QUANTUM SPIN DYNAMICS OF RARE-EARTHS IONS B. Barbara, W. Wernsdorfer, E. Bonet, L. Thomas (IBM), I. Chiorescu (FSU), R. Giraud (LPN) Laboratory Louis Néel, CNRS, Grenoble Collaborations with other groups B. Malkin (Kazan) A.M. Tkachuk (S t Petersburg) H. Suzuki (Tsukuba) D. Gatteschi (Florence) A. Müller (Bielefeld) D. Mailly (LPN, Marcoussis)
Nanometer scale S = 5 3 6 Single Molecule Magnetic Protein Cluster Nanoparticle nm 2 nm 3 nm 2 nm
The molecules are regularly arranged in the crystal
Mn2acetate Mn(III) S=2 Mn(IV) S=3/2 Total Spin =
SINGLE MOLECULE «MAGNET» Energy barrier in zero field (symmetrical) H= - DS z2 -BS z 4 - E(S +2 + S -2 ) - C(S +4 + S -4 ) Simplified picture of an isolated spin: Landau-Zener model S, m-n > S, -m > Energy S,-S+2> Thermally activated tunneling S,S-2> ² - P S,-S+> S,-S> spin down Ground state tunneling S Z S,S-> S,S> spin up P energy S, -m > magnetic field S, m-n > If applied field // -M non-symmetrical barrier New resonances at gµ B H n = nd Molecular magnets (S. Miyashita) large spins give extremely small splittings Tunneling probability: P LZ = exp[-π( /ħ) 2 /γc] c = dh/dt
Tunneling of magnetization in Mn 2 -ac: «Technical» hysteresis loop + resonant tunneling Steps at H n =45.n mt,5 M/M S -,5 - -3-2 - 2 3 B L (T).5K.6K.9K 2.4K ICM 94 Barbara et al, JMMM (995); NATO-ASI, QTM 94 ed. Gunther and Barbara; Thomas et al Nature (996); Friedman et al, PRL (996);. Slow quantum spin dynamics of molecule magnets.
Crossover From Classical to Quantum Regime (Mn 2 -ac) n= 5, 4,5 4, 3,5 3, 2,5 2,,5,,5, B n n=9 n=8 n=7 n=6 n=5 n=4 n=3 n=2 n= n= Ground-state Tunneling (Spin-Bath),5,,5 2, 2,5 3, 3,5 4, T (K) Classical (Thermal Activation) Activated Tunneling (Phonon Bath) Measured ( ) and Calculated ( ) Resonance Fields Barbara et al, JMMM 4-44, 89 (995) and J. Phys. Jpn. 69, 383 (2) Paulsen, et al, JMMM 4-44, 379 (995); NATO, Appl. Sci. 3, Kluwer (995)
Resonance width and tunnel window Effects of magnetic couplings and hyperfine Interactions Data points and calculated lines Level Scheme Inhomogeneous dipolar broadening and the electronic spin-bath 4 B n (T) 5, - - 4,5 4, 3,5 3, 9-9- 9-2 8-8- 8-2 7-7- 7-2 6-6- 6-2,4,6,8,,2,4 T(K) 2 E (K) - -2-3 (n-p) : -S+p S-n-p 6-2 6-6- 7-2 7-7- 8-2 8-8- 9-2 - 9- - 9-3, 3,5 4, 4,5 5, B (T) Chiorescu et al, PRL, 83, 947 (999) Barbara et al, J. Phys. Jpn. 69, 383 Homogeneous (2) broadening of the tunnel Kent et al, EPL, 49, 52 (2) window by nuclear spins: Wernsdorfer et al, PRL (999) Prokofiev and Stamp (998) dm / db 3 2 n=8 T=.95 K 3,75 3,8 3,85 3,9 3,95 4, 4,5 4, 4,5 B (T) 8-8- t =5s -6-5 t =s 8-6 t =2s 6-6 t =4s 4-6 -.5.5-7 -.4 -.2.2.4.6.8 µ H(T) Γ sqrt (s - ) -5 2-5 t =s M in = -.2 M s Weak HF coupling: Broadens the tunnel window ( 5 ) Decoherence mechanisms
Landau-Zener model For For an an ensemble isolated of spin spins energy H= - DS z2 -BS z4 - E(S +2 + S -2 ) - C(S +4 + S -4 ) - gµ B S z H z /2,/2> /2,-/2> S, m-n > ² S, -m > -P - P _ hω H = 2 +(2µ B B ) 2 Tunneling probability: P LZ = exp[-π( /ħ) 2 /γc] c = dh/dt Single Molecule Magnets: large spins very small tunnel splittings: ~ (E/D) -2S P energy /2,-/2> S, -m > magnetic field, P /2,/2> S, m-n > applied field very small tunnel probabilities: P LZ ~ 2 /c ~ (E/D) 4S / - c P PS ~ ( 2 /ω )e - ξ /ξ Larger tunneling rate Strong decoherence
energy /2,/2> /2,-/2> /2,-/2> _ hω H = 2 +(2µ B B ) 2, applied field V 5, a molecule with S=/2 Dipolar interactions 3 times smaller but I=7/2 -P P /2,/2> M/M s.5 -.5 Absorption of sub-centimetric waves.4 K GHz. T/s - -.6 -.4 -.2.2.4.6 µ H (T) M ( B) M ( B) eq 2Γ( B) τ = + 2Γ( B) τ period: ms s. ms.5 ms ms 2 ms 3 ms Γ Max ~ 5 s - I. Chiorescu, W. Wernsdorfer, A. Müller, H. Boggë, and B. Barbara et al, PRL (2) W. Wernsdorfer, D.Mailly, A. Müller, and B. Barbara, EPL, 24
Gaussian absorption lines W. Wernsdorfer, D.Mailly, A. Müller, and B. Barbara, EPL, 24 Important broadening by nuclear spins and other molecule spins Loss of coherence Ω R ~ γb ~ 3 khz << /τ 2 ~ γσ ~.2 GHz Rabi oscillations, require much larger b. N = B Max /2πσ = γbτ 2 /2π ~2 Precession ~ 2 turns ( ) [ ] + + = t b B b B b P 2 2 2 2 2 2 2 ) ( 2 sin ) ( ) ( ) ( γ ν γ γ ω γ γ ) ( ) ( 4 2 L B L f b γ ν γ π = = Γ
Tunneling of the angular momentum of Isolated Rare-earths ions (ensemble measurements of «paramagnetic» ions) An extention of the slow quantum dynamics studies of SMM to the cases of strong spin-orbit and hyperfine coupling.2 % Ho 3+ in substitution of Y 3+ In YLiF 4 Tetragonal symmetry (Ho in S4); (J = L+S = 8; g J =5/4) Dipolar interactions ~ 2 mk << 2 mk (levels separation)
CF levels and energy barrier of Ho 3+ in Y.998 Ho.2 LiF 4 Strong mixing H CF = B + 4 4 4 4 2O2 + B4O4 + B6 O6 + B4O4 B6 O6 a) E (K) Barrier short-cuts 5 5-5 -4 b) -8-2 -6 E (K) - -5-2 -6-4 -2 2 4 6 <J z > -2-24 -9-6 -3 3 6 9 µ H z (T) R. Giraud, W. Wernsdorfer, D. Mailly, A. Tkachuk, and B. Barbara, PRL, 87, 5723- (2) Singlet excited state + Doublet ground-state + Large τ (Orbach process) B2 =.66 K, B4 = -3.253 mk, B44 =- 42.92 mk, B6 =-8.4mK, B64 =- 87.3mK Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (978); N.I. Agladze et al, PRL, 66, 477 (99) Energy barrier ( ~ K)
Hysteresis loop of weakly interacting Ho 3+ ions in YLiF 4 Comparison with Mn2-ac Many steps! M/M S,5 -,5 - -3-2 - 2 3 B L (T).5K.6K.9K 2.4K,,5, -,5 -, M/M S 2 mk 5 mk 5 mk dh/dt=.55 mt/s 3 2-8 -4 4 8 2 µ H z (mt) /µ dm/dh z (/T) n= dh/dt > n=2 n= n=- n=3-2 2 4 6 8 L.Thomas, F. Lionti, R. Ballou, R. Sessoli, R. Giraud, W. Wernsdorfer, D. Mailly, A.Tkachuk, D. Gatteschi,and B. Barbara, Nature, 996. and B. Barbara, PRL, 2 Steps at B n = 45.n (mt) Nuclear spins Steps at B n = 23.n (mt) Tunneling of Mn 2 -ac Molecules Tunneling of Ho 3+ ion
Quasi-Ising CF Ground-state + Hyperfine Interactions H = H CF-Z + A{J z I z +(J + I - + J - I + )/2} The ground-state doublet 2(2 x 7/2 + ) = 6 states E (K) -78,5-79, -79,5-8, I = 7/2-7/2-2 -5 - -5 5 5 2 µ H z (mt) -7/2 g J µ B H n = n.a/2-5/2-5/2 7/2 7/2 5/2 3/2,,5, -,5 -, M/M S 2 mk 5 mk 5 mk 3 2-8 -4 4 8 2 µ H z (mt) /µ dm/dh z (/T) n= dh/dt > n=2 n= n=- n=3-2 2 4 6 8 A = 38.6 mk, Linewidth ~ mk ~ Dip. Int. Avoided Level Crossings between Ψ, I z > and Ψ +, I z > if I= (I z -I z )/2= odd Co-Tunneling of electronic and nuclear momenta: Electro-nuclear entanglement (2-bodies)
Application of a transverse magnetic field: (slow sweeping field: sample at the cryostat temperature)..5. -.5 -. M/M S H T = mt H T =3 mt H T =5 mt H T =9 mt H T =7 mt H T =5 mt H T =3 mt H T = mt H T =9 mt H T =7 mt T = 3 mk v =.6 mt/s -75-5 -25 25 5 75 µ H z (mt) Acceleration of quantum dynamics the remanent magnetization vanishes Quantum fluctuations destroy the local moment Transition from «Classical» to Quantum Paramagnet (QPT) Nature of the mixing: entangled electro-nuclear states
,,5, -,5 4 2 a) M/M S 5 mk.3 5 mk T/s.3 T/s E (K) -79, -79,5-8 -6-4 -2 2 4 6 8 n -, -3-2 - 2 3 µ H z (mt) 6 b) n= 8 n = 6 /µ dm/dh z (/T) Additional steps at fields: H n = (23/2).n (mt) (single Ho 3+ tunneling being at avoided level crossings at H n = 23.n mt) n= µ H n (mt) 4 24 8 2-6 -2-8 2 6 2 24-8, -5-75 75 5 225 µ H z (mt) 6 linear fit µ H n = n x 23 mt n = 7 integer n half integer n n = 8 n = 9 dh/dt< Giraud et al, PRL 87, 5723 (2) Simultaneous tunneling of Ho 3+ pairs due to dipolar interactions (4-bodies entanglement) Two Ho 3+ Hamiltonian avoided level -2-5 - -5 crossings at H n = (23/2).n H (m
Ac susceptibility (SQUID measurements) Single-ion and dipolar-bias Tunneling Co-tunneling Tunneling rates and ac measurement frequency R. Giraud, A. Tkachuk, B. Barbara, PRL, 23.
a) -78.5 Energy (K) -79. -79.5-8. I = 7/2...... m I =+5/2 m I =+7/2 m I =+7/2 m I =+5/2 Single-ion level structure E n = E ± gµ B H n Tunneling: gµ B H n = (n -n)α/2 Co-tunneling: gµ B H n =(n -n+/2)α/2 (A= Ho hyperfine constant) b) Energy (K) -2-2 n=-9 H n=-8 n=3/2 z (Oe) -357-358 -359-36 n = n = n = 2-359.6 n = /2 n = 3/2-36. 2 3 4 5-2 - 2 H z (Oe) 2 3 4 Energy (K) 5 6 7 H bias 8 H z (Oe) Two-ions Level structure Electronic Spin-bath: Co-tunneling Biais tunneling Diffusive tunneling Nuclear spin-bath (Li, F, Y): Linewidths R. Giraud, A. Tkachuk, and B. Barbara PRL (23).
Ho-dimer satellites in the EPR signal in 7 LiYF4 (% Ho): Bias-tunneling transitions only Boris Malkin group, Kazan 2 3 4 ν=75 GHz 2+3 2 3 2 3 4 2 5-3 ν=25 GHz,2,4 Magnetic field (T),4,6 Magnetic field (T) In the 7 Li.% sample the width of single ions ~3.5 mt and of dimers ~ 2mT G. Shakurov, B. Malkin, B. Barbara, Appl. Magn. Res. 25
Toy model of two coupled effective spins, with g z /g x >> H/J = Σ ij S i z S j z + ασ ij (S i + S j - + S j + S i - )/2 + βσ ij (S i + S j + + S j - S i - ) with α = (J x + J y )/4J β = (J x -J y )/4J Diffusive tunneling Co-tunneling This is why dipolar interactions induce co-tunneling
Direct check of hyperfine sublevels from EPR In Ho:YLiF 4 (B. Malkin group) J 24 frequency GHz 3,5 3, Hyperfine sublevels of Ho 3+ ion in LiYF 4 22 2 8 6 Energy (GHz) 2,5 2,,5,,5, -,5-75 -2 m= Γ 2 Γ 34 ν, GHz 8 6 4 7LiYF4:Ho.% B B c 4 2 8 6 4 2,8 3,2 3,6 4, 4,4 4,8 5,2 5,6 6, 6,4 6,8 7,2 7,6 Magnetic field (koe) -225-25 m=2,,5,,5,2,25,3 Magnetic field B (T) 2 2 3 4 5 6 7 B, KGs G. Shakurov, B. Malkin, B. Barbara, Appl. Magn. Res. 25
Direct observation of levels repulsions Hyperfine sublevels ( m=2) in the EPR spectra 7 ν (GHz) 2.9 2.8 2.55 2.35 2.25,,5,,5 Magnetic field B (T) G. Shakurov, B. Malkin, B.Barbara, Appl. Magn. Res. 25
9 F_NMRM. J. Graf, A. Lascialfari, F. Borsa, A. M. Tkachuk, and B. Barbara (cond-ma.2 (b) /T (msec - ).. 8 2 4 6 H z (m T) Phenomenological fit: /T = B 2 W/ [W 2 + (ω N - ) 2 ], = [.3x 8 (H-23n) 2 + n2 ] /2 with n ~2 mk, Levels broadening at crossing is extremely small (~ 2 mt): Decoherence strongly suppressed : possible to measure directly level repulsion
Case of a metallic matrix: Ho 3+ ions in Y.999 Ho. Ru 2 Si 2.5 n=2 n= n=.4 K M/M s.36 mt/s.68 mt/s.34 mt/s.7 mt/s -.5 - -.8 -.4.4.8 µ H (T) These steps come from tunneling transitions of J+I of single Ho 3+ ions, in a sea of free electrons. B. Barbara, R. Giraud, W. Wernsdorfer, D. Mailly, A. Tkachuk, H. Suzuki, ICM-Rome, JMMM (24)
CONCLUSION Molecular magnets Coexistence of classical hysteresis loop and resonant quantum tunneling non-adiabatic Landau-Zener (single-ion picture) Observation of tunneling made possible by environmental spins (nuclear spins) Spin tunneling asssited by photons (photons bath) Strong decohrence by environmental spins (nuclear spins) Highly diluted Ho 3+ in LiYF 4 Tunneling of the total angular momentum J = L+S of Ho 3+ single ions two-bodies entanglement Quasi-isolated Ho 3+ ions: J and I tunnel simultaneously (in a metal also: Ho in YSi 2 Ru 2 ). Relevant quantum number of Ho 3+ is not J but I+J (Kramers, QPT ). Co-tunneling, bias-tunneling, spin-diffusion in Ho 3+ dimmers four-bodies entanglements, Co-tunneling of dimmers is observed. Crucial role of the anisotropic character of dipolar interactions. Microscopic basis for the study of QPT (concentrated systems) and coherent quantum dynamics.. Molecular magnets with Rare-Earths R-E Double-Deckers also show single-ion tunneling on electro-nuclear states (M. Ruben)
Some perspectives Higher order many-body tunneling and decoherence by the environment (quantum phase transitions) Spin-echo experiment and Rabi oscillations on electronic states of - Molecular magnets (intra-molecules hyperfine interactions ~ mk) - Entangled E-N pairs of Ho 3+ (dipolar interactions, hyperfine interactions ~ mk) Metallic systems : Decoherence by free carriers on spin tunneling in meta Injection of polarized spins.. (Tunneling, Kondo, Heavy fermions, Spintronics) Spin qubits manipulated by photons
Manipulating the exchange interactions between two spins Qubit de spins coupled by the injection of an electron and manipulated by transfert of photo-electrons Photon hν 3 e- Photon hν Photon hν 2 Far infra-red : variations of charges (S) Sub-centimeter: variations of spin projections(m S ) Collaborations: J. Bonvoisin e t C. Joachim (CEMES, Toulouse F. Ciontu et Ph. Jorrand (IMAG, Grenoble) Remerciements: J.P. Sutter, M. Kahn.
MANY THANKS FOR YOUR ATTENTION!
Mesoscopic Magnetism Nano-magnetism Tunneling of narrow DWs in highly anisotropic systems (Dy 3 Al 2, SmCoCu) Nanomagnetism on distributions (deposited clusters, BaFeO nanoparticles ) Single-particles nano-magnetism (micro-squids) nanoparticles, clusters ( nm): thermally activated reversal (SW, NB models ) Quantum nano-magnetism single molecule magnets ( nm): reversal by tunneling, slow quantum dynamics with different environments: inhomogeneous and homogeneous baths (spin, phonon, radiation, free carriers...) Sub-nanometer scale Atomic magnets (Quantum) Rare-Earths ions (. nm): Crucial role of nuclear spins