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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
Gauss’ Law:
s : permittivity (F/cm) : charge density (C/cm3)
Poisson’s Equation
E(x) E(x+x)
x
area A
EE130/230M Spring 2013 Lecture 7, Slide 2
xAAxAxx ss )()(
sdx
d
sx
xxx
)()(
Charge Density in a Semiconductor• Assuming the dopants are completely ionized:
= q (p – n + ND – NA)
EE130/230M Spring 2013 Lecture 7, Slide 3
Work Function
M: metal work function S: semiconductor work function
0: vacuum energy level
EE130/230M Spring 2013 Lecture 7, Slide 4
Metal-Semiconductor Contacts
There are 2 kinds of metal-semiconductor contacts: • rectifying
“Schottky diode”
• non-rectifying“ohmic contact”
EE130/230M Spring 2013 Lecture 7, Slide 5
Ideal M-S Contact: M < S, p-type
MBp GE
Band diagram instantly after contact formation:
Equilibrium band diagram:Schottky Barrier Height:
EE130/230M Spring 2013 Lecture 7, Slide 6
qVbi = Bp– (EF – Ev)FB
W
p-typesemiconductor
Bp
EE130/230M Spring 2013 Lecture 7, Slide 7
Ideal M-S Contact: M > S, p-typep-typesemiconductor
Equilibrium band diagram:
Band diagram instantly after contact formation:
Ideal M-S Contact: M < S, n-type
EE130/230M Spring 2013 Lecture 7, Slide 8
Band diagram instantly after contact formation:
Equilibrium band diagram:
Ideal M-S Contact: M > S, n-type
MBn
Band diagram instantly after contact formation:
Equilibrium band diagram:Schottky Barrier Height:
EE130/230M Spring 2013 Lecture 7, Slide 9
qVbi = Bn– (Ec – EF)FB
W
n
Effect of Interface States on Bn
• Ideal M-S contact:Bn =M –
• 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
EE130/230M Spring 2013 Lecture 7, Slide 10
M
Bn
• Bn tends to increase with increasing metal work function
Metal Er Ti Ni W Mo PtM (eV) 3.12 4.3 4.7 4.6 4.6 5.6Bn (eV) 0.44 0.5 0.61 0.67 0.68 0.73Bp (eV) 0.68 0.61 0.51 0.45 0.42 0.39
Schottky Barrier Heights: Metal on Si
EE130/230M Spring 2013 Lecture 7, Slide 11
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.
Silicide ErSi1.7 TiSi2 CoSi2 NiSi WSi2 PtSi
M (eV) 3.78 4.18 4.6 4.65 4.7 5Bn (eV) 0.3 0.6 0.64 0.65 0.65 0.84Bp (eV) 0.8 0.52 0.48 0.47 0.47 0.28
Schottky Barrier Heights: Silicide on Si
EE130/230M Spring 2013 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
EE130/230M Spring 2013 Lecture 7, Slide 13
Electrostatics• Poisson’s equation:
• The solution is:
ss
DqN
x
xWqN
x D s
xdxxV )(
EE130/230M Spring 2013 Lecture 7, Slide 14
Depletion Width, W
2
D
bis
qN
VW
2
02xW
K
qNxV
S
D
At x = 0, V = -Vbi
• W decreases with increasing ND
EE130/230M Spring 2013 Lecture 7, Slide 15
Summary: Schottky Diode (n-type Si)
Eo
MSi
Bn
W
qVbi = Bn – (Ec – EFS)FB
n-type Simetal
Ev
EF
Ec
M >S
2
D
bis
qN
VW
Depletion width
EE130/230M Spring 2013 Lecture 7, Slide 16
Summary: Schottky Diode (p-type Si)
M
Si
Bp
W
qVbi = Bp– (EF – Ev)FB
p-type Simetal
Ev
EF
Ec
Eo
M <S
EE130/230M Spring 2013 Lecture 7, Slide 17
2
A
bis
qN
VW
Depletion width