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Semiconductor Junctions The following energy diagrams are for the potential energy U = q = e of an electron in the electrostatic potential . The Fermi level EF is given for low temperatures. When two neutral semiconductor pieces are put into contact, electrons flow across the boundary to reach lower energy levels. Neutral dopants become ionized and build up an electrostatic potential (“space charge”). This process stops when the two Fermi levels coincide. 1a) Semiconductor – Semiconductor: pn – Junction Separated p- and n-type semiconductors: n-type After forming the pn - junction: p-type CB CB EF Ionized Donors CB EF EF VB Ionized Acceptors VB VB Electrons flow downhill from n to p until the Fermi levels EF line up. 1b) Semiconductor – Semiconductor: Two separate semiconductors: n-type Depletion: EF mid-gap Heterojunction After forming a junction: intrinsic Ionized Donors Electrons EF Band Offset Energy U= e = Electrostatic Potential 1 Modulation Doping: Achieve high electron density without scattering of electrons by dopants. Dopants are on one side of the junction, electrons on the other. Achieves record mobilities of >106 cm2/Vs in GaAlAs /GaAs. Used for creating a two-dimensional electron gas (see Lecture 35). 2) Semiconductor – Metal: Schottky Barrier Separate semiconductor and metal: After forming a junction: CB Schottky Barrier (n-type) EF CB EF EF VB Electrons trapped in interface states VB + Depletion width ~ 1/ND+ 3) Metal Oxide Semiconductor (MOS): For a p-type semiconductor = nMOS (conducting via n-type carriers using inversion) Flatband Accumulation Inversion Charging of a Capacitor Metal Ins. Semicond. EF EF EF VGate: Field Effect Transistor (FET) see also p. 5. VGate: 2 Calculating Carrier Densities and Band Diagrams A. Homogeneous Semiconductor, Static Equilibrium Two variables: nh = Hole Density, ne = Electron Density Two equations: 1. Charge Neutrality: ND+ = Ionized Donor Density NA = Ionized Acceptor Density nh ne + ND+ NA = 0 Determines the position of the Fermi level in the gap. 2. Mass Action: nh ne = const. = [nint(T)]2 nint = Intrinsic Carrier Density Chemistry analog: H+ + OH H2O h+ + e photon Recombination rate nh ne Generation rate exp[Eg/kBT] (e recombines with h+) (thermal photon creates e h+ pair) In equilibrium the rates for the forward and back reaction are equal. Driving up nh reduces ne , because the extra holes take away electrons by recombination. B. Inhomogeneous Semiconductor (Junctions), Static Equilibrium Charge neutrality is gone. The space charge induced by the flow of carriers across a junction decays over a Debye screening length ~ 1/n . It is an analog to the ThomasFermi screening length in metals and the London penetration depth in superconductors. One needs to solve two equations simultaneously (= self-consistently) by iteration: 1. Poisson Equation: (z) (z) 2. Fermi Dirac Distribution: (z) (z) 1. d2/dz2 = (z) / 0 2. (z) = + e n+(z) e n(z) n(z) = fe(E) D[E+U(z)] dE n+(z) = [1fe(E)] D+[E+U(z)] dE fe(E) = 1 / {exp[(EEF)/kBT] 1} = Total Charge Density = Electrostatic Potential U(z) = q(z) = Potential Energy q = Charge D (E) = Density of States for Conduction Band + Acceptors D+(E) = Density of States for Valence Band + Donors 3 C. Junction with Bias Voltage V, Dynamic Equilibrium Reverse Bias: Forward Bias: + I + e eV/kT Two opposing currents, matched at V=0 V 1 Generation (Drift) Current: Minority carriers are generated thermally and drift down the barrier, independent of V. I = 1 + e eV/kT Recombination (Diffusion) Current: Majority carriers from dopants diffuse against the barrier (EgeV), depends exponentially on V. Rectifying Diode Reverse Bias Zero Bias Forward Bias + + Photodiode Solar Cell Light Emitting Diode (LED) + Photon creates e h+ pair Reverse Bias + + eh+ pair recombines into a photon Forward Bias 4 Field Effect Transistor (FET) nMOS: Two Cuts: Off: VGate 0 Gate Source n Drain n Channel p On: VGate > 0 Inversion This cut: Source Channel Drain = back-to-back diodes Other cut: Gate Oxide Channel (p. 2) CMOS (Complementary Metal Oxide Semiconductor) Back-to-back nMOS and pMOS with common gate. Low power consumption, draws current only during switching. pMOS in out out in 1 nMOS in 2 Inverter NAND 5 DRAM (Dynamic Random Access Memory) Single field effect transistor + storage capacitor. Needs to be refreshed due to leakage current. CCD (Charge-Coupled-Device) Array of MOS capacitors that store light-induced charge and pass it on along the array for readout. Dark Illuminated High voltage on pixel 2. H Readout by pulsing pixel 3 to high voltage, which transfers charge from pixel 2 to pixel 3. HEMT (High Electron Mobility Transistor) = MODFET (Modulation-Doped FET) Modulation doping achieves high mobility in the channel by putting the dopants into an adjacent layer and let the electrons spill over. Fastest transistors (GaAlAs/GaAs, SiGe/Si). Quantum Well Laser A semiconductor with smaller band gap is inserted at the pn junction of a LED (see slide). Both electrons and holes are trapped in the narrow-gap semiconductor. That enhances the probability for an electron to meet a hole and recombine into a photon. If the quantum well is narrow enough (single digit nm), the bands become quantized in the direction perpendicular to the well, and their density of states gets concentrated. That enhances the number of electron-hole pairs that are able to contribute to a narrow laser line. 6