ECE-305: Spring 2018 Exam 2 Review

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1 ECE-305: Spring 018 Exam Review Pierret, Semiconductor Device Fundamentals (SDF) Chapter 3 (pp ) Chapter 5 (pp ) Professor Peter Bermel Electrical and Computer Engineering Purdue University, West Lafayette, IN USA /13/018 Bermel ECE 305 S18 1

2 Key topics to review Minority carrier diffusion equation Band structures PN junctions /13/018 Bermel ECE 305 S18

3 Semiconductor equations: key cases p t D p d p dx p t p G L /13/018 Bermel ECE 305 S18 3

4 How to solve (some) Exam problems Step 1: From material information (semiconductor, doping, etc.), calculate carrier densities, Fermi level, etc. Start with the majority carriers, =, =. Then get the other carrier from = Step : Use band-diagram to calculate potential profile, electric field, = /, or = /, and =, etc. For homogenous semiconductor with a battery attached, = /. Step 3: Decide if this is drift-related problem (resistivity, velocity, mobility, etc.), or a diffusion related problem (light turning on-off, etc.) Step 4A: For a drift-problem use = +. For, you may be given a number, or table, or diffusion coefficient, etc. Learn how to read such a table. Step 4B: For a diffusion problem, read carefully for clues to simplify the minority carrier equation. /13/018 Bermel ECE 305 S18 4

5 How to solve equations Step 4B: Two general types of minority diffusion problem. i) Determine if electron or the hole is the minority carrier. ii) If holes are the minority carriers, write the equation: p t D p d p dx p t p G L iii) iv) If steady-state, drop the time-derivative. If transient, keep the time derivative. If spatially uniform, drop the diffusion term. Without light, drop the generation term. If the region is very short, drop the recombination term. Choose the solutions from the following table. Use the boundary conditions to complete solution. /13/018 Bermel ECE 305 S18 5

6 How to solve equations Transient p t D p d p dx p t p G L Steady State =, 0 = d Δ + Δ solution Δ = G + Boundary condition for B: Concentration before light was turned on? solution Δ = + + If, Δ = + + BC to determine A and B: Concentration at leftmost and rightmost points /13/018 Bermel ECE 305 S18 6

7 eq. energy band diagram E F E F 1) Begin with E F ) Draw the E-bands where you know the carrier density 3) Electrostatic potential by flipping E-band upside down. 4) E-field from slope 5) n(x), p(x) from the E-band diagram 6) rho(x) from n(x) and p(x) 7) diffusion current from (5) or from (6) E C x E C ref qv x E x 1 q de C x dx /13/018 Bermel ECE 305 S18 7

8 energy band diagram E E C x E C qv x E C E F E i E V de C x dx q dv x dx qe x x n x 0 x x p x /13/018 Bermel ECE 305 S18 8

9 Short-cut to Band-diagram Neutral Space Charge Neutral ND N A Vacuum level 1 E C E V E F /13/018 is equivalent to solving the Poisson equation Bermel ECE 305 S18 9

10 p-n Junction Devices Symbols N A N P N D Finding hotspot /13/018 Bermel ECE 305 S18 10

11 What is a Diode good for.? solar cells GaAs lasers Organic LED Avalanche Photodiode GaN lasers image.google.com /13/018 Bermel ECE 305 S18 11

12 carrier densities vs. x log 10 nx,log 10 px n 0N N D p 0P N A p 0N n i N D n 0 p n i N A N x n x p x P /13/018 Bermel ECE 305 S18 1

13 the depletion approximation N r r qn D P de dx r x K S e 0 x n x p x r qn A d V Se 0 D A K q p n N N dx N D x n N A x p /13/018 Bermel ECE 305 S18 13

14 Depletion Regions in Homojunctions Neutral N Space Charge D N A Neutral x n x p N D x n N A x p x n kse 0 q N D N A N N V A D bi qv bi qndx qn n Ax k e k e s p 0 s 0 x p kse 0 q N A ND N N V A D bi Can you solve the same problem for a hetero-junction? 14 /13/018 Bermel ECE 305 S18

15 Key results for PN junctions é W K Se 0 ê ë q E é 0 ê ë qv bi K s e 0 N A N D VN D N A V bi V ù bi ú û N D N A N A N D ù ú û 1/ 1/ E 0 V bi W ( ) N E x n x p x P W x n x P x n N A N A N D W V bi k T B q ln N N D A n i N D x n N A x P x p N D N A N D W /13/018 Bermel ECE 305 S18 V x x p ò x E x dx 15

16 Built-in Potential: heterojunctions qv bi 1 1 E g, qv bi E g, 1 1 qv i E 1 1 b g, N N E N AND k T ln 1 N N e A D g, kbt ln kbt ln 1 NV, NC, 1 B E g, / k B V, C, 1 T /13/018 Bermel ECE 305 S18 16

17 Interface Boundary Conditions: heterojunctions D E = (D/kεo) x n x n x p x p position position D K e E(0 ) K e E(0 ) D E K K (0 ) E(0 ) Displacement is continuous across the interface, but field need not be.. /13/018 Bermel ECE 305 S

18 equilibrium e-band diagram E qv bi E C E F V A 0 I 0 E F E V W x x n /13/018 Bermel ECE 305 S18 18 x p

19 e-band diagram under forward bias E V A 0 E C E F V bi V A V 0 V A > 0 E V W x x n The applied voltage drops across the junction, but /13/018 Bermel ECE 305 S18 x p 1

20 QFL s split E V 0 E C F n V bi V A V A > 0 F n > F p qv A F p E V W x x n /13/018 Bermel ECE 305 S18 x p

21 e-band diagram under reverse bias E V bi V A V bi V R V 0 E C F p V A < 0 F n F n < F p E V W /13/018 x 3 p x n x

22 one-sided junction N E x V A < 0 P N D >> N A x n x p >> x n V A > 0 x p x /13/018 Bermel ECE 305 S18 4

23 key points (one-sided NP junctions) V bi k T B q ln N N D A n i é W K Se 0 ê ë qn A V bi V A ù ú û 1/ W µ V bi V A W µ 1 N A E 0 V bi V A W E 0 µ V bi V A E 0 µ N A /13/018 Bermel ECE 305 S18 5

24 Applying a Bias: Poisson Equation qv bi E C -E F E F -E V q(v bi -V) -qv E C -F n F p -E V /13/018 Bermel ECE 305 S18 6

25 Flat Quasi-Fermi Level up to Junction E C E V E C J n J p E V /13/018 Bermel ECE 305 S18 7

26 One Sided Minority Diffusion Can calculate current anywhere, let us solve the problem where it is the easiest q(v bi -V) Steady state Acceptor doped dj n 1 n r n t q dx g n -V F p -E V J n qn E n dn qdn dx 0 n d n D dx /13/018 Bermel ECE 305 S18 8

27 Boundary Conditions n( x 0 ) n e p( x 0 ) n e i i ( F E ) n ( F E ) p i i n(0 ) n(0 ) n(0 ) n N i A V e G qv A 1 V G 0 ( Fn Fp ) i i np n e n e qv A p(0 ) N A q(v bi -V A ) ni qva n(0 ) e N A -V A F p F n N A /13/018 Bermel ECE 305 S18 9

28 Right Boundary Condition n( x W ) i n( x W ) 0 p p n N A E C V E V /13/018 Bermel ECE 305 S18 30

29 Example: One Sided Minority Diffusion d n 0 D dx N n( x, t) C Dx V /13/018 x W, n( x W ) 0 C DW p p p ni 0', ( 0) qv 1 A x n x e C N n i (, ) qv A x n x t e 1 1 N A W A p Bermel ECE 305 S18 31

30 Electron & Hole Fluxes n i ( ) qv A n x e 1 1 N A x W p J qn E qd n N N N dn qdn ni qv A J n qdn e 1 dx W N x0 p A n F n F p dp qdp ni qv A J p qdp e 1 dx W N x0' n D p /13/018 Bermel ECE 305 S18 3

31 Exam Equation Sheet /13/018 Bermel ECE 305 S18 33

32 Exam Equation Sheet /13/018 Bermel ECE 305 S18

33 Exam Fall 016 /13/018 Bermel ECE 305 S18 35

34 Exam Fall 016 /13/018 Bermel ECE 305 S18 36

35 Exam Fall 016 /13/018 Bermel ECE 305 S18 37

36 Exam Fall 016 /13/018 Bermel ECE 305 S18 38

37 Exam Fall 016 /13/018 Bermel ECE 305 S18 39

38 Exam Fall 016 /13/018 Bermel ECE 305 S18 40

39 Review Questions 1) If you apply negative bias to a terminal, which direction does the band move? ) What is the difference between Fermi & Quasi-Fermi levels? 3) How can we get away with solving just the MCDE in certain cases? 4) What are the most basic parameters of a p-n junction, that can be used to calculate everything else? /13/018 Bermel ECE 305 S18 41

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