Fermi Surface of Metals

Dr. Michael Sutherland Cavendish Quantum Materials Group

QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page1of22


OutlineandGoals

ReviewofFermiDiracStaFsFcs,theFreeelectronGas.

Reviewofelementarybandstructuretheory,nearlyfreeelectronsmodels.

ExamplesofrealbandstructuresandrealFermisurfaces

OverviewofExperimentalTechniquesforprobingtheFermiSurface(QuantumOscillaFons,ARPES)

Detailedlistforfurtherreading


TheFermiDiracDistribuFon

ElectronsarefermionsparFcleswithhalfintegerspinthatobeythePauliExclusionPrinciple:notwofermionsmayhaveexactlythesamesetofquantumnumbers.

ForasystemofidenFcalfermions,theprobabilitythatasingleparFclestatewithenergyEisoccupiedisgivenby

inthisexpressionisthechemicalpotenFal,oRendenedastheenergywherefD(E,T)=.

isanimportantenergyscalethatismaterialdependent.AtT=0,wedenetheFermienergyEFbyEF=(T=0).

AtT=0,theFermienergyisthedividinglinebetweenlledandunlledquantumstates.

TheenergydependenceoffD(E,T)changesdramaFcallyasafuncFonofT.

When/kT>>1(lowtemperatures)thefuncFonresemblesastepfuncFoncenteredatE=EFand

When/kT


TheSommerfeldmodelofametalusesFermiDiracStaFsFcstotakeintoaccountthequantumnatureofelectrons.WeignorethedetailsoftheatomicpotenFal.

EachelectronsaFsesthefreeparFcleSchrodingerequaFonwithperiodicboundarycondiFonswithperiodL:

with

Thus,thereisonedisFncttripletofquantumnumberskx,ky,kzforthevolumeelement(2/L)3,andTWOelectronscanlleachstate(accounFngforspin).

ForelectronsinanionicsolidofaveragepotenFalV0theenergyisgivenby

(shiRzeroofenergyupbyaconstantV0)

Inatypicalmetalwehavemanyfreeelectrons.Theyoccupystateswiththelowestenergyrst,thenllprogressivelyhigherenergystates.

IfwehaveNelectrons,thevolumeofkstateslledisasphereofradiuskF:

k(r) = exp(ik r)(1)

1

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)

1

Image: Sutton, McGill Physics

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)

Vkspace =43pik3F = 2 N

(2piL

)3(3)

kF = (3pi2n)13

(4)

1

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)


(2piL

)3(3)

kF = (3pi2n)13 , n N/L3

(4)

1

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)


(2piL

)3(3)

kF = (3pi2n)13 , n N/L3

(4)

E(k) = V0 + h2k2

2me=

h2k2

2me

1

TheFreeElectronFermiGasII


TheFreeElectronFermiGasII

Thecorrespondingelectronenergywhenk=kFissimply

ThisleadstotheinterpretaFonthattheFermisurfaceisasurfaceofconstantenergyEFinkspace.

Formanymetals,theFermienergyisveryhighcomparedtothethermalenergyatroomtemperature,kBT

ItisoRenusefultodenetheFermitemperatureasTF=EF/kB.Since300K


TheNearlyFreeElectronModelI

Inrealmetals,onemusttakeintoaccounttheperiodicityofthelapce,whichmodiesthepotenFalandhencethewavefuncFonsoluFonstotheSchrodingerequaFon.

Theproblemiseasiesttotackleinreciprocalspace.ForarealspacelapcewithlapcetranslaFonvectors

Recallthatwemaydenethereciprocallapcevectorsby:

TheperiodicityofthepotenFalV(r)allowustowrite:

ThewavefuncFonsoluFonsofthefreeelectroncase(planewaves)arealsomodiedtoreecttheperiodicityofthepotenFal.

HenceBlochstheorem:

whereisafuncFonthathastheperiodicityofthepotenFalandareFouriercoecients.

WecanwritedowntheSchrodingerequaFonusingtheseresults.

ThesoluFonforthisequaFonofcoecientsgivesusthewavefuncFonsandenergystatesoftheelectronsinthepotenFal.

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)


(2piL

)3(3)

kF = (3pi2n)13 , n N/L3

(4)

E(k) = V0 + h2k2

2me=

h2k2

2me(5)

EF =h2kF 2

2me=

h2

2me(3pi2n)

23

(6)

eiGnx Gn = npi/a (7)

(8)

k(r) = uk(r)eikr(9)

1

1

k(r) = exp(ik r)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . .

Vkspace = 43pik3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k)=-V0 +h2k2

2me= h

2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23

eiGnx Gn = npi/a

k(r) = uk(r)eikr

k(x) = uk(x)eikx =Ck,nAe

i(k+Gn)x

T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. n1, n2, n3 primitive lattice vectors

G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors

1

k(r) = exp(ik r)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . .


(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k)=-V0 +h2k2

2me= h

2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23

eiGnx Gn = npi/a

k(r) = uk(r)eikr


i(k+Gn)x

T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors


1

k(r) = exp(ik r)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . .


(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k)=-V0 +h2k2

2me= h

2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23

eiGnx Gn = npi/a

k(r) = uk(r)eikr


i(k+Gn)x



1

k(r) = exp(ik r)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . .


(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k)=-V0 +h2k2

2me= h

2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23

eiGnx Gn = npi/a

k(r) = uk(r)eikr


i(k+Gn)x



1

k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik

3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23

eiGnx Gn = npi/ak(r) = uk(r)eikr

k(x) = uk(x)eikx =

Ck,nAe

i(k+Gn)x



V (r) =G

VGeiGr

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0


TheNearlyFreeElectronModelII

ConsidertheperiodicpotenFalin1DforillustraFon

Forthesestatestheenergybecomes:

TheeectisthatanenergygapopensupneartheBZedge.Therearenoallowedstatesinthatgap.

TheposiFonofEFwithrespecttogapdetermineswhetherasystemisaninsulator,semiconductor,ormetal.

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)


(2piL

)3(3)

kF = (3pi2n)13 , n N/L3

(4)

E(k) = V0 + h2k2

2me=

h2k2

2me(5)

EF =h2kF 2

2me=

h2

2me(3pi2n)

23

(6)


(8)

k(r) = uk(r)eikr(9)

1

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

Image: Sutton, McGill Physics

John Ellis Cambridge McGill Physics

Foragivenstatewithwavevectork,onlystateswithk=kG,k2GwillcontributeFouriercomponentstotheoverallwavefuncFon.

Forastatewithsmallk,thestateswithk=kG,k2Gareallmuchhigherinenergy.ForasmallpotenFalthesestatesdonotmixstrongly,andhencethedispersionrelaFonresemblesthefreeelectroncase.

ForkstatesneartheBZboundary(/ain1D)thestatewithk=k/aisverycloseinenergyandhenceaectsthedispersionrelaFon(degenerateperturbaFontheory)

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12


TheNearlyFreeElectronModelIII

OnecanunderstandthisgapasarisingfromcontribuFonstothewavefuncFonfromtwostandingwaveswithk - /a.OnehasamaximumchargedensityneartheBZedge(higherenergy)andonehasaminimumchargedensitythere(lowerenergy)

Image: John Ellis, Cambridge

HigherorderharmonicsofthepotenFalVGwilllinkstatesathigherenergiesandgivegapsathighBZedges.

SinceanykvectorinahigherBZmaybemappedbyareciprocallapcevectorGintotherstBZ,itisoRenconvenienttofoldtheenergybandbackintothe1stBZ.Thisisknownasthereducedzonescheme.

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

TheapplicaFonofaforce(sayelectromoFve)changesthewavepacketenergybyanamount

ForanelectriceldofstrengthE,wehave


Wavepackets:semiclassicalmodel

AknowledgeofthebandstructureofamaterialgivesadeepinsightintotheelectronicproperFes.ThedispersionrelaFongivesknowledgeoftheeecFvemassandgroupvelocity:

WheretheeecFvemasstensorisdenedby

InpracFcethismeansthatthevelocityisinverselyproporFonaltotheslopeofthebandwhenitcrossestheFermilevel,andthemassisinverselyproporFonaltothesecondderivateof.

Thusat,shallow,bandstendtocontainslowmoving,heavyelectrons.

Considerthegroupvelocityvgofawavepacketofelectrons,madeupasasuperposiFonofwavefuncFons. Image: John Ellis, Cambridge

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

d#

dt= F vg = dk

dt$k#(k) = hdk

dt

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

d#

dt= F vg = dk

dt$k#(k) = hdk

dt

#(k) = #(k0) +1

2(h(k k0))m1(h(k k0))

mij1 =

1

h2$ki $kj #(k)

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

d#

dt= F vg = dk

dt$k#(k) = hdk

dt

#(k) = #(k0) +1

2(h(k k0))m1(h(k k0))

mij1 =

1

h2$ki $kj #(k)

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

d#

dt= F vg = dk

dt$k#(k) = hdk

dt

#(k) = #(k0) +1

2(h(k k0))m1(h(k k0))

mij1 =

1

h2$ki $kj #(k)

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

d#

dt= F vg = dk

dt$k#(k) = hdk

dt

#(k) = #(k0) +1

2(h(k k0))m1(h(k k0))

mij1 =

1

h2$ki $kj #(k)

1


3F = 2 N

(2piL

)3kF = (3pi2n)

13 , n N/L3

E(k) = V0 + h2k22me = h2k2

2me

EF =h2kF 2

2me= h

2

2me(3pi2n)

23


k(r) = uk(r)eikr =

k

Ckeikr



V (r) =G

VGeiGr

(h2k2

2m E

)Ck +

G

VGCkG = 0

#k =12

(Ek + Ekpi/a

)[(

EkEkpi/a2

)2+ |Vpi/a|2

] 12

vg =1

h$k#(k)

d#

dt= F vg = dk

dt$k#(k) = hdk

dt

#(k) = #(k0) +1

2(h(k k0))m1(h(k k0))

mij1 =

1

h2$ki $kj #(k)

3

fD(E, T ) =1

e(E)/kBT + 1dk

dt= 1

heE

TheenFreFermisphereisshiRedbyauniformamount(dependsonscaveringrate).

Fromknowledgeofthebandstructure,itispossibletoesFmateconducFvity.


Bandstructure2Ddivalentmetal

Considera2Ddivalentmetal.Thereare2electronsperunitcell,soweshouldexpectafullband,andhenceaninsulatoreg.diamond.

Howeversomedivalentmetals(Ca)aremetallic.Whyisthis?Metalsneednearbyemptystatesinordertopropagatecharge.

Considerthecaseofa2DsquarelapcepotenFal.ThefreeelectronFS(acircle)hasthesameareaastherstBZ,andhencespillsovertothesecondBZ.

Images: Singleton, Band theory of solids

HoweverturningonapotenFalraisestheenergyofthestatesneartheedgesoftheBZandlowersthemnearthecorners.HenceelectronsaretransferredfromthesecondtorstBZ,resulFnginametal.

Inthereducedzonescheme,thisresultsinsmallFSpockets,bothholeandelectronlike1

stBZ2ndBZ Freeelectron

circle

1stBZ 2ndBZ

1stBZ 2ndBZPotenFalturnedon

Electrons

Holes


Copperisa3DmonovalentmetalwithconguraFon[Ar]3d104s1.[Ar]3d10electronsgiverisetoFghtlyboundbands,farbelowEF.One4selectrononeFSsheet.

fcccrystallapce(bccreciprocallapce).TherstBrillouinzone(BZ)isatruncatedpolyhedra

TheshortestdistancetotheBZedgeisalongthedirecFon.UsingthelapceparametersforCu,inthefreeelectronapproximaFonkF/k=0.903(shouldnottouchzoneedge)

HowevertheexperimentallydeterminedFStouchestheBZedgealongthedirecFon.ThisarisesfrommixingofstatesofsimilarenergiesneartheBZedge.

Images: S. Blundell Oxford

k(r) = exp(ik r)(1)

kx, ky, kz = 0; 2pi/L; 4pi/L; . . . (2)


(2piL

)3(3)

kF = (3pi2n)13 , n N/L3

(4)

E(k) = V0 + h2k2

2me=

h2k2

2me(5)

EF =h2kF 2

2me=

h2

2me(3pi2n)

23

(6)


(8)

k(r) = uk(r)eikr(9)

1

BandstructureCopper


Formanysimplediandtrivalentmetals,theFSmaybeschemaFcallyconstructedbyfollowingthesesteps:

1. Drawafreeelectronspherebasedonthenumberofvalenceelectrons

2. SuperimposethisontotherstfewBrillouinzones.

3. WheretheFSmeetstheBZboundarywherethebandgapopensup,splitandroundothesurface.

4. TranslatetheresulFngsecFonsbackintotherstBZ,usingtheperiodicityofkspace

Image: University of Florida

GeneralrulesforconstrucFngFermisurfaces

Evenusingtheserules,simplemetalsmayhaveexceedinglycomplexFermisurfaces.Thesecanbediculttocalculate,especiallyforthosematerialswithdandfbandswhichtendtobeveryat.

Eg.Rhenium,hexagonalcrystalstructurewith6(!)bandscrossingtheFermilevel.

AnexcellentwebresourceistheperiodictableofFermisurfacesfoundontheUniveristyofFloridaswebsite:

www.phys.u.edu/fermisurface/


FermiSurfaceofSr2RuO4

ItissomeFmesusefultothinkofrealspaceorbitalswhenconstrucFngFermisurfaces.ThisapproachiscalledtheFghtbindingapproximaFon

Sr2RuO4isalayeredruthenatematerialthatisalsoanunconvenFonalsuperconductoratlowtemperature.StronglyanisotropicconducFon(AB>>Chencequasi2Dmetal)

Tetragonalcrystalstructure.Thedyz(dxz)statehasweakx(y)dependence.WethusexpecttheFSformedbythesestatestobeaatsheetperpendiculartothex(y)direcFon.Wheretheyintersect,theypinchotoformcylinderscenteredaboutthecenterandcornersoftheBZ.

ThedxystatehasweakzdependenceandformsacylinderinthecenteroftheBZ.

Image: C. Bergemann, Cambridge

4delectronshybridizetoformthreelowenergystatesdxydyzdxzwhicharepopulatedby4electrons.

Inreciprocalspace,theelectroncantakeanyvalueofkz,butthevalueofkxandkyarequanFzedsuchthat:

with:

TheseorbitsformLandautubesinkspace


Inverycleanmetals,inlargemagneFceldsandatlowtemperatures,anoscillatorycomponentisobservedintransport,thermodynamicandmagneFcmeasurements.

Thefrequency,angularandtemperaturedependenceofthesignalcanbeusedtocharacterizetheFermisurface.

ConsideraeldalongthezdirecFon.TheelectronexperiencesaLorentzforce:

IfthescaveringFmeislarge,theelectroncanundergomanycyclotronorbitsorfrequency

Image: I. Shiekin, Grenoble

ExperimentalprobesoftheFermisurfaceQuantumOscillaFons

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

Allowedkspaceorbits(Landautubes)

H

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

InrealexperimentsseveraldierentperiodiciFesareoRenobserved.HowdowemakequanFtaFvesenseofthem?


Theresultisthatthedensityofelectronstatesg(E)acquiresastronglypeakedenergydependence(eachpeakcorrespondingtoaLandautube):

IfthemagneFceldisvaried,thepeaksshiRinenergy.IfapeakcrossestheFermienergy,thestateispopulated,thenempFesasitmovesaway.

ThisleadstooscillaFonsthatareperiodicininversemagneFc,1/B.

ExperimentalprobesoftheFermisurfaceQuantumOscillaFonsII

B1(Tesla1)Fouriertransformofdata(twofrequencies)

SrFe2As2(amagneFcmetal)

MagneFzaFon

InrealexperimentsseveraldierentperiodiciFesareoRenobserved.HowdowemakequanFtaFvesenseofthem?

BohrSommerfeldquanFzaFoncondiFon(seeKivel):

wherepisthecanonicalmomentum:

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr = (n+ 1

2)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =h

eB

(n+

1

2

)

TheLorenzforcetellsushowtheareasoftherealspaceandreciprocalspaceorbitsarerelated

Wecancombingthesetoshowthat

SonallywecanobtainarelaFonbetweenthefrequencyoftheoscillaFonsandtheareaofthereciprocalspaceorbits

ThisisknownastheOnsagerrelaFonandshowshowoscillaFonstudiesmayactasacaliperoftheFermisurface.

ThefrequencyisproporFonaltotheextremalcrosssecFonalareaperpendiculartotheappliedeld.


Applyingthisyields

UsingStokestheoremonthesecondterm:

whereisthemagneFcux.Usingq=efortheelectronandcollecFngtermsgives:

ShowingthatthemagneFcuxisquanFzed.

Sincetheuxisjusttheeldthreadingtherealspaceorbit,ofareaArealwecanwrite:

ExperimentalprobesoftheFermisurfaceQuantumOscillaFonsIII

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr = (n+ 1

2)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr = (n+ 1

2)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =h

eB

(n+

1

2

)

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip


OrbitsmaybeobservedenclosinglledorunlledporFonsoftheFermisurface.

ExperimentalprobesoftheFermisurfaceQuantumOscillaFonsIV

ThetemperaturedependenceoftheamplitudeisdeterminedbyconsideringthetemperaturedependenceoftheFermiDiracfuncFon.

TheanalyFcexpressionis:


OscillaFonsmaybeobservedinmanyphysicalquanFFes,forinstanceresisFvity(theShubnikovdeHaaseect)orinmagneFzaFon(thedeHaasvanAlpheneect).

AtypicalmeasurementofthedHvAeectinvolvesmeasurementofthedierenFalsuscepFlbiltywithcounterwoundpickupcoils,suchthatthepickupvoltageis

InaddiFontothefrequency,theamplitudeoftheoscillaFonsmaytellusagreatdeal.

WhereisthescaveringrateandCisthecyclotronfrequency.TheelddependencecangiveinformaFononscavering.

ExperimentalprobesoftheFermisurfaceQuantumOscillaFonsV

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip

Ef =h2k2f2m

= Ei + h kf = ki

V = dM

dB

dB

dt

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip

Ef =h2k2f2m

= Ei + h kf = ki

V = dM

dB

dB

dt

Amp. B1e pic

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip

Ef =h2k2f2m

= Ei + h kf = ki

V = dM

dB

dB

dt

Amp. B1e picAmp.

sinh()

= 14.7m!T/B

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip

Ef =h2k2f2m

= Ei + h kf = ki

V = dM

dB

dB

dt

Amp. B1e picAmp.

sinh()

= 14.7m!T/B

Image: C. Bergemann, Cambridge

Twofrequencies,correspondingtoneckandbellyorbits(extremalorbits).

AngulardependencegivessuccessivecrosssecFonalcuts,perpendiculartoappliedeld.

Weaktemperaturedependence(lightmasses).


ExampleofaquantumoscillaFonstudy:Ag5Pb2O6nearlyfreeelectronsuperconductor

Neckorbit(lowfrequency)

Bellyorbit(highfrequency)

1stBrillouinzone

UseconservaFonofmomentumandenergytoworkouttheiniFalstateofthetheelectron

Momentumparalleltothesurfaceisconserved


OtherProbesoftheFermiSurface:ARPES

IniFalElectronstate

Finalmeasuredenergyandmomentum

AngularResolvedPhotoEmissionSpectroscopyispowerfultechniquethatusesthephotoelectriceecttoprobetheelectronicstatesnearthesurfaceofasample.

AnincomingphotonofenergyejectselectronofmomentumkF

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip

Ef =h2k2f2m

= Ei + h kf = ki

2

E =h2k2Z2m

+ (!+ 1/2)hc

c =eB

m

F = eB v

p dr =

(n+

1

2

)h

p = mv +q

cA

p dr = hk dr+ q

c

A dr

A dr ="A d =

n =(n+

1

2

)hc

e

Areal =hc

eB

(n+

1

2

)

Arecip =(e

h

)2B2Areal

1

Bn=

2pie

hcArecip

(n+

1

2

)

1

B=

1

Bn+1 1Bn

=2pie

h

1

Arecip

Ef =h2k2f2m

= Ei + h kf = ki


OtherProbesoftheFermiSurface:ARPESII

Dierentkstates(detectorangles)

FermilevelsetbycalibraFngtoastandard

Occupiedstates

Example:Sr2RuO4(again!)

Images: A. Damascelli, UBC


EssenFalFurtherReading

Thepreceedingnotesgiveaverybriefoverviewofelectronicbandstructure,Fermisurfaces,andtheexperimentalprobesusedtomeasurethem.Youshouldreadwidelyonthesubjectandbecomfortablewiththemainideas.

FreeElectronModel:basicconceptsinIntroducFontoSolidStatePhysics9thedKivel,Chapter6.

NearlyFreeelectronModel:BandTheoryandElectronicProperFesofSolidsJ.Singleton,OxfordMasterSeriesinCondensedMaverPhysics,Chapter3(2001).

QuantumoscillaFonsandthedeHaasvanAlphenEect:J.Singleton,chapter8(goodintroducFon)

MagneFcOscilaFonsinMetalsD.Shoenberg,CambridgeUniversityPress,(1984).

ARPEStechniquesProbingtheFermiSurfaceofCorrelatedElectronSystemsA.DamascelliPhysicaScripta.Vol.T109,6174,2004

Fermi Surface of Metals

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