View
90
Download
6
Category
Preview:
Citation preview
Microelectronics I
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
Microelectronics I : Introduction to the Quantum Theory of Solids
Isolated single atom (ex; Si)
electronenergy
Quantized energy level (quantum state)
1s
2s2p
3s3p
+ n=1
n=2
n=3
Crystal (~1020 atom)
electronenergy + + …. = ?
x 1020
1s
2s2p
3s3p
1s
2s2p
3s3p
1s
2s2p
3s3p
Microelectronics I : Introduction to the Quantum Theory of Solids
Si Crystal
Tetrahedral structureDiamond structure
Tetrahedral structure
energy
Valence band
conduction band
Energy gap, Eg=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 2atom 1
energy
1s
Isolated atom
z
x x
1s
z
yx
Distance from center
Pro
babi
lity
dens
ity
yx x
1s 1s
Wave function of two atom electron overlap
interaction
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
Microelectronics I : Introduction to the Quantum Theory of Solids
Regular periodic arrangement of atom (crystal)
ex: 1020 atomsTotal 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
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
Microelectronics I : Introduction to the Quantum Theory of Solids
Distance from center
Pro
babi
lity
dens
ity
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.
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
3s3p
energy
Sp3 hybrid orbital
Reform 4 equivalent states � 4 equivalent bond (symmetric)
Microelectronics I : Introduction to the Quantum Theory of Solids
Si SiSi
SiSi
energy
+ + + +
energy
filled
empty
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
filledValence band
4 x 1022 states/cm3
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
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
kE
2
22h
=
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
En=3m
kE
nLm
E
2
2
22
2
22
h
h
=
=
π
nL
k
=
π
E
E-k diagram for electron in crystal? The Kronig-Penney Model
x=Lx=0
En=1
En=2
kπ/L 2π/L
Microelectronics I : Introduction to the Quantum Theory of Solids
The Kronig-Penney Model
+ + + +r
erV
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
Microelectronics I : Introduction to the Quantum Theory of Solids
Schrodinger equation (E < V0)
Region I 0)()( 2
2
2
=+∂
∂x
x
xI
I ϕαϕ
Region II 0)()( 2
2
2
=−∂
∂x
x
xII
II ϕβϕ
2
2 2
h
mE=α
2
02 )(2
h
EVm −=β
�Potential periodically changes
)()( LxVxV +=
jkxexUx )()( =ϕ
)()( LxUxU +=
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(
bUaU
UU
III
III
−=
= Continuous wave function
)()(
)0()0(
''
''
bUaU
UU
III
III
−=
=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
22
kLabab =⋅+⋅−
αβαβαβ
αβ
B � 0, V0 � ∞ Approximation for graphic solution
)cos()cos()sin(
2
0 kaaa
abamV=+
α
α
α
h
)cos()cos()sin(' kaa
a
aP =+ α
α
α2
0'
h
bamVP =
Gives relation between k, E(from α) and V0
Microelectronics I : Introduction to the Quantum Theory of Solids
)cos()sin(
)( ' aa
aPaf α
α
αα +=
Left side
)cos()( kaaf =α
Right side
Value must be between -1 and 1
Allowed value of αa
Microelectronics I : Introduction to the Quantum Theory of Solids
mE
mE
2
2
22
2
2
h
h
α
α
=
=
Plot E-k
Discontinuity of E
Microelectronics I : Introduction to the Quantum Theory of Solids
)2cos()2cos()cos()( ππα nkankakaaf ==+==
Right side
Shift 2πShift 2π
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
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
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
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
iieJ
1
Drift current density, [A/cm3]
n; no. of electron per unit volume in the conduction band
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
Microelectronics I : Introduction to the Quantum Theory of Solids
From relation of E and k
mdk
Ed
m
kE
2
2
2
22
2
h
h
=
=
Mass of electron, mMass of electron, m
=
2
2
2
dk
Edm
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*
Microelectronics I : Introduction to the Quantum Theory of Solids
E versus k curve
EFree 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
Microelectronics I : Introduction to the Quantum Theory of Solids
4. Concept of 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
Microelectronics I : Introduction to the Quantum Theory of Solids
Metals, Insulators and semiconductor
Conductivity,
σ (S/cm)MetalSemiconductorInsulator
10310-8
Conductivity; no of charged particle (electron @ hole)
1. Insulatorcarrier
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
Microelectronics I : Introduction to the Quantum Theory of Solids
2. Metal
e
full
Partially fillede
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
Microelectronics I : Introduction to the Quantum Theory of Solids
from E-k curve , 1. Energy gap, Eg2. 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
AB
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
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)
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)
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”
Microelectronics I : Introduction to the Quantum Theory of Solids
Density of states (DOS)
Eh
mEg
3
2/3)2(4)(
π=
�A function of energy�As 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 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
dEEh
meV
×=
××
××=
=
−
−
−
∫
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> 0KConduction 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
Microelectronics I : Introduction to the Quantum Theory of Solids
CEEh
mEg −=
3
2/3)2(4)(
π
Energy
Ec
Microelectronics I : Introduction to the Quantum Theory of Solids
1. Hole as carrier
Empty state
e
eConduction 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
EEh
mEg 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 between Ec and Ec+kT at T=300K
Solution;
3
2/3)2(4+
−= ∫ dEEEh
mg
kTEc
Cnπ
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
kTh
m
h
n
Ec
C
π
π
Mn; mass of electron
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
mg
Ev
v
pπMp; 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
kTh
m
h
p
kTEv
v
π
π
Mp; mass of hole
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
−=
EEEfF
1)(
−+
=
kT
EEEf
F
F
exp1
)(
EF; Fermi energy
�Fermi 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 electrons and some energy states below EF will be empty
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
fCC
2. At Ec+kT
)3.0(0259.01
1
−−+
+
=eVEE
fCC
61032.9
0259.0
3.01
1
)3.0(1
−×=
+
=
−−+
kT
eVEE CC
61043.3
0259.0
3259.01
1
)3.0(0259.01
−×=
+
=
−−++
kT
eVEE CC
Electron needs higher energy to be at higher energy states. The probability of electron at Ec+kT lower than at Ec
Microelectronics I : Introduction to the Quantum Theory of Solids
−+
=
kT
EEEf
F
F
exp1
1)( electron
Hole?
The probability that states are being empty is given by
−+
−=−
kT
EEEf
F
F
exp1
11)(1
Microelectronics I : Introduction to the Quantum Theory of Solids
Approximation when calculating fF
−+
=
kT
EEEf
F
F
exp1
1)(
When E-EF>>kT
−
≈EE
EfF
F
exp
1)(
Maxwell-Boltzmann approximation
kT
Fexp Maxwell-Boltzmann approximation
Approximation is valid in this range
Recommended