Presented by Chanam Lee
August 22 nd

Carrier Properties
State and Carrier Distributions
Equilibrium Carrier Concentration
Carrier Concentration for the Quantum Well
Devices

Carrier Movement in Free Space
Carrier Movement Within the Crystal
Intrinsic Carrier Concentration
Extrinsic n-Type Semiconductor
Extrinsic p-Type Semiconductor

Two-dimensional representation of an
Individual Si atom.
Elemental semiconductors
Valence III IV V
B
Al
Ga
C
Si
Ge
P
As
Represents each valence electron

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
F = − qE = m
0 dv 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 m
F = − qE = m * n
Material
Si
Ge
GaAs dv dt m * n
/ m
0
1.18
0.55
0.066
F = − qE = m * p dv dt m * p
/ m
0.81
0.36
0.52
0
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 = n i
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 5 th 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

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

Number of cm 3
States
/ eV g c
( E ) = m * n
2 m * n
π 2
( E
3
− E c
)
, E ≥ E c g c
( E ) dE represents the number of
/ cm the energy range between E and E + dE g v
( E ) = m * p
2 m * p
( E v
π 2 3
− E )
, E ≤ E v g v
( E ) dE represents the number of
/ cm energy range between E and E + dE g c
( E c
) = g v
( E v
) = 0

Fermi Function (I)
How many of the states at the energy E will be filled with an electron f ( E ) =
1 + e ( E
1
− E
F
) / 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
If
E = E
F
, f ( E
F
) = 1 / 2
E f
≥
(
E
E )
F
≈
+ 3 k
B exp[(
T
E
,
F
− E ) / k
B
T ]
If E ≤ f (
E
E
F
)
−
≈
3
1 k
−
B
T , exp[( E − E
F
) / k
B
T ]
At T=0K (above), No occupation of states above EF and complete occupation of states below E
F
At T>0K (below), occupation probability is reduced with increasing energy f(E=E
F
) =
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
Density of
States n-type: more electrons than
Holes
Intrinsic:
Equal number of electrons and holes p-type: more holes than electrons
Occupancy factors
Carrier
Distribution

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 E
F

Formulas for n and p n =
E top
E c g c
( E ) f ( E ) dE p =
E v
E bottom g v
( E )[ 1 − f ( E )] dE n = m * n
π 2
2 m * n
3
E top
E c
1 +
( e
E
( E
− E c
− E
F
) /
) kT dE
N c
= 2
2 π m * n kT h 2
3 / 2
, N v
= 2
2 π m * p kT h 2
3 / 2
Letting when
η =
E − kT
E c
E = E c and
, η = 0
Let E
Top
→ ∞
η c
=
E
F
− kT
E c n = m * n
2 m * n
π 2
( kT ) 3 / 2
3
E
∞
E
0
1 +
η 1 / e ( η
2
− η c
) d η
F
1 / 2
( η c
) n ≡ N c p ≡ N v
2
π
F
1 / 2
2
π
F
1 / 2

Degenerate vs. Non-degenerate
Semiconductor
If E
F
< E c
− 3 k
B
T , f ( E ) =
F
1 / 2
1 +
= e
1
( η
π
− η c
2 e
)
(
≅
E
F e − ( η − η c
− E c
) / kT
) n = N c e ( E
F
− E c
) / k
B
T , p = N v e ( E v
− E
F
) / k
B
T
If E
F
> E v
+ 3 k
B
T , F
1 / 2
=
2
π e ( E v
− E
F
) / kT

Alternative Expressions for n and p n
0
= N c e ( E
F
− E c
) / k
B
T p
0
= N v e ( E v
− E
F
) / k
B
T
When n=n i
, E
F
= E i
, then n i
= N c e ( E i
− E c
) / k
B
T
N c
= n i e ( E c
− E i
) / k
B
T ,
= N v e ( E v
− E i
) / k
B
T
N v
= n i e ( E i
− E v
) / k
B
T n p
0
0
=
= n n i i e ( e (
E c
E i
− E i
− E v
+ E
F
+ E v
−
− E c
E
F
) / k
B
T
) / k
B
T
=
= n n i i e e (
( E i
E
F
−
− E i
E
F
)
) / k
B
T
/ k
B
T n
0 p
0
= n i
2

n i and the np Product n i
= N c e ( E i
− E c
) / k
B
T = N v e ( E v
− E i
) / k
B
T n i
2 = N c
N v e − ( E c
− E v
) / k
B
T = N c
N v e
− E g
/ k
B
T n i
N c
N v e
− E g
/ 2 k
B
T

For uniformly doped semiconductor:
Charge must be balanced under equilibrium conditions otherwise charge would flow qp
qn
qN −
A
qN +
D
0
Thermally generated + assume ionization of dopant addition all dopant sites

( p − N
A
) + ( N
D
− n n
( n i
2
2 n
−
− n (
N
N
A
D
) +
−
( N
D
N
A
) −
− n n i
2
)
) =
=
=
0
0
0 n =
N
D
−
2
N
A +
N
D
−
2
N
A
2
+ n i
2
1 / 2
, p =
N
A
−
2
N
D +
N
A
−
2
N
D
2
+ n i
2
1 / 2 np = n i
2

N
N −
A
=
1 + g
A
(
N
E
A
A
− E
F
) / kT
, N +
D
=
1 + g
D
(
N
E
D
F
− E
D
) / kT
N −
A
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 g
D
=2 to account for the spin degeneracy at the donor sites g
A 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 g
A
=4

Determination of E
F
(Intrinsic Material) n = N c e ( E i
− E c
) / k
B
T = N v e ( E v
− E i
) / k
B
T = p
N c e ( E i
− E c
) / k
B
T = N v e ( E v
− E i
) / k
B
T
E i
=
E c
+
2
E v + k
B
T
2 ln
N v
N c
N c
= 2
2 π m * n h 2 kT
3 / 2
, N v
= 2
2 π m * p kT h 2
3 / 2
N v
N c
= m * p m * n
1 / 2
E i
=
E c
+
2
E v +
3 k
4
B
T ln m * p m * n

Determination of E
F
(Extrinsic Material) n = n i e ( E
F
− E i
) / k
B
T
E
F
− E i
= kT ln n n i
, p = n i e ( E i
− E
F
) / k
B
T
= − kT ln p n i or for N
D
>> N
A and N
D
>> n i
E
F
− E i
= kT ln
N
D n i or for N
A
>> N
D and N
A
>> n i
E
F
− E i
= − kT ln
N n i
A

Determination of E
F
(Extrinsic Material)
Fermi level positioning in Si at room temperature as a function of the doping concentration. Solid E
F 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

3D
2D g ( E ) = m
π 2
2 mE
3 g =
π 2 m
3 L
Z
Energy Dependent
Energy Independent

Carrier Concentration – 2D n =
E top
E c g c
( E ) f ( E ) dE n =
π m * n
2 kT
3 L z i ln 1
[
+ e ( E
FC
− E i
) / kT
] p =
π m * p kT
2 3 L z i ln 1
[
+ e ( E
FV
− E i
) / kT
]

Charge Neutrality
P hh
+ P lh
= N
Γ
+ N
Χ
+ N
L

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, J
R
., Quantum Well Lasers , Academic
Press, 1993, chapter 1, 7

Effective Mass of Holes - 3D g v
( E ) = m * p
= m * hh
2 m * p
( E v
π 2 3
− E )
2 m * hh
π 2
( E v
3
− E )
+ m * lh
2 m * lh
π 2
( E v
3
− E )
( m * p
) 3 / 2 = ( m * hh
) 3 / 2 + ( m * lh
) 3 / 2 m * p
= (
[ m * hh
) 3 / 2 + ( m * lh
) 3 / 2
2
]
3 /