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1 . Lecture #8
OUTLINE
• Poisson’s Equation
• Work Function
• Metal-Semiconductor Contacts
– equilibrium energy-band diagram
– depletion-layer width
Reading: Chapter 5.1.2,14.1, 14.2
Spring 2003 EE130 Lecture 8, Slide 1
Poisson’s Equation
Gauss’s Law:
area A
εs : permittivity (F/cm)
(x) (x+∆x)
ρ : charge density (C/cm3)
∆x
Spring 2003 EE130 Lecture 8, Slide 2
1
2 . Charge Density in a Semiconductor
• Assuming the dopants are completely ionized:
ρ = q (p – n + ND – NA)
Spring 2003 EE130 Lecture 8, Slide 3
Work Function
Ε0: vacuum energy level
ΦM: metal work function ΦS: semiconductor work function
Spring 2003 EE130 Lecture 8, Slide 4
2
3 . Metal-Semiconductor Contacts
There are 2 kinds of metal-semiconductor contacts:
• rectifying
“Schottky diode”
• non-rectifying
“ohmic contact”
Spring 2003 EE130 Lecture 8, Slide 5
Ideal MS Contact: ΦM > ΦS, n-type
Band diagram instantly
after contact formation:
Schottky
Equilibrium Barrier :
band diagram:
Φ Bn = Φ M − χ
Spring 2003 EE130 Lecture 8, Slide 6
3
4 . Ideal MS Contact: ΦM < ΦS, n-type
Band diagram instantly
after contact formation:
Equilibrium
band diagram:
Φ Bn = Φ M − χ
Spring 2003 EE130 Lecture 8, Slide 7
Ideal MS Contact: ΦM < ΦS, p-type
metal p-type Si
Eo
χSi
Ec
ΦM
EF
Ev
ΦBp qVbi = ΦBp– (EF – Ev)FB ΦBp = χ + EG - ΦM
W
Spring 2003 EE130 Lecture 8, Slide 8
4
5 . Effect of Interface States on ΦBn
metal n-type Si
ΦM > ΦS • Ideal MS contact:
ΦBn = ΦM – χ
Eo
χSi • Real MS contacts:
ΦM
A high density of
qVbi = ΦB – (Ec – EF)FB allowed energy
ΦBn states in the
Ec
EF
band gap at the
MS interface pins
EF to the range
0.4 eV to 0.9 eV
Ev below Ec
W
Spring 2003 EE130 Lecture 8, Slide 9
Schottky Barrier Heights: Metal on Si
Metal Mg Ti Cr Ni W Mo Pd Au Pt
Φ Bn (eV) 0.4 0.5 0.61 0.61 0.67 0.68 0.77 0.8 0.9
Φ Bp (eV) 0.61 0.5 0.51 0.42 0.3
Φ M (eV) 3.7 4.3 4.5 4.7 4.6 4.6 5.1 5.1 5.7
• ΦBn tends to increase with increasing metal work function
Spring 2003 EE130 Lecture 8, Slide 10
5
6 . Schottky Barrier Heights: Silicide on Si
Silicide ErSi1.7 HfSi MoSi2 ZrSi2 TiSi2 CoSi2 WSi2 NiSi2 Pd2Si PtSi
Φ Bn (V) 0.28 0.45 0.55 0.55 0.61 0.65 0.67 0.67 0.75 0.87
Φ Bp (V) 0.45 0.55 0.49 0.45 0.45 0.43 0.43 0.35 0.23
Silicide-Si interfaces are more stable than metal-silicon
interfaces. 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.
Spring 2003 EE130 Lecture 8, Slide 11
The Depletion Approximation
The semiconductor is depleted of mobile carriers to a depth W
⇒ In the depleted region (0 ≤ x ≤ W ):
ρ = q (ND – NA)
Beyond the depleted region (x > W ):
ρ=0
Spring 2003 EE130 Lecture 8, Slide 12
6
7 . Electrostatics
∂ ρ qN D
• Poisson’s equation: = ≅
∂x εs εs
• The solution is: (x ) = − qN D (W − x )
εs
V ( x ) = − ∫ ( x′)dx′
Spring 2003 EE130 Lecture 8, Slide 13
Depleted Layer Width, W
− qN D
V (x ) = (W − x )2
2K Sε 0
At x = 0, V = -Vbi
2ε sVbi
⇒ W=
qN D
• W decreases with increasing ND
Spring 2003 EE130 Lecture 8, Slide 14
7
8 . Equilibrium Current Flow – Qualitative
VA = 0
IM•→S = -Is IS•→M = Is
Ec
EF
Ev
Spring 2003 EE130 Lecture 8, Slide 15
Current Flow with D.C. Bias – Qualitative
VA > 0 VA < 0
Fermi level splits into two levels (EFM and EFS) separated by qVA
Spring 2003 EE130 Lecture 8, Slide 16
8
9 .W Dependence on D.C. Bias (n-type Si)
2ε s (Vbi − VA )
W=
qN D
• W increases with increasing –VA
• W decreases with increasing ND
Spring 2003 EE130 Lecture 8, Slide 17
W Dependence on D.C. Bias (p-type Si)
2ε s (VA + Vbi )
W=
qN A
• W increases with increasing VA
• W decreases with increasing NA
Spring 2003 EE130 Lecture 8, Slide 18
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