1. SNS Experimental FacilitiesOak Ridge X0000910/arb Spin dynamics in cuprate superconductors T. E. Mason Spallation Neutron Source Project Harrison Hot Springs Dec. 9, 2000 CIAR Superconductivity Program

2. SNS Experimental FacilitiesOak Ridge X0000910/arb 2 excitations characterized by  ´´(Q,  )  a measure of absorption at (Q,  ). neutron scattering measures: S(Q,  ) ~  ´´(Q,  ) [n(  )+1]. note: Q  0, recover uniform susceptibility. the proportionality constant involves magnetic moment direction and form factor. Neutron Scattering and Spin Fluctuations

3. SNS Experimental FacilitiesOak Ridge X0000910/arb 3  o (Q,  ) for Metals Excitations are electron-hole pairs Lindhard susceptibility: As T  0 states near  F dominate Note: NMR relaxation rate:

4. SNS Experimental FacilitiesOak Ridge X0000910/arb 4 Facilities Neutron scattering measurements were carried out using TAS6 (RITA) at Risø, IN20 at the ILL, and MARI at ISIS.

5. SNS Experimental FacilitiesOak Ridge X0000910/arb 5 (La,Nd) 2-x Sr x CuO 4 Pure La 2 CuO 4 is an insulating antiferromagnet with quasi-two- dimensional magnetic interactions Doping with Sr (or Ba) suppresses T N, introduces holes in the CuO 2 planes, and leads to superconductivity (maximum T c =39 K for x~0.15) Vaknin et al (1987)

6. SNS Experimental FacilitiesOak Ridge X0000910/arb 6 Paramagnetic Critical Scattering Antiferromagnetic correlations in La 2 CuO 4 above T N are well described by renormalized classical model (see Keimer et al (1992) and Birgeneau et al (1995))

7. SNS Experimental FacilitiesOak Ridge X0000910/arb 7 La 2 CuO 4 Spin Wave Response The magnetic excitations of undoped La 2 CuO 4 are well described by (renormalized) classical spin wave theory for the 2D spin 1/2 Heisenberg antiferromagnet (Hayden et al, 1991) (Hayden et al, 1990)

8. SNS Experimental FacilitiesOak Ridge X0000910/arb 8 Stripe Ordering Tranquada et al have shown that static, long range ordering of spin and charge occurs in (La,Nd) 2-x Sr x CuO 4 pinned to the LTT structural distortion at x=1/8.

9. SNS Experimental FacilitiesOak Ridge X0000910/arb 9 Effect of Doping The high energy magnetic excitations in nearly optimally doped La 2-x Sr x CuO 4 retain the characteristics of the antiferromagnet: – slightly softened maximum energy – same periodicity with broader momentum distribution

10. SNS Experimental FacilitiesOak Ridge X0000910/arb 10 Energy Integrated Response The correlation length extracted from S(Q) decreases from 6.2 Å in La 2 CuO 4 (T=295 K) to 3.7 Å (=a o ) in La 1.86 Sr 0.14 CuO 4 (17 K) however the bulk of the spin fluctuations are still AF in nature.

11. SNS Experimental FacilitiesOak Ridge X0000910/arb 11 Local Susceptibility A new energy scale (~25 meV) is present in the metallic sample

12. SNS Experimental FacilitiesOak Ridge X0000910/arb 12 Low Energy Excitations in the Metallic State For metallic compositions the low energy response has shifted away from the commensurate (  ) position along the ( ,0) direction. The peaks are well defined (  >a o /  x). 0.5 ( , 0) x=0.075x=0.14 (Yamada et al, 1998) 1 meV, 12 K2 meV, 35 K (>T c )

13. SNS Experimental FacilitiesOak Ridge X0000910/arb 13 Normal State Energy Dependence As the frequency is increased the peaks become less well defined. The response is qualitatively quite similar to that of the spin density wave system Cr, above T N.

14. SNS Experimental FacilitiesOak Ridge X0000910/arb 14 Increased Energy, Temperature Have Similar Effects A combination of polarized and unpolarized measurements have permitted a reliable determination of the Q and  dependence of the magnetic response over a wide range of temperature.

15. SNS Experimental FacilitiesOak Ridge X0000910/arb 15 Temperature Dependence The magnetic intensity drops off rapidly with T. The peak susceptibility varies as 1/T 2 between T c (=35 K) and 350 K. This trend is interrupted by superconductivity: below T c the response is suppressed. The inverse length scale extracted from resolution corrected fits to the lineshape increases systematically with increasing T or .

16. SNS Experimental FacilitiesOak Ridge X0000910/arb 16 The inverse length scale which characterizes the peak width at a given energy and temperature is well described by: ,T Scaling In the T,   0 limit    o =0.034Å for x=0.14 and 0.06 Å for x=0.17 The fact that  and  enter with the same exponent implies: z = 1 where z is the dynamical exponent. Together with the 1/T 2 susceptibility this implies  =1. The inset shows  ´´ P /  vs T varying with an exponent of 3: for z=1 this implies  =0. Ambiguity because of  o.

17. SNS Experimental FacilitiesOak Ridge X0000910/arb 17 Quantum Criticality Taken together these results reveal that La 1.86 Sr 0.14 CuO 4 is close to a quantum critical point characterized by exponents z=1,  =0. These exponents are consistent with expectations for the QCP associated with 2D insulating magnets (Sachdev and Ye, 1992; Chubukov et al, 1994). Alternatively z=1,  =1 would be expected for 1D quantum antiferromagnets (Luther & Peschel, 1975). The similarity of the dynamic fluctuations to the patterns observed in the ordered stripe phases for Nd doped sample suggests a connection.

18. SNS Experimental FacilitiesOak Ridge X0000910/arb 18 Link to Commensurate Stripe Instability? The observation that the residual  o for La 1.83 Sr 0.17 CuO 4 is larger points to lower doping... The low energy length scale extracted from studies at various doping levels becomes anomalously large near x=1/8: the concentration for which commensurate stripe order occurs nearby in phase space and for which short range structural features have been observed in La 2-x (Ba,Sr) x CuO 4.

19. SNS Experimental FacilitiesOak Ridge X0000910/arb 19 Recap - Normal State Insulator, Antiferromagnet – spin waves La 2 CuO 4 (x=0) Insulator, MFL broad, commensurate response La 2-x (Ba,Sr) x CuO 4 (x=0.05) Metal incommensurate response La 2-x Sr x CuO 4 (x=0.14, 0.17) Temperature and energy dependence of  ´´ for the metallic samples suggests proximity to T=0 QCP.

20. SNS Experimental FacilitiesOak Ridge X0000910/arb 20 YBa 2 Cu 3 O 6+x Spin Dynamics Spin dynamics of antiferromagnetically ordered Y123 also well described by (renormalized) classical spin wave theory, including bi- layer coupling (see work by Tranquada et al, Rossat-Mignod et al, and Hayden et al)

21. SNS Experimental FacilitiesOak Ridge X0000910/arb 21 Superconducting YBa 2 Cu 3 O 7-x Low energy, commensurate [Q=(  ), acoustic mode] response in the normal state of four different compositions of YBa 2 Cu 3 O 7- x measured at 100 K. As the doping is increased the feature at   ) broadens and weakens, and there is very little normal state response at the commensurate position for the overdoped sample. From Bourges et al. (1998).

22. SNS Experimental FacilitiesOak Ridge X0000910/arb 22 Incommensurate fluctuations in YBa 2 Cu 3 O 7-x Images of the magnetic scattering from YBa 2 Cu 3 O 6.6 above and below Tc at 34 and 24.5 meV in the two dimensional reciprocal space of the CuO 2 planes. At the lower energy (e,f) an incommensurate response, described by the model shown in d, appears at the positions noted in the schematic map, a. The resonance appears at the  ) position (b,c) in the superconducting state. From Mook et al. (1998a).

23. SNS Experimental FacilitiesOak Ridge X0000910/arb 23 The (  ) resonance Variation of the  ) resonance energy with superconducting transition temperature, T c. From Bourges (1998). A similar feature is also seen in BSSCO (Keimer et al)

24. SNS Experimental FacilitiesOak Ridge X0000910/arb 24 The resonance in underdoped Y123 Temperature dependence of the 35 meV resonance in YBa 2 Cu 3 O 6.6 with temperature. A broadened response at  ) persists in the normal state for underdoped compositions. From Mook et al. (1998b).

25. SNS Experimental FacilitiesOak Ridge X0000910/arb 25 Superconductivity in LSCO Superconductivity suppresses the low energy response (below ~ 8 meV) and enhances the higher energy response The extent of the suppression is sample/composition dependent Data shown is for x=0.16, further from QCP than x=0.14 sample - more metallic, better sc

26. SNS Experimental FacilitiesOak Ridge X0000910/arb 26 Changes Induced by Superconductivity are Significant Suppression at low T is complete – Consistent with x=0.15 (Yamada et al) Higher energy response shows different Q dependence in sc state – Response at incommensurate point is sharper

27. SNS Experimental FacilitiesOak Ridge X0000910/arb 27 Momentum Dependence Analysis of the momentum dependence of the inelastic response reveals that: – magnetic “gap” does not vary with Q - 6.7 meV – inverse lifetime or broadening of the gap is momentum dependent with a minimum at the incommensurate wavevector – the incommensurate peak in the real part of the susceptibility is reduced by superconductivity

28. SNS Experimental FacilitiesOak Ridge X0000910/arb 28 Recap - Superconductivity The wavevector independence of the spin gap is in contrast to the nodal structure seen in the charge channel for high T c cuprates – this deviation of the behavior of spin and charge may be taken as evidence of spin-charge separation – at the very least it implies simple (non-interacting) models of the effects of d- wave superconductivity on the susceptibility are inadequate Suppression of  ´ implies sc competes with stripe instability

29. SNS Experimental FacilitiesOak Ridge X0000910/arb 29 Recap Absence of low energy spin response along the ( ,  ) direction is not expected in simple models with nodal quasiparticles. Although statistics limit the bound on low , low T signal the very strong effects at higher energies, including Q- independence of spin gap are well established and visible, even in the raw data. – Enhancement and sharpening in Q for  > 8 meV. – Minima in inverse lifetime at the incommensurate points. ( ,  ) resonance observed in other cuprates, notably Y123, not found in single layer La214, appears above T c for underdoped Y123. Incommensurate spin fluctuations seem to be common feature at least for “underdoped” compositions.