties of Engineering Materials, Wiley (December 21, 1999).