Review: Depletion Region Recombination

Transcription

1 Review: Depletion Region Recombination Flat Quasi-Fermi Level (QFL) Requires rate limiting recombination is slow compared to thermal diffusion Assuming: k n ~k p = σ v, Flat QFL We get for recombination in the bulk at position x, U (x)=n óv dr T n(x)p(x) - n n(x)+p(x) i Note, Vapp < 0 here

2 Review: Depletion Region Recombination n(x)p(x)=n i Exp(-qV app/kt) n(x)p(x) - n U dr(x)=ntóv n(x)+p(x) n( x) p( x ) = constant for flat QFL and constant V app i -Substitute definitions and divide top and bottom by n(x)p(x) U (x)= dr N óvn {Exp(-qV /kt)-exp(qv /kt)} T i app app n(x) p(x) + p(x) n(x)

3 Review: Depletion Region Recombination U (x)= dr app N óvn sinh(-qv T i app n(x) p(x) + p(x) n(x) /kt) Or, for V >3kT T i app U (x)= Maximum occurs when n(x) = p(x) dr T i U dr,max (x)= Exp - N óvn Exp(-qV n(x) + p(x) N óvn! qvapp " # $ % kt & p(x) n(x) /kt)

4 New Material: Recall we desire U TOTAL Continue with U dr,max, recalling we want to solve for U TOTAL, as W' U TOTAL =! U(x) dx 0 We need definition for n(x) and p(x) You can show that n(x)=n mexp[qå(x-x m)/kt] p(x)=p Exp[-qå(x-x )/kt] m å = Electric Field Nm = concentration of e - for max U, n(xm)=p(xm) Xm= position where n=nm m In region 1, x<xm Exp (-) so n(x)<n(xm) More band bending In region, x>xm Exp (+) so n(x)>n(xm) Less band bending

5 Calculating U TOTAL : Back to Math Substituting in for n(x) and p(x) N óvn Exp(-qV /kt) T i app U(x)= n m Exp[qå(x-x m )/kt] p m Exp[-qå(x-x m )/kt] + p Exp[-qå(x-x )/kt] n Exp[qå(x-x )/kt] m m m m Use Exp(x) =Exp(x) and n m= p Exp(-x) m N óvn Exp(-qV /kt) T i app U(x)= Exp[qå(x-x m )/kt]+exp[-qå(x-x m )/kt]

6 Use definition of cosh x = U MAX U(x)= cosh[qå(x-x m )/kt] Calculating U TOTAL x x e +e, and formula for U MAX W' W' UMAX U TOTAL =! U(x) dx =! dx cosh[qå(x-x 0 0 m)/kt] For normally doped Si with Kn~Kp (assumed) U MAX is strongly peaked away from W, so we can extend the integral from W.

7 ! " Calculating U TOTAL U MAX U TOTAL = dx cosh[qå(x-x 0 m)/kt] After a change of variables, Recognizing that: V å= = cm V +V bi W app! D kt U TOTAL = U qå D ktw U = U TOTAL q(v +V )!"#"$ bi app # ratio of thermal spreading to carrier positioning. Thermal Voltage 0 dx " = cosh[x] MAX MAX

8 U TOTAL Substituting for U MAX, we obtain D ktw(n óv)n! qv " U = T i Exp - app TOTAL # $ q(v +V ) kt bi app % &!"""#"""$!""#""$ "Jo" term Notice Vapp though Important, A = not like thermionic emission A = : There are e - s and h + s recombining. It is as if you lose half of the voltage to the other carrier.

9 U TOTAL : Everyone Does it Differently Note: Nate and Mary quote a different value. TOTAL W' U = U(x) dx = Box Area! 0 kt Box Width = W q(v +V ) bi app!"#"$ Dimensionless: ratio thermal to applied volage kt 1 U = U W TOTAL MAX q (V +V app) bi D J dr and UTOTAL off by : O(1)

10 Quasi-Neutral Region Ionized donor atoms neutralized by injection of electrons into CB. p(x) has not changed: before injection of electrons, p(w)<p(w ) despite flat bands Concentration gradient causing holes to diffuse away from W into the bulk. Need to understand Diffusion/Recombination/Generation equations Assume No Generation

11 Diffusion-Recombination Recombination-Diffusion = 0 Diffusion:! Co Fick's First Law: Flux=-D ( x, t ) o! x! (, ) (, ) Fick's Second Law: C o x t! =D C o x t! t o! x Putting it together: " p p(x)-p o( x) " p( x) = 0 = - D " t! o p!"#"$ " x Excess Carrier lifetime Need Boundary Conditions

12 app Diffusion-Recombination Assume no recombination in D.R. or at S.S. V < 0, so more holes at W' as E suggest i F,p At x=w': n(w')p(w') = n Exp(-qV n(w') = n since E app /kt) = E o F,n F,o Boundary Conditions for Bulk D-R stats: p(w') = poexp(-qv app/kt) p(!) = p o

13 After the math: Diffusion-Recombination! (x-w') " p(x) = p o+p o[exp(-qv app/kt)-1]exp - # L p $ % & L p= Diffusion Length = Dpôp Does it work? D = R? recall, D=D o! You tell me.! p( x) x

14 Diffusion-Recombination Current = -q Flux +q Flux hole electrons! p(x) Current= -q Flux hole W' =-qdp! x W' We only have p(x), so lets take region where flux only due to holes: at x=w # p p! o (x-w') " = - [Exp(-qV app/kt)-1]exp - Evaluate at x=w # x Lp $ Lp % & ' n n p J = qd o [Exp(-qV /kt)-1] ; n p n p i i p L app o o =, B/R i o =! p no N D qdpn i J = [Exp(-qV app/kt)-1] B/R LpN D!"#"$ J o

15 k k (n p - n n p ) U =N b b i bulk T k n(n +n )+k p(p +p ) b 1 b 1 -! n -! p U= =! t! t Schockley-Read-Hall -" n =N k n ' (1-f T)- N kn f T(D.O.S.in CB,empty) " t T n b T -" p ' = N kpp f T! N k p (1! f T)(D.O.S.in VB,filled) " t T b T k ' k n, k ' n = n p = kpp 1 1 n = N Exp[-(E -E T )/kt] 1 C C p = N V Exp[-(E T -E V )/kt] 1 Trapped states can be caused by metal impurities, oxygen, or dopants

16 Schockley-Read-Hall Units for U = cm bulk 6 cm -6 (cm ) -3 s -3-1 =cm s 3 cm -3 (cm ) s cm 7 k= óv = cm ; v th =10 cm/s; s ó =!r =!(3*10 )! 10 trap -8 3 k n =vthó trap! 10 cm /s! =lifetime of carrier 1 ô n = k N n T

17 Schockley-Read-Hall LLI: Lower Level Injection of Electrons and holes from photons For n-type: 15 Än<<n b(10 ) 5 Äp<<p b(10 ) HLI: Generate more electrons and holes than We had in our lattice: " # E C =Än=Äp>n b n= f(! )DOS; Using Boltzmann Statistics, f(! )=Exp[-(E-E )/ kt ] n=än+n p=äp+p b,o b,o F

18 Schockley-Read-Hall (Än+nb,o)( Äp+p b,o) - n ÄnÄp + pb,oän + n b,oäp + pb,onb,o- n U = i = i bulk ô p(än+n b,o+n )+ô n(äp+p b,o+p ) ô p(än+n b,o+n )+ô n(äp+p b,o+p ) Assuming LLI of n-type p <<! n =! p << n! b, o "#$#%!, ) (10 ) (10 ) (10 b o U is dominated by " p*n term in numerator bulk b,o and! p n in denominator b,o Leaving us with : Äp U = bulk ô p We assumed ô n! ô p

19 Schockley-Read-Hall Äp! n! p LLI: U = = " = " bulk ô p! t! t Solving yields: p= " pexp(-t/! p ) + p o For HLI, same assumptions: HLI decays twice as slowly HLI: U = bulk Äp ôp LLI: Wait for hole carriers to recombine HLI: hole & e- carriers must recombine twice as long