Chapter3 introduction to the quantum theory of solids
KMMonir52
5,015 views
52 slides
Dec 10, 2016
Slide 1 of 52
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
About This Presentation
solid states physics
Slide Content
Microelectronics I
Chapter 3: Introduction to the
Quantum Theory of Solids
Chapter 3: Introduction to the
Quantum Theory of Solids
Microelectronics I : Introduction to the Quantum Theory of Solids
Chapter 3 (part 1)
1. Formation of allowed and forbidden
energy band
k-space
diagram
(Energy
-
wave number diagram)
Qualitative and quantitative
discussion
Kronig-Penney model
(Energy
-
wave number diagram)
2. Electrical conduction in solids
Drift current, electron effective mass, concept of hole
Energy band model
Chapter 3 (part 1)
1. Formation of allowed and forbidden
energy band
k-space
diagram
(Energy
-
wave number diagram)
Qualitative and quantitative
discussion
Kronig-Penney model
(Energy
-
wave number diagram)
2. Electrical conduction in solids
Drift current, electron effective mass, concept of hole
Energy band model
Microelectronics I : Introduction to the Quantum Theory of Solids
Isolated single atom (ex; Si)
electron
energy
Quantized energy level
(quantum state)
1s
2s
2p
3s
3p
+
n=1
n=2
n=3
Crystal (~10
20
atom)
electron
energy++ ….=
?
x 10
20
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
Isolated single atom (ex; Si)
electron
energy
Quantized energy level
(quantum state)
1s
2s
2p
3s
3p
+
n=1
n=2
n=3
Crystal (~10
20
atom)
electron
energy++ ….=
?
x 10
20
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
Microelectronics I : Introduction to the Quantum Theory of Solids
Si Crystal
Tetrahedral structure
Diamond structure
Tetrahedral structure
energy
Valence band
conduction band
Energy gap, E
g=1.1 eV
Formation of
energy band
and
energy gap
Si Crystal
Tetrahedral structure
Diamond structure
Tetrahedral structure
energy
Valence band
conduction band
Energy gap, E
g=1.1 eV
Formation of
energy band
and
energy gap
Microelectronics I : Introduction to the Quantum Theory of Solids What happen if 2 identical atoms approach each other ?
r
atom 2
atom 1
energy
1s
Isolated atom
z
x
x
1sz
y
x
Distance from center
Probability density
y
x
x
1s1s
Wave function of two atom electron overlap
interaction
r
atom 2
atom 1
energy
1s
Isolated atom
z
x
x
1sz
y
x
Distance from center
Probability density
y
x
x
1s1s
Wave function of two atom electron overlap
interaction
Microelectronics I : Introduction to the Quantum Theory of Solids
r
atom 2
atom 1
∞When the atoms are far apart
(r=∞), electron from different
atoms can occupy same
energy level.
E
1s,atom 1 =E
1s, atom 2
∞As the atoms approach each
other,
energy level splits
energy
1s
other,
energy level splits E
1s,atom 1 ≠E
1s, atom 2
r
a
energy
≠interaction between two overlap wave function
≠Consistent with Pauli exclusion principle
a ; equilibrium interatomic distance
r
atom 2
atom 1
∞When the atoms are far apart
(r=∞), electron from different
atoms can occupy same
energy level.
E
1s,atom 1 =E
1s, atom 2
∞As the atoms approach each
other,
energy level splits
energy
1s
other,
energy level splits E
1s,atom 1 ≠E
1s, atom 2
r
a
energy
≠interaction between two overlap wave function
≠Consistent with Pauli exclusion principle
a ; equilibrium interatomic distance
Microelectronics I : Introduction to the Quantum Theory of Solids
Regular periodic arrangement of atom (crystal)
ex: 10
20
atoms
Total number of quantum states
do not change when forming a
system (crystal)
energy
1s
10
20
energy levels
a
energy
“energy band”
dense allowed energy levels
Regular periodic arrangement of atom (crystal)
ex: 10
20
atoms
Total number of quantum states
do not change when forming a
system (crystal)
energy
1s
10
20
energy levels
a
energy
“energy band”
dense allowed energy levels
Microelectronics I : Introduction to the Quantum Theory of Solids
energy
M
10
20
energy state
1 eV
Consider
M
10
20
energy state
MEnergy states are equidistant
Energy states are separated by 1/10
20
eV =
10
-20
eV
(Almost)
continuous energy states
within energy band
energy
M
10
20
energy state
1 eV
Consider
M
10
20
energy state
MEnergy states are equidistant
Energy states are separated by 1/10
20
eV =
10
-20
eV
(Almost)
continuous energy states
within energy band
Microelectronics I : Introduction to the Quantum Theory of Solids
Distance from center
Probability density
energy
2s
1s
atom 2
atom 1
1s
2s
r
atom 2
atom 1
energy
1s
a
2s
“there is no energy level”
forbidden band →
energy gap, Eg
∞As the atoms are brought together,
electron from 2s will interact. Then electron
from 1s.
Distance from center
Probability density
energy
2s
1s
atom 2
atom 1
1s
2s
r
atom 2
atom 1
energy
1s
a
2s
“there is no energy level”
forbidden band →
energy gap, Eg
∞As the atoms are brought together,
electron from 2s will interact. Then electron
from 1s.
Microelectronics I : Introduction to the Quantum Theory of Solids
Si: 1s(2), 2s(2), 2p(6),
3s(2), 3p (2)
14 electrons
Ex;
Tightly bound to
nucleus
Involved in
chemical reactions
energy
energy
3s
3p
energy
Sp3 hybrid orbital
Reform 4 equivalent states c4 equivalent bond (symmetric)
Si: 1s(2), 2s(2), 2p(6),
3s(2), 3p (2)
14 electrons
Ex;
Tightly bound to
nucleus
Involved in
chemical reactions
energy
energy
3s
3p
energy
Sp3 hybrid orbital
Reform 4 equivalent states c4 equivalent bond (symmetric)
Microelectronics I : Introduction to the Quantum Theory of Solids
Si
Si
Si
Si
Si
energy
+
+++
energy
filled
empty
Si
Si
Si
Si
Si
energy
+
+++
energy
filled
empty
Microelectronics I : Introduction to the Quantum Theory of Solids
Si crystal (10
22
atoms/cm
3
)
filled
empty
energy
conduction band
Energy gap, E
g=1.1 eV
energy
filled
Valence band
4 x 10
22
states/cm
3
Si crystal (10
22
atoms/cm
3
)
filled
empty
energy
conduction band
Energy gap, E
g=1.1 eV
energy
filled
Valence band
4 x 10
22
states/cm
3
Microelectronics I : Introduction to the Quantum Theory of Solids
Forbidden band →
band gap, E
allowed band
Actual band structure“calculated by quantum mechanics”
→
band gap, E
G
allowed band
Forbidden band →
band gap, E
allowed band
Actual band structure“calculated by quantum mechanics”
→
band gap, E
G
allowed band
Microelectronics I : Introduction to the Quantum Theory of Solids
Quantitative discussion Determine the relation between energy of electron(E) , wave number (k)
∞Relation of E and k for free electron
2
2
Ψ(x,t)= exp ( j(
k
x-ωt))
E
m
k
E
2
2
2
h
=
Continuous value of E
K-space diagram
k
E
Quantitative discussion Determine the relation between energy of electron(E) , wave number (k)
∞Relation of E and k for free electron
2
2
Ψ(x,t)= exp ( j(
k
x-ωt))
E
m
k
E
2
2
2
h
=
Continuous value of E
K-space diagram
k
E
Microelectronics I : Introduction to the Quantum Theory of Solids
E-k diagram for electron in quantum well
E
n=3
mk
E
n
L m
E
2
2
2 2
2
2
2
h
h
=
=
π
n
L
k
=
π
E
E-k diagram for electron in crystal?The Kronig-Penney Model
x=L
x=0
E
n=1
E
n=2
k
π/L
2π/L
E-k diagram for electron in quantum well
E
n=3
mk
E
n
L m
E
2
2
2 2
2
2
2
h
h
=
=
π
n
L
k
=
π
E
E-k diagram for electron in crystal?The Kronig-Penney Model
x=L
x=0
E
n=1
E
n=2
k
π/L
2π/L
Microelectronics I : Introduction to the Quantum Theory of Solids
The Kronig-Penney Model
+
+
+
+
r
e
rV
0
2
4
)(
πε−
=
Periodic potential
V
0
IIIIII II IIIIII
Potential
well
tunneling
Periodic potential
Wave function overlap
-ba
L
Determine a relationship between k, E and V
0
The Kronig-Penney Model
+
+
+
+
r
e
rV
0
2
4
)(
πε−
=
Periodic potential
V
0
IIIIII II IIIIII
Potential
well
tunneling
Periodic potential
Wave function overlap
-ba
L
Determine a relationship between k, E and V
0
Microelectronics I : Introduction to the Quantum Theory of Solids
Schrodinger equation (E < V
0
)
Region I
0)(
)(
2
2
2
= +
∂
∂
x
x
x
I
I
ϕα
ϕ
Region II
0)(
)(
2
2
2
= −
∂
∂
x
x
x
II
II
ϕβ
ϕ
2
2
2
h
mE
=
α
2
0 2
) (2
h
E Vm
−
=
β
MPotential periodically changes
) ( )(LxV xV
+
=
jkx
exU x)( )(=
ϕ
) ( )(LxU xU
+
=
Wave function
amplitude
k; wave number [m
-1
]
Phase of the wave
Bloch theorem
Schrodinger equation (E < V
0
)
Region I
0)(
)(
2
2
2
= +
∂
∂
x
x
x
I
I
ϕα
ϕ
Region II
0)(
)(
2
2
2
= −
∂
∂
x
x
x
II
II
ϕβ
ϕ
2
2
2
h
mE
=
α
2
0 2
) (2
h
E Vm
−
=
β
MPotential periodically changes
) ( )(LxV xV
+
=
jkx
exU x)( )(=
ϕ
) ( )(LxU xU
+
=
Wave function
amplitude
k; wave number [m
-1
]
Phase of the wave
Bloch theorem
Microelectronics I : Introduction to the Quantum Theory of Solids
Boundary condition
)( )(
)0( )0(
b U a U
U U
II I
II I
− =
=
Continuous wave function
)( )(
)0( )0(
' '
' '
b U a U
U U
II I
II I
− =
=
Continuous first derivative
Boundary condition
)( )(
)0( )0(
b U a U
U U
II I
II I
− =
=
Continuous wave function
)( )(
)0( )0(
' '
' '
b U a U
U U
II I
II I
− =
=
Continuous first derivative
Microelectronics I : Introduction to the Quantum Theory of Solids
From Schrodinger equation, Bloch theorem and boundary condition
) cos( ) cos( ) cosh( ) sin( ) sinh(
2
2 2
kL a b a b= ⋅ + ⋅
−
α β α β
αβ
α β
B →0, V
0→∞Approximation for graphic solution
) cos( ) cos(
) sin(
2
0
ka a
a
a ba mV
= +
α
α
α
h
) cos( ) cos(
) sin(
'
ka a
a
a
P= +
α
α
α
2
0 '
h
ba mV
P=
Gives relation between k, E(from α) and V
0
From Schrodinger equation, Bloch theorem and boundary condition
) cos( ) cos( ) cosh( ) sin( ) sinh(
2
2 2
kL a b a b= ⋅ + ⋅
−
α β α β
αβ
α β
B →0, V
0→∞Approximation for graphic solution
) cos( ) cos(
) sin(
2
0
ka a
a
a ba mV
= +
α
α
α
h
) cos( ) cos(
) sin(
'
ka a
a
a
P= +
α
α
α
2
0 '
h
ba mV
P=
Gives relation between k, E(from α) and V
0
Microelectronics I : Introduction to the Quantum Theory of Solids
) cos(
) sin(
)(
'
a
a
a
P af
α
α
α
α
+ =
Left side
) cos( )(ka af
=
α
Right side
Value must be
between -1 and 1
Allowed value of αa
) cos(
) sin(
)(
'
a
a
a
P af
α
α
α
α
+ =
Left side
) cos( )(ka af
=
α
Right side
Value must be
between -1 and 1
Allowed value of αa
Microelectronics I : Introduction to the Quantum Theory of Solids
m
E
mE
22
2 2
2
2
h
h
α
α
=
=
Plot E-k Discontinuity of E
m
E
mE
22
2 2
2
2
h
h
α
α
=
=
Plot E-k Discontinuity of E
Microelectronics I : Introduction to the Quantum Theory of Solids
) 2 cos( ) 2 cos( ) cos( )(
π
π
α
n ka n ka ka af
=
=
+
=
=
Right side
Shift 2
π
Shift 2
π
) 2 cos( ) 2 cos( ) cos( )(
π
π
α
n ka n ka ka af
=
=
+
=
=
Right side
Shift 2
π
Shift 2
π
Microelectronics I : Introduction to the Quantum Theory of Solids
Allowed energy band Forbidden energy band
From the Kronig-Penney Model (1 dimensional periodic potential function)
Allowed energy band
Allowed energy band
Forbidden energy band
Forbidden energy band
First Brillouin zone
Allowed energy band Forbidden energy band
From the Kronig-Penney Model (1 dimensional periodic potential function)
Allowed energy band
Allowed energy band
Forbidden energy band
Forbidden energy band
First Brillouin zone
Microelectronics I : Introduction to the Quantum Theory of Solids
energy
conduction band
-
Electrical condition in solids
1. Energy band and the bond model
Valence band
Energy gap, E
g=1.1 eV
+
MBreaking of covalent bond
MGeneration of positive and negative charge
energy
conduction band
-
Electrical condition in solids
1. Energy band and the bond model
Valence band
Energy gap, E
g=1.1 eV
+
MBreaking of covalent bond
MGeneration of positive and negative charge
Microelectronics I : Introduction to the Quantum Theory of Solids
E versus k energy band
conduction band
T = 0 KT > 0 K
When no external force is applied, electron and “empty state” distributions are
symmetrical with k
Valence band
E versus k energy band
conduction band
T = 0 KT > 0 K
When no external force is applied, electron and “empty state” distributions are
symmetrical with k
Valence band
Microelectronics I : Introduction to the Quantum Theory of Solids 2. Drift current
MCurrent; diffusion current and
drift current
When Electric field is applied
EE
dE = F dx = F v dt
“Electron moves to higher empty state”
kk
E
No external force
∑
=
υ −=
n
i
i
e J
1
Drift current density,[A/cm
3
]
n; no. of electron per unit volume in the conductio n band
MCurrent; diffusion current and
drift current
When Electric field is applied
EE
dE = F dx = F v dt
“Electron moves to higher empty state”
kk
E
No external force
∑
=
υ −=
n
i
i
e J
1
Drift current density,[A/cm
3
]
n; no. of electron per unit volume in the conductio n band
Microelectronics I : Introduction to the Quantum Theory of Solids 3. Electron effective mass
F
ext
+ F
int
= ma
MElectron moves differently in the free space and in the crystal (periodical potential)
External forces
(e.g; Electrical field)
Internal forces
(e.g; potential)
+=
mass
acceleration
Internal forces
F
ext
= m*a
External forces
(e.g; Electrical field)
Internal forces (e.g; potential)
=
Effective mass
acceleration
Effect of internal force
F
ext
+ F
int
= ma
MElectron moves differently in the free space and in the crystal (periodical potential)
External forces
(e.g; Electrical field)
Internal forces
(e.g; potential)
+=
mass
acceleration
Internal forces
F
ext
= m*a
External forces
(e.g; Electrical field)
Internal forces (e.g; potential)
=
Effective mass
acceleration
Effect of internal force
Microelectronics I : Introduction to the Quantum Theory of Solids
From relation of E and k
m
dk
Ed
m
k
E
2
2
2
2 2
2
h
h
=
=
Mass of electron, m Mass of electron, m
=
2
2
2
dk
Ed
m
h
Curvature of E versus k curve
E versus k curve
Considering effect of internal force (periodic potentia l)
m from eq. above is
effective mass, m*
From relation of E and k
m
dk
Ed
m
k
E
2
2
2
2 2
2
h
h
=
=
Mass of electron, m Mass of electron, m
=
2
2
2
dk
Ed
m
h
Curvature of E versus k curve
E versus k curve
Considering effect of internal force (periodic potentia l)
m from eq. above is
effective mass, m*
Microelectronics I : Introduction to the Quantum Theory of Solids
E versus k curve
E
Free electron
Electron in crystal A
Electron in crystal B
k
MCurvature of E-k depends on the medium that electron moves in
Effective mass changes
m*
Am*
B m>>
Ex; m*
Si=0.916m
0, m*
GaAs=0.065m
0m
0; in free space
E versus k curve
E
Free electron
Electron in crystal A
Electron in crystal B
k
MCurvature of E-k depends on the medium that electron moves in
Effective mass changes
m*
Am*
B m>>
Ex; m*
Si=0.916m
0, m*
GaAs=0.065m
0m
0; in free space
Microelectronics I : Introduction to the Quantum Theory of Solids 4. Concept of hole
Electron fills the empty state
Positive charge
empty the state
“Hole”
Electron fills the empty state
Positive charge
empty the state
“Hole”
Microelectronics I : Introduction to the Quantum Theory of Solids
When electric field is applied,
hole
electron
I
Hole moves in same direction as an applied field
When electric field is applied,
hole
electron
I
Hole moves in same direction as an applied field
Microelectronics I : Introduction to the Quantum Theory of Solids
Metals, Insulators and semiconductor
Conductivity,
σ (S/cm)
Metal
Semiconductor
Insulator
10
3
10
-8
Conductivity; no of charged particle (electron @ hole)
1.
Insulator
carrier
1.
Insulator
e
Big energy gap, Eg
empty
full
MNo charged particle can contribute to
a drift current
MEg; 3.5-6 eV
Conduction
band
Valence
band
Metals, Insulators and semiconductor
Conductivity,
σ (S/cm)
Metal
Semiconductor
Insulator
10
3
10
-8
Conductivity; no of charged particle (electron @ hole)
1.
Insulator
carrier
1.
Insulator
e
Big energy gap, Eg
empty
full
MNo charged particle can contribute to
a drift current
MEg; 3.5-6 eV
Conduction
band
Valence
band
Microelectronics I : Introduction to the Quantum Theory of Solids
2. Metal
e full
Partially filled
e
No energy gap
Many electron for
conduction
e
3. Semiconductor
e
Almost full
Almost empty
Conduction
band
Valence
band
Eg; on the order of 1 eV
MConduction band; electron
MValence band; hole
T> 0K
2. Metal
e full
Partially filled
e
No energy gap
Many electron for
conduction
e
3. Semiconductor
e
Almost full
Almost empty
Conduction
band
Valence
band
Eg; on the order of 1 eV
MConduction band; electron
MValence band; hole
T> 0K
Microelectronics I : Introduction to the Quantum Theory of Solids
from E-k curve ,
1. Energy gap, Eg
2. Effective mass, m*
Q. 1;
Eg=1.42 eV
Calculate the wavelength and Calculate the wavelength and energy of photon released when
electron move from conduction band
to valence band? What is the color
of the light?
from E-k curve ,
1. Energy gap, Eg
2. Effective mass, m*
Q. 1;
Eg=1.42 eV
Calculate the wavelength and Calculate the wavelength and energy of photon released when
electron move from conduction band
to valence band? What is the color
of the light?
Microelectronics I : Introduction to the Quantum Theory of Solids
Q. 2;
E (eV)
k(Å
-1
)
0.1
0.7
0.07
A
B
Effective mass of the two electrons?
Q. 2;
E (eV)
k(Å
-1
)
0.1
0.7
0.07
A
B
Effective mass of the two electrons?
Microelectronics I : Introduction to the Quantum Theory of Solids
Extension to three dimensions
[110]
1 dimensional model (kronig-Penney Model)
1 potential pattern
[100]
direction
[110] direction
Different direction
Different potential patterns
E-k diagram is given by a function of the direction in the crystal
Extension to three dimensions
[110]
1 dimensional model (kronig-Penney Model)
1 potential pattern
[100]
direction
[110] direction
Different direction
Different potential patterns
E-k diagram is given by a function of the direction in the crystal
Microelectronics I : Introduction to the Quantum Theory of Solids
E-k diagram of Si
MEnergy gap; Conduction band minimum –
valence band maximum
Eg= 1 eV
MIndirect bandgap;
Maximum valence band and minimum
conduction band do not occur at the same k
Not suitable for optical device application
(laser)
E-k diagram of Si
MEnergy gap; Conduction band minimum –
valence band maximum
Eg= 1 eV
MIndirect bandgap;
Maximum valence band and minimum
conduction band do not occur at the same k
Not suitable for optical device application
(laser)
Microelectronics I : Introduction to the Quantum Theory of Solids
E-k diagram of GaAs MEg= 1.4 eV
MDirect band gap
suitable for optical device application (laser) (laser)
MSmaller effective mass than Si.
(curvature of the curve)
E-k diagram of GaAs MEg= 1.4 eV
MDirect band gap
suitable for optical device application (laser) (laser)
MSmaller effective mass than Si.
(curvature of the curve)
Microelectronics I : Introduction to the Quantum Theory of Solids
Current flow in semiconductor
∝
Number of carriers (electron @ hole)
MHow to count number of carriers,n?
If we know
1.
No. of energy states
Assumption; Pauli exclusion principle
1.
No. of energy states
2. Occupied energy states
Density of states (DOS)
The probability that energy states is
occupied
“Fermi-Dirac distribution function”
n = DOS x “Fermi-Dirac distribution function”
Current flow in semiconductor
∝
Number of carriers (electron @ hole)
MHow to count number of carriers,n?
If we know
1.
No. of energy states
Assumption; Pauli exclusion principle
1.
No. of energy states
2. Occupied energy states
Density of states (DOS)
The probability that energy states is
occupied
“Fermi-Dirac distribution function”
n = DOS x “Fermi-Dirac distribution function”
Microelectronics I : Introduction to the Quantum Theory of Solids
Density of states (DOS)
E
h
m
Eg
3
2/3
)2( 4
)(
π
=
MA function of energy
MAs energy decreases available quantum states decreases
Derivation; refer text book
Density of states (DOS)
E
h
m
Eg
3
2/3
)2( 4
)(
π
=
MA function of energy
MAs energy decreases available quantum states decreases
Derivation; refer text book
Microelectronics I : Introduction to the Quantum Theory of Solids
Solution
Calculate the density of states per unit volume with ene rgies between 0 and 1 eV
Q
.
1
2
/
3
1
0
)
2
(
4
)(
m
dE Eg N
eV
eV
=
∫
π
3 21
2/3 19
3 34
2/3 31
1
0
3
2
/
3
/
10
5.
4
) 10 6.1(
3
2
) 10 625.6(
) 10 11.9 2( 4
)
2
(
4
cm
states
dEE
h
m
eV
×
=
×
×
× ×
=
=
−
−
−
∫
π
π
Solution
Calculate the density of states per unit volume with ene rgies between 0 and 1 eV
Q
.
1
2
/
3
1
0
)
2
(
4
)(
m
dE Eg N
eV
eV
=
∫
π
3 21
2/3 19
3 34
2/3 31
1
0
3
2
/
3
/
10
5.
4
) 10 6.1(
3
2
) 10 625.6(
) 10 11.9 2( 4
)
2
(
4
cm
states
dEE
h
m
eV
×
=
×
×
× ×
=
=
−
−
−
∫
π
π
Microelectronics I : Introduction to the Quantum Theory of Solids
Extension to semiconductor
Our concern; no of carrier that contribute to conduction (flow of current)
Free electron or hole
1. Electron as carrier
e
T> 0K
Conduction band
Can freely moves
ee
band
Valence
band
Ec
Ev
Electron in conduction band contribute to conduction
Determine the DOS in the conduction band
Extension to semiconductor
Our concern; no of carrier that contribute to conduction (flow of current)
Free electron or hole
1. Electron as carrier
e
T> 0K
Conduction band
Can freely moves
ee
band
Valence
band
Ec
Ev
Electron in conduction band contribute to conduction
Determine the DOS in the conduction band
Microelectronics I : Introduction to the Quantum Theory of Solids
C
E E
h
m
Eg− =
3
2/3
)2( 4
)(
π
Energy Ec
C
E E
h
m
Eg− =
3
2/3
)2( 4
)(
π
Energy Ec
Microelectronics I : Introduction to the Quantum Theory of Solids
1. Hole as carrier
Empty
state
ee
Conduction
band
Valence
band
Ec
Ev
freely freely moves
hole in valence band contribute to conduction
Determine the DOS in the valence band
1. Hole as carrier
Empty
state
ee
Conduction
band
Valence
band
Ec
Ev
freely freely moves
hole in valence band contribute to conduction
Determine the DOS in the valence band
Microelectronics I : Introduction to the Quantum Theory of Solids
E E
h
m
Eg
v
− =
3
2/3
)2( 4
)(
π
Energy
Ev
E E
h
m
Eg
v
− =
3
2/3
)2( 4
)(
π
Energy
Ev
Microelectronics I : Introduction to the Quantum Theory of Solids
Q1; Determine the total number of energy states in Si bet ween Ec and Ec+kT at
T=300K
Solution;
3
2/3
) 2( 4
+
− =
∫
dE E E
hm
g
kT Ec
C
n
π
M
n
; mass of electron
3 19
2/3 19
3 34
2/3 31
2/3
3
2/3
3
10
12
.
2
) 10 6.1 0259 .0(
3
2
) 10 625.6(
) 10 11.9 08.1 2( 4
)(
3
2 ) 2( 4
−
−
−
−
×
=
× ×
×
× × ×
=
=
∫
cm
kT
h
m
h n
Ec
C
π
π
M
n
; mass of electron
Q1; Determine the total number of energy states in Si bet ween Ec and Ec+kT at
T=300K
Solution;
3
2/3
) 2( 4
+
− =
∫
dE E E
hm
g
kT Ec
C
n
π
M
n
; mass of electron
3 19
2/3 19
3 34
2/3 31
2/3
3
2/3
3
10
12
.
2
) 10 6.1 0259 .0(
3
2
) 10 625.6(
) 10 11.9 08.1 2( 4
)(
3
2 ) 2( 4
−
−
−
−
×
=
× ×
×
× × ×
=
=
∫
cm
kT
h
m
h n
Ec
C
π
π
M
n
; mass of electron
Microelectronics I : Introduction to the Quantum Theory of Solids
Q2; Determine the total number of energy states in Si bet ween Ev and Ev-kT at
T=300K
Solution;
3
2/3
) 2( 4
− =
∫
dEE E
hm
g
Ev
v
p
π
M
p
; mass of hole
3 18
2/3 19
3 34
2/3 31
2/3
3
2/3
3
10
92
.
7
) 10 6.1 0259 .0(
3
2
) 10 625.6(
) 10 11.9 56.0 2( 4
)(
3
2 ) 2( 4
−
−
−
−
−
×
=
× ×
×
× × ×
=
=
∫
cm
kT
h
m
h p
kT Ev
v
π
π
M
p
; mass of hole
Q2; Determine the total number of energy states in Si bet ween Ev and Ev-kT at
T=300K
Solution;
3
2/3
) 2( 4
− =
∫
dEE E
hm
g
Ev
v
p
π
M
p
; mass of hole
3 18
2/3 19
3 34
2/3 31
2/3
3
2/3
3
10
92
.
7
) 10 6.1 0259 .0(
3
2
) 10 625.6(
) 10 11.9 56.0 2( 4
)(
3
2 ) 2( 4
−
−
−
−
−
×
=
× ×
×
× × ×
=
=
∫
cm
kT
h
m
h p
kT Ev
v
π
π
M
p
; mass of hole
Microelectronics I : Introduction to the Quantum Theory of Solids
The probability that energy states is occupied “Fermi-Dirac distribution function” MStatistical behavior of a large number of electrons
MDistribution function
−
=
E
E
E
f
F
1
)
(
−
+
=
kT
E
E
E
f
F
F
exp 1
)
(
E
F; Fermi energy
MFermi energy;
Energy of the highest occupied quantum state
The probability that energy states is occupied “Fermi-Dirac distribution function” MStatistical behavior of a large number of electrons
MDistribution function
−
=
E
E
E
f
F
1
)
(
−
+
=
kT
E
E
E
f
F
F
exp 1
)
(
E
F; Fermi energy
MFermi energy;
Energy of the highest occupied quantum state
Microelectronics I : Introduction to the Quantum Theory of Solids
For temperature above 0 K, some electrons jump to higher energy level.
So some energy states above EF will be occupied by electro ns and some
energy states below EF will be empty
For temperature above 0 K, some electrons jump to higher energy level.
So some energy states above EF will be occupied by electro ns and some
energy states below EF will be empty
Microelectronics I : Introduction to the Quantum Theory of Solids
Q; Assume that E
Fis 0.30 eV below Ec. Determine the probability of a st ates being
occupied by an electron at Ec and at Ec+kT (T=300K)
Solution;
1. At Ec
)
3
.
0
(
1
1
−
−
+
=
eV
E
E
f
C
C
2. At Ec+kT
)
3.
0
(
0259
.
0
1
1
−
−
+
+
=
eV
E
E
f
C
C
6
10 32.9
0259 .0
3.0
1
1
)
3
.
0
(
1
−
× =
+
=
−
−
+
kT
eV
E
E
C
C
6
10 43.3
0259 .0
3259 .0
1
1
)
3.
0
(
0259
.
0
1
−
× =
+
=
−
−
+
+
kT
eV
E
E
C
C
Electron needs higher energy to be at higher energy st ates. The probability
of electron at Ec+kT lower than at Ec
Q; Assume that E
Fis 0.30 eV below Ec. Determine the probability of a st ates being
occupied by an electron at Ec and at Ec+kT (T=300K)
Solution;
1. At Ec
)
3
.
0
(
1
1
−
−
+
=
eV
E
E
f
C
C
2. At Ec+kT
)
3.
0
(
0259
.
0
1
1
−
−
+
+
=
eV
E
E
f
C
C
6
10 32.9
0259 .0
3.0
1
1
)
3
.
0
(
1
−
× =
+
=
−
−
+
kT
eV
E
E
C
C
6
10 43.3
0259 .0
3259 .0
1
1
)
3.
0
(
0259
.
0
1
−
× =
+
=
−
−
+
+
kT
eV
E
E
C
C
Electron needs higher energy to be at higher energy st ates. The probability
of electron at Ec+kT lower than at Ec
Microelectronics I : Introduction to the Quantum Theory of Solids
−
+
=
kT
E E
Ef
F
F
exp 1
1
)(
electron
Hole?
The probability that states are being empty is given by
−
+
−= −
kT
E E
Ef
F
F
exp 1
1
1)( 1
−
+
=
kT
E E
Ef
F
F
exp 1
1
)(
electron
Hole?
The probability that states are being empty is given by
−
+
−= −
kT
E E
Ef
F
F
exp 1
1
1)( 1
Microelectronics I : Introduction to the Quantum Theory of Solids
Approximation when calculating f
F
−
+
=
kT
E E
Ef
F
F
exp 1
1
)(
When E-E
F>>kT
−
≈
E E
Ef
F
F
exp
1
)(
Maxwell
-
Boltzmann approximation
kT
F
exp
Maxwell
-
Boltzmann approximation
Approximation is valid in this range
Approximation when calculating f
F
−
+
=
kT
E E
Ef
F
F
exp 1
1
)(
When E-E
F>>kT
−
≈
E E
Ef
F
F
exp
1
)(
Maxwell
-
Boltzmann approximation
kT
F
exp
Maxwell
-
Boltzmann approximation
Approximation is valid in this range
Tags
Categories
ScienceDownload
Download Slideshow Get the original presentation fileQuick Actions
Statistics
- Views 5,015
- Slides 52
- Favorites 8
- Age 3285 days
Related Slideshows
23
Earthquakes_Type of Faults_Science G8.pptx
OctabellFabila1
34 views
15
Quiz #1 Science 10 in the first quarter for jhs
HendrixAntonniAmante
34 views
9
Astronomy history from long ago till doday
ssuserbd9abe
34 views
9
Great history of astronomy from long ago till today
ssuserbd9abe
31 views
20
EARTHQUAKE-DRILL.powerpoint.............
chalobrido8
34 views
9
History of astronomy from old times to the present times
ssuserbd9abe
33 views