Dr.

Michael Sutherland
Cavendish Quantum Materials Group

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

Page1of22

OutlineandGoals

ReviewofFermiDiracStaFsFcs,theFreeelectronGas.
Reviewofelementarybandstructuretheory,nearly
freeelectronsmodels.
ExamplesofrealbandstructuresandrealFermi
surfaces
OverviewofExperimentalTechniquesforprobingthe
FermiSurface(QuantumOscillaFons,ARPES)
Detailedlistforfurtherreading

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

Page2of22

1
1
1
2e 1
=

=
B
Bn+1 Bn
h
Arecip

h
k
TheFermiDiracDistribuFon
E =
= E +h

f

2 2
f

2m
kf " = ki"

ElectronsarefermionsparFcleswithhalf
integerspinthatobeythePauliExclusion
dM dB
V
=

Principle:notwofermionsmayhaveexactlythe
dB dt

samesetofquantumnumbers.
Amp. B 1 e c

ForasystemofidenFcalfermions,the
Amp.
sinh()
probabilitythatasingleparFclestatewith
= 14.7m! T /B
energyEisoccupiedisgivenby
1
fD (E, T ) = (E)/k T
B
e
+1
inthisexpressionisthechemicalpotenFal,
oRendenedastheenergywherefD(E,T)=.
isanimportantenergyscalethatismaterial
dependent.AtT=0,wedenetheFermienergy
EFbyEF=(T=0).
AtT=0,theFermienergyisthedividingline
betweenlledandunlledquantumstates.

TheenergydependenceoffD(E,T)changes
dramaFcallyasafuncFonofT.
When/kT>>1(lowtemperatures)the
funcFonresemblesastepfuncFoncenteredat
E=EFand
When/kT<<1(hightemperatures)the
funcFonissmearedout.

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

Page3of22

TheFreeElectronFermiGasII

Image: Sutton, McGill Physics

TheSommerfeldmodelofametalusesFermi
DiracStaFsFcstotakeintoaccountthequantum
natureofelectrons.Weignorethedetailsofthe
atomicpotenFal.
(r) = exp(ik r)
k

EachelectronsaFsesthefreeparFcle
k (r) = exp(ik r)
SchrodingerequaFonwithperiodicboundary
condiFonswithperiodL:
kx , ky , kz = 0; 2/L; 4/L; . . . (1)
(2)
k (r) = exp(ik !r) "
with
3

4 3
2
Vkspacekx=, kyk
=
2

N
,
k
=
0;
2/L;
4/L; . . .
3 Fz
L

(1)

Thus,thereisonedisFncttripletofquantum
numberskx,ky,kzforthevolumeelement(2/L)3,
andTWOelectronscanlleachstate(accounFng
1
forspin).
kF = (3 2 n) 3 , n N/L3
(3)
Forelectronsinanionicsolidofaverage
potenFalV0theenergyisgivenby

2 k2
h
h2 k 2
(3)
E(k) = V0 +
=

(shiRzeroofenergyupbya
2me
2me
constantV0)

k (r) = exp(ik r)

Inatypicalmetalwehavemanyfree
k (r) = exp(ik r)
(2)
kx , ky , kz = 0; 2/L; 4/L; . . .
electrons.Theyoccupystateswiththelowest
energyrst,thenllprogressivelyhigher
! "3
energystates.
4 3
2
Vkspace = kF = 2 N
3 0; 2/L; 4/L;
L ...
kx , ky , kz =
IfwehaveNelectrons,thevolumeofkstates
lledisasphereofradiuskF:
Vkspace

4
= kF3 = 2 N
3
1

2
L

kF = (3 2 n) 3 , n N/L3

(4)
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/

"3

kF = (3 2Page4of22
n) 3
1

kF = (3 2 n) 3 , n N/L3

Image: J. Ellise, McGill

TheFreeElectronFermiGasII
2 k2
h

2 k2
h

Thecorrespondingelectronenergywhenk=k
E(k) = V0 +
=
Fis
2m
2m
e
e
simply

(5)

2
2 kF 2
h
h2
EF =
=
(3 2 n) 3
2me
2me

ThisleadstotheinterpretaFonthattheFermi
surfaceisasurfaceofconstantenergyEFink
space.

material

Fermi
Fermi
energyEF temperatureTF

Na

3.24eV

38000K

Formanymetals,theFermienergyisveryhigh
comparedtothethermalenergyatroom
temperature,kBT

Mg

7.0eV

82000K

Fe

11.1eV

130000K

ItisoRenusefultodenetheFermitemperature
asTF=EF/kB.Since300K<<TF,theFermifuncFon
inequaFon[1]tellsusthefollowing:

Cu

7.0eV

81000K

MgB2*

0.3eV

3200K

Whitedwarf
stars*

0.3MeV

3X109K

3He*

0.1meV

1.5K

i)wecanusuallyfocusontheelectronswithina
smallenergywindowkBTaboutEF.
ii)EF

*Considerwhythesematerialshavesuch
dierentvaluesforEFthanmoreordinarymetals.

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

Page5of22

(3
n) 3 , n4/L;
N/L
kxk
, kFy , k=
0; 2/L;
. .!
. "3 =
z =
(r)
=V
exp(ik
r)4 k 3
kspace
Vkspace

kF 3= (3
n
F n)2 , N
2
3 k
= 43 k
=
2

N
F
kx , ky ,!kz"=
0; L2/L; 4/L; . . .
3
2=

1
3

"3
2
LN/L3

2
2 kF 2
h
h2
EF =
=
(3 2 n) 3
2me
2me

! "3
exp(ik r)
Vkspace = 43 kF3 = 2h2 kN2 2
24 2 3
21
L
h
k
3h2 k2
2
h2(6)

k2
V
=
k
=
2

N
3, n

(r)
=
exp(ik

r)
kspace
k
=
(3
n)

N/L
=
E(k)
=
V
+
F
k
+
=
F
3
L
0
; 2/L; 4/L; .E(k)=-V
..
0
2me
2me
2me3
e
! "3 kF = (3 2 n) 13 ,2m
2
2
n

N/L
h2 k F 2

2
3
EF = 2me = 2me (3 n)
2
3
kF3 = 2 N 2
(3
, n N/L
kkxF, k=
0; 2/L;
4/L;
...
y, k
z = n)
L
1
3
h2 k 2
h2 k 2

3 , n=
2 = (3 2 n)
2E(k)
k
V
N/L
=
+
F
h2 k F 2

2
0
2m
2me
!
"
3
e
EF = 4 2m
=2hk2k2m2e (3
3
h

h2 k2 n)
2 iG2n x
2
2
h3 k+

I
nrealmetals,onemusttakeintoaccountthe
h

k
h

2
V
=
k
=
2

N
E(k)=-V
=
F
e = Gn =
n/a
kspace 0 3+ 0
1
=2m
F
3 ThewavefuncFonsoluFonsofthefreeelectron
(3
n)
3 E(k)=-V
2
2m
2m
e L 2me EF = 2m
2me ikr
e
n) 3 , n N/L
eiGn x Gn = n/a
k (r)
u
(r)e
periodicityofthelapce,whichmodiesthe
2 2
2=
2
k
h k

case(planewaves)arealsomodiedtoreect
= h2mke
2
3E(k) = V0 + 2m
e
k
=
(3
n)
,
n

N/L
h

k
h

2
F
iG
x
potenFalandhencethewavefuncFonsoluFons
2 (3 n)
#
EFe = n2m h2G
=
2
2 2
2
kFn2m
h2

=
n/a
theperiodicityofthepotenFal.
h2 k2Fn)
hiG

ikr
nx 2
3

E
=
=
(3
3= n/a
2 2
2 2
e
G
E
=
=
(3
n)

(r)
=
u
(r)e
=
Ck eikr
F
n
k
k
F
k
2me
2me 2me
2me
V0 + h2mke = htotheSchrodingerequaFon.
k
2me
h k

h k

k (r) = uk (r)eikr
(8)
E(k)=-V
0 + 2m = 2m
2
iG
x
h2

HenceBlochstheorem:
e
Gn = n/a
= 2me (3 2 n) 3
ikr
iG
x
#
n
nx
T = n1 a1 + n2 a2 + n3 a3
=
Theproblemiseasiesttotackleinreciprocal
ekh(r)
Gnhu=
n/a
k (r)e
ikr
eiG#
Gn = n/a
k
2

(r)
=
u
(r)e
=
Ck eikr
EF = 2m = 2m (3
k
k

ikxn)
i(k+G
)x integers. a1 , a2 , a3 primitive
n
,
n
,
n
lattice
vectors
ikr
n
ikr
1
2
3
k(x)
ukk(r)e
(x)e#k (r)
= =
space.Forarealspacelapcewithlapce
ukC
(r)e
=u
k,n Ae
k (r)=
k

ikx
i(k+G )x
ikr

(x)
=
u
(x)e
=
C
Ae
=
n/a
k
k
k,n
n

2whereisafuncFonthathastheperiodicity
(r)m=
translaFonvectors
G = m 1 b1 + m 2 b
k+
3 b3uk (r)e
ikr
#
eiGx (r)
Gn =
n/a
ikr
=
u
(r)e
ikr
k
k
T
=
n
a
+
n
a
+
n
a
)e
1
1
2
2
3
3

(r)
=
u
(r)e
=
Ck eikrlattice vectors
m
,
m
,
m

integers.
b
,
b
,
b
primitive
reciprocal
(9)
#
k ofthepotenFalandareFouriercoecients.
k
1
2
3
1 2 3
ikx
i(k+G
n )x
kT
(x)
uk1(x)e
Ae
T
an21a+
a32
+Cnk,n
k vectors
==
n=
n=
23 a
3na13, n2 , n3 integers. a1 , a2 , a3 primitive lattice
1 a1n+
2 +n

TheNearlyFreeElectronModelI
1
3

F
e

2 2

2
F 3
e

2 2

2
3

2 2

2 2

2
3

(r) = u (r)eikr
1 k (x)
2 AeG
3
k (x)
=2uk (x)e
= Ck,n

=3 uk (x)e
=Ae 1Ck,n

k
#
#
nn1 , n,2kn
, n3,
aikx
vectors
1 , a2 , a3i(k+G
i(k+G
ikx
n )x
n integers.
integers.
nprimitive
, nn)x , nlattice
primitive

lattice vectors

b3
V (r) =
VG eiGr
W3ecanwritedowntheSchrodingerequaFon
T
= n1 a1 + n2=a2m+1 bn13+
a3m2 b2 +m
GT
= m1 b1 a
+ m2 b2 +am3+
b3
G lattice vectors
m2 , m3a1, aintegers.
b1usingtheseresults.
, b2 , blattice
reciprocal
1 1+n
2 2 , nn33
1 ,b
3 primitive
Recallthatwemaydenethereciprocallapce
n21m
,n
integers.
vectors
2 , a3 primitive
G==nm
b +
b +a3m
m
m1 ,Tm=
m3a +integers.
lattice vectors
3 reciprocal
3
2, n
1 , b22, b32primitive
n1a 1+ nba

2 2
3 3 a , a , a primitive lattice vectors
n1 , n2 , 1n31 integers.
$ 2
%
1 2 3
vectorsby:
2
integers.
a
,
a
lattice
vectors
m
,
m

integers.
b
,
b
,
b
primitive
reciprocal
lattice
vectors
1 , n12,, n
3
1
2 , a3 primitive
1 + n2 a2 + n3 an
3m
2
3
1
2
3
h

k
G = m1 b1 + m2 b2 + m3 b3

(r)
E =C#
iGr
k +V eV
G CkG = 0
ntegers. a1 , a2 , a3 G
primitive
lattice
vectors
V
G
m
b
+
m
b
+
m
b
G
= m=
b
+
m
b
+
m
b
2m
1 1 1 21 2 m 2
3
32 , m 3
3
,
m

integers.
b
,
b
,
b
primitive
reciprocal
lattice
vectors
G
1
2
3
1 2 3
mm
, m3,mintegers.
b1 , b2 , b3 primitive
latticereciprocal
vectors
1 , m,2m
integers.
b , b , breciprocal
primitive

m3 b31

b1 + m2 b2 +
integers. b1 , b2 , b3 primitive reciprocal lattice vectors

TheperiodicityofthepotenFalV(r)allowusto
write:
#
VG eiGr

V (r) =

lattice vectors

ThesoluFonforthisequaFonofcoecients
#
V
(r)
=
VG eiGr
givesusthewavefuncFonsandenergystatesof
G
theelectronsinthepotenFal.
1

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

Page6of22

k (r) = exp(ik r)
eiGn x. . .Gn = n/a
kx , ky , kz = 0; 2/L; 4/L;
! "3
ikr
2 = uk (r)e
V
= 4 k 3 = 2 Nk (r)

TheNearlyFreeElectronModelII
kspace

Image: Sutton, McGill Physics

(6)

kF = (3 2 n) 3 , n N/L3

k (r) = u0k (r)e

2me

2
2 kF 2
h
h2
EF =
=
(3 2 n) 3
2me
2me
#
ikr
ikr

(r)
=
u
(r)e
=
C
e
k
k
k
T=n a +n a +n a
1 1

2 2

n1 , n2 , n3 integers.

ConsidertheperiodicpotenFalin1DforillustraFon

3 3
k
a1 , a2 , a3

prim

T = n1 a1 + n2 a2 + n3 a3
2 k2
h
h2 k 2

=
2me n1 , 2m
n2e, n3 integers. a1 , a2 , a3
2
2
3

G = m1 b1 + m2 b2 + m3 b3
primitive
lattice vectors
h2 k F 2

h2

m
,
m
,
m
integers. b1 , b2 , b3 pr
1
2
3
EF = 2me = 2me (3 n)
iGn x
e
G
n = n/a
G = m1 b1 + m2 b2 + m3 b3
m1 , m2 , m3 integers. b1 , b2 , b3 primitive reciprocal lattice vecto
iGn x
e
Gn = n/a
(8)
k (r) = uk (r)eikr
#
V
(r)
=
VG eiGr
$
#
Foragivenstatewithwavevectork,onlystateswith
G
h

k (r) = uk (r)eikr =
Ck eikr
E(k) = V0 +

John Ellis Cambridge McGill Physics

k=kG,k2GwillcontributeFourier
componentstotheoverallwavefuncFon.

k (r) = uk (r)e
T = n1 a1 + n2 a2 + (9)
n3 a3

ikr

$ 2
h
k2

E Ck +

VG CkG =
2m !
&!
G "
n
,
n
,
n

integers.
a
,
a
,
a
primitive
lattice
vectors
Ek
1
2
3
1
2
3
Forastatewithsmallk,thestateswithk=kG,
#k = 21 Ek + Ek/a
Forthesestatestheenergybecomes:
'1
k2Gareallmuchhigherinenergy.Forasmall
" &! E E
"2
G = m 1 b 1 + m 2 b2 + m 3 b 3 !
2
k
k/a
1
2
#
=
E
+
E

+
|V
|
k
potenFalthesestatesdonotmixstrongly,andhence
k/a
/a
m1 , m2 , m3 integers. b1 , b2 ,kb3 primitive
reciprocal
lattice vectors
2
2

thedispersionrelaFonresemblesthefreeelectron
case.
ForkstatesneartheBZboundary(/ain1D)the
statewithk=k/aisverycloseinenergyand
henceaectsthedispersionrelaFon(degenerate
perturbaFontheory)

Theeectisthatanenergygapopensupnear
#
V
(r)
=
VG eiGr
theBZedge.Therearenoallowedstatesinthat
G
gap.
TheposiFonofEFwithrespecttogap
determineswhetherasystemisaninsulator,
1
semiconductor,ormetal.

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

Page7of22

k (r) = uk (r)e

TheNearlyFreeElectronModelIII

Image: John Ellis, Cambridge

Onecanunderstandthisgapasarisingfrom
contribuFonstothewavefuncFonfromtwo
standingwaveswithk - /a.Onehasamaximum
chargedensityneartheBZedge(higherenergy)and
onehasaminimumchargedensitythere(lower
energy)
HigherorderharmonicsofthepotenFalVGwilllink
statesathigherenergiesandgivegapsathighBZ
edges.
SinceanykvectorinahigherBZmaybemappedby
areciprocallapcevectorGintotherstBZ,itis
oRenconvenienttofoldtheenergybandbackinto
the1stBZ.Thisisknownasthereducedzone
scheme.

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

T = n1 a1 + n2 a2 + n3 a3
n1 , n2 , n3 integers. a1 , a2 , a3 prim

G = m1 b1 + m2 b2 + m3 b3
m1 , m2 , m3 integers. b1 , b2 , b3 pr

#k =

1
2

"

Ek + Ek/a

Page8of22

&!
Ek

 E1 Ck +
VG CkG = 0Ek Ek/a
2
#
h
2 k$
2m
#
=
E
+
E

#
k
k
k/a
1
G
20
2
2 E2G C=k%m
+ b V+
=
iGr
GC
kG
#
m
b
+
m
b
v
=
$
#(k)
V
(r)
=
V
e
h

k
1
1
2
2
3
3
g
k
G
2m
G

+ |V/a |2

E GCk&+
VG CkG = 0 h
'1
m2m
b"12 , b2 , b3 primitive
reciprocal lattice vectors
" 3 !integers.
1 , m2 , m
2
G
E
E
k
k/a
1
2
1
Ek +"E
+ |V/a |
k/a ' 1
!
" #&k!=
2
2

T
heapplicaFonofaforce(sayelectromoFve)
v
=
$k #(k)
2
$
%
2
g
E
E
k 2 k/a
1
2
2
&!
'1
#
h

d#
dk
dk
E
+
E

+
|V
|
h

k
!
"
"
k
k/a
/a
#
2
2
2 Ek Ek/a 2
changesthewavepacketenergybyanamount
2 C=
F=
v0g =
$k #(k)
==
h

E Ck +
+ |V/aV|G
kG
#k = 21 Ek + Ek/a
V
(r)
VG eiGr
1
2
dt
dt
dt
2m
G
vg = $k #(k)
Aknowledgeofthebandstructureofamaterial
G
h

1
d#
dk
dk
v
=
$
#(k)
=
F

v
=

$
#(k)
=
h

g
k
givesadeepinsightintotheelectronicproperFes.
1
g
k
$ 2
%
!
" &! E E
"2
g = ' 21$k #(k)
1
vh
dt #
dt
dt
2
1
k
k/a
2
h
kk0 ))m (
h
=
#(k)
#(k
)
+
(
h
(k

h
(k

k
))
1
Ek + Ek/aThedispersionrelaFongivesknowledgeofthe

+
|V
|
0
0
d#
dk
dk
/a
2
Eh Ck +fD (E,
VGTC
=0 T
) kG
= (E)/k
= F vg2=
$
k #(k) =F
oranelectriceldofstrengthE,wehave
B
2m
e
+1
eecFvemassandgroupvelocity:
dt
dt
dt
d#
dk
dk
G
dk
= d#
F vg = 1 dk
$k #(k) = h

dk
1
= F vdt
$k #(k) =

dt
dth
g =
1
1
#(k)
&
'
=

eE
1
dt vg = $
dt
dt
k
"
! $
"#(k)
2
$k1
2
ij 1 = E
kik/a
h
1 !#(k) = #(k0m
E
dt
h

j (
2 kk
2
)
+
(
h
(k

))m
h
(k

k
))
0|
02
#k = 2 Ek + Ek/a 2 h
+ |V/a
!

Wavepackets:semiclassicalmodel

1
1 k0 ))m1 (
#(k) =#(k)
#(k=
+ )(
h
(k
(k(h(kk
0 ) #(k
0 ))
+
(
h(k k0 ))mh1
k0 ))
0
2
1
2
1
1
1
mij = 2 $ki $kj #(k)vg = $k #(k)
WheretheeecFvemasstensorisdenedby
h

1
1
1
1
(r) = exp(ik r)
mij m=ij 2=
$ki2 $
#(k)
k
$kij $kj #(k)
h
h

kz = 0; 2/L; 4/L; . . .

ce

"3

d#
dt
proporFonaltotheslopeofthebandwhenitcrosses

2
= 34 kF3 = 2
NInpracFcethismeansthatthevelocityisinversely
= F vg =
L

theFermilevel,andthemassisinversely
= (3 2 n) 3 , n
N/L3
1

k) = V0 +

2 kF 2
h
2me

dk
dk
$k #(k) = h

dt
dt

2 k2
h
2me
2

h2

(3
2me

1
2

proporFonaltothesecondderivateof.
#(k) = #(k0 ) + (
h(k k0 ))m1 (
h(k k0 ))
=

2 k2
h
2me

2
TheenFreFermisphereisshiRedbya
n)
Thusat,shallow,bandstendtocontainslow
3
1
uniformamount(dependsonscaveringrate).
moving,heavyelectrons.
mij 1 =
2 $ki $kj #(k)

Gn x

Gn = n/a
Considerthegroupvelocityvgofawavepacketof
= uk (r)eikr

electrons,madeupasasuperposiFonofwave
#
k (r) = uk (r)eikr =
Ck eikr
funcFons.
Image: John Ellis, Cambridge

Fromknowledgeofthebandstructure,itis
possibletoesFmateconducFvity.

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

= n1 a1 + n2 a2 + n3 a3
n3 integers. a1 , a2 , a3 primitive lattice vectors

Page9of22

Bandstructure2Ddivalentmetal

PotenFalturnedon

1stBZ

2ndBZ

Considera2Ddivalentmetal.Thereare2electrons
perunitcell,soweshouldexpectafullband,and
henceaninsulatoreg.diamond.
Howeversomedivalentmetals(Ca)aremetallic.
Whyisthis?Metalsneednearbyemptystatesin
ordertopropagatecharge.
Considerthecaseofa2DsquarelapcepotenFal.
ThefreeelectronFS(acircle)hasthesameareaas
therstBZ,andhencespillsovertothesecondBZ.
1stBZ

2ndBZ

HoweverturningonapotenFalraisesthe
energyofthestatesneartheedgesoftheBZ
andlowersthemnearthecorners.Hence
electronsaretransferredfromthesecondto
rstBZ,resulFnginametal.

Electrons

Holes

1stBZ

2ndBZ

Freeelectron
circle

Inthereducedzonescheme,thisresultsin
smallFSpockets,bothholeandelectronlike
Images: Singleton, Band theory of solids

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

Page10of22

Vkspace
(3)

4
= kF3 = 2 N
3

2
L

BandstructureCopper

Images: S. Blundell Oxford

(4)

kF = (3 2 n) 3 , n N/L3
Copperisa3Dmonovalentmetalwith
conguraFon[Ar]3d104s1.[Ar]3d10electronsgive
risetoFghtlyboundbands,farbelowEF.One4s
electrononeFSsheet.
2 k2
h
h2 k 2
E(k) = V0 +
=
fcccrystallapce(bccreciprocallapce).Therst
2me
2m
Brillouinzone(BZ)isatruncatedpolyhedra e

(5)

2
2 kF 2
h
h2
EF =
=
(3 2 n) 3
2me
2me

(6)

eiGn x
(8)

Gn = n/a

TheshortestdistancetotheBZedgeisalong
the<111>direcFon.Usingthelapce
parametersforCu,inthefreeelectron
approximaFonkF/k<111>=0.903(shouldnot
(7)
touchzoneedge)
HowevertheexperimentallydeterminedFS
touchestheBZedgealongthe<111>
direcFon.Thisarisesfrommixingofstatesof
similarenergiesneartheBZedge.

k (r) = uk (r)eikr
(9) QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/

Page11of22

GeneralrulesforconstrucFng
Fermisurfaces
Formanysimplediandtrivalentmetals,theFS
maybeschemaFcallyconstructedbyfollowing
thesesteps:
1. Drawafreeelectronspherebasedonthe
numberofvalenceelectrons
2. SuperimposethisontotherstfewBrillouin
zones.
3. WheretheFSmeetstheBZboundarywhere
thebandgapopensup,splitandroundo
thesurface.
4. TranslatetheresulFngsecFonsbackintothe
rstBZ,usingtheperiodicityofkspace
Evenusingtheserules,simplemetalsmayhave
exceedinglycomplexFermisurfaces.Thesecanbe
diculttocalculate,especiallyforthosematerials
withdandfbandswhichtendtobeveryat.

Image: University of Florida

Eg.Rhenium,hexagonalcrystal
structurewith6(!)bandscrossingthe
Fermilevel.

Anexcellentwebresourceisthe
periodictableofFermisurfacesfound
ontheUniveristyofFloridaswebsite:
www.phys.u.edu/fermisurface/

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

Page12of22

FermiSurfaceofSr2RuO4

Image: C. Bergemann, Cambridge

ItissomeFmesusefultothinkofrealspace
orbitalswhenconstrucFngFermisurfaces.This
approachiscalledtheFghtbindingapproximaFon
Sr2RuO4isalayeredruthenatematerialthatis
alsoanunconvenFonalsuperconductoratlow
temperature.StronglyanisotropicconducFon
(AB>>Chencequasi2Dmetal)

Tetragonalcrystalstructure.
Thedyz(dxz)statehasweakx(y)
dependence.WethusexpecttheFSformedby
thesestatestobeaatsheetperpendicularto
thex(y)direcFon.Wheretheyintersect,they
pinchotoformcylinderscenteredaboutthe
centerandcornersoftheBZ.
4delectronshybridizetoformthreelow
energystatesdxydyzdxzwhicharepopulated
by4electrons.

Thedxystatehasweakzdependenceand
formsacylinderinthecenteroftheBZ.

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

Page13of22

ExperimentalprobesoftheFermi
surfaceQuantumOscillaFons
Inverycleanmetals,inlargemagneFcelds
andatlowtemperatures,anoscillatory
componentisobservedintransport,
thermodynamicandmagneFcmeasurements.

h
2 kZ2
E=
+ (! + 1/2)
hc
Thefrequency,angularandtemperature
2m
dependenceofthesignalcanbeusedto
characterizetheFermisurface.

eB
m
ConsideraeldalongthezdirecFon.The
c =

electronexperiencesaLorentzforce:

h
2 kZ2F = eB v
E=
+ (! + 1/2)
hc
2m
IfthescaveringFmeislarge,theelectron
canundergomanycyclotronorbitsor
eB
frequencyc =

F = eB v

Image: I. Shiekin, Grenoble

2
Allowed
kspace
orbits
(Landau
tubes)

2
Inreciprocalspace,theelectroncantakeany
valueofkz,butthevalueofkxandkyare
quanFzedsuchthat:
h
2 kZ2
E=
+ (! 2+ 21/2)
hc
h
kZ
2m
E=
+ (! + 1/2)
hc
2m
eB
with:
c =
eB
m
c =
m
TheseorbitsformLandautubesinkspace

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

Page14of22

SrFe2As2(amagneFcmetal)

ExperimentalprobesoftheFermi
surfaceQuantumOscillaFonsII
Theresultisthatthedensityofelectron
statesg(E)acquiresastronglypeakedenergy
dependence(eachpeakcorrespondingtoa
MagneFzaFon
Landautube):

IfthemagneFceldisvaried,thepeaksshiR
inenergy.IfapeakcrossestheFermienergy,
thestateispopulated,thenempFesasit
movesaway.

B1(Tesla1)

Fouriertransform

Inrealexperimentsseveral
ofdata(two
h
2 kZ2
dierentperiodiciFesareoRen
E=
+ (! + 1/2)
hcfrequencies)
2m
observed.Howdowemake
h
2 kZ2
E=
+ (! + 1/2)
hc
quanFtaFvesenseofthem?
2m
eB

=
c
InrealexperimentsseveraldierentperiodiciFes
m
eB
areoRenobserved.HowdowemakequanFtaFve
c =
m
senseofthem?
F = eB v

BohrSommerfeldquanFzaFoncondiFon(see
F = eB v
"
#
Kivel):
!
1
p dr = n +
h
!
21
p dr = (nq+ )h
wherepisthecanonicalmomentum:
p = mv + A2
cq
ThisleadstooscillaFonsthatareperiodicin
!
!
p = mv + Aq !
inversemagneFc,1/B.
c
! k dr +
A dr
! p dr = h
cq !
! p dr = h
! k dr +Page15of22
A dr
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/
c
A dr = " A d =
!

A dr =

" A d =

E=

c =+ (! + 1/2)hc
F = eBm v
2m

eB
ExperimentalprobesoftheFermi

=
F
=
eB
v
!
m 1
2 2
h
k
+ 1/2)
)hhc
surfaceQuantumOscillaFonsIII
E =p drZ =
+ (n
(! +
2
2m
c

F = eB q v1
p

dr
= (n
p = mv
+ +A )h
Applyingthisyields
c 2
eB
!
!

=
c " q q#!
!
m+
mv
A1 h A dr
p dr =pph
= dr
k= +
dr
nc+
2c !
!!
!!
q
q
pA
dr
==
h
pF==
kmv
dr
+
v d A
+
UsingStokestheoremonthesecondterm:
dr
"eB

AA
= dr
c
c
!!

!!

q!
A
dr ==h
p dr
"
k
+ d
A dr
"drA
# =
!
1c
!

p dr! = n +
h
2
A dr = " A d =
whereisthemagneFcux.Usingq=efor
q
p = mv + A
theelectronandcollecFngtermsgives:
c#
!
!"
1 hc
q!

=
n
+
p dr =n h
k dr2 + e A dr
c#
!
! h "
1
A
=
n
+
real
A

dr
=
"

d =
ShowingthatthemagneFcuxisquanFzed.
eB
2
"
#
Sincetheuxisjusttheeldthreadingthe
1 hc
n = n +realwecanwrite:
realspaceorbit,ofareaA

p =dr =
n+
h
A dr
! " A d =

2
A dr =

1
1
1
2e 1
=

=
B
Bn+1 Bn
h
Arecip

2
"A
q d =
# A
p ="mv +
1 chc

=
n
+
n
!
!
!
" "2 # e q#
TheLorenzforcetellsushowtheareasofthe
p dr = h
hc
k dr
1 +1hc A dr

=
n
+
A
=
n
+ c
n
realspaceandreciprocalspaceorbitsarerelated
real
!
!eB 2 2e
# =
A dr = "hce"#"2 2A 1d
=
B+
Areal
recip=
AAreal
n
h

eB " 2 #
1
2e
"
#2 # 1
"
=
e
1n +hc
h
cArecip
nn =
ABrecip
Wecancombingthesetoshowthat
=
n + B 2 A2real
h
2 e
""
##
1
2e
hc
11
A= =
nn++
Bn realh
cArecip
eB
22
" #2
e
SonallywecanobtainarelaFonbetweenthe
Arecip =
B 2 Areal
h

frequencyoftheoscillaFonsandtheareaofthe
"
#
1
1
2e
reciprocalspaceorbits
=
n+
Bn
h
cArecip
2

ThisisknownastheOnsagerrelaFonandshows
howoscillaFonstudiesmayactasacaliperof
theFermisurface.

e
ThefrequencyisproporFonaltotheextremal
#
hc
1
crosssecFonalareaperpendiculartotheapplied
Areal =
n+
eB
2
eld.
" #2
e
Arecip =
B 2 Areal
h

QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/
"
#
Page16of22
1
1
2e
=
n+
Bn
h
cArecip
2
"

ExperimentalprobesoftheFermi
surfaceQuantumOscillaFonsIV
OrbitsmaybeobservedenclosinglledorunlledporFonsoftheFermisurface.

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

Page17of22

n = n
"+

21 hce
n = n +
"
#
hc "2 e 1#
ExperimentalprobesoftheFermi
Areal = hc n + 1
Areal =eB n + 2
eB
2
surfaceQuantumOscillaFonsV
" " ##22
ee
Arecip
B 22A
Arecip
= = h B
Areal
real
h
"
#
OscillaFonsmaybeobservedinmany
1 #
1
2e "
1
= 2e
n+ 1
physicalquanFFes,forinstanceresisFvity(the
B
=n h cArecip n +2
ShubnikovdeHaaseect)orinmagneFzaFon
Bn1
h
1cArecip1 2e 2 1

A dr =

" A d =

"

1 hc
n = n +
2 e
Thetemperaturedependenceoftheamplitudeis
"
#
"
#
1 hc
determinedbyconsideringthetemperature
hc
1
n = n +
dependenceoftheFermiDiracfuncFon.
2 e Areal = eB n + 2
"
#
" #2
hc
1
e
Areal =
n+ A
=
B 2 Areal
recip
eB
2
h

" #2
"
#
e
2
1
2e
1
Arecip =
B Areal =
n+
h

B
h

cA
2
n
recip
=

=
(thedeHaasvanAlpheneect).
"
#
h
Arecip1
1 B 1Bn+1 1Bn 2e
1
2e
1
1
1
1
2e 1
=
n
+
=

=
2 2
=

B
h

cA
2
B
Bn+1h kf Bn
h
Arecip
n
recip
AtypicalmeasurementofthedHvAeect
B
Bn+1 Bn
h
Arecip
Ef =
= Ei + h

2m
involvesmeasurementofthedierenFal
1
1
1
2e 1 2 2
2 2
h

k
=

f kf " = ki"
h
k
suscepFlbiltywithcounterwoundpickupcoils,
B
Bn+1 Bn
h
fA
Ef =
= Ei + h

E
=recip f = Ei + h

2m
suchthatthepickupvoltageis
2m
h
2 kf2
kf " =dM
ki"dB
Ef =
= Ei + h
kf " = ki"
V =
2m
dB dt
kf " = ki"
dM dB
InaddiFontothefrequency,theamplitudeof
dM dB
V
=

V =
theoscillaFonsmaytellusagreatdeal.
dB dt
dB dt
dM dB
1

V
=

Amp. B e c
Image: C. Bergemann, Cambridge
dB dt
Amp. B 1 e c

1
Amp.

Amp. B e c
TheanalyFcexpressionis:
sinh()
WhereisthescaveringrateandCisthe

cyclotronfrequency.Theelddependencecan
Amp.
= 14.7m! T /B
sinh()
giveinformaFononscavering.
= 14.7m! T /B
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/
Page18of22

ExampleofaquantumoscillaFon
study:Ag5Pb2O6nearlyfree
electronsuperconductor

Neckorbit(low
frequency)

Bellyorbit(high
frequency)
1stBrillouinzone

Twofrequencies,correspondingtoneckandbelly
orbits(extremalorbits).
Angulardependencegivessuccessivecross
secFonalcuts,perpendiculartoappliedeld.
Weaktemperaturedependence(lightmasses).

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

Page19of22

F = eB v

OtherProbesoftheFermi
Surface:ARPES
AngularResolvedPhotoEmission
Spectroscopyispowerfultechniquethatuses
thephotoelectriceecttoprobethe
electronicstatesnearthesurfaceofasample.
Anincomingphotonofenergyejects
electronofmomentumkF

q
p
=
mv
+
A
"
#
!
1
Final
c
p dr
!
! = n+ 2 h!
measured
q
p dr = h
k drq +
A energyand
dr
p = mv + A c
momentum
c
!!

A
dr==h
p dr
!

!!

A dr =

q!
k"
dr+A d
A =
dr
c

" A d =
"
#

1 hc
#
1 2 hc e
n = n + "
#
hc "2 e #1
Areal = hc n +
1
Areal = eB n + 2
eB #
2
"
" e#2 2
e
2 2A
Arecip
Arecip =
=
BB
Arealreal
h
h

""
# #
1 IniFalElectron
2e
1
1 1 = 2e
n + state
=
n
+
Bn
h
cArecip
2
B
h

cA
2
n
recip
UseconservaFonofmomentumandenergyto
n = " n +

1
1
1
2e 1

=
workouttheiniFalstateofthetheelectron
1 B B1n+1 B1n
h
2e
Arecip 1

=
2 2
Bn+1
B
h
Arecip
n
h
kf

Ef =

h
22m
kf2

= Ei + h

Ef =

f "= E
i"i + h
Momentumparalleltothesurfaceisconserved
2m
k

=k

kf " = ki"
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/

Page20of22

OtherProbesoftheFermi
Surface:ARPESII

Images: A. Damascelli, UBC

Occupied
states

Dierentk
states
(detector
angles)
Fermilevel
setby
calibraFngto
astandard

Example:Sr2RuO4(again!)

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

Page21of22

EssenFalFurtherReading
Thepreceedingnotesgiveaverybriefoverview
ofelectronicbandstructure,Fermisurfaces,and
theexperimentalprobesusedtomeasurethem.
Youshouldreadwidelyonthesubjectandbe
comfortablewiththemainideas.

ARPEStechniques
ProbingtheFermiSurfaceofCorrelated
ElectronSystemsA.DamascelliPhysica
Scripta.Vol.T109,6174,2004

FreeElectronModel:basicconceptsin
IntroducFontoSolidStatePhysics9thedKivel,
Chapter6.
NearlyFreeelectronModel:
BandTheoryandElectronicProperFesofSolids
J.Singleton,OxfordMasterSeriesinCondensed
MaverPhysics,Chapter3(2001).
QuantumoscillaFonsandthedeHaasvanAlphen
Eect:
J.Singleton,chapter8(goodintroducFon)
MagneFcOscilaFonsinMetalsD.Shoenberg,
CambridgeUniversityPress,(1984).
QMsmallgrouplecturenotes:www.qm.phy.cam.ac.uk/sutherland/teaching/

Page22of22