Critical and Glassy Spin Dynamics in Non-Fermi-Liquid Heavy-Fermion Metals
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1 Critical and Glassy Spin Dynamics in Non-Fermi-Liquid Heavy-Fermion Metals D. E. MacLaughlin Department of Physics University of California Riverside, California U.S.A. Leiden p.1
2 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Leiden p.2
3 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): Leiden p.2
4 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, Leiden p.2
5 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, Leiden p.2
6 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, slow spin dynamics, disorder. Leiden p.2
7 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, slow spin dynamics, disorder. Comparison of muon relaxation with resistivity & specific heat in NFL systems. Leiden p.2
8 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, slow spin dynamics, disorder. Comparison of muon relaxation with resistivity & specific heat in NFL systems. Local (single-ion or cluster) vs. cooperative dynamics, Leiden p.2
9 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, slow spin dynamics, disorder. Comparison of muon relaxation with resistivity & specific heat in NFL systems. Local (single-ion or cluster) vs. cooperative dynamics, Critical and (or?) glassy dynamics. Leiden p.2
10 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, slow spin dynamics, disorder. Comparison of muon relaxation with resistivity & specific heat in NFL systems. Local (single-ion or cluster) vs. cooperative dynamics, Critical and (or?) glassy dynamics. Conclusions and questions. Leiden p.2
11 Behavior of spin fluctuations at low temperatures in non-fermi-liquid (NFL) f -electron heavy-fermion metals. Muon spin relaxation (µsr): the technique, time-field scaling, low-frequency divergences, slow spin dynamics, disorder. Comparison of muon relaxation with resistivity & specific heat in NFL systems. Local (single-ion or cluster) vs. cooperative dynamics, Critical and (or?) glassy dynamics. Conclusions and questions. Reference: DEM et al., J. Phys. Cond. Matter 16, S4479 (2004). Leiden p.2
12 Breakdown of Fermi liquid paradigm in a number of metals, Leiden p.3
13 Breakdown of Fermi liquid paradigm in a number of metals, including many paramagnetic heavy-fermion f -electron materials (v. Löhneysen 96, Coleman 99, Schofield 99, Bernhoeft 01, Stewart 01, Varma et al. 02,... ). Leiden p.3
14 Breakdown of Fermi liquid paradigm in a number of metals, including many paramagnetic heavy-fermion f -electron materials (v. Löhneysen 96, Coleman 99, Schofield 99, Bernhoeft 01, Stewart 01, Varma et al. 02,... ). Attention focused on quantum critical phenomena associated with a T = 0 phase transition. Coleman et al. 01 Leiden p.3
15 Breakdown of Fermi liquid paradigm in a number of metals, including many paramagnetic heavy-fermion f -electron materials (v. Löhneysen 96, Coleman 99, Schofield 99, Bernhoeft 01, Stewart 01, Varma et al. 02,... ). Attention focused on quantum critical phenomena associated with a T = 0 phase transition. Quantum rather than thermal critical fluctuations, Coleman et al. 01 Leiden p.3
16 Breakdown of Fermi liquid paradigm in a number of metals, including many paramagnetic heavy-fermion f -electron materials (v. Löhneysen 96, Coleman 99, Schofield 99, Bernhoeft 01, Stewart 01, Varma et al. 02,... ). Attention focused on quantum critical phenomena associated with a T = 0 phase transition. Quantum rather than thermal critical fluctuations, New low-lying excitations not Fermi-liquid quasiparticles. Coleman et al. 01 Leiden p.3
17 Muon Spin Relaxation in NFL metals Leiden p.4
18 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date Leiden p.4
19 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; Leiden p.4
20 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are Leiden p.4
21 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and Leiden p.4
22 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and inhomogeneous (broad spatial distributions of coupling strengths and/or fluctuation amplitudes), but Leiden p.4
23 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and inhomogeneous (broad spatial distributions of coupling strengths and/or fluctuation amplitudes), but cooperative: Leiden p.4
24 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and inhomogeneous (broad spatial distributions of coupling strengths and/or fluctuation amplitudes), but cooperative: Form of correlation function is homogeneous, so Leiden p.4
25 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and inhomogeneous (broad spatial distributions of coupling strengths and/or fluctuation amplitudes), but cooperative: Form of correlation function is homogeneous, so not a simple distribution of fluctuation rates. Leiden p.4
26 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and inhomogeneous (broad spatial distributions of coupling strengths and/or fluctuation amplitudes), but cooperative: Form of correlation function is homogeneous, so not a simple distribution of fluctuation rates. Properties suggest glassy dynamical behavior, but Leiden p.4
27 Muon Spin Relaxation in NFL metals Results: in f-electron NFL metals studied to date low-frequency spin fluctuations at low temperatures are singular [with a (very) low-frequency cutoff] as ω 0; In disordered NFL materials spin fluctuations are slow (enhanced low-frequency spectral weight) and inhomogeneous (broad spatial distributions of coupling strengths and/or fluctuation amplitudes), but cooperative: Form of correlation function is homogeneous, so not a simple distribution of fluctuation rates. Properties suggest glassy dynamical behavior, but no spin freezing for T 20 mk. Leiden p.4
28 Longitudinal-field µsr Leiden p.5
29 Longitudinal-field µsr A spin-polarized muon from a meson factory (PSI, TRIUMF,... )... S µ µ + p µ Schematic diagram of longitudinal-field µsr. Leiden p.5
30 Longitudinal-field µsr A spin-polarized muon from a meson factory (PSI, TRIUMF,... )... is stopped in the sample. A clock is started. S µ µ + p µ M H sample start stop CLOCK Schematic diagram of longitudinal-field µsr. Leiden p.5
31 Longitudinal-field µsr A spin-polarized muon from a meson factory (PSI, TRIUMF,... )... is stopped in the sample. A clock is started. Muon β decay µ + e + + ν e + ν µ. Decay time distributed; average 2.2 µs. Positron emitted preferentially in direction of muon spin. S µ µ + p µ M start pe CLOCK e + stop sample Schematic diagram of longitudinal-field µsr. H Leiden p.5
32 Longitudinal-field µsr A spin-polarized muon from a meson factory (PSI, TRIUMF,... )... is stopped in the sample. A clock is started. Muon β decay µ + e + + ν e + ν µ. Decay time distributed; average 2.2 µs. Positron emitted preferentially in direction of muon spin. Decay time and positron direction are recorded. S µ µ + p µ M start pe BP CLOCK e + stop (after J. Brewer & J. Sonier) sample H FP Schematic diagram of longitudinal-field µsr. Leiden p.5
33 Longitudinal-field µsr A spin-polarized muon from a meson factory (PSI, TRIUMF,... )... is stopped in the sample. A clock is started. Muon β decay µ + e + + ν e + ν µ. Decay time distributed; average 2.2 µs. Positron emitted preferentially in direction of muon spin. Decay time and positron direction are recorded. Experiment repeated 10 7 times. S µ µ + p µ M start pe BP CLOCK e + stop (after J. Brewer & J. Sonier) sample H FP Schematic diagram of longitudinal-field µsr. Leiden p.5
34 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). Leiden p.6
35 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). P µ relaxes ( 0) due to local fields at muon sites; dominated by dynamic relaxation due to f -spin fluctuations. Asymmetry (%) Time (µs) Leiden p.6
36 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). P µ relaxes ( 0) due to local fields at muon sites; dominated by dynamic relaxation due to f -spin fluctuations. Asymmetry (%) Time (µs) Relaxation rate fluctuation noise power at muon Zeeman frequency ω µ. Leiden p.6
37 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). P µ relaxes ( 0) due to local fields at muon sites; dominated by dynamic relaxation due to f -spin fluctuations. Asymmetry (%) Time (µs) Relaxation rate fluctuation noise power at muon Zeeman frequency ω µ. Determined by longitudinal field H P µ : ω µ = γ µ H. Leiden p.6
38 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). P µ relaxes ( 0) due to local fields at muon sites; dominated by dynamic relaxation due to f -spin fluctuations. Asymmetry (%) Time (µs) Relaxation rate fluctuation noise power at muon Zeeman frequency ω µ. Determined by longitudinal field H P µ : ω µ = γ µ H. Extremely low frequency (MHz) spectroscopy. Leiden p.6
39 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). P µ relaxes ( 0) due to local fields at muon sites; dominated by dynamic relaxation due to f -spin fluctuations. Asymmetry (%) Time (µs) Relaxation rate fluctuation noise power at muon Zeeman frequency ω µ. Determined by longitudinal field H P µ : ω µ = γ µ H. Extremely low frequency (MHz) spectroscopy. Ideally suited to study extremely low-frequency NFL excitations! Leiden p.6
40 Obtain time histogram of (positron asymmetry) (muon spin polarization P µ ). P µ relaxes ( 0) due to local fields at muon sites; dominated by dynamic relaxation due to f -spin fluctuations. Asymmetry (%) Time (µs) Relaxation rate fluctuation noise power at muon Zeeman frequency ω µ. Determined by longitudinal field H P µ : ω µ = γ µ H. Extremely low frequency (MHz) spectroscopy. Ideally suited to study extremely low-frequency NFL excitations! A local probe sums over all q. Leiden p.6
41 Sample-average asymmetry relaxation function G(t, H) given (in motionally narrowed limit) by Leiden p.7
42 Sample-average asymmetry relaxation function G(t, H) given (in motionally narrowed limit) by G(t, H) { exp[ W (H)t] (homogeneous) dr exp[ W (r, H)t] (inhomogeneous) ; Leiden p.7
43 Sample-average asymmetry relaxation function G(t, H) given (in motionally narrowed limit) by G(t, H) { exp[ W (H)t] (homogeneous) dr exp[ W (r, H)t] (inhomogeneous) ; (Distributed) muon relaxation rate W (r, H) related to local spin autocorrelation function q(t) = S(r, t) S(r, 0) : Leiden p.7
44 Sample-average asymmetry relaxation function G(t, H) given (in motionally narrowed limit) by G(t, H) { exp[ W (H)t] (homogeneous) dr exp[ W (r, H)t] (inhomogeneous) ; (Distributed) muon relaxation rate W (r, H) related to local spin autocorrelation function q(t) = S(r, t) S(r, 0) : W (r, H) 0 dt q(t)e iω µt, ω µ = γ µ H. Leiden p.7
45 Sample-average asymmetry relaxation function G(t, H) given (in motionally narrowed limit) by G(t, H) { exp[ W (H)t] (homogeneous) dr exp[ W (r, H)t] (inhomogeneous) ; (Distributed) muon relaxation rate W (r, H) related to local spin autocorrelation function q(t) = S(r, t) S(r, 0) : W (r, H) 0 dt q(t)e iω µt, ω µ = γ µ H. FD theorem dynamic susceptibility; contact with neutron scattering: Leiden p.7
46 Sample-average asymmetry relaxation function G(t, H) given (in motionally narrowed limit) by G(t, H) { exp[ W (H)t] (homogeneous) dr exp[ W (r, H)t] (inhomogeneous) ; (Distributed) muon relaxation rate W (r, H) related to local spin autocorrelation function q(t) = S(r, t) S(r, 0) : W (r, H) 0 dt q(t)e iω µt, ω µ = γ µ H. FD theorem dynamic susceptibility; contact with neutron scattering: W (r, H) T χ (r, ω µ )/ω µ, ω µ = γ µ H. Leiden p.7
47 Time-field scaling Leiden p.8
48 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Leiden p.8
49 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Leiden p.8
50 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Coefficient V (r) spatially distributed but exponent α the same for all spins. Leiden p.8
51 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Coefficient V (r) spatially distributed but exponent α the same for all spins. Then Leiden p.8
52 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Coefficient V (r) spatially distributed but exponent α the same for all spins. Then G(t, H) dr exp[ V (r)(t/ωµ)] α Leiden p.8
53 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Coefficient V (r) spatially distributed but exponent α the same for all spins. Then G(t, H) dr exp[ V (r)(t/ωµ)] α = G(t/H α ) (ω µ = γ µ H). Leiden p.8
54 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Coefficient V (r) spatially distributed but exponent α the same for all spins. Then G(t, H) dr exp[ V (r)(t/ωµ)] α = G(t/H α ) (ω µ = γ µ H). This is time-field scaling. First seen in LF-µSR in spin-glass AgMn, T > T g (Keren et al. 96). Leiden p.8
55 Time-field scaling Near a critical point (thermal or quantum) expect q(t) = t y f(t/τ) (scaling form). Then W (r, H) = V (r)/ωµ α, α = 1 y, for ωτ 1. Coefficient V (r) spatially distributed but exponent α the same for all spins. Then G(t, H) dr exp[ V (r)(t/ωµ)] α = G(t/H α ) (ω µ = γ µ H). This is time-field scaling. First seen in LF-µSR in spin-glass AgMn, T > T g (Keren et al. 96). No need to assume any particular form for G(t)! Leiden p.8
56 Time-field scaling in NFL UCu 4 Pd Leiden p.9
57 Time-field scaling in NFL UCu 4 Pd LF-µSR relaxation functions in UCu 5 x Pd x, x = 1.0 and 1.5 (not shown), obey timefield scaling (DEM et al. 01). Asymmetry (%) 20 UCu 4 Pd T = 0.5 K 10 H = 21 Oe Time (µs) t /H 0.35 LF-µSR scaling in UCu 4 Pd (DEM et al. 01, 02). Leiden p.9
58 Time-field scaling in NFL UCu 4 Pd LF-µSR relaxation functions in UCu 5 x Pd x, x = 1.0 and 1.5 (not shown), obey timefield scaling (DEM et al. 01). Exponents consistent with χ (ω) from neutron scattering (Aronson et al. 95). Asymmetry (%) Oe H = 21 Oe UCu 4 Pd T = 0.5 K Time (µs) t /H 0.35 LF-µSR scaling in UCu 4 Pd (DEM et al. 01, 02). Leiden p.9
59 Time-field scaling in NFL UCu 4 Pd LF-µSR relaxation functions in UCu 5 x Pd x, x = 1.0 and 1.5 (not shown), obey timefield scaling (DEM et al. 01). Exponents consistent with χ (ω) from neutron scattering (Aronson et al. 95). Observed rates in NFL systems never very large; evidence against f -spin freezing (static moment 10 3 µ B ). Asymmetry (%) Oe 68 Oe H = 21 Oe UCu 4 Pd T = 0.5 K Time (µs) t /H 0.35 LF-µSR scaling in UCu 4 Pd (DEM et al. 01, 02). Leiden p.9
60 Time-field scaling in NFL UCu 4 Pd LF-µSR relaxation functions in UCu 5 x Pd x, x = 1.0 and 1.5 (not shown), obey timefield scaling (DEM et al. 01). Exponents consistent with χ (ω) from neutron scattering (Aronson et al. 95). Observed rates in NFL systems never very large; evidence against f -spin freezing (static moment 10 3 µ B ). Asymmetry (%) Oe 250 Oe 68 Oe H = 21 Oe UCu 4 Pd T = 0.5 K Time (µs) t /H 0.35 LF-µSR scaling in UCu 4 Pd (DEM et al. 01, 02). Leiden p.9
61 Time-field scaling in NFL UCu 4 Pd LF-µSR relaxation functions in UCu 5 x Pd x, x = 1.0 and 1.5 (not shown), obey timefield scaling (DEM et al. 01). Exponents consistent with χ (ω) from neutron scattering (Aronson et al. 95). Observed rates in NFL systems never very large; evidence against f -spin freezing (static moment 10 3 µ B ). Asymmetry (%) Oe 630 Oe 250 Oe 68 Oe H = 21 Oe UCu 4 Pd T = 0.5 K Time (µs) t /H 0.35 LF-µSR scaling in UCu 4 Pd (DEM et al. 01, 02). Leiden p.9
62 Comparison between NFL systems Leiden p.10
63 Comparison between NFL systems LF-µSR relaxation behavior at low temperatures: Leiden p.10
64 Comparison between NFL systems LF-µSR relaxation behavior at low temperatures: CePtSi 1 x Ge x, UCu 5 x Pd x, UCu 5 x Pt x *: relaxation subexponential ( inhomogeneous) and strong. G(t)/G(0) 1 CePtSi UCu 4 Pd *not shown Time (µs) LF-µSR relaxation functions G(t) at low temperatures in NFL materials (DEM et al. 03). Leiden p.10
65 Comparison between NFL systems LF-µSR relaxation behavior at low temperatures: CePtSi 1 x Ge x, UCu 5 x Pd x, UCu 5 x Pt x *: relaxation subexponential ( inhomogeneous) and strong. CeNi 2 Ge 2, YbRh 2 Si 2, CeCu 5.9 Au 0.1,* Ce(Ru 0.5 Rh 0.5 ) 2 Si 2 *: relaxation nearly exponential ( homogeneous) and much weaker. *not shown G(t)/G(0) 1 CeNi 2 Ge 2 YbRh 2 Si 2 CePtSi UCu 4 Pd Time (µs) LF-µSR relaxation functions G(t) at low temperatures in NFL materials (DEM et al. 03). Leiden p.10
66 LF-µSR, resistivity, and specific heat Leiden p.11
67 LF-µSR, resistivity, and specific heat Compare LF-µSR parameters with other properties in a number of NFL systems: Leiden p.11
68 LF-µSR, resistivity, and specific heat Compare LF-µSR parameters with other properties in a number of NFL systems: Effect of disorder? Leiden p.11
69 LF-µSR, resistivity, and specific heat Compare LF-µSR parameters with other properties in a number of NFL systems: Effect of disorder? Compare with residual resistivities ρ(0). Leiden p.11
70 LF-µSR, resistivity, and specific heat Compare LF-µSR parameters with other properties in a number of NFL systems: Effect of disorder? Compare with residual resistivities ρ(0). Materials-dependent differences in fluctuation energy scales? Leiden p.11
71 LF-µSR, resistivity, and specific heat Compare LF-µSR parameters with other properties in a number of NFL systems: Effect of disorder? Compare with residual resistivities ρ(0). Materials-dependent differences in fluctuation energy scales? Compare with low-temperature specific heat coefficients γ(t ) as measures of these scales. (Choose T = 1 K.) Leiden p.11
72 LF-µSR asymmetry data well fit by stretched exponential relaxation function Leiden p.12
73 LF-µSR asymmetry data well fit by stretched exponential relaxation function G(t) = exp[ (Λt) K ] G(t)/G(0) 1 CeNi 2 Ge 2 YbRh 2 Si 2 CePtSi UCu 4 Pd Time (µs) LF-µSR relaxation functions G(t) at low temperatures in NFL materials (DEM et al. 03). Leiden p.12
74 LF-µSR asymmetry data well fit by stretched exponential relaxation function G(t) = exp[ (Λt) K ] (Parameterization only; no ab initio justification). G(t)/G(0) 1 CeNi 2 Ge 2 YbRh 2 Si 2 CePtSi UCu 4 Pd Time (µs) LF-µSR relaxation functions G(t) at low temperatures in NFL materials (DEM et al. 03). Leiden p.12
75 LF-µSR asymmetry data well fit by stretched exponential relaxation function G(t) = exp[ (Λt) K ] (Parameterization only; no ab initio justification). Λ 1 = characteristic 1/e time for relaxation, G(t)/G(0) 1 CeNi 2 Ge 2 YbRh 2 Si 2 CePtSi UCu 4 Pd Time (µs) LF-µSR relaxation functions G(t) at low temperatures in NFL materials (DEM et al. 03). Leiden p.12
76 LF-µSR asymmetry data well fit by stretched exponential relaxation function G(t) = exp[ (Λt) K ] (Parameterization only; no ab initio justification). Λ 1 = characteristic 1/e time for relaxation, K < 1 measure of spread in rates (broad distribution reduced K). G(t)/G(0) 1 CeNi 2 Ge 2 YbRh 2 Si 2 CePtSi UCu 4 Pd Time (µs) LF-µSR relaxation functions G(t) at low temperatures in NFL materials (DEM et al. 03). Leiden p.12
77 Correlation of normalized Λ and K with residual resistivity: with increasing ρ(0) Leiden p.13
78 Correlation of normalized Λ and K with residual resistivity: with increasing ρ(0) Λ increases disorder shifts fluctuation power spectrum to low frequencies; Normalized Λ (10 3 [A 6 µs 1 ]) CeNi 2 Ge 2 YbRh 2 Si 2 CeCu 5.9 Au 0.1 CeRhRuSi 2 CePtSi UCu 3.5 Pd 1.5 UCu 4 Pd (a) ρ(0) (µω cm) Λ Leiden p.13
79 Correlation of normalized Λ and K with residual resistivity: with increasing ρ(0) Λ increases disorder shifts fluctuation power spectrum to low frequencies; K decreases from 1 (exponential) disorder increases spread in rates. Normalized Λ (10 3 [A 6 µs 1 ]), K CeNi 2 Ge 2 YbRh 2 Si 2 CeCu 5.9 Au 0.1 CeRhRuSi 2 CePtSi UCu 3.5 Pd 1.5 UCu 4 Pd (a) ρ(0) (µω cm) Λ K Leiden p.13
80 Correlation of normalized Λ and K with residual resistivity: with increasing ρ(0) Λ increases disorder shifts fluctuation power spectrum to low frequencies; K decreases from 1 (exponential) disorder increases spread in rates. Normalized Λ (10 3 [A 6 µs 1 ]), K CeNi 2 Ge 2 YbRh 2 Si 2 CeCu 5.9 Au 0.1 CeRhRuSi 2 CePtSi UCu 3.5 Pd 1.5 UCu 4 Pd ρ(0) (µω cm) Λ K (a) Correlation is good (smooth). Leiden p.13
81 Correlation of normalized Λ and K with residual resistivity: with increasing ρ(0) Λ increases disorder shifts fluctuation power spectrum to low frequencies; K decreases from 1 (exponential) disorder increases spread in rates. Correlation is good (smooth). Correlation with γ(1 K) is poor. Normalized Λ (10 3 [A 6 µs 1 ]), K CeNi 2 Ge 2 YbRh 2 Si 2 CeCu 5.9 Au 0.1 CeRhRuSi 2 CePtSi UCu 3.5 Pd 1.5 UCu 4 Pd ρ(0) (µω cm) γ(1k) (J mol 1 K 2 ) Λ Λ K K (a) (b) Leiden p.13
82 Correlation of normalized Λ and K with residual resistivity: with increasing ρ(0) Λ increases disorder shifts fluctuation power spectrum to low frequencies; K decreases from 1 (exponential) disorder increases spread in rates. Correlation is good (smooth). Correlation with γ(1 K) is poor. differences in relaxation rates not due to materialdependent energy scales. Normalized Λ (10 3 [A 6 µs 1 ]), K CeNi 2 Ge 2 YbRh 2 Si 2 CeCu 5.9 Au 0.1 CeRhRuSi 2 CePtSi UCu 3.5 Pd 1.5 UCu 4 Pd ρ(0) (µω cm) γ(1k) (J mol 1 K 2 ) Λ Λ K K (a) (b) Leiden p.13
83 Local or cooperative dynamics? Leiden p.14
84 Local or cooperative dynamics? Distributed low-temperature rate, but Leiden p.14
85 Local or cooperative dynamics? Distributed low-temperature rate, but Unique value of time-field scaling exponent α for all f spins. Leiden p.14
86 Local or cooperative dynamics? Distributed low-temperature rate, but Unique value of time-field scaling exponent α for all f spins. Form of spin correlation function S(r, t) S(r, 0) the same for all spins. Leiden p.14
87 Local or cooperative dynamics? Distributed low-temperature rate, but Unique value of time-field scaling exponent α for all f spins. Form of spin correlation function S(r, t) S(r, 0) the same for all spins. Strongly suggests Leiden p.14
88 Local or cooperative dynamics? Distributed low-temperature rate, but Unique value of time-field scaling exponent α for all f spins. Form of spin correlation function S(r, t) S(r, 0) the same for all spins. Strongly suggests homogeneous dynamics (but inhomogeneous fluctuation amplitudes and hyperfine couplings), Leiden p.14
89 Local or cooperative dynamics? Distributed low-temperature rate, but Unique value of time-field scaling exponent α for all f spins. Form of spin correlation function S(r, t) S(r, 0) the same for all spins. Strongly suggests homogeneous dynamics (but inhomogeneous fluctuation amplitudes and hyperfine couplings), cooperative rather than single-ion or independent-cluster behavior. Leiden p.14
90 Local or cooperative dynamics? Distributed low-temperature rate, but Unique value of time-field scaling exponent α for all f spins. Form of spin correlation function S(r, t) S(r, 0) the same for all spins. Strongly suggests homogeneous dynamics (but inhomogeneous fluctuation amplitudes and hyperfine couplings), cooperative rather than single-ion or independent-cluster behavior. Evidence against inhomogeneous local single-ion or cluster dynamics (distributed fluctuation rates) of Kondo disorder/ Griffiths-phase pictures. Leiden p.14
91 Conclusions Leiden p.15
92 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Leiden p.15
93 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Field dependence of relaxation rate Leiden p.15
94 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Field dependence of relaxation rate gives χ (ω µ ), ω µ = γ µ H; Leiden p.15
95 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Field dependence of relaxation rate gives χ (ω µ ), ω µ = γ µ H; divergence of χ (ω)/ω down to very low (but not zero) frequencies for both ordered and disordered systems. Leiden p.15
96 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Field dependence of relaxation rate gives χ (ω µ ), ω µ = γ µ H; divergence of χ (ω)/ω down to very low (but not zero) frequencies for both ordered and disordered systems. Disorder gives rise to much stronger low-frequency fluctuations than homogeneous quantum criticality. Leiden p.15
97 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Field dependence of relaxation rate gives χ (ω µ ), ω µ = γ µ H; divergence of χ (ω)/ω down to very low (but not zero) frequencies for both ordered and disordered systems. Disorder gives rise to much stronger low-frequency fluctuations than homogeneous quantum criticality. correlation function same form for all spins cooperative rather than local (single-ion or cluster) dynamics. Leiden p.15
98 Conclusions LF-µSR relaxation in ordered and disordered f -electron NFL materials: Field dependence of relaxation rate gives χ (ω µ ), ω µ = γ µ H; divergence of χ (ω)/ω down to very low (but not zero) frequencies for both ordered and disordered systems. Disorder gives rise to much stronger low-frequency fluctuations than homogeneous quantum criticality. correlation function same form for all spins cooperative rather than local (single-ion or cluster) dynamics. Glassy behavior in disordered systems? Leiden p.15
99 Questions Leiden p.16
100 Questions Are these slow glassy spin fluctuations Leiden p.16
101 Questions Are these slow glassy spin fluctuations ordered-system quantum critical fluctuations strongly modified by disorder, or Leiden p.16
102 Questions Are these slow glassy spin fluctuations ordered-system quantum critical fluctuations strongly modified by disorder, or new quantum spin glass excitations created by disorder? Leiden p.16
103 Questions Are these slow glassy spin fluctuations ordered-system quantum critical fluctuations strongly modified by disorder, or new quantum spin glass excitations created by disorder? Do glassy NFL spin dynamics depend universally on residual resistivity? What does this mean? Leiden p.16
104 Questions Are these slow glassy spin fluctuations ordered-system quantum critical fluctuations strongly modified by disorder, or new quantum spin glass excitations created by disorder? Do glassy NFL spin dynamics depend universally on residual resistivity? What does this mean? How are fluctuations slowed down? Is frustration involved? Leiden p.16
105 Questions (continued) Leiden p.17
106 Questions (continued) What is significance of observation χ (ω) ω 1 α (α > 0) down to MHz frequencies? Leiden p.17
107 Questions (continued) What is significance of observation χ (ω) ω 1 α (α > 0) down to MHz frequencies? Ordinarily χ (ω) ω. When can this break down? Leiden p.17
108 Questions (continued) What is significance of observation χ (ω) ω 1 α (α > 0) down to MHz frequencies? Ordinarily χ (ω) ω. When can this break down? Can we understand low-frequency cutoff frequencies 100 khz? Leiden p.17
109 Questions (continued) What is significance of observation χ (ω) ω 1 α (α > 0) down to MHz frequencies? Ordinarily χ (ω) ω. When can this break down? Can we understand low-frequency cutoff frequencies 100 khz? Relation between scaling exponent in ordered (YbRh 2 Si 2, α 1; CeCu 5.9 Au 0.1, α 0.75 [from neutron scattering and NMR]) and disordered (UCu 4 Pd, α 0.3) NFL systems? Leiden p.17
110 Questions (continued) What is significance of observation χ (ω) ω 1 α (α > 0) down to MHz frequencies? Ordinarily χ (ω) ω. When can this break down? Can we understand low-frequency cutoff frequencies 100 khz? Relation between scaling exponent in ordered (YbRh 2 Si 2, α 1; CeCu 5.9 Au 0.1, α 0.75 [from neutron scattering and NMR]) and disordered (UCu 4 Pd, α 0.3) NFL systems?... Leiden p.17
111 Collaborators: M. S. Rose B.-L. Young = ; Univ. of California, Riverside O. O. Bernal Cal. State Univ., Los Angeles R. H. Heffner LANL, JAERI Tokai G. D. Morris LANL, TRIUMF K. Ishida Osaka Univ., U.C. Riverside, Kyoto Univ. G. J. Nieuwenhuys Kamerlingh Onnes Lab., Leiden Univ. J. E. Sonier Simon Fraser Univ., TRIUMF M. B. Maple Univ. of California, San Diego 9 G. R. Stewart = Univ. of Florida B. Andraka ; K. Heuser Uni. Augsburg 9 O. Trovarelli >= C. Geibel MPI-CPS, Dresden >; F. Steglich Special thanks: µsr user support groups at the Paul Scherrer Institute (D. Herlach, C. Baines, A. Amato) and TRIUMF (S. Kreitzman, B. Hitti, D. Arseneau); M. Aronson, A. Castro Neto, P. Coleman, V. Dobrosavljević, G. Kotliar, A. Millis, E. Miranda, R. Osborn, Q. Si, C. Varma, R. Walstedt. Supported in part by U.S. NSF (UCR, CSULA, UCSD), Ministry of Education, Sport, Science, and Technology of Japan (Osaka), Netherlands NWO and FOM (Leiden), Canadian NSERC (Simon Fraser), and U.S. DOE (Los Alamos, U. Florida). Leiden p.18
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