Question

A piece of p-ty pe semiconductor is used as a Hall effect device. The device has a thickness of \(d_{\text {thick }}=1 \mathrm{~mm}\). It is placed in an external magnetic field of \(|\vec{B}|=10^{-5} \frac{\mathrm{Wb}}{\mathrm{cm}^{2}}\). A Hall voltage of \(5 \mu \mathrm{V}\) is measured when a current of \(3 \mathrm{~mA}\) is applied. Calculate \(p,\) the charge (hole) concentration in units \(\frac{1}{\mathrm{~cm}^{3}}\).

Short answer

The charge (hole) concentration \( p \) is \( 3.75 \times 10^{20} \, \text{cm}^{-3} \).

Step-by-step solution

Understanding the Hall Effect Formula

The Hall effect relates the measured Hall voltage \( V_H \), the magnetic field \( B \), the thickness \( d \), the current \( I \), and the charge (hole) concentration \( p \) through the formula:\[ V_H = \frac{IB}{peq} \]where \( q \) is the charge of an electron (\( q = 1.6 \times 10^{-19} \, C \)). We aim to solve for \( p \).

Solving for p

Rearrange the equation to solve for \( p \):\[ p = \frac{IB}{eV_H} \]Insert the given values: \( I = 3 \, \text{mA} = 3 \times 10^{-3} \, \text{A} \), \( B = 10^{-5} \, \text{Wb/cm}^2 = 10^{-1} \, \text{T} \) (since 1 Wb/cm² = 10 T), and Hall voltage \( V_H = 5 \times 10^{-6} \, \text{V} \).

Calculating Charge Concentration

Now substitute the known values:\[ p = \frac{(3 \times 10^{-3}) \times (10^{-1})}{(1.6 \times 10^{-19}) \times (5 \times 10^{-6})} \]Calculate each part and solve for \( p \).

Final Calculation

First calculate the numerator and the denominator separately:Numerator: \( 3 \times 10^{-4} \)Denominator: \( 8 \times 10^{-25} \)Divide to solve: \( p = \frac{3 \times 10^{-4}}{8 \times 10^{-25}} \)Calculate: \( p = 3.75 \times 10^{20} \, \text{cm}^{-3} \).

Key concepts

Semiconductors

Semiconductors are materials that have conductivity between conductors and insulators. This characteristic makes them uniquely valuable for electronic devices. Unlike conductors, which allow electricity to flow easily, and insulators, which block it, semiconductors can control electrical current.
In semiconductors, electrical conductivity is determined by the presence of charge carriers. These can be either electrons or holes. Holes are essentially the absence of an electron in the atomic structure, creating a "positive" charge.
A p-type semiconductor specifically has more holes than electrons. They are created by introducing certain types of impurities, which replace some of the atoms in the semiconductor crystal, allowing more acceptance of electrons, thereby increasing the number of holes. This manipulation is done through a process called doping.

Charge Concentration

Charge concentration in a semiconductor refers to the quantity of charge carriers per unit volume. It is often expressed in units of \( \text{cm}^{-3} \) to indicate the number of carriers per cubic centimeter.
In the context of the Hall effect, understanding charge concentration helps in determining how effectively a semiconductor material can conduct electricity when a current is applied. Higher concentration means more available charge carriers and, consequently, higher conductivity.
Measuring charge concentration using the Hall effect involves a Hall voltage resulting from an external magnetic field applied to the semiconductor. With known current and magnetic field values, the Hall voltage can be measured, and using the Hall effect equation, the charge concentration can be calculated with great precision. This is crucial for designing and optimizing semiconducting devices, particularly in creating components like transistors and diodes.

Magnetic Field

A magnetic field is an invisible field that applies a force to charged particles moving within it. It is characterized by its magnetic induction, denoted by \(|\vec{B}|\), often measured in Teslas (T) or Weber per square centimeter \(\left( \text{Wb/cm}^2 \right)\).
This force is crucial in generating the Hall effect. When a current flows through a semiconductor placed in a magnetic field, a measurable Hall voltage is created perpendicular to both the current and the magnetic field. This phenomenon allows the determination of charge carrier concentration.
Magnetic fields are indispensable for many applications beyond the Hall effect, including magnetic resonance imaging (MRI) and electric transformers. In semiconductor technology, control over magnetic fields can significantly advance device functionality, enabling the creation of more efficient and compact electronic components.