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1 Overview of Silicon Device Physics Dr. David W. Graham West Virginia University Lane Department of Computer Science and Electrical Engineering
1
Overview of Silicon Device Physics
Dr. David W. Graham
West Virginia UniversityLane Department of Computer Science and Electrical Engineering
2
Silicon
Nucleus
Valence Band
Energy Bands(Shells)
Si has 14 Electrons
Silicon is the primary semiconductor used in VLSI systems
At T=0K, the highest energy band occupied by an electron is called the valence band.
Silicon has 4 outer shell / valence electrons
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Energy Bands
• Electrons try to occupy the lowest energy band possible
• Not every energy level is a legal state for an electron to occupy
• These legal states tend to arrange themselves in bands
Allowed Energy States
Disallowed Energy States
Increasing Electron Energy
}
}
Energy Bands
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Energy Bands
Valence Band
Conduction Band
Energy Bandgap
Eg
EC
EV
Last filled energy band at T=0K
First unfilled energy band at T=0K
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Band Diagrams
Eg
EC
EV
Band Diagram RepresentationEnergy plotted as a function of position
EC Conduction band Lowest energy state for a free electron
EV Valence band Highest energy state for filled outer shells
EG Band gap Difference in energy levels between EC and EV
No electrons (e-) in the bandgap (only above EC or below EV) EG = 1.12eV in Silicon
Increasing electron energy
Increasing voltage
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Intrinsic Semiconductor
Silicon has 4 outer shell / valence electrons
Forms into a lattice structure to share electrons
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Intrinsic Silicon
EC
EV
The valence band is full, and no electrons are free to move about
However, at temperatures above T=0K, thermal energy shakes an electron free
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Semiconductor PropertiesFor T > 0K
Electron shaken free and can cause current to flow
e–h+
• Generation – Creation of an electron (e-) and hole (h+) pair
• h+ is simply a missing electron, which leaves an excess positive charge (due to an extra proton)
• Recombination – if an e- and an h+ come in contact, they annihilate each other
• Electrons and holes are called “carriers” because they are charged particles – when they move, they carry current
• Therefore, semiconductors can conduct electricity for T > 0K … but not much current (at room temperature (300K), pure silicon has only 1 free electron per 3 trillion atoms)
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Doping
• Doping – Adding impurities to the silicon crystal lattice to increase the number of carriers
• Add a small number of atoms to increase either the number of electrons or holes
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Periodic Table
Column 4 Elements have 4 electrons in the Valence Shell
Column 3 Elements have 3 electrons in the Valence Shell
Column 5 Elements have 5 electrons in the Valence Shell
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Donors n-Type Material
Donors• Add atoms with 5 valence-band
electrons• ex. Phosphorous (P)• “Dontates an extra e- that can freely
travel around• Leaves behind a positively charged
nucleus (cannot move)• Overall, the crystal is still electrically
neutral• Called “n-type” material (added
negative carriers)• ND = the concentration of donor
atoms [atoms/cm3 or cm-3]~1015-1020cm-3
• e- is free to move about the crystal (Mobility n ≈1350cm2/V)
+
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Donors n-Type Material
Donors• Add atoms with 5 valence-band
electrons• ex. Phosphorous (P)• “Donates” an extra e- that can freely
travel around• Leaves behind a positively charged
nucleus (cannot move)• Overall, the crystal is still electrically
neutral• Called “n-type” material (added
negative carriers)• ND = the concentration of donor
atoms [atoms/cm3 or cm-3]~1015-1020cm-3
• e- is free to move about the crystal (Mobility n ≈1350cm2/V)
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n-Type Material
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Shorthand Notation Positively charged ion; immobile Negatively charged e-; mobile;
Called “majority carrier” Positively charged h+; mobile;
Called “minority carrier”
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Acceptors Make p-Type Material
––
h+
Acceptors• Add atoms with only 3 valence-
band electrons• ex. Boron (B)• “Accepts” e– and provides extra h+
to freely travel around• Leaves behind a negatively
charged nucleus (cannot move)• Overall, the crystal is still
electrically neutral• Called “p-type” silicon (added
positive carriers)• NA = the concentration of acceptor
atoms [atoms/cm3 or cm-3]• Movement of the hole requires
breaking of a bond! (This is hard, so mobility is low, μp ≈ 500cm2/V)
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Acceptors Make p-Type Material
Acceptors• Add atoms with only 3 valence-
band electrons• ex. Boron (B)• “Accepts” e– and provides extra h+
to freely travel around• Leaves behind a negatively
charged nucleus (cannot move)• Overall, the crystal is still
electrically neutral• Called “p-type” silicon (added
positive carriers)• NA = the concentration of acceptor
atoms [atoms/cm3 or cm-3]• Movement of the hole requires
breaking of a bond! (This is hard, so mobility is low, μp ≈ 500cm2/V)
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p-Type Material
Shorthand NotationNegatively charged ion; immobilePositively charged h+; mobile;
Called “majority carrier”Negatively charged e-; mobile;
Called “minority carrier”
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The Fermi Function
f(E)
1
0.5
EEf
The Fermi Function• Probability distribution function (PDF)• The probability that an available state at an energy E will be occupied by an e-
E Energy level of interestEf Fermi level
Halfway point Where f(E) = 0.5
k Boltzmann constant= 1.38×10-23 J/K= 8.617×10-5 eV/K
T Absolute temperature (in Kelvins)
kTEE feEf
1
1
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Boltzmann Distribution
f(E)
1
0.5
EEf
~Ef - 4kT ~Ef + 4kT
kTEE feEf
kTEE f If
Then
Boltzmann Distribution• Describes exponential decrease in the density of particles in thermal equilibrium with a potential gradient
• Applies to all physical systems• Atmosphere Exponential distribution of gas molecules• Electronics Exponential distribution of electrons• Biology Exponential distribution of ions
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Band Diagrams (Revisited)
Eg
EC
EV
Band Diagram RepresentationEnergy plotted as a function of position
EC Conduction band Lowest energy state for a free electron Electrons in the conduction band means current can flow
EV Valence band Highest energy state for filled outer shells Holes in the valence band means current can flow
Ef Fermi Level Shows the likely distribution of electrons
EG Band gap Difference in energy levels between EC and EV
No electrons (e-) in the bandgap (only above EC or below EV) EG = 1.12eV in Silicon
Ef
f(E)10.5
E
• Virtually all of the valence-band energy levels are filled with e-
• Virtually no e- in the conduction band
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Effect of Doping on Fermi LevelEf is a function of the impurity-doping level
EC
EV
Ef
f(E)10.5
E
n-Type Material
• High probability of a free e- in the conduction band• Moving Ef closer to EC (higher doping) increases the number of available
majority carriers
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Effect of Doping on Fermi LevelEf is a function of the impurity-doping level
EC
EV
Ef
p-Type Material
• Low probability of a free e- in the conduction band• High probability of h+ in the valence band• Moving Ef closer to EV (higher doping) increases the number of available
majority carriers
f(E)10.5
E
f(E)10.5
E
Ef1
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Thermal Motion of Charged Particles
• Applies to both electronic systems and biological systems
• Look at drift and diffusion in silicon
• Assume 1-D motion
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DriftDrift → Movement of charged particles in response to an external field (typically an
electric field)
E
E-field applies forceF = qE
which accelerates the charged particle.
However, the particle does not accelerate indefinitely because of collisions with the lattice (velocity saturation)
Average velocity<vx> ≈ -µnEx electrons< vx > ≈ µpEx holes
µn → electron mobility→ empirical proportionality constant
between E and velocityµp → hole mobility
µn ≈ 3µp
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DriftDrift → Movement of charged particles in response to an external field (typically an
electric field)
E-field applies forceF = qE
which accelerates the charged particle.
However, the particle does not accelerate indefinitely because of collisions with the lattice (velocity saturation)
Average velocity<vx> ≈ -µnEx electrons< vx > ≈ -µpEx holes
µn → electron mobility→ empirical proportionality constant
between E and velocityµp → hole mobility
µn ≈ 3µp
Current Density
qpEJ
qnEJ
pdriftp
ndriftn
,
,
q = 1.6×10-19 C, carrier densityn = number of e-
p = number of h+
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DiffusionDiffusion → Motion of charged particles due to a concentration gradient
• Charged particles move in random directions
• Charged particles tend to move from areas of high concentration to areas of low concentration (entropy – Second Law of Thermodynamics)
• Net effect is a current flow (carriers moving from areas of high concentration to areas of low concentration)
dx
xdpqDJ
dx
xdnqDJ
pdiffp
ndiffn
,
,q = 1.6×10-19 C, carrier densityD = Diffusion coefficientn(x) = e- density at position xp(x) = h+ density at position x
→ The negative sign in Jp,diff is due to moving in the opposite direction from the concentration gradient
→ The positive sign from Jn,diff is because the negative from the e- cancels out the negative from the concentration gradient
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Einstein Relation
Einstein Relation → Relates D and µ (they are not independent of each other)
q
kTD
UT = kT/q→ Thermal voltage= 25.86mV at room temperature≈ 25mV for quick hand approximations → Used in biological and silicon applications
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p-n Junctions (Diodes)
p-n Junctions (Diodes)
• Fundamental semiconductor device
• In every type of transistor
• Useful circuit elements (one-way valve)
• Light emitting diodes (LEDs)
• Light sensors (imagers)
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p-n Junctions (Diodes)
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p-type n-type
Bring p-type and n-type material into contact
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p-n Junctions (Diodes)
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• All the h+ from the p-type side and e- from the n-type side undergo diffusion→ Move towards the opposite side (less concentration)
• When the carriers get to the other side, they become minority carriers• Recombination → The minority carriers are quickly annihilated by the large number
of majority carriers• All the carriers on both sides of the junction are depleted from the material leaving
• Only charged, stationary particles (within a given region)• A net electric field This area is known as the depletion region (depleted of carriers)
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Depletion Region
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Charge Density
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Depletion Region
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(x)
qND
-qNA
x
Cha
rge
Den
sity
The remaining stationary charged particles results in areas with a net charge
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Electric Field• Areas with opposing charge densities creates an E-field
• E-field is the integral of the charge density
• Poisson’s Equation
ε is the permittivity of Silicon
x
(x)
qND
-qNA
Ch
arg
e D
en
sity
E
x
Ele
ctric
Fie
ld
x
dx
dE
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Potential
x
(x)
qND
-qNA
Ch
arg
e D
en
sity
E
x
Ele
ctric
Fie
ld
xEdx
d
x
bi
Po
ten
tial
• E-field sets up a potential difference
• Potential is the negative of the integral of the E-field
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Band Diagram
x
(x)
qND
-qNA
Ch
arg
e D
en
sity
E
x
Ele
ctric
Fie
ld
x
bi
Po
ten
tial
• Line up the Fermi levels• Draw a smooth curve to connect them
EC
EfEV
Ba
nd
Dia
gra
m
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p-n Junction Band Diagram
p n
VA
EC
EfEV
p-type n-type
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p-n Junction – No Applied Bias
p n
VA
If VA = 0
EC
EfEV
EC
EfEV
• Any e- or h+ that wanders into the depletion region will be swept to the other side via the E-field
• Some e- and h+ have sufficient energy to diffuse across the depletion region
• If no applied voltageIdrift = Idiff
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p-n Junction – Reverse Biased
p n
VA
If VA < 0
• Barrier is increased• No diffusion current occurs (not sufficient energy to cross the barrier)
• Drift may still occur• Any generation that occurs inside the depletion region adds to the drift current
• All current is drift current
Reverse Biased
EC
EfEV
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p-n Junction – Forward Biased
p n
VA
If VA < 0
• Barrier is reduced, so more e- and h+ may diffuse across
• Increasing VA increases the e- and h+ that have sufficient energy to cross the boundary in an exponential relationship (Boltzmann Distributions)→Exponential increase in diffusion current
• Drift current remains the same
Forward Biased
EC
EfEV
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p-n Junction Diode
10 TA nUVeII
Combination of drift and generation
Diffusion Drift
q
kTUT → Thermal voltage = 25.86mV
2
1n
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p-n Junction Diode
0
00 1
I
eIeII
TA
TA
nUVnUV for VA > 0
for VA < 0
I
-I0VA
1
1
0
0
TA
TA
nUV
nUV
eI
I
eII
0
0
0
lnln
lnlnln
lnln
InU
VI
IeI
eI
I
T
A
nUV
nUV
TA
TA
ln(I)
ln(I0)
VA
nkT
q
nUT
1
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Curve Fitting Exponential Data (In MATLAB)
TA nUVeII 0
Curve Fitting Exponential Data (In MATLAB)
• Given I and V (vectors of data)• Use the MATLAB functions
•polyfit – function to fit a polynomial (find the coefficients)•polyval – function to plot a polynomial with given coefficients and x values
[A] = polyfit(V,log(I),1);% polyfit(independent_var,dependent_var,polynomial_order)% A(1) = slope% A(2) = intercept
[I_fit] = polyval(A,V);% draws the curve-fit line
