2/10/14 Things you should know when you leave…
Key Questions
ECE 340
Lecture 10 : Drift of Carriers
in Electric Fields
•  How do carriers move through a
semiconductor?
•  How does applying an electric
field change the motion?
•  What is the drift current?
•  What is the mobility and what
does it mean physically?
Class Outline:
• Conductivity and Mobility
M.J. Gilbert
ECE 340 – Lecture 10
Conductivity and Mobility
Conductivity and Mobility
But this situation is very boring…
Carriers are not sitting, what are they doing?
•  The carriers are in constant motion!
•  Conduction at T = 0 K
–  For each electron moving there is
another moving with equal and
opposite momentum.
–  Examine the current density:
E
–  Thermal motion of carriers may be viewed as a random
walk.
–  The carriers are interacting with many different things:
•  Lattice vibrations – Increases with temperature
•  Other electrons – Increases with carrier concentrations
•  Impurities – Increases with doping concentrations
N
J = (− q )∑ vi = 0
i
•  Promote an electron from the valence
band to the conduction band
–  Now there is an imbalance as each +k
state is not balanced by a –k state.
–  Look at the current density:
k
N
J = (− q )∑ vi − (− q )v j
i
M.J. Gilbert
Still boring!!
ECE 340 – Lecture 10
M.J. Gilbert
ECE 340 – Lecture 10
1 2/10/14 Conductivity and Mobility
Conductivity and Mobility
They move randomly through the semiconductor…
So they move quickly, so what?
So, how fast do they move?
EKinetic =
1 * 2
m v th
2
EThermal =
d
kbT
2
We have three degrees of freedom in our system:
3
1
kbT = m*vth2
2
2
Let’s see how far they go before they scatter…
Define the characteristic time between collisions:
Define the characteristic length of
thermal motion (mean free path):
(in meters)
τ c ≈ 10 −14 ~ 10 −13 s
(
3kbT
3 × 0.26eV × 1.6 ×10 −19 J / eV
vth =
=
m*
0.26 × 9.11×10 −31 kg
In between collisions, the carriers
M.J. Gilbert
λ ≡ vthτ c
(in seconds)
Plug in numbers characteristic of silicon at room
temperature:
For silicon at room-temperature…
acquire a large velocity!!
τc
m
s
λ ≈ 1− 10nm
vth ≈ 10 5
)
2.3 ×10 7 cm / s
ECE 340 – Lecture 10
M.J. Gilbert
Carriers scatter
many times in a
typical device!!
ECE 340 – Lecture 10
Conductivity and Mobility
Conductivity and Mobility
Still, on average, the carriers don’t go anywhere!
How can we describe this action??
-q
Net current in any direction is zero! So let’s apply an electric field to our
semiconductor…
• The net motion of the group of electrons is in the –x
direction in response to the applied electric field.
• Individually, this may not be true but, on average, it is
true.
-q
Let’s examine the momentum in the x-direction, px.
The force of the field on the n electrons is:
dp
− nqEx = x
dt
E (V/m)
•  Now the mobile charges will be
accelerated by the electric field.
•  This superimposes a direction on
the random walk.
•  Because of scattering, carriers
do not achieve constant
acceleration.
M.J. Gilbert
Net force on
carriers
F = −qE
F = qE
ECE 340 – Lecture 10
Electrons
Holes
Field
-q
-q
Ex
• Due to collisions, the net acceleration is
balanced by the net deceleration.
• Net rate of change in momentum is zero.
For random collisions, there will be a constant probability of collision. Consider N0
electrons at time t0:
# of electrons NOT having undergone collisions
−
dN (t ) 1
= N (t )
dt
τc
M.J. Gilbert
N (t ) = N 0 e −t /τ c
ECE 340 – Lecture 10
2 2/10/14 Conductivity and Mobility
Conductivity and Mobility
But we know that the velocity will be randomized every τc!
With this information, we can define the drift current…
Rate of change of the
momentum due to collisions:
dp x
dt
=−
collisions
I
px
τc
We already know that the sum of these effects should be zero…
−
px
τc
− nqEx = 0 Average momentum per electron
px =
px
= −qτ c Ex
n
Ex
We want to define a current, or charge per unit time crossing of observation
orientated normal to the direction of current flow.
J ndrift = − qn vdn = qnµ n E
J pdrift = qp vdp = qpµ p E
• Here we have taken an expectation value which operates over the entire ensemble
of electrons.
• This indicates that the electrons have an average net velocity in the –x
direction.
+
holes
p
qτ E
v x = *x = ± c* x
Drift Velocity
electrons
mn , p
mn , p
M.J. Gilbert
ECE 340 – Lecture 10
M.J. Gilbert
Conductivity and Mobility
Using the definitions for the hole and electron
drift currents:
J
drift
n
= − qn vdn = qnµ n E
J
drift
p
= qp vdp = qpµ p E
The electron and hole mobility then becomes:
qτ c
2mn*
qτ
µ p = c*
2m p
In units of:
M.J. Gilbert
Electron Mobility
Hole Mobility
cm 2
V ⋅s
ECE 340 – Lecture 10
Conductivity and Mobility
And from the definition of the current, we can define the mobility…
µn =
Proportional to:
• Carrier drift velocity
• Carrier concentration
• Carrier charge
What can we say about the
mobility in general?
So what effects the mobility?
µn =
qτ c
2mn*
µp =
qτ c
2m*p
ECE 340 – Lecture 10
Electric
Field
Why can’t we apply Newton’s 2nd law in a semiconductor?
•
•
•
• Refers to the ease with
which carriers move through a
host crystal.
Silicon
The effective mass effects the mobility!
Complex scattering
Complex interactions
F = −qE = m0
dv
dt
F = −qE = mn*
dv
dt
In a semiconductor, we cannot use classical laws!
–  For overall motion – NO!
–  For motion in-between scattering – NO!
•
We defined a new “effective” mass which incorporated all of the complicated interactions.
•
Lighter carrier mass leads to higher mobility.
M.J. Gilbert
ECE 340 – Lecture 10
3 2/10/14 Conductivity and Mobility
Conductivity and Mobility
Remember: Which effective mass do I need?
kz
ml = 0.98 m0
mt = 0.19 m0
What else effects the mobility?
kz
Silicon mobility at 300 K
ml
mt
ky
ky
mt
kx
•
kx
Use the “density of states” effective mass for calculating:
•  Density of states
•  Equilibrium carrier concentrations
2
1
*
mdos
= M 3c (m1m2 m3 )3 m0
-------
• Use the “conductivity” effective mass for calculating
• Binding energies
3
*
mcond
=
• Electronic structure
1
1
1
+
+
m1 m2 m3
M.J. Gilbert
ECE 340 – Lecture 10
Lattice Scattering
M.J. Gilbert
Ionized Impurity
Scattering
ECE 340 – Lecture 10
Conductivity and Mobility
Mobility and Conductivity
Scattering effects the mobility…
How can we obtain the conductivity and resistivity?
•  For low doping levels,
the mobility is limited
by collisions with the
lattice.
•  For medium to high
doping levels the
mobility is limited by
interactions with
ionized impurities.
•  For low
temperatures, the
mobility is limited by
interactions with
other electrons.
M.J. Gilbert
Let’s write down an expression for the total drift current:
J drift = J ndrift + J pdrift
Plug in what we know about the electron and hole drift currents:
J drift = q(nµn + pµ p )E
• This assumes that both electrons and holes participate in the conduction, if one
or the other is dominant then the drift current associated with the minority
carrier can be neglected.
• Hey, this looks like Ohm’s law!
J = σE =
ECE 340 – Lecture 10
E
ρ
M.J. Gilbert
ρ=
1
σ
=
1
q(nµn + pµ p )
⎡ 1 ⎤
⎣ Ω ⋅ cm ⎥⎦
σ = conductivity ⎢
ρ = resistivity[Ω ⋅ cm]
ECE 340 – Lecture 10
4