Semiconductors

woundcallousΗμιαγωγοί

1 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

100 εμφανίσεις

Semiconductors

Physics 355

computers


air bags



Palm pilots



cell phones



pagers



DVD players




TV remotes



satellites



fiber networks



switches



photocells



Peltier

refrigerators



thermoelectric generators



lasers



aerospace electronics




CD players



televisions



flat panel displays



clocks



amplifiers



logic

circuits


temperature sensors



calculators



millimeter
-
wave radar




synthesizers



portable computer drives



digital cameras



etc.

Diamond


In the diamond structure, the
carbon atoms are arranged on an
fcc
-
type lattice with a total of 16
electrons per primitive cell.



The valence band and 7 lower bands are full, leaving no
electrons in the conduction band.

Diamond

Electrons may be thermally
activated to jump a gap. At
room temperature,
k
B
T
is
only 0.026 eV. To jump the
energy gap, the electron
requires very high
temperatures. So, diamond
is an excellent insulator.


ρ

= 10
18


-
m

Graphite

ρ

= 9

-
m

Silicon


Silicon has the diamond structure.


There are 14 electrons per primitive
cell.


Gap is only 1.12 eV, however.

Now there is a small (but finite) chance for a few
electrons to be thermally excited from valence band to
conduction band.

Silicon

Effective Mass Revisited


An electron moving in the solid under the
influence of the crystal potential is subjected
to an electric field.


We expect an external field to accelerate the
electron, increasing
E
and
k
and change the
electron’s state.


Effective Mass Revisited

Effective Mass Revisited


eE = F

Effective Mass Revisited


This relates the curvature of the band to the “effective
mass.”


One can show that a free electron “band” gives an effective
mass equal to the rest mass of an electron.


Electrons in a crystal are accelerated in response to an
external force just as though they were free electrons with
effective mass
m
e
.


U
sually ,
m
e

< m
0
.



Effective Mass Revisited

Effective Mass Revisited

in multiples of the free electron mass

m
0

= 9.11


10

31

kg

Experimental Measurement



Traditionally effective masses were measured using cyclotron
resonance, a method in which microwave absorption of a
semiconductor immersed in a magnetic field goes through a sharp peak
when the microwave frequency equals the cyclotron frequency.





In recent years effective masses have more commonly been
determined through measurement of band structures using techniques
such as angle
-
resolved photoemission or, most directly, the de Haas
-
van Alphen effect.



Effective masses can also be estimated using the coefficient
g

of the
linear term in the low
-
temperature electronic specific heat at constant
volume
C
v
. The specific heat depends on the effective mass through
the density of states at the Fermi level and as such is a measure of
degeneracy as well as band curvature.

Electrons & Holes

In a semiconductor, there are two charge carriers:


Electrons

(conduction band)


negative mass


negative charge


Holes

(valence band)


positive mass,


positive charge

Electrons & Holes

For the electrons occupying the vacant
states,




(Negative!) and the electrons will move in
same direction as electric field (wrong
way!)

Carrier Concentration

To calculate the carrier concentrations in energy bands we
need to know the following parameters:


The
distribution of energy states
or levels as a function of
energy within the energy band,
D
(

).


The
probability
of each of these states being occupied by
an electron,
f
(

).


A band is shown for a one
-
dimensional crystal. The
square represents an initially empty state in an otherwise
filled band. When an electric field is applied, the states
represented by arrows successsively become empty as
electrons make transitions.

The band is completely filled except for a state marked by
a square. Except for the electron represented as a circle,
each electron can be paired with another, so the sum of
their crystal momentum vanishes. The total crystal
momentum for the band and the crystal momentum of the
hole are both
ħk.

The empty state and the unpaired electron for two times
are shown when an electric field is applied. The change in
momentum is in the direction of the field.

Conduction Band Carrier Concentration

For


>>

F
, the Boltzmann distribution approximates the F
-
D

distribution:

which is valid for the tail end of the distribution.

Conduction Band Carrier Concentration

Conduction Band

Carrier Concentration


The hole distribution is related to the electron distribution,
since a hole is the absence of an electron.



Valence Band Carrier Concentration


The holes near the top of the valence band behave like
particles with effective mass
m
h
; and the density of states is



Equilibrium Relation


Multiply
n

and
p

together:





The product is constant at a given temperature.


It is also independent of any impurity concentration at a
given temperature. This is because any impurity that adds
electrons, necessarily fills holes.


This is important in practice, since we could reduce the
total carrier concentration
n

+
p

in an impure crystal via the
controlled introduction of suitable impurities


such
reduction is called
compensation
.

Intrinsic Semiconductors


F


c


v

Extrinsic Semiconductors


Extrinsic semiconductors: we can add impurities
to make a material semiconducting (or to change
the properties of the gap).


There are 2 types of extrinsic semiconductors:




p
-
type
and

n
-
type


These are materials which have mostly hole
carriers (
p
) or electron carriers (
n
).


These give you ways of modifying the band gap
energies (important for electronics, detectors, etc).

Extrinsic Semiconductors:
n

type



Add a small amount of phosporus (P:
3s
2
3p
3
) to Silicon (Si: 3s
2
3p
2
)
(generally, a group V element to a
group IV host) P replaces a Si atom
and it donates an electron to the
conduction band (P is called the donor
atom). The periodic potential is
disrupted and we get a localized energy
level,

D
.




This is an n
-
type semiconductor


more electrons around that
can be mobile; and the Fermi energy is closer to the
conduction band.

Extrinsic Semiconductors:
n

type

Phosphorus provides an extra electron.



C




D

= 45 meV

So, its easy for the donor electrons to enter
the conduction band at room temperature.

This means that at room temperature
n

N
D
.

This is called complete ionization (only true if
n
i

<< N
D
). Therefore, by doping Si crystal with
phosphorus, we increase the free electron
concentration.

At low temperature, these extra electrons get
trapped at the donor sites (no longer very mobile)
-

the dopant is frozen out.


Next suppose Si atom is replaced with Boron (B: 2s
2
2p) to
Silicon (Si: 3s
2
3p
2
). Again, we have a perturbed lattice and


a localized E
-
level created.


Boron is missing an electron and accepts an


electron from valence band, creating a hole.


Therefore doping with B increases hole


concentration. We call this
p
-
type doping,


the electron concentration
n

is reduced.



F

moves closer to


V
.

Extrinsic Semiconductors:
p

type


F


c


v


A

Extrinsic Semiconductors

Extrinsic Semiconductors

Boron in Silicon

Mass Action Law


Valid for both intrinsic and extrinsic semiconductors.



It is important in devices to control
n
and
p
concentrations and to suppress the influence of the
intrinsic concentration.



These equations are important in establishing upper
limits in semiconductor operating temperature.



We generally require
n
i

<< (minimum doping density)
and, practically, this means we need doping
concentrations above 10
14

cm

3
.

Semiconductors