president universityerwin sitompulsdp 4/1 dr.-ing. erwin sitompul president university lecture 4...
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President University Erwin Sitompul SDP 4/1
Dr.-Ing. Erwin SitompulPresident University
Lecture 4
Semiconductor Device Physics
http://zitompul.wordpress.com
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Electron kinetic energyIn
crea
sing
ele
ctro
n en
ergy
Ec
EvHole kinetic energy In
crea
sing
hol
e en
ergy
c referenceP.E. E E
Potential vs. Kinetic EnergyChapter 3 Carrier Action
Ec represents the electron potential energy:
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Ec
Ev
E
x
Band BendingChapter 3 Carrier Action
Until now, Ec and Ev have always been drawn to be independent of the position.
When an electric field E exists inside a material, the band energies become a function of position.
• Variation of Ec with position is called “band bending”
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c reference
1( )V E Eq
qVP.E.
VdV
dx
E
c v i1 1 1dE dE dE
q dx q dx q dx E
Band BendingChapter 3 Carrier Action
The potential energy of a particle with charge –q is related to the electrostatic potential V(x):
• Since Ec, Ev, and Ei differ only by an additive constant
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DiffusionChapter 3 Carrier Action
Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion (Brownian Motion).
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1-D Diffusion ExampleChapter 3 Carrier Action
Thermal motion causes particles to move into an adjacent compartment every τ seconds.
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e
N|diff N
dnqD
dxJ P|diff P
dpqD
dxJ
Diffusion CurrentsChapter 3 Carrier Action
• D is the diffusion coefficient [cm2/sec]
n
x
Current flowElectron flow
hp
x
Current flowHole flow
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N N|drift N|diff n N dn
q n qDdx
E J J J
N P J J J
P P|drift P|diff p P dp
q p qDdx
EJ J J
Total CurrentsChapter 3 Carrier Action
Drift current flows when an electric field is applied. Diffusion current flows when a gradient of carrier concentration
exist.
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N n N 0dn
q n qDdx
EJ
Current Flow Under Equilibrium ConditionsChapter 3 Carrier Action
In equilibrium, there is no net flow of electrons or :
N 0,J P 0J
The drift and diffusion current components must balance each other exactly.
A built-in electric field of ionized atoms exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient.
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F cC cE E kTN dEdne
dx kT dx
F c
CE E kTn N e
Ev(x)
Ec(x)
EF
dn qn
dx kT E
Current Flow Under Equilibrium ConditionsChapter 3 Carrier Action
Consider a piece of non-uniformly doped semiconductor:
n-type semiconductor
Decreasing donor concentration
• Under equilibrium, EF inside a material or a group of materials in intimate contact is not a function of position
cdEn
kT dx
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N
n
D kT
q
Similarly, P
p
D kT
q
N n N 0dn
q n qDdx
EJ
n N 0q
qn qn DkT
E E
Einstein Relationship between D and Chapter 3 Carrier Action
But, under equilibrium conditions, JN = 0 and JP = 0
• Einstein Relationship
Further proof can show that the Einstein Relationship is valid for a non-degenerate semiconductor, both in equilibrium and non-equilibrium conditions.
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P p
kTD
q
1 eV1 V
1 e
191 eV 1.602 10 J
Example: Diffusion CoefficientChapter 3 Carrier Action
What is the hole diffusion coefficient in a sample of silicon at 300 K with p = 410 cm2 / V.s ?
2 1 125.86 meV410 cm V s
1e
2cm25.86 mV 410
V s
210.603 cm /s
• Remark: kT/q = 25.86 mV at room temperature
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Recombination–Generation Chapter 3 Carrier Action
Recombination: a process by which conduction electrons and holes are annihilated in pairs.
Generation: a process by which conduction electrons and holes are created in pairs.
Generation and recombination processes act to change the carrier concentrations, and thereby indirectly affect current flow.
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Band-to-Band R–G Center Impact Ionization
EG
c1 dE
q dxE
ET: trap energy level
Generation ProcessesChapter 3 Carrier Action
Release of energy
• Due to lattice defects or unintentional impurities
• Also called indirect generation
• Only occurs in the presence of large E
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Band-to-Band R–G Center Auger
Collision
Recombination ProcessesChapter 3 Carrier Action
• Rate is limited by minority carrier trapping
• Primary recombination way for Si
• Occurs in heavily doped material
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Ev
Ec
Ec
EvGaAs, GaN(direct semiconductors)
Si, Ge(indirect
semiconductors)
PhotonPhoton
Phonon
Direct and Indirect SemiconductorsChapter 3 Carrier Action
E-k Diagrams
• Little change in momentum is required for recombination
• Momentum is conserved by photon (light) emission
• Large change in momentum is required for recombination
• Momentum is conserved by mainly phonon (vibration) emission + photon emission
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0nnn
0ppp
, 0n p
Equilibrium valuesDeviation from
equilibrium values
Excess Carrier ConcentrationsChapter 3 Carrier Action
Positive deviation corresponds to a carrier excess, while negative deviations corresponds to a carrier deficit.
Values under arbitrary conditions
pn Charge neutrality condition:
, 0n p
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Often, the disturbance from equilibrium is small, such that the majority carrier concentration is not affected significantly:
For an n-type material
For a p-type material
0 0, p n n n
0 0, n p p p
“Low-Level Injection”Chapter 3 Carrier Action
However, the minority carrier concentration can be significantly affected.
0p p
0n n • Low-level injection condition
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p TR
pc N p
t
G G-equilibrium
p p
t t
NT : number of R–G centers/cm3
Cp : hole capture coefficient
Indirect Recombination RateChapter 3 Carrier Action
Suppose excess carriers are introduced into an n-type Si sample by shining light onto it. At time t = 0, the light is turned off. How does p vary with time t > 0?
Consider the rate of hole recombination:
In the midst of relaxing back to the equilibrium condition, the hole generation rate is small and is taken to be approximately equal to its equilibrium value:
R-equilibrium
p
t
p T 0c N p
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R G R G
p p p
t t t
p TR G p
p pc N p
t
n TR G n
n nc N n
t
pp T
1
c N where
where nn T
1
c N
Indirect Recombination RateChapter 3 Carrier Action
The net rate of change in p is therefore:
p T p T 0c N p c N p p T 0c N p p
• For holes in n-type material
• For electrons in p-type material
Similarly,
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pp T n T
1 1 nc N c N
Minority Carrier LifetimeChapter 3 Carrier Action
The minority carrier lifetime τ is the average time for excess minority carriers to “survive” in a sea of majority carriers.
The value of τ ranges from 1 ns to 1 ms in Si and depends on the density of metallic impurities and the density of crystalline defects.
The deep traps originated from impurity and defects capture electrons or holes to facilitate recombination and are called recombination-generation centers.
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16 30 10 cmp
p n
Example: PhotoconductorChapter 3 Carrier Action
Consider a sample of Si doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s
a) What are p0 and n0?
2i
00
nn
p
210
16
10
10 4 310 cm
b) What are Δn and Δp?
LG 20 610 10 14 310 cm• Hint: In steady-state,
generation rate equals recombination rate
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Example: PhotoconductorChapter 3 Carrier Action
Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s.
c) What are p and n?
d) What are np product?
• Note: The np product can be very different from ni
2 in case of perturbed/agitated semiconductor
0p p p 16 1410 10 16 310 cm
0n n n 4 1410 10 14 310 cm
16 1410 10np 30 310 cm 2in
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2i
R G R G p 1 n 1( ) ( )
n npp n
t t n n p p
• ET : energy level of R–G center
Net Recombination Rate (General Case)Chapter 3 Carrier Action
For arbitrary injection levels and both carrier types in a non-degenerate semiconductor, the net rate of carrier recombination is:
T i( )1 i E E kTn n e
i T( )1 i
E E kTp n e
where
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1.(4.27)Problem 3.12, from (a) until (f), for Figure P3.12(a) and Figure P3.12(f),Pierret’s “Semiconductor Device Fundamentals”.
Chapter 2 Carrier Action
Homework 3
2.(5.28)The electron concentration in silicon at T = 300 K is given by
Deadline: 10 February 2011, at 07:30.
16 3( ) 10 exp cm18
xn x
where x is measured in μm and is limited to 0 ≤ x ≤ 25 μm. The electron diffusion coefficient is DN = 25 cm2/s and the electron mobility is μn = 960 cm2/(Vs). The total electron current density through the semiconductor is constant and equal to JN = –40 A/cm2. The electron current has both diffusion and drift current components.
Determine the electric field as a function of x which must exist in the semiconductor. Sketch the function.