NMR: Examples. Vesna Mitrović, Brown University. Boulder Summer School, 2008

Transcription

1 NMR: Examples Vesna Mitrović, Brown University Boulder Summer School, 2008

2 To Remember Static NMR Spectrum Measurements Local Magnetic Field Probability Distribution ω n = γ n H loc = γ n (H 0 + H hf ) H hf = n A n,k S k Width of an NMR spectrum Distribution of S z (r) K(T ) χ (q =0, ω 0) Shift of an NMR spectrum Magnetic susceptibility In metals: K(T ) N(E F ) T 1 1 χ (q, ω 0) T 1 2 χ (q, ω 0)

3 2D Electron Gas - T1 Contrast Imaging MBE heterostructures M242 and M layers: 25 nm 185 nm 30 nm 250 nm GaAs Ga 0.7 Al 0.3 As Si delta doping Ga 0.9 Al 0.1 As e - density (S.E. Barrett and collaborators, Yale University) (M. Horvatic & C. Bertier and collaborators, GHMFL,Grenoble)

4 2D Electron Gas - T1 Contrast Imaging MBE heterostructures M242 and M layers: E 2D Electron Gas in H0 z : B = 0 B >> 0 25 nm 185 nm 30 nm 250 nm eb /m* GaAs Ga 0.7 Al 0.3 As Si delta doping Ga 0.9 Al 0.1 As e - density n 0 = eb /h gµ Β Β n n (E) LLL filling factor ν = 1 (S.E. Barrett and collaborators, Yale University) (M. Horvatic & C. Bertier and collaborators, GHMFL,Grenoble)

5 2D Electron Gas - T1 Contrast Imaging MBE heterostructures M242 and M layers: E 2D Electron Gas in H0 z : B = 0 B >> 0 25 nm 185 nm 30 nm 250 nm eb /m* GaAs Ga 0.7 Al 0.3 As Si delta doping Ga 0.9 Al 0.1 As e - density n 0 = eb /h gµ Β Β n n (E) LLL filling factor ν = 1 [Fig: S. Melinte, PhD thesis (2001)] (S.E. Barrett and collaborators, Yale University) (M. Horvatic & C. Bertier and collaborators, GHMFL,Grenoble)

6 Static Shift Measurements : NMR line-shift ( 69,71 Ga) N N K S P = = 2 N + N n k average spin polarisation S z k GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference 100 layers: 30 nm 250 nm GaAs Ga 0.7 Al 0.3 As Si delta doping Ga 0.9 Al 0.1 As (GHMFL,Grenoble)

7 Static Shift Measurements : NMR line-shift ( 69,71 Ga) N N K S P = = 2 N + N n k average spin polarisation S z k GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference (GHMFL,Grenoble)

8 Static Shift Measurements : NMR line-shift ( 69,71 Ga) N N K S P = = 2 N + N n k average spin polarisation S z k GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference 30 nm 250 nm barriers Quantum Wells close to Si δ-doping (GHMFL,Grenoble)

9 Static Shift Measurements : NMR line-shift ( 69,71 Ga) K S P = N N N + N = 2 n k S z k P(gµ B H/k B T ) 0 average spin polarisation What if g -> 0? GaAs QWs: 2D electrons shift K S AlGaAs barriers: no electrons reference 100 layers: 30 nm 250 nm GaAs Ga 0.7 Al 0.3 As Si delta doping Ga 0.9 Al 0.1 As (GHMFL,Grenoble)

10 Static Shift Measurements : 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 ) 0 GaAs QWs: 2D electrons shift K S What if g -> 0? AlGaAs barriers: no electrons reference (GHMFL,Grenoble)

11 ν = 1 state - elementary excitations? Localized holes and electrons Skyrmion Anti-skyrmion Energy gap & size (s) S.E. Barrett et al., Phys. Rev. Lett. 74, 5112 (1995) P. Khandelwal et al., Phys. Rev. Lett. 86, 5353 (2001) g = E z E c

12 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 0 s To vary skyrmion size tune g : g 0 - hydrostatic pressure - Al 0.13 Ga 0.87 As - QW (g theory = ; g GaAs = 0.44)

13 Large Skyrmions in Al 0.13 Ga 0.87 As (g 0) Quantum Wells Magnetotransport measurements in Al 0.13 Ga 0.87 As QW (S. P. Shukla et al., PRB 61, 4469 (2000)) s 50 Small g sample: 30 QWs g theory = g GaAs = 0.44 n 2D = cm -2 Problem: How to separate the signals? P(gµ B H/k B T ) 0 m 0 = cm 2 / Vs QW: 24 nm of Al 0.13 Ga 0.87 As 1.0 barrier ----> 71 Ga, ν = 1 barriers: nm of Al 0.35 Ga 0.65 As 0.8 T = 65 mk Magnitude [a. u.] Frequency [MHz]

14 Separate overlapping QW and barriers signal? Small tip angle technique : M(t), pulse, signal π/2 t = 0 π t = τ M (t=2τ) t = 2τ repeat t = T R time S N = f ( ) TR, Tip Angle T 1

15 Separate overlapping QW and barriers signal? Small tip angle technique : M(t), pulse, signal π/2 t = 0 π t = τ M (t=2τ) t = 2τ repeat t = T R time S N = f ( ) TR, Tip Angle T 1

16 Separate overlapping QW and barriers signal? Small tip angle technique : M(t), pulse, signal π/2 t = 0 π t = τ M (t=2τ) t = 2τ repeat t = T R time % 90 % S N = f ( ) TR, Tip Angle T 1 Tip Angle θ [rad] % 70 % 60 % 50 % 40 % 30 % 20 % % T R / T 1

17 Separate overlapping QW and barriers signal? Small tip angle technique : S N = f ( ) TR, Tip Angle T 1 Tip Angle θ [rad] % 90 % 80 % 70 % 60 % 50 % 40 % 30 % 20 % 10 % T R / T 1

18 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] % 90 % 80 % 70 % 60 % 50 % 40 % 30 % 20 % Amplitude [a. u.] T = 65 mk barriers QW % T R / T Frequency [MHz] Pulse Length [µs] 2 µs = π/2 [V. Mitrović et al., 2007]

19 Magnitude [a. u.] Ga, ν = 1 QW contribution only (barriers subtracted) T = 65 mk 130 mk 208 mk 315 mk 375 mk 450 mk 612 mk 700 mk 908 mk "P = 1" Frequency [KHz]

20 QW signal Magnitude [a. u.] Ga, ν = 1 QW contribution only (barriers subtracted) Frequency [KHz] 20 T = 65 mk 130 mk 208 mk 315 mk 375 mk 450 mk 612 mk 700 mk 908 mk "P = 1" Frequency [KHz] Frequency shift [KHz] Line Width (FWHM) Average Line Position Polarization Temperature [K] g -0.1 g GaAs at low-t : "negative", small, strongly inhomogeneous polarization

21 Inhomogeneous Systems Stripe phase (Tranquada et al., Nature 1995) (T. Imai and collaborators, MIT & McMaster University) (M. Julien and collaborators, UJF & GHMFL,Grenoble) (C.Hamel, N. Curro and collaborators, LANL)

22 What s going on in LSCO? «Inc.» AF LRO (stripe pattern) Suzuki et al. Kimura et al. Wakimoto et al. Glassy freezing Cluster spin-glass Niedermayer et al., Julien et al.? Enhanced AF order from vortex cores Katano et al. Lake et al. Inhomogneneity & Phase Separation Singer et al.

23 Spatial inhomogeneities (Distribution of T 1 values) La 1.88 Sr 0.12 CuO 4 (T c = 30 K) 139 La NMR T = 4.9 K Case 1. Spatially resolved measurement Case 2. Not resolved phenomenological fit exponent α quantifies disorder strength Case 3. Not resolved convolute with distribution function deduce parameters Single T 1 : M 0 M M = k c k e b k t T 1

24 Analysis of Spatial Inhomogeneities I 1. phenomenological fit: exponent α quantifies disorder strength good qualitative analysis 2. Convolute with distribution function of T 1 Single T 1 α α α M α (t, T1 1 ) = e 28 t T e 15 t T e 6 t T e M G (t) = ( π/2 σ log ) 1 t T 1 α. e 2 (log R 1 log T 1 1 ) 2 /σ 2 log Mα=1 (t, R 1 ) d(log R 1 ) M G (t) = P (log R 1 log T 1 1 ) M α=1 (t, R 1 ) d(log R 1 )

25 Analysis of Spatial Inhomogeneities I 1. phenomenological fit: exponent α quantifies disorder strength good qualitative analysis 2. Convolute with distribution function of T 1 Single T 1 α α α M α (t, T1 1 ) = e 28 t T e 15 t T e 6 t T e M G (t) = ( π/2 σ log ) 1 t T 1 α. e 2 (log R 1 log T 1 1 ) 2 /σ 2 log Mα=1 (t, R 1 ) d(log R 1 ) La 1.88 Sr 0.12 CuO 4 (T c = 30 K) 139 La NMR M G (t) = P (log R 1 log T 1 1 ) M α=1 (t, R 1 ) d(log R 1 ) T = 4.9 K

26 T 1 Distribution: x = 12 % (T c = 30 K) Probability distribution K La 1.88 Sr 0.12 CuO 4 80 K T c = 30 K 59 K 45 K 38 K 32 K 30 K 25 K 20 K 18 K 15 K 13 K 10 K 9 K 7 K 6 K 5 K 3.2 K 2.3 K -2 0 T g from µsr = 20 K 2 log[t 1-1 ] 4 6 T 1-1 [s -1 ] ( T 1 1 x = T c = 30 K T [K] )average ( T1 1 ) α α σ log Inhomogeneities develop below ~ 80 K

27 Probing spin dynamics with T 1 Time fluctuations of the hyperfine field 1 = γ 2 n T 2 A 1 + ( ) 2 S(0)S(t) e iω nt dt J(ω n ) Simple Model: 1 S(0)S(t) = S 2 e t /τ c BPP T 1 τ c 1 + ω n 2 τ c 2 Slowing down of fluctuations τ c T 1-1 enhancement until maximum when τ c -1 = ω n

28 T 1 Distribution: x = 12 % (T c = 30 K) Probability distribution K La 1.88 Sr 0.12 CuO 4 80 K T c = 30 K 59 K 45 K 38 K 32 K 30 K 25 K 20 K 18 K 15 K 13 K 10 K 9 K 7 K 6 K 5 K 3.2 K 2.3 K -2 0 T g from µsr = 20 K 2 log[t 1-1 ] 4 6 T 1-1 [s -1 ] ( T 1 1 x = T c = 30 K T [K] )average ( T1 1 ) α α σ log Inhomogeneities develop below ~ 80 K

29 Spatial inhomogeneities (Vortex States) Case 1. Spatially resolved measurement Case 2. Not resolved phenomenological fit exponent α quantifies disorder strength Case 3. Not resolved convolute with distribution function deduce parameters

30 Real Space Magnetic Field Distribution in the Vortex Lattice

31 NMR Microscopy Field Probability Distribution ~ NMR Spectrum M. Takigawa et al., PRL 83, 3057 (1999) -R. Wortis et al., PRB 61, (2000) -D. Morr and R. Wortis, PRB 61, R882 (2000) -N. J. Curro et al., PRB 62, 3473 (2000)

32 Normal State vs. Low Temperature Spectra Data Calculation H 0 = 4 T T 50 mk Magnitude [A.U.] H[T]

33 Normal State vs. Low Temperature Spectra Data Calculation H 0 = 4 T T 50 mk Magnitude [A.U.] H[T]

34 Spectra - f(t R )

35 Spectra - f(t R )

36 Low Temperature T 1 vs. H 0 & H int 0.8 T = 11K T 1-1 [s -1 ] H 0 13 T 23 T 37 T H int [T]

37 Low Temperature T 1 vs. H 0 & H int 0.8 T = 11K T 1-1 [s -1 ] H 0 13 T 23 T 37 T H int [T]

38 Low Temperature T 1 vs. H 0 & H int 0.8 T = 11K T 1-1 [s -1 ] H 0 13 T 23 T 37 T H int [T]

39 Low Temperature T 1 vs. H 0 & H int 0.8 T = 11K T 1-1 [s -1 ] H 0 13 T 23 T 37 T H int [T]

40 Outside the Cores T 1-1 increases with increasing H 0 T 1-1 increases with increasing H int, i.e. on approaching vortex core (TT 1 ) -1 = Constant T 1 lower then inside

41 Low Energy Excitations s-wave Low energy excitations bound to the core region and and occupy a fraction ~ B\H c2 (Caroli-deGennes-Matricon States). C. Caroli, P. G. degennes, J. Matricon, J., Phys. Lett. 9, 307, (1964) H. F. Hess et al., PRL 62, 214, (1989) d-wave Low energy excitations extended along nodal directions. G. E. Volovik, JETP Lett. 58, 469 (1993) Nature of the core states? I. Maggio-Aprile et al., PRL 75, 2754 (1995) Ch. Renner Ch. et al., PRL 80, 3603 (1998) S. H. Pan et al., PRL 85, 1536 (2000) J. E. Hoffman et al., Science 295, 452 (2002)

42 Outside the Cores - Energy Spectrum of Nodal QP T 1 1 N i (E)N f (E) In the nodal region quasiparticle DOS varies linearly on energy T 1 depends on the product of initial and final QP energies QP energy depends on temperature, applied field, and internal field E k = ɛ 2 k + 2 k ± 1 2 γ e H 0 + v f (k) p s = E T ±Z +D

43 Low Temperature T 1 vs. H 0 & H int 0.8 T = 11K T -1-1 T 1 [s -1 1 [s -1 ] ] H 0 H 0 13 T T T 37 T H int int [T]

44 Outside the Cores - Energy Spectrum of Nodal QP T 1 1 N i (E)N f (E) E i E f E k = E T ± Z + D T 1 1 ~ D 2 Z 2 F 1 T 1 1 E T + D i + Z E T + D f Z T 1 1 ~ D 2 + Z 2 F 3

45 Outside the Cores - Energy Spectrum of Nodal QP T 1 1 ~ D 2 + Z 2 F T = 11K 0.6 T 1-1 [s -1 ] 0.4 H T 23 T 37 T F 3 dominant scattering process H int [T] BUT q ~ (π, π) required => QP are AF correlated

46 Spatial inhomogeneities (FFLO) 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 0 2πξ 2 E P = E c H P c 2 = 1 2 χ nh 2 0 = 1 2 N(0) 2 χ n = 1 2 (gµ B) 2 N 0 2 gµ B Relative importance of the two effects described by the Maki Parameter: α = 2 Horb c 2 H P c 2

47 Basic Idea: The FFLO State Pauli pair breaking dominates over the orbital effects ( α ) 2 Horb C 2 HC P > Formation of a new pairing state with finite center-of-mass momentum can reduce the paramagnetic pair-breaking effect. The critical field can be further enhanced! P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964); A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964). BCS ( k, k ) FFLO ( k, k + q ) q 2µ BH u F

48 Basic Idea: The FFLO State Pauli pair breaking dominates over the orbital effects ( α ) 2 Horb C 2 HC P > Formation of a new pairing state with finite center-of-mass momentum can reduce the paramagnetic pair-breaking effect. The critical field can be further enhanced! P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964); A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964).

49 The FFLO State Finite q breaks spatial symmetry. SC order parameter oscillates in real space. (FF) ( r) = q e i q r!(r) (LO) More generally: ( r) = q cos ( q r) 0 1/q ( r) = m m e iq m r Y. Matsuda & H. Shimahara

50 The FFLO State Finite q breaks spatial symmetry. SC order parameter oscillates in real space. (FF) ( r) = q e i q r!(r) (LO) ( r) = q cos ( q r) 0 More generally: ( r) = m m e iq m r 1/q 2D thin film Y. Matsuda & H. Shimahara

51 FFLO SC Imbalanced Spin Populations T=0 k' E k Nature and stability of SC phase with population imbalanced? E F ~ µ B B FFLO q = k' - k g(e) P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964); A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964). Phases with gapless excitations G. Sarma, J. Phys. Chem. Solids 24, 1029 (1963); breached pair SC - W. V. Liu and F. Wilczek, PRL. 90, (2003); R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004). Mixed Phases of SC and Normal P. F. Bedaque, H. Caldas, G. Rupak PRL 91, (2003); H. Caldas, PRA 69, (2004); J. Carlson, S. Reddy, PRL 95, (2005); M. W. Zwierlein et al., Science 311, 492 (2006).

52 Microscopic Probes? Order Parameter Text d-wave Magnetization Text s-wave s-wave d-wave d-wave LDOS (spin-up)text s-wave LDOS (spin-up; low energy peak)text d-wave Q. Wang et al., PRL 96, (2006)

53 The NMR Probe Image of Local Magnetic Field Probability Distribution Local Magnetic susceptibility (LDOS)!(r) 1D 0 1/q d-wave D Magnitude [A.U.] Hint [T]

54 The NMR Probe Image of Local Magnetic Field Probability Distribution Local Magnetic susceptibility (LDOS) M. Ichioka and K. Machida, PRB 76, (2007).

55 Pseudogap Shastry-Mila-Rice form factors for HTS, Physica C 157, 561 (1989). 1 ν T 1z T = C q F β Im χ β (q, ω nn ) ω nn Examine magnetic field responce of the pseudogap in different regions of the Brillouin zone => spin gap vs pairing gap?

56 Pseudogap Explore magnetic field dependece of T1 T~ T* T~ T* T~ Tc T~ Tc MMP - (Millis 1990) χ(q, ω) =χ AF + χ FL = 1 4 j αξ 2 µ 2 B 1+ξ 2 (q Q j ) 2 iω/ω SF + χ 0 1 iπω/γ q (π, π) q (0, 0) Q i =(π ± δ, π ± δ)

57 SF-MMP: lim ω 0 1 T 1 T = χ (q, ω) ω k B χ(q, ω) =χ AF + χ FL = 1 4 2µ 2 B 2 q F c (q) 1 4 Pseudogap j j αξ 2 µ 2 B 1+ξ 2 (q Q j ) 2 iω/ω SF + αξ(t ) 2 µ 2 B /ω SF [1 + ξ(t ) 2 (q Q j ) 2 ] 2 + χ 0π Γ χ 0 1 iπω/γ ξ(t )=ξ 0 T x (T x + T ) 17 ( T1 T) a) Spin-Gap: 0.2 ω 1 SF [ tanh ( (T Tp ) c 1 )] [ξ 2 0 ξ(t )2 ] 63 ( T1 T) b) Temperature [K]

58 Pseudogap AF correlations (q (π,π)) supprressed below T* away from (q (π,π)) pseudo gap shows magnetic field dependence (H0 < 10 T) => pairing fluctuations - precursory effetc to SC 0.5 a) 17 ( T1 T) χ (q (0, 0)) 63 ( T1 T) χ (q (π, π)) b) Temperature [K]

59 NMR in SC K χ(q =0, ω = 0) N(E F ) For singlet SC: Yoshida (1958) Curro(2005)

60 NMR in SC 1 χ (q, ω = 0) N i (E F )N f (E F ) T 1 d-wave: s-wave: d-wave: 1 T 3 T 1 s-wave: Habel-Slichter Peak (u & v - coherence factors)