Lecture #6

OUTLINE

• Carrier scattering mechanisms

• Drift current

• Conductivity and resistivity

• Relationship between band diagrams & V, Read: Section 3.1

EE130 Lecture 6, Slide 2Spring 2007

Dominant scattering mechanisms:

1. Phonon scattering (lattice scattering)

2. Impurity (dopant) ion scattering

2/32/1

1

velocityermalcarrier thdensityphonon

1

TTTphononphonon

Phonon scattering mobility decreases when T increases:

= q / m

Mechanisms of Carrier Scattering

Tvth

EE130 Lecture 6, Slide 3Spring 2007

_+

- -Electron

Boron Ion Electron

Arsenic Ion

DADA

thimpurity NN

T

NN

v

2/33

There is less change in the electron’s direction of travel if the electron zips by the ion at a higher speed.

Impurity Ion Scattering

EE130 Lecture 6, Slide 4Spring 2007

Matthiessen's Rule

• The probability that a carrier will be scattered by

mechanism i within a time period dt is

where i is the mean time between scattering events due to mechanism i

The probability that a carrier will be scattered within a

time period dt is

impurityphonon

impurityphonon

111

111

i i

dt

i

dt

EE130 Lecture 6, Slide 5Spring 2007

1E14 1E15 1E16 1E17 1E18 1E19 1E20

0

200

400

600

800

1000

1200

1400

1600

Holes

Electrons

Mo

bili

ty (

cm2 V

-1 s

-1)

Total Impurity Concenration (atoms cm-3)Total Doping Concentration NA + ND (cm-3)

Mobility Dependence on Doping

EE130 Lecture 6, Slide 6Spring 2007

Temperature Effect on Mobility

impurityphonon

impurityphonon

111

111

EE130 Lecture 6, Slide 7Spring 2007

vd t A = volume from which all holes cross plane in time t

p vd t A = # of holes crossing plane in time t

q p vd t A = charge crossing plane in time t

q p vd A = charge crossing plane per unit time = hole current

Hole current per unit area J = q p vd

Drift Current

EE130 Lecture 6, Slide 8Spring 2007

Jp,drift = qpvdn = qppJn,drift = –qnvdn = qnn

Jdrift = Jn,drift + Jp,drift = =(qnn+qpp)

Conductivity of a semiconductor is qnn + qpp

Resistivity 1 /

Conductivity and Resistivity

(Unit: ohm-cm)

EE130 Lecture 6, Slide 9Spring 2007

n-type

p-type

Resistivity Dependence on Doping

For n-type material:

nqn 1

For p-type material:

pqp 1

Note: This plot does not apply for compensated material!

EE130 Lecture 6, Slide 10Spring 2007

Electrical Resistance

where is the resistivity

Resistance Wt

L

I

VR (Unit: ohms)

V

+_

L

tW

I

homogeneously doped sample

EE130 Lecture 6, Slide 11Spring 2007

Consider a Si sample doped with 1016/cm3 Boron.What is its resistivity?

Answer:

NA = 1016/cm3 , ND = 0 (NA >> ND p-type)

p 1016/cm3 and n 104/cm3

Example

cm 4.1)450)(10)(106.1(

11

11619

ppn qpqpqn

EE130 Lecture 6, Slide 12Spring 2007

Example: Dopant Compensation

Consider the same Si sample, doped additionallywith 1017/cm3 Arsenic. What is its resistivity?

Answer:

NA = 1016/cm3, ND = 1017/cm3 (ND>>NA n-type)

n 9x1016/cm3 and p 1.1x103/cm3

cm 12.0)600)(109)(106.1(

11

11619

npn qnqpqn

EE130 Lecture 6, Slide 13Spring 2007

Consider a Si sample doped with 1017cm-3 As.How will its resistivity change when the temperature is increased from T=300K to T=400K?

Solution:

The temperature dependent factor in (and therefore ) is n. From the mobility vs. temperature curve for 1017cm-3, we find that n decreases from 770 at 300K to 400 at 400K. As a result, increases by

Example: Temperature Dependence of

93.1400

770

EE130 Lecture 6, Slide 14Spring 2007

Potential vs. Kinetic Energy

electron kinetic energyin

crea

sing

ele

ctro

n en

ergy

Ec

Evhole kinetic energy

incr

easi

ng h

ole

ener

gy

referencecP.E. EE Ec represents the electron potential energy:

EE130 Lecture 6, Slide 15Spring 2007

N-+ –

0 .7 V

Si

E

x

Ec(x)

Ef(x)

Ev(x)

E

0.7 V

-

+

V(x)

0.7 V

x0

(a)

(b )

(c)

+ –

0 .7 V

Si

E

x

Ec(x)

Ef(x)

Ev(x)

E

0.7 V

-

+

V(x)

0.7 V

x0

(a)

(b )

(c)

Electrostatic Potential, V

• The potential energy of a particle with charge -q is related to the electrostatic potential V(x):

)(1

creference EEq

V

qVP.E.

EE130 Lecture 6, Slide 16Spring 2007

N-+ –

0 .7 V

Si

E

x

Ec(x)

Ef(x)

Ev(x)

E

0.7 V

-

+

V(x)

0.7 V

x0

(a)

(b )

(c)

+ –

0 .7 V

Si

E

x

Ec(x)

Ef(x)

Ev(x)

E

0.7 V

-

+

V(x)

0.7 V

x0

(a)

(b )

(c)

Electric Field,

dx

dE

qdx

dV c1

• Variation of Ec with position is called “band bending.”

EE130 Lecture 6, Slide 17Spring 2007

Carrier Drift (Band Diagram Visualization)

Ec

Ev

EE130 Lecture 6, Slide 18Spring 2007

Summary• Carrier mobility varies with doping

– decreases w/ increasing total concentration of ionized dopants

• Carrier mobility varies with temperature– decreases w/ increasing T if lattice scattering is dominant– decreases w/ decreasing T if impurity scattering is dominant

• The conductivity of a semiconductor is dependent on the carrier concentrations and mobilities

• Ec represents the electron potential energy

Variation in Ec(x) variation in electric potential V

Electric field

• E - Ec represents the electron kinetic energy

= qnn + qpp

dx

dE

dx

dE vc