Quantum Tunneling of Magnetization in Molecular Magnets. Department of Physics, New York University. Tutorial T2: Molecular Magnets, March 12, 2006

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1 Quantum Tunneling of Magnetization in Molecular Magnets ANDREW D. KENT Department of Physics, New York University Tutorial T2: Molecular Magnets, March 12, Outline 1. Introduction Nanomagnetism Quantum Tunneling of Magnetization Initial discoveries 2. Magnetic Interactions, Energy Scales, Spin Hamiltonian 3. Experimental Techniques Micro-magnetometry High Frequency EPR 4. Resonant Quantum Tunneling of Magnetization Thermally activated, thermally assisted and pure Crossover between regimes Berry Phase and Landau-Zener Method 5. High Spin Superposition States SMM Ni 4 : microwave spectroscopy and magnetometry Photon induced transitions between superposition states Decoherence rate Longitudinal (energy) relaxation times 6. Perspectives 2

2 Nanomagnetism multi-domain single-domain Size: 100 nm nm 1-10 nm 0.1 nm Thin-films Nano-particles Single-molecules Atoms Spin, S = ~20 ~1 r M r M P r j 3 Quantum Tunneling of Magnetization +m -m U Thermal Quantum T c = U/k B B(0) Quantum Tunneling Thermal relaxation (over the barrier) Magnetic Field Relaxation rate T c Temperature also, Enz and Schilling, van Hemmen and Suto (1986) 4

3 Magnetic Bistability in a Molecular Magnet Nature 1993, and Sessoli et al., JACS 1993 Magnetic hysteresis at 2.8 K and below (2.2 K) S=10 ground state spin 5 Quantum Tunneling in Single Molecule Magnets Single crystal studies of Mn12: L. Thomas, et al. Nature 383, 145 (1996) Susceptibility studies: J.M. Hernandez, et al. EPL 35, 301 (1996) 6

4 Single Molecule Magnets! Molecules with a large spin ground state (S~10)! Large (Ising-like) uniaxial magnetic anistropy! Single crystals: ordered 3D arrays of weakly interacting (almost identical) molecules! Well defined discrete set of magnetic quantum states! Chemical control of quantum energy levels Molecule Spin Molecule Symmetry Magnetic Anisotropy Intermolecular Interactions Basic Properties! Individual molecules can be magnetized and exhibit magnetic hysteresis! Quantum tunneling of the magnetization 7 SMMs Fundamental studies of nanomagnets Study of resonant QTM (Friedman, et al; Thomas et al. CNRS 1995) Magnetization reversal and relaxation - Crossover from classical thermal activation to quantum tunneling (Sangregorio, et al. 1997, Bokacheva and ADK, PRL 2000) - Modeling from a microscopic point of view (Pederson et al., Harmon et al.) Interference effects in magnetic quantum tunneling (W. Wernsdorfer and R. Sessoli, Science 1999) Effects of nuclear spins on quantum tunneling Study of decoherence of quantum systems Potential Applications: Magnetic Data Storage (1 bit/molecule in 2D: 100 tera-bit/in 2 ) Magnetic Cooling Millimeter and sub-millimeter wave devices Quantum information storage (Leuenberger and Loss 2001) 8

5 Quantum information storage and quantum computing 9 Magnetic Interactions Energy scales Exchange Magnetic anisotropy Dipolar Hyperfine Spin-Hamiltonian 10 10

6 First SMM: Mn 12 -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 Single Crystal S 4 site symmetry Tetragonal lattice a=1.7 nm, b=1.2 nm Strong uniaxial magnetic anisotropy (~60 K) Weak intermolecular interactions (~0.1 K) Micro-Hall magnetometer 11 Intra-molecular Exchange Interactions S=2 J 1 S=3/2 " J 2 J 3 J 1 J 2 J 3 J 4 J 1 ~ 215 K J 2,J 3 ~ 85 K J 4 ~ 45 K H = # <ij> J 4 r J ij S i " S r j + # i r S i " D i " S r i +... (2S 1 +1) 8 (2S 2 +1) 4 =10 8! S=10 and 2S+1=21 12

7 Magnetic Anisotropy and Spin Hamiltonian Spin Hamiltonian H = "DS z 2 " gµ B r S # r H S z m >= m m > E m = Dm 2 (Ising-like) Uniaxial anisotropy 2S+1 spin levels z up x down y U=DS 2 13 Experimental Techniques Magnetometry Micro-Hall Effect Magnetometry Micro-Squid Spectroscopy EPR Neutron Scattering NMR Specific Heat 14 14

8 Micromagnetometry µ-hall Effect! µ-squid B sample B sample 1 µm Hall bars 1 to 10 µm Josephson Junctions Based on Lorentz Force Measures magnetic field Large applied in-plane magnetic fields (>20 T) Broad temperature range Single magnetic particles! Based on flux quantization! Measures magnetic flux! Applied fields below the upper critical field (~1 T)! Low temperature (below T c )! Single magnetic particles! Ultimate sensitivity ~1 µ B Ultimate sensitivity ~10 2 µ B see, A. D. Kent et al., Journal of Applied Physics 1994 W. Wernsdorfer, JMMM $ High Frequency EPR S. Hill, UF, Gainsville Cylindrical TE01n (Q ~ ) f = 16 # 300 GHz Single crystal 1! 0.2! 0.2 mm 3 T = 0.5 to 300 K, µ o H up to 45 tesla We use a Millimeter-wave Vector Network Analyzer (MVNA, ABmm) as a spectrometer M. Mola et al., Rev. Sci. Inst. 71, 186 (2000) 16

9 Single-crystal, high-field/frequency EPR Reminder: field//z z, S 4 -axis H z m s represents spin- projection along the molecular 4-fold axis Magnetic dipole transitions (%m s = ±1) - note frequency scale! EPR measures level spacings directly, unlike magnetometry methods 17 Energy level diagram for D < 0 system, B//z B // z-axis of molecule 18

10 HFEPR for high symmetry (C3v) Mn4 cubane; S = 9 /2 Cavity transmission (arb. units - offset) [Mn 4 O 3 (OSiMe 3 )(O 2 CEt) 3 (dbm) 3 ]! 5 / 2 to! 3 / 2! 7 / 2 to! 5 / 2! 9 / 2 to! 7 / 2 f = 138 GHz! 3 / 2 to! 1 / 2! 1 / 2 to 1 / 2 24 K 18 K 14 K 8 K 6 K 4 K Magnetic field (tesla) 19 Fit to easy axis data - yields diagonal crystal field terms 20

11 Single-crystal, high-field/frequency EPR Rotate field in xy-plane and look for symmetry effects H ˆ = DS ˆ 2 z + H ˆ r T + gµ B B " S r # E(m s ) = $ " D 2 ± H T % & ' ( m 2 s + gµ B m s B s z, S 4 -axis H xy In high-field limit (gµ B B > DS), m s represents spin-projection along the applied field-axis 21 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/g! 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 22

12 Resonant Quantum Tunneling of Magnetization H L = kh R kh R < H L < (k+1)h R H L = (k+1)h R onresonance offresonance onresonance on-resonance off-resonance on-resonance QT on (fast relaxation) QT off (slow relaxation) Resonant Quantum Tunneling of Magnetization Relaxation processes in SMMs Magnetic relaxation at high temperature Thermal activation (over the barrier) z up y U=U 0 (1-H/H o ) 2 x down Magnetic field M/M s! o H z (T) 24

13 Resonant Quantum Tunneling of Magnetization Relaxation processes in SMMs Magnetic relaxation at intermediate temperature Thermally assisted tunneling z up U=U 0 (1-H/H o ) 2 x down y U " TAT = f (T) k=6 k=7 k=8 Magnetic field H z 25 Resonant Quantum Tunneling of Magnetization Relaxation processes in SMMs Magnetic relaxation at low temperature Pure quantum tunneling z up y U=U 0 (1-H/H o ) 2 x down Magnetic field M/M s Field sweep rate T/min 0.2 T/min 0.1 T/min T/min 0.02 T/min k= Magnetic hysteresis k=6 k= Applied Field (Tesla) 26

14 Crossover from Thermally Assisted to Pure QTM H = "DS z 2 " gµ B S z H z " gµ B S x H x Tunnel splitting on resonance $ " m,m' # D H ' x & ) % ( H A m*m' Upper levels Large splitting Low Boltzmann population Relaxation pathway = f(t) Lowest levels Small splitting High Boltzman population % m,m Theory of thermally assisted tunneling: Villian (1997), Leuenberger & Loss (2000) and Chudnovsky & Garanin (1999) Crossover: Chudnovsky & Garanin, PRL Energy/(DS 2 ) 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 = H z /H o K. Mertes et al. PRB

15 Experiments on the Crossover to Pure QTM in Mn12-acetate dm/dh versus Hz Schematic: dominant levels as a function of temperature Energy/(DS 2 ) m =6 m =7 m =8 Thermal ~1 K m =9 ~0.1 K m =10 n = Quantum H z /H o L. Bokacheva, ADK, M. Walters, PRL High Spin Superposition States SMM Ni4; microwave spectroscopy and magnetometry Photon induced transition between superposition states Decoherence rate Longitudinal (energy) relaxation times 30

16 Coherent QTM: Microwave Spectroscopy and Magnetometry & o Limit S" A" Second condition: insures dephasing by phonons and nuclear spins is small E o typical energy width of nuclear spin multiplet " " o >> # >> kt, E o First condition: consider just 2 lowest energy states Predictions (Stamp) for dephasing rate due to nuclear spins in an applied field Γ φ = E o (Chudnovsky) spin-phonon interaction: universal lower bound on decoherence τ1 1 = Γ 1 = S2 2 ω 3 12π h 2 coth( hω ρc 5 t 2kT ) 2 31 Experiment Study: Magnetization Dynamics Induced by Microwaves (cw and pulsed) Photon induced transition between superposition states combined with magnetization measurements Expectation S" A" % M/M s H L Monitor spin-state populations while performing microwave spectroscopy Pulsed and CW microwave fields High magnetic sensitivity: ~10 5 spins/hz 1/2 High magnetic fields Time resolved magnetic measurements (~GHz) 32

17 Description of SMM Ni 4 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 S = 4 m=0 m =+1 m =+2 m =-1 m =-2 m =+3 m =-4 U = 12K T exp ~ 0.4 K m =-4 m =-3 '5.5 K 33 Description of SMM Ni 4 High frequency EPR: Stephen Hill, UF - Gainesville, FL H = "DS z 2 + C S S " 4 ( ) " gµ B S z H z " gµ B S x H x D L = K D U = K B 0 4 = K B 4 4 = K g z = 2.30 g x = g y = 2.23 Magnetic field (tesla) Cavity transmission (arb. units) Magnetic field (tesla) f = 101 GHz Angle (degrees) Multiple but narrow peaks molecular environments with slightly different D values 34

18 Experiment A( " S( T = 0.4K z up > HT down > 1 ( up"+ down" ) 2 1 A" = ( up"# down" ) 2 S" = M/Ms Expectation Calculated expectation H! L 35! Experimental Setup Vector Network Analyzer He3 cryostat continuous wave pulsed microwave switch pulses Pattern generator Superconducting magnet 36

19 Results: Photon Induced Transitions between Superposition States Magnetization with microwaves 0.0 H L (T) T = 0.45 K "T < 0.01 K in the peaks 37 Results: Photon Induced Transitions between Superposition States up( down( down( up( Level Repulsion: Observation of quantum superposition of high spin states with opposite magnetization 38

20 Results: Photon Induced Transitions between Superposition States Curvature 39 Approach to saturation and Ni 4 micro-environments As a function of mw power cw radiation H T = 3.2 T f = 39.4 GHz P loss-in-coax = 15 db µ We observe variations up to 90% for higher fields and lower frequencies 40

21 Results: Transverse relaxation rate () 2 ) decoherence () * ) Decoherence time lower bound H T = 3.2 T f = 39.4 GHz 41 Results: Longitudinal relaxation rate () 1 ) Pulsed mw experiments "M ' (T ON - T OFF ) P P B 1 P B 2 A 42

22 Results: Longitudinal relaxation rate () 1 ) Pulsed radiation experiments 43 Longitudinal relaxation effects in cw experiments Energy relaxation increases with freq. 44

23 Spin-Phonon Relaxation Pulsed radiation experiments τ 1 Relaxation rate: 1 = Γ 1 = S2 2 ω 3 12π hρc 5 t coth( hω 2kT ) Chudnovsky, PRL 2004 c t = 10 3 m/s, f = /h = 20 GHz τ 1 = 10 3 sec Upper limit on the relaxation time! coupling Phonon-laser effect: Γ L ω 3 Chudnovsky and Garanin, PRL 2004 Other levels important? 45 Summary (Ni 4 ) Observation of energy splittings between low-lying superpositions of high spin-states. Direct measurement of the magnetization combined with microwave spectroscopy Lower bound for the decoherence time ) * > 0.5 ns Similar to that observed in the Mn 4 -dimer through EPR measurements a system with strong intermolecular exchange interactions Determination of the longitudinal relaxation times ) ~ s Characterized ) as a function of longitudinal field and frequency 46

24 Perspectives Coherent oscillations of the magnetization; Rabi and Spin Echo experiments t +/2 delay + t delay +/2 High radiation power is needed for measurable Rabi periods Ex: +-pulse,h 1 ST pulse = +, H 1 =1 µt, T pulse = 5 µs H 1 =100 µt, T pulse = 50 ns Open questions also relate to collective effects: Intermolecular magnetic interactions---exchange and dipolar Radiation and phonon fields! Magnetic Relaxation Enhancement (Cavity) and Superradiance in Mn 12 -acetate Tejada et al., Phys. Rev. B, 2003 and Appl. Phys. Lett. B, 2004 Predicted by Chudnovsky and Garanin - > L Coherent radiation. = N. photon 47