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www.nanohub.org

NCN

Lecture7:EnergyBandsinRealCrystals

MuhammadAshraful Alamalam@purdue.edu

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Outline

1) Ekdiagram/constant

energy

surfaces

in

3D

solids

aracter zat ono agram: an gap

3) Characterizationof

Ekdiagram:

Effective

Mass

4) Conclusions

Reference:Vol.6,Ch.3(pages7177)

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ElectronicStates

OriginalPeriodic

Structure

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Brillouin ZoneinCubicLattice

CubicLattice Reciprocal

Lattice Brillouin Zone

a

2 aa 0

Follow W-S algorithm, but

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now or rec proca att ce

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EkalongXDirection

Ge

7

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EkDiagram

3valence

bands

(light

hole,

heavy

hole,

split

off)

valencebandsneark=0isessentiallyE~k2

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M n mamaynot eatzonecenter

(Ge:

8

L

valleys,

Si:

6

X

valleys,

and

GaAs:

valleys)

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AnalogyforEkDiagram

on ours o ens y .

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, ,

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ConstantEsurfaceforConductionBand

)( 2322

21 kkBAkEE c +++= )( 23

22

21 kkkAEE c +++=

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ConstantEsurface

2 2 2

1 2 3c= + + + 1 2 3c=

2

2

1 1 E=

ij i j

1 1 1 1= = = =

1 1 1 10

22 BA= = = =

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11 22 33( )ijm m m m i j

2 2

11 22 33)(

ijm m im m j

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FourvalleysinsideBZforGermanium

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Outline

1) Ekdiagram/constant

energy

surfaces

in

3D

solids

arac er za ono agram: an gap

3) Characterizationof

Ekdiagram:

Effective

Mass

4) Conclusions

Reference:Vol.6,Ch.3(pages7177)

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MeasurementofBandGap

E

4

bandsthat

have

filled

and

empty

states

n

bsorpti

1 E23

a

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TemperaturedependentBandGap

E

4

( ) ( )0G G E T E T

=

+

1

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Outline

1) Ekdiagram/constant

energy

surfaces

in

3D

solids

aracter zat ono agram: an gap

3) Characterizationof

Ekdiagram:

Effective

Mass

4) Conclusions

Reference:Vol.6,Ch.3(pages7177)

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MeasurementofEffectiveMass

E

4

1

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MeasurementofEffectiveMass

0=24GHz

(fixed)

Bfieldvariablekz

IoutIin

00 0

q qm

B B* = =

kxky

0vm

kz

Iout-I

in

22BBcon Bval kxky

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DerivetheCyclotronFormula 0

2

qm*

B

v

=

Foranparticlein(xy)planewithBfieldinzdirection,

2m*B B

= =

B

0

0 0

z zr

qB r =m* 0

0

2 2r m*

qB

= =

0

0

0

2 m*

qB

=

= =

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0 0m*

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Conclusions

1) Ekdiagram/constantenergysurfacesaresimplewaysto

.

arisefrom

the

solution

of

Schrodinger

equation

in

periodiclattice.

2) Ekdiagramandenergybandscontainequivalent

. ,constructtheother.

xper menta

measurementsare

ey

to

ma ng

sure

t at

thetheoreticalcalculationsarecorrect.Wewilldiscuss

theminthenextclass.

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