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1 UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences EECS 130 Spring 2008 Professor Chenming Hu Midterm I - Solutions Name: SID: Grad/Undergrad: Closed book. One sheet of notes is allowed. There are 12 pages of this exam including this page. Problem 1 15 Problem 2 30 Problem 3 25 Problem 4 30 Total 100

2 Physical Constants Electronic charge q C Permittivity of vacuum ε F cm -1 Relative permittivity of silicon ε Si /ε Boltzmann s constant k x 10-5 ev/ K or J K -1 Thermal voltage at T = 300K kt/q V Effective density of states N c 2.8 x cm -3 Effective density of states N v 1.04 x cm -3

3 Problem 1 [15pts] [5pts] (a) Draw qualitatively the Fermi-Dirac distribution at T=300K, T=150K, and T 0K. f(e) 1 T = 300K T = 150K T ~ 0K E f E

4 [10pts] (b) Quantum wells are often used in applications such as semiconductor lasers. In a quantum well, electrons are confined in a thin slab of material, as shown below nm 10 nm 1000 nm The density-of-states, D(E) in an ideal quantum-well is step-like: D(E): states / ev / nm 3 At T 0K, there are states in this system, mark the location of Fermi Level, E f? (In this problem, there is no need to consider spin explicitly since it is already taken into account in D(E).) At 0K, all the states below E f are filled with electrons; all the states above E f are empty. Therefore, Ef N = V i n= V i f( E) D( E) de = V i D( E) de Ec Ec This means the number of electrons in the system, N, equals the area under the D(E) curve up to E f, times the volume of the semiconductor, V. 7 3 V = 10nm 1000nm 1000nm= 10 nm. To determine the location of E f, we divide the system in two regions and calculates the number of states in these regions. 7 (1) 0.005eV < E < 0.02eV 0.4 ( ) 10 = 60000( states) = ( states) (2) 0.02eV < E < 0.045eV ( ) electrons will completely fill region (1); are left for region (2). Region (2) will only be partially filled. The precise location of E f : E 0.02eV = ( f ) E f = 0.03eV Ef ev

5 Problem 2 [30pts] [10pts] (a) A P-N junction is fabricated by the process sequence shown below. The cartoons show cross sectional views of the wafer after each processing step. Fill in the names of each process steps shown below: 1. (Starting Wafer) 5. etching. 2. thermal oxidation. 6. resist strip. SiO 2 3. photoresist spin-coating photoresist 7. ion implantation. As Ions 4. lithography. 8. Diffusion at 1100C for 60min Process Available: ion implantation, sputtering, resist strip, etching, lithography, chemical mechanical polishing, epitaxy, thermal oxidation

6 [5pts] (b) The starting wafer is P-type and doped with N A =10 15 cm -3. Find the depletion region width. (You may assume N d >> N a and Φ bi =0.85 V) The implanted N-type region is usually much more heavily doped than the background P-doping (10 15 cm -3 ), this is why we can assume N d >> N a. We may use the lightly-doped side doping concentration to calculate W dep. 14 2εφ si bi F / cm 0.85V 4 Wdep = = = cm= 1.05μm qna C 10 cm [5pts] (c) Find the depletion region width when a reverse bias of V R = 4.15V is applied. (Also assume N d >> N a ) The depletion region is widened under reverse bias. 14 ( φ ) 2ε + V F / cm 5V si bi R 4 Wdep = = = cm= 2.56 m qna C 10 cm μ [10pts] (d) If the diode area is 0.01cm 2 1, plot 2 C dep and the slope of the curve, respectively? versus V R. What are the x-intercept The depletion capacitance can be calculated using the parallel-plate capacitor formula with plate separation W dep. C dep εsi A qnaεsi A = = W 2 2 ( φ + V ) dep bi R Arrange terms: 2 φ + V = C qn ε A ( ) 1 bi R 2 2 dep a si (Here we also assume N d >> N a ) x-intercept = -Φbi = φ slope: bi 2 2 = qnaε sia C 10 cm F / cm 0.01cm ( ) = = FC i F iv 1/C dep 2 Capacitance data Slope = 2/qNε s A 2 Increasing reverse bias V r R

7 Problem 3. [35pts] The carrier profiles in a forward biased PN diode are shown below. The majority carrier densities for both P and N-side are given. Also shown is the minority carrier density on the P-side. 10 N cm P [5pts] (a) Calculate the bias voltage. From the above graph you can find the doping density. 10, 10 From the minority carrier profile, / 10 /

8 [5pts] (b) Find the minority electron and hole current density 0 and 0. Use 12 /, 35 /, 0.01 and / / [5pts] (c) Add the minority hole concentration profile on the N-side and show N cm P

9 [5pts] (d) Redraw if the electron diffusion constant is 4 times larger. Assume all other conditions are the same. cm P If the diffusion constant is increased by a factor of 4, the diffusion length is doubled. So the minority carrier decay length will double. [5pts] (e) Redraw if 510 cm donors are added to the P-side and show 0. Assume all other conditions are the same. cm P

10 [5pts] (f) Redraw the applied bias voltage is reduced by Assume all other conditions are the same. cm P 10 With a 60mV reduction the minority carrier concentration is reduced by a factor of [5pts] (h) Draw the excess minority carrier distribution if the diode is reverse biased at ~1. N cm P With a reverse bias the excess minority carrier concentration at the edge of the deletion region will be

11 Problem 4. [20pts] You are given an N-type piece of silicon with /. Assume 300 and there are no acceptor dopants, i.e. 0. cm [5pts] (a) Given that is constant, what is the electron current density? When the Fermi level is constant, the system is in equilibrium. So there will be no current flowing. The diffusion current and drift current will cancel each other out.,, 0 [5pts] (b) Find the electron diffusion current density,. Use 30 /., / 910 / / 910 / 10 / / (or / )

12 [5pts] (c) Derive an expression for the electron drift current density, in terms of., Using Poisson s equation and 1 Therefore, / /, 910 / 9 10 / / 9 10 / 10 / / / / [2pts] (d) Using the total electron current density, find the value of. From,, , / / / [3pts] (e) Numerically verify Einstein s relationship with the values you ve calculated above / / 0.026

### UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences. Professor Chenming Hu.

UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences EECS 130 Spring 2009 Professor Chenming Hu Midterm I Name: Closed book. One sheet of notes is

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