Download - Fermi Surface of Metals
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Dr. Michael Sutherland Cavendish Quantum Materials Group
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page1of22
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OutlineandGoals
ReviewofFermiDiracStaFsFcs,theFreeelectronGas.
Reviewofelementarybandstructuretheory,nearlyfreeelectronsmodels.
ExamplesofrealbandstructuresandrealFermisurfaces
OverviewofExperimentalTechniquesforprobingtheFermiSurface(QuantumOscillaFons,ARPES)
Detailedlistforfurtherreading
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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
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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)
Vkspace =43pik3F = 2 N
(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)
Vkspace =43pik3F = 2 N
(2piL
)3(3)
kF = (3pi2n)13 , n N/L3
(4)
E(k) = V0 + h2k2
2me=
h2k2
2me
1
TheFreeElectronFermiGasII
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TheFreeElectronFermiGasII
Thecorrespondingelectronenergywhenk=kFissimply
ThisleadstotheinterpretaFonthattheFermisurfaceisasurfaceofconstantenergyEFinkspace.
Formanymetals,theFermienergyisveryhighcomparedtothethermalenergyatroomtemperature,kBT
ItisoRenusefultodenetheFermitemperatureasTF=EF/kB.Since300K
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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)
Vkspace =43pik3F = 2 N
(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; . . .
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. a1, a2, a3 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; . . .
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. a1, a2, a3 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; . . .
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. a1, a2, a3 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; . . .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
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
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TheNearlyFreeElectronModelII
ConsidertheperiodicpotenFalin1DforillustraFon
Forthesestatestheenergybecomes:
TheeectisthatanenergygapopensupneartheBZedge.Therearenoallowedstatesinthatgap.
TheposiFonofEFwithrespecttogapdetermineswhetherasystemisaninsulator,semiconductor,ormetal.
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 , 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 = 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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
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
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
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QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page8of22
TheNearlyFreeElectronModelIII
OnecanunderstandthisgapasarisingfromcontribuFonstothewavefuncFonfromtwostandingwaveswithk - /a.OnehasamaximumchargedensityneartheBZedge(higherenergy)andonehasaminimumchargedensitythere(lowerenergy)
Image: John Ellis, Cambridge
HigherorderharmonicsofthepotenFalVGwilllinkstatesathigherenergiesandgivegapsathighBZedges.
SinceanykvectorinahigherBZmaybemappedbyareciprocallapcevectorGintotherstBZ,itisoRenconvenienttofoldtheenergybandbackintothe1stBZ.Thisisknownasthereducedzonescheme.
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
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
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page9of22
Wavepackets:semiclassicalmodel
AknowledgeofthebandstructureofamaterialgivesadeepinsightintotheelectronicproperFes.ThedispersionrelaFongivesknowledgeoftheeecFvemassandgroupvelocity:
WheretheeecFvemasstensorisdenedby
InpracFcethismeansthatthevelocityisinverselyproporFonaltotheslopeofthebandwhenitcrossestheFermilevel,andthemassisinverselyproporFonaltothesecondderivateof.
Thusat,shallow,bandstendtocontainslowmoving,heavyelectrons.
Considerthegroupvelocityvgofawavepacketofelectrons,madeupasasuperposiFonofwavefuncFons. Image: John Ellis, Cambridge
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
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(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
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$k#(k)
1
k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik
3F = 2 N
(2piL
)3kF = (3pi2n)
13 , n N/L3
E(k) = V0 + h2k22me = h2k2
2me
EF =h2kF 2
2me= h
2
2me(3pi2n)
23
eiGnx Gn = npi/ak(r) = uk(r)eikr
k(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
vg =1
h$k#(k)
d#
dt= F vg = dk
dt$k#(k) = hdk
dt
1
k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik
3F = 2 N
(2piL
)3kF = (3pi2n)
13 , n N/L3
E(k) = V0 + h2k22me = h2k2
2me
EF =h2kF 2
2me= h
2
2me(3pi2n)
23
eiGnx Gn = npi/ak(r) = uk(r)eikr
k(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
vg =1
h$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 $kj #(k)
1
k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik
3F = 2 N
(2piL
)3kF = (3pi2n)
13 , n N/L3
E(k) = V0 + h2k22me = h2k2
2me
EF =h2kF 2
2me= h
2
2me(3pi2n)
23
eiGnx Gn = npi/ak(r) = uk(r)eikr
k(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
vg =1
h$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 $kj #(k)
1
k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik
3F = 2 N
(2piL
)3kF = (3pi2n)
13 , n N/L3
E(k) = V0 + h2k22me = h2k2
2me
EF =h2kF 2
2me= h
2
2me(3pi2n)
23
eiGnx Gn = npi/ak(r) = uk(r)eikr
k(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
vg =1
h$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 $kj #(k)
1
k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik
3F = 2 N
(2piL
)3kF = (3pi2n)
13 , n N/L3
E(k) = V0 + h2k22me = h2k2
2me
EF =h2kF 2
2me= h
2
2me(3pi2n)
23
eiGnx Gn = npi/ak(r) = uk(r)eikr
k(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
vg =1
h$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 $kj #(k)
1
k(r) = exp(ik r)kx, ky, kz = 0; 2pi/L; 4pi/L; . . .Vkspace = 43pik
3F = 2 N
(2piL
)3kF = (3pi2n)
13 , n N/L3
E(k) = V0 + h2k22me = h2k2
2me
EF =h2kF 2
2me= h
2
2me(3pi2n)
23
eiGnx Gn = npi/ak(r) = uk(r)eikr
k(r) = uk(r)eikr =
k
Ckeikr
T = n1a1 + n2a2 + n3a3n1, n2, n3 integers. a1, a2, a3 primitive lattice vectors
G = m1b1 +m2b2 +m3b3m1,m2,m3 integers. b1, b2, b3 primitive reciprocal lattice vectors
V (r) =G
VGeiGr
(h2k2
2m E
)Ck +
G
VGCkG = 0
#k =12
(Ek + Ekpi/a
)[(
EkEkpi/a2
)2+ |Vpi/a|2
] 12
vg =1
h$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.
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page10of22
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
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page11of22
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)
Vkspace =43pik3F = 2 N
(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
BandstructureCopper
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page12of22
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/
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page13of22
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
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page14of22
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?
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page15of22
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.
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page16of22
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
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page17of22
OrbitsmaybeobservedenclosinglledorunlledporFonsoftheFermisurface.
ExperimentalprobesoftheFermisurfaceQuantumOscillaFonsIV
-
ThetemperaturedependenceoftheamplitudeisdeterminedbyconsideringthetemperaturedependenceoftheFermiDiracfuncFon.
TheanalyFcexpressionis:
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page18of22
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).
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page19of22
ExampleofaquantumoscillaFonstudy:Ag5Pb2O6nearlyfreeelectronsuperconductor
Neckorbit(lowfrequency)
Bellyorbit(highfrequency)
1stBrillouinzone
-
UseconservaFonofmomentumandenergytoworkouttheiniFalstateofthetheelectron
Momentumparalleltothesurfaceisconserved
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page20of22
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
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page21of22
OtherProbesoftheFermiSurface:ARPESII
Dierentkstates(detectorangles)
FermilevelsetbycalibraFngtoastandard
Occupiedstates
Example:Sr2RuO4(again!)
Images: A. Damascelli, UBC
-
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/ Page22of22
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