Transcript Energy Band Diagrams - Lane Department of Computer Science

Overview of Silicon
Device Physics
Dr. David W. Graham
West Virginia University
Lane Department of Computer Science and Electrical Engineering
1
Silicon
Silicon is the primary semiconductor used in VLSI systems
Si has 14 Electrons
Energy Bands
(Shells)
Valence Band
Nucleus
Silicon has 4 outer shell /
valence electrons
At T=0K, the
highest energy
band occupied by
an electron is
called the valence
band.
2
Energy Bands
}
Increasing
Electron
Energy
}
Disallowed
Energy
States
Allowed
Energy
States
• 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
Energy Bands
3
Energy Bands
EC
Conduction Band
First unfilled energy
band at T=0K
Eg
EV
Energy
Bandgap
Valence Band
Last filled energy
band at T=0K
4
Band Diagrams
Increasing electron energy
EC
Eg
EV
Increasing voltage
Band Diagram Representation
Energy 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
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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
The valence band is full, and
no electrons are free to move
about
EC
EV
However, at temperatures
above T=0K, thermal energy
shakes an electron free
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Semiconductor Properties
For T > 0K
Electron shaken free and can
cause current to flow
h+
e–
• 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)
8
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 3
Elements have 3
electrons in the
Valence Shell
Column 4
Elements have 4
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 mn ≈1350cm2/V)
+
11
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 mn ≈1350cm2/V)
n-Type Material
+
–
+ –
+ –+
–
+ +
+–
+
–
+ –
+ –
+
–
– + –+
+ –+
–
+ +–
–
+
–
+
–
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 valenceband 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
p-Type Material
•
–
+
– +
+
– +–
+
– –
+
+
–
–
+
–
–
+
– +
+
–
–
–
–
+
+
– –+
+
–
+
–
+
Shorthand Notation
– Negatively charged ion; immobile
+ Positively charged h+; mobile;
Called “majority carrier”
– Negatively charged e-; mobile;
Called “minority carrier”
•
•
•
•
•
•
•
Acceptors
Add atoms with only 3 valenceband 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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The Fermi Function
The Fermi Function
• Probability distribution function (PDF)
• The probability that an available state at
an energy E will be occupied by an e-
f(E)
1
f E  
1
1 e
E  E f  kT
E  Energy level of interest
Ef  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)
0.5
Ef
E
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Boltzmann Distribution
If E  E f  kT
f(E)
Then
f E   e

 EE f
 kT
1
0.5
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
Ef
~Ef - 4kT
E
~Ef + 4kT
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Band Diagrams (Revisited)
E
EC
Ef
Eg
EV
Band Diagram Representation
Energy 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
0.5
1
f(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 Level
Ef is a function of the impurity-doping level
n-Type Material
E
EC
Ef
EV
0.5
1
f(E)
• 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 Level
Ef is a function of the impurity-doping level
p-Type Material
1  f E 
E
EC
Ef
EV
0.5
1
f(E)
• 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
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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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Drift
Drift → Movement of charged particles in response to an external field (typically an
electric field)
E-field applies force
F = 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 velocity
µp → hole mobility
E
µn ≈ 3µp
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Drift
Drift → Movement of charged particles in response to an external field (typically an
electric field)
E-field applies force
F = 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 velocity
µp → hole mobility
Current Density
J n,drift  mn qnE
J p ,drift  m p qpE
q = 1.6×10-19 C, carrier density
n = number of ep = number of h+
µn ≈ 3µp
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Diffusion
Diffusion → 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)
dn x 
dx
dp  x 
  qD p
dx
J n ,diff  qDn
J p ,diff
q = 1.6×10-19 C, carrier density
D = Diffusion coefficient
n(x) = e- density at position x
p(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 ecancels 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)
D
kT

m q
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)
p-type
+
–
+
–
+
–
–
+
–
+
–
+
+ –
– +
+ –
– +
+ –
– +
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
–
+
–
+
–
+
n-type
–
+
–
+
–
+
+
–
+
–
+
–
– + –
+ – +
– + –
+ – +
– + –
+ – +
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
Bring p-type and n-type material into contact
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p-n Junctions (Diodes)
p-type
+
–
+
–
+
–
–
+
–
+
–
+
+ –
– +
+ –
– +
+ –
– +
+
–
+
–
+
–
–
+
–
+
–
+
n-type
+
–
+
–
+
–
–
+
–
+
–
+
–
+
–
+
–
+
+
–
+
–
+
–
– + –
+ – +
– + –
+ – +
– + –
+ – +
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
Depletion Region
• 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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Charge Density
p-type
+
–
+
–
+
–
Charge Density
(x)
–
+
–
+
–
+
+ –
– +
+ –
– +
+ –
– +
+
–
+
–
+
–
–
+
–
+
–
+
n-type
+
–
+
–
+
–
–
+
–
+
–
+
–
+
–
+
–
+
+
–
+
–
+
–
– + –
+ – +
– + –
+ – +
– + –
+ – +
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
Depletion Region
qND
x
-qNA
The remaining stationary charged particles results in areas with a net charge
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Electric Field
Electric Field
Charge Density
(x)
qND
x
-qNA
E
x
• Areas with opposing charge
densities creates an E-field
• E-field is the integral of the
charge density
• Poisson’s Equation
dE   x 

dx

ε is the permittivity of Silicon
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Potential
Electric Field
Charge Density
(x)
qND
x
-qNA
E
x
Potential

• E-field sets up a potential
difference
• Potential is the negative of the
integral of the E-field
d
  E x 
dx
bi
x
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Band Diagram
qND
x
-qNA
E
Electric Field
Charge Density
(x)
x
Potential

bi
x
Band Diagram
EC
• Line up the Fermi levels
• Draw a smooth curve to connect
them
Ef
EV
31
p-n Junction Band Diagram
VA
p
n
EC
Ef
EV
p-type
n-type
32
p-n Junction – No Applied Bias
VA
If VA = 0
p
EC
Ef
EV
n
• 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 voltage
Idrift = Idiff
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p-n Junction – Reverse Biased
VA
If VA < 0
p
Reverse Biased
EC
Ef
EV
n
• 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
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p-n Junction – Forward Biased
VA
If VA < 0
p
Forward Biased
EC
Ef
EV
n
• Barrier is reduced, so more eand h+ may diffuse across
• Increasing VA increases the eand 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
35
p-n Junction Diode

I  I0 e

V A nUT
1
Diffusion
Drift
Combination of drift
and generation
UT 
kT
→ Thermal voltage = 25.86mV
q
1
n
2
36
p-n Junction Diode

I  I0 e
V A nUT
I 0eVA nUT
1  
  I0

for VA > 0
for VA < 0
ln(I)
I
1
q

nU T nkT
VA
-I0

I  I0 e
VA
V A nUT
I
 eVA
I0
nUT
ln(I0)

1
1
 I 
ln    ln eVA nUT
 I0 
ln I   ln eVA nUT  ln I 0 




VA
ln I  
 ln I 0 
nU T
37
Curve Fitting Exponential Data (In MATLAB)
Curve Fitting Exponential Data (In MATLAB)
I  I 0 eVA
nUT
• 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
38