Thermodynamics of multicaloric effects

Here, we briefly recall the main thermodynamic relations that are required to understand multicaloric responses in materials that display more than one ferroic property. Let’s assume the likely scenario of two external fields Y ₁ and Y ₂, which are associated with their two conjugate ferroic variables X ₁ and X ₂, respectively. For example, a magnetic field μ₀H and an electric field E would be associated with sample magnetization M and polarization P, respectively. We will restrict our discussion to isothermal changes of entropy ΔS, and we will assume equilibrium thermodynamics.

Given that entropy is a state function, it will undergo path-independent changes that do not depend on whether the driving fields are applied simultaneously, or in some sequence. We therefore express the entropy change arising due to a change in both applied fields by following any specific path,^{Reference Planes, Castán and Saxena30} e.g.,:

(1)$$\Delta S\left[ {\left( {0,0} \right) \to \left( {{Y_1},{Y_2}} \right)} \right] = \Delta S\left[ {(0,0) \to ({Y_1},0)} \right] + \Delta S\left[ {({Y_1},0) \to ({Y_1},{Y_2})} \right].$$

The first term on the right is the monocaloric effect due to the application of field Y ₁ at Y ₂ = 0, while the second term on the right is the monocaloric effect due to the application of field Y ₂ at constant Y ₁. One may then express the two terms via the appropriate Maxwell relations to obtain:

(2)$\Delta S\left[ {\left( {0,0} \right) \to \left( {Y_1 ,Y_2 } \right)} \right] = \mathop \smallint \limits_0^{Y_1 } \left( {{{\partial X_1 } \over {\partial T}}} \right)_{Y_1^\prime ,Y_2 = 0} {\rm{d}}Y_1^\prime + \mathop \smallint \limits_0^{Y_2 } \left( {{{\partial X_2 } \over {\partial T}}} \right)_{Y_1 Y_2^\prime } {\rm{d}}Y_2^\prime .$

Expanding the second term on the right of Equation 2,^{Reference Planes, Castán and Saxena30,Reference Stern-Taulats31} the expression becomes:

(3)$\Delta S\left[ {\left( {0,0} \right) \to \left( {Y_1 ,Y_2 } \right)} \right] = \mathop \smallint \limits_0^{Y_1 } \left( {{{\partial X_1 } \over {\partial T}}} \right)_{Y_1^\prime ,Y_2 = 0} {\rm{d}}Y_1^\prime + \mathop \smallint \limits_0^{Y_2 } \left( {{{\partial X_2 } \over {\partial T}}} \right)_{Y_1 = 0,Y_2^\prime } {\rm{d}}Y_2^\prime + \mathop \smallint \limits_0^{Y_1 } \mathop \smallint \limits_0^{Y_2 } {{\partial {\rm{\chi }}_{12} } \over {\partial T}}{\rm{d}}Y_1^\prime {\rm{d}}Y_2^\prime .$

In contrast to Equation 2, the entropy change is now expressed as the sum of two monocaloric responses that are determined while the inactive field is zero, and a cross-coupling term that is significant if the cross-susceptibility,

(4)$${\rm{\chi }}_{12} = \left( {{{\partial X_1 } \over {\partial Y_2 }}} \right)_{T,Y_1 } = \left( {{{\partial X_2 } \over {\partial Y_1 }}} \right)_{T,Y_2 } ,$$

exhibits a notable temperature dependence. (This cross-susceptibility represents the magnetoelectric coupling coefficient^{Reference Eerenstein, Mathur and Scott10} when applying magnetic and electric fields.)

Equation 3 shows that the multicaloric response of a material is not given by the sum of each monocaloric response while the inactive field is zero, unless $\partial {\chi _{12}}/\partial T = 0$. This may be appreciated by considering the diagram in Figure 2, where the field variables are magnetic field and pressure. First, it is necessary to observe that path independence implies that one may achieve the same entropy change (A → D) either by applying pressure at zero magnetic field (A → C) and then magnetic field at finite pressure (C → D), or by applying magnetic field at zero pressure (A → B) and then pressure at finite magnetic field (B → D). It then follows that the magnitude of a caloric effect driven by one field variable depends on the magnitude of the other field variable, due to the cross-coupling. For example, the magnetocaloric effects are different in magnitude when driven at zero pressure (A → B) or finite pressure (C → D).

Figure 2.

Schematic variation of isothermal entropy with pressure and magnetic field (χ₁₂ ≠ 0). Multicaloric entropy changes can be achieved by applying pressure and magnetic field either simultaneously or sequentially. For example, the entropy change associated with pathway A → D can be achieved in either one step, two steps (by applying first pressure [A → C] and then field [C → D], or by applying first field [A → B] and then pressure [B → D]), or multiple steps.

Multicaloric examples

The ability to vary more than one field gives access to a wider range of parameter space, such that a given phase transition may become accessible with lower fields in a broader window of temperatures. This may be appreciated by considering the (anhysteretic) phase diagrams in Figure 3a–b where the field variables are again magnetic field and pressure. For example, finite pressure permits the α → β transition in Figure 3a to be isothermally driven using relatively small magnetic fields. More generally, the signs of both field changes should be chosen in order to drive the transition in the same direction. For example, the α → β transition in Figure 3b is favored by increasing magnetic field and reducing pressure.

Figure 3.

Schematic phase diagrams for materials at finite temperatures, in which transitions between a high-temperature α phase and a low-temperature β phase are associated with (a) conventional magnetocaloric effects (dT/dH > 0) and conventional barocaloric effects (dT/dp > 0), and (b) conventional magnetocaloric effects (dT/dH > 0) and inverse barocaloric effects (dT/dp < 0). The sloping pink planes denote the (anhysteretic) phase boundary.

The hysteresis that is, in practice, associated with first-order phase transitions can be manipulated by judiciously using a secondary applied field. This strategy can shift hysteresis from the primary field to the secondary field,^{Reference Liu, Gottschall, Skokov, Moore and Gutfleisch12} or fully undrive caloric effects that cannot be fully undriven by the removal of the primary field alone. An example of the former scenario is explained in Figure 4, again using magnetic field and pressure as the field variables. The energy dissipated during the multicaloric cycle shown (red dashed line) is given to a good approximation by |∆p∆v|, where ∆p = p ₂ – p ₁ and ∆v is the difference in specific volumes for the two phases (Figure 4b).^{Reference Stern-Taulats, Castán, Planes, Lewis, Barua, Pramanick, Majumdar and Mañosa23} Experimentally, these types of multicaloric cycles have been realized using Fe-Rh^{Reference Stern-Taulats, Castán, Planes, Lewis, Barua, Pramanick, Majumdar and Mañosa23} and La-Fe-Mn-Si.^{Reference Lovell, Bez, Boldrin, Nielsen, Smith, Bahl and Cohen32}

Figure 4.

Multicaloric trajectories. (a) The red dashed line (A → B → C → D → A) includes isobaric magnetization-field legs at pressures p ₁ (light blue dashed and solid lines) and p ₂ (dark blue dashed and solid lines). Projections of this trajectory along the pressure and field axes (gray solid lines, located for convenience at the plot edge) help understand the hysteresis. The projection along the pressure axis shows that magnetic hysteresis can be traded for pressure hysteresis by increasing the pressure before removing the magnetic field. The resulting dissipation is seen by plotting (b) the corresponding changes of volume versus pressure. For simplicity, we assume that the magnetization and volume do not change when the pressure is varied (B → C and D → A).

The pressure and magnetic field variables were chosen in Figures 2–4 because most multicaloric studies focus on magnetocaloric effects that are modified by hydrostatic pressure.^{Reference Gama, Coelho, de Campos, Carvalho, Gandra, von Ranke and de Oliveira33–Reference Albertini, Kamarád, Arnold, Pareti, Villa and Righi37} For example, the peak magnetocaloric effects in Mn-Ni-Si-based compounds^{Reference Samanta, Lepkowski, Saleheen, Shankar, Prestigiacomo, Dubenko, Quetz, Oswald, McCandless, Chan, Adams, Young, Ali and Stadler36} and Mn-rich Ni-Mn-Ga Heusler alloys^{Reference Albertini, Kamarád, Arnold, Pareti, Villa and Righi37} are increased by hydrostatic pressure, whereas there is a decrease in the peak magnetocaloric effects in Gd-Si-Ge^{Reference Carvalho, Alves, de Campos, Coelho, Gama, Gandra, von Ranke and Oliveira34} and Ni-Mn-Co-Sb Heusler alloys.^{Reference Nayak, Suresh, Nigam, Coelho and Gama38} There are also reports on elastocaloric effects that are modified by either magnetic or electric fields.^{Reference Castillo, Soto-Parra, Matutes-Aquino, Ochoa-Gamboa, Planes, Mañosa, González-Alonso, Stipcich, Romero, Ríos-Jara and Flores-Zúñiga24,Reference Chauhan, Patel and Vaish39} In Heusler alloys, the elastocaloric refrigerant capacity can be increased by ∼20% using magnetic fields of μ₀H = 1 T.^{Reference Castillo, Soto-Parra, Matutes-Aquino, Ochoa-Gamboa, Planes, Mañosa, González-Alonso, Stipcich, Romero, Ríos-Jara and Flores-Zúñiga24} By contrast, the elastocaloric properties of single-crystal ferroelectric oxides are hardly affected by an electric field.^{Reference Chauhan, Patel and Vaish39}

There are still few reports on caloric effects driven by both magnetic and electric fields. However, it should be possible to electrically control the magnetocaloric effect in multiferroic heterostructures via strain.^{Reference Cong, Wang, Cao, Liu, Liu and Du18,Reference Moya, Hueso, Maccherozzi, Tovstolytkin, Podyalovskii, Ducati, Phillips, Ghidini, Hovorka, Berger, Vickers, Defay, Dhesi and Mathur40–Reference Liu, Phillips, Mattana, Bibes, Barthélémy and Dkhil42} For, example a small multicaloric response has been demonstrated using ribbons of the Heusler material Ni-Co-Mn-In on a ferroelectric substrate of (1–x)Pb(Mg_1/3Nb_2/3)-xPbTi_0.28O₃,^{Reference Cong, Wang, Cao, Liu, Liu and Du18} where an electric field modified the peak temperature of the inverse magnetocaloric effects. Similar results were also obtained^{Reference Liu, Phillips, Mattana, Bibes, Barthélémy and Dkhil42} in FeRh-films on BaTiO₃ substrates.

Outlook

The use of more than one field variable opens up what may be considered to be a new field of research, just as the use of both magnetic and electric fields has opened up the field of magnetoelectrics following the large separate bodies of work on ferromagnetic and ferroelectric materials. The study of multicaloric materials and effects is now particularly timely given the confluence of research in magnetocaloric, electrocaloric, and mechanocaloric effects. This confluence is manifested in both review articles^{Reference Fähler, Rößler, Kastner, Eckert, Eggeler, Emmerich, Entel, Müller, Quandt and Albe5–Reference Moya, Kar-Narayan and Mathur7,Reference Crossley, Mathur and Moya43} and international conferences. In addition to a cross-fertilization of expertise, one can anticipate a blossoming of multicaloric research itself, yielding both novel science and possible applications. Consequently, we can say with confidence that the field of multicalorics will continue to be greater than the sum of the parts.