Giant elastocaloric effect at low temperatures in TmVO₄ and implications for cryogenic cooling

We adopt a coordinate system in which unprimed coordinates refer to the principle crystal axes, and primed coordinates are rotated by 45 degrees about the c-axis.

Stress applied along the [110] crystallographic direction induces a combination of symmetric and antisymmetric (i.e. shear) strains. Using a reference frame in which $x^{\prime }$ is along the stress direction, the total observed elastocaloric effect, $\mathrm{\Delta }T/\mathrm{\Delta }{\epsilon }_{x^{\prime }{x}^{\prime }}$ reflects contributions from both the symmetric and antisymmetric strains. Since the elastic stiffness tensor of TmVO${}_4$ is known (24), measurement of ${\epsilon }_{x^{\prime }{x}^{\prime }}$ under conditions of uniaxial stress is sufficient to determine all of the terms in the strain tensor, and simple coordinate transformations can be used to go between unprimed and primed coordinate systems. Details associated with strain relaxation effects are discussed in SI Appendix.

Of the two symmetry-decomposed strains that the sample experiences as a consequence of the [110] stress, antisymmetric strain has a much larger effect on the entropy, and hence on the measured elastocaloric temperature oscillations. Shear strain, ${\epsilon }_{\mathit{xy}}$, couples to the quadrupole moments bilinearly (i.e. $H_{\mathit{eff}}\sim {\epsilon }_{\mathit{xy}}{P}_{\mathit{xy}}$, where $P_{\mathit{xy}}$ is the operator that describes the $\mathit{xy}$ quadrupole, $P_{\mathit{xy}}=1/2(J_x{J}_y+J_y{J}_x)$, and hence has a large effect on the entropy at low temperatures. In contrast, symmetric strain [both $1/2({\epsilon }_{x^{\prime }{x}^{\prime }}+{\epsilon }_{y{}^{\prime }y{}^{\prime }})$ and ${\epsilon }_{\mathit{zz}}$] does not split the CEF groundstate above $T_Q$, and consequently has a much smaller effect on the observed total elastocaloric effect. The small contribution to the elastocaloric temperature oscillations that arises from symmetric strains can be readily identified based on symmetry grounds (SI Appendix), and subtracted from the total signal to yield the part of the ECE that arises purely from $\mathit{xy}$ shear: $\partial T/\partial {\epsilon }_{\mathit{xy}}$. This quantity is plotted as a function of temperature for several representative DC offset strains in Fig. 3. These data were taken by sweeping temperature while a fixed bias voltage was applied to the outer stacks of the CS100 strain cell. Strain values are listed at 10 K; subtle changes in strain with temperature are accounted for in the subsequent analysis and are used here only as labels.

Inspection of Fig. 3 reveals several key features. First, the observed ECE is odd with respect to the bias strain. That is, the sign of the ECE is either positive or negative depending on the sign of the bias strain. This is a natural consequence of antisymmetric deformations for which the AC component either adds or subtracts from the DC bias strain. Second, the data reveal a characteristic “fishtail” shape, in which the maximum value is observed close to 3 K, reflecting the slope of the entropy contours indicated in Fig. 2. Finally, the magnitude of the temperature oscillations is seen to be very large. For example, for a bias strain of only $4·{10}^{-4}$ (0.04 %), the amplitude of the temperature oscillation (normalized by the amplitude of the AC strain oscillation, which is $5·{10}^{-5}$) is approximately 700 K in dimensionless units of strain, or equivalently 7 K per percent of strain.

The data shown in Fig. 3 can be replotted, now as a function of strain, ${\epsilon }_{\mathit{xy}}$; this is shown in Fig. 4. Because the strain is recorded for every temperature, these curves do not suffer from the same concerns about strain changing as a function of temperature. Data are shown for a range of temperatures, each a different color. For comparison, the AC strain amplitude is also indicated. As can be readily seen, for small strains, $\partial T/\partial {\epsilon }_{\mathit{xy}}$ varies linearly with ${\epsilon }_{\mathit{xy}}$, and indeed, as mentioned above, is an odd function of ${\epsilon }_{\mathit{xy}}$.

Despite the large values of the ECE that we observe, these values are uniformly approximately a factor of five smaller than theoretical values due to imperfect adiabaticity of the experiment. Finite element simulations confirm the imperfect adiabaticity, which can be corrected for by multiplying by a constant, temperature-independent, factor (SI Appendix, Fig. S16). Doing so does not change any of the subsequent conclusions, including the temperature and strain dependence of the entropy, and the temperature dependence of the quadrupole strain susceptibility.

The above data can be used to reconstruct the entropy landscape. Starting from the measured heat capacity as a function of temperature, simple integration with respect to temperature yields $S(T,{\epsilon }_{\mathit{xy}}=0)$. Because the experiment measures $\partial T/\partial {\epsilon }_{\mathit{xy}}$ for adiabatic conditions, a process of numerical iteration can be used to sequentially build up the entire entropy landscape. The result of performing this process is shown in Fig. 5 with the entropy shown as a colorscale. Here we have included the factor of five mentioned above in order to capture the full entropy. This entropy analysis can be compared with the anticipated theoretical result, using the known Hamiltonian and magnetoelastic coupling coefficients (25). Starting from the measured zero-strain heat capacity, the obtained entropy landscape (Fig. 5B) closely matches that determined from the Hamiltonian (Fig. 5A).

Further analysis of the data presented above yields an estimate of the quadrupole strain susceptibility. To understand this, it is helpful to briefly consider the thermodynamic quantity that is being measured. Specifically,

where we have used a Maxwell relation to relate the strain derivative of the entropy to the temperature derivative of the elastic modulus $c_{66}$, and $C_{\epsilon }$ is the heat capacity under constant strain. We use the volumetric heat capacity in our subsequent analysis to allow for a direct comparison with ultrasound measurements. For materials that undergo a continuous, $\mathit{xy}$-symmetric quadrupolar (or more generally nematic^*) phase transition, the temperature dependence of $c_{66}$ is given, quite generally, by $c_{66}(T)=c_{66}^0-{\lambda }^2{\chi }_{B_{2g}}$, where $c_{66}^0$ is a “bare” (unrenormalized) elastic stiffness, $\lambda$ is the coupling constant that relates strain and the order parameter (i.e. the free energy contains a term $\mathrm{\Delta }F\sim \lambda {\epsilon }_{B_{2g}}{\mathrm{\Psi }}_{B_{2g}}$), and ${\chi }_{B_{2g}}=\partial P_{\mathit{xy}}/\partial {\epsilon }_{\mathit{xy}}$ is the quadrupole strain susceptibility (17). In other words, the measured adiabatic elastocaloric coefficient is proportional to the temperature derivative of the quadrupole strain susceptibility. And because this is measured at a nonzero frequency, it is technically a dynamic susceptibility (albeit, in this case, strictly in the DC limit given the measurement frequency of 211 Hz). All that is required to obtain this quantity is to extract the slope of $\partial T/\partial {\epsilon }_{\mathit{xy}}$ with respect to ${\epsilon }_{\mathit{xy}}$ (i.e. the slope of the data shown in Fig. 4) and multiply by the measured heat capacity. In the absence of heat capacity measurements performed in situ, we use data obtained from a standard relaxation technique for which subtle differences in strain conditions can affect the heat capacity (SI Appendix, Fig. S12) and consequently the extracted susceptibility at the lowest temperatures. To compensate for strain effects, we have corrected the heat capacity to better match the low-temperature ultrasound data (SI Appendix, Fig. S14). There are no differences above 6 K and the low-temperature upturn remains a robust feature regardless of subtle differences in the heat capacity.

Fig. 6 shows the result of performing this analysis, with details described in SI Appendix. As can be seen, the data diverge toward low temperature. More specifically, the data can be very well fit to $A{(T-\theta )}^2$, the anticipated functional form assuming ${\chi }_{B_{2g}}$ follows Curie–Weiss behavior. It is precisely this diverging quadrupole susceptibility that permits the large elastocaloric effect and supports our proposal to use adiabatic decompression of such materials for cryogenic cooling.

Because of the relationship between the ECE and $\partial c_{66}/\partial T$, we can also compare our measurement to direct measurements of the elastic stiffness tensor, in this instance from pulse-echo ultrasound measurements. For a carefully chosen experimental geometry (Materials and Methods and SI Appendix), the speed of sound obtained from these pulse-echo measurements yields an estimate for the specific combination of stiffness coefficients ($c_{66}+c_{11}+c_{12}$). Since the singular behavior associated with ${\chi }_{B_{2g}}$ only affects $c_{66}$, any weak temperature dependence associated with $c_{11}$ and $c_{12}$ is overwhelmed by that of $c_{66}$, and thus taking the temperature derivative of this quantity yields a result that is predominantly determined by $\partial {\chi }_{B_{2g}}/\partial T$. The agreement between the ECE and pulse-echo measurements [and earlier estimates of $c_{66}$ (24)] is remarkable, as is the agreement with the anticipated Curie–Weiss functional form.

M.P.Z., P.M.H., E.W.R., B.J.R., and I.R.F. designed research; M.P.Z., M.S.I., P.M.H., and Y.L. performed research; P.M. contributed new reagents/analytic tools; M.P.Z., M.S.I., P.M.H., L.Y., J.A.W.S., and Y.L. analyzed data; and M.P.Z., B.J.R., and I.R.F. wrote the paper.

The authors declare no competing interest.