Lecture 4

Lecture 4

OUTLINE• Semiconductor Fundamentals (cont’d)

– Properties of carriers in semiconductors– Carrier drift

• Scattering mechanisms• Drift current

– Conductivity and resistivity

Reading: Pierret 3.1; Hu 1.5, 2.1-2.2

Mobile Charge Carriers in Semiconductors

• Three primary types of carrier action occur inside a semiconductor:

– Drift: charged particle motion under the influence of an electric field.

– Diffusion: particle motion due to concentration gradient or temperature gradient.

– Recombination-generation (R-G)

Lecture 4, Slide 2EE130/230A Fall 2013

Electrons as Moving Particles

F = (-q)E = moa F = (-q)E = mn*a

where mn* is the conductivity effective mass

In vacuum In semiconductor

Lecture 4, Slide 3EE130/230A Fall 2013

R.F. Pierret, Semiconductor Fundamentals, Figure 2.9

Conductivity Effective Mass, m*Under the influence of an electric field (E-field), an electron or a hole is accelerated:

electrons

holes

acceleration qE–mn---------=

*nm

qa

*pm

qa

Lecture 4, Slide 4

Si Ge GaAsmn*/mo 0.26 0.12 0.068

mp*/mo 0.39 0.30 0.50

mo = 9.110-31 kg

Electron and hole conductivity effective masses

EE130/230A Fall 2013

How to Measure the Effective Mass

Cyclotron Resonance Technique:

Centripetal force = Lorentzian force

• fcr is the Cyclotron resonance frequency, which is independent of v and r.

• Electrons strongly absorb microwaves of that frequency.

By measuring fcr , mn can be found.

qvBr

vmn 2

nm

qBrv

ncr m

qB

r

vf

22

C.Hu, Modern Semiconductor Devices for Integrated Circuits, Fig. 1-15

Lecture 4, Slide 5EE130/230A Fall 2013

Carrier Scattering• Mobile electrons and atoms in the Si lattice are always in

random thermal motion.– Electrons make frequent collisions with the vibrating atoms

“lattice scattering” or “phonon scattering” – increases with increasing T

• Other scattering mechanisms:– deflection by ionized impurity atoms– deflection due to Coulombic force between carriers

“carrier-carrier scattering” – only significant at high carrier concentrations

• The net current in any direction is zero, if no E-field is applied.

123

45

electron

Lecture 4, Slide 6EE130/230A Fall 2013

Thermal Velocity, vth

Average electron kinetic energy 2*

2

1

2

3thnvmkT

cm/s103.2m/s103.2

kg101.926.0

J/eV)106.1(eV026.033

75

31

19

*

nth m

kTv

Lecture 4, Slide 7EE130/230A Fall 2013

Carrier Drift• When an electric field (e.g. due to an externally applied voltage)

exists within a semiconductor, mobile charge-carriers will be accelerated by the electrostatic force:

12

3

45

electron

EElectrons drift in the direction opposite to the E-field net current

Because of scattering, electrons in a semiconductor do not undergo constant acceleration. However, they can be viewed as quasi-classical particles moving at a constant average drift velocity vdn

Lecture 4, Slide 8EE130/230A Fall 2013

Carrier Drift (Band Model)

Ec

Ev

Lecture 4, Slide 9EE130/230A Fall 2013

Electron Momentum• With every collision, the electron loses momentum

• Between collisions, the electron gains momentum–qEmn

mn ≡ average time between electron scattering events

dnnvm*

Lecture 4, Slide 10

Conservation of momentum |mn*vdn | = | qEmn|

EE130/230A Fall 2013

Carrier Mobility, |vdn| = qEmn / mn* ≡ nE

n [qmn / mn*] is the electron mobility

p [qmp / mp*] is the hole mobility

Similarly, for holes: |vdp|= qEmp / mp* pE

Lecture 4, Slide 11

Si Ge GaAs InAsn (cm2/Vs) 1400 3900 8500 30,000

p (cm2/Vs) 470 1900 400 500

Electron and hole mobilities for intrinsic semiconductors @ 300K

For electrons:

EE130/230A Fall 2013

Example: Drift Velocity Calculationa) Find the hole drift velocity in an intrinsic Si sample for E = 103 V/cm.

b) What is the average hole scattering time?

vdp = pE

q

m

m

q ppmp

p

mpp

*

*

Lecture 4, Slide 12

Solution:

a)

b)

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Mean Free Path• Average distance traveled between collisions

mpthvl

Lecture 4, Slide 13EE130/230A Fall 2013

Mechanisms of Carrier ScatteringDominant scattering mechanisms:

1. Phonon scattering (lattice scattering)2. Impurity (dopant) ion scattering

2/32/1

1

velocityermalcarrier thdensityphonon

1

TTTphononphonon

Phonon scattering limited mobility decreases with increasing T:

= q / m Tvth

Lecture 4, Slide 14EE130/230A Fall 2013

Impurity Ion Scattering

DADA

thimpurity NN

T

NN

v

2/33

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

Lecture 4, Slide 15

Ion scattering limited mobility increases with increasing T:

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Matthiessen's Rule

Probability that a carrier will be scattered by any mechanism

within a time period dt is

impurityphononimpurityphonon 111

111

i i

dt

i

dt

Lecture 4, Slide 16

• The probability that a carrier will be scattered by mechanism i

within a time period dt is

i ≡ mean time between scattering events due to mechanism i

EE130/230A Fall 2013

Mobility Dependence on DopingCarrier mobilities in Si at 300K

Lecture 4, Slide 17EE130/230A Fall 2013

Mobility Dependence on Temperature

impurityphonon 111

Lecture 4, Slide 18EE130/230A Fall 2013

Velocity Saturation• At high electric field, carrier drift velocity saturates:

The saturation velocity, vsat , is the maximum drift velocity

Lecture 4, Slide 19

Siin holesfor cm/s 106

Siin sonfor electr cm/s 1086

6

satv

J. Bean, in High-Speed Semiconductor Devices, S.M. Sze (ed.), 1990

EE130/230A Fall 2013

Hole Drift Current Density, Jp,drift

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

p vdp t A = number of holes crossing plane in time t

q p vdp t A = hole charge crossing plane in time t

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

Hole drift current per unit area Jp,drift = q p vdpLecture 4, Slide 20EE130/230A Fall 2013

R.F. Pierret, Semiconductor Fundamentals, Figure 3.3

Conductivity and Resistivity

Lecture 4, Slide 21

)( npdrift qnqpJ

)( ,, ndriftnpdriftp qnJqpJ

npdriftndriftpdrift qnqpJJJ ,,

• In a semiconductor, both electrons and holes conduct current:

np qnqp • The conductivity of a semiconductor is– Unit: mho/cm

1

• The resistivity of a semiconductor is– Unit: ohm-cm

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Resistivity Dependence on Doping

Lecture 4, Slide 22

For n-type material:

nqn 1

For p-type material:

pqp 1

Note: This plot (for Si) does not apply to compensated material (doped with both acceptors and donors).

EE130/230A Fall 2013

R.F. Pierret, Semiconductor Fundamentals, Figure 3.8

Electrical Resistance

where is the resistivity

Resistance Wt

L

I

VR [Unit: ohms]

V+ _

L

tW

I

uniformly doped semiconductor

Lecture 4, Slide 23EE130/230A Fall 2013

Example: Resistivity CalculationWhat is the resistivity of a Si sample doped with 1016/cm3 Boron?

Answer:

cm 4.1)450)(10)(106.1(

11

11619

ppn qpqpqn

Lecture 4, Slide 24EE130/230A Fall 2013

Example: Compensated Doping

cm 93.0)750)(109)(106.1(

11

11619

npn qnqpqn

Consider the same Si sample doped with 1016/cm3 Boron, and additionally doped with 1017/cm3 Arsenic. What is its resistivity?Answer:

Lecture 4, Slide 25EE130/230A Fall 2013

Example: T Dependence of

Consider a Si sample doped with 1017 As atoms/cm3. How will its resistivity change when T is increased from 300K to 400K?

93.1400

770

Lecture 4, Slide 26

Answer: The temperature dependent factor in (and therefore ) is n.

From the mobility vs. temperature curve for 1017 cm-3, we find that n decreases from 770 at 300K to 400 at 400K.

Thus, increases by

EE130/230A Fall 2013

Summary• Electrons and holes can be considered as quasi-classical

particles with effective mass m*

• In the presence of an electric field E, carriers move with average drift velocity vd = E , is the carrier mobility– Mobility decreases w/ increasing total concentration of ionized dopants – Mobility is dependent on temperature

• decreases w/ increasing T if lattice scattering is dominant• decreases w/ decreasing T if impurity scattering is dominant

• The conductivity () hence the resistivity () of a semiconductor is dependent on its mobile charge carrier concentrations and mobilities

Lecture 4, Slide 27

np qnqp 1

EE130/230A Fall 2013