INTEGRATED CIRCUITS

EENG7715 – HW Set #1

Updated: 11/01/2015 Page 1

Homework Set #1 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. [20 points]

HINT: de Broglie wavelength of electron is given by 𝜆 = ℎ 𝑚𝑚

, where ℎ is Planck’s constant, 𝑚 is the mass

of an electron and 𝑣 is its velocity. Also use the kinetic energy relationship 𝐸 = 1 2 𝑚𝑣2. 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. [12 points]

Problem 3: Schematically show the number of electrons in the various subshells of an atom with the electronic shell structure 1s22s22p4 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? [12 points]

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 [10 points]

(b) 2.9 x 1016 / cm3 arsenic atoms and 2.7 x 1016 / cm3 boron atoms [10 points]

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

Problem 5: An n-type sample of silicon has a uniform density 𝑁𝑑 = 1016 atoms / cm 3 of arsenic and a p-

type silicon sample has 𝑁𝑎 = 1015 atoms / cm 3 of boron. For each semiconductor material determine

the following:

(a) The temperature at which the intrinsic concentration 𝑛𝑖 exceeds the impurity density by a factor of 20. See Table 1.4 for 𝑛𝑖(𝑇). [15 points]

HINT: A nonlinear equation in T will result for each of the semiconductor type. You need to solve the equations iteratively by plugging in 𝑇 = 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));

EENG7715 – HW Set #1

Updated: 11/01/2015 Page 2

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

[5 points] (c) The Fermi level referred to the valence-band edge 𝐸𝑚 in each material at 300 K. [8 points]

(d) The Fermi level if both types of impurities are present in a single sample. [8 points]