QMGreensite_merged

16.1. THEKRONIG-PENNYMODEL 257

ButaccordingtoBloch’stheorem,intheregion−a<x< 0 (regionII),

ψII(x) = e−iKaψI(x+a)

= e−iKa[Asink(x+a)+Bcosk(x+a)] (16.21)

Nowweapplycontinuityofthewavefunctionatx= 0 toget

B=e−iKa[Asin(ka)+Bcos(kb)] (16.22)

Foradeltafunctionpotential,wefoundlastsemesteradiscontinuityintheslopeof
thewavefunctionatthelocation(x=0)ofthedeltafunctionspike,namely

(

dψ

dx

)

|!

−

(

dψ

dx

)

|−!

=−

2 mg

h ̄^2

ψ(0) (16.23)

andthisconditiongivesus

kA−e−iKak[Acos(ka)−Bsin(ka)]=−

2 mg

h ̄^2

B (16.24)

Solvingeq. (16.22)forA,

A=

eiKa−cos(ka)

sin(ka)

B (16.25)

insertingintoeq. (16.24)andcancellingBonbothsidesoftheequationleadsfinally,
afterafewmanipulations,to

cos(Ka)=cos(ka)−

mg

̄h^2 k

sin(ka) (16.26)

Thisequationdeterminesthepossiblevaluesofk, andthereby, viaE= ̄h^2 k^2 / 2 m,
thepossibleenergyeigenvaluesofanelectroninaperiodicpotential.
Now comes the interesting point. The parameter K can take on the values
2 πn/Na,andcos(Ka)variesfromcos(Ka)=+1(n=0)downtocos(Ka)= − 1
(n=N/2),andbackuptocos(Ka)≈+1(n=N−1). So thelefthandside is
alwaysintherange[− 1 ,1]. Ontheotherhand,therighthandsideisnotalwaysin
thisrange,andthatmeanstherearegapsintheallowedenergiesofanelectronina
periodicpotential. ThisisshowninFig. [16.2],wheretherighthandsideof(16.26)
isplotted. Valuesof kforwhichthecurve isoutside therange[− 1 ,1]correspond
toregionsofforbiddenenergies,knownasenergygaps,whilethevalueswherethe
curveisinside the[− 1 ,1] rangecorrespondtoallowedenergies, knownas energy
bands. Thestructure of bandsandgapsis indicatedinFig. [16.3]; eachof the
closelyspacedhorizontallinesisassociatedwithadefinitevalueofK.
InthecasethatwehaveM>N non-interactingelectrons,eachoftheelectrons
mustbeinanenergystatecorrespondingtoalineinoneoftheallowedenergybands.
Thelowestenergystatewouldnaivelybethatof allelectronsinthelowestenergy