Struktur Pita Dan Sifat Listrik Bahan [2015.10]

7/24/2019 Struktur Pita Dan Sifat Listrik Bahan [2015.10]

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STRUKTUR PITA DANSTRUKTUR PITA DANSIFAT LISTRIK BAHANSIFAT LISTRIK BAHAN

(Pertemuan 10)(Pertemuan 10)


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CHAPTER 2CHAPTER 2

ENERGY BANDS AND EFFECTIEENERGY BANDS AND EFFECTIE!ASS!ASS

Pr"#$ Dr$ Be &re G'NLPr"#$ Dr$ Be &re G'NL

Semiconductors, insulators and metalsSemiconductors, insulators and metalsSemiconductorsSemiconductorsInsulatorsInsulators

MetalsMetalsThe concept of effective massThe concept of effective mass


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Semiconductors, Insulators and Metals

The electrical properties of metals andinsulators are well known to all of us.

Everyday experience has already taught

us a lot aout the electrical properties ofmetals and insulators.

!ut the same cannot e said aout"semiconductors#.

$hat happens when we connect a

attery to a piece of a silicon%would it conduct well ? orwould it act like an insulator ?


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The name "semiconductor# implies that it conducts

somewhere etween the two cases &conductors or

insulators'

(onductivity )

*metals +--/0cm

*insulators + -011 /0cm

The conductivity () of a

semiconductor (S/C) lies

between these two

extreme cases.

S/C


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.)) The !and Theory of Solids ))..)) The !and Theory of Solids )).

The electrons surroundinga nucleus have certain well0

defined energy0levels.

Electrons don2t like to have

the same energy in thesame potential system.

The most we could get

together in the same

energy0level was two,provided that they had

opposite spins. This is

called Pauli ExclusionPauli Exclusion

Principle.Principle.1 2 !!!!!!"

Number of atoms

Allowedband

Forbiddenband

Forbiddenband

Allowedband

Allowedband


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The difference in energy etween each of these smallerlevels is so tiny that it is more reasonale to considereach of these sets of smaller energy0levels as eingcontinuous bandsof energy, rather than considering theenormous numer of discrete individual levels.

Each allowed bandallowed bandis seperated from another one y aforbidden bandforbidden band.

Electrons can e found in allowed bandsallowed bandsut they can

not e found in forbidden bandsforbidden bands.


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.)) (34(543TI67.)) (34(543TI67

Consider 1 cm3 of Silicon. How many atoms does this contain ?

Solution)

The atomic mass of silicon is 18. g which contains 3vagadro2s numer of atoms.

3vagadro2s numer 7 is 9.-1 x -1:atomsmol .

The density of silicon) 1.: x -:kgm:

so cm:of silicon weighs 1.: gram and so contains

This means that in a piece of silicon just one cubic centimeter involume , each electron energy-level has split up into 4.9 ! "#$$

smaller levels %

2#22$.%2 1%

2.# . 1%

2'.1

atoms

=


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Both full and empty bands do not partake in electrical conduction.

.)) Semiconductor, Insulators, (onductors )).

Fullband

All energy levels

areoccupied byelectrons

&mptyband

All energy levels are

empty ' no electrons(


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.)) Semiconductor energy ands at low temperature ))..)) Semiconductor energy ands at low temperature )).

3t low temperatures the valanceand is full, and the conductionand is empty.

;ecall that a full and can notconduct, and neither can an emptyand.

3t low temperatures, sc2s do notconduct, they ehave like insulators.

The thermal energythermal energyof the electronssitting at the top of the full and ismuch lower than that of the EgEg atatlow temperatureslow temperatures.

Forbiddenenergy gap )&g*

&mptyconductionband

Fullvalanceband

Electron

ener

gy


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(onduction Electron )(onduction Electron )

3ssume some kind of energy isprovided to the electron &valenceelectron' sitting at the top of thevalance band.valance band.

This electron gains energy from theapplied field and it would like tomove into higher energy states.

This electron contriutes to theconductivityconductivity and this electron is

called as a conduction electronconduction electron.

3t --


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$hen enough energyenergy is supplied to

the ee00 sitting at the top of the

valance and, ee00 can make a

transition to the ottom of the

conduction and.

$hen electron makes such a

transition it leaves ehind a missingmissing

electron state.electron state.

This missing electron state is called

as a hole.hole.

>ole ehaves as a positive chargepositive chargecarrier.carrier.

Magnitude of its charge is the same

with that of the electron ut with an

opposite sign.

Semiconductor energy ands at room temperatureSemiconductor energy ands at room temperature

Forbiddenenergy gap )&g

Fullvalanceband

&mptyconductionband

e e e eener*y


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(onclusions(onclusions $$

>oles contriute to current in valance bandvalance band &?!' as e02sare ale to create current in conduction bandconduction band&(!'.

>ole is notnot a free particle. It can only exist within thecrystal. 3 hole is simply a vacant electron state.

3 transition results an e@ual numer of e0 in (! and

holes in ?!. This is an important property of intrinsic,orundoped s/cs. Aor extrinsic, or doped, semiconductorsthis is no longer true.


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!ipolar &two carrier' conduction!ipolar &two carrier' conduction

fter transitionfter transition, the

valance and is now no

longer full, it ispartly filled

and may conduct electriccurrent.

The conductivityconductivity is due to

oth electrons and holes,and this device is called a

bipolar conductorbipolar conductor or

bipolar device.bipolar device.

occu+ied

+alanceand'partly lled

band(Electron

energy

em+ty

After transitionAfter transition


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!hat kind of excitation mechanism can cause an e" to make a transition from the

top of the valance band #$%& to the minimum or bottom of the conduction band

#'%& ?

Thermal energy B Electrical field B

Electromagnetic radiation B

Answer ,

To have a partly field band configuration in a s/c ,To have a partly field band configuration in a s/c ,

one must use one of these excitation mechanisms.one must use one of these excitation mechanisms.

-*

artly filled

C

artly filled

0

-ner*y band dia*ram of a

s/c at a finite tem+erature.


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0Thermal Energy )0Thermal Energy )(hermal energy) kx ().:8 x -01: C< x :-- < )1D me?

Excitation rate) constantx exp#"Eg / k(&

3lthough the thermal energy at room temperature, *(,is very small, i.e.i.e.1Dme?, a few electrons can e promoted to the '%.

Electrons can be promoted to the '% by means of thermal energy.Electrons can be promoted to the '% by means of thermal energy.

This is due to the exponential increase of excitation rate with increasingtemperature.

Excitation rate is a strong function of temperature.


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Aor low fields, this mechanism doesn2t promote electrons to the(! in common sc2s such as Si and a3s.

3n electric field of -8 ?m can provide an energy of the order of e?. This field is enormous.

10 Electric field )10 Electric field )

+o , the use of the electric field as an excitation+o , the use of the electric field as an excitationmechanism is not useful way to promote electrons inmechanism is not useful way to promote electrons in

s/cs.s/cs.


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:0 Electromagnetic ;adiation ):0 Electromagnetic ;adiation )

# ' 1.2($.$2 1% ) (# 1% / ) / ( ) ( )(in )

cE h h x J s x x m s m E eV

m

= = = =

h $.$2 x 1%# s

c # x 1%' m/s1 e01.$x1%1&

1.2Silicon 1.1 ( ) 1.1

1.1

gfor E eV m m = = = n

To promote electrons from V to ! "ilicon , the wavelengthTo promote electrons from V to ! "ilicon , the wavelength

of the photons must #.#of the photons must #.#$$m or lessm or less

"ear

infrared


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The converse transition can alsohappen.

3n electron in (! recomines with

a hole in ?! and generate a

photon. The energy of the photon will e in

the order of Eg.

If this happens in a direct and0gap

sc, it forms the asis of 4EF2s and43SE;S.

e

+hoton

0alance and

Conduction and


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The magnitude of the and gapdetermines the differences etweeninsulators, scGs and metals.

The excitation mechanism of thermalis not a useful way to promote anelectron to (! even the meltingtemperature is reached in aninsulator.

Even very high electric fields is alsounale to promote electrons acrossthe and gap in an insulator.

Insulators )Insulators )

CB(com+letely em+ty)

VB(com+letely full)

-*3several electron volts-*3several electron volts

4ide band *a+s between 0 and C


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Metals )Metals )

C

0

C

0

7o gap etween valance bandvalance bandand conduction bandconduction band

Touching ?! and (! 6verlapping ?! and (!

These two ands

looks like as if partly

filled ands and it is

known that partly

filled ands conducts

well.

This is the reason

why metals have high

conductivity.


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The (oncept of Effective Mass )The (oncept of Effective Mass )

omparingomparingAree e* in vacuum

3n e* in a crystal

In an electric field

moH. x -0:

Aree electron mass

In an electric field

In a crystal

m H B

mJ effective mass

If the same magnitude of electric fieldis applied tooth electrons in vacuum and inside the crystal,the electronswill accelerate at a different rate fromeach other due to the existence of differentpotentials inside the crystal.

The electron inside the crystalhas to try to makeits own way.

So the electrons inside the crystal will have adifferent mass than that of the electron in vacuum.

This altered mass is called as an effective-mass.effective-mass.


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$hat is the expression for$hat is the expression for mm++

Karticles of electrons and holes ehave as a wave under certain

conditions. So one has to consider the de !roglie wavelength to link

partical ehaviour with wave ehaviour.

Kartical such as electrons and waves can e diffracted from the

crystal Lust as 0rays .

(ertain electron momentum is not allowed y the crystal lattice. This

is the origin of the energy and gaps.

sin2dn =

n the order of the diffraction

5 the wavelen*th of the 6ray

d the distance between +lanes7 the incident an*le of the 6ray beam

dn 2/ (1)


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The energy of the free e0

is related to the k

free e0 mass , m-

is the propogation constant

dn 2/

%

2/

The waves are standing waves

The momentum is

%& /

(1)

(2)

y means of e0uations '"( and'$(certain e- momenta are notallowed

by the crystal. The velocity ofthe electron at thesemomentum values is 1ero.

The energy of the free electron

can e related to its momentum

mE

&

2

2

/ h

&/

2

12 2 (2 )

22 2

2 2

2 2E

m

%h hEm m

%

=

= =

h

2

h/

momentum

k

Energy

E versus k diagram is a paraola.

Energy is continuous with k, i,e, all

energy &momentum' values are allowed.

E versus k diagram

or

Energy versus momentum diagrams


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To find effective mass , mmJJ

$e will take the derivative of energyenergywith respect to k k

2

2 2

2

2 2

28

dE %

d% m

d E

md%

m

d E d%

=

=

=

h

h

h

Chan*em'm'

instead ofmm

This formula is the effective masseffective massof

an electron inside the crystal.

0m-m- is determined y the curvature of the E0k curve

0 m-m- is inversely proportional to the curvature


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Firect and indirect0and gap materials )

Aor a direct"band gap materialdirect"band gap material, the

minimum of the conduction and and

maximum of the valance and lies at the

same momentum, k, values.

$hen an electron sitting at the ottom of

the (! recomines with a hole sitting at

the top of the ?!, there will e no change

in momentum values.

Energy is conserved y means of emittinga photon, such transitions are called as

radiative transitions.

Firect0and gaps/c9s (e.*. :a;s< =n< ;l:a;s)

e

B

CBE

k


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Aor an indirect0and gapmaterial% the

minimum of the (!and maximum of

the ?!lie at different k0values.

$hen an e0

and hole recomine in anindirect0and gap sc, phonons must

e involved to conserve momentum.

Indirect0and gaps/c9s (e.*. Si and :e)

B

CB

E

k

e

KhononKhonon

3toms virate aout their meanposition at a finite temperature.These

virations produce virational waves

inside the crystal.

Khonons are the @uanta of these

virational waves. Khonons travel with

a velocity of sound . Their wavelength is determined y the

crystal lattice constant. Khonons can

only exist inside the crystal.

Eg


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The transition that involves phonons without producing photons are

called nonradiative #radiationless& transitions.

These transitions are oserved in an indirect band gap sc and

result in inefficient photon producing.

So in order to have efficient 4EF2s and 43SE;2s, one should

choose materials having direct and gaps such as compound sc2s

of a3s, 3la3s, etcN


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or as, calculate a typical #band gap& photon energy and momentum, and

compare this with a typical phonon energy and momentum that might be expectedwith this material.

.)) (34(543TI67.)) (34(543TI67

photon phonon

E&photon' H Eg&a3s' H .O: ev

E&photon' H h ) hc / 0

cH :x-8 msec

K H h P hH9.9:x-0:O

C0sec

P &photon'H .1O, .O: H -.88 Qm

K&photon' H h P H R.D: x -018 kg0msec

E&phonon'H h ) hvs/ 0

H hvs/ a1

P &phonon' +a- H lattice constant HD.9Dx-0-m

$sH Dx-: msec & velocity of sound'

E&phonon' H hvs,a-H-.-:R e?

K&phonon'H h P H h a-H .Rx-01O kg0msec


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Khoton energy H .O: e?

Khonon energy H :R me?

Khoton momentum H R.D: x -018 kg0msec

Khonon momentum H .R x -01O kg0msec

Khotons carry large energies ut negligile amount of momentum.Khotons carry large energies ut negligile amount of momentum.

6n the other hand, phonons carry very little energy ut significant6n the other hand, phonons carry very little energy ut significant

amount of momentum.amount of momentum.


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Kositive and negative effective mass

The sign of the effective mass is determineddirectly from the sign of the curvature of the E0kcurve.

The curvature of a graph at a minimum point is apositive @uantity and the curvature of a graph at amaximum point is a negative @uantity.

Karticles&electrons' sitting near the minimum

have apositive effective mass.

Karticles&holes' sitting near the valence andmaximum have anegative effective mass.

3 negative effective mass implies that a particle

will go the wrong way! when an extrernal forceis applied.

Firect0and gaps/c9s (e.*. :a;s< =n< ;l:a;s)

e

B

CBE

k

2 2

28m

d E d% =

h


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1

2

%

2

#

1

a3s

(onduction

and

?alanceand

%

EH-.:

Eg

>111? >1%%? k

Ener

gy&e?'

1

2

%

2

#

1

Si

(onduction

and

?alance

and

%

Eg

>111? >1%%? k

Ener

gy&e?'

Energy and structures of asas and +i+i


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1

2

%

2

#

1

a3s

(onduction

and

?alanceand

%

EH-.:

Eg

>111? >1%%? k

Ener

gy&e?'

Energy and structure of asas

and *a+ is the smallest ener*y

se+aration between the valence

and conduction band ed*es.

The smallest ener*y difference

occurs at the same momentum

value

@irect band *a+ semiconductor


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1

2

%

2

#

1

Si

(onduction

and

?alanceand

%

Eg

>111? >1%%? k

Ener

gy&e?'

Energy and structure of +i+i

The smallest ener*y *a+ is

between the to+ of the 0 at A%

and one of the C minimal awayfrom A%

=ndirect band *a+ semiconductor

Band structure of ;l:a;s

B-ffective masses of C satellites

BDeavy and li*hthole masses in

0


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EEgg kk

EEEEEE

-&re.t-&re.ttran/&t&"ntran/&t&"n


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EEggkk

EEEE

-&re.t-&re.ttran/&t&"ntran/&t&"n


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EEggkk

EE


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EEggindirectindirect

transitiontransitionkk

EE


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EEggindirectindirect

transitiontransitionkk

EE

Struktur Pita Dan Sifat Listrik Bahan [2015.10]

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