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EECS 105 Spring 2004, Lecture 19 Lecture 19: Review, PN junctions, Fermi levels, forward bias Prof J. S. Smith Department of EECS University of California, Berkeley 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. Department of EECS University of California, Berkeley 1 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Electrons and Holes z Electrons in silicon can be in a number of different states: Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Electrons and Holes z Electrons in silicon can be in a number of different states: Empty states Fermi level ↕ Band gap ↕ In thermal equilibrium, at each location the electrons will fill the Full States states up to a particular level Department of EECS University of California, Berkeley 2 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Fermi function z In thermal equilibrium, the probability of occupancy of any state is given by the Fermi function: 1 F(E) = E−E f 1+ e kT z At the energy E=Ef the probability of occupancy is 1/2. z At high energies, the probability of occupancy approaches zero exponentially z At low energies, the probability of occupancy approaches 1 Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Exponential approximation (electrons) z In semiconductors, the Fermi energy is usually in the band gap, far from either the conduction band or the valence band (compared to kT). z For the conduction band, since the exponential is much larger than 1, we can use the approximation: ⎛ E−E ⎞ ⎜ f ⎟ 1 1 −⎜ ⎟ F(E) = ≈ = e ⎝ kT ⎠ E−E f E−E f 1+ e kT e kT Department of EECS University of California, Berkeley 3 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Electrons z Under this approximation, we can integrate over the conduction band states, and we can write the result as: E−E − f kT n = Nce Where Nc is a number, called the effective density of states in the conduction band Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Exponential approximation (holes) z For the valence band band, since the exponential is much smaller than 1, we can use the approximation: ⎛ E−E ⎞ ⎜ f ⎟ 1 ⎜ ⎟ F(E) = ≈1− e⎝ kT ⎠ E−E f 1+ e kT 1 since ≈1− x (for x small) 1+ x z 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 ⎞ ⎜ ⎟ e⎝ kT ⎠ Department of EECS University of California, Berkeley 4 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Holes z Under this approximation, we can also integrate over the conduction band states, and we can write the result as: E−E f kT p = NV e z Where Nv is a number, called the effective density of states in the valence band Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Intrinsic concentrations z In thermal equilibrium, the Fermi energy must be the same everywhere, including the Fermi energy for the electrons and the holes, so: E−E E−E E f − f −2 f kT kT kT 2 pn = NV e Nce = NV Nce = ni (T ) 2 z We call this constant n i ( T ) because in a neutral, undoped semiconductor p = n = ni Department of EECS University of California, Berkeley 5 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) But from these intrinsic, equilibrium, neutral results develop many useful approximations. Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Rapid thermalization z 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 z If they don’t, its called a Hot Carrier Effect z 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 6 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Minority Carriers z 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 z In an N type material, for example, holes can be injected raising n 2 p > i 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 ni Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith φn and φp In neutral silicon which is doped with an acceptor at a density far above the intrinsic carrier concentration: E−E − f kT n ≈ Nd = Nce Which we can rewrite as: φ − n0 kT n ≈ Nd = nie And similarly: qφ p 0 kT p ≈ Na = nie Since electrons are negative, potentials come out to be the negative of energy: Energy = qV = (−e)φ Department of EECS University of California, Berkeley 7 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Reference: intrinsic These equations use intrinsic silicon as a reference so the point where φn=0 and φp=0→ q(φ =0) − n q(φ p =0) n = n e kT = n and kT i i p = nie = ni ←Conduction band→ φ n ← φ=0, intrinsic → (−)φ p ←Valence band→ N type, doped with P type, doped with Donors (fixed positive ions) Acceptors (fixed negative ions) Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith P and N regions in thermal Eq. z Remember, though, that in thermal equilibrium it is the Fermi levels that are the same everywhere (−)φ p φ n P type, doped with Acceptors (fixed negative ions) N type, doped with Donors (fixed positive ions) Department of EECS University of California, Berkeley 8 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Thermal equilibrium again z In thermal equilibrium, the Fermi energy is the same everywhere, and the same for electrons and holes, so at a particular point, φn = φ p = φ(x, y, z) ⎛ qφ (x, y,z) ⎞ −⎜ ⎟ ⎝ kT ⎠ n = nie qφ (x, y,z) kT p = nie 2 z And of course we get back np = n i everywhere which is always the case in equilibrium Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Quasi-Fermi levels z Since when we are not in thermal equilibrium we can still use the Fermi function for electrons and holes, but with different Fermi energies, we call these new energies quasi-Fermi energies On the band diagram, it would look like this: Department of EECS University of California, Berkeley 9 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Quasi-Fermi levels: Band edge diagram z When we draw a band edge diagram out of equilibrium, we need to draw a different Fermi level (quasi-Fermi level) for the electrons and holes z This, for example, is what the band edge diagram would look like for a forward biased PN diode Quasi Fermi level for electrons φ P } φ Exactly how this N Taper looks depends Quasi Fermi level for holes on diffusion and recombination Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Built-in field z In thermal equilibrium, the PN diode has a potential difference for electrons, and a potential difference for holes, and an electric field that both see, with zero voltage appearing at the contacts, because the contacts are at the voltage of the Fermi level, not the conduction band on both sides or the valence band on both sides. z The built in potential which is the energy change that an electron in the conduction band would see is equal to q ⎛ n ⎞ q ⎛ p ⎞ ⎜ ⎟ ⎜ ⎟ φbi = φN −φP = log⎜ ⎟ + log⎜ ⎟ kT ⎝ ni ⎠ kT ⎝ ni ⎠ φ P z φ The same potential N change is seen by a hole in the valence band Department of EECS University of California, Berkeley 10 EECS 105 Spring 2004, Lecture 19 Prof. J. S. Smith Transport z The net motion of electrons and holes through silicon was calculated to be the sum of drift and diffusion for each: dn J = J dr + J diff = qnµ E + qD n n n n n dx dp J = J dr + J diff = qnµ E + qD p p p p p dx D kT z Where = for each carrier type µ q Department of EECS University of California, Berkeley EECS 105 Spring 2004, Lecture 19 Sidebar: Prof.
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