Editing Electron mobility (section)

===Hall mobility===
{{main|Hall effect}}

[[File:Hall Effect Measurement Setup for Holes.png|right|frame|Hall effect measurement setup for holes]]
[[File:Hall Effect Measurement Setup for Electrons.png|right|frame|Hall effect measurement setup for electrons]]

Carrier mobility is most commonly measured using the [[Hall 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 the ''x''-direction and a [[magnetic field]] is applied in the ''z''-direction. The resulting Lorentz force will accelerate the electrons (''n''-type materials) or holes (''p''-type materials) in the (−''y'') direction, according to the [[right hand rule]] and set up an electric field ''ξ<sub>y</sub>''. As a result there is a voltage across the sample, which can be measured with a [[High impedance|high-impedance]] voltmeter. This voltage, ''V<sub>H</sub>'', is called the [[Hall effect|Hall voltage]]. ''V<sub>H</sub>'' is negative for ''n''-type material and positive for ''p''-type material.

Mathematically, the [[Lorentz force]] acting on a charge ''q'' is given by

For electrons:
<math display="block">\mathbf F_{Hn} = -q(\mathbf v_n \times \mathbf B_z)</math>

For holes:
<math display="block">\mathbf F_{Hp} = +q(\mathbf v_p \times \mathbf B_z)</math>

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 electrons,

<math display="block">\mathbf F_y = (-q)\xi_y + (-q)[\mathbf v_n \times\mathbf B_z] = 0</math>

<math display="block">\Rightarrow -q\xi_y + qv_xB_z = 0</math>

<math display="block"> \xi_y = v_xB_z</math>

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

The [[Electric current|electron current]] ''I'' is given by <math>I = -qnv_xtW</math>. Sub ''v''<sub>''x''</sub> into the expression for ''ξ''<sub>''y''</sub>,

<math display="block">\xi_y = -\frac{IB}{nqtW} = +\frac{R_{Hn}IB}{tW}</math>

where ''R<sub>Hn</sub>'' is the Hall coefficient for electron, and is defined as
<math display="block">R_{Hn} = -\frac{1}{nq}</math>

Since <math>\xi_y = \frac{V_H}{W}</math>
<math display="block">R_{Hn} = -\frac{1}{nq} = \frac{V_{Hn}t}{IB}</math>

Similarly, for holes
<math display="block">R_{Hp} = \frac{1}{pq} = \frac{V_{Hp}t}{IB}</math>

From the Hall coefficient, we can obtain the carrier mobility as follows:
<math display="block">\begin{align}
\mu_n &= \left(-nq\right) \mu_n \left(-\frac{1}{nq}\right) \\
&= -\sigma_n R_{Hn} \\
&= -\frac{\sigma_n V_{Hn} t}{IB}
\end{align}</math>

Similarly,
<math display="block">\mu_p = \frac{\sigma_p V_{Hp}t}{IB}</math>

Here the value of ''V<sub>Hp</sub>'' (Hall voltage), ''t'' (sample thickness), ''I'' (current) and ''B'' (magnetic field) can be measured directly, and the conductivities ''σ''<sub>n</sub> or ''σ''<sub>p</sub> are either known or can be obtained from measuring the resistivity.