Introduction optoelectronic devices in semiconductor

Introduction to optoelectronic

1. BITS Pilani
Pilani Campus
M Tech Microelectronics,
WILP
Prof. Kranthi Kumar Palavalasa.
Assistant Professor
Department of EEE (WILP)

2. BITS Pilani
Pilani Campus
MEL ZG512,
Optoelectronic Devices, Circuit &
Systems
Lecture No-1.
Credits: NJIT ECE-271 Dr. S. Levkov

3. BITS Pilani, Pilani Campus
Pallab Bhattacharya, “Semiconductor Optoelectronic Devices,” 2nd edition,
Pearson
J. Wilson and J.Haukes, “Opto Electronics – An Introduction”, Prentice Hall of India Pvt.
Ltd.,
 New Delhi, 1995.
A. Yariv and P. Yeh , “Photonics - Optical Electronics in Modern Communications,”
S. O. Kasap, “Optoelectronics and Photonics: Principles and Practices,”
Prentice-Hall, 2001.
B. Streetman and S. Banerjee, “Solid State Electronic Devices,” 6th edition,
Pearson/Prentice Hall, 2006
Text Book(s)
Reference Book(s) & other resources

4. BITS Pilani, Pilani Campus
Evaluation Scheme:
No Name Type DurationWeightDay, Date, Session, Time
EC-1Assignment-1 Open Book  10% September 1-10, 2024
  Assignment-2 Open Book  10% October 10-20, 2024
  Assignment 3 Open book  10%
November 1-10, 2024
EC-2Mid-Semester TestClosed Book2 hours30%
Saturday, 21/09/2024 (EN)
EC-3Comprehensive ExamOpen Book 2 ½
hours
40%
Saturday, 30/11/2024 (EN)

5. BITS Pilani, Pilani Campus
1.Characterize resistivity of insulators, semiconductors, and conductors.
2.Develop covalent bond and energy band models for semiconductors.
3.Understand band gap energy and intrinsic carrier concentration.
4.Explore the behavior of electrons and holes in semiconductors.
5.Discuss acceptor and donor impurities in semiconductors.
6.Learn to control the electron and hole populations using impurity doping.
7.Understand drift and diffusion currents in semiconductors.
8.Explore low-field mobility and velocity saturation.
9.Discuss the dependence of mobility on doping level.
Chapter Goals
Chap 2 - 5

6. BITS Pilani, Pilani Campus
Electronic materials fall into three categories (WRT resistivity):
–Insulators  > 10
5
-cm (diamond  = 10
16
)
–Semiconductors 10
-3
<  < 10
5
-cm
–Conductors  < 10
-3
-cm (copper  = 10
-6
)
Elemental semiconductors are formed from a single type of atom of column IV,
typically Silicon.
Compound semiconductors are formed from combinations of elements of column III
and V or columns II and VI.
Germanium was used in many early devices.
Silicon quickly replaced germanium due to its higher bandgap energy, lower cost, and
ability to be easily oxidized to form silicon-dioxide insulating layers.
Solid-State Electronic
Materials
Chap 2 - 6

7. BITS Pilani, Pilani Campus
Bandgap is an energy range in a solid where no electron states can exist. It
refers to the energy difference between the top of the valence band and
the bottom of the conduction band in insulators and semiconductors
Semiconductor Materials
Chap 2 - 7

8. BITS Pilani, Pilani Campus
Semiconductor Materials
(cont.)
Chap 2 - 8
Semiconductor
Bandgap
Energy E
G
(eV)
Carbon (diamond) 5.47
Silicon 1.12
Germanium 0.66
Tin 0.082
Gallium arsenide 1.42
Gallium nitride 3.49
Indium phosphide 1.35
Boron nitride 7.50
Silicon carbide 3.26
Cadmium selenide 1.70

9. BITS Pilani, Pilani Campus
Covalent Bond Model
Chap 2 - 9
Silicon diamond lattice unit
cell.
Corner of diamond lattice
showing four nearest
neighbor bonding.
View of crystal
lattice along a crystallographic
axis.
• Silicon has four electrons in the outer shell.
• Single crystal material is formed by the covalent bonding of each
silicon atom with its four nearest neighbors.

10. BITS Pilani, Pilani Campus
Silicon Covalent Bond Model
(cont.)
Chap 2 - 10
Silicon atom

11. BITS Pilani, Pilani Campus
Silicon Covalent Bond Model
(cont.)
Chap 2 - 11
Silicon atom Silicon atom
Covalent bond

12. BITS Pilani, Pilani Campus
Silicon Covalent Bond Model
(cont.)
Chap 2 - 12
Silicon atom Covalent bonds in silicon

13. BITS Pilani, Pilani Campus
Silicon Covalent Bond Model
(cont.)
Chap 2 - 13
•Near absolute zero, all bonds are complete

•Each Si atom contributes one electron to
each of the four bond pairs
•The outer shell is full, no free electrons,
silicon crystal is an insulator
•What happens as the temperature increases?

14. BITS Pilani, Pilani Campus
Silicon Covalent Bond Model
(cont.)
Chap 2 - 14
•Increasing temperature adds energy to the
system and breaks bonds in the lattice,
generating electron-hole pairs.
•The pairs move within the matter forming
semiconductor
•Some of the electrons can fall into the
holes – recombination.
•Near absolute zero, all bonds are complete

•Each Si atom contributes one electron to
each of the four bond pairs
•The outer shell is full, no free electrons,
silicon crystal is an insulator

15. BITS Pilani, Pilani Campus
The density of carriers in a semiconductor as a function of temperature and
material properties is:
E
G
= semiconductor bandgap energy in eV (electron volts)
k = Boltzmann’s constant, 8.62 x 10
-5
eV/K
T = absolute termperature, K
B = material-dependent parameter, 1.08 x 10
31
K
-3
cm
-6
for Si
Bandgap energy is the minimum energy needed to free an electron by
breaking a covalent bond in the semiconductor crystal.
Intrinsic Carrier Concentration
Chap 2 - 15
2 3 -6
exp cm
G
i
E
n BT
kT
 
 
 
 

16. BITS Pilani, Pilani Campus
Electron density is n (electrons/cm
3
)
and for intrinsic material n = n
i.
Intrinsic refers to properties of pure
materials.
n
i
≈ 10
10
cm
-3
for Si
The density of silicon atoms is n
a
≈
5x10
22
cm
-3

Thus at a room temperature one
bond per about 10
13
is broken
Intrinsic Carrier Concentration
(cont.)
Chap 2 - 16

17. BITS Pilani, Pilani Campus
•A vacancy is left when a covalent bond is broken.
•The vacancy is called a hole.
•A hole moves when the vacancy is filled by an electron from a nearby
broken bond (hole current).
•The electron density is n (n
i for intrinsic material)
•Hole density is represented by p.
•For intrinsic silicon, n = n
i = p.
•The product of electron and hole concentrations is pn = n
i
2
.
•The pn product above holds when a semiconductor is in thermal
equilibrium (not with an external voltage applied).
Electron-hole concentrations
Chap 2 - 17

18. BITS Pilani, Pilani Campus
Charged particles move or drift under the influence of the applied field.
The resulting current is called drift current.
Electrical resistivity  and its reciprocal, conductivity , characterize current flow in
a material when an electric field is applied.
Drift current density is
j = Qv [(C/cm
3
)(cm/s) = A/cm
2
]
j= current density, (Coulomb charge moving through a unit area)
Q= charge density, (Charge in a unit volume)
v= velocity of charge in an electric field.
Note that “density” may mean area or volumetric density, depending on the context.
Drift Current
Chap 2 - 18

19. BITS Pilani, Pilani Campus
At low fields, carrier drift velocity v (cm/s) is proportional to electric field E
(V/cm). The constant of proportionality is the mobility, :
v
n
= - 
n
E and v
p
= 
p
E , where
v
n
and v
p
- electron and hole velocity (cm/s),

n and 
p - electron and hole mobility (cm
2
/Vs)

n ≈ 1350 (cm
2
/Vs), 
p ≈ 500 (cm
2
/Vs),
Hole mobility is less than electron since hole current is the result of multiple
covalent bond disruptions, while electrons can move freely about the
crystal.
Mobility
Chap 2 - 19

20. BITS Pilani, Pilani Campus
At high fields, carrier velocity saturates and places upper limits on the speed
of solid-state devices.
Velocity Saturation
Chap 2 - 20

21. BITS Pilani, Pilani Campus
Given drift current and mobility, we can calculate resistivity (Q is
the charge density) :
j
n
drift
= Q
nv
n = (-qn)(- 
nE) = qn 
nE A/cm
2
j
p
drift
= Q
p
v
p
= (+qp)(+ 
p
E) = qp 
p
E A/cm
2
j
T
drift =
j
n
+ j
p
= q(n 
n
+ p 
p
)E = E
This defines electrical conductivity:
 = q(n 
n
+ p 
p
) (cm)
-1
Resistivity  is the reciprocal of conductivity:
 = 1/ (cm)
Intrinsic Silicon Resistivity
Chap 2 - 21


E
j
T
drift

V/cm
A/cm
2
 cm







22. BITS Pilani, Pilani Campus
Problem: Find the resistivity of intrinsic silicon at room temperature and classify it as
an insulator, semiconductor, or conductor.
Solution:
Need: Resistivity  and classification.
Assumptions: assume “room temperature” with n
i = 10
10
/cm
3
.
Analysis: Charge density of electrons is Q
n
= -qn
i
and for holes is Q
p
= +qn
i
. Thus:

 = (1.60 x 10
-10
)[(10
10
)(1350) + (10
10
)(500)] (C)(cm
-3
)(cm
2
/Vs)
= 2.96 x 10
-6
(cm)
-1 --->
 = 1/ = 3.38 x 10
5
cm
Recalling the classification in the beginning, intrinsic silicon is near the low end of
the insulator resistivity range
Conclusions: Resistivity has been found, and intrinsic silicon is a poor insulator.
Example: Calculate the
resistivity of intrinsic silicon
Chap 2 - 22
 = q(n 
n
+ p

p
)

23. BITS Pilani, Pilani Campus
The interesting properties of semiconductors emerges when impurities are
introduced.
Doping is the process of adding very small well controlled amounts of
impurities into a semiconductor.
Doping enables the control of the resistivity and other properties over a wide
range of values.
For silicon, impurities are from columns III and V of the periodic table.
Semiconductor Doping
Chap 2 - 23

24. BITS Pilani, Pilani Campus
Phosphorous (or other column V element) atom replaces silicon atom in
crystal lattice.
Since phosphorous has five outer shell electrons, there is now an ‘extra’
electron in the structure.
Material is still charge neutral, but very little energy is required to free the
electron for conduction since it is not participating in a bond.
Donor Impurities in Silicon
Chap 2 - 24
A silicon crystal doped by a pentavalent element
(f. i. phosphorus). Each dopant atom donates a free
electron and is thus called a donor. The doped
semiconductor becomes n type.
q
q
e

25. BITS Pilani, Pilani Campus
Boron (column III element) has been added to silicon.
There is now an incomplete bond pair, creating a vacancy for an electron.
Little energy is required to move a nearby electron into the vacancy.
As the ‘hole’ propagates, charge is moved across the silicon.
Acceptor Impurities in Silicon
Chap 2 - 25
A silicon crystal doped with a trivalent impurity (f.i.
boron). Each dopant atom gives rise to a hole, and the
semiconductor becomes p type.
Vacancy
q
q
e

26. BITS Pilani, Pilani Campus
Acceptor Impurities – Hole
propagation
Chap 2 - 26
Hole is propagating through the silicon.

27. BITS Pilani, Pilani Campus
Acceptor Impurities – Hole
propagation
Chap 2 - 27
Hole is propagating through the silicon.
e
Hole

28. BITS Pilani, Pilani Campus
Acceptor Impurities – Hole
propagation
Chap 2 - 28
Hole is propagating through the silicon.
Hole

29. BITS Pilani, Pilani Campus
Acceptor Impurities – Hole
propagation
Chap 2 - 29
Hole is propagating through the silicon.
e

30. BITS Pilani, Pilani Campus
In doped material, the electron and hole concentrations are no longer equal.
If n > p, the material is n-type.
If p > n, the material is p-type.
The carrier with the largest concentration is the majority carrier, the smaller
is the minority carrier.
N
D = donor impurity concentration
N
A = acceptor impurity concentration atoms/cm
3
Charge neutrality requires q(N
D + p - N
A - n) = 0:
positive charge: p (holes) + N
D
(ionized donors)
negative charge: n (electrons) + N
D
(ionized acceptors)
It can also be shown that pn = n
i
2
, even for doped semiconductors in thermal
equilibrium.
Doped Silicon Carrier Concentrations
(how to calculate)
Chap 2 - 30

31. BITS Pilani, Pilani Campus
Substituting p = n
i
2
/n into q(N
D + p - N
A - n) = 0 yields n
2
- (N
D - N
A)n - n
i
2
= 0.
Solving for n
For (N
D - N
A) >> 2n
i, n  (N
D - N
A) .
n-type Material
Chap 2 - 31

n
(N
DN
A)(N
DN
A)
2
4n
i
2
2
and p
n
i
2
n

32. BITS Pilani, Pilani Campus
Similar to the approach used with n-type material we find the following
equations:
For (N
A
- N
D
) >> 2n
i
, p  (N
A
- N
D
) .
We find the majority carrier concentration from charge neutrality and find the
minority carrier concentration from the thermal equilibrium relationship.
p-type Material
Chap 2 - 32

p
(N
AN
D)(N
AN
D)
2
4n
i
2
2
and n
n
i
2
p

33. BITS Pilani, Pilani Campus
Majority carrier concentrations are established at manufacturing time and are
independent of temperature (over practical temp. ranges).
However, minority carrier concentrations are proportional to n
i
2
, a highly
temperature dependent term.
For practical doping levels (dopant concentration usually is quite larger then
n
i
):
n  (N
D
- N
A
) for n-type material
p  (N
A - N
D) for p-type material.
Typical doping ranges are 10
14
/cm
3
to 10
21
/cm
3
.
Practical Doping Levels
Chap 2 - 33

34. BITS Pilani, Pilani Campus
Mobility and Resistivity in
Doped Semiconductors
Chap 2 - 34
•Impurities degrade mobility
(different size disrupt the lattice,
atoms ionized – electrons scatter
) – see on the left.
•However, doping vastly increases
the density of majority carriers
 dramatically decreases
resistivity despite the lower
mobility.
• = q
n
(N
D
– N
A
) for n-type
• = q
p
(N
A
– N
D
) for p-type

35. BITS Pilani, Pilani Campus
In practical semiconductors, it is quite useful to create carrier concentration
gradients by varying the dopant concentration and/or the dopant type
across a region of semiconductor.
This gives rise to a diffusion current resulting from the natural tendency of
carriers to move from high concentration regions to low concentration
regions.
Diffusion current is analogous to a gas moving across a room to evenly
distribute itself across the volume.
Diffusion Current
Chap 2 - 35

36. BITS Pilani, Pilani Campus
Diffusion Current (cont.)
Chap 2 - 36
A bar of silicon (a) into which holes are injected, thus
creating the hole concentration profile along the x axis,
shown in (b). The holes diffuse in the positive
direction of x and give rise to a hole-diffusion current
in the same direction.
If the electrons are injected and the electron-
concentration profile shown is established in a
bar of silicon, electrons diffuse in the x
direction, giving rise to an electron-diffusion
current in the negative -x direction.

37. BITS Pilani, Pilani Campus
Carriers move toward regions of lower concentration, so diffusion current
densities are proportional to the negative of the carrier gradient.
Diffusion Current (cont.)
Chap 2 - 37
Diffusion currents in the
presence of a concentration
gradient.
2
2
A/cm )(
A/cm )(
x
n
qD
x
n
Dqj
x
p
qD
x
p
Dqj
nn
diff
n
pp
diff
p






















Diffusion current density equations

38. BITS Pilani, Pilani Campus
D
p and D
n are the hole and electron diffusivities with units cm
2
/s. Diffusivity
and mobility are related by Einsteins’s relationship:
The thermal voltage, V
T = kT/q, is approximately 25 mV at room
temperature. We will encounter V
T many times throughout this course.
Diffusion Current (cont.)
Chap 2 - 38

D
n

n

kT
q

D
p

p
V
TThermal voltage
D
n
nV
T , D
p
pV
T

39. BITS Pilani, Pilani Campus
Total current is the sum of drift and diffusion current:
Chap 2 - 39
Total Current in a Semiconductor


T
n n n
T
p p p
j q n qD
j
E
E
n
x
q
x
q
p
p D






 
 
1
1

T
n n T
T
p p T
n
j q n E V
n x
p
j q p E V
p x






 
 
 
 
 
 
 
 
In the following sections, we will use
these equations, combined with
Gauss’ law, (E)=Q, to calculate
currents in a variety of
semiconductor devices.
Example here
Rewriting using Einstein’s relationship (D
p
= 
n
V
T
),

40. BITS Pilani, Pilani Campus
Semiconductor Energy Band
Model
Chap 2 - 40
Semiconductor energy
band model. E
C
and E
V

are energy levels at the
edge of the conduction
and valence bands.
Electron participating in
a covalent bond is in a
lower energy state in the
valence band. This
diagram represents 0 K.
What happens as
temperature increases?

41. BITS Pilani, Pilani Campus
Semiconductor Energy Band
Model
Chap 2 - 41
Semiconductor energy
band model. E
C
and E
V

are energy levels at the
edge of the conduction
and valence bands.
Electron participating in
a covalent bond is in a
lower energy state in the
valence band. This
diagram represents 0 K.
Thermal energy breaks
covalent bonds and
moves the electrons up
into the conduction
band.

42. BITS Pilani, Pilani Campus
Energy Band Model for a
Doped Semiconductor
Chap 2 - 42
Semiconductor with donor or n-type
dopants. The donor atoms have free
electrons with energy E
D
. Since E
D
is
close to E
C, (about 0.045 eV for
phosphorous), it is easy for electrons
in an n-type material to move up into
the conduction band.

43. BITS Pilani, Pilani Campus
Energy Band Model for a
Doped Semiconductor
Chap 2 - 43
Semiconductor with donor or n-type
dopants. The donor atoms have free
electrons with energy E
D
. Since E
D
is
close to E
C, (about 0.045 eV for
phosphorous), it is easy for electrons
in an n-type material to move up into
the conduction band and create
negative charge carriers.
Semiconductor with acceptor or p-type
dopants. The acceptor atoms have
unfilled covalent bonds with energy
state E
A
. Since E
A
is close to E
V
, (about
0.044 eV for boron), it is easy for
electrons in the valence band to move
up into the acceptor sites and complete
covalent bond pairs, and create holes –
positive charge carriers.

44. BITS Pilani, Pilani Campus
At absolute zero all the electronic states of the valence band are full and those of
conduction band are empty
Classically all electrons have zero energy at 0
0
K (i.e., practically insulator. When temp is
increased then electrons jump from VB to CB) But
Quantum Mechanically all electrons are not having zero energy at 0
0
K
The maximum energy that electrons may posses at 0
0
k is the Fermi energy E
F
Quantum mechanically electrons actually have energies extending from 0 to E
F at 0
0
K
FERMI LEVEL AND EFFECT OF
TEMPERATURE ON INTRINSIC SC
For intrinsic semiconductors like silicon and
germanium, the Fermi level is essentially
halfway between the valence and conduction
bands

45. BITS Pilani, Pilani Campus
Particles seek to occupy the lowest energy state possible.
Therefore, electrons in a solid will tend to fill the lowest energy states first.
At T=0 , every low-energy state is occupied, right up to the Fermi level, but no states
are filled at energies greater than E
F .
"Fermi level" is the term used to describe the top of the collection of electron energy
levels at absolute zero temperature
FERMI LEVEL
Illustration of the Fermi function for zero
temperature. All electrons are stacked neatly
below the Fermi level.

46. BITS Pilani, Pilani Campus
The Fermi energy is a concept in quantum mechanics referring to the energy of the
highest occupied quantum state in a system of fermions at absolute zero
temperature.
The Fermi level is an energy that pertains to electrons in a semiconductor. It is the
chemical potential μ that appears in the electrons' Fermi-Dirac distribution
function.
The Fermi-Dirac distribution also called the "Fermi function," is a fundamental
equation expressing the behaviour of mobile charges in solid materials
FERMI ENERGY

47. BITS Pilani, Pilani Campus
The increase in conductivity with temperature can be modeled in terms of the Fermi
function, which allows one to calculate the population of the conduction band
Fermi factor: how many of the energy states in the VB and CB will be occupied at
different temperatures OR
The Fermi function: the probability that a state is occupied.
FERMI FACTOR or Fermi function is the probability that an electron occupies the state
of a given energy (E) under thermal equilibrium. It is a function of temperature and
energy
The probability that the particle
will have an energy E is
At absolute zero, the probability is equal to 1 for energies less than the Fermi energy
and zero for energies greater than the Fermi energy.
FERMI FACTOR OR FERMI
FUNCTION
KT
EE
F
e
EF



1
1
)(

48. BITS Pilani, Pilani Campus
F
EE
Case-I When T=0
0
K , then for
0
1
1
)( 



e
EF
1
1
1
)( 



e
EF
F
EEand for
Case-II When t=T
0
K , then at E=E
F

2
1
1
1
)(
0



e
EF

49. BITS Pilani, Pilani Campus
When the temperature is not 00K but some higher value [T=10000K], then some covalent
bonds will be broken and some electrons will be available in CB.
The Fermi energy level , EF , is the energy at which the probability of occupancy is exactly
1/2 for temperatures greater than zero.
Illustration of the Fermi function for
temperatures above zero. Some electrons drift
above the Fermi level. Their density at higher
energies is proportional to the Fermi function.

50. BITS Pilani, Pilani Campus

51. BITS Pilani, Pilani Campus
Fermi Function
kTEE
F
e
Ef
/)(
1
1
)(



•Probability that an available state at energy E is occupied:
•E
F is called the Fermi energy or the Fermi level
There is only one Fermi level in a system at equilibrium.
If E >> E
F
:
If E << E
F :
If E = E
F
:

52. BITS Pilani, Pilani Campus
Effect of Temperature on f(E)

53. BITS Pilani, Pilani Campus
Boltzmann Approximation
Probability that a state is empty (i.e. occupied by a hole):
kTEE
F
F
eEfkTEE
/)(
)( ,3 If


kTEE
F
F
eEfkTEE
/)(
1)( ,3 If


kTEEkTEE
FF
eeEf
/)(/)(
)(1



54. BITS Pilani, Pilani Campus
•Obtain n(E) by multiplying g
c
(E) and f(E)
Equilibrium Distribution of Carriers
Energy band
diagram
Density of
States, g
c
(E)
Probability of
occupancy, f(E)
Carrier
distribution, n(E)× =
cnx.org/content/m13458/latest

55. BITS Pilani, Pilani Campus
Obtain p(E) by multiplying g
v
(E) and 1-f(E)
Energy band
diagram
Density of
States, g
v
(E)
Probability of
occupancy, 1-f(E)
Carrier
distribution, p(E)× =
cnx.org/content/m13458/latest

56. BITS Pilani, Pilani Campus
Equilibrium Carrier Concentrations
•By using the Boltzmann approximation, and
extending the integration limit to , we obtain

band conduction of top


c
E
c
(E)f(E)dEgn
2/3
2
*
, /)(
2
2 where










h
kTm
NeNn
DOSn
c
kTEE
c
Fc

•Integrate n(E) over all the energies in the conduction
band to obtain n:

57. BITS Pilani, Pilani Campus
•By using the Boltzmann approximation, and
extending the integration limit to -, we obtain
  1
band valenceof bottom

vE
v dEf(E)(E)gp
2/3
2
*
, /)(
2
2 where










h
kTm
NeNp
DOSp
v
kTEE
v
vF

•Integrate p(E) over all the energies in the valence
band to obtain p:

58. BITS Pilani, Pilani Campus
Intrinsic Carrier Concentration
  
2 / /)(
/)( /)(


i
kTE
vc
kTEE
vc
kTEE
v
kTEE
c
neNNeNN
eNeNnp
Gvc
vFFc





2/kTE
vci
G
eNNn


Si Ge GaAs
N
c
(cm
-3
) 2.8 × 10
19
1.04 × 10
19
4.7 × 10
17
N
v
(cm
-3
)1.04 × 10
19
6.0 × 10
18
7.0 × 10
18
Effective Densities of States at the Band Edges (@ 300K)

59. BITS Pilani, Pilani Campus
In an intrinsic semiconductor, n = p = n
i
and E
F
= E
i
n(n
i, E
i) and p(n
i, E
i)
/)(
/)(
kTEE
ic
kTEE
ci
ic
ic
enN
eNnn




/)(
/)(
kTEE
iv
kTEE
vi
vi
vi
enN
eNnp




/)( kTEE
i
iF
enn


/)( kTEE
i
Fi
enp



60. BITS Pilani, Pilani Campus
To find E
F
for an intrinsic semiconductor, use the fact that n
= p:
Intrinsic Fermi Level, E
i
2
ln
4
3
2
ln
22
*
,
*
,
/)( /)(
vc
DOSn
DOSpvc
i
i
c
vvc
F
kTEE
v
kTEE
c
EE
m
mkTEE
E
E
N
NkTEE
E
eNeN
vFFc




























61. BITS Pilani, Pilani Campus
n-type Material
Energy band
diagram
Density of
States
Probability
of occupancy
Carrier
distributions

62. BITS Pilani, Pilani Campus
Example: Dopant Ionization
Probability of non-ionization 
meVE
n
N
kTEE
c
c
cF 150ln 






kTEE
FD
e
/)(
1
1



Consider a phosphorus-doped Si sample at 300K with N
D
= 10
17
cm
-3
.
What fraction of the donors are not ionized?
Hint: Suppose at first that all of the donor atoms are ionized.
017.0
1
1
26/)45150(



 meVmeVmeV
e

63. BITS Pilani, Pilani Campus
p-type Material
Energy band
diagram
Density of
States
Probability
of occupancy
Carrier
distributions

64. BITS Pilani, Pilani Campus
Non-degenerately Doped Semiconductor
•Recall that the expressions for n and p were derived using
the Boltzmann approximation, i.e. we assumed
kTEEkTE
cFv
33 
E
c
E
v
3kT
3kT
E
F
in this range
The semiconductor is said to be non-degenerately doped in this case.

65. BITS Pilani, Pilani Campus
•If a semiconductor is very heavily doped, the
Boltzmann approximation is not valid.
In Si at T=300K: E
c
-E
F
< 3k
B
T if N
D
> 1.6x10
18
cm
-3
E
F
-E
v
< 3k
B
T if N
A
> 9.1x10
17
cm
-3
The semiconductor is said to be degenerately doped in this
case.
•Terminology:
“n+”  degenerately n-type doped. E
F
 E
c
“p+”  degenerately p-type doped. E
F
 E
v
Degenerately Doped Semiconductor