Towards mechanomagnetics in elastic crystals: insights from [Cu(acac)$_2$]

TL;DR

This study predicts that bending an elastic crystal of uacac significantly alters its magnetic properties, with potential implications for mechanomagnetic applications, though practical use remains limited due to low ordering temperatures.

Contribution

It introduces the concept of mechanomagnetics in elastic crystals and shows how bending dramatically affects magnetic ordering temperatures in uacac.

Findings

Bending increases the magnetic ordering temperature by 24 orders of magnitude.

uacac is an almost perfect Tomonaga-Luttinger liquid when unbent.

Geometric frustration suppresses magnetic ordering and enhances sensitivity to bending.

Abstract

We predict that the magnetic properties of \cuacac, an elastically flexible crystal, change dramatically when the crystal is bent. We find that unbent \cuacac\ is an almost perfect Tomonaga-Luttinger liquid. Broken-symmetry density functional calculations reveal that the magnetic exchange interactions along the chains is an order of magnitude larger than the interchain exchange. The geometrically frustrated interchain interactions cannot magnetically order the material at any experimentally accessible temperature. The ordering temperature ($T_N$), calculated from the chain random phase approximation, increases by approximately 24 orders of magnitude when the material is bent. We demonstrate that geometric frustration both suppresses $T_N$ and enhances the sensitivity of $T_N$ to bending. In \cuacac, $T_N$ is extremely sensitive to bending, but remains too low for practical applications, even when bent. Partially frustrated materials could achieve the balance of high $T_N$ and good sensitivity to bending required for practical applications of mechanomagnetic elastic crystals.

Tables

Table 1. Table 1: The ordering temperatures in units of J ∥ subscript 𝐽 parallel-to J_{\parallel} of various quasi-one-dimensional molecular crystals found in the literature along with the value predicted for [ [ [ Cu(acac) ] 2 {}_{2}]  in this paper. A lower value of T N / J ∥ subscript 𝑇 𝑁 subscript 𝐽 parallel-to T_{N}/J_{\parallel} indicates a material closer to the 1D limit.

Material	$T_N/J_{\parallel }$	Ref.
Cu3(CO3)2(OH)2	0.19	26
Cs2CuCl4	0.15	27
KCuF3	0.10	21; 28; 29
$[$Cu(pz)(pyO)2(H2O)${}_2{}^{}]$ (PF6)2	0.03	30
$[$Cu(pz)(NO3)${}_2{}^{}]$	0.01	23
$[$Cu(acac)${}_2{}^{}]$	${10}^{-33}$	This work

Table 2. Table 2: Heisenberg exchange ( J i ​ j subscript 𝐽 𝑖 𝑗 J_{ij} ) parameters for the unbent structure of [ [ [ Cu(acac) ] 2 {}_{2}]   determined with BS-DFT. J ∥ subscript 𝐽 parallel-to J_{\parallel} and J ⟂ 1 subscript 𝐽 perpendicular-to absent 1 J_{\perp 1} are antiferromagnetic and J ⟂ 2 subscript 𝐽 perpendicular-to absent 2 J_{\perp 2} is ferromagnetic. The distances between Cu atoms for each dimer are also reported. Axes are shown in Fig. 2 .

	Direction	Cu$\leftrightarrow$Cu (Å)	$J_{ij}/k_B$ (K)
$J_{\parallel }$	$\pm 𝒃$	4.643	0.75
$J_{⟂1}$	$\pm 𝒃/2\pm \left(𝒂+𝒄\right)$	7.818	0.04
$J_{⟂2}$	$\pm 𝒃/2\pm \left(𝒂-𝒄\right)$	8.133	-0.10

Equations

$${\mathcal{H}}_{\mathrm{Heisenberg}}=\sum _{ij}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j,$$

$$J_{ij}=2\frac{E_{ij}^{\mathrm{BS}}-E_{ij}^{\mathrm{T}}}{⟨S^2{⟩}_{ij}^{\mathrm{BS}}-⟨S^2{⟩}_{ij}^{\mathrm{T}}},$$

$$T_N\approx \Lambda e^{-2.68R_J^2},$$

$$T_N^{\mathrm{cubic}}\approx 0.56\frac{J_{\parallel }}{k_B{R}_J}\sqrt{\log \left(\frac{\Lambda }{T_N}\right)},$$

Full text

Introduction

Crystal adaptronics is a new and exciting field, bolstered by the recent discovery of elastically flexible molecular crystals.Chen et al. (2014); Ghosh et al. (2015a, b); Hayashi and Koizumi (2016); Worthy et al. (2018); Hayashi and Koizumi (2018); Ahmed et al. (2018) These crystals can be bent without irreversibly changing their structure. The mechanism by which the molecules can elastically slip past each other is beginning to be understood.Reddy et al. (2006); Mishra et al. (2016); Saha et al. (2018); Brock et al. (2018); Zakharov and Boldyreva (2019) However, there are limited examples of the modification of functional properties; the most successful so far being mechanochromism.Hayashi and Koizumi (2016); Hayashi et al. (2017, 2018)

In this paper, we explore the possible changes in magnetic properties induced by bending $[$Cu(acac)${}_{2}]$ (acac$=$acetylacetonate), a recently discovered elastically flexible crystal. Brock et al. (2018); Worthy et al. (2018) We discuss how the geometry of the crystal leads to these changes in the hope of motivating a search for elastic crystals with similar geometry, but larger exchange interactions.

$[$Cu(acac)${}_{2}]$ is an extremely well known material. It is a commercially available reactant used in numerous organic and organometallic syntheses and is often made in undergraduate chemistry laboratories. Worthy et al.Worthy et al. (2018) published atomically resolved structural information across bent samples, providing the opportunity to use first-principles calculations to model how its magnetic properties change as the material is bent.

We find that, apart from being elastic, $[$Cu(acac)${}_{2}]$ has exotic quantum magnetic properties – it is an almost perfect quasi-one-dimensional magnet. The frustrated geometry of the crystal lattice enhances this low dimensionality and also leads to extreme sensitivity of the magnetic properties to bending. $[$Cu(acac)${}_{2}]$’s partnership of elasticity and geometrical frustration lead to it being an excellent prototype for applications for elastic crystals. We predict that the change in geometry of $[$Cu(acac)${}_{2}]$, brought on by bending, will lead to its magnetic ordering temperature changing by approximately 24 orders of magnitude. This demonstrates the possibility of using elastic flexible crystals to passively sense small deformations or flexures with extremely high precision.

Passive flex sensors often operate with a change in electrical resistivity. They are useful for measuring physical activity or joint movement in the human body, for facilitating human-computer interactions, for monitoring machines, and for measurement devices (for example measuring the curvature of a small surface).Saggio et al. (2016) A material with dramatic magnetic changes caused by bending, such as $[$Cu(acac)${}_{2}]$, could also be used for these purposes down to the micrometer scales. The magnetic ordering temperature, which can be detected via the concomitant divergence in the magnetic susceptibility, can change by many orders of magnitude; such devices could have sensitivities far exceeding those of resistive devices.

The behavior of flexible quantum magnets, a new field opened by the discovery of elastic crystals, allows one to examine many new questions of fundamental importance. Low dimensional magnetic crystals display fascinating quantum phenomena. Particularly, one-dimensional materials exhibit fractionalized excitations and strong quantum fluctuations that prohibit long range magnetic order.Balents (2010); Savary and Balents (2017) Spin-1/2 one-dimensional Heisenberg chains are described by Tomonaga-Luttinger liquid (TLL) theory. Tomonaga (1950); Luttinger (1963); Haldane (1981) An important prediction of TLL theory is that there will be a continuum of low-energy excitations, which are indeed observed in neutron scattering experiments. Lake et al. (2005) Quasi-one-dimensional crystals contain weak interchain interactions, which become significant at low temperatures and lead to Néel ordering below a certain temperature, $T_{N}$. These materials can be understood as weakly coupled chains. However, at low enough temperatures, interchain interactions eventually cause long-range magnetic order.

Copper II molecular crystals are well known for their exotic magnetic properties. Landee and Turnbull (2013) One of the best examples of a quasi-one-dimensional molecular crystal is copper pyrazine dinitrate, [Cu(pz) (NO3)2] (pz$=$pyrazine), which orders magnetically at 0.107 K Lancaster et al. (2006) and was recently shown to exhibit 1D quantum criticality. Breunig et al. (2017) Its magnetic low dimensionality has been confirmed with density functional theory calculations, which give an interchain coupling of $J_{\perp}=0.0044J_{\parallel}$, where $J_{\parallel}$ is the intrachain coupling.Jornet-Somoza et al. (2010) The extent to which a material is 1D can be quantified with $T_{N}/J_{\parallel}$. Table 1 shows some of the lowest values found to date.

Below, we demonstrate that unbent $[$Cu(acac)${}_{2}]$ is an almost perfect TLL that does not order magnetically at any experimentally accessible temperature. We establish this through a combination of first principles electronic structure calculations and quantum many-body theory, revealing that the presence of geometrical frustration in the lattice (see Fig. 1) causes two major effects: (i) $[$Cu(acac)${}_{2}]$’s extreme magnetic one-dimensionality and (ii) the significant change the Néel temperature, $T_{N}$, when the material is bent.

We parametrize a Heisenberg Hamiltonian via broken-symmetry density functional theory (BS-DFT), Noodleman (1981); Mouesca (2014) which reveals three significant exchange couplings between neighboring molecules, $J_{\parallel}$, $J_{\perp 1}$, and $J_{\perp 2}$ (shown in Fig. 2). The magnitude of the exchange coupling along the crystallographic $b$-axis ($J_{\parallel}$) is much larger than the couplings in the other directions, indicating that $[$Cu(acac)${}_{2}]$ can be modelled as weakly coupled Heisenberg spin-1/2 chains. The interchain couplings, $J_{\perp 1}$ and $J_{\perp 2}$, are both geometrically frustrated (see Figs. 1 and 2), maintaining $[$Cu(acac)${}_{2}]$ in the 1D limit.

We use the chain random phase approximation (CRPA) Bocquet et al. (2001) to predict the Néel temperature, magnetic susceptibility and dynamical structure factor of the unbent crystal. The measured susceptibility is in good agreement with our calculations. When the crystal is bent, the ratio of intra to interchain couplings changes significantly. This leads to a change in Néel temperature of 24 orders of magnitude, demonstrating the dramatic potential of mechanomagnetics.

Computational Details and Theoretical Methods

We use the unbent and bent $[$Cu(acac)${}_{2}]$ crystal structures from Worthy et al.Worthy et al. (2018) Three nearest neighbor exchange pathways are shown in Fig. 2. In terms of the crystallographic axes we label $J_{\parallel}$ to be along $b$. The four nearest neighbour interactions in the $\pm\left(b/2\pm\left(a+c\right)\right)$ directions are equal (by symmetry) and we label them $J_{\perp 1}$. Similarly, we label the four nearest neighbour interactions along $\pm\left(b/2\pm\left(a-c\right)\right)$ as $J_{\perp 2}$.

When the crystal is bent, the lattice parameters change approximately linearly as a function of position across the bend.Worthy et al. (2018) On the inside of the bend, the $b$-axis is compressed while the $a$ and $c$ axes are stretched. Conversely, on the outside of the bend, the $b$-axis stretches while $a$ and $c$ are compressed. The $\beta$ angle increases approximately linearly from the outside to the inside of the bend. However, the individually measured atomic coordinates are not as precise as those from bulk crystals due to the small effective sample size. We therefore created a linearized set of lattice parameters using crystallographic data for two bends with different radii of curvature, $r_{c}=1.2\,$mm and $r_{c}=3.2\,$mm. We then used these parameters to produce a new smooth set of model structures assuming constant intra-molecular bond lengths. Details of this process and a plot of the lattice parameters across each bend, including our linear fits, are given in the Supplementary Information.

We parametrize a Heisenberg model,

[TABLE]

where $\bm{S}_{i}$ is the spin operator on the $i$th molecule and $J_{ij}$ are the exchange coupling constants. The sign of $J$ indicates an antiferromagnetic ($J\!>\!0$) or ferromagnetic ($J\!<\!0$) interaction.

We calculate the exchange couplings, $J_{ij}$, within $[$Cu(acac)${}_{2}]$ using broken-symmetry density functional theory (BS-DFT),Noodleman (1981); Mouesca (2014) along with the Yamaguchi spin decontamination procedure.Yamaguchi et al. (1988) In this approach,

[TABLE]

where $E^{\mathrm{T}}_{ij}$ is the triplet energy of the isoloated dimer containing molecules $i$ and $j$, and $E^{\mathrm{BS}}_{ij}$ is the energy of the broken-symmetry state on that same dimer. $\langle S^{2}\rangle^{\mathrm{BS}}_{ij}$ and $\langle S^{2}\rangle^{\mathrm{T}}_{ij}$ are the corresponding expectation values of the spin operator, $S^{2}$. Calculations were perfomed in Gaussian09 Frisch et al. with the uB3LYP functional Becke (1993); Stephens et al. (1994) and using the LANL2DZDunning Jr. and Hay (1977); Hay and Wadt (1985a); Wadt and Hay (1985); Hay and Wadt (1985b) (for Cu) and 6-31+G*Ditchfield et al. (1971); Hehre et al. (1972); Hariharan and Pople (1973); Francl et al. (1982) basis sets with an SCF convergence criterion of $10^{-10}$ a.u. Benchmarking of $J_{ij}$ using different basis sets and functionals is discussed in the Supplementary Information.

The dynamical magnetic susceptibility for a single Heisenberg chain can be calculated from a combination of the Bethe ansatz and quantum field theory techniques. Bethe (1931); Yang and Yang (1966); Schulz and Bourbannais (1983); Eggert et al. (1994); Schulz (1986); Affleck (1998); Barzykin (2000); Lukyanov (1998); Luther and Peschel (1974); Tsvelik (2003) Within the chain random phase approximation (CRPA), the full three-dimensional dynamical susceptibility is a function of the interchain coupling, $J_{\perp}$ (see Eq. S.3).Scalapino et al. (1975); Schulz (1996); Essler et al. (1997); Bocquet et al. (2001) The CRPA susceptibility is valid above the Néel temperature, $T_{N}$. Generically, one expects an RPA treatment to overestimate $T_{N}$. However, the geometrical frustration in $[$Cu(acac)${}_{2}]$ enhances the range of validity of this approximation; the CRPA has been compared with numerical methods and found to be accurate for $|J_{\perp}|<0.1\,J_{\parallel}$ on a geometrically unfrustrated lattice Yasuda et al. (2005) and $|J_{\perp}|<0.7\,J_{\parallel}$ for a frustrated lattice.Starykh et al. (2010) One can determine $T_{N}$ by considering the condition for a zero frequency pole in the CRPA expression for the dynamical susceptibility. Details of this calculation are given in the Supplementary Information.

We use the CRPA to predict a number of experimentally measurable properties of $[$Cu(acac)${}_{2}]$. We fit the CRPA, using the Bonner-Fisher chain susceptibilityBonner and Fisher (1964); Estes et al. (1977); Ami et al. (1995) to the experimental bulk susceptibility above 2 K.Moreno et al. (2013) We then predict the low-temperature CRPA susceptibility ($T_{N}<T<1.5$ K) with the temperature dependent bulk susceptibility of a single antiferromagnetic Heisenberg chain, calculated numerically by Eggert et al.Eggert et al. (1994); Egg The bulk magnetic susceptibility will diverge, undergoing a second order phase transition, at $T_{N}$. The dynamical structure factor (measured in inelastic neutron-scattering experiments) can also be calculated with the CRPA susceptibility (see S.19). Details of the experimental predictions are also given in the Supplementary Information.

Results and Discussion

Unbent Crystal. The three distinct BS-DFT nearest neighbor exchange interactions in the unbent crystal, along with their crystallographic directions, are reported in Table 2. All longer range interactions that we calculated are smaller than the accuracy limit of our DFT results.

The exchange coupling ratios, $J_{\perp 1}/J_{\parallel}=0.06,\ J_{\perp 2}/J_{\parallel}=-0.13,$ indicate a low dimensionality in the magnetic degrees of freedom in $[$Cu(acac)${}_{2}]$. In the limit $J_{\perp 1}=J_{\perp 2}=0$, one has independent Heisenberg chains which are Tomonaga-Luttinger liquids (TLLs) at low temperatures. Haldane (1980) However, when there are interactions between chains (i.e. $J_{\perp 1},J_{\perp 2}\neq 0$), the TLL will undergo a phase transition into a Néel ordered state below a critical temperature, $T_{N}$.

Using the CRPA susceptibility (details given in the Supplementary Information), we find that the Néel temperature of $[$Cu(acac)${}_{2}]$ is given by

[TABLE]

where $R_{J}=J_{\parallel}/(|J_{\perp 1}|+|J_{\perp 2}|)$ is the ratio of the intrachain coupling to the interchain couplings (see Table 2 and Figure 2) and $\Lambda=24.27J_{\parallel}/k_{B}$ is a non-universal parameter.Barzykin (2001) Evaluating Eq. 3 for the unbent $[$Cu(acac)${}_{2}]$ crystal gives $T_{N}\approx 1\times 10^{-33}$ K. Thus, we predict that the unbent crystal of $[$Cu(acac)${}_{2}]$ will be magnetically disordered down to the lowest experimentally reachable temperatures – experimentally $[$Cu(acac)${}_{2}]$ will appear as an almost perfect TLL. To highlight the extreme one-dimensionality of $[$Cu(acac)${}_{2}]$ compared to other materials, one can make the striking comparison of $T_{N}/J_{\parallel}\approx 10^{-33}$ to the other materials in Table 1.

Given the form of Eq. 3, it is clear that $T_{N}$ is very sensitive to $R_{J}$; because $T_{N}$ decays exponentially as a function of $R_{J}^{2}$, a small change in $R_{J}$ leads to a dramatic change in $T_{N}$. This extreme sensitivity is caused by the geometry of $[$Cu(acac)${}_{2}]$; the interchain interactions are geometrically frustrated, as illustrated in Fig. 2. It is instructive to compare the $T_{N}$ calculated above with that of an unfrustrated analogue – a cubic lattice where $J_{\perp 1}$ and $J_{\perp 2}$ are the same magnitude as in $[$Cu(acac)${}_{2}]$, but their directions are perpendicular to $J_{\parallel}$ (see Fig. 1a). The same CRPA calculation as above then results in

[TABLE]

which yields $T_{N}^{\mathrm{cubic}}\approx 0.23J_{\parallel}/k_{B}\approx 0.17\,\mathrm{K}$ using the parameters in Table 2. This is 32 orders of magnitude higher than $T_{N}$ for the frustrated $[$Cu(acac)${}_{2}]$ lattice. Moreover, in contrast to the exponential dependence of Eq. 3, $T_{N}^{\mathrm{cubic}}$ is proportional to $1/R_{J}$ – it is larger and less sensitive to small changes in the value of $R_{J}$. This will be important when we discuss the bent crystals.

The large contrast between geometrically frustrated and unfrustrated interactions is also demonstrated in previous work on the 2D anisotropic triangular lattice Heisenberg model, for Cs2CuCl4 in particular.Starykh et al. (2010); Bocquet et al. (2001)

We predict that the experimental properties of $[$Cu(acac)${}_{2}]$ will closely mimic an isolated spin-1/2 Heisenberg chain, displaying properties of an almost perfect TLL. The CRPA prediction of the bulk magnetic susceptibility using the exact 1D theory is limited to the low-temperature regime studied by Eggert et al. Eggert et al. (1994) Conversely, the bulk susceptibility of $[$Cu(acac)${}_{2}]$ has only been measured above 2 K, with no magnetic ordering detected. Moreno et al. (2013) Therefore, to compare our prediction with experiment, we first fit the CRPA using the Bonner-Fisher susceptibility of a single spin chain, which is successful in other materials at higher temperatures.Bonner and Fisher (1964); Estes et al. (1977); Ami et al. (1995) We set $J_{\parallel}=0.75$ K (our BS-DFT result) and found that the best fit corresponded to $J_{\perp 1}+J_{\perp 2}=0.14$ K, in reasonable agreement our BS-DFT results for the interchain couplings. We used this value of $J_{\perp 1}+J_{\perp 2}$ to parametrize our low-temperature prediction. More details of the fit are given in the Supplementary Information. Fig. 3(a) shows the Bonner-Fisher fit to the experimental data from Moreno et al.Moreno et al. (2013) and our prediction of the low temperature magnetic susceptibility. The agreement is exceptional.

The dynamical structure factor for $[$Cu(acac)${}_{2}]$ has not been measured. Our dynamical structure factor prediction in Fig. 3(b) (measurable via neutron scattering experiments) was calculated with our BS-DFT exchange parameters (Table 2). It shows a slight asymmetry, which is absent for a TLL in an isolated Heisenberg chain at low temperatures. There are no adjustable parameters in this prediction.

Bent Crystal. Our BS-DFT results across the bent crystals of $[$Cu(acac)${}_{2}]$ are shown in Figure 4, where we plot $J_{\parallel}$, $J_{\perp 1}$, and $J_{\perp 2}$ as ratios of the unbent parameters across each of the bent crystals. We calculate that the interchain coupling changes by over 20% as a consequence of the crystal distortion.

Figure 5 shows $R_{J}^{2}$ and $T_{N}$ across the bent crystals. The change in geometry brought on by bending the $[$Cu(acac)${}_{2}]$ crystals causes a significant change in magnetic behaviour at different points across the bend; a small change in $R_{J}^{2}$ causes a very large change in the ordering temperature. In the most bent crystal, this means a change in $T_{N}$ of 24 orders of magnitude from one side of the bend to the other.

When the lattice is strained by bending, this causes a simultaneous, but opposite, change in $J_{\parallel}$ and the perpendicular couplings, $J_{\perp 1}$ and $J_{\perp 2}$, relative to the center of the bent crystal – illustrated in Fig. 6. On the inside of the bend, the distance between copper atoms along the chain is smaller than in the center, due to the compression of the lattice along the $b$-axis, leading to a relative increase in $J_{\parallel}$. Whereas, the distance between the chains increases because the lattice is expanded along the $a$ and $c$ axes relative to the center, decreasing $J_{\perp 1}$ and $J_{\perp 2}$. Both of these processes independently decrease $T_{N}$. On the outside of the bend, the opposite effect occurs; the $b$-axis is elongated causing $J_{\parallel}$ to decrease and the $a$ and $c$ axes are compressed causing the interchain couplings to increase, leading to an increase in $T_{N}$.

The transition at $T_{N}$ is an antiferromagnetic transition, which could be detected with the divergence of the magnetic susceptibility. We predict that, in a bent crystal, the bulk magnetic susceptibility would be a superposition of single chain transitions resulting from the different $T_{N}$ values at different points across the crystal.

Geometric frustration plays a vital role in this dramatic change in $T_{N}$ across the bend; the extreme sensitivity of $T_{N}$ to the changes in the crystal described above is due to the exponential dependence of $T_{N}$ on $R_{J}$ (Eq. 3). If the lattice was cubic, the Néel temperature would have stronger proportionality to $R_{J}$ (Eq. 4), and one would not observe such a dramatic change in $T_{N}$ (also, the unbent crystal would have a much larger $T_{N}$).

Conclusions

In conclusion, we predict that the magnetic ordering temperature of elastically flexible $[$Cu(acac)${}_{2}]$ changes dramatically when the material is bent. The unbent crystal will behave, experimentally, like an almost perfect TLL (i.e. uncoupled 1D spin chains). When the sample is bent perpendicular to the chain direction, the crystal geometry changes in such a way to maximally affect the value of the Néel temperature, $T_{N}$. A stretched crystal with a change in exchange couplings of 20% has a theoretical ordering temperature of 0.01 mK, which is 24 orders of magnitude higher than the unbent crystal, with a Néel temperature of $\sim 10^{-33}$ K. This change in $T_{N}$ across a bend would be experimentally evidenced by measuring the bulk susceptibility. The interchain interactions only weakly renormalize the properties of $[$Cu(acac)${}_{2}]$ relative to a single Heisenberg chain. This is due in part to the weakness of the interchain couplings, $J_{\perp 1}$ and $J_{\perp 2}$, but mostly to the presence of geometric frustration in the lattice; geometric frustration leads to the exponential suppression of the Néel temperature, stabilizing the Tomonaga-Luttinger spin-liquid phase. Our results provide a powerful proof-of-principle demonstration that magnetic interactions can be controlled via bending flexible crystals. We have demonstrated the possibility of using elastic flexible magnetic crystals to passively sense small deformations, curvatures, or flexures with extremely high precision by detecting the divergence of the magnetic susceptibility in the sample.

$[$Cu(acac)${}_{2}]$ has a Néel temperature that is highly sensitive to bending, but its extreme geometric frustration means that $k_{B}T_{N}$ is many orders of magnitude smaller than the magnetic exchange interactions. Therefore, simply increasing the exchange couplings would not be expected to lead to experimentally accessible Néel temperatures. Rather, as highlighted in Table 1, the extreme geometrical frustration of $[$Cu(acac)${}_{2}]$ is actually responsible for the low $T_{N}$. This suggests that an incompletely frustrated material may open the door to mechanomagnetics at experimentally accessible temperatures. However, our results show that there is a trade-off. Geometrical frustration also enhances the sensitivity of $T_{N}$ to bending. Unfrustrated coupling leads to a higher, measurable $T_{N}$ but lowers its sensitivity. Therefore, partial geometrical frustration, e.g., imperfectly triangular couplings perpendicular to the chain, could provide a balance with both a high $T_{N}$ and a strong sensitivity to bending. We hope that this insight will play a key role in the future search and design of elastically flexible mechanomagnetic crystals.

Acknowledgement

The authors thank Jack Clegg, Arnaud Grosjean, Amie Khosla, and Ross McKenzie for helpful conversations. We are indebted to Sebastian Eggert for making the data from Ref. 49 available at Ref. 65. This work was supported by the Australian Research Council through Grants No. FT130100161 and DP160100060.