1. The approach of nanomagnets to thermal equilibrium F. Luis, F. Bartolomé, J. Bartolomé, J. Stankiewicz, J. L. García- Palacios, V. González, and L. M. García Instituto de Ciencia de Materiales de Aragón, Zaragoza, Spain F. Petroff, V. Cross, and H. Jaffrès Unité Mixte de Physique, CNRS-Thales, Orsay, France F. L. Mettes, M. Evangelisti, and L. J. de Jongh Kamerlingh Onnes Laboratory, Leiden University, The Netherlands Kyoto 2003

2. I Single nanomagnet: Anisotropy and its microscopic origin U = KV “up”“down” III Material: Dipolar interactions II Spin-bath interactions: phonons, electrons, decoherence Thermal bath T  coherence Spin-lattice relaxation

3. Outline of the talk Magnetic relaxation in Co clusters Size-dependent anisotropy and orbital magnetism Influence of a Cu layer on K and L Dipolar interactions and magnetic relaxation Spin-lattice relaxation of single-molecule magnets Non-linear susceptibility in the thermally activated regime Spin-lattice relaxation in the quantum regime Long range dipolar order

4. Surface anisotropy of Co clusters prepared by sequential deposition (Orsay) No trace of oxidation fcc crystal structure Good control of the average diameter between 0.7 and 6 nm Co Al 2 O 3 Si t Co =0.1 - 1 nm t Al 2 O 3 = 3 nm t Co =0.1 - 1 nm

5. Size distribution approximately independent of D Clusters of 30 to 4000 atoms Co 5 5 Co 147 Co 561 Co 2057

6. Activation energy: effective anisotropy M. I. Shliomis and V. I. Stepanov, Adv. Chem. Phys. 87, 1 (1994) bulk

7. e-e- K and L sensitive to the matrix (metallic or insulating) surrounding the cluster Surface anisotropy and orbital magnetic moment L S (m L  – m L|| ) L K   S-O LS Electron confinement enhanced L Surface anisotropic L P. Bruno, Phys. Rev. B 39, 865 (1989)

8.  D  = 2.6 nm XMCD study of the orbital moment Sum rules m L /m S Circularly polarized X-rays L 2,3 edges of Co Fluorescence and total electron yield B < 5 T ++ -- ++ --

9. Bulk Co: m L /m S = 0.097 m L m L A m S m S  D = bulk + The orbital magnetic moment increases as D decreases In bulk L = 0.15  B at the surface L  0.39  B L surface L bulk

10. K/L at the surface ~ 10(K/L) in bulk L becomes much more anisotropic at the surface (m L  – m L|| ) L K 

11. Effect of a metallic layer (Cu) 1.5 nm Clusters covered by a thin layer of Cu Samples with and without Cu show approximately the same equilibrium magnetic response Same cluster size distribution

12. but larger blocking temperature and larger orbital magnetic moment

13. Co Cu e-e- L becomes larger at the Co/Cu interface Agrees with experiments on Co/Cu layers (M.Tischer et al., Phys. Rev. Lett. (1995)) Modified DOS by hibridization with the Cu conduction band? (Wang et al. J. Mag. Mag. Mater., 237 (1994))

14. Interactions and magnetic relaxation Self organized growth of the clusters in 3D Babonneau et al., Appl. Phys. Lett. 76, 2892 (2000) Control over dipolar interactions Number of layers N Interlayer separation     

15. Series of samples: N = 1, 2, 3,..., 20 prepared under identical conditions The size distribution is almost independent of N

16. Experimental results The average U increases one layer30 layers

17. The blocking temperature increases almost linearly with the number of nearest neighbours F. Luis et al., Phys. Rev. Lett. 88, 217205 (2002) U = K s S + A N 

18. z 1 2 3 11 33 22 Theoretical model Inspired in Dormann model J.L. Dormann et al., J. Phys. C 21, 2015 (1988)  dominated by largest particles Nearest neighbors fluctuate rapidly Interaction energy is continuously minimized  =  0 exp U + E dip kBTkBT

20. The anisotropy is two orders of magnitude larger than in bulk and it is mainly determined by the atoms located at the surface: U = K s S The enhanced K is related to an increase of the orbital moment L at the surface L at the surface is much more anisotropic than in bulk  (K/L) surface  10 (K/L) bulk K and L can be enhanced by embedding the clusters in a metallic (Cu) matrix: potential for applications Dipolar interactions slow-down the relaxation process: U = K s S + AN nn Conclusions (Co clusters)

21. Single-molecule magnets D. Gatteschi et al., Science 265, 1054 (1994) Large intramolecular exchange interactions Net spin S Intermediate situation between paramagnetic atoms and magnetic nanoparticles Mn 12 Quantum world Classical world

22. ZFS 7 – 14 K Anisotropy H single = -DS z 2 – E(S x 2 – S y 2 ) Giant spin model: anisotropy and quantum tunnelling Tunnelling U z S

23. Slow relaxation towards thermal equilibrium Thermal bath (lattice) Phonon-induced transitions between levels Fast intrawell transitions    10 -7 s  C m 0 Fe 8 Slow interwell transitions:  >>    C m eq – C m 0 No equilibrium when  >  e C m = C m 0 e -  /  + C m eq (1 – e -  /   ee Equilibrium No Equilibrium H = H single + H spin-lattice

24. Dipole-dipole interactions J. F. Fernández and J. Alonso, Phys. Rev. B 62, 53 (2000) Large molecular spins Super-exchange interactions can be neglected Fast spin-lattice relaxation (low anisotropy) D  0.01 K Mn 6 S = 12 T c ~ 0.1 – 0.5 K long-range order (B dip ) 2,1 H = H single + H spin-lattice + H dipolar

25. Dipolar ferromagnet T c = 0.17 K A. Morello, et al. Phys. Rev. Lett. 90, 017206 (2002). Equilibrium experiments down to very low T T < T c T > T c

26. U kBTkBT  =  0 exp Resonant tunneling via excited states (T > 1 K) Multilevel Orbach process (Pauli Master equation)  <<   >   Tunnelling blocked by dipolar and hyperfine stray magnetic fields U ee TBTB U k B ln(  e /  0 ) T B = F. Luis, J. Bartolomé, and J. F. Fernández, Phys. Rev. B 57, 505 (1998)

27. Non-linear susceptibility of Mn 12 clusters M =  0 H –  3 H 3 +... Gives information on Equilibrium: magnetic anisotropy Non-equilibrium: spin-bath interaction damping  2/L2/L 00 2/L2/L 00 J. García-Palacios and P. Svedlindh, Phys. Rev. Lett. 85, 3724 (2000)

28. Third harmonic:  (3  ) Second order coefficient in  (  ) =   (  ) - 3   (  ) H 2 +... Experimental determination of  3 There are two possibilities h ac sufficiently small not to induce any extra nonlinearity The same qualitative behavior in the classical limit

29. A story of two Mn 12 molecular crystals U = 65 K for both compoundsSame anisotropy  0 = 3×10 -8 sMn 12 acetateDifferent spin-lattice interaction  0 = 1.5×10 -8 s Mn 12 2-Cl benzoate (benzoate)  2 (acetate) < 10 -3

30. Results Calculated Weak dependence on  0 Opposite signs!!! Experimental ?

31. Classical:  2  /  H 2 < 0 Quantum tunnelling:  2  /  H 2 > 0 The classical  3 should be recovered at high fields > 0.1  coherence <  0 Suitable method to ascertain if relaxation takes place via QT Explanation: quantum non-linearity

32. Application to more complex systems: natural ferritin D = 7 nm S  100 Tejada et al (1997): QT?Yes Mamiya et al (2002): QT? No Classical relaxation near T B

33. Spin-lattice relaxation in the quantum regime (T < 1 K) >> k B T ? ×    Tunnelling induced by a fluctuating bias (Prokof’ev and Stamp, Phys. Rev. Lett. 80, 5794 (1998))    Two-level system  Spin reversal but... No relaxation of energy Thermal bath (lattice) (Fernández and Alonso, Phys. Rev. Lett. 91, 047202 (2003))

34. Experiments: time-dependent specific heat (Leiden) (spin-lattice relaxation time  ) (“relaxation” or “experimental” time  e Adjustable: 0.1 – 1000 seconds) C =  e /R

35. Relaxation towards (ordered) equilibrium via quantum tunnelling: Mn 4 S = 9/2 U = 14.5 K H = – DS z 2 – E(S x 2 – S y 2 ) R Symmetry of the cluster R = Cl - (OAc) 3 (dbm) 3  = 10 -7 K R = (O 2 CC 6 H 4 -p-Me) 4 (dbm) 3  = 10 -4 K!!

36. Relaxation rate: time-dependent specific heat: Mn 4 Cl  becomes independent of T below 1 K: incoherent tunnelling Five order of magnitude faster than predicted for known processes!

37. Conclusions (single-molecule magnets) Quantum tunnelling provides a mechanism for relaxation to equilibrium for all T: High T: resonant tunnelling via excited states Low T: incoherent tunnelling mediated by phonons and nuclear spins (challenge for theoreticians!) Long-range magnetic ordering induced purely by dipolar interactions: Mn 6 (isotropic) and Mn 4 (Ising) Large quantum non-linear susceptibility

38. Collaborations: Samples Fe 8 J. Tejada, Departamento de Física Fonamental, Universitat de Barcelona, Spain Mn 4, Mn 6 G. Aromí, Universidad de Barcelona, Spain G. Christou, N. Aliaga, University of Florida, USA Mn 12 D. Gatteschi, Department of Chemistry, University of Florence, Italy 57 Fe 8 R. Sessoli, Department of Chemistry, University of Florence, Italy Theory J. F. Fernández, Instituto de Ciencia de Materiales de Aragón, CSIC-Universidad de Zaragoza, Spain

39. Arigato Thank you! ¡Gracias!

40. Tunnelling via lower lying states? BB Leaves the symmetry intact Increases  for all ±m doublets Tranverse magnetic field m = ±10 m = ±9 m = ±8 m = ±7 m = ±6 

41. T B = U/k B ln(t/  0 ) decreases No blocking at all when B  > 1.7 T !  tunnelling via the ground state

42. Direct measurement of the tunnel splitting: Fe 8 kBTkBT F. Luis, F. L. Mettes, J. Tejada, D. Gatteschi, and L. J. de Jongh, Phys. Rev. Lett. 85, 4377 (2000)

43.  tun     0.5 ns   1 ms (phonons) B  = 0 (  )  classical states m = -10 or m = +10 A mesoscopic Schrödinger cat B  = 3 T (  )  Quantum superpositions + - 

44. Common pehnomenon at the atomic scale Protons in hydrogen bonds Ammonium molecule But hard to conciliate with our macroscopic intuition Possible technological applications: quantum computing

45. The role of nuclear spin bath Quantum tunnelling of the electronic spin is forbidden by destructive interference when S = 9/2 (Loss et al., von Delft et al., 1992) at zero field  = 0 Hyperfine interactions with nuclear spins can break the degeneracy H = – DS z 2 – E(S x 2 – S y 2 ) + A hf (I x S x + I y S y + I z S z ) I S For Mn nuclei I = 5/2   0 Tunnelling

46. Experimental study: giant isotope effect in Fe 8 Spin of Fe nuclei: Natural  I = 0 57 Fe  I = 1/2

47. F. Luis, J. Bartolomé, and J. F. Fernández, Phys. Rev. B 57, 505 (1998)  coherence <<  0 > 0.1

48. T B (20 s) < T B (1 s)