Editing Electron mobility (section)

==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 (solid-state physics)|effective mass]]) is [[scattering]] time, i.e. how long the carrier is [[ballistic transport|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 [[crystallographic defect|defect]] scattering.<ref name=Singh>{{cite book|author=Singh|title=Electronic Devices And Integrated Circuits|url=https://books.google.com/books?id=2aqtlybkFE0C&pg=PA77|access-date=1 March 2011|publisher=PHI Learning Pvt. Ltd.| isbn=978-81-203-3192-1|pages=77–|year=2008}}</ref>

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.<ref name="sct">Ferry, David K. Semiconductor transport. London: Taylor & Francis, 2000. {{ISBN|0-7484-0865-7}} (hbk.), {{ISBN|0-7484-0866-5}} (pbk.)</ref>

===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.<ref name=sct />

===Lattice (phonon) scattering===
At any temperature above [[absolute zero]], the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed [[phonon]]s. 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.<ref name=sct />

===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.<ref name=sct />

===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.<ref name=sct /><ref name="ssp">Ibach, Harald.; Luth, Hans. Solid-state physics : an introduction to principles of materials science / Harald Ibach, Hans Luth. New York: Springer, 2009. -(Advanced texts in physics) {{ISBN|978-3-540-93803-3}}</ref><ref name="bulusu">{{cite journal | doi = 10.1016/j.spmi.2008.02.008 | volume=44 | title=Review of electronic transport models for thermoelectric materials | year=2008 | journal=Superlattices and Microstructures | pages=1–36 | last1 = Bulusu | first1 = A. | issue=1 | bibcode=2008SuMi...44....1B}}.</ref><ref name="pallab">Bhattacharya, Pallab. Semiconductor optoelectronic devices / Pallab Bhattacharya. Upper Saddle River (NJ): Prentice-Hall, 1997. {{ISBN|0-13-495656-7}} (nid.)</ref><ref name="Takeda">Y. Takeda and T.P. Pearsall, "Failure of Mattheissen's Rule in the Calculation of Carrier Mobility and Alloy Scattering Effects in Ga0.47In0.53As", Electronics Lett. 17, 573-574 (1981).</ref>

===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.<ref name=sct />

===Electron–electron scattering===
Due to the Pauli exclusion principle, electrons can be considered as non-interacting if their density does not exceed the value 10<sup>16</sup>~10<sup>17</sup> cm<sup>−3</sup> or electric field value 10<sup>3</sup> 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.<ref name=sct /><ref name=ssp /><ref name=bulusu />

===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:<ref>{{cite book| author1=Peter Y. Yu|author2=Manuel Cardona|title=Fundamentals of Semiconductors: Physics and Materials Properties| url=https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA205+|access-date=1 March 2011|date=30 May 2010|publisher=Springer| isbn=978-3-642-00709-5|pages=205–}}</ref>
<math display="block">\mu = \frac{q}{m^*}\overline{\tau}</math>
where ''q'' is the [[elementary charge]], ''m''* is the carrier [[effective mass (solid-state physics)|effective mass]], and {{overline|''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):

<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 τ<sub>impurities</sub> 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.<ref name=Takeda/>  The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability.<ref name=sct /><ref name=ssp /><ref name=pallab /> 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.<ref>{{cite book|author1=Antonio Luque|author2=Steven Hegedus|title=Handbook of photovoltaic science and engineering|url=https://books.google.com/books?id=u-bCMhl_JjQC|access-date=2 March 2011|date=9 June 2003|publisher=John Wiley and Sons|isbn=978-0-471-49196-5|page=79, eq. 3.58}} [http://www.knovel.com/web/portal/browse/display?_EXT_KNOVEL_DISPLAY_bookid=1081 weblink (subscription only)]</ref>

===Temperature dependence of mobility===

{| class="wikitable" style="float:right; text-align:center;"
|+ Typical temperature dependence of mobility<ref name=BVZ>[http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_7.htm Chapter 2: Semiconductor Fundamentals] {{Webarchive|url=https://web.archive.org/web/20090121102148/http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_7.htm |date=2009-01-21 }}. Online textbook by B. Van Zeghbroeck]</ref>
! 
! Si
!Ge
!GaAs
|-
! Electrons
| ∝T <sup>−2.4</sup>
| ∝T <sup>−1.7</sup>
| ∝T <sup>−1.0</sup>
|-
! Holes
| ∝T <sup>−2.2</sup>
| ∝T <sup>−2.3</sup>
| ∝T <sup>−2.1</sup>
|}

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 [[Chemical polarity|non-polar]] semiconductors, such as silicon and germanium, is dominated by [[Phonon|acoustic phonon]] interaction. The resulting mobility is expected to be proportional to ''T''&nbsp;<sup>−3/2</sup>, while the mobility due to optical phonon scattering only is expected to be proportional to ''T''&nbsp;<sup>−1/2</sup>. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.<ref name=BVZ/>

As <math display="inline">\frac{1}{\tau }\propto \left \langle v\right \rangle\Sigma </math>, where <math>\Sigma </math> is the scattering cross section for electrons and holes at a scattering center and <math>\left \langle v\right \rangle</math> 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 <math>\left \langle v\right \rangle\sim\sqrt{T}</math>.

For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section Σ<sub>ph</sub> 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 <math>{\Sigma }_\text{def}\propto {\left \langle v\right \rangle}^{-4}</math>. 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 <math>\left \langle v\right \rangle</math>, to be for scattering from acoustic phonons <math>{\mu }_{ph}\sim T^{-3/2}</math> and from charged defects <math>{\mu }_\text{def}\sim T^{3/2}</math>.<ref name=ssp /><ref name=pallab />

The effect of ionized impurity scattering, however, ''decreases'' with increasing temperature because the average thermal speeds of the carriers are increased.<ref name=Singh/> 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.