# Materials Science and Engineering/Doctoral review questions/Semiconductor Device Fundamentals

## Semiconductors

A semiconductor is a solid material that has electrical conductivity in between that of a conductor and that of an insulator; it can vary over that wide range either permanently or dynamically. Semiconductors are tremendously important in technology. Semiconductor devices, electronic components made of semiconductor materials, are essential in modern electrical devices. Examples range from computers to cellular phones to digital audio players. Silicon is used to create most semiconductors commercially, but dozens of other materials are used as well.

### Material Properties

Semiconductors' intrinsic electrical properties are often permanently modified by introducing impurities by a process known as doping. Usually, it is sufficient to approximate that each impurity atom adds one electron or one "hole" (a concept to be discussed later) that may flow freely. Upon the addition of a sufficiently large proportion of impurity dopants, semiconductors will conduct electricity nearly as well as metals. Depending on the kind of impurity, a doped region of semiconductor can have more electrons or holes, and is named N-type or P-type semiconductor material, respectively. Junctions between regions of N- and P-type semiconductors create electric fields, which cause electrons and holes to be available to move away from them, and this effect is critical to semiconductor device operation. Also, a density difference in the amount of impurities produces a small electric field in the region which is used to accelerate non-equilibrium electrons or holes.

In most semiconductors, when electrons lose enough energy to fall from the conduction band to the valence band (the energy levels above and below the band gap), they often emit light, a quantum of energy in the visible electromagnetic spectrum. This photoemission process underlies the light-emitting diode (LED) and the semiconductor laser, both of which are very important commercially. Conversely, semiconductor absorption of light in photodetectors excites electrons to move from the valence band to the higher energy conduction band, thus facilitating detection of light and vary with its intensity. This is useful for fiber optic communications, and providing the basis for energy from solar cells.

#### Materials

Semiconductors may be elemental materials such as silicon and germanium, or compound semiconductors such as gallium arsenide and indium phosphide, or alloys such as silicon germanium or aluminium gallium arsenide.

#### Composition

The materials chosen as suitable dopants depend on the atomic properties of both the dopant and the material to be doped. In general, dopants that produce the desired controlled changes are classified as either electron acceptors or donors. A donor atom that activates (that is, becomes incorporated into the crystal lattice) donates weakly-bound valence electrons to the material, creating excess negative charge carriers. These weakly-bound electrons can move about in the crystal lattice relatively freely and can facilitate conduction in the presence of an electric field. (The donor atoms introduce some states under, but very close to the conduction band edge. Electrons at these states can be easily excited to conduction band, becoming free electrons, at room temperature.) Conversely, an activated acceptor produces a hole. Semiconductors doped with donor impurities are called n-type, while those doped with acceptor impurities are known as p-type. The n and p type designations indicate which charge carrier acts as the material's majority carrier. The opposite carrier is called the minority carrier, which exists due to thermal excitation at a much lower concentration compared to the majority carrier.

For example, the pure semiconductor silicon has four valence electrons. In silicon, the most common dopants are IUPAC group 13 (commonly known as group III) and group 15 (commonly known as group V) elements. Group 13 elements all contain three valence electrons, causing them to function as acceptors when used to dope silicon. Group 15 elements have five valence electrons, which allows them to act as a donor. Therefore, a silicon crystal doped with boron creates a p-type semiconductor whereas one doped with phosphorus results in an n-type material.

#### Purity

Semiconductors with predictable, reliable electronic properties are necessary for mass production. The level of chemical purity needed is extremely high because the presence of impurities even in very small proportions can have large effects on the properties of the material. A high degree of crystalline perfection is also required, since faults in crystal structure (such as dislocations, twins, and stacking faults) interfere with the semiconducting properties of the material. Crystalline faults are a major cause of defective semiconductor devices. The larger the crystal, the more difficult it is to achieve the necessary perfection. Current mass production processes use crystal ingots between four and twelve inches (300 mm) in diameter which are grown as cylinders and sliced into wafers.

Because of the required level of chemical purity and the perfection of the crystal structure which are needed to make semiconductor devices, special methods have been developed to produce the initial semiconductor material. A technique for achieving high purity includes growing the crystal using the Czochralski process. An additional step that can be used to further increase purity is known as zone refining. In zone refining, part of a solid crystal is melted. The impurities tend to concentrate in the melted region, while the desired material recrystalizes leaving the solid material more pure and with fewer crystalline faults.

#### Electronic Structure

There are three popular ways to describe the electronic structure of a crystal. The first starts from single atoms. An atom has discrete energy levels. When two atoms come close each energy level splits into an upper and a lower level, whereby they delocalize across the two atoms. With more atoms the number of levels increases, and groups of levels form bands. Semiconductors contain many bands. If there is a large distance between the highest occupied state and the lowest unoccupied space, then a gap will likely remain between occupied and unoccupied bands even after band formation.

A second way starts with free electrons waves. When fading in an electrostatic potential due to the cores, due to Bragg reflection some waves are reflected and cannot penetrate the bulk, that is a band gap opens. In this description it is not clear, while the number of electrons fills up exactly all states below the gap.

A third description starts with two atoms. The split states form a covalent bond where two electrons with spin up and spin down are mostly in between the two atoms. Adding more atoms now is supposed not to lead to splitting, but to more bonds. This is the way silicon is typically drawn. The band gap is now formed by lifting one electron from the lower electron level into the upper level. This level is known to be anti-bonding, but bulk silicon has not been seen to lose atoms as easy as electrons are wandering through it. Also this model is most unsuitable to explain how in graded hetero-junction the band gap can vary smoothly.

### Crystal Growth

#### Czochralski process

The Czochralski process is a method of crystal growth used to obtain single crystals of semiconductors (e.g. silicon, germanium and gallium arsenide), metals (e.g. palladium, platinum, silver, gold), salts and some man made, (or "lab") gemstones.

The most important application may be the growth of large cylindrical ingots, or boules, of single crystal silicon. High-purity, semiconductor-grade silicon (only a few parts per million of impurities) is melted down in a crucible , which is usually made of quartz. Dopant impurity atoms such as boron or phosphorus can be added to the molten intrinsic silicon in precise amounts in order to dope the silicon, thus changing it into n-type or p-type extrinsic silicon. This influences the electrical conductivity of the silicon. A seed crystal, mounted on a rod, is dipped into the molten silicon. The seed crystal's rod is pulled upwards and rotated at the same time. By precisely controlling the temperature gradients, rate of pulling and speed of rotation, it is possible to extract a large, single-crystal, cylindrical ingot from the melt. This process is normally performed in an inert atmosphere, such as argon, and in an inert chamber, such as quartz.

While the largest silicon ingots produced today are 400 mm in diameter and 1 to 2 metres in length, 200 mm and 300 mm diameter crystals are standard industrial processes. Thin silicon wafers are cut from these ingots (typically about 0.2 - 0.75 mm thick) and can be polished to a very high flatness for making integrated circuits, or textured for making solar cells. Other semiconductors, such as gallium arsenide, can also be grown by this method, although lower defect densities in this case can be obtained using variants of the Bridgeman technique.

When silicon is grown by the Czochralski method the melt is contained in a silica (quartz) crucible. During growth the walls of the crucible dissolve into the melt and Czochralski silicon therefore contains oxygen impurities with a typical concentration of 1018cm − 3. Perhaps surprisingly, oxygen impurities can have beneficial effects. Carefully chosen annealing conditions can allow the formation of oxygen precipitates. These have the effect of trapping unwanted transition metal impurities in a process known as gettering. Additionally, oxygen impurities can improve the mechanical strength of silicon wafers by immobilising any dislocations which may be introduced during device processing. It has experimentally been proved in the 1990s that the high oxygen concentration is also beneficial for radiation hardness of silicon particle detectors used in harsh radiation environment ( eg. CERN's LHC/S-LHC projects) Therefore, radiation detectors made of Czochralski- and Magnetic Czochralski-silicon are considered to be promising candidates for many future high-energy physics experiments. However, oxygen impurities can react with boron in an illuminated environment, such as experienced by solar cells. This results in the formation of an electrically active boron–oxygen complex that detracts from cell performance. Module output drops by approximately 3% during the first few hours of light exposure.

Occurrence of unwanted instabilities in the melt can be avoided by investigating and visualizing the temperature and velocity fields during the crystal growth process.

The process is named after Polish scientist Jan Czochralski, who discovered the method in 1916 while investigating the crystallization rates of metals.

## Carrier Modeling

### Carrier Properties

#### Charge

##### Electrons

Electrons have an electric charge of −1.6021765 × 10−19 coulomb, a mass of 9.11 × 10−31 kg based on charge/mass measurements and a relativistic rest mass of about 0.511 MeV/c². The mass of the electron is approximately 1/1836 of the mass of the proton. The common electron symbol is e−.[1] Electron mean lifetime is >4.6x1026 years, see Particle decay.

According to quantum mechanics, electrons can be represented by wavefunctions, from which a calculated probabilistic electron density can be determined. The orbital of each electron in an atom can be described by a wavefunction. Based on the Heisenberg uncertainty principle, the exact momentum and position of the actual electron cannot be simultaneously determined. This is a limitation which, in this instance, simply states that the more accurately we know a particle's position, the less accurately we can know its momentum, and vice versa.

##### Holes

An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics and chemistry. The concept describes the lack of an electron. It is different from the positron, which is the antimatter duplicate of the electron.

The electron hole was introduced into calculations for the following two situations:

• If an electron is excited into a higher state it leaves a hole in its old state. This meaning is used in Auger electron spectroscopy (and other x-ray techniques), in computational chemistry, and to explain the low electron-electron scattering-rate in crystals (metals, semiconductors).
• In crystals, band structure calculations lead to an effective mass for the charge carriers, which can be negative. Inspired by the Hall effect, Newton's law is used to attach the negative sign onto the charge.

#### Effective Mass

In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron me (9.11×10-31 kg).

##### Definition

When an electron is moving inside a solid material, the force between other atoms will affect its movement and it will not be described by Newton's law. So we introduce the concept of effective mass to describe the movement of electron in Newton's law. The effective mass can be negative or different due to circumstances.

Effective mass is defined by analogy with Newton's second law ${\displaystyle {\vec {F}}=m{\vec {a}}}$. Using quantum mechanics it can be shown that for an electron in an external electric field E:

${\displaystyle a={{1} \over {\hbar ^{2}}}\cdot {{d^{2}\varepsilon } \over {dk^{2}}}qE}$

where ${\displaystyle a}$ is acceleration, ${\displaystyle \hbar =h/2\pi }$ is reduced Planck's constant, ${\displaystyle k}$ is the wave number (often called momentum since ${\displaystyle k}$ = ${\displaystyle p/\hbar }$ for free electrons), ${\displaystyle \epsilon (k)}$ is the energy as a function of ${\displaystyle k}$, or the dispersion relation as it is often called. From the external electric field alone, the electron would experience a force of ${\displaystyle {\vec {F}}=q{\vec {E}}}$, where q is the charge. Hence under the model that only the external electric field acts, effective mass ${\displaystyle m^{*}}$ becomes:

${\displaystyle m^{*}=\hbar ^{2}\cdot \left[{{d^{2}\varepsilon } \over {dk^{2}}}\right]^{-1}}$

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept.

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Effective mass, being generally dependent on direction (with respect to the crystal axes), is a tensor. However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Effective mass can only be understood with quantum mechanics.

#### Carrier Numbers in Intrinsic Material

An intrinsic semiconductor, also called an undoped semiconductor or i-type semiconductor, is a pure semiconductor without any significant dopant species present. The number of charge carriers is therefore determined by the properties of the material itself instead of the amount of impurities. In intrinsic semiconductors the number of electrons and the number of holes are equal. n = p.

The conductivity of intrinsic semiconductors can be due to crystal defects or to thermal excitation. In an intrinsic semiconductor the number of electrons in the conduction band is equal to the number of holes in the valence band. An example is Hg0.8Cd0.2>Te at room temperature.

##### Doping

In semiconductor production, doping refers to the process of intentionally introducing impurities into an extremely pure (also referred to as intrinsic) semiconductor in order to change its electrical properties. The impurities are dependent upon the type of semiconductor. Lightly and moderately doped semiconductors are referred to as extrinsic. A semiconductor which is doped to such high levels that it acts more like a conductor than a semiconductor is called degenerate.

Some dopants are generally added as the (usually silicon) boule is grown, giving each wafer an almost uniform initial doping. To define circuit elements, selected areas (typically controlled by photolithography)[1] are further doped by such processes as diffusion[2] and ion implantation, the latter method being more popular in large production runs due to its better controllability.

The number of dopant atoms needed to create a difference in the ability of a semiconductor to conduct is very small. Where a comparatively small number of dopant atoms are added (of the order of 1 every 100,000,000 atoms) then the doping is said to be low, or light. Where many more are added (of the order of 1 in 10,000) then the doping is referred to as heavy, or high. This is often shown as n+ for n-type dopant or p+ for p-type doping. A more detailed description of the mechanism of doping can be found in the article on semiconductors.

### State and Carrier Distributions

#### Density of States

In statistical and condensed matter physics, Density of states (DOS) is a property that quantifies how closely packed energy levels are in a quantum-mechanical system. It is usually denoted with one of the symbols g, ρ, n, or N. It is a function g(E) of the internal energy E, such that the expression g(E) dE represents the number of states with energies between E and E+dE. It can also be written as a function of the angular frequency ω, which is proportional to the energy. The density of states is used extensively in condensed-matter physics, where it can refer to electron, photon, or phonon energy levels in a crystalline solid. In crystalline solids, there are often energy ranges where the density of electron states is zero, which means that the electrons cannot be excited to these energies. The density of states also occurs in Fermi's golden rule, which describes how fast quantum-mechanical transitions occur in the presence of a perturbation.

 ${\displaystyle g_{c}(E)={\frac {m_{n}^{*}{\sqrt {2m_{n}^{*}(E-E_{c})}}}{\pi ^{2}\hbar ^{3}}}}$


#### Fermi Function

In statistical mechanics, Fermi-Dirac statistics is a particular case of particle statistics developed by Enrico Fermi and Paul Dirac that determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. In other words, it is a probability of a given energy level to be occupied by a fermion.

More generally, Fermi-Dirac statistics means that the total wavefunction of fermions must be antisymmetric under an exchange of every pair of fermions (that is, if one exchanges any fermion with another, the wavefunction gets an overall minus sign).

Fermions are particles which are indistinguishable and obey the Pauli exclusion principle, i.e., no more than one particle may occupy the same quantum state at the same time. Fermions have half-integral spin. Statistical thermodynamics is used to describe the behaviour of large numbers of particles. A collection of non-interacting fermions is called a Fermi gas.

 ${\displaystyle n_{i}={\frac {g_{i}}{e^{(\epsilon _{i}-\mu )/kT}+1}}}$


### Equilibrium Carrier Concentrations

#### Formulas of n and p

${\displaystyle n=\int _{E_{c}}^{E_{top}}g_{c}(E)f(E)dE}$

${\displaystyle p=\int _{E_{bottom}}^{E_{v}}g_{v}(E)[1-f(E)]dE}$

 ${\displaystyle n=N_{C}{\frac {2}{\sqrt {\pi }}}F_{1/2}(\eta _{c})}$
${\displaystyle p=N_{V}{\frac {2}{\sqrt {\pi }}}F_{1/2}(\eta _{V})}$

 ${\displaystyle n=N_{C}e^{(E_{F}-E_{C})/kT}}$
${\displaystyle p=N_{V}e^{(E_{V}-E_{F})/kT}}$

 ${\displaystyle n=n_{i}e^{(E_{F}-E_{i})/kT}}$
${\displaystyle p=n_{i}e^{(E_{i}-E_{F})/kT}}$


#### ni and the np Product

 ${\displaystyle n_{i}={\sqrt {N_{C}N_{V}}}e^{-E_{G}/2kT}}$

 ${\displaystyle np=n_{i}^{2}}$


#### Charge Neutrality Relationship

 ${\displaystyle p-n+N_{D}^{+}-N_{A}^{-}=0}$


With total ionization of dopant atoms:

 ${\displaystyle p-n+N_{D}-N_{A}=0\;}$


#### Carrier Concentration Calculations

 ${\displaystyle n={\frac {N_{D}-N_{A}}{2}}\left(\left({\frac {N_{D}-N_{A}}{2}}\right)^{2}+n_{i}^{2}\right)^{1/2}}$

 ${\displaystyle p={\frac {n_{i}^{2}}{n}}}$
${\displaystyle {\frac {N_{A}-N_{D}}{2}}\left(\left({\frac {N_{A}-N_{D}}{2}}\right)^{2}+n_{i}^{2}\right)^{1/2}}$


#### Determination of EF

In intrinsic material,

${\displaystyle n=p\;}$

${\displaystyle N_{C}e^{(E_{i}-E_{c})/kT}=N_{V}e^{(E_{v}-E_{i})/kT}}$

${\displaystyle E_{i}={\frac {E_{c}+E_{v}}{2}}+{\frac {kT}{2}}\ln \left({\frac {N_{V}}{N_{C}}}\right)}$

${\displaystyle {\frac {N_{V}}{N_{C}}}=\left({\frac {m_{p}^{*}}{m_{n}^{*}}}\right)^{3/2}}$

 ${\displaystyle E_{i}={\frac {E_{c}+E_{v}}{2}}+{\frac {3}{4}}kT\ln \left({\frac {m_{p}^{*}}{m_{n}^{*}}}\right)}$

 ${\displaystyle E_{F}-E_{i}=kT\ln(N_{D}/n_{i})\;}$
${\displaystyle E_{i}-E_{F}=kT\ln(N_{A}/n_{i})\;}$


## Carrier Action

### Drift

The mobile charged particles within a conductor move constantly in random directions. In order for a net flow of charge to exist, the particles must also move together with an average drift rate. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the direction of the electric field.

 ${\displaystyle I_{P|drift}=qpv_{d}A\;}$
${\displaystyle \mathbf {J} _{P|drift}=qp\mathbf {v} _{d}}$
${\displaystyle \mathbf {J} _{P|drift}=q\mu _{p}pE}$


#### Mobility

Units: ${\displaystyle cm^{2}/V\cdot sec}$

#### Resistivity

${\displaystyle E=\rho \mathbf {J} }$

${\displaystyle \mathbf {J} _{drift}=\mathbf {J} _{N|drift}+\mathbf {J} _{P|drift}}$ ${\displaystyle \mathbf {J} _{drift}=q(\mu _{n}n+\mu _{p}p)E}$

 ${\displaystyle \rho ={\frac {1}{q(\mu _{n}n+\mu _{p}p)}}}$


#### Band Bending

 ${\displaystyle V=-{\frac {1}{q}}(E_{c}-E_{ref})}$

 ${\displaystyle E={\frac {1}{q}}{\frac {dE_{c}}{dx}}={\frac {1}{q}}{\frac {dE_{v}}{dx}}={\frac {1}{q}}{\frac {dE_{i}}{dx}}}$


### Diffusion

#### Diffusion Current

${\displaystyle F=-D\nabla \eta }$

 ${\displaystyle \mathbf {J} _{P|diff}=-qD_{p}\nabla p}$
${\displaystyle \mathbf {J} _{N|diff}=-qD_{N}\nabla n}$


#### Total Current

${\displaystyle \mathbf {J} _{P}=\mathbf {J} _{P|drift}+J_{P|diff}}$

 ${\displaystyle \mathbf {J} _{P}=q\mu _{p}pE-qD_{P}\nabla p}$


${\displaystyle \mathbf {J} _{N}=\mathbf {J} _{N|drift}+J_{N|diff}}$

 ${\displaystyle \mathbf {J} _{P}=q\mu _{n}nE-qD_{N}\nabla n}$


${\displaystyle \mathbf {J} =\mathbf {J} _{N}\mathbf {J} _{P}}$

#### Einstein Relationship

 ${\displaystyle {\frac {D_{N}}{\mu _{n}}}={\frac {kT}{q}}}$


## Recombination-Generation

Like other solids, semiconductor materials have electronic band structure determined by the crystal properties of the material. The actual energy distribution among the electrons is described by the Fermi energy and the temperature of the electrons. At absolute zero temperature, all of the electrons have energy below the Fermi energy; but at non-zero temperatures the energy levels are randomized and some electrons have energy above the Fermi level.

In semiconductors the Fermi energy lies in the middle of a forbidden band or band gap between two allowed bands called the valence band and the conduction band. The valence band, immediately below the forbidden band, is normally very nearly completely occupied. The conduction band, above the Fermi level, is normally nearly completely empty. Because the valence band is so nearly full, its electrons are not mobile, and cannot flow as electrical current.

However, if an electron in the valence band acquires enough energy to reach the conduction band, it can flow freely among the nearly empty conduction band energy states. Furthermore it will also leave behind an electron hole that can flow as current exactly like a physical charged particle. Carrier generation describes processes by which electrons gain energy and move from the valence band to the conduction band, producing two mobile carriers; while recombination describes processes by which a conduction band electron loses energy and re-occupies the energy state of an electron hole in the valence band.

In a material at thermal equilibrium generation and recombination are balanced, so that the net charge carrier density remains constant. The equilibrium carrier density that results from the balance of these interactions is predicted by thermodynamics. The resulting probability of occupation of energy states in each energy band is given by Fermi-Dirac statistics.

### R-G Statistics

#### Photogeneration

${\displaystyle I=I_{0}e^{-\alpha x}\;}$

${\displaystyle {\frac {\partial n}{\partial t}}|_{\mbox{light}}={\frac {\partial p}{\partial t}}|_{\mbox{light}}=G_{L}(x,\lambda )}$

${\displaystyle G_{L}(x,\lambda )=G_{L0}e^{-\alpha x}\;}$

${\displaystyle \tau _{p}={\frac {1}{c_{p}N_{T}}}}$

${\displaystyle \tau _{n}={\frac {1}{c_{n}N_{T}}}}$

${\displaystyle {\frac {\partial p}{\partial t}}|_{i-thermalR-G}=-{\frac {\Delta p}{\tau _{p}}}}$

#### Equations of State

${\displaystyle {\frac {\partial n}{\partial t}}={\frac {\partial n}{\partial t}}|_{drift}+{\frac {\partial n}{\partial t}}|_{diff}+{\frac {\partial n}{\partial t}}|_{thermalR-G}+{\frac {\partial n}{\partial t}}|_{otherprocesses}}$

 ${\displaystyle {\frac {\partial n}{\partial t}}={\frac {1}{q}}\nabla \cdot \mathbf {J} _{N}{\frac {\partial n}{\partial t}}|_{thermalR-G}+{\frac {\partial n}{\partial t}}|_{otherprocesses}}$


#### Minority Carrier Diffusion Equations

 ${\displaystyle {\frac {\partial \Delta n_{p}}{\partial t}}=D_{N}{\frac {\partial ^{2}\Delta n_{p}}{\partial x^{2}}}-{\frac {\Delta n_{p}}{\tau _{n}}}+G_{L}}$


### Diffusion Lengths

${\displaystyle L_{P}={\sqrt {D_{P}\tau _{P}}}}$

## pn Junction Electrostatics

### Preliminaries

#### Poisson's Equation

${\displaystyle \nabla \cdot E={\frac {\rho }{K_{S}\epsilon _{0}}}}$

If there is a spherically symmetric Gaussian charge density ${\displaystyle \rho (r)}$:

${\displaystyle \rho (r)={\frac {Q}{\sigma ^{3}{\sqrt {2\pi }}^{3}}}\,e^{-r^{2}/(2\sigma ^{2})},}$

where Q is the total charge, then the solution φ (r) of Poisson's equation with ${\displaystyle \varepsilon _{r}=1}$,

${\displaystyle {\nabla }^{2}\varphi =-{\rho \over \varepsilon _{0}}}$,

is given by

${\displaystyle \varphi (r)={1 \over 4\pi \varepsilon _{0}}{\frac {Q}{r}}\,{\mbox{erf}}\left({\frac {r}{{\sqrt {2}}\sigma }}\right)}$.

where erf(x) is the error function. This solution can be checked explicitly by a careful manual evaluation of ${\displaystyle {\nabla }^{2}\varphi }$. Note that, for r much greater than σ, erf(x) approaches unity and the potential φ (r) approaches the point charge potential ${\displaystyle {1 \over 4\pi \varepsilon _{0}}{Q \over r}}$, as one would expect. Furthermore the erf function approaches 1 extremely fast as its argument increase; in practice for r > 3σ the relative error is smaller than 1/1000.

#### Assume that the dopants are totally ionized

 ${\displaystyle \rho =q(p-n+N_{D}-N_{A})\;}$


#### Built-in Potential

 ${\displaystyle V_{b}i={\frac {kT}{q}}\ln \left({\frac {N_{A}N_{D}}{n_{i}^{2}}}\right)}$


### Quantitative Electrostatic Relationships

#### Depletion Width

${\displaystyle x_{n}=\left({\frac {2K_{s}\epsilon _{0}}{q}}{\frac {N_{A}}{N_{D}(N_{A}+N_{D})}}V_{bi}\right)^{1/2}}$

${\displaystyle x_{p}={\frac {N_{D}x_{n}}{N_{A}}}}$

${\displaystyle W=x_{n}+x_{p}=\left({\frac {2K_{s}\epsilon _{0}}{q}}\left({\frac {N_{A}+N_{D}}{N_{A}N_{D}}}\right)V_{bi}\right)^{1/2}}$

#### Depletion Width with Applied Bias

${\displaystyle x_{p}=\left({\frac {2K_{s}\epsilon _{0}}{q}}{\frac {N_{D}}{N_{A}(N_{A}+N_{D})}}(V_{bi}-V_{A})\right)^{1/2}}$

${\displaystyle W=\left({\frac {2K_{s}\epsilon _{0}}{q}}\left({\frac {N_{A}+N_{D}}{N_{A}N_{D}}}\right)(V_{bi}-V_{A})\right)^{1/2}}$