Hall Effect Measuring

Setup

A voltage $V_L$ is applied to a semiconductor causing electrons to drift with a velocity $v_d$ leading to a current $I_L$ to flow.

If a magnetic field $B$ is applied perpendicular to the plane, the Lorentz force $F=q{v}_d\ast B$ acts on the electrons causing them to move towards one side.

This gives rise to a perpendicular electric field, the Hall field $E_H$ which opposes the Lorentz force and stops further electrons from being deflected.

In equilibrium the force due to the Hall field field equals the Lorentz force, hence: $F=q{E}_H=qvd\ast B\to E_H=v_{d}B$ (since vectors are perpendicular)

$$V_H=E_{H}w=\frac{B}{ndq}{I}_L\text{where }\frac{B}{ndq}\text{is hall resistance}$$

or

$$V_H=R_H\frac{B}{d}{I}_L\text{where }{R}_H=\frac{1}{nq}\text{ (Hall Coefficient)}$$

Thus, by measuring $V_H$ and $I_L$ (or $V_L$) as a function of $B$, we can determine the carrier density, $n$

NB: The sign of $V_H$ also tells us the type of carrier (electron or hole)

So by measuring $R_H$ we can determine the carrier density $n$ and by measuring the resistance we can also determine conductivity $\sigma$ as $\sigma =\frac{1}{\rho }$ and $\rho =R\frac{A}{l}$

Using this we can determine mobility $\mu$ since $\sigma =ne\mu$

We also note here that $V_H=E_{H}w=v_{d}Bw$ i.e. the magnitude of the Hall voltage depends on the magnetic field, the width of the material (i.e. physical size) and the drift velocity

From before, $v_d\propto 𝜇$ so materials with a high mobility will give rise to higher Hall voltages

Temperature Dependance

$$np=4{\left(\frac{\sqrt{m_e{m}_h}k}{2h^2\pi }\right)}^3{T}^3\exp \left(-\frac{E_g}{kT}\right)$$