Semiconductor Applications

ECE 4339 L. Trombetta

ECE 4339: Physical Principles of Solid State Devices

Len Trombetta

Summer 2006

Chapter 3:

Carrier Action

Goal: To understand what these
equations are for.

ECE 4339 L. Trombetta

Initially the mobility of electrons and holes increases linearly with electric field, but then tapers
off and saturates at high fields. Often, modern devices operate at the saturated velocity.

ECE 4339 L. Trombetta

We will be reading from this chart (and the one on the next page) a lot.

ECE 4339 L. Trombetta

ECE 4339 L. Trombetta

ECE 4339 L. Trombetta

ECE 4339 L. Trombetta

ECE 4339 L. Trombetta

ECE 4339 L. Trombetta

Relationship between energy and potential
for an electron energy band diagram.

increasing electron
energy

increasing electron
potential

E
C

E
V

A

B

D
V = V
B

-

V
A

+

-

D
E

The electron moves to the left (down the hill and toward
lower energy) under the action of the potential.

D
E =
-
q
D
V

Note that the right
-
hand side of the figure is at a more negative potential and has a higher energy
than the left
-
hand side. This is simple electrostatics: the electron has higher energy toward the
right because like
-
charges repel: the electron doesn’t want to be in a “negative” region.

ECE 4339 L. Trombetta

V =
-
E/q



㴠
-

摖/摸

ECE 4339 L. Trombetta

Fig. 3.10 (previous slide) provides more detail on the relationships between
energy, voltage, and electric field as they relate to the energy band diagram. Note
that the vertical axis gives total electron energy, which is the sum of kinetic and
potential energy. The relevant equations for an electron are…

potential energy

kinetic energy

electric field

We have defined a reference that is completely arbitrary. We can put E
ref

anywhere because
we will always deal only with differences in potential and energy. That’s why we don’t have to
put values on our vertical axis: it doesn’t matter where E = 0 happens to be.

Kinetic energy is the difference between the total energy E
-

E
ref

and the potential energy.

Electric field is the negative of the gradient of the potential (E&M). The reference energy E
ref

is constant, so the gradient of E
ref

is 0. Also, E
C
, E
V

and E
i

all “track” one another so it
makes no difference which one we use for the gradient.

ECE 4339 L. Trombetta

If N
D

is a function of distance (i.e.,
not constant), and if no current is
flowing, then…

…we will get an electric field from
the non
-
uniform carrier density.

ECE 4339 L. Trombetta

Some definitions…

Generation
:

the production of electron hole pairs

Recombination
:

the removal of electron hole pairs

Equilibrium
:

a condition in which the energy of the system is determined only
by the temperature. (For semiconductors this means there is no current flowing
or light shining anywhere.)

Steady State
:

a condition in which the every process (recombination, for
example) is balanced by an inverse process (generation). A system in steady
state is not necessarily in equilibrium.

ECE 4339 L. Trombetta

Recombination/Generation processes

Here we look at the ways in which electrons and holes can be generated or recombine. This will
be important in understanding how current flows or how semiconductors respond to light.

ECE 4339 L. Trombetta

Bottom Line:

If a semiconductor is not in equilibrium (which will happen if current is
flowing or the semiconductor is exposed to light), R/G processes will come in to play
to try to return the system to equilibrium.

ECE 4339 L. Trombetta

In Figure 3.15 b and e, E
T

is the energy of a “trap”; the situation is similar to a donor or
acceptor, except that the energy level is far from the band edges (whereas donor/acceptor
levels are close to the band edges). This figure shows some of the elements that generate
traps in Si, and indicates their energy levels relative to the band edges.

ECE 4339 L. Trombetta

E
g

E
g

eV

m
-
1

Recall our formula for the energy of a
“free” QM particle of wavenumber k:

Here is the formula applied to electrons and holes,
assumed to be free, in a semiconductor.

But electrons and holes aren’t
quite

free
in semiconductors. Here is a more
accurate picture for two cases: “direct”
and “indirect” semiconductors.

m
n
* = 0.25

m
p
* = 0.10

ECE 4339 L. Trombetta

Electron/hole
recombination here
results in light output.

Electron/hole
recombination here
results in heat output.

ECE 4339 L. Trombetta

E
C

E
V

E
C

E
V

h
n

> E
G

h
n

< E
G

a

large: material is opaque

Simple picture of absorption/transmission

More complete picture of absorption/transmission

x

x

a

small: material is transparent

ECE 4339 L. Trombetta

Large input impedance

Idea: R
S

(semiconductor resistance) is large unless light is shining on it, in which case it is small. In the
dark, v
L

will be small, but during a light flash, it will rise (see “scope” trace). As excess carriers recombine,
R
S

goes back up, with a time constant given by the minority carrier lifetime.

ECE 4339 L. Trombetta

G
L

is the rate at which excess carriers are being generated (cm
-
3
s
-
1
). This equation says that when we shine a
light on the sample beginning at t = 0, the excess carriers buildup at a rate determined by the minority
carrier lifetime

p
. After the light has been on for a long time (several time constants), the total excess hole
density is G
L

p
.

ECE 4339 L. Trombetta

If we “flash” the light, we can get watch minority carriers decay exponentially, again with a time constant
given by the minority carrier lifetime. In this figure we imagine flashing the light repeatedly.

ECE 4339 L. Trombetta

Here the illumination is constant, but it hits only at the end of the sample. We are now looking at how the
excess carrier concentration decays into the bulk of the sample. This happens because carriers diffuse and
recombine at the same time.

ECE 4339 L. Trombetta