ECE-305: Fall 2016 Minority Carrier Diffusion Equation (MCDE) Professor Peter Bermel Electrical and Computer Engineering Purdue University, West Lafayette, IN USA pbermel@purdue.edu Pierret, Semiconductor Device Fundamentals (SDF) Chapter 3 (pp. 122-132) 1
minority carrier diffusion equation (hole continuity equation) p t d dx J px q G R L p (1D, generation by light) p t d dx qd p dp dx q G p L t p (low-level injection, no electric field) p t D p d 2 p dx 2 p t p G L (D p spatially uniform) 2
How to solve (some) Exam 2 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 2: 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. 9/21/2016 Bermel ECE 305 F16 3
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 2 p dx 2 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. 9/21/2016 Bermel ECE 305 F16 4
How to solve equations Transient p t D p d 2 p dx 2 p t p G L Steady State =, 0 = d Δ + Δ solution Δ = G + Boundary condition for B: Concentration before light was turned on? 9/21/2016 solution Δ = + + If, Δ = + + BC to determine A and B: Concentration at leftmost and rightmost points Bermel ECE 305 F16 5
Example #1: Reading the problem Step 1: Determine carrier densities. a) The majority carrier is? b) the majority carrier concentration is? c) The minority carrier concentration is? d) How did they get the diffusion coefficient? e) Special words: uniformly doped. 9/21/2016 Bermel ECE 305 F16 6
Example #1: Solution P-type / equilibrium E C n 0 n 2 i 10 3 cm -3 p 0 p 0 n i e E i E F k B T 10 17 10 10 e E i E F k B T E i E V p 0 10 17 cm -3 E F E F E i 0.41 ev Steady-state, uniform generation, no spatial variation 7
Example #1: Reading the problem Step 3) What type of problems are we talking about voltage or light related? Step 4B ) Key words: steady-state, with light, uniform generation. ii) If I write p t D p d 2 p dx 2 p t p G L for MCDE, would I be right? iii) Which approximate equation should I choose: =, solution Δ = G + 0 = d Δ + Δ solution Δ = + + 8
Example #1: Solution P-type / out of equilibrium F p E i 0.41 ev E C E i n 0 n 2 i 10 3 cm -3 p 0 E V p 0 10 17 cm -3 F n F p n» n n i e F n E i k B T 10 14 10 10 e F n E i k B T F n E i 0.24 ev Steady-state, uniform generation, no spatial variation 9
Example #1: Solution n x nx G L t n 0 = d Δ + Δ Δ = + + n G L t n 10 20 10 6 10 14 x 0 x Steady-state, uniform generation, no spatial variation 10
Example #2 Now turn off the light. Transient, no generation, no spatial variation iii) Which approximate equation should I choose? =, solution Δ = G + 0 = d Δ + Δ Solution: Δ = + + iv) What is my boundary condition: Just before, light was turned on for a long-time, before it is turned off. Δ (0 ) = = Δ (0 ) 11
Example #2 nt n0 10 14 nx n0e t/t n t 0 n t 0 t 12
Example #2 F p E i 0.41 ev nt» nt n i e F n t 10 14 e t/t n 1010 e F n t F n E i k B T E i k B T t E i k B T ln10 4 k B T t t n E C E i E V F n F p transient, no generation, no spatial variation 13
Example #3: One sided Minority Diffusion Steady state, no generation/recombination Acceptor doped; sample short compared to diffusion length Metal contact n x', t 0 1 J n 1 dj n rn t q dx x a 0 N N N g N dn qnm E qd dx 2 d n 0 D N 2 dx 9/21/2016 Bermel ECE 305 F16 14
Example #3: One sided Minority Diffusion 0 DN 2 d n dx 2 n x', t 0 Metal contact n( x, t) C Dx' x a 0 x a, n( x ' a) 0 C Da x 0', n( x ' 0') C n( x, t) n( x 0') 1 9/21/2016 x ' a Bermel ECE 305 F16 x 15
Example #4 Steady-state, sample long compared to the diffusion length. No generation. n x 0 fixed 10 12 cm -3 Step 3) What type of problems are we talking about? Step 4B ) Key words: steady-state, without light, long device p d ii) If I write 2 p D p p G for MCDE, would I be right? t dx 2 L t p iii) Which approximate equation should I choose: =, solution Δ = G + 0 = d Δ + Δ 16 solution Δ = + + 18
Example #4 nx n0 nx n0e x/l n L n D n t n << L x 0 nx 0 x x L 200 mm Steady-state, sample long compared to the diffusion length. 17
Example #4 Steady-state, sample is 5 micron long. No generation. nx 0 10 12 cm -3 n x 5 mm 0 Which approximate equation should I choose: =, solution Δ = G + 0 = + solution Δ = + + or Δ = + + What are my boundary conditions? 18
Example #4 nx 0 10 12 cm -3 n x L D 0 nx Ax B nx 0 10 12 cm -3 nx n 0 1 x L L 0 n x 19
Example #5 Steady-state, sample is 30 micrometers long. No generation. nx 0 10 12 cm -3 nx 30 mm 0 fixed n t D p d 2 n dx 2 t n G L 0 D p d 2 n dx 2 t n 0 1) Simplify the MCDE 2) Solve the MCDE 3) Deduce F p from Δp d 2 n dx 2 L n 28 mm n L n 0 L n º D n t n L 30 mm 20
Example #5 Steady-state, sample is 30 micrometers long. No generation. nx 0 10 12 cm -3 nx 30 mm 0 fixed d 2 n dx 2 n L n 0 nx Ae x/l n Be x/l n 1) Simplify the MCDE 2) Solve the MCDE 3) Deduce F p from Δp n0 A B 10 12 nl Ae L/L n Be L/L n 0 21
Example #5 nx n0 L n n x n 0 D n t n» L sinh éë L x / L n ù û sinh L / L n nx L 0 x x 0 x L Steady-state, sample neither long nor short compared to the diffusion length. 9/21/2016 Bermel ECE 305 F16 22
Spatially-dependent MCDE example summary Length scale Solution type Long ( ) Decaying exponentials Short ( ) Linear Intermediate ( ~ ) Hyperbolic functions Notes: is length of region where MCDE applies =,, is the diffusion length for carriers 23