Band diagram


In solid-state physics of semiconductors, a band diagram is a diagram plotting various key electron energy levels 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. 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, 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. 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.

Anatomy

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.
While a basic band diagram only shows electron energy levels, often a band diagram will be decorated with further features.
It is common to see cartoon depictions of the motion in energy and position of an electron as it drifts, is excited by a light source, or relaxes from an excited state.
The band diagram may be shown connected to a circuit diagram showing how bias voltages are applied, how charges flow, etc.
The bands may be colored to indicate filling of energy levels, or sometimes the band gaps will be colored instead.

Energy levels

Depending on the material and the degree of detail desired, a variety of energy levels will be plotted against position:
When looking at a band diagram, the electron energy states in a material can curve up or down near a junction. This effect is known as band bending. It does not correspond to any physical bending. Rather, 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.
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.
The reason for the charge imbalance is that, although a homogeneous material is charge neutral everywhere, there is no such requirement for interfaces.
Practically all types of interface develop a charge imbalance, though for different reasons:
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 or ohmic. The degree of band bending depends on the relative Fermi levels and carrier concentrations of the materials forming the junction. In an n-type semiconductor the band bends upward, while in p-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.