Quantum Tunneling and Magnetization Dynamics Quantum Tunneling and - PowerPoint PPT Presentation

The European School on Magnetism Targoviste (Romania) August 22rd – September 2nd, 2011 Quantum Tunneling and Magnetization Dynamics Quantum Tunneling and Magnetization Dynamics in Low Dimensional Systems in Low Dimensional Systems Andrea CORNIA Department of Chemistry and INSTM University of Modena and Reggio Emilia via G. Campi 183, I– 41100 MODENA (Italy) Website: www.corniagroup.unimore.it E–mail: acornia@unimore.it 1

The European School on Magnetism Targoviste (Romania) August 22rd – September 2nd, 2011 Mr. k B T 2

Outline • Quantum spins and magnetic anisotropy • Boosting up molecular spin: from individual ions to high- spin clusters • Slow magnetic relaxation in high-spin clusters: thermal activation vs. quantum tunneling effects • Back to single ions: rare-earth complexes • Glauber dynamics: a glance at Single-Chain Magnets • Summary 3

The essence of quantum spins z z z z z  2, -1   2, -2  x x x x x y y y y y  2, +2   2, +1   2, 0  EF = eigenfunction; EV = eigenvalue 4

A key ingredient: magnetic anisotropy • A perfectly isolated electronic spin would show no preference for specific directions is space and its response to external perturbations would be perfectly isotropic* • SPIN ORBIT COUPLING (SOC) makes the spin sensitive to the environment and to molecular structure • In a spherical environment, even in the presence of SOC the spin remains isotropic and the  S , M S  states are exactly isoenergetic • A non-spherical environment lifts (partially or totally) the degeneracy of the  S , M S  states (ZERO FIELD SPLITTING, ZFS) z z x x y M S = ± 2, ± 1, 0 y spherical non-spherical *dipolar interactions may generate anisotropy in multispin systems 5

The D D tensor • Being S an angular momentum, its components change sign upon time reversal • The Hamiltonian must be invariant upon time reversal: terms describing ZFS can contain only even powers of spin components ( S 2 , S x S y , S 4 , etc.) z z • Usually, the leading terms are 2 ° powers of spin components (“second-order” terms); they are described by the D tensor, which is a real symmetric traceless 3x3 matrix z ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ  2     2  . . . H D S D S S D S S D S S D S ZFS xx x xy x y xz x z yx y x yy y  ˆ    S   D D D x   x xx xy xz ˆ ˆ ˆ ˆ ˆ ˆ       ( ) S S S D D D S   S D S x y z yx yy yz y       ˆ D D D   y S   zx zy zz z • D xx x 2 + D xy xy + D xz xz + D yx yx + D yy y 2 + ... = 1 is the equation of a general ellipsoid, which has three orthogonal principal axes 6

D and E parameters z • If the reference frame is chosen along the principal axes, the new tensor D ’ is diagonal x ˆ ˆ ˆ ˆ ˆ ˆ         2  2  2 H S D S D S D S D S ZFS xx x yy y zz z • By convention, the diagonal elements are re-defined in terms of y the so-called axial ( D ) and rhombic ( E ) ZFS parameters        0 0 3 0 0 D E D     xx       D 0 0 0 3 0   D D E   yy     0 0 2 3   0 0  D D zz to give rhombic ZFS axial ZFS parameter parameter         ˆ ˆ ˆ ˆ ˆ ˆ ˆ    2    2  2  2    2  2 3 3 2 3 [ 1 ] ( ) 1 H D E S D E S D S D S S S E S S 3 ZFS x y z z x y • For E = 0 only, the  S , M S  are EFs of the ZFS hamiltonian, with the following EVs     ]    2 [ 1 1 E M D M S S 3 ZFS S S 7

Easy-axis and hard-axis anisotropies High-spin Mn 3+ ( S = 2) High-spin Fe 3+ ( S = 5/2) with D < 0 with D >0 z z hard-axis anisotropy easy-axis anisotropy M S = ±5/2 M S = 0 (10/3) D -2 D M S = ±1 - D 6| D| 4| D| M S = ±3/2 -(2/3) D -(8/3) D 2 D M S = ±1/2 M S = ±2 2  2        [ 2 ] [ 35 12 ] E M D M E M D M ZFS S S ZFS S S TOTAL SPLITTING for half-integer S for integer S             2  ( )    2 1 1 0 E E S D S E E S D S 2 4 ZFS ZFS ZFS ZFS 8

Large metal ion clusters low pH H 2 O high pH [Fe(H 2 O) 6 ] 3+ [Fe n (OH) x (O) y (H 2 O) z ] 3n-x-2y Fe(OH) 3  carboxylates 9 R. E. P. Winpenny, Dalton Trans. 2002, 1

Large metal ion clusters [Fe 19 (OH) 14 (O) 6 (H 2 O) 12 ( metheidi ) 10 ] connecting ligands terminal ligands metheidi Iron “crusts” 10 J. C. Goodwin, et al., J. Chem. Soc. Dalton Trans. 2000, 1835

[Mn 12 O 12 (OAc) 16 (H 2 O) 4 ]·2AcOH·4H 2 O 4 || c S Mn II (OAc) 2 • 4H 2 O + KMn VII O 4 60% v/v AcOH/H 2 O [Mn 12 O 12 (OAc) 16 (H 2 O) 4 ] ·2AcOH·4H 2 O (80% ) (8Mn III + 4Mn IV ) ·2AcOH·4H 2 O Oxygen Manganese(IV) ( s = 3/2) Mn 12 acetate Carbon Manganese(III) ( s = 2) Hydrogen 11 T. Lis, Acta Crystallogr. B. 1980, 36 , 2042

[Mn 12 O 12 (OAc) 16 (H 2 O) 4 ]·2AcOH·4H 2 O Tetragonal Space Group I4 12 T. Lis, Acta Crystallogr. B. 1980, 36 , 2042

How large? Mn 84 vs. a Co nanoparticle 4.2 nm 3.0 nm 1.2 nm Co 13 A. J. Tasiopoulos, et al., Angew. Chem. Int. Ed. 2004, 43 , 2117

The physics of Mn 12 acetate in a nutshell S = 10,  M T = 55.0 emu K mol -1 limit of uncoupled spins 31.5 emu K mol -1  M T H || c T = 2 K H  c S = 10 (Giant Spin) (= 8  2-4  3/2)   20  B (Giant Magnetic Moment) Easy-axis Anisotropy 14 R. Sessoli, D. Gatteschi, A. Caneschi, M. A. Novak, Nature 1993, 365 , 141

Magnetic anisotropy in Mn 12 acetate c = z M S = 0 M S = 10 M S = ± 5 M S = ± 6 M S = ± 7 U M S = ± 8 M S = 0 M S = ± 9 M S = ± 10 2 – 110/3] ZFS ( M S ) = D [ M S M S = -10 E 2  47 cm -1 U = | D | S D = -0.47 cm -1 from EPR U / k B  68 K E = 0 15 R. Sessoli, D. Gatteschi, A. Caneschi, M. A. Novak, Nature 1993, 365 , 141

Evidence for an energy barrier from magnetization decay from AC susceptibility (1-270 Hz) H || c   U U U eff / k B = 61 K      0  eff    eff ln ln exp   0 k T k T  0 = 2.1  10 -7 s   B B Arrhenius Law  (2.1 K) = 8.7 ·10 5 s (10 d) 16 R. Sessoli, D. Gatteschi, A. Caneschi, M. A. Novak, Nature 1993, 365 , 141

Single Molecule Magnets S = 10 U M S for integer S for half-integer S             2      2  0 ( ) 1 1 U E E S D S U E E S D S 2 4 ZFS ZFS ZFS ZFS 1 2 ˆ  1      3 , 1 , 0 [ ( 0 ) ( 1 )] S V S E E  0 c  4 5 s p ZFS ZFS  s 17 D. Gatteschi, R. Sessoli, J. Villain, Molecular Nanomagnets , Oxford Univ. Press, 2006

Evidence for Quantum Tunneling dipivaloylmethane OH OH OH R S = 5 ground state H 3 L = 2-R-2-hydroxymethyl- -1,3-propanediol Iron(III) ( s = 5/2) Oxygen Fe 4 (OMe) 6 ( dpm ) 6 Carbon 18

19 A library of ligands

The breakdown of Arrhenius law  =  0 exp( U eff / k B T ) (  7 h) H U eff / k B = 15.7(2) K  0 = 3.5(5)  10 -8 s +3 +3 -3 D = -0.433(2) cm -1 +4 -3 E = 0.014(2) cm -1 +4 -4 U / k B = (| D |/ k B ) S 2 M S = +5 -4 = 16.0 K M S = +5 -5 -5 20 A. Cornia et al. , Inorg. Chim. Acta 2008, 361 , 3481

Axial S = 5 in a longitudinal field   ˆ ˆ ˆ ˆ ˆ         2 [ 1 ] 1 H H H D S S S gH S 0 3 ZFS Zee z B z      2     0  Energy [ 1 ] 1 M D M S S gHM 3 S S B S tunnel splitting D = -0.4 cm -1 g = 2  = 0  M S = 10  0 H r = n | D |/( g  B ) = 0, ±0.43 T, ±0.86 T, ... Resonant Quantum Tunnelling 21

What promotes quantum tunneling? • The occurrence of tunneling requires the presence of spin Hamiltonian terms that DO NOT COMMUTE with Ŝ z and mix  S , M S  states with different values of M S • Such terms are, for instance, rhombic anisotropy terms permitted by the molecular structure, and transverse magnetic fields arising from dipolar or hyperfine interactions and from misalignement of crystal domains   ˆ ˆ ˆ ˆ ˆ ˆ            2 2 2 [ 1 ] ( ) 1 H D S S S gH S E S S gH S 0 0 3 z B z x y B x x ˆ ˆ ˆ ˆ ˆ ˆ rhombic transverse     ( ) / 2 ( ) / 2 S S S S S S i     x y anisotropy Zeeman term ˆ ˆ ˆ ˆ 2  2    2 2 ( ) 1  M S =  2  M S =  1 S S S S  2 x y ˆ      , ( 1 ) ( 1 ) , 1 S S M S S M M S M  S S S S • These terms may act in sinergy (see below) • The EVs need to be calculated by numerical diagonalization of the representative of Ĥ on the  S , M S  basis. 22 D. Gatteschi, R. Sessoli , Angew. Chem. Int. Ed. 2003, 42 , 268