3.225 17 Hydrogenic Model • For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as an orbiting electron around a net fixed charge • We can estimate t
Trang 13.225 11
Density of Thermally Promoted of Carriers
∫
∞
=
c
E
dE E g E f
Density of electron states per volume per dE Fraction of states occupied at a particular temperature
Number of electrons per
volume in conduction
band
( E E e dE
e
m
E
E
g T
k
E
g
b
F ∞ −
∫ −
2
1 h
π
Since
2
0
2
1 − = π
∞
∫ x e x dx
E T
k
E b
g
b
F
e
e T k
m
n
−
= 2
3
2
*
2
2 h
π
NC
(
(
T k E E
e
e E
E
E
T
k E
F
b
F ≈ − >>
+
=
−
−
1
1 ) (
T
k E
E
C b
g
F
e N
n
−
=
1
2
2
*
2
2
2
1
)
g = −
h
π
© E Fitzgerald-1999
)
)
)
)
• A similar derivation can be done for holes, except the density of states for holes is used
• Even though we know that n=p, we will derive a separate expression
anyway since it will be useful in deriving other expressions
Density of Thermally Promoted of Carriers
( 2 1
2
3
2
*
2
2
2
1
)
v = −
h
0
E f f dE
E g E f
∞
−
T
k
E b
F
e T k
m
p
−
= 2
3
2
*
2
2 h
π
T
k
E
v b
F
e N
p
−
= )
Trang 23.225 13
Thermal Promotion
• Because electron-hole pairs are generated, the Fermi level is approximately in the middle of the band gap
• The law of mass action describes the electron and hole
populations, since the total number of electron states is fixed in the system
+
=
4
3 2 gives
e
h
b
g
m T k E E p
n
Since me* and mh* are close and in the ln term, the Fermi level sits about in the center of the band gap
E
v
e
b
g
e m
m T
k n n
3 2
2 2
or
−
=
=
h
π
© E Fitzgerald-1999 )
Law of Mass Action for Carrier Promotion
E h
e
b
g
e m
m T
k np
n
−
=
3
*
*
3
2
2
2
4 h
E V C
g
e N N
n
−
=
2
;
• Note that re-arranging the right equation leads to an expression similar to a chemical reaction, where Eg is the barrier
• NCNV is the density of the reactants, and n and p are the products
•
+
′
→
N
V
C
V
C i T
k
E
V
n
e N
N
=
=
−
of the other carrier will lead to a dominant carrier type in the material
semiconductor
)
Trang 33.225 15
Intrinsic Semiconductors
• Conductivity at any temperature is determined mostly by the size of the band gap
• All intrinsic semiconductors are insulating at very low temperatures
*
2
*
2
h
h
e
e h
e
m
pe
m
ne pe
Recall:
E h e
g
e e
int
−
∝ +
σ
• One important note: No matter how pure Si is, the material will always be a
poor insulator at room T
• As more analog wireless applications are brought on Si, this is a major issue
for system-on-chip applications
This can be a measurement
© E Fitzgerald-1999
+ )
Extrinsic Semiconductors
• Adding ‘correct’ impurities can lead to controlled domination of one carrier type – n-type is dominated by electrons
– p-type if dominated by holes
• Adding other impurities can degrade electrical properties
Impurities with close electronic
structure to host
Impurities with very different electronic structure to host
x
x
x
x
Ge
Si
P
x
x
x
x
x
x
x
x
Au
Si
deep level
Ec
Ev
Ec
Ev
Ec
Ev
ED
EDEEP
-+
Trang 43.225 17
Hydrogenic Model
• For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as
an orbiting electron around a net fixed charge
• We can estimate the energy to free the carrier into the conduction band or valence band
by using a modified expression for the energy of an electron in the H atom
2 2 2
2
4 13 6
me
E
o
ε
2
* 2 2 2 2
2
4
* 2
2
2
8
8
2
2
ε ε
ε
ε
ε
m
m n n
h
e
m n
h
me
E
r
o
e
e
o
(in eV)
• Thus, for the ground state n=1, we can see already that since ε is on the order of 10, the binding energy of the carrier to the center is <0.1eV
• Expect that many carriers are then thermalize at room T
• Experiment:
• B acceptor in Si: 046 eV
• P donor in Si: 0.044 eV
• As donor in Si: 0.049
© E Fitzgerald-1999
The Power of Doping
• Can make the material n-type or p-type: Hydrogenic impurities are nearly fully ionized at room temperature
– ni 2 for Si: ~1020cm-3
– Add 1018cm-3 donors to Si: n~Nd
– n~1018cm-3, p~102 (ni /Nd)
• Can change conductivity drastically
– 1 part in 107 impurity in a crystal (~1022cm-3 atom density)
– 1022*1/107=1015 dopant atoms per cm-3
– n~1015, p~1020/1015~105
σ/σi~(p+n)/2ni~n/2ni~105!
Impurities at the ppm level drastically change the conductivity
(5-6 orders of magnitude)
Trang 53.225 19
Expected Temperature Behavior of Doped Material (Example:n-type)
• 3 temperature regimes
ln(n)
1/T
Intrinsic Extrinsic
Freeze-out
© E Fitzgerald-1999
Contrasting Semiconductor and Metal Conductivity
• Semiconductors
– changes in n(T) can dominate over τ
– as T increases, conductivity increases
• Metals
– n fixed
– as T increases, τ decreases, and conductivity decreases
σ = ne τ
m
2
Trang 63.225 21
• Metals and majority carriers in semiconductors
– τ is the scattering length
– Phonons (lattice vibrations), impurities, dislocations,
and grain boundaries can decrease τ
1 1 1 1
gb disl impur
τ
τ
1
1
=
=
=
i i
i
i i th
th
i
i
N
l
N v
v
l
σ
σ
number of scatterers per volume, and l is the average distance before collisions
The mechanism that will tend to dominate the scattering will be the mechanism with the shortest l (most numerous), unless there is a large difference in the cross-sections
Example: Si transistor, τphonon dominates even though τimpur gets worse with scaling
© E Fitzgerald-1999
Estimate of T dependence of conductivity
• τ ~l for metals
• τ ~l/vth for semiconductors
• First need to estimate l=1/N σ
2
1
x N
l
ion
ion ion
ph
π
σ
σ
∝
=
x=0
∫
∫
∞ +
∞
−
+∞
∞
−
Ψ
Ψ
Ψ
Ψ
=
dx
dx
x
x
*
2
*
1
2
−
=
=
kT
e E x
h
h
Average energy of harmonic oscillator
Trang 73.225 23
Estimate of T dependence of conductivity
1
1
2
−
=
=
−
=
=
T
kT
e
k
E
k
e E x
k
θ
ω
θ θ
ω
ω
h
h
h
Therefore, <x2> is proportional to T if T large compared to θ:
T N v x N v
v
l
T
x
l
T
x T
e
ion F ion
F
F cond
T
π
π τ
µ
σ
σ
θ
θ
1
1
1 1
1
1
2
2
2
=
=
≈
∝
∝
∝
∝
∝
∝
+
≈
For a metal:
For a semiconductor, remember that the carriers at the band edges are classical-like:
2
3
2
1
*
1
3
−
∝
∝
=
T
T
m
kT
l
v
l
th
3
*
−
∝
m
e τ
µ
© E Fitzgerald-1999
Example: Electron Mobility in Ge
µ~T-3/2 if phonon dominated (T-1/2 from vth, T-1 from x-section σ
At higher doping, the
ionized donors are the
dominate scattering
mechanism
Trang 83.225 1
• Minority carriers (e.g electrons (minority carrier)in p-type material with
majority holes
τis the time to recombination: recombination time
– means for system to return to equilibrium after perturbation, e.g by
illumination
Ec
Ev
τ , l
Recombination
x E
© E Fitzgerald-1999
Generation
• Deep levels in semiconductors act as carrier traps and/or enhanced
recombination sites
Ec
Ev
Recombination through deep level
Edeep
Generation and Recombination
• Generation
– photon-induced or thermally induced, G=#carriers/vol.-sec
– e.g g = P/hν
– Gois the equilibrium generation rate
• Recombination
– R=# carriers/vol.-sec
– Rois the equilibrium recombination rate, balanced by Go
• Net change in carrier density:
– dn/dt = G - R = G - (n - n0) / τ = G -∆n / τ
– Under steady state illumination: dn/dt = 0
– np(0) = np0+ G τ
Αfter turning off illumination:
– np(0) = np0+ G τ e -t/τ
© H.L Tuller, 2001
g τ
t
np(t)
np0
np(0)
τ
Trang 93.225 3
Key Processes: Drift and Diffusion
Electric Field: Drift
Concentration Gradient: Diffusion
E en J A env
I
E ep J A epv
I
e e
d
e
h h
d
h
µ
µ
=
=
=
=
;
;
n eD
J
p eD
J
e
e
h
h
∇
=
∇
−
=
n eD E en
J
p eD E ep
J
e e
eTOT
h h
hTOT
∇ +
=
∇
−
=
µ
µ
© E Fitzgerald-1999
Electrochemical Potential ϕ
µ
η j = j + z jq
j j
j = µ0 + kT ln c
µ
=
ϕ
x q z
j
j
j
∂
∂
−
x
c qD
z
x
j j j
j
∂
∂
−
∂
∂
= σ ϕ
Note: = 0 Under equilibrium conditions
∂
∂
x
j
η
Electrochemical Potential ⇒ EF Chemical Potential
Electrostatic Potential
Trang 103.225 5
Continuity Equations
• For a given volume, change in carrier concentration in time is related to J
G
R TOT
G
R TOT
G R diff drift
t
p
t
p
J e
t
p
t
n
t
n
J e
t
n
t
n
t
n
t
n
t
n
t
n
∂
∂ +
∂
∂
−
⋅
∇
−
=
∂
∂
∂
∂ +
∂
∂
−
⋅
∇
=
∂
∂
∂
∂ +
∂
∂
−
∂
∂ +
∂
∂
=
∂
∂
1
1
1-D,
G
R
x
p
D
x
E p
t
p
G
R
x
n
D
x
E
n
t
n
h
h
e
e
+
−
∂
∂ +
∂
∂
−
=
∂
∂
+
−
∂
∂ +
∂
∂
=
∂
∂
2
2
2
2
µ
µ
© E Fitzgerald-1999
Minority Carrier Diffusion Equations
• In many devices, carrier action outside E-field controls properties > minority carrier devices
• Only diffusion in these regions
e
h
h
e
n
R
p
R
G
R
x
p
D
t
p
G
R
x
n
D
t
n
τ
τ
∆
−
=
∆
−
=
+
−
∂
∂
=
∂
∂
+
−
∂
∂
=
∂
∂
type,
-p
in
type,
-n
in
2
2
2
t
n
t
n
t
n
t
∂
∆
∂
≈
∂
∆
∂ +
∂
∂
=
∂
∂ therefore
material type
-n
in
material type
-p in
G
2
2
2
2
G
p
x
p
D
t
p
n
x
n
D
t
n
h
h
e
e
+
∆
−
∂
∆
∂
=
∂
∆
∂
+
∆
−
∂
∆
∂
=
∂
∆
∂
τ
τ