nearly free electron model and Bloch theorem.pdf

Fermi energy and fermi surface both are related to describe crystal structure

1. SolidStatePhysicsNEARLYFREEELECTRON
MODEL(Contd)
Lecture19A.H.Harker
PhysicsandAstronomy
UCL

2. 7.3 Anexactly-solublemodel
Weknowfromsecond-yearquantummechanicsthatsquarewellpo-
tentials are quite easy to deal with. The Kronig-Penney model is
basedonthis.
For details of the calculation, see for example KittelIntroduction to
SolidStatePhysics.
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3. Wecanseenthegapsintheenergyspectrum–regionsofenergyin
whichtherearenoallowedstates.
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4. Thefreeelectronapproximationremainsagoodapproximationwell
awayfromtheedgesoftheBrillouinzone–onlywave-vectorscloseto
amultipleof=aaremixedtogetherandhavetheirenergiesaltered
bytheperiodicpotential.
Translationalsymmetryisnotessentialforproducingabandgap–
amorphoussolidsalsohavebandgaps.
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5. 7.4 Sketchingenergybands
7.4.1 Theemptylattice
Imaginerstthattheperiodiccrystalpotentialisvanishinglysmall.
Then we want to impose periodic structure without distorting the
freeelectrondispersioncurves.Wenowhave
E(k)=E(k+G);
whereGisareciprocallatticevector.
We can use theextended zone scheme(left) or displace all the seg-
mentsofthedispersioncurvebackintotherstBrillouinzone(right).
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6. 7.4.2 Thenearlyfreeelectron
Modify the free electron picture by opening up small gaps near the
zoneboundaries.
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7. 7.5 Consequencesoftheenergygap
7.5.1 Densityofstates
The number of allowedkvalues in a Brillouin zone is equal to the
number of unit cells in the crystal.Proof:in one dimension, with
periodicboundaryconditions,
g(k)=
L
2
;
whereListhelengthofthecrystal,sothenumberofstatesinaBril-
louinzoneis
N=
Z
=a
=a
g(k)dk=
L
2
Z
=a
=a
dk=
L
a
;
butawas the size of the real space unit cell, soNis the number of
unit cells in the crystal. The same argument holds in two or three
dimensions. Note that we get the number of unit cells – only for a
monatomicunitcellisthisthesameasthenumberofatoms.
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8. So, taking spin degeneracy into account, a Brillouin zone contains
2Nallowedelectronstates.
7.5.2 Statesinonedimension
In the insulator, there is an energy gap between the occupied and
unoccupiedstates. Forametal,theremaybeoverlap(b)ornot(c).
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9. Draw the dispersion relation with slight distortions near the zone
boundary
ineithertheextendedzonescheme(left)orthereducedzonescheme
(right). Notethatstatesfurtherfromtheoriginintheextendedzone
schemecanalsoberepresentedashigherbandsinthereducedzone
scheme.
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10. Forfreeelectrons,theconstantenergysurfacesarecircular.
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11. For a monovalent element, the volume of the Fermi surface is half
thatoftheBrillouinzone:
sothatitisfreetobedisplacedbyanelectriceld–afree-electron-
likemetallicsystem.
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12. With the crystal potential, the energy inside therst Brillouin zone
islowerclosetothezoneboundary. SotheFermisurfaceisextended
towardsthezoneboundaryasitgetsclose.
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13. DependingonthedirectionintheBrillouinzone,wemaygotolarger
k(largerE)beforethestatesareperturbed. 13

14. Consider a divalent metal in two dimensions. The area ofk-space
needed to accommodate all the electrons is equal to the area of the
rstBrillouinzone.Wecanseethattheredstatesinthesecondband
willstarttobelled.
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15. Ifthegapissmall,thelledstateswillbeinboththerstandsecond
zones.
Thiswillbeametal.
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16. Takealargerenergygap
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17. Superposethecurves
Foralargegap, thewholeoftherstzonewillbelled. Thisgives
an insulator because if we apply a eld to increase an electron'sk
vector, electrons at the zone boundary will be Bragg reected back
totheothersideofthezone–therewillbenonettdriftvelocity. We
onlygetcurrentifwecanexcitesomeelectronsintoahigherenergy
band.
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18. Aninsulator.
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19. Bragg reection is a natural consequence of the periodic nature of
theenergyink-space,andthefactthat
group velocity=
d!
dk
=
1
~
dE
dk
:
Oncrossingthezoneboundary,thephasevelocitychangesdirection:
theelectronisreected.
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20. 7.5.3 SketchinganearlyfreeelectronFermisurface
Startwiththesphere,anddistortitneartheedgesofthezone.
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21. 7.5.4 TypicalFermisurfacesin3D
TheBrillouinzoneistakenasthereciprocalspaceWigner-Seitzcell.
FCClattice,BCCreciprocallattice
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22. BCClattice,FCCreciprocallattice
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23. Thealkalimetalsareonlyslightlydistortedfromspheres.
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24. Thenoblemetalsareconnectedink-space.
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25. Forpolyvalentmaterials,theFermisurfacesgetmorecomplicated.
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26. 7.5.5 Effectsofeldsonelectronsinbands
Electriceld: asimplepicturewillshowhowtheFermisurfaceina
partly-lledzonewillbeshifted:
Anettcurrentow(reallyarisingfromhvi,nothki,soaconductor.
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27. Magneticeld:
~
dk
dt
=evB:
ThechangeinkisperpendiculartobothvandB–theelectronstays
ontheconstantenergysurface.
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28. Nearthetopofaband:
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29. TheelectronsareBraggreectedattheedgesoftheBrillouinzone.
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30. Theelectronsorbit,ink-space,theoppositewayroundoccupiedor
unoccupied states. The behaviour looks like that of an oppositely
chargedparticle–ahole.
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31. Whatabouttheelectronsinthesecondzoneinametallicsystem?
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32. Redrawing,usingtheperiodicnatureofthesystem:
Thisiselectron-likebehaviour.
Therecanbeabalancebetweenelectron-likeandhole-likebehaviour
–hencethestrangeHallcoefcientsofthepolyvalentmetals.
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