6.012 - Electronic Devices and Circuits Lecture 5 - p-n Junction Injection and Flow - Outline • Review Depletion approximation for an abrupt p-n junction Depletion charge storage and depletion capacitance (Rec. Fri.) qDP(vAB) = – AqNApxp = – A[2εq(φb-vAB){NApNDn/(NAp+NDn)}]1/2 Cdp(VAB) ≡ ∂ qDP/∂ vAB|VAB = A[εq{NApNDn/(NAp+NDn)}/2(φb-VAB)]1/2 • Biased p-n Diodes Depletion regions change (Lecture 4) Currents flow: two components – flow issues in quasi-neutral regions (Today) – boundary conditions on p' and n' at -xp and xn (Lecture 6) • Minority carrier flow in quasi-neutral regionsThe importance of minority carrier diffusion Boundary conditions Minority carrier profiles and currents in QNRs – Short base situations – Long base situations – Intermediate situations Clif Fonstad, 9/24/09 Lecture 5 - Slide 1x-xp xn -xp -qNAp qNDn The Depletion Approximation: an informed first estimate of ρ(x) Assume full depletion for -xp < x < xn, where xp and xn are two unknowns yet to be determined. This leads to: ! "(x) =0#qNApqNDn0 for for for forx < #xp#xp< x < 00 < x < xnxn< x$ % & & ' & & ρ(x)Integrating the charge once gives the electric field ! E(x) =0 for x < "xp"qNAp#Six + xp( ) for " xp< x < 0qNDn#Six " xn( ) for 0 < x < xn0 for xn< x$ % & & & ' & & & Ε(x) n E(0) = -qNApxp/εSi = -qNDnxn/εSi Clif Fonstad, 9/24/09 Lecture 5 - Slide 2 x xThe Depletion Approximation, cont.: Integrating again gives the electrostatic potential: φ(x) φn -xp x xn φp equation relating our unknowns, xn and xp: -xp Ε(x)x ! NApxp= NDnxnxn 1 E(0) = -qNApxp/εSi = -qNDnxn/εSi Requiring that the potential be continuous at x = 0 givesus our second relationship between xn and xp: ! "(x) ="p for x < #xp"p+qNAp2$Six + xp( )2 for - xp< x < 0"n#qNDn2$Six # xn( )2 for 0 < x < xn"n for xn< x% & ' ' ' ( ' ' ' Insisting E(x) is continuous at x = 0 yields our first φ(0) = φp + qNApxp2/2εSi = φn − qNDnxn2/2εSi ! "p+qNAp2#Sixp2="n$qNDn2#Sixn22 Clif Fonstad, 9/24/09 Lecture 5 - Slide 3Comparing the depletion approximation with a full solution: Example: An unbiased abrupt p-n junction with NAp= 1017 cm-3, NDn = 5 x 1016 cm-3 Charge E-field Potential nie±qφ(x)/kT po(x), no(x) Clif Fonstad, 9/24/09 Lecture 5 - Slide 4 Courtesy of Prof. Peter Hagelstein. Used with permission.Depletion approximation: Applied bias Forward bias, vAB > 0: φ vAB -wp wn-xp 0 xn (φb-vAB) x No dropin wire No dropat contact No dropin QNR No dropin QNR No dropat contact No dropin wire In a well built diode, all the appliedvoltage appears as a change in thethe voltage step crossing the SCL -wp x wn-xp 0 xn vAB φ (φb-vAB) Reverse bias, vAB < 0: Note: With applied bias we are no longer in thermal equilibrium so it is no longer true that n(x) = ni eqφ(x)/kT and p(x) = ni e-qφ(x)/kT. Clif Fonstad, 9/24/09 Lecture 5 - Slide 5The Depletion Approximation: Applied bias, cont. Adding vAB to our earlier sketches: assume reverse bias, vAB < 0 ρ(x) xn -xp -qNAp qNDn w xnxp Ε(x)x x ! w =2"Si#b$ vAB( )qNAp+ NDn( )NApNDn! xp=NDnwNAp+ NDn( ), xn=NApwNAp+ NDn( )xn -xp |Epk| ! "#=#b$ vAB and #b=kTqlnNDnNApni2! Epk=2q"b# vAB( )$SiNApNDnNAp+ NDn( )φ(x) xn -xp (φb -vAB) x Clif Fonstad, 9/24/09 Lecture 5 - Slide 6The Depletion Approximation: comparison cont. Example: Same sample, reverse biased -2.4 V Charge E-field Potential nie±qφ(x)/kT p (x) n (x) Clif Fonstad, 9/24/09 Lecture 5 - Slide 7 Courtesy of Prof. Peter Hagelstein. Used with permission.The Depletion Approximation: comparison cont. Example: Same sample, forward biased 0.6 V Charge E-field Potential nie±qφ(x)/kT p (x) n (x) Clif Fonstad, 9/24/09 Lecture 5 - Slide 8Courtesy of Prof. Peter Hagelstein. Used with permission.The value of the depletion approximation The plots look good, but equally important is that 1. It gives an excellent model for making hand calculationsand gives us good values for quantities we care about: • Depletion region width• Peak electric field • Potential step 2. It gives us the proper dependences of these quantities onthe doping levels (relative and absolute) and the bias voltage. Apply bias; what happens? Two things happen 1. The depletion width changes• (φb - vAB) replaces φb in the Depletion Approximation Eqs. 2. Currents flow • This is the main topic of today’s lecture Clif Fonstad, 9/24/09 Lecture 5 - Slide 9xn -xp qA qB ( = -q A) -qNAp qNDn Depletion capacitance: Comparing depletion charge stores with a parallel plate capacitor ρ(x) ρ(x) d/2 -d/2 qA qB( = -qA) xx Depletion region charge storeParallel plate capacitor ! qA,PP= A"dvABCpp(VAB) #$qA,PP$vABvAB=VAB=A"dMany similarities; important differences. ! qA,DP(vAB) = "AqNApxpvAB( )= "A 2q#Si$b" vAB[ ]NApNDnNAp+ NDn[ ]Cdp(VAB) %&qA,DP&vABvAB=VAB= Aq#Si2$b" VAB[ ]NApNDnNAp+ NDn[ ]=A#Siw(VAB)Clif Fonstad, 9/24/09 Lecture 5 - Slide 10Bias applied, cont.: With vAB ≠ 0, it is not true that n(x) = ni eqφ(x)/kT and p(x) = ni e-qφ(x)/kT because we are no longer in TE. However, outside of the depletion region things are in quasi-equilibrium, and we can define local electrostatic potentials for which the equilibrium relationships hold for the majority carriers, assuming LLI. Forward bias, vAB > 0: In this region n(x) ≈ ni eqφQNRn/kT Reverse bias, vAB < 0: -wp x wn-xp 0 xn vAB φQNRn (φb-vAB) φQNRp vAB vAB vAB x -wp wn0 xn vAB φQNRn (φb-vAB) φQNRp vAB vAB -xp vAB In this region p(x) ≈ ni e-qφQNRp/kT Clif Fonstad, 9/24/09 Lecture 5 - Slide 11Current Flow qφ Unbiased junction Population inequilibrium with barrier qφ Forward bias on junction Barrier lowered so carriers to left can cross over it. qφ Reverse bias on junction Barrier raised so the few carriers on top spill back down it. Clif Fonstad, 9/24/09 Lecture 5 - Slide 12 x x xCurrent flow: finding the relationship between iD and vAB There are two pieces to the problem: • Minority carrier flow in the QNRs is what limits the current. • Carrier equilibrium across the SCR determines n'(-xp) and p'(xn), the boundary conditions of the QNR minority carrier flow problems. Ohmic Uniform p-type Uniform n-type -wp -xp0 xn wn x p n Ohmic contact contact A B iD + -vAB Quasineutral
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