Spin Tunneling and Inversion Symmetry E NRIQUE DEL B ARCO Department of Physics – UCF Orlando QCPS II Vancouver

E NRIQUE DEL B ARCO, C HRISTOPHER R AMSEY (UCF) S TEPHEN H ILL ( NHMFL and Physics Department, FSU – Tallahassee ) S ONALI J. S HAH, C HRISTOPHER C. B EEDLE AND D AVID N. H ENDRICKSON (Chemistry Department, UCSD – La Jolla-San Diego) P HILIP C.E. S TAMP AND I GOR T UPITSYN (PITP-Physics, UBC, Vancouver) Nature Physics 4, (2008) Spin Tunneling and Inversion Symmetry

THE MOLECULE [Mn 12 (Adea) 8 (CH 3 COO) 14 ]·7CH 3 CN Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004) /2 S=7

MAGNETIZATION - QTM T c ~0.3K T B ~0.9K S = 7 D = 0.4K T = 0.9K S = 7, D = 0.4 K 50  m wheel axis HLHL m S =

MAGNETIZATION - QTM S = 7, D = 0.4 K Ms =7Ms =7 Ms =6Ms =6 Ms =5Ms =5 H ? HLHL HTHT

[Mn 12 (Adea) 8 (CH 3 COO) 14 ]·7CH 3 CN Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004) /2 S=7 THE MOLECULE

THE MOLECULE [Mn 12 (Adea) 8 (CH 3 COO) 14 ]·7CH 3 CN d* d d d dd d d d d d d avg ~3.17Å d*~3.49Å J ~2-5 cm -1 J* <<J Foguet-Albiol, D. et al., Angew. Chem. Int. Edn 44, 897–901 (2005) Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004).

THE MOLECULE d* Rumberger et al., Inorg. Chem. 43, 6531–6533 (2004). 7/2 7/2 [Mn 12 (Adea) 8 (CH 3 COO) 14 ]·7CH 3 CN

EXCHANGE-COUPLED SPINS QUANTUM TUNNELING BTW. STATES OF DIFFERENT SPIN LENGH

QUANTUM INTERFERENCE HARD BERRY PHASE INTERFERENCE OF TWO COUPLED TUNNELING SPINS BERRY PHASE INTERFERENCE OF TWO COUPLED TUNNELING SPINS HLHL  HTHT

NEW TOPOLOGICAL EFFECT Quantum Tunneling Spin Fe 8 : Wernsdorfer & Sessoli, Science (1999) Mn 12 : del Barco et al., PRL (2003) Mn 12 -tBuAc: da Silva Neto et al., (2008) EXPERIMENTTHEORY Loss et al., PRL (1992) Von Delft et al., PRL (1992) Garg, EPL (1993) Coupled Tunneling Spins SINGLE SPIN Classical spin precession i.e. Wagh et al., PRL (1998) Pancharatnam (1956) (light interference) Berry (1984) (quantal systems) Aharanov and Anandan (1987) (generalization Hilbert space). INTERACTING SPINS Classical coupled-spins precession Sjoqvist, PRA (2000) THEORY (??) Mn 12 wheel: Ramsey et al., Nature Physics (2008) EXPERIMENT

SYMMETRY RULES ANTI-SYMMETRIC TERM NEEDED Dzyaloshinskii–Moriya interaction NOT ALLOWED ON A DIMER MODEL with INVERSION SYMMETRY

SYMMETRY RULES 7/2 7/2 Wernsdorfer, PRB (2008) a - Dimer model identically used in a Mn 6 wheel (CI) b - DM interaction used to explain results Wernsdorfer, PRL (2008) a - Dimer model used in an “identical” Mn 12 wheel b – DM interaction used to explain results Rejected by NP: See our response in arXiv: Wernsdorfer, arXiv: v1,v2,v3 a - Dimer model not valid

SYMMETRY RULES 7/2 7/2 Wernsdorfer, PRB (2008) a - Dimer model identically used in a Mn 6 wheel (CI) b - DM interaction used to explain results Wernsdorfer, PRL (2008) a - Dimer model used in an “identical” Mn 12 wheel b – DM interaction used to explain results Rejected by NP: See our response in arXiv: Wernsdorfer, arXiv: v1,v2,v3 a - Dimer model not valid Wernsdorfer-justification: 1)Disorder 2)Local DM interactions are not forbidden del Barco et al., PRL (2009) 1)Disorder 2)Local DM interactions are not forbidden

SYMMETRY RULES /2 d1d1 center of inversion middle point 7/2 middle point center of inversion D = 0

SYMMETRY RULES 7/2 middle point center of inversion D  0 parallel to z-axis (Ramsey, Nature Physics) D  0 tilted (Wernsdorfer, PRL) The Hamiltonian of the coupled half-wheels: Each half-wheel: Exchange coupling: Symmetric exchange: Antisymmetric exchange (DM interaction):

SYMMETRY RULES /2 d1d1 center of inversion middle point D *  * D  x y z

SYMMETRY RULES /2 d1d1 center of inversion middle point D  x y z D  x y z

SYMMETRY RULES H H Center of Inversion

SYMMETRY RULES /2 d1d1 center of inversion middle point D  x y z D  x y z 2 2 3/2 center of inversion middle point (d’ >J) (d,J)(d,J) D  x y z D  x y z D’ ’ ’’ x y z ’ ’’ x y z

SYMMETRY RULES The Hamiltonian of 4 coupled quarter-wheels: Each quarter-wheel: Exchange coupling: Antisymmetric exchange (DM interaction): Symmetric exchange: Center of inversion symmetry imposes: k = 1(A) is degenerate 2 2 3/2 center of inversion middle point (d’ >J) (d,J)(d,J) D  x y z D  x y z D’ ’ ’’ x y z ’ ’’ x y z

SYMMETRY RULES  

In a centro-symmetric molecule local DM-interactions MUST BE related by inversion symmetry and DO NOT BREAK THE DEGENERACY BETWEEN LEVELS OF OPPOSITIVE PARITY independently of how complex the Hamiltonian is because PARITY (good quantum number) MUST BE CONSERVED

SYMMETRY RULES DM-interactions are important in S = 1/2 systems ONLY SOURCE OF DEGENERACY BREAKING but never mix states of opposite parity in a system with inversion symmetry (Kagome lattice – weak ferromagnetism) when inversion symmetry is not present BOTH SYMMETRIC and ANTISYMMETRIC INTERACTIONS CAN BREAK DEGENERACIES E. del Barco, S. Hill and D. N. Hendrickson, Phys. Rev. Lett. in press (2009) E. del Barco et al., In preparation

/2 d1d1 center of inversion middle point D  x y z D  x y z  Dipolar fields? (Philip?)

CONCLUSIONS Quantum superposition of states with different spin length in a SMM New topological effect: Quantum phase interference of two coupled tunneling spins Local DM interactions in a centro-symmetric SMM do not break the degeneracy between states of opposite parity

Del Barco Lab Low temperature nanomagnetism Single-molecule magnets FM thin films and nanowires Nanoparticles Low temperature nanotransport Molecular spintronics Single-electron transistors Low-dimensional systems i.e. graphene, nanowires, i.e. graphene, nanowires, nanoparticles, molecules,… nanoparticles, molecules,… Physics collaborations Stephen Hill (NHMFL-FSU) Masa Ishigami, Robert Peale, Lee Chow (UCF) Agustin Camon, Fernando Luis (UZ-Spain) Javier Tejada (UB-Spain) Oliver Waldmann (U.Freiburg-Germany) Andrew Kent (NYU) XiXiang Zhang (KAUST) Eduardo Mucciolo, Michael Leuenberger (UCF) Philip Stamp, Igor Tupitsyn (UBC-Canada) Chemistry collaborations David Hendrickson (UCSD) George Christou (UF) Eugenio Coronado (UV-Spain) Florenzio Hernandez (UCF) Joel Miller (UU)

[Mn 12 (Adea) 8 (CH 3 COO) 14 ].7CH 3 CN S = 7/2 + 7/2 [Mn 12 (Edea) 8 (CH 3 CH 2 COO) 14 ] S = 7 [Mn 12 (Edea) 8 (CH 3 COO) 2 (CH 3 CH 2 COO) 12 ] S = 7 d* d d d d d d d*/d avg = 1.100d*/d avg = 1.093d*/d avg = <>SISTER MOLECULES J*/J avg >><<