# Solid-State Heating Using the Multicaloric Effect in Multiferroics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{T,H}= (∂P/∂H)

_{T,E}= α. Then the electrically (1) and magnetically (2) induced multicaloric effects in a given multiferroic system are expressed as [22]:

_{0}is the magnetic permeability of vacuum, ε

_{0}is the dielectric permittivity of vacuum, C is the specific heat capacity of the system per unit volume as C = T(∂S/∂T)

_{H,E}, and χ

^{m}and χ

^{e}are the susceptibilities of the magnetic and polar phase, respectively. The full derivation of relations (1) is given in references [22,23].

_{E,H}> 0) for adiabatic polarization/magnetization, or negative (ΔT

_{E,H}< 0) for adiabatic depolarization/demagnetization, respectively. A closer examination of relations (1) indicates that they are very similar to those describing the electrocaloric and magnetocaloric effects, except that the multicaloric effects contain additional terms, (α

_{e}/( µ

_{0}·χ

^{m})·(∂M/∂T) and (α

_{m}/(ε

_{0}·χ

^{e})·(∂P/∂T). These additional terms result from the magnetoelectric coupling in multiferroics, and they can further enhance the thermal effect, especially in strongly coupled multiferroics. For large multicaloric coupling terms, we expect a significant increase in the total temperature change. In fact, according to (1), ΔT can increase indefinitely with increasing the α coupling coefficient. This, of course, is not possible, thus, the following question arises:

What is the maximum predicted temperature change in the multicaloric effect?

_{e}·(ε

_{0}·χ

^{e})

^{−1}· (∂P/∂T) and (∂P/∂T) = α

_{m}· (µ

_{0}· χ

^{m})

^{−1}· (∂M/∂T). Using these expressions, we obtained the following simplified relations of the electrically and magnetically induced multicaloric effects:

_{f}− E

_{i}and ΔH = H

_{f}− H

_{i}. Equation (2) shows clearly the enhancement of the electrocaloric and magnetocaloric effects in the case of multicaloric effect in multiferroics, with the additional contribution to ΔT given by the magnetoelectric caloric coupling term (α

^{2}/(µ

_{0}· ε

_{0}· χ

^{m}χ

^{e}) + 1). Indeed, in the particular case of a material that does not display any magnetoelectric coupling, or it is not multiferroic (α

_{e}= α

_{m}= 0), Equation (2) simply describes the electrocaloric and magnetocaloric effects.

^{2}≤ (µ

_{0}· ε

_{0}· χ

^{m}χ

^{e}) [11]. A more recent derivation of this limit was given here [37]. This implies that the magnetoelectric caloric coupling term always takes fractional values, α

^{2}/(µ

_{0}· ε

_{0}· χ

^{m}χ

^{e}) ≤ 1. Placing this condition in Equation (2), an upper limit can be established for the maximum ΔT expected for the electrically or magnetically induced multicaloric effect, which is twice the temperature change expected for the equivalent electrocaloric or magnetocaloric effects induced by the same excitation fields, in case no magnetoelectric coupling exists. This offers a significant enhancement of the temperature change in a multicaloric effect relative to single caloric effects.

_{A}. The system is also initially assumed in thermal equilibrium. Since the operation temperature T

_{A}is selected so that the cooling agent is in a para-ferroic state, then at this state, magnetic and electric dipole moments are thermally activated and undergo random fluctuations in a para-multiferroic state (Figure 1a). Imposing an adiabatic state, a single excitation field H or E is applied to the multiferroic system. The effect of the field application is to align the magnetic and electric dipole moments, essentially resulting in a transition from disorder (high entropy state) to ordered multiferroic (low entropy state) (transition A–B, Figure 1b). Hence, the decrease in the entropy of the system under adiabatic conditions will increase the overall temperature of the system to T

_{B}= T

_{A}+ ΔT. This additional temperature could be reduced back to the initial temperature via a heat sink. In this process, the applied E or H field is maintained constant, preventing the magnetic and electric dipoles from reabsorbing heat. The required operating temperature usually dictates the nature of the heat sink, and it is usually a fluid coolant such as water for room temperature operation or a cryogenic liquid for cryogenic cooling. The transition B–C in Figure 1b corresponds to the system returning to the initial equilibrium temperature T

_{A}of the heat sink. Using a thermal switch to break the contact with the heat sink, the system returns to adiabatic conditions, and the total entropy remains constant again. Simultaneously, the applied H or E field is switched off, corresponding to transition C–D (Figure 1b). The field removal initiates an adiabatic demagnetization and depolarization process, causing the magnetic and electric moments to absorb heat as they relax back to equilibrium. Since entropy increases again, the adiabatic condition is fulfilled by decreasing the temperature of the refrigerant to a value lower than the temperature of the heat sink, i.e., T

_{D}= T

_{A}− ΔT. The transition D–A in Figure 1b corresponds to the multiferroic refrigerant being placed in thermal contact with the environment being refrigerated, ending the cooling cycle. The solid-state cooling technology and its thermodynamic cycle are well established and essentially applied identically to all the solid-state caloric effects, with the only difference being the caloric material itself and the corresponding excitation force/field.

_{A}(Figure 1c,d). Upon applying an H or E field, forcing the magnetic spins and electric dipole moments to align, reducing the entropy of the system, the multiferroic’s temperature increases to T

_{B}= T

_{A}+ ΔT (transition A–B, Figure 1c,d). While the applied excitation field is still on, the excess temperature is transferred to the environment via a heat sink. For room temperature heating applications, the heat exchange/sink is typically water circulated in contact with the multiferroic heating element. In this process, the system returns to the initial equilibrium temperature T

_{A}given by the heat sink (transition B–C, Figure 1c,d). Maintaining thermal contact with the heat sink, the applied H or E field is switched off (transition C–A, Figure 1c,d), creating, in effect, an isothermal demagnetization and depolarization process, which causes the spins and electric dipoles to exchange heat with the environment, at constant temperature T

_{A}. The multiferroic heating element is then subjected to another field application, and the whole cycle is repeated.

_{c}

^{m}≈ T

_{c}

^{e}≈ 300 K), the ∂M/∂T, ∂P/∂T and the total entropy change are most significant at around 300 K. This property combined with a large enough magnetoelectric coupling coefficient, would result in significant ΔT changes. In terms of room temperature heating applications for domestic use, this is interesting as it suggests that a domestic heating system operating on the proposed multicaloric heating principle would only require a temperature change of around ΔT = 10 K, in order to ensure that it maintains a constant working temperature of the environment ideal for habitation. Assuming that active multiferroic elements displaying large magnetoelectric coupling effects at room temperature are developed, the heating systems operating on this principle would undoubtedly become a reality.

_{0.5}Ta

_{0.5}O

_{3}(PST). Using the data presented in Nair et al. [40], we have constructed an idealized model to investigate the potential output heating power, electrical input power, and the resulting coefficient of performance (CoP) for a system operating between 10 °C and 60 °C, these being the respective vales for T

_{A}and T

_{B}of Figure 1d. As it is typical of currently known electrocaloric materials, the temperature change in the application of an electric field is low; for PST, the maximum is ~4 K at 305 K with 15.8 V µm

^{−1}electric field [40]. In order to increase the temperature differential between points T

_{A}and T

_{B}(Figure 1d), we have considered a 13 PST multilayer system. The resulting thermal output power and electrical input power are presented in Figure 2 as a function of the speed at which the thermodynamic cycle in Figure 1d can be achieved. The resulting coefficient of performance, i.e., the difference in output power to input power, is ~3, giving an efficiency of ~300%. This result highlights the high efficiency/CoP due to the ability to extract heat from the low-temperature end of the system in the same way that heat pumps extract heat from their surroundings.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagrams of the proposed cooling (

**a**,

**b**adapted from ref. [22]) and heating cycles of the solid-state multicaloric effect. (

**a**) Schematic of the four stages cooling cycle (A–D) showing the entropy change due to relaxation of the magnetic moments (black arrows) and electric dipoles (ovals); (

**b**) The corresponding Brayton cooling cycle; (

**c**) Schematic of the three stages heating cycle (A–C); (

**d**) The corresponding thermodynamic heating cycle.

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**MDPI and ACS Style**

Vopson, M.M.; Fetisov, Y.K.; Hepburn, I.
Solid-State Heating Using the Multicaloric Effect in Multiferroics. *Magnetochemistry* **2021**, *7*, 154.
https://doi.org/10.3390/magnetochemistry7120154

**AMA Style**

Vopson MM, Fetisov YK, Hepburn I.
Solid-State Heating Using the Multicaloric Effect in Multiferroics. *Magnetochemistry*. 2021; 7(12):154.
https://doi.org/10.3390/magnetochemistry7120154

**Chicago/Turabian Style**

Vopson, Melvin M., Yuri K. Fetisov, and Ian Hepburn.
2021. "Solid-State Heating Using the Multicaloric Effect in Multiferroics" *Magnetochemistry* 7, no. 12: 154.
https://doi.org/10.3390/magnetochemistry7120154