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


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


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


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


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


Carrier Drift (Band Model)

Ec

Ev


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


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


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:

EE130/230A Fall 2013

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

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Mobility Dependence on DopingCarrier mobilities in Si at 300K


Mobility Dependence on Temperature

impurityphonon 111


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


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

EE130/230A Fall 2013

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


Electrical Resistance

where is the resistivity

Resistance Wt

L

I

VR [Unit: ohms]

V+ _

L

tW

I

uniformly doped semiconductor


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


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:


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

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Lecture 4