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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
Poisson’s Equation
area A
Gauss’ Law:
 s ( x  Dx) A   s ( x) A  DxA
E(x)        E(x+Dx)
Dx

 ( x  Dx)   ( x)                             s : permittivity (F/cm)
Dx          s                         : charge density (C/cm3)

d  


dx   s

EE130/230M Spring 2013          Lecture 7, Slide 2
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
E0: vacuum energy level

FM: metal work function             FS: semiconductor work function

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: FM > FS, n-type

Band diagram instantly
after contact formation:

qVbi = FBn– (Ec – EF)FB
Equilibrium band diagram:                n

Schottky Barrier Height:
F Bn  F M  
W

EE130/230M Spring 2013     Lecture 7, Slide 6
Ideal M-S Contact: FM < FS, n-type

Band diagram instantly
after contact formation:

Equilibrium band diagram:

EE130/230M Spring 2013     Lecture 7, Slide 7
Ideal M-S Contact: FM < FS, p-type
p-type
semiconductor

Band diagram instantly
after contact formation:

Equilibrium band diagram:
Schottky Barrier Height:    FBp
F Bp    EG  F M                                qVbi = FBp– (EF – Ev)FB
W

EE130/230M Spring 2013     Lecture 7, Slide 8
Ideal M-S Contact: FM > FS, p-type
p-type
semiconductor
Band diagram instantly
after contact formation:

Equilibrium band diagram:

EE130/230M Spring 2013     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
FM                                            the band gap at the M-S
interface “pins” EF to be
FBn
within the range 0.4 eV
to 0.9 eV below Ec

EE130/230M Spring 2013   Lecture 7, Slide 10
Schottky Barrier Heights: Metal on Si

Metal         Er     Ti       Ni          W      Mo     Pt
FM (eV)      3.12   4.3       4.7         4.6    4.6    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

EE130/230M Spring 2013          Lecture 7, Slide 11
Schottky Barrier Heights: Silicide on Si

Silicide ErSi1.7 TiSi2   CoSi2      NiSi   WSi2   PtSi
FM (eV) 3.78 4.18       4.6 4.65 4.7    5
FBn (eV) 0.3      0.6 0.64 0.65 0.65 0.84
FBp (eV) 0.8      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.

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:               qN D
 
x  s s

• The solution is:        x      qN D
s
W  x 

V x     ( x)dx

EE130/230M Spring 2013                Lecture 7, Slide 14
Depletion Width, W
 qN D
V x           W  x 2
2K S 0

At x = 0, V = -Vbi

2 sVbi
 W
qN D

• W decreases with increasing ND

EE130/230M Spring 2013                  Lecture 7, Slide 15
Summary: Schottky Diode (n-type Si)
metal       FM > FS     n-type Si

Eo

Si
FM

FBn                 qVbi = FBn – (Ec – EFS)FB
Ec
EF

Depletion width

2 sVbi
Ev   W
W
qN D
EE130/230M Spring 2013             Lecture 7, Slide 16
Summary: Schottky Diode (p-type Si)
metal        FM < FS   p-type Si
Eo

Si

Ec
FM

EF
Ev   Depletion width
FBp                 qVbi = FBp– (EF – Ev)FB           2 sVbi
W
W
qN A
EE130/230M Spring 2013           Lecture 7, Slide 17

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