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Chapter 27: Problem 52

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

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

Step by step solution

01

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
02

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
03

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: \(n = \frac{IB}{V_Hqe}\)
04

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\(^{-3}\).

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Most popular questions from this chapter

It would be mathematically possible, for a region with zero current density, to define a scalar magnetic potential analogous to the electrostatic potential: \(V_{B}(\vec{r})=-\int_{\vec{r}_{0}}^{\vec{r}} \vec{B} \cdot d \vec{s},\) or \(\vec{B}(\vec{r})=-\nabla V_{B}(\vec{r}) .\) However, this has not been done. Explain why not.

In your laboratory, you set up an experiment with an electron gun that emits electrons with energy of \(7.50 \mathrm{keV}\) toward an atomic target. What deflection (magnitude and direction) would Earth's magnetic field \((0.300 \mathrm{G})\) produce in the beam of electrons if the beam is initially directed due east and covers a distance of \(1.00 \mathrm{~m}\) from the gun to the target? (Hint: First calculate the radius of curvature, and then determine how far away from a straight line the electron beam has deviated after \(1.00 \mathrm{~m}\).)

A helium leak detector uses a mass spectrometer to detect tiny leaks in a vacuum chamber. The chamber is evacuated with a vacuum pump and then sprayed with helium gas on the outside. If there is any leak, the helium molecules pass through the leak and into the chamber, whose volume is sampled by the leak detector. In the spectrometer, helium ions are accelerated and released into a tube, where their motion is perpendicular to an applied magnetic field, \(\vec{B},\) and they follow a circular orbit of radius \(r\) and then hit a detector. Estimate the velocity required if the orbital radius of the ions is to be no more than \(5 \mathrm{~cm},\) the magnetic field is \(0.15 \mathrm{~T}\) and the mass of a helium- 4 atom is about \(6.6 \cdot 10^{-27} \mathrm{~kg}\). Assume that each ion is singly ionized (has one electron less than the neutral atom). By what factor does the required velocity change if helium- 3 atoms, which have about \(\frac{3}{4}\) as much mass as helium- 4 atoms, are used?

A charged particle moves under the influence of an electric field only. Is it possible for the particle to move with a constant speed? What if the electric field is replaced with a magnetic field?

A particle with a charge of \(20.0 \mu \mathrm{C}\) moves along the \(x\) -axis with a speed of \(50.0 \mathrm{~m} / \mathrm{s}\). It enters a magnetic field given by \(\vec{B}=0.300 \hat{y}+0.700 \hat{z},\) in teslas. Determine the magnitude and the direction of the magnetic force on the particle.
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