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?

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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