Homework Set

Problem 1: What is the de Broglie wavelength (in Angstroms) of an electron at 100 eV? What is the wavelength for electrons at 15 keV, which is typical of electron microscopes? Comparing this to visible light, comment on the advantage of an electron microscope.

HINT: de Broglie wavelength of electron is given by ?? = h/??v', where h is Planck's constant, ?? is the mass of an electron and ?? is its velocity. Also use the kinetic energy relationship ?? = 1/2????². Note that 1 eV = 1.602 x 10^-19 J.

Problem 2: Calculate the first three energy levels for an electron in a quantum well of width 12.5 nm with infinite walls.

Problem 3: Schematically show the number of electrons in the various subshells of an atom with the electronic shell structure 1s² 2s² 2p⁴ and an atomic weight of 21. Indicate how many protons and neutrons exist in the nucleus. Is this atom capable of creating covalent bonds with other atoms and why?

Problem 4: Find the equilibrium electron and hole concentrations and the location of the Fermi level for silicon at 27^oC if the silicon contains the following concentrations of shallow dopant atoms:

(a) 2.5x10¹⁵ / cm³ boron atoms

(b) 2.9 x 10¹⁶ / cm³ arsenic atoms and 2.7 x 10¹⁶ / cm³ boron atoms

HINT: Use Eqs. (1.1.26) and (1.1.27) from text

Problem 5: An n-type sample of silicon has a uniform density ???? = 10¹⁶ atoms / cm³ of arsenic and a ptype silicon sample has ??_?? = 10¹⁵ atoms / cm³ of boron. For each semiconductor material determine the following:

(a) The temperature at which the intrinsic concentration ???? exceeds the impurity density by a factor of 20. See Table 1.4 for ????(??).

HINT: A nonlinear equation in T will result for each of the semiconductor type. You need to solve the equations iteratively by plugging in ?? = 1000 on the right hand side of the equations to start with. Stop when new calculated values are not changing significantly. You may use a MATLAB code similar to the following to solve the nonlinear equation:

% Solving nonlinear equation iteratively
% T = 7014/(1.5 * T - 0.5)
T(1) = 1000;
for n = 2:7
T(n)= 7014/(1.5 * log(T(n-1)) - 0.5);
end
fprintf('%d %d %d %d %d %d %d\n', round(T));

(b) The equilibrium minority-carrier concentrations at 300 K. Assume full ionization of impurities.

(c) The Fermi level referred to the valence-band edge ??_?? in each material at 300 K.

(d) The Fermi level if both types of impurities are present in a single sample.