Question

A high electron mobility transistor (HEMT) controls large currents by applying a small voltage to a thin sheet of electrons. The density and mobility of the electrons in the sheet are critical for the operation of the HEMT. HEMTs consisting of AlGaN/GaN/Si are being studied because they promise better performance at higher powers, temperatures, and frequencies than conventional silicon HEMTs can achieve. In one study, the Hall effect was used to measure the density of electrons in one of these new HEMTs. When a current of \(10.0 \mu\) A flows through the length of the electron sheet, which is \(1.00 \mathrm{~mm}\) long, \(0.300 \mathrm{~mm}\) wide, and \(10.0 \mathrm{nm}\) thick, a magnetic field of \(1.00 \mathrm{~T}\) perpendicular to the sheet produces a voltage of \(0.680 \mathrm{mV}\) across the width of the sheet. What is the density of electrons in the sheet?

Short answer

Answer: The density of electrons in the sheet is approximately \(1.22 \times 10^{20}\) m\(^{-3}\).

Step-by-step solution

Write down the given information

We are given the following information: - Current, \(I = 10.0 \mu A = 10.0 \times 10^{-6}\) A - Length, \(l = 1.00\) mm - Width, \(e = 0.300\) mm - Thickness, \(t = 10.0\) nm - Magnetic field, \(B = 1.00\) T - Hall voltage, \(V_H = 0.680\) mV

Convert units into SI units

To solve the problem, we need to ensure that all quantities are in the proper SI units. We will convert the given measurements into meters: - Length, \(l = 1.00 \times 10^{-3}\) m - Width, \(e = 0.300 \times 10^{-3}\) m - Thickness, \(t = 10.0 \times 10^{-9}\) m - Hall voltage, \(V_H = 0.680 \times 10^{-3}\) V

Rearrange the Hall voltage formula to solve for n

We are given the Hall voltage formula as: \(V_H = \frac{IB}{nqe}\) We need to solve for the density of electrons (\(n\)), so we will rearrange the formula: \(n = \frac{IB}{V_Hqe}\)

Substitute the given values into the formula and solve for n

Now we will substitute the given values and constants into the formula: \(n = \frac{(10.0 \times 10^{-6})(1)}{(0.680 \times 10^{-3})(1.6 \times 10^{-19})(0.300 \times 10^{-3})}\) Calculate the value of \(n\): \(n \approx 1.22 \times 10^{20}\) m\(^{-3}\) The density of electrons in the sheet is \(1.22 \times 10^{20}\) m\(^{-3}\).

Key concepts

Electron Mobility

Electron mobility is a fundamental aspect of how electrons move through a material, especially within semiconductors. It describes how quickly electrons can travel when subjected to an electric field. A higher electron mobility means that electrons move more easily and respond more swiftly to a voltage. This is essential for electronic devices, as it directly affects their speed and efficiency.

When voltage is applied, electrons accelerate, and their mobility is influenced by factors such as temperature, impurities, and the material's intrinsic properties. In semiconductors like gallium nitride (GaN), high electron mobility is particularly crucial because it allows devices to handle more power efficiently.

• Increased electron mobility results in faster signal processing.
• It also reduces energy losses, improving device performance.
• High mobility is beneficial in establishing strong current flows with minimal voltage applications.

Understanding mobility helps in optimizing materials and designs for electronic components, like transistors, where speed and efficiency are paramount.

High Electron Mobility Transistor (HEMT)

High Electron Mobility Transistors (HEMTs) are advanced semiconductor devices known for their excellent performance under high-frequency and high-power conditions. They utilize materials that support high electron mobility, such as AlGaN/GaN, to deliver superior characteristics compared to conventional transistors.

HEMTs incorporate a unique heterostructure that creates a two-dimensional electron gas (2DEG), facilitating extremely low resistance paths for electron flow. This structure enhances electron mobility, allowing the device to operate efficiently at higher speeds. Such features make HEMTs ideal for applications in wireless communication and radar technology.

• The 2DEG minimizes scattering, maintaining high speeds for electron travel.
• HEMTs support robust operation under thermal stress and high frequencies.
• The use of alloys like AlGaN/GaN offers improved performance attributes.

Incorporating these materials into HEMTs allows electronic design engineers to push the boundaries of what is possible in terms of signal processing speed and power management.

Electron Density

Electron density is a measure of how many electrons are present per volume unit in a material. It plays a pivotal role in determining a device's electrical properties and functionality. In a high electron mobility transistor (HEMT), knowing the electron density helps understand how efficiently a current can be transported across the device.

Achieving the ideal electron density is crucial for maximizing a HEMT's performance, as it affects both conductivity and the overall mobility of electrons. The electron density can vary widely based on the material properties and the applied voltage.

• A proper balance in electron density ensures effective current flow.
• Lower densities might lead to inadequate current, affecting the device performance.
• Optimal density conditions can enhance device efficiency and power handling capabilities.

Calculating electron density accurately, as demonstrated through exercises utilizing the Hall effect, gives deep insights into optimizing semiconductor structures for better electronic device performance.

Magnetic Field

A magnetic field is an invisible force field that exerts a force on moving charges, such as electrons in semiconductors. When applied perpendicular to an electron flow in materials like those used in HEMTs, it gives rise to the Hall effect, a phenomenon that can be harnessed to measure important properties like electron density.

In the context of a HEMT, understanding the interaction between electron flow and magnetic fields is crucial for characterizing and improving device performance. The Hall effect is particularly insightful as it provides a non-invasive method to assess how electrons interact within the material.

• Helps in measuring carrier concentration and mobility through the Hall voltage.
• Provides essential insights into the semiconductor's electronic properties.
• Enables designers to optimize HEMT designs for better efficiency.

By strategically using magnetic fields, engineers can gain valuable data that supports the development and refinement of high-performance electronics based on semiconducting materials.