1. Microscopic origin of the adiabatic change of the magnetization Hans De Raedt Applied Physics-Computational Physics, Materials Science Centre, University of Groningen,The Netherlands deraedt@phys.rug.nlderaedt@phys.rug.nl ; http://www.compphys.orghttp://www.compphys.org Material for this talk taken from work done in collaboration with S. Miyashita (Tokyo), K. Michielsen (Groningen), V.V. Dobrovitski & B.N. Harmon (Ames Lab) M.I. Katsnelson (Uppsula)

2. Outline Introduction Microscopic mechanism(s) for energy-level repulsions V 15 (and Mn 12 ) Coherence of oscillations of the magnetization Estimating the decoherence (T 2 ) time NMR spin echo experiments on 29 Si in Si powders Computer simulation

3. Introduction At very low temperatures, the magnetization-versus applied field curve of single-molecule magnets such as Mn 12 and V 15 exhibit unusual features. V 15 K 6 [V 15 As 6 O 42 (H 2 O)]. 8H 2 O Mn 12 (CH 3 C00) 16 (H 2 O) 4 O 12 ]. 2CH 3 C00H. 4H 2 O Mn 12

4. Introduction The position and size of the steps contains information about the details of the energy- level structure of the interacting magnetic moments in the molecule. B. Barbara et al., JMMM 200 (1999) 167 Mn 12

5. Introduction The position of the steps (values of the applied magnetic field) can often be described by very simple models of one (Mn 12 ) or three (V 15 ) spins. I. Chiorescu et al., JMMM 221 (2000) 103 V 15

6. Introduction At very low temperatures and for very slow sweeps of the applied field, the change of the magnetization at a step is related to the energy splitting of the quantum states with different magnetization. Landau-Zener-Stückelberg (LZS) transition (Miyashita 1996). E h EE

7. Introduction Theoretical understanding of the magnetization dynamics of the single-molecule magnets requires detailed knowledge of the energy levels, their degeneracy and whether levels cross or repel at particular values of the applied magnetic field. What are the microscopically relevant interactions? The calculation of energy levels/splittings of these systems is a very challenging computational problem. Large Hilbert space, (nearly) degenerate eigenvalues, large differences in energy scales, … Lanczos + full orthogonalization Chebyshev-polynomial-based projector method Full exact diagonalization

8. Modeling magnetic properties of single-molecule magnets Electronic structure: Ab initio, LSDA+U,… Yields estimates for exchange interactions … Many-spin models Not “many-body” but still fairly large number of degrees of freedom Hilbert space dimension: 10 000 – 100 000 000 Single-spin models Hilbert space dimension: 2 – 100

9. 15xS = 1/2 frustrated antiferromagnet: Hilbert space dimension = 32768 Full diagonalization impossible! V 15 molecule V1V1 V2V2 V2V2 14 13 12 11 10 15 9 7 8 1 2 3 4 5 6 J2J2 J1J1 J4J4 J5J5 J6J6 J3J3 J J’ J” V1V1 x y z V3V3 Dzyaloshinsky-Moriya interaction

10. V 15 model parameters (1) D.W. Boukhvalov et al, JAP, 93 (2002) 7080 All J’s from LDA+U calculation experiment

11. V 15 model parameters (2) N.P. Konstantinidis and D. Coffey, PRB 66 (2001) 014408 J = -800K, J 1 = J’ = -225K, J 2 = J” = -350K DM interaction: D x 1,2 = D y 1,2 = D z 1,2 = 40K Use rotational symmetry of the hexagons

12. V 15 model parameters (3) I. Rudra et al, J.Phys.C 13 (2001) 11717 J = -800K, J 1 = J’ = -54.4K, J 2 = J” = -160K DM interaction: D x 1,2 = D y 1,2 = D z 1,2 = 40K Use rotational symmetry of the hexagons

13. V 15 Energy levels of lowest eight states may be represented by a S=1/2 model of spins on the triangle M z =1/2  3/2 M z =-1/2  1/2 I. Chiorescu et al., JMMM 221 (2000) 103

14. V 15 : Adiabatic transitions At h = 0 there is a level CROSSING. Near h = 0 there is a level REPULSION No LZS transition from M z = 1/2 to 3/2, in disagreement with experiment I. Chiorescu et al., JMMM 221 (2000) 103 ?

15. V 15 : Energy level diagram near h=0 Level repulsion is here, not at h=0! No level repulsions at all

16. Three-spin model for V 15 Dzyaloshinsky-Moriya interaction + symmetry of the triangle NO adiabatic transition at all H-sweep

17. Three-spin model for V 15 : dependence on H-direction Level repulsion is here, not at h=0!

18. Microscopic origin of the energy gaps ? Dzyaloshinsky-Moriya interaction breaks rotational symmetry and seems therefore to be a good candidate to explain the origin of energy gaps in the single-molecule magnets. Our calculations for 3- and 15-spin V 15 models demonstrate that the energy gaps are anisotropic with respect to the direction of the applied field. Only for special directions of the applied field, adiabatic changes of the magnetization are possible. Adopting the extended (Kaplan-Shekhtman-Entin-Wohlman- Aharony) form of the interaction does not change these conclusions. Alternative mechanisms … h  E ?

19. Summary The Dzyaloshinsky-Moriya interaction yields level splittings that depend on the direction of the applied magnetic field Does not seem compatible with present experimental findings For Mn 12 we find that DM + local anisotropy (in x- y-z direction) does not yield observable energy gaps. Microscopic origin of energy gaps remains an open question

20. Quantum dynamics of the magnetization: Decoherence Energy level repulsions: oscillations of the spins Presently very relevant for experimental demonstration of qubit operation Questions: What does the observation of oscillations mean in terms of coupling of the spins to their environment Decoherence, T 2

21. Concrete example Spin-1/2 objects  interacting with a “bath” of spin- 1/2 objects . “Simple” model: In general, the interactions between the spins are often anisotropic Quantum dynamics of the subsystem: Magnetization: Reduced density matrix: Variance on the subsystem entropy: Characterizes “mixing” of the state: decoherence

22. Simulation of two S=1/2 spins coupled to a S=1/2 bath Model: Random J n Initial state of subsystem: Initial state of bath: random We observe: fast decoherence followed by a resurrection of magnetization oscillation a very slow decay of the amplitude of these oscillations Magnetization Variance of the entropy Magnetization envelope S 1 S 2 correlations V.V. Dobrovitski et al, Phys. Rev. Lett. 90, 210401 (2003) L=14

23. Simulation of two S=1/2 spins coupled to a S=1/2 bath Model: Initial state of subsystem: Fast decoherence, followed by oscillations with a very slow decay of the amplitude Spin-spin correlation vanishes fast Presence of oscillations in an observable of the subsystem does not say much about decoherence It is necessary to measure CORRELATION L=20

24. What is the decoherence time of 29 Si in silicon* ? 29 Si in silicon: dilute dipolar-coupled spins in a solid Model Hamiltonian (rotating frame)  i : random magnetic shift a ij =-2b ij : dipolar interaction between spins on random lattice positions *A.E. Dementyev et al, Phys. Rev B 68, 153302 (2003)

25. NMR on 29 Si in silicon* Standard NMR measurement of T 2 : Hahn echo sequence 90 X -(TE/2)-180 Y -(TE/2)-ECHO TE = variable delay time Carr-Purcell-Meiboom-Gill (CMPG) sequence 90 X -{(TE/2)-180 Y -(TE/2)-ECHO} repeat n times Common wisdom: CPMG sequence should not excite echos beyond T 2 (Hahn sequence) *A.E. Dementyev et al, Phys. Rev B 68, 153302 (2003)

26. NMR on 29 Si in silicon* Experiments* at T=4.2K: Hahn echo signal TE=1.12ms TE=2.65ms *A.E. Dementyev et al, Phys. Rev B 68, 153302 (2003) CMPG echo signal

27. Computer simulation of NMR on 29 Si in silicon (1) 1. Make Si crystallites containing L ( 29 Si) spins (= 29 Si atoms) 2. Determine dipolar interactions,… 3. Solve the time-dependent Schrödinger equation for the CPMG or Hahn pulse sequence 4. Average over many (10-100) crystallites

28. Computer simulation of NMR on 29 Si in silicon (2) Si crystallites with L ( 29 Si) spins ( 29 Si atoms), determine dipolar interactions etc. Yellow: 29 Si (4.67% n.a) Red: Selected as one of the L spins

29. Computer simulation of NMR on 29 Si in silicon (3) Simulation results (L = 20; 10 samples) We set H dip-dip =0 during the pulses: “perfect pulses” Only possible in computer simulation

30. Physical picture: Spin k feels a random field  k and interacts with a “bath” of other spins via a dipole-dipole interaction. This interaction leads to de-phasing:T 2 T 2 as obtained from different pulse sequences (e.g. Hahn or CMPG) may be substantially different. Simulation results give a clue to understand these results: During the 180 Y pulses, the interaction modifies the time evolution and this has to be taken into account.

31. Conclusion

32. Algorithms Exact diagonalization: Test purposes, small systems only Suzuki product-formula-approach: Pair-product and XYZ decomposition Second-order and fourth-order accuracy in the time step Short iterative Lanczos method of order N:, Chebyshev polynomial approach

33. Algorithms: Comparison Exact Diagonalization Chebyshev Polynomial Suzuki Second- order Pair Suzuki Fourth- order Pair Suzuki Second- order XYZ Suzuki Fourth- order XYZ Short Iterative Lanczos of order N=5(10) Conservation of probability = unconditional stability - MP - Error and CPU time on a W2000 Athlon XP 1900+. Example: 12 S=1/2, t=20J 0 - MP - 0.22 10 -3 0.45 10 -8 0.71 10 -1 0.18 10 -4 0.13 10 -5 ( - MP- ) 6793.11.8s2.8s9.8s1.1s5.7s17.3s (34.6s) Time-dependent H-- - MemoryO(D 2 )O(D) O(ND) CPU (D=Hilbert space dimension) O(D 3 )O(t)O(t 3/2 )O(t 3/4 )O(t 3/2 )O(t 3/4 )O(N 2 Dt/t) Overall (for a reasonable level of time-integration and accuracy) Essential for testing purposes Not useful for large problems Most efficient to reach long times Not accurate enough Useful for isotropic cases Not accurate enough Useful for very anisotropic cases Non-trivial dependence of performance on N and time-step t

34. Test case: Mn 12 Model reduction: 4xS = 1/2 + 4xS = 2 Hilbert space dimension = 10 000 Compare full exact diagonalization of 10000x10000 matrices with results of more efficient (in terms of computer time and/or memory) algorithms J1J1 J2J2 J3J3 J4J4 13 5 7 2 4 6 8 J J’ x y

35. Test case: Mn 12 Ferromagnetic interactions J, J’ + Dzyaloshinsky-Moriya interactions B. Barbara et al., JMMM 177 (1998) 1324 Fit J, J’, K z, D i,j to neutron scattering, EPR and magnetic susceptibility data 13 5 7 2 4 6 8 J J’ x y M.I. Katsnelson et al., PRB 59 (1999) 6919

36. Mn 12 : Energy levels Lanczos+full orthogonalization results Model parameters: J = 23.8K J’ = 79.2K K z = 5.72K D z = 22K D y = 0 D z = 10K M z  9 M z  10 M z  8

37. Alternative Mechanism: Hyperfine Interaction Three-spin model Very slow H-field sweep: All three LZS transitions Energy-level diagram J = -10, A = 1,     N = 0.001

38. Simulation results:one S=1/2 coupled to an Ising spin Bath Magnetization Variance of the entropy Magnetization envelope V.V. Dobrovitski, H.A. De Raedt, M.I. Katsnelson, and B.N. Harmon, arXiv: quant-ph/0112053