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Energy Bands in SolidsEnergy Bands in Solids

Physics 355Physics 355

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Consider the available energies for electrons in the materials.

As two atoms are brought

close together, electrons

must occuy different

energies due to Pauli

E!clusion rincile.

"nstead of having discrete

energies as in the case of

free atoms, the available

energy states form bands.

Conductors, "nsulators, and Semiconductors

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#ree Electron #ermi $as#ree Electron #ermi $as

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==

m

kE

2

22

#or free electrons, the

wavefunctions are lane

waves%

)rk()r(

ie

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Band $a

&one boundary

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'doed('thermally

e!cited(

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)

*rigin of the Band $a

+o get a standing wave at the boundaries, you cantae a linear combination of two lane waves%

axiikx

axiikx

ee

ee

/2

/1

=

=

=+=+= +a

xee axiaxi

cos2//21

/ /

1 2 2 sini x a i x a x

e e ia

= = =

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*rigin of the Band $a

Electron -ensity

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*rigin of the Band $a

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Bloch #unctions

#eli! Bloch showed that the actualsolutions to the Schrdinger e/uation for

electrons in a eriodic otential must have

the secial form%

where uhas the eriod of the lattice, that

is

)rk(kk )r()r( = ieu

)Tr()r( kk +=uu

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0ronig1Penney 2odel

a+b4 b a a+b

U!4

x

U0

+he wave e/uation can be solved when the otential is

simle... such as a eriodic s/uare well.

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0ronig1Penney 2odel

=+ )(2 2

22xU

dx

d

m

Region I - where 0 < x< aand U= 0

The eigenfunction is a linear comination of !lane wa"estra"eling oth left and right#

The energ$ eigne"alue is#

iKxiKx BeAe +=

m

K

2

22

=

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Region II - where b< x< 0 and U= U0%ithin the arrier& the eigenfunction looks like this

and

QxQx DeCe +=

m

QU

2

22

0

=

0ronig1Penney 2odel

=+ )(2 2

22xU

dx

d

m

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0ronig1Penney 2odel

a+b4 b a a+b

U!4

x

U0

+o satisfy 2r. Bloch, the solution in region """

must also be related to the solution in region "".

III III

)( baik

IIIII e

+

=

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0ronig1Penney 2odel

A,B,C, and - are chosen so that both the wavefunctionand its derivative with resect toxare continuous at

thex6 and a.

Atx6 ...

Atx = a...

)()( DCQBAiKDCBA = +=+

)( baikIIIII e

+=

)()( baikQbQbiKaiKa eDeCeBeAe + +=+

( ) ( ) )( baikQbQbiKaiKa eDeCeQBeAeiK + =

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0ronig1Penney 2odel

)(coscoscoshsinsinh2

22

bakKaQbKaQbQK

KQ+=+

+7esult for E8 U%

+o obtain a more convenient form 0ronig and Penney considered the

case where the otential barrier becomes a delta function, that is, the

case where Uis infinitely large, over an infinitesimal distance b, but the

roduct U0bremains finite and constant.

02 UQ

and also goes to infinity as U. +herefore%

222

0

)('im QKQ

U

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0ronig1Penney 2odel9hat haens to the roduct Qbas Ugoes to infinity:

; bbecomes infinitesimal as Ubecomes infinite.

; >

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0ronig1Penney 2odel

=2

2baQP kaKaKaKaP coscossin =+

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0ronig1Penney 2odel

= 3 ka ka

Plot of energy versus

wavenumber for the

Kronig-Penney Potential,

with P = !".

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Crucial to the

conduction rocess

is whether or notthere are electrons

available for

conduction.

Conductors, "nsulators, and Semiconductors

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Conductors, "nsulators, and Semiconductors

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Conductors, "nsulators, and Semiconductors

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Conductors, "nsulators, and Semiconductors

'doed('thermally

e!cited(