Quantum dynamics in Single-Molecule Magnets

Quantum dynamics in Single-Molecule Magnets Wolfgang Wernsdorfer Laboratoire de Magnétisme Louis Néel C.N.R.S. - Grenoble S = 10 2 to 10 6 S = 1/2 to 30

permanent magnets macroscopic micron particles Magnetic structures nanoparticles clusters atomic molecular clusters atoms S = 10 2 0 10 1 0 10 8 10 6 10 5 10 4 10 3 10 2 10 1 multi - domains single - domains spins 1 mm 20 nm 3 nm 1 nm

Magnetization reversal in magnetic structures permanent magnets macroscopic micron particles nanoparticles clusters atomic molecular clusters atoms S = 10 2 0 10 1 0 10 8 10 6 10 5 10 4 10 3 10 2 10 1 multi - domains nucleation, propagation and annihilation of domain walls single - domains uniform rotation, curling, etc. spins quantum tunneling, interference, coherence 1 1 1 M / M S 0 M / M S 0 M / M S 0 Fe 8 1 K 0.7K 0.1K -1-1 -1-40 -20 0 20 40 µ 0 H(mT) -100 0 100 µ 0 H(mT) -1 0 1 µ 0 H(T)

Magnetization reversal in magnetic structures permanent magnets macroscopic micron particles nanoparticles clusters atomic molecular clusters atoms S = 10 2 0 10 1 0 10 8 10 6 10 5 10 4 10 3 10 2 10 1 multi - domains nucleation, propagation and annihilation of domain walls single - domains uniform rotation, curling, etc. spins quantum tunneling, interference, coherence Classical magnetism Micromagnetics Landau Lifshitz Gilbert equation Quantum magnetism Schrödinger equation Operator formalism Path intergrals ab-initio calulations etc.

Interactions in magnetic quantum systems (decoherence) Photons Conduction electrons Phonons Magnetic quantum system - spin - molecule - nanoparticle - spin chain Spin bath - nuclear spins - paramagnetic spins - intermolecular interactions etc.

Lis, 1980 Single-molecule magnets (SMM) Giant spins Müller, 1993 Mn 12 S = 10 V 15 S = 1/2 Ni 12 S = 12 Mn 84 S 6 Fe 8 S = 10 Christou, 2004 Winpenny, 1999 Wiegart, 1984

Micro-SQUID magnetometry particle stray field B 1 µm fabricated by electron beam lithography (D. Mailly, LPN, Marcoussis - Paris) sensitivity : 10-4 Fo 10 2-10 3 µ B i.e. (2 nm) 3 of Co 10-18 - 10-17 emu Josephson junctions A. Benoit, CRTBT, 1989

Roadmap of the micro-squid technique

Crystal of SMMs

crystal Micro-SQUID array B crystal size > few µm 10-12 to 10-17 emu temperature 0.03-7 K field < 1.4 T and < 20 T/s rotation of field transverse field several SQUIDs at different positions 50 µm

Micro-magnetometry µ-hall Effect µ-squid B sample B sample 1 µm Hall bars 1 to 10 µm Jospheson Junctions Based on Lorentz Force Measures magnetic field V H =!I ne M Large applied in-plane magnetic fields (>20 T) Broad temperature range Single magnetic particles Ultimate sensitivity ~10 2 µ B 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

Outline I. A simple tunnel picture (Mn 12, Fe 8 ) Giant spin model Landau Zener tunneling Berry phase II. Coupling with environment

Mn 12 -tbuac [Mn 12 O 12 (O 2 CCH 2 Bu t ) 16 (CH 3 OH) 4 ].CH 3 OH Mn III : s = 2 Mn IV : s = 3/2

Mn 12 -Ac Comparison between Mn 12 -tbuac [Mn 12 O 12 (O 2 CCH 3 ) 16 (H 2 O) 4 ].2CH 3 CO 2 H.4H 2 O [Mn 12 O 12 (O 2 CCH 2 Bu t ) 16 (CH 3 OH) 4 ].CH 3 OH not aligned well aligned!! larger unit cell less disorder

Giant spin approximation (Mn 12 ) S = 10 Mn III : s = 2 Mn IV : s = 3/2

Giant spin approximation Example: Mn12-ac Hilbert space: (2x2 + 1) 8 (2x3/2 + 1) 4 10 8 Mn 3+ 160 Mn 4+ J 1 J 2 J 4 J 3 J 1-215 K J 2 J 3-86 K J 4 < 43 K Energy(K) 120 80 40 0 30 K S = 10 D = 0.63 K S = 9 D' = 0.57 K -10-8 -6-4 -2 0 2 4 6 8 10 quantum number M I. Tupitsyn (2000)

Spin Hamiltonian: Giant spin model " = #D S z 2 # B S z 4 +" tr # gµ B r S r H (2S + 1) energy states: M = -S, -S+1,, S Energy levels: Zeeman diagram Energy H = 0 energy 8-10! -10-5 0 5 10 quantum number M -10 magnetic field 8 with S = 10, D = 0.56 K, B = 1.5 mk

Tunneling probability at an avoided level crossing Landau-Zener model (1932) S, m' > S, m > energy! 1 P 1 - P $ P =1" exp "c #2 ' & ) %& dh /dt() S, m > magnetic field S, m' > c = " 2h gµ B m # m' µ 0 L. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C. Zener, Proc. R. Soc. London, Ser. A 137, 696, (1932); E.C.G. Stückelberg, Helv. Phys. Acta 5, 369 (1932); S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 (1995); V.V. Dobrovitski and A.K. Zvezdin, Euro. Phys. Lett. 38, 377 (1997); L. Gunther, Euro. Phys. Lett. 39, 1 (1997); G.Rose and P.C.E. Stamp, Low Temp. Phys. 113, 1153 (1999); M. Leuenberger and D. Loss, Phys. Rev. B 61, 12200 (2000); M. Thorwart, M. Grifoni, and P. Hänggi, Phys. Rev. Lett. 85, 860 (2000);

0.1 K Application of Landau-Zener tunneling Mn 12 S = 10 1 2 3 4-7 5-8 -9-10 10 9 8 6 7 " = #D S z 2 # B S z 4 +" tr # gµ B r S r H with D = 0.56 K, B = 1.5 mk

Interactions in magnetic quantum systems (decoherence) Photons Conduction electrons Phonons Magnetic quantum system - spin - molecule - nanoparticle - chain Spin bath - nuclear spins - paramagnetic spins - intermolecular interactions etc.

Landau-Zener tunneling & temperature Mn 12 S = 10 1 2 3 4-7 5-8 -9-10 10 9 8 6 7 " = #D S z 2 # B S z 4 +" tr # gµ B r S r H with D = 0.56 K, B = 1.5 mk

Field derivative of M(H) at different temperatures Mn 12 -tbuac W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Temperature dependence of the peak positions of dm/dh Mn 12 -tbuac 5 4 (10:0) (10:1) (10:2) (10:3) (9:0) (9:1) (9:2) (9:3) (8:0) (8:1) (8:2) (8:3) (7:0) (7:1) 2 mt/s µ 0 H z (T) 3 2 (7:2) (7:3) (6:0) (7:4) (6:1) (6:2) (6:3) (6:4) (5:3) (5:4) (5:5) (4:3) (4:4) (4:5) 1 (2:4) (2:5) (2:6) (3:4) (3:5) (3:6) (1:5) (1:6) (1:7) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 T (K) W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Mn 12 -Ac Comparison between Mn 12 -tbuac I. Chiorescu, B. Barbara, et al., Phys. Rev. Lett. 85, 4807 (2000) W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Mn 12 -Ac Comparison between Mn 12 -tbuac A. D. Kent, et al., EPL 49, 521 (2000) and J. Appl. Phys. 87, 5493 (2000) W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Mn 12 -Ac Comparison between Mn 12 -tbuac K. M. Mertes, M. P. Sarachik, et al. Phys. Rev. B 65, 212401 (2002) W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437

Mn 12 -Ac Comparison between Mn 12 -tbuac sharp crossover L. Bokacheva, A. D. Kent, et al., PRL 85, 4803 (2000) smooth crossover I. Chiorescu, B. Barbara, et al., PRL 85, 4807 (2000) W. Wernsdorfer, M. Murugesu, G. Christou, cond-mat/0508437 sharp crossover in agreement with theory E. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett. 79, 004469 (1997)

M/M S 1 0.5 0-0.5-1 40 mk v=140 mt/s v=70 mt/s v=14 mt/s v=2.8 mt/s 0-1 -0.5 0 0.5 1 µ 0 H(T) Application of Landau-Zener tunneling Fe 8 S = 10-10 Energy (K) -20-30 -7 7-8 8-9 9-10 -40-1 -0.5 0 0.5 1 µ 0 H z (T) 10! = "D S 2 z + E S 2 2 r ( x " S y ) + gµ B S H r with S = 10, D = 0.27 K, E = 0.046K A.-L. Barra et al. EPL (1996)

Giant spin Hamiltonian of Fe 8! = "D S 2 z + E S 2 2 r ( x " S ) y + gµ B S H r easy axis Z S x H x + S y H y + S z H z j H trans Y X hard axis easy plan YZ

Hysteresis loops at different transverse fields 1-0.9-0.92 H t r a n s = 0 T M/M S 0.5 0-0.5-1 -0.94-0.96-0.98-1 d H z /dt = 1.4 mt/s -0.04-0.02 0 0.02 0.04 dh z /dt = 14 mt/s 0.056 T 0.112 T 0.196 T H trans = 0.196 T 0.112 T 0.056 T 0.000 T -0.2 0 0.2 0.4 0.6 0.8 µ 0 H z (T)

Quantum phase interference (Berry phase) in single-molecule magnets Tunnel splitting!(10-7 K) 10!! 90!! 50!! 20 easy axis Z!! 7 1 j H trans!! 0 X M = -10 10 hard axis 0.1 0 0.2 0.4 0.6 0.8 1 Transverse field (T) 1.2 1.4 easy plan YZ W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999) Y

Transverse field dependence of tunnel splitting (operator formalism) Tunnel splitting!(10-7 K) 10 1! " 90! " 50! " 20 M = -10 -> 10! " 7! " 0 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Magnetic transverse field (T) Tunnel spitting!(10-7 K) 10 1 0.1! = 90 50 30 20 10 0 0.2 0.4 0.6 0.8 1 1.2 Magnetic tranverse field (T) 5 0 W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999)

Path integrals (Feynman) Path-integral partition function: where L E is the Euclidean magnetic Lagrangian related to the real-time Lagrangian L through L E = - L (t -it ) extremal trajectories that minimize the Euclidian action, at T = 0 Z A destructive interference occures whenever the shaded area is kp/s, for odd k. X! H B Y A. Garg, Europhys. Lett. 22, 205 (1993)

Transverse field dependence of tunnel splitting (path integrals formalism) Z 10 7 A 10 5! = 90 50 30! Y! (10-8 K) 1000 10 20 10 5 X H B 0.1 0.001 0 0.5 1 1.5 2 2.5 3 µ 0 H t r a n s ( T )! = "D S 2 z + E S 2 2 r ( x " S ) y + gµ B S H r 0 A. Garg, Europhys. Lett. 22, 205 (1993)

Quantum phase interference and spin parity in Mn 12 [Mn 12 ] -e S = 19/2 [Mn 12 ] -2e S = 10 1.9 K 10-2 10-1 1.8 K 1.2 K!M/M s 10-2 1.7 K 1.6 K!M/M s 10-3 1.1 K 10-3 1.5 K 10-4 1.0 K 10-4 -1.2-0.8-0.4 0 0.4 0.8 1.2 µ 0 H (T) 10-5 -1.2-0.8-0.4 0 0.4 0.8 1.2 µ 0 H (T) W. Wernsdorfer, N. E. Chakov, G. Christou, PRL 95, 037203 (2005)

2004 Spin ground states of Mn based SMMs 25 20 2004 Mn 25 S 15 Mn 18 10 Mn 4 Mn 12 5 Mn 2 Mn 9 Mn30 Mn 70 Mn 84 0 1 2 3 4 5 7 10 20 30 50 100 number of Mn-ions

Anisotropy barriers of Mn based SMMs 2004 70 60 50 2004 Mn 12!E (K) 40 30 20 Mn 4 Mn 9 Mn 18 Mn 70 10 0 Mn 2 Mn Mn 84 30 Mn 25 1 2 3 4 5 7 10 20 30 50 100 number of Mn-ions

Quantum Tunneling of Magnetization in Lanthanide Single-Molecule Magnets Bis(phthalocyaninato)terbium Naoto Ishikawa, Department of Applied Chemistry, Chuo University, Tokyo 0.5 nm E (K) 600 K 1.5 nm N.Ishikawa, et al., J.Phys.Chem. A 106, 9543 (2002) J. Am.Chem.Soc. 125, 8694 (2003) Inorg.Chem. 42, 2440 (2003) J.Phys.Chem. A 107, 9543 (2003) J. Phys.Chem.B 108, 11265 (2004) H (T)

Quantum Tunneling of Magnetization in Lanthanide Single-Molecule Magnets Bis(phthalocyaninato)terbium Naoto Ishikawa, Department of Applied Chemistry, Chuo University, Tokyo 1 0.5 0.14 T/s 2% Tb M/M s 0 E (K) 600 K -0.5-1 0.04 K 2.0 K 7 K -1.2-0.8-0.4 0 0.4 0.8 1.2 µ 0 H (T) H (T) N. Ishikawa, M. Sugita, W. Wernsdorfer, Angew. Chem. Int. Ed. 44,2 (2005) N. Ishikawa, M. Sugita, W. Wernsdorfer, J. Am. Chem. Soc. 127, 3650 (2005)

H = Zeeman + LF term + A hf J I + P quad {I z 2 + (1/3)I(I+1)} M/M s - 643.8-644.0-644.2-644.4-644.6 1 0.5 0-60 - 40-20 0 20 40 60 0.04 K 2% Tb A hf =0.0173cm -1 P quad =0.010cm - 1 :avoided crossing occurs by off-diagonal LF term :avoided crossing occurs by transverse field Others :avoided crossing does not occur by either LF term nor transverse field -0.5 0.280 T/s 0.140 T/s 0.070 T/s 0.035 T/s 0.017 T/s 0.008 T/s 0.004 T/s 0.002 T/s 0.001 T/s -1-0.06-0.04-0.02 0 0.02 0.04 0.06 µ 0 H (T) N. Ishikawa, M. Sugita, W. Wernsdorfer, Angew. Chem. Int. Ed. 44,2 (2005) N. Ishikawa, M. Sugita, W. Wernsdorfer, J. Am. Chem. Soc. 127, 3650 (2005)

Exchange-biased quantum tunnelling in a supramolecular dimer of single-molecule magnets 9/2-9/2-9/2 J J 9/2 W. Wernsdorfer, N. Aliaga-Alcalde, D. N. Hendrickson & G. Christou Nature 416, 406 (28 March 2002)

Single-chain magnets (SCM) Mn 2 Ni - chain of S = 3 units R. Clérac, H. Miyasaka, M. Yamashita, and C. Coulon, J. Am. Chem. Soc.124, 12837 (2002). See talk of L. Bogani Monday, 16:15 Quantum nucleation in a single-chain magnet WW et al. cond-mat/0509309

Quantum computing in molecular magnets Michael N. Leuenberger & Daniel Loss NATURE, 410, 791 (2001) implementation of Grover's algorithm storage unit of a dynamic random access memory device. fast electron spin resonance pulses can be used to decode and read out stored numbers of up to 10 5 with access times as short as 0.1 nanoseconds.

Photon assisted tunneling Absorption of circular polarized microwaves H = 0 0 Energy M = +1 hw tunneling -10-5 0 5 10 quantum number m M = -1 hw Energy (K) -10-20 -30 PRB 68, 220407(R) (2003) -7-8 -9 h!!m = ±1-10 10-40 -1-0.5 0 0.5 1 µ 0 H z (T) 7 8 9

Photon assisted tunneling Absorption of circular polarized microwaves 115 GHz PRB 68, 220407(R) (2003)

Absorption of circular polarized microwaves 1 0.5 60 mk 115 GHz 0.007 T/s (115 GHz) PRB 68, 220407(R) (2003) M/M s 0-0.5 P/P 0 = -1-1 -0.5 0 0.5 1 µ 0 H z (T) 0 0.119 0.151 0.190 0.237 0.256 0.320 0.458

Conclusion Beginning: Mn 12 -ac L. Thomas, B. Barbara, et al., Nature (1996) T. Lis, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 36, 2042 (1980) R. Sessoli, D. Gatteschi, M. Novak, D. Hendrikson, G. Christou, et al. (1989-93) M. Novak, C. Paulsen, B. Barbara, et al. (1994) J. Friedman, M. Sarachik, et al., PRL (1996) L. Thomas, B. Barbara et al., Nature (1996)

Followed by: >250 systems (in our group) Mn, Mn 2, Mn 3, Mn 4, [Mn 4 ] 2, Mn 5, Mn 6, Mn 7, Mn 8, Mn 9, Mn 10, Mn 11, Mn 12, Mn 13, Mn 16, Mn 18, Mn 21, Mn 22, Mn 24, Mn 26, Mn 30, Mn 70, Mn 84 Fe 2, Fe 3, Fe 4, Fe 5, Fe 6, Fe 7, Fe 8, Fe 10, Fe 11, Fe 13, Fe 17/19, Fe 19, Fe 30 Ni 4, Ni 5, Ni 6, Ni 8, Ni 12, Ni 21, Ni 24 Co 4, Co 5, Co 6, Co 7, Co 10 Co 2 Gd 2, Co 2 Dy 2, Cr 12, CrNi 6, CrNi 2, CrCo 3, Fe 10 Na 2, Fe 2 Ni 3, Mn 2 Dy 2, Mn 2 Nd 2, V 15, Ho, Fe 2 Ho 2, Mn 11 Ln 4,... Only few of these molecules are SMMs!! S = 0, 1/2, 1, 3/2, 2, 5/2, 4, 9/2, 5,. 51/2

Development of molecular Spin-Electronics APS March Meeting 2004

Mesoscopic Physics 4 nm Mn 30 Mn 4 Mn 12 Mn 84 N 1 10 100 1000 Quantum world Classical world A. J.Tasiopoulos, A. Vinslava, W. Wernsdorfer, K. A.Abboud, and G. Christou, Angew. Chem. Int. Ed., 43, 2117 (2004)

Collaborations (Physics) L. Thomas PhD 1996: Mn 12 -ac F. Lionti PhD 1997: Mn 12 -ac, Fe 17/19 I. Chiorescu PhD 2000: Mn 12 -ac, V 15 R. Giraud PhD 2002: Ho 3+ C. Thirion PhD 2003: nanoparticles, GHz R. Tiron PhD 2004: [Mn 4 ] 2 S. Bahr PhD started 2005: GHz K. Petukhov post-doc 2004-5: GHz F. Balestro, E. Bonet, W. Wernsdorfer, B. Barbara, LLN, CNRS, Grenoble T. Ohm PhD 1998: Fe 8 V. Villar PhD 2001: Fe 8, chaines E. Lhotel PhD 2004: chaines V. Bouchiat, C. Paulsen, A. Benoit, CRTBT, CNRS, Grenoble L. Sorace, post-doc 2003: GHz A.-L. Barra, LCMI - CNRS, Grenoble J. Villain, CEA, Grenoble D. Mailly, LPN, CNRS, Marcoussis

Winpenny, 2003 Collaborations (Chemistry) Group of G. Christou, Dept. of Chemistry, Florida Group of R. Sessoli, D. Gatteschi, Univ. de Firenze, Italie Group of A. Cornia, Univ. de Modena, Italie Group of R.E.P. Winpenny, Univ. de Manchester, UK Group of E. Brechin, Univ. de Manchester, UK Group of T. Mallah, Orsay Group of V. Marvaud, Univ. P. et M. Curie, Paris Group of A. Müller, Univ. de Bielefeld, Germany Group of A. Powell, Univ. de Kahlsruhe, Germany Group of D. Hendrickson, Dept. of Chemistry, San Diego Group of E. Coronado, Univ. de Valence, Spain Group of D. Luneau, Univ. of Lyon, France Group of G. Royal, Univ. J. Fourier, Grenoble Group of R. Clerac & C. Coulon, Univ. Bordeaux, Pessac Group of H. Miyasaka, Tokyo Metropolitan Uni. Group of M. Verdaguer, Univ. P. et M. Curie, Paris Group of M. Julve, Univ. de Valence, Spain SMMs SCMs Mn 84 Christou, 2004