Semiconductor carrier mobility

Mobility is a measure of how easily a carrier moves in a particular material. It is often represented as μ, and is defined as the ratio of carrier velocity in the field direction (drift velocity, v_d) to the magnitude of the electric field, ζ:

$\mu = |\frac{V_d}{\zeta}|$

Mobility is normally expressed in centimeters squared per volt-second ((cm²)/(V∙s)).
In semiconductor, carrier mobility applies to both electrons and holes. The electron and hole mobilizes are dependent on the impurity concentrations of donors and acceptors, on temperature, and on whether the carriers are minority or majority carriers. (Majority carriers are electrons in n-type material and holes in p-type materials.) Besides, drift velocity and thus carrier mobility is influenced by carrier scattering and the electric field.

Doping concentration dependence

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), the mobility in silicon is often characterized by the empirical formula:

$\mu = \mu_o + \frac{\mu_1}{1+(\frac{N}{N_\text{ref}})^\alpha}$

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 equat
ion becomes: Majority carriers:

$\mu_n(N_D) = 65 + \frac{1265}{1+(\frac{N_D}{8.5\times10^{16}})^{0.72}}$

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

Minority carriers :

$\mu_n(N_A) = 232 + \frac{1180}{1+(\frac{N_A}{8\times10^{16}})^{0.9}}$

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

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

Relationship between carrier scattering and mobility

Recall that by definition, mobility is dependent on the drift velocity. The carrier drift velocity is influenced by scattering events, i.e., change in
direction and/or energy of a carrier by collision with a particle, such as an ionized impurity atom or a phonon.
Ionized impurity scattering： The impurities (donors and acceptors) 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 will the mean free time between collisions, a
nd the smaller the mobility.
Lattice (phonon) scattering： At any temperature 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 collide with an electron (or hole) and sc
atter it. The energy distribution of phonons and the concentration of phonons at a given energy depend on the amplitude of the pressure wave, and thus on temperature. With increasing temperature, the stronger lattice vibrations cause an increase in the concentration of phonons and thus increased scattering or reduced mobility.
Influence of scattering on carrier mobility:
Consider the force on an electron due to the electric field:

$F = -q\zeta = m_e^*\frac{dv}{dt}$

Rearranging,

$dv = -\frac{q\zeta}{m_e^*}dt$

Integrating both sides,

$v(t) - v(t_o) = -\frac{q\zeta}{m_e^*}(t-t_o)$

where t_o is the time of the previous
collision (scattering event). This gives us the velocity reached by a particular electron just before its next collision at time t. If we consider a large number of collisions and take the average,

$\langle v(t) - v(t_o)\rangle = \frac{q\zeta}{m_e^*}\langle t-t_o\rangle$

We define the drift velocity of electron as $v_{dn} = \langle v(t) - v(t_o)\rangle$ , which is the velocity gained after the previous collision. Also de
fine $\overline{t_n} = \langle t-t_o\rangle$ , the mean time between collisions. Thus,

$v_{dn} = -\frac{q\zeta}{m_e^*}\overline{t_n}$

From definition of μ_n, i.e. (v_dn = − μ_nζ) and the above expression, electron mobility can be expressed as:

$\mu_n = \frac{q\overline{t_n}}{m_e^*}$

Similarly, hole mobility can be expressed as:

$\mu_p = \frac{q\overline{t_p}}{m_h^*}$

Carrier mobility for any given scattering mechanism depends on the carrier effective mass and on the scattering time associated with that scattering mechanism. The scattering time is dependent on the frequency of carrier collisions with other "particle". The scattering times for carrier-ion and for carrier-phonon collisions are independent, by Matthiessen's rule,

$\frac{1}{t} = \frac{1}{t_i}+\frac{1}{t_l}$

$\frac{1}{\mu} = \frac{1}{\mu_i}+\frac{1}{\mu_l}$

where the substricpts i and l refer to ionized impurity and lattice (phonon) scattering respectively.

Temperature dependence

With increasing temperature, phonon concentration increases and causes increased scattering. Thus the carrier mob
ility due to lattice scattering decreases. 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 1.
The effect of ionized impurity scattering, however, decreases with increasing temperature because the average thermal speed
s of the carriers are increase. Thus,
the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced. In another word, the mobility due to ionized impurity scattering increases with increasing temperature.
These two effects operate simultaneously on the carriers. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates.

High-field effects

At low fields, the drift velocity v_d is proportional to ξ, so mobility μ is constant. This value of μ is called the low-field mobility.
As the field is increased, the mean free time between collisions $\overline{t}$ decreases with increasing ξ, resulting in reduced mobility. As a result, v_d increases sub-linearly and appears to approach a limiting velocity v_sat. 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 saturation results from a process called optical phonon scattering. At high fields, carriers are accelerated enough to gain sufficient kinetic energy between collisions that, when they collide with the lattice, they can impart enough energy to create an optical phonon. The probability that a carrier having such high energy will create a phonon is quite high, and thus the maximum carrier velocity is limited by the relation:

$\frac{m^*v_{max}^2}{2} \approx E_{phonon (opt.)}$

where E_phonon(opt.) is the optical phonon energy and m* is the carrier effective mass in the direction of ξ. The value of E_{phonon (opt.)} is 0.063 eV for Si and 0.034 eV for GaAs and Ge.
For more highly doped semiconductors, the scattering time is reduced, with a corresponding reduction in the low-field mobility and the saturation velocity. Because of the dependence of $\overline{t}$ on impurity concentration and temperature, v_max also depends on these parameters.

Measurement of mobility (Hall mobility)

Carrier mobility is most commonly measured through Hall mobilities, making use of Hall Effect.
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 resulted Lorenz's force will accelerates the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to the left 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 positive for n-type material and negative for p-type material.
Mathematically, the Lorentz force acting on a charge Q is given by
For electrons:

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

For holes:

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

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 electron,

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

$\Rightarrow -q\zeta_y + qv_xB_z = 0$

$\therefore \zeta_y = v_xB_z$

For electron, the field points in the +y direction, and for holes, it points in the −y direction.
The electron current I is given by I = − qnv_xtW. Sub v_x into the expression for ξ_y,

$\zeta_y = -\frac{IB}{nqtW} = +\frac{R_{Hn}IB}{tW}$