Decoherence in molecular magnets: Fe 8 and Mn 12

Transcription

1 Decoherence in molecular magnets: Fe 8 and Mn 12 I.S. Tupitsyn (with P.C.E. Stamp) Pacific Institute of Theoretical Physics (UBC, Vancouver) Early 7-s: Fast magnetic relaxation in rare-earth systems (Dy 3 Al 2, SmCo 3.5 Cu 1.5 ) Quantum Tunneling Phenomenon Early 9-e: Single-molecule magnets (SMM) Quantum Relaxation in crystals of SMM Fe 8 : S = 1 Mn 12 : S = 1 Present days: Quantum Coherent Oscillations? More than 1 systems are synthesized these days; S =, 1/2, 1, 3/2,, 51/2,?

2 Each molecule contains a core of magnetic ions, characterized by nonzero electronic spins s i (5/2 in Fe 8, 3/2 and 2 in Mn 12 ), surrounded by various atoms with nonzero, or zero nuclear spins. At low-t all s i are strongly coupled together forming the so called Central, or Giant Spin S. Fe 8, T < 1 K, S=1 The states with positive and negative S Z are separated by the potential barrier Energy 65K in Mn 12 25K in Fe 8 Mn 12, T < 4 K, S=1

3 The Central Spin Hamiltonian Low-T all electronic spins are strongly coupled together Fe 8 : T < 1 K (S=1) Mn 12 : T < 4 K (S=1) Fe 8 (2S+1 states) 1 Energy (K) µ H z (T)

4 Quantum Tunneling Classically, to go from one potential minimum to another, system can only activate over the top of the barrier. Quantum-mechanically, however, system can pass through the classically forbidden region - Quantum Tunneling. Anticrossing of levels in Fe 8 1 Energy (K) Δ m,n µ H z (T) ) It is characterized by the tunneling matrix element Δ m,n between the initial and α the final states Δ =< f V i, where is non-diagonal, like. > The tunneling splitting is then 2Δ m,n. The higher the barrier, the smaller Δ m,n. Its value can be changed by applying the transverse field H. V ) S ±

5 Can the tunneling splitting be measured? Yes, if it is not too small Experiment: Tunneling splitting 2Δ o between two lowest states in Fe 8 as a function of transverse magnetic field. o W. Wernsdorfer and R. Sessoli, 1999

6 Z Tunneling splitting: Theory Δ S iπs iπs, = Δ e + e = 2Δo cos( πs) S o H = ϕ H trans Y 1x1 1x1-2 Fe 8 9 o X 1x1-4 Δο(K) 1x o o 7 o 1x1-8 o H <H C (two ways) H >H C (one way) 1x H (T) H Δ S, S = 2Δo cos( π S + iπh X / TX + πhy / TY )

7 Very low-t limit only two lowest states in both systems are occupied Each molecule can be modeled as a Two Level System. This model works, however, only if Δ o << Ω o (Ω o is the gap to the first excited state). Single TLS, no environment Two solutions: Energy Symmetric: Antisymmetric: S> = u > + v > A> =-v > + u > E S,A = 2ε; ε=(δ o2 +ξ 2 ) 1/2 Time-evolution: < e -iht/ћ > 2 Δo P = sin 2 ( εt 2 ε / h), Ω o S> oscillations do not decay A>

8 Very low-t limit real SMMs By applying transverse magnetic field one can create symmetric S> and antisymmetric A> states separated by the gap 2Δ o (H ). By applying then microwave pulse one can mix up S> and A> states and create the one-well states Z ± > = ( S> ± A>)/2 1/2, initiating oscillations between them. Can these oscillations be coherent in a real SMM? If yes, for how long coherence can last? Z ± > are not eigenstates of H S oscillations DECAY - INTRINSIC DECOHERENCE What in the environment in SMMs, i.e., what are the sources of DECOHERENCE?

9 Environment (1) Interaction with the nuclear spin bath: γ (1) k l k m k (2) Spin-phonon interaction: γ (2) k 2 β k where the sum is performed over all the terms allowed by symmetry. Example: (3) Pair-wise interaction with another molecules: exchange and dipolar interactions

10 Nuclear spin bath γ (1) k l k m k 6 2 β k γ (2) k H Number Fe ω (MHz) k

11 Br 79 Br 13 C 5 4 C Number 3 Number ω (MHz) k ω (MHz) k 5 4 N 14 N 17 O 5 4 O Number 3 Number ω (MHz) k ω (MHz) k

12 Nuclear spin bath Fe 8, H Z = E o -S 8 irons, 12 hydrogens, 8 bromines, 18 nitrogens, 36 carbons and 23 oxygens S 2Δ o symmetric Interaction with the nuclear spin bath leads to the spread of each electronic energy level and the half-width of the distribution of states, E o, describes the static properties of the nuclear spin bath. Knowing positions of all the ions in the molecule, it is easy to calculate E o. Eo (K) Fe 8 56 Fe 8 S z -5 antisymmetric Hx (T) 56 Fe 8D P.C.E. Stamp and I.S. Tupitsyn, PRB 69 (24) Hx (T)

13 Nuclear spin bath How E o can be measured? As it has been shown (Prokof ev and P.C.E. Stamp, PRL 8 (1998)), due to interactions with the nuclear spin bath the short-time low-t relaxation in crystals of magnetic molecules follows the square-root law and during the relaxation the hole in the dipolar fields distribution is growing. The shape of this hole is Lorentzian and its short-time half-width is E o (I.S. Tupitsyn, P.C.E. Stamp and N.V. Prokof ev, PRB 69 (24)). W. Wernsdorfer et al., PRL 82 (1999)

14 Nuclear spin bath E o in Fe 8 and Mn 12 comparison with experimental results of Wernsdorfer et. al. PRL 82 (1999); PRL 84 (2); and Europhys Lett. 47 (1999). Mn 12 Fe 8 Fe 8

15 Phonon bath Considering all the terms allowed by symmetry of problem, we can keep only the dominant ones. These can be filtered by studying the field dependence of the spin part of H sp-ph. For H =H Y in Fe 8 and for H =H X in Mn 12 these are: η i =D

16 Coherence Window Decoherence in many solid-state systems is anomalously high. At the same time, it has been shown (P.C.E. Stamp and I.S. Tupitsyn, PRB 69 (24)), that in magnetic insulators there is a transverse field region, where the phonon and nuclear spin mediated decoherence is drastically reduced. Such a coherence window opens up around some critical field, where the total nuclear spin bath and phonon dimensionless decoherence rate reaches its minimum. (N.V. Prokof ev and P.C.E. Stamp, cond-mat/654; P.C.E. Stamp and I.S. Tupitsyn, PRB 69 (24)); (A. Morello, P.C.E. Stamp, and I.S. Tupitsyn, PRL 97 (26))

17 Coherence Window Fe 8 -SMM Number of coherent oscillations Q ~ 1/γ φ

18 Coherence Window Mn 12 -SMM Number of coherent oscillations Q ~ 1/γ φ

19 Ensembles of SMMs 3) Pair-wise interaction with another molecules: V dd ij r r r r μ ( )( o geμ r r Si ij S j B ( r) = S 3 3 is j 2 4π rij rij ij ) r ij α Uniform precession --> q= magnon In a transverse magnetic field the oscillations are equivalent to a uniform spin precession along the field directions, i.e., to a q= magnon. Scattering of the q= mode off thermal magnons leads to a decay of oscillations. The corresponding decay time can be both measured and calculated.

20 Scattering of the q= mode off thermal magnons The lowest order processes that, in principle, can conserve both energy and momentum here are 4-magnon processes: does not change # of magnons T 2 q q' q-q' δ(ω o +ω q -ω q -ω q-q ) 2 in - 2 out Σ T 1 q+q' δ(ω o +ω q +ω q -ω q+q ) 3 in - 1 out change # of magnons

21 Fe 8 sample averaged rates (triclinic lattice, spherical sample) Fe 8 Except at very low T, dipolar decoherence completely dominates over nuclear and phonon decoherence, unless H Y >2.7 T. A. Morello, P.C.E. Stamp, and I.S. Tupitsyn, PRL 97 (26)