254 CHAPTER16. LIVEWIRESANDDEADSTARS
16.1 The Kronig-Penny Model
A crystalline solid isa regular array of atoms, and, at first sight, conductionof
electricityisamystery: ifelectronsareboundtoatoms,howispossibleforthemto
movethroughthesolidundertheinfluenceofasmallelectricfield? Theansweris
thatinacrystal, notallof theelectronsareactuallyboundtotheatoms;infact,
someoftheelectronsinthemetalbehavemorelikeagasoffreeparticles,albeitwith
somepeculiarcharacteristicswhichareduetotheexclusionprinciple.
Tounderstandhowelectronsinacrystalcanactasagas,itisusefultosolvefor
theelectronenergyeigenstatesinahighlyidealizedmodelofasolid,knownasthe
Kronig-Pennymodel,whichmakesthefollowingsimplifications:
S1.Thesolidisone-dimensional,ratherthanthree-dimensional. TheNatomsare
spacedadistanceafromoneanother.Inorderthattherearenospecialeffects
attheboundaries,weconsiderasolidhasnoboundaryatall,byarrangingthe
atomsinacircleasshowninFig.[16.1].
S2. InsteadofaCoulombpotential,thepotentialofthen-th atomisrepresented
byadelta-functionattractivepotentialwell
Vn(x)=−gδ(x−xn) (16.2)
wherexnisthepositionofthen-thatom.
S3.Interactionsbetweenelectronsinthe1-dimensionalsolidareignored.
Obviously, these areprettydrastic simplifications. Theimportantfeature of this
model, whichitshares withrealisticsolids, isthat the electronsaremoving ina
periodicpotential. Forpurposesof understandingtheexistenceofconductivity, it
istheperiodicityofthepotential,notitspreciseshape(orevenitsdimensionality)
whichisthecrucialfeature.
Arrangingtheatomsinacircle,asinFig.[16.1],meansthatthepositionvariable
isperiodic,likeanangle.Justasθ+ 2 πisthesameangleasθ,sothepositionx+L
isthesamepositionasx,where
L=Na (16.3)
isthelengthofthesolid. Letthepositionof then-thparticlebexn =na, n=
0 , 1 ,...,N−1,thepotentialthenhastheform
V(x)=−g
N∑− 1
n=0
δ(x−na) (16.4)
Itsclearthatthepotentialsatisfies
V(x+a)=V(x) (16.5)