μsr Studies on Magnetism and Superconductivity

The 14 th International Conference on Muon Spin Rotation, Relaxation and Resonance (μsr217) School (June 25-3, 217, Sapporo) μsr Studies on Magnetism and Superconductivity Y. Koike Dept. of Applied Physics, Tohoku University Spring Summer Sapporo Sendai Winter Tokyo Autumn

1. Introduction to μsr 2. μsr in magnetic materials 3. μsr studies in high-t c superconductors (1) Magnetic anomalies * Antiferromagnetic ordered phase * 1/8 anomaly and stripe order * Impurity-induced magnetism * Undoped superconductivity in T -cuprates (2) Inhomogeneity of internal field * Penetration depth in type-ii superconductors (3) Muon Knight shift * Pseudo gap 4. μsr vs neutron scattering and NMR

Muon μ elementary particle * mass: m μ ~ 2 m e ~1/9m p * charge: μ + or μ - * spin: S = 1/2 * magnetic moment: m = γ μ ћs γ μ = gyromagnetic ratio =ge/(2m μ c) =2π 13.55 khz/g for μ + * life time = 2.2 μsec

A spin-polarized positive muon μ + is injected, stops near a negative ion, feels the internal magnetic field H, makes a Larmor precession. H Freq. ω = γ μ H Period T = 2π/ω Cu 2+ μ + O 2- =2π/(γ μ H)

Decay of μ + life time = 2.2 μsec μ + e + + ν e + ν μ muon positron two neutrinos μ + e + e + is emitted at the moment of the decay in the direction of μ + spin in the highest probability.

In the case of no internal magnetic field e + sample μ + (S=1/2) spin-polarized!! Backward counters B(t) Asymmetry A(t) = F(t) B(t) F(t) + B(t) Forward counters F(t) G Z (t): polarization rate of muon spins = A G Z (t) 1 t

In the case of uniform internal magnetic field e + Backward counters B(t) Asymmetry A(t) = sample F(t) B(t) F(t) + B(t) H Forward counters F(t) G Z (t) = A G Z (t) 1 μ + (S=1/2) spin-polarized T = 2π/(γ μ H) t

In the case of inhomogeneous internal field e + Backward counters B(t) H sample Forward counters F(t) G Z (t) μ + (S=1/2) spin-polarized Asymmetry A(t) = F(t) B(t) F(t) + B(t) = A G Z (t) 1 damping t

e + sample μ + B(t) F(t)

Magnetism Various internal field Magnetic order Ferromagnetic Antiferromagnetic Spiral Magnetic disorder Spin glass static Paramagnetic dynamical (time-dependent) at high T

Paramagnetic state Electron spins are fast fluctuating Time-averaged internal H due to electron spins = Internal H due to nuclear spins: slowly fluctuating static random internal H: a few gauss 3 orders of magnitude smaller than H due to electron spins nuclear spin electron spin + - + ++

Muon spin relaxation due to nuclear spins Random small internal H: assumed as gaussian distribution G Z (t) = 1/3 + (2/3)(1 - Δ 2 t 2 )exp(-δ 2 t 2 /2) Δ:FWHM of internal H γ μ G Kubo-Toyabe function Z (t) 1 1/3 1/3 Slow depolarization 3/ 3/Δ t

G Z (t) = 1/3 + (2/3)(1 - Δ 2 t 2 )exp(-δ 2 t 2 /2) G Z (t) Δ:FWHM of internal H γ μ Kubo-Toyabe function 1 1/3 1/3 3/ 3/Δ t 1/3 tail 1/3 tail A muon makes no precession due to the z-component of internal field parallel to the initially polarized direction. G Z (t) G Z (t) Precessions due to the x- and y-components of internal field are damped. 1 1-1 z-comp. t t x-comp. y-comp.

Development of magnetic correlation Electron spin fluctuation slows down A muon feels random slowly fluctuating large internal H Small internal H due to nuclear spins is ignored. nuclear spin + + ++ electron spin G Z (t) - 1 Fast depolarization Disappearance of 1/3 tail Z-component of internal field is also fluctuating. t

Ferromagnetic state A(t) = e + F(t) B(t) F(t) + B(t) = A G Z (t) sample μ + B(t) F(t) G Z (t) G Z (t) G Z (t) 1 1 1 t t t -1

Ferromagnetic state polycrystal G Z (t) multi-site for muons bad quality of a sample G Z (t) 1 1 damping 1/3 1/3 t t

Antiferromagnetic state polycrystal one site for muons G Z (t) multi-site for muons G Z (t) 1 1 damping 1/3 1/3 t t

Spin glass state G Z (t) G Z (t) 1 Strong damping 1 1/3 1/3 t t

Muon Spin Relaxation (μsr) Sample 2.2μsec e + in zero external magnetic field μ + beam B (t) Counter A() = 1 F (t) Counter A(t) = 1 e + F A ( t ) F High T: paramagnetic ( t ) ( t ) B ( t ) = Polarization rate of muon spins B ( t ) development of Low T: magneric order magnetic correlation Asym. nuclear spin electron spin Time random small internal field Gaussian Exponential Oscillation Asym. Time random large internal field Asym. Time homogeneous internal field

T-(La,Sr) 2 CuO 4 High-T c Cuprates Hole-Doped Several 1 kinds Electrond-Doped 3 kinds T- (Ca,Na) 2 CuO 2 Cl 2 T*-(Nd,Ce,Sr) T -(Nd,Ce) 2 CuO 4 (Sr,Ln)CuO 2 CuO 4 2 Cu O YBa 2 Cu 3 O 7 Bi 2 Sr 2 CaCu 2 O x Bi 2 Sr 2 Ca 2 Cu 3 O x Blocking layer T - Li x Sr 2 CuO 2 Br 2 Blocking layer CuO 2 plane

Electron-doped system T (K) 3 Hole-doped system x AF SC T N 2 1 underdoped overdoped regime regime optimally doped 1/8 anomaly SC.2.1.1.2 Electron doping Hole doping T c x For the determination of T N, magnetic susceptibility is not available. There is little change in the susceptibility at T N, because a 2-dim. spin-correlation is developed at high temperatures. μsr is very useful for the determination of T N. doped electron Antiferromagnetic Mott Insulator doped hole Cu 2+ spin (s=1/2)

N. Nishida et al., Jpn. J. Appl. Phys. 26 (1987) L1856 YBa 2 Cu 3 O 7

Electron-doped system T (K) 3 Hole-doped system x SC AF T N 2 1 underdoped overdoped regime regime optimally doped 1/8 anomaly SC.2.1.1.2 Electron doping Hole doping μsr is very useful for the determination of T N!! Sintered samples are available for μsr measurements!! -- A single crystal is not necessary. T c x doped electron Antiferromagnetic Mott Insulator doped hole Cu 2+ spin (s=1/2)

1/8 Anomaly in the La214 System La 2-x Ba x CuO 4 La 2-x Sr x CuO 4 Zn % Tc Tc Zn 1% Moodenbaugh et al., PRB 38, 4596 (1988). Kumagai et al., JMMM 76&77, 61 (1988). Koike et al., Solid State Commun. 82, 889 (1992). The TLT structure or Zn doping the 1/8 anomaly. The origin of the 1/8 anomaly was not clarified for a while.

I. Watanabe, K. Kawano, K. Kumagai, K. Nishiyama, K. Nagamine J. Phys. Soc. Jpn. 61 (1992) 358 Zero-field μsr in La 2-x Ba x CuO 4 (x=1/8) I. Watanabe La 2-x Ba x CuO 4 Tc x = p = 1/8 Confirmation of a long-range magnetic order 1/8 anomaly Sintered samples are available for μsr measurements!!

Neutron Elastic Scattering Stripe order of holes and spins!! La 1.6-x Nd.4 Sr x CuO 4 (x=.12) single crystal SDW CDW hole domain Spin domain Cu 2+ spin CuO 2 plane La 1.6-x Nd.4 Sr x CuO 4 Tranquada et al., Nature 375, 561 (1995) PRB 54, 7489 (1996). Tc

x=.1 x=.115 x=.13 x =.1 x =.115 x =.13 1.5 y = y = y = ZF-μSR in La 2-x Sr x Cu 1-y Zn y O 4 1. T c.115.1.13 I. Watanabe T. Adachi y =.1 p y = Nomalized Asymmetry.5. 1..5. 1..5. 1..5. 1..5. 1..5. 1..5. 1..5. Zn operates to pin dynamical y =.7 stripes y =.7 y =.7 1. Zn 7%.5 18516(R) (22). the formation of static stripes!!. y =.1 y =.1 y =.1 1. Watanabe et al., Phys. Rev. B, 65, Adachi et al., Phys. Rev. B, 69, 18457 (24). 2 K 15 K 6 K 2 K.5 y =.25 y =.5 y =.75 y =.1 y =.2 y =.3 y =.5. 1 2 Time ( sec ) y =.25 y =.5 y =.75 y =.1 y =.2 y =.3 y =.5 Static stripes are non-superconducting. 1 2 Time ( sec ) 2 K 15 K y =.25 y =.5 y =.75 y =.1 y =.2 y =.3 y =.5 6 K 2 K 1 2 3 Time ( sec ) Zn-free Zn.25% Zn.5% Zn.75% Zn 1% Zn 2% Zn 3% Zn 5% Zn 1%

Static Stripe Dynamical Stripe Non-superconducting Superconducting Dynamical stripes may play an important role in the appearance of high-t c superconductivity!! (Theory by Emery and Kiverson)

T-(La,Sr) 2 CuO 4 High-T c Cuprates Hole-Doped Electrond-Doped T- (Ca,Na) 2 CuO 2 Cl 2 T*-(Nd,Ce,Sr) 2 CuO 4 Bi2212 system 3 kinds T -(Nd,Ce) 2 CuO 4 (Sr,Ln)CuO 2 Bi 2 Sr 2 Ca 1-x Y x Cu 2 O 8+δ 1 Bi 2 Sr 2 Ca 1-x Y x Cu 2 O 8+ YBa 2 Cu 3 O 7 Bi 2 Sr 2 CaCu 2 O x Bi 2 Sr 2 Ca 2 Cu 3 O x Tc (K) 5 T - Li x Sr 2 CuO 2 Br 2 p=1/8.6.5.4.3.2.1 x (Y)

M. Akoshima, T. Noji, Y. Ono, Y. Koike Phys. Rev. B 57 (1998) 7491 1 Bi 2 Sr 2 Ca 1-x Y x (Cu 1-y Zn y ) 2 O 8+ M. Akoshima I. Watanabe, M. Akoshima, Y. Koike, K. Nagamine, Phys. Rev. B 6 (1999) R9955 Tc (K) 5.6.5.4.3 p ~ 1/8 x (Y).2 y= y=.1 y=.2 y=.25 y=.3.1 1/8 anomaly is universal!! The dynamical stripes may be universal in the high-t c s. The dynamical stripes may be important for the high-t c.

La 2-x Sr x CuO 4 Noramlized Asymmetry 1.5 1..5 LSCO 2 K 8 K 5 K 4 K 3 K 2 K.2.4.6.8 1. Time (μsec) x =.8 x =.115 x =.2 5 2 K 6 K 4 K 2 K.2.4.6.8 1 Time (μsec) 2, 1, 6, 2 K T. Adachi et al., PRB (24, 28). Y. Tanabe et al., PRB (211)..2.4.6.8 1 Time (μsec) T c (K) 4 3 2 1 La 2-x Sr x CuO 4 SC K. M. Suzuki.1.2.3 p (per Cu) 36

Fe-substituted La 2-x Sr La x2-x CuO Sr x4 Cu 1-y Fe y O 4 Noramlized Asymmetry Noramlized Asymmetry 1.5 1..5 1.5 1..5 2 K 8 K 5 K 4 K 3 K 2 K.2.4.6.8 1. Time (μsec) 2 K 1K 8 K 7 K 6 K 2 K LSCO x =.8 x =.115 x =.2 T c (K) 5 Fe 1% p =.7 4 3 2 1.2.4.6.8 1. Time (μsec) 2 K 6 K 4 K 2 K.2.4.6.8 1 Time (μsec) La 2-x Sr x CuO 4 2 K 15 K 12 K 1 K SC 2 K.1.2.3.2.4.6.8 1. Time (μsec) p (per Cu) 2, 1, 6, 2 K T. Adachi et al., PRB (24, 28). Y. Tanabe et al., PRB (211)..2.4.6.8 1 Time (μsec) p =.115 p =.2 2 K 12 K 1 K 6 K 2 K.2.4.6.8 1. 37 Time (μsec)

Normalized Asymmetry 2.4 2.2 2 1.8 1.6 1.4 1.2 1.8.6.4.2 La 2-x Sr x Cu 1-y Fe y O 4 y =.1, T = 2K p = x-y =.5.7.9.115.13.15.17.2.22.225 Muon spin precession K. M. Suzuki et al., Phys. Rev. B 86 (212) 14522 No precession (static shortrange order).5 1 time (μsec) Distinct magnetic states between underdoped and overdoped regimes 38

Antiferromagnetic state 1 Homogeneous field Inhomogeneous field Spin glass state 1 1 t t Very inhomogeneous field 1/3 t

Normalized Asymmetry 2.4 2.2 2 1.8 1.6 1.4 1.2 1.8.6.4.2 La 2-x Sr x Cu 1-y Fe y O 4 y =.1, T = 2K.5 1 Underdoped regime time (μsec) p = x-y =.5.7.9.115.13.15.17.2.22.225 Fe Muon spin precession K. M. Suzuki et al., Phys. Rev. B 86 (212) 14522 No precession (static shortrange order) Strong electron-correlation Fermi liquid Weak electron-correlation Overdoped regime Fe Stripe order of Cu spins and holes pinned by Fe Spin glass of Fe spins due to RKKY int. 4

Electron-doped system T (K) 3 Hole-doped system x AF SC T N 2 1 underdoped overdoped regime regime optimally doped 1/8 anomaly SC.2.1.1.2 Electron doping Hole doping T c x Nd 2 CuO 4 -type (Tʼ) K 2 NiF 4 -type (T) Excess Oxygen!! O Nd/Ce O La/Sr Cu Cu

Electron-doped system T (K) 3 Hole-doped system x O. Matsumoto, M. Naito et al., Physica C 469, 924 (29) Film sample SC T N 2 1 underdoped overdoped regime regime optimally doped 1/8 anomaly SC.2.1.1.2 Electron doping Hole doping T c x Nd 2 CuO 4 -type (Tʼ) K 2 NiF 4 -type (T) Excess Oxygen!! O Nd/Ce O La/Sr Cu Cu

Low-temperature synthesis of T -La 1.8-x Eu.2 CuO 4 T-La 1.8-x Eu.2 CuO 4 N-La 1.8-x Eu.2 CuO 3.5 T -La 1.8-x Eu.2 CuO 4+ T-La 1.8-x Eu.2 CuO 4 by the conventional solid-state reaction Removal of oxygen oxidation CaH 2 O 2 225ºC, 24 h 4ºC,12 h Reductant To remove Excess Oxygen, Reduction annealing at 7ºC for 24 h in vacuum.4.2 La 1.8 Eu.2 CuO 4+ Excess Oxygen!! O Nd/Ce Cu Powdery bulk sample undoped SC (1-3 emu/g) -.2 -.4 -.6 -.8-1. -1.2 H = 1 Oe Zero-field cooling 5 1 15 2 25 3 T (K) T. Takamatsu et al., Appl. Phys. Express 5, 7311 (212)

Electron-doped system T (K) 3 Hole-doped system x O. Matsumoto, M. Naito et al., Physica C 469, 924 (29) Film sample SC T N 2 1 underdoped overdoped regime regime optimally doped 1/8 anomaly SC.2.1.1.2 Electron doping Hole doping T -La 1.8-x Eu.2 CuO 4 bulk sample Nd 2 CuO 4 -type (Tʼ) T c K 2 NiF 4 -type (T) x Excess Oxygen!! O Nd/Ce O La/Sr Cu Cu

35 3 La 1.8 Eu.2 CuO 4+ μsr in T -La 1.8-x Eu.2 CuO 4 T. Adachi et al., J. Phys. Soc. Jpn. 85, 114716 (216) As-grown 35 3 La 1.8 Eu.2 CuO 4+ 7 o C-reduced (T c = 15 K) Raw Asymmetry (%) 25 2 15 1 T = 2 K 1 K Raw Asymmetry (%) 25 2 15 1 T = 2 K 5 5 K 3 K T = 1.6 K 4 K 1 K Zero field 1 2 3 4 5 6 Time ( sec) Spin correlation is developed at low T. AF long-range order (T N =39K) T. Adachi 5 1 K 2 K T = 1.6 K 4 K 15 K 3 K 1 K Zero field 1 2 3 4 5 6 Time ( sec) Spin correlation is developed at low T. Short-range magnetic order (T N =2K) Volume fraction : 1% Coexistence of superconductivity and short-range magnetic order

μsr in longitudinal external magnetic field Longitudinal field e + (LF-μSR) sample μ + G Z (t) B(t) Internal field F(t) Longitudinal field ~1 x Internal field G Z (t) 1 1 t t

G Z (t) A(t) Spins are fluctuating 1 1 GA(t) Z (t) 1 LF 1/3 t 3/ t t t (b) (c) G Z (t) 1 A(t) Spins are static 1 GA(t) Z (t) 1 LF 1/3 t t t 1/3 t

Raw Asymmetry (%) 35 3 25 2 15 1 T. Adachi et al., J. Phys. Soc. Jpn. 85, 114716 (216) Longitudinal-field 35 La 1.8 Eu.2 CuO 4+ LF μsr in T -La 1.8-x Eu.2 CuO 4 7 o C-reduced (T c = 15 K) Raw Asymmetry (%) 3 25 2 15 1 La 1.8 Eu.2 CuO 4+ 7 o C-reduced (T c = 15 K) T = 2 K 5 T = 1.6 K LF = G 2 G 5 G Longitudinal field 1 G 3 G 1 G 1 2 3 4 5 6 7 8 Time ( sec) Static order at T=1.6K T. Adachi 5 1 K 2 K T = 1.6 K 4 K 15 K 3 K 1 K Zero field 1 2 3 4 5 6 Time ( sec) Spin correlation is developed at low T. Short-range magnetic order (T N =2K) Volume fraction : 1% Coexistence of superconductivity and short-range magnetic order

μsr in transverse external magnetic field Transverse field e + (TF-μSR) sample μ + B(t) Paramagnetic: T > T c Field is homogeneous F(t) Transverse field >> internal field G Z (t) 1 t r -1

Penetration of magnetic filed Magnetic field in a type-ii superconductor H c1 < H < H c2 vortex vortex λ Inhomogeneous field Damping of precession vacuum Damping rate λ -2 n s /m* n s : superconducting electron density superconductor

TF-μSR Damping rate λ -2 n s /m* n s : superconducting electron density Damping rate (μsec -1 )

Y. J. Uemura et al. Phys. Rev. Lett. 62 (1989) 2317 T c Damping rate λ -2 n s /m* n s : superconducting electron density T c n s : not explained in terms of BCS theory Bose-Einstein condensation-like as in the case of superfluidity in liquid He 4 Strong-coupling superconductivity

Electron-doped system T (K) 3 T N 2 Hole-doped system x SC AF 1 Pseudo gap SC.2.1.1.2 Electron doping Hole doping T c x Pseudo gap: Decrease in the density of states at E F Knight shift of NMR, T 1, ARPES, incoherent preformed pair charge order stripe doped electron Antiferromagnetic Mott Insulator d density wave (staggered flux phase) doped hole Cu 2+ spin (s=1/2)

TF-μSR in a paramagnetic metal Transverse field e + Conduction electron spins zero field B(t) sample H F(t) Polarization of conduction electron spins μ + ω = γ μ H γ μ (H + χ Pauli para H) T=2π/ω shorten Knight shift χ Pauli para Density of states at the Fermi level Transverse field G Z (t) 1-1 T = 2π/(γ μ H) t

Hole-Doped Bi 1.76 Pb.36 Sr 1.89 CuO 6+δ M. Miyazaki et al., Phys. Rev. B 94, 115123 (216) FFT spectrum from TF-μSR TF = 5T T = 2-3 K 3 25 Bi 1.76 Pb.35 Sr 1.89 CuO 6+ M. Miyazaki T c (K) OPT 2 SUD OD 15 Optimum 1 HUD HOD 5 HLD LD NSOD.5.1.15.2.25.3 p (per Cu) Two kinds of internal field Two kinds of muon site!!

Hole-Doped Bi 1.76 Pb.36 Sr 1.89 CuO 6+δ M. Miyazaki et al., Phys. Rev. B 94, 115123 (216) FFT spectrum from TF-μSR TF = 5T T = 2-3 K M. Miyazaki Change of internal field with decreasing T Development of pseudo gap

Temperature (K) 3 25 2 15 1 5 Comparison with other probes La-Bi221 (Bi,Pb)221 T * SR T c.8.12.16.2 Hole Concentration p (1/CuO 2 ) T * SR : (Bi,Pb)221 T c ( SR) : (Bi,Pb)221 T * c : (Bi,Pb)221 T c (Res.) : (Bi,Pb)221 T * NMR 1/T 1 T : La-Bi221 T * NMR K S : La-Bi221 T c (NMR) : La-Bi221 T * ARPES : La-(Bi,Pb)221 T c (ARPES) : La-(Bi,Pb)221 Observation of development of pseudo gap Knight shift of μsr (insensitive to sample quality) Knight shift of NMR (sensitive to sample quality)

Neutron scattering μsr NMR Magnetism detailed magnetic detection of structure magnetic anomaly (long-range order) (long-range order) (short-range order) Time ~1-12 sec 1-11 ~1-6 sec ~1-6 sec window more dynamic static more static Sample single crystal single crystal detection (large-sized) poly-crystal of signal robust to disorder weak to disorder μsr : powerful for the study of magnetic systems and strongly correlated electron systems complementary to neutron scattering and NMR

1. Introduction to μsr Thank you!! 2. μsr in magnetic materials 3. μsr studies in high-t c superconductors (1) Magnetic anomalies * Antiferromagnetic ordered phase * 1/8 anomaly and stripe order * Impurity-induced magnetism * Undoped superconductivity in T -cuprates (2) Inhomogeneity of internal field * Penetration depth in type-ii superconductors (3) Muon Knight shift * Pseudo gap 4. μsr vs neutron scattering and NMR