High E Field Transport BW: Sect. 8.10, p 198YC, Sect. 5.4; S, Sect. 4.13; + Outside sources

“Low Field”  Ohm’s “Law” holds J  σE or vd  μE 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 The Boltzmann Equation, at least in the relaxation time approximation, is no longer valid.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 “HOT CARRIER” Problem 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

 The “NON-EQUILIBRIUM 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.

Field Effect Transistors Hot & non-equilibrium carriers & their effects are important for some devices: Laser Diodes Gunn Oscillators Field Effect Transistors Others…

This depends on the E field & on the material Under what conditions can it be assumed that the carrier distribution function is the quasi-equilibrium 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 carrier-phonon & 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

A Non-Equilibrium Carrier Distribution Must be Used. 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 non-linear and/or chaotic mechanical systems.

Some “Hot” Charge Carrier Properties Just Some of the interesting, observed non- ohmic 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 non- ohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: Velocity Saturation at high enough E: happens for ALL materials.

Some “Hot” Charge Carrier Properties Just Some of the interesting, observed non- ohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: Velocity Saturation at high enough E: happens for ALL materials. 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 non-ohmic behavior at high E fields. The drift velocity vd vs. electric field E at high E: Velocity Saturation at high enough E: happens for ALL materials. Negative Differential Resistance (NDR) or Negative Differential Mobility (NDM) at high enough E: happens only for SOME materials, like GaAs. Gunn Effect at high enough E:

Considerable research 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.

“Velocity Saturation”. 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: For low fields, E  ~ 103 V/cm, vd is linear in E. The mobility can then be defined vd  μE  Ohm’s “Law” holds. For higher E: vd  a constant, vsat. This is called “Velocity Saturation”.

“Negative Differential Resistance” 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