Carrier Action: Motion, Recombination and Generation.

Semiconductor

Nov 1, 2013 (4 years and 8 months ago)

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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).