Laboratoire National des Champs Magnétiques Intenses Toulouse

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/

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

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…

Outline







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

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

II. ARPES

Synchroton SOLEIL

http://www.synchrotron-soleil.fr/

From IR to RX

II. ARPES

http://arpes.phys.tohoku.ac.jp/contents/calendar-e. html

II. ARPES

A. Damascelli, http://www.physics.ubc.ca/~quantmat/ARPES/PRESENTATIONS/Talks/ARPES_Intro.pdf

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


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

2) Abrupt potential change along z ⇒ is not conserved across the surface⊥kr


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


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


II. ARPES Non-interacting case


II. ARPES Interacting systems


II. ARPES Example: Quasi-2D overdoped cuprate

A. Ino et al, PRB’02

Outline







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)

David Shoenberg (1911 - 2004)

The experimental pioneer …

III.1 History

and his friend the other post-doc

L.D. Landau (1908 – 1968)

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/

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

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

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=

⇔

⇔

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

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)

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

⇒

−∝ 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 ?)

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

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

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

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

24 MW 300 l/s

DC field installation LNCMI Grenoble


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

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=

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

III.4 Fermiology

<111>

A.S. Joseph et al, Phys. Rev’66

<100>

<110>

Dog’s bone Lemon

4-corner Rosetta6-corner Rosetta

III.4 FermiologyARPES in Cu

P. Aebi et al, Surface Science’94

III.4 Fermiology

3D 2D Q2D

Quasi 2D case

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

III.4 FermiologySr2RuO4

A. P. Mackenzie et al, PRL’96

β

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



III.4 FermiologySr2RuO4: Angular dependence of the amplitude of QO

αααα ββββ

γγγγ




III.4 FermiologyARPES in Sr 2RuO4

First measurements give results different from band structure calculations!

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 !

Outline







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

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

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

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

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


?underdoped

ARPES⇒ Fermi arcs

IV. Hot topics

IV. Hot topicsARPES in underdoped HTSC

Ca2-xNaxCuO2Cl2 K. Shen et al., Science’05

PSEUDOGAP


?underdoped

ARPES⇒ Fermi arcs

IV. Hot topics

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

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

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

Degeneracy lift

Example of Fermi surface reconstruction (commensurate case)

e.g. competing order- AF order (π, π)- d-density wave

IV. Hot topics

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

underdoped overdoped

Topological change in FS:Broken symmetry

?

IV. Hot topics

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)

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

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

AMRO

• Two-axis rotation probe

Br

Θ → Polar angle

ϕ → Azimutal angle

• Steady magnetic fields

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

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

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

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

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


o LCMI (1992)

o LNCMP (1990)

o HLD (2006)o HFML (2003)

o continue

o pulsé

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)


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