Transcript Part V

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time approximation, is no longer valid.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