From the atom to magnetic nanoparticles Edgar Bonet Laboratoire Louis Néel CNRS – Grenoble Brasov, september 2003 S = 10 2 to 10 6 S = 1/2 to ~ 30
Magnetic scales S = 1 10 10 2 10 3 10 4 10 5 10 6 10 8 10 10 10 20 giant spin single - domain multi - domain quantum tunneling, uniform rotation nucleation, propagation and quantization curling annihilation of domain walls quantum interference 1 1 1 Fe 8 S 0.7K S S 0 0 0 M/M M/M 0.1K M/M 1K -1 -1 -1 -1 0 1 -100 0 100 -40 -20 0 20 40 µ 0 H(T) µ 0 H(mT) µ 0 H(mT)
Giant spin molecules Mn 12 (S = 10) • Single crystals • high intra-molecular couplings • low inter-molecular couplings V 15 (S = ½) Ni 12 (S = 12) Collection of identical quantum systems Fe 8 (S = 10)
Giant spin model
Giant spin model
Landau-Zener tunneling • Oscillation time |S, m> |S, m’> • crossing time |S, m’> |S, m> • keeps the same state • follows energy level
Landau-Zener tunneling |S, m’> |S, m> • general result for a single level crossing • solution of the Schroedinger equation |S, m> |S, m’> L. Landau , Phys. Z. Sowjetunion 2 , 46 (1932); C. Zener , Proc. R. Soc. London, Ser. A 137 , 696, (1932); E.C.G. Stückelberg , Helv. Phys. Acta 5 , 369 (1932); S. Miyashita , J. Phys. Soc. Jpn. 64 , 3207 (1995); V.V. Dobrovitski and A.K. Zvezdin , Euro. Phys. Lett. 38 , 377 (1997); L. Gunther , Euro. Phys. Lett. 39 , 1 (1997); G.Rose and P.C.E. Stamp, Low Temp. Phys. 113, 1153 (1999); M. Leuenberger and D. Loss , Phys. Rev. B 61, 12200 (2000); M. Thorwart, M. Grifoni, and P. Hänggi , Phys. Rev. Lett. 85, 860 (2000); …
1 40 mK Magnetization 0.5 steps M/M S 0 Fe 8 S = 10 v=140 mT/s -0.5 v=70 mT/s v=14 mT/s v=2.8 mT/s -1 -1 -0.5 0 0.5 1 0 µ 0 H(T) -10 Energy (K) -20 -7 7 -8 8 -30 -9 9 -10 10 -40 with S = 10, D = 0.27 K, E = 0.046K -1 -0.5 0 0.5 1 A.-L. Barra et al. EPL (1996) µ 0 H z (T)
Spin-parity dependent quantum tunneling Kramers theorem: No matter how unsymmetric the crystal field, a system possessing an odd number of electrons must have a ground state that is at least doubly degenerate, even in the presence of crystal fields and spin-orbit interactions H. A. Kramers, Proc. Acad. Sci. Amsterdam 33, 959 (1930) Mesoscopic systems: M. Enz and R. Schilling R.,J.Phys.C ,19 (1986) L711 J.L. Van Hemmen and S. Süto, Europhys. Lett. 1, 481 (1986) D. Loss, D.P. DiVincenzo, and G. Grinstein, Phys. Rev. Lett., 69, 3232 (1992) J. von Delft and C. L. Hendey, Phys. Rev. Lett., 69, 3236 (1992) ∆ (a.u.) 10 1 0 H trans (a.u.)
Spin-parity dependent quantum tunneling S = 9/2 H a = 4.6 T ∆ 0 = 1.9*10 -7 K S = 8 H a = 5.1 T ∆ 0 = 0.28*10 -7 K S = 10 H a = 4.0 T ∆ 0 = 0.94*10 -7 K Environnemental effects • hyperfine interaction (nuclear spins) Phys. Rev. B 65, • dipolar interaction between molecules 180403 (2002) • exchange interaction between molecules etc.
Quantum phase interference (Berry phase) in single-molecule magnets easy axis ϕ - 50° ϕ - 20° 10 ϕ - 90° Tunnel splitting ²(10 -7 K) Z ϕ - 7° 1 H trans Y ϕ ϕ - 0° X hard axis M = -10 10 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 easy plane YZ Transverse field (T)
Quantum phase interference (Berry phase) in single-molecule magnets W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999) Theory: A. Garg, Europhys. Lett. 22, 205 (1993) easy axis ϕ - 50° ϕ - 20° 10 ϕ - 90° Tunnel splitting ²(10 -7 K) Z ϕ - 7° 1 Y ϕ - 0° X hard axis M = -10 10 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 easy plane YZ Transverse field (T)
Quantum phase interference (Berry phase) in single-molecule magnets W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999) Theory: A. Garg, Europhys. Lett. 22, 205 (1993) easy axis ϕ - 50° ϕ - 20° 10 ϕ - 90° Tunnel splitting ²(10 -7 K) Z ϕ - 7° H trans 1 Y ϕ - 0° X hard axis M = -10 10 0.1 easy plane 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Transverse field (T)
Parity of level crossings 100 M = -10 8 M = -10 9 10 -7 K) ² (10 M = -10 10 1 0.1 -1 -0.5 0 0.5 1 µ 0 H x (T) W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999)
Intermolecular interactions ( dipolar and exchange) Mn 4 spin chains, etc. Mn 12 ac (SB1) SMM MM “ideal” 0 0.001 0.01 0.1 1 10 100 J/D ? doped Fe 6 [Mn 4 ] 2 Fe 8 Fe 5 Ga
Molecular dimers R. Tiron,W. Wernsdorfer, C. Thirion, R. Giraud, E. Bonet, B. Barbara (LLN, CNRS, Grenoble, France), A. Benoit (CRTBT, CNRS, Grenoble, France), D. Mailly (LPN, CNRS, Marcoussis, France), N. Aliaga, S. Bhaduri, C. Boskovic, C. Canada, M. Soler, G. Christou (Dept. of Chemistry, Uni. of Florida, USA), E. Yang, E. M. Rumberger, D. N. Hendrickson (Dept. of Chemistry, Uni. of California at San Diego, USA)
Single molecule vs. Dimer H = H 1 + H 2 + J r r r r 2 + H i trans + g µ B µ 0 H i = − D S i , z S S S H 1 2 i (2S i + 1) energie states (2S 1 + 1)(2S 2 + 1) energie states S i = 9/2 : 10 levels S i = 9/2 : 100 levels m i = -S i , -S i +1, …, S i m 1 = -S 1 , -S 1 +1, …, S 1 m 2 = -S 2 , -S 2 +1, …, S 2
Zeeman Diagram for the S = 9/2 dimer H = H 1 + H 2 + J r r S S 1 2 100 energy states (m 1 ,m 2 ) r r 2 + H i trans + g µ B µ 0 H i = − D S i , z S H i
Exchange bias fixed bias
Anisotropy & intermolecular coupling -15 1 (9/2,9/2) (9/2,7/2) 0.04 K -20 0.14 T/s (-9/2,5/2) 0.5 x 1 x 2 -25 Energy(K) (-7/2,9/2) M/M s (-9/2,7/2) 0 -30 (-9/2,9/2) -35 -0.5 x 1 x 2 -40 (-9/2,-9/2) -1 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 -1 -0.5 0 0.5 1 µ 0 H z (T) µ 0 H (T) D = g * µ B /k B * X 1 = 2 * 0.928/1.38 * 0.58 = 0.75 K J tot = g * µ B /k B * X 2 / S = 2 * 0.928/1.38 * 0.34 / 4.5 = 0.10 K D = anisotropy constant; J tot = coupling constant
Tunneling in 4 the dimer 3 5 2 Transitons ( 1) (-9/2,-9/2) ? (-9/2, 9/2) ; 1 (2) (-9/2,-9/2) ? (-9/2, 7/2) relaxes ? (-9/2, 9/2) ; (3) (-9/2, 9/2) ? ( 9/2, 9/2) ; (4) (-9/2,-9/2) ? (-9/2, 5/2) relaxes ? (-9/2, 9/2) ; (5) (-9/2, 9/2) ? ( 7/2, 9/2) relaxes ? ( 9/2, 9/2). (1) and (3) are symmetric relative to the origin; 1 2 3 4 5
3 NA3 NA3 (O 2 CH 2 Cl 2 ) 2x Mn 2x Mn 4 O O 3 Cl Cl 4 (O ) 3 (py) (py) 3 3 NA11 NA11 2x Mn Mn 4 O O 3 Cl Cl 4 (O (O 2 CEt) CEt) 3 (py) (py) 3 2x -10 (9/2,7/2) (9/2,9/2) -15 -20 Energy (K) (-9/2,5/2) -25 (-7/2,9/2) (-9/2,7/2) -30 (-9/2,9/2) -35 5 2 3 1 4 -40 3 4 1 2 5 (-9/2,-9/2) -45 -1 -0.5 0 0.5 1 µ 0 H z (T) 1 1 NA11 0.04 K 0.04 K 0.5 0.5 M/M s M/M s 0 0 0.280 T/s 0.560 T/s 0.140 T/s -0.5 0.140 T/s -0.5 0.035 T/s 0.035 T/s 0.008 T/s 0.008 T/s 0.004 T/s 0.002 T/s -1 -1 -1 -0.5 0 0.5 1 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 µ 0 H (T) µ 0 H (T) Inter- Inter -molecular coupling is stronger in NA11 than in NA3; molecular coupling is stronger in NA11 than in NA3; Easier to resolve resonances (2) from (3) and (4) from (5) Easier to resolve resonances (2) from (3) and (4) from (5)
0.6 1/2 S = 1/2 Absorption of 0.4 0.2 microwaves Energy (K) h � 0 ν -0.2 -0.4 V 15 S = 1/2 -1/2 -0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 µ 0 H (T) 1 1 0.04 K 0.04 K 11 GHz 0.5 0.5 0.001 T/s period: 10 ms M/M s M/M s 0 0 0 s 0.1 ms 0.5 ms 1 ms -0.5 -0.5 2 ms 3 ms -1 -1 -0.6 -0.6 -0.4 -0.4 -0.2 -0.2 0 0 0.2 0.2 0.4 0.4 0.6 0.6 µ 0 H (T) µ 0 H (T)
Frequency dependence of the absorption of microwaves in V 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.04 K 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s M/M s 2.5 GHz 3.5 GHz 11 GHz 10 GHz 16 GHz 15 GHz 14 GHz 12 GHz 13 GHz 8 GHz 5 GHz 4 GHz 7 GHz 6 GHz 3 GHz 9 GHz 2 GHz 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T) µ 0 H (T)
1 D = 0.5 K Reducing 0 intermolecular -1/2 1/2 -1 Energy (K) couplings -2 -3/2 3/2 -3 -4 Fe 6 wheels: S = 0 -5/2 5/2 -5 -0.4 -0.2 0 0.2 0.4 µ 0 H z (T) 1 0.04 K 0.5 Doping with Ga M/M s 0 Fe 5 Ga : S = 5/2 0.280 T/s 0.140 T/s -0.5 0.070 T/s 0.035 T/s 0.017 T/s -1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 µ 0 H z (T)
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