CRYOGENIC ELASTOCALORIC HEAT PUMP AND ASSOCIATED METHODS

Description

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent Application No. 63/730,077, filed Dec. 10, 2024, which is incorporated herein by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under contract FA9550-20-1-0252 awarded by the Air Force Office of Scientific Research and under contract 2232515 awarded by the National Science Foundation. The Government has certain rights in the invention.

BACKGROUND

Low-temperature cooling in the kelvin and sub-kelvin regimes is of considerable interest because of its application in low-temperature research and, more recently, for efficient cooling of quantum computers. The workhorse method of dilution refrigeration relies on the availability of ³He, a scarce and costly resource, as well as requiring complex gas handling systems. Magnetocaloric cooling is simpler, but requires large magnetic fields, which may be incompatible with certain applications and cannot be cycled rapidly. Alternate methods of cryogenic cooling are consequently of particular interest.

SUMMARY

Until recently, elastocaloric cooling in the cryogenic regime has been largely unexplored, in part because methods to measure the elastocaloric effect at cryogenic temperatures had not yet been developed, and in part because of the lack of candidate materials. Despite these challenges, elastocaloric cryogenic cooling using piezoelectric actuators offers several distinct advantages, including fast response times, localized cooling, and small space requirements. The elastocaloric effect may also provide unique insights into the entropy landscape and phases present in quantum materials, further motivating the development of new low-temperature techniques and methodologies, in addition to exploring candidate materials with suitably large responses for cryogenic applications [1-6].

Interest in elastocaloric cooling has primarily focused on refrigeration and air conditioning near room temperature, and has typically utilized the latent heat associated with first-order phase transitions. However, there is also considerable potential to employ elastocaloric effects for cooling at much lower, cryogenic temperatures. Low-temperature cooling of materials using pressure and strain has been explored previously within the context of materials that couple to these quantities either partially or indirectly. Here, we explore the potential of adiabatic decompression of Jahn-Teller active materials, for which strain couples directly (bilinearly) to the active quadrupolar degrees of freedom, for cryogenic cooling. The requisite large elastocaloric effect in this case is provided by the presence of a large quadrupole strain susceptibility, which is implicit in such materials in the temperature regime above any associated phase transitions. The cooling effect is analogous to that achieved by adiabatic demagnetization in paramagnetic materials and does not directly rely on the presence of a phase transition.

The present embodiments include systems and methods for cryogenic cooling that utilize the elastocaloric effect in quantum materials. In some embodiments, a cryogenic elastocaloric heat pump includes an elastocaloric working material, a piezoelectric transducer mechanically coupled to the working material, a first thermal switch configured to selectively establish thermal connection between the elastocaloric working material and a sample stage, and a second thermal switch configured to selectively establish thermal connection between the working material and a hot stage. The elastocaloric working material may be a low-temperature Jahn-Teller material, a low-temperature nematic material, or a low-temperature quadrupolar material. In any case, the elastocaloric working material exhibits a large change in entropy in response to strain within the relevant cryogenic temperature range.

During operation, the cryogenic elastocaloric heat pump executes a thermodynamic cycle that includes compression and decompression, where the piezoelectric transducer applies mechanical strain to the working material to induce temperature changes. The thermal switches are electronically controlled to enable heat transfer between the sample stage and the hot stage. The system transfers heat from the cold stage to the hot stage, thereby reducing the temperature of the cold stage without requiring additional cryogenic fluids or large magnetic fields.

In some embodiments, a method uses the cryogenic elastocaloric heat pump to cool the sample stage by cyclically compressing and decompressing the working material and strategically opening and closing the thermal switches to effectuate heat transfer. The disclosed systems and methods are particularly suited for applications requiring efficient cooling in the kelvin and sub-kelvin regimes, including quantum computing and low-temperature research.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a cryogenic elastocaloric heat pump undergoing adiabatic decompression, in some embodiments.

FIG. 2 shows the cryogenic elastocaloric heat pump of FIG. 1 undergoing isothermal compression, in some embodiments.

FIG. 3 shows one example of a thermodynamic cycle executed using the cryogenic elastocaloric heat pump of FIG. 1, in some embodiments.

FIG. 4 shows a plot of entropy calculated for an elastocaloric working material composed of TmVO₄ as a function of temperature for different strains.

FIG. 5 shows an experimental setup to measure a cryogenic elastocaloric response of an elastocaloric working material composed of TmVO₄.

FIG. 6 illustrates a measurement workflow for characterizing the elastocaloric response of an elastocaloric working material.

FIG. 7 includes a representative pulse-and-response plot, with both cooling and heating curves, where the rise time of the strain is much shorter than the time constants associated with thermal relaxation.

FIG. 8 shows a thermal circuit modelling heat flow between a sample of an elastocaloric working material, a thermally connected bath, and a thermally connected thermometer.

FIG. 9 shows a representative elastocaloric response as a function of time for different initial strains.

FIG. 10 shows a representative elastocaloric response as a function of time for different bath temperatures for an elastocaloric working material subject to a moderate strain pulse.

FIG. 11 shows a representative elastocaloric response as a function of time for different bath temperatures for an elastocaloric working material subject to a large strain pulse.

FIG. 12 shows a calculated entropy landscape of an elastocaloric working material as a function of strain and temperature to illustrate how the elastocaloric effect of an elastocaloric working material has larger magnitudes for certain bath temperatures.

FIG. 13 shows a plot of thermal relaxation time of an elastocaloric material as a function of temperature.

FIG. 14 shows one example of a large elastocaloric response as a function of time for a particular set of parameters.

FIG. 15 shows a representative change in temperature of an elastocaloric working material as a function of bath temperature.

FIG. 16 illustrates an experimental setup to measure a cryogenic elastocaloric response of an elastocaloric working material composed of TmVO₄.

FIG. 17 shows a picture of the experimental setup depicted in FIG. 16.

FIG. 18 shows a coordinate system relative to the principal crystal axes used for determining values of strain.

FIG. 19 displays AC elastocaloric response of an elastocaloric working material as a function of frequency for different temperatures.

FIG. 20 shows measurements of the thermal relaxation time of an elastocaloric working material obtained from both an AC and DC elastocaloric responses of the elastocaloric working material.

FIG. 21 shows full time dependence of a representative elastocaloric response.

FIG. 22 shows representative elastocaloric response as a function of time for different bath temperatures for an elastocaloric working material subject to a large strain pulse.

FIG. 23 shows a plot of a long-time curve of a representative elastocaloric response.

FIG. 24 shows thermal relaxation time of an elastocaloric working material plotted as a function of time according to a logistic decay function.

FIG. 25 shows calculated elastocaloric response as a function of time based on the thermal relaxation time in FIG. 24.

FIG. 26 shows thermal relaxation time of an elastocaloric working material plotted as a function of time according to a thermal-evolution model.

FIG. 27 shows a calculated elastocaloric response as a function of time based on the thermal relaxation time in FIG. 26.

FIG. 28 shows a plot of voltage applied across a piezoelectric stack as a function of time for different time constants for a DC elastocaloric response.

FIG. 29 shows a plot of voltage applied across a piezoelectric stack as a function of time for different time constants for an alternating-current elastocaloric response.

DETAILED DESCRIPTION

The elastocaloric effect is a phenomenon in which the application of mechanical strain to a material induces a change in the material's temperature. A cryogenic elastocaloric material is a material that exhibits the elastocaloric effect at cryogenic temperatures. A heat pump operates to transfer thermal energy from a cold stage to a hot stage, thereby cooling the cold stage. In some embodiments, a cryogenic elastocaloric heat pump includes an elastocaloric working material, a first thermal switch configured to selectively establish a thermal connection between the working material and a sample stage, a second thermal switch configured to selectively establish a thermal connection between the working material and a hot stage, and a piezoelectric transducer mechanically coupled to the working material. During operation, the cryogenic elastocaloric heat pump executes a sequence of steps to transfer heat from the cold stage to the hot stage. The cold stage is also referred to herein as a “sample stage” since it is thermally coupled to the sample to be cryogenically cooled.

In embodiments, a cryogenic elastocaloric heat pump transfers heat from a cold stage to a hot stage. Heat pumps operate by inducing heat flow from a lower-temperature region to a higher-temperature region. For cryogenic applications, the hot stage may be maintained at a temperature above approximately 1 K (e.g., 1.5 K or 2 K) using conventional cooling technology (e.g., a pumped helium system). The elastocaloric heat pump then removes heat from the cold stage and transfers it to the hot stage, thereby reducing the temperature of the cold stage. This heat pump operation does not require any cryogenic fluids.

FIG. 1 shows a cryogenic elastocaloric heat pump 100, in accordance with some of the present embodiments. The cryogenic elastocaloric heat pump 100 includes an elastocaloric working material 102 (labeled “EWM” in FIG. 1), a first thermal switch 104 thermally connected to a sample stage 114, and a second thermal switch 106 thermally connected to a hot stage 116.

The elastocaloric working material 102 may be adiabatically compressed or decompressed to respectively increase or decrease a temperature of the elastocaloric working material 102.

Compression and decompression may be achieved using a piezoelectric transducer mechanically coupled to the elastocaloric working material 102. In some embodiments, the cryogenic elastocaloric heat pump 100 includes a controller to implement the compression and decompression processes by electrically driving the piezoelectric transducer to apply corresponding stresses to the elastocaloric working material 102.

FIG. 1 also shows a strain ε_xy applied to the elastocaloric working material 102 plotted as a function of a temperature T of the elastocaloric working material 102. The decompression process of the elastocaloric working material 102 may occur adiabatically, in which case the process flows along an adiabat of constant entropy S₁ from point A to point B, as shown in the plot of FIG. 1. The adiabaticity of the decompression process is enabled by opening (i.e., thermally disconnecting) the first thermal switch 104 and the second thermal switch 106. Another adiabat of constant entropy S₂ is also shown on the plot for clarity.

FIG. 2 shows the cryogenic elastocaloric heat pump 100 of FIG. 1 undergoing isothermal compression, in accordance with some of the present embodiments. The elastocaloric working material 102 may undergo a compression process at constant temperature to flow from point C to point A, as depicted in the plot of FIG. 2. This isothermal compression process is enabled by closing the second thermal switch 106 to allow heat to flow from the elastocaloric working material 102 to the hot stage 116. In this example, the transferred heat was generated by compressing the elastocaloric working material 102.

An ideal candidate for the elastocaloric working material 102 exhibits a large change in entropy in response to strain within the relevant cryogenic temperature range. Representative materials include, but are not limited to, low-temperature Jahn-Teller compounds, such as thulium vanadate (TmVO₄). Due to the pronounced dependence of entropy on strain, adiabatic changes to the strain state (i.e., performed quickly enough to minimize heat exchange with the environment) produce significant temperature changes in the material. For example, an elastocaloric-material sample of TmVO₄ may exhibit approximately 2.5 K of adiabatic elastocaloric cooling when starting from a bath temperature near 5 K, with potential for larger effects under more-optimized conditions.

In the present embodiments, the elastocaloric working material 102 undergoes cyclic compression and decompression, thereby completing a thermodynamic cycle. Compression is achieved by electrically actuating one or more piezoelectric elements that are mechanically coupled to the elastocaloric working material 102. These piezoelectric elements may be thermally connected to a heat sink at the hot stage 116 while the elastocaloric working material 102 is thermally isolated from external influences. During each cycle, the elastocaloric working material 102 is heated upon compression and cooled during decompression, which facilitates the transfer of heat from the sample stage 114 to the hot stage 116.

The thermal switches 104 and 106 may be electronically controlled, enabling precise thermal management of the cryogenic elastocaloric heat pump 100. By adjusting the states of these switches, thermal connections are selectively established or broken between the elastocaloric working material 102 and either the sample stage 114 or the hot stage 116, as dictated by the thermodynamic cycle. For example, the thermal switch 106 may be electronically set to a closed state to permit heat transfer between the elastocaloric working material 102 and the hot stage 116. Conversely, the thermal switch 106 may set to an open state to prevent any heat transfer between the elastocaloric working material 102 and the hot stage 116.

The operation of the thermal switches 104 and 106 may be precisely controlled based on temperature differentials. The thermal switch 104 leading to the sample stage 114 is closed only when the elastocaloric working material 102 becomes colder than the sample stage 114. This configuration allows heat to flow from the sample stage 114 into the elastocaloric working material 102, effectively extracting heat from (i.e., cooling) the sample stage 114. In contrast, the thermal switch 106 connecting to the hot stage 116 is closed exclusively when the temperature of the elastocaloric working material 102 exceeds that of the hot stage 116. In this state, heat flows from the elastocaloric working material 102 to the hot stage 116, thereby enabling heat to be transferred from the sample stage 114 to the hot stage 116.

By continuously cycling the processes of compressing and decompressing the elastocaloric working material 102, and by strategically opening and closing the thermal switches 104 and 106, heat is repeatedly transferred from the sample stage 114 to the hot stage 116. This cyclical operation is central to the cooling effect, as it systematically reduces the temperature of the sample stage 114 through repeated cycles of heat transfer.

Multiple realizations for the thermal switches 104 and 106 are possible. One approach involves a mechanical contact between two efficient thermal conductors, where a vacuum gap is created and subsequently closed through the use of an electrically actuated piezoelectric mechanism. Alternatively, a superconducting thermal switch may be implemented. In this case, the switch is controlled by inducing transitions in the superconductor's state, for example, through application or removal of a magnetic field. Other types thermal switches may be used without departing from the scope hereof.

The fundamental properties required of the elastocaloric working material 102 may be straightforwardly specified. Various elastocaloric working materials are available that satisfy these requirements and function effectively within the described system (e.g., the cryogenic elastocaloric heat pump 100). Furthermore, ongoing and future research may focus on optimizing the choice of elastocaloric working material for specific applications, such as achieving a particular base temperature or accommodating predetermined ranges of strain. As the concept is not limited to a single material, various elastocaloric working materials may be used for the cryogenic elastocaloric heat pump 100 without departing from the scope described herein.

FIG. 3 shows one example of a thermodynamic cycle 300 executed using the cryogenic elastocaloric heat pump 100 of FIG. 1, in accordance with some of the present embodiments. The thermodynamic cycle 300 includes an initial thermalization stage 101, an isothermal compression stage 103, an adiabatic decompression stage 105, and a cooling of sample stage 107. These stages are also shown in a strain-temperature plot. A refrigeration cycle cooling the sample stage 114 is one example of the thermodynamic cycle 300.

During the initial thermalization stage 101, the thermal switches 104 and 106 are open and closed, respectively, allowing the elastocaloric working material 102 to thermalize with the hot stage 116. During the isothermal compression stage 103, the elastocaloric working material 102 is isothermally compressed, thereby transferring heat generated by the compressed elastocaloric working material 102 to the hot stage 116. During the adiabatic decompression stage 105, the elastocaloric working material 102 is adiabatically decompressed while the thermal switches 104 and 106 are open, thereby reducing a temperature of the elastocaloric working material 102. During the cooling of sample stage 107, the thermal switches 104 and 106 are closed and open, respectively, thereby allowing heat to be transfer from the sample stage 114 to the elastocaloric working material 102 to decrease a temperature of the sample stage 114.

Cryogenic elastocaloric heat pumps (e.g., the cryogenic elastocaloric heat pump 100 of FIG. 1) that include low-temperature Jahn-Teller systems are especially appropriate for cryogenic cooling via an adiabatic decompression process for two reasons. First, they preserve a large amount of entropy at low temperatures, owing to a degeneracy of crystal electric field (CEF) levels. Second, they have a large magnetoelastic coupling, which allows for the ground state, and thus the entropy, to be easily tuned with strain. Our focus of this study is not to propose a particular material for which low-temperature cooling is optimized, but rather to experimentally verify the entropy landscape of an archetypical material in this class, TmVO₄. In so doing, we also demonstrate an especially suitable and effective method to measure a quantity of fundamental interest in the study of quadrupole order, the dynamic quadrupole-strain susceptibility.

TmVO₄ undergoes a cooperative Jahn-Teller phase transition at T_Q=2.15 K to a ferroquadrupolar ordered state. The phase transition is continuous, and the crystal symmetry changes from tetragonal to orthorhombic, corresponding to a spontaneous ε_xy shear strain with quadrupolar order of the same xy symmetry. As has been demonstrated, in the paraquadrupolar regime (i.e., for temperatures above the phase transition), the material has a large quadrupole-strain susceptibility with respect to xy strains, much like a ferromagnetic material has a large magnetic susceptibility above its Curie temperature. In terms of entropy, above T_Q, each Tm ion contributes k_Bln2 due to the crystal field groundstate, which is a non-Kramers doublet. This entropy is removed at the phase transition, which splits the ground state doublet. To the extent that relevant terms in the Hamiltonian are known, it is possible to estimate the entropy as a function of temperature and strain (see FIG. 4 below). Of particular significance, the large quadrupole-strain susceptibility means that the entropy may be substantially reduced above T_Q by application of modest shear strains (dotted curve in FIG. 4). Such large changes in entropy at low temperatures in response to modest applied strains imply the possibility of a sequence of isothermal compression (arrow from point A to point B in FIG. 4) and adiabatic decompression (arrow from point B to point C in FIG. 4) to achieve a substantial cooling effect for temperatures above T_Q. Such large temperature changes (ΔT/T˜1 under adiabatic conditions, or equivalently ΔS/S˜1 under isothermal conditions) suggests a significant the elastocaloric effect in this material.

FIG. 4 shows a plot 400 of entropy calculated for TmVO₄ as a function of temperature for different strains, ε_xy=0 and 5·10⁻³. Entropy is expressed per formula unit (f.u.). A 4f entropy contribution is calculated from a Hamiltonian using coefficients from published literature while a phonon contribution is determined from heat capacity experiments. In the absence of externally induced strain (ε_xy=0), the entropy falls rapidly from approximately k_Bln2 above T_Q towards zero at zero temperature. Inducing a modest xy strain of just 5·10⁻³ leads to a substantial reduction of the entropy for all temperatures below approximately 15 K, while also removing the phase transition. An example of a possible thermodynamic cycle is indicated with arrows. The cycle consists of an isothermal path along which ε_xy is increased (A→B) and an adiabatic path along which ε_xy is decreased (B→C). A thermalization step may be included from C→A to produce a full cycle. The figure demonstrates that a large cooling effect may, at least in principle, be realized if adiabatic conditions are maintained during the decompression part of the cycle.

Certain embodiments may be used for experimental confirmation of an entropy profile predicted for TmVO₄ and to demonstrate a corresponding large elastocaloric response at cryogenic temperatures. Using an AC elastocaloric measurement protocol that is representative of dynamic processes implementable in cooling systems, the disclosed measurements further provide a direct determination of the dynamic quadrupole-strain susceptibility of the material.

Measurements may be best understood by considering an estimated entropy landscape shown in a plot in FIG. 5. Under adiabatic conditions, appropriate for sufficiently rapid strain oscillations, the material moves back and forth along a contour of constant entropy in response to an externally induced AC strain, resulting in a concomitant temperature oscillation at the same frequency (double-sided arrow in the plot in FIG. 5 between points A and B). As temperature is lowered toward the ferroquadrupolar phase transition, the constant entropy curves decreases in slope, yielding a larger amplitude temperature oscillation for a given amplitude strain oscillation. This is the regime in which the entropy is dominated by the 4f electrons, in contrast to the higher temperature regime (above approximately 15 K), where phonons, which have a much smaller strain dependence, dominate the total entropy. The cooling effect in the exemplary material of TmVO₄ may thus be optimal when restricted to temperatures below approximately 15 K.

The elastocaloric response of the elastocaloric material may be measured to map out the entropy in a temperature-strain space by cooling the elastocaloric-material sample and inducing an xy strain in the TmVO₄ single crystal via stress applied along a tetragonal [110] crystallographic direction (see schematic of heat pump in FIG. 3). In simpler terms, stress is applied along the diagonal of the tetragonal a-a plane, pulling or pushing at the “corners,” causing an xy-symmetric deformation. To apply this stress, a commercially available Razorbill CS100 cell may be used. The elastocaloric-material sample may be mounted with the tetragonal [110] crystallographic direction aligned along the displacement direction and the temperature response of the elastocaloric-material sample may be measured with respect to a stress-induced AC strain using, for example, a RuO₂ temperature sensor connected to a Wheatstone bridge. Simultaneously, an offset strain may be applied and the temperature of the elastocaloric-material sample may be varied so as to traverse the temperature-strain (T-ε_xy) entropy landscape, as illustrated in FIG. 5.

FIG. 5 shows an experimental setup 500 to measure a cryogenic elastocaloric response of an elastocaloric working material composed of TmVO₄. The schematic diagram illustrates an elastocaloric working material being a single crystal of TmVO₄ which is held between two titanium mounting plates. The two titanium mounting plates are actuated via a piezoelectric transducer to apply a stress to the TmVO₄ crystal. The crystal is cut such that the [110] crystallographic direction lies along the direction of the applied stress. Both alternating-current (AC) and direct-current (DC) voltages may be applied across different piezoelectric stacks to control the distance between the titanium mounting plates, thus controlling the strain the material experiences in response to the applied stress. The strain oscillations produce temperature oscillations that are measured by a RuO₂ thermometer. Typical elastocaloric-material sample dimensions of the TmVO₄ crystal include 1.6 mm in length (of which approximately 0.9 mm “active area” extends between the titanium plates) by 0.3 mm width by 0.05 mm thickness.

FIG. 5 also includes a strain-temperature plot showing a calculated entropy landscape of TmVO₄ as a function of temperature T and strain ε_xy, shown here as a heat map with isentropic curves shown as dashed lines (e.g., isentropic curves along lines of constant entropy S₁ and S₂). Isentropic curves increase from left to right, i.e., S₂>S₁. Three broad regions may be identified. At high temperatures, phonons dominate the entropy, and isentropic curves are nearly vertical; cooling potential is low in this region. At low temperatures and larger strains, the isentropic curves become almost radial lines (emanating from the origin), as the 4f contribution to the entropy outweighs the phonon contribution; potential for cooling by adiabatic decompression is optimized in this region. Finally, below T_Q=2.15 K, a first-order transition separates a region in which the material is homogeneously strained (above and to the right of the solid black line), from a region in which domains of oppositely oriented orthorhombic regions are energetically favored (below and to the left of the solid black line). Neglecting the small contribution to the total entropy that arises from the domain walls, the total entropy does not exhibit a strain dependence within this region below T_Q, providing a practical limit to the lowest possible temperature that could be achieved via adiabatic cooling for realistic strains. The double-sided arrow (exaggerated in magnitude for clarity) indicates the AC elastocaloric-cooling-effect method that we employ to experimentally determine the entropy landscape.

Demonstration of Elastocaloric Cooling at Cryogenic Temperatures

This section and the following sections demonstrate that the cryogenic elastocaloric heat pump 100 of FIG. 1 is enabled by utilizing an appropriate elastocaloric working material. One prerequisite for achieving elastocaloric cooling at cryogenic temperatures is the use of a material in which an isothermal change in strain produces a substantial change in entropy. Consequently, materials that exhibit a low-temperature cooperative Jahn-Teller effect are suitable candidates [7](although other mechanisms are also possible [8]). In certain embodiments, the elastocaloric working material includes a Jahn-Teller compound, a nematic material, or a quadrupolar material. As an illustrative example, thulium vanadate (TmVO₄) is a well-characterized Jahn-Teller material that demonstrates a large entropy change in response to strain. This property enables TmVO₄ to exhibit a pronounced elastocaloric effect suitable for cryogenic cooling applications. Accordingly, TmVO₄ is employed in certain embodiments as an archetypal elastocaloric working material to realize elastocaloric cooling at temperatures below approximately 30 K.

TmVO₄ is an electrically insulating compound that exhibits a second-order ferroquadrupolar phase transition at approximately T_Q=2.15 K [9, 10]. This electronic ordering is accompanied by a structural transition from a tetragonal phase (T>T_Q, space group I4₁/amd) to an orthorhombic phase (T<T_Q, space group Fddd) [11]. The coupled electronic-structural instability is characteristic of the cooperative Jahn-Teller effect, and TmVO₄ is widely recognized as an archetypal material for this phenomenon [9,10].

In the tetragonal phase, the crystal electric field ground state of the Tm³⁺ ion is a non-Kramers doublet [12]. The quadrupole moment associated with this doublet couples linearly to strain of matching symmetry. In the ordered state, the doublet splits spontaneously, releasing approximately k_Bln(2) of entropy per Tm³⁺ ion [13-15]. Above T_Q, the doublet also splits linearly in response to externally applied strain, enabling direct control of a substantial entropy reservoir (approximately k_Bln(2) per Tm³⁺ ion) using an xy-symmetric strain, ε_xy. Under adiabatic conditions, the temperature of TmVO₄ may therefore be modulated by an externally driven antisymmetric strain.

The relevant elastic mode, c₆₆, is exceptionally soft at low temperatures (approximately 16 GPa) and further softens as the phase transition is approached [16], reducing the risk of sample fracture during strain application. These characteristics make TmVO₄ an excellent candidate for elastocaloric cooling in the cryogenic regime.

Techniques for measuring low-temperature elastocaloric responses have only recently been developed. One approach employs an AC technique in which strain is oscillated over a small range at a frequency sufficiently high to thermally decouple an elastocaloric-material sample (e.g., TmVO₄) from the apparatus used to induce the strain, permitting measurement of the adiabatic response [17]. Typical AC strain amplitudes are on the order of 10⁻⁵, producing temperature changes of approximately 1 mK. By varying the DC bias strain, the entropy as a function of strain and temperature may be extracted, confirming the suitability of TmVO₄ for elastocaloric cooling [7].

To demonstrate significant elastocaloric cooling at low temperatures requires a method that more closely resembles that which would be used for cooling purposes, with much larger changes in strain and temperature. Akin to processes employed at room temperature [18, 19], the method may include a strain load/unload (or pulse) technique. Accordingly, a strain load/unload (pulse) technique may be employed, analogous to processes used for room-temperature elastocaloric cooling [18, 19]. This approach enables substantial cooling and allows time-domain analysis of the elastocaloric response, in contrast to AC techniques that operate in the frequency domain. The hardware configuration closely resembles that used for AC measurements [17], allowing direct comparison of results obtained from both techniques on the same elastocaloric-material sample and mounting arrangement which may provide key insights into the associated thermal relaxation time constants and how they manifest in each measurement (see Appendices A and B).

Experimental Methods

In certain embodiments, single crystals of TmVO₄ are grown via a flux technique [20, 21]. Elastocaloric-material samples of TmVO₄ were cut and polished to render long, thin rectangular parallelepiped bars, with the long axis along the tetragonal [110] crystallographic direction. Uniaxial stress applied along that direction yields a combination of xy-symmetry shear strain, ε_xy, together with a small amount of symmetry-preserving strains, (ε_xx+ε_yy)/2 and ε_zz [22]. Because c₆₆ is soft compared to

1 2 ⁢ ( c 1 ⁢ 1 + c 2 ⁢ 2 ) ,

nearly all of the strain (approximately 90%, which varies slightly as a function of temperature) is xy strain (see Appendix A and supplementary material of Ref. [7] for more details). A coordinate system is adopted in which unprimed coordinates refer to the [100], [010], and [001] crystallographic directions, while primed coordinates refer to the stress direction, rotated by 45 degrees about the z-axis (see Appendix A). Strain in the x′ direction (i.e., along [110]), ε_x′x′, was measured in the experiment, and is reported in figures including strain transfer corrections (see Appendix A). The elastocaloric-material sample (e.g., the elastocaloric working material 102 of FIG. 1) was strained using a Razorbill CS100 strain cell. The elastocaloric-material sample was suspended between two pairs of sample plates to which it was epoxied. Piezoelectric stacks (e.g., a piezoelectric transducer) controlled the strain that the elastocaloric-material sample experienced, and the temperature of the elastocaloric-material sample was measured using a RuO₂ thin film thermometer. The strain that the elastocaloric-material sample experienced was measured using a capacitive sensor integrated on the strain cell. More details on the experimental setup may be found in Appendix A.

Results

FIG. 6 illustrates an exemplary measurement workflow for characterizing the elastocaloric response of an elastocaloric working material. The measurement workflow timeline of the measurements, with the strain represented by the line labeled c, is not to scale. The horizontal t-axis represents the flow of time during the measurement and is labeled with the different stages of the experiment (also not to scale). In certain implementations, the elastocaloric working material (e.g., the elastocaloric-material sample of TmVO₄) is first initialized at a selected temperature and subjected to a predetermined bias strain. Data acquisition is then initiated, and a short-duration strain pulse is applied to the elastocaloric working material. The resulting temperature change ΔT_t associated with the elastocaloric effect is recorded during this capture interval.

Following the initial measurement, a longer-duration strain pulse (approximately one second) is applied over multiple cycles to determine the change in strain associated with the short-duration strain pulse. This calibration step enables an accurate correlation between strain input and an associated temperature response. The procedure is repeated for each data point in the dataset to generate a comprehensive characterization of the elastocaloric effect under varying strain and temperature conditions.

FIG. 7 includes a representative pulse-and-response plot, with both cooling and heating curves, where the rise time of the strain (˜500 μs) is much shorter than the time constants associated with thermal relaxation. Representative data (upper panel) of the approximate strain, ε_x′,x′, experienced by the elastocaloric-material sample is plotted alongside (lower panel) the temperature of the elastocaloric-material sample relative to the bath read out via an attached thermometer, both as a function of time. In the lower panel, data points represent experimental data and the dashed line represents a fit (R²>0.99) to the model described by Eqn. 2.

FIG. 8 shows a thermal circuit 800 modelling thermal relaxation times of a sample of an elastocaloric working material thermally connected to a thermometer and a thermal bath. Assuming that the time variation in the temperature of the elastocaloric working material (e.g., the elastocaloric working material 102 of FIG. 1) is solely due to thermal conduction to the thermal bath (e.g., the hot stage 116 of FIG. 1), a simple thermal model may be used [8, 17]. FIG. 8 displays a schematic of this heat-flow model which is governed by time constants τ_t and τ_s. The time evolution of a temperature T_s(t) of the elastocaloric-material sample may be modeled as:

T s ( t ) = T bath + Δ ⁢ T s ⁢ e - t τ s , ( 1 )

and a temperature T_t(t) of the thermometer may be modeled as:

T t ( t ) = T bath + τ s τ t - τ s ⁢ Δ ⁢ T s ( e - t τ s + e - t τ t ) . ( 2 )

Here, T_bath is a temperature of the thermal bath, τ_t(:=C_s/G_s) is a thermal relaxation time constant between the elastocaloric-material sample and the thermal bath (i.e., a characteristic time describing how long it takes for the elastocaloric-material sample to return to the bath temperature T_bath), τ_t (:=C_t/G_t) is a thermal relaxation time constant between the thermometer and the elastocaloric-material sample (i.e., a characteristic time describing how long it takes the elastocaloric-material sample and the thermometer to equilibrate), ΔT_s is an intrinsic change in temperature of the elastocaloric-material sample caused by a change in strain (e.g., the change in temperature of the elastocaloric-material sample due to the elastocaloric effect), C_s and G_s are a heat capacity and a thermal conductance of the elastocaloric-material sample, and C_t and G_t are a heat capacity and a thermal conductance of the thermometer, respectively. Necessarily, C_s>>C_t. The representative curve shown in FIG. 7 fits well to Eqn. 2 with an excellent R² value of 0.997. Note that the elastocaloric-material sample as defined in this model is representative of the elastocaloric working material 102 of FIG. 1 and the thermal bath is representative of the hot stage 116 of FIG. 1.

FIG. 9 shows a representative elastocaloric response of an elastocaloric working material as measured by a temperature change, ΔT_t, of a thermometer. The elastocaloric response may be obtained across the ε_xy-T landscape to track systematic changes as a function of each independent variable. For example, the elastocaloric cooling response at 4 K with a strain pulse of Δε_x′x′=2·10⁻⁴ for different values of the initial strain

ε x ′ ⁢ x ′ ( 0 )

is shown in FIG. 9. As the magnitude of compressive strain that the elastocaloric-material sample experiences increases, the cooling of the elastocaloric-material sample initially increases, then decreases once the strain goes beyond

ε x ′ ⁢ x ′ ( 0 ) = - 1.8 · 10 - 3 .

The time constant associated with thermal relaxation from the elastocaloric-material sample to the bath increases slightly as compressive strain is increased.

A more drastic change in the elastocaloric response is evident when temperature is varied, as shown in FIG. 10. While at high temperatures relative to the phase transition temperature (8 K), the elastocaloric response is small and grows considerably until it reaches a maximum at approximately 3.6 K, which agrees with previous AC elastocaloric measurements [7]. As the elastocaloric-material sample is cooled further, the elastocaloric response begins to shrink and the thermal relaxation time of the elastocaloric-material sample increases dramatically, with significant temperature changes lasting for hundreds of milliseconds.

In an effort to achieve the largest temperature changes possible, the strain pulse amplitude was increased nearly ten fold to approximately 1.8·10⁻³. The elastocaloric response was then measured at different temperatures, and the results are displayed in FIG. 11. Once again, the response is initially small and increases as the bath temperature decreases, albeit here the response peaks at approximately 5 K. The peak realized temperature change of the thermometer is just over 900 mK. As the bath temperature is further cooled, the elastocaloric response decreases, though a clear shoulder-like feature appears, deviating from the functional form predicted by the heat flow model given in Eqn. 2. The initial, sharp increase in temperature for each curve is labeled with a star, and the end of the shoulder-like feature for each curve is labeled by a flower. The feature is shown in all curves from 4 K down to 2.4 K and grows considerably in size. This feature is even subtly present in FIG. 10 in the data obtained at 2.1 K.

The failing of Eqn. 2 originates from the absence of a time or temperature dependence of the thermal time constants in the model. This is most pertinent for the elastocaloric-material sample time constant (τ_s), as the specific heat [13-15] and thermal conductivity [23, 24] of TmVO₄ vary dramatically as a function of temperature near the phase transition, with the former varying more of the two. The specific heat of TmVO₄ grows significantly as the temperature is lowered to T≈T_Q, although the exact temperature dependence depends on the strain state of the material [13-15], while the thermal conductivity has a small but noticeable kink [23, 24].

As a visual aid, the process of elastocaloric cooling in TmVO₄ using the strain load/unload technique is shown in FIG. 12 using the calculated entropy landscape of TmVO₄ [7]. Here, for simplicity, we only show the entropy as a function of one component of the strain tensor, ε_xy, this component dominates the total strain value (see Appendix A), but nevertheless this figure should only be used for qualitative comparison with experimental data, for the total temperature change when symmetric strains are also present. Under an adiabatic change in strain, the material follows an isentrope, which will dictate its change in temperature. As the material cools via its own elastocaloric effect, it enters the ordered state, increasing its thermal time constant dramatically. In addition, in the ordered state, the entropy of the material does not change as a function of applied strain, except due to changes arising from domain walls. Hence, the cooling effect stops, and the isentrope in FIG. 12 becomes vertical as the material enters the ordered state. Once the material is in this ordered state and the change in strain has ceased, heat leaks into the material from the environment. The deeper the material is in the ordered state (i.e., the left arrow is deeper in the ordered state compared to the right arrow in FIG. 12), the longer it will be in the ordered state, and the longer it will have a larger time constant, reinforcing the effect. In FIG. 12, this occurs because one elastocaloric process starts at a lower temperature than the other. This agrees with FIG. 11 in which the shoulder-like feature persists for a longer time when the elastocaloric-material sample is at a lower temperature, indicating that the elastocaloric-material sample takes a longer time to warm into the unordered state. A more quantitative representation of this is present in FIG. 13, the estimated elastocaloric-material sample time constant from specific heat (see supplementary material of Ref. [7]) and thermal conductivity data [23]. This quantity increases by four orders of magnitude from 8 K to 2 K, the sharpest increase being near the phase transition, bolstering the notion that the shoulder in FIG. 14 originates from a dramatic change in the time constant of the elastocaloric-material sample.

The curves for which there is no shoulder-like feature may be fit to the functional form given in Eqn. 2. The largest elastocaloric cooling response from the dataset displayed in FIG. 11 is fit in FIG. 14. Plotted alongside the fit is a calculated temperature of the elastocaloric-material sample as a function of time from Eqn. 1. The extracted value of the change in elastocaloric-material sample temperature, ΔT_s, is approximately 2.36 K, nearly half of the elastocaloric-material sample temperature of 5 K, a very large cooling effect. Although the curves in FIG. 11 below 5 K may not be adequately fit by Eqn. 2, ΔT_s may still be estimated by realizing that elastocaloric cooling stops abruptly once the temperature of the elastocaloric-material sample reaches the phase transition temperature (i.e. the isentrope hits the solid black line in FIG. 12) and also assuming that the thermometer is well thermalized with the elastocaloric-material sample. The estimation of ΔT_s for the dataset shown in FIG. 11 is plotted in FIG. 15 using both the fitting method for curves taken with T≥5.1 K and the intersection with phase boundary method for curves taken with T<5.1 K. The change in elastocaloric-material sample temperature peaks near 5 K. More importantly, the relative order of magnitude of the quantities agree, ensuring that the methods are reasonable and that the elastocaloric-material sample is indeed cooling into the ordered state when the bath temperature is below 5 K.

Discussion

The large change in temperature demonstrated in TmVO₄ of 2.36 K at 5 K under a change in strain of ε_x′x′=1.8·10⁻³, which corresponds to a change in stress of approximately 14 MPa using an elastic constant of 8 GPa [16], may be compared against the results from other materials that exhibit a significant temperature change under uniaxial pressure/strain at low temperatures. One classic material that exhibits elastocaloric cooling at cryogenic temperatures is OH⁻ embedded KCl [25]. The maximum cooling observed in OH⁻ embedded KCl was 0.15 K at 1.4 K with ΔP=2.3 MPa, though there were experimental issues as a result of frictional heating. Additionally, although OH⁻ embedded KCl may have slightly more entropy release per dipole than per 4f ion in TmVO₄ (k_B(ln6−ln2) versus k_Bln2), the cooling effect of OH⁻ embedded KCl is limited by the number of constituents, which is at most 0.5% [26], compared to TmVO₄, where every Tm³⁺ ion contributes to the change in entropy (k_Bln2 per formula unit). More recently, other rare earth-based materials, including Ce₃Pd₂₀Ge₆ [8, 27], CeSb [27,28], HoAs [27], and EuNi₂(Si_0.15Ge_0.85)₂[27] have been measured and all exhibit a smaller cooling effect at cryogenic temperatures compared to the measurements of TmVO₄ presented here thanks to the strong strain-quadrupole coupling in the latter.

In a similar manner to what is done for room temperature elastocaloric cooling, figures of merit are important metrics by which materials may be compared. The most common figures of merit at room temperature are specific power (W/g) and power (W) [29], which are based on temperature change and thermal resistance. However, at cryogenic temperatures, minimizing the space taken by a cooling agent, and therefore maximizing its volumetric specific cooling power (W/cm³), is imperative. The maximum volumetric specific cooling power, using the maximum change in temperature of the thermometer ΔT_t and the estimated thermal resistance (see Appendix F for more details) [30] and converting via the density, is 0.34 W/cm³ in the 0.19 mg elastocaloric-material sample of TmVO₄ at 5 K.

The specific cooling power, however, is an engineering-specific metric (since it relies on thermal resistance and not thermal resistivity); the geometry of the system will dictate it. In addition, energy efficiency is not as vital of a concern as is the ability to cool to the sub-kelvin regime with little material. As an alternative, we introduce two material-specific metrics for elastocaloric materials used in cryogenic applications, the power gradient and energy density. The power gradient, in units of W/m, indicates how easily heat may be transferred to the cooling agent, and is calculated via the product of the inferred temperature change of the elastocaloric-material sample and the thermal conductivity, κ_sΔT_s. The energy density, in units of J/cm³, demonstrates the energy reservoir present when cooling, and is taken to be the product of the inferred temperature change and the volumetric specific heat, C₃ΔT_s. Using the reported thermal conductivity and heat capacity, the largest power gradient observed in the measurements presented here is 0.30 W/cm at 5 K, which has a corresponding energy density of 2.6·10⁻³ J/cm³. More information on the figures of merit may be found in Appendix F.

The theoretical entropy landscape, which was shown in FIG. 12 for qualitative purposes, may be compared against the inferred cooling shown in FIG. 14. Not including elastocaloric cooling from symmetry-preserving strain and assuming a rough symmetry decomposition of 90% (see Appendix A), the expected cooling at 5 K given

ε x ′ ⁢ x ′ ( 0 ) ≈ - 2.7 · 10 - 3

and Δε_x′x′≈1.8·10⁻³ is approximately 3 K because the isentrope hits the phase boundary. Another more useful comparison is the amount of strain required to produce the same cooling, which is approximately Δε_x′x′≈1.3·10⁻³, indicating the realized elastocaloric cooling is of the correct order of magnitude as predicted by the Hamiltonian.

Although a large cooling effect was observed in this study, there is significant room for improvement in elastocaloric cooling at cryogenic temperatures. To further the use of quantum materials for elastocaloric cooling at cryogenic temperatures, both the technical and materials aspects of the problem must be addressed.

In principle, access to larger strains, both in terms of larger offset strains and larger changes in strains such that one has access to more points and paths in ε-T space, will yield a larger elastocaloric effect. In addition, better thermalization with the device to be cooled (e.g., in this study, a thermometer), and less thermalization with the bath, will produce longer hold-times and a greater thermal transfer. For practical implementation of a cryogenic cooling device that relies on elastocaloric cooling, the primary challenges will be associated with the thermal flow (including heat sinking of piezoelectric stacks, vibrations, and appropriate heat switches) as well as the geometry of the elastocaloric-material sample and the associated trade-offs between more working material and the ability to apply necessary strain and force. Although these constraints are not insignificant, the engineering problem is, in principle, able to be optimized using finite element analysis. Nevertheless, our measurements here serve as a proof-of-principle that even without these improvements, there is a material, or rather a class of materials, which may be explored and optimized to produce the most favorable cooling conditions.

Jahn-Teller materials, of which TmVO₄ is a specific realization, are a class of materials for which a large amount of entropy is preserved to low temperatures via degenerate crystal electric field levels. Ideally, the degeneracy may be lifted easily with strain because of direct, linear coupling between the electronic states and strain; the stronger this coupling, the larger the elastocaloric effect. Similar to adiabatic demagnetization, the best candidates for elastocaloric cooling to sub-kelvin temperatures are those that remain paramagnetic (or rather paranematic) to 0 K and have rapidly growing, indeed diverging, (Curie-Weiss-like) susceptibility as T→0 K. This prevents isentropes from ‘running into’ the ordered phase, which prevents further cooling.

To search for candidate materials, AC elastocaloric effect, ultrasound, or any other method that are congruent to these measurements via Maxwell relations, are excellent; they are able to determine the strain susceptibility of a candidate materials. Spectroscopic probes including NMR [31-33] and Raman scattering [34] that are also sensitive to rotational symmetry down to low temperatures are also insightful. Overall, however, measuring the AC elastocaloric effect is especially convenient, as the same tools may be used to perform the measurements presented in this study.

Aside from improvements, the AC elastocaloric effect has been shown to be an invaluable tool for better understanding quantum materials, and the strain load/unload technique may similarly be used. In a complementary way, the strain load/unload technique offers real-time information about the elastocaloric response of a material, which could reveal interesting relaxation behavior that may be difficult to observe otherwise.

CONCLUSION

In this study, a strain load/unload technique was used to measure the elastocaloric response of the Jahn-Teller insulator TmVO₄. The results are consistent with expectations based on the entropy landscape established from previous AC measurements [7] and the known temperature dependence of the specific heat [13-15] and thermal conductivity [23, 24]. The data obtained provide direct realization of the predicted giant elastocaloric response that may otherwise only be inferred from AC techniques [7]. This load/unload technique also clearly reveals the influence of the ordered phase and the changes in the time constant of the elastocaloric-material sample. Finally, the largest change in temperature of the elastocaloric-material sample was measured to be approximately 2.36 K at 5 K, a significant step in achieving viable cryogenic cooling using the elastocaloric effect. With rapid cycling, the absence of a magnetic field, and compactness, elastocaloric cooling using quantum materials, particularly Jahn-Teller materials, has tremendous potential as an alternative cooling method for cryogenic cooling applications.

Appendix A

As described in the main text, single crystals of TmVO₄ were grown using a molten flux technique. The single crystals were cut and polished, such that the [110] direction was the long-axis of the crystal, in a geometry optimal for strain measurements. The elastocaloric-material sample was suspended and epoxied between two pairs of titanium mounting plates with DEVCON 2-ton and 5-minute epoxy, with the [110] axis aligned along the stress direction. A very thin layer of AngstromBond from Fiber Optic Center was added to the surface of the crystal to make the elastocaloric-material sample much less susceptible to cracking or breaking during the measurement. 5-minute epoxy was used to initially affix the elastocaloric-material sample to the sample plates, 2-ton was used to fill the space between the sample plates (which requires a longer curing time while providing stronger bonding), and Angstrom Bond was used because of its low viscosity. The sample plates were affixed to a CS100 Razorbill strain cell, which was used to control the strain that the material experiences via piezoelectric stacks. A thin piece of cigarette paper was used to prevent epoxy from flowing down the holes in the bottom sample plates.

A RuO₂ thermometer (Mouser Electronics) thermometer was attached to a gold heat pipe (0.002 inch diameter, sourced from California Fine Wire Co.) that was glued to the surface of the elastocaloric-material sample using DuPont 4929N silver paste prior to the application of Angstrom Bond, to allow for the largest possible thermal conductance between the thermometer and the TmVO₄ crystal. The heat-pipe thermally connects the thermometer to the elastocaloric-material sample to prevent any elastoresistance signal contamination in the elastocaloric response of the thermometer. The thermometer is electrically connected to a Wheatstone bridge in which each of the resistors in the four arms of the bridge comprise RuO₂ from the same batch to minimize variation. The bridge is affixed to the surface the strain cell, while thermally disconnected from the TmVO₄ crystal. This configuration allows for excellent cancellation of the large resistance of the thermometer attached to the elastocaloric-material sample, reducing the voltage input and enhancing signal-to-noise. In order to achieve even better cancellation due to slight mismatches in the values of the resistors, an adjustable resistor was added in parallel at room temperature to manually tune the background signal as close to zero as possible.

FIG. 16 illustrates a schematic of the experimental setup and FIG. 17 displays a picture of the mounted elastocaloric-material sample. Primed and unprimed axes refer to lab and crystal axes, respectively. Specifically, unprimed x, y, and z axes are aligned along the crystallographic [100], [010] and [001] axes. Since stress is applied along the [110] direction, a rotated (primed) coordinate system is used in the lab frame to describe the elastocaloric-material sample deformation. Hence, the x′ axis lies along the [110] crystallographic direction (see FIG. 18).

All measurements in this study were performed in a Quantum Design 14T PPMS using a custom-built strain-cell probe. The instrument setup for these experiments was identical to that used previously for AC elastocaloric measurements [17], except for a few key differences. First, as stated before, a RuO₂ thermometer was used to measure the temperature change of the elastocaloric-material sample instead of a thermocouple. Second, because the elastocaloric-material sample experienced pulses of strain rather than an oscillating strain, a function generator (SIGLENT SDG6022X Function Generator) was amplified by a high voltage amplifier (TEGAM Dual Channel High-Voltage Amplifier Model 2350) to generate the strain pulses. The strain pulse was sent to the outer stack of the Razorbill cell (varying between 40 V and 400 V), while the inner stack was controlled by a separate voltage source and remained constant during each pulse (usually ±200 V). The function generator was set to pulse mode with the fall edge set to 0.5 ms and the pulse width set to 200 ms. It was found empirically that a shorter fall edge produced a spurious voltage response in the TEGAM. Finally, the lock-in amplifier that measured the elastocaloric response of the material (SR860 Lock-In Amplifier) was configured much differently to previous AC measurements [17]. It referenced an external frequency from the current source (100 μA, 1.453 kHz) to the Wheatstone bridge, had the sync filter enabled, and was configured with a time constant of 100 μs. To obtain the data, the Data Capture feature of the SR860 Lock-In Amplifier was used in one-shot mode with n=4, yielding a data acquisition rate of 20.3 kHz.

AC elastocaloric effect measurements were also performed, as detailed later in the appendices. These measurements were completed in the same manner as previous AC elastocaloric measurements in TmVO₄ [7], with a frequency of 223 Hz and peak-to-peak AC strain amplitude of 4.7·10⁻⁵.

A capacitance sensor on the Razorbill CS100 Strain Cell was used to measure changes in separation across the cell as a proxy for changes in strain of the elastocaloric-material sample (ε_x′x′). The capacitance was read-out using a Keysight E4980AL LCR Meter. Zero strain was determined by examining the point at which the AC elastocaloric response arising from antisymmetric strain was zero; this was done at a sufficiently high temperature that effects arising from symmetry-preserving strain are negligible (T>8 K) in the exact same manner as previous AC elastocaloric effect measurements on TmVO₄ (see supplementary material of Ref. [7]). The same finite element analysis strain transfer values, as had been determined alongside previous strain measurements on TmVO₄ (see supplementary of Ref [7]), were used here. This temperature-dependent correction, which ranges from about 65% to 90% strain transfer, was included in the ε_x′x′ calculations throughout the manuscript. In principle, although not reported here, symmetry decomposition of E_x′x′ to its symmetric ((ε_xx+ε_yy)/2) and antisymmetric (ε_xy) components is also possible; this would be completed by transforming between unprimed and primed coordinates and using the known elastic moduli to determine the Poisson ratio (see supplementary of ref. [7]). However, this decomposition is unnecessary here because we are interested in the total elastocaloric response arising from both symmetry strains.

Appendix B

Both the strain load/unload technique (as shown in the main text) and the AC strain technique [17] provide information on the time constants in the thermal model (FIG. 8 in the main text). To compare these two methods, AC elastocaloric measurements were also performed on the same elastocaloric-material sample used for strain load/unload measurements. FIG. 19 displays the AC elastocaloric response of the material as a function of frequency, also called the thermal transfer function, for different temperatures. In the thermal transfer function, the low-frequency tail and high frequency tail reveal τ_s and τ_t, respectively. The magnitude of the thermal transfer function may be fit to the functional form [17]:

T t ( ω ) = T max ( 1 ω ⁢ τ s - ω ⁢ τ t ) 2 + ( 1 + C t C s + τ t τ s ) 2 ( B1 )

FIG. 19 also includes the fits to the above functional form. The shape of the thermal transfer function changes as the temperature is lowered towards the phase transition; a characteristic ‘plateau’ feature emerges because thermal transfer is maximized.

To compare the two methods, FIG. 20 shows the obtained values of τ_s from both the AC elastocaloric effect and the strain load/unload technique used in the main text as a function of temperature. The agreement is reasonable, establishing similar temperature dependence and corroborating the two measurements.

Appendix C

As established earlier in the appendices, the strain the elastocaloric-material sample experienced was controlled by applying a potential difference across piezoelectric stacks (−200 to 200 V). These piezoelectric stacks may be thought of as effective capacitors, and, under a sudden change in voltage, heat with a finite power. Specifically, the power dependence of an ideal capacitor is:

P ⁡ ( t ) = V ⁡ ( t ) ⁢ d ⁢ V ⁡ ( t ) d ⁢ t ⁢ C . ( C1 )

With significant changes in the potential (up to ±400 V in this work), the piezoelectric stacks may undergo non-negligible increases in temperature.

FIG. 21 shows the full time dependence of an elastocaloric response using the largest changes in strain used for this study. There was a spurious, low-amplitude response that persisted for a long time. This response also possessed the same sign for both the load and unload portions of the strain pulse. Here, piezoelectric stack heating of the Wheatstone bridge, which is used to measure the change in temperature of the elastocaloric-material sample, manifested as a cooling signature in the elastocaloric response.

FIG. 22 displays the same figure as FIG. 11 in the main text but without background subtraction. Although the heating response varied as a function of temperature, all but the amplitude appeared to remain the same. So, to more clearly present the shoulder-like feature, the long-time response of the 7 K curve, as shown in FIG. 23, was used to remove the background from all of the curves by scaling and then subtracting to finally obtain FIG. 11 in the main text. Note that the background subtraction does not change the location of the shoulder-like feature and solely removes the broad, extrinsic response.

Appendix D

In an effort to reproduce the shoulder-like feature in FIG. 11 in the main text, different methods were used. A first empirical attempt was made by assuming that the elastocaloric-material sample time constant, as a function of time, behaved with a functional form described by logistic decay as shown in FIG. 24:

f ⁡ ( t ) = C 1 + e - k ⁡ ( t - t 0 ) , ( D1 )

where, C, k, and t₀ are independent parameters. This method was intended to model the qualitative behavior of the time constant of the elastocaloric-material sample. Using this and Eqn. 2 from the main text, the calculated, convolved elastocaloric response is displayed in FIG. 25 (using τ_t=3.1·10⁻³ s, τ_s=1.2·10⁻² s, and ΔT_s '₂ 0.6 K). This method of calculation captured much of the qualitative behavior of the shoulder-like feature.

Another attempt was made using the time constant as a function of temperature shown in FIG. 13 in the main text. This method used the temperature of the material as feedback for the selected time constant. In this model, after an initial change in temperature, the elastocaloric-material sample relaxed and the time constant was adjusted based on the temperature at that time. This then produced the time constant as a function of temperature, shown in FIG. 26, and the resulting elastocaloric response, in FIG. 27. This method did not produce as flat a feature as is in the data. This could be because of inaccuracies in the estimation of the thermal time constant of the elastocaloric-material sample, which is the quotient of two experimentally determined quantities.

Appendix E

Considering a simple circuit with a resistor, capacitor, and voltage source, the potential across the capacitor depends on the RC time constant, τ_RC, present. Given that the potential supplied to the piezoelectric stacks is a pulse with amplitude A and has a rise time of w, the charge Q on the capacitor satisfies:

Q P ⁢ Z ⁢ T ′ ( ω ) ⁢ R + Q P ⁢ Z ⁢ T ( ω ) C = { 0 t < 0 At 0 ≤ t < w Aw t ≥ w . ( E1 )

Solving this for Q and computing the potential across the piezoelectric stacks, the solution is:

V P ⁢ Z ⁢ T = ⁢ { 0 t < - ACR ⁡ ( e - t EC - 1 ) + At 0 ≤ t < w ACR ⁡ ( 1 - e - w RC ) ⁢ e - t RC + Aw t ≥ w . ( E2 )

The supplied voltage and potential across the piezoelectric stacks are plotted in FIG. 28 for different time constants. The time constant corresponding to the experimental setup used in this study is z=8.7-10-s s. Even for the largest changes in voltage supplied, the RC time constant is short enough such that there is no significant change in strain within a meaningful time window, albeit a small time difference between the supplied and actual potential. At the time scales and temperatures present in these measurements, creep and hysteresis are not expected [35]. No other effects are expected aside from those arising from electromechanical resonance, which would lead to a spurious voltage response and was not evident in our measurements.

The same may also be done for an AC supply, which is pertinent for the AC elastocaloric measurements performed. The charge on the capacitor is given by:

V P ⁢ Z ⁢ T ′ ( ω ) ⁢ R + V P ⁢ Z ⁢ T ( ω ) C = sin ⁢ ω ⁢ t , ( E3 )

and the potential across the capacitor is:

V P ⁢ Z ⁢ T = sin ⁢ ω ⁢ t C 2 ⁢ R 2 ⁢ ω 2 + 1 - CR ⁢ ω ⁢ cos ⁢ ω ⁢ t C 2 ⁢ R 2 ⁢ ω 2 + 1 . ( E4 )

For frequencies near the one used for the AC elastocaloric measurements presented here (223 Hz), there is a negligible change in the waveform across the piezoelectric stacks for τ=8.7·10⁻⁵ s, as shown in FIG. 29.

Appendix F

The volumetric specific cooling power (0.34 W/cm³) was calculated by using the maximum value of the cooling power stated in the literature [30] along with the value of ΔT_t at 5 K (0.90 K) and the thermal resistance of the elastocaloric-material sample (obtained with the dimensions of the elastocaloric-material sample and the thermal conductivity given in the literature [23]):

Q ˙ = Δ ⁢ T 2 ⁢ π ⁢ R therm , ( F1 )

and subsequently dividing by the mass of the elastocaloric-material sample (0.19 mg sample), the thermal resistance (R_therm) and converting with the density of TmVO₄ (6.02 g/cm³ [36]).

The power gradient and energy density were calculated by taking the product of ΔT_s at 5 K (2.36 K) and the thermal conductivity [23] and the specific heat (see supplementary material of Ref. [7]), respectively.

Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.

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