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ECE 340 Lecture 6 : Intrinsic and

Extrinsic Material I

Class Outline:

• Effective Mass • Intrinsic Material • Extrinsic Material M.J. Gilbert ECE 340 – Lecture 6

Things you should know when you leave…

Key Questions • What is the physical meaning of

the effective mass • What does a negative effective

mass mean? • What is intrinsic material? • What is thermal equilibrium? • What is extrinsic material? • How does doping work?

M.J. Gilbert ECE 340 – Lecture 6

Effective Mass

At the end of lecture 5, we talked about effective mass…

Electric Field

Electric Field

•  In a vacuum, we can apply Newton’s second law:

•  In a semiconductor, we cannot. –  For overall motion – NO! –  For motion in-between

scattering – NO! •  We defined a new “effective”

mass which incorporated all of the complicated interactions.

dtdvmqEF 0=−=

dtdvmqEF n

*=−=

M.J. Gilbert ECE 340 – Lecture 6

Effective Mass

We even defined the effective mass…

22

2*

/ dkEdm !

=

We can define the effective mass as:

Nevertheless, two questions remain: 1.  Where does this definition come

from?

2.  What does it mean physically?

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M.J. Gilbert ECE 340 – Lecture 6

Effective Mass

Let’s begin to think about where effective mass comes from… Start with the energy-wavevector (dispersion) relation for free electrons:

mkEk 2

22!=

Now look at the equation of motion for how electrons move in an energy band in an electric field.

Suppose that the wavepacket is made of wavefunctions near a particular k.

Ψ(x)

k’ k

The wavepacket is moving with some group velocity, vg:

dkdEvg !

1=

All of the information of the effects of the crystal on the motion of the electron are in the dispersion relation.

E

(6.1)

(6.2)

M.J. Gilbert ECE 340 – Lecture 6

Effective Mass

What are the forces that the electron is experiencing?

Ψ(x)

k’ k

E vg

How much work is the field doing on the electron?

tveEE gfield δδ −= (6.3)

We observe that by using eq. 6.2…

kvkdkdEE gδδδ !=⎟

⎠

⎞⎜⎝

⎛= (6.4)

Combine eqns. 6.3 and 6.4 to arrive at an external force that is exerted on the electrons by the applied electric field.

FeEdtdkwhere

teE

k

field

field

=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

!

!

,

δδ

FdkEd

dtdk

dkEd

dkdtEd

dtdvg

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛==

2

2

2

2

22

1

1

!

!!

*

1m

Newton’s 2nd law!

M.J. Gilbert ECE 340 – Lecture 6

Effective Mass

Simple Example… Consider a simple cosine approximation to the band:

( ) ( ) ⎟⎠

⎞⎜⎝

⎛=−=2

sincos121 2 kaWkaWkE

•  Sample parameters –  W (Band Width) ~ 5 eV –  a (lattice spacing) ~ 0.5

nm

aπ−

aπ

( )eVkE

0

5

What are the group velocity and the effective mass?

Group velocity: ( ) ( ) ( )kaaWdkkdEkvg sin

21

!!== vg(k)

M.J. Gilbert ECE 340 – Lecture 6

Effective Mass

The group velocity goes to zero!! What about the effective mass?

•  The effective mass becomes negative! –  States of positive mass occur near the bottom of the bands

due to positive band curvature. –  States of negative mass occur at the top of bands.

•  Physically, it means that on going from k to k+Δk the momentum transfer to the lattice from the electron is larger than that of the momentum transfer from the applied force to the electron. –  As we approach Bragg reflection at the edge, when we

increase the wavevector we can get an overall decrease in the forward momentum.

aπ−

aπ

( ) ( )kaWam

km sec22

0

2* !

=

Effective mass:

0.3

-0.3

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M.J. Gilbert ECE 340 – Lecture 6

Intrinsic Material

Intrinsic Material is pure with no additional contaminants…

•  At T = 0 K, there is no energy in the system. –  All of the covalent bonds are satisfied. –  Valence band is full and conduction band is empty.

•  At T > 0 K, thermal energy breaks bonds apart –  Crystal lattice begins to vibrate and exchange energy with

carriers. –  Electrons leave the valence band to populate the conduction band.

T = 0 K

T = 300 K

M.J. Gilbert ECE 340 – Lecture 6

Intrinsic Material

But there are more processes at work…

•  Generation – Break up of a covalent bond to form an electron and

a hole. – Requires energy from thermal, optical, mechanical or

other external sources. – Supply of bonds to break is virtually inexhaustible.

•  Atomic density >> # of electrons or # of holes.

Generation Rate:

⎟⎠

⎞⎜⎝

⎛⋅

+++=scm

GGGG mechoptth 3

1...

M.J. Gilbert ECE 340 – Lecture 6

Intrinsic Material

Since we are in thermal equilibrium, there must be an opposite process…

•  Recombination –  Formation of a bond by bringing together and electron and a hole. –  Releases energy in the form of thermal or optical energy. –  Recombination events require the presence of 1 electron and 1

hole. –  These events are most likely to occur at the surfaces of

semiconductors where the crystal periodicity is broken.

Recombination Rate:

⎟⎠

⎞⎜⎝

⎛⋅

•∝scm

pnR 3

1

• N – number of electrons • P – number of holes

M.J. Gilbert ECE 340 – Lecture 6

Intrinsic Material

In the steady state…

•  The generation rate must be balanced by the recombination rate.

•  Important consequence is that for a given semiconductor the np product depends only on the temperature.

=

20000 inpnRG =⇒= inpnpn ==⇒= 0000

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M.J. Gilbert ECE 340 – Lecture 6

Intrinsic Material

Putting numbers to the intrinsic concentrations…

•  For silicon –  5 x 1022 atoms/cm3

–  4 bonds per atom –  2 x 1023 bonds/cm3

–  ni (300 K) ~ 1010 cm-3

–  1 broken bond per 1013 bonds.

Silicon ni ~ 1010 cm-3

Germanium ni ~ 2 x 1013 cm-3

GaAs ni ~ 2 x 106 cm-3

M.J. Gilbert ECE 340 – Lecture 6

Extrinsic Semiconductors

The great strength of semiconductors…

•  We can change their properties many orders of magnitude by introducing the proper impurity atoms.

•  Which columns add – Electrons? – Holes?

•  What about impurities?

M.J. Gilbert ECE 340 – Lecture 6

Extrinsic Materials

How does a donor work?

Silicon (Si) 4 valence electrons

Phosphorous (P) 5 valence electrons

M.J. Gilbert ECE 340 – Lecture 6

Extrinsic Materials

How does an acceptor work?

Silicon  (Si)      4  valence  electrons  

Boron  (B)      3  valence  electrons  

Si!

B  

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M.J. Gilbert ECE 340 – Lecture 6

Extrinsic Materials

In general, we can modify the materials properties with the introduction of immobile impurity atoms…

•  We can –  Selectively create

regions of n and p. •  Needed for CMOS.

–  Modify the conductivity over several orders of magnitude.

– Manipulate the number of conduction electrons over 5 orders of magnitude.

M.J. Gilbert ECE 340 – Lecture 6

Extrinsic Materials

How tightly bound is the extra electron or hole?

•  We can use the Bohr’s hydrogen model to get an idea.

•  Electrons move in Si and not in a vacuum. –  Different relative

permittivity. •  The electron mass must be

represented by the effective mass

r

Donor Acceptor

e-

h+

( ) 220

2

4*

32 !r

nB

qmEεεπ

−=

Donor in Si P As Sb

Binding energy (eV) 0.045 0.054 0.039

Acceptor in Si B Al Ga In

Binding energy (eV) 0.045 0.067 0.072 0.16

M.J. Gilbert

Extrinsic Materials

Visualizing donors on the band diagram…

Ed

Ea Ev

Ec

Ev

Ec

Δx

Let’s take a look at Silicon with Phosphorus impurity atoms:

Ed

Ev

Ec

Eg = 1.12 eV

0.045 eV

ECE 340 – Lecture 6 M.J. Gilbert

Extrinsic Material

Remember the intrinsic concentrations…

•  For silicon –  5 x 1023 atoms/cm3

–  4 bonds per atom –  2 x 1023 bonds/cm3

–  ni (300 K) ~ 1010 cm-3

–  1 broken bond per 1013 bonds.

Silicon ni ~ 1010 cm-3

Germanium ni ~ 2 x 1013 cm-3

GaAs ni ~ 2 x 106 cm-3

ECE 340 – Lecture 6

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M.J. Gilbert

Extrinsic Materials

Revisiting the effect of temperature…

T = 0 K T = 50 K T = 300 K

ECE 340 – Lecture 6 M.J. Gilbert

Extrinsic Material

Commonly used terms: •  Dopants – specific impurity atoms that are added to semiconductors in controlled amounts for

the express purpose of increasing either the electron or hole concentrations.

•  Intrinsic semiconductor – undoped semiconductor; extremely pure semiconductor sample containing an insignificant amount of impurity atoms; a semiconductor whose properties are native to the material.

•  Extrinsic semiconductor – doped semiconductor; a semiconductor whose properties are controlled by added impurity atoms.

•  Donor – impurity atom that increases the electron concentration; n-type dopant.

•  Acceptor – impurity atom that increases the hole concentration; p-type dopant.

•  N-type material – a donor doped material; a semiconductor containing more electrons than holes.

•  P-type material – an acceptor doped material; a semiconductor containing more holes than electrons.

•  Majority carrier – the most abundant carrier in a given semiconductor sample; electrons in n-type and holes in p-type.

•  Minority carrier – the least abundant carrier in a given semiconductor sample; electrons in p-type and holes in n-type.

ECE 340 – Lecture 6