Zero-field quantum tunneling relaxation of the molecular spin in Fe₈ observed by ⁵⁷Fe Mössbauer spectrometry

The modern study of physical phenomena in nanomaterials allows exploring quantum effects that give rise to new macroscopical properties. Magnetism is, with no doubt, one of the fundamental properties of matter. Engineering efforts to acquire organic materials with properties that are usually associated with classical inorganic materials have long been forced during the last few years. Molecular magnets are one of the evidences of such a development. They are ensembles of magnetic ions surrounded with organic non-magnetic ligands. The spins of the metal ions are, at sufficiently low temperature, strongly coupled together by the exchange interaction that forces individual spins to align rigidly resulting in a macrospin (of the order of S = 10). The study of the spin properties, restricted to the microscopic world, is of major interest because it is in the origin of magnetic moments in metal atoms and covalent bonds in some light atoms. In many ways, molecular magnets behave like classical tiny magnets in the sense that after being magnetized by an external magnetic field they retain the magnetization for a long time. Consequently they exhibit magnetic hysteresis [1]. In that sense, these molecules follow classical laws of magnetism. However, molecular magnets show also quantum features; indeed, the hysteresis loops present steps at regular intervals of the applied magnetic field, which is a clear evidence of quantum tunneling of the magnetic moment [2–6].

The number of single molecular magnets identified and studied for the last decades has significantly increased and it is still growing with the aim of finding compounds with larger barrier heights. Among the large set of synthesized molecular magnets, Mn₁₂ and Fe₈ have been the most studied because of their relative easy preparation, large molecular spin, and large magnetic anisotropy. Fe₈ was initially prepared by Wieghardt et al. [7]. In the Fe₈ molecule, two edge-sharing FeO₆ octahedra connect two identical units of three vertex-sharing FeO₃N₃ octahedra. Its whole charge is +8, which is neutralized in the crystal by the necessary bromine anions that fill the space between the complex molecules, as well as some water molecules. As seen by Mössbauer spectrometry and other techniques, the iron atoms are in the high-spin ferric state. The antiferromagnetic coupling of the 8 $\text{Fe}^{3+}\ (S=5/2)$ cations makes them assemble, at very low temperature, in such a manner that they give rise to a high spin S = 10 molecule with an anisotropy barrier of about 27 K (see fig. 1).

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The main motivation of this work is to explore the possibility to observe quantum magnetic relaxation of the magnetization through Mössbauer spectrometry as previously evidenced by ac-susceptibility measurements [8] and by magnetization measurements [9]. In the first case, a peak appears in the real part of the ac magnetic susceptibility measured as a function of applied dc field; the quantum relaxation is even clearer as the frequency of the measurements increases and the peaks shift. The relaxation time τ is given by an Arrhenius law $\tau_0\exp[U/k_BT]$, where U is the anisotropy energy barrier and $\tau_0$ is the characteristic time of the system (for a Fe₈ compound at zero external magnetic field, $U \sim27\ \text{K}$ and $\tau_0\sim 2\cdot10^{-7}\ \text{s}$) [4]. At 2.5 K the overcoming barrier frequency, Γ, (i.e., $\tau=1/\Gamma$) is about 500 Hz giving a threshold for the measuring time to probe under-barrier magnetization dynamics (i.e., lower measuring frequencies cause that spins behave as superparamagnets [10]). Proving of magnetization tunneling through hysteresis curves is usually done at measuring times much larger than a second, implying a working temperature lower than 1.5 K.

Mössbauer spectra at low temperatures, i.e., 4.2 K, show complete splitting of the ⁵⁷Fe nuclear levels (Zeeman effect) in agreement with rather blocked Fe magnetic moments. When the temperature increases, the Mössbauer spectra consist of a progressive sum of a broken degeneracy sextet for the blocking state and a quadrupolar doublet attributed to the paramagnetic state, with small values of quadrupole splittings for high-spin ferric ions. In order to observe a relaxing state in Fe₈, it would be necessary that the relaxing time (e.g., flipping time) was similar or larger than the measuring time. The lifetime of the excited nuclear state ⁵⁷Fe establishes the characteristic probe time for the Mössbauer technique, $\tau_m\sim10^{-9}\text{-}10^{-7}\ \text{s}$. Thus spin-dynamic processes at this time scales can be analyzed. Thermally activated spin dynamics has been studied [11] together with superparamagnetic transitions and values for the Arrhenius prefactor were obtained [12]. A temperature dependence of the Mössbauer spectra in a Fe₈ cluster was studied in [12] and the transition probabilities were discussed.

As we know the time scale of the Mössbauer technique, we can set the experimental conditions that could allow observation of tunneling of magnetization in the Fe₈ molecular magnet. We need to increase the temperature to populate spin levels with large tunneling rates but keeping at the same time the temperature low enough so that the superparamagnetic spins —the spins overcoming the barrier— do not dominate the dynamics. Additionally, to compare dynamics due to tunnel transitions with dynamics due to superparamagnetic transitions we have to apply magnetic fields to bias the energy levels. These biasing fields are not expected to introduce changes in Mössbauer spectra because they are much smaller than hyperfine felds.

In a first approximation the magnetic Hamiltonian of a Fe₈ crystal is

$$\begin{equation} \mathcal{H} = - DS_z^2 + E(S_x^2-S_y^2)-g\mu_B \ \mathbf{H}\cdot\mathbf{S}, \end{equation} \tag{ 1 }$$

where the parameters of the easy, D, and hard, E, anisotropy have been extensively measured through HF-EPR [13] and INS [14]. We consider hereafter the values $D=0.27\ \text{K}$ and $E=0.029\ \text{K}$. The eigenstates of the above Hamiltonian, $\psi_m={|{m}\rangle}$ (m is their projection along the z-axis), correspond to the spin states of the system and the eigenvalues $\mathcal{E}_m=-Dm^2$ to their energies. The height of the energy barrier separating positive and negative projections along the anisotropy axis is $U=DS^2$. The energy barrier reflects the bi-stability of the system. Energies for the Hamiltonian of eq. (1), with E = 0 are degenerated and have, unless $\mathcal{E}_0$, two corresponding eigenstates, ${|{\pm m}\rangle}$ (see fig. 1). Longitudinal applied fields, h_z, break the degeneracy of the Hamiltonian of eq. (1) as the spin states with z projections m > 0 will decrease its corresponding energies $(h_z>0)$ while the ones with m < 0 will increase it. At particular values of the longitudinal field (resonant fields), the energies of some spin states, m and $m'$ with $|m|\neq|m'|$ also coincide.

The spin projections S_x,y of the spin operator S do not commute with S_z and consequently with the Hamiltonian. Thus, the understanding of the effect of either transverse fields or transverse anisotropies involve a more sophisticated mathematical analysis. The most interesting point of transverse terms is however that they break the degeneracy at resonant fields. The eigenstates of the system are no longer the states corresponding to the spin projections along the easy direction; the corresponding eigenstates are a mixing of the positive and negative spin projections (see fig. 2).

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At non-zero temperature the given number of spins in the states ${|{m}\rangle}$ follows a Boltzmann distribution and is proportional to $\exp\left[\frac{- \mathcal{E}_i}{k_BT}\right]$. At high temperatures the magnetization is thermally activated and the time the molecules remain in one state (up or down) is much shorter than the measuring times. The magnetic moments of the particles flip leading to thermal equilibrium, $\mathcal{M}\propto1/T$, behaving as a paramagnet [10]. At low temperatures, $\frac{U}{k_BT}\gg 1$ the magnetization dynamics due to overbarrier transitions can be simplified and described by equations for the number of molecules (spins) in the left and right wells,

$$\begin{equation} \dot{n}_{\pm}=\pm\Gamma_-n_+\mp\Gamma_+n_-=\Gamma (n_{\pm}^{\text{eq}}-n_{\pm}), \end{equation} \tag{ 2 }$$

where $\Gamma_\pm$ is the scape rate from the positive/negative wells and $\Gamma= \Gamma_++\Gamma_-$.

Using the fact that the escape rates are

$$\begin{equation} \begin{array}{rcl} \Gamma_-&=&\displaystyle\Gamma_0\exp\left[-\frac{U}{k_BT}\right],\\ \Gamma_+&=&\displaystyle\Gamma_0\exp\left[-\frac{U+\Delta \mathcal{E}}{k_BT}\right], \end{array} \end{equation} \tag{ 3 }$$

where U is the energy barrier and $\Delta \mathcal{E}$ the energy difference between the two wells, one gets the expression for the frequency Γ,

$$\begin{equation} \Gamma=\Gamma_0 \exp\left[-\frac{U}{k_BT}\right]\left(1+\exp\left[-\frac{\Delta \mathcal{E}}{k_BT}\right]\right). \end{equation} \tag{ 4 }$$

When the transition rate between the two wells is low compared with the measuring time, the magnetization is blocked; only transitions within one well are allowed. The fraction of paramagnetic spins can be therefore calculated as $\Gamma/t_{\text{meas}}$.

In addition to the classical thermal relaxation described above, there is tunneling between spin states. We could obtain the splitting values, $\Delta_{m,m'}$, by diagonalizing the Hamiltonian of eq. (1) or by treating the transverse anisotropy as a perturbation (see ref. [15]). For the case of zero longitudinal magnetic field where the resonances occur between levels m and $-m$ the equation yields

$$\begin{equation} \Delta_{m} = D\frac{2^{(3-2|m|)}}{[(-m-1)!]^{2}}\frac{(S-m)!}{(S+m)!} \left( \frac{a}{4D}\right)^{|m|}. \end{equation} \tag{ 5 }$$

The splitting increases in several orders of magnitude each time we climb up one level towards the top of the barrier. Larger transverse fields increase also the splitting [10].

The tunnel rate of quantum transition between two resonant levels is given by a combination of the tunnel rate $\Delta_{m,m'}/\hbar$ and the amplitude of each level $\gamma_{m}$. Recent experiments pointed out an additional inhomogeneous broadening at the resonances [16]. Considering $\gamma_m=\Gamma_0$ one gets [17]

$$\begin{equation} \Gamma_{m,m'}=\frac{\Delta_{m',m'}^2}{\Gamma_0\hbar^2}. \end{equation} \tag{ 6 }$$

Figure 2 schematically plots the symmetric states of the Hamiltonian of eq. (1) labeled with their energies, E_i and their tunnel rates $\Gamma_{m,m'}$ at zero applied field. The first six quasi-degenerated states are essentially combinations of the ${|{m}\rangle}$ and ${|{-m}\rangle}$ states, being the levels $m=6,5$ and 4 the ones with tunnel frequencies faster or equal to Mössbauer measuring times. Higher energy levels have a strong mixing of states.

Low temperatures do not populate the states $m=6, 5$ and 4 where tunnel rates are of the order of the measuring technique. On the other hand, high temperatures largely increase the fraction of superparamagnetic spins. We need to account for variations in the superparamagnetic population as a function of the longitudinal applied magnetic field. We measured the magnetization curves of a crystal of Fe₈ at two different temperatures (with a SQUID magnetometer) and is plotted in fig. 3. We also plotted in fig. 3 the computed frequency, Γ, with the same parameters we fitted the magnetization curves for a temperature of 12 K (right-hand-side axis). At this temperature, such frequency is of the order of the MHz, which represents about 10% of the total spin population in the superparamagnetic regime if measuring at a time of the order of 10⁻⁷.

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Single crystals of Fe₈ were prepared following the Wieghardt [7] procedure but using iron chloride obtained by acid attack of a ⁵⁷Fe enriched iron metal foil. Epoxy embedded ⁵⁷Fe enriched Fe₈ crystals were oriented in a 5 T magnetic field at room temperature up to the glue consolidation. The magnetic easy axis of the crystals was along the axis of the resulting solid disk. The disk was mounted in a cryostat equipped with a superconducting magnet where the external magnetic field is oriented parallel to the γ-beam and irradiated by a ⁵⁷Co source diffused into a Rh matrix moving in a constant accelerated mode. Good statistics spectra were obtained at a temperature well stabilized for several days at $12.0\ (\pm0.1)\ \text{K}$ in the presence of external fields increasing from 0 up to 1.65 T with a step of 0.11 T within a single batch of measurements. Thus, any misalignment between the field and the mean easy axis of the sample remained constant along the Mössbauer data acquisition. Measured spectra were reproducible and presented no dependence on the field history —the spin dynamics of Fe₈ behaves paramagnetically 12 K at the time scale of the varying fields.

Figure 4 compares two in-field Mössbauer spectra at 12 K with $\mu_0H = 0$ and 0.11 T. The two sextets are extremely similar: they exhibit broadened and asymmetrical lines. They have to be described by means of at least two components although a better agreement comes from using several magnetic components, which can be attributed to the different iron sites of the molecule. In addition, the intensities of the intermediate lines (2 and 5) indicate that the Fe magnetic moments are slightly canted. One may conclude that the sample contains magnetically rather ordered crystals along the easy axis. Different fitting models can be established from the superimposition of several magnetic sextets. However, it is important to emphasize that the evolution of hyperfine parameters or mean values of hyperfine parameters vs. external field prevents from a clear interpretation of relaxation phenomena.

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Consequently, we preferred to plot the difference between two consecutive spectra after normalization of the absorption area: one is depicted in fig. 4 (other spectra differences are similar). We notice here that the zero-field curve presents larger absorption at zero velocity (which is compensated by negative absorption satellites); this is indicative that the amount of flipping spins has increased. The total number of paramagnetic spins is not expected to vary significantly with the used values for the applied magnetic field (see curve of the frequency in fig. 3). We thus attribute the increase of the signal at zero velocity to the magnetization tunneling.

Subtraction of both spectra allows detection of small differences between the two curves (see inset in fig. 4), which surprisingly appear physically meaningful and just in the sense of the expected results for the designed experiment, i.e. the difference show that we have been able to detect the tunneling spins through the levels $m\ge4$ at H = 0. The small field $(\mu_0 H=0.11\ \text{T})$ forbids magnetization reversal because of the absence of aligned resonance levels. The calculus presented above describes about a 10% of the spin population being superparamagnetic whereas the fraction of spins populating higher levels where tunneling is allowed is of about 20%. Therefore, we cannot expect the relaxing area that blocks with the small field to be above 10%. The measured area is of about 2%, which agrees with the order of the expected results. The discrepancy between the calculated 10% and the measured 2% might come from the fact that a small misalignment between the applied field and the crystal's easy axis could cause some tunneling from the higher energy spin states even at applied fields that do not correspond to a resonance field. We also neglected in the calculations dipolar fields that the sample's shape could originate and eventually modify the values for the tunneling rates. The discrepancy also suggests that the description of the giant spin model might fail to fully describe the dynamics of the Fe₈ molecular magnet system at high temperatures, specially regarding the dynamics of the higher energy levels.

In conclusion we have designed and performed an experiment combined with a modeling based on the differences of Mössbauer spectra to find evidence for the tunneling of the magnetic moment in molecular magnets through Mössbauer spectrometry. We could observe this effect in the Fe₈ molecular magnet at about $T = 12\ \text{K}$ by measuring an intensity variation in Mössbauer spectra recorded under magnetic fields only separated by a 0.11 T step. Our results are conclusive in the sense that we observed a tunneling effect. However, the measured magnitude challenges the validity of the existing Giant spin model for describing the spin dynamics when high energy spin levels are populated.

FM acknowledges support from Catalan Government through COFUND-FP7. JT and FM also acknowledge support from MAT2011-23698. EM acknowledges 2014SGR1643 and MEC (ENE2012-36368).