Exploring Magnetic Molecules with Classical Spin Dynamics Methods

Transcription

1 Exploring Magnetic Molecules with Classical Spin Dynamics Methods 362. Wilhelm und Else Heraeus-Seminar, Bad Honnef Advances and Prospects in Molecular Magnetism Christian Schröder University of Applied Sciences, Bielefeld Ames Laboratory, Iowa State University, Ames, Iowa, USA 1

2 No success without Collaborators! Synthesis: Achim Müller, Thorsten Glaser (U. Bielefeld) Lee Cronin (U. Glasgow) Paul Kögerler (Ameslab) Richard Winpenny (U. Manchester) Theory: Marshall Luban (Ameslab, ISU), Jürgen Schnack, Heinz-Jürgen Schmidt (U. Osnabrück), Experiment: Hiroyuki Nojiri (Pulsed Magnetic Field Facility, ESR) (Okayama U.) Ferdinando Borsa (NMR), Stephen Nagler (Neutrons) (Ameslab, Los Alamos, Oak Ridge) Computer Science: Dave Turner (Scalable Computer Lab, Ameslab) Glenn Luecke (ISU Mathematics) Carl Chang (ISU Computer Science) Contents Classical spin dynamics in 21 seconds Our tool box Recent successes The fancy molecule: {Mo 72 Fe 30 } Discovering novel anomalous magnetic features in {Mo 72 Fe 30 } Exciting physical properties of new molecular structures -or- physicist s wishful thinking: How chemists could make us happy Summary & conclusion 2

3 Classical Spin Dynamics in 21 Seconds Essentially, there are two families of classical spin dynamics methods in use Monte Carlo methods Heat bath coupling methods (and combinations of both) How do they differ? Looking from a bird s perspective Monte Carlo methods: Randomly change the configuration of a system and accept of reject it according to a given probability function. Do this many times! On average this leads to thermal equilibrium properties. Heat bath coupling methods: Couple additional degrees of freedom (which imitate the heat bath!) to the original equations of motion. Integrate these modified equations and sample the time evolution. On average this leads to thermal equilibrium properties (if the system is ergodic!). Our Tool Box Heat bath coupling vs. Monte Carlo Both approaches have a different flavor! The first approach preserves the dynamics (i.e. by modifying the equations of motion to include heat bath interaction) The second one does (very fast!) state space sampling Both approaches provide very powerful tools to calculate magnetic properties for any structure, any system size and any purpose! (however it depends on what you are ready to pay! ) 3

4 Classical Spin Dynamics in 21 Seconds Heat bath coupling methods Stochastic Methods = red arrow = brown arrow = blue arrow = yellow arrow = green arrow where we use Heisenberg exchange interaction between moments Our Tool Box What do we calculate? Classical ground state properties Experimentally accessible thermodynamic properties Magnetization Magnetic Susceptibility Magnetic contribution to the specific heat Time dependent equilibrium spin correlation functions Calculation of NMR relaxation times T -1 1 Neutron scattering cross sections Time dependent non-equilibrium properties e.g. M(B,T) where B=B(t)! 4

5 Classical Ground State Properties Simple spin systems E.g. three spins, coupled antiferromagnetically at very low temperatures Magnetic Frustration! relative angle [deg.] Energy [arb. units] 3.0x x x x x x10 5 2x10 5 3x10 5 4x10 5 5x10 5 6x x10 5 2x10 5 3x10 5 4x10 5 5x10 5 6x10 5 time steps [arb. units] Classical Ground State Properties Ground state of the classical antiferromagnetic dodecahedron angles between nearest neighbor spins * H.-J. Schmidt, M. Luban, J. Phys. A: Math. Gen. 36, (2003) time [arb. units] * 5

6 Classical Ground State Properties Ground state of the classical antiferromagnetic icosahedron angles between nearest neighbor spins time [arb. units] * * H.-J. Schmidt, M. Luban, J. Phys. A: Math. Gen. 36, (2003) Classical Dynamic Properties Modified Spin Waves and Solitons in Classical Spin Rings (H.-J. Schmidt, C. Schröder, M. Luban, to be published) 6

7 Static Thermal Properties The fancy molecule {Mo 72 Fe 30 }! (sloppily just called Fe 30 )* 30 paramagnetic Fe 3+ ions (S = 5/2) embedded on the vertices of an icosidodecahedron Hilbert space dim. 1/3 of Avogadro s number! *A. Müller et al., Angew. Chem (figures by P. Kögerler) Static Thermal Properties to be continued T [K] A. Müller, M. Luban, C. Schröder, R. Modler, P. Kögerler, M. Axenovich, J. Schnack, P. Canfield, S. Bud ko, N. Harrison, Classical and Quantum Magnetism in Giant Keplerate-type Magnetic Molecules, ChemPhysChem, Vol. 2, pp. 517,

8 A Recent Discovery What happens if we put a classical model system for {Mo 72 Fe 30 } into a slowly up-ramping magnetic field B at T 0 and sample the magnetization M(B)? B B sat = 17.7 Tesla Magnetic Field B [Tesla] O.k.! Now, let s do some basic algebra: What would dm/db look like? A Recent Discovery Surprise! But this is just theory, isn t it? 8

9 A Recent Discovery C. Schröder et al., Phys. Rev. Lett. 94 (2005) A Recent Discovery So, what s the physical origin of that minimum??? There is no analytical way to derive it for the icosidodecahedron! Why? The effect is only visible at T > 0 and nobody can calculate the classical or quantum partition function for such a system! But we can understand the dip by looking at small building blocks: One can decompose the icosidodecahedron into 20 triangles with the following properties Each bond between nearest neighbor spins belongs to exactly one triangle Each spin is at the vertex of exactly two triangles O.k., back to the roots! Let s have a look at the magnetic properties of a triangle 9

10 A Recent Discovery Aha! But again, what s the physical origin of that minimum??? There are 2 effects which in combination give a clear explanation of the minimum within a classical framework! A Recent Discovery Question: What do you expect for dm/db at B sat /3??? B sat /3 If additional configurations become accessible, fluctuations should be enhanced and so dm/db! 1. effect: For T 0, in the immediate vicinity of B sat /3 a family of collinear up-updown spin configurations becomes energetically competitive with a degenerate continuous family of spin configurations (of umbrella type) of lowest energy. 10

11 A Recent Discovery J>J B sat /3 A Recent Discovery T=0 11

12 A Recent Discovery T>0 Question: What is dm/db for the plateau? 2. effect: The family of up-up-down configurations behaves magnetically stiff due to geometrical constraints, i.e. for T 0 it shows dm z /db 0! A Recent Discovery The combination of both effects causes the minimum! The up-up-down configurations contribute to the total dm tot /db with dm uud /db 0 and hence dm tot /db is reduced where M has its plateau! Now, one can go to the limit of J/J 1: The plateau shrinks to a point at B sat /3, i.e. a minimum in the vicinity of B sat /3 occurs! B 12

13 A Recent Discovery In fact, the triangle is the basic building block for a class of structures within the family of the Archimedian Polyhedra* Octahedron Cuboctahedron Icosidodecathedron *Polyhedra with regular polygon faces of identical or different type. [source: Steven Dutch, U. Wisconsin] A Recent Discovery C. Schröder et al., Phys. Rev. Lett. 94 (2005)

14 A Recent Discovery Moreover... The quantum analog for the classical octahedron and cuboctahedron also show a minimum at B sat /3! Quantum cuboctahedron s=1 C. Schröder et al., Phys. Rev. Lett. 94 (2005) Yet Another Exciting Structure! Now, what about the magnetic properties of other frustrated polytopes? Let s put an isotropic AF icosahedron* into a magnetic field: B *which is a Platonic Solid (i.e. polyhedra with regular polygon faces of identical type only) 14

15 Yet Another Exciting Structure! Surprise! For T=0 the system shows signs of a 1 st order phase transition! Yet Another Exciting Structure! By simulation we find different spin configuration families! 15

16 Yet Another Exciting Structure! The T=0 properties can be obtained by exact analytical treatment! There exists a family of possible ground states ( 4-θ-family ) which provides a local minimum of energy E 1 (M) for 0 M A different family of possible ground states ( decagon family ) provides a local minimum of energy E 2 (M) for M 12 but is of lower energy than the 4-θfamily for M = M 0 > The absolute minimum E(M) is the minimum of both curves and a non-convex function which translates to a jump in the magnetization M=M 2 -M 1 at B c (slope of the tangent) Yet Another Exciting Structure! The total energies of the phases, E 1 (M) BM 1 and E 2 (M) BM 2, are equal for B=B c. According to the Ehrenfest classification* the phase transition is of 1 st order! An interesting (high field) phase is born The decagon family *According to Ehrenfest s classification scheme, a first order transition is characterized by a discontinuous first derivative of the free energy. At T=0 such transitions can happen in a finite system, i.e., without thermodynamic limit. For quantum systems, such transitions are known as quantum phase transitions. 16

17 Yet Another Exciting Structure! Surprise! The decagon family is metastable for T=0! Yet Another Exciting Structure! Would finite temperature destroy the metastability??? up cycle 17

18 Yet Another Exciting Structure! NO! The system remains in the metastable phase! up & down cycle Yet Another Exciting Structure! By numerical simulation one can determine the lifetime of the metastable phase at finite temperatures Survival probability P s (t) for the metastable decagon phase subject to an external field B/B sat =0.27 for temperatures k B T/J = 0.025, 0.015, 0.005, (left to right) C. Schröder et al., Phys. Rev. Lett. 94 (2005)

19 Yet Another Exciting Structure! Moreover... The quantum analog shows the transition as well! T=0 M=2 C. Schröder et al., Phys. Rev. Lett. 94 (2005) Yet Another Exciting Structure! Conclusion: We found a natural born and robust magnetic switch! H H crit =

20 Yet Another Exciting Structure! Surprise! (at least for a physicist) There is already a magnetic molecule on the road to an icosahedron! The (magnetically tridiminished) icosahedron {Fe9} E. I. Tolis, C. Schröder, J. Raftery, G. Rajaraman, G. A. Timco, H. Nojiri, L. P. Engelhardt, R. E. P. Winpenny, and M. Luban, (2005) submitted to Angew. Chem. A bold perspective... 20

21 Summary Experiments on {Mo 72 Fe 30 } show an anomalous feature in dm/db in the vicinity of B sat /3. By using classical spin dynamics we find that field-induced competitive spin configurations lead to such an effect. The same effect would be detectable in antiferromagnetic molecules with an octahedral and cuboctahedral structure*. C. Schröder, H. Nojiri, J. Schnack, P. Hage, M. Luban, P. Kögerler, Competing Spin Phases in Geometrically Frustrated Magnetic Molecules, Phys. Rev. Lett. 94 (2005) For the simple icosahedron ( pure Heisenberg model of antiferromagnetically coupled classical spins with only one exchange constant!) we discovered a first order metamagnetic phase transition revealing metastability and hysteresis! C. Schröder, H.-J. Schmidt, J. Schnack, M. Luban, Metamagnetic phase transition of the antiferromagnetic Heisenberg icosahedron, Phys. Rev. Lett. 94 (2005) *Theoretical studies of the classical Heisenberg antiferromagnet on the kagome lattice show that dm/db has a pronounced minimum at one-third of B sat (see M. E. Zhitomirsky, Phys. Rev. Lett. 88, (2002)) Conclusion Don t underestimate the value of classical spin dynamics methods! Numerical simulations based on classical spin dynamics serve as an excellent tool to supplement exact and approximate quantum methods. Moreover, classical numerical simulations allow us to explore magnetic molecules very efficiently which has lead to the discovery of a variety of new and surprising physical phenomena. 21

22 Thank you for your attention! A theorist s dream of a chemist s laboratory! High ground state degeneracy Signs of extreme Frustration in {Mo 72 Fe 30 } 22

23 The Primary Suspect II Layered {Mo 72 Fe 30 }* {Mo 72 Fe 30 } balls arranged on a square lattice ball-to-ball distance shrinks during synthesis process Formation of additional oxygen bridges which connect each ball with its 4 nearest neighbors. Susceptibility measurements suggest a strong antiferromagnetic ball-toball interaction! But how strong??? *A. Müller et al., Solid State Sciences, Angew. Chem (figure by P. Kögerler) Determination of Static Thermal Properties* *Chr. Schröder et al., to be published 23

24 Yet Another Exciting Structure! The system does NOT follow the lowest energy path on the down cycle! Yet Another Exciting Structure! How about other ways to destabilize the system? Random variations of the exchange parameter J up to +/- 10% 24