Quantum Oscillations, Magnetotransport and the Fermi Surface of cuprates Cyril PROUST

Quantum Oscillations, Magnetotransport and the Fermi Surface of cuprates Cyril PROUST Laboratoire National des Champs Magnétiques Intenses Toulouse

Collaborations D. Vignolles B. Vignolle C. Jaudet J. Levallois A. Audouard L. Taillefer N. Doiron-Leyraud D. LeBœuf J-B Bonnemaison M. Nardone D. Bonn B. Ramshaw J. Day R. Liang K. Behnia W. Hardy N. Hussey A. Carrington A. Bangura J. Fletcher R. A. Cooper M. M. J. French University of St Andrews A.P. Mackenzie

Outline Introduction to quantum oscillations Quantum oscillations on both sides of the phase diagram The case for an electron pocket Cartography of the electron pocket Fermi surface reconstruction scenarios

Quantum theory E = Ez + E = h kz m + hωc n + 1 ω = c qb m c m n( E) = πv h 3/ hω Density of states c n=0 1 E hω ( n + 0.5) c 3D E n(ε ) B=0 ε / hω c ε F Oscillation of most electronic properties hω c Magnetization: de Haas-van Alphen (dhva) k z Resistivity: Shubnikov-de Haas (SdH)

ω = c eb * m Temperature / Disorder effects on quantum oscillations k B T / Quantum theory Γ = h τ Low T measurements hω c >k B T Need high quality single crystals ε F h ω > h τ c ω c τ > 1 hω c k z

Quantum theory Lifshitz-Kosevich theory (1956) R, M R T R D R S sin π F B γ F B R R R T D S = = h πq AF B X sh( X ) where X Onsager relation A F = 14.694 Tm / B m * 14.694 TDmc π = exp = exp TD = B µ B * = cos π * mbg m g b c h πk B τ Extremal area Cyclotron mass Dingle temperature (mean free path) Spin factor (spin zero) Direct measure of the Fermi surface extremal area (but number of orbits? location in k-space?)

The overdoped case: Tl Ba CuO 6+δ Fermi surface of overdoped Tl-01 φ π 0 F = A k B Photo credit: N. Hussey Oscillation amplitude < 1% (1 mω)! B. Vignolle et al, Nature 08

The overdoped case: Tl Ba CuO 6+δ F=18100 ± 50 T Onsager relation : A = π k k F Luttinger theorem : φ π 0 F = A k k F =7.4 ± 0.05 nm -1 (65 % of the FBZ) Ak F n = = (π ) φ 0 AMRO k F =7.35 ± 0.1 nm -1 Hussey et al, Nature 03 ARPES Platé et al, PRB 05 Carrier density: n=1.3 carrier /Cu atom (n=1+p with p=0.3) Electronic specific heat: m * =5 ± 0.5 m 0 For overdoped polycristalline Tl-01: γ = 7 ± 1mJ/mol.K All the numbers are in excellent agreement with - in-field probes: AMRO, Hall effect - zero field probes: ARPES, thermodynamic el πn k 3h A B * γ el = m = 7 ± a 0.5mJ/mol.K (Loram et al, Physica C 94)

Quantum oscillations in UD YBa Cu 3 O 6.5 Shubnikov - de Haas φ π 0 F = A k de Haas van Alphen D. Vignolles Piezoresistif cantilever N. Doiron-Leyraud et al, Nature 07 Frequency : F = (540 ± 4) T Mass : m* = (1.76 ± 0.07) m 0 Closed pocket and coherent QP at low T C. Jaudet et al, PRL 08

Quantum oscillations in UD YBa Cu 4 O 8 Frequency : F = (660 ± 30) T Mass : m* = (.7 ± 0.3) m 0 A. Bangura et al, PRL 08 See also E.A Yelland et al, PRL 08

Quantum oscillations in HTSC underdoped YBa Cu 3 O 6.5 overdoped Tl Ba CuO 6+δ 4 R (mω) 0 B Aimantation (u.a.) 0-1.7 1.8 1.9.0 100 / B (T -1 ) N. Doiron-Leyraud et al, Nature 07 φ π 0 F = A k -4 1.7 1.8 1.9.0 100 / B (T -1 ) B. Vignolle et al, Nature 08

Implication of quantum oscillations YBa Cu 3 O 6.5 Frequency : F = (530 ± 0) T A k = 1.9 % of 1 st Brillouin zone Tl Ba CuO 6+δ Frequency : F = (18100 ± 50) T A k = 65 % of 1 st Brillouin zone Amplitude de Fourier (u.a.) 1.0 n=0.038 carrier/cu Χ Radical 0.5 change of the carrier density 0.0 0 1 10 15 0 5 F (kt) Γ n = F φ 0 n=1.3 carrier /Cu R H (mm 3 /C) 0 0-0 -40 Hall effect (LeBoeuf et al, Nature 07) y=6.67 0 5 50 75 100 15 150 T (K) y=6.5 Tl-01 Electron pocket C. Jaudet et al, Physica B 09

Scenarios for the Fermi surface Small Pockets LDA Doped Mott Insulator e.g. 4 nodal hole pockets Competing order e.g. FS reconstruction spin SC

Doped Mott insulator scenario four-nodal hole pockets e.g. doped Mott insulator ARPES in Na-CCOC K. Shen et al., Science (005) BUT Negative Hall effect (electron like) Luttinger theorem for D FS: Ak F n = = (π ) φ 0 YBa Cu 3 O 6.5 F = 530 T 0.15 carriers per planar Cu atom!!! (0.1) YBa Cu 4 O 8 F = 660 T n = 0.19 carriers per planar Cu atom!!! (0.14)

Scenarios for the Fermi surface Small Pockets LDA Doped Mott Insulator e.g. 4 nodal hole pockets Competing order e.g. FS reconstruction spin SC

Fermi surface reconstruction AF / d-dw order (Rice, Chakravarty) (Field induced) SDW (Sachdev, Harrison) Stripes (Millis and Norman, Vojta) CDW / SDW Q=(π, π) Q=(π±δ, π) Q=(3π/4, π) Fermi surface reconstruction scenarios electron pocket at the anti-node

Hall and Seebeck coefficients y=6.5 Tl-01 Hall y=6.67 LeBoeuf et al, Nature 07 y=6.67 Seebeck Chang et al, PRL 10

Two bands model in Y48 P. Rourke et al, PRB 10

π-shift in Rxx and Rxy R xy (Ω) 0.00-0.03 0.08 1.5K.1K.5K 3K 3.5K 4.K R / R(45T) 1.4 1. 1.0 T=1.5 K R xx -R xy R xx (Ω) 0.00 30 40 50 60 C. Jaudet et al, Physica B 09 B (T) 0.8 40 45 50 55 60 B (T)

Angle-dependence of QO in YBa Cu 3 O 6.6 NO F β frequency! Angle dependence Temperature dependence B. Ramshaw et al, to be published in Nature Physics

Angle-dependence of QO in YBa Cu 3 O 6.6 Bilayer splitting + Warping c-axis coherence F = * 4mt eh F=90 T t ~1.3 mev (15 K)

Spin zero phenomena Staggered moments have a large component along the field direction Not compatible with a pure AF scenario! B. Ramshaw et al, to be published in Nature Physics

Location of the electron pocket c-axis Resistivity in YBCO t t ( k) = x ya 4 [ cos( k a) cos( k )] σ c = 4 e ct 4 πh m * τ c K. Takenaka et al, PRB 94 Scenario: Charge confinement in the CuO plane (ρ c as T 0)?

c-axis magnetoresistance in UD YBCO Tc=57 K Tc=61 K Tc=66 K B. Vignolle et al, unpublished

Two-band model: ( ) ( ) ( ) ( ) 0 1 ) ( B B B R R B R R B e h e h e h e e h h e h e h β α ρ µ µ µ µ µ µ µ µ µ µ ρ + + = + + + + + + = ρ c (0): extrapolated zero-field resistivity B. Vignolle et al, unpublished Extrapolation of the magnetoresistance

Temperature dependence of ρ c (0) Cross-over: Incoherence Coherence T coh =7 ± 5 K for p=0.109 Good agreement with t ~15 K from QO Sr RuO 4 3D Fermi Surface in underdoped YBCO! B. Vignolle et al, unpublished

Phase diagram 50 YBa Cu 3 O y 00 T * T (K) 150 100 T onset 50 T c T coh 0 0.05 0.10 0.15 Hole doping p B. Vignolle et al, unpublished

Electron pocket at the anti-node 1.4 T=1.5 K R xx R / R(45T) 1. 1.0 -R xy 0.8 40 45 50 55 60 B (T)

Doping dependence of QO c-axis magnetoresistance in underdoped YBa Cu 3 O y

Doping dependence of the Hall effect No sign change below p~0.08 0 R H / R H ( 100 K ) - -4-6 -8-10 p = 0.1 45 T p = 0.107 54 T p = 0.10 50 T p = 0.088 50 T p = 0.085 50 T p = 0.083 50 T p = 0.078 55 T -1 0 0 40 60 80 100 T [K] D. LeBoeuf et al, arxiv: 1009.078

Phase diagram Lifshitz transition B. Vignolle et al, unpublished

Lifshitz transition Spin density wave with (π, π) reconstruction 0.08 Doping Stripe scenario

Stripe scenario Only one QO frequency

Metal-insulator cross-over Stripe scenario In-plane resistivity From Y. Ando et al, PRL 04 D. LeBoeuf et al, arxiv: 1009.078

In-plane anisotropy Stripe scenario Y. Ando et al, PRL 0

Conclusion Evidence of closed and coherent FS in UD YBCO and OD Tl-01 Small pockets vs large orbit Evidence of electron pocket Reconstruction of the FS Topological change in FS: Broken symmetry If pseudogap causes reconstruction results in overdoped Tl01 argue for a QCP hidden by the SC dome Restoration of the coherence along c-axis at low T in underdoped YBCO 3D Fermi surface Fermi liquid behavior