ECE 340 Lecture 9 Temperature Dependence of Carrier Concentrations
Key Questions ECE 340 What is the physical meaning of Lecture 6...
Transcript of Key Questions ECE 340 What is the physical meaning of Lecture 6...
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ECE 340 Lecture 6 : Intrinsic and
Extrinsic Material I
Class Outline:
• Effective Mass • Intrinsic Material • Extrinsic Material M.J. Gilbert ECE 340 – Lecture 6
Things you should know when you leave…
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 ECE 340 – Lecture 6
Effective Mass
At the end of lecture 5, we talked about effective mass…
Electric Field
Electric Field
• In a vacuum, we can apply Newton’s second law:
• In a semiconductor, we cannot. – For overall motion – NO! – For motion in-between
scattering – NO! • We defined a new “effective”
mass which incorporated all of the complicated interactions.
dtdvmqEF 0=−=
dtdvmqEF n
*=−=
M.J. Gilbert ECE 340 – Lecture 6
Effective Mass
We even defined the effective mass…
22
2*
/ dkEdm !
=
We can define the effective mass as:
Nevertheless, two questions remain: 1. Where does this definition come
from?
2. What does it mean physically?
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M.J. Gilbert ECE 340 – Lecture 6
Effective Mass
Let’s begin to think about where effective mass comes from… Start with the energy-wavevector (dispersion) relation for free electrons:
mkEk 2
22!=
Now look at the equation of motion for how electrons move in an energy band in an electric field.
Suppose that the wavepacket is made of wavefunctions near a particular k.
Ψ(x)
k’ k
The wavepacket is moving with some group velocity, vg:
dkdEvg !
1=
All of the information of the effects of the crystal on the motion of the electron are in the dispersion relation.
E
(6.1)
(6.2)
M.J. Gilbert ECE 340 – Lecture 6
Effective Mass
What are the forces that the electron is experiencing?
Ψ(x)
k’ k
E vg
How much work is the field doing on the electron?
tveEE gfield δδ −= (6.3)
We observe that by using eq. 6.2…
kvkdkdEE gδδδ !=⎟
⎠
⎞⎜⎝
⎛= (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.
FeEdtdkwhere
teE
k
field
field
=−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
!
!
,
δδ
FdkEd
dtdk
dkEd
dkdtEd
dtdvg
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛==
2
2
2
2
22
1
1
!
!!
*
1m
Newton’s 2nd law!
M.J. Gilbert ECE 340 – Lecture 6
Effective Mass
Simple Example… Consider a simple cosine approximation to the band:
( ) ( ) ⎟⎠
⎞⎜⎝
⎛=−=2
sincos121 2 kaWkaWkE
• Sample parameters – W (Band Width) ~ 5 eV – a (lattice spacing) ~ 0.5
nm
aπ−
aπ
( )eVkE
0
5
What are the group velocity and the effective mass?
Group velocity: ( ) ( ) ( )kaaWdkkdEkvg sin
21
!!== vg(k)
M.J. Gilbert ECE 340 – Lecture 6
Effective Mass
The group velocity goes to zero!! What about the effective mass?
• The effective mass becomes negative! – 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.
• 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.
aπ−
aπ
( ) ( )kaWam
km sec22
0
2* !
=
Effective mass:
0.3
-0.3
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M.J. Gilbert ECE 340 – Lecture 6
Intrinsic Material
Intrinsic Material is pure with no additional contaminants…
• 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.
T = 0 K
T = 300 K
M.J. Gilbert ECE 340 – Lecture 6
Intrinsic Material
But there are more processes at work…
• 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.
Generation Rate:
⎟⎠
⎞⎜⎝
⎛⋅
+++=scm
GGGG mechoptth 3
1...
M.J. Gilbert ECE 340 – Lecture 6
Intrinsic Material
Since we are in thermal equilibrium, there must be an opposite process…
• 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.
Recombination Rate:
⎟⎠
⎞⎜⎝
⎛⋅
•∝scm
pnR 3
1
• N – number of electrons • P – number of holes
M.J. Gilbert ECE 340 – Lecture 6
Intrinsic Material
In the steady state…
• The generation rate must be balanced by the recombination rate.
• Important consequence is that for a given semiconductor the np product depends only on the temperature.
=
20000 inpnRG =⇒= inpnpn ==⇒= 0000
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M.J. Gilbert ECE 340 – Lecture 6
Intrinsic Material
Putting numbers to the intrinsic concentrations…
• For silicon – 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.
Silicon ni ~ 1010 cm-3
Germanium ni ~ 2 x 1013 cm-3
GaAs ni ~ 2 x 106 cm-3
M.J. Gilbert ECE 340 – Lecture 6
Extrinsic Semiconductors
The great strength of semiconductors…
• 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
How does a donor work?
Silicon (Si) 4 valence electrons
Phosphorous (P) 5 valence electrons
M.J. Gilbert ECE 340 – Lecture 6
Extrinsic Materials
How does an acceptor work?
Silicon (Si) 4 valence electrons
Boron (B) 3 valence electrons
Si!
B
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M.J. Gilbert ECE 340 – Lecture 6
Extrinsic Materials
In general, we can modify the materials properties with the introduction of immobile impurity atoms…
• We can – Selectively create
regions of n and p. • Needed for CMOS.
– Modify the conductivity over several orders of magnitude.
– Manipulate the number of conduction electrons over 5 orders of magnitude.
M.J. Gilbert ECE 340 – Lecture 6
Extrinsic Materials
How tightly bound is the extra electron or hole?
• We can use the Bohr’s hydrogen model to get an idea.
• Electrons move in Si and not in a vacuum. – Different relative
permittivity. • The electron mass must be
represented by the effective mass
r
Donor Acceptor
e-
h+
( ) 220
2
4*
32 !r
nB
qmEεεπ
−=
Donor in Si P As Sb
Binding energy (eV) 0.045 0.054 0.039
Acceptor in Si B Al Ga In
Binding energy (eV) 0.045 0.067 0.072 0.16
M.J. Gilbert
Extrinsic Materials
Visualizing donors on the band diagram…
Ed
Ea Ev
Ec
Ev
Ec
Δx
Let’s take a look at Silicon with Phosphorus impurity atoms:
Ed
Ev
Ec
Eg = 1.12 eV
0.045 eV
ECE 340 – Lecture 6 M.J. Gilbert
Extrinsic Material
Remember the intrinsic concentrations…
• For silicon – 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.
Silicon ni ~ 1010 cm-3
Germanium ni ~ 2 x 1013 cm-3
GaAs ni ~ 2 x 106 cm-3
ECE 340 – Lecture 6
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M.J. Gilbert
Extrinsic Materials
Revisiting the effect of temperature…
T = 0 K T = 50 K T = 300 K
ECE 340 – Lecture 6 M.J. Gilbert
Extrinsic Material
Commonly used terms: • 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 n-type and holes in p-type.
• Minority carrier – the least abundant carrier in a given semiconductor sample; electrons in p-type and holes in n-type.
ECE 340 – Lecture 6