Lecture #15Linearly Graded JunctionBiased PN JunctionsEffect of Bias on Electrostaticspn Junction Electrostatics, VA 0Slide 6Peak Electric FieldCurrent Flow - QualitativeCurrent Flow in a pn Junction DiodeSlide 10Ideal Diode Analysis: AssumptionsIdeal Diode Analysis: ApproachCarrier Concentrations at –xp, xn“Law of the Junction”Excess Carrier Concentrations at –xp, xnExample: Carrier InjectionExcess Carrier DistributionSlide 18pn Diode I-V CharacteristicSlide 20Diode Saturation Current I0SummaryEE130 Lecture 15, Slide 1Spring 2007Lecture #15OUTLINE• pn junction I-V characteristicsReading: Chapter 6.1NOTE:•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 if NA net acceptor doping on p-side (NA-ND)p-side ND net donor doping on n-side (ND-NA)n-sideEE130 Lecture 15, Slide 2Spring 2007Linearly Graded JunctionEE130 Lecture 15, Slide 3Spring 2007Biased PN JunctionsNote that VA should be significantly smaller than Vbi(Otherwise, we cannot assume low-level injection)EE130 Lecture 15, Slide 4Spring 2007Effect of Bias on ElectrostaticsEE130 Lecture 15, Slide 5Spring 2007pn Junction Electrostatics, VA 0•Built-in potential Vbi (non-degenerate doping):•Depletion width W :2lnlnlniDAiDiAbinNNqkTnNqkTnNqkTVDAAbisnpNNVVqxxW11)(2WNNNxDADp WNNNxDAAnEE130 Lecture 15, Slide 6Spring 2007•Electric field distribution (x)•Potential distribution V(x))()0( that NoteAbiDADVVNNNV EE130 Lecture 15, Slide 7Spring 2007Peak Electric Field•For a one-sided junction:thereforeAbiVVWdx )0(21 sAbiAbiVVqNWVV22)0()(2AbisVVqNW EE130 Lecture 15, Slide 8Spring 2007Current Flow - QualitativeEE130 Lecture 15, Slide 9Spring 2007Current 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 carriersEE130 Lecture 15, Slide 10Spring 2007•Current density J = Jn(x) + Jp(x)•J is constant throughout the diode, but Jn(x) and Jp(x) vary with positiondxndqDnqdxdnqDnqxJnnnnn)()(dxpdqDpqdxdpqDpqxJppppp)()(EE130 Lecture 15, Slide 11Spring 2007Ideal 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 & Jp are constant inside the depletion region0 ,0 dxdJdxdJpnEE130 Lecture 15, Slide 12Spring 2007Ideal 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 Jn at x=-xp and Jp at x=xn J(VA) = Jn(VA)|x=-xp + Jp(VA )|x=xn dxndqDVxJpnAn)(),(dxpdqDVxJnpAp)(),(EE130 Lecture 15, Slide 13Spring 2007Carrier Concentrations at –xp, xnn-sidep-sideConsider the equilibrium (VA = 0) carrier concentrations:A20A0)()(NnxnNxpippppD20D0)()(NnxpNxninnnnIf low-level injection conditions prevail in the quasi-neutralregions when VA 0, thenA)( NxpppD)( NxnnnEE130 Lecture 15, Slide 14Spring 2007“Law of the Junction”The voltage VA applied 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/)(iiNkTFEenp/)(iPikTqVenpn/2iA kTFFkTEFkTFEPNiNPieneenpn/)(2i/)(/)(2i EE130 Lecture 15, Slide 15Spring 2007Excess Carrier Concentrations at –xp, xn 1)( )()(/A2/0A/2AAAAkTqVippkTqVpkTqVippppeNnxnenNenxnNxpn-sidep-side 1)( )()(/D2/0D/2DAAAkTqVinnkTqVnkTqVinnnneNnxpepNenxpNxnEE130 Lecture 15, Slide 16Spring 2007Example: Carrier InjectionA pn junction has NA=1018 cm-3 and 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 15, Slide 17Spring 2007Excess 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 ppLxLxneAeAxp/'2/'1)'(NEW: x’’ 0 0 x’EE130 Lecture 15, Slide 18Spring 2007From the x = boundary condition, A1 = 0.From the x = xn boundary condition, Therefore,Similarly, we can derive0' ,)1()'(/'/xeepxppALxkTqVnonppLxLxneAeAxp/'2/'1)'(0'' ,)1()''(/''/xeenxnnALxkTqVpop)1(/2kTqVnoAepAEE130 Lecture 15, Slide 19Spring 2007pLxkTVqnppnppeepLDqdxxpdqDJ'0)1(')'(AnLxkTVqpnnpnneenLDqdxxndqDJ''0)1('')''(Apn Diode I-V Characteristicn-side:p-side:)1(ADA2i00kTVqppnnxpxnxxpxxneNLDNLDqnJJJJJJnpEE130 Lecture 15, Slide 20Spring 2007)1(0kTVqAeIIAnnDppiNLDNLDAqnI20EE130 Lecture 15, Slide 21Spring 2007Diode Saturation Current I0•I0 can vary by orders of magnitude, depending on the semiconductor material•In an asymmetrically doped pn junction, the term associated with the more heavily doped side is negligible:–If the p side is much more heavily doped,–If the n side is much more heavily doped,AnnDppiNLDNLDAqnI20DppiNLDAqnI20AnniNLDAqnI20EE130 Lecture 15, Slide 22Spring 2007Summary•The
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