ECE331_Wi06_hw4_sol

ECE331_Wi06_hw4_sol - ECE331 Homework#4 Solution 1 Explain...

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Unformatted text preview: ECE331 Homework #4 Solution 1. Explain the concept of electron effective mass. At what scenario the effective mass is the actual electron mass? Use equations if necessary. Effective mass is the term used to describe the total effect of forces outside and inside of a crystal lattice. In reference to the band diagram, effective mass is inversely proportional to the band curvature. That is, for high band curvature the effective mass is low, while for low curvature the effective mass is high. Also, at the band minima the effective mass is positive, and at the band maxima (the top of the valence band for example) the effective mass is negative. Negative effective electron mass is the positive effective mass of a hole. Effective mass will equal the electron mass only if there are no interior crystal forces, thus only in free space will the effective mass equal the free 2 electron mass. One equation for effective mass is: m * = 2 ∂E ∂k 2. Sketch the band diagram of semiconductors, metals, and insulators at 0 K. What is the difference of a semiconductor and an insulator? (Sketch of semiconductor, insulator, and metal band diagram at 0K, The gap in an insulator should be large, short for semiconductor, and overlapping for a metal. See Fig.3-4 on page 68 of 6th Streetman book.) The bandgap of an insulator (>3.4eV) is much greater than the bandgap of a semiconductor (<3.4eV). 3. Explain the concept of intrinsic semiconductor and extrinsic semiconductor? What are the common dopants for Si (n-type and p-type)? An intrinsic semiconductor is a perfect crystal with no impurities or defects. An extrinsic semiconductor is a semiconductor that has been doped so that at equilibrium (no outside force or bias) the carrier concentrations are different than the intrinsic concentrations. Common p-type dopants are elements with three valence electrons. (Boron) Common n-type dopants are elements with five valence electrons. (Phosphorus, Arsenic) 4. Explain the concept of direct and indirect semiconductors. Use E-K diagram if necessary. In a direct bandgap semiconductor the conduction band minima is directly over the valence band maxima at the Γ point (k=0) in the band diagram. 1 In an indirect bandgap semiconductor the conduction band minima is not directly over the valence band maxima in the band diagram. 5. Streetman, P. 115, Question 2. (a) The E-k diagram on the X-valley has a lower curvature than that on the Γ-valley, thus the effective mass on the X-valley is larger than that on the Γ-valley. (b) The velocity magnitude in the Γ-valley is greater than that on the X-valley. 6. Streetman, P. 116, Question 3 (20 points). (a) GaAs has the lowest electron effective mass because GaAs has a higher curvature in the Γ-valley. (b) GaAs produces photons more efficiently through electron-hole combination because the conduction band minimum and the valence maximum are both at k=0. (c) Since the bandgap of GaAs is 1.43 eV, the energy of the emitted photon is 1.43 eV. hc The wavelength of the emitted photons is E = hν = λ −15 8 hc 4.136 × 10 (eV ⋅ s ) × 3 × 10 (m / s ) λ = hν = = = 0.868µm , which is infrared. E 1.43(eV ) (d) Si has 6 equivalent conduction band minimum at X along the 6 equivalent <100> directions. GaAs has 1 equivalent conduction band minimum. (Fig.1) Fig. 1. Shapes of constant energy surfaces near the minimum of the conduction band in Ge, Si, and GaAs 2 ...
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