Weakly Interacting Molecular Spins in On‐Surface Synthesized Nanoclusters on a Graphene Oxide Nanosheet

With an average diameter of ≈2 nm, on‐surface synthesized amino‐ferrocene nanoclusters are chemisorbed onto a graphene oxide nanosheet, where their Fe ions are in an S = 5/2 high‐spin state. In this two‐dimensional (2D) nanomaterial, the molecular spins in a given nanocluster are weakly magnetic dipole interacting. It generates spin correlations and slow dynamics accessible by magnetic susceptibility and Mössbauer spectroscopy at a temperature where the magnetic anisotropy is negligible. Magnetic simulations show that minimizing their magnetic dipole energies produces spatially entangled structures of the spin orientations under thermal fluctuations at T ≲ 15 K and that the structures behave like a spin liquid. The competition between the formation of the structures and their thermal destruction generates slow dynamics. The ability to create emergent functionalities from dense stacks of weakly interacting magnetic molecules in a 2 nm space paves the way for new designs of an ultra‐compact building block and a functional component in 2D spintronic and neuromorphic devices.


Introduction
Engineering the magnetic interactions between molecular spins plays an essential role in the chemical design and arrangement of molecular magnets, magnetic molecules, and their assemblies required for the integration of nanoscale spintronic and neuromorphic devices.On the one hand, molecular magnets have the ability to maintain their spin orientations through magnetic anisotropy and intramolecular exchange interactions, [1,2] and a DOI: 10.1002/aelm.202300347variety of magnets have been synthesized for molecular spintronics and quantum computing, acting as information storage devices [3,4] and, for those with long spin coherence, as building blocks for quantum computation. [5,6]ome magnetic molecules have another ability to self-assemble into various nano-and micro-scale structures, [7,8] which can lead to the development of new functionalities not accessible to single molecules through structuring in combination with the intrinsic properties of the molecules themselves.The aminoferrocene molecules used in this study (Figure 1A) can self-assemble into nanoclusters on a graphene oxide nanosheet via on-surface chemical reaction. [9]We have measured the magnetic properties of densely stacked weakly interacting molecular spins in a nanocluster with a diameter of ≈2 nm and discussed those properties by combining theoretical simulations and a systematic analysis of the experimental data.
The amino-ferrocene-based assemblies, referred to as nanoclusters in this study, are chemisorbed on the surface of a graphene oxide (GO) nanosheet, forming a 2D material.The electronic states of the 3d electrons of the Fe ion in ferrocene or ferrocene-derivatives are dominated by the ligand field. [10]Inside the nanocluster, the Fe ion of a given amino-ferrocene molecule has unpaired electrons due to the through-bond electron transfer process between the Fe ion and the GO, [9] which behaves as a localized spin due to the geometry of the ferrocene unit (Figure 1A).Magnetic dipole interactions between localized spins in molecular assemblies are generally much weaker than intramolecular interactions and magnetic anisotropy in the molecule, because the dipole energy approximated as m 2 /r 3 is equivalent to only 0.0006 K assuming a magnetic moment m of 1  B and a distance r between them of 1 nm.However, in assemblies of small magnets with large magnetic moments, magnetic dipole interactions have been reported to play a crucial role, for example, in the magnetic ordering of high-spin molecular cluster magnet crystals [7,8] and in the modulation of magnetic relaxation in magnetic nanoparticle assemblies. [11,12]Here, the nearest dipole interactions dominate the formation of these properties.In our amino-ferrocene-based nanoclusters, the d m = 0.28 nm distance between neighboring localized spins [9] (Figure 1A) is much shorter than the d m between neighboring magnetic moments in Fe 12 cluster magnet crystals (d m ≈ 1.4 nm) [7] and in nanoparticle assemblies (d m = 2-10 nm), [11,12] leading to the completely different situation that all molecular spins within the nanocluster are weakly connected by magnetic dipole fields of the order of 10 2 -10 3 Oe.The interactions in the nanoclusters differ from those treated in a mean-field-based theory. [13]In addition, the controlled on-surface self-organization of our nanoclusters can produce a uniform nanocluster distribution with a relatively large inter-cluster distance, [9] resulting in a surface structure close to an isolated magnetic nanocluster distribution on an insulating GO nanosheet.Here, we analyze the behavior of densely packed molecular spins resulting from such unique magnetic interactions in nearly 0D nanoclusters by measuring the magnetic susceptibility, magnetization, 57 Fe Mössbauer spectrum and by simulating the distribution of the spin orientations by Markov-chain Monte Carlo methods.We also elucidate the mechanism of the slow dynamics generated in this 2D material by comparing it with that studied in spin glasses , [14,15] interacting magnetic nanoparticles , [11,12] and frustrated magnets. [16,17]

Results and Discussion
The deviation from the paramagnetic Fe ions in the aminoferrocene (AFc) molecules forming the nanoclusters on a GO nanosheet (an AFc-GO sheet) was estimated by accounting static magnetization and magnetic susceptibility.On the GO nanosheets, resulting in an average inter-cluster distance 〈d〉 of 10 nm (Figure 1A), the amino-ferrocene-based nanoclusters were synthesized for 5 h by our on-surface chemical reaction technique (Experimental Section). [9]The small black spots in a transmission electron microscope (TEM) image (Figure 1B) and the small white spots in a scanning transmission microscope (STEM) image (Figure S1, Supporting Information) of a chemically reacted GO nanosheet correspond to amino-ferrocenebased nanoclusters on a GO nanosheet (Figure S1, Supporting Information).The diameter distribution of the AFc-GO sheets (〈d〉 = 10 nm) shows an average diameter of ≈2 nm (inset of Figure 1B).As calculated from the magnetization of the AFc-GO sheets and of the pristine GO nanosheets, [9] the magnetization M AFc of the molecules on the GO nanosheets is plotted in Figure 1b as a function of  B H/k B T, where  B is the Bohr magneton, H is the applied magnetic field, k B is the Boltzmann constant, and T is the temperature.Characterizing the paramagnetic properties of a material as a function of field and temperature, [18] the Brillouin B S ( B H/k B T) and Langevin L S ( B H/k B T) functions are plotted as solid and dotted lines using the spin state numbers S = 5/2 or S = 1/2 (Figure 1C; Figure S2, Supporting Information), respectively.At small  B H/k B T values, the experimental data are well fitted using S = 5/2.However, at low tem- perature and high field, the experimental data deviated from the lines, meaning that a change in the energy levels of the 3d electrons under the applied field is not expressed simply by the Zeeman splitting, [18] as  B H/k B T increases.This is different from superparamagnetic materials, for example, ferromagnetic nanoparticles [19] and molecular magnets. [20,21]Note that this deviation was observed at T = 50 and 100 K, where the thermal energy is certainly larger than the magnetic interactions between the neighboring Fe ions.
Under a small magnetic field and for T > 20 K, the static susceptibility  M, AFc of our nanomaterial follows the Curie-Weiss law: [18]  M, AFc = C/(T − Θ) (Figure 1D).The constant C = g 2  2 B S(S + 1)∕3k B W FD [18] is the slope of a linear fit to the experimental data, where g is the g-factor and W FD is the mass of the molecule.Assuming g = 2, C gives a magnetic moment g B S = 5.1  B , confirming that the Fe ions are in the high-spin state (S = 5/2) rather than the low-spin state (S = 1/2) (Figure 1A).Note that this value was underestimated in our previous study [9] because it was determined in a magnetic field of 70 kOe and in an unsaturated M-H loop at T = 2 K.The constant Θ was −2.9 K, indicating that there were negative interactions, as predicted by mean-field theory. [18]ow-temperature zero-field 57 Fe Mössbauer spectroscopy [22,23] confirms that the Fe ion in the amino-ferrocene molecules changed from a non-magnetic S = 0 state to a S = 5/2 high-spin state when assembled by on-surface chemical reaction on a GO nanosheet.At 5 K, the Mössbauer spectrum of a pristine amino-ferrocene powder indicated that the Fe ions here are in the Fe 2+ S = 0 nonmagnetic state (Figure 2A). [11]On the other hand, for the amino-ferrocene molecules assembled on a GO nanosheet surface, a magnetic hyperfine splitting was clearly seen in the spectra (Figure 2B).This is due to the change in the ionized state of the Fe ions caused by the charge transfer. [9]As the temperature increased, thermal disturbances attenuated the peaks in the spectra.The onset temperature to observe a splitting was ≈15 K and the separation between the peaks did not depend on temperature (Figure 2B).This is different from the splitting temperature dependence observed for superparamagnetic particles, whose magnetic properties are dominated by magnetic anisotropy. [24]At 5 K, the parameters characterizing the 3d electrons in the Fe ions were determined by taking the best fit of the spectrum (Figure S3, Supporting Information).As summarized in Table S1 (Supporting Information) and at 5 K, the 470-400 kOe hyperfine magnetic fields for an AFc-GO sheet again confirm that the Fe ions are in an S = 5/2 Fe 3+ high-spin state and not in the S = 1/2 low spin state. [22]At 5 K, ≈82% of the Fe ions on the AFc-GO sheets have a large hyperfine field corresponding to S = 5/2.About 18% of the Fe ions on the GO nanosheets are in the S = 0 non-magnetic state.Some of the amino-ferrocene molecules have not chemically reacted with the GO surface and do not contribute to the magnetic properties of the chemically reacted sheets.
The magnetic properties of a mononuclear organometallic molecule are generally characterized by a magnetic anisotropy resulting from zero-field splitting at its metal site. [25]There are few experimental reports on the magnetic anisotropy of ferrocene molecules because individual ferrocene molecules do not have unpaired 3d electrons. [11]Assembled on a GO nanosheet, Fe 3+ ion 3d electrons are in a high S = 5/2 spin state and the electronic cloud of the 3d electrons is spherically distributed (no orbitalangular momentum). [26]Therefore, the spin-orbit interactions normally at the origin of magnetic anisotropy are too small to produce the hyperfine splitting observed in the Mössbauer spectra at T = 5-15 K (Figure 2B).First-principles calculations on MnCp 2 molecule, which has the same number of 3d electrons as the amino-ferrocene molecule, gives a magnetic anisotropy of ≈0.2 meV in the direction perpendicular to the planes holding the cyclopentadienyl rings in MnCp 2 . [27]An anisotropy energy K = 0.2 meV leads to a freezing temperature T B = 0.09 K using T B ≈ K/25k B . [19]This temperature is much lower as compared to the onset temperature (≈ 15 K) in the splitting (Figure 2B).Thus, magnetic anisotropy is not at the origin of this splitting.Furthermore, from the electronic and geometric structures of the ferrocene molecule [28,29] and from the arrangement of aminoferrocene molecules within the nanocluster (Figure 1A), there is no direct chemical coordination between neighboring Fe ions, suggesting that the bond-mediated interactions between Fe ions are weak to explain the properties generated at T ≲ 15 K. Therefore, the splitting originates from magnetic dipole interactions between Fe ions densely arranged within a nanocluster.
We can also represent the distribution of spin orientations at Fe ion sites within the nanocluster by using an assembly of classical spins at the positions of Fe ions within a given nanocluster (Figure 3A).The lateral distance between neighboring Fe ions was taken as the experimental value (0.28 nm). [9]The vertical distance was assumed to be 0.4 nm, which is 20% longer than the distance between two cyclopentadienyl rings of a given ferrocene molecule (0.33 nm). [29]We assumed that these localized spins have classical magnetic moments of 5 μ B .Magnetic screening [30] and magnetic interactions through spin-polarized carriers [31] are negligible due to the insulating GO nanosheet.A zero-field Markov-chain Monte Carlo simulation [32] of the assembly shows that there is a monotonic decrease in the average total dipole energy with decreasing temperature (Figure 3B).The snapshots of spin orientations and dipole energy at the sites are shown in Figure 3C-F.The vortex structure of spin orientations was formed at T = 0.01 K by minimizing the dipole energy while reducing the thermal perturbations.As the temperature increases, the structure becomes thermally perturbed, and the low dipole energy regions (colored red, orange, and sand, as shown in Figure 3C-F) become smaller in lateral size.The energy change is due to the formation of the entangled structures of the spin orientations in such a way that the total dipole energy is reduced.The term "entangled structure" refers here to the "entangled polymers" below the glass transition temperature. [33]At T < 0.2 K, vortex structures of the spin orientations are formed.At T ≳ 0.2 K, spatially entangled structures of the spin orientations are formed, and they are simply estimated by the size of the low dipole energy regions.Note that there is here a large change in the total dipole energy at T ≲ 15 K (Figure 3B) consistent with the temperature at which splitting started in the Mössbauer spectra (Figure 2B).
The model simulation elucidated several magnetic properties generated by the dense stacking of weakly interacting molecular spins in the nanocluster.The magnetization M AFc of the molecule in AFc-GO sheets (〈d〉 = 10 nm) under a static magnetic field was well fitted by an average magnetization calculated for the assembly in the field region (1 kOe ≲ H ≲ 5 kOe), where the applied field is comparable to the magnetic dipole fields within the nanocluster (Figure 4A).The deviation of the data from the Brillouin curves with S = 5/2 in Figure 1C is mainly caused by the interactions between the spin orientations.In addition, this entanglement acts to resist spin alignment in the direction of the field, producing a negative Θ in the Curie-Weiss law (Figure 1D).Simulation of an assembly (S = 5/2) with vertical distance l v of 0.20, 0.21, and 0.25 nm showed a large decrease in the average dipole energy at T ≲ 15 K (Figure 4B).It should be noted that the dipole energy of an assembly (l v of 0.2 nm) composed of magnetic moments of 1  B (S = 1/2) did not change at T ≳ 2 K (Figure 4B), confirming that the large magnetic moments of the molecules play an important role in the substantial decrease of the energy at T ≲ 15 K. Since the thermal fluctuations of the spin orientations are weakened by the entangled structure of the spin orientations, the temperature at which the spin fluctuations are strongly reduced is similar to the temperature at which the dipole energy is strongly reduced by the formation of the entangled structures (Figure 4C).
The model simulations also revealed features characterizing the stability of the spin orientation distribution in the assembly.The spin orientations and dipole energy at the sites at T = 0.01 K were nearly the same regardless of the Monte Carlo steps (MCS) (Figure 4D), indicating that this entangled structure, namely the vortex structure, was stable.As the temperature increased, the site distribution of spin orientations and dipole energy changed against the progress of the MCS (Figure 4E,F), keeping the total dipole energy close to the average value.The low dipole energy domain (colored red, orange, and sand, as shown in Figure 4E,F) was formed by minimizing the energy.As the simulation progressed, the domain frequently changed its position within the nanocluster at T = 0.5 and 10 K (Figure 4E,F) and behaved like a liquid, suggesting that there are many possible spin orientation distributions in the assembly with the same energy (Figure S4, Supporting Information).This is similar to many degenerate states in spin-glass materials [14,15] and spin liquid states in frustrated magnets. [16,17]As the temperature was further increased, sites with low or high dipole energy were distributed almost in isolation and changed positions frequently (Figure S4B, Supporting Information).
At low temperatures, magnetic dipole interactions in the dense stacking of molecular spins in the nanocluster give rise to two characteristic phenomena: spin correlations and slow dynamics.For AFc-GO sheets with 〈d〉 = 10 nm, the imaginary component ″ AFc of the dynamic susceptibility of the molecule emerged at T ≲ 11-12 K (Figure 5A).From the fluctuation-dissipation theorem, [34] this suggests that spin correlations occur in this temperature range.Since there is no imaginary component for the dynamic susceptibility of pristine GO nanosheets (Figure S5, Supporting Information), these correlations occur within the nanocluster.The spin correlations are due to the formation of the entangled structures of the spin orientations (Figure 3C-E) 3B-D, which are maintained for a long time due to magnetic friction, [35] an effect neglected in the simulation.The field-cooled (FC) and zero-field-cooled (ZFC) static susceptibility  M, AFc (T) curves of the AFc-GO sheets (〈d〉 = 10 nm) separate at a temperature below the freezing temperature T B of ≈11 K (Figure 5B), indicating the onset of a slow magnetization relaxation analogous to that found in spin glass materials, [14,15] ferromagnetic particles [12,19] and single-molecule magnets. [36]The frequency-dependent ′ AFc (Figure 5A) is typical of a slow dynamics. [14]The origin of this slow dynamics in our AFc-GO sheets results from a competition between the formation of the entangled structures by magnetic dipole interactions and their destruction by thermal perturbations.Although the dipole interaction between the nearest spins in the nanocluster is equivalent to 0.7 K, the contributing dipole interactions of all spins, which are connected due to the short intermolecular distance and the magnetic moment of 5 μ B , generate the entangled structures of spin orientations essentially at T ≲ 15 K (Figure 3B).In the Mössbauer spectra (Figure 2B), the hyperfine splitting at T ≲ 15 K also originates from these slow dynamics because the relaxation time of the magnetization at T ≲ T B is much longer than the response time of Mössbauer spectroscopy (10 −8 -10 −12 s). [22]he relaxation process was further analyzed to extract the parameters characterizing non-equilibrium relaxation of weakly interacting spins in the nanocluster.The residual magnetization M(t) of AFc-GO sheets (〈d〉 = 10 nm) was generated by the removal of the field after field-cooling (500 Oe) to each temperature and decayed slowly (Figure S6A, Supporting Information).Field-cooling of AFc-GO sheets forced the spin orientations to align with the direction of the field.The removal of the field caused a non-equilibrium distribution of spin orientations, which was observed as the residual magnetization.The relaxation process was analyzed using a stretched exponential form: [14,37] where  is the relaxation time, M(0) is the residual magnetization, and 0 ≦ n ≦ 1. Fitting the data yielded a relaxation time (T) as a function of temperature and n of ≈0.67.This relaxation time is assumed to follow a classical thermal activation process, that is, an Arrhenius process: [14,38] where  0 is a prefactor characterizing the relaxation time scale and E A,FC is the height of the energy barrier blocking the decay of the residual magnetization for fieldcooled AFc-GO sheets.A linear fit of the plot ln((T)) versus T (Figure 5C) supports this assumption, yielding E A, FC /k B = 11 K and  0 = 1.6 × 10 4 s.The strength of the energy barrier that maintains the residual magnetization against thermal activation is equivalent to the degree of spatial entanglement of the spin orientations.
The two processes: the alignment of spins by the applied magnetic field and the subsequent removal of the field to create a non-equilibrium spin orientation distribution, were applied to AFc-GO sheets (〈d〉 = 10 nm) after zero-field-cooling (ZFC).By applying the field to the zero-field-cooled sheets for 2 h, the random distribution of the spin orientations after the ZFC was changed to the distribution whose spin orientations were partially aligned with the direction of the applied field.Removal of the field caused spin orientations in the non-equilibrium distribution, resulting in a residual magnetization that slowly decayed (Figure S6B, Supporting Information).Its relaxation was analyzed by the same method applied to the field-cooled sample (Figure S7, Supporting Information).The blocking energy barrier E A,ZFC /k B (≲4 K) and the residual magnetization M(0) (≲0.08 emu g −1 ) generated in the zero-field-cooled AFc-GO sheets were much smaller than E A,FC /k B (=11 K) and M(0) (≈0.15 emu g −1 ) of the field (500 Oe)-cooled sheets (Figure 5D), indicating the difference in the number of molecules forming the entangled structures within the nanocluster.Note that E A, ZFC depends on the field applied after ZFC (Figure 5D).A larger magnetic field applied to the zero-field-cooled AFc-GO sheets induces a larger number of spins to align their orientations with the direction of the field, resulting in a larger deviation from the random distribution and, consequently, a higher energy barrier after removal of the field.This result demonstrates that the non-equilibrium distribution and the residual magnetization, namely, the entangled structures of the spin orientations in the nanocluster, can be tuned by varying the intensity of the applied field for the AFc-GO sheets whose spin orientations are randomly distributed.

Conclusion
This study shows that weakly interacting molecular spins in a nanocluster with an average diameter of ≈2 nm behave like a spin liquid, which is confirmed by the results that the present system exhibits slow dynamics and spin correlations below the freezing temperature and that the spin orientation distribution has so many degenerate states.Analysis of the Mössbauer spectrum and magnetic susceptibility shows that the magnetic state of the Fe ions in the nanocluster is in the 5/2 high-spin state.The creation of unique functionalities such as slow dynamics, spin liquid-like behavior, and tunable non-equilibrium spin orientation distribution in 2 nm space on a surface provides new insights into low-dimensional magnetism and spintronics.The design and synthesis of chemically functionalized nanosheets by arranging magnetic molecules through on-surface chemistry will lead to major advances in the development of 2D materials and 2D spintronic devices.

Experimental Section
Chemical Synthesis: One milliliter of GO solution (4 mg mL −1 ) was mixed with 50 mL of N,N-dimethylformamide (DMF) in a round-bottomed flask.The solution was stirred while adding amino-ferrocene powder (20 mg) and coupling agents (EDC-HCl, HOBt, and trimethylpyridine) at 0 °C for the first 3 h to reduce the overreaction, and then stirred at 24 °C for 2 h.The on-surface chemical reaction produced amino-ferrocene-based nanoclusters with an average inter-cluster distance of 10 nm on the GO nanosheet.The as-synthesized products were separated from the supernatant liquid by using an ultrafast centrifuge after repeated washing, first twice with DMF and then with ethanol to remove unreacted chemicals on the sheets, followed by drying.
Magnetization Measurement: The static magnetization M of aggregated AFc-GO sheets weighing 5-30 mg was measured using a superconducting quantum interference device (SQUID) magnetometer (MPMS-7T, Quantum Design).The static magnetic susceptibility  M,AFc was calculated as  M,AFc = M AFc /H by using the applied static magnetic field H.The dynamic susceptibility was measured at zero static magnetic field by using a SQUID magnetometer (MPMS-1T, Quantum Design) at an alternating field (amplitude ΔH = 1 or 0.5 Oe) with driving frequency f = 2-500 Hz.The slow dynamics of AFc-GO sheets (〈d〉 = 10 nm) weere estimated by measuring the residual magnetization obtained by removing the field after the sheets were slowly cooled to the temperature under the field of 500 Oe.The residual magnetization of the zero-field-cooled sheets was obtained by the following three steps: 1) zero-field-cooling slowly to a target temperature; 2) applying a magnetic field for 2 h at the temperature; 3) removing the field at each temperature.The change in the residual magnetization was then measured as a function of time at each temperature.Mössbauer spectroscopic measurements were performed using a conventional transmission geometry with the 57 Fe excitation energy of 14.4 keV to see the magnetic states of the Fe ions in the AFc-GO sheets.The source velocity was calibrated using pure -Fe films.A sample with a diameter of ≈14 mm was formed by random stacking of AFc-GO sheets, and it was dried and covered with thin Kapton sheets.For the measurements, the sample was sandwiched between thin, pure Al films to keep the temperature distribution uniform.The sample, with a total weight of Fe atoms of ≈20 mg, was placed on a stage of a low-temperature cryostat.The spectra were fitted with magnetically-split sextets with Voigt profiles (convolution of Lorentzian and Gaussian functions) and non-magnetic doublets with Lorentzian function profiles.
Markov-Chain Monte Carlo Simulation: The Monte Carlo algorithm generated a sequence of spin orientations as follows: 1) The initial state of the assembly (Figure 3a) was a spin configuration where each spin orientation was random.2) The program randomly selected a molecular site within the assembly and changed its spin orientation to the direction denoted by  and  in polar coordinates (0 ≦  ≦  and 0 ≦  ≦ 2) by using random numbers.After calculating the energy difference ΔE between the spin configurations before and after the spin orientation change, the Boltzmann factor exp( − ΔE/k B T), where k B is the Boltzmann constant and T is the temperature of the assembly, was used to estimate the probability P of the spin orientation change according to a Markov chain process.3) The program randomly selected a fraction R with equal probability over the interval [0,1].Then it performed the orientation change if R ≤ P, where P is calculated as P = x/(1 + x) ; x = exp( − ΔE/k B T).After repeated iterations of steps ( 2) and (3), the system was at or near equilibrium.This method is called the heat-bath method. [28]The initial configurations (600 Monte Carlo steps (MCS)) in the simulation were removed for the calculation of the average value because the configurations were out of thermodynamic equilibrium.

Figure 1 .
Figure 1.a) Schematic illustration of an amino-ferrocene molecule, approximate energy splitting of the 3d orbitals in the Fe 2+ (S = 0), Fe 3+ (S = 1/2), and Fe 3+ (S = 5/2) ions of a ferrocene unit, and schematic drawing of a nanocluster consisting of amino-ferrocene molecules with an average inter-cluster distance 〈d〉 of 10 nm on a graphene oxide (GO) nanosheet.b) A transmission electron microscope (TEM) image of an AFc-GO sheet (〈d〉 = 10 nm).The inset shows the diameter distribution of the AFc-GO sheets.c) Magnetization M AFc of molecules of AFc-GO sheets (〈d〉 = 10 nm), Brillouin function (solid lines), and Langevin function (dotted lines) plotted against μ B H/k B T. d) Inverse of the static susceptibility of the molecule  M,AFc for AFc-GO sheets (〈d〉 = 10 nm) versus temperature.The solid line is from the Curie-Weiss law.

Figure 3 .
Figure 3. a) Schematic illustration of the fundamental unit of Fe ions of amino-ferrocene molecules (left panel) and the positions of Fe ions within the nanocluster (right panel).b) Plot of the average total dipole energy of the spin assembly, calculated using a zero-field Markov-chain Monte Carlo method, versus temperature.Snapshots of spin orientations at sites of the assembly (top), spin orientations (left), and dipole energy (right) at sites of the layers forming the assembly calculated at c) T = 0.01, d) 0.5, e) 10, and f) 200 K under a zero field.

Figure 4 .
Figure 4. a) Calculated magnetization M AFc of the molecule plotted versus applied field at T = 10, 20, 50 K (open circles) compared to experimental values (solid line).b) Average dipole energy of assemblies (l c = 0.20, 0.21, 0.25 nm, S = 5/2) and assembly (l c = 0.20 nm, S = 1/2) versus temperature.c) Standard deviation (STD) of the dipole energy of the assemblies (l c = 0.20 nm, S = 1/2, 5/2) versus temperature.Snapshots of spin orientations and dipole energy at sites of the bottom layer of the assembly (l c = 0.20 nm, S = 5/2) versus each MCS at temperature T of d) 0.01, e) 0.5, and f) 10 K.

Figure 5 .
Figure 5. a) Dynamic susceptibility (real component: ′ AFc , imaginary component: ′′ AFc ) of the molecules of AFc-GO sheets (〈d〉 = 10 nm) as a function of temperature T. b) Field-cooled (FC) and zero-field-cooled (ZFC) static susceptibilities  M of the molecules plotted against temperature, showing a separation between the curves at the freezing temperature T B .c) Logarithmic plot of the relaxation time (T) of AFc-GO sheets (〈d〉 = 10 nm) in nonequilibrium spin orientations obtained after field-cooling (red open circle) and zero-field-cooling (orange and blue open circles) against the inverse of temperature.d) Blocking energy barrier E A,ZFC /k B in the relaxation of the residual magnetization by applying the static field H to zero-field-cooled AFc-GO sheets (〈d〉 = 10 nm) for 2 h and removing the field.