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.
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. \mu = \frac{q}{m^*}\overline{\tau} 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): \frac{1}{\mu} = \frac{1}{\mu_{\rm impurities}} + \frac{1}{\mu_{\rm lattice}}. where
μ is the actual mobility, \mu_{\rm impurities} is the mobility that the material would have if there was impurity scattering but no other source of scattering, and \mu_{\rm lattice} 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 \frac{1}{\mu} = \frac{1}{\mu_{\rm impurities}} + \frac{1}{\mu_{\rm lattice}} + \frac{1}{\mu_{\rm defects}} + \cdots. Matthiessen's rule can also be stated in terms of the scattering time: \frac{1}{\tau} = \frac{1}{\tau_{\rm impurities}} + \frac{1}{\tau_{\rm lattice}} + \frac{1}{\tau_{\rm defects}} + \cdots . 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 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 \left \langle v\right \rangle\Sigma , where \Sigma is the scattering cross section for electrons and holes at a scattering center and \left \langle v\right \rangle 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 \left \langle v\right \rangle\sim\sqrt{T}. For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section Σph 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 {\Sigma }_\text{def}\propto {\left \langle v\right \rangle}^{-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 \left \langle v\right \rangle, to be for scattering from acoustic phonons \mu_\text{ph} \sim T^{-3/2} and from charged defects {\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 ==