1EE130 Lecture 12, Slide 1Spring 2003Lecture #12OUTLINE– pn Junctions• narrow-base diode• charge-control modelReading: Finish Chapter 6EE130 Lecture 12, Slide 2Spring 2003Carrier Concentration Profiles2EE130 Lecture 12, Slide 3Spring 2003Narrow or Short-Base Diode• We have the following boundary conditions:• With the following coordinate system:• Then, the solution is of the form:0)''( →=∆cnxxp)1()(/−=∆kTqVnonnAepxpppLxLxeAeAxp/2/1)(′−′+=∆NEW:x’’ 0 0 x’0 x’cEE130 Lecture 12, Slide 4Spring 2003Applying the boundary conditions, we have:• So, we have:• Note: () ()'///'/'/0'0 ,)1()'(''''cLxLxLxxLxxkTqVnnxxeeeeepxpPcPcPcPcA<<−−−=∆−−−−pcpcLxLxneAeAAAp/2/121''0)0(−+=+=∆()2sinhξξξ−−=ee3EE130 Lecture 12, Slide 5Spring 2003•So:• If we write:• Thenwhere()[][]'''/0'0 ,/sinh/'sinh)1()'(cPcPckTqVnnxxLxLxxepxpA<<−−=∆)1('0−=kTVqAeII()()PcPcDiPPLxLxNnLDqAI/sinh/cosh'0''2=()2coshξξξ−+=eeEE130 Lecture 12, Slide 6Spring 2003()0 as sinh →→ξξξNote:()0 as 1cosh2→+→ξξξ• If xc’ << LP:and4EE130 Lecture 12, Slide 7Spring 2003Narrow Base Diode: I-V EquationLetandThen,region type-p of width region type-n of width ≡≡PNWW()1/2−′+′=kTqVAPNDNPiAeNWDNWDqnJNpPPPnNNLxWWLxWW<<−≡′<<−≡′()()11/0/2−=−′+′=kTqVkTqVAPNDNPiAAeIeNWDNWDqAnIEE130 Lecture 12, Slide 8Spring 2003Current Flow in a One-Sided pn Junction• Note that the diode current is dominated by the term associated with the more lightly doped side:p+n diode:pn+diode:i.e. current flowing across junction is dominated by carriers injected from the more heavily doped side sidenshort siden long )( 220−′−=≅DNPiDPPinPNWDqAnNLDqAnxII sidepshort sidep long )(220−′−=−≅APNiANNipNNWDqAnNLDqAnxII5EE130 Lecture 12, Slide 9Spring 2003Excess Carrier Profiles: Limiting CasesLong base (xc’ → ∞):() ()pAPcPcpPcpPcAPcPcPcPcALxkTqVnLxLxLxLxLxLxkTqVnLxLxLxxLxxkTqVnneepeeeeeeepeeeeepxp/'/0///'//'//0///'/'/0)1( )1( )1()'(''''''''−−−−−−−−−≅−−−=−−−=∆EE130 Lecture 12, Slide 10Spring 20032. Short base (xc’ → 0):∆pnis a linear function of xÆ Jpis constant (no recombination)()[][]()−−=−−=−−=∆'/0''/0''/0'1)1(//')1(/sinh/'sinh)1()'(AAckTqVnPcPckTqVnPcPckTqVnnxxepLxLxxepLxLxxepxpA6EE130 Lecture 12, Slide 11Spring 2003Minority-Carrier Charge Storage• When VA>0, excess minority carriers are stored in the quasi-neutral regions of a pn junction:PnnxnPLxpqAdxxpqAQn)( )(∆=∆=∫∞NppxpNLxnqAdxxnqAQp)( )(−∆−=∆−=∫−∞−EE130 Lecture 12, Slide 12Spring 2003Derivation of Charge Control Model• Consider a forward-biased pn junction. The total excess hole charge in the n quasi-neutral region is:• The minority carrier diffusion equation is (without GL):• Since the electric field is very small,• Therefore:∫∞∆=nxnPdxtxpqAQ ),(pnnPnpxpDtpτ∆−∂∆∂=∂∆∂22xpPPnqDJ∂∆∂−=pnPnpqxJtpqτ∆−∂∂−=∂∆∂ )(7EE130 Lecture 12, Slide 13Spring 2003• Integrating over the n quasi-neutral region:• Furthermore, in a p+n junction:•So:∆−−=∆∂∂∫∫∫∞∞∞nPnpnxnpJxJPxndxpqAdJAdxpqAtτ1)()(DIFFnPnPPJxJPixAJxAJAJdJAPnp≅=+∞−=−∫∞)()()()()(pPDIFFPQidtdQτ−=(Long Base Diode)EE130 Lecture 12, Slide 14Spring 2003Charge Control ModelWe can calculate pn-junction current in 2 ways:1. From slopes of ∆np(-xp) and ∆pn(xn) 2. From steady-state charges QN, QPstored in each excess-minority-charge distribution:pPnPnPQxIxAJτ)()( ==⇒nNpNQxIτ)( Similarly,−=−0τ)( =−=pPnPPQxAJdtdQ8EE130 Lecture 12, Slide 15Spring 2003Charge Control Model for Narrow Base• For a narrow-base diode, replace τpand/or τnby the minority-carrier transit time τtr– time required for minority carrier to travel across the quasi-neutral region– For holes on narrow n-side:– Similarly, for electrons on narrow p-side:()PNPPtrNnnPnPPPNnnWxnPDWIQWxpqADdxpdqADAJIWxpqAdxxpqAQNn2τ )()(21)(2′==⇒′∆=∆−==′∆=∆=∫()NPtrDW2τ2′=EE130 Lecture 12, Slide 16Spring 2003Summary: Narrow (Short) Base Diode• If the width of the quasi-neutral region (e.g. xc’) is notmuch larger than the minority-carrier diffusion length (e.g. LP), then the solution to the minority-carrier diffusion equation is•If xc’ < 0.1LP: ′−′∆=∆PcPcnnnLxLxxxpxpsinhsinh)()'(′′−∆≅∆cnnnxxxpxp 1)()'(9EE130 Lecture 12, Slide 17Spring 2003•If LP>> xc’, negligible recombination occurs⇒ hole current is constant throughout n-type region• Compare this to the hole current contribution in the long-base
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