laboratoire national des champs magnétiques intenses toulouse
TRANSCRIPT
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Laboratoire National des Champs Magnétiques Intenses
Toulouse
Fermi Surface part II: measurements
Cyril PROUST
Periodic Table of the Fermi Surfaces of Elemental S olids
http://www.phys.ufl.edu/fermisurface/
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Outline
I. Why and how to measure a Fermi surface ?
II. Angular Resolved Photoemission Electron Spectroscopy (ARPES)
III. Quantum oscillations (QO)1) History2) Theory
a) Semiclassical theorya) Landau levels quantificationb) Lifshitz-Kosevich theoryc) High magnetic field phenomena
3) High magnetic fields facilities4) Fermiology
IV. Hot topics1) Phase transition2) High Tc superconductors
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I. Why and how to measure a Fermi surface
1D
2D
3D
EF ~ 1-10 eV
Typical excitations (∆V, ∆T) ~meV
( ))cos()cos(2)( bkaktkE yx +−=r
FS measurements
Global properties: Cv, χPauli, RH, ∆ρ/ρ...
Topographic properties: ARPES, AMRO, QO
2D:
Comparison with band structure calculations, effect of interactions, phase transitions…
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Outline
I. Why and how to measure a Fermi surface ?
II. Angular Resolved Photoemission Electron Spectroscopy (ARPES)
III. Quantum oscillations (QO)1) History2) Theory
a) Semiclassical theorya) Landau levels quantificationb) Lifshitz-Kosevich theoryc) High magnetic field phenomena
3) High magnetic fields facilities4) Fermiology
IV. Hot topics1) Phase transition2) High Tc superconductors
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II. ARPES Photoelectric effect:
1886: 1st experimenal work by Hertz
1905: Theory by Einstein
φυ −−= Bkin EhE
Work function of thesurface (Potential barrier)
Binding energy of theelectron in the solid
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II. ARPES
X. J. Zhou et al, Treatise of HTSC
Angular Resolved PhotoEmission Spectroscopy
Vacuum Conservation laws Solid
Ekin
K
EB
kν
φυ
hif
Bkin
kkk
EhErrr
=−
−−=
Angular ↔ Momentum resolved
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II. ARPES
Synchroton SOLEIL
http://www.synchrotron-soleil.fr/
From IR to RX
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II. ARPES
http://arpes.phys.tohoku.ac.jp/contents/calendar-e. html
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II. ARPES
A. Damascelli, http://www.physics.ubc.ca/~quantmat/ARPES/PRESENTATIONS/Talks/ARPES_Intro.pdf
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II. ARPES Conservation laws
Energy of the photoelectron outside the solid
m
KEkin
2
22h=
One match the free-electron parabolas inside and outside the solid to obtain k inside the solid
φυ −−= Bkin EhE
Φvaccum
Bottom valence band (E0)EB
Efinal
EB, E0 and Efinal are referenced to EF
Ekin is referenced to Evaccum
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II. ARPES Conservation laws
νhif kkkrrr
=− Ultraviolet (hν < 100 eV) ⇒ khν=2π/λ=0.05 Å-1
2π/a=1.5 Å-1 (a=4 Å)
0=− if kkrr
or
Gkk if
rrr=−
Umklapp ⇒ Shadow bands or superstructures
1) The surface does not perturb the translational symmetry in the x-y plane:
//kr
is conserved (within )//Gr
θsin21
//// kinmEKkh
==
A. Damascelli et al, RMP’03
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2) Abrupt potential change along z ⇒ is not conserved across the surface⊥kr
II. ARPES Conservation laws
But determination of needed for 3D system to map E(k)⊥kr
Hyp: Nearly free electron description for the final bulk Bloch states
( )0
22//
2
0
22
22)( E
m
kkE
m
kkE f −+=−= ⊥
rrh
rhr
φ+= kinf EE θ22//
2
sin2
kinEm
k =r
hand
( )02cos2
1VEmk kin +=⊥ θ
h
φ+= 00 EV
A. Damascelli et al, RMP’03
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II. ARPES 2D case
If 0≈⊥iv ⇒
When k// is completely determined (2D), ARPES lineshape can be directly interpreted as lifetime
Γi, Γf → inverse lifetime of photoelectron and photoholevi, vf → group velocities ( )⊥⊥ ∂∂= kEv ii /h
A. Damascelli et al, RMP’03
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II. ARPES Non-interacting case
A. Damascelli et al, RMP’03
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II. ARPES Interacting systems
A. Damascelli et al, RMP’03
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II. ARPES Example: Quasi-2D overdoped cuprate
A. Ino et al, PRB’02
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Outline
I. Why and how to measure a Fermi surface ?
II. Angular Resolved Photoemission Electron Spectroscopy (ARPES)
III. Quantum oscillations (QO)1) History2) Theory
a) Semiclassical theorya) Landau levels quantificationb) Lifshitz-Kosevich theoryc) High magnetic field phenomena
3) High magnetic fields facilities4) Fermiology
IV. Hot topics1) Phase transition2) High Tc superconductors
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III.1 History1930 de Haas-van Alphen / Shubnikov-de Haas effects
W.J. de Haas
(1878-1960)
P.M. van Alphen
(1906-1967)
Bismuth
105 M/H
H
L.V. Shubnikov
(1901-1937)
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David Shoenberg (1911 - 2004)
The experimental pioneer …
III.1 History
and his friend the other post-doc
L.D. Landau (1908 – 1968)
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III.1 History
1950 – 70: Two decades of mapping 3D Fermi surfaces of ‘simple’ metals.
Periodic Table of the Fermi Surfaces of Elemental S olids
http://www.phys.ufl.edu/fermisurface/
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III.2 TheorySemiclassical theory
Fdt
kd r
h
r1=
)(1
kv k
r
h
r ε∇=
( )elEBVqdt
kd rrrr
h +×=⇒
Real space Momentum space
ky
kx
E
E+δE
χr
χ&r
Br
dtdd ..2 χχ &=Α
χχ dEdE .∇=r
ρχ&rh
r=∇ Ewith
dEqB
d2
h
& =Α TdEqB
d .2
h=Α cT ωπ /2=⇒ with
dA
dEBqc .
22
h
πω =
c
cm
qB=ωzk
cdE
dAm
&
h
=π2
2
⇒
χχχχ and ρρρρ are the projection of k and r in the plane ⊥ to B
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III.2 Theory
Onsager relation
hnrdp )(. γ+=∫rr
Aqkhprrr += ∫ ∫Α+×=∫ ⊥ ρρρρ rrrrrrr
dqdBqdp ...
∫ ∫∫+×−=∫ ⊥ dSnBqdBqdp ...rrrrrrr ρρρ
)( γφ += ne
hn
2
Φ=h
eB
BA nn
( )γπ += neB
Anh
2
Bohr-Sommerfeld condition:
Φ−=Φ+Φ−=∫ ⊥ qqqdp 2. ρrr
with
Onsager relation
γ=0.5 for free electrons
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III.2 Theory
( )5.02 += neB
Anh
πOnsager relation ( ) ( )5.0222 +=+ neB
kk yxh
ππ
B=0
Density of states
)n(ε
c/ ωε h
Landau tubes
Oscillation when An=AF(kz) ( )5.02 += neB
AFh
π
( )Be
An F 1
25.0
πh=+ Oscillation periodic in 1/B with
e
AF F
π2
h=
⇔
⇔
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III.2 Theory
• Free electrons in high magnetic fields ⇒ Landau levels (LL)
++=+= ⊥2
1
2
22
nm
kEEE c
zz ωhh
( )2
2
1Aqp
mH
r−= ⇒
3D
( ) )(2
1
2)(
22
02
222
xxxmm
px
m
kE c
xz ϕωϕ
−+=
− h
)0;;0( BxA =r
(jauge de Landau)
Equation of a harmonic oscillator with pulsation ωc and orbits centred atqB
kx
yh=0
Solutions:
• Degeneracy of each Landau level gL: xLx << 00h
xy
qBLk <<0
Quantum theory
y
xL
L
qBLg
π2/
h=
Bq
LLg yxLh
=
⇒
⇒
• Density of states (1 LL): n(Ez)=2*gL*n1D(Ezn)z
zzDE
mLEn
12)(
5.0
21
=h
where
For a given E ∑+−
=∞
=0
2/3
2 )5.0(
122)(
nc
cnE
mVEn
ωωπ
hh
h
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kz
εF
cωh
E3D
B=0
Density of states
)n(ε
c/ ωε h
III.2 Theory
++=+= ⊥2
1
2
22
nm
kEEE c
zz ωhh
c
cm
qB=ω ∑+−
=∞
=0
2/3
2 )5.0(
122)(
nc
cnE
mVEn
ωωπ
hh
h
⇓Oscillation of most electronic properties
Magnetization: de Haas-van Alphen (dHvA)
Resistivity: Shubnikov-de Haas (SdH)
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III.2 Theory
kz
εF
cωh
Temperature / Disorder effects on quantum oscillations
*cm
eBω =
kBT / τh=Γ
• Low T measurements
• Need high quality single crystals
τω hh >c
1>τωc
TkBc >ωh
⇒
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−∝ γB
F2πsRR∆M∆R, DT inRS
Dingle temperature(mean free path)
Extremal area
Cyclotron mass
III.2 TheoryLifshitz-Kosevich theory (1956)
T≠≠≠≠0
p=1
Onsager relation ⇒ AF
⇒ m*
⇒
−=
×−=
×==
=
Bexp
694.14exp
/694.14 where)(
2
µπ
B
mTR
BTmXXsh
XR
B
A
πqB
F
cDD
cT
Fh
τπ B
Dk
T2
h=
= gmR *
bS2
cosπ
⇒ gmb
*
Direct measure of the Fermi surface extremal area (but number of orbits ? location in k-space ?)
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B
B10 20 30 40 50 60
-3
-2
-1
0
∆R /
R
B (T)
(4)
(1) or (2)
(3)
(1)+(3)
Extremal Area
(4)
(1)
(3)
F=500 T
F=250 T
F=230 T
III.2 Theory
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III.2 TheoryEnergy scales
For kF ≈ 7 nm-1 (large FS)
eB
kr F
c
h=
−=l
crπexp
)/sinh(
/
0
0
BTmu
BTmuR
c
cT =Dingle term
(the evil term!)
Temperature damping term
• kBT=0.09 meV/K ⇒ 1 meV=11.6 K • gµBB=0.12 x B meV / T
TmeVc [email protected]=ωh
TmeVBm
eBc /12.0 ×== hhω•
gµBB=4.6 meV @ 40 T
• cyclotron orbits
kF=7 nm-1 ⇒ rc=100 nm @ 40 T
kF=1.3 nm-1 ⇒ rc=14 nm @ 40 T
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III.2 TheoryEffect of interactions
Electrons in Fermi gas at T=0 Electrons in Fermi liquid at T=0
Sus
c. (
arb.
uni
ts)
13
110 me
70 me60 me
50 me
70 me
Am
plitu
de (
arb.
uni
ts)
F (kilotesla)0 2 8 10
UPt3
L. Taillefer et al, J. Magn. Magn. Mater’87
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III.3 High magnetic fields lab.
0.00 0.05 0.10 0.15 0.200
10
20
30
40
50
60
B (
T)
t (s)
Bmax
= 61 T
tmontée
= 25 ms
tdescente
= 110 ms
Decreasing:
~ 100 – 250 ms
Numerical lock-inHigh speed digitizer R
Piezoresistif cantilever
BMrr
×=τ
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24 MW 300 l/s
DC field installation LNCMI Grenoble
III.3 High magnetic fields lab.
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III.4 Fermiology
Alkali metals:
Li: 1s2s1
Na: [Ne]3s1
K: [Ar]4s1
Rb: [Kr]5s1
Cs: [Xe]6s1
1 conduction electron (body-centered cubic)
32
3 2
3 a
kn F ==
π
akF
π2620.0=
aN
π2707.0=Γ
⇒ The sphere is inside of the first Brillouin zone
Potassium
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III.4 FermiologyPotassium
Aexp=(1.74 ±0.02) 1016 cm-2
Atheo=1.748 1016 cm-2 (free electron)
Deviation from the sphere
Fixed magnetic field and rotation
∆A ~ 10-4 A
e
AF F
π2
h=
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III.4 FermiologyNoble Metals: Cu, Ag, Au (f.c.c.)
Li: 1s2s1 -Na: [Ne]3s1 -K: [Ar]4s1 Cu: [Ar]3d104s1
Rb: [Kr]5s1 Ag: [Kr]4d105s1
Cs: [Xe]6s1 Au: [Xe]4f145d106s1
d-bands
Free electron models:
FS=sphere inside the FBZ
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III.4 Fermiology
<111>
A.S. Joseph et al, Phys. Rev’66
<100>
<110>
Dog’s bone Lemon
4-corner Rosetta6-corner Rosetta
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III.4 FermiologyARPES in Cu
P. Aebi et al, Surface Science’94
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III.4 Fermiology
3D 2D Q2D
Quasi 2D case
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III.4 Fermiology
D. J. Singh, PRB’95
Band structure calculations
Sr2RuO4: a Quasi-2D Fermi liquid (school case…)
3 sheets of FSα hole like
β, γ electron like
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III.4 FermiologySr2RuO4
A. P. Mackenzie et al, PRL’96
β
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III.4 Fermiology
∆
−∝ )tan(cos
2γBcos
F2πsRR∆M∆R, 00DT θ
θπ
θckJ
B
FJinR FS
∆F
C. Bergemann et al, PRL’00
C. Bergemann et al, Advances in Physics’03
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III.4 FermiologySr2RuO4
C. Bergemann et al, Advances in Physics’03
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III.4 FermiologySr2RuO4: Angular dependence of the amplitude of QO
αααα ββββ
γγγγ
C. Bergemann et al, Advances in Physics’03
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III.4 FermiologySr2RuO4
C. Bergemann et al, Advances in Physics’03
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III.4 FermiologyARPES in Sr 2RuO4
First measurements give results different from band structure calculations!
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III.4 FermiologyARPES in Sr 2RuO4
BUT surface atomic reconstruction seen by STM (Matzdorf et al. Science’00)
Solution: Sample cleaved at 180 K ⇒ surface-related features are suppressed !
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Outline
I. Why and how to measure a Fermi surface ?
II. Angular Resolved Photoemission Electron Spectroscopy (ARPES)
III. Quantum oscillations (QO)1) History2) Theory
a) Semiclassical theorya) Landau levels quantificationb) Lifshitz-Kosevich theoryc) High magnetic field phenomena
3) High magnetic fields facilities4) Fermiology
IV. Hot topics1) Phase transition2) High Tc superconductors
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IV. Hot topicsQO across the metagnetic transition in Sr 3Ru2O7
βδ γ1
α1α2
Paramagnet close to ferromagnetism (small moment)
J.F. Mercure et al, PRB10
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IV. Hot topicsQO across the metagnetic transition in CeRh 2Si2 0 5 10 15 20 25 30
-4
-2
0
2
4
6
8
CeRh2Si
2
T = 50 mKB // [001]
Tor
que
(arb
. uni
ts)
µ0H (Tesla)
12 16 20 24 28 32
-5
0
5
... and above
Below the transition
Osc
illa
tory
tor
que
(arb
. uni
ts)
Field (Tesla)
0 5 10 15 200
2
4
6
8
Fabove
=16 kT
Fbelow
=1.8 kT
Am
plit
ude
(arb
. uni
ts)
dHvA frequency (kT)
10-25 T (below the transition) 27.5-32 T (x10, above)
W. Knafo et al, PRB10I. Sheikin et al, unpublished
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IV. Hot topicsQO in high T c superconductors
PSEUDOGAP FERMIliquid
Platé et al, PRB’05
Tl2Ba2CuO6+δp~0.25
Hussey et al, Nature’03
Mott insulator
½ fillingstrong e-e repulsion
AF ground state
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Photo credit: N. Hussey
Oscillation amplitude < 1% (1 m ΩΩΩΩ) !
B. Vignolle et al, Nature’08
kAF2
0
2πφ=
IV. Hot topicsOverdoped Tl 2Ba2CuO6+δδδδ
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F=18100 ± 50 T
Hussey et al, Nature’03
AMRO
kF=7.35 ±±±± 0.1 nm -1
65 % of the FBZ
kAF2
0
2πφ=Onsager relation :
⇒ Carrier density: n=1.3 carrier /Cu atom (n=1+p with p=0.3)
02)2(
2
φπFA
n k ==Luttinger theorem :
⇒ kF=7.42 ± 0.05 nm-12Fk kA π=
2el mJ/mol.K27γ ±=For overdoped polycristalline Tl-2201: (Loram et al, Physica C’94)
*
2
22BA
el m3
akπNγ
h= ⇒
2el mJ/mol.K16γ ±=Electronic specific heat:
Effective mass : m*= (4.1 ± 1) m0BTmX
Xsh
XR cT /694.14
)( ×== ⇒
Mean free path : ldHvA≈320 Å (ltransp≈670 Å)
−=
l
h
Be
kR FT
πexp ⇒
IV. Hot topicsOverdoped Tl 2Ba2CuO6+δδδδ
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PSEUDOGAP FERMIliquid
?underdoped
ARPES⇒ Fermi arcs
IV. Hot topics
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IV. Hot topicsARPES in underdoped HTSC
Ca2-xNaxCuO2Cl2 K. Shen et al., Science’05
PSEUDOGAP
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PSEUDOGAP FERMIliquid
?underdoped
ARPES⇒ Fermi arcs
IV. Hot topics
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IV. Hot topics
underdopedYBa2Cu3Oy
Shubnikov - de Haas de Haas – van Alphen
Piezoresistif cantilever
D. Vignolles
C. Jaudet et al, PRL’08
N. Doiron-Leyraud et al, Nature’07
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IV. Hot topics
1.7 1.8 1.9 2.0-4
0
4
Aim
anta
tion
(u.a
.)100 / B (T-1)kAF
20
2πφ=
underdopedYBa2Cu3O6.5
overdopedTl2Ba2CuO6+δδδδ
B
1.7 1.8 1.9 2.0
-2
0
2
∆R (
mΩ
)
100 / B (T-1)
B. Vignolle et al, Nature’08N. Doiron-Leyraud et al, Nature’07
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IV. Hot topics
YBa2Cu3O6.5
Frequency : F = (530 ± 20) T
Ak = 5.1 nm-2
= 1.9 % of 1st Brillouin zone
Tl2Ba2CuO6+δδδδ
Frequency : F = (18100 ± 50) T
Ak = 173.0 nm-2
= 65 % of 1st Brillouin zone
n=1.3 carrier /Cun=0.038 carrier/Cu
0φF
n =
Radical change of the carrier density
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Degeneracy lift
Example of Fermi surface reconstruction (commensurate case)
e.g. competing order- AF order (π, π)- d-density wave
IV. Hot topics
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2 hole pockets
FSdH=990 T
nh=0.138 hole/Cu atom
1 electron pocket
ne=0.038 electron/Cu atom
FSdH=530 T
Fermi surface reconstruction (commensurate case)
e.g. competing order- AF order (π, π)- d-density wave
Higher mobility at low T
n=nh-ne=p=0.1
…
IV. Hot topics
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underdoped overdoped
Topological change in FS:Broken symmetry
?
IV. Hot topics
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ReferencesARPES:- A. Damascelli, Z-X Shen and Z. Hussain, " Angle-resolved photoemission spectroscopy of the cuprate superconductors", Rev. Mod. Phys. 75, 473 (2003)
- A. Damascelli, Z-X Shen and Z. Hussain, " Probing the Electronic Structure of Complex Systems by ARPES", Physica Scripta. 109, 61 (2004)
- S. Hüfner, ‘‘Photoelectron Spectroscopy,’’ (Springer-Verlag, Berlin, 1995)
- http://www.physics.ubc.ca/~quantmat/ARPES/PRESENTATIONS/talks.html
- http://www-bl7.lbl.gov/BL7/who/eli/SRSchoolER.pdf
Quantum oscillations:
- D. Shoenberg, "Magnetic oscillations in metals" (Ed. Cambridge Monographs on Physics)
- W. Mercouroff, "La surface de Fermi des métaux" (Ed. Masson)
- C. Bergemann, A. Mackenzie, S. Julian, D. Forsythe and E. Ohmichi, " Quasi-two-dimensional Fermi liquid properties of the unconventional superconductor Sr2RuO4", Advances in Physics 52, 639 (2003)
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I. Why and how to measure a Fermi surface
TEgkT
UC FBv ×=
∂∂= )(
3
2π∫=FE
dEEfEnEU0
)()(
22
*
)(πh
FF
kmEg =
Global properties
• Specific heat where
• Pauli susceptibility )(2
2
FB
Pauli Eggµχ =
• Hall effectnqB
Rxy
H
1==ρ
• Magnetoresistance ∫==SF
24π
S.dkδvevneJ
rrrrr
Eeτ
kδr
h
r=
ES.dv4π
τeJ
SF3
2 rrr
h
r
∫=
wherekδr
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AMRO
Work at 2D and for simple Fermi surface
Angular dependence of the MagnetoResistance Oscilla tions
At particular angles (‘Yamaji angles’)
01 =
∂∂=
z
zk
Ev
h
Semi-classical effectC. Bergemann et al, Advances in Physics’03
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AMRO
• Two-axis rotation probe
Br
Θ → Polar angle
ϕ → Azimutal angle
• Steady magnetic fields
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AMRO
Max. of the magnetoresistance when )4/1()tan(// ±=Θ ikc i π
Projection of kF in the plane ⊥ to B
0 20 40 60 80 100 120 140 160 180
ϕ =
R/R
Θ=9
0°, B
=15
T
Θ (°)
100
110
120
130
140
160150
170180
90
0 20 40 60 80 100 120 140 160 1802000
4000
6000
8000
R (
Ω)
Θ (°)
70 80 90 100 110
8000
8100
8200
8300
8400
R (
Ω)
Θ (°)
Quasi-2D organic metal:(BED0-TTF)2ReO4H2O
C. Proust et al, Synthetic Metals’99
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AMRO
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
3
4
5
6
7
8
index
-tan (Θ)
Quasi-2D organic metal: (BED0-TTF)2ReO4H2O
0.0
0.1
0.2
0
20
40
60
80100
120
140
160
180
200
220
240
260 280
300
320
340
0.0
0.1
0.2
ϕ
k// (A-1)
)4/1()tan(// ±=Θ ikc i π
C. Proust et al, Synthetic Metals’99
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III.2 Theory
Spin splitting: BgE Bµ=∆
Free electron: cm
eBB
m
eE ωhh
h ===∆002
*2
No change of the frequency but additional damping factor due to phase shift
=
∆= gmE
Rc
S 02
cos2cosπ
ωπh
Particular case: spin zero phenomena
Assume)cos(
)( 00 θ
θ mm =
0)cos(2
cos 0 =
=
θπ gm
RS
when ( )12
cos 0
+=
k
gmkθ
0 10 20 30 40 50 60 70 800
1
2
3
4
ampl
itude
(u.
a.)
Θ (°)
Spin splitting in Quantum oscillations
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III.2 TheoryMagnetic breakdown (Cohen&Falicov 1961)
0 5 10 15 20 25 30 350
2
4
6
8
10
12
14
R /
RB
=0 -
1
B (T)
κ-(BEDT-TTF)2Cu(NCS)2
Probability p of MB
−=B
Bp 02 exp
F
c
Ee
mB
h
2
0
∆∝where
T=1.5 K
In strong magnetic fields, electron can tunnel from one orbit to another
∆ → ‘Energy gap’ (forbidden band) between the two orbits
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III.2 TheoryLifshitz-Kosevich theory (1956)
TB
M
,µ
∂Ω∂−= ( )∑ −=Ω
electronsallFEEwhere Thermodynamic grand potentialT=0
∫ ∑
−++=Ω
=
0
0 0
22
2)5.0(
2
κω
πmn
nzFc
z dkEnm
kqBh
h
h
LL crosses EF
e
AF F
π2
h=
C. Bergemann, Thesis
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III.3 High magnetic fields lab.
o LCMI (1992)
o LNCMP (1990)
o HLD (2006)o HFML (2003)
o continue
o pulsé
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State of the art technical performances
Superconducting: USA: 33,8 T (NHMFL) (Commercial: 23 T Bruker)
Resistif : USA: 45,5 T (NHMFL, hybride)
Japan: 38,9 T (TML, hybride)
Europe: 35 T (LNCMI)
Pulsed : USA: 89,8 T (NHMFL)
Japan: 82 T (Osaka)
Europe: 87 T (HLD)
III.3 High magnetic fields lab.