EECS 105 Spring 2004 Lecture 19 EECS 105 Spring 2004 Lecture 19 Prof J S Smith Context The first part of this lecture is a review of electrons and holes in silicon z Fermi levels and Quasi Fermi levels z Majority and minority carriers z Drift z Diffusion And we will apply these to z Diode Currents in forward and reverse bias chapter 6 z BJT Bipolar Junction Transistors in the next lecture Lecture 19 Review PN junctions Fermi levels forward bias Prof J S Smith Department of EECS EECS 105 Spring 2004 Lecture 19 University of California Berkeley Department of EECS University of California Berkeley Prof J S Smith EECS 105 Spring 2004 Lecture 19 Prof J S Smith Electrons and Holes z Electrons and Holes Electrons in silicon can be in a number of different states z Electrons in silicon can be in a number of different states Empty states Fermi level In thermal equilibrium at each location the electrons will fill the states up to a particular level Department of EECS University of California Berkeley Department of EECS Band gap Full States University of California Berkeley 1 EECS 105 Spring 2004 Lecture 19 Prof J S Smith Fermi function z z z z Prof J S Smith Exponential approximation electrons In thermal equilibrium the probability of occupancy of any state is given by the Fermi function 1 F E E E f 1 e EECS 105 Spring 2004 Lecture 19 z z kT In semiconductors the Fermi energy is usually in the band gap far from either the conduction band or the valence band compared to kT For the conduction band since the exponential is much larger than 1 we can use the approximation At the energy E Ef the probability of occupancy is 1 2 At high energies the probability of occupancy approaches zero exponentially At low energies the probability of occupancy approaches 1 F E 1 1 e E E f 1 kT E E f e e E E f kT kT Department of EECS University of California Berkeley Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 19 Prof J S Smith EECS 105 Spring 2004 Lecture 19 Prof J S Smith Electrons z Exponential approximation holes Under this approximation we can integrate over the conduction band states and we can write the result as E E f n Nce z kT For the valence band band since the exponential is much smaller than 1 we can use the approximation E E f kT 1 F E 1 e E E f 1 e Where Nc is a number called the effective density of states in the conduction band Department of EECS kT 1 1 x for x small 1 x Since we are counting holes as the absence of an electron we have the probability of not having an electron in a state E E f kT e since z University of California Berkeley Department of EECS University of California Berkeley 2 EECS 105 Spring 2004 Lecture 19 Prof J S Smith EECS 105 Spring 2004 Lecture 19 Holes z Intrinsic concentrations Under this approximation we can also integrate over the conduction band states and we can write the result as E E f p NV e Prof J S Smith z In thermal equilibrium the Fermi energy must be the same everywhere including the Fermi energy for the electrons and the holes so kT pn NV e z Where N is a number v called the effective density of states in the valence band z E E f kT Nce E E f NV N c e kT 2 We call this constant ni undoped semiconductor T 2 Ef kT ni2 T because in a neutral p n ni Department of EECS University of California Berkeley Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 19 Prof J S Smith EECS 105 Spring 2004 Lecture 19 Prof J S Smith Non Neutral Non Equilibrium Our devices z won t be in thermal equilibrium or they wouldn t do anything interesting z They mostly won t be undoped z They might not be neutral such as in a depletion zone Rapid thermalization z z z But from these intrinsic equilibrium neutral results develop many useful approximations Department of EECS University of California Berkeley Even 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 T If they don t its called a Hot Carrier Effect In 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 Department of EECS University of California Berkeley 3 EECS 105 Spring 2004 Lecture 19 Prof J S Smith EECS 105 Spring 2004 Lecture 19 Minority Carriers z z n and p This 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 equilibrium In an N type material for example holes can be injected raising ni2 In neutral silicon which is doped with an acceptor at a density far above the intrinsic carrier concentration n N d Nce n E E f n N d ni e n 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 2 The np product can be reduced below as in a reverse biased junction i kT Which we can rewrite as p Prof J S Smith And similarly n 0 kT p N a ni e q p 0 kT Since electrons are negative potentials come out to be the negative of energy Energy qV e Department of EECS University of California Berkeley Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 19 Prof J S Smith EECS 105 Spring 2004 Lecture 19 Prof J S Smith Reference intrinsic P and N regions in thermal Eq These equations use intrinsic silicon as a reference so the point where n 0 and p 0 n ni e q n 0 kT n ni p ni e q p 0 kT ni Conduction band 0 intrinsic N type doped with Donors fixed positive ions Department of EECS and p Valence band P type doped with Acceptors fixed negative ions University of California Berkeley z Remember though that in thermal equilibrium it is the Fermi levels that are the same everywhere n p P type doped with Acceptors fixed negative ions N type doped with Donors fixed positive ions Department of EECS University of California Berkeley 4 EECS 105 Spring 2004 Lecture 19 Prof J S Smith EECS 105 Spring …
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