Question

The figure shows schematically a setup for a Hall effect measurement using a thin film of zinc oxide of thickness \(1.50 \mu \mathrm{m}\). The current, \(i\), across the thin film is \(12.3 \mathrm{~mA}\), and the Hall potential, \(V_{\mathrm{H}},\) is \(-20.1 \mathrm{mV}\) when a magnetic field of magnitude \(B=0.900 \mathrm{~T}\) is applied perpendicular to the current flow.. a) What are the charge carriers in the thin film? [Hint: They can be either electrons with charge \(-e\) or electron holes (missing electrons) with charge \(+e .]\) b) Calculate the density of charge carriers in the thin film.

Short answer

a) The type of charge carriers in the thin film of zinc oxide are electrons, as indicated by the negative Hall potential (-20.1mV) and the negative Hall coefficient (-3.03x10^-4 m^3/C). b) The density of charge carriers in the thin film of zinc oxide is approximately 2.07x10^22 m^-3.

Step-by-step solution

Understand the Hall effect

The Hall effect is the production of a voltage difference (Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current. By analyzing the sign of the Hall potential (voltage), we can determine whether the charge carriers are positive or negative.

Identify charge carriers' sign

We are given a Hall potential of -20.1mV, which means the Hall voltage is negative. This indicates that the charge carriers are electrons with charge -e.

Calculate the Hall coefficient

The Hall coefficient (\(R_H\)) can be calculated using the formula: $$ R_H = \frac{V_H d}{i B} $$ Where \(V_H\) is the Hall potential, \(d\) is the thickness of the film, \(i\) is the current, and \(B\) is the magnetic field. Using the given data: \(V_H = -20.1 mV = -20.1\times10^{-3} V\) \(d = 1.50 \mu m = 1.50\times10^{-6} m\) \(i = 12.3 mA = 12.3\times10^{-3} A\) \(B = 0.900 T\) Plugging the values into the formula, we get: $$ R_H = \frac{(-20.1\times10^{-3} V)(1.50\times10^{-6} m)}{(12.3\times10^{-3} A)(0.900 T)} $$

Calculate the Hall coefficient (R_H)

Calculating R_H, we get: $$ R_H = -3.03\times10^{-4} \frac{m^3}{C} $$ The Hall coefficient is negative, which also confirms the charge carriers are electrons.

Calculate the density of charge carriers

The density of charge carriers (\(n\)) can be calculated using the relation between the Hall coefficient and charge carrier density: $$ R_H = \frac{1}{n e} $$ We know the charge of an electron is \(e = 1.6\times10^{-19} C\). Solving for \(n\), we have: $$ n = \frac{1}{R_H e} $$ Substituting the values, we get: $$ n = \frac{1}{(-3.03\times10^{-4} \frac{m^3}{C})(1.6\times10^{-19} C)} $$

Find the density of charge carriers

Calculating the density, we get: $$ n \approx 2.07\times10^{22} m^{-3} $$ So, the density of charge carriers in the thin film of zinc oxide is approximately \(2.07\times10^{22} m^{-3}\).

Key concepts

Hall voltage

Imagine you are in a moving car and suddenly turn sharply – you feel a force pushing you to the side. Similarly, when electrons move through a material and a magnetic field is applied perpendicular to their direction of motion, they experience a sideways force. This phenomenon is called the Hall effect. The result is a build-up of charge on the sides of the material, creating a voltage difference known as the Hall voltage. This voltage is crucial because it tells us the type of charge carrier and, indirectly, their concentration within the material.

For instance, if the Hall voltage comes out negative in a Hall effect measurement, this indicates that negatively charged electrons are the dominant charge carriers moving opposite to the 'positive' current direction. In contrast, a positive Hall voltage would suggest that the charge carriers are holes, which are essentially the absence of electrons that behave like positive charges. This subtle distinction is pivotal for understanding semiconductor behavior and tailoring materials for specific electronic applications.

Charge carrier density

In the microscopic alleys of a material, charge carriers – either electrons or holes – bustle about, carrying electrical current. The charge carrier density is like a census of these movers, counting how many there are in a given volume. Higher density means more carriers are available to transport charge, directly impacting the electrical conductivity of the material.

The charge carrier density is not merely a number; it serves as a fundamental parameter in designing and analyzing semiconductors and other electronic materials. Understanding density allows engineers to modify and control electrical properties for various technologies, including transistors, solar cells, and sensors. In practical terms, calculating the charge carrier density involves measurements such as the Hall effect, providing a gateway to the invisible, bustling world within materials.

Hall coefficient

The Hall coefficient is a navigator's tool, guiding us through the charged sea of carriers within a material. This coefficient encapsulates the response of a material to applied electric and magnetic fields, presenting itself through the Hall voltage. A negative Hall coefficient confirms that electrons are the primary carriers, navigating the paths of the material like traffic on a highway. Conversely, a positive Hall coefficient reveals that holes are at the wheel.

More than just a sign, the magnitude of the Hall coefficient is inversely proportional to the charge carrier density. This relationship turns the Hall coefficient into a powerful diagnostic tool, revealing not only the type of carriers present but also their concentration. The calculation steps typically involve the material's thickness, current, magnetic field strength, and observed Hall voltage. By quantifying the Hall coefficient, we can unlock a detailed map of the electrical characteristics intrinsic to the material.