2_1_EE2 Midterm-Solutions

2_1_EE2 Midterm-Solutions - EE2 Midterm Spring 2009 EE 2...

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Unformatted text preview: EE2 Midterm Spring 2009 EE 2 Midterm Spring 2009 Name:____________________________________ 1 EE2 Midterm Spring 2009 EE 2 Midterm Spring 2009 Name:____________________________________ Student ID#: ____________________________ A. You are allowed to bring in one 8 ½” x 11” sheet of paper of notes. B. Please put all the work on the test paper. C. The back page of the exam has some information you may find useful. D. Show your work! Provide clear explanations and show calculations. 2 EE2 Midterm Spring 2009 1. (10 points) What is the probability that a hole will exist at 4kT below the Fermi level (assuming that energy level does not fall inside the band gap) fhole ( E ) = 1 − 1 1+ e E − EF kT = 1− 1 1+ e −4 kT kT = 1− 1 = 1 − 0.982 ≈ 0.018 1 + e −4 2. (3 points) Is there bipolar or unipolar conduction in a p‐type semiconductor? unipolar 3. (3 points) Can a full valence band contribute to conduction? No 4. (5 points) You have material 1, intrinsic silicon, and material 2, silicon doped with 1016 Boron and 1016 Arsenic. Which one is more conductive and why? intrinsic is more conductive, b/c doped semi has scattering and therefore lower mobility 5. (3 points) In general, would you expect semiconductors or insulators to have a larger bandgap? insulators 6. (3 points) Everything else being equal, would you rather use a direct bandgap or indirect bandgap semiconductor for a solar cell? direct bandgap 7. (20 points) In our universe, the density of states in the conduction band is 2 ⎛ m⎞2 given by N ( E ) = 2 ⎜ 2 ⎟ E , where E is the distance from Ec in the picture π ⎝ ⎠ − EF 3 e kT below. Assume we lived in some alternate universe where N ( E ) = kT Derive an expression for the carrier concentration for the semiconductor below, and simplify it as much as possible. Note: The conduction band spans the energy from EC to Ec,top. You may assume Ec as your reference (Ec = 0)), and you may assume that Maxwell‐ Boltzmann statistics are valid. 3 EE2 Midterm Spring 2009 !"$%&'# ! "# fm − b ( E ) = e e kT N (E) = kT EC ,top − ( E − EF ) kT − EF n= Ec EC ,top − EF ∫ ∫ 0 N ( E ) f ( E ) dE = e kT e kT − ( E − EF ) kT EC ,top ∫ 0 N ( E ) f ( E ) dE −E e kT dE = − e kT kT −E EC ,top − EC ,top kT EC ,top n= dE = ∫ 0 = 1− e 0 n ≈ 1 is ok, because Maxwell-Boltzmann statistics imply that we will have very few carriers at higher energies 8. (20 points) Early in the quarter, we found a wave function solution to the time‐independent Schrödinger equation for an infinite potential well to be 2 ⎛ nπ x ⎞ ψ ( x) = sin ⎜ . Let’s say there was some different condition that ⎝L⎟ ⎠ L resulted in the wave function given below. Calculate what the parameter A is for the new wave function. 4 EE2 Midterm Spring 2009 ψ ( x ) = Ax ψ ( x) = A( L − x) ψ ( x) = 0 0≤x≤ L 2 ()*+" $%#&'" L ≤x≤L 2 otherwise !" #&'" #" *" ,-./012.3"4562-1/789"8:";.<5":=92789">1<59" 19"5?=.7896"@8"@05"35A" ∞ −∞ L ∫ (ψ ( x )) 2 dx = 1 ∫ (ψ ( x )) 0 L 2 2 dx = 1 symmetry 2 ∫ (ψ ( x )) dx = 1 2 0 L /2 Ax 3 2 ∫ ( Ax ) dx = 2 3 0 2 L 2 =1 0 A 2 L3 =1 3* 8 12 A= L3 9. (10 points) Assume the visible spectrum of light is 400 nm < λ < 700 nm. What range of energy photons does this represent? c = fλ c f= λ c ω = 2π f = 2π λ c E = ω = 2π λ E ( λ = 400 nm ) = 3.10 {eV} 2 E ( λ = 700 nm ) = 1.77 {eV} 3.10 < E < 1.77 {eV} 5 EE2 Midterm Spring 2009 10. (15 points) You compare the conductivity of a piece of n‐type silicon and find it to be 1,000,000x that of a piece of intrinsic silicon. What is the doping concentration of the n‐type silicon (in units 1/cm3)? (Note#1: you may use the mobility values given on the last page of the exam. Note#2: you are at room temperature and the n‐type material has shallow dopants, so you may assume they are completely ionized) σ intrinsic = qni µn + µ p ( ) σ intrinsic = 4.4 × 10 −6 σ n − type σ n − type = qnµn n= ⎧1⎫ ⎨ ⎬ ⎩ Ωcm ⎭ = 4.4 × 10 −6 * 1, 000, 000 ⎧1⎫ ⎨ 3⎬ ⎩ cm ⎭ σ n − type qµ n = 1.83 × 1016 11. (5 points) If you take a room temperature semiconductor and increase the temperature, will the conductivity increase, decrease, or stay the same? Why? Increase, because more electrons and holes will be generated with the increased thermal energy. 6 EE2 Midterm Spring 2009 from PhD Comics by Jorge Cham = 6.582 × 10 −16 {eV * s} ni ( room temperature ) = 1.4 × 1010 speed of light ≡ c = 3 × 10 8 ⎧m ⎫ ⎨⎬ ⎩s⎭ ⎧1⎫ ⎨ 3⎬ ⎩ cm ⎭ ω = 2π f q = 1.6 × 10 −19 {C} mobility, µ ⎧ cm 2 ⎫ µn = 1500 ⎨ ⎬ ⎩ Vs ⎭ ⎧ cm 2 ⎫ µ p = 458 ⎨ ⎬ ⎩ Vs ⎭ 7 ...
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This note was uploaded on 03/29/2011 for the course ELECTRICAL EE2 taught by Professor A during the Spring '09 term at UCLA.

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