Carrier Modeling

1
Carrier Modeling
2
Quantization Concept
plank constant
Core electrons
Valence electrons
3
Periodic Table of the Elements
4
Quantization Concept

• The Shell Model

L shell with two sub shells
Nucleus
s
1
K
s
2
L
p
2
1s22s22p2 or He2s22p2

• The shell model of the atom in which the
  electrons are confined to live within certain
  shells and in sub shells within shells.

5
Quantization Concept
Stable orbit has radius r0
z
z
y
y
x
x

e
1s orbital
2px orbital
v
z
z
r
o
y
y

e
x
x
2py orbital
2pz orbital (ml 0)
The planetary model of hydrogen atom in which the
negatively charged electron orbits the positively
charged nucleus.
Orbitals
6
Atomic Bonding

• Bonding forces in Solids

1. Ionic bonding (such as NaCl)
2. Metallic bonding (all metals)
3. Covalent bonding (typical Si)
4. Van der Waals bonding (water)
5. Mixed bonding (GaAs, ZnSe, ionic covalent)

7
Energy Band Formation
Allowed energy levels of an electron acted on by
the Coulomb potential of an atomic nucleus.
Splitting of energy states into allowed bands
separated by a forbidden energy gap as the atomic
spacing decreases the electrical properties of a
crystalline material correspond to specific
allowed and forbidden energies associated with an
atomic separation related to the lattice constant
of the crystal.
8
Energy Band Formation

• Broadening of allowed energy levels into allowed
  energy bands separated by forbidden-energy gaps
  as more atoms influence each electron in a solid.

One-dimensional representation
Two-dimensional diagram in which energy is
plotted versus distance.
9
Energy Band Formation
Energy Bandgap where no states exist
Pauli Exclusion Principle
Only 2 electrons, of spin/-1/2, can occupy the
same energy state at the same point in space.
As atoms are brought closer towards one another
and begin to bond together, their energy levels
must split into bands of discrete levels so
closely spaced in energy, they can be considered
a continuum of allowed energy.

• Strongly bonded materials small interatomic
  distances.
• Thus, the strongly bonded materials can have
  larger energy bandgaps than do weakly bonded
  materials.

10
Energy Band Formation (Si)

• Energy levels in Si as a function of
  inter-atomic spacing

The 2N electrons in the 3s sub-shell and the 2N
electrons in the 3p sub-shell undergo sp3
hybridization.
conduction band (empty)
valence band (filled)
The core levels (n1,2) in Si are completely
filled with electrons.
11
Energy Band Formation
12
Energy Band Formation

• Energy band diagrams.

N electrons filling half of the 2N allowed
states, as can occur in a metal.
A completely empty band separated by an energy
gap Eg from a band whose 2N states are completely
filled by 2N electrons, representative of an
insulator.
13
Metals, Semiconductors, and Insulators
Ef
Ef
Metal
Semiconductor

• Allowed electronic-energy-state systems for metal
  and semiconductors.
• States marked with an X are filled those
  unmarked are empty.

14
Metals, Semiconductors, and Insulators

• Typical band structures of Metal

Electron Energy,
E
Free electron
Vacuum
E
0
3s Band
level
2
p
Band
3
p
Overlapping energy bands
3
s
2
p
2
s
Band
2
s
Electrons
1
s
1
s
SOLID
ATOM

• In a metal the various energy bands overlap to
  give a single band of energies that is only
  partially full of electrons.
• There are states with energies up to the vacuum
  level where the electron is free.

15
Electron Motion in Energy Band
Current flowing
E 0
E ? 0

• Electron motion in an allowed band is analogous
  to fluid motion in a glass tube with sealed ends
  the fluid can move in a half-filled tube just as
  electrons can move in a metal.

16
Electron Motion in Energy Band
E 0
E ? 0

• No fluid motion can occur in a completely filled
  tube with sealed ends.

17
Energy Band Formation

• Energy band diagrams.

• Energy-band diagram for a semiconductor showing
  the lower edge of the conduction band Ec, a donor
  level Ed within the forbidden gap, and Fermi
  level Ef, an acceptor level Ea, and the top edge
  of the valence band Ev.

18
Electron Motion in Energy Band

• Fluid analogy for a semiconductor

• No flow can occur in either the completely filled
  or completely empty tube.

• Fluid can move in both tubes if some of it is
  transferred from the filled tube to the empty
  one, leaving unfilled volume in the lower tube.

19
Metals, Semiconductors, and Insulators

• Typical band structures at 0 K.

Insulator
Semiconductor
Metal
20
Material Classification based on Size of Bandgap

• Ease of achieving thermal population of
  conduction band determines whether a material is
  an insulator, metal, or semiconductor.

21
Metals, Semiconductors, and Insulators

• Range of conductivities exhibited by various
  materials.

I
n
s
u
l
a
t
o
r
s
S
e
m
i
c
o
n
d
u
c
t
o
r
s
C
o
n
d
u
c
t
o
r
s
M
a
n
y

c
e
r
a
m
i
c
s



S
u
p
e
r
c
o
n
d
u
c
t
o
r
s
A
l
u
m
i
n
a
D
i
a
m
o
n
d
I
n
o
r
g
a
n
i
c

G
l
a
s
s
e
s
M
e
t
a
l
s
M
i
c
a
P
o
l
y
p
r
o
p
y
l
e
n
e
D
e
g
e
n
e
r
a
t
e
l
y
S
o
d
a

s
i
l
i
c
a

g
l
a
s
s
P
V
D
F
doped Si
A
l
l
o
y
s
B
o
r
o
s
i
l
i
c
a
t
e
P
E
T
P
u
r
e

S
n
O
2
I
n
t
r
i
n
s
i
c

S
i
A
m
o
r
p
h
o
u
s
T
e
A
g
G
r
a
p
h
i
t
e
N
i
C
r
S
i
O
I
n
t
r
i
n
s
i
c

G
a
A
s
2
A
s
S
e
2
3
-
3
-
6
3
0
-
9
-
1
2
-
1
5
-
1
8
9
1
2
6
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Conductivity (?m)-1
22
Energy Band Diagram

• E-k diagram, Bloch function.

PE of the electron around an isolated atom
When N atoms are arranged to form the crystal
then there is an overlap of individual electron
PE functions.
PE of the electron, V(x), inside the crystal is
periodic with a period a.

• The electron potential energy PE, V(x), inside
  the crystal is periodic with the same periodicity
  as that of the crystal, a.
• Far away outside the crystal, by choice, V 0
  (the electron is free and PE 0).

23
Energy Band Diagram

• E-k diagram, Bloch function.

Schrödinger equation
Periodic Potential
Periodic Wave function Bloch Wavefunction

• There are many Bloch wavefunction solutions to
  the one-dimensional crystal each identified with
  a particular k value, say kn which act as a kind
  of quantum number.
• Each ?k (x) solution corresponds to a particular
  kn and represents a state with an energy Ek.

24
Energy Band Diagram

• E-k diagram of a direct bandgap semiconductor

• The E-k curve consists of many discrete points
  with each point corresponding to a possible
  state, wavefunction ?k (x), that is allowed to
  exist in the crystal.
• The points are so close that we normally draw the
  E-k relationship as a continuous curve. In the
  energy range Ev to Ec there are no points ?k
  (x), solutions.

25
Energy Band Diagram

• The energy is plotted as a function of the wave
  number, k, along the main crystallographic
  directions in the crystal.

Ge
Si
GaAs
The bottom axis describe different directions of
the crystal.
26
Energy Band Diagram

• E-k diagram

E
E
E
C
B
I
n
d
i
r
e
c
t

B
a
n
d
g
a
p
,

E
g
E
C
B
c
D
i
r
e
c
t

B
a
n
d
g
a
p
C
B
P
h
o
t
o
n
E
E
E
g
E
c
c
E
P
h
o
n
o
n
k
r
v
c
b
E
E
v
v
V
B
k
V
B
V
B
v
b
k
k
k

k
k
k



S
i

w
i
t
h

a

r
e
c
o
m
b
i
n
a
t
i
o
n

c
e
n
t
e
r
G
a
A
s

S
i
In GaAs the minimum of the CB is directly above
the maximum of the VB. direct bandgap
semiconductor.
Recombination of an electron and a hole in Si
involves a recombination center.
In Si, the minimum of the CB is displaced from
the maximum of the VB. indirect bandgap
semiconductor
27
Direct and Indirect Energy Band Diagram
(a) Direct transition with accompanying photon
emission. (b) Indirect transition via defect
level.
28
Energy Band

• A simplified energy band diagram with the highest
  almost-filled band and the lowest almost-empty
  band.

vacuum level
? electron affinity
conduction band edge
valence band edge
29
Metals vs. Semiconductors

• Pertinent energy levels

Metal
Semiconductor

• Only the work function is given for the metal.
• Semiconductor is described by the work function
  qFs, the electron affinity q?s, and the band gap
  (Ec Ev).

30
Metals, Semiconductors, and Insulators

• Typical band structures of Semiconductor

C
o
v
a
l
e
n
t

b
o
n
d
S
i

i
o
n

c
o
r
e

(

4
e
)
E
l
e
c
t
r
o
n

e
n
e
r
g
y
,

E
c
E

c
C
o
n
d
u
c
t
i
o
n

B
a
n
d

(
C
B
)
E
m
p
t
y

o
f

e
l
e
c
t
r
o
n
s

a
t

0

K
.
E
c
B
a
n
d

g
a
p



E
g
E
v
V
a
l
e
n
c
e

B
a
n
d

(
V
B
)
F
u
l
l

o
f

e
l
e
c
t
r
o
n
s

a
t

0

K
.
0
A simplified two dimensional view of a region of
the Si crystal showing covalent bonds.
The energy band diagram of electrons in the Si
crystal at absolute zero of temperature.
31
Electrons and Holes
Electrons Electrons in the conduction band that
are free to move throughout the crystal. Holes
Missing electrons normally found in the valence
band
(or empty states in the valence band that would
normally be filled).
These particles carry electricity. Thus, we
call these carriers
32
Electrons and Holes

• Generation of Electrons and Holes

E
l
e
c
t
r
o
n

e
n
e
r
g
y
,

E
c
E

c
C
B
E
u
c
h

gt

E

F
r
e
e

e
g
h
u
h
o
l
e
E
g

e

H
o
l
e

h
E
v
V
B
0
Each line between Si-Si atoms is a valence
electron in a bond. When a photon breaks a Si-Si
bond, a free electron and a hole in the Si-Si
bond is created.
A photon with an energy greater then Eg can
excite an electron from the VB to the CB.
33
Effective Mass (I)

• An electron moving in respond to an applied
  electric field.

E
E
within a Vacuum
within a semiconductor crystal

• It allow us to conceive of electrons and holes as
  quasi-classical particles and to employ classical
  particle relationships in semiconductor crystals
  or in most device analysis.

34
Carrier Movement Within the Crystal
Density of States Effective Masses at 300 K
Ge and GaAs have lighter electrons than Si
which results in faster devices
35
Effective Mass (II)

• Electrons are not free but interact with periodic
  potential of the lattice.
• Wave-particle motion is not as same as in free
  space.

Curvature of the band determine m. m is
positive in CB min., negative in VB max.
36
Effective Mass Approximation

• The motion of electrons in a crystal can be
  visualized and described in a quasi-classical
  manner.
• In most instances
• The electron can be thought of as a particle.
• The electronic motion can be modeled using
  Newtonian mechanics.
• The effect of crystalline forces and quantum
  mechanical properties are incorporated into the
  effective mass factor.
• m gt 0 near the bottoms of all bands
• m lt 0 near the tops of all bands
• Carriers in a crystal with energies near the top
  or bottom of an energy band typically exhibit a
  constant (energy-independent) effective mass.
• near band edge

37
Covalent Bonding
38
Covalent Bonding
39
Band Occupation at Low Temperature (0 K)
40
Band Occupation at Low Temperature (0 K)
41
Band Occupation at Low Temperature (0 K)
42
Band Occupation at Low Temperature (0 K)
43
Band Occupation at Low Temperature (0 K)
44
Band Occupation at Low Temperature (0 K)
45
Impurity Doping

• The need for more control over carrier
  concentration

• Without help the total number of carriers
  (electrons and holes) is limited to 2ni.
• For most materials, this is not that much, and
  leads to very high resistance and few useful
  applications.
• We need to add carriers by modifying the crystal.
• This process is known as doping the crystal.

46
Concept of a Donor Adding extra Electrons
47
Concept of a Donor Adding extra Electrons
48
Concept of a Donor Adding extra Electrons
49
Binding Energies of Impurity

• Hydrogen Like Impurity Potential (Binding
  Energies)

• Effective mass should be used to account the
  influence of the periodic potential of crystal.
• Relative dielectric constant of the semiconductor
  should be used (instead of the free space
  permittivity).

Electrons in donor atoms Holes in
acceptor atoms
Binding energies in Si 0.03 0.06 eV Binding
energies in Ge 0.01 eV
50
Concept of a Donor Adding extra Electrons
Band diagram equivalent view
51
Concept of a Donor Adding extra Electrons

• n-type Impurity Doping of Si

E
l
e
c
t
r
o
n

E
n
e
r
g
y
C
B

A
s
E

c
e

0
.
0
5

e
V
E




A
s
A
s
A
s
A
s
d
D
i
s
t
a
n
c
e

i
n
t
o
E
x
v
c
r
y
s
t
a
l
6
A
s

a
t
o
m

s
i
t
e
s

e
v
e
r
y

1
0

S
i

a
t
o
m
s
The four valence electrons of As allow it to bond
just like Si but the 5th electron is left
orbiting the As site. The energy required to
release to free fifth- electron into the CB is
very small.
Energy band diagram for an
n
-type Si doped
with 1 ppm As. There are donor energy levels
just
around As sites.
below Ec

52
Concept of a Donor Adding extra Electrons

• Energy Band Diagram in an Applied Field

• Energy band diagram of an n-type semiconductor
  connected to a voltage supply of V volts.
• The whole energy diagram tilts because the
  electron now has an electrostatic potential
  energy as well.
• Current flowing

53
Concept of a Acceptor Adding extra Holes
All regions of material are neutrally charged
One less bond means the neighboring Silicon is
left with an empty state.
One less bond means the acceptor is
electrically satisfied.
54
Hole Movement
Empty state is located next to the Acceptor
55
Hole Movement
Another valence electron can fill the empty state
located next to the Acceptor leaving behind a
positively charged hole.
56
Hole Movement
57
Hole Movement
The positively charged hole can move throughout
the crystal. (Really it is the valance electrons
jumping from atom to atom that creates the hole
motion)
58
Hole Movement
59
Hole Movement
Region around the hole has one
less electron and thus is positively charged.
The positively charged hole can move throughout
the crystal. (Really it is the valance electrons
jumping from atom to atom that creates the hole
motion)
60
Concept of a Acceptor Adding extra Holes
Band diagram equivalent view
61
Concept of a Acceptor Adding extra Holes

• p-type Impurity Doping of Si

E
l
e
c
t
r
o
n

e
n
e
r
g
y
6
B

a
t
o
m

s
i
t
e
s

e
v
e
r
y

1
0

S
i

a
t
o
m
s
E
x
D
i
s
t
a
n
c
e
c
i
n
t
o

c
r
y
s
t
a
l


B
B

B

B
E
a

0
.
0
5

e
V

h
E
v
V
B
Boron doped Si crystal. B has only three valence
electrons. When it substitute for a Si atom one
of its bond has an electron missing and therefore
a hole.
Energy band diagram for a p-type Si crystal doped
with 1 ppm B. There are acceptor energy levels
just above Ev around B- site. These acceptor
levels accept electrons from the VB and therefore
create holes in the VB.
62
Intrinsic, n-Type, p-Type Semiconductors

• Energy band diagrams

C
B
E
E
E
c
c
c
E
F
n
E
F
i
E
F
p
E
E
E
v
v
v
V
B
p-type semiconductors
n-type semiconductors
Intrinsic semiconductors

• In all cases, npni2
• Note that donor and acceptor energy levels are
  not shown.

63
Heavily Doped n-Type, p-Type Semiconductors
E
C
B
C
B
E
Impurities forming a band
n
F
E
E
c
c
g
(
E
)
E
E
v
v
E
F
p
V
B
Degenerate p-type semiconductor
Degenerate n-type semiconductor. Large number of
donors form a band that overlaps the CB.
64
Impurity Doping
65
Impurity Doping
Valence Band
Valence Band
66
Impurity Doping

• Position of energy levels within the bandgap of
  Si for
• common dopants.

67
Summary of Important terms and symbols
Bandgap Energy Energy required to remove a
valence electron and allow it to freely
conduct. Intrinsic Semiconductor A native
semiconductor with no dopants. Electrons in the
conduction band equal holes in the valence band.
The concentration of electrons (holes) is the
intrinsic concentration, ni. Extrinsic
Semiconductor A doped semiconductor. Many
electrical properties controlled by the
dopants, not the intrinsic semiconductor. Donor
An impurity added to a semiconductor that adds an
additional electron not found in the native
semiconductor. Acceptor An impurity added to a
semiconductor that adds an additional hole not
found in the native semiconductor. Dopant
Either an acceptor or donor. N-type material
When electron concentrations (nnumber of
electrons/cm3) exceed the hole concentration
(normally through doping with donors). P-type
material When hole concentrations (pnumber of
holes/cm3) exceed the electron concentration
(normally through doping with acceptors). Majority
carrier The carrier that exists in higher
population (i.e. n if ngtp, p if pgtn) Minority
carrier The carrier that exists in lower
population (i.e. n if nltp, p if pltn) Other
important terms Insulator, semiconductor, metal,
amorphous, polycrystalline, crystalline (or
single crystal), lattice, unit cell, primitive
unit cell, zincblende, lattice constant,
elemental semiconductor, compound semiconductor,
binary, ternary, quaternary, atomic
density, Miller indices