Download Carrier Concentrations

Carrier
Concentrations
Presented by Chanam Lee
August 22nd
Carrier Concentration
Carrier Properties
State and Carrier Distributions
Equilibrium Carrier Concentration
Carrier Concentration for the Quantum Well
Devices
Carrier Properties
Carrier Movement in Free Space
Carrier Movement Within the Crystal
Intrinsic Carrier Concentration
Extrinsic n-Type Semiconductor
Extrinsic p-Type Semiconductor
Electronic Materials
Two-dimensional representation of an
Individual Si atom.
Elemental semiconductors
Valence
Represents each valence electron
III
IV
V
B
C
Al
Si
P
Ga
Ge
As
Semiconductors
When Si (or Ge & GaAs) atoms contact other
Si atoms, they form a tetrahederal
2D representation of lattice structure:
Electronic Materials
Two dimensions
Three dimensions
Carrier Movement in Free Space
Newton’s second law
dv
F = −qE = m0
dt
Carrier Movement Within the Crystal
Electrons moving inside a semiconductor crystal will collide with
semiconductor atoms ==> behaves as a “wave” due to the quantum
mechanical effects
The electron “wavelength” is perturbed by the crystals periodic
potential
Carrier Movement Within the Crystal
m n* (electron effective mass)
m *p (hole effective mass)
dv
F = − qE = m
dt
dv
F = − qE = m
dt
*
p
*
n
Material
mn* / m0
m*p / m0
Si
1.18
0.81
Ge
0.55
0.36
GaAs
0.066
0.52
Density of States Effective Masses at 300 K
Intrinsic Carrier Concentration
Contains an insignificant
concentration of impurity atoms
Under the equilibrium conditions,
for every electron is created, a
hole is created also
n = p = ni
As temperature is increased, the
number of broken bonds (carriers)
increases
As the temperature is decreased,
electrons do not receive enough
energy to break a bond and
remain in the valence band.
Extrinsic n-Type Semiconductor
Donors (Group V): The 5th in a
five valence electrons is readily
freed to wander about the
lattice at room temperature
There is no room in the valence
band so the extra electron
becomes a carrier in the
conduction band
Does NOT increase the number
of hole concentration
Extrinsic p-Type Semiconductor
Acceptors (Group III) : three
valence electrons readily
accept an electron from a
nearby Si-Si bond
Completing its own bonding
creates a hole that can wander
about the lattice
Does NOT increase the number
of electron concentration
State and Carrier Distribution
How the allowed energy states are distributed in energy
How many allowable states were to be found at any given
energy in the conduction and valence bands?
Essential component in determining carrier distributions and
concentration
Density of States
Fermi Function
Dopant States
Density of States:
Number of
cm 3
gc (E) =
States
mn* 2mn* ( E − Ec )
π
2
3
,
/
eV
E ≥ Ec
g c ( E )dE represents the number of
conduction band states/ cm 3 lying in
the energy range between E and E +
dE
gv (E) =
m *p 2m *p ( Ev − E )
π
2
3
, E ≤ Ev
g v ( E )dE represents the number of
3
valence band states / cm lying in the
energy range between E and E + dE
g c ( Ec ) = g v ( Ev ) = 0
Fermi Function (I)
How many of the states at the energy E will be filled
with an electron
f ( E) =
1
1 + e ( E −EF ) / kT
f(E), under equilibrium conditions, the probability that an
available state at an energy E will be occupied by an
electron
1- f(E), under equilibrium conditions, the probability that
an available state at an energy E will NOT be occupied
by an electron
Fermi Function (II)
If E = E F , f ( E F ) = 1 / 2
If E ≥ EF + 3k BT ,
f ( E ) ≈ exp[( EF − E ) / k BT ]
If E ≤ E F − 3k BT ,
f ( E ) ≈ 1 − exp[( E − EF ) / k BT ]
At T=0K (above), No occupation of states
above EF and complete occupation of
states below EF
At T>0K (below), occupation probability is
reduced with increasing energy f(E=EF) =
1/2 regardless of temperature.
At higher temperatures, higher energy states can be occupied, leaving
more lower energy states unoccupied (1-f(E))
Fermi Function (III)
Dopant States
n-type: more
electrons than
Holes
Intrinsic:
Equal number
of electrons
and holes
p-type: more
holes than
electrons
Density of
States
Occupancy
factors
Carrier
Distribution
Equilibrium Carrier Concentration
Formulas for n and p
Degenerate vs. Non-degenerate Semiconductor
Alternative Expressions for n and p
ni and the np Product
Charge Neutrality Relationship
Carrier Concentration Calculations
Determination of EF
Formulas for n and p
n=
n=
Etop
Ec
mn* 2mn*
π2
Ec
E − Ec
kT
E = Ec
when
n=
Etop
3
Letting η =
Let
p=
g c ( E ) f ( E ) dE
ETop
*
n
m
( E − Ec )
dE
( E − E F ) / kT
1+ e
and ηc =
Ebottom
Nc = 2
E F − Ec
kT
gv (E)[1− f (E)]dE
2πm kT
h
*
n
2
3/ 2
, Nv = 2
2πm*p kT
h2
, η =0
→ ∞
*
n
2 3
2m ( kT )
π
Ev
n ≡ Nc
3/ 2
E∞
E0
η
1/ 2
1+ e
(η −η c )
F1 / 2 (η c )
dη
p ≡ Nv
2
π
2
π
F1 / 2 (η c )
F1 / 2 (η v )
3/ 2
Degenerate vs. Non-degenerate
Semiconductor
E F < E c − 3k B T , f ( E ) =
If
1
≅ e −(η −η c )
(η −η c )
1+ e
F1/ 2 (η c ) =
π
2
e ( EF − Ec ) / kT
n = N c e ( E F − Ec ) / k B T , p = N v e ( E v − E F ) / k B T
If
E F > Ev + 3k BT , F1/ 2 (η v ) =
π
2
e ( Ev − EF ) / kT
Alternative Expressions for n and p
n0 = N c e
p0 = N v e ( E v − E F ) / k B T
( E F − Ec ) / k B T
When n=ni, EF = Ei, then
ni = N c e( Ei − Ec ) / kBT = N v e( Ev − Ei ) / k BT
N c = ni e
( E c − Ei ) / k B T
, N v = ni e
( Ec −Ei + EF −Ec ) / kBT
n0 = nie
( Ei − E v ) / k B T
( EF −Ei ) / kBT
= ni e
p0 = ni e ( Ei − Ev + Ev − EF ) / k BT = ni e ( Ei − EF ) / k BT
n0 p0 = ni
2
ni and the np Product
ni = N c e
( Ei − Ec ) / k BT
n = Nc Nve
2
i
= N ve
− ( Ec − Ev ) / k B T
( E v − Ei ) / k BT
= Nc Nve
ni = N c N v e
− E g / 2 k BT
− E g / k BT
Charge Neutrality Relationship
For uniformly doped semiconductor:
Charge must be balanced under equilibrium conditions otherwise
charge would flow
−
A
+
D
qp − qn − qN + qN = 0
Thermally generated + assume ionization of
dopant addition
all dopant sites
Carrier Concentration Calculations
( p − N A ) + ( N D − n) = 0
ni2
( − N A ) + ( N D − n) = 0
n
n 2 − n( N D − N A ) − ni2 = 0
n=
ND − N A
+
2
ND − N A
2
1/ 2
2
+ ni2
,p=
np = n i
2
N A − ND
+
2
N A − ND
2
1/ 2
2
+ ni2
+
D
Relationship for N and N
N A− =
NA
1+ g A
( E A − E F ) / kT
, N D+ =
−
A
ND
1+ gD
( E F − E D ) / kT
The degeneracy factors account for the possibility of electrons
with different spin, occupying the same energy level (no electron
with the same quantum numbers can occupy the same state)
Most semiconductor gD=2 to account for the spin degeneracy at
the donor sites
gA is 4 due to the above reason combined with the fact that
there are actually 2 valence bands in most semiconductors
Thus, 2 spins x 2 valance bands makes gA=4
Determination of EF (Intrinsic Material)
n = Nce( Ei −Ec ) / kBT = Nve( Ev −Ei ) / kBT = p
N c e ( Ei − E c ) / k B T = N v e ( E v − Ei ) / k B T
Ei =
Ec + Ev k BT
N
+
ln v
2
2
Nc
2πm kT
Nc = 2
h
*
n
2
3/ 2
, Nv = 2
2πm kT
h
*
p
2
3/ 2
m *p
Nv
=
Nc
mn*
1/ 2
m*p
Ec + Ev 3k BT
Ei =
+
ln *
mn
2
4
Determination of EF (Extrinsic Material)
n = ni e
( E F − Ei ) / k B T
,
p = ni e
( Ei − E F ) / k B T
n
p
E F − Ei = kT ln
= − kT ln
ni
ni
or
for ND >>NA and ND >>ni
N
E F − Ei = kT ln D
ni
or
for NA >>ND and NA >>ni
E F − Ei = − kT ln
NA
ni
Determination of EF (Extrinsic Material)
Fermi level positioning in Si at room temperature as a function of
the doping concentration. Solid EF lines were established using Eq.
(previous page)
Carrier Concentration for the
Quantum Well Devices
Density of States 3D vs. 2D
Carrier Concentration – 2D
Charge Neutrality
Density of States 3D vs. 2D
3D
g (E) =
2D
g=
m 2mE
π
2
3
m
π
2
3
LZ
Energy Dependent
Energy Independent
Carrier Concentration – 2D
n=
Etop
Ec
g c ( E ) f ( E ) dE
mn* kT
n= 2 3
π Lz
p=
i
m*p kT
π
2
3
Lz
[
ln 1 + e ( EFC − Ei ) / kT
[
ln 1 + e
i
( E FV − Ei ) / kT
]
]
Charge Neutrality
Phh + Plh = N Γ + N Χ + N L
References
Robert F. Pierret, Semiconductor Fundamentals
(VOLUME I), Addison-Wesley Publishing Company,
1988, chapter 2
Robert F. Pierret, Advanced Semiconductor
Fundamentals (VOLUME VI), Addison-Wesley
Publishing Company, 1987, chapter 4
Ben G. Streetman and Sanjay Banerjee, Solid State
Electronic Devices, Prentice Hall, Inc., 2000, chapter 3
Peter S. Zory, JR., Quantum Well Lasers, Academic
Press, 1993, chapter 1, 7
Effective Mass of Holes - 3D
gv (E) =
=
m*p 2m*p ( Ev − E )
π2
3
*
*
mhh
2mhh
( Ev − E)
π
2 3
+
mlh* 2mlh* ( Ev − E)
π2
3
* 3/ 2
(m *p ) 3 / 2 = (mhh
) + (mlh* ) 3 / 2
[
m = (m )
*
p
* 3/ 2
hh
]
* 3/ 2 2 / 3
lh
+ (m )