Multicaloric Effect: Fundamentals

• The multicaloric effect is a phenomenon where materials exhibit field-induced entropy and temperature changes via coupled order parameters under multiple external stimuli.
• It employs cross-coupling between properties like magnetization, polarization, and strain, leading to enhanced caloric responses compared to single-field effects.
• Research in this area spans thermodynamic formulations, experimental studies in multiferroic and metamagnetic systems, and innovative applications in solid-state cooling and heating.

Searching arXiv for recent and foundational papers on the multicaloric effect to support the encyclopedia article. The multicaloric effect denotes a caloric response in which a material exhibits field-induced entropy change and adiabatic temperature change under more than one external stimulus, or under coupled action of multiple order parameters responsive to a single stimulus. In the literature, the term has been used in two partially overlapping senses. In multiferroics, it has often referred to the adiabatic temperature change induced by a single electric or magnetic field in a material with coupled polarization and magnetization, so that the directly driven caloric response is supplemented by a cross-coupled contribution mediated by magnetoelectric coupling (Vopson, 2016, Vopson et al., 2021). In metamagnetic or ferroelastic systems, it has also come to denote combined or sequential caloric operation under multiple external fields—most prominently magnetic field and mechanical stress—when both fields act on the same first-order transition and thereby produce a larger or broader useful response than a single field alone (Gràcia-Condal et al., 2020, Hou et al., 2021). More recent work has sharpened this distinction by separating multifield caloric effects from single-field, multi-order responses, identifying the genuinely two-field contribution as an additional entropy term beyond the ordinary magnetocaloric and electrocaloric parts (Ino et al., 25 Jul 2025).

1. Definitions and thermodynamic formulations

The general thermodynamic language of multicaloricity is built from generalized displacements $X_i$ and conjugate fields $x_i$, for example

$X_i=M,\;P,\;V,\;\varepsilon,\ldots$

and

$x_i=H,\;E,\;\sigma,\ldots$

with generalized susceptibilities

$$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$

A multicaloric response is then associated with field-driven entropy changes in systems where multiple $X_i$ are coupled (Vopson, 2016).

In the multiferroic formulation defended by Vopson, the adiabatic temperature change contains a direct caloric term and an additional cross-coupled term proportional to a generalized coupling coefficient $\alpha_{ij}$, so that the applied field $x_j$ alters not only its conjugate order parameter $X_j$ but also another coupled $X_i$ (Vopson, 2016). In the electric–magnetic case, the structure defended in the reply is

$x_i$0

$x_i$1

A related 2021 treatment of multiferroic heating uses the Gibbs differential

$x_i$2

together with linear magnetoelectric coupling

$x_i$3

and derives simplified forms in which the multiplicative coupling factor

$x_i$4

enhances the ordinary electrocaloric or magnetocaloric response, subject to the thermodynamic bound

$x_i$5

so that the enhancement factor is at most $x_i$6 (Vopson et al., 2021).

A more recent thermodynamic clarification distinguishes the total isothermal entropy change under simultaneous electric and magnetic fields,

$x_i$7

from its decomposition into magnetic, electric, and genuinely multifield terms,

$x_i$8

with

$x_i$9

In this usage, $X_i=M,\;P,\;V,\;\varepsilon,\ldots$0 is the additional entropy uniquely associated with cross-correlation between fields, not merely the coexistence of two monocaloric effects (Ino et al., 25 Jul 2025).

For first-order transitions driven by generalized fields, the review literature emphasizes a generalized Clausius–Clapeyron form

$X_i=M,\;P,\;V,\;\varepsilon,\ldots$1

where $X_i=M,\;P,\;V,\;\varepsilon,\ldots$2 and $X_i=M,\;P,\;V,\;\varepsilon,\ldots$3 with the usual sign convention for pressure–volume work (Hou et al., 2021). This relation underlies much of the practical interpretation of magnetocaloric, electrocaloric, elastocaloric, and barocaloric behavior in multicaloric candidates.

2. Conceptual development and the multiferroic controversy

The multicaloric effect was first proposed in multiferroics in 2012, but the early theoretical formulation became controversial because it appeared to predict changes in one order parameter at constant value of the corresponding externally applied field, while simultaneously attributing those changes to variation of that same field (Vopson, 2016, Starkov et al., 2015). The rebuttal by Starkov and coauthors argued that the original derivation confused dependent and independent variables and improperly combined special-case constitutive relations valid under mutually incompatible held-fixed conditions (Starkov et al., 2015).

The contested relations concerned the magnetoelectric response

$X_i=M,\;P,\;V,\;\varepsilon,\ldots$4

and the ordinary magnetic response

$X_i=M,\;P,\;V,\;\varepsilon,\ldots$5

from which the original treatment inferred

$X_i=M,\;P,\;V,\;\varepsilon,\ldots$6

The rebuttal argued that this inference is illegitimate if one carefully tracks which variables are fixed, because the full linear constitutive laws should instead be written as

$X_i=M,\;P,\;V,\;\varepsilon,\ldots$7

and the simplified relations arise only under distinct special constraints (Starkov et al., 2015).

Vopson’s reply accepted that the original derivation appears contradictory if written incautiously, but defended the concept by introducing a distinction between external fields and magnetoelectrically induced internal fields. The argument rewrites the magnetic differential as involving an externally applied field $X_i=M,\;P,\;V,\;\varepsilon,\ldots$8 and an induced internal field $X_i=M,\;P,\;V,\;\varepsilon,\ldots$9, leading to

$x_i=H,\;E,\;\sigma,\ldots$0

and similarly for the converse induced electric field,

$x_i=H,\;E,\;\sigma,\ldots$1

On this reading, the controversial cross terms are retained, but only as effects of induced internal fields rather than externally controlled ones (Vopson, 2016).

This exchange established a lasting division in the literature. One strand treats multicaloricity in multiferroics as a single-field, multi-order effect mediated by magnetoelectric coupling (Vopson, 2016, Vopson et al., 2021). Another strand, especially in metamagnetic Heuslers and FeRh, uses the term for cooperative or sequential operation under multiple external fields acting on the same first-order transition (Gràcia-Condal et al., 2020, Amirov et al., 25 Feb 2025). The later terminology of MOCE and MFCE was proposed precisely to reduce this ambiguity (Ino et al., 25 Jul 2025).

3. Material classes and physical mechanisms

The multicaloric effect is strongest in materials where multiple order parameters are strongly coupled and at least one accessible field can drive a large entropy-bearing phase transition. The review literature identifies several major classes: magnetostructural compounds, metamagnetic Heusler alloys, FeRh-based systems, multiferroic oxides and ceramics, and composite architectures that transduce one field into another (Hou et al., 2021).

In multiferroics, the relevant mechanism is magnetoelectric coupling between magnetic and electric subsystems. In PFN-PMW,

$x_i=H,\;E,\;\sigma,\ldots$2

the same single-phase bulk ceramic exhibits both electrocaloric and magnetocaloric responses over a broad sub-room-temperature range, with maximum $x_i=H,\;E,\;\sigma,\ldots$3 K at $x_i=H,\;E,\;\sigma,\ldots$4 K under $x_i=H,\;E,\;\sigma,\ldots$5 kOe and maximum $x_i=H,\;E,\;\sigma,\ldots$6 K at $x_i=H,\;E,\;\sigma,\ldots$7 K under $x_i=H,\;E,\;\sigma,\ldots$8 kV/cm (Ursic et al., 2016). This demonstrates coexistence of both caloric channels in one material, although not a directly measured simultaneous dual-field enhancement.

In KDP-type ferroelectrics, the multicaloric response emerges because both the longitudinal electric field $x_i=H,\;E,\;\sigma,\ldots$9 and the shear stress $$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$0 are conjugate to the same proton-ordering state. The calculated adiabatic temperature change is obtained from

$$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$1

and

$$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$2

Within that model, the combined response is stronger than either the electrocaloric or piezocaloric effect alone, and at temperatures far from the transition it can exceed the simple sum of the separate shifts (Vdovych et al., 2015).

In metamagnetic Heusler alloys, especially Ni–Mn–In derivatives, the physical basis is a first-order martensitic transition between low-magnetization martensite and high-magnetization austenite. Magnetic field stabilizes austenite, while uniaxial compressive stress stabilizes martensite. Because both fields act on the same transition, cooperative use of field application and stress removal can access more of the latent heat than either field alone (Gràcia-Condal et al., 2020). This same physics also underlies the proposed exploiting-hysteresis cycle, where field-induced austenite is retained after unloading because of hysteresis and the reverse transformation is later triggered by stress (Gràcia-Condal et al., 2020, Pfeuffer et al., 2021).

In FeRh, the first-order AFM–FM transition is intrinsically multicaloric because it couples magnetization, volume, and stress. The transition changes electronic structure and density of states, with $$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$3 states/eV per FeRh formula unit in the AFM state and $$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$4 in the FM state, and is accompanied by about a $$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$5 density decrease on going to the FM phase (Zarkevich et al., 2017). This allows magnetocaloric, barocaloric, and elastocaloric responses to be viewed as different field projections of the same entropy-carrying transition.

A related near-room-temperature example is the MnNiSi-based alloy

$$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$6

especially $$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$7, which shows giant barocaloric and magnetocaloric effects at the same magnetostructural transition. The maximum barocaloric entropy change is

$$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$8

for $$\chi_i=\left(\frac{\partial X_i}{\partial x_i}\right).$$9 kbar, while the magnetocaloric entropy change is

$X_i$0

for $X_i$1 T (Samanta et al., 2016). This is one of the clearest demonstrations of a single material supporting two giant caloric channels in the same temperature range.

4. Representative systems and quantitative behavior

The diversity of multicaloric behavior is best seen across representative material systems.

Selected examples

System	Stimuli / channels	Representative result
Ni$X_i$2Mn$X_i$3In$X_i$4	Magnetic field + uniaxial stress	$X_i$5, $X_i$6 K under combined loading (Gràcia-Condal et al., 2020)
Fe$X_i$7Rh$X_i$8	Magnetic field + uniaxial tension	$X_i$9 K at $\alpha_{ij}$0 K for $\alpha_{ij}$1 T + $\alpha_{ij}$2 MPa (Amirov et al., 25 Feb 2025)
PFN-PMW	Electrocaloric + magnetocaloric coexistence	$\alpha_{ij}$3 K at $\alpha_{ij}$4 K; $\alpha_{ij}$5 K at $\alpha_{ij}$6 K (Ursic et al., 2016)
Y$\alpha_{ij}$7CoMnO$\alpha_{ij}$8	Magnetic and electric caloric channels in a multiferroic	$\alpha_{ij}$9 J kg$x_j$0 K$x_j$1, estimated total $x_j$2 K (Murthy et al., 2014)
$x_j$3	Electric + magnetic cross-correlation	$x_j$4 mJ kg$x_j$5 K$x_j$6 at $x_j$7 T, $x_j$8 kV/cm (Ino et al., 25 Jul 2025)

In the direct multicaloric study of Ni$x_j$9Mn$X_j$0In$X_j$1, the combined path of stress removal from $X_j$2 MPa and magnetic-field application yields

$X_j$3

$X_j$4

exceeding both the magnetocaloric-only maximum of $X_j$5 at $X_j$6 T and the elastocaloric response from $X_j$7 MPa alone (Gràcia-Condal et al., 2020). At the practically important field of $X_j$8 T, the single-field magnetocaloric response is only about $X_j$9–$X_i$0, whereas combining $X_i$1 T with removal of $X_i$2 MPa yields up to $X_i$3 at $X_i$4 K and $X_i$5 K (Gràcia-Condal et al., 2020).

In Fe$X_i$6Rh$X_i$7, direct measurements under combined magnetic field and uniaxial tension give

$X_i$8

for $X_i$9 T at $x_i$00 K,

$x_i$01

for $x_i$02 MPa at $x_i$03 K, and

$x_i$04

for co-application of $x_i$05 T and $x_i$06 MPa at $x_i$07 K (Amirov et al., 25 Feb 2025). The combined effect exceeds either mono-caloric effect, but the authors explicitly note that it is not equal to the arithmetic sum of the two separate responses.

In multiferroic Y$x_i$08CoMnO$x_i$09, the main magnetic entropy change is

$x_i$10

at $x_i$11 K for $x_i$12 T, with relative cooling power $x_i$13–$x_i$14, while the electric entropy change inferred from pyroelectric data is

$x_i$15

near the same temperature (Murthy et al., 2014). The estimated magnetically induced multicaloric temperature change is $x_i$16 K, but the magnetoelectric correction is unfavorable because the magnetoelectric coefficient is negative, reducing the total below the pure magnetic contribution (Murthy et al., 2014).

The 2025 magnetoelectric study of $x_i$17 is distinctive because it isolates the genuinely two-field entropy term. For $x_i$18 T and $x_i$19 kV/cm, the peak cross-correlation entropy is

$x_i$20

around $x_i$21 K, about $x_i$22 of the peak magnetocaloric entropy change (Ino et al., 25 Jul 2025). The dominant contribution comes not from the linear magnetoelectric coefficient but from the temperature derivative of magnetic-order-induced polarization $x_i$23, which suggests that magnetic phase transitions can generate larger multifield entropy than ordinary linear magnetoelectricity (Ino et al., 25 Jul 2025).

5. Hysteresis, path dependence, microstructure, and screening

Multicaloric performance is governed not only by intrinsic entropy change but by hysteresis, kinetics, microstructure, and cyclic durability.

In Ni–Mn–In, the field-induced reverse martensitic transformation exhibits strong sweep-rate dependence. Simultaneous pulsed-field measurements of adiabatic temperature change and strain in Ni$x_i$24Mn$x_i$25In$x_i$26 showed

$x_i$27

when the reverse transformation is fully induced at $x_i$28 K, but also revealed an apparent delay at the end of the transformation for sweep rates above $x_i$29, attributed to annihilation of retained martensite (Pfeuffer et al., 2021). This raises the field hysteresis and the saturation field, which is particularly relevant for exploiting-hysteresis multicaloric cycles.

Microstructure is equally decisive. In ternary Ni–Mn–In, suction-cast microstructures with favorable texture can strongly reduce critical transformation stresses, while grain size affects failure and magnetic-field-induced transition dynamics. In that system, combining a $x_i$30 T magnetic field with a moderate sequential stress of $x_i$31 MPa increases the maximum cyclic effect by more than $x_i$32 to $x_i$33 K relative to the magnetic-field-only cyclic response (Pfeuffer et al., 2021). This shows that outstanding multicaloric performance at moderate fields is inseparable from microstructural control.

Fe-doped Ni–Mn–In provides a more explicit microstructural roadmap. In dual-phase Ni–Mn–In–Fe, a coherent, strongly Fe-enriched and In-depleted secondary $x_i$34-phase forms at grain boundaries once the matrix Fe content reaches its solubility limit of about $x_i$35 at.%. In the optimized Fe6 state, this architecture suppresses intergranular fracture, preserves a large magnetocaloric effect of about $x_i$36 and $x_i$37 K in about $x_i$38 T, and enables an elastocaloric effect of $x_i$39 K for more than $x_i$40 cycles without structural or functional degradation (Pfeuffer et al., 2021). This corresponds to an increase of cyclic stability by more than three orders of magnitude relative to single-phase Ni–Mn–In–(Fe) (Pfeuffer et al., 2021). The same design logic appears in Gd-precipitate engineering, where modest magnetocaloric degradation is traded for major mechanical strengthening, though that study does not directly demonstrate multicaloric cycling (Scheibel et al., 2023).

FeRh highlights another practical issue: the measured multicaloric response depends strongly on geometry and loading protocol. In Fe$x_i$41Rh$x_i$42, a plate-with-holes tension geometry introduces heterogeneous stress distributions, confirmed by FEM and strain-gauge measurements, which distort elastocaloric and multicaloric measurements and can even reverse the apparent sign of $x_i$43 relative to homogeneous expectations (Amirov et al., 25 Feb 2025). This suggests that direct multicaloric measurements require especially careful control of mechanical boundary conditions.

For discovery and screening, reliable thermodynamic estimators are essential. In FeRh, the AFM–FM transition temperature can be estimated from the zero-K enthalpy difference $x_i$44 through

$x_i$45

with $x_i$46 K in good agreement with the observed $x_i$47 K (Zarkevich et al., 2017). The total entropy change at the transition is

$x_i$48

with a substantial lattice contribution

$x_i$49

which cannot be captured reliably by linear-response phonons near anharmonic instability (Zarkevich et al., 2017). This work treats FeRh as a benchmark for multicaloric screening and warns that harmonic methods can mis-rank candidate materials near the instabilities that make caloric behavior large (Zarkevich et al., 2017).

6. Applications, device concepts, and unresolved issues

Multicaloric materials are motivated by solid-state cooling and, more recently, solid-state heating. The review literature frames them as candidate technologies for higher energy efficiency and reduced greenhouse-gas emissions, extending the caloric concept beyond single-field magnetocaloric or electrocaloric operation (Hou et al., 2021). The main potential advantages are larger or broader operating windows, reduced magnitude of an expensive field by adding a cheaper secondary field, and the possibility of exploiting coupled transitions inaccessible to monocaloric cycles (Hou et al., 2021).

A notable proposed application is the exploiting-hysteresis cycle in metamagnetic Heuslers. In this sequence, magnetic field first drives the martensite-to-austenite transition, hysteresis preserves the austenitic state after field removal, and uniaxial stress later restores martensite (Gràcia-Condal et al., 2020, Pfeuffer et al., 2021). This can reduce magnet exposure time and permanent-magnet volume, but it also requires precise management of hysteresis and transformation kinetics.

Composite architectures extend multicaloricity by transduction. The perspective literature discusses magneto-elastocaloric composites such as Terfenol-D/Cu-Al-Mn, in which magnetostriction converts magnetic field into strain, allowing $x_i$50 up to $x_i$51 K under only $x_i$52 T (Hou et al., 2021). Thin-film heterostructures such as FeRh/BaTiO$x_i$53 or FeRh/PMN-PT use electric-field-induced strain to tune the FeRh phase transition and nominal hysteresis (Hou et al., 2021). A plausible implication is that multicaloric engineering may increasingly rely on composite transduction rather than only intrinsic single-phase multifunctionality.

The same perspective formalizes materials-level coefficients of performance for single-field and two-field Stirling cycles: $x_i$54

$x_i$55

emphasizing that multicaloric gains depend not only on increased $x_i$56 but also on entropy generation from irreversibility (Hou et al., 2021). This suggests that nominal suppression of hysteresis by adding a non-conjugate field is only genuinely beneficial if dissipated energy is reduced rather than merely shifted to another generalized force.

The heating literature extends the concept beyond refrigeration. A three-stage multicaloric heating cycle has been proposed for multiferroics: adiabatic field application, heat delivery at constant field, and isothermal field removal (Vopson et al., 2021). Using a PST-based electrocaloric surrogate with $x_i$57 K at $x_i$58 K under $x_i$59, a 13-layer idealized system yields an estimated $x_i$60, with the suggestion that a true multiferroic implementation could approach twice the single-caloric $x_i$61 in the thermodynamic upper limit (Vopson et al., 2021).

Several misconceptions remain recurrent. One is that multicaloric enhancement must always be additive or synergistic; in fact, coupled terms can reduce the net response if signs are unfavorable, as in Y$x_i$62CoMnO$x_i$63 (Murthy et al., 2014). Another is that large equilibrium entropy change guarantees useful cyclic performance; the FeRh and Ni–Mn–In studies show that geometry, kinetic delay, and hysteresis can dominate actual operation (Pfeuffer et al., 2021, Amirov et al., 25 Feb 2025). A third is that coexistence of two monocaloric effects in one compound automatically constitutes a measured multifield enhancement; PFN-PMW shows coexistence convincingly, but not a direct simultaneous dual-field effect (Ursic et al., 2016).

Current evidence therefore places the multicaloric effect in a nuanced position. It is a real and experimentally accessible thermodynamic phenomenon across several material classes, with direct multifield enhancement demonstrated most clearly in magneto-mechanical systems such as Ni–Mn–In and FeRh (Gràcia-Condal et al., 2020, Amirov et al., 25 Feb 2025). In multiferroics, the concept is theoretically well developed and increasingly quantified, but the separation between single-field multi-order responses and genuinely two-field entropy cross-correlation remains important (Vopson, 2016, Ino et al., 25 Jul 2025). The principal open issues are not the existence of multicaloricity as such, but the optimization of hysteresis, field generation, fatigue resistance, and device integration under realistic cyclic conditions (Hou et al., 2021).