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ECE 271
Electronic Circuits I
Topic 2
Semiconductors Basics
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 1
Chapter Goals
•
•
•
•
•
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Characterize resistivity of insulators, semiconductors, and conductors.
Develop covalent bond and energy band models for semiconductors.
Understand band gap energy and intrinsic carrier concentration.
Explore the behavior of electrons and holes in semiconductors.
Discuss acceptor and donor impurities in semiconductors.
Learn to control the electron and hole populations using impurity
doping.
• Understand drift and diffusion currents in semiconductors.
• Explore low-field mobility and velocity saturation.
• Discuss the dependence of mobility on doping level.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 2
The Inventors of the Integrated Circuit
Jack Kilby
NJIT ECE-271 Dr. S. Levkov
Andy Grove, Robert Noyce, and
Gordon Moore with Intel 8080 layout.
Chap 2 - 3
Solid-State Electronic Materials
• Electronic materials fall into three categories (WRT resistivity):
– Insulators
 > 105 -cm (diamond  = 1016 )
– Semiconductors 10-3 <  < 105 -cm
– Conductors
 < 10-3 -cm (copper  = 10-6 )
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 4
Solid-State Electronic Materials
• Electronic materials fall into three categories (WRT resistivity):
– Insulators
 > 105 -cm (diamond  = 1016 )
– Semiconductors 10-3 <  < 105 -cm
– Conductors
 < 10-3 -cm (copper  = 10-6 )
• Elemental semiconductors are formed from a single type of atom
of column IV, typically Silicon.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 5
Solid-State Electronic Materials
• Electronic materials fall into three categories (WRT resistivity):
– Insulators
 > 105 -cm (diamond  = 1016 )
– Semiconductors 10-3 <  < 105 -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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 6
Solid-State Electronic Materials
• Electronic materials fall into three categories (WRT resistivity):
– Insulators
 > 105 -cm (diamond  = 1016 )
– Semiconductors 10-3 <  < 105 -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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 7
Solid-State Electronic Materials
• Electronic materials fall into three categories (WRT resistivity):
– Insulators
 > 105 -cm (diamond  = 1016 )
– Semiconductors 10-3 <  < 105 -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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 8
Solid-State Electronic Materials (cont)
• 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
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 9
Semiconductor Materials (cont.)
Semiconductor
Bandgap
Energy EG (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
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 10
Covalent Bond Model
• 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.
Silicon diamond lattice unit
cell.
NJIT ECE-271 Dr. S. Levkov
Corner of diamond lattice
showing four nearest
neighbor bonding.
View of crystal
lattice along a crystallographic
axis.
Chap 2 - 11
Silicon Covalent Bond Model (cont.)
Silicon atom
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 12
Silicon Covalent Bond Model (cont.)
Covalent bond
Silicon atom
NJIT ECE-271 Dr. S. Levkov
Silicon atom
Chap 2 - 13
Silicon Covalent Bond Model (cont.)
Silicon atom
NJIT ECE-271 Dr. S. Levkov
Covalent bonds in silicon
Chap 2 - 14
Silicon Covalent Bond Model (cont.)
•
•
•
•
What happens as the temperature
increases?
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
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 15
Silicon Covalent Bond Model (cont.)
•
•
•
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
NJIT ECE-271 Dr. S. Levkov
•
•
•
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.
Chap 2 - 16
Intrinsic Carrier Concentration
• The density of carriers in a semiconductor as a function of temperature
and material properties is:
 EG 
n  BT exp  

kT


2
i
3
cm-6
• EG = 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 1031 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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 17
• Electron density is n
(electrons/cm3) and for
intrinsic material n = ni.
• Intrinsic refers to
properties of pure
materials.
• ni ≈ 1010 cm-3 for Si
• The density of silicon
atoms is na ≈ 5x1022 cm-3
• Thus at a room
temperature one bond per
about 1013 is broken
NJIT ECE-271 Dr. S. Levkov
Intrinsic carrier density (cm-3)
Intrinsic Carrier Concentration (cont.)
Chap 2 - 18
Electron-hole concentrations
• A vacancy is left when a covalent bond is broken.
• The vacancy is called a hole.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 19
Electron-hole concentrations
• 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).
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 20
Electron-hole concentrations
• 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 (ni for intrinsic material)
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 21
Electron-hole concentrations
• 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 (ni for intrinsic material)
• Hole density is represented by p.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 22
Electron-hole concentrations
• 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 (ni for intrinsic material)
• Hole density is represented by p.
• For intrinsic silicon, n = ni = p.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 23
Electron-hole concentrations
• 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 (ni for intrinsic material)
• Hole density is represented by p.
• For intrinsic silicon, n = ni = p.
• The product of electron and hole concentrations is pn = ni2.
• The pn product above holds when a semiconductor is in
thermal equilibrium (not with an external voltage applied).
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 24
Drift Current
•
•
Charged particles move or drift under the influence of the applied field.
The resulting current is called drift current.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 25
Drift Current
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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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 26
Drift Current
•
•
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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/cm3)(cm/s) = A/cm2]
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 27
Drift Current
•
•
•
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/cm3)(cm/s) = A/cm2]
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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 28
Mobility
• At low fields, carrier drift velocity v (cm/s) is proportional
to electric field E (V/cm). The constant of proportionality
is the mobility, :
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 29
Mobility
• At low fields, carrier drift velocity v (cm/s) is proportional
to electric field E (V/cm). The constant of proportionality
is the mobility, :
• vn = - nE and vp = pE , where
• vn and vp - electron and hole velocity (cm/s),
• n and p - electron and hole mobility (cm2/Vs)
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 30
Mobility
• At low fields, carrier drift velocity v (cm/s) is proportional
to electric field E (V/cm). The constant of proportionality
is the mobility, :
• vn = - nE and vp = pE , where
• vn and vp - electron and hole velocity (cm/s),
• n and p - electron and hole mobility (cm2/Vs)
• n ≈ 1350 (cm2/Vs), p ≈ 500 (cm2/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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 31
Velocity Saturation
At high fields, carrier
velocity saturates and
places upper limits on
the speed of solid-state
devices.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 32
Intrinsic Silicon Resistivity
• Given drift current and mobility, we can calculate
resistivity (Q is the charge density) :
jndrift = Qnvn = (-qn)(- nE) = qn nE A/cm2
jpdrift = Qpvp = (+qp)(+ pE) = qp pE A/cm2
jTdrift = jn + jp = q(n n + p p)E = E
This defines electrical conductivity:
 = q(n n + p p) (cm)-1
Resistivity  is the reciprocal of conductivity:

E
 = 1/ (cm)
 

NJIT ECE-271 Dr. S. Levkov
jTdrift


V /cm



cm

A /cm 2

Chap 2 - 33
Example: Calculate the resistivity of
intrinsic silicon
Problem: Find the resistivity of intrinsic silicon at room temperature and
classify it as an insulator, semiconductor, or conductor.
Solution:
•
•
•
Known Information and Given Data: The room temperature motilities. For intrinsic
silicon, the electron and hole densities are both equal to ni.
Unknowns: Resistivity  and classification.
Assumptions: assume “room temperature” with ni = 1010/cm3.
•
Analysis: Charge density of electrons is Qn = -qni and for holes is Qp = +qni. Thus:
 = (1.60 x 10-10)[(1010)(1350) + (1010)(500)] (C)(cm-3)(cm2/Vs)
= 2.96 x 10-6 (cm)-1 --->  = 1/ = 3.38 x 105 cm
 = q(n n + p p)
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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 34
Semiconductor Doping
• The interesting properties of semiconductors emerges
when impurities are introduced.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 35
Semiconductor Doping
• 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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 36
Semiconductor Doping
• 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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 37
Semiconductor Doping
• 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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 38
Donor Impurities in Silicon
•
•
•
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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 39
Donor Impurities in Silicon
q
•
•
•
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.
NJIT ECE-271 Dr. S. Levkov
q
e
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.
Chap 2 - 40
Acceptor Impurities in Silicon
•
•
•
•
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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 41
Acceptor Impurities in Silicon
•
•
•
•
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.
NJIT ECE-271 Dr. S. Levkov
q
e
q
Vacancy
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.
Chap 2 - 42
Acceptor Impurities – Hole propagation
Hole is propagating through the silicon.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 43
Acceptor Impurities – Hole propagation
e
Hole
Hole is propagating through the silicon.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 44
Acceptor Impurities – Hole propagation
Hole
Hole is propagating through the silicon.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 45
Acceptor Impurities – Hole propagation
e
Hole is propagating through the silicon.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 46
Doped Silicon Carrier Concentrations
(how to calculate)
•
In doped material, the electron and hole concentrations are no longer equal.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 47
Doped Silicon Carrier Concentrations
(how to calculate)
•
•
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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 48
Doped Silicon Carrier Concentrations
(how to calculate)
•
•
•
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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 49
Doped Silicon Carrier Concentrations
(how to calculate)
•
•
•
•
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.
ND = donor impurity concentration
NA = acceptor impurity concentration atoms/cm3
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 50
Doped Silicon Carrier Concentrations
(how to calculate)
•
•
•
•
•
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.
ND = donor impurity concentration
NA = acceptor impurity concentration atoms/cm3
Charge neutrality requires q(ND + p - NA - n) = 0:
positive charge: p (holes) + ND (ionized donors)
negative charge: n (electrons) + ND (ionized acceptors)
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 51
Doped Silicon Carrier Concentrations
(how to calculate)
•
•
•
•
•
•
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.
ND = donor impurity concentration
NA = acceptor impurity concentration atoms/cm3
Charge neutrality requires q(ND + p - NA - n) = 0:
positive charge: p (holes) + ND (ionized donors)
negative charge: n (electrons) + ND (ionized acceptors)
It can also be shown that pn = ni2, even for doped semiconductors in thermal
equilibrium.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 52
Doped Silicon Carrier Concentrations
(how to calculate)
•
•
•
•
•
•
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.
ND = donor impurity concentration
NA = acceptor impurity concentration atoms/cm3
Charge neutrality requires q(ND + p - NA - n) = 0:
positive charge: p (holes) + ND (ionized donors)
negative charge: n (electrons) + NA (ionized acceptors)
It can also be shown that pn = ni2, even for doped semiconductors in thermal
equilibrium.
Explanation. The rate of e/h recombination is Cnp (kind of a number of possibilities of each electron to
recombine with each hole). At the thermal equilibrium, rate of e/h recombination is equal to the rate of e/h
pairs creation, thus np is the constant for certain temperature.
Since creation recombination is the thermal process (depends on temperature, not doping), np should be
the same as for intrinsic material, so np = ni pi = ni2.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 53
n-type Material
• Substituting p = ni2/n into q(ND + p - NA - n) = 0
yields n2 - (ND - NA)n - ni2 = 0.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 54
n-type Material
• Substituting p = ni2/n into q(ND + p - NA - n) = 0
yields n2 - (ND - NA)n - ni2 = 0.
• Solving for n
(N D  N A )  (N D  N A ) 2  4n i2
n i2
n
and p 
2
n

NJIT ECE-271 Dr. S. Levkov
Chap 2 - 55

n-type Material
• Substituting p = ni2/n into q(ND + p - NA - n) = 0
yields n2 - (ND - NA)n - ni2 = 0.
• Solving for n
(N D  N A )  (N D  N A ) 2  4n i2
n i2
n
and p 
2
n
• For (ND - NA) >> 2ni, n  (ND - NA) .
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 56
p-type Material
• Similar to the approach used with n-type material we find
the following equations:
(N A  N D )  (N A  N D ) 2  4n i2
ni2
p
and n 
2
p

NJIT ECE-271 Dr. S. Levkov
Chap 2 - 57
p-type Material
• Similar to the approach used with n-type material we find
the following equations:
(N A  N D )  (N A  N D ) 2  4n i2
ni2
p
and n 
2
p
• For (NA - ND) >> 2ni, p  (NA - ND) .

NJIT ECE-271 Dr. S. Levkov
Chap 2 - 58
p-type Material
• Similar to the approach used with n-type material we find
the following equations:
(N A  N D )  (N A  N D ) 2  4n i2
ni2
p
and n 
2
p
• For (NA - ND) >> 2ni, p  (NA - ND) .

• We find the majority carrier concentration from charge
neutrality and find the minority carrier concentration from
the thermal equilibrium relationship.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 59
Practical Doping Levels
• Majority carrier concentrations are established at
manufacturing time and are independent of temperature
(over practical temp. ranges).
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 60
Practical Doping Levels
• Majority carrier concentrations are established at
manufacturing time and are independent of temperature
(over practical temp. ranges).
• However, minority carrier concentrations are proportional
to ni2, a highly temperature dependent term.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 61
Practical Doping Levels
• Majority carrier concentrations are established at
manufacturing time and are independent of temperature
(over practical temp. ranges).
• However, minority carrier concentrations are proportional
to ni2, a highly temperature dependent term.
• For practical doping levels (dopant concentration usually is
quite larger then ni):
n  (ND - NA) for n-type material
p  (NA - ND) for p-type material.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 62
Practical Doping Levels
• Majority carrier concentrations are established at
manufacturing time and are independent of temperature
(over practical temp. ranges).
• However, minority carrier concentrations are proportional
to ni2, a highly temperature dependent term.
• For practical doping levels:
n  (ND - NA) for n-type material
p  (NA - ND) for p-type material.
• Typical doping ranges are 1014/cm3 to 1021/cm3.
NJIT ECE-271 Dr. S. Levkov
Example here
Chap 2 - 63
Mobility and Resistivity in
Doped Semiconductors
• 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 (ND – NA) for n-type
•  = qp (NA – ND) for p-type
Example here
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 64
Diffusion Current
• 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.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 65
Diffusion Current (cont.)
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.
NJIT ECE-271 Dr. S. Levkov
If the electrons are injected and the electronconcentration 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.
Chap 2 - 66
Diffusion Current (cont.)
• Carriers move toward regions of
lower concentration, so diffusion
current densities are proportional to
the negative of the carrier gradient.
j
diff
p
jndiff
p
 p 
 ( q ) D p     qD p
A/cm 2
x
 x 
n
 n 
 (q ) Dn      qDn
A/cm 2
x
 x 
Diffusion current density equations
NJIT ECE-271 Dr. S. Levkov
Diffusion currents in the
presence of a concentration
gradient.
Chap 2 - 67
Diffusion Current (cont.)
• Dp and Dn are the hole and electron diffusivities with units
cm2/s. Diffusivity and mobility are related by Einsteins’s
relationship:
kT D p


 VT  Thermal voltage
n
q
p
Dn
Dn   n VT , D p   p VT
• The thermal voltage, VT = kT/q, is approximately 25 mV
at room
temperature. We will encounter VT many times

throughout this course.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 68
Total Current in a Semiconductor
• Total current is the sum of drift and diffusion current:
n
j  q n nE  qDn
x
p
T
j p  q  p pE  qD p
x
T
n
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 69
Total Current in a Semiconductor
• Total current is the sum of drift and diffusion current:
n
j  q n nE  qDn
x
p
T
j p  q  p pE  qD p
x
T
n
Rewriting using Einstein’s relationship (Dp = nVT),
1 n

j  q n n  E  VT

nx


1  p
T
j p  q  p p  E  VT

p x

T
n
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
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 70
Semiconductor Energy Band Model
Semiconductor energy
band model. EC and EV
are energy levels at the
edge of the conduction
and valence bands.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 71
Semiconductor Energy Band Model
What happens as
temperature increases?
Semiconductor energy
band model. EC and EV
are energy levels at the
edge of the conduction
and valence bands.
NJIT ECE-271 Dr. S. Levkov
Electron participating in
a covalent bond is in a
lower energy state in the
valence band. This
diagram represents 0 K.
Chap 2 - 72
Semiconductor Energy Band Model
Semiconductor energy
band model. EC and EV
are energy levels at the
edge of the conduction
and valence bands.
NJIT ECE-271 Dr. S. Levkov
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.
Chap 2 - 73
Energy Band Model for a Doped
Semiconductor
Semiconductor with donor or n-type
dopants. The donor atoms have free
electrons with energy ED. Since ED is
close to EC, (about 0.045 eV for
phosphorous), it is easy for electrons
in an n-type material to move up into
the conduction band.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 74
Energy Band Model for a Doped
Semiconductor
Semiconductor with donor or n-type
dopants. The donor atoms have free
electrons with energy ED. Since ED is
close to EC, (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.
NJIT ECE-271 Dr. S. Levkov
Semiconductor with acceptor or p-type
dopants. The aaacceptor atoms have
unfilled covalent bonds with energy
state EA. Since EA is close to EV, (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.
Chap 2 - 75
Integrated Circuit Fabrication Overview
Top view of an integrated pn diode.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 76
Integrated Circuit Fabrication (cont.)
(a) First mask exposure, (b) post-exposure and development of photoresist, (c) after
SiO2 etch, and (d) after implantation/diffusion of acceptor dopant.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 77
Integrated Circuit Fabrication (cont.)
(e) Exposure of contact opening mask, (f) after resist development and etching of contact
openings, (g) exposure of metal mask, and (h) After etching of aluminum and resist removal.
NJIT ECE-271 Dr. S. Levkov
Chap 2 - 78