1
Carrier Action: Motion, Recombination
and Generation.
What happens
after
we figure out
how many electrons and holes are
in the semiconductor?
2
Carrier Motion I
Described by 2 concepts:
•
Conductivity:
s
(or resistivity:
r = 1/s
)
•
Mobility:
m
Zero Field movement:
Random
–
over all e

Thermal
–
Energy
Distribution.
Motion
Electrons are
scattered by
impurities,
defects etc.
What happens when you apply a force?
3
Carrier Motion II
Apply a force:
Electrons
accelerate
:

n
0
qE
x
=dp
x
/dt
{from F=ma=d(mv)/dt}
Electrons
decelerate
too.
•
Approximated as a viscous damping force
(much like wind on your hand when driving)
dp
x
=

p
x
dt/
t
{dt = time since last “randomizing
collision” and
t
= mean free time between
randomizing collisions.}
Net result: deceration =
dp
x
/dt =

p
x
/
t
4
Carrier Motion III
Acceleration=Deceleration in steady state.
•
dp
x
/dt(accel) + dp
x
/dt(decel) = 0
•

n
0
qE
x

p
x
/
t
= 0.
Algebra:
•
p
x
/n
0
=

q
t
E
x
= <p
x
>
•
But
•
<p
x
> = m
n
*<v
x
> Therefore:
5
Currents
“Current density” (J) is just the amount of
charge passing through a unit area per unit
time.
J
x
= (

q)(n
0
)<v
x
> in C/(s m
2
) or A/m
2
= +(
qn
0
m
n
)E
x
for e

’s acting alone.
=
s
n
E
x
(defining
e

conductivity
)
If both electrons and holes are present:
6
Current, Resistance
How do we find:
•
current (I)? We integrate J.
•
resistance (R)?
•
Provided
r
, w, t are all
constants along the x

axis.
t
w
L
V
x
7
Mobility changes …
Although it is far too simplistic we use:
m
n
= q
t
/m
n
*
t
depends upon:
•
# of scatter centers (impurities, defects etc.)
More doping => lower mobility (see Fig. in books)
More defects (worse crystal) => smaller mobility too.
•
The lattice temperature (vibrations)
Increased temp => more lattice movement => more
scattering =>
smaller
t
and smaller
m
.
t
is the “mean free time.”
m
n
* is the “effective mass.”
(depends on material)
m
Increasing
Doping
8
Mobility Changes II
Mobility is also a function of the electric field
strength (E
x
) when E
x
becomes large. (This leads
to an effect called “velocity saturation.”)
<v
x
>
E
x
(V/cm)
10
5
cm/s
10
6
cm/s
10
7
cm/s
10
2
10
3
10
4
10
5
10
6
V
sat
At ~10
7
cm/s, the carrier KE becomes
the same order of magnitude as k
B
T.
Therefore: added energy tends to warm
up the lattice rather than speed up the
carrier from here on out. The velocity
becomes constant, it “saturates.”
Here
m
is constant (low fields). Note constant
m
=> linear plot.
9
What does E
x
do to our Energy Band Diagram?
Drift currents depend upon the electric
field. What does an electric field do to our
energy band diagrams?
It “bends” them or causes slope in E
C
, E
V
and E
i
.
We can show this.
•
Note:
E
electron
= Total E
= PE + KE
How much is PE vs. KE???
E
electron
E
C
E
V
E
g
e

h
+
10
Energy Band Diagrams in electric fields
E
C
is the lower edge for potential energy (the
energy required to break an electron out of a
bonding state.)
Everything above E
C
is
KE
then.
PE always has to have a
reference! We’ll choose
one arbitrarily for the
moment. (E
REF
= Constant)
Then PE = E
C

E
REF
We also know: PE=

qV
E
electron
E
C
= PE
E
V
= PE
E
g
e

h
+
KE
KE
E
REF
PE
11
Energy Band Diagrams in electric fields II
Electric fields and voltages are related by:
E =

V (or in 1

D E=

dV/dx)
•
So: PE = E
C

E
REF
=

qV or V =

(E
C

E
REF
)/q
•
E
x
=

dV/dx =

d/dx{

(E
C

E
REF
)/q} or
E
x
= +(1/q) dE
C
/dx
12
Energy Band Diagrams in electric fields III
The Electric Field always points into the
rise in the Conduction Band, E
C
.
What about the Fermi level? What
happens to it due to the Electric Field?
E
electron
E
C
E
V
E
g
E
REF
E
i
E
x
13
Another Fermi

Level Definition
The Fermi level is a measure of the
average energy or “electro

chemical
potential energy” of the particles in the
semiconductor. THEREFORE:
The FERMI ENERGY has to be a constant
value at equilibrium. It can not have any
slope (gradients) or discontinuities at all.
The Fermi level is our real

life E
REF
!
14
Let’s examine this constant E
F
Note: If current flows => it is
not equilibrium and E
F
must be
changing.
In this picture, we have
no
connections.
Therefore I=0 and
it is still equilibrium!
Brings us to a good question:
•
If electrons and holes are moved
by E
x
, how can there be NO
CURRENT here??? Won’t E
x
move the electrons => current?
The answer lies in the concept
of “Diffusion”. Next…
E
electron
E
C
E
V
E
F
Semiconductor
E
i
E
x
+ V

Looks
N

type
Looks
P

type
E
x
15
Diffusion I
Examples:
•
Perfume,
•
Heater in the corner (neglecting convection),
•
blue dye in the toilet bowl.
What causes the motion of these particles?
•
Random thermal motion coupled with a density
gradient. ( Slope in concentration.)
16
Green dye in a fishbowl …
If you placed green dye in a fishbowl, right in the center,
then let it diffuse, you would see it spread out in time until
it was evenly spread throughout the whole bowl. This
can be modeled using the simple

minded motion
described in the figure below. L

bar is the “mean
(average) free path between collisions” and
t
the mean
free time. Each time a particle collides, it’s new direction
is randomly determined. Consequently, half continue
going forward and half go backwards.
x
Dye Concentration
0 1 2 3

3

2

1
32
16
16
8
8
8
8
4
4
8
8
4
4
17
Diffusion II
Over a large scale, this would look more
like:
t=0
t
1
t
2
t
3
t
equilibrium
Let’s look more in depth
at this section of the
curve.
18
Diffusion III
What kind of a particle movement does Random Thermal
motion (and a concentration gradient) cause?
n(x)
x

axis
Bin
(1)
Bin
(2)
Bin
(0)
n
b1
n
b2
n
b0
It causes net motion from
large concentration regions
to small concentration
regions.
Line with slope:
Half of
e

go left
half go
right.
19
Diffusion IV
Net number of electrons crossing x
0
is:
•
Number going right: 0.5*n
b1
*ℓ*A
•
Minus Number going left: 0.5*n
b2
*ℓ*A
•
Net is = 0.5*ℓ*A*(n
b1

n
b2
)
•
(note ℓ*A=volume of a bin.)
Flux = # of particles crossing a plane per unit time and
unit area. Symbol is:
f
f
=
0.5*ℓ*A*(n
b1

n
b2
)
(
t
= mean free time.)
t
*A
Or
f
=
0.5*ℓ
(n
b1

n
b2
)
t
20
Diffusion V
Using the fact that slope (dn/dx) =

(n
b1

n
b2
)/ℓ gives:
f
=

0.5*ℓ
2
dn
or
f
=

D
n
*dn/dx (electrons)
t
dx
or
f
=

D
p
*dp/dx (holes)
Now: When charges move we get current. Consequently,
the current density is directly related to the particle flux.
The equations are:
•
(electrons)
(holes)
21
Diffusion VI
Let’s look at an example:
n(x)
J(x)
x
x
dn/dx = 0 here
The electrons are diffusing
out of the center and
toward the edges.
22
Currents round

up
So now we know that our total currents
have 2 components:
•
DRIFT
–
due to any electric field we apply
•
DIFFUSION
–
due to any (dp/dx, dn/dx) we
apply and thermal motion.
23
Answering that old question
How can we have an electric
Field and still have no current?
(Still have J = 0?)
Diffusion must
balance Drift!
Example:
E
electron
E
C
E
V
E
F
Semiconductor
E
i
E
x
+ V

Looks
N

type
Looks
P

type
E
x
24
Einstein Relationship
We next remember: p=n
i
exp((E
i

E
F
)/k
B
T)
Plugging this into our equation for the electric
field and noting that dE
F
/dx = 0 … we get
The Einstein Relationships.
These are very useful. You will never find a table
for both D
p
and
m
p
as a result of these. Once you
have
m
, you have D too, by this relationship.
25
A sanity check
Pretend we have:
What will be the fluxes
and currents?
x
E
x
n(x)
p(x)
Holes
Mechanism
Electrons
Diffusion
Flux (
f
)
Current Density (J)
Drift
Flux (
f
)
Current Density (J)
26
Recombination
–
Generation I
Generation (G)
: How e

and h
+
are produced or
created.
Recombination (R)
:
How e

and h
+
are
destroyed or removed
At equilibrium: r = g and
since the generation
rate is set by the
temperature, we write it
as: r = g
thermal
The concepts are
visually seen in the
energy band diagram
below.
E
C
E
V
E
e
x
G
R
h
v
h
v
27
Recombination
–
Generation II
Recombination must
depend upon
•
the # of electrons: n
o
•
the # of holes: p
o
(If no e

or h
+
, nothing can
recombine!)
From the chemical reaction
•
e

+ h
+
→ Nothing
we can know that
•
r = α
r
n
o
p
o
= α
r
n
i
2
= g
thermal
When the
temperature is
raised
•
g
thermal
increases
Therefore
•
n
i
must increase
too!
The recombination
“rate coefficient”
28
Recombination
–
Generation III
A variety of recombination mechanisms exist:
E
C
E
V
x
G
R
h
v
h
v
E
C
E
V
x
G
R
E
C
E
V
x
G
R
E
e
E
e
E
e
Direct, Band to Band
Auger
Indirect via R

G centers
R

G Center Energy Level
29
GaAs band structure produced by J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976)
using an empirical Pseudo

potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).
E
g
–
The
Band Gap
Energy
GaAs is a
Direct
Band Gap
Semiconductor
Direct
recombination
of electrons
with holes
occurs. The
electrons fall
from the bottom
of the CB to the
VB by giving
off a photon!
30
GaAs band structure produced by W. R. Frensley, Professor of EE @ UTD
using an empirical Pseudo

potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).
31
Silicon band structure produced by J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976)
using an empirical Pseudo

potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).
E
g
–
The
Band Gap
Energy
Si is an
Indirect
Band Gap
Semiconductor
Only indirect
recombination
of electrons
with holes
occurs. The
electrons fall
from the bottom
of the CB into
an R

G center
and from the
R

G center to the
VB. No photon!
32
Silicon band structure produced by W. R. Frensley, Professor of EE @ UTD
using an empirical Pseudo

potential method see also: Cohen and Bergstrasser, Phys. Rev. 141, 789 (1966).
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