ECE 440: Lectures 8-9
Carrier Mobility and Drift
Let’s recap the 5-6 major concepts so far:
Memorize a few things, but recognize many.
(why? semiconductors require lots of approximations)
Why all the fuss about the abstract concept of EF?
Consider (for example) joining an n-doped piece of Si with a
p-doped piece of Ge. How does the band diagram look?
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So far, we’ve learned the effects of temperature and doping
on carrier concentrations.
But no electric field = not useful = boring materials.
The secret life of C-band electrons (or V-band holes): They
are essentially free to move around at finite temperature &
doping. So what do they do?
Instantaneous velocity given by thermal energy:
Scattering time (with what?) is of the order ~ 0.1 ps.
So average distance travelled between scattering: L ~
But on average, this electron goes: _________________
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So turn ON an electric field:
F = ± qE
F = m*a

a=
Between collisions, carriers accelerate along E field:
vn(t) = ant =
vp(t) = apt =
for electrons
for holes
In the energy band picture this looks like:
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On average, velocity is randomized again every 𝛕C.
So average velocity in E-field is:
v=
We call the proportionality constant mobility:
µn,p =
This is a very important result‼!
(what are the units?)
What are the roles of mn,p and 𝝉C:
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Then for electrons:
vn = -µnE
And for holes:
vp = µpE
Mobility is a measure of ease of carrier drift in E-field.
 If m↓ “lighter” particle means µ…
 If 𝛕C ↑ means longer time between collisions, so µ…
Mobilities of some undoped (intrinsic) semiconductors at
room temperature.
m n (cm2/V·s)
m p (cm2/V·s)
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Si
1400
Ge
3900
GaAs
8500
InAs
30000
470
1900
400
500
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What does mobility (through 𝛕C) depend on?
 Lattice scattering (host lattice, e.g. Si, Ge, vibrations)
 Ionized impurity (dopant atom) scattering
 Electron-electron or electron-hole scattering
 Interface scattering
Which ones are more likely to depend on temperature?
Qualitatively, how?
Strongest scattering, i.e. lowest mobility dominates.
Contributions to mobility add up in parallel:
1
m
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
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Qualitatively:
Experimental data for crystalline silicon:
Source: http://www.ioffe.rssi.ru/SVA/NSM/Semicond/Si/electric.html
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Again, qualitatively we expect the mobility to decrease with
total additional impurities (ND+NA):
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Linear scale, from the ECE 440 course web site:
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Ex: What is the hole drift velocity at room temperature in
silicon, in a field E = 1000 V/cm? What is the average time
and distance between collisions?
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More generally, the velocity-field relationship:
(here, for electrons in silicon)
Low-field slope = mobility
High-field effect = velocity saturation due to very strong
lattice scattering. Any additional energy from the E-field is
transferred to the lattice (phonons) rather than increasing
the carrier velocity.
Result: constant velocity (current) at very high fields!
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Now that we know the velocity-field relationship, how can
we calculate current flow?
Net velocity of charged particles  electric current
Drift current density
∝
net carrier drift velocity
∝
carrier concentration
∝
carrier charge
(charge crossing plane of area A in time t)
J ndrift  qnvdn  qnm n E
J pdrift  qpvdp  qpm p E
Check units:
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Check signs:
Total drift current:
J drift  J ndrift  J pdrift  q(nmn  pm p ) E
Has the form of Ohm’s Law!
Current density:
J E 
E

Current:
I  JA 
This is very neat. We derived Ohm’s Law from basic
considerations (electrons, holes) in a semiconductor.
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Resistivity of a semiconductor:

1


1
q(nm n  pm p )
What about when n ≫ p? (n-type doped sample)
What about when p ≫ n? (p-type doped sample)
Drift and resistance:
L
1 L
R

wt  wt
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Experimentally, for silicon at room temperature:
This is absolutely essential to illustrating our control over a
semiconductor’s resistivity via doping.
Notes:
 This plot does not apply to compensated material (with
both n-type and p-type dopant impurities).
 This applies at relatively low fields (at high field we
must be a little careful with velocity saturation).
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