Project Description
Theoretical framework
To a first approximation, the spin Hamiltonian of nanometer scale magnetic particles and magnetic clusters in a magnetic field H is written as
H = -D Sz2 - g μB S . H + H’ + Hdis,
where
- D is the magnetic anisotropy constant,
- S is the spin of the particle/molecule,
- H’ stands for other anisotropy terms,
- Hdis represents the interaction of the spin system with the environment, and
- z refers to the easy axis direction.
The first term in this Hamiltonian generates spin levels Sz inside each wellseparated by an energy D (2Sz - 1), and in zero magnetic field the spin levels in the two wells (which are separated by an energy barrier U = D S2) are degenerate. Thesymmetry-violating terms in the Hamiltonian inducing tunneling are those associated to the transverse component of both the magnetic field and magnetic anisotropy.
Although many of the mesoscopic magnets have very large spin values and/or small magnetic anisotropy, being therefore unlikely to detect the discrete structure of the spin projection on the easy axis direction, the case of antiferromagnetic mesoscopic particles and molecular clusters is different, as the gap between the spin levels corresponding to the different Sz values, -S < Sz < S, may be of the order of 10 K in temperature units.
When applying a magnetic field, it can be chosen to be either parallel (Hpar) orperpendicular (Hperp) to the easy axis:
- The application of Hpar reduces the barrier height between the two possible orientations of the spin.
- The effect of Hperp increases the spin tunneling rate between the degenerateSz levels.
In other words, the values of temperature, Hpar, and Hperp determine the levels between which resonant spin tunneling occurs.
- At sufficiently low temperature, the populations of the excited levels are reduced to exponentially small values and for Hpar = 0 the only possible tunneling process would occur between the two lowest states Sz = S and Sz = -Swhich may be denoted respectively as the the “yes” and “not” states.
- By increasing Hperp, the rate of tunneling between the “yes” and “not” states increases as a consequence of their mixing and gives rise to thesymmetric and antisymmetric combinations.
- At Hperp = 0, tunneling between the two degenerate lowest states is suppressed and the two levels entering the discussion for the resonance experiments are those corresponding to Sz = S and Sz = S - 1 lying in the same potential well.
- Depending on the values of Hperp, the frequency of the resonant tunneling transitions between the degenerate levels of the two wells which can lead to superposition of the two spin levels may range from Hz to THz.
In other words, the attraction of working with mesoscopic magnets and very low temperature is that we have two different sets of quantum states with anenergy gap between them of
- 100 mK (in the case of quantum splitting), to
- 10 K (in the case of uniform precession of the magnetic moment),
which may be used to study quantum coherence phenomena.
The parameters entering the spin Hamiltonian of both samples may be obtained fromEPR and magnetic measurements:
- the value of the barrier height, U = D S2, may be estimated from the blocking temperature, 25Tb = D S2, and
- the anisotropy field, Ha, defined as the field where the demagnetisation curve branches from the initial magnetisation curve, is Ha = 2DS.
That is, by performing zero field cooled (ZFC) and field cooled (FC) magnetisation measurements and isothermal magnetisation vs field measurements at very low temperature, it is possible to estimate both the spin and the anisotropy term of the spin Hamiltonian.
