Download • To understand PN junction’s IV characteristics, Charge Carriers in Semiconductor

Charge Carriers in Semiconductor
• To understand PN junction’s IV characteristics,
it is important to understand charge carriers’
behavior in solids, how to modify carrier
densities, and different mechanisms of charge
flow.
After notes by Wentai Liu- UCSC
Silicon - Semiconductor
Conductor – low resistivity
Insulator – high resistivity
Semiconductor intermediate
• Si has four valence electrons. Therefore, it can form
covalent bonds with four of its neighbors.
• When temperature goes up, electrons in the covalent
bond can become free.
Electron-Hole Pair Interaction
• With free electrons breaking off covalent bonds, holes
are generated.
• Holes can be filled by absorbing other free electrons, so
effectively there is a flow of charge carriers.
• Conduction electrons are negatively charged
• Holes are positively charged.
• Concentration of holes and electrons can be modulated,
for example, by impurity dopants
Free Electron Density in Intrinsic Silicon
• Eg, or bandgap energy determines the amount of energy
to break off an electron from its covalent bond.
• There exists an exponential relationship between the
free-electron density and bandgap energy.
− Eg
ni = 5.2 × 10 T exp
electrons / cm 3
2 kT
ni (T = 300 0 K ) = 1.08 × 1010 electrons / cm 3
15
3/ 2
ni (T = 600 0 K ) = 1.54 × 1015 electrons / cm 3
Extrinsic by Doping
• Pure Si can be doped with other elements to change its
electrical properties.
• If Si is doped with P
(phosphorous), each P atom
can contribute a conduction
electron, then it has more
electrons, or becomes N-type
Donor – Group V
• If Si is doped with B
(boron), each B atom can
contribute a hole. Then it
has more holes, or
becomes P-type
Acceptor – Group III
Summary of Charge Carriers
• Potential barriers (due to bands that are bended) across a junction can block
the carriers from going from one side to the other side when the barrier is
higher in energy than the highest (in energy) available carrier that wants to
diffuse or drift across the barrier. External fields lower or increase these
potential barriers letting more or less carriers from the distribution through.
• There is more than 1 free electron (hole) in a semiconductor. They cannot
have all the same energy, thus the available free electrons (n) or holes (p) will
be distributed in energy.
How to use energy band diagrams?
Remember two features
Fermi-Dirac distribution and the Fermi-level
Density of states tells us how many states exist at a given energy
E. The Fermi function f(E) specifies how many of the existing
states at the energy E will be filled with electrons. The function
f(E) specifies, under equilibrium conditions, the probability that an
available state at an energy E will be occupied by an electron. It is
a probability distribution function.
EF = Fermi energy or Fermi level
k = Boltzmann constant = 1.38 1023 J/K
= 8.6  105 eV/K
T = absolute temperature in K
1
Fermi-Dirac distribution: Consider T  0 K
For E > EF :
f ( E  EF ) 
1
 0
1  exp ()
For E < EF :
f ( E  EF ) 
1
 1
1  exp ()
E
EF
0
1
f(E)
2
Fermi-Dirac distribution: Consider T > 0 K
If E = EF then f(EF) = ½
E  EF  3kT
If
then
 E  EF 
exp 
  1
 kT 
Thus the following approximation is valid:
  ( E  EF ) 
f ( E )  exp 

kT


i.e., most states at energies 3kT above EF are empty.
If
E  EF  3kT
then
 E  EF 
exp 
  1
 kT 
 E  EF 
f ( E )  1  exp 

 kT 
So, 1f(E) = Probability that a state is empty, decays to zero.
Thus the following approximation is valid:
So, most states will be filled.
kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small
in comparison.
3
Temperature dependence of Fermi-Dirac distribution
4
5
Equilibrium distribution of carriers
Distribution of carriers = DOS probability of occupancy
= g(E) f(E)
(where DOS = Density of states)
Total number of electrons in CB (conduction band) =
n0  
E top
EC
g C ( E ) f ( E ) dE
Total number of holes in VB (valence band) =
p0  
EV
E Bottom
g V ( E ) 1  f ( E )  dE
6
Electron and Hole Density
• The product of electron and hole densities is ALWAYS equal
to the square of intrinsic electron density regardless of doping
levels.
np = ni
2
Majority Carriers :
p ≈ NA
Minority Carriers :
n
n≈ i
NA
Majority Carriers :
n ≈ ND
Minority Carriers :
n
p≈ i
ND
2
2
Charge Transport Mechanism - Drift
• The process in which charge particles move because of an
electric field is called drift.
• Charge particles will move at a velocity that is proportional
to the electric field.
→
dQ n
d ( x ( − W . h . n . q ))
=
dt
dt
= − v ⋅ W ⋅ h ⋅ n ⋅ q = J nW .h
→
In =
vh = µ p E
→
→
ve = − µ n E
Total drift current
J tot = µ n E ⋅ n ⋅ q + µ p E ⋅ p ⋅ q
= q( µ n n + µ p p) E
Velocity Saturation
µ=
µ0
1 + bE
vsat =
v =
µ0
b
µ0
E
µ0 E
1+
vsat
• A topic treated in more advanced courses is velocity
saturation.
• In reality, velocity does not increase linearly with electric
field. It will eventually saturate to a critical value.
Charge Transport Mechanism - Diffusion
• Charge particles move from a region of high concentration
to a region of low concentration. It is analogous to an every
day example of an ink droplet in water.
• Diffusion current is proportional to the gradient of charge
(dn/dx) along the direction of current flow.
• Its total current density consists of both electrons and holes.
dn
I = AqDn
dx
dn
J n = qDn
dx
dp
J p = − qD p
dx
dn
dp
J tot = q ( Dn
− Dp )
dx
dx
Einstein’s Relation
D
kT
=
µ
q
• While the underlying physics behind drift and diffusion
currents are totally different, Einstein’s relation provides a
mysterious link between the two.