Spin Dynamics in Single Molecule Magnets Andrew D. Kent Department of Physics, New York University Collaborators: Gregoire de Loubens, Enrique del Barco Stephen Hill Dmitry Garanin Myriam Sarachik, Yosi Yeshurun Vladimir Krymov, Gary Gerfen George Christou, David Hendrickson Sponsor 1
Outline I. Introduction Quantum Tunneling of Magnetization Single molecule magnets II. Mn12-acetate Spin Hamiltonian Crossover from Thermally Assisted to Pure Quantum Tunneling III. Spin-Relaxation in Ni4 EPR and micromagnetometry on a chip Spin-phonon relaxation mechanisms IV. Summary and Perspectives 2
Quantum Tunneling of Magnetization +m -m U Thermal Quantum T c = U/k B B(0) Thermal relaxation (over the barrier) Quantum Tunneling Magnetic Field Relaxation rate T c Temperature 3 also, Enz and Schilling, van Hemmen and Suto (1986)
Single Molecule Magnets Physics Mn 84 S=6 SMM Characteristics Molecules High spin ground state Uniaxial anisotropy Single crystals Synthesized in solution Modified chemically Peripheral ligands Oxidized/reduced Soluble = Bonded to surfaces -Individual molecule can be magnetized and exhibit magnetic hysteresis 1993 -Quantum Tunneling of Magnetization 1995 -Quantum Phase Interference 1999 -Quantum Coherence 2008 4
First SMM: Mn12-acetate [Mn 12 O 12 (O 2 CCH 3 ) 16 (H 2 O) 4 ].2CH 3 COOH.4H 2 O 8 Mn 3+ S=2 4 Mn 4+ S=3/2 Magnetic Core Competing AFM Interactions Ground state S=10 Organic Environment 2 acetic acid molecules 4 water molecules S 4 site symmetry Single Crystal Tetragonal lattice a=1.7 nm, b=1.2 nm Strong uniaxial magnetic anisotropy (~60 K) Weak intermolecular dipole interactions (~0.1 K) Micro-Hall magnetometer 5
Resonant Quantum Tunneling of Magnetization H = DS z 2 gµ B S z H z S z m>= m m > E m = Dm 2 gµ B H z m with Zeeman term Resonance fields where antiparallel spin projections are coincident, H k =kd/gµ B, levels m and m ; k=m+m z up y U=U 0 (1-H/H o ) 2 x down Magnetic field Anisotropy Field: H A = 2DS gµ B 6
Resonant Quantum Tunneling of Magnetization H L = kh R kh R < H L < (k+1)h R H L = (k+1)h R -10-9 -8 7 8 on-resonance 9 10-8 -9-10 off-resonance 7 8 9 10-8 7-9 8-10 9 on-resonance 10 on-resonance off-resonance on-resonance QT on (fast relaxation) QT off (slow relaxation) -0.5 0 0.5 1.0 1.5 2.0 H L (T ) 7
Thermal to Quantum Crossover in Magnetization Relaxation ln Γ Quantum Tunneling Thermal activation crossover ln Γ Top E T 0 1/S T ln Γ E T 0 1/S T T 0 T Bottom T 0 T T 0 T Second order crossover Larkin and Ovchinnikov '83 First order crossover Chudnovsky '92 8 U(x) = x 2 + x 4 x 2 ± x 3 U(x) = x 2 x 4 ± x 5 Chudnovsky and Garanin '97 Uniaxial nanomagnets for small transverse magnetic fields!!!
Energy/(DS 2 ) 1.0 0.5 0.0 D=0.548(3) K g z =1.94(1) H0= D/gzµB = 0.42 T B=1.17(2) 10-3 K (EPR: Barra et al., PRB 97) Energy relative to the lowest level in the metastable well m =6 m =7 m =8 m =9 m =10 n = 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 9 H z /H o K. Mertes et al. PRB 2001
Experiments on the Crossover to Pure QTM in Mn12-acetate dm/dh versus Hz Schematic: dominant levels as a function of temperature 1.0 m =6 Energy/(DS 2 ) 0.5 0.0 m =7 m =8 m =9 m =10 n = 0 1 2 3 4 5 6 7 8 9 Thermal ~1 K ~0.1 K Quantum 0 2 4 6 8 10 H z /H o ADK et al. EPL 2000 L. Bokacheva, PRL 2001 K. Mertes et al. PRB 2001 W. Wernsdorfer et al. PRL 2006
Spin Relaxation between Low Lying Spin-States of Ni4 H = DSz 2 + H T + gµ BH S D>0 S =4 Ω o S A First condition: consider just 2 lowest energy states Second condition: mainly the ground state is populated E o typical energy width of nuclear spin multiplet What are the spin-relaxation mechanisms at low temperature? Is it possible to measure the direct single-spin-phonon relaxation rate in SMMs? Ω o > kt, E o Chudnovsky: Universal lower bound on the decoherence rate: all parameters can be measured 11 τ 1 1 =Γ 1 = S2 2 ω 3 12π h 2 ρc 5 t coth( hω 2kT ) Stamp: for dephasing rate due to nuclear spins Γ φ = E 2 o
Collective Spin Phonon Relaxation» Spin and phonon DOF are strongly coupled n undamped phonons n o =0.1 B = N S /N Γ Microscopic model of the bottleneck, Garanin PRB 2007, 2008» Phonon wavelength is much greater than the lattice spacing λ > L Coherent radiation Γ = N Γ photon Chudnovsky & Garanin, PRL 2004 12
Description of Ni 4 Crystals I41/a space group H = DS 2 z + C S 4 4 ( + + S ) gµ B S z H z gµ B S x H x [Ni(hmp)(t-BuEtOH)Cl] 4 S4 site symmetry No solvate molecules! No nuclear spin on Ni sites* *at 1 % level-- 61 Ni S = 4 m=0 m =+1 m =-1 m =+2 m =-2 m =+3 m =-3 U = 12 K 5.5 K m =-4 m =-4 T exp ~ 0.4 K 13
Experiment Magnetization dynamics induced by microwaves 14
Experiment Magnetization dynamics induced by microwaves 15 E. del Barco, ADK, et al. PRL 2004
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EPR Spectroscopy: Level Dispersion Experiment: mapping in (H z, H T ) Experiment: mapping in (Hz and HT) Direct Diagonalization of the Hamiltonian Direct diagonalization of the Hamiltonian D A = 0.757 K (D B = 0.785 K) B = 7.9 10-3 K, C = 3.25 10-5 K g z =2.3, g x =g y =2.23 Good agreement with HF EPR Linewidth T 2 * ~ 0.2 ns (inhomogeneous Multiple but narrow peaks: molecular environments with slightly different D values broadening, lower bound for the coherence time) Analysis of peak amplitude: Fermi golden-rule 17
P abs, M and the Relaxation Rate in Steady State Λ absorption rate vs. relaxation rate Γ Steady State b a P abs = Λ ωn o m b m a N o = Population difference N 0 = N a N b In equilibrium N eq 0 = N S tanh N eq o 1+2Λ/Γ ( E 2k B T ) Energy balance Γ = (m b m a ) M P abs ω 0 M = N a m a + N b m b = M eq + M Simultaneous measurement of P abs and M yields Γ 18
Simultaneous CW Absorption and Magnetization Measurements -14 β Energy (K) P abs (nw) -16-18 -20-22 4 3 2 1 T = 0.39 K f = 27.8 GHz P in = 1.8 mw H = 3.1 T T µw α b a Δ H 0.1 T T 2 > 0.25 ns 0 0.4 ΔM/M s (%) 0.2 0.0-0.2-0.4-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 H z (T) 19
Temperature Dependence M eq /M s (%) 40 30 20 10 0 4 30 GHz 10 GHz 2-state model M eq = m a + m b exp ( ω 0 /(k B T )) M s S(1 + exp ( ω 0 /(k B T ))) P abs (nw) 3 2 1 P abs = ω 0 ( gµb h ac ) 2 a S z b 2 2Γ 2 ( ) ω0 N S tanh 2k B T ΔM/M s (%) 0 1.0 0.5 0.0 0.0 0.4 0.8 1.2 1.6 2.0 T T T (K) H =2.55T, H z =0.03T f = 10 GHz, P in = 100 nw H =3.1T, H z =0.11T f = 27.8 GHz, P in = 1 mw Relaxation rate Γ = (m b m a ) P abs M ω 0 20
Temperature Dependence of Relaxation Rate 10-4 10-3 T (s) 10-2 10-3 21 Moderate increase with increasing T below ~0.8 K Exponential increase at high-t (>1 K)
Temperature Dependence of Relaxation Rate comparison with direct spin-phonon relaxation form 10 5 Direct process: 60 cotanh(hf/(2k B T)) (10 GHz) 10 4 70 cotanh(hf/(2k B T)) (27.8 GHz) Experimental data: H T =2.55T, H z =0.03T H T =3.1T, H z =0.11T Γ (s -1 ) 10 3 10 2 Direct process (Chudnovsky & Garanin 2005) 10 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1/T (K -1 ) Γ sp,t = QS 2 2 ω 0 (gµ B H ) 2 12πE 4 t E t =(ρv 5 t 3 ) 1/4 = 76 K Q = 1 for ( ω0 coth 2k B T SH z <H ) 22
Relaxation Mechanisms Γ T = Γ ( ) ph coth 2 ω0 + B ω0 2k B T i ( ) Ei Γ 0i exp. k B T 30 GHz 10 GHz l ph 500 µm ṅ ω0 = Γ sp [n ω0 + (2n ω0 1) p ω0 ] ṗ ω0 = Γ ph ( p eq ω0 p ω0 ) Bω0 ṅ ω0 B ω0,t = B ω0 tanh 2 ( ω 0 /(2k B T )) 23
Relaxation Mechanisms 30 GHz 10 GHz 24
Summary The energy splittings between low-lying superpositions of high spin-states of Ni4 has been measured spectroscopically Energy relaxation has been measured in a cw experiment in a low-excitation limit by combining magnetic and microwave data acquired simultaneously The temperature dependence of the energy relaxation rate in Ni4 has been measured at two different frequencies. The data show that the spin relaxation is dominated by a phonon bottleneck at low temperatures and occurs by an Orbach mechanism at high temperature. To eliminate the phonon bottleneck requires B ω0 Γ sp,t Γ ph, T ph 10 ns Submicron scale crystals!!! Coherence is expected to play an important role in the collective spin-phonon relaxation in SMM crystals: The fact that the DME calculations+phonon bottleneck work so well to explain the data is somewhat surprising. References: G. de Loubens et al., Europhysics Letters 2008 & JAP 2007 25
Single Molecule Magnets Physics -Individual molecule can be magnetized and exhibit magnetic hysteresis 1993 -Quantum Tunneling of Magnetization 1995 -Quantum Phase Interference 1999 -Quantum Coherence 2008 Open Questions/Research Directions -Understanding mechanism of decoherence and finding means to mitigate environmental effects -Addressing individual Mn 84 molecules, on surfaces (STM) between electrodes S=6(quantum dots) & by magnetometry -Electronic transport properties: coupling between electronic and spin-dof -Collective effects in spin-relaxation: superradiance... b m b 26 a m a