B. Barbara, R. Giraud, I. Chiorescu*, W. Wernsdorfer, Lab. Louis Néel, CNRS, Grenoble. Collaborations with other groups: D. Mailly (Marcoussis) D. Gatteschi (Florence) A. Müller (Bielefeld) G. Christou (Gainsville) A.M. Tkachuk (S t Petersburg) S. Miyashita (Tokyo) * Present adress Delft University of Technology Quantum Magnetism: from large spin molecules to single ions

Magnetization reversal in nanoparticles Large (submicrometer), Small (nanometer) Magnetic tunneling in molecules Large spin molecules (Mn 12 -ac, Fe 8 ) Tunneling, Berry phases, Quantum dynamics Low spin molecules (V 15 ) Adiabatic LZS with and without dissipation Case of nearly isolated Ions Rare-earth ions: (Ho 3+ in Y Ho LiF 4, Y Ho Cu 2 Si 2 ) Entangled electro-nuclear states, co-tunneling OUTLINE

Particles from micrometers to 100 nanometers Obtained by: Lithography, Electro-deposition Measurements: Micro-Squids 100 nm 50 nm x 1  m 1  m x 2  m Small ellipseLarge ellipseNanowire MULTI – DOMAIN: nucleation, pinning, propagation and annihilation of domain walls SINGLE - DOMAIN Single Nucleation Curling

Nanometer scale NanoparticleCluster 20 nm3 nm1 nm2 nm Magnetic ProteinSingle Molecule 50S =

The molecules are regularly arranged in the crystal

Mn(IV) S=3/2 Mn(III) S=2 Total Spin =10 Mn12acetate

Barrier in Zero Field H= - DS z 2 - BS z 4 - E(S S - 2 ) - C(S S - 4 ) + g  B S x H x Thermally activated tunneling Multi-Orbach process Thermal Activation Mn 12 -ac : D = 0.56 K, E = 5 mK, B = 1.18 mK, C = K, Fe 8 : D = 0.23 K, E = - 47 mK, B = 0.03 mK, Tunnel splitting 

Resonant Tunneling in Mn 12 -ac in Large // Fields Top of the barrier Ground-state Avoided level crossing Energy scheme in B// H= - DS z 2 - BS z 4 - E(S S - 2 ) - C(S S - 4 ) + g  B SH z - Landau-Zener Mechanism - Resonance fields

Dynamics: Landau-Zener Transition (isolated system) Tunneling Probability : General result for a single level crossing Solution of the Schrödinger equation 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). P=1 – exp[-  (  /ħ) 2 /  c] where c = dH/dt P ~1 when  pass  c  ~  os  = ħ/  

Resonance Width and Tunnel Window Effects of Magnetic couplings and Hyperfine Interactions Chiorescu et al, PRL, 83, 947 (1999) Barbara et al, J. Phys. Jpn. 69, 383 (2000) Kent et al, EPL, 49, 521 (2000) Inhomogeneous broadening of Two resonances: Dipolar fields… Data points and calculated linesLevel Scheme Homogeneous broadening of nuclear spins: Tunnel window Wernsdorfer et al, PRL (1999) Prokofiev and Stamp (1998 )

Measured and Calculated Resonance Fields above 0.4 K QT TA Three Regimes Barbara et al, ICM Warsaw (1994); JMMM , 1891 (1995); JMMM-200, (1999) Paulsen, et al, JMMM , 379 (1995); NATO, Appl. Sci. 301, Kluwer (1995) TAQT T c-o T blocking

Scaling of quantum dynamics of Mn 12 -ac M/M s = f (-t/  (H,T) ) Strong deviations from exponential to square root relaxation below the cross-over temperature (~2K)  ~ 1 to  ~ 0.5 Prokofiev, Stamp, PRL 80, 5794 (1998) Thomas et al, J. Low Temp. Phys. 113 (1998); PRL 83, 298 (1999). Weak tunnel splitting distribution preserves square-root relaxation (  and t are separable variables)

In a Transverse Field Chiorescu et al, PRL, 83, 947 (1999); Barbara et al, J. Phys. Jpn. 69, 383 (2000) Calculated Energy Spectrum Measured relaxation

Quantum phase interference (Berry phase) in Fe 8 Wernsdorfer and Sessoli, Science 284, 133 (1999)  2 ~  -1 Prokofiev and Stamp PRL 80, 5794 (1998)

Parity Effect: Odd vs. Even Resonances W. Wernsdorfer and R. Sessoli, Science, 1999.

V 15 : a Gapped Spin ½ Molecule (D H =2 15 ) Dzyaloshinsky-Moriya interactions: H DM = -  D ij S i xS j The Multi-Spin Character of the Molecule (15 spins) + Time Reversal Symmetry  = 0 (Kramers Theorem) Exchange interactions: Antiferromagnetic ~ several 10 2 K Müller, Döring, Angew. Chem. Intl. Engl., 27, 171 (1988) Anisotropy of g-factor: ~ 0.6% Ajiro et al, J..Low. Temp. Phys. to appear (2003) Barra et al,J. Am. Chem. Soc. 114, 8509 (1992)

More details on the D-M gap of multi-spin systems Calculation of energy spectra with antisymmetrical interactions: H = J  S i S j -  ij S ix S jz - g  B B z  S iz  ij =  ji  =0 (2 Kramers doublets)  ij ≠  ji  ≠ 0 (2 pairs of singlets) Miyashita, Nagaosa, Prog. Theo. Phys.106, 533 (2001). Barbara et al, cond-mat / v1 and Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002).

LZS transition at finite Temperature At low sweeping rate / strong coupling to the cryostat (  small) n 1 /n 2 = exp (  /kTs) Phonon-bath  bottleneck model: h    ±  Abragam, Bleaney, 1970; Chiorescu et al, M(t)/M(  ) = x(t) is given by -t/  B =x(t) – x(0) + ln [(x(t) –1) /[(x(0)-1)]  Bott =(  /  2 )th 2 (  /2kT ) Nuclear spin-bath  level broadening:  30mK (Stamp, Prokofiev, 1998). 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, (2000); … Spin-phonons transitions (dissipation)  Irreversible M(H) n1n1 n2n2

Low sweeping rate / Strong coupling to the cryostat « Non-Isolated V 15 » : A two-level system with dissipation Butterfly hysteresis loop LZS transition at Finite Temperature (dissipative)  1 ~  botl >  meas Hysteresis (≠Orbach process). Measured Calculated Chiorescu et al, PRL 84, 3454 (2000) M(H): Irreversible Equilibrium (Reversible) M(H)=M s th{(  2 +H 2 ) 1/2 /2kT}

Landau-Zener transition at « Zero Kelvin » Fast sweeping rate / Weak coupling to the cryostat (  large).... But out of equilibrium. n 1 /n 2 = exp (  /kTs)   Ground-State  M(H) Nuclear Spin-Bath  Level broadening  30mK  Overlap near zero field. No spin-phonon transition (no dissipation)  Reversible M(H)... n1n1 n2n2 (Stamp, Prokofiev, 1998).   Bott =(  /  2 )th 2 (  /2kT) >>  meas

  80 mK Adiabatic Landau-Zener Spin Rotation « Isolated V 15 » : A two-level system « without dissipation » M(H) = dE(H)/dH= (1/2)(g  B ) 2 H/[(  2 +(g  B H) 2 )] 1/2 Fast sweeping rate / Weak coupling to the cryostat V 15 M(H) : Reversible and out of equilibrium Chiorescu et al, submitted to PRB, Cond-mat / v1 Barbara et al, Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002),   80 mK Adiabatic Landau-Zener Spin Rotation

Relaxation Experiments Outside and Within the Mixing Region (   Fit of M(t) to the Bottleneck model   B (B,T). Phonon bath Barbara et al, Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002). Inside  : Outside  : Fit not good;  B << calculated value Good fit;  B (B,T) ~ calculated value Spin-bath (nuclear spins) Phonon bath

A new direction…Mesoscopic Physics of Rare-earth Ions : Ho 3+ in Y Ho LiF 4 Entangled electro-nuclear states, co-tunneling,… Dipolar interactions between different Ho 3+ a few ~ mT H CF-Z = -B 2 0 O B 4 0 O B 4 4 O B 6 0 O B 6 4 O g J  B JH B l m : acurately determined by high resolution optical spectroscopy Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (1978); N.I. Agladze et al, PRL, 66, 477 (1991) Tetragonal symmetry (Ho in S4) J = L+S = 8; g J =5/4

CF energy barriers: a comparizon between Mn 12-ac and Ho 3+ Short-cuts Lowest energy levels Giraud et al, PRL, 87, (2001) ground-state: Ising doublet Mn 12 -ac S = 10 D = 0.56 K, E ~ 5 mK, B = 1.18 mK, C = mK Ho 3+ J = 8 B20 = K, B40 = mK, B44 = mK, B60 =-8.41mK, B64 = mK   

Hysteresis loop of Ho 3+ ions in YLiF 4 Thomas et al, Nature (1996) Giraud et al, PRL, 87, (2001) Steps at Bn = 450.n (mT) Steps at B n = 23.n (mT) Tunneling of Mn 12 -ac Molecules Tunneling of Ho 3+ ion Comparison with Mn12-ac

Ising CF Ground-state + Strong Hyperfine Interactions H = H CF-Z + A.I.J Avoided Level Crossings between |  , I z  and |  +, I z ’  if  I= (I z -I z ’ )/2= odd -7/2 7/2 5/2 3/2 -7/2 Co-Tunneling of Electronic and Nuclear Spins: Electro-nuclear entanglement

Acceleration of slow quantum dynamics associated with co-tunneling of I and J in a transverse field: Fast increase of the splitting of entangled electro-nuclear states of single Ho 3+ ions. dB/dt = 0.55mT/s dB/dt ~ 1 mT/s

In fast ssweeping field Two différent relaxation regimes LiY 1-x Ho x F 4 (x~0.1% at.) Slow ( ~ 1 mT/s) Fast ( ~ 1 T/s) Thermodynamical and thermal equilibrium Out of thermodynamical equilibrium

Fast sweeping rate... a different regime 50 mK 0.3 T/s Giraud et al, PRL 87, (2001) Additional steps at fields: B n = (23/2).n (mT) Tunneling, Co-tunneling and Cross-spin reversal of Ho 3+ pairs 50 mK 0.3 T/s. Hysteresis from bottleneck (  B ) and barrier (  1 ) T Bl ~ 200 mK > Tmea= 50mK. Cross-spin relaxations  2 << Time-scale  <<  1 = C.exp(10/T) (Orbach) -½, I z ½, I’ z

Single ion tunneling Co-tunneling of two ions Cross-tunneling Unambiguous observation of different types of tunneling Ac-susceptibility at high temperature R. Giraud and B. Barbara, to be published

Exchange-biased tunnelling between two molecules (Mn 4 dimer) W. Wernsdorfer et al, Nature 416, 406 (2002) S=9 S=0 For a dimer

 +>  -> A B C A A’ B C Energy H > 0 H < 0  + +>  - ->  + -> +  - +>  + -> -  - +> Bias tunneling and co-tunneling Simple 2-spins model:singlet / triplet B. Barbara, News and Views, Nature to appear

Last new direction: Effect of free electrons on tunneling … Ho 3+ in YRu 2 Si 2 (Hiro Susuki, Tsukuba)

Ho in YRu 2 Si 2 (Same matrix as in CeRu 2 Si 2 ) Non-trivial behaviour when v decreases

Ho in YRu 2 Si 2 (Same matrix as in CeRu 2 Si 2 ) Continuous M(H)… and jumps at well defined fields !

Tunneling seems possible, in the presence of free electrons ! Ho 3+ in YRu 2 Si 2 Hiro Susuki (Tsukuba) Y 1-  Ho  Ru 2 Si 2  ~ 0.1% Same hyperfine constant. Tunneling of electro- nuclear states in a metal…simple cross-spin relaxations… Y Ho LiF 4

Conclusion Evidence for tunneling of single ions of rare-earth Ho 3+ Avoided level crossings result from crystal field and hyperfine interactions Entangled electro-nuclear states Ho 3+, Mn 4 pairs: Cross-spin and Spin-phonon transitions co-tunneling Tunneling in metals ? Ho 3+ * Adiabatic LZS transition with or without dissipation. * Multi-spin molecule spin ½ gap V 15 Resolved resonance lines Quantum classical Quantum Dynamics Mn 12 -ac Berry Phases, Quantum Dynamics

Hysteresis loop in very slow applied field

Conclusion Evidence for tunneling of single ions of rare-earth Ho 3+ Avoided level crossings result from crystal field and hyperfine interactions Entangled electro-nuclear states Ho 3+, Mn 4 pairs: Cross-spin and Spin-phonon transitions co-tunneling Ho 3+ * Adiabatic LZS transition with or without dissipation. * Multi-spin molecule spin ½ gap V 15 Resolved resonance lines Quantum classical Quantum Dynamics Mn 12 -ac Berry Phases, Quantum Dynamics

Collaborations with other groups: D. Mailly (Bagneux) A. Caneschi, R. Sessoli, D. Gatteschi (Florence) A. Müller, H. Bögge (Bielefeld) G. Christou (Gainsville)

Evidence for a Mn 12 -like hysteresis loop Steps at B n = 23 n (mT) Tunneling of Ho 3+ ions Giraud et al, PRL, 87, (2001)

Exchange-biased quantum tunnelling between two molecules (Mn 4 dimer) W. Wernsdorfer et al, Nature 416, 406 (2002)

Collaborations with other groups: D. Mailly (Bagneux) A. Caneschi, R. Sessoli, D. Gatteschi (Florence) A. Müller, H. Bögge (Bielefeld) G. Christou (Gainsville) A.M. Tkachuk (St Petersburg)

Origin of the Gap in the Spin ½ V 15 Molecule: The Multi-Spin Character of the Molecule. Time Reversal Symmetry  =0 (Kramers Theorem) V 15 : 15 spins ½ with AF couplings Hilbert Space Dimension D H =2 15 Ground-state: spin ½ with fourfold degeneracy (two doublets). Dzyaloshinsky-Moriya interactions: H DM = -  D ij S i xS j J=2.445 K  = 80mK, if D=50 mK (D~J  g /g~80mK)

(SMM)

Tunnelling in a dimer of Mn 4 molecule S=S 1 +S 2 with S 1 = S 2 = 9/2 H = H 1 + H 2 + JS 1 S 2 H 1 and H 2 : single spin Hamiltonians (2S 1 +1)(2S 2 +2) = 100, instead of 10 for single Mn 4 molecule or 21 (10 8 ) in single Mn 12 molecule

Selection rules: Tunneling for odd n also Mn 12 -ac Barbara et al JMMM-200, 1999.

Increasing the sweeping rate: broader hysteresis loop and additional steps Sample temperature increases. Tunneling relaxation decreases. __ 50 mK

Low energy spectrum of a pair of Ho 3+ ions (including nuclear spins) Single Ho 3+ ions B n = 23.n (mT) Pair of Ho 3+ ions B n = (23/2).n (mT) (3x8) 2 = 576 states

Resonant Tunneling in Mn 12 -ac in Large // Fields Chiorescu et al, PRL, 83, 947 (1999); Kent et al, EPL, 49, 521 (2000) Barbara et al, JMMM-200, « Magnetism beyond 2000 » (1999). Top of the barrier Ground-state Avoided level crossing Energy scheme in B// Hysteresis loop in B//

Outline Large spin molecules ( Mn 12 -ac, Fe 8 ) Tunneling, Berry phases, Quantum dynamics Low spin molecules ( V 15 ) Adiabatic LZS with and without dissipation Rare-earth ions: ( Ho 3+ in Y Ho LiF 4 ) Entangled electro-nuclear states, co-tunneling

Measured Tunnel Splitting experimental calculated with D = -0.29, E = 0.046, C = -2.9x10 -5 K W. Wernsdorfer and R. Sessoli, Science, 1999.

Crossover From Classical to Quantum Regime, Mn12-ac Activated Tunneling Measured ( ) and Calculated ( ) Resonance Fields Barbara et al, JMMM , 1891 (1995) and J. Phys. Jpn. 69, 383 (2000) Classical Thermal Activation T blocking Ground-state Tunneling T c-o All the resonances tunneling

Quantum Tunneling in Mn 12 -ac and resonant tunneling on oriented grains, powder Decreasing of relaxation near zero field Log  Barbara et al, ICM’94, JMMM, , 1824(1995) Paulsen et al,Proc., QTM’94, NATO ASI, Vol 301, 189, (1995) Barbara et al, ICM’94, JMMM , 1824 (1995) Friedman et al, PRL, 76, 20 (1996) TCTC QT TA

Resonant Tunneling of Magnetization in Mn 12 -ac Single Crystal Hysertesis loop Magnetic relaxation L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara Nature, 383, 145 (1996).

Resonant Tunneling in Mn 12 -ac in Large // Fields Chiorescu et al, PRL, 83, 947 (1999); Kent et al, EPL, 49, 521 (2000) Barbara et al, JMMM-200, « Magnetism beyond 2000 » (1999). Hysteresis loop in B//

Nature of the ground-state of the V 15 molecule Diagonalization of the 15-Spin ½ Hamiltoninan H =  J ij S i S j (I. Tupitsyn, H. de Raedt) 200 calculated levels. The 8 levels lowest levels frustrated 3-spins ½ triangle. Measurements of M(H) and  (T) confirm this picture. Effective hamiltoninan: H = |J |  (S 1 S 2 + S 2 S 3 + S 3 S 4 ) – g  B B(S 1 + S 2 + S 3 )

M(H) (Measured) Lowes Energy Levels (Calculated) Chiorescu et al, JMMM, 221, 103 (2000); JAP 87, 5496 (2000). Barbara et al, Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002); cond/mat v1. S=1/2 S=3/2 S= -1/2 S=1/2 S=1/2 S=3/2 Spin reversal within the « three spins » molecule V 15, at equilibrium (no barrier) A model system for the adiabatic Landau-Zener model : two limiting cases

The Low spin Molecule V 15 S=1/2, D H =2 15 Anisotropy of g-factor: ~ 0.6% Ajiro et al, J..Low. Temp. Phys. to appear (2003) Barra et al,J. Am. Chem. Soc. 114, 8509 (1992) Exchange interactions: Antiferromagnetic ~ several 10 2 K Müller, Döring, Angew. Chem. Intl. Engl., 27, 171 (1988)