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! –Any formalism discussed was for “Low Fields” only. “Low Field”  Ohm’s “Law” holds J  σE or v d  μE For “High Enough” fields, Ohm’s “Law” breaks down! –In semiconductors this field is around E   10 4 V/cm  We need to do transport theory at High E Eield. 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. 1. 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 The Boltzmann Equation, at least in the relaxation time approximation, is no longer valid.

There are two common types of non-equilibrium situations: 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, assume that the carriers are at a temperature T e (the “carrier temperature”) which is different than the lattice temperature T (T e >> 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 T e, rather than the lattice temperature T  The “HOT CARRIER” Problem

The 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! 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

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  τ e (τ e is not necessarily = the relaxation time τ from the low field transport problem). τ e is the time for the “thermalization” of the carriers (due to carrier-phonon & carrier-carrier scattering). Some optical measurements in GaAs: –If n > ~10 18 cm -3, carrier-carrier scattering will be the dominant scattering mechanism & τ e  s (1 fs) –For lower n, carrier-phonon scattering dominates & τ e  τ (the carrier-phonon scattering time)  s s

In addition, carriers will have a finite lifetime τ c because of electron-hole recombination. τ c  average electron-hole recombination time At high defect densities, defects (deep levels) can shorten carrier the lifetime τ c too. A rough approximation is that, if τ c < τ e 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” is valid.

Some “Hot” Charge Carrier Properties The non-ohmic behavior of the drift velocity v d vs. electric filed E at high E: –Velocity Saturation at high enough E (for all materials) –Negative Differential Resistance (NDR) or Negative Differential Mobility (NDM) at high enough fields (only for some materials, like GaAs). –The Gunn Effect (only for some materials, like GaAs). Possible topics: 1. The general “hot” carrier problem 2. Impact ionization & electrical breakdown 3. The “Lock-on” effect in GaAs. This is related to the research of 2 of my PhD students: Samsoo Kang, 1998, Ken Kambour, 2003.

As was just mentioned, for high enough fields, the drift velocity v d vs. electric field E relationship is non-ohmic (non-linear)! For all materials, the following is true: 1. For low fields, E  ~ 10 3 V/cm, v d is linear in E. The mobility can then be defined v d  μE  Ohm’s “Law” holds. 2. For higher E: v d  a constant, v sat. This is called “velocity saturation”. For direct gap materials, like GaAs: v d vs. E peaks before saturation & decreases again, after which it finally saturates. Because of this peak, there are regions in the v d vs. E relationship that have: dv d /dE < 0 (for high E) This effect is called “Negative Differential Resistance” or “Negative Differential Mobility” or “Negative Differential Conductivity”