Rotating Magnetocaloric Effect in an Anisotropic Two-Dimensional

Article pubs.acs.org/IC

Rotating Magnetocaloric Effect in an Anisotropic Two-Dimensional CuII[WV(CN)8]3− Molecular Magnet with Topological Phase Transition: Experiment and Theory Piotr Konieczny,*,† Robert Pełka,† Dominik Czernia,‡ and Robert Podgajny*,§ †

Institute of Nuclear Physics PAN, Radzikowskiego 152, 31-342 Kraków, Poland Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland § Faculty of Chemistry, Jagiellonian University, Gronostajowa 2, 30-387 Kraków, Poland

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‡

S Supporting Information *

ABSTRACT: Conventional (MCE) and rotating (RMCE) magnetocaloric effects have been explored in the twodimensional (2D) coordination polymer {(tetren)H5)0.8CuII4[WV(CN)8]4·7.2H2O}n (WCu-t; tetren = tetraethylenepentamine). The unusual magnetostructural properties were exploited, including the bilayered Prussian Blue like coordination skeleton and the XY easy-plane magnetic anisotropy based on the in-plane correlation between WV and CuII spins of 1/2, underlying the Berezinskii−Kosterlitz− Thouless (BKT) topological phase transition to the longrange-ordered state at TC = 33 K. The magnetic properties were studied on single crystals along the H∥ac easy plane and H∥b hard axis. The maximal entropy change for MCE for easy-plane geometry at 38.0 K and the magnetic field change μ0ΔH = 7.0 T reached ∼4.01 J K−1 kg−1. The strong magnetic anisotropy was used to study the RMCE in which the maximal entropy change was observed at 35.5 K for 7.0 T, attaining 1.81 J K−1 kg−1. Moreover, easy-plane anisotropy introduces the inverse magnetocaloric effect for H∥b, which enhances the RMCE by up to 47%. This observation was confirmed by a theoretical investigation considering the XY model using a molecular field and cluster variational method in the pair approximation approach, dedicated to the bilayered systems with the adequate nearest neighbor number z = 5 and spin S = 1/2.

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nets,24,25 chiral magnets,26 and others, offering multilevel control of the state of the matter. Research on 2D systems is particularly interesting.10,27−38 From the point of view of basic research, the phase diagram and the role of intralayer anisotropy in 2D compounds get much of the attention.10,27,33,37,39,40 It is known that only in strongly anisotropic 2D systems (like Ising, 2D XY, etc.) the phase transition to ordered state does occur at TC ≠ 0. In the case of 2D XY, the topological phase transition takes place because of bonding of vortex−antivortex pairs below a critical temperature TBKT, where the BKT stands for Berezinskii−Kosterlitz− Thouless.41−43 For this type of topological phase transition, David J. Thouless, F. Duncan M. Halden, and J. Michael Kosterlitz received the Nobel Prize in Physics in 2016.44 On the other hand, layered structures are of significant importance in application as alternative sources of magnetic multilayers, which are currently strongly employed in electronics. In our opinion, 2D molecular compounds are also promising materials for magnetic cooling technology via

INTRODUCTION Modern coordination chemistry provides a wide scope of functional metallo-organic molecular platforms revealing different coordination dimensionalities, from discrete molecules (0D) through chain (1D) and layered (2D) substructures to fully extended (3D) systems.1−15 Apart from the fundamental role of local atomic/molecular features of interest (size, symmetry, electron transfer, chirality, electronic properties, etc.), which are important for the desired properties, in many cases the target functionality emerges in accordance with their coordination dimensionality. Along the structure−magnetism correlation line, the related fundamental and applicationoriented low-dimensional properties are slow magnetic relaxation,11,16−19 low-temperature magnetocaloric effects,14,20 or recent studies of spin coherence correlations in quantum gate-like systems due to very weak intramolecular interactions.21 The high-dimensional (2D and 3D) systems previously studied, mostly due to the long-range magnetic ordering phenomena, reappeared to molecular magnetism with a new glare of multifunctional molecular magnetic materials including intercalation-tuned layered hydroxide systems,10 magnetic sponges,22 3D porous magnets,23 spin-crossover photomag© 2017 American Chemical Society

Received: July 28, 2017 Published: September 15, 2017 11971

DOI: 10.1021/acs.inorgchem.7b01930 Inorg. Chem. 2017, 56, 11971−11980

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Inorganic Chemistry

analogue network cut, a significant 2D XY magnetic anisotropy (with the ac easy plane and the b hard axis; Figure 1) and a BKT topological phase transition at TC ≈ 33 K were observed.31,39,54,55 The 2D magnetic character of WCu-t was recently evidenced by the combination of single-crystal measurements (bulk magnetization, neutron flipping ratio, electron magnetic resonance, and neutron diffraction) and theoretical modeling [exchange-charge model of the crystal field, dipolar energy, and density functional theory (DFT) calculations].40 The observed behavior was explained in terms of the different strengths of the local ferromagnetic interaction represented by Jeq and Jax, Jeq > 10Jax, acquired from the quantum-chemical computation of the density of states and magnetic exchange coupling along the CuII−NC−WV linkages (Figure 1, bottom).39,40 We expect that much stronger anisotropy in WCu-t compared to that in MnNb (Figure 2) will have a strong impact on the RMCE, yielding a significantly larger entropy change.

rotating magnetocaloric effect (RMCE). Conventional magnetocaloric effect (MCE) is a widely known and studied magnetothermodynamic process, in which the temperature of the magnetic material is altered because of a change in the magnetic field, for instance, by moving the magnetic material in and out of the magnetic field.45−48 Recently, a new type of MCE has been developed, the RMCE, in which rotation of an anisotropic magnetic material in a constant field is used to change the temperature.49 The main advantages of RMCE over MCE are simple construction49,50 and fast cooling cycles51,52 (high frequency of operation) and, consequently, high efficiency. In our previous work, we have studied the RMCE in a 2D cyanido-bridged MnII−NbIV (MnNb) molecular ferrimagnet with easy-plane anisotropy.53 In this type of anisotropy, the magnetic moments, which preferably are lying within the easy plane, can be distorted by applying a magnetic field perpendicular to this layer and therefore increasing the magnetic disorder. This phenomenon, where the external magnetic field increases the magnetic entropy, is known as the inverse magnetocaloric effect, and we have shown it can be used to enhance the RMCE. However, the observed magnitude of ΔSR (change of the magnetic entropy via rotation) was modest because of a relatively weak anisotropy. To address the above motivation, we investigated the RMCE in the unique 2D cyanido-bridged network (tetrenH5)0.8{CuII[WV(CN)8]4}· 7.2H2O}n (WCu-t; Figure 1, top).29 Along the double-layered coordination skeletons reminiscent of the Prussian Blue

Figure 2. Isothermal magnetization at 2.0 K for single crystals of WCu-t (red circles) and MnNb (blue triangles). The hard axes (H∥b in the case of WCu-t and a*∥H for MnNb) are marked by empty points, while the full points represent the data for easy planes (H∥ac in the case of WCu-t and bc∥H for MnNb).

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EXPERIMENTAL SECTION

Materials and Syntheses. Tetraethylenepentamine pentahydrochloride (tetren·5HCl) was purchased from Sigma-Aldrich. CuCl2· 2H2O was purchased from Idalia. K4[W(CN)8] was prepared according to the literature method.56 To obtain the solid Na3[W(CN)8]·4H2O, we oxidized K4[W(CN)8] by KMnO4 in the aqueous acidic media (2 M HNO3), and then we precipitated Ag3[W(CN)8] using AgNO3. Finally, metathesis with an aqueous solution of NaCl gave the solution, from which the yellow crystals of Na3[W(CN)8]· 4H2O were obtained. The long-time air-durable plate-like single crystals of (tetrenH5)0.8{CuII4[WV(CN)8]4}·7.2H2O (WCu-t) of a size and quality appropriate for single-crystal magnetic measurements were obtained using a slow diffusion method in an H-tube according to the literature method.29 The conditions were set to ensure the free selfassembly of CuII aqua complexes, [W(CN)8]3− anions, and fully protonated tetrenH55+ in an acidic media (pH 1.5, HCl). The identity of the samples was confirmed by elemental analysis, IR spectroscopy, single-crystal X-ray diffraction, and preliminary test magnetic measurements (Figures S2−S4). Magnetic Properties. Magnetic measurements were carried out with a MPMS XL magnetometer from Quantum Design. The oriented crystals were mounted on a plastic plate with Apiezon grease (Figure S1). The mass of all mounted crystals was ∼0.5 mg. The plate with crystals was attached to a sample holder to obtain two orientations of the crystals: ac∥H and b∥H (where H is the applied field and a, b, and c are the crystallographic directions). The temperature dependences of

Figure 1. Crystal structure and magnetic anisotropy of WCu-t (all projections along the c direction): bilayered coordination architecture with a schematic illustration of the weak AF interlayer interactions zJ′ (top); local molecular structure with a scheme of the local magnetic exchange interactions along the cyanide bridges (bottom left); illustration of the easy plane and hard axis projected onto a platelike single crystal morphology (bottom right) according to the general order of the magnetic interaction strength Jeq > 10Jax ≫ |zJ′|.39,40 Color code: W, pink; Cu, orange; CCN, gray; NCN, pale blue; tetrenH55+ and H2O layers, yellow. 11972


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Inorganic Chemistry zero-field-cooled and field-cooled magnetization were measured under an applied field of 50 Oe in the temperature range of 2.0−50.0 K. Isothermal magnetization in a full magnetic field range, −7.0 to +7.0 T, was collected at 2.0 K. The magnetocaloric effect was calculated from magnetization data measured in the temperature range of 2.0−80.0 K, where the applied field was changed from 7.0 T to 0 T for each temperature to eliminate the influence from remanence. All of the magnetic measurements were performed for both orientations of the crystals. The diamagnetic corrections were determined by fitting the experimental data.

difference between Tpeak for both orientations rises. Another feature worth noting is the inverse magnetocaloric effect, which is observed only for the hard-axis geometry showing the largest absolute values for the highest variations of the magnetic field μ0ΔH = 6-0 T and 7-0 T: ΔSbmin ∼ −0.2 J K−1 kg−1. As was mentioned before, this inverse effect is possible because of easyplane anisotropy and the perpendicular orientation of the field to the easy plane (H∥b geometry). The differences between the easy-plane and hard-axis geometries of WCu-t motivated us to study the RMCE, which could be obtained simply by rotating the crystals around an axis within the ac plane by 90° in a constant field. The magnetic entropy change related to the rotation can be expressed then by ΔSR = ΔSac − ΔSb, where ΔSac and ΔSb stand for the magnetic entropy changes for H∥ac and H∥b, respectively. Figure 4 shows the temperature dependence of

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RESULTS AND DISCUSSION Experimental Insight. To evaluate the magnetocaloric effect, we used an indirect method and isothermal magnetization measurements collected in a wide temperature range of 2.0−80.0 K for H∥ac and H∥b orientations (Figure S5). The resultant temperature dependences of ΔS for both geometries were calculated by using the Maxwell relationship ΔSm(T,ΔH) = ∫ H0 max[∂M(T,H)/∂T[H dH. The results are shown in Figure 3

Figure 4. Magnetic entropy change related to the rotation by 90° in the constant field μ0H of single crystals of WCu-t from the H∥ac position to the H∥b position.

ΔSR. The ΔSR(T) curves reveal double maxima for fields up to 6.0 T and only one peak for the 7.0 T case (Figure S7). These rather untypical shapes correspond to different field dependences of Tpeak (Figure S8) for the easy plane and hard axis. The single maximum of ΔSR in 7 T is the result of the coincidence of Tpeak for both geometries, while for lower fields, there is a clear mismatch between them (for instance, in 3 T; Figure S9). Let us also note that the discrepancy between the positions of ΔSmax results in a small inverse RMCE for temperatures in which ΔSm for the hard axis is higher than that for the easy plane. Moreover, the inverse MCE detected in the H∥b orientation enhances the RMCE over the magnetic entropy change in the easy plane in a way described in our previous work.53 The temperature range and magnitude of this enhancement (up to 47% for 2.0 T at 8.0 K) of the RMCE are marked by the green field in Figure 5. Moreover, this figure also depicts the consequences of a mismatch between ΔSac and ΔSb, which led to the inverse RMCE (marked by the blue field in Figure 5). For fields up to 2.0 T, the magnitude of the inverse RMCE is larger than that of the inverse MCE for the hard axis. Although the size of these effects is still insufficient for commercial use, we have shown that a proper tuning of the magnetic anisotropy allows one to manipulate the direction of the magnetic entropy change. It is certain that magnetic systems with a comparable magnetic anisotropy but with ions contributing relatively higher magnetic moments will strongly improve the performance of the RMCE. Nevertheless, ΔSR observed for WCu-t has a magnitude comparable to that of the previously reported samples of dysprosium acetate tetrahydrate, [{Dy(OAc)3(H2O)2}2]·4H2O.51

Figure 3. Temperature dependence of the magnetic entropy change of WCu-t for (a) H∥b (hard axis) and (b) H∥ac (easy plane). The values of the magnetic field change are shown in the figure. The scales on both axes in parts a and b are the same.

for the applied field changes μ0ΔH = 0.5-0, 1-0, 2-0, 3-0, 4-0, 50, 6-0, and 7-0 T. When the ΔSm(T) curves are examined, clear differences between the H∥ac and H∥b orientations are apparent. The most obvious one is the maximum value of the magnetic entropy change ΔSmax, which for H∥ac is the largest for the biggest change of the magnetic field (Figure S6). For instance, for field variation μ0ΔH = 7-0 T K, ΔSmax for the ac plane is ΔSmax‑ac = 4.01 J K−1 kg−1 at Tpeak‑ac = 38.0 K, whereas for the hard axis, it is only ΔSmax‑b = 2.24 J K−1 kg−1 at Tpeak‑ac = 38.5 K. Another distinct property is the field dependence of the peak temperature (Tpeak) corresponding to the maximum value of ΔSm for a given field change (Figure S8). It turns out that only for the maximum field change (7-0 T) is Tpeak similar for H∥ac and H∥b, and as the change of field decreases, the 11973


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in mind that RMCE refrigerators will operate in a different manner than the MCE ones, and a direct comparison is misleading. In particular, the repetition rate of the cooling cycles in rotary magnetic refrigerators can be higher because no change of the magnetic field amplitude is needed. Theoretical Insight. To get insight into the precise cause of the appearance of the inverse MCE for the compound studied, we have analyzed a simplified model defined by the following Hamiltonian: Ĥ = −J ∑ (Six̂ Sjx̂ + Siŷ Sjŷ ) + gμB H ∑ Siẑ ij

Figure 5. Ratio of ΔSR and ΔSac as a function of the temperature for different magnetic fields. Values above 100% marked with a greenish background indicate the excess of RMCE over MCE for the ac∥H geometry, whereas those below 0% are marked by a bluish background.

i

(1)

where the first term corresponds to the spin−spin coupling, which is assumed to be anisotropic of the XY type (no coupling of the z components of the spins is present). The coupling is further assumed to be confined to the nearest neighbors, which is expressed by the summation over the nearest-neighbor spin pairs denoted symbolically by ⟨ij⟩. The spins reside on the bilayer system, i.e., two interacting monoatomic planes, each one of simple quadratic structure, which fixes unambiguously the coordination number of that lattice at z = 5 (no other regular structure with this number can be proposed). The second term is the Zeeman coupling with the external magnetic field H oriented perpendicular to the bilayer structure. J denotes the coupling constant, which is assumed to be ferromagnetic (J > 0), g is the spectroscopic factor, and μB denotes the Bohr magneton. To conform to the particular compound under study, all spins are assumed to be of magnitude 1/2. Our first approach was based on the standard molecular-field (MF) approximation. The strategy of this approximation consists of neglecting the correlations of the spins; i.e., one assumes expressions of the type (Sî − ⟨Sî ⟩)(Ŝj − ⟨Sĵ ⟩) = 0, where the angle brackets denote a quantum-mechanical and thermal average. Then the system of interacting spins can be shown to be described by a decoupled system, where the spin− spin interaction is replaced by the Zeeman coupling with an effective HMF given by the formula

Another important issue is the refrigerant capacity (RC), which informs us about the amount of heat transferred between the cold and hot sink in one cycle (assuming 100% Carnot efficiency).57 The RC was calculated as ∫ TTHCΔSm(T) dT, where TH and TC are the temperatures corresponding to the width at half-maximum of the signal of the hot and cold sinks, respectively. The field dependence of RC for WCu-t is shown in Figure 6, where the result for the easy-plane

Figure 6. Field dependence of the RC corresponding to the easy plane (red circles), hard axis (blue triangles), and rotation (green squares). Dotted lines are guides for the eyes.

HMF = −

zJ [⟨Sx̂ ⟩, ⟨Sŷ ⟩] gμ B

(2)

where the square brackets are to denote the assembly of two inplane components of the MF vector. Then the whole system is fully represented by a single spin governed by the MF Hamiltonian

geometry dominates over the whole magnetic field range. The RC for the RMCE is the smallest one up to ∼5.5 T, but above this field, it exceeds that for the hard-axis geometry. There is very scarce coverage of the RC values in molecular compounds in the literature. However, in many cases, it is possible to estimate the magnitude of the RC from the usually reported plots of the temperature dependence of the isothermal magnetic entropy change. A quick inspection reveals that the molecular magnets including lanthanide58−62 ions display the highest values of the RC on the order of 300−400 J/kg, while those based solely on the transition-metal ions63−67 show more modest RC values on the order of 50−150 J/kg (for the field change of μ0ΔH = 5 T). This substantial difference is related to the size of the constituent spin values, for which the 4f elements are the well-known record holders. In the case under study, where only the transition-metal ions CuII and WV are involved, the RC values fall onto the lower limit (∼40 J/kg), which is unsurprising for a compound comprising exclusively the possibly smallest spins of 1/2. Nevertheless, we have to keep

ĤMF = gμB (HMFxSx̂ + HMFySŷ + HSẑ )

(3)

which is a 2 × 2 Hermitean matrix. In eq 3, a scalar term of the form

Ng 2μB 2 2zJ

|HMF|2 has been neglected because it will not affect

the statistical distribution of the physical observables, although it is crucial for the correct expression of the system energy. The quantum-mechanical averages of the spin operators corresponding to the thermal equilibrium at temperature T are given by the formula ̂ )a = x , y , z ⟨Sâ ⟩ = Tr(Sâ ρMF

(4)

where the density matrix ρ̂MF = exp(−βĤ MF)/Tr[exp(−βĤ MF)] with β = 1/kBT (kB is the Boltzmann constant) can be readily calculated. The results read 11974


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Inorganic Chemistry 1 2

⟨Sâ ⟩ = −

relation in its integral form to calculate the isothermal entropy change due to the field change from Hi to Hf: ⎤ ⎡1 tanh⎢ βgμB |HMF|2 + H2 ⎥ ⎦a = x , y ⎣ 2 |HMF| + H HMFa

×

2

ΔS(ΔH = Hf − Hi) Hf ⎛ ∂M ⎞ ⎜ z ⎟ dH = Hi ⎝ ∂T ⎠ H

2

∫

(5)

1 ⟨Sẑ ⟩ = − 2

⎡1 ⎤ tanh⎢ βgμB |HMF|2 + H2 ⎥ ⎣ ⎦ 2 |HMF| + H H 2

1 = R 2

2

(6)

⎤ ⎡1 tanh⎢ βgμB |HMF|2 + H2 ⎥ = 1 ⎦ ⎣ 2 2 2 |HMF| + H zJ

mz =

hMF =

gμB |HMF| J

,

h=

gμ B H J

(8)

⎤ ⎡1 h 2 + h2 /t ⎥ 2 hMF 2 + h2 = z tanh⎢ ⎦ ⎣ 2 MF

(9)

(10)

Mz = 1/2NAgμB

h hMF 2

⎤ ⎡1 tanh⎢ hMF 2 + h2 /t ⎥ ⎦ ⎣2 + h2

The magnetization values mz have been calculated for a dense array of (t,h) values and then used in the discretized version of eq 10, where we set hi = 0 and hf=h (see Figure 8 for the

which in dimensionless variables kBT , J

i

⎛ ∂mz ⎞ ⎜ ⎟ dh ⎝ ∂h ⎠h

(11)

(7)

t=

hf

where the z component of the magnetization is given by the formula

Equations 2 and 5 yield a consistency condition, which is an implicit equation for |HMF|: 2gμB

∫h

reads

The fact that only the absolute value of HMF (or respectively hMF) is obtained is associated with the XY symmetry of the model; i.e., the spins in the ordered state (hMF ≠ 0) can assume any direction within the bilayer (xy) plane. Equation 9 was solved numerically for an array of (t,h) values (see Figure 7). It

Figure 8. Isothermal entropy change corresponding to the field change hi = 0 → hf = h, with field perpendicular to the bilayer (green) as obtained within the MF approximation. Within the red area, coinciding with the phase space region where the in-plane spontaneous order exists (hMF ≠ 0), the entropy change vanishes.

results). One can see that around tc = 1.25 the entropy change −ΔS displays a maximum, while within the phase space area where the in-plane spontaneous order exists (marked in red), it vanishes. Thus, the MF approach is inconclusive with regard to the appearance of the inverse MCE below the transition point, which was observed experimentally. It may be that if one duly accounts for the quantum correlations in this spin system, which the MF approximation completely neglects, the inverse MCE will be present. Hence, we decided for a more elaborate approach, called the cluster variational method in the pair approximation (CVMPA), which to some extent takes the spin−spin correlations into account. This strategy has been successfully employed in an array of spin 1/2 systems ranging from a bilayer68,69 through multilayers with antiferromagnetic interlayer coupling70 up to a diluted Heisenberg ferromagnet with interaction anisotropy71 in order to assess possible configurations in the phase space and/or estimate the thermal properties of the studied systems. The CVMPA method is based on a self-consistent determination of the Gibbs energy from which thermodynamic properties are subsequently derived. The Gibbs energy is given by the general formula

Figure 7. Temperature and field dependences of hMF. The in-plane ordered state (hMF ≠ 0) is destabilized by the external magnetic field h acting in the direction perpendicular to the bilayer.

can be seen that an in-plane ordered state (hMF ≠ 0) is destabilized by the external magnetic field h acting in the direction perpendicular to the bilayer. Thus, the temperature point where the spontaneous order appears shifts toward lower temperatures with increasing field values. For h = 0, the critical temperature tc = 1.25. Given the experimentally observed value of the transition temperature Tc = 33 K, this yields an estimate for the in-plane coupling constant J = 26.4 K. The MF approximation is known to overestimate the critical temperature; therefore, the value of 26.4 K sets genuinely a lower bound on the value of the coupling constant J. Our major goal is to estimate the magnetocaloric effect of the system. To this end, we employ the thermodynamic Maxwell 11975


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G = ⟨Ĥ ⟩ − ST

Because of the XY symmetry (rotational symmetry around the z axis) of the studied system, one can without loss of generality assume that Λy = Λy′ = 0, i.e., consider the case with the direction of the spontaneously ordered spins fixed along the x axis. Then, it is easy to find that the condition in eq 22 is satisfied when

(12)

where ⟨Ĥ ⟩ is the thermodynamic mean value of the Hamiltonian (magnetic enthalpy) and S is entropy of the system. Upon averaging of the Hamiltonian given in eq 1, one arrives at the formula ⟨Ĥ ⟩ = −

NzJ ̂ ̂ ⟨SixSjx + Siŷ Sjŷ ⟩ + NgμB H⟨Szî ⟩ 2

λx =

(13)

where N denotes the total number of spins (lattice sites). The enthalpy thus includes the pair correlation function (the first term in eq 13) as well as the single-site average (the second term in eq 13). These averages are calculated using the singlesite and pair density operators ρ̂1 and ρ̂2, respectively. They are defined by the formula ρα̂ = exp[β(Gα − Ĥ α)]α= 1,2

(14)

∂G(t , h , η) =0 ∂η

(15)

̂

(16)

Nz S = NS1 + (S2 − 2S1) 2

Sα = −kB⟨ln ρα̂ ⟩ = −kBTr(ρα̂ ln ρα̂ )α= 1,2

= (z − 1)η

(24)

(18)

similar to the case of the MF approximation, the external magnetic field h perpendicular to the bilayer destabilizes the spontaneously ordered state (in-plane order); i.e., the temperature at which the ordered state sets in shifts toward lower temperatures with increasing magnetic field. By contrast, a small phase space area comprising the lowest temperature and field values appears, where the MF parameter η vanishes. This suggests the possibility for a quantum critical transition to occur in this region. For h = 0, the critical temperature tc = 0.849, which is substantially lower than the value obtained previously in the MF approximation. Given the experimentally observed value of the transition temperature TC = 33 K, this yields an improved estimate for the in-plane coupling constant J = 38.9 K. The present treatment is again approximate, and the above value may be deemed as the lower bound on the true value of the coupling constant J. The value of J found here falls below the average intrabilayer exchange integral estimated in ref 39 by considering a model of bilayers coupled weakly by the dipole− dipole forces (J = 77.5 K) as well as the value of the equatorial exchange coupling derived from the DFT calculations performed for the exchange network of the WCu-t unit cell

(19)

Using eqs 12, 13, and 17−19 and demanding that the terms include the spin averages, one arrives at the final expression for the Gibbs free energy G z = G2 − (z − 1)G1 N 2

(20)

with additional consistency conditions

(z − 1)Λb = z Λb′

J

Figure 9. Temperature and field dependences of the MF (variational) parameter η.

where S1 and S2 are the single-site and pair entropies, respectively, which can be calculated using the single-site and pair density operators using

1 ̂ ⟨Sia + Sjâ ⟩2 2

gμ B Λ x ′

(17)

While G1 can be established analytically by explicitly diagonalizing the 2 × 2 single-site density matrix ρ̂1, the determination of G2 requires numerical calculations because the 4 × 4 pair density matrix ρ̂2 cannot be diagonalized explicitly. The entropy calculation can be carried out in approximation only because one is not able to diagonalize the density operator for the whole system. In the cumulant method, with accuracy to the second-order cumulants, the entropy reads

⟨Siâ ⟩1 =

λx′ =

The condition in eq 24 was solved numerically for an array of temperature and field values with an additional check of the sufficient condition for the minimum of the Gibbs energy; i.e., ∂2G/∂η2 > 0. The result is shown in Figure 9. One can see that,

Λa and Λa′ (where a = x, y) denote the MF parameters to be determined later. The single-site Gibbs energy G1 and the twosite Gibbs energy G2 are determined from the normalization conditions Trρ̂α = 1 (α = 1, 2), yielding Gα = −kBT ln[Tre−βHα]

and

where λx and λx′ are dimensionless counterparts of Λx and Λx′, respectively, and η is a dimensionless parameter that can be interpreted as the MF originating from the single nearestneighbor site. The Gibbs energy is thus a function of three dimensionless variables G = G(t,h,η). Requiring that the system is in a stable equilibrium state at a given temperature t and external field h, one must impose the necessary extremum condition for the Gibbs energy with respect to parameter η, i.e.

Ĥ 2 = −J(Six̂ Sjx̂ + Siŷ Sjŷ ) + gμB [Λx′(Six̂ + Sjx̂ ) + Λ y′ (Siŷ + Sjŷ ) + H(Siẑ + Sjẑ )]

J

= zη

(23)

where H1̂ = gμB (ΛxSix̂ + Λ ySiŷ + HSiẑ )

gμ B Λ x

(a = x , y , z )

(b = x , y )

(21) (22)

where angle brackets with subscripts 1 and 2 denote the averaging by the single site ρ̂1 and pair density operator ρ̂2, respectively. 11976


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Inorganic Chemistry (Jeq = 8.5 meV = 98.6 K).40 These discrepancies are most probably due to the fact that the model considered here assumes the perfect XY exchange anisotropy (no coupling of the z components of the spins is present in the Hamiltonian), while in the true system, the neglected z component of the exchange coupling may not vanish or may even be comparable to the in-plane components.40 One can then easily imagine that the presence of the spin coupling in the z direction would work as a destabilizing factor for the XY alignment of the spins and thus would require a larger magnitude of J to initiate the transition at the experimentally observed temperature. Upon calculation of the variational parameter η, the consistency conditions in eq 21 have been checked numerically. For a = y, the conditions are satisfied trivially because the magnetization in the y direction was assumed to vanish (Λy = Λy′ = 0). For a = x, the conditions are satisfied showing insignificant numerical errors. However, in the case of the z component of the spins (a = z), the numerical calculations revealed a substantial discrepancy localized in the small phasespace area comprising the lowest temperature and field values, where the MF parameter η was found to vanish (see Figure S10 for the temperature and field dependences of the difference 1 Δ⟨Sẑ ⟩ ≡ 2 ⟨Siẑ + Sjẑ ⟩2 − ⟨Siẑ ⟩1). First, this means that the approximate scheme breaks down in that part of the phase space and the ensuing results should be treated carefully there. Second, this indicates that for the lowest temperature and field values the phase stabilized in the studied system requires a more subtle theoretical approach. In particular, it should account simultaneously for ⟨Ŝiz + Ŝjz⟩2 = 0 and ⟨Ŝiz⟩1 ≠ 0, found in the present approximation and suggesting some kind of a dimerized state. To calculate the isothermal entropy change, we use eq 10, where the dimensionless magnetization mz is now given by the formula mz =

⎤ ⎡1 2 2 tanh⎢ z η + h2 /t ⎥ ⎦ ⎣ 2 zη +h

the pair spin−spin correlations were accounted for, MCE seems to depend crucially on the quantum spin fluctuations in this region of the phase space, where an ordered state is present. One can see from Figure 10 that around tc = 0.849 the entropy change −ΔS displays a maximum like in the case of the MF approximation. Unlike the MF approximation, the CVMPA approach reveals a second maximum located around t ≈ 0.03 corresponding to as low a temperature as T ≈ 1.2 K, which is why we failed to observe it experimentally. Finally and most importantly, for intermediate values of temperature (between 0.03 and tc = 0.849), the inverse MCE appears. Figure 11 shows the temperature dependence of the

Figure 11. Temperature dependence of the isothermal entropy change corresponding to the field change hi = 0 → hf = h for a selected set of field values (indicated in the plot).

isothermal entropy change −ΔS for a selected set of field values. It can be seen from Figure 11, as well as Figure 10, that when the field is increasing, the depression associated with the inverse MCE first deepens, attains a maximum for hmax ≈ 2, and then diminishes. Simultaneously, the temperature point where the inverse MCE sets in shifts smoothly toward lower temperatures. The dimensionless field hmax ≈ 2.0 corresponds to the field Hmax ≈ 58 T, which is far beyond the standard experimental capabilities. Figure S11 shows the experimental and calculated data for the isothermal entropy change corresponding to the field change in the direction perpendicular to the bilayers (hard axis) superimposed in a single plot using the reduced variables t and h as defined in eq 8. Let us note that the experimental field range (μ0ΔH = 0−7 T) covers only a very narrow interval of the reduced field variable (h = 0−0.24). It is apparent that the agreement between both sets of data is only qualitative: there is a peak around the transition temperature and a negative depression below the transition. Both features do also appear in the calculated data, but this is so for much larger values of the reduced field change h. However, let us stress that our major aim of the theoretical considerations was not a faithful modeling of the studied compound but a possible demonstration that the anisotropic spin−spin interactions favoring the in-plane alignment of the spins (proven experimentally for the studied compound) do lead to the occurrence of the inverse MCE.

h

2 2

2

(25)

Figure 10 shows the results of the calculations. In comparison to the previous result obtained within the MF

Figure 10. Isothermal entropy change corresponding to the field change hi = 0 → hf = h, with the field perpendicular to the bilayer (green) as obtained within the CVMPA approach.

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approximation (see Figure 8), there is a dramatic change of the calculated signal. First of all, in no region of the phase space does the isothermal entropy change vanish. Remembering that in the MF approximation the spin−spin correlations were completely neglected while in the CVMPA approach at least

CONCLUSIONS AND OUTLOOK In summary, we have studied the magnetocaloric effect in the bilayered cyanido-bridged compound {(tetren)H5)0.8CuII4[WV CN8]4·7.2H2O}n, which undergoes the BKT transition. The 11977


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strong magnetic anisotropy between the easy plane (H∥ac) and hard axis (H∥b) was used to investigate the RMCE. Despite the fact that the studied compound is based on ions with small magnetic moments, the observed entropy change in the RMCE has a magnitude comparable to that of dysprosium acetate tetrahydrate, [{Dy(OAc)3(H2O)2}2]·4H2O, and it is 1 order of magnitude larger than that in the 2D cyanido-bridged MnII− NbIV compound reported previously. To understand and estimate the inverse MCE, we have investigated a theoretical model with XY anisotropy with the MF and CVMPA approaches. The major conclusion is that the anisotropic spin−spin interactions favoring the in-plane alignment of the spins can lead to the inverse MCE. We believe that in the case of the compound under study the through-space dipole−dipole interactions serve as a natural origin of such an anisotropic coupling. The next step to find a highly efficient cryogenic rotating molecular magnetic cooler able to compete with the expensive He-3 refrigerators will be to test the compounds fulfilling the following conditions: (1) TC around 2 K; (2) strong magnetic anisotropy to achieve a large magnetic entropy change via rotation; (3) easy-plane character of the anisotropy to obtain the inverse MCE and thus to enhance RMCE below TC; (4) large magnetic moments coupled by ferromagnetic exchange. We suggest also an alternative approach to modulating the magnetic anisotropy of WCu-t, that is, by exploiting the anion−π approach. The underlying supramolecular interaction involves the electrostatic attraction between anions (or anionic groups) and positive potential above and below the specifically decorated or modified aromatic rings, e.g., with electronwithdrawing groups, heteroatoms, or assisting positive charge. Until now, the anion−π interactions were of interest from the viewpoint of organic catalysis,72 selective anion binding,73 biological activity,74 and others.75−77 Very recently, it was pointed out that they can significantly change the image of weak intermolecular magnetic interactions.78 It was also indicated that such interactions may also involve anionic complexes,79−81 including the polycyanide ones presented by some of us.81 Considering that above, the {CuII[WV(CN)8]}− bilayers (see Figure 1) can be treated as polymeric anionic complexes with triads of CN− ligands directed outside the layer and are able to form anion−π synthons using their nitrogen ending atoms, known to collect the majority of the accessible negative charge. We believe that by introducing different planar cations derived from the aromatic rings or ring systems and equipped with additional bulky groups we could modify the interplanar distances and interlayer magnetic interactions, thus modifying the RMCE characteristics.

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Piotr Konieczny: 0000-0002-0024-9557 Robert Pełka: 0000-0002-9796-250X Dominik Czernia: 0000-0003-3201-3765 Robert Podgajny: 0000-0001-7457-6799 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Polish National Science Center (Grant DEC-2013/11/N/ST8/01267). R.P. is thankful for financial support from the Polish National Science Centre within the OPUS-8 Project 2014/15/B/ST5/02098.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.7b01930.

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REFERENCES

Experimental and structural details and additional magnetic data (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. 11978


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