Insolid-state physics, theelectron mobility characterises how quickly anelectron can move through ametal orsemiconductor when pushed or pulled by anelectric field. There is an analogous quantity forholes, calledhole mobility. The termcarrier mobility refers in general to both electron and hole mobility.

Electron and hole mobility arespecial cases ofelectrical mobility of charged particles in a fluid under an applied electric field.

When an electric fieldE is applied across a piece of material, the electrons respond by moving with an average velocity called thedrift velocity,${\displaystyle v_d}$. Then the electron mobilityμ is defined as$${\displaystyle v_d=\mu E.}$$

Electron mobility is almost always specified in units ofcm²/(V⋅s). This is different from theSI 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 oftransistors 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 whenvelocity saturation occurs. It can be determined by theHall effect, or inferred from transistor behavior.

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Without any applied electric field, in a solid,electrons andholesmove 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 (calledballistic transport). However, in a solid, the electron repeatedly scatters offcrystal 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 thedrift 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 themean free path), or for a very short time (as short as themean free time). In these cases, drift velocity and mobility are not meaningful.

The electron mobility is defined by the equation:$${\displaystyle v_d={\mu }_{e}E.}$$where:

• E is themagnitude of theelectric field applied to a material,
• v_d is themagnitude of the electron drift velocity (in other words, the electron driftspeed) 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 ism/s, and the SI unit of electric field isV/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.

Starting withNewton'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 theeffective 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 themean 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.

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

We have previously derived the relationship between electron mobility and current density$${\displaystyle J=J_e+J_h=(en{\mu }_e+ep{\mu }_h)E}$$NowOhm'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$${\displaystyle \sigma =e(n{\mu }_e+p{\mu }_h)}$$

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 byFick's law:$${\displaystyle F=-D_{\text{e}}\mathrm{\nabla }n}$$where:

• F is flux.
• D_e is thediffusion 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 theEinstein relation. For a classical system (e.g. Boltzmann gas), it reads:$${\displaystyle D_{\text{e}}=\frac{{\mu }_{\text{e}}{k}_{\mathrm{B}}T}{e}}$$where:

• k_B is theBoltzmann constant
• T is theabsolute temperature
• e is the electric charge of an electron

For a metal, described by a Fermi gas (Fermi liquid), quantum version of the Einstein relation should be used. Typically, temperature is much smaller than the Fermi energy, in this case one should use the following formula:$${\displaystyle D_{\text{e}}=\frac{{\mu }_{\text{e}}{E}_F}{e}}$$where:

• E_F is the Fermi energy

Typical electron mobility at room temperature (300 K) in metals likegold,copper andsilver is 30–50 cm²/(V⋅s). Carrier mobility in semiconductors is doping dependent. Insilicon (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.^[1]

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),^[2]carbon nanotubes (100,000 cm²/(V⋅s) at room temperature)^[3] and freestandinggraphene (200,000 cm²/(V⋅s) at low temperature).^[4]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.^[5]

At low fields, the drift velocityv_d is proportional to the electric fieldE, so mobilityμ is constant. This value ofμ is called thelow-field mobility.

As the electric field is increased, however, the carrier velocity increases sublinearly and asymptotically towards a maximum possible value, called thesaturation velocityv_sat. For example, the value ofv_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 ofdoping 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 calledoptical phonon scattering. At high fields, carriers are accelerated enough to gain sufficientkinetic 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:$${\displaystyle \frac{m^{\ast }{v}_{\text{emit}}^2}{2}\approx \hslash {\omega }_{\text{phonon (opt.)}}}$$whereω_phonon(opt.) is the optical-phonon angular frequency and m* the carrier effective mass in the direction of the electric field. The value ofE_{phonon (opt.)} is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half ofv_emit, because the electron starts at zero velocity and accelerates up tov_emit in each cycle.^[13] (This is a somewhat oversimplified description.^[13])

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

Recall that by definition, mobility is dependent on the drift velocity. The main factor determining drift velocity (other thaneffective mass) isscattering time, i.e. how long the carrier isballistically 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, anddefect scattering.^[14]

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 statek to k', while emitting or absorbing a phonon of wave vectorq. 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.^[15]

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 asionized 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 themean 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.^[15]

At any temperature aboveabsolute zero, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termedphonons. 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 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.^[15]

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.^[15]

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.^{[15][16][17][18][19]}

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.^[15]

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.^[15][16][17]

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:^[20]$${\displaystyle \mu =\frac{q}{m^{\ast }}\overline{\tau }}$$whereq is theelementary charge,m* is the carriereffective mass, andτ 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.

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 byAugustus Matthiessen in 1864):

$${\displaystyle \frac{1}{\mu }=\frac{1}{{\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}.}$$whereμ 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:$${\displaystyle \frac{1}{\tau }=\frac{1}{{\tau }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\tau }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}+\frac{1}{{\tau }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s}}}+\cdots .}$$whereτ 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.^[19] The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability.^[15][16][18] 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.^[21]

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 innon-polar semiconductors, such as silicon and germanium, is dominated byacoustic phonon interaction. The resulting mobility is expected to be proportional toT ^−3/2, while the mobility due to optical phonon scattering only is expected to be proportional toT ^−1/2. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.^[22]

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 toT. 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${\displaystyle {\mu }_{\text{def}}\sim T^{3/2}}$.^[16][18]

The effect of ionized impurity scattering, however,decreases with increasing temperature because the average thermal speeds of the carriers are increased.^[14] 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.

While in crystalline materials electrons can be described by wavefunctions extended over the entire solid,^[23] this is not the case in systems with appreciable structural disorder, such aspolycrystalline oramorphous semiconductors.Anderson suggested that beyond a critical value of structural disorder,^[24] electron states would belocalized. 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.

Mott later developed^[25] 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, termedmultiple 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 anArrhenius relationship in such a system:

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

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. TheUrbach Energy can be used as a proxy for activation energy in some systems.^[26]

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 bytunnelling for one site to another, in a process calledvariable range hopping. In the original theory of variable range hopping, as developed by Mott and Davis,^[27] 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,^[28] and${\displaystyle \alpha }$ is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown^[27] 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.

Carrier mobility is most commonly measured using theHall 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 thex-direction and amagnetic field is applied in thez-direction. The resulting Lorentz force will accelerate the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to theright hand rule and set up an electric fieldξ_y. As a result there is a voltage across the sample, which can be measured with ahigh-impedance voltmeter. This voltage,V_H, is called theHall voltage.V_H is negative forn-type material and positive forp-type material.

Mathematically, theLorentz force acting on a chargeq is given by

For electrons:$${\displaystyle {\mathbf{F}}_{Hn}=-q({\mathbf{v}}_n\times {\mathbf{B}}_z)}$$

For holes:$${\displaystyle {\mathbf{F}}_{Hp}=+q({\mathbf{v}}_p\times {\mathbf{B}}_z)}$$

In steady state this force is balanced by the force set up by the Hall voltage, so that there is nonet force on the carriers in they direction. For electrons,

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

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

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

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

Theelectron currentI is given by${\displaystyle I=-qn{v}_{x}tW}$. Subv_x into the expression forξ_y,

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

whereR_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:

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

Here the value ofV_Hp (Hall voltage),t (sample thickness),I (current) andB (magnetic field) can be measured directly, and the conductivitiesσ_n orσ_p are either known or can be obtained from measuring the resistivity.

The mobility can also be measured using afield-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.^[29] (SeeMOSFET for a description of the different modes or regions of operation.)

In this technique,^[29] 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 slopem_sat is measured. Then the mobility is:$${\displaystyle \mu =m_{\text{sat}}^2\frac{2L}{W}\frac{1}{C_i}}$$whereL andW are the length and width of the channel andC_i is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:$${\displaystyle I_D=\frac{\mu C_i}{2}\frac{W}{L}(V_{GS}-V_{th}{)}^2.}$$whereV_th is the threshold voltage. This approximation ignores theEarly effect (channel length modulation), among other things. In practice, this technique may underestimate the true mobility.^[30]

In this technique,^[29] the transistor is operated in the linear region (or "ohmic mode"), where V_DS is small and${\displaystyle I_D\propto V_{GS}}$ with slopem_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.^[30]

Electron mobility may be determined from non-contact laserphoto-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.^[31][32]

Electron mobility can be calculated from time-resolvedterahertz probe measurement.^[33][34]Femtosecond laser pulses excite the semiconductor and the resultingphotoconductivity is measured using a terahertz probe, which detects changes in the terahertz electric field.^[35]

A proxy for charge carrier mobility can be evaluated using time-resolved microwave conductivity (TRMC).^[36] 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.

Thecharge 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 theexperimental 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 theempirical relationship:^[37]$${\displaystyle \mu ={\mu }_o+\frac{{\mu }_1}{1+{\left(\frac{N}{N_{\text{ref}}}\right)}^{\alpha }}}$$whereN is the doping concentration (eitherN_D orN_A), andN_ref and α are fitting parameters. Atroom temperature, the above equation becomes:

Majority carriers:^[38]$${\displaystyle {\mu }_n(N_D)=65+\frac{1265}{1+{\left(\frac{N_D}{8.5\times {10}^{16}}\right)}^{0.72}}}$$$${\displaystyle {\mu }_p(N_A)=48+\frac{447}{1+{\left(\frac{N_A}{6.3\times {10}^{16}}\right)}^{0.76}}}$$

Minority carriers:^[39]$${\displaystyle {\mu }_n(N_A)=232+\frac{1180}{1+{\left(\frac{N_A}{8\times {10}^{16}}\right)}^{0.9}}}$$$${\displaystyle {\mu }_p(N_D)=130+\frac{370}{1+{\left(\frac{N_D}{8\times {10}^{17}}\right)}^{1.25}}}$$

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

• Speed of electricity

• semiconductor glossary entry for electron mobilityArchived 2009-01-04 at theWayback Machine
• Resistivity and Mobility Calculator from the BYU Cleanroom
• Online lecture-Mobility from an atomistic point of view

• Physical quantities
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• Semiconductors
• Electric and magnetic fields in matter
• MOSFETs

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