1. Thermodynamics of phase formation in Sr 3 Ru 2 O 7 Andy Mackenzie University of St Andrews School of Physics and Astronomy University of St Andrews, UK PITP Toronto 2008

2. M. Allan 1, F. Baumberger 1, R.A. Borzi 1, J.C. Davis 1,3,4, J. Farrell 1, S.A. Grigera 1, J. Lee 4, Y. Maeno 5, J.F. Mercure 1, R.S. Perry 1,2, A. Rost 1, Z.X. Shen 6, A. Tamai 1, A. Wang 3 University of St Andrews Collaborators 1 University of St Andrews; 2 University of Edinburgh; 3 Cornell University; 4 Brookhaven National Laboratory 5 Kyoto University; 6 Stanford University

3. Contents 1. Introduction – materials and terminology 2. Metamagnetic quantum criticality and low-frequency dynamical susceptibility in slightly dirty Sr 3 Ru 2 O 7. 4. Magnetocaloric effect as a probe of the ‘entropic landscape’ 5. Spectroscopic imaging of conductance oscillations around scattering centres: a dynamics-to-statics transducer. 6. Conclusions 3. Phase formation in ultra-pure Sr 3 Ru 2 O 7

4. M H Metamagnets and the vapour-liquid transition Mapping between both systems P, T,  H, T, M T P Critical end-point 1 st order liquid vapour T H H

5. u T h Metamagnets and Quantum Critical Points Critical end-point 1 st order Important difference with water: The transition can be tuned to T=0. Large majority of real itinerant metamagnets are first order at T = 0 even after best effort to tune. See e.g. T. Goto et al., Physica B 300, 167 (2001)

6. S.A. Grigera, R.A. Borzi, A.P. Mackenzie, S.R. Julian, R.S. Perry & Y. Maeno, Phys. Rev. B 67, 214427 (2003). Experimental phase diagram of “clean” Sr 3 Ru 2 O 7 0 20 40 60 80 100 0 200 400 600 800 1000 1200 1400 5 6 7 8 F i e l d [ t e s l a ] T e m p e r a t u r e [ m K ] a n g l e f r o m a b [ d e g r e e s ] Plane defined by maxima of imaginary part T* inferred from maximum in real part of a.c. susceptibility Quantum critical end-point  c-axis (90)

8. 1.2 0.8 0.4 0.0 T (K) 4.5 5.5 6.5 Field (tesla) 1.2 0.8 0.4 0.0 T (K) 4.5 5.5 6.5 Field (tesla) T* = 1.25K  = 0 (H // ab) x 10 1.2 0.8 0.4 0.0 T (K) 4.9 6.4 7.9 Field (tesla) 1.2 0.8 0.4 0.0 T (K) 4.9 6.4 7.9 Field (tesla) T* = 1.05K  = 40° x 10 Constructing the experimental phase diagram

9. 1.2 0.8 0.4 0.0 T (K) 5.5 6.5 7.5 Field (tesla) 1.2 0.8 0.4 0.0 T (K) 5.5 6.5 7.5 Field (tesla)  = 60° T* = 0.55K x 0.5 x 10 1.2 0.8 0.4 0.0 T (K) 6.5 7.5 8.5 Field (tesla) 1.2 0.8 0.4 0.0 T (K) 6.5 7.5 8.5 Field (tesla)  = 90° (H // c) x 10 T* < 0.1K No evidence of first-order behaviour for H // c

10.  = 0 (H // ab) 1.2 0.8 0.4 0.0 T (K) 4.9 6.4 7.9 Field (tesla) 1.2 0.8 0.4 0.0 T (K) 4.9 6.4 7.9 Field (tesla)  = 40° Evidence for very slow dynamics Why are the global maxima so weak? Large changes at amazingly low frequency  max (10 -6 m 3 /mol Ru) 2 4 0 123 f (kHz)

11. 7.98.18.37.7  o H (T) 0.4 0.8 1.2 0 T(K) - Resistivity: d  /dH and d 2  /dT 2 - Susceptibility:  ’ and  ’’ - Magnetostriction: (H) - Magnetisation Approach to criticality ‘cut off’ by a new phase in highest purity samples ( ~ 3000 Å) S.A.Grigera et al., Science 306, 1154 (2004) P. Gegenwart et al., Phys. Rev. Lett. 96, 136402 (2006) R.A. Borzi et al., Science 315, 214 (2007) Phase lines bound a region with pronounced resistive anisotropy: ‘electronic nematic’ properties

12. 7.98.18.37.7  o H (T) 0.4 0.8 1.2 0 T(K) “The wrong shape” usually: “dome” here: “muffin” first order phase trasitions? -> Clausius-Clapeyron The H-T Phase diagram S inside bigger than S outside S>S> S<S<

13.  S < 0 →  T > 0 Entropy H 1 H 2 Temperature S T 1 T 2 SS TT How to “measure the entropy”

14. Copper Ring CuBe Springs Kevlar Strings (35 @ 17μm) Silver Platform with sample on other side Thermometer (Resistor) 2 cm Our experimental setup (Andreas Rost) High level of control possible via tunable thermal link; easy system to model.

15. H [T] T [mk] Metamagnetic crossover seen in susceptibility Sharper features associated with first order transitions Sample raw Magnetocaloric Effect data from Sr 3 Ru 2 O 7 ‘Signs’ of changes confirm that entropy is higher between the two first order transitions than outside them. Under fully adiabatic conditions

16. μ 0 H [T] T [mK] T=150mK Magnetocaloric quantum oscillations 1 0 ΔT [mK] μ 0 H [T] 8.5 9 9.5 10 10.5 11 11.5 Measurement noise level: 25 μK / √Hz

17. μ 0 H [T] T [mK] T=150mK Magnetocaloric quantum oscillations 1 0 ΔT [mK] 1/μ 0 H [T -1 ] 0.09 0.1 0.11 Measurement noise level: 25 μK / √Hz

18. Preliminary conclusions from magnetocaloric effect (MCE) work on Sr 3 Ru 2 O 7

19. MCE confirms our prior identification of first-order lines as equilibrium phase transitions Entropy is indeed higher between the lines than either side of them. ‘Phase’ seems to be characterised by ‘quenching’ of Preliminary conclusions from magnetocaloric effect (MCE) work on Sr 3 Ru 2 O 7

20. μ 0 H [T] T [K] Taking the next step: the ‘entropic landscape’

21. μ 0 H [T] T [K] S/T [J/mol K 2 ] Taking the next step: the ‘entropic landscape’ 0.12 0.17 0.22 0.27

22. μ 0 H [T] T [K] S/T [J/mol K 2 ] Taking the next step: the ‘entropic landscape’ μ 0 H [T] 0.12 0.17 0.22 0.27

23. μ 0 H [T] T [K] S/T [J/mol K 2 ] Taking the next step: the ‘entropic landscape’ μ 0 H [T] 0.12 0.17 0.12 0.17 0.22 0.27

24. μ 0 H [T] T [K] S/T [J/mol K 2 ] Taking the next step: the ‘entropic landscape’ μ 0 H [T] 0.12 0.17 0.22 0.27

25. μ 0 H [T] T [K] μ 0 H [T] T [K] S/T [J/mol K 2 ] Taking the next step: the ‘entropic landscape’ 0.12 0.17 0.22 0.27

26. Power Law Fit To Specifc Heat (C(H)-C(5T))/ T Field [T] Fitequation Fitrange 5 T to 7.1 T Resulting Parameters a = 0.004(1) b = -0.99(5) c = -0.012(2) 3.544.555.566.577.588.5 0 0.02 0.04 0.06 0.08 0.1 data fitted curve

27. dHvA and STM QPI and ARPES: Fermi velocities in Sr 3 Ru 2 O 7 of 10 km/s and below: suppressed from LDA values by at least a factor of 20: direct observation of d-shell heavy fermions.   k kFkF -kF-kF q = 2k F  =  F q < 2k F q > 2k F Spatially resolved conductance oscillations around scattering centres: a dynamics–to–statics transducer

28. Conclusions Sr 3 Ru 2 O 7 can be tuned towards a quantum critical metamagnetic transition. If this is done in ultra-pure crystals (mfp > 3000Å) a new phase forms before the quantum critical point is reached. The magnetocaloric effect, if measured with care in a calibrated system, can give a comprehensive picture of the entropy evolution near QCPs. Material with slight disorder shows strongly frequency- dependent low T susceptibility; situation in pure material still needs to be investigated.

29. μ 0 H [T] T [K] S/T [J/mol K 2 ]

30. ΔS (J/K) Field (T) 0.09 0.05 0 7 7.5 8 8.5 T = 600 mK

31. Consider the ferromagnetic superconductor URhGe Superconductivity at low T, B Metamagnetic transition due to spin reorientation deep in ferromagnetic state Metamagnetic QCP? D. Aoki, I Sheikin, J Flouquet & A. Huxley, Nature 413, 613 (2001)

32. In URhGe the new phase in the vicinity of the metamagnetic QCP is superconducting Re-entrant superconductivity! F. Lévy, I. Sheikin, V. Hardy & A. Huxley, Science 309, 1343 (2005). Perspective: A.P. Mackenzie & S.A. Grigera, ibid p. 1330

33. F. Lévy, I. Sheikin & A. Huxley, Nature Physics 3, 461 (2007) Potentially more than ‘just’ interesting basic science: 25 T insufficient to destroy superconductivity although T c < 0.5 K!

34. Pronounced resistive anisotropy in a region of phase space bounded by low T 1 st order phase transitions J  H  J // H  HH J HH J R.A. Borzi, S.A. Grigera, J. Farrell, R.S. Perry, S. Lister, S.L. Lee, D.A. Tennant, Y. Maeno & A.P. Mackenzie, Science 315, 214 (2007)  T = 100 mK

35.  (  cm)  ac (arb. Units) T = 100 mK

37. Example of magneto-thermal oscillation with field aligned to c-axis H [T]

38. University of St Andrews Structure chi(T) and refto Shinichi etc. Basic bulk properties of Sr 3 Ru 2 O 7 At low temperature and low applied magnetic field, it is an anisotropic Fermi liquid (  c /  ab  100). S.I. Ikeda, Y. Maeno, S. Nakatsuji, M. Kosaka and Y. Uwatoko, Phys. Rev. B 62, R6089 (2000). Low-T susceptibility is remarkably isotropic and T-independent: strongly enhanced Pauli paramagnet on verge of ferromagnetism?

39. Ruthenates: electronic structure considerations d shelltet. cryst. fieldfilling & hybridisation

40. d shelltet. cryst. fieldfilling & hybridisation Cu 2+ 3d 9 Ruthenates: electronic structure considerations

41. d shelltet. cryst. fieldfilling & hybridisation Ru 4+ 4d 4 Ruthenates: electronic structure considerations

42. Intermediate Report 23 rd September 2008

43. Entropy Change Decreasing T (S(H)-S(5T))/T as a function of H Different temperatures are offset for clarity

44. (S(H)-S(5T))/T [J / mol K^2] H [T] T [K] Entropy Surface

45. H [T] T [K] (S(H)-S(5T))/T [J / mol K^2] Entropy Surface

46. T [k] H [T] Entropy Surface

47. Entropy Change Decreasing T (S(H)-S(5T))/T as a function of H Different temperatures are offset for clarity

48. 45678910111213 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 Field [T] Entropy change (S(H)-S(5T))/ T For better comparison I will choose 4 traces at T= (230mK, 400mK,900mK,1450mK) Entropy Change

49. 45678910111213 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 Field [T] Entropy change (S(H)-S(5T))/ T 230mK For better comparison I will choose 4 traces at T= (230mK, 400mK,900mK,1450mK) 400mK 900mK 1450mK Entropy Change

50. 45678910111213 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 456789 0 0.02 0.04 0.06 0.08 0.1 Field [T] Entropy change (S(H)-S(5T))/ T Field [T] Entropy Change On the right these curves are plot without offset

51. Comparison (C(H)-C(5T))/T vs (S(H)-S(5T))/T 456789 0 0.02 0.04 0.06 0.08 0.1 Field [T] Entropy change (S(H)-S(5T))/ T The curve in blue is (C(H)-C(5T))/T at 250mK. The fact that its amplitude is identical to the measured entropy change confirms that up 7.1T the system behaves like a Fermi Liquid. 44.555.566.577.588.59 0 0.02 0.04 0.06 0.08 0.1

52. Power Law Fit To Specifc Heat (C(H)-C(5T))/ T Field [T] Fitequation Fitrange 5 T to 7.1 T Resulting Parameters a = 0.004(1) b = -0.99(5) c = -0.012(2) 3.544.555.566.577.588.5 0 0.02 0.04 0.06 0.08 0.1 data fitted curve

53. Isentropes dS=0 T [K] H [T]

54. Si /Rosch Paper In a Fermi Liquid: Definition of Magnetocaloric Effect: Assume Power Law: I.e. it is a general result that is independent of the power law the entropy itself follows!

55. Si /Rosch vs Millis/Grigera/… Si / Rosch (on the unorder (high field) side Millis / Grigera / … around Both assume that the dynamical dimension is z=3 and the real dimension is d=2 for a ferromagnetic QCEP in 2 dimensions but they mention different critical exponent for the specific heat coefficient But: I need to check that these calculations have been done for constant number and not constant chemical potential…

56. Antisymmetrise the up and down sweep Old way: First integrate each trace and then smooth New way:First smooth the signals and then integrate T H H Δ T 0 ΔSΔS ΔSΔS ΔSΔS 0H H H H T T