Transcript Lecture 7

Lecture 7
OUTLINE
• Poisson’s equation
• Work function
• Metal-Semiconductor Contacts
– Equilibrium energy band diagrams
– Depletion-layer width
Reading: Pierret 5.1.2, 14.1-14.2; Hu 4.16
Poisson’s Equation
area A
Gauss’ Law:
 s ( x  Dx) A   s ( x) A  DxA
 ( x  Dx)   ( x) 
Dx
E(x+Dx)
Dx
s : permittivity (F/cm)
 : charge density (C/cm3)

s

d


dx
s
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E(x)
Lecture 7, Slide 2
Charge Density in a Semiconductor
• Assuming the dopants are completely ionized:
 = q (p – n + ND – NA)
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Lecture 7, Slide 3
Work Function
E0: vacuum energy level
FM: metal work function
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FS: semiconductor work function
Lecture 7, Slide 4
Metal-Semiconductor Contacts
There are 2 kinds of metal-semiconductor contacts:
• rectifying
“Schottky diode”
• non-rectifying
“ohmic contact”
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Lecture 7, Slide 5
Ideal M-S Contact: FM < FS, p-type
p-type
semiconductor
Band diagram instantly
after contact formation:
Equilibrium band diagram:
Schottky Barrier Height:
F Bp    EG  F M
FBp
qVbi = FBp– (EF – Ev)FB
W
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Lecture 7, Slide 6
Ideal M-S Contact: FM > FS, p-type
p-type
semiconductor
Band diagram instantly
after contact formation:
Equilibrium band diagram:
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Lecture 7, Slide 7
Ideal M-S Contact: FM < FS, n-type
Band diagram instantly
after contact formation:
Equilibrium band diagram:
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Lecture 7, Slide 8
Ideal M-S Contact: FM > FS, n-type
Band diagram instantly
after contact formation:
Equilibrium band diagram:
qVbi = FBn– (Ec – EF)FB
n
Schottky Barrier Height:
FBn  FM  
W
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Lecture 7, Slide 9
Effect of Interface States on FBn
• Ideal M-S contact:
FBn = FM – 
• Real M-S contacts:
A high density of
allowed energy states in
the band gap at the M-S
interface “pins” EF to be
within the range 0.4 eV
to 0.9 eV below Ec
FM
FBn
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Lecture 7, Slide 10
Schottky Barrier Heights: Metal on Si
•
Metal
FM (eV)
Er
3.12
Ti
4.3
Ni
4.7
W
4.6
Mo
4.6
Pt
5.6
FBn (eV)
0.44
0.5
0.61
0.67
0.68
0.73
FBp (eV)
0.68
0.61
0.51
0.45
0.42
0.39
FBn tends to increase with increasing metal work function
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Lecture 7, Slide 11
Schottky Barrier Heights: Silicide on Si
Silicide ErSi1.7 TiSi2
CoSi2
NiSi
WSi2
PtSi
FM (eV) 3.78 4.18
FBn (eV) 0.3
FBp (eV) 0.8
4.6 4.65 4.7
5
0.6 0.64 0.65 0.65 0.84
0.52 0.48 0.47 0.47 0.28
Silicide-Si interfaces are more stable than metal-silicon
interfaces and hence are much more prevalent in ICs.
After metal is deposited on Si, a thermal annealing
step is applied to form a silicide-Si contact. The term
metal-silicon contact includes silicide-Si contacts.
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Lecture 7, Slide 12
The Depletion Approximation
The semiconductor is depleted of mobile carriers to a depth W
 In the depleted region (0  x  W ):
 = q (ND – NA)
Beyond the depleted region (x > W ):
=0
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Lecture 7, Slide 13
Electrostatics
• Poisson’s equation:
• The solution is:

 qN D
 
x
s
s
 x   
qN D
s
W  x 
V x      ( x)dx
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Lecture 7, Slide 14
Depletion Width, W
 qN D
W  x 2
V x  
2K S 0
At x = 0, V = -Vbi
2 sVbi
 W
qN D
• W decreases with increasing ND
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Lecture 7, Slide 15
Summary: Schottky Diode (n-type Si)
metal
FM > FS
n-type Si
Eo
Si
FM
qVbi = FBn – (Ec – EFS)FB
FBn
Ec
EF
Depletion width
Ev
W
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Lecture 7, Slide 16
2 sVbi
W
qN D
Summary: Schottky Diode (p-type Si)
metal
FM < FS
p-type Si
Eo
Si
Ec
FM
EF
Ev
FBp
qVbi = FBp– (EF – Ev)FB
W
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Lecture 7, Slide 17
Depletion width
2 sVbi
W
qN A