principles of semiconductor devices-l11

8/8/2019 Principles of Semiconductor Devices-L11

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Alam ECE-606 S09 1

www.nanohub.org

NCN

EE-606: Solid State DevicesLecture 11: Equilibrium Statistics

Muhammad Ashraful [email protected]


2/20

Outline

Alam ECE-606 S09 2

1) Law of mass-action & intrinsic concentration

2) Statistics of donors and acceptor levels

3) Conclusion

Reference: Vol. 6, Ch. 4 (pages 110-120)


3/20

Fermi-Level for Intrinsic Semiconductors

Alam ECE-606 S09 3

2

=

gE

i

F

V

i

Cn eE

NE

N

FE

( ) ( )

1

2 2

+

= =

= +

c i v i E E E E

Gi

V

C

VCn p e e

E E ln

N

N

N

N

E

k

3

2

2

= =

= gV

i

E

i CN

p nn eNn


4/20

However

Alam ECE-606 S09 4

Intrinsic concentration too small ..

We need to do something toincrease carrier concentration or

conductivity of semiconductors

This is done by doping.


5/20

Looking ahead: Carrier-Density w/Doping

Alam ECE-606 S09 5

0+ + = D AN p n dV N

0+ + =ADp n NN

1 10

42

++ =

+

F D

F

A B

C

F

B F B

B

V D

( E E )

A( E E ) / k T ( E E )

( E

/ k T

E ) / k T C / k TVN

NN

eee

Ne

A bulk material must be charge neutral over all

Further if the material is spatially homogenous

Let us see how the formula come about


6/20

Outline

Alam ECE-606 S09 6

1) Law of mass-action and intrinsic concentration

2) Statistics of donor and acceptor levels

3) Conclusion


7/20

Characteristics of Donor Atoms

Alam ECE-606 S09 7

(D)

r0

ET


8/20

Localized vs. Delocalized States

Alam ECE-606 S09 8

2N states/per-band (with spin)

2N-2 states/per-band (with spin)


9/20

Statistics of Donor Levels

Alam ECE-606 S09 9

0 0 0 0 1

1 0 1 1

0 1 1 1

i F

B

i F

B

i i i

( E E )

k T

( E E )

k T

u / d E N P

/ / Z

/ e Z

/ e Z

1/1 x x x

=

i i F B F B

i i F

ii

B

( E N E ) / k T ( E ) / k T

i ( E N E ) /

N

k T

i

Ee eP

e Z

Coulomb interactionforbids this configuration


10/20

Statistics of Donor Levels

Alam ECE-606 S09 10

0 0 0 0 1

0 1 1 1

1 0 1 1

i F

B

i F

B

i i i

( E E )

k T

( E E )

k T

u / d E N P

/ / Z

/ e Z

/ e Z

00

00

00 01 102

1 1

1 2 1

= = =+ + + +i F B F i B( E E ) / k T ( E E ) / k T

P / Zf

P P P / Z e / Z e

00

1 11 1

1 11

2

2

= =+

+F i B

i F B

( E E )/ k T ( E E ) / k T

fe

e

Prob. that the donor

is empty (charged)

Prob. that the donor

is filled with at least

one electron (neutral)

Note the extra factor .


11/20

Coulomb Exclusion for Band Electrons?

Alam ECE-606 S09 11

0 0 0 1

1 1 1

i F

B

( E E

i

)

i

T

i

k

state E N P

/ Z

e Z

i i F B i i F B

i i F B

( E N E ) / k T ( E N E ) / k T

i ( E N E ) / k T

i

e eP

e Z

=

Ei

E=0

E=2

E=4


12/20

Localized vs. Band Electrons

12

E1 2/Lx

E2 4/Lx

E3 6/Lx

E4 8/Lx

E5 10/Lx

E6 12/Lx

E1 2p/(Lx/2)

E2 4/(Lx/2)

E3 6/(Lx/2)

Lx

Lx/2 Lx/2

Two electrons (even withopposite spin) can notbe at the same positionand same energy becauseof electrostatic repulsion

Band electrons (withopposite spin) need not

be at the same position,so they can share occupysame energy level.

When we divide space by a factor of 2, the

number of states (e.g. 6 here) does not change.


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Acceptor Atoms

Alam ECE-606 S09 13

State [1] . Hole present N-1 charges

State [0] Hole filled . N charges


14/20

Statistics of Acceptor Levels in Si and Ge

14

1. Each atom contributes 2 states (up & down spin)

to a band, therefore a band has 2N states.

2. Every time a host atom is replaced by a impurity

atom, 2 states are disappear per a band and

appear as localized states (sort of).

3. Therefore an acceptor atom close to hh and lh

bands removes four states from those bands.

4. Because of Coulomb interaction only 1 hole can

seat in these 4 states: the states are

0000, 0001, 0010, 0100, 1000.

5. One now uses Pi to compute the occupation of

acceptors.

from lh

from hh

EC

EV


15/20

Number and Energy Considerations

Alam ECE-606 S09 15

4N-4 States

In HH/LH bands0000 1000 0100 0010 0001

1) [0000] is the charged state as it has N electrons, but N-1 protons.

2) Single hole configuration [0001] is uncharged, as we have N-1

electrons, and N-1 protons same is true for [0010], [0100], [1000]

states.

3) Going from [0000] to [0001] states, the number of electrons goesdown by 1 (Ni=-1).

4) Going from [0000] to [0001] states energy goes down by EA,

because one electron is no longer occupying the high energy level at

EA.


16/20

Statistics of Acceptor Levels

Alam ECE-606 S09 16

0000 1000 0100 0010 00010 0

0000

1F B

i i F B

( xE ) / k T

( E N E ) / k T

i

eP

e Z

=

1

0001 0010 0100 1000

A F B A F B

i i F B

( E ( )E ) / k T ( E E ) / k T

( E N E ) / k T

i

e eP P P P

e Z

= = = =

Steps 3 & 4

0000

0000

1000 0100 0010 0001

1

1 4

= =+ + + + A F B( E E ) / k T

Pf

P P P P e


17/20

Filled and empty Donor/Acceptor Levels

Alam ECE-606 S09 17

00

1

1 2

= =+ F i B

D D ( E E )/ k T N f N

e

[ ]0000

=filled

A A A N N N f

2N-2 states

4N-4 States

In HH/LH bands

(Two holes can not seat together)

0000 1000 0100 0010 0001

00 10 01 +emptyD DN N

1

1 4

=

+A F B

A ( E E ) / k T N

e


18/20

Distributions are physical .

Alam ECE-606 S09 18

1

=+ F D B

DD ( E / T

D

E ) k g

Nfe

1 1

= =+ +F D B F

'B D B

D DD ( E E ) / k k T ET ( E ) / k T

N Nf

e e e

Degeneracy factor

Effective donor level


19/20

Summary

Alam ECE-606 S09 19

0ADN p n dV N+ + + =

0AD

Nn Np + + + =

10

12 4

F V B C F B

F D A F BB

( E E ) / k T ( E E ) / k T A

( E E ) / V AD

( E E ) k/ k TT

Ne e

eN N

N

e

+ + + =

A bulk material must be charge neutral over all

Further if the material is spatially homogenous


20/20

Conclusions

Alam ECE-606 S09 20

Intrinsic concentration of electrons in a semiconductor is

relatively small.

The statistics of dopant levels are different because they are

close to the conduction band and because they are localized.

Influenced by pair-wise coulomb repulsion.

The statistics of donor and acceptor atoms are also different,

because donor levels couple with single conduction band,while acceptor levels couple with two valence bands.

principles of semiconductor devices-l11

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