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