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2/2/14 Things you should know when you leave…
ECE 340
Lecture 6 : Intrinsic and
Extrinsic Material I
Class Outline:
• Effective Mass
• Intrinsic Material
• Extrinsic Material
Key Questions
•  What is the physical meaning of
the effective mass
•  What does a negative effective
mass mean?
•  What is intrinsic material?
•  What is thermal equilibrium?
•  What is extrinsic material?
•  How does doping work?
M.J. Gilbert
Effective Mass
Effective Mass
At the end of lecture 5, we talked about effective mass…
We even defined the effective mass…
•  In a vacuum, we can apply
Newton’s second law:
Electric
Field
Electric
Field
F = −qE = m0
We can define the effective mass
as:
dv
dt
m* =
•  In a semiconductor, we
cannot.
ECE 340 – Lecture 6
dv
dt
!2
d 2 E / dk 2
Nevertheless, two questions remain:
–  For overall motion – NO!
–  For motion in-between
scattering – NO!
•  We defined a new “effective”
mass which incorporated all of
the complicated interactions.
F = −qE = mn*
M.J. Gilbert
ECE 340 – Lecture 6
1.  Where does this definition come
from?
2.  What does it mean physically?
M.J. Gilbert
ECE 340 – Lecture 6
1 2/2/14 Effective Mass
Effective Mass
Let’s begin to think about where effective mass comes from…
What are the forces that the electron is experiencing?
Start with the energy-wavevector (dispersion) relation
for free electrons:
! 2k 2
Ek =
2m
(6.1)
k’
Now look at the equation of motion for how electrons
move in an energy band in an electric field.
k
⎛ eE field
⎝ !
δk = −⎜⎜
(6.2)
ECE 340 – Lecture 6
M.J. Gilbert
Effective Mass
approximation to the band:
–  W (Band Width) ~ 5 eV
–  a (lattice spacing) ~ 0.5
nm
dt
d 2 E 1 ⎛ d 2 E dk ⎞
⎟
= ⎜
!dkdt ! ⎜⎝ dk 2 dt ⎟⎠
⎛ 1 d 2 E ⎞
⎟ F
= ⎜⎜ 2
2 ⎟
⎝ ! dk ⎠
=
Newton’s 2nd law!
1
m*
ECE 340 – Lecture 6
The group velocity goes to zero!! What about the effective mass?
1
⎛ ka ⎞
E (k ) = W (1 − cos ka ) = W sin 2 ⎜ ⎟
2
⎝ 2 ⎠
0.3
E (k )
eV
5
Effective mass:
0
π
a
−π
a
-0.3
•  The effective mass becomes negative!
1 dE (k ) aW
Group velocity: vg (k ) =
=
sin (ka ) − π
! dk
2!
a
ECE 340 – Lecture 6
m* (k ) =
2! 2
sec(ka )
m0 a 2W
–  States of positive mass occur near the bottom of the bands
due to positive band curvature.
–  States of negative mass occur at the top of bands.
What are the group velocity and the effective mass?
M.J. Gilbert
dv g
Effective Mass
Simple Example… Consider a simple cosine
•  Sample parameters
⎞
⎟⎟δt
⎠
where,
dk
!
= −eE field = F
dt
All of the information of the effects of the crystal on the motion of the electron
are in the dispersion relation.
M.J. Gilbert
(6.4)
Combine eqns. 6.3 and 6.4 to arrive at an external force that is exerted on the
electrons by the applied electric field.
The wavepacket is moving with some group
velocity, vg:
k’
We observe that by using eq. 6.2…
⎛ dE ⎞
⎟δk = !v g δk
⎝ dk ⎠
E
1 dE
vg =
! dk
k
δE = ⎜
Suppose that the wavepacket is made of wavefunctions near a particular k.
Ψ(x)
How much work is the field doing on the
vg electron?
δE = −eE field vg δt
(6.3)
E
Ψ(x)
vg(k)
π
a
•  Physically, it means that on going from k to k+Δk the
momentum transfer to the lattice from the electron is
larger than that of the momentum transfer from the
applied force to the electron.
–  As we approach Bragg reflection at the edge, when we
increase the wavevector we can get an overall decrease in the
forward momentum.
M.J. Gilbert
ECE 340 – Lecture 6
2 2/2/14 Intrinsic Material
Intrinsic Material
Intrinsic Material is pure with no additional contaminants…
But there are more processes at work…
T=0K
Generation Rate:
⎛ 1 ⎞
G = Gth + Gopt + Gmech + ...⎜ 3 ⎟
⎝ cm ⋅ s ⎠
T = 300 K
•  At T = 0 K, there is no energy in the system.
–  All of the covalent bonds are satisfied.
–  Valence band is full and conduction band is empty.
•  At T > 0 K, thermal energy breaks bonds apart
–  Crystal lattice begins to vibrate and exchange energy with
carriers.
–  Electrons leave the valence band to populate the conduction band.
M.J. Gilbert
ECE 340 – Lecture 6
•  Generation
–  Break up of a covalent bond to form an electron and
a hole.
–  Requires energy from thermal, optical, mechanical or
other external sources.
–  Supply of bonds to break is virtually inexhaustible.
•  Atomic density >> # of electrons or # of holes.
M.J. Gilbert
Intrinsic Material
Intrinsic Material
Since we are in thermal equilibrium, there must be an opposite process…
In the steady state…
=
Recombination Rate:
⎛ 1 ⎞
R ∝ n • p⎜ 3 ⎟
⎝ cm ⋅ s ⎠
• N – number of electrons
• P – number of holes
•  Recombination
–  Formation of a bond by bringing together and electron and a hole.
–  Releases energy in the form of thermal or optical energy.
–  Recombination events require the presence of 1 electron and 1
hole.
–  These events are most likely to occur at the surfaces of
semiconductors where the crystal periodicity is broken.
M.J. Gilbert
ECE 340 – Lecture 6
ECE 340 – Lecture 6
•  The generation rate must be balanced by
the recombination rate.
G0 = R0 ⇒ n0 p0 = ni2
n0 = p0 ⇒ n0 = p0 = ni
•  Important consequence is that for a
given semiconductor the np product
depends only on the temperature.
M.J. Gilbert
ECE 340 – Lecture 6
3 2/2/14 Intrinsic Material
Extrinsic Semiconductors
Putting numbers to the intrinsic concentrations…
The great strength of semiconductors…
Silicon
ni ~ 1010 cm-3
Germanium
ni ~ 2 x 1013 cm-3
•  For silicon
GaAs
ni ~ 2 x 106 cm-3
M.J. Gilbert
– 
– 
– 
– 
– 
5 x 1022 atoms/cm3
4 bonds per atom
2 x 1023 bonds/cm3
ni (300 K) ~ 1010 cm-3
1 broken bond per 1013
bonds.
ECE 340 – Lecture 6
Extrinsic Materials
How does a donor work?
•  We can change
their properties
many orders of
magnitude by
introducing the
proper impurity
atoms.
•  Which columns
add
–  Electrons?
–  Holes?
•  What about
impurities?
M.J. Gilbert
ECE 340 – Lecture 6
Extrinsic Materials
Phosphorous (P)
5 valence electrons
Silicon (Si)
4 valence electrons
How does an
acceptor work?
Silicon (Si) 4 valence electrons Boron (B) 3 valence electrons Si!
B M.J. Gilbert
ECE 340 – Lecture 6
M.J. Gilbert
ECE 340 – Lecture 6
4 2/2/14 Extrinsic Materials
Extrinsic Materials
In general, we can modify the materials properties with the introduction of
immobile impurity atoms…
How tightly bound is the extra electron or hole?
•  We can
–  Selectively create
regions of n and p.
•  Needed for CMOS.
• 
Acceptor in Si
Binding energy (eV)
P
0.045
B
0.045
M.J. Gilbert
Extrinsic Materials
EB = −
As
0.054
Al
0.067
Acceptor
*
n
4
mq
2
32π 2 (ε 0ε r ) ! 2
Sb
0.039
Ga
0.072
r
In
0.16
ECE 340 – Lecture 6
Extrinsic Material
Visualizing donors on the band diagram…
Ec
Remember the intrinsic concentrations…
Ec
Ed
Ev
Ea
Ev
Silicon
ni ~ 1010 cm-3
Let’s take a look at Silicon with Phosphorus impurity atoms:
Ec
Ed
Eg = 1.12 eV
GaAs
ni ~ 2 x 106 cm-3
ECE 340 – Lecture 6
Germanium
ni ~ 2 x 1013 cm-3
•  For silicon
0.045 eV
Ev
M.J. Gilbert
Donor
The electron mass must be
represented by the
effective mass
Donor in Si
Binding energy (eV)
ECE 340 – Lecture 6
Δx
e-
–  Different relative
permittivity.
–  Modify the
conductivity over
several orders of
magnitude.
–  Manipulate the
number of
conduction
electrons over 5
orders of
magnitude.
M.J. Gilbert
•  We can use the Bohr’s
hydrogen model to get an
idea.
•  Electrons move in Si and not
in a vacuum.
h+
M.J. Gilbert
– 
– 
– 
– 
– 
5 x 1023 atoms/cm3
4 bonds per atom
2 x 1023 bonds/cm3
ni (300 K) ~ 1010 cm-3
1 broken bond per 1013
bonds.
ECE 340 – Lecture 6
5 2/2/14 Extrinsic Materials
Extrinsic Material
Revisiting the effect of temperature…
Commonly used terms:
T = 0 K
M.J. Gilbert
T = 50 K
ECE 340 – Lecture 6
T = 300 K
• 
Dopants – specific impurity atoms that are added to semiconductors in controlled amounts for
the express purpose of increasing either the electron or hole concentrations.
• 
Intrinsic semiconductor – undoped semiconductor; extremely pure semiconductor sample
containing an insignificant amount of impurity atoms; a semiconductor whose properties are
native to the material.
• 
Extrinsic semiconductor – doped semiconductor; a semiconductor whose properties are
controlled by added impurity atoms.
• 
Donor – impurity atom that increases the electron concentration; n-type dopant.
• 
Acceptor – impurity atom that increases the hole concentration; p-type dopant.
• 
N-type material – a donor doped material; a semiconductor containing more electrons than holes.
• 
P-type material – an acceptor doped material; a semiconductor containing more holes than
electrons.
• 
Majority carrier – the most abundant carrier in a given semiconductor sample; electrons in ntype and holes in p-type.
• 
Minority carrier – the least abundant carrier in a given semiconductor sample; electrons in ptype and holes in n-type.
M.J. Gilbert
ECE 340 – Lecture 6
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