Band diagram

Last updated September 25, 2023

The inner workings of a light emitting diode, showing circuit (top) and band diagram when a bias voltage is applied (bottom). PnJunction-LED-E.svg — The inner workings of a light emitting diode, showing circuit (top) and band diagram when a bias voltage is applied (bottom).

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.^[1] 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.^[2]

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 (as resulting from a momentum-dependent band structure).

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 (or electron hole) 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:

E_F or μ: Although it is not a band quantity, the Fermi level (total chemical potential of electrons) is a crucial level in the band diagram. The Fermi level is set by the device's electrodes. For a device at equilibrium, the Fermi level is a constant and thus will be shown in the band diagram as a flat line. Out of equilibrium (e.g., when voltage differences are applied), the Fermi level will not be flat. Furthermore, in semiconductors out of equilibrium it may be necessary to indicate multiple quasi-Fermi levels for different energy bands, whereas in an out-of-equilibrium insulator or vacuum it may not be possible to give a quasi-equilibrium description, and no Fermi level can be defined.
E_C: The conduction band edge should be indicated in situations where electrons might be transported at the bottom of the conduction band, such as in an n-type semiconductor. The conduction band edge may also be indicated in an insulator, simply to demonstrate band bending effects.
E_V: The valence band edge likewise should be indicated in situations where electrons (or holes) are transported through the top of the valence band such as in a p-type semiconductor.
E_i: The intrinsic Fermi level may be included in a semiconductor, to show where the Fermi level would have to be for the material to be neutrally doped (i.e., an equal number of mobile electrons and holes).
E_imp: Impurity energy level. Many defects and dopants add states inside the band gap of a semiconductor or insulator. It can be useful to plot their energy level to see whether they are ionized or not.^[3]
E_vac: In a vacuum, the vacuum level shows the energy $-e\phi$ , where $\phi$ is the electrostatic potential. The vacuum can be considered as a sort of insulator, with E_vac playing the role of the conduction band edge. At a vacuum-material interface, the vacuum energy level is fixed by the sum of work function and Fermi level of the material.
Electron affinity level: Occasionally, a "vacuum level" is plotted even inside materials, at a fixed height above the conduction band, determined by the electron affinity. This "vacuum level" does not correspond to any actual energy band and is poorly defined (electron affinity strictly speaking is a surface, not bulk, property); however, it may be a helpful guide in the use of approximations such as Anderson's rule or the Schottky–Mott rule.

Band bending

When looking at a band diagram, the electron energy states (bands) 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 (spatial) bending. Rather, band bending refers to the local changes in electronic structure, 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 (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 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.^{[ citation needed ]}

Related Research Articles

A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. Its conducting properties may be altered in useful ways by introducing impurities ("doping") into the crystal structure. When two differently doped regions exist in the same crystal, a semiconductor junction is created. The behavior of charge carriers, which include electrons, ions, and electron holes, at these junctions is the basis of diodes, transistors, and most modern electronics. Some examples of semiconductors are silicon, germanium, gallium arsenide, and elements near the so-called "metalloid staircase" on the periodic table. After silicon, gallium arsenide is the second-most common semiconductor and is used in laser diodes, solar cells, microwave-frequency integrated circuits, and others. Silicon is a critical element for fabricating most electronic circuits.

In solid-state physics, the work function is the minimum thermodynamic work needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electric fields in the vacuum. The work function is not a characteristic of a bulk material, but rather a property of the surface of the material.

The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by µ or E_F for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties; how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics.

The electron affinity (E_ea) of an atom or molecule is defined as the amount of energy released when an electron attaches to a neutral atom or molecule in the gaseous state to form an anion.

A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal–semiconductor junction. Schottky barriers have rectifying characteristics, suitable for use as a diode. One of the primary characteristics of a Schottky barrier is the Schottky barrier height, denoted by Φ_B. The value of Φ_B depends on the combination of metal and semiconductor.

A p–n junction is a boundary or interface between two types of semiconductor materials, p-type and n-type, inside a single crystal of semiconductor. The "p" (positive) side contains an excess of holes, while the "n" (negative) side contains an excess of electrons in the outer shells of the electrically neutral atoms there. This allows electric current to pass through the junction only in one direction. The p- and n-type regions creating the junction are made by doping the semiconductor, for example by ion implantation, diffusion of dopants, or by epitaxy.

In solid-state physics, the electronic band structure of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have.

An ohmic contact is a non-rectifying electrical junction: a junction between two conductors that has a linear current–voltage (I–V) curve as with Ohm's law. Low-resistance ohmic contacts are used to allow charge to flow easily in both directions between the two conductors, without blocking due to rectification or excess power dissipation due to voltage thresholds.

Capacitance–voltage profiling is a technique for characterizing semiconductor materials and devices. The applied voltage is varied, and the capacitance is measured and plotted as a function of voltage. The technique uses a metal–semiconductor junction or a p–n junction or a MOSFET to create a depletion region, a region which is empty of conducting electrons and holes, but may contain ionized donors and electrically active defects or traps. The depletion region with its ionized charges inside behaves like a capacitor. By varying the voltage applied to the junction it is possible to vary the depletion width. The dependence of the depletion width upon the applied voltage provides information on the semiconductor's internal characteristics, such as its doping profile and electrically active defect densities.^, Measurements may be done at DC, or using both DC and a small-signal AC signal, or using a large-signal transient voltage.

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.

Band offset describes the relative alignment of the energy bands at a semiconductor heterojunction.

In solid-state physics, a metal–semiconductor (M–S) junction is a type of electrical junction in which a metal comes in close contact with a semiconductor material. It is the oldest practical semiconductor device. M–S junctions can either be rectifying or non-rectifying. The rectifying metal–semiconductor junction forms a Schottky barrier, making a device known as a Schottky diode, while the non-rectifying junction is called an ohmic contact.

In bulk semiconductor band structure calculations, it is assumed that the crystal lattice 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, 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.

Photoelectrochemistry is a subfield of study within physical chemistry concerned with the interaction of light with electrochemical systems. It is an active domain of investigation. One of the pioneers of this field of electrochemistry was the German electrochemist Heinz Gerischer. The interest in this domain is high in the context of development of renewable energy conversion and storage technology.

This article provides a more detailed explanation of p–n diode behavior than is found in the articles p–n junction or diode.

In physics, the field effect refers to the modulation of the electrical conductivity of a material by the application of an external electric field.

In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in which electrons are normally present at absolute zero temperature, while the conduction band is the lowest range of vacant electronic states. On a graph of the electronic band structure of a semiconducting material, the valence band is located below the Fermi level, while the conduction band is located above it.

In a basic Schottky-junction (Schottky-barrier) solar cell, an interface between a metal and a semiconductor provides the band bending necessary for charge separation. Traditional solar cells are composed of p-type and n-type semiconductor layers sandwiched together, forming the source of built-in voltage. Due to differing energy levels between the Fermi level of the metal and the conduction band of the semiconductor, an abrupt potential difference is created, instead of the smooth band transition observed across a p-n junction in a standard solar cell, and this is a Schottky height barrier. Although vulnerable to higher rates of thermionic emission, manufacturing of Schottky barrier solar cells proves to be cost-effective and industrially scalable.

In semiconductor physics, the flat band potential of a semiconductor defines the potential at which there is no depletion layer at the junction between a semiconductor and an electrolyte or p-n-junction. This is a consequence of the condition that the redox Fermi level of the electrolyte must be equal to the Fermi level of the semiconductor and therefore preventing any band bending of the conduction and valence band. An application of the flat band potential can be found in the determining the width of the space charge region in a semiconductor-electrolyte junction. Furthermore, it is used in the Mott-Schottky equation to determine the capacitance of the semiconductor-electrolyte junction and plays a role in the photocurrent of a photoelectrochemical cell. The value of the flat band potential depends on many factors, such as the material, pH and crystal structure of the material

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. 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. The use of semiconductor junctions sparked the computer revolution in 1990. Devices such as the diode, the transistor, the photocell and many more still play an important role in technology.

References

↑ "The energy band diagram of the Metal-Oxide-Silicon (MOS) Capacitor". ecee.colorado.edu. Archived from the original on 2020-07-27. Retrieved 2017-11-05.
↑ "Schottky Barrier Basics". academic.brooklyn.cuny.edu. Retrieved 2017-11-05.
↑ "Doped Semiconductors". hyperphysics.phy-astr.gsu.edu. Retrieved 2017-11-05.

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

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[1] "The energy band diagram of the Metal-Oxide-Silicon (MOS) Capacitor". ecee.colorado.edu. Archived from the original on 2020-07-27. Retrieved 2017-11-05.

[2] "Schottky Barrier Basics". academic.brooklyn.cuny.edu. Retrieved 2017-11-05.

[3] "Doped Semiconductors". hyperphysics.phy-astr.gsu.edu. Retrieved 2017-11-05.

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