The role of anharmonic phonons in under-barrier spin relaxation of single molecule magnets

Abstract

The use of single molecule magnets in mainstream electronics requires their magnetic moment to be stable over long times. One can achieve such a goal by designing compounds with spin-reversal barriers exceeding room temperature, namely with large uniaxial anisotropies. Such strategy, however, has been defeated by several recent experiments demonstrating under-barrier relaxation at high temperature, a behaviour today unexplained. Here we propose spin-phonon coupling to be responsible for such anomaly. With a combination of electronic structure theory and master equations we show that, in the presence of phonon dissipation, the relevant energy scale for the spin relaxation is given by the lower-lying phonon modes interacting with the local spins. These open a channel for spin reversal at energies lower than that set by the magnetic anisotropy, producing fast under-barrier spin relaxation. Our findings rationalize a significant body of experimental work and suggest a possible strategy for engineering room temperature single molecule magnets.

Figure 1. Phonon-induced spin relaxation for an S=1 system.

(a) Displays a Lorentzian and a Dirac-like phonons' density of states. The spin energy barrier profile is pictured in b, where the ground state of energy E₀ is separated from the spin-flip state of energy E₁ by a barrier U₀=E₂−E₀. The spin relaxation occurs by exciting the spin system to the state of energy E₂ via absorption of a phonon. In the standard Orbach's process, the phonon is resonant with the spin excitation energy and its spectral function, A(E), is a δ-function (black bar in a). In contrast, when one considers a finite phonon lifetime (red curve in a), the phonon does not need to be resonant with the spin levels. In c, we report the logarithm of the relaxation time τ against the temperature T scaled by the excitation energy U₀/k_B. The inset reports the qualitative behaviour of the phonon linewidth, Δ, as function of the temperature, where T* represents the temperature above which the anharmonic effects start to be important. The black symbols describe the Arrhenius behaviour expected from the standard Orbach process, while the solid lines represent the expected behaviour for anharmonic crystals in three different regimes.

Figure 2. The S=2 [(tpa^Ph)Fe]⁻ SMM.

In a, we show the optimized molecular structure and in b the corresponding energy diagram. Atoms colour code: Fe=pink, N=blue, C=green and H=white. The red arrow lying along the pseudo C₃ molecular symmetry axis shows the magnetization easy axis direction. The energy diagram presents five spin states and it has been calculated with the molecule orientated with its easy axis along the external magnetic field. The first four states are only slightly non-degenerate (∼0.5–1 cm⁻¹) and their energy difference could not be appreciated in the figure. The green arrow represents the direct relaxation pathway and the blue arrow represents the Orbach's relaxation mechanism.

Figure 3. [(tpa^Ph)Fe]⁻ ab initio spin dynamics results.

Calculated temperature dependence of the relaxation time, τ. Black dots are for simulations where all the relaxation processes are considered. The green triangles and the green line represent the experimental results as taken from ref. . The blue and red lines represent calculations performed, respectively, by considering only transitions between the quasi-degenerate doublet, and by completely neglecting them (that is, by considering only relaxation processes activated through the excited states). Arrhenius's fits performed for the T<5 K and for 5 K<T<10 K cases return effective barriers of 19.7 cm⁻¹ and 55.7 cm⁻¹, respectively. In the inset, we show the relaxation time in the high-T range calculated using only a limited number of phonon modes when all the relaxation processes are included. The red line with squares describes the simulation done with only five modes, the blue line with diamonds corresponds to the simulation done with 15 modes and the black line with dots corresponds to the full-phonon spectra case.