Lecture #9Derivation of Continuity EquationSlide 3Derivation of Minority Carrier Diffusion EquationSlide 5Carrier Concentration NotationSimplifications (Special Cases)ExampleSlide 9Minority Carrier Diffusion LengthQuasi-Fermi LevelsExample: Quasi-Fermi LevelsSlide 13SummarySlide 15Lecture #9OUTLINE• Continuity equations• Minority carrier diffusion equations• Minority carrier diffusion length• Quasi-Fermi levelsRead: Sections 3.4, 3.5EE130 Lecture 9, Slide 2Spring 2007Derivation of Continuity Equation•Consider carrier-flux into/out-of an infinitesimal volume:JN(x) JN(x+dx)dxArea A, volume Adx AdxnAdxxJAxJqtnAdxnNN)()(1EE130 Lecture 9, Slide 3Spring 2007nNNNNnxxJqtndxxxJxJdxxJ)(1 )()()(LpPLnNGpxxJqtpGnxxJqtn )(1 )(1ContinuityEquations:EE130 Lecture 9, Slide 4Spring 2007Derivation of Minority Carrier Diffusion Equation•The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers.•Simplifying assumptions:–The electric field is small, such that in p-type material in n-type material–n0 and p0 are independent of x (uniform doping)–low-level injection conditions prevailxnqDxnqDnqJNNnNxpqDxpqDpqJPPpPEE130 Lecture 9, Slide 5Spring 2007•Starting with the continuity equation for electrons: LnNLnNLnNGnxnDtnGnxnnqDxqtnnGnxxJqtn 1 )(12200EE130 Lecture 9, Slide 6Spring 2007Carrier Concentration Notation•The subscript “n” or “p” is used to explicitly denote n-type or p-type material, e.g.pn is the hole (minority-carrier) concentration in n-type materialnp is the electron (minority-carrier) concentration in n-type material•Thus the minority carrier diffusion equations areLpnnPnLnppNpGpxpDtpGnxnDtn 2222EE130 Lecture 9, Slide 7Spring 2007Simplifications (Special Cases)•Steady state:•No diffusion current:•No R-G:•No light:0 0 tptnnp0 02222xpDxnDnPpN0 0 pnnppn0 LGEE130 Lecture 9, Slide 8Spring 20072PP22LpDpxpnpnnLP is the hole diffusion length:pPPDLExample•Consider the special case:–constant minority-carrier (hole) injection at x=0–steady state; no light absorption for x>0pnnPpxpD2200)0(nnpp EE130 Lecture 9, Slide 9Spring 2007The general solution to the equationiswhere A, B are constants determined by boundary conditions:Therefore, the solution is 2P22LpxpnnPPLxLxnBeAexp//)( 0 0)( Bpn00 )0(nnnpApp PLxnnAepxp/0)(EE130 Lecture 9, Slide 10Spring 2007•Physically, LP and LN represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated.•Example: ND=1016 cm-3; p = 10-6 sMinority Carrier Diffusion LengthEE130 Lecture 9, Slide 11Spring 2007•Whenever n = p 0, np ni2. However, we would like to preserve and use the relations:•These equations imply np = ni2, however. The solution is to introduce two quasi-Fermi levels FN and FP such thatQuasi-Fermi Levels /)( kTEEiiFenn /)( kTEEiFienp /)( kTEFiiNenn /)( kTFEiPienpiiPnpkTEF lniiNnnkTEF lnEE130 Lecture 9, Slide 12Spring 2007Example: Quasi-Fermi LevelsConsider a Si sample with ND = 1017 cm-3 and n = p = 1014 cm-3.What are p and n ?What is the np product ?EE130 Lecture 9, Slide 13Spring 2007•Find FN and FP :iiPnpkTEF lniiNnnkTEF lnEE130 Lecture 9, Slide 14Spring 2007Summary•The continuity equations are established based on conservation of carriers, and therefore are general:•The minority carrier diffusion equations are derived from the continuity equations, specifically for minority carriers under certain conditions (small E-field, low-level injection, uniform doping profile):LpPLnNGpxxJqtpGnxxJqtn )(1 )(1LpnnPnLnppNpGpxpDtpGnxnDtn 2222EE130 Lecture 9, Slide 15Spring 2007•The minority carrier diffusion length is the average distance that a minority carrier diffuses before it recombines with a majority carrier:•The quasi-Fermi levels can be used to describe the carrier concentrations under non-equilibrium conditions:pPPDLnNNDLiiPnpkTEF lniiNnnkTEF
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