Conduction: Problems and Solutions - Electronic Properties of Materials | Deepak Rajput

Electronic Properties of Materials: Conduction - Problems and Solutions

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Na is a monovalent metal (BCC) with a density of 0.9712 g cm^-3. Its atomic mass is 22.99 g mol^-1. The drift mobility of electrons in Na is 53 cm²(V-s)^-1. [solution available]
1. Consider the collection of conduction electrons in the solid. If each Na atom donates one electron to the electron sea, estimate the mean separation between the electrons. (Note: If n is the concentration of particles, then the particles's mean separation d = 1/n^1/3.)
2. Estimate the mean separation between an electron (e^-) and a metal ion (Na⁺), assuming that most of the time the electron prefers to be between two neighbouring Na⁺ ions. What is the approximate Coulombic interaction energy (in eV) between an electron and an Na⁺ ion?
3. How does this electron/metal-ion interaction energy compare with the average thermal energy per particle, according to the kinetic molecular theory of matter? Do you expect the kinetic molecular theory to be applicable to the conduction electrons in Na? If the mean electron/metal-ion interaction energy is of the same order of magnitude as the mean KE of the electrons, what is the mean speed of electrons in Na? Why should the mean kinetic energy be comparable to the mean electron/metal-ion interaction energy?
4. Calculate the electrical conductivity of Na and compare this with the experimental value of 2.1 x 10⁷ (ohm-m)^-1 and comment on the difference.

Electron drift mobility in indium has been measured to be 6 cm²V^-1s^-1. The room temperature resistivity of In is 8.37 x 10^-8 ohm-m, and its atomic mass and density are 114.82 amu and 7.31 g cm^-3, respectively. [solution available]
1. Based on the resistivity value, determine how many free electrons are donated by each In atom in the crystal. How does this compare with the position of In in the Periodic Table (Group IIIB)?
2. If the mean speed of conduction electrons in In is 1.74 x 10⁸ cm/s, what is the mean free path?
3. Calculate the thermal conductivity of In. How does this compare with the experimental value of 81.6 W (m-K)^-1?

Consider a thin insulating disk made of mica to electrically insulate a semiconductor device from a conducting heat sink. Mica has thermal conductivity of 0.75 W (m-K)^-1. The disk thickness is 0.1 mm, and the diameter is 10 mm. What is the thermal resistance of the disk? What is the temperature drop across the disk if the heat current through it is 25 W? [solution available]

Consider a coaxial cable that has a copper core conductor and polyethylene dielectric with the following properties: Core conductor resistivity of 19 n-ohm m, core radius of 4 mm, dielectric thickness of 3.5 mm, dielectric thermal conductivity of 0.3 W (m-K)^-1. The outside temperature is 25^oC. The cable is carrying a current of 500A. What is the temperature of the inner conductor? [solution available]

Consider a rectangular sample, a metal or an n-type semiconductor, with a length L, width W, and thickness D. A current I is passed along L, perpendicular to the cross-sectional area WD. The face W X L is exposed to a magnetic field intensity B. A voltmeter is connected across the width to read the Hall voltage V_H. [solution available]
1. Show that the Hall voltage recorded by the voltmeter us V_H = IB/Den
2. Consider a 1-micron-thick strip of gold layer on an insulating substrate that is a candidate for a Hall probe sensor. If the current through the film is maintained at constant 100 mA. What is the magnetic field that can be recorded per micro-volt of Hall voltage?

Consider Matthiessen's rule applied to a thin film. Show that, very approximately, the product of the thermal coefficient of resistivity (TCR) and the resistivity of the thin film is equivalent to the product of the bulk TCR and resistivity. [solution will be posted soon]