1Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19Lecture 19: Review, PN junctions,Fermi levels, forward biasProf J. S. SmithDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithContextThe first part of this lecture is a review of electrons and holes in silicon:zFermi levels and Quasi-Fermi levelszMajority and minority carrierszDrift zDiffusionAnd we will apply these to:zDiode Currents in forward and reverse bias (chapter 6)zBJT (Bipolar Junction Transistors) in the next lecture.Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithElectrons and HoleszElectrons in silicon can be in a number of different states:Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithElectrons and HoleszElectrons in silicon can be in a number of different states:Fermi levelFull States In thermal equilibrium,at each location theelectrons will fill thestates up to a particular levelEmpty states↕ Band gap ↕2Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithFermi functionzIn thermal equilibrium, the probability of occupancy of any state is given by the Fermi function:zAt the energy E=Efthe probability of occupancy is 1/2. zAt high energies, the probability of occupancy approaches zero exponentiallyzAt low energies, the probability of occupancy approaches 1kTEEfeEF−+=11)(Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithExponential approximation (electrons)zIn semiconductors, the Fermi energy is usually in the band gap, far from either the conduction band or the valence band (compared to kT).zFor the conduction band, since the exponential is much larger than 1, we can use the approximation:⎟⎟⎠⎞⎜⎜⎝⎛−−−−=≈+=kTEEkTEEkTEEfffeeeEF111)(Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithElectronszUnder this approximation, we can integrate over the conduction band states, and we can write the result as:Where Ncis a number,called the effectivedensity of states in the conduction bandkTEEcfeNn−−=Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithExponential approximation (holes)zFor the valence band band, since the exponential is much smaller than 1, we can use the approximation:zSince we are counting holes as the absence of an electron, we have the probability of not having an electron in a state:⎟⎟⎠⎞⎜⎜⎝⎛−−−≈+=kTEEkTEEffeeEF 111)(small)(for x 111 since x x −≈+⎟⎟⎠⎞⎜⎜⎝⎛−kTEEfe3Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithHoleszUnder this approximation, we can also integrate over the conduction band states, and we can write the result as:zWhere Nvis a number,called the effectivedensity of states in the valence bandkTEEVfeNp−=Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithIntrinsic concentrationszIn thermal equilibrium, the Fermi energy must be the same everywhere, including the Fermi energy for the electrons and the holes, so: zWe call this constant because in a neutral, undoped semiconductor )(22TneNNeNeNpnikTEcVkTEEckTEEVfff===−−−−innp ==)(2TniDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithNon Neutral, Non EquilibriumOur devices:zwon’t be in thermal equilibrium (or they wouldn’t do anything interestingzThey mostly won’t be undopedzThey might not be neutral (such as in a depletion zone)But from these intrinsic, equilibrium, neutral results develop many useful approximations.Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithRapid thermalizationzEven if a semiconductor is not in thermal equilibrium, electrons and holes can exchange energy with each other and the lattice so quickly that they mostly remain in a thermal distribution at a temperature TzIf they don’t, its called a Hot Carrier EffectzIn the absence of hot carrier effects, both the electron and hole occupancies will be given by the Fermi function, but the distributions for the electrons and the holes may not be given by the same Fermi energy!4Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithMinority CarrierszThis is because electrons take a relatively long time to recombine with holes, and that electrons in the valence band to jump to the conduction band and form an electron hole pair, so the relative number of electrons and holes can diverge far from equilibriumzIn an N type material, for example, holes can be injectedraising –The carrier type which there are fewer of are called minority carriers, in this case the holes. –Strangely enough, as we will see, the minority carriers often dominate the transport through a device.–Minority carriers aren’t so important for FET’s, so they are called majority carrier devices–The np product can be reduced below , as in a reverse biased junctionnnpi2>2inDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. Smithφnand φpIn neutral silicon which is doped with an acceptor at a density far above the intrinsic carrier concentration:Which we can rewrite as:And similarly:Since electrons are negative, potentials come out to be the negative of energy:kTEEcdfeNNn−−=≈kTidnenNn0φ−=≈kTqiapenNp0φ=≈φ)(Energy eqV −==Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithReference: intrinsicThese equations use intrinsic silicon as a reference so the point where φn=0 and φp=0→ikTqinennn===−)0(φikTqinenpp=== )0(φand←Conduction band→←Valence band→nφpφ)(−N type, doped withDonors (fixed positive ions)P type, doped withAcceptors (fixed negative ions)← φ=0, intrinsic →Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 19 Prof. J. S. SmithP and N regions in thermal Eq.zRemember, though, that in thermal equilibrium it is the Fermi levels that are the same everywherenφpφ)(−N type, doped withDonors (fixed positive ions)P type, doped withAcceptors
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