NYU An Introduction to Quantum Tunneling of the Magnetization and Magnetic Ordering in Single Molecule Magnets. Andrew D. Kent

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1 An Introduction to Quantum Tunneling of the Magnetization and Magnetic Ordering in Single Molecule Magnets Andrew D. Kent Department of Physics, New York University 1

2 Outline I. Introduction Quantum tunneling of magnetization Initial discoveries Single molecule magnets II. Magnetic Interactions in SMMs Energy scales, Spin Hamiltonians Mn12-acetate III. Resonant Quantum Tunneling of Magnetization Thermally activated, thermally assisted and pure Crossover between regimes Quantum phase interference IV. Experimental Techniques Micromagnetometry SQUIDs and Nano-SQUIDs EPR V. Collective Effects Magnetic ordering Magnetic deflagration 2

3 Magnetic Nanostructures Image from, W. Wernsdorfer, Advances in Chemical Physics 2001 and ArXiv:

4 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 4 also, Enz and Schilling, van Hemmen and Suto (1986)

5 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

6 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) 2009 Boulder School, Lecture 3

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

7 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 Quantum Tunneling of Magnetization Quantum Phase Interference Crossover Thermal Assisted to Pure QTM Quantum Coherence Random-Field Ising Ferromagnetism

8 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) 8 Micro-Hall magnetometer

9 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 9

10 Magnetic Anisotropy and Spin Hamiltonian Spin Hamiltonian H = DS z 2 gµ B S 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 10

11 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 M/M s Magnetic field µ o H z (T) 11

12 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 12

13 Resonant Quantum Tunneling of Magnetization H L = kh R kh R < H L < (k+1)h R H L = (k+1)h R on-resonance off-resonance on-resonance 10 on-resonance off-resonance on-resonance QT on (fast relaxation) QT off (slow relaxation) H L (T ) 13

14 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 Γ TAT = f (T) k=6 k=7 k=8 Magnetic field 14 H z

15 Resonant Quantum Tunneling of Magnetization Relaxation processes in SMMs Magnetic relaxation at low temperature Pure quantum tunneling z up U=U 0 (1-H/H o ) 2 y x Magnetic field down M/M s Magnetic hysteresis Field sweep rate 0.4 T/min 0.2 T/min 0.1 T/min 0.05 T/min 0.02 T/min k=7 k=8-0.5 k= Applied Field (Tesla)

16 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 Upper levels Large splitting Low Boltzmann population Δ m,m' D H x H A m m' Relaxation pathway = f(t) Lowest levels Small splitting High Boltzman population Δ m,m Theory of thermally assisted tunneling: Villian (1997), Leuenberger & Loss (2000) and Chudnovsky & Garanin (1999) Crossover: Chudnovsky & Garanin, PRL

17 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 17 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!!!

18 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 2001

19 m(t) NYU 1.10 K K 0.4 this crossover should lead to a better understanding of the mechanisms of tunneling in Mn12. This work was supported by NSF-INT (Grant No ) and NYU. Experiments on the Crossover to Pure QTM in Mn12-acetate 1.29 K 0.2 VOLUME 85, NUMBER 22 E.TM.T Chudnovsky and J. Tejada,27 Macroscopic P H Y 1.40 S I CK A L R E V I E[1] W See LE ERS NOVEMBERTunnel2000 ing of the Magnetic Moment (Cambridge University Press, Cambridge, UK, 1997), Chap. 7. We the measured crossoverj.temperature K) [2] note J. R.that Friedman, M. P. Sarachik, Tejada, and ('1.1 R. Ziolo, oo θ=0 is significantly higher than predicted (0.6 K) [24]. This θ=35 Phys. Rev. Lett. 76, 3830 (1996); L. Thomas, F. Lionti, R. n=6 may Ballou, be due to intrinsicr. mechanism promoting tunneling n=4 D.an Gatteschi, Sessoli, and B. Barbara, Nature 0.99 K in Mn such as a transverse anisotropy. Further studies of 12 (London) 383, 145 (1996). this shouldt.lead to C. a better understanding of the [3] crossover C. Sangregorio, Ohm, Paulsen, R. Sessoli, and D K 0.6 mechanisms tunneling in Mn. (1997). Gatteschi,ofPhys. Rev. Lett. 78, K work D.was supported by NSF-INT (Grant [4]This R. Sessoli, Gatteschi, A. Caneschi, and M. A. Novak, 0.2 Nature (London) 365, 141 (1993). No ) and NYU. 0.4 [5] M. Novak and R. Sessoli, in Quantum Tunneling of Mag1.29 K K 0.60 netization-qtm 94, edited by L. Gunther and B. Barbara (Kluwer, Dordrecht, 1995), p. 171; B. Barbara et al., J. [1] See E. M. Chudnovsky and J. Tejada, Macroscopic Tunnel , 1825 (1995). Magn. Magn. Mater K ing of the Magnetic Moment (Cambridge University Press, m =6 [6] J. M. Hernandez et al., Europhys. Lett. 35, 301 (1996) K Cambridge, UK, 1997), Chap. 7. 0, 133R.(1999). [7] W. Wernsdorfer and R. Sessoli, Science 284and [2] J. R. Friedman, M. P. Sarachik, J. Tejada, Ziolo, 0.4 b) o m =7, 2453 (1998). [8] S. Hill et al., Phys. Rev. Lett. 80 z Time (seconds) θ=35 Phys. Rev. Lett. 76, 3830 (1996); L. Thomas, F. Lionti, R., 8819 (1997). Nature [9] M. Hennion et al., Phys. B 56and n=4 Ballou, D. Gatteschi, R. Rev. Sessoli, B. Barbara, FIG. 4. Relaxation of the magnetization vs time at different [10] A. L. Barra, D., Gatteschi, and R. Sessoli, Phys. Rev. B 56, m =8 145 (1996). (London) 383 temperatures0.3 for (a) n! 6, u! 0±, showing a crossover to a (1997). 0.5 [3] 8192 C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D. quantum regime at approximately 1 K, and (b) n! 4, u! 35±,, 628 (1999). [11] I. MirebeauPhys. et al.,rev. Phys. Rev (1997). Gatteschi, Lett. 78,Lett. showing no temperature independent regime. m!t" is a reduced (1999). [12] Y. Zhong etd. al.,gatteschi, J. Appl. m =9 Phys. 85, 5636and [4] R. Sessoli, A. Caneschi, M. A. Novak, magnetization: 0.2 m!t"! #Ms 2 M!t"$%2Ms. In (a) the five [13] E. M. Chudnovsky and, 141 D. A.(1993). Garanin, Phys. Rev. Lett. 79, Nature (London) 365 curves below 0.74 K overlap (0.56, 0.58, 0.63, 0.68, and (1997). [5] 4469 M. Novak and R. Sessoli, in Quantum Tunneling of Mag0.74 K). These curves can be fit with m!t"! m0 exp#!2t%t"b $, 11 L. 847Gunther (1999); and G.-H.B.Kim, Eu[14] G.-H. Kim, Phys. Rev.edited B m =10 59,by netization-qtm 94, Barbara where m0! , t!! " Ks, b!, 216 (2000); H. J. W. Müller-Kirsten, D. K. rophys The fit overlaps the data. In (b) the unmarked (Kluwer,Lett. Dordrecht, 1995), p. 171; B. Barbara et al., 4J. n = 0Rev. B160, Park, J. M.Mater. S. Rana, Phys. curves from top to bottom correspond to T! 0.68, 0.70, 0.75,, 1825 (1995). (1999), and Magn.and Magn therein. 0.83, 0.91, and 0.95 K. [6] references J. M. Hernandez et al., Europhys. Lett. 35, 301 (1996) K [15] The analogy to phase is a purely one as 133 (1999). [7] W. Wernsdorfer and R.transitions Sessoli, Science 284,formal discussed in [13]. The first-order, second-order terminol, 2453 (1998). [8] S. Hill et al., Phys. Rev. Lett. 80 temperature it changes significantly. This temperature cor0 2 due to4larkin Time (seconds) for the escape crossover, 8819 (1997). [9] ogy M. Hennion et al.,rate Phys. Rev. B is56originally responds to the crossover temperature seen in Fig. 3 [16]. and R. Sessoli, Phys. Rev. B 56, FIG. 4. Relaxation of the magnetization vs time at different [10] and A. L.Ovchinnikov Barra, D. Gatteschi, consistent with pure QT. In contrast, for the peak n!to4,a ±, showing a crossover temperatures for (a) n! 6, u! 0 [16] A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 59, 8192 (1997). uquantum! 35±,regime the magnetization relaxation at approximately 1 K, andrate (b) changes n! 4, usignifi! 35±, (1984). et al., Phys. Rev. Lett. 83, 628 (1999). z [11] 420 I. Mirebeau cantly temperature decreases in them!t" entire showingasnothe temperature independent regime. is a studied reduced, 512(1999). (2000). [17] Kentetetal., al.,j.europhys. Lett. [12] A. Y. D. Zhong Appl. Phys. 85, magnetization: m!t" curves! #Ms 2 (a) the exfive s. a In range. Relaxation canm!t"$%2m be fit by stretched (2000). [18] Wernsdorfer et al., Lett. Phys. 50, 552 [13] W. E. M. Chudnovsky and Europhys. D. A. Garanin, Rev. Lett. 79, curves below 0.74 K overlap 0.58, b0.63, 0.68,band ponential function m!t"! m0(0.56, exp#2!t%t" $, where [19] A. D. Kent, S. von Molnar, S. Gider, and D. D. Awschalom, 4469 (1997). b& 0.74 K). These curves can be fit with m!t"! m0 exp#!2t%t" $, 6656 (1994). Appl. Phys. 76,Rev. 0.4 form of relaxation has pre, (1999); G.-H. Kim, Eu[14] J. G.-H. Kim, Phys. B 59 where0.6. m0 This! , t!! been 0.15" observed s, b! (1980). D. K. [20] T. Lis, Acta Sect.H.BJ.36 W., 2042 Müller-Kirsten, rophys. Lett. Crystallogr. 51, 216 (2000); viously in FeThe in Mn [21], Inalthough it is not overlaps the (b) the unmarked 8 [3]fitand 12 data. and Park, and J. M. Rana, Phys. Rev. B 60, 6662 [21] L. Thomas, A. S. Caneschi, and B. Barbara, Phys.(1999), Rev. Lett. curves from understood top to bottom[22]. correspond to T! 0.68, 0.70, 0.75, completely references therein., 2398 (1999) , 0.91, and 0.95 In summary, we K. have presented new low temperature [15] N. The to phase is a Phys. purely Rev. formal one80 as, [22] V.analogy Prokof ev and P.transitions C. E. Stamp, Lett. magnetic studies of thermally assisted and pure quantum discussed in [13]. The first-order, second-order terminol5794 (1998); E. M. Chudnovsky, Phys. Rev. Lett. 84, 5676 temperature changes significantly. temperature tunneling in itmn crossoverthis between these cortwo 12. The ogy for the escape rate crossover due to, 5677 (2000); N. V. Prokof ev and P. C.isE.originally Stamp, ibid., 84Larkin respondswas to found the crossover temperature in Fig. 3 regimes to be either abrupt or seen gradual, dependand Ovchinnikov [16]. C. Paulsen, and R. Sessoli, ibid., (2000); W. Wernsdorfer, consistent pure QT. contrast, of forapplied the peak n! 4, ing on the with magnitude and In orientation magnetic [16] 84 A., I.5678 Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 59, (2000). ± u! 35, the longitudinal magnetization relaxation rate changes signififield. Higher and transverse fields broaden the 420A.(1984). [23] D. Garanin, X. Martinez Hidalgo, and E. M. Chudcantly as the temperature in the entirewe studied consistent with adecreases recent model [23]. have 2012,crossover, Orlando, Tutorial Session, Sunday, October 7,novsky, 2012 [17] A. D. Kent et al., 49, 512 (2000). 639 (1998). Phys. Rev.Europhys. B 57, 13 Lett. range. Relaxation curves can be fit by a stretched ex(2000). [18] L. W.Bokacheva, WernsdorferA.etD.al., Europhys. Lett. 50, 552Polyhedron also shown that below the crossover temperature the mag[24] Kent, and M. A. Walters, ponential function m!t"! m0 exp#2!t%t"b $, where b & [19] A. D. Kent, S. von Molnar, S. Gider, and D. D. Awschalom, 0 1 a) 0.4 b) 0.56 K T 0.74 K 0.83 K Schematic: dominant levels as a function of temperature m(t) m(t) M(t) m(t) dm/dh versus H c Energy/(DS2) c Thermal ~1 K ~0.1 K H /H o ADK et al. EPL 2000 L. Bokacheva, PRL 2001 K. Mertes et al. PRB 2001 W. Wernsdorfer et al. PRL ICMM Quantum

20 Tunneling Selection Rules H = DS 2 z gµ B S z H z + H A Form of H A determined by the site symmetry of the molecule Fe 8 Mn 12 -acetate 180 o 90 o C 2 -site symmetry (rhombic) S 4 -site symmetry (tetragonal) H A = E(S 2 x S 2 y) H A = C(S S 4 ) 20

21 Hysteresis Loops as a Function of Transverse Magnetic Field Landau-Zener Transitions P 1 1 P P = e = 2 /(2 v) v =2Sgµ B db z /dt 21

22 Oscillations and Parity Effect in the Tunnel Splitting Spin-parity effects in QTM: Loss et al., 1992 von Delft & Henley, 1992 Predicted for a biaxial system by Garg 1992! 22

23 Quantum Phase Interference in Fe8 H = DS 2 z + E(S 2 x S 2 y) gµ B S x H x H T H T 23

24 Micromagnetometry µ-hall Effect B sample µ-squid 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 24 see, A. D. Kent et al., Journal of Applied Physics 1994 W. Wernsdorfer, JMMM 1995

25 Nano-SQUID 25

High Frequency EPR S. Hill, HMFML & FSU θ Cylindrical TE01n (Q ~ 10 4-10 5 ) f = 16 300 GHz Single crystal 1 0.2 0.2 mm 3 T = 0.

26 High Frequency EPR S. Hill, HMFML & FSU θ Cylindrical TE01n (Q ~ ) f = GHz Single crystal 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 26 M. Mola et al., Rev. Sci. Inst. 71, 186 (2000)

27 Single-crystal, high-field/frequency EPR S. Hill, HMFML & FSU Reminder: field//z z, S 4 -axis H z m s represents spin- projection along the molecular 4-fold axis Magnetic dipole transitions (Δm s = ±1) - note frequency scale! EPR measures level spacings directly, unlike magnetometry methods 27

28 Energy level diagram for D < 0 system, B//z S. Hill, HMFML & FSU B // z-axis of molecule 28

29 HFEPR for high symmetry (C3v) Mn4 cubane; S = 9/2 Cavity transmission (arb. units - offset) S. Hill, HMFML & FSU [Mn 4 O 3 (OSiMe 3 )(O 2 CEt) 3 (dbm) 3 ] 3 / 2 to 1 / 2 1 / 2 to 1 / 2 5 / 24 K 2 to 3 / 2 18 K 14 K 7 / 8 K 2 to 5 / 2 6 K 9 / 2 to 7 / 2 f = 138 GHz 4 K Magnetic field (tesla) 29

30 Fit to easy axis data - yields diagonal crystal field terms 30

Collective Effects in Single Molecule Magnets Magnetic Molecules in crystals interact with one another through magnetic dipole and exchange interactions Dipolar Interactions H ij = µ 0 4 r 3 3( m i

31 Collective Effects in Single Molecule Magnets Magnetic Molecules in crystals interact with one another through magnetic dipole and exchange interactions Dipolar Interactions H ij = µ 0 4 r 3 3( m i ˆr ij )( m j ˆr ij ) m i m j H dip = 1 2 i=j H ij m = S = gµ B / Interaction strength is small but the interaction is long range E dip = µ 0 (gµ B S) 2 /V cell ' 0.1 K for Mn 12 Long range order at low temperature (e.g., FM or AFM state) Magnetic ground state depends on the crystal structure and crystal shape 31

32 Quantum Fluctuations and Long-Range Order Mn12-acetate Single Crystals (bcc tetragonal lattice of molecules) A ferromagnetic phase was predicted: -Fernandez and Alonso, PRB Garanin and Chudnovsky, PRB 2008 Neutron scattering data shows low-t ferromagnetic order: -Luis et al., PRL 2005 Expect Mn12 to be an example of a transverse field Ising system Interacting Ising spins in a transverse field H = X ij J ij S z i S z j h X i S x i H =0 T H P. Subedi, ADK, B. Wen, M. P. Sarachik, Y. Yeshurun, A. J. Millis, S. Mukherjee and G. Christou, PRB 2012 B. Wen, Pradeep Subedi, Y. Yeshurun, M. Sarachik, ADK, A. J. Millis, PRB 2010 Myriam Sarachik, TUL 4 (Tuesday afternoon) 32

33 Susceptibility Randomness Theoretical Model Theory vs Experiment Conclusion NYU eld (H ) = T/s Quantum Fluctuations and Long-Range Order T = 2 K Mn12-acetate Single Crystals (bcc tetragonal lattice of molecules) H!! 067 T/s = T/s T = 2 K H! ! Energy, E m m Introduction Setup Susceptibility Randomness Theoretical Model Theory vs Experiment Conclusion! m m Magnetic Field, Hz neracies 0 by 1mixing 0 2 the 3 states χ 1 (arb. units) ying H 1 H 0 T 1 T 2 T 3 T 4 T 5 T T (K) Pure-system Random-field P. Subedi, ADK, B. Wen, M. P. Sarachik, Y. Yeshurun, A. J. Millis, S. Mukherjee and G. Christou, PRB 2012 Energy, E breaks the symmetry and lifts the Phase Diagram & Curie-Weiss Temperature m m breaks the symmetry and lifts the Applying H degeneracies by mixing the states. Increasing H the relaxation Curie-Weiss process m law towards thermal equilibrium. m c FM = T TCW 0.4 Magnetic Field, Hz 0.8 promotes quantum tunneling, accelerating T CW (K) Experimental Pure-system MFA Random-field MFA asing H B. promotes Wen, Pradeep Subedi, quantum Y. Yeshurun, tunneling, M. Sarachik, ADK, accelerating A. J. Millis, PRB 2010 elaxation process towards thermal equilibrium. 8 H (T) ` PM 27

34 Magnetic Deflagration Magnetic Molecules in crystals also couple through photons and phonons a y Using transverse field both energies Y. Suzuki, et al., Phys. Rev. Lett. 95, (2005) & A. Hernandez-Mınguez, Phys.Rev. Lett. 95, (2005) MF-06: Pradeep Subedi, Quantum deflagration in Mn12 a transverse magnetic field MC-09: Jonathan Friedman, Collective Coupling of Fe8 SMM to a resonant cavity 34

35 Summary I. Introduction Quantum tunneling of magnetization Initial discoveries Single molecule magnets II. Magnetic Interactions in SMMs Energy scales, Spin Hamiltonians Mn12-acetate III. Resonant Quantum Tunneling of Magnetization Thermally activated, thermally assisted and pure Crossover between regimes Quantum phase interference IV. Experimental Techniques Micromagnetometry SQUIDs and Nano-SQUIDs EPR V. Collective Effects Magnetic ordering Magnetic deflagration 35

Spins Dynamics in Nanomagnets. Andrew D. Kent

Spins Dynamics in Nanomagnets. Andrew D. Kent Spins Dynamics in Nanomagnets Andrew D. Kent Department of Physics, New York University Lecture 1: Magnetic Interactions and Classical Magnetization Dynamics Lecture 2: Spin Current Induced Magnetization