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www.nanohub.org

NCN

Lecture6:EnergyBands(continued)

MuhammadAshraful Alamalam@purdue.edu

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Outline

1) Propertiesofelectronicbands

2) Ekdiagramandconstantenergysurfaces

3) Conclusions

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. , .

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ElectronandHolefluxes:Filled/EmptyBands

E

30

i

qJ = =

( )i filled

0maxk

( )2 i

i filled

JL

= 0

0

min

i i

kL L

= =

Filled and empty bandscarr no current !

F

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ElectronandHoleFluxes:PartiallyFilledBands

E

E

( )3 i

i filledL=

( )

i i

all i empty

q q

L L = +

( )

2 i

i filled

qJ

L=

3( )

i

i empty

q

L=

ve chargemovingwithve mass2

Alam ECE606S09 4k

+ve c argemov ngw t +ve mass

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what good is effective mass ?

*E

k

2

2 2m* dk

=

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k-k

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Outline

ropert eso e ectron c an s

2) Ekdiagramandconstantenergysurfaces

3) Conclusions

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SolutionSpace:Brillouin Zone

E

p

xReal

kKlattice2

p

4

p

2

p

4

p

k

7

pp

p

p

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GeneralrulesforBrillouin Zone

1)

Define

reciprocal

lattice

with

the

following

vectors

.

2 2 2 x y z

c c a ak k k

a b c a b c a b c

= = =

i i i

2) UseWignerSeitzalgorithmtofindtheunitcell

.

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WignerSeitzMethodforReciprocalSpace

Primitivecellinrealspace Unitcellinreciprocallattice

b ky

a kx

b z z a

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x y

a b z a b z = =

i i

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Brillouin Zone forOnedimensionalSolids

Reals ace

p

Re lacin

st

(a+b)by

p

p

p

kx2

p

2

p

Ekdiagram

kx

pp 0

10

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Ekdiagramin2Dsolids

a

ERealspace

1st BZ k

b

ky

a

a

b

xky

Ekdiagram

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b

x

0

y

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ConstantEnergysurfacein2D

ky E

1st BZ

a

a

kx

b

E1

Ekdiagram

kxky

xy

1

ab

kyk1

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.

Surface kx

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Conclusion

1) Electronscanonlysitinspecificenergybands.Effectivemassesand

.

2) Effectivemassisnotafundamentalconcept.Therearesystemsfor

.

3) Of allthepossiblebands,onlyafewcontributetoconduction.Theseare

oftencalledconductionandvalencebands.

4) For2D/3Dsystems,energybandsareoftendifficulttovisualize.Ek

diagramsalongspecificdirectionandconstantenergysurfacesfor

specificbands

summarize

such

information.

5) Mostofthepracticalproblemscanonlybeanalyzedbynumerical

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so ut on.