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In solid-state physics of semiconductors, a band diagram is a diagram plotting various key electron energy levels (Fermi level and nearby energy band edges) as a function of some spatial dimension, which is often denoted x. These diagrams help to explain the operation of many kinds of semiconductor devices and to visualize how bands change with position (band bending). The bands may be coloured to distinguish level filling. A band diagram should not be confused with a band structure plot. In both a band diagram and a band structure plot, the vertical axis corresponds to the energy of an electron. The difference is that in a band structure plot the horizontal axis represents the wave vector of an electron in an infinitely large, homogeneous material (a crystal or vacuum), whereas in a band diagram the horizontal axis represents position in space, usually passing through multiple materials. Because a band diagram shows the changes in the band structure from place to place, the resolution of a band diagram is limited by the Heisenberg uncertainty principle: the band structure relies on momentum, which is only precisely defined for large length scales. For this reason, the band diagram can only accurately depict evolution of band structures over long length scales, and has difficulty in showing the microscopic picture of sharp, atomic scale interfaces between different materials (or between a material and vacuum). Typically, an interface must be depicted as a "black box", though its long-distance effects can be shown in the band diagram as asymptotic band bending. The vertical axis of the band diagram represents the energy of an electron, which includes both kinetic and potential energy. The horizontal axis represents position, often not being drawn to scale. Note that the Heisenberg uncertainty principle prevents the band diagram from being drawn with a high positional resolution, since the band diagram shows energy bands (as resulting from a momentum-dependent band structure).

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Band diagram

Band bending

In solid-state physics, band bending refers to the process in which the electronic band structure in a material curves up or down near a junction or interface. It does not involve any physical (spatial) 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.

Anderson's rule

Anderson's rule is used for the construction of energy band diagrams of the heterojunction between two semiconductor materials. Anderson's rule states that when constructing an energy band diagram, the vacuum levels of the two semiconductors on either side of the heterojunction should be aligned (at the same energy). It is also referred to as the electron affinity rule, and is closely related to the Schottky–Mott rule for metal–semiconductor junctions. Anderson's rule was first described by R. L. Anderson in 1960.

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Charge transition levels of nitrogen dangling bonds at Si/SiO2 interfaces: A first-principles study

Alfredo Pasquarello, Peter Broqvist, Philipp Dahinden

The defect levels of the nitrogen dangling bond at the Si/SiO2 interface are determined through a density-functional approach. The composition grading at the interface is modeled through crystalline and amorphous models of stoichiometric SiO2, nitrided SiO2, and substoichiometric silicon oxides. The relevant charge transition levels are first determined with respect to the band edges of the parent oxides within a semilocal density-functional scheme. Through the use of band edge shifts and band offsets previously obtained with hybrid functionals, the calculated defect levels are then positioned with respect to a band diagram of the Si/SiO2 interface which shows good agreement with the experimental one. We find that the 0/- charge transition level locates within the Si band gap. The level locates close to the Si valence band for nitrogen atoms in the stoichiometric oxide, but is found to rise across the silicon band gap as the environment of the nitrogen atoms becomes more silicon rich. The latter raise is accompanied by a stabilization of the incorporated nitrogen.

2010