# Band diagram

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]

Band diagram for p–n junction at equilibrium. The depletion region is shaded.
The inner workings of a light emitting diode, showing circuit (top) and band diagram when a bias voltage is applied (bottom).
Band diagram for Schottky barrier at equilibrium.
Band diagram for semiconductor heterojunction at equilibrium.

## 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:

• EF 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.
• EC: 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.
• EV: 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.
• Ei: 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).
• Eimp: 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]
• Evac: In a vacuum, the vacuum level shows the energy ${\displaystyle -e\phi }$, where ${\displaystyle \phi }$ is the electrostatic potential. The vacuum can be considered as a sort of insulator, with Evac 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 the energy offset of a semiconductor's band structure near a junction, due to space charge effects. Because the common way to visualize the is to draw bands on an energy vs. distance plot.

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 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.

## References

1. ^ "The energy band diagram of the Metal-Oxide-Silicon (MOS) Capacitor". ecee.colorado.edu. 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.
• James D. Livingston, Electronic Properties of Engineering Materials, Wiley (December 21, 1999).
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.

Backward diode

In semiconductor devices, a backward diode (also called back diode) is a variation on a Zener diode or tunnel diode having a better conduction for small reverse biases (for example –0.1 to –0.6 V) than for forward bias voltages.

The reverse current in such a diode is by tunneling, which is also known as the tunnel effect.

Bipolar junction transistor

A bipolar junction transistor (bipolar transistor or BJT) is a type of transistor that uses both electron and hole charge carriers. In contrast, unipolar transistors, such as field-effect transistors, only use one kind of charge carrier. For their operation, BJTs use two junctions between two semiconductor types, n-type and p-type.

BJTs are manufactured in two types, NPN and PNP, and are available as individual components, or fabricated in integrated circuits, often in large numbers.

The basic function of a BJT is to amplify current. This allows BJTs to be used as amplifiers or switches, giving them wide applicability in electronic equipment, including computers, televisions, mobile phones, audio amplifiers, industrial control, and radio transmitters.

Doping (semiconductor)

In semiconductor production, doping is the intentional introduction of impurities into an intrinsic semiconductor for the purpose of modulating its electrical, optical and structural properties. The doped material is referred to as an extrinsic semiconductor. A semiconductor doped to such high levels that it acts more like a conductor than a semiconductor is referred to as a degenerate semiconductor.

In the context of phosphors and scintillators, doping is better known as activation. Doping is also used to control the color in some pigments.

Electron affinity

In chemistry and atomic physics, the electron affinity (Eea) of an atom or molecule is defined as the amount of energy released or spent when an electron is added to a neutral atom or molecule in the gaseous state to form a negative ion.

X + e− → X− + energyIn solid state physics, the electron affinity for a surface is defined somewhat differently (see below).

Electronic band structure

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energies that an electron within the solid may have (called energy bands, allowed bands, or simply bands) and ranges of energy that it may not have (called band gaps or forbidden bands).

Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).

Fermi level

The Fermi level chemical potential for electrons (or electrochemical potential for electrons) of a body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by µ or EF

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.

In band structure theory, used in solid state physics to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a 50% probability of being occupied at any given time.

The position of the Fermi level with the relation to the band energy levels is a crucial factor in determining electrical properties.

The Fermi level does not necessarily correspond to an actual energy level (in an insulator the Fermi level lies in the band gap), nor does it require the existence of a band structure.

Nonetheless, the Fermi level is a precisely defined thermodynamic quantity, and differences in Fermi level can be measured simply with a voltmeter.

Field effect (semiconductor)

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 a metal, the electron density that responds to applied fields is so large that an external electric field can penetrate only a very short distance into the material. However, in a semiconductor the lower density of electrons (and possibly holes) that can respond to an applied field is sufficiently small that the field can penetrate quite far into the material. This field penetration alters the conductivity of the semiconductor near its surface, and is called the field effect. The field effect underlies the operation of the Schottky diode and of field-effect transistors, notably the MOSFET, the JFET and the MESFET.

Heterojunction

A heterojunction is the interface that occurs between two layers or regions of dissimilar crystalline semiconductors. These semiconducting materials have unequal band gaps as opposed to a homojunction. It is often advantageous to engineer the electronic energy bands in many solid-state device applications, including semiconductor lasers, solar cells and transistors ("heterotransistors") to name a few. The combination of multiple heterojunctions together in a device is called a heterostructure, although the two terms are commonly used interchangeably. The requirement that each material be a semiconductor with unequal band gaps is somewhat loose, especially on small length scales, where electronic properties depend on spatial properties. A more modern definition of heterojunction is the interface between any two solid-state materials, including crystalline and amorphous structures of metallic, insulating, fast ion conductor and semiconducting materials.

In 2000, the Nobel Prize in physics was awarded jointly to Herbert Kroemer of the University of California, Santa Barbara, California, USA and Zhores I. Alferov of Ioffe Institute, Saint Petersburg, Russia for "developing semiconductor heterostructures used in high-speed-photography and opto-electronics".

MOSFET

The metal-oxide-semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET) is a type of field-effect transistor (FET), most commonly fabricated by the controlled oxidation of silicon. It has an insulated gate, whose voltage determines the conductivity of the device. This ability to change conductivity with the amount of applied voltage can be used for amplifying or switching electronic signals. A metal-insulator-semiconductor field-effect transistor or MISFET is a term almost synonymous with MOSFET. Another synonym is IGFET for insulated-gate field-effect transistor.

The basic principle of the field-effect transistor was first patented by Julius Edgar Lilienfeld in 1925.

The main advantage of a MOSFET is that it requires almost no input current to control the load current, when compared with bipolar transistors (bipolar junction transistors/BJTs). In an enhancement mode MOSFET, voltage applied to the gate terminal increases the conductivity of the device. In depletion mode transistors, voltage applied at the gate reduces the conductivity.The "metal" in the name MOSFET is sometimes a misnomer, because the gate material can be a layer of polysilicon (polycrystalline silicon). Similarly, "oxide" in the name can also be a misnomer, as different dielectric materials are used with the aim of obtaining strong channels with smaller applied voltages.

The MOSFET is by far the most common transistor in digital circuits, as billions may be included in a memory chip or microprocessor. Since MOSFETs can be made with either p-type or n-type semiconductors, complementary pairs of MOS transistors can be used to make switching circuits with very low power consumption, in the form of CMOS logic.

Metal-induced gap states

In bulk semiconductor band structure calculations, it is assumed that the crystal lattice (which features a periodic potential due to the atomic structure) 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 (by thermal evaporation, for example), 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.

Multi-junction solar cell

Multi-junction (MJ) solar cells are solar cells with multiple p–n junctions made of different semiconductor materials. Each material's p-n junction will produce electric current in response to different wavelengths of light. The use of multiple semiconducting materials allows the absorbance of a broader range of wavelengths, improving the cell's sunlight to electrical energy conversion efficiency.

Traditional single-junction cells have a maximum theoretical efficiency of 33.16%. Theoretically, an infinite number of junctions would have a limiting efficiency of 86.8% under highly concentrated sunlight.Currently, the best lab examples of traditional crystalline silicon (c-Si) solar cells have efficiencies between 20% and 25%, while lab examples of multi-junction cells have demonstrated performance over 46% under concentrated sunlight. Commercial examples of tandem cells are widely available at 30% under one-sun illumination, and improve to around 40% under concentrated sunlight. However, this efficiency is gained at the cost of increased complexity and manufacturing price. To date, their higher price and higher price-to-performance ratio have limited their use to special roles, notably in aerospace where their high power-to-weight ratio is desirable. In terrestrial applications, these solar cells are emerging in concentrator photovoltaics (CPV), with a growing number of installations around the world.Tandem fabrication techniques have been used to improve the performance of existing designs. In particular, the technique can be applied to lower cost thin-film solar cells using amorphous silicon, as opposed to conventional crystalline silicon, to produce a cell with about 10% efficiency that is lightweight and flexible. This approach has been used by several commercial vendors, but these products are currently limited to certain niche roles, like roofing materials.

Photoelectrochemical cell

Photoelectrochemical cells, or PECs, are solar cells that produce electrical energy or hydrogen in a process similar to the electrolysis of water.

Photoelectrochemistry

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.

P–n junction

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 electrical current to pass through the junction only in one direction. The p-n junction is created by doping, for example by ion implantation, diffusion of dopants, or by epitaxy (growing a layer of crystal doped with one type of dopant on top of a layer of crystal doped with another type of dopant). If two separate pieces of material were used, this would introduce a grain boundary between the semiconductors that would severely inhibit its utility by scattering the electrons and holes.p–n junctions are elementary "building blocks" of semiconductor electronic devices such as diodes, transistors, solar cells, LEDs, and integrated circuits; they are the active sites where the electronic action of the device takes place. For example, a common type of transistor, the bipolar junction transistor, consists of two p–n junctions in series, in the form n–p–n or p–n–p; while a diode can be made from a single p-n junction. A Schottky junction is a special case of a p–n junction, where metal serves the role of the p-type semiconductor.

Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.Classical physics, the physics existing before quantum mechanics, describes nature at ordinary (macroscopic) scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.

Quantum mechanics differs from classical physics in that energy, momentum, angular momentum and other quantities of a bound system are restricted to discrete values (quantization); objects have characteristics of both particles and waves (wave-particle duality); and there are limits to the precision with which quantities can be measured (uncertainty principle).Quantum mechanics gradually arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle.

Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.

Resonant-tunneling diode

A resonant-tunneling diode (RTD) is a diode with a resonant-tunneling structure in which electrons can tunnel through some resonant states at certain energy levels. The current–voltage characteristic often exhibits negative differential resistance regions.

All types of tunneling diodes make use of quantum mechanical tunneling.

Characteristic to the current–voltage relationship of a tunneling diode is the presence of one or more negative differential resistance regions, which enables many unique applications. Tunneling diodes can be very compact and are also capable of ultra-high-speed operation because the quantum tunneling effect through the very thin layers is a very fast process. One area of active research is directed toward building oscillators and switching devices that can operate at terahertz frequencies.

Schottky barrier

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 ΦB (see figure). The value of ΦB depends on the combination of metal and semiconductor.Not all metal–semiconductor junctions form a rectifying Schottky barrier; a metal–semiconductor junction that conducts current in both directions without rectification, perhaps due to its Schottky barrier being too low, is called an ohmic contact.

Theory of solar cells

The theory of solar cells explains the process by which light energy in photons is converted into electric current when the photons strike a suitable semiconductor device. The theoretical studies are of practical use because they predict the fundamental limits of a solar cell, and give guidance on the phenomena that contribute to losses and solar cell efficiency.

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