Download Overview of Silicon Device Physics

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
5
Intrinsic Semiconductor
Silicon has 4 outer shell /
valence electrons
Forms into a lattice structure
to share electrons
6
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
7
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
9
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
10
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 µn ≈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 µn ≈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”
12
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)
13
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)
14
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
15
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
16
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
17
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
18
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
19
Thermal Motion of Charged Particles
• Applies to both electronic systems and
biological systems
• Look at drift and diffusion in silicon
• Assume 1-D motion
20
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> ≈ -µ nEx electrons
< vx > ≈ µ pEx holes
µ n → electron mobility
→ empirical proportionality constant
between E and velocity
µ p → hole mobility
E
µ n ≈ 3µ p
21
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> ≈ -µ nEx electrons
< vx > ≈ -µ pEx holes
µ n → electron mobility
→ empirical proportionality constant
between E and velocity
µ p → hole mobility
Current Density
J n ,drift = µ n qnE
J p ,drift = µ p qpE
q = 1.6×10-19 C, carrier density
n = number of ep = number of h+
µ n ≈ 3µ p
22
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)
J n ,diff
J p ,diff
dn( x )
= qDn
dx
dp( x )
= −qD p
dx
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
23
Einstein Relation
Einstein Relation → Relates D and µ (they
are not independent of each other)
D
kT
=
µ
q
UT = kT/q
→ Thermal voltage
= 25.86mV at room temperature
≈ 25mV for quick hand approximations
→ Used in biological and silicon applications
24
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)
25
p-n Junctions (Diodes)
p-type
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
–
+
–
+
–
+
n-type
–
+
–
+
–
+
+
–
+
–
+
–
– + –
+ – +
– + –
+ – +
– + –
+ – +
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
Bring p-type and n-type material into contact
26
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)
27
Charge Density
p-type
+
–
+
–
+
–
Charge Density
ρ(x)
–
+
–
+
–
+
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
–
+
–
+
–
+
n-type
+
–
+
–
+
–
–
+
–
+
–
+
–
+
–
+
–
+
+
–
+
–
+
–
– + –
+ – +
– + –
+ – +
– + –
+ – +
+
–
+
–
+
–
–
+
–
+
–
+
+
–
+
–
+
–
Depletion Region
qND
x
-qNA
The remaining stationary charged particles results in areas with a net charge
28
Electric Field
Electric Field
Charge Density
ρ(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 ρ ( x )
=
dx
ε
ε is the permittivity of Silicon
29
Potential
Electric Field
Charge Density
ρ(x)
qND
x
-qNA
E
x
Potential
ψ
• E-field sets up a potential
difference
• Potential is the negative of the
integral of the E-field
dψ
= − E (x )
dx
Φ bi
x
30
Band Diagram
qND
x
-qNA
E
Electric Field
Charge Density
ρ(x)
x
Potential
ψ
Φ 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
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
EC
Ef
EV
• If no applied voltage
Idrift = Idiff
33
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
34
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 nU T
−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 nU T
 I 0 eV A nUT
−1 ≈ 
 − I0
)
for VA > 0
for VA < 0
ln(I)
I
q
1
=
nU T nkT
VA
-I0
(
I = I 0 eVA
I
= eVA
I0
ln(I0)
VA
nU T
nU T
)
−1
−1
(
)
I 
ln  = ln eV A nUT
 I0 
ln(I ) = ln eV A nU T + ln (I 0 )
(
ln(I ) =
)
VA
+ ln(I 0 )
nU T
37
Curve Fitting Exponential Data (In MATLAB)
Curve Fitting Exponential Data (In MATLAB)
I ≈ I 0 eVA
nU T
• 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