Elastic relaxation behavior, magnetoelastic coupling, and order-disorder processes in multiferroic metal-organic frameworks

Resonant ultrasound spectroscopy has been used to analyze magnetic and ferroelectric phase transitions in two multiferroic metal-organic frameworks (MOFs) with perovskite-like structures [(CH${}_3$)${}_2$NH${}_2$]M(HCOO)${}_3$ (DMA[M]F, M $=$ Co, Mn). Elastic and anelastic anomalies are evident at both the magnetic ordering temperature and above the higher temperature ferroelectric transition. Broadening of peaks above the ferroelectric transition implies the diminishing presence of a dynamic process and is caused by an ordering of the central DMA ([(CH${}_3$)${}_2$NH${}_2$]${}^{+}$) cation which ultimately causes a change in the hydrogen bond conformation and provides the driving mechanism for ferroelectricity. This is unlike traditional mechanisms for ferroelectricity in perovskites which typically involve ionic displacements. A comparison of these mechanisms is made by drawing on examples from the literature. Small elastic stiffening at low temperatures suggests weak magnetoelastic coupling in these materials. This behavior is consistent with other magnetic systems studied, although there is no change in $Q^{-1}$ associated with magnetic order-disorder, and is the first evidence of magnetoelastic coupling in MOFs. This could help lead to the tailoring of MOFs with a larger coupling leading to magnetoelectric coupling via a common strain mechanism.

Figure 1

Building block of DMAMnF showing the DMA cation disordered over three positions in the center of the ReO${}_3$ type cavity of the Mn-formate framework. Mn is shown in purple, O in red, C in grey, H in white, and N in blue. Note the disorder of the central nitrogen over three positions. Hydrogens associated with the central cation are omitted for clarity.

Figure 2

Selected RUS spectra of DMACoF displayed as a stack plot. The $y$ axis should be the amplitude in volts, but the spectra have been offset in proportion to the temperature at which they were collected and the axis is therefore labeled as temperature. Note the disappearance of most peaks above $\sim$170 K.

Figure 3

Values of $f^2$ (red crosses) and $Q^{-1}$ (blue circles) for DMACoF sample 1 and $Q^{-1}$ for sample 2 (green circles). Linear baselines and fits of Eq. (1) to $Q^{-1}$ data shown as solid lines. Fit coefficients for sample 1: $Q_m^{-1}=$ 0.025, $T_m=$ 200 K, $r_2\left(\beta \right)=1$, and $E_{\mathrm{a}}=20$ kJ/mol; and for sample 2: $Q_m^{-1}=$ 0.046, $T_m=$ 211 K, $r_2\left(\beta \right)=1$, and $E_{\mathrm{a}}=25$ kJ/mol. Temperatures of phase transitions are displayed as vertical black lines.

Figure 4

Low temperature values of $f^2$ (crosses) and $Q^{-1}$ (open circles) for DMACoF (red) and DMAMnF (black) as well of fits of Eq. (2) as solid lines. Fit coefficients for DMACoF are $a_1=1.6335\times {10}^{11}$, $a_2=-1.3136\times {10}^8$, and ${\theta }_{\mathrm{s}}=26.244$ K, and for DMAMnF are $a_1=3.2065\times {10}^{12}$, $a_2=-5.8553\times {10}^{17}$, and ${\theta }_{\mathrm{s}}=12.462$ K. Transition temperatures are shown as vertical lines. Note the change of scale on the left axis.

Figure 5

(a) Enlargement of low temperature variations of $f^2$ for several peaks of DMACoF scaled to appear in the range $(159.2–160.2)\times {10}^9$ Hz${}^2$. (b) Low temperature variation of $f^2$ for several peaks of DMAMnF scaled to appear in the range $(340–346)\times {10}^{12}$ Hz${}^2$. Magnetic transition temperatures are displayed as solid black lines. Peaks are labelled with their frequency values at $\sim$5 K.

Figure 6

Stack plot of RUS spectra for DMAMnF, with peaks analyzed in Fig. 7 labeled, and spectra offset on the $y$ axis in proportion to the temperature at which they were collected. Some spectra are omitted for clarity.

Figure 7

Variation with temperature of $f^2$ (open symbols) and $Q^{-1}$ (closed symbols) for peaks labeled in Fig. 6. Phase transitions are displayed as black solid lines. Data were scaled so as to overlap at $\sim$10 K.

Figure 8

(a) Fit of Eq. (3) to heat capacity data from Jain et al.[32] in the temperature region 0–30 K, and (b) the excess entropy associated with magnetic ordering in DMAMnF.