Q20Engineering Physics

Discovered by Edwin Hall in 1879, the Hall effect is a fundamental galvanomagnetic phenomenon that provides profound insights into the electrical conduction mechanisms of metals and semiconductors. It is the definitive experimental method used to determine the exact sign of the charge carriers (whether they are negative electrons or positive 'holes'), to calculate the charge carrier density (concentration), and to evaluate the drift mobility of those carriers.

1. The Principle of the Hall Effect

Consider a flat, rectangular slab of a conducting material (like a strip of metallic Sodium). A constant direct current $I$ is passed through it along the x-axis. This means charge carriers are drifting along the slab with a drift velocity $v_d$. Now, a strong, uniform magnetic field $B$ is applied perpendicularly to the slab, along the z-axis.

According to the Lorentz force law, any charged particle moving through a magnetic field experiences a deflecting magnetic force ($F_m$) that is perpendicular to both its velocity and the magnetic field. $F_m=q({\vec{v}}_d\times \vec{B})$ For a current flowing in the +x direction: - If the charge carriers are positive (holes), they are drifting in the +x direction. The magnetic force pushes them toward the bottom edge of the slab (-y direction). - If the charge carriers are negative (electrons), they are physically drifting in the -x direction to create a +x current. By the right-hand rule (and accounting for the negative charge), the magnetic force also pushes them toward the bottom edge of the slab (-y direction).

Because the charge carriers are deflected to one side, they accumulate on that edge, leaving a depletion (and thus an opposite net charge) on the top edge. This separation of charges creates an internal, transverse electric field across the width of the slab, pointing along the y-axis. This is the Hall Electric Field ($E_H$).

This transverse electric field exerts an electric force ($F_e=q{E}_H$) on the moving charges that acts in the exact opposite direction to the deflecting magnetic force. As charge continues to accumulate on the edges, the Hall electric field grows stronger. Almost instantly, an equilibrium is reached where the opposing electric force exactly cancels the magnetic force, and the charge carriers once again flow straight down the slab.

At equilibrium: $F_e=F_m⟹q{E}_H=q{v}_{d}B⟹E_H=v_{d}B$.

If the width of the slab is $w$, the potential difference created across the width is the Hall Voltage ($V_H$): $V_H=E_H\times w=v_{d}Bw$

2. Derivation of the Hall Coefficient ($R_H$) and Carrier Density ($n$)

We relate the drift velocity $v_d$ to macroscopic, measurable quantities. The current density $J$ (current $I$ divided by cross-sectional area $A=w\times t$, where $t$ is thickness) is given by: $J=nq{v}_d$ Where $n$ is the number of charge carriers per unit volume (carrier density) and $q$ is the elementary charge ($e=1.6\times 10^{-19}\text{ C}$).

Substituting $v_d=\frac{J}{nq}$ into the equilibrium electric field equation ($E_H=v_{d}B$): $E_H=\left(\frac{J}{nq}\right)B⟹E_H=\left(\frac{1}{nq}\right)JB$

The quantity $\left(\frac{1}{nq}\right)$ is a fundamental property of the material and is defined as the Hall Coefficient ($R_H$). $R_H=\frac{1}{nq}$

The Hall coefficient relates the generated transverse electric field to the product of the applied current density and magnetic field. $E_H=R_{H}JB⟹R_H=\frac{E_H}{JB}$

By experimentally measuring the Hall voltage $V_H$, we calculate $E_H$ ($V_H/w$). Knowing $J$ and $B$, we find $R_H$. Once $R_H$ is known, the carrier density $n$ is easily calculated: $n=\frac{1}{R_{H}q}$

3. Application to Sodium and Mobility Calculation

Why is the sign important? The sign of the Hall voltage (and thus the Hall coefficient $R_H$) directly indicates the sign of the charge carriers. When the Hall effect experiment is performed on a strip of Sodium metal, the measured Hall voltage is conclusively negative. This elegantly proves that in alkali metals like sodium, the electrical conduction is entirely carried out by negatively charged particles (electrons), not positive ions or holes. Therefore, $q=-e$, making $R_H$ mathematically negative.

Calculating Carrier Mobility ($\mu$): Mobility ($\mu$) is a measure of how easily a charge carrier can move through a solid lattice when pulled by an electric field. It is defined as the drift velocity per unit electric field: $\mu =\frac{v_d}{E}$.

The electrical conductivity $\sigma$ of a material is given by: $\sigma =nq\mu$

We know from the Hall effect that $R_H=\frac{1}{nq}$, which rearranges to $nq=\frac{1}{R_H}$. Substituting this into the conductivity equation yields an incredibly useful relationship: $\sigma =\left(\frac{1}{R_H}\right)\mu ⟹\mu =\sigma R_H$

Thus, by performing a simple Hall effect experiment to find $R_H$, and conducting a standard four-probe resistance measurement to find the conductivity $\sigma$, engineers can instantly calculate the microscopic drift mobility $\mu$ of the electrons inside the solid sodium lattice.