Hall effect is directly caused by a current passing through a conducting sample in a magnetic field \(\vec{B}\). The conducting sample may have either positive or negative charge carriers (charges that are free to move in a conductor). There will be no potential difference in the direction perpendicular to the current when there is no magnetic field applied. However, if a magnetic field \(\vec{B}\) is present in the direction perpendicular to the sample, a Lorentz force (see Eq.\ref{eq:righthand}) will be produced which will initially induce a movement of the current (charge carriers). In this way, carriers with opposite charges will accumulate on opposite sides of the sample by the Right Hand Rule. Once the electric force produced by the oppositely charged sides balance the magnetic force, the current will not curve anymore. The resulting voltage across the two sides when \(F_M=F_E\) is the Hall voltage. The sign of the carrier decides whether the carrier will curve upward or downward. In other words, the sign of the carrier is directly related to the sign of the Hall voltage.

N-type Germanium and P-type Germanium semiconductors have been used in this lab. Carriers are mainly electrons (\(q=-e\)) in N-type semiconductors and mainly holes (\(q=+e\)) in P-type semiconductors. Electrons move opposite to the applied current while holes move along with it. According to Equation \ref{eq:righthand}: \[\label{eq:righthand} \vec{F}=q \vec{v}\times \vec{B}\] With the same \(+\vec{I}\), the direction of the force will be the same for both positive and negative charge carriers. Therefore, carriers will accumulate on the same side for either type of carrier. The voltage difference resulted from this cumulation of charge on one side of the sample is the Hall voltage. Due to the different signs of the charge carriers (\(\pm e\)), the sign of the Hall voltage will be different.

Carrier density (Melissinos 2003) is an important concept in Hall Effect, since it determines the magnitude of \(V_{H}\), and its sign also reflects the type of carrier present. It represents the number of carriers per unit volume, by definition, and mathematically can be calculated using Equation \ref{eq:Hallcoefficient} and Equation \ref{eq:density}: \[\label{eq:Hallcoefficient} R_{H}=\frac{V_{H}\cdot t}{I\cdot B}\] where \(R_{H}\) is the Hall coefficient, \(V_{H}\) is the measured Hall voltage, \(t\) is the thickness of the sample, \(I\) is the supplied current to the sample and \(B\) is the magnetic field. \[\label{eq:density} n=\frac{1}{R_{H}e}\] where \(n\) is the carrier density and \(e\) is the elementary charge., which equals to \(1.60\cdot10^{-19}\) coulombs.