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Ming-Chang Lee, Integrated Photonic Devices
Introduction to Solid State Physics
Class: Integrated Photonic DevicesTime: Fri. 8:00am ~ 11:00am.
Classroom: 資電206Lecturer: Prof. 李明昌(Ming-Chang Lee)
Ming-Chang Lee, Integrated Photonic Devices
Electrons in An Atom
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Ming-Chang Lee, Integrated Photonic Devices
Electrons in An Atom
Ming-Chang Lee, Integrated Photonic Devices
Electrons in Two atoms
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Ming-Chang Lee, Integrated Photonic Devices
Analogy to A Coupled Two-Pendulum System
f f
spring
f1
f2
Ming-Chang Lee, Integrated Photonic Devices
One-dimensional Kronig-Penney Model
Pote
ntia
lEn
ergy
Pote
ntia
lEn
ergy
Electron
Electron
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Ming-Chang Lee, Integrated Photonic Devices
Solutions of the S.E. in Kronig-Penny Model
Periodic Potential Well
( ) 0)(2
22
=Ψ−+Ψ∇− ExUm
Schrödinger’s equation
Bloch Theorem
a+b
)()exp()( xuxjkx ⋅=Ψ)()]([ xUbaxU =++If
Then
solution
E-k diagram
)()]([ xubaxu =++where
Ming-Chang Lee, Integrated Photonic Devices
Density of States as a function of k and E
• For electrons near the bottom of the band, the band itself form a pseudo-potential well. The well bottom lies at Ec and the termination of the band at the crystal surfaces forms the walls of the well
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Ming-Chang Lee, Integrated Photonic Devices
Density of States as a function of k and E
z
• The density of states near the band edges can therefore be equal to the density of states available to a particle of mass m*
in a three dimension box with the dimension of the crystal
2 2 22
2 2 2 0kx y zϕ ϕ ϕ ϕ∂ ∂ ∂+ + + =
∂ ∂ ∂
0 , 0 , 0 x a y b z c< < < < < <
22 /k mE≡
2 2
2kEm
=or
(a)
(Time-independent S.E.)( ) 0U x = except boundary
Boundary
Ming-Chang Lee, Integrated Photonic Devices
Density of States as a function of k and E
( , , ) ( ) ( ) ( )x y zx y z x y zϕ ϕ ϕ ϕ=
22 22
2 2 2
1 1 1 0yx z
x y x
dd d kdx dy dz
ϕϕ ϕϕ ϕ ϕ
+ + + =
( , , ) sin sin sinx y zx y z A k x k y k zϕ = ⋅ ⋅
2 2 2 2x y zk k k k= + +
; ;yx zx y z
nn nk k ka b c
ππ π= = = , , 1, 2, 3,...x y zn n n = ± ± ±
where
22
2
1 constantxx
x
d kdx
ϕϕ
= = −
• To solve the equation, we employ the separation of variables technique; that is
(b)
• Substitute (b) to (a)
(Solution)
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Ming-Chang Lee, Integrated Photonic Devices
Density of States as a function of k and E
- Densities of states in k-spaceA unit cell of volume:
( )( )( )a b cπ π π
• The larger the a,b and c, the smaller the unit cell volume
Ming-Chang Lee, Integrated Photonic Devices
Density of States as a function of k and E
• How many states are within k and k+dk?
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1( ) (4 ) 28
Energy states with k abck dk k dkbetween k and k dk
σ ππ
≡ = ⋅ ⋅ ⋅ ⋅ +
Redundancy
spin up and down
Density of states
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Ming-Chang Lee, Integrated Photonic Devices
Density of States as a function of k and E
• How many states are within E and E+dE?
2 2
2kEm
=
From the previous equation,
( ) ( )E dE k dkσ σ= ( ) ( ) dkE kdE
σ σ=
2dE kdk m
= 2 3
2( ) ( ) m mEE abcσπ
=
Density of states per unit volume
2 3 2 3
2 2( ) ( ) / ( ) /m mE m mEE E V abc Vρ σπ π
= = =
V abc=
Therefore
Ming-Chang Lee, Integrated Photonic Devices
Electrons and Holes in Semiconductors
Electric Field
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Ming-Chang Lee, Integrated Photonic Devices
Electron Distribution Functions
−+
=
kTEE
EfFexp1
1)(
The probability density function of electron residing in energy level E can be represented by Fermi-Dirac distribution
where EF: Fermi Energy Level
Ming-Chang Lee, Integrated Photonic Devices
Carrier Density Distribution of Intrinsic Semiconductors
=×
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Ming-Chang Lee, Integrated Photonic Devices
Metal, Insulator, and Semiconductor
Metal Metal Insulator Semiconductor
Ming-Chang Lee, Integrated Photonic Devices
Work Function
• Work function is the minimum energy required to enable an electron to escape from the surface of solid.
• Work function φis the energy difference between Fermi level and the vacuum level.
• In semiconductors, it is more usual to use the electron affinity χ, defined as the energy difference between the bottom of the conduction band and the vacuum level.
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Ming-Chang Lee, Integrated Photonic Devices
Extrinsic Semiconductors
Ming-Chang Lee, Integrated Photonic Devices
Carrier Density Distribution of Extrinsic Semiconductors
n-type
p-type
=×
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Ming-Chang Lee, Integrated Photonic Devices
p-n Junction
(a). Initially separated p-type and n-type semiconductor;
(b) the energy band distortion after the junction is formed;
(c) the space charge layers of ionized impurity atoms within the depletion region W; and
(d) the potential distribution at the junction.
Ming-Chang Lee, Integrated Photonic Devices
Contact Potential on p-n junction
• Contact potential V0
The electron concentration in the conduction band of p-type side as
exp p pC Fp c
E En N
kT
− = −
exp n nC Fn c
E En N
kT −
= −
Similarly the electron concentration in the n-type side is
Since the Fermi level is constant. n pF F FE E E= =
0lnp n
nC C
p
nE E kT eVn
− = =
0 ln n
p
nkTVe n
=
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Ming-Chang Lee, Integrated Photonic Devices
Minority Carrier Distribution
• At the temperature in the range of 100K ≤ T ≤ 400K, the majority carrier concentrations are equal to the doping levels, that is and , n dn N=
p aP N=
0 ln a d
i
N NkTVe n
=
(recall ) 2
inp n=
• Carrier concentration difference between p- and n-type semiconductor
0expp neVn nkT
= −
0expn peVp pkT
= −
and
Ming-Chang Lee, Integrated Photonic Devices
Current flow in a forward-biased p-n junction
• The junction is said to be forward biased if the p region is connected to the positive terminal of the voltage source
• The external voltage is dropped across the depletion region and lower down the potential barrier by (V0-V). So the diffusion current becomes larger than the drift current. There is a net current from the p to the n region.
• The Fermi levels are no longer aligned across the junction in light of the external voltage.
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Ming-Chang Lee, Integrated Photonic Devices
Current flow in a forward-biased p-n junction
Carrier densities
• Once the majority carriers flow across depletion region, they become minority carriers. The minority concentrations near the junction rise to new value np’ and pn’. The majority carrier concentration is almost unchanged unless a large current injection
• Due to the minority concentration gradient, the injected current diffuses away from the junction. The nonlinear gradient indicates minority holes (electrons) are recombined with electrons (holes) that are replenished by external voltage source.
Ming-Chang Lee, Integrated Photonic Devices
Current flow in a forward-biased p-n junction
• The injected carrier concentration at the edge of depletion region
0( )' expp ne V Vn nkT
− = −
0( )' expn pe V Vp pkT
− = −
0expn peVp pkT
= −
(a)
Since
−=kTeVnn np
0expand
Then
=kTeVpp nn exp'
=kTeVnn pp exp'and
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Ming-Chang Lee, Integrated Photonic Devices
Current flow in a forward-biased p-n junction
As we noted in previous slides, the excess minority carrier concentration will decrease due to recombination
( ) (0)exp( )h
xp x pL
∆ = ∆ −
where ( ) '( )n np x p x p∆ = −
Therefore, from (a)
(0) exp 1neVp pkT
∆ = −
Ming-Chang Lee, Integrated Photonic Devices
Current flow in a forward-biased p-n junction
• The diffusion current
(0)exp( )hh
h h
eD xJ pL L
= ∆ −
At x = 0
exp( ) 1hh n
h
eD eVJ pL kT
= −
Similarly, for electron diffusion current at the depletion edge
exp( ) 1ee p
e
eD eVJ nL kT
= − The total current
0 exp 1eVJ JkT
= − 0
h en p
h e
D DJ e p nL L
= +
where
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Ming-Chang Lee, Integrated Photonic Devices
Current flow in a reverse-biased p-n junction
0 exp 1eVJ JkT
= − V is negative
Ming-Chang Lee, Integrated Photonic Devices
I-V curves of p-n junction
I-V Curve
Reversed Bias Forward Bias
0 exp 1eVJ JkT
= −
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Ming-Chang Lee, Integrated Photonic Devices
Zener Breakdown
•Tunnelling of carriers across the depletion region and the process is independent of temperature•Doping densities must be high to occur before avalanching •Breakdown current is independent of voltage
Ming-Chang Lee, Integrated Photonic Devices
Metal-semiconductor Junction --- SchottkyContact
A schottky barrier formed by contacting a metal to an n-type semiconductor with the metal having the larger work function: band diagrams (a) before and (b) after the junction is formed.
sm φφ >
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Ming-Chang Lee, Integrated Photonic Devices
Work Function of Metal
Ming-Chang Lee, Integrated Photonic Devices
Electron Affinity of Semiconductor
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Ming-Chang Lee, Integrated Photonic Devices
Metal-semiconductor Junction --- SchottkyContact
The metal to n-type semiconductor junction shown under (a) forward bias and (b) reverse bias.
Ming-Chang Lee, Integrated Photonic Devices
Metal-semiconductor Junction --- OhmicContact
sm φφ <
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Ming-Chang Lee, Integrated Photonic Devices
Summary of Metal-semiconductor Junction Contact
Schottly
Ming-Chang Lee, Integrated Photonic Devices
p-N heterojunction
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Ming-Chang Lee, Integrated Photonic Devices
n-P heterojunction
Ming-Chang Lee, Integrated Photonic Devices
Double Heterojunctions
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Ming-Chang Lee, Integrated Photonic Devices
Carriers Confined in Double Heterojuncitons
It is extremely efficient to enhance electron-hole recombination
Ming-Chang Lee, Integrated Photonic Devices
Fundamentals of Electromagnetic Waves
• E and B are fundamental fields • D and H are derived from the response of the medium• How many variables in Electromagnetic Wave?
Electrical field
Electrical displacement
Magnetic field
Magnetic induction
),( trE
),( trD
),( trH
),( trB
V/m
C/m2
A/m
webers/m2
space time