2. 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
3. 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 (~1020 atom)
electron
energy + + …. = ?
x 1020
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
4. Microelectronics I : Introduction to the Quantum Theory of Solids
Si Crystal
Tetrahedral structure
Diamond structure
Tetrahedral structure
energy
Valence band
conduction band
Energy gap, Eg=1.1 eV
Formation of energy band and energy gap
5. Microelectronics I : Introduction to the Quantum Theory of Solids
What happen if 2 identical atoms approach each other ?
r
atom 2atom 1
energy
1s
Isolated atom
z
x x
1s
z
y
x
Distance from center
Probabilitydensity
y
x x
1s 1s
Wave function of two atom electron overlap
interaction
6. Microelectronics I : Introduction to the Quantum Theory of Solids
r
atom 2atom 1
When the atoms are far apart
(r=∞), electron from different
atoms can occupy same
energy level.
E1s,atom 1 =E1s, atom 2
As the atoms approach each
other, energy level splits
energy
1s
other, energy level splits
E1s,atom 1 ≠E1s, atom 2
ra
energy
interaction between two overlap wave function
Consistent with Pauli exclusion principle
a ; equilibrium interatomic distance
7. Microelectronics I : Introduction to the Quantum Theory of Solids
Regular periodic arrangement of atom (crystal)
ex: 1020 atoms
Total number of quantum states
do not change when forming a
system (crystal)energy
1s
1020 energy levels
a
energy
“energy band”dense allowed energy levels
8. Microelectronics I : Introduction to the Quantum Theory of Solids
energy
1020 energy state
1 eV
Consider
1020 energy state
Energy states are equidistant
Energy states are separated by 1/1020 eV = 10-20 eV
(Almost) continuous energy states within energy band
9. Microelectronics I : Introduction to the Quantum Theory of Solids
Distance from center
Probabilitydensity
energy
2s
1s
atom 2atom 1
1s
2s
r
atom 2atom 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.
10. 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
energyenergy
3s
3p
energy
Sp3 hybrid orbital
Reform 4 equivalent states 4 equivalent bond (symmetric)
11. Microelectronics I : Introduction to the Quantum Theory of Solids
Si Si
Si
Si
Si
energy
+ + + +
energy
filled
empty
12. Microelectronics I : Introduction to the Quantum Theory of Solids
Si crystal (1022 atoms/cm3)
filled
empty
energy
conduction band
Energy gap, Eg=1.1 eV
energy
filled
Valence band
4 x 1022 states/cm3
13. 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, EG
allowed band
14. 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
22
Ψ(x,t)= exp ( j(kx-ωt))
E
m
k
E
2
22
h
=
Continuous value of E
K-space diagram
k
E
15. Microelectronics I : Introduction to the Quantum Theory of Solids
E-k diagram for electron in quantum well
En=3
m
k
E
n
Lm
E
2
2
22
2
22
h
h
=
=
π
n
L
k
=
π
E
E-k diagram for electron in crystal? The Kronig-Penney Model
x=Lx=0
En=1
En=2
k
π/L 2π/L
16. Microelectronics I : Introduction to the Quantum Theory of Solids
The Kronig-Penney Model
+ + + +
r
e
rV
0
2
4
)(
πε
−
=
Periodic potential
V0
I II I I III II II II
Potential
well
tunneling
Periodic potential
Wave function overlap
-b a
L
Determine a relationship between k, E and V0
17. Microelectronics I : Introduction to the Quantum Theory of Solids
Schrodinger equation (E < V0)
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
02 )(2
h
EVm −
=β
Potential periodically changes
)()( LxVxV +=
jkx
exUx )()( =ϕ
)()( LxUxU +=
Wave function
amplitude
k; wave number [m-1]
Phase of the wave
Bloch theorem
18. Microelectronics I : Introduction to the Quantum Theory of Solids
Boundary condition
)()(
)0()0(
bUaU
UU
III
III
−=
= Continuous wave function
)()(
)0()0(
''
''
bUaU
UU
III
III
−=
=
Continuous first derivative
19. Microelectronics I : Introduction to the Quantum Theory of Solids
From Schrodinger equation, Bloch theorem and boundary condition
)cos()cos()cosh()sin()sinh(
2
22
kLabab =⋅+⋅
−
αβαβ
αβ
αβ
B 0, V0 ∞ Approximation for graphic solution
)cos()cos(
)sin(
2
0
kaa
a
abamV
=+
α
α
α
h
)cos()cos(
)sin('
kaa
a
a
P =+ α
α
α
2
0'
h
bamV
P =
Gives relation between k, E(from α) and V0
20. Microelectronics I : Introduction to the Quantum Theory of Solids
)cos(
)sin(
)( '
a
a
a
Paf α
α
α
α +=
Left side
)cos()( kaaf =α
Right side
Value must be
between -1 and 1
Allowed value of αa
21. Microelectronics I : Introduction to the Quantum Theory of Solids
m
E
mE
2
2
22
2
2
h
h
α
α
=
=
Plot E-k
Discontinuity of E
22. Microelectronics I : Introduction to the Quantum Theory of Solids
)2cos()2cos()cos()( ππα nkankakaaf ==+==
Right side
Shift 2πShift 2π
23. 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
24. 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, Eg=1.1 eV
+
Breaking of covalent bond
Generation of positive and negative charge
25. Microelectronics I : Introduction to the Quantum Theory of Solids
E versus k energy band
conduction band
T = 0 K T > 0 K
When no external force is applied, electron and “empty state” distributions are
symmetrical with k
Valence band
26. Microelectronics I : Introduction to the Quantum Theory of Solids
2. Drift current
Current; diffusion current and drift current
When Electric field is applied
E E
dE = F dx = F v dt
“Electron moves to higher empty state”
k k
ENo external force
∑=
υ−=
n
i
ieJ
1
Drift current density, [A/cm3]
n; no. of electron per unit volume in the conduction band
27. Microelectronics I : Introduction to the Quantum Theory of Solids
3. Electron effective mass
Fext + Fint = ma
Electron 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
Fext = m*a
External forces
(e.g; Electrical field)
Internal forces
(e.g; potential)
= Effective mass acceleration
Effect of internal force
28. Microelectronics I : Introduction to the Quantum Theory of Solids
From relation of E and k
mdk
Ed
m
k
E
2
2
2
22
2
h
h
=
=
Mass of electron, mMass 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 potential)
m from eq. above is effective mass, m*
29. 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
Curvature of E-k depends on the medium that electron moves in
Effective mass changes
m*A m*Bm> >
Ex; m*Si=0.916m0, m*GaAs=0.065m0 m0; in free space
30. Microelectronics I : Introduction to the Quantum Theory of Solids
4. Concept of hole
Electron fills the empty state
Positive charge empty the state
“Hole”
31. 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
32. Microelectronics I : Introduction to the Quantum Theory of Solids
Metals, Insulators and semiconductor
Conductivity,
σ (S/cm)
MetalSemiconductorInsulator
103
10-8
Conductivity; no of charged particle (electron @ hole)
1. Insulator
carrier
1. Insulator
e
Big energy gap, Eg
empty
full
No charged particle can contribute to
a drift current
Eg; 3.5-6 eV
Conduction
band
Valence
band
33. 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
Conduction band; electron
Valence band; hole
T> 0K
34. 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 andCalculate the wavelength and
energy of photon released when
electron move from conduction band
to valence band? What is the color
of the light?
35. 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?
36. 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
37. Microelectronics I : Introduction to the Quantum Theory of Solids
E-k diagram of Si
Energy gap; Conduction band minimum –
valence band maximum
Eg= 1 eV
Indirect bandgap;
Maximum valence band and minimum
conduction band do not occur at the same k
Not suitable for optical device application
(laser)
38. Microelectronics I : Introduction to the Quantum Theory of Solids
E-k diagram of GaAs
Eg= 1.4 eV
Direct band gap
suitable for optical device application
(laser)(laser)
Smaller effective mass than Si.
(curvature of the curve)
39. Microelectronics I : Introduction to the Quantum Theory of Solids
Current flow in semiconductor ∝ Number of carriers (electron @ hole)
How 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”
40. Microelectronics I : Introduction to the Quantum Theory of Solids
Density of states (DOS)
E
h
m
Eg 3
2/3
)2(4
)(
π
=
A function of energy
As energy decreases available quantum states decreases
Derivation; refer text book
41. Microelectronics I : Introduction to the Quantum Theory of Solids
Solution
Calculate the density of states per unit volume with energies between 0 and 1 eV
Q.
12/3
1
0
)2(4
)(
m
dEEgN
eV
eV
= ∫
π
321
2/319
334
2/331
1
0
3
2/3
/105.4
)106.1(
3
2
)10625.6(
)1011.92(4
)2(4
cmstates
dEE
h
m
eV
×=
×
×
××
=
=
−
−
−
∫
π
π
42. 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
e
e band
Valence
band
Ec
Ev
Electron in conduction band contribute to conduction
Determine the DOS in the conduction band
43. Microelectronics I : Introduction to the Quantum Theory of Solids
CEE
h
m
Eg −= 3
2/3
)2(4
)(
π
Energy
Ec
44. Microelectronics I : Introduction to the Quantum Theory of Solids
1. Hole as carrier
Empty
state
e
e
Conduction
band
Valence
band
Ec
Ev
freelyfreely
moves
hole in valence band contribute to conduction
Determine the DOS in the valence band
45. Microelectronics I : Introduction to the Quantum Theory of Solids
EE
h
m
Eg v −= 3
2/3
)2(4
)(
π
Energy
Ev
46. Microelectronics I : Introduction to the Quantum Theory of Solids
Q1;
Determine the total number of energy states in Si between Ec and Ec+kT at
T=300K
Solution;
3
2/3
)2(4
+
−= ∫ dEEE
h
m
g
kTEc
C
nπ
Mn; mass of electron
319
2/319
334
2/331
2/3
3
2/3
3
1012.2
)106.10259.0(
3
2
)10625.6(
)1011.908.12(4
)(
3
2)2(4
−
−
−
−
×=
××
×
×××
=
=
∫
cm
kT
h
m
h
n
Ec
C
π
π
Mn; mass of electron
47. Microelectronics I : Introduction to the Quantum Theory of Solids
Q2;
Determine the total number of energy states in Si between Ev and Ev-kT at
T=300K
Solution;
3
2/3
)2(4
−= ∫ dEEE
h
m
g
Ev
v
pπ
Mp; mass of hole
318
2/319
334
2/331
2/3
3
2/3
3
1092.7
)106.10259.0(
3
2
)10625.6(
)1011.956.02(4
)(
3
2)2(4
−
−
−
−
−
×=
××
×
×××
=
=
∫
cm
kT
h
m
h
p
kTEv
v
π
π
Mp; mass of hole
48. Microelectronics I : Introduction to the Quantum Theory of Solids
The probability that energy states is occupied
“Fermi-Dirac distribution function”
Statistical behavior of a large number of electrons
Distribution function
−
=
EE
EfF
1
)(
−
+
=
kT
EE
Ef
F
F
exp1
)(
EF; Fermi energy
Fermi energy;
Energy of the highest occupied quantum state
49. 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 electrons and some
energy states below EF will be empty
50. Microelectronics I : Introduction to the Quantum Theory of Solids
Q;
Assume that EF is 0.30 eV below Ec. Determine the probability of a states being
occupied by an electron at Ec and at Ec+kT (T=300K)
Solution;
1. At Ec
)3.0(
1
1
−−
+
=
eVEE
f
CC
2. At Ec+kT
)3.0(0259.0
1
1
−−+
+
=
eVEE
f
CC
6
1032.9
0259.0
3.0
1
1
)3.0(
1
−
×=
+
=
−−
+
kT
eVEE CC
6
1043.3
0259.0
3259.0
1
1
)3.0(0259.0
1
−
×=
+
=
−−+
+
kT
eVEE CC
Electron needs higher energy to be at higher energy states. The probability
of electron at Ec+kT lower than at Ec
51. Microelectronics I : Introduction to the Quantum Theory of Solids
−
+
=
kT
EE
Ef
F
F
exp1
1
)( electron
Hole?
The probability that states are being empty is given by
−
+
−=−
kT
EE
Ef
F
F
exp1
1
1)(1
52. Microelectronics I : Introduction to the Quantum Theory of Solids
Approximation when calculating fF
−
+
=
kT
EE
Ef
F
F
exp1
1
)(
When E-EF>>kT
−
≈
EE
Ef
F
F
exp
1
)(
Maxwell-Boltzmann approximation
kT
F
exp Maxwell-Boltzmann approximation
Approximation is valid in this range