NMR in Strongly Correlated Electron Systems

Transcription

1 NMR in Strongly Correlated Electron Systems Vesna Mitrović, Brown University Journée Claude Berthier, Grenoble, September 211

2 C. Berthier, M. H. Julien, M. Horvatić, and Y. Berthier, J. Phys. I France 6, 225 (1996). M. H. Julien, T. Fehér, M. Horvatić, C. Berthier, O. N. Bakharev, P. Ségransan, G.Collin, and J. F. Marucco, Phys. Rev. Lett. 84, 3422 (2).

3 2D Electron Gas MBE heterostructures M242 and M28 1 layers: E 2D Electron Gas in H z : B = B >> 25 nm 3 nm 185 nm 25 nm eb /m* GaAs Ga.7 Al.3 As Si delta doping Ga.9 Al.1 As e - density n = eb /h gµ Β Β n n (E) LLL filling factor ν = 1 S.E. Barrett and collaborators, Yale University - OP-NMR [Fig: S. Melinte, PhD thesis (21)] C. Bertier and collaborators, GNHMFL,Grenoble - NMR

4 2D Electron Gas NMR line-shift ( 69,71 Ga) N N K S P = = 2 N + N n average spin polarisation k S z k GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference 1 layers: 3 nm 25 nm GaAs Ga.7 Al.3 As Si delta doping Ga.9 Al.1 As K S (R i ) P(R i )ρ e (R i ) (GNHMFL,Grenoble)

5 2D Electron Gas NMR line-shift ( 69,71 Ga) N N K S P = = 2 N + N n average spin polarisation k S z k GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference K S (R i ) P(R i )ρ e (R i ) (GNHMFL,Grenoble)

6 2D Electron Gas NMR line-shift ( 69,71 Ga) N N K S P = = 2 N + N n average spin polarisation k 3 nm S z k 25 nm GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference barriers Quantum Wells K S (R i ) P(R i )ρ e (R i ) close to Si δ-doping (GNHMFL,Grenoble)

7 2D Electron Gas NMR line-shift ( 69,71 Ga) K S P = N N N + N = 2 n average spin polarisation k S z k P(gµ B H/k B T ) if g GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference 1 layers: 3 nm 25 nm GaAs Ga.7 Al.3 As Si delta doping Ga.9 Al.1 As (GHMFL,Grenoble) S. Melinte, et al., Phys. Rev. B 64, (21).

8 ν =1 state - elementary excitations? Localized holes and electrons Skyrmion Anti-skyrmion S.L. Sondhi et al., Phys. Rev. B 47, (1993). H. A. Fertig et al., Phys. Rev. B 5, 1118 (1994). Energy gap & size (s) S.E. Barrett et al., Phys. Rev. Lett. 74, 5112 (1995). P. Khandelwal et al., Phys. Rev. Lett. 86, 5353 (21). g = E z E c

9 Skyrmion Size (S) Energy gap & size (s = N o of reversed spins within an excitation) g = E z E c Limits # of spin-flips Favors FM ordering E z = gµ B B E c = e 2 εl B g s To vary skyrmion size tune g : g - hydrostatic pressure - Al.13 Ga.87 As - QW (g theory = -.4 ; g GaAs =.44)

10 Large Skyrmions in Al.13 Ga.87 As (g ) Quantum Wells Magnetotransport measurements in Al.13 Ga.87 As QW (S. P. Shukla et al., PRB 61, 4469 (2)) s 5 Small g sample: 3 QWs g theory = -.4 g GaAs =.44 n 2D = cm -2 Problem: How to separate the signals? P(gµ B H/k B T ) m = 3 cm 2 / Vs QW: 24 nm of Al.13 Ga.87 As 1. barrier ----> 71 Ga, ν = 1 barriers: nm of Al.35 Ga.65 As.8 T = 65 mk Magnitude [a. u.] Frequency [MHz]

11 Separate overlapping QW and barriers signal? Small tip angle technique : M(t), pulse, signal π/2 t = π t = τ M (t=2τ) t = 2τ repeat t = T R time S N = f TR, Tip Angle T 1

12 Separate overlapping QW and barriers signal? Small tip angle technique : M(t), pulse, signal π/2 t = π t = τ M (t=2τ) t = 2τ repeat t = T R time S N = f TR, Tip Angle T 1

13 Separate overlapping QW and barriers signal? Small tip angle technique : M(t), pulse, signal π/2 t = π t = τ M (t=2τ) t = 2τ repeat t = T R time % 9 % S N = f TR, Tip Angle T 1 Tip Angle θ [rad] % 7 % 6 % 5 % 4 % 3 % 2 %.4 1 % T R / T 1

14 Separate overlapping QW and barriers signal? Small tip angle technique : S N = f TR, Tip Angle T 1 GOOD!!! very different T 1 for QW and barriers Pulse power dependence : Tip Angle θ [rad] % 9 % 8 % 7 % 6 % 5 % 4 % 3 % 2 % Amplitude [a. u.] T = 65 mk barriers QW.4 1 % T R / T Frequency [MHz] Pulse Length [µs] 2 µs =.75 π/2 [V. Mitrović et al., 27]

15 QW signal Magnitude [a. u.] Ga, ν = 1 QW contribution only (barriers subtracted) T = 65 mk 13 mk 28 mk 315 mk 375 mk 45 mk 612 mk 7 mk 98 mk "P = 1" Frequency [KHz] Frequency shift [KHz] Line Width (FWHM) Average Line Position Polarization Frequency [KHz] Temperature [K] 1. g -.1 g GaAs at low-t : "negative", small, strongly inhomogeneous polarization

16 Superconductor in a Magnetic Field The destruction of SC by a magnetic field I. Orbital Effect II. Pauli paramagnetism Abrikosov Vortex Lattice Cooper-pairs Breaking G-L Equation: H orb c 2 = Φ 2πξ 2 E P = E c H P c 2 = 2 gµ B STM images obtained at NIST Relative importance of the two effects described by the Maki Parameter: α = 2 Horb c 2 Hc P χ nh 2 = 1 2 N() 2 χ n = 1 2 (gµ B) 2 N

17 The FFLO State Pauli pair breaking dominates over the orbital effects α 2 Horb C 2 HC P > a novel SC phase - FFLO is predicted to form FFLO * Cooper pairs acquire finite momentum, q 2µ B H/v F & (r) varies periodically in space (r) 1/q BCS k, k FFLO k, k + q q 2µ BH u F P. Fulde and R. A. Ferrell, Phys. Rev. 135, A55 (1964); A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964).

18 The FFLO State Pauli pair breaking dominates over the orbital effects α 2 Horb C 2 HC P > a novel SC phase - FFLO is predicted to form FFLO * Cooper pairs acquire finite momentum, q 2µ B H/v F & (r) varies periodically in space (r) 1/q P. Fulde and R. A. Ferrell, Phys. Rev. 135, A55 (1964); A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964).

19 FFLO SC Imbalanced Spin Population T= E Nature and stability of SC phase with population imbalanced? k' k E F FFLO P. Fulde and R. A. Ferrell, Phys. Rev. 135, A55 (1964); A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964). ~ µ B B q = k' - k Phases with gapless excitations g(e) G. Sarma, J. Phys. Chem. Solids 24, 129 (1963); breached pair SC - W. V. Liu and F. Wilczek, PRL. 9, 472 (23); R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (24). Mixed Phases of SC and Normal P. F. Bedaque, H. Caldas, G. Rupak PRL 91, 2472 (23); H. Caldas, PRA 69, 6362 (24); J. Carlson, S. Reddy, PRL 95, 641 (25); M. W. Zwierlein et al., Science 311, 492 (26).

20 FFLO Observed? The conditions for the FFLO to stabilize are very stringent: H orb c 2 Type-II SC with very high are required so that the Pauli paramagnetic limit can be reached. In other words the FFLO state can form only in materials with large Maki parameter ( α>1.8). Materials with high purity are required (ξ l) since impurities destroy the FFLO state. Materials with anisotropic Fermi surface or gap function are more likely to observe an FFLO phase since such anisotropy favors its stabilization. The dimensionality of the system can also be crucial. Low dimensionality proves to be beneficial.

21 CeCoIn5 Heavy-Fermion SC : ( Tc = 2.3 K) CeCoIn5 H orb c 2 1 ξ 2 (m ) 2 Very Clean Material : very high. l 3 nm ξ 3 Å Ce Co In1 In2 Anisotropic SC gap function : d x 2 y 2 symmetry with line nodes. Layered structure. quasi-2d electronic nature Rich ground for interesting physics: Unconventional SC, Quantum Criticality, Interplay of SC and Magnetism

22 Phase Diagram 12 1 H [1] H [T] H [T] H [1] T T [K] [K] A. Bianchi et al., PRL 91, 1874 (23)

23 Phase Diagram 12 1 H [1] H [T] H [T] H [1] T T [K] [K] A. Bianchi et al., PRL 91, 1874 (23)

24 Phase Diagram H [T] H [T] FFLO? H [1] H [1] Pauli pair breaking dominates over the orbital effects Other reports supporting the FFLO: Thermal conductivity Magnetization Penetration depth Ultrasound velocity NMR α 2 Horb C 2 HC P > T T [K] [K] A. Bianchi et al., PRL 91, 1874 (23) & a novel SC phase - FFLO is predicted to form Cooper pairs acquire finite (r) momentum, q 2µ B H/u F (r) varies periodically in space. 1/q

25 Phase Diagram H [T] H [T] FFLO? mag. order! H [1] H [1] Pauli pair breaking dominates over the orbital effects Other reports supporting the FFLO: Thermal conductivity Magnetization Penetration depth Ultrasound velocity NMR α 2 Horb C 2 HC P > T T [K] [K] A. Bianchi et al., PRL 91, 1874 (23) & a novel SC phase - FFLO is predicted to form Cooper pairs acquire finite (r) momentum, q 2µ B H/u F (r) varies periodically in space. 1/q NMR and Neutron Diffraction reveal long-range magnetism nature of SC, character of magnetic order, driving mechanism of their coexistence?

26 NMR Measurements Ce Co In1 In2 H [1]

27 NMR Measurements Ce Co In1 In2 H [1]

28 NMR Measurements Ce Co In1 In2 H [1]

29 NMR Measurements Ce Co In1 In2 H [1]

30 NMR Measurements Ce Co In1 In2 H [1] FFLO * (r) varies periodically in space ξ 35 Å LRO a 5 Å Same features on all 3 India SC Different features on different In sites MAGNETISM

31 Phase Diagram by NMR IC + FFLO? H* Normal H [T] 1 9 esc H c2 8 lfsc Temperature [K] G. Koutroulakis et al., PRL (21).

32 Phase Diagram by NMR IC + FFLO? H* Normal H [T] 1 9 esc H c2 8 lfsc Temperature [K] G. Koutroulakis, M. D. Stewart, Brown University G. Koutroulakis et al., PRL (21). M. Horvatić, C. Berthier, Laboratoire National des Champs Magnétiques Intenses, CNRS G. Lapertot, G. Knebel, J. Flouquet, SPSMS, CEA, Grenoble CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

33 SC & AF G. Knebel et al., J. Phys. Soc. Jpn. 77, (28). (28). V. F. Mitrovic et al., Physica B 43, 986 (28). Ce Co In1 In AF FFLO N H [T] 9 8 SC Temperature [mk]

34 Phase Diagram by NMR H [T] IC + FFLO?.2 H* Normal esc lfsc Temperature [K] H c Magnitude [arb.units] IC In(1) In(1) In(2 ) ac ( H o ) H o = T T T T T T 1.95 T 1.76 T 1.69 T 1.63 T SC T~ 7 mk In (2 ) ac G. Koutroulakis et al., PRL 14, 871 (21) 9.67 T 9.43 T f - f [MHz] res 1.36 T 9.94 T 9.9 T 7.99 T

35 Phase Diagram by NMR H [T] IC + FFLO?.2 H* Normal esc lfsc Temperature [K] H c Magnitude [arb.units] IC In(1) In(1) In(2 ) ac ( H o ) H o = T T T T T T 1.95 T 1.76 T 1.69 T 1.63 T SC T~ 7 mk In (2 ) ac G. Koutroulakis et al., PRL 14, 871 (21) 9.67 T 9.43 T f - f [MHz] res 1.36 T 9.94 T 9.9 T 7.99 T

36 Phase Diagram by NMR IC + FFLO? H* Normal H [T] 1 9 esc H c2 8 lfsc Temperature [K]

37 Phase Diagram by NMR H [T] IC + FFLO?.2 H* Normal esc lfsc Temperature [K] H c Magnitude [arb.units] IC In(1) T~ 7 mk In(2 ) ac ( H o ) In (2 ) ac 9.67 T 9.43 T f - f [MHz] res H o = T T 1.36 T 9.94 T 9.9 T 7.99 T Rapid loss of spin coherence Strong AF fluctuations present

38 Phase Diagram by NMR H [T] IC + FFLO?.2 H* Normal esc lfsc Temperature [K] H c Magnitude [arb.units] IC In(1) T~ 7 mk In(2 ) ac ( H o ) In (2 ) ac 9.67 T 9.43 T f - f [MHz] res H o = T T 1.36 T 9.94 T 9.9 T 7.99 T Rapid loss of spin coherence Strong AF fluctuations present Significant line broadening Enhanced susceptibility around defects

39 Phase Diagram by NMR H [T] IC + FFLO?.2 H* Normal esc lfsc Temperature [K] H c Magnitude [arb.units] IC In(1) T~ 7 mk In(2 ) ac ( H o ) In (2 ) ac 9.67 T 9.43 T f - f [MHz] res H o = T T 1.36 T 9.94 T 9.9 T 7.99 T Rapid loss of spin coherence Strong AF fluctuations present Significant line broadening Enhanced susceptibility around defects New Phase: possibly a FFLO in the presence of strong AF fluctuations Y. Yanase & M. Sigrist, J. Phys. Soc. Jpn. 78, (29).