Electron mobility

In solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor when pushed or 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, <math> v_d</math>. Then the electron mobility μ is defined as

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

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

The electron mobility is defined by the equation:

where:

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

The hole mobility is defined by a similar equation:

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:

where:

a is the acceleration between collisions.
F is the electric force exerted by the electric field, and
<math>m_e^* </math> is the effective mass of an electron.

Since the force on the electron is −eE:

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

<math display="block">v_d = a \tau_c = -\frac{e\tau_c}{m_e^*}E,</math> where <math>\tau_c</math> 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

<math display="block">v_d = -\mu_e E,</math> where <math>\mu_e = \frac{e\tau_c}{m_e^*}</math>

Similarly, for holes we have

<math display="block">v_d = \mu_h E,</math> where <math>\mu_h = \frac{e\tau_c}{m_h^*}</math>

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 <math>-e v_d</math>, so that the total current density due to electrons is given by:

Using the expression for <math>v_d</math> gives

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

where p is the hole concentration and <math>\mu_h</math> the hole mobility.

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

Relation to conductivity

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

Now Ohm's law can be written in the form

<math display="block">J=\sigma E</math>

where <math>\sigma</math> is defined as the conductivity. Therefore we can write down:

<math display="block">\sigma=en\mu_e+ep\mu_h</math>

which can be factorised to

<math display="block">\sigma=e(n\mu_e+p\mu_h)</math>

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:

<math display="block">F=-D_\text{e}\nabla n</math>

where:

F is flux.
De is the diffusion coefficient or diffusivity
<math>\nabla n</math> is the concentration gradient of electrons

The diffusion coefficient for a charge carrier is related to its mobility by the Einstein relation. For a classical system (e.g. Boltzmann gas), it reads:

<math display="block">D_\text{e} = \frac{\mu_\text{e} k_\mathrm{B} T}{e}</math>

where:

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

For a metal, described by a Fermi gas (Fermi liquid), the 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:

where:

EF is the Fermi energy

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.

{| class="wikitable sortable"

|+ List of highest measured mobilities

!rowspan="2"| Material

!colspan="2"| Mobility (cm2/(V⋅s))

! Electron

! Hole

|AlGaAs/GaAs heterostructures

|35,000,000

|Freestanding graphene

|200,000

|Cubic boron arsenide (c-BAs)

|1,600

|>1000

|43

|Amorphous silicon

|~1

Electric field dependence and velocity saturation

At low fields, the drift velocity vd 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 vsat. For example, the value of vsat 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:

<math display="block">\frac{m^* v_\text{emit}^2}{2} \approx \hbar \omega_\text{phonon (opt.)}</math>

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

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.

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.

<math display="block">\mu = \frac{q}{m^*}\overline{\tau}</math>

where q is the elementary charge, m* is the carrier effective mass, and 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):

<math display="block">\frac{1}{\mu} = \frac{1}{\mu_{\rm impurities + \frac{1}{\mu_{\rm lattice.</math>

where μ is the actual mobility, <math>\mu_{\rm impurities}</math> is the mobility that the material would have if there was impurity scattering but no other source of scattering, and <math>\mu_{\rm lattice}</math> 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

<math display="block">\frac{1}{\mu} = \frac{1}{\mu_{\rm impurities + \frac{1}{\mu_{\rm lattice + \frac{1}{\mu_{\rm defects + \cdots.</math>

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

<math display="block">\frac{1}{\tau} = \frac{1}{\tau_{\rm impurities + \frac{1}{\tau_{\rm lattice + \frac{1}{\tau_{\rm defects + \cdots .</math>

where τ is the true average scattering time and τimpurities 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.

Temperature dependence of mobility

{| class="wikitable" style="float:right; text-align:center;"

|+ 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. 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 <math>E_{C}</math>, 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:

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

<math display="block">\mu=\mu_{0}\exp\left(-\frac{E_\text{A{k_\text{B}T}\right)</math>

where <math>\mu_{0}</math> is a mobility prefactor, <math>E_\text{A}</math> is activation energy, <math>k_\text{B}</math> is the Boltzmann constant, and <math>T</math> 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 <math>P_{ij}</math>, of an electron hopping from one site <math>i</math>, to another site <math>j</math>, depends on their separation in space <math>r_{ij}</math>, and their separation in energy <math>\Delta E_{ij}</math>.

<math display="block">P_{ij} = P_{0} \exp\left(-2\alpha r_{ij} - \frac{\Delta E_{ij{k_\text{B} T}\right)</math>

Here <math>P_{0}</math> is a prefactor associated with the phonon frequency in the material, and <math>\alpha</math> is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown (See MOSFET for a description of the different modes or regions of operation.)

Using saturation mode

In this technique,

Using the linear region

In this technique,

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)

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 <math>\phi\Sigma\mu=\phi(\mu_{e}+\mu_{h})</math> can be evaluated, where <math>\phi</math> is the carrier generation yield (between 0 and 1), <math>\mu_{e}</math> is the electron mobility and <math>\mu_{h}</math> is the hole mobility. <math>\phi\Sigma\mu</math> 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. <math>10^{18}\mathrm{cm}^{-3} </math> and up), the mobility in silicon is often characterized by the empirical relationship:

<math display="block">\mu = \mu_o + \frac{\mu_1}{1 + \left(\frac{N}{N_\text{ref\right)^\alpha}</math>

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

Majority carriers:

<math display="block">\begin{align}

\mu_n(N_D) &= 65 + \frac{1265}{1+ \left(\frac{N_D}{8.5\times10^{16\right)^{0.72 \\[1ex]

\mu_p(N_A) &= 48 + \frac{447}{1+ \left(\frac{N_A}{6.3\times10^{16\right)^{0.76

\end{align}</math>

Minority carriers:

<math display="block">\begin{align}

\mu_n(N_A) &= 232 + \frac{1180}{1+ \left(\frac{N_A}{8\times10^{16\right)^{0.9 \\[1ex]

\mu_p(N_D) &= 130 + \frac{370}{1+ \left(\frac{N_D}{8\times10^{17\right)^{1.25

\end{align}</math>

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

References

External links

semiconductor glossary entry for electron mobility
Resistivity and Mobility Calculator from the BYU Cleanroom
Online lecture- Mobility from an atomistic point of view

Introduction

Drift velocity in an electric field

Definition and units

Derivation

Relation to current density

Relation to conductivity

Relation to electron diffusion

Examples

Electric field dependence and velocity saturation

Ionized impurity scattering

Inelastic scattering

Matthiessen's rule

Temperature dependence of mobility

Multiple trapping and release

Variable Range Hopping

Using saturation mode

Using the linear region

Terahertz mobility

Time resolved microwave conductivity (TRMC)

Doping concentration dependence in heavily-doped silicon

See also

References

External links