Recent Developments in Quantum Dynamics of Spins
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1 Recent Developments in Quantum Dynamics of Spins 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) R. Winpenny ( Manchester ) G. Christou (Gainsville) A.M. Tkachuk (St Petersburg) S. Miyashita (Tokyo) *Present address : Post-Doc at Delft University of Technology, Delft * Present address: Post-Doc at Laboratoire de Spectrométrie Physique, Grenoble
2 OUTLINE Magnetic tunneling in molecule magnets Non-adiabatic and adiabatic Landau-Zener spin reversal A new direction for this field: Quantum dynamics of rare-earths ions (diluted in matrices with different symmetries, insulating, metallic, semi-conducting) -Magnetic tunneling rare - earth ion magnets Ho 3+ in Y.998 Ho.2 LiF 4 (Tunneling of single Ho 3+ ions, pairs of ions, role of nuclear spins) Magnetic tunneling in a metal Y.999 Ho.1 Ru 2 Si 2 (connections with Kondo systems, heavy fermions and spintronics?)
3 Mn12acetate Mn(III) S=2 Mn(IV) S=3/2 Total Spin =1 ~ 1 2 electrons ~5 electro
4 Barrier in Zero Field H= - DS z2 -BS z 4 - E(S +2 + S -2 ) - C(S +4 + S -4 ) + gµ B S x H x Mn 12 -ac : D =.56 K, E = 5±2 mk, B = 1.18 mk, C = K, Fe 8 : D =.23 K, E = - 47 mk, B =.3 mk, Energy S,-S+2> Thermally activated tunneling S,S-2> Multi-Orbach process S,-S+1> S,S-1> S,-S> spin down Ground state tunneling S Z S,S> spin up Tunnel splitting
5 Tunneling of Magnetization in Mn 12 -ac Single Molecule magnet (large barrier) 1,5 M/M S -, B L (T) 1.5K 1.6K 1.9K 2.4K Thomas et al, Nature (1996) Steps at B n = 45.n (mt)
6 Resonance Width and Tunnel Window Effects of Magnetic couplings and Hyperfine Interactions Data points and calculated lines Level Scheme Inhomogeneous broadening of Two resonances: Dipolar fields 4 B n (T) 5, ,5 4, 3,5 3, E (K) (n-p) : -S+p S-n-p dm / db n=8 T=.95 K 3,75 3,8 3,85 3,9 3,95 4, 4,5 4,1 4,15 B (T) ,4,6,8 1, 1,2 1,4 T(K) 3, 3,5 4, 4,5 5, B (T) Chiorescu et al, PRL, 83 (1999) Barbara et al, J. Phys. Jpn.(2)
7 Measured and Calculated Resonance Fields Mn 12 -ac n=1 n=9 5, 4,5 Three Regimes n=8 4, n=7 3,5 n=6 B n (T) n=4 n=3 n=2 n=1 3, 2,5 2, 1,5 1,,5 QT T c-o TAQT T blocking TA n=,,5 1, 1,5 2, 2,5 3, 3,5 4, T (K) Barbara et al, ICM Warsaw (1994); JMMM , 1891 (1995); JMMM-2, (1999) Paulsen, et al, JMMM , 379 (1995); NATO, Appl. Sci. 31, Kluwer (1995)
8 Scaling of quantum relaxation in Mn 12 -ac M/M s = f(t/τ(h,t)) M/ Ms β ~.5 to β ~ K 1.7 K 1.8 K M/Ms 2. K 2.1 K 2.2 K 2.3 K 2.4 K 2.5 K 2.6 K 2.7 K 2.8 K K 1.9 K 1 t 1/2 (s 1/2 2 3 ) t (s) Thomas et al, J. Low Temp. Phys. 113 (1998); PRL 83, 298 (1999). Square-root relaxation: Prokofiev, Stamp, PRL 8, 5794 (1998) (Weak tunnel splitting distribution preserves square-root relaxation)
9 V 15 : a Gapped Spin ½ Molecule, «no barrier» (D H =2 15 ) Intra-Molecule exchange interactions: ~ 1 2 K Müller et al (1988) Inter-Molecule exchange interactions: ~ 1 mk 6 Anisotropy of g-factor: ~.6% Barra et al (1992), Ajiro et al (23) 4 4 energy (K) = 8 mk S=3/2 S=1/ magnetic field (T) Multi-spins character Antisymmetrical interactions Large gap (no barrier) Miyashita and Nagaosa, Prog. Theo. Phys.16, 533 (21). Barbara et al, Prog. Theo. Phys. Jpn, Supp. 145, 357 (22). cond-mat / v1
10 Low sweeping rate / Strong coupling to the cryostat Adiabatic Landau-Zener with spin-phonons transitions Measured 1, (c) (s-p bottleneck) Calculated M(H): Irreversible M (µ B ) M (µ B ),8,6,4,2, 1, (d),8,6,4,2 Equilibrium (Reversible) M(H)=M s th{( 2 +H 2 ) 1/2 /2kT},15 T S =T ph (K),1,5 T = 1 mk.14 T/s.7T/s 4.4 mt/s T=.1 K B, (T) -,6 -,3,,3,6,,,1,2,3,4,5,6,7 B (T) Chiorescu et al, PRL 84, 3454 (2) LZS transition at Finite Temperature (dissipative) energy 1 1/2,1/2> 1/2,-1/2> 1/2,-1/2> 1-P _ hω H = 2 +(2µ B B ) 2 P 1/2,1/2>, applied field τ botl > τ meas Butterfly hysteresis loop (Bottleneck)
11 Adiabatic Landau-Zener without spin-phonons transitions At fast sweeping rate / weak coupling to the cryostat τ meas << τ 1 < τ Bott no spin-phonon transition M(H) : Reversible (Ground-state only) Adiabatic Landau-Zener Spin Rotation 1, S, m-n > S, -m > M/M S,8,6,4,2 M(H) : Irreversible (Bottleneck) α = 13 V 15 6 mk.14 T/s.14 mt/s,,,2,4,6,8 1, B (T) Chiorescu et al, PRB (22); Cond-mat / v1 Barbara et al, Prog. Theo. Phys. Jpn. (22) 1 P energy S, -m > ² S, m-n > magnetic field M(H) = de(h)/dh E= [ 2 +(gµ B H) 2 ] 1/2 1 - P 8 mk
12 Relaxation measurements and fit to the Bottleneck model (Phonon bath),3,25,2 Inside Spin-bath B =.14 T.15 K τ H : fit=551s / th=1323s.5 K τ H : fit=157s / th=8716s,8,7,6,5 Outside Phonon bath M/M S,15,1,5, t (s) Faster relaxation: Levels overlap near zero field (nuclear spin fluctuations). Interactions between molecule spins M/M S S, m-n >,4,3,2 B =.7 T.15 K τ H : fit=97s / th=997s,1.5 K τ H : fit=3883s / th=3675s, Spin bath S, -m > 1 - P t (s) E(H)= H/M(H) Obtained from Adiabatic M(H) measurements E(H) 8 mk 1 P energy S, -m > magnetic field Chiorescu, et al PRB (22). Barbara et al, Prog. Theo. Phys. Jpn (22) ² Spin-Phonon bath S, m-n > Direct determination of E(H) by Absorption of electromagnetic radiations (decoherence)
13 A new direction Mesoscopic Physics of Rare-earth Ions: Ho 3+ in Y.998 Ho.2 LiF 4 Entangled electro-nuclear states, co-tunneling, Tetragonal symmetry (Ho in S4) J = L+S = 8; g J =5/4 Dipolar interactions between different Ho 3+ a few ~ mt H CF-Z = -B 2 O 2 - B 4 O 4 -B 4 4 O 4 4 -B 6 O 6 -B 64 O g J µ B JH B lm : 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)
14 CF energy barriers: a comparizon between Mn 12-ac and Ho 3+ Ho 3+ J = 8 B2 =.66 K, B4 = mk, B44 = mk, B6 =-8.41mK, B64 = mK Mn 12 -ac S = 1 D =.56 K, E ~ 5 mk, B = 1.18 mk, C =.3 mk a) E (K) 15-4 b) E (K) 1-8 Energy S,-S+2> S,S-2> S,-S+1> S,S-1> S,-S> Ground state tunneling S,S> spin down spin up S Z <J z > µ (T) Short-cuts Lowest energy levels Giraud et al, PRL, 87, (21) Strong mixing ground-state: Ising-like doublet
15 Hysteresis loop of Ho 3+ ions in YLiF 4 Comparison with Mn12-ac M/M S 1,5 -, B L (T) 1.5K 1.6K 1.9K 2.4K 1,,5, -,5-1, M/M S 2 mk 15 mk 5 mk dh/dt=.55 mt/s µ (mt) 1/µ dm/d (1/T) n=1 dh/dt > n=2 1 n= n=-1 n= Thomas et al, Nature (1996) Giraud et al, PRL, 87, (21) Steps at B n = 45.n (mt) Steps at B n = 23.n (mt) Tunneling of Mn 12 -ac Molecules Tunneling of Ho 3+ ion
16 Ising CF Ground-state + Strong Hyperfine Interactions H = H CF-Z + A{J z I z + (J + I - + J - I + )/2} E (K) -178,5-179, -179,5-18, -7/ ½, -7/2-1/2, 5/2 I = 7/2 µ (mt) -7/2 5/2 7/2 7/2 3/2 1,,5, -,5-1, M/M S 2 mk 15 mk 5 mk µ (mt) 1/µ dm/d (1/T) n=1 dh/dt > n=2 1 n= n=-1 n= 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 Spins: Electro-nuclear entanglement
17 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. M/M S 1,,5, -,5-1, T = 5 mk < 5 mt 5 mt 1 mt 2 mt -179,6-179,8-18, -18, µ (mt) E (K) dh/dt < M/M S =1 mt =3 mt =5 mt =19 mt =17 mt =15 mt =13 mt =11 mt =9 mt =7 mt T = 3 mk v =.6 mt/s µ (mt) db/dt =.55mT/s db/dt ~ 1 mt/s
18 In fast sweeping field LiY 1-x Ho x F 4 (x~.1% at.) Two différent relaxation regimes Slow ( ~ 1 mt/s) ~Thermodynamical equilibrium τ mes >> τ Bot > τ 1 Fast ( ~ 1 T/s) Out of thermodynamical equilibrium τ mes ~ τ 2 << τ 1 < τ Bot M/M S =1 mt =3 mt =5 mt =19 mt =17 mt =15 mt =13 mt =11 mt =9 mt =7 mt T = 3 mk v =.6 mt/s µ (mt) M/M S T = 5 mk v =.28 T.s ajustement linéaire 18 µ H n = n*23 mt n entier -18 n demi-entier n µ H n (mt) µ (mt)
19 Fast sweeping rate... Phonons bottleneck: T phon = T spins >> T cryost 1,,5, -,5 4 2 a) M/M S 5 mk.3 5 mk T/s.3 T/s -1, n µ (mt) 6 b) n= 8 n = 6 1/µ dm/d (1/T) n=1 µ H n (mt) µ (mt) 6 linear fit µ H n = n x 23 mt -12 integer n -18 half integer n n = 7 n = 8 n = 9 dh/dt< Giraud et al, PRL 87, (21). Hysteresis from bottleneck (τ B ) and barrier (τ 1 ) Additional steps at fields: B n = (23/2).n (mt) Roles of phonons diffusion, Spin diffusion, electromagnetic irradiation T Bl ~ 2 mk > Tmea= 5mK. Cross-spin relaxations τ 2 << Time-scale << τ 1 = C.exp(1/T) (Orbach) Single-ion tunneling Co-tunneling ( M ) Spin-diffusion ( M=) Bias-tunneling ( M -½, I z ½, I z High spin degeneracy
20 Clear observation of different types of tunneling by ac-susceptibility at high temperature 2.5x1-6 Co-tunneling of two ions (spin bath) 1.3x1-6 χ'' (emu/g). Single ion tunneling (phonon bath) T = 1.75 K f = 81 Hz µ (G) Biais-tunneling (weak dipolar interactions) (spin-bath) Giraud et al, PRL 87, (21), and to be published
21 Exchange-biased tunnelling between two molecules (Mn 4 dimer) S=9 S= For a dimer W. Wernsdorfer et al, Nature 416, 46 (22) (B. Barbara, News and Views, Nature, 2 Jan. 23)
22 The toy picture of two coupled effective spins 1/2 First Hamiltoninan for RVB, with < α < 1 (Fasekas and Anderson, 1974): H= J Σ ij S i z S j z + Jα Σ ij (S i + S j - + S j + S i - )/2 For two effective spins, with g z >> 1 and an applied field: H= J S 1 z S 2 z + Jα (S 1 + S S 2 + S 1 - )/2 - g(s 1 + S 2 )H Diagonalisation for H=, J>, the usual limits: + + > --> J/2 + + > + - > + - +> --> + - > - +> Ising (α =) Heisenberg (α =1) + -> - -+>
23 Effects of anisotropic interactions ( < α < 1),, H x + + > --> + + > 2g z --> x( + - > + - +>) + y( + + > + - ->) ( + + > - - ->) 2(g x H x ) 2 /J(1-α) J/2 J/2 + - > + - +> + - > + - +> J/2 y( + - > + - +>) - x( + + > + - ->) αj + - > - - +> αj + -> - -+> αj + -> - -+> αj - 2(g x H x ) 2 /J(1-α) H= =, H x ; x and y = f(h x, α) E E αj=j x + J y βj= J x -J y Exchange-biais Co-tunneling For co-tunneling : transverse field or lower symmetry : H= J S z 1 S z 2 + (J x +J y )(S S 2 + S + 2 S - 1 )/2 + (J x J y )( S + 1 S S - 2 S - 1 )/4 -g z (S z 1 + S z 2 )H Dipolar interactions co-tunneling in insulating systems (multi-spin tunneling)
24 After insulating sytems: metallic or semi-conducting systems. Effect of free electrons or carriers on tunneling ~1% Ho 3+ in YRu 2 Si 2 single crystal (same matrix as for Ce in YRu 2 Si 2 ) (grown by Hiro Suzuki, Tsukuba) Same Ising doublet ground-state as in LiYF 4 : if tunneling, then drived by hyperfine interactions 1. T = 4 mk v=14 mt/s v=2.2 mt/s M/M S.5. v=7 mt/s v=35 mt/s v=17 mt/s 1..5 T = 4 mk H x =14 mt -.5 v=8.8 mt/s v=4.4 mt/s µ (mt) Continuous (distribution) and then abrupt ~ 4-45 mt
25 Ho.1 Y.999 Ru 2 Si 2 at slower sweeping field 1. T = 4 mk T = H x 4 =14 mk mt.5 v=2.2 mt/s v=.14 mt/s n = 1 M/M S. v=.7 mt/s v=1.1 mt/s -.5 v=.55 mt/s v=.27 mt/s µ (mt) Still abrupt, but M decreases with vanishes between v =.27 and.14mt/s Two resonances, n=1 and n=2 appear in this interval
26 Ho.1 Y.999 Ru 2 Si 2 between v =.27 and.7mt/s 1 n=2 M/M s.5 n=1.4 K.136 mt/s.68 mt/s.34 mt/s.17 mt/s µ H (T) The resonances n=1 and n=2 are clearly visible and independent of the sweeping field. The rest of M(H) is continuous
27 Tunneling in the presence of free electrons is evidenced! in Y 1-ε Ho ε Ru 2 Si 2 ε ~.1% Tunneling of electro-nuclear states in a metal, In a «sea» of many body tunneling events -179,5 E (K) -18, M/M S -,5 v =.11 mt/s -1, M/M s 1,,5, a ) b ) µ (m T) µ (m T) Ho.1 Y.999 Ru 2 Si.4 K mt/s.68 mt/s.34 mt/s.17 mt/s µ H (T) The resonances fields of Ho 3+ ions, in YLiF 4 and YCu 2 Si 2 are the same Y.998 Ho.2 LiF 4 Same resonance fields Y 1-ε Ho ε Ru 2 Si 2 ε ~.1%
28 CONCLUSION SHORT TERM PERSPECTIVES Tunneling and quantum spin dynamics studied in: -Large spin molecules (Large D, large Orbach time, small tunneling gap: mesoscopic and non-adiabtic LZ ) -Low spin molecules ( low D: large tunneling gap: non-mesoscopic and adiabatic LZ, multi-spin character) In both cases the respective roles of the spin-bath and of the phonons bath, more or less understood -Now study of the coherence with electromagnetic radiations (first studies in molecular systems with electromagnetic radiations). -Till now the effect of free carriers on tunneling was not considered on the experimental side New project on rare-earth ions in metals and semi-conductors - Insulating first:: Evidence of tunneling of single rare-earth ions (ensemble measurements) Role of electro-nuclear interactions (CF+HF) Evidence of many body tunneling effects (co-tunneling, spin-diffusion, dipolar-bias tunneling): Crucial role of anisotropinc intercations : importance of dipolar interactions. also of antisymmetrical interactions (DM) in molecules and disordered sustems (no IC). Now search for higher order many-body effect in the presence of strong bias fields, and study of spin-phonons and spins photons transitions in many-body tunneling systems. Roles of the phonons and electrmagnetic radiations baths (also the spin-bath). -Projects on molecular qbits drived by photons in molecular and rare-earths systems (quality of crystal much bette, and large choice of symmetries of the 4f schell and tghe the matrix; nature of the matrix (insulating, metal, semicnductors ). -Last results on rare-eart in a metal: First evidence of tunnelling of single ions and multispin effects in a metal with RKKY + dipolar interactions.. (connections with Kondo systems, heavy fermions and spintronics)
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