Electronic Properties of Materials: Conduction - Problems and Solutions


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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 Omegam, and its atomic mass and density are 114.82 amu and 7.31 g cm-3, respectively.

  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?
      





Solution  
  1. In this part, we are asked to determine the number of free electrons donated by each In atom in the crystal. In other words, we are asked to determine the valency, which is the total number of conduction electrons per cm3 divided by the total number of atoms per cm3. Given are the electrical resistivity = 8.37 x 10-8 Omegam (or 8.37 x 10-6 Omegacm) and the electron drift mobility = 6 cm2V-1s-1. The reciprocal of resistivity gives the conductivity: 1/resistivity = conductivity (sigma) = 1/(8.37 x 10-6) = 11.947 x 104 siemens per centimeter (or mho per centimeter).
    We know that sigma = en{mu_d}, where sigma is the electrical conductivity and {mu_d} is the drift mobility of the electrons.
    Plugging the values in the above equation gives the electron concentration (n) = 1.244 x 1023 per cm3.

    Now we need to determine the total number of Indium atoms per cm3.
    7.31 g of Indium has a volume of 1 cm3.
    114.82 g of Indium has 6.023 x 1023 atoms.
    Therefore, 7.31 g of Indium has (7.31 x 6.023 x 1023)/114.82 = 3.835 x 1022 atoms
    It means 1 cm3 of Indium has 3.835 x 1022 atoms

    Thus, the valency of Indium is = 1.244 x 1023/3.835 x 1022 = 3.24 (i.e., approximately 3)

    Indium is placed in Group IIIB in the periodic table along with boron, aluminum, and gallium, which means its valency is 3. The above calculations attest the position of Indium in the periodic table.

  2. The mean free path is the product of the mean speed of electrons (given to be 1.74 x 108 cm/s) and the mean free time (i.e, the relaxation time tau). The equation that relates the drift mobility and the mean free time is {mu_d} = taue/m, where m is the mass of the electron and e the electron charge. Plugging the known values gives the mean free time tau = 3.413 x 10-15 s. Kindly note that here we need to keep the unit of mobility in SI units, i.e., m2V-1s-1 and not cm2V-1s-1 because we are using the mass of electron and the electron charge in SI units.
    Now we can determine the value of the mean free path = mean free time x mean speed = 3.413 x 10-15 s x 1.74 x 108 cm/s = 5.939 x 10-7 cm or 5.939 nm.

  3. To determine the thermal conductivity of Indium using its electrical conductivity, we need to use the Wiedemann-Franz-Lorenz law. The Wiedemann-Franz-Lorenz law states that the ratio of the thermal conductivity (kappa) and the electrical conductivity (sigma) of a metal is propotional to the temperature.
    i.e., kappa/sigma is propotional to T
    Or, kappa/sigma = LT, where L is the propotionality constant (referred to as the Lorenz number) given by the equation L = {{{pi^2}{{kappa_B}^2}}/{3{e^2}}}. The value of the Lorenz number is 2.44 x 10-8 WOmegaK-2.
    Plugging the values gives kappa = 86.87 W (m-K)-1. The experimental value is 81.6 W (m-K)-1. Thus, we can say that the theoretical value is within 6% of the experimental value.