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High E Field Transport BW: Sect. 8.10, p 198YC, Sect. 5.4; S, Sect. 4.13; + Outside sources • All transport phenomena discussed so far: – We’ve treated only “Low Field” effects! – Formalism discussed was for “Low Fields” only. “Low Field” Ohm’s “Law” holds J σE or vd μE • For “High Enough” fields Ohm’s “Law” breaks down! • In semiconductors, this field is around E 104 V/cm To understand this, we need to do transport theory at High E Eields!!!! This is difficult & highly computational. Transport Theory at High E Fields This is difficult because of: • The VERY fast rate at which carriers gain energy at high E fields. • There is always energy gain from the field at some rate. • There is always energy loss to lattice at some rate (mainly due to carrier-phonon & carrier-carrier scattering). In “Ordinary” (low E) Transport, The Energy gain rate from the field The Energy loss rate to the lattice. • This is a steady state (almost equilibrium) situation. – We derived Ohm’s “Law” assuming steady state. – If there is no steady state, then Ohm’s “Law” will be violated! • In situations with no steady state, Ohm’s “Law” is violated. • This happens in any material at high enough E! – In this case: The energy gain rate from the field >>> The energy loss rate to the lattice. • In this case, the charge carriers & the lattice are neither in thermal equilibrium nor in a steady state situation. It is a highly non-equilibrium situation. The carrier distribution function is highly non-equilibrium. The concept of temperature is no longer strictly valid! The Boltzmann Equation, at least in the relaxation time approximation, is no longer valid. • The two common types of non-equilibrium situation: 1. The carriers are in thermal equilibrium with each other, but NOT with lattice. This is often approximated as a quasi-equilibrium situation: • In this case, it is assumed that the carriers are at a temperature Te (the “carrier temperature”) which is different than the lattice temperature T (Te >> T). • If this is the case, then an approximation for the carrier distribution function is that it has an equilibrium form (Maxwell-Boltzmann or Fermi-Dirac) but at a temperature Te, rather than the lattice temperature T The “HOT CARRIER” Problem • Second common type of non-equilibrium situation: 2. The carriers are at such high energies (due to the extreme high E) that they are no longer in thermal equilibrium even with each other! This is a truly non-equilibrium situation! Rigorously, even the concept of “Carrier Temperature”makes no sense. The “NON-EQUILIBRIUM CARRIER Problem” – We will talk almost exclusively about case 1, where a carrier temperature is a valid concept. • Hot & non-equilibrium carriers & their effects are important for some devices: Laser Diodes Gunn Oscillators Field Effect Transistors Others… • Under what conditions can it be assumed that the carrier distribution function is the quasiequilibrium one, so that the carrier temperature concept can be used? • This depends on the E field & on the material • It depends on various time scales: – A useful time for this is the time it takes for the non-equilibrium distribution to relax to equilibrium The thermal relaxation time τe (τe is not necessarily = the relaxation time τ from the low field transport problem). τe = time for the “thermalization” of the carriers (due to carrierphonon & carrier-carrier scattering). • Consider, for example, some optical measurements in GaAs: – If n > ~1018 cm-3, carrier-carrier scattering will be the dominant scattering mechanism & τe 10-15 s (1 fs) – For lower n, carrier-phonon scattering dominates & τe τ (the carrier-phonon scattering time) 10-11 s - 10-12 s • In addition, carriers will have a finite lifetime τc because of electron-hole recombination. τc average electron-hole recombination time • At high enough defect densities, defects (deep levels) can shorten carrier the lifetime τc too. • A rough approximation is that, if τc < τe Then A Non-Equilibrium Carrier Distribution Must be Used. • Hot & Non-Equilibrium Carriers have properties which are Very Different in comparison with those of equilibrium carriers! • Some properties are Very Strange if you think linearly or if you think “Ohmically” ! That is, they are strange if you are used to thinking in the linear regime where Ohm’s “Law” holds. A Side Comment • Consideration of these high field effects is somewhat analogous to considering nonlinear and/or chaotic mechanical systems. Some “Hot” Charge Carrier Properties • Just Some of the interesting, observed nonohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: Some “Hot” Charge Carrier Properties • Just Some of the interesting, observed nonohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: 1. Velocity Saturation at high enough E: happens for ALL materials. Some “Hot” Charge Carrier Properties • Just Some of the interesting, observed nonohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: 1. Velocity Saturation at high enough E: happens for ALL materials. 2. Negative Differential Resistance (NDR) or Negative Differential Mobility (NDM) at high enough E: happens only for SOME materials, like GaAs. Some “Hot” Charge Carrier Properties • Just Some of the interesting, observed nonohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: 1. Velocity Saturation at high enough E: happens for ALL materials. 2. Negative Differential Resistance (NDR) or Negative Differential Mobility (NDM) at high enough E: happens only for SOME materials, like GaAs. 3. Gunn Effect at high enough E: happens only for SOME materials, like GaAs. Some Possible Topics Considerable research still needs to be done on high E field effects! 1. The general “Hot” Carrier Problem 2. Impact Ionization 3. Electrical Breakdown 4. The “Lock-on” Effect in GaAs. Related to the research of 2 of my PhD students: Samsoo Kang, 1998 Ken Kambour, 2003. • As we mentioned, for high enough E fields, the drift velocity vd vs. electric field E relationship is non-ohmic (non-linear)! • For all materials, the following is true: 1. For low fields, E ~ 103 V/cm, vd is linear in E. The mobility can then be defined vd μE Ohm’s “Law” holds. 2. For higher E: vd a constant, vsat. This is called “Velocity Saturation”. • For direct bandgap materials, like GaAs: vd vs. E peaks before saturation & decreases again, after which it finally saturates. • Because of this peak, there are regions in the vd vs. E relationship that have: dvd/dE < 0 (for high enough E) This effect is called “Negative Differential Resistance” or “Negative Differential Mobility” or “Negative Differential Conductivity” Transport Processes in Transistors