Electron mobility

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electron_mobility

In solid-state physics, the electron mobility characterises how quickly an electron can move through a metal or semiconductor when pulled by an electric field. There is an analogous quantity for holes, called hole mobility. The term carrier mobility refers in general to both electron and hole mobility.

Electron and hole mobility are special cases of electrical mobility of charged particles in a fluid under an applied electric field.

When an electric field E is applied across a piece of material, the electrons respond by moving with an average velocity called the drift velocity, ${\displaystyle v_d}$. Then the electron mobility μ is defined as

$${\displaystyle v_d=\mu E.}$$

Electron mobility is almost always specified in units of cm²/(V⋅s). This is different from the SI unit of mobility, m²/(V⋅s). They are related by 1 m²/(V⋅s) = 10⁴ cm²/(V⋅s).

Conductivity is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each. For semiconductors, the behavior of transistors and other devices can be very different depending on whether there are many electrons with low mobility or few electrons with high mobility. Therefore mobility is a very important parameter for semiconductor materials. Almost always, higher mobility leads to better device performance, with other things equal.

Semiconductor mobility depends on the impurity concentrations (including donor and acceptor concentrations), defect concentration, temperature, and electron and hole concentrations. It also depends on the electric field, particularly at high fields when velocity saturation occurs. It can be determined by the Hall effect, or inferred from transistor behavior.

Introduction

Drift velocity in an electric field

Main article: Drift velocity

Without any applied electric field, in a solid, electrons and holes move around randomly. Therefore, on average there will be no overall motion of charge carriers in any particular direction over time.

However, when an electric field is applied, each electron or hole is accelerated by the electric field. If the electron were in a vacuum, it would be accelerated to ever-increasing velocity (called ballistic transport). However, in a solid, the electron repeatedly scatters off crystal defects, phonons, impurities, etc., so that it loses some energy and changes direction. The final result is that the electron moves with a finite average velocity, called the drift velocity. This net electron motion is usually much slower than the normally occurring random motion.

The two charge carriers, electrons and holes, will typically have different drift velocities for the same electric field.

Quasi-ballistic transport is possible in solids if the electrons are accelerated across a very small distance (as small as the mean free path), or for a very short time (as short as the mean free time). In these cases, drift velocity and mobility are not meaningful.

Definition and units

See also: Electrical mobility

The electron mobility is defined by the equation:

$${\displaystyle v_d={\mu }_{e}E.}$$

where:

• E is the magnitude of the electric field applied to a material,
• v_d is the magnitude of the electron drift velocity (in other words, the electron drift speed) caused by the electric field, and
• µ_e is the electron mobility.

The hole mobility is defined by a similar equation:

$${\displaystyle v_d={\mu }_{h}E.}$$

Both electron and hole mobilities are positive by definition.

Usually, the electron drift velocity in a material is directly proportional to the electric field, which means that the electron mobility is a constant (independent of the electric field). When this is not true (for example, in very large electric fields), mobility depends on the electric field.

The SI unit of velocity is m/s, and the SI unit of electric field is V/m. Therefore the SI unit of mobility is (m/s)/(V/m) = m²/(V⋅s). However, mobility is much more commonly expressed in cm²/(V⋅s) = 10⁻⁴ m²/(V⋅s).

Mobility is usually a strong function of material impurities and temperature, and is determined empirically. Mobility values are typically presented in table or chart form. Mobility is also different for electrons and holes in a given material.

Derivation

Starting with Newton's Second Law:

$${\displaystyle a=F/m_e^{\ast }}$$

where:

• a is the acceleration between collisions.
• F is the electric force exerted by the electric field, and
• ${\displaystyle m_e^{\ast }}$ is the effective mass of an electron.

Since the force on the electron is −eE:

$${\displaystyle a=-\frac{eE}{m_e^{\ast }}}$$

This is the acceleration on the electron between collisions. The drift velocity is therefore:

$${\displaystyle v_d=a{\tau }_c=-\frac{e{\tau }_c}{m_e^{\ast }}E,}$$

where ${\displaystyle {\tau }_c}$ is the mean free time

Since we only care about how the drift velocity changes with the electric field, we lump the loose terms together to get

$${\displaystyle v_d=-{\mu }_{e}E,}$$

where ${\displaystyle {\mu }_e=\frac{e{\tau }_c}{m_e^{\ast }}}$

Similarly, for holes we have

$${\displaystyle v_d={\mu }_{h}E,}$$

where ${\displaystyle {\mu }_h=\frac{e{\tau }_c}{m_h^{\ast }}}$ Note that both electron mobility and hole mobility are positive. A minus sign is added for electron drift velocity to account for the minus charge.

Relation to current density

The drift current density resulting from an electric field can be calculated from the drift velocity. Consider a sample with cross-sectional area A, length l and an electron concentration of n. The current carried by each electron must be ${\displaystyle -e{v}_d}$, so that the total current density due to electrons is given by:

$${\displaystyle J_e=\frac{I_n}{A}=-en{v}_d}$$

Using the expression for ${\displaystyle v_d}$ gives

$${\displaystyle J_e=en{\mu }_{e}E}$$

A similar set of equations applies to the holes, (noting that the charge on a hole is positive). Therefore the current density due to holes is given by

$${\displaystyle J_h=ep{\mu }_{h}E}$$

where p is the hole concentration and ${\displaystyle {\mu }_h}$ the hole mobility.

The total current density is the sum of the electron and hole components:

$${\displaystyle J=J_e+J_h=(en{\mu }_e+ep{\mu }_h)E}$$

Relation to conductivity

We have previously derived the relationship between electron mobility and current density

$${\displaystyle J=J_e+J_h=(en{\mu }_e+ep{\mu }_h)E}$$

Now Ohm's Law can be written in the form

$${\displaystyle J=\sigma E}$$

where ${\displaystyle \sigma }$ is defined as the conductivity. Therefore we can write down:

$${\displaystyle \sigma =en{\mu }_e+ep{\mu }_h}$$

which can be factorised to

$${\displaystyle \sigma =e(n{\mu }_e+p{\mu }_h)}$$

Relation to electron diffusion

In a region where n and p vary with distance, a diffusion current is superimposed on that due to conductivity. This diffusion current is governed by Fick's Law:

$${\displaystyle F=-D_{\text{e}}\mathrm{\nabla }n}$$

where:

• F is flux.
• D_e is the diffusion coefficient or diffusivity
• ${\displaystyle \mathrm{\nabla }n}$ is the concentration gradient of electrons

The diffusion coefficient for a charge carrier is related to its mobility by the Einstein relation:

$${\displaystyle D_{\text{e}}=\frac{{\mu }_{\text{e}}{k}_{\mathrm{B}}T}{e}}$$

where:

• k_B is the Boltzmann constant
• T is the absolute temperature
• e is the electric charge of an electron

Examples

Typical electron mobility at room temperature (300 K) in metals like gold, copper and silver is 30–50 cm²/ (V⋅s). Carrier mobility in semiconductors is doping dependent. In silicon (Si) the electron mobility is of the order of 1,000, in germanium around 4,000, and in gallium arsenide up to 10,000 cm²/ (V⋅s). Hole mobilities are generally lower and range from around 100 cm²/ (V⋅s) in gallium arsenide, to 450 in silicon, and 2,000 in germanium.

Very high mobility has been found in several ultrapure low-dimensional systems, such as two-dimensional electron gases (2DEG) (35,000,000 cm²/(V⋅s) at low temperature), carbon nanotubes (100,000 cm²/(V⋅s) at room temperature) and freestanding graphene (200,000 cm²/ V⋅s at low temperature). Organic semiconductors (polymer, oligomer) developed thus far have carrier mobilities below 50 cm²/(V⋅s), and typically below 1, with well performing materials measured below 10.

List of highest measured mobilities [cm²/ (V⋅s)]

Material	Electron mobility	Hole mobility
AlGaAs/GaAs heterostructures	35,000,000
Freestanding Graphene	200,000
Carbon nanotubes	79,000
Cubic boron arsenide (c-BAs)	1,600
Crystalline silicon	1,400	450
Polycrystalline silicon	100
Metals (Al, Au, Cu, Ag)	10-50
2D Material (MoS2)	10-50
Organics	8.6	43
Amorphous silicon	~1

Electric field dependence and velocity saturation

Main article: Velocity saturation

At low fields, the drift velocity v_d is proportional to the electric field E, so mobility μ is constant. This value of μ is called the low-field mobility.

As the electric field is increased, however, the carrier velocity increases sublinearly and asymptotically towards a maximum possible value, called the saturation velocity v_sat. For example, the value of v_sat is on the order of 1×10⁷ cm/s for both electrons and holes in Si. It is on the order of 6×10⁶ cm/s for Ge. This velocity is a characteristic of the material and a strong function of doping or impurity levels and temperature. It is one of the key material and semiconductor device properties that determine a device such as a transistor's ultimate limit of speed of response and frequency.

This velocity saturation phenomenon results from a process called optical phonon scattering. At high fields, carriers are accelerated enough to gain sufficient kinetic energy between collisions to emit an optical phonon, and they do so very quickly, before being accelerated once again. The velocity that the electron reaches before emitting a phonon is:

$${\displaystyle \frac{m^{\ast }{v}_{\text{emit}}^2}{2}\approx \hslash {\omega }_{\text{phonon (opt.)}}}$$

where ω_phonon(opt.) is the optical-phonon angular frequency and m* the carrier effective mass in the direction of the electric field. The value of E_{phonon (opt.)} is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half of v_emit, because the electron starts at zero velocity and accelerates up to v_emit in each cycle. (This is a somewhat oversimplified description.)

Velocity saturation is not the only possible high-field behavior. Another is the Gunn effect, where a sufficiently high electric field can cause intervalley electron transfer, which reduces drift velocity. This is unusual; increasing the electric field almost always increases the drift velocity, or else leaves it unchanged. The result is negative differential resistance.

In the regime of velocity saturation (or other high-field effects), mobility is a strong function of electric field. This means that mobility is a somewhat less useful concept, compared to simply discussing drift velocity directly.

Relation between scattering and mobility

Recall that by definition, mobility is dependent on the drift velocity. The main factor determining drift velocity (other than effective mass) is scattering time, i.e. how long the carrier is ballistically accelerated by the electric field until it scatters (collides) with something that changes its direction and/or energy. The most important sources of scattering in typical semiconductor materials, discussed below, are ionized impurity scattering and acoustic phonon scattering (also called lattice scattering). In some cases other sources of scattering may be important, such as neutral impurity scattering, optical phonon scattering, surface scattering, and defect scattering.

Elastic scattering means that energy is (almost) conserved during the scattering event. Some elastic scattering processes are scattering from acoustic phonons, impurity scattering, piezoelectric scattering, etc. In acoustic phonon scattering, electrons scatter from state k to k', while emitting or absorbing a phonon of wave vector q. This phenomenon is usually modeled by assuming that lattice vibrations cause small shifts in energy bands. The additional potential causing the scattering process is generated by the deviations of bands due to these small transitions from frozen lattice positions.

Ionized impurity scattering

Semiconductors are doped with donors and/or acceptors, which are typically ionized, and are thus charged. The Coulombic forces will deflect an electron or hole approaching the ionized impurity. This is known as ionized impurity scattering. The amount of deflection depends on the speed of the carrier and its proximity to the ion. The more heavily a material is doped, the higher the probability that a carrier will collide with an ion in a given time, and the smaller the mean free time between collisions, and the smaller the mobility. When determining the strength of these interactions due to the long-range nature of the Coulomb potential, other impurities and free carriers cause the range of interaction with the carriers to reduce significantly compared to bare Coulomb interaction.

If these scatterers are near the interface, the complexity of the problem increases due to the existence of crystal defects and disorders. Charge trapping centers that scatter free carriers form in many cases due to defects associated with dangling bonds. Scattering happens because after trapping a charge, the defect becomes charged and therefore starts interacting with free carriers. If scattered carriers are in the inversion layer at the interface, the reduced dimensionality of the carriers makes the case differ from the case of bulk impurity scattering as carriers move only in two dimensions. Interfacial roughness also causes short-range scattering limiting the mobility of quasi-two-dimensional electrons at the interface.

Lattice (phonon) scattering

At any temperature above absolute zero, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed phonons. Like electrons, phonons can be considered to be particles. A phonon can interact (collide) with an electron (or hole) and scatter it. At higher temperature, there are more phonons, and thus increased electron scattering, which tends to reduce mobility.

Piezoelectric scattering

Piezoelectric effect can occur only in compound semiconductor due to their polar nature. It is small in most semiconductors but may lead to local electric fields that cause scattering of carriers by deflecting them, this effect is important mainly at low temperatures where other scattering mechanisms are weak. These electric fields arise from the distortion of the basic unit cell as strain is applied in certain directions in the lattice.

Surface roughness scattering

Surface roughness scattering caused by interfacial disorder is short range scattering limiting the mobility of quasi-two-dimensional electrons at the interface. From high-resolution transmission electron micrographs, it has been determined that the interface is not abrupt on the atomic level, but actual position of the interfacial plane varies one or two atomic layers along the surface. These variations are random and cause fluctuations of the energy levels at the interface, which then causes scattering.

Alloy scattering

In compound (alloy) semiconductors, which many thermoelectric materials are, scattering caused by the perturbation of crystal potential due to the random positioning of substituting atom species in a relevant sublattice is known as alloy scattering. This can only happen in ternary or higher alloys as their crystal structure forms by randomly replacing some atoms in one of the sublattices (sublattice) of the crystal structure. Generally, this phenomenon is quite weak but in certain materials or circumstances, it can become dominant effect limiting conductivity. In bulk materials, interface scattering is usually ignored.

Inelastic scattering

During inelastic scattering processes, significant energy exchange happens. As with elastic phonon scattering also in the inelastic case, the potential arises from energy band deformations caused by atomic vibrations. Optical phonons causing inelastic scattering usually have the energy in the range 30-50 meV, for comparison energies of acoustic phonon are typically less than 1 meV but some might have energy in order of 10 meV. There is significant change in carrier energy during the scattering process. Optical or high-energy acoustic phonons can also cause intervalley or interband scattering, which means that scattering is not limited within single valley.

Electron–electron scattering

Due to the Pauli exclusion principle, electrons can be considered as non-interacting if their density does not exceed the value 10¹⁶~10¹⁷ cm⁻³ or electric field value 10³ V/cm. However, significantly above these limits electron–electron scattering starts to dominate. Long range and nonlinearity of the Coulomb potential governing interactions between electrons make these interactions difficult to deal with.

Relation between mobility and scattering time

A simple model gives the approximate relation between scattering time (average time between scattering events) and mobility. It is assumed that after each scattering event, the carrier's motion is randomized, so it has zero average velocity. After that, it accelerates uniformly in the electric field, until it scatters again. The resulting average drift mobility is:

$${\displaystyle \mu =\frac{q}{m^{\ast }}\overline{\tau }}$$

where q is the elementary charge, m* is the carrier effective mass, and τ is the average scattering time.

If the effective mass is anisotropic (direction-dependent), m* is the effective mass in the direction of the electric field.

Matthiessen's rule

Normally, more than one source of scattering is present, for example both impurities and lattice phonons. It is normally a very good approximation to combine their influences using "Matthiessen's Rule" (developed from work by Augustus Matthiessen in 1864):

$${\displaystyle \frac{1}{\mu }=\frac{1}{{\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}.}$$

where µ is the actual mobility, ${\displaystyle {\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}$ is the mobility that the material would have if there was impurity scattering but no other source of scattering, and ${\displaystyle {\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}$ is the mobility that the material would have if there was lattice phonon scattering but no other source of scattering. Other terms may be added for other scattering sources, for example

$${\displaystyle \frac{1}{\mu }=\frac{1}{{\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}+\frac{1}{{\mu }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s}}}+\cdots .}$$

Matthiessen's rule can also be stated in terms of the scattering time:

$${\displaystyle \frac{1}{\tau }=\frac{1}{{\tau }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\tau }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}+\frac{1}{{\tau }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s}}}+\cdots .}$$

where τ is the true average scattering time and τ_impurities is the scattering time if there was impurity scattering but no other source of scattering, etc.

Matthiessen's rule is an approximation and is not universally valid. This rule is not valid if the factors affecting the mobility depend on each other, because individual scattering probabilities cannot be summed unless they are independent of each other. The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability. For example, lattice scattering alters the average electron velocity (in the electric-field direction), which in turn alters the tendency to scatter off impurities. There are more complicated formulas that attempt to take these effects into account.

Temperature dependence of mobility

Typical temperature dependence of mobility

	Si	Ge	GaAs
Electrons	∝T −2.4	∝T −1.7	∝T −1.0
Holes	∝T −2.2	∝T −2.3	∝T −2.1

With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in non-polar semiconductors, such as silicon and germanium, is dominated by acoustic phonon interaction. The resulting mobility is expected to be proportional to T ^−3/2, while the mobility due to optical phonon scattering only is expected to be proportional to T ^−1/2. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.

As $\frac{1}{\tau }\propto ⟨v⟩\mathrm{\Sigma }$, where ${\displaystyle \mathrm{\Sigma }}$ is the scattering cross section for electrons and holes at a scattering center and ${\displaystyle ⟨v⟩}$ is a thermal average (Boltzmann statistics) over all electron or hole velocities in the lower conduction band or upper valence band, temperature dependence of the mobility can be determined. In here, the following definition for the scattering cross section is used: number of particles scattered into solid angle dΩ per unit time divided by number of particles per area per time (incident intensity), which comes from classical mechanics. As Boltzmann statistics are valid for semiconductors ${\displaystyle ⟨v⟩\sim \sqrt{T}}$.

For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section Σ_ph is determined from the square of the average vibrational amplitude of a phonon to be proportional to T. The scattering from charged defects (ionized donors or acceptors) leads to the cross section ${\displaystyle {\mathrm{\Sigma }}_{\text{def}}\propto {⟨v⟩}^{-4}}$. This formula is the scattering cross section for "Rutherford scattering", where a point charge (carrier) moves past another point charge (defect) experiencing Coulomb interaction.

The temperature dependencies of these two scattering mechanism in semiconductors can be determined by combining formulas for τ, Σ and ${\displaystyle ⟨v⟩}$, to be for scattering from acoustic phonons ${\displaystyle {\mu }_{ph}\sim T^{-3/2}}$ and from charged defects ${\displaystyle {\mu }_{\text{def}}\sim T^{3/2}}$.

The effect of ionized impurity scattering, however, decreases with increasing temperature because the average thermal speeds of the carriers are increased. Thus, the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced.

These two effects operate simultaneously on the carriers through Matthiessen's rule. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates, and the actual mobility reaches a maximum at an intermediate temperature.

Disordered Semiconductors

Density of states of a solid possessing a mobility edge, ${\displaystyle E_C}$.

While in crystalline materials electrons can be described by wavefunctions extended over the entire solid, this is not the case in systems with appreciable structural disorder, such as polycrystalline or amorphous semiconductors. Anderson suggested that beyond a critical value of structural disorder, electron states would be localized. Localized states are described as being confined to finite region of real space, normalizable, and not contributing to transport. Extended states are spread over the extent of the material, not normalizable, and contribute to transport. Unlike crystalline semiconductors, mobility generally increases with temperature in disordered semiconductors.

Multiple trapping and release

Mott later developed the concept of a mobility edge. This is an energy ${\displaystyle E_C}$, above which electrons undergo a transition from localized to delocalized states. In this description, termed multiple trapping and release, electrons are only able to travel when in extended states, and are constantly being trapped in, and re-released from, the lower energy localized states. Because the probability of an electron being released from a trap depends on its thermal energy, mobility can be described by an Arrhenius relationship in such a system:

Energy band diagram depicting electron transport under multiple trapping and release.

$${\displaystyle \mu ={\mu }_0\exp \left(-\frac{E_{\text{A}}}{k_{\text{B}}T}\right)}$$

where ${\displaystyle {\mu }_0}$ is a mobility prefactor, ${\displaystyle E_{\text{A}}}$ is activation energy, ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, and ${\displaystyle T}$ is temperature. The activation energy is typically evaluated by measuring mobility as a function of temperature. The Urbach Energy can be used as a proxy for activation energy in some systems.

Variable Range Hopping

At low temperature, or in system with a large degree of structural disorder (such as fully amorphous systems), electrons cannot access delocalized states. In such a system, electrons can only travel by tunnelling for one site to another, in a process called variable range hopping. In the original theory of variable range hopping, as developed by Mott and Davis, the probability ${\displaystyle P_{ij}}$, of an electron hopping from one site ${\displaystyle i}$, to another site ${\displaystyle j}$, depends on their separation in space ${\displaystyle r_{ij}}$, and their separation in energy ${\displaystyle \mathrm{\Delta }{E}_{ij}}$.

$${\displaystyle P_{ij}=P_0\exp \left(-2\alpha r_{ij}-\frac{\mathrm{\Delta }{E}_{ij}}{k_{B}T}\right)}$$

Here ${\displaystyle P_0}$ is a prefactor associated with the phonon frequency in the material, and ${\displaystyle \alpha }$ is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown to be:

$${\displaystyle \mu ={\mu }_0\exp \left(-{\left[\frac{T_0}{T}\right]}^{-1/(d+1)}\right)}$$

where ${\displaystyle {\mu }_0}$ is a mobility prefactor, ${\displaystyle T_0}$ is a parameter (with dimensions of temperature) that quantifies the width of localized states, and ${\displaystyle d}$ is the dimensionality of the system.

Measurement of semiconductor mobility

Hall mobility

Main article: Hall effect

 

Hall effect measurement setup for holes

 

Hall effect measurement setup for electrons

Carrier mobility is most commonly measured using the Hall effect. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement").

Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the x-direction and a magnetic field is applied in the z-direction. The resulting Lorentz force will accelerate the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to the right hand rule and set up an electric field ξ_y. As a result there is a voltage across the sample, which can be measured with a high-impedance voltmeter. This voltage, V_H, is called the Hall voltage. V_H is negative for n-type material and positive for p-type material.

Mathematically, the Lorentz force acting on a charge q is given by

For electrons:

$${\displaystyle {\overrightarrow{F}}_{Hn}=-q({\overrightarrow{v}}_n\times {\overrightarrow{B}}_z)}$$

For holes:

$${\displaystyle {\overrightarrow{F}}_{Hp}=+q({\overrightarrow{v}}_p\times {\overrightarrow{B}}_z)}$$

In steady state this force is balanced by the force set up by the Hall voltage, so that there is no net force on the carriers in the y direction. For electrons,

$${\displaystyle {\overrightarrow{F}}_y=(-q){\overrightarrow{\xi }}_y+(-q)[{\overrightarrow{v}}_n\times {\overrightarrow{B}}_z]=0}$$

$${\displaystyle \Rightarrow -q{\xi }_y+q{v}_x{B}_z=0}$$

$${\displaystyle {\xi }_y=v_x{B}_z}$$

For electrons, the field points in the −y direction, and for holes, it points in the +y direction.

The electron current I is given by ${\displaystyle I=-qn{v}_{x}tW}$. Sub v_x into the expression for ξ_y,

$${\displaystyle {\xi }_y=-\frac{IB}{nqtW}=+\frac{R_{Hn}IB}{tW}}$$

where R_Hn is the Hall coefficient for electron, and is defined as

$${\displaystyle R_{Hn}=-\frac{1}{nq}}$$

Since ${\displaystyle {\xi }_y=\frac{V_H}{W}}$

$${\displaystyle R_{Hn}=-\frac{1}{nq}=\frac{V_{Hn}t}{IB}}$$

Similarly, for holes

$${\displaystyle R_{Hp}=\frac{1}{pq}=\frac{V_{Hp}t}{IB}}$$

From the Hall coefficient, we can obtain the carrier mobility as follows:

$${\displaystyle \begin{array}{rl}{\mu }_n& =\left(-nq\right){\mu }_n\left(-\frac{1}{nq}\right)\\ & =-{\sigma }_n{R}_{Hn}\\ & =-\frac{{\sigma }_n{V}_{Hn}t}{IB}\end{array}}$$

Similarly,

$${\displaystyle {\mu }_p=\frac{{\sigma }_p{V}_{Hp}t}{IB}}$$

Here the value of V_Hp (Hall voltage), t (sample thickness), I (current) and B (magnetic field) can be measured directly, and the conductivities σ_n or σ_p are either known or can be obtained from measuring the resistivity.

Field-effect mobility

See also: MOSFET

Not to be confused with Wien effect.

The mobility can also be measured using a field-effect transistor (FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement").

The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements. (See MOSFET for a description of the different modes or regions of operation.)

Using saturation mode

In this technique, for each fixed gate voltage V_GS, the drain-source voltage V_DS is increased until the current I_D saturates. Next, the square root of this saturated current is plotted against the gate voltage, and the slope m_sat is measured. Then the mobility is:

$${\displaystyle \mu =m_{\text{sat}}^2\frac{2L}{W}\frac{1}{C_i}}$$

where L and W are the length and width of the channel and C_i is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:

$${\displaystyle I_D=\frac{\mu C_i}{2}\frac{W}{L}(V_{GS}-V_{th}{)}^2.}$$

where V_th is the threshold voltage. This approximation ignores the Early effect (channel length modulation), among other things. In practice, this technique may underestimate the true mobility.

Using the linear region

In this technique, the transistor is operated in the linear region (or "ohmic mode"), where V_DS is small and ${\displaystyle I_D\propto V_{GS}}$ with slope m_lin. Then the mobility is:

$${\displaystyle \mu =m_{\text{lin}}\frac{L}{W}\frac{1}{V_{DS}}\frac{1}{C_i}.}$$

This equation comes from the approximate equation for a MOSFET in the linear region:

$${\displaystyle I_D=\mu C_i\frac{W}{L}\left((V_{GS}-V_{th})V_{DS}-\frac{V_{DS}^2}{2}\right)}$$

In practice, this technique may overestimate the true mobility, because if V_DS is not small enough and V_G is not large enough, the MOSFET may not stay in the linear region.

Optical mobility

Electron mobility may be determined from non-contact laser photo-reflectance technique measurements. A series of photo-reflectance measurements are made as the sample is stepped through focus. The electron diffusion length and recombination time are determined by a regressive fit to the data. Then the Einstein relation is used to calculate the mobility.

Terahertz mobility

Electron mobility can be calculated from time-resolved terahertz probe measurement. Femtosecond laser pulses excite the semiconductor and the resulting photoconductivity is measured using a terahertz probe, which detects changes in the terahertz electric field.

Time resolved microwave conductivity (TRMC)

Main article: Time resolved microwave conductivity

A proxy for charge carrier mobility can be evaluated using time-resolved microwave conductivity (TRMC). A pulsed optical laser is used to create electrons and holes in a semiconductor, which are then detected as an increase in photoconductance. With knowledge of the sample absorbance, dimensions, and incident laser fluence, the parameter ${\displaystyle \varphi \mathrm{\Sigma }\mu =\varphi ({\mu }_e+{\mu }_h)}$ can be evaluated, where ${\displaystyle \varphi }$ is the carrier generation yield (between 0 and 1), ${\displaystyle {\mu }_e}$ is the electron mobility and ${\displaystyle {\mu }_h}$ is the hole mobility. ${\displaystyle \varphi \mathrm{\Sigma }\mu }$ has the same dimensions as mobility, but carrier type (electron or hole) is obscured.

Doping concentration dependence in heavily-doped silicon

The charge carriers in semiconductors are electrons and holes. Their numbers are controlled by the concentrations of impurity elements, i.e. doping concentration. Thus doping concentration has great influence on carrier mobility.

While there is considerable scatter in the experimental data, for noncompensated material (no counter doping) for heavily doped substrates (i.e. ${\displaystyle {10}^{18}{\mathrm{c}\mathrm{m}}^{-3}}$ and up), the mobility in silicon is often characterized by the empirical relationship:

$${\displaystyle \mu ={\mu }_o+\frac{{\mu }_1}{1+{\left(\frac{N}{N_{\text{ref}}}\right)}^{\alpha }}}$$

where N is the doping concentration (either N_D or N_A), and N_ref and α are fitting parameters. At room temperature, the above equation becomes:

Majority carriers:

$${\displaystyle {\mu }_n(N_D)=65+\frac{1265}{1+{\left(\frac{N_D}{8.5\times {10}^{16}}\right)}^{0.72}}}$$

$${\displaystyle {\mu }_p(N_A)=48+\frac{447}{1+{\left(\frac{N_A}{6.3\times {10}^{16}}\right)}^{0.76}}}$$

Minority carriers:

$${\displaystyle {\mu }_n(N_A)=232+\frac{1180}{1+{\left(\frac{N_A}{8\times {10}^{16}}\right)}^{0.9}}}$$

$${\displaystyle {\mu }_p(N_D)=130+\frac{370}{1+{\left(\frac{N_D}{8\times {10}^{17}}\right)}^{1.25}}}$$

These equations apply only to silicon, and only under low field.

Gravitational singularity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Gravitational_singularity 

 

Animated simulation of gravitational lensing caused by a Schwarzschild black hole passing in a line-of-sight planar to a background galaxy. Around and at the time of exact alignment (syzygy) extreme lensing of the light is observed.

A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete.

Gravitational singularities are mainly considered in the context of general relativity, where density would become infinite at the center of a black hole without corrections from Quantum Mechanics, and within astrophysics and cosmology as the earliest state of the universe during the Big Bang. Physicists are undecided whether the prediction of singularities means that they actually exist (or existed at the start of the Big Bang), or that current knowledge is insufficient to describe what happens at such extreme densities.

General relativity predicts that any object collapsing beyond a certain point (for stars this is the Schwarzschild radius) would form a black hole, inside which a singularity (covered by an event horizon) would be formed. The Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner. The termination of such a geodesic is considered to be the singularity.

The initial state of the universe, at the beginning of the Big Bang, is also predicted by modern theories to have been a singularity. In this case, the universe did not collapse into a black hole, because currently-known calculations and density limits for gravitational collapse are usually based upon objects of relatively constant size, such as stars, and do not necessarily apply in the same way to rapidly expanding space such as the Big Bang. Neither general relativity nor quantum mechanics can currently describe the earliest moments of the Big Bang, but in general, quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths.

Interpretation

Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the ultraviolet catastrophe, re-normalization, and instability of a hydrogen atom predicted by the Larmor formula.

In classical field theories including special relativity, but not general relativity, one can say that a solution has a singularity at a particular point in spacetime where certain physical properties become ill defined, with spacetime serving as a background field to locate the singularity. A singularity in general relativity, on the other hand, is more complex because spacetime itself becomes ill defined, and the singularity is no longer part of the regular spacetime manifold. In general relativity, a singularity cannot be defined by "where" or "when".

Some theories, such as the theory of loop quantum gravity, suggest that singularities may not exist. This is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. The idea can be stated in the form that due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance.

Types

There are different types of singularities, each with different physical features which have characteristics relevant to the theories from which they originally emerged, such as the different shape of the singularities, conical and curved. They have also been hypothesized to occur without event horizons, structures which delineate one spacetime section from another in which events cannot affect past the horizon; these are called naked.

Conical

A conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the coordinate system is used.

An example of such a conical singularity is a cosmic string and a Schwarzschild black hole.

Curvature

A simple illustration of a non-spinning black hole and its singularity

Solutions to the equations of general relativity or another theory of gravity (such as supergravity) often result in encountering points where the metric blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.

An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, spacetime at the event horizon is regular. The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly smooth. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i.e. ${\displaystyle R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }}$, which is diffeomorphism invariant, is infinite.

While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a Kerr black hole, the singularity occurs on a ring (a circular line), known as a "ring singularity". Such a singularity may also theoretically become a wormhole.

More generally, a spacetime is considered singular if it is geodesically incomplete, meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time (t=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature.

Naked singularity

Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the naked singularity look like a black hole.

Disappearing event horizons exist in the Kerr metric, which is a spinning black hole in a vacuum, if the angular momentum (${\displaystyle J}$) is high enough. Transforming the Kerr metric to Boyer–Lindquist coordinates, it can be shown that the coordinate (which is not the radius) of the event horizon is, ${\displaystyle r_{\pm }=\mu \pm {\left({\mu }^2-a^2\right)}^{1/2}}$, where ${\displaystyle \mu =GM/c^2}$, and ${\displaystyle a=J/Mc}$. In this case, "event horizons disappear" means when the solutions are complex for ${\displaystyle r_{\pm }}$, or ${\displaystyle {\mu }^2<a^2}$. However, this corresponds to a case where ${\displaystyle J}$ exceeds ${\displaystyle G{M}^2/c}$ (or in Planck units, ${\displaystyle J>M^2}$); i.e. the spin exceeds what is normally viewed as the upper limit of its physically possible values.

Similarly, disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole if the charge (${\displaystyle Q}$) is high enough. In this metric, it can be shown that the singularities occur at ${\displaystyle r_{\pm }=\mu \pm {\left({\mu }^2-q^2\right)}^{1/2}}$, where ${\displaystyle \mu =GM/c^2}$, and ${\displaystyle q^2=G{Q}^2/\left(4\pi {ϵ}_0{c}^4\right)}$. Of the three possible cases for the relative values of ${\displaystyle \mu }$ and ${\displaystyle q}$, the case where ${\displaystyle {\mu }^2<q^2}$ causes both ${\displaystyle r_{\pm }}$ to be complex. This means the metric is regular for all positive values of ${\displaystyle r}$, or in other words, the singularity has no event horizon. However, this corresponds to a case where ${\displaystyle Q/\sqrt{4\pi {ϵ}_0}}$ exceeds ${\displaystyle M\sqrt{G}}$ (or in Planck units, ${\displaystyle Q>M}$); i.e. the charge exceeds what is normally viewed as the upper limit of its physically possible values. Also, actual astrophysical black holes are not expected to possess any appreciable charge.

A black hole possessing the lowest ${\displaystyle M}$ value consistent with its ${\displaystyle J}$ and ${\displaystyle Q}$ values and the limits noted above; i.e., one just at the point of losing its event horizon, is termed extremal.

Entropy

Further information: Black hole, Hawking radiation, and Entropy

Before Stephen Hawking came up with the concept of Hawking radiation, the question of black holes having entropy had been avoided. However, this concept demonstrates that black holes radiate energy, which conserves entropy and solves the incompatibility problems with the second law of thermodynamics. Entropy, however, implies heat and therefore temperature. The loss of energy also implies that black holes do not last forever, but rather evaporate or decay slowly. Black hole temperature is inversely related to mass. All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, which means they will gain energy on net by absorbing this radiation. They cannot begin to lose energy on net until the background temperature falls below their own temperature. This will occur at a cosmological redshift of more than one million, rather than the thousand or so since the background radiation formed.

Ultra-high vacuum

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Ultra-high_vacuum 

Ultra-high vacuum (UHV) is the vacuum regime characterised by pressures lower than about 100 nanopascals (1.0×10⁻⁷ Pa; 1.0×10⁻⁹ mbar; 7.5×10⁻¹⁰ Torr). UHV conditions are created by pumping the gas out of a UHV chamber. At these low pressures the mean free path of a gas molecule is greater than approximately 40 km, so the gas is in free molecular flow, and gas molecules will collide with the chamber walls many times before colliding with each other. Almost all molecular interactions therefore take place on various surfaces in the chamber.

UHV conditions are integral to scientific research. Surface science experiments often require a chemically clean sample surface with the absence of any unwanted adsorbates. Surface analysis tools such as X-ray photoelectron spectroscopy and low energy ion scattering require UHV conditions for the transmission of electron or ion beams. For the same reason, beam pipes in particle accelerators such as the Large Hadron Collider are kept at UHV.

Overview

Maintaining UHV conditions requires the use of unusual materials for equipment. Useful concepts for UHV include:

• Sorption of gases
• Kinetic theory of gases
• Gas transport and pumping
• Vacuum pumps and systems
• Vapour pressure

Typically, UHV requires:

• High pumping speed — possibly multiple vacuum pumps in series and/or parallel
• Minimized surface area in the chamber
• High conductance tubing to pumps — short and fat, without obstruction
• Use of low-outgassing materials such as certain stainless steels
• Avoid creating pits of trapped gas behind bolts, welding voids, etc.
• Electropolishing of all metal parts after machining or welding
• Use of low vapor pressure materials (ceramics, glass, metals, teflon if unbaked)
• Baking of the system to remove water or hydrocarbons adsorbed to the walls
• Chilling of chamber walls to cryogenic temperatures during use
• Avoiding all traces of hydrocarbons, including skin oils in a fingerprint — gloves must always be used

Hydrogen and carbon monoxide are the most common background gases in a well-designed, well-baked UHV system. Both Hydrogen and CO diffuse out from the grain boundaries in stainless steel. Helium could diffuse through the steel and glass from the outside air, but this effect is usually negligible due to the low abundance of He in the atmosphere.

Measurement

Pressure

Main article: Pressure measurement

Measurement of high vacuum is done using a nonabsolute gauge that measures a pressure-related property of the vacuum, for example, its thermal conductivity. See, for example, Pacey. These gauges must be calibrated. The gauges capable of measuring the lowest pressures are magnetic gauges based upon the pressure dependence of the current in a spontaneous gas discharge in intersecting electric and magnetic fields.

UHV pressures are measured with an ion gauge, either of the hot filament or inverted magnetron type.

Leak rate

In any vacuum system, some gas will continue to escape into the chamber over time and slowly increase the pressure if it is not pumped out. This leak rate is usually measured in mbar L/s or torr L/s. While some gas release is inevitable, if the leak rate is too high, it can slow down or even prevent the system from reaching low pressure.

There are a variety of possible reasons for an increase in pressure. These include simple air leaks, virtual leaks, and desorption (either from surfaces or volume). A variety of methods for leak detection exist. Large leaks can be found by pressurizing the chamber, and looking for bubbles in soapy water, while tiny leaks can require more sensitive methods, up to using a tracer gas and specialized Helium mass spectrometer.

Outgassing

Main article: Outgassing

Outgassing is a problem for UHV systems. Outgassing can occur from two sources: surfaces and bulk materials. Outgassing from bulk materials is minimized by selection of materials with low vapor pressures (such as glass, stainless steel, and ceramics) for everything inside the system. Materials which are not generally considered absorbent can outgas, including most plastics and some metals. For example, vessels lined with a highly gas-permeable material such as palladium (which is a high-capacity hydrogen sponge) create special outgassing problems.

Outgassing from surfaces is a subtler problem. At extremely low pressures, more gas molecules are adsorbed on the walls than are floating in the chamber, so the total surface area inside a chamber is more important than its volume for reaching UHV. Water is a significant source of outgassing because a thin layer of water vapor rapidly adsorbs to everything whenever the chamber is opened to air. Water evaporates from surfaces too slowly to be fully removed at room temperature, but just fast enough to present a continuous level of background contamination. Removal of water and similar gases generally requires baking the UHV system at 200 to 400 °C (392 to 752 °F) while vacuum pumps are running. During chamber use, the walls of the chamber may be chilled using liquid nitrogen to reduce outgassing further.

Bake-out

Main article: Bake-out

In order to reach low pressures, it is often useful to heat the entire system above 100 °C (212 °F) for many hours (a process known as bake-out) to remove water and other trace gases which adsorb on the surfaces of the chamber. This may also be required upon "cycling" the equipment to atmosphere. This process significantly speeds up the process of outgassing, allowing low pressures to be reached much faster. After baking, to prevent humidity from getting back into the system after it is exposed to atmospheric pressure, a nitrogen gas flow that creates a small positive pressure can be maintained to keep the system dry.

System design

Pumping

There is no single vacuum pump that can operate all the way from atmospheric pressure to ultra-high vacuum. Instead, a series of different pumps is used, according to the appropriate pressure range for each pump. In the first stage, a roughing pump clears most of the gas from the chamber. This is followed by one or more vacuum pumps that operate at low pressures. Pumps commonly used in this second stage to achieve UHV include:

• Turbomolecular pumps (especially compound pumps which incorporate a molecular drag section and/or magnetic bearing types)
• Ion pumps
• Titanium sublimation pumps
• Non-evaporable getter (NEG) pumps
• Cryopumps
• Diffusion pumps, especially when used with a cryogenic trap designed to minimize backstreaming of pump oil into the systems.

Turbo pumps and diffusion pumps rely on supersonic attack upon system molecules by the blades and high speed vapor stream, respectively.

Airlocks

See also: Airlock

To save time, energy, and integrity of the UHV volume an airlock or load-lock vacuum system is often used. The airlock volume has one door or valve, such as a gate valve or UHV angle valve, facing the UHV side of the volume, and another door against atmospheric pressure through which samples or workpieces are initially introduced. After sample introduction and assuring that the door against atmosphere is closed, the airlock volume is typically pumped down to a medium-high vacuum. In some cases the workpiece itself is baked out or otherwise pre-cleaned under this medium-high vacuum. The gateway to the UHV chamber is then opened, the workpiece transferred to the UHV by robotic means or by other contrivance if necessary, and the UHV valve re-closed. While the initial workpiece is being processed under UHV, a subsequent sample can be introduced into the airlock volume, pre-cleaned, and so-on and so-forth, saving much time. Although a "puff" of gas is generally released into the UHV system when the valve to the airlock volume is opened, the UHV system pumps can generally snatch this gas away before it has time to adsorb onto the UHV surfaces. In a system well designed with suitable airlocks, the UHV components seldom need bakeout and the UHV may improve over time even as workpieces are introduced and removed.

Seals

See also: Vacuum flange

Metal seals, with knife edges on both sides cutting into a soft, copper gasket are employed. This metal-to-metal seal can maintain pressures down to 100 pPa (7.5×10⁻¹³ Torr). Although generally considered single use, the skilled operator can obtain several uses through the use of feeler gauges of decreasing size with each iteration, as long as the knife edges are in perfect condition. For SRF cavities, indium seals are more commonly used in sealing two flat surfaces together using clamps to bring the surfaces together. The clamps need to be tightened slowly to ensure the indium seals compress uniformly all around.

Material limitations

Many common materials are used sparingly if at all due to high vapor pressure, high adsorptivity or absorptivity resulting in subsequent troublesome outgassing, or high permeability in the face of differential pressure (i.e.: "through-gassing"):

• The majority of organic compounds cannot be used:
  • Plastics, other than PTFE and PEEK: plastics in other uses are replaced with ceramics or metals. Limited use of fluoroelastomers (such as Viton) and perfluoroelastomers (such as Kalrez) as gasket materials can be considered if metal gaskets are inconvenient, though these polymers can be expensive. Although through-gassing of elastomerics can not be avoided, experiments have shown that slow out-gassing of water vapor is, initially at least, the more important limitation. This effect can be minimized by pre-baking under medium vacuum. When selecting O-rings, permeation rate and permeation coefficients need to be considered. For example the penetration rate of nitrogen in Viton seals is 100 times lower than the penetration of nitrogen in silicon seals, which impacts the ultimate vacuum that can be achieved. 
  • Glues: special glues for high vacuum must be used, generally epoxies with a high mineral filler content. Among the most popular of these include asbestos in the formulation. This allows for an epoxy with good initial properties and able to retain reasonable performance across multiple bake-outs.
• Some steels: due to oxidization of carbon steel, which greatly increases adsorption area, only stainless steel is used. Particularly, non-leaded and low-sulfur austenitic grades such as 304 and 316 are preferred. These steels include at least 18% chromium and 8% nickel. Variants of stainless steel include low-carbon grades (such as 304L and 316L), and grades with additives such as niobium and molybdenum to reduce the formation of chromium carbide (which provides no corrosion resistance). Common designations include 316L (low carbon), and 316LN (low carbon with nitrogen), which can boast a significantly lower magnetic permeability with special welding techniques making them preferable for particle accelerator applications. Chromium carbide precipitation at the grain boundaries can render a stainless steel less resistant to oxidation.
• Lead: Soldering is performed using lead-free solder. Occasionally pure lead is used as a gasket material between flat surfaces in lieu of a copper/knife edge system.
• Indium: Indium is sometimes used as a deformable gasket material for vacuum seals, especially in cryogenic apparatus, but its low melting point prevents use in baked systems. In a more esoteric application, the low melting point of Indium is taken advantage of as a renewable seal in high vacuum valves. These valves are used several times, generally with the aid of a torque wrench set to increasing torque with each iteration. When the indium seal is exhausted, it is melted and reforms itself and thus is ready for another round of uses.
• Zinc, cadmium: High vapor pressures during system bake-out virtually preclude their use.
• Aluminum: Although aluminum itself has a vapor pressure which makes it unsuitable for use in UHV systems, the same oxides which protect aluminum against corrosion improve its characteristics under UHV. Although initial experiments with aluminum suggested milling under mineral oil to maintain a thin, consistent layer of oxide, it has become increasingly accepted that aluminum is a suitable UHV material without special preparation. Paradoxically, aluminum oxide, especially when embedded as particles in stainless steel as for example from sanding in an attempt to reduce the surface area of the steel, is considered a problematic contaminant.
• Cleaning is very important for UHV. Common cleaning procedures include degreasing with detergents, organic solvents, or chlorinated hydrocarbons. Electropolishing is often used to reduce the surface area from which adsorbed gases can be emitted. Etching of stainless steel using hydrofluoric and nitric acid forms a chromium rich surface, followed by a nitric acid passivation step, which forms a chromium oxide rich surface. This surface retards the diffusion of hydrogen into the chamber.

Technical limitations:

• Screws: Threads have a high surface area and tend to "trap" gases, and therefore, are avoided. Blind holes are especially avoided, due to the trapped gas at the base of the screw and slow venting through the threads, which is commonly known as a "virtual leak". This can be mitigated by designing components to include through-holes for all threaded connections, or by using vented screws (which have a hole drilled through their central axis or a notch along the threads). Vented Screws allow trapped gases to flow freely from the base of the screw, eliminating virtual leaks and speeding up the pump-down process.
• Welding: Processes such as gas metal arc welding and shielded metal arc welding cannot be used, due to the deposition of impure material and potential introduction of voids or porosity. Gas tungsten arc welding (with an appropriate heat profile and properly selected filler material) is necessary. Other clean processes, such as electron beam welding or laser beam welding, are also acceptable; however, those that involve potential slag inclusions (such as submerged arc welding and flux-cored arc welding) are obviously not. To avoid trapping gas or high vapor pressure molecules, welds must fully penetrate the joint or be made from the interior surface, otherwise a virtual leak might appear.

UHV manipulator

A UHV manipulator allows an object which is inside a vacuum chamber and under vacuum to be mechanically positioned. It may provide rotary motion, linear motion, or a combination of both. The most complex devices give motion in three axes and rotations around two of those axes. To generate the mechanical movement inside the chamber, three basic mechanisms are commonly employed: a mechanical coupling through the vacuum wall (using a vacuum-tight seal around the coupling: a welded metal bellows for example), a magnetic coupling that transfers motion from air-side to vacuum-side: or a sliding seal using special greases of very low vapor pressure or ferromagnetic fluid. Such special greases can exceed USD $400 per kilogram. Various forms of motion control are available for manipulators, such as knobs, handwheels, motors, stepping motors, piezoelectric motors, and pneumatics. The use of motors in a vacuum environment often requires special design or other special considerations, as the convective cooling taken for granted under atmospheric conditions is not available in a UHV environment.

The manipulator or sample holder may include features that allow additional control and testing of a sample, such as the ability to apply heat, cooling, voltage, or a magnetic field. Sample heating can be accomplished by electron bombardment or thermal radiation. For electron bombardment, the sample holder is equipped with a filament which emits electrons when biased at a high negative potential. The impact of the electrons bombarding the sample at high energy causes it to heat. For thermal radiation, a filament is mounted close to the sample and resistively heated to high temperature. The infrared energy from the filament heats the sample.

Typical uses

Ultra-high vacuum is necessary for many surface analytic techniques such as:

• X-ray photoelectron spectroscopy (XPS)
• Auger electron spectroscopy (AES)
• Secondary ion mass spectrometry (SIMS)
• Thermal desorption spectroscopy (TPD)
• Thin film growth and preparation techniques with stringent requirements for purity, such as molecular beam epitaxy (MBE), UHV chemical vapor deposition (CVD), atomic layer deposition (ALD) and UHV pulsed laser deposition (PLD)
• Angle resolved photoemission spectroscopy (ARPES)
• Field emission microscopy and Field ion microscopy
• Atom Probe Tomography (APT)

UHV is necessary for these applications to reduce surface contamination, by reducing the number of molecules reaching the sample over a given time period. At 0.1 millipascals (7.5×10⁻⁷ Torr), it only takes 1 second to cover a surface with a contaminant, so much lower pressures are needed for long experiments.

UHV is also required for:

• Particle accelerators The Large Hadron Collider (LHC) has three UH vacuum systems. The lowest pressure is found in the pipes the proton beam speeds through near the interaction (collision) points. Here helium cooling pipes also act as cryopumps. The maximum allowable pressure is 1×10⁻⁶ pascals (1.0×10⁻⁸ mbar)
• Gravitational wave detectors such as LIGO, VIRGO, GEO 600, and TAMA 300. The LIGO experimental apparatus is housed in a 10,000 cubic metres (350,000 cu ft) vacuum chamber at 1×10⁻⁷ pascals (1.0×10⁻⁹ mbar) in order to eliminate temperature fluctuations and sound waves which would jostle the mirrors far too much for gravitational waves to be sensed.
• Atomic physics experiments which use cold atoms, such as ion trapping or making Bose–Einstein condensates.

While not compulsory, it can prove beneficial in applications such as:

• Molecular beam epitaxy, E-beam evaporation, sputtering and other deposition techniques.
• Atomic force microscopy. High vacuum enables high Q factors on the cantilever oscillation.
• Scanning tunneling microscopy. High vacuum reduces oxidation and contamination, hence enables imaging and the achievement of atomic resolution on clean metal and semiconductor surfaces, e.g. imaging the surface reconstruction of the unoxidized silicon surface.
• Electron-beam lithography