1EE130 Lecture 11, Slide 1Spring 2003Lecture #11OUTLINE• pn Junctions – reverse breakdown– ideal diode analysis» current flow (qualitative) » minority carrier distributionsReading: Chapter 6EE130 Lecture 11, Slide 2Spring 2003A Note of Caution• Typically, pn junctions in IC devices are formed by counter-doping. The equations derived in class (and in the textbook) can be readily applied to such diodes ifNA≡ net acceptor doping on p-side (NA-ND)p-sideND≡ net donor doping on n-side (ND-NA)n-side2EE130 Lecture 11, Slide 3Spring 2003pn Junction Electrostatics, VA≠ 0• Built-in potential Vbi (non-degenerate doping):• Depletion width W :=+=2lnlnlniDAiDiAbinNNqkTnNqkTnNqkTV+−=+=DAAbisnpNNVVqxxW11)(2εWNNNxDADp+=WNNNxDAAn+=EE130 Lecture 11, Slide 4Spring 2003• Electric field distribution ε(x)• Potential distribution V(x))()0( that NoteAbiDADVVNNNV −+=3EE130 Lecture 11, Slide 5Spring 2003Peak Electric Field• For a one-sided junction:thereforeAbiVVWdx −==∫)0(21 εε() ()sAbiAbiVVqNWVVεε−≅−=22)0()(2AbisVVqNW −≅εEE130 Lecture 11, Slide 6Spring 2003A Zener diode is designed to operate in the breakdown mode.VIVBRPNARForward CurrentSmall leakageCurrent(a)3.7VR(b)ICZener diodeJunction Breakdown4EE130 Lecture 11, Slide 7Spring 2003•If Vreverse= -VAis so large such that the peak electric field exceeds a critical value εcrit, then the junction will break down (large reverse current will flow)• Thus, the reverse bias at which breakdown occurs is bicritsBRVqNV −=22εε()sBRbicritVVqNεε+=2EE130 Lecture 11, Slide 8Spring 2003if VBR>> Vbiεcritincreases slightly with N:For 1014cm-3< N < 1018cm-3, 105V/cm <εcrit< 106V/cmqNVcritsBR22εε≈Avalanche Breakdown MechanismSmall E-field:High E-field:5EE130 Lecture 11, Slide 9Spring 2003Dominant breakdown mechanism when both sides of a junction are very heavily doped.EcEvVA= 0:EvEcEmpty StatesFilled States-Tunneling (Zener) Breakdown Mechanism VA< 0:bicritsBRVqNV −=22εεV/cm 106≈critεTypically, VBR< 5 V for Zener breakdownEE130 Lecture 11, Slide 10Spring 2003Empirical Observations of VBR• VBRdecreases with increasing N• VBRdecreases with decreasing EG6EE130 Lecture 11, Slide 11Spring 2003Breakdown Temperature Dependence• For the avalanche mechanism: VBRincreases with increasing T– Mean free path decreases• For the tunneling mechanism: VBRdecreases with increasing T– Flux of valence-band electrons available for tunneling increasesEE130 Lecture 11, Slide 12Spring 2003Current Flow in a pn Junction Diode• When a forward bias (VA>0) is applied, the potential barrier to diffusion across the junction is reduced– Minority carriers are “injected” into the quasi-neutral regions => ∆np> 0, ∆pn> 0 • Minority carriers diffuse in the quasi-neutral regions, recombining with majority carriers7EE130 Lecture 11, Slide 13Spring 2003• Current density J = Jn(x) + Jp(x)• J is constant throughout the diode, but Jn(x) and Jp(x) vary with positiondxndqDnqdxdnqDnqxJnnnnn)()(∆+=+=εε µµdxpdqDpqdxdpqDpqxJppppp)()(∆−=−=εε µµEE130 Lecture 11, Slide 14Spring 2003Ideal Diode Analysis: Assumptions• Non-degenerately doped step junction• Steady-state conditions• Low-level injection conditions prevail in the quasi-neutral regions• Recombination-generation is negligible in the depletion regioni.e. Jn& Jpare constant inside the depletion region0 ,0 ==⇒dxdJdxdJpn8EE130 Lecture 11, Slide 15Spring 2003Ideal Diode Analysis: Approach• Solve the minority-carrier diffusion equations in quasi-neutral regions to obtain ∆np(x,VA),∆pn(x,VA)– apply boundary conditions •p-side: ∆np(-xp), ∆np(-∞)•n-side: ∆pn(xn), ∆pn(∞)• Determine minority-carrier current densities in quasi-neutral regions• Evaluate Jnat x=-xp and Jpat x=xn J(VA) = Jn(VA)|x=-xp+ Jp(VA )|x=xndxndqDVxJpnAn)(),(∆=dxpdqDVxJnpAp)(),(∆−=EE130 Lecture 11, Slide 16Spring 2003Carrier Concentrations at –xp, xnn-sidep-sideConsider the equilibrium (VA= 0) carrier concentrations:A20A0)()(NnxnNxpipppp=−=−D20D0)()(NnxpNxninnnn==If low-level injection conditions prevail in the quasi-neutralregions when VA≠0, thenA)( Nxppp=−D)( Nxnnn=9EE130 Lecture 11, Slide 17Spring 2003“Law of the Junction”The voltage VAapplied to a pn junction falls mostly acrossthe depletion region (assuming that low-level injection conditions prevail in the quasi-neutral regions).We can draw 2 quasi-Fermi levels in the depletion region:kTEFenn/)(iiN−=kTFEenp/)(iPi−=kTqVenpn/2iA =kTFFkTEFkTFEPNiNPieneenpn/)(2i/)(/)(2i −−−==EE130 Lecture 11, Slide 18Spring 2003Excess Carrier Concentrations at –xp, xn()1)( )()(/A2/0A/2AAAA−=−∆==−=−kTqVippkTqVpkTqVippppeNnxnenNenxnNxpn-sidep-side()1)( )()(/D2/0D/2DAAA−=∆===kTqVinnkTqVnkTqVinnnneNnxpepNenxpNxn10EE130 Lecture 11, Slide 19Spring 2003Example: Carrier InjectionA pn junction has NA=1018 cm-3and ND=1016 cm-3. The applied voltage is 0.6 V.Question: What are the minority carrier concentrations at the depletion-region edges?Answer:Question: What are the excess minority carrier concentrations?Answer:-312026.06.0cm 10100)( =×==− eenxnkTVqpoppA-314026.06.04cm 1010)( =×== eepxpkTVqnonnA-31212cm 1010010)()( =−=−−=−∆poppppnxnxn-314414cm 101010)()( =−=−=∆nonnnnpxpxpEE130 Lecture 11, Slide 20Spring 2003Excess Carrier Distribution• From the minority carrier diffusion equation:• We have the following boundary conditions:• For simplicity, we will develop a new coordinate system:• Then, the solution is of the form:0)( →∞∆np)1()(/−=∆kTqVnonnAepxp222pnppnnLpDpdxpd ∆=∆=∆τppLxLxneAeAxp/'2/'1)'(−+=∆NEW: x’’ 0 0 x’11EE130 Lecture 11, Slide 21Spring 2003From the x = ∞ boundary condition, A1= 0.From the x = xnboundary condition, Therefore,Similarly, we can derive0' ,)1()'(/'/>−=∆−xeepxppALxkTqVnonppLxLxneAeAxp/'2/'1)'(−+=∆0'' ,)1()''(/''/>−=∆−xeenxnnALxkTqVpop)1(/2−=kTqVnoAepAEE130 Lecture 11, Slide 22Spring 2003pLxkTVqnppnppeepLDqdxxpdqDJ'0)1(')'(A−−=∆−=nLxkTVqpnnpnneenLDqdxxndqDJ''0)1('')''(A−−=∆−=pn Diode I-V Characteristicn-side:p-side:)1(ADA2i00−+=+=+==′=′′=−=kTVqppnnxpxnxxpxxneNLDNLDqnJJJJJJnp12EE130 Lecture 11, Slide 23Spring 2003)1(0−=kTVqAeII+=AnnDppiNLDNLDAqnI20EE130 Lecture 11, Slide 24Spring
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