Modelling the non-equilibrium low-temperature magnetic cooling effect in Mn₁₂ clusters

Abstract

Based on the theoretical framework recently developed by some of us, we predict and justify the possibility of a nonequilibrium magnetic cooling effect in the Mn₁₂ family of clusters by considering a monocrystalline sample of the prototypical single-molecule magnet Mn₁₂Ac as a representative example. In contrast to the quasi-static processes underlying the conventional magnetocaloric effect (MCE), we address a dynamic regime involving sudden magnetic field quenching. The proposed cooling mechanism is determined by the relaxation kinetics arising after a sudden change in the spin Hamiltonian that generates a nonequilibrium population distribution within the spin subsystem and therefore does not rely on the standard equilibrium entropy cycles. During the subsequent restoration of thermal equilibrium, heat is redistributed between the phonon bath and the spin degrees of freedom. Under appropriate conditions, this relaxation-driven process results in cooling of the lattice. The central result is that the strong easy-axis magnetic anisotropy associated with a significant magnetization reversal barrier, features typically considered detrimental to conventional magnetocaloric cooling, becomes advantageous in the nonequilibrium regime. These properties enhance both the magnitude of the cooling effect and the practical feasibility of the sudden-quench approach. This study therefore broadens the potential cryogenic applicability of single-molecule magnets by identifying a cooling mechanism that operates precisely in the parameter range where classical magnetocaloric approaches are least efficient.

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1. Introduction

Molecular nanomagnets and single-molecule magnets (SMMs) have garnered significant attention over the past decades, primarily due to their potential applications in high-density data storage and quantum information processing.^1–7 Among them, a famous dodecanuclear manganese cluster [Mn₁₂O₁₂(CH₃COO)₁₆(H₂O)₄], commonly referred to as Mn₁₂Ac, represents the prototypical SMM.^1,2,4–9 Numerous Mn₁₂ derivatives with the general formula [Mn₁₂O₁₂(RCOO)₁₆(H₂O)_x]Y (where Y denotes the solvent molecule) have been synthesized and structurally characterized.¹ Examples include R = CH₂CH₃, x = 3, Y = 4H₂O^10,11 and R = CH₂C(CH₃)₂, x = 4, Y = 4H₂O, as well as many other derivatives are presented in ref. 1–3.

Let us first summarize the main structural features of the Mn₁₂ family^1–3 predetermining their functional properties, which are the subject of this article. The Mn₁₂Ac cluster contains 12 manganese ions surrounded by sixteen acetate groups (Fig. 1). The core comprises a tetrahedron of four Mn^IV ions (S = 3/2) and an outer ring of eight Mn^III ions (S = 2), which are linked together by oxo groups and acetate bridging ligands. The crystal of Mn₁₂Ac has tetragonal symmetry (space group I4̄) and the dodecanuclear cluster has S₄ symmetry. The crystal structure also includes four solvate water molecules and two CH₃COOH molecules that link the Mn₁₂Ac clusters.^1,2 The eight S = 2 spins of the Mn^IV ions are coupled ferromagnetically as well as the four S = 3/2 spins of the Mn^IV ions. Antiferromagnetic exchange interactions between ions with different oxidation states lead to the formation of a ferrimagnetic structure with a total molecular spin of S = 10 in the ground state. This molecule exhibits pronounced easy axis-type magnetic anisotropy, which arises primarily from the local anisotropies caused by the Jahn–Teller distortions at the eight Mn(III) sites. The molecular easy axis of magnetization is parallel to the tetragonal c-axis of the crystal lattice and normal to the disk-like core of the molecule. This anisotropy gives rise to a bistable magnetic ground state in which M_S = +10 and M_S = −10 substates are separated by a substantial magnetization reversal barrier, which is responsible for the slow relaxation of magnetization at low temperatures. Besides pure scientific interest, Mn₁₂Ac and its derivatives are attractive due to their potential applications in spintronics, quantum computing and ultra-dense data storage devices.

Fig. 1 Molecular structure of Mn₁₂Ac viewed along the crystallographic c-axis (the S₄ axis that is the easy axis of magnetization). Atom colors: violet, Mn^III; green, Mn^IV; grey, C; and red, O. H atoms are omitted for clarity.

In addition to their relevance for information storage and quantum computing, SMMs have been intensively investigated as potential materials for cryogenic magnetic refrigeration based on the magnetocaloric effect (MCE).^12–30 The conventional MCE relies on the entropy change achieved during the isothermal demagnetization process and the temperature change occurring in the course of the adiabatic demagnetization process. Theoretical and experimental studies indicate that an ideal molecular refrigerant should possess a large spin ground state combined with minimal magnetic anisotropy, thereby maximizing the magnetic entropy change upon demagnetization.^23,25,28 An exception to this general trend is the rotational magnetocaloric effect (RMCE),^31,32 in which the magnetization and demagnetization are achieved by mechanical rotations of the sample in a constant external magnetic field, with the effect being favored by magnetic anisotropy.

Apart from the RMCE scenario, however, the magnetic anisotropy intrinsic to SMMs such as Mn₁₂Ac is generally considered a hindrance to magnetic refrigeration. The zero-field splitting (ZFS) of spin levels reduces the magnetic entropy change in the low-temperature regime and thus limits the efficiency of conventional MCE-based cooling cycles. In the present work, we adopt a different theoretical perspective that exploits precisely the feature usually regarded as detrimental, namely, strong magnetic anisotropy, to achieve magnetic cooling. We investigate the nonequilibrium thermal processes occurring in Mn₁₂Ac crystals subjected to fast (sudden) magnetic field quenching. Unlike conventional MCE, which assumes a quasi-static reversible process where the system remains close to thermodynamic equilibrium, the present approach focuses on a regime in which the magnetic field sweep rate significantly exceeds the spin–lattice relaxation rate.

The magnetothermal effect induced by fast field quenching was previously studied for 3d-metal-based mononuclear complexes in ref. 33 and 34 as well as for the exchange-coupled clusters in ref. 35 and 36. For the mononuclear complexes, it was shown that the relaxation of the spin system from a nonequilibrium state created by the sudden field change to a new equilibrium state is accompanied by a heat flow from the phonon bath to the spin subsystem, provided that the system exhibits easy-axis-type anisotropy. This energy transfer results in the cooling of the crystal lattice.

Under such conditions, the strong easy-axis-type magnetic anisotropy of Mn₁₂Ac is expected to play a dual constructive role. First, it determines the specific energy level pattern required for the cooling mechanism. Second, it ensures sufficiently long relaxation times at low temperatures, making the “sudden” quenching approximation experimentally feasible. Using the theoretical framework and purely model-based treatment recently developed by some of us,^33–36 in this article, we predict and justify the possibility of a nonequilibrium magnetic cooling effect in the Mn₁₂ family of clusters by considering a monocrystalline sample of the prototypical single-molecule magnet Mn₁₂Ac as a representative example.

We demonstrate that fast field quenching can induce absorption of the heat by the spin subsystem and hence significant cooling of the crystal lattice in Mn₁₂Ac. This nonequilibrium magnetic cooling effect represents a magnetothermal phenomenon that is quite distinct from the MCE and offers a new perspective on the use of highly anisotropic SMMs in low-temperature physics.

2. Theoretical approach

We focus on the low-temperature nonequilibrium thermal processes that are expected in Mn₁₂Ac upon fast (sudden) magnetic field quenching. The isotropic exchange interaction in this cluster is known to stabilize the ground state with spin S = 10. Since this state is well isolated from the excited states, at low temperatures, one can consider Mn₁₂Ac as a magnetic particle with S = 10. This is the so-called “giant spin approximation” often used to describe the low-temperature properties of the clusters with strong ferromagnetic exchange. Then, the cluster is approximately described by the following spin Hamiltonian:

This Hamiltonian, eqn (1), includes the axial component of the ZFS tensor and the Zeeman interaction, where D is the axial ZFS parameter defined for the S = 10 state, Ŝ_Z is the Z-component of the spin operator, g_∥ is the parallel component of the g-tensor, μ_B is the Bohr magneton, and B_Z is the magnetic field strength in the Z-direction. We use the following D and g_∥ values reported for Mn₁₂Ac: D = −0.5 cm⁻¹⁴ and g_∥ = 2.05.⁸

In writing eqn (1), we neglected the higher-order ZFS terms, which are several orders of magnitude weaker than the axial ZFS term in eqn (1) and are not important for the present consideration. For negative D, which is the case here, the system exhibits easy-axis-type magnetic anisotropy and possesses a magnetization reversal barrier responsible for the SMM properties of Mn₁₂Ac. We will deal with the single crystal sample and analyze the case when the magnetic field is directed along the easy axis Z.

The eigenvalues of the Hamiltonian, eqn (1), are obtained as follows:

where −S ≤ M_S ≤ S. At B_Z = 0, the energy pattern consists of ten non-Kramers doublets with M_S = ±10, ±9…±1 and one singlet with M_S = 0 (the highest in energy). In an applied magnetic field, each doublet is split into two Zeeman sublevels, as shown in Fig. 2, in which the energy of the ground state is regarded as the reference energy. An important feature of this energy pattern (as will be clear from the subsequent discussion) is that the decrease of the field leads to the diminution of all energy gaps separating the ground state from the excited ones.

Fig. 2 Energy levels of Mn₁₂Ac as a function of the axial magnetic field. The M_S-values for the Zeeman sublevels of the ground and first excited non-Kramers doublets are indicated. The energy of the ground state |M_S = −10〉 is chosen as the reference energy.

We analyze the thermal processes arising upon a fast (sudden) change of the magnetic field from its initial value Bⁱ_Z to the final value B^f_Z, where B^f_Z < Bⁱ_Z. We consider the case when the sample under study, consisting of the S = 10 clusters and the phonon bath, is isolated from the external world with an adiabatic envelope. Under these conditions, heat exchange is allowed only between the spin and phonon subsystems within the sample, rather than between the sample and the environment. It is also assumed that the field direction remains unchanged in the course of the switching event. At the initial field Bⁱ_Z, the S = 10 cluster (spin subsystem) is assumed to be in thermal equilibrium with the phonon bath, and thus, it is fully characterized by the set of Boltzmann populations of the spin states, which are the eigenstates of the spin Hamiltonian, eqn (1), in which one should set B_Z = Bⁱ_Z. The Boltzmann population of the k^th state is thus defined as follows:

where E_k(Bⁱ_Z) is the k^th eigenvalue of the spin Hamiltonian, eqn (1), evaluated at B_Z = Bⁱ_Z (k = 1, 2,…21), k_B is the Boltzmann constant, and T_i is the initial temperature of the sample. Upon rapidly changing the field from Bⁱ_Z to B^f_Z, the thermal equilibrium proves to be broken down. Provided that the field change occurs much faster than the spin–lattice relaxation, the populations of the states do not have time to adapt to the new energy pattern corresponding to the field B^f_Z. This means that immediately after the fast field change, the populations remain the same as the initial equilibrium populations n_k(Bⁱ_Z); however, they cease to be the equilibrium ones. Strictly speaking, this statement is valid only because M_S is a good quantum number and hence a fast change of the field does not induce any transitions between different Zeeman states.

The fact that the field is quenched suddenly also means that the temperature of the phonon subsystem has no time to change, and hence, immediately after the field change, the temperature of the phonon bath remains the same as at the beginning of the process, i.e., it is equal to T_i. Then, the molar internal energy acquired by the complex immediately after the field change is thus expressed in terms of the following sum of the products of eigenvalues E_k(B^f_Z) of the spin Hamiltonian, eqn (1), evaluated at the final field B_Z = B^f_Z, and the initial equilibrium populations n_k(Bⁱ_Z):

where N_A is the Avogadro constant.

The second stage of the considered nonequilibrium magnetothermal process is the relaxation of the system to a new (final) equilibrium state. During this stage, the spin and phonon subsystems exchange heat in order to reach equilibrium. Since the sample (spin subsystem + phonon subsystem) is isolated from the environment, the relaxation should result in a temperature change, ΔT = T_f − T_i, where T_f is the final temperature established in the sample as a result of relaxation. The final equilibrium state is characterized by the set of Boltzmann populations:

and the molar internal energy

In the considered case, where the sample is isolated from the environment, the following heat-balance equation can be written as:

where C_ph is the heat capacity of the phonons. For the low-temperature range considered here, the Debye law appears to be a good approximation:where θ_D is the Debye temperature. For Mn₁₂Ac, θ_D = 40.9 K.⁹ Note that the number of oscillating units in eqn (8) is assumed to be equal to the number of spins in eqn (4) and (6), which seems to be a reasonable assumption for the long-wavelength acoustic phonons contributing to the Debye phonon heat capacity, where each Mn₁₂Ac cluster can be regarded as oscillating as a whole. By using eqn (4) and (6) and performing the integration, one can present eqn (7) in the following final form:

To find the final temperature T_f (or, alternatively, the temperature change ΔT = T_f − T_i) as a function of the initial temperature T_i and the initial and final magnetic fields Bⁱ_Z and B^f_Z, one should solve eqn (9). However, some preliminary conclusions concerning the sign of the thermal effect (heating versus cooling) can be derived even prior to solving this equation, just based on the analysis of the field dependences of the energy levels shown in Fig. 1. Stabilization of the excited states upon decreasing the field leads to underpopulation of the excited states immediately after the fast field change, which means that, in order to reach the equilibrium state, heat should flow from the phonon bath to the spin subsystem in the course of relaxation, which should lead to the diminution of the temperature. Such a nonequilibrium magnetic cooling effect has already been mentioned for much simpler systems with D < 0 representing mononuclear paramagnetic complexes with S = 1 (e.g., Ni(II) complexes).^33,34 Such a magnetothermal behavior appears to be a common feature of all magnetic molecules with easy-axis-type anisotropy. Below, we support the conclusion regarding the possibility of low-temperature magnetic cooling in Mn₁₂Ac based on the numerical solution of eqn (9).

3. Results and discussion

The theoretical framework described above is valid under the condition of fast (“sudden”) magnetic field variation. This implies that, during the field-quenching process, the spin and phonon subsystems do not have sufficient time to exchange heat. Quantitatively, this requires that the field-switching time t_q be significantly shorter than the spin-relaxation time τ. In the present analysis, it is assumed that t_q is one order of magnitude shorter than τ. We consider a field decrease of 1 T as the upper limit of the change in the field. Then, by considering a sweep rate of about 10 T s⁻¹ (a relatively fast but experimentally achievable rate), we estimate the upper limit of the quenching time as t^max_q = 0.1 s. Then, to meet the conditions, the relaxation time should be no less than 1 s. This restriction defines the temperature range where the sudden-field quenching approach is applicable.

It is important to emphasize that Mn₁₂ clusters exhibit exceptionally slow spin dynamics at low temperatures, with relaxation times spanning many orders of magnitude. In the thermally activated regime that excludes extremely low temperatures (approximately T ≳ 3–5 K for many compounds), the temperature dependence of the relaxation time is described by the Arrhenius law:

where τ₀ is the pre-exponential factor and U_b is the magnetization reversal barrier. For the representative example of Mn₁₂Ac, these parameters are known to be τ₀ = 2.1 × 10⁻⁷ s and U_b = 70.3 K.⁴ These values are typical of a broad class of Mn₁₂ derivatives, as discussed above. In order to justify the use of Mn₁₂Ac as a representative example, it is instructive to note that closely related members of the [Mn₁₂O₁₂(RCOO)₁₆(H₂O)₄] family exhibit Arrhenius parameters of comparable magnitude. Similar barrier heights have been reported for several carboxylate-substituted derivatives with an S = 10 ground state and axial anisotropy of the Mn₁₂ core. More generally, high-frequency EPR and AC susceptibility studies across a broad series of Mn₁₂ derivatives indicate that the dominant relaxation barrier in the thermally activated regime typically lies between ∼50 and 70 K, with pre-exponential factors commonly in the range 10⁻⁸–10⁻⁶ s.¹ For example, Mn₁₂ complexes incorporating naphthalene–carboxylate ligands exhibit effective barriers in the range U_b = 53–61 K, as determined from AC susceptibility measurements in the Arrhenius regime.³⁷ These values remain of the same order of magnitude as those for Mn₁₂Ac, despite the substantially bulkier peripheral ligands. Likewise, dichloroacetate derivatives such as (PPh₄)₂[Mn₁₂O₁₂(O₂CCHC_l2)₁₆(H₂O)₄] display thermally activated relaxation with effective barriers in the range U_b ≈ 18–32 K, depending on the crystal form and solvation.³⁸ Finally, studies on Mn₁₂–stearate [Mn₁₂O₁₂(CH₃(CH₂)₁₆CO₂)₁₁(CH₃CO₂)₅(H₂O)₄] show, for the bulk material, activation energies U_b in the range of 60 K to 62 K and τ₀ that is typically around 10⁻⁸ to 10⁻⁴ s, demonstrating that changing the carboxylate ligand can modulate the relaxation behavior.³⁹ Thus, while ligand substitution, solvation, and crystal packing can modulate the effective barrier and relaxation prefactor, the Arrhenius parameters of Mn₁₂Ac may reasonably be regarded as representative of the high-temperature over-barrier relaxation regime in this family. The estimates given so far provide a good justification for the sudden-quench approximation adopted in the present analysis.

Fig. 3 shows the temperature dependence of the relaxation time for Mn₁₂Ac. To fulfill the condition of fast quenching, the relaxation times should satisfy the inequality τ ≥ 1 s. This defines the maximum temperature at which the fast-field quenching approach is still valid as T_max ≈ 4.6 K. Below, we show that the key features of the magnetothermal behavior of Mn₁₂Ac fit well into this allowed temperature range.

Fig. 3 Temperature dependence of the relaxation time for Mn₁₂Ac. Red line marks τ = 1 s, which is the minimum relaxation time for which the fast field-switching approach is applicable.

Fig. 4 shows the dependence of ΔT on the initial temperature T_i evaluated for Mn₁₂Ac at two values of the initial field Bⁱ_Z and different values of the final field B^f_Z by solving the heat-balance equation, eqn (9). One can see that the temperature change ΔT is always negative, which means that sudden magnetic field quenching causes cooling of the system, in accordance with the above qualitative arguments, according to which a sudden decrease of the field produces underpopulation of the excited spin states of Mn₁₂Ac, which gives rise to heat flow from the phonons to the spin subsystem. In the low-temperature limit, ΔT tends to zero because, in this limit, only the ground state is populated. Additionally, ΔT approaches zero in the high-temperature limit, both because the phonon heat capacity increases rapidly with the increase of temperature and due to the fact that, in the high-temperature limit, all states are equally populated, which precludes the heat exchange between the two subsystems. Between these two limits, the function ΔT(T_i) passes through a minimum corresponding to the most efficient cooling. It is seen from the comparison of the plots in Fig. 4a and b that, by increasing the initial magnetic field, one can reinforce the cooling effect; however, for both initial fields, the effect remains rather weak provided that the field is switched off completely (curves with B^f_Z = 0 T in Fig. 4a and b). The effect can be enhanced by several orders of magnitude if the field quenching is incomplete (compare the minimum ΔT values calculated for B^f_Z = 0 T and 0.1 T). Note that, at weak final fields, (ΔT)_min decreases with the increase of B^f_Z and then starts to decrease when the final field becomes strong enough, as can be seen by comparing the curves calculated for B^f_Z = 0.1 T and 0.3 T.

Fig. 4 Dependences of ΔT on the initial temperature T_i evaluated at B^f_Z = 0.5 T (a) and 1 T (b) for different B^f_Z values shown in the plots.

To get more details about the effects of the initial and final fields on the magnetic cooling effect, we plotted in Fig. 5 the dependences of the minimum temperature change (ΔT)_min and the initial temperature (T_i)_min at which the minimum temperature change is reached on B^f_Z calculated for the two Bⁱ_Z values. In these plots, B^f_Z varies from 0 to Bⁱ_Z. It follows from Fig. 5 that the cooling effect is stronger at higher initial fields. It is also observed that both (T_i)_min and (ΔT)_min depend on B^f_Z non-monotonically. A very small deviation of B^f_Z from zero leads to an abrupt decrease in the temperature ensuring maximal cooling (Fig. 5a and b) and also to a pronounced enhancement of the cooling effect (Fig. 5c and d). In contrast, for stronger deviations of B^f_Z from zero, the increase of B^f_Z gives rise to a gradual increase in the values of (T_i)_min and (ΔT)_min.

Fig. 5 Dependences of (T_i)_min (a and b) and (ΔT)_min (c and d) on the final magnetic field B^f_Z evaluated for the following two values of the initial field: B^f_Z = 0.5 T (a and c) and Bⁱ_Z = 1 T (b and d).

The main conclusion, which can be drawn by inspecting Fig. 4 and 5, is that, for Mn₁₂Ac, one can predict a pronounced magnetic cooling effect of a few K in the temperature range T < 4.6 K, where the conditions of sudden field quenching are fulfilled at a quite achievable sweep rate of the field of about 10 T s⁻¹.

At this stage, it is appropriate to compare the non-equilibrium magnetothermal effect under study with the conventional MCE, which occurs under the conditions that the magnetic field change occurs so slowly that thermal equilibrium is maintained at any moment in time during the process of changing the field. Then, the temperature change under the adiabatic conditions can be evaluated based on the fact that an adiabatic process is isentropic, such that

S_tot(B^f_Z, T_f) = S_tot(Bⁱ_Z, T_i). (11)

In this equation, the total entropy of the system is defined as follows:

S_tot(B_Z, T) = S_m(B_Z, T) + S_ph(T), (12)

where

is the entropy of the spin subsystem, and

is the entropy of the phonon subsystem defined within the Debye model. By solving eqn (12), one can determine the temperature change ΔT occurring upon adiabatic demagnetization.

In Fig. 6, one can see a comparison of the temperature dependence of ΔT evaluated for the non-equilibrium process with that obtained for the quasistatic process (i.e., for the conventional MCE) by solving eqn (11). It is seen that the MCE and non-equilibrium effect provide comparable temperature changes of several K at low temperatures. Although the MCE proves to be slightly stronger than the non-equilibrium magnetothermal effect, a significant advantage of the latter effect is that, at the slow relaxation exhibited by Mn₁₂Ac, it allows much faster temperature changes, which can be important for cryogenic applications.

Fig. 6 Temperature change ΔT evaluated as a function of the initial temperature T_i at Bⁱ_Z = 0.5 T and B^f_Z = 0.1 T for the non-equilibrium process (solid line) and the quasistatic process occurring in the MCE (dashed line).

It is worth making some remarks regarding the experimental feasibility and possible realization of the cooling mechanism proposed in this article. The non-equilibrium cooling mechanism is based on three key conditions: (i) sufficiently fast magnetic field quenching, (ii) slow enough (under the conditions specified so far) spin–lattice relaxation, and (iii) a rather good thermal isolation of the sample. All three requirements can be realistically fulfilled experimentally in Mn₁₂-based SMMs.

First, magnetic field sweep rates on the order of 1–10 T s⁻¹ are experimentally accessible. Such rates have been achieved in pulsed-field experimental facilities and have been employed in studies of molecular nanomagnets, particularly their magnetization dynamics.^8,40,41 Fast local magnetic field variations can be realized using microcoil- or micro-SQUID-based techniques developed for studies of nanomagnets and SMMs.^40–42

Thus, for the practical implementation of fast quenching of the magnetic field, no new specific experimental techniques are required.^43,44

Second, the required separation of time scales (τ_switch ≪ τ) is naturally fulfilled in Mn₁₂ systems in the low-temperature regime. As discussed above, the relaxation times exceed 1 s typically below 4.5 K and increase rapidly upon further cooling, reaching values of many seconds or even longer in the magnetic blocking regime. These time scales are well established from AC susceptibility and magnetization-relaxation measurements^8,45 with the use of the experimental setup as previously mentioned.

Third, detection of the predicted temperature changes, of the order of several kelvin, can in principle be achieved using sensitive fast-response thermometry combined with microcalorimetric techniques. Experimental approaches of this type are routinely employed in studies of magnetocaloric effects in molecular materials.^4,16 Realization of the proposed protocol would require weak thermal exchange between the sample and the environment in order to fulfil adiabatic conditions on the time scale of the field sweep and subsequent relaxation. Although achieving ideal adiabatic insulation is the most challenging experimental part of the problem, in practice, such conditions can be achieved using traditional low-temperature calorimetric methods. Typically, the sample is mounted on a platform connected to the thermostat via an insulating thermal connection (including, for example, thin insulating supports or low-thermal-conductivity wires) and placed in a vacuum environment to suppress convective heat dissipation. Under these conditions, which can be treated as quasi-adiabatic on the experimental time scale, heat exchange occurs mainly between the spin and phonon subsystems.

Along with the common methods used in molecular magnetism, some advanced experimental techniques that could be useful in the problem of non-equilibrium low-temperature magnetic cooling effects are also worth mentioning. Nuclear magnetic resonance (NMR) is an excellent probe of local spin dynamics and has been extensively used to study Mn₁₂Ac.⁴⁶ The method monitors the relaxation of nuclear spins ¹H or ⁵⁵Mn, which are coupled to the electronic spins of the Mn12Ac cluster. By measuring the time dependence of the relaxation time T₁ after field quenching, one can detect the evolution of the spin temperature. The muon spin rotation/relaxation (μSR)⁴⁶ technique seems to be promising because it is very sensitive to slow spin dynamics and has been successfully used in Mn₁₂Ac for studying spin relaxation. Less common, but effective, are optical methods focused on the study of spin dynamics such as magneto-optical methods (magnetic circular dichroism and the Faraday effect), which have already been applied to Mn₁₂Ac.

4. Conclusions

In this article, we have presented a theoretical analysis of the nonequilibrium magnetothermal processes that can be expected in a single-crystal sample of the SMM Mn₁₂Ac upon fast (sudden) quenching of the external magnetic field applied along the easy axis of magnetization. We demonstrate that such rapid field variation can induce cooling of the sample. In contrast to the conventional MCE, where magnetic anisotropy is generally considered a detrimental factor hindering efficient magnetic cooling, the strong easy-axis anisotropy of Mn₁₂Ac is the key prerequisite for the implementation of the proposed cooling mechanism.

The cooling effect originates from the mismatch between the initial spin populations, which remain the same during the fast field change, and the new energy level structure formed under the final magnetic field. The subsequent relaxation of the spin system toward equilibrium requires energy absorption from the phonon bath, leading to a decrease in the lattice temperature under adiabatic conditions.

Numerical analysis based on solving the heat-balance equation reveals several important features of this process. First, the easy-axis-type anisotropy in Mn₁₂Ac determines the direction of heat flow from the lattice subsystem to the spin subsystem during the relaxation after sudden field switching, which leads to lattice cooling. A significant cooling effect of up to several kelvin is predicted, confirming the potential of Mn₁₂Ac as a low-temperature refrigerant operating in this specific dynamic mode. Second, a non-trivial dependence of the final temperature on the final magnetic field has been deduced. Particularly, it has been found that incomplete field quenching can be more efficient than switching the field off completely. This can be attributed to the optimization of energy gaps between the ground and excited states, which maximizes the heat intake by the spin system.

The considered approach is applicable in the low-temperature range (below approximately 4.6 K for Mn₁₂Ac), where the spin–lattice relaxation time is sufficiently long (τ > 1 s) to satisfy the conditions of sudden field quenching at experimentally achievable sweep rates of ∼10 T s⁻¹. The presented arguments regarding the experimental feasibility and possible implementation of the cooling mechanism in single-molecule magnets based on Mn₁₂ show that the proposed scheme is quite realistic. We hope that the presented predictions could stimulate specifically oriented experimental studies of the fast-field magnetothermal effects in SMMs.

Overall, the present study provides a new perspective on the utilization of highly anisotropic molecular clusters in cryogenic applications. We also show that by shifting from quasi-static thermodynamic cycles to non-equilibrium dynamic protocols, SMMs with high relaxation barriers can serve as effective cooling agents, complementing the traditional low-anisotropy molecular refrigerants.

Conflicts of interest

There are no conflicts to declare.

Abbreviations

SMM	Single-molecule magnet
MCE	Magnetocaloric effect
RMCE	Rotational magnetocaloric effect
Mn12Ac	Mn12-acetate
ZFS	Zero-field splitting

Data availability

Numerical data and Wolfram Mathematica programs used for the calculation of the thermocaloric effect characteristics are available from the corresponding authors upon request.

Acknowledgements

A. P., V. B., D. K. and S. A. acknowledge financial support from the Russian Science Foundation (Project No. 25-13-00010).

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