EE130 Lecture 10, Slide 1Spring 2007

Lecture #10

OUTLINE

• Poisson’s Equation

• Work function

• Metal-Semiconductor Contacts– equilibrium energy-band diagram– depletion-layer width

Read: Chapter 5.1.2,14.1, 14.2

EE130 Lecture 10, Slide 2Spring 2007

Gauss’s Law:

s : permittivity (F/cm) : charge density (C/cm3)

Poisson’s Equation

E(x) E(x+x)

x

area A

EE130 Lecture 10, Slide 3Spring 2007

Charge Density in a Semiconductor

• Assuming the dopants are completely ionized:

= q (p – n + ND – NA)

EE130 Lecture 10, Slide 4Spring 2007

Work Function

M: metal work function S: semiconductor work function

0: vacuum energy level

EE130 Lecture 10, Slide 5Spring 2007

There are 2 kinds of metal-semiconductor contacts: • rectifying

“Schottky diode”

• non-rectifying“ohmic contact”

Metal-Semiconductor Contacts

EE130 Lecture 10, Slide 6Spring 2007

EE130 Lecture 10, Slide 7Spring 2007

Ideal MS Contact: M > S, n-type

MBn

Band diagram instantly after contact formation:

Equilibrium band diagram:

Schottky Barrier :

EE130 Lecture 10, Slide 8Spring 2007

Ideal MS Contact: M < S, n-type

Band diagram instantly after contact formation:

Equilibrium band diagram:

EE130 Lecture 10, Slide 9Spring 2007

EE130 Lecture 10, Slide 10Spring 2007

Ideal MS Contact: M < S, p-type

M

Si

Bp

W

qVbi = Bp– (EF – Ev)FB

p-type Simetal

Ev

EF

Ec

Eo

Bp =+ EG -M

EE130 Lecture 10, Slide 11Spring 2007

Effect of Interface States on Bn

Eo

MSi

Bn

W

qVbi = B – (Ec – EF)FB

n-type Simetal

Ev

EF

Ec

M >S

• Ideal MS contact:Bn =M –

• Real MS contacts: A high density of

allowed energy states in the band gap at the MS interface pins EF to the range 0.4 eV to 0.9 eV below Ec

EE130 Lecture 10, Slide 12Spring 2007

Bn tends to increase with increasing metal work function

Metal Er Ti Ni W Mo Pt

M (eV) 3.12 4.3 4.7 4.6 4.6 5.6

Bn (eV) 0.44 0.5 0.61 0.67 0.68 0.73

Bp (eV) 0.68 0.61 0.51 0.45 0.42 0.39

Schottky Barrier Heights: Metal on Si

EE130 Lecture 10, Slide 13Spring 2007

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.

Silicide ErSi1.7 TiSi2 CoSi2 NiSi WSi2 PtSi

M (eV) 3.78 4.18 4.6 4.65 4.7 5

Bn (eV) 0.3 0.6 0.64 0.65 0.65 0.84

Bn (eV) 0.8 0.52 0.48 0.47 0.47 0.28

Schottky Barrier Heights: Silicide on Si

EE130 Lecture 10, Slide 14Spring 2007

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 Lecture 10, Slide 15Spring 2007

Electrostatics

• Poisson’s equation:

• The solution is:

ss

DqN

x

xWqN

x D s

xdxxV )(

E

E

E

EE130 Lecture 10, Slide 16Spring 2007

Depleted Layer Width, W

2

D

bis

qN

VW

2

02xW

K

qNxV

S

D

At x = 0, V = -Vbi

• W decreases with increasing ND

EE130 Lecture 10, Slide 17Spring 2007

Summary: Schottky Diode (n-type Si)

Equilibrium (VA = 0)-> EF continuous,

constant

Bn =M –

Eo

MSi

Bn

W

qVbi = Bn – (Ec – EF)FB

n-type Simetal

Ev

EF

Ec

M >S

2

D

bis

qN

VW

Depletion width:

EE130 Lecture 10, Slide 18Spring 2007

Summary: Schottky Diode (p-type Si)

M

Si

Bp

W

qVbi = Bp– (EF – Ev)FB

p-type Simetal

Ev

EF

Ec

Eo

M <S

Equilibrium (VA = 0)-> EF continuous,

constant

Bp =+ EG -M

2

A

bis

qN

VW

Depletion width: