Beyond the Giant Spin Approximation: The view from EPR

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1 Beyond the Giant Spin Approximation: The view from EPR Simple is Stephen Hill, NHMFL and Florida State University At UF: Saiti Datta, Jon Lawrence, Junjie Liu, Erica Bolin better In collaboration with: Enrique del Barco and John Henderson, U. Central Florida David Hendrickson, Patrick Feng, Chris Beedle, En-Che Yang, UCSD Euan Brechin, Costas Milios, Ross Inglis, University of Edinburgh George Christou, Theocharis Stamatatos, University of Florida

2 Beyond the Giant Spin Approximation: Outline The view from EPR Simple is Simple is better The Giant Spin Approximation why/why not use it? Simple molecules; consider single-ion physics The view from EPR, with some examples: Interplay between exchange and anisotropy Origin of fourth-order (+ higher) anisotropy SMM properties as a result of anisotropic exchange Spin control - raising the barrier in SMMs Symmetry and QTM selection rules Summary and conclusions

3 The giant spin approximation Mn Mn 12 Simplest effective model: Mn(III) uniaxial anisotropy Mn(IV) 2 ˆ ( m ˆ 2 DS ) = s ( D 0) z ms E H = < E ms Oxygen Spin projection - m s Energy E 4 E 5 S = 10 E 6 E 7 S = 2 E 4 E 5 S = 3/2 E 6 E 7 "up" "down" S = (8 2) (4 3/2) E 10 S = 10 Magnetic anisotropy bistability, hysteresis E 8 E 9 E 8 E 9 E 10

4 Energy "up" Quantum effects at the nanoscale (S( = 10) E ms E 4 E 5 E 6 E 7 E 8 E 9 Spin projection - m s 2{ "up" "down" } ± 1 E 4 E 5 E 6 E 7 E 8 E 9 Break axial symmetry: "down" ˆ 2 = DS ˆ ˆ z + HT H [Ĥ T,Ŝ z ] 0 Contains Ŝ x and Ŝ y. m s not good quantum # Mixing of m s states Quantum tunneling (of m s ) through barrier E 10 ο (very small) E 10 Tunnel splitting Tunnel splitting a measure of tunneling rate

5 Why use a giant spin approximation? Mn 12 S = 10 Matrix dimension J s irrelevant (apparently)!! S = 9 Ignores ( ) higher-lying states S = 11 S = 10 Full But calculation what is for the Mn physical 12 produces origin matrix of of dimension parameters Even after major approximation: dimension is obtained 4 Multiple from exchange EPR coupling and parameters other experiments (J( s ); anisotropy (LS- coupling); particularly different oxidation those and different that cause symmetry MQT? sites.

6 To answer this.....study simpler molecules r ˆ t H = J sˆ sˆ + d sˆ + e sˆ sˆ + B g sˆ 2 ( 2 2 ) ij i j i zi i xi yi µ B i i i j i i Ni II 4 S 4 symmetry III Mn 3 Mn III R3 III Mn 6 (2S + 1) 4 = 81 (2S + 1) 3 = 125 Centrosymmetric (2S + 1) 6 = Ni 4 : E.-C. Yang et al., Inorg. Chem. 44, 3836 (2005); A. Wilson et al., PRB 74, R (2006). Mn 3 : P. Feng et al., Inorg. Chem. 46, 8126 (2008); T. Stamatatos et al., JACS 129, 9484 (2007). Mn 6 : C. Milios et al., JACS 129, (2007); S. Datta et al., DOI: /j.poly

7 To answer this.....study simpler molecules S = 4 r ˆ t H = J sˆ sˆ + d sˆ + e sˆ sˆ + B g sˆ Can also study the Zn analog doped with Ni II to obtain single-ion ion parameters 2 ( 2 2 ) ij i j i zi i xi yi µ B i i i j i i Ni II 4 Zn S 4 symmetry Zn Zn Frequency (GHz) En-Che Yang et al., Inorg. Chem. 44, 3836 (2005) Wilson et al., PRB 74, R (2006); Datta et al., PRB 76, (2007); Lawrence et al., Inorg. Chem. 47, 1965 (2008); Cao et al., PRL 100, (2008) T = 2.5 K B//c d = 5.30(5) cm 1 e = ±1.20(1) cm 1 Magnetic field (tesla) Angle-dependent single-crystal HFEPR of 2% Ni-doped Zn complex Obtain matrices and orientations for S = 1 Ni II Only 2 nd order anisotropy for S = 1

8 To answer this.....study simpler molecules S = 4 r ˆ t H = J sˆ sˆ + d sˆ + e sˆ sˆ + B g sˆ 2 ( 2 2 ) ij i j i zi i xi yi µ B i i i j i i II Ni 4 S 4 symmetry Matrix = (2S + 1) 4 = S = Magnetic field (tesla) Interplay between isotropic (exchange) and anisotropic (single- ion spin-orbit) admixes S = 3, 2, etc.. to the S = 4 state Causes S-mixing corrections to zfs ~(Ŝ 2 z ) n, i.e. 4 th, 6 th order... Energy (cm -1 ) S ~ 2, etc.. S = 3 J Wilson et al., PRB 74, R (2006); Datta et al., PRB 76, (2007); Lawrence et al., Inorg. Chem. 47, 1965 (2008); Cao et al., PRL 100, (2008). d = 4.72 cm -1 e = ±1.19 cm -1 J = 5.9 cm -1 Tilt = 15 o

9 To answer this.....study simpler molecules This Giant physics spin Hamiltonian: is responsible Ĥ S=4 = DŜ z2 + for B 40 Ôthe 40 + BQTM 44 Ô 44 + in gµ B Ni B Ŝ 4 Interplay between isotropic (exchange) and anisotropic (single- ion spin-orbit) admixes 0 S = 3, 2, etc.. 4 to the S = 4 state Causes S-mixing B = 4 10 cm 4 Energy (cm -1 ) -60 S = Magnetic field (tesla) mixing corrections to zfs ~(Ŝ 2 z ) n, i.e. 4 th, 6 th order D = cm -1 B 0 4 = cm-1 S ~ 2, etc.. S = 3 B and B 1/ J 4 4 J Wilson et al., PRB 74, R (2006); Datta et al., PRB 76, (2007); Lawrence et al., Inorg. Chem. 47, 1965 (2008); Cao et al., PRL 100, (2008). d = 4.72 cm -1 e = ±1.19 cm -1 J = 5.9 cm -1 Tilt = 15 o

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11 Same idea, different physics Zn Co 3 S 4 symmetry Zn Co Zn Zn Magnetic Field (tesla) a) y 1 y 2 y 3 b) Angle (degrees) Lawrence et al., Polyhedron 26, 2299 (2007); Yang et al., J. Appl. Phys 91, 7382 (2002). y y z x z x φ B xy x z y y 4 g x = 2.0 g y = 2.2 y B θ z g z = 7.8 g y = GHz 2 K Effective S' = 1/2 ground state with highly anisotropic g tensor Lower Kramers doublet of S = 3/2 state No single-ion ion zero-field field-splitting (no zero-field anisotropy) g = 6.6 x

12 Co 4 shows some characteristics of a SMM Cavity transmission (arb. units offset) B//c Frequency (GHz) Frequency (GHz) g ~ 7.4 Zero-field splitting g ~ Magnetic field (tesla) Magnetic field (tesla) + Magnetization hysteresis (below 1 K) J. Liu et al., Polyhedron; DOI: /j.poly Lawrence et al., Polyhedron 26, 2299 (2007); Yang et al., J. Appl. Phys 91, 7382 (2002).

13 SMM properties from anisotropic exchange Frequency (GHz) H = J Sˆ Sˆ + α ( Sˆ Sˆ + Sˆ Sˆ ) + µ B r t g Sˆ i< j i Toy model: 4 spin-1/ S T = 0 S T = 1 iz jz ix jx iy jy B i i S T = 2 Magnetic field (tesla) S T = 0 S T = 1 α = 0 1/ 2 J = 200 α = 2 θ = 58 J 1 J 2 S T = Magnetic field (tesla) g z = 7.8 g xy = 2.0 J. Liu et al., DOI: /j.poly ; R. A. Klemm et al., PRB 77, (2008).

14 Spin control: targeted ligand-imposed imposed distortion III Mn 3 S = 6 or S = 2 S = 12 or S = 4 III Mn 6 Molecule Torsion angle α / J/cm -1 S eff /K EPR /K [Mn 6 O 2 (Me-sao) 6 (O 2 CCPH 3 ) 2 (EtOH) 4 ] [Mn 6 O 2 (Et-sao) 6 (O 2 CCMe 3 ) 2 (EtOH) 4 (H 2 O) 2 ] [Mn 6 O 2 (Et-sao) 6 (O 2 CPh) 2 (EtOH) 4 (H 2 O) 2 ] [Mn 6 O 2 (Et-sao) 6 (O 2 CPhMe 2 ) 2 (EtOH) 4 (H 2 O) 2 ] 25.5, 29.7, , 36.9, , 38.2, , 39.1, / / Mn 3 : P. Feng et al., Inorg. Chem. 46, 8126 (2008); T. Stamatatos et al., JACS 129, 9484 (2007). Mn 6 : C. Milios et al., JACS 129, (2007); S. Datta et al., DOI: /j.poly

15 Spin control: raising the magnetic barrier r t H J s s d s B g s ˆ 2 = ˆ ˆ ˆ ˆ ij i j + i zi + µ B i i i j i i Projection technique (strong coupling, J >> d S ): Total uniaxial anisotropy : D tot = α 1 d 1 + α 2 d 2 + α 3 d 3 S = 6, f = 137 GHz D S=6 = 3/11 d s SMM Mn 3 D S=2 = 69/49 d s SMM FRUSTRATION! Barrier height: [ S=6 / S=2 ] = Ruiz et al. Chem. Commun., (2008) O. Waldmann Inorg. Chem., 46, (2007)

16 Spin control: raising the magnetic barrier r t H J s s d s B g s ˆ 2 = ˆ ˆ ˆ ˆ ij i j + i zi + µ B i i i j i i Projection technique (strong coupling, J >> d S ): 5 S = 12 6 J 2 J 2 J 1 J 3 J 3 4 D S=12 = 3/23 d S J 1 1 J 2 J particle case 2 J 2 J 2 Ruiz et al. Chem. Commun., (2008) O. Waldmann Inorg. Chem., 46, (2007) 5 J 1 J 3 J 3 S = J 1 1 J 2 J 2 Case #1: D S=4 = 21/25 d S for the coupling scheme: (((((S 1 +S 2 )+S 4 )+S 5 )-S 3 )-S 6 ) Case #2: D S=4 = 2967/4235 d S for the coupling scheme: (((((S 1 +S 2 )-S 3 )+S 4 )+S 5 )-S 6 ) D S=12 /D S=4 = ; 12 / 4 =

17 Spin control: raising the magnetic barrier f = 285 GHz S = 4 S = 12 f = 331 GHz D S=4 = 1.27(2) cm 1 B 40 = +1.3(3) 10 6 cm 1 D S=12 /D S=4 = 0.28; 12 / 4 = 2.6 D S=12 = 0.360(5) cm 1 B 40 = 5.7(5) 10 6 cm 1

18 Spin control: the barrier definitely increases r t H J s s d s B g s ˆ 2 = ˆ ˆ ˆ ˆ ij i j + i zi + µ B i i i j i i Exact diagonalization (dimension ) 0.22 Weak exchange D 12 /D Strong exchange limit J/d S D S=12 /D S=4 = 0.28; 12 / 4 = 2.6

19 Mn 3, R3, S = 6, no solvent r ˆ t H = J sˆ sˆ + d sˆ + e sˆ sˆ + B g sˆ 2 ( 2 2 ) ij i j i zi i xi yi µ B i i i j i i solvent spectacularspectacular SMM T = 300 mk α = 2 T/min k = 3 M/M s k = 1 k = k = 0 H ~ 0.85 T H (T) Chiral SMM: [NEt 4 ] 3 [Mn 3 Zn 2 (salox) 3 O(N 3 )X 2 ] with X = Cl/Br Clear observation of spin-selection selection rules;likely due to lack of disorder Feng et al., Inorg. Chem. 47, 8610 (2008); Feng et al., Inorg. Chem. (in press) (2009).

20 Mn 3, R3, S = 6, no solvent spectacular spectacular SMM r ˆ t H = J sˆ sˆ + d sˆ + e sˆ sˆ + B g sˆ 2 ( 2 2 ) ij i j i zi i xi yi µ B i i i j i i m S = 6, k =2 k = m S 6,3 E/k B (K) k =1 k =0 (S +6 + S -6 ) H L (T) -6 +4? s = 2 d = 4.2 K J = 4.88 K g = 2 e = 0.9 K θ = 8.5 o Tilting JT axes + dipolar fields explains all behavior Chirality B 43 = S z (S +3 + S -3 )

21 Summary and conclusions Simple is Simple low-nuclearity clusters provide remarkable insights into the physics of SMMs better The interplay between exchange and anisotropy causes spin state mixing which, in turn, gives rise to terms in the Giant Spin Hamiltonian that do not otherwise arise naturally It is this physics that is behind much of the lowtemperature quantum dynamics in SMMs

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