Spatio-temporal solid-state electrocaloric effect exceeding twice the adiabatic temperature change

In an all-solid-state electrocaloric arrangement, an absolute temperature change which exceeds twice the electrocaloric adiabatic temperature change is locally realized, using just the distributed thermal capacitances and resistances and spatio-temporal distributed electric field control. First, simulations demonstrate surface temperature changes up to four times (400%) the electrocaloric adiabatic temperature change for several implementations of all-solid state distributed element configurations. Then, experimentally, an all-solid-state assembly is built from commercial electrocaloric capacitors with two independently-controlled parts, and the measured surface temperature change was 223% of the adiabatic electrocaloric temperature change, which clearly exceeds twice the adiabatic temperature change and verifies the practical feasibility of the approach. This allows a significant increase of the maximum temperature difference per stage in cascaded and thermal switch-based electrocaloric heat pumps, which was previously limited by the adiabatic electrocaloric temperature change (100%) under no-load conditions. Distributed thermal element simulations provide insight in the spatio-temporal temperatures within the all-solid-state electrocaloric element. Since only the distributed thermal capacitance and resistance is used to boost the temperature change, the maximum absolute temperature change occurs only in parts of the all-solid-state element, for example close to the surfaces. A trade-off of the approach is that the required electrocaloric capacitance increases more than the gained boost of the absolute temperature change, reducing the power density and electrical efficiency in heat pump systems. Nevertheless, the proposed approach enables to simplify electrocaloric heat pumps or to increasing the achievable temperature span, and might also improve other electrocaloric applications.


Background
The electrocaloric effect [1] is a reversible temperature change ∆T AD under adiabatic conditions in special dielectrics resulting from electric field change ∆E, and promising to realize efficient and emission-free (zero-global-warming-potential) [2] solid-state heat pumps for cooling or heating applications. The adiabatic temperature change ∆T AD of most electrocaloric materials [3] is typically very low (single digit Kelvins) compared to the high temperature span ∆T SPAN = T H − T C (between the hot and cold site temperatures T H and T C ), which is required for example in domestic heat pump applications [4] (above 30 K). Nevertheless, the material performance [5] of some electrocaloric materials is very high which theoretically allows to reach a very high heat pump system coefficient of performance (COP) [6], relative to the Carnot-limit, possibly outperforming today's vapor compression systems [7]. Also, the capacitive nature of electrocaloric elements requires only a dynamic charging and discharging current, but no static current such as the Peltier element. Most of the stored energy in the electrocaloric capacitance can furthermore be recovered [8]. All together, it is thus predicted that the electrocaloric effect might enable a competitive high heat pump system COP [9] in future. While the Peltier-effect results in a static temperature gradient across the unsymmetrical layer stack and thus allows a simple all-solid-state and permanent series connection (cascade) [10] to increase the overall temperature span, the electrocaloric effect is a dynamic effect resulting from an alternating electrical field change. Thus, alternating (thermal/mechanical) connection of the electrocaloric element to either the heat source or sink is typically required to achieve unidirectional heat flow by separation of the cooling and heating phases within a full cycle. To realize this thermal separation in order to increases the overall heat pump temperature span, thermal switches (or diodes) were proposed for electrocaloric systems [11] and are often used in published electrocaloric heat pump prototypes (review in [12,13]) to cascade many electrocaloric elements in series. Even though thermal diodes might also be realized with passive mechanical structures, for example unidirectional check valves in a heat-pipe system [14], still, the complexity of such heat pump systems is increased by the high number of required stages and thermal switches or diodes. A reduction of the number of required thermal switches, or an increase of the achievable temperature change per stage thus is desired.

State-of-the-art
To increase the overall temperature span in caloric heat pump systems, cascading of many electrocaloric elements (also called multi-stage systems) using thermal diodes or switches is known since decades [11], and most electrocaloric prototypes implement such cascades. Under no thermal load conditions (and without additional heat regenerators), the cascading of a number N of stages allows to realize a maximum temperature span ∆T SPAN ⩽ N∆T AD which is up to N-times the electrocaloric adiabatic temperature change. With thermal loading (useful heat pump operation) the temperature span is further reduced, a typical value for Carnot-like cycles with the highest relative COP is often at around half of ∆T AD per stage [15][16][17], which is half of the maximum no-load temperature span on the heat pump system level.
A four-stage prototype with electrocaloric polymers and electrostatic actuation (effectively realizing eight thermal switches) is presented in [18], a nine-stage prototype using mechanical actuation of electrocaloric multilayer ceramic capacitors on printed circuit boards (effectively realizing ten thermal switches) in [16]. A two-stage prototype using multifunctional cantilevers (electrocaloric ceramics) was presented in [19]. A two-stage prototype using commercial electrocaloric capacitors and a simple linear servo actuator was presented in [20] to demonstrate Carnot-like thermodynamic cycles. Also many one-stage systems were published, where the temperature span of the system is below the adiabatic temperature change, for example in [8] where electrical energy recovery was demonstrated in addition to the heat pump prototype. Cascading with the need for thermal control devices is also known from magnetocalorics [15,21,22] and elastocalorics [23]. In all cases, the temperature difference per stage was limited to the adiabatic electrocaloric temperature change.
It should be mentioned, that another approach to increase the temperature span of (electrocaloric) heat pumps is to use an additional thermal regeneration process. For example, in [24] two rotating electrocaloric rings in a single-stage system (the voltage is only applied or removed at the hot or cold end) allow heat regeneration between many separate electrocaloric parts, resulting in a regeneration factor of around three, which clearly exceeds one times ∆T AD of a single-stage system without regeneration. This approach of heat regeneration however is an independent second universal approach which can and should be added to heat pumps to improve their performance. It is even used in fully passive thermal systems, for example such as in counter-flow heat exchangers (without heat pump functionality). These kind of regenerator however also requires thermal switches. In this work, a similar regenerator approach is partly also used (in the cases for above 2 times the adiabatic temperature change), but the difference is that in this work the regenerator is permanently attached to the electrocaloric element such that no additional thermal switches for the regeneration process are required. In [25], a 13 K temperature span was achieved by electrocaloric regenerators using electrocaloric PST ceramics, which under adiabatic conditions have an electrocaloric temperature change per element below 5.5 K [26]. In [27], an electrocaloric tube with three individually controlled parts using sawtooth voltages instead of rectangular pulses, and a moving fluid was used to realize an increased temperature effect.
In [28], an arrangement of electrocaloric elements was presented which uses two independentlycontrolled layers, which is similar to the distributed electric field control used in this work. The work [28] also discusses arrangements where two electrocaloric layers are permanently attached to either the heat sink or heat source, such that only one thermal control device is required. However, the work [28] aims to increase the cooling power and the temperature difference per stage is still limited to below one times the adiabatic temperature change.
In contrast to the state-of-the-art, this work aims to increase the temperature difference per stage (where one stage is an all-solid-state arrangement) beyond one times the adiabatic temperature change, using the distributed geometry of the electrocaloric elements with independently-controlled regions.
Another ongoing research direction (in contrast to cascading or regeneration) tries to exploit non-linear and multi-parameter dependent material properties to improve the electrocaloric element or system performance using so-called solid-state cooling lines [29][30][31]. Even though this approach seems very promising to realize full solid-state electrocaloric systems, a comprehensive experimental demonstration or high performance has not yet been demonstrated (to the best of the authors' knowledge and understanding). Unidirectional heat flow in an all-solid electrocaloric heat pump without moving parts has also been postulated [32], but the calculated cooling power (and temperature span) was much lower than that of a conventional electrocaloric heat pump [28]. Another approach to combine at least two caloric effects (multicalorics [33]) is also beyond the scope of this work.

Problem
Due to the typically very low electrocaloric adiabatic temperature change ∆T AD , state-of-the-art electrocaloric heat pump systems (prototypes) require a cascading approach and/or heat regenerators to achieve a large temperature span. Today, such systems are complex and require many thermal switches. As a consequence, most of today's prototypes use only a few (two to ten) stages for demonstration purposes such that the temperature span of electrocaloric heat pump prototypes still lacks behind the requirement for real applications.
There exist many approaches to realize thermal control elements for caloric energy conversion [34], but they all have some disadvantages. Thermal diodes and switches with high performance (for example the contrast ratio of a switched thermal resistance) are difficult to realize. For example, solid state thermal diodes only have a low ratio between on-state and off-state thermal conductivity. To achieve higher thermal switching ratios, often slow mechanically actuated systems are used instead of solid-state thermal control devices to realize thermal switches with good performance. However, the improved thermal switching ratio of such mechanical systems typically comes at the expense of a limited cycle (actuation) frequency, which limits the absolute thermal power of the system. Thermal control devices are reviewed in [34].
Even if cascading or regeneration is used, one fundamental problem remains: in each stage, the maximum achievable (no-load) temperature difference is typically limited to below one times the electrocaloric temperature change. Therefore, an all-solid-state solution, or at least a reduction of the number of required thermal control devices, or an increase of the temperature change in each stage of a cascade is still desired. This work aims to increase the achievable temperature difference per stage significantly beyond one times ∆T AD .

Approach
This work's approach is to use the spacial distributed thermal resistance and capacitance of an electrocaloric element as a thermal low-pass filter. With independent electrical field control in at least two parts of one all-solid-state (permanently attached) EC element, now two regions of different temperatures can be spatially and temporally established within one element. By suitable field control (e.g. phase shifted control), it is possible to change the temperature within one (first) part of the EC element by a fraction of ∆T AD (e.g. half of ∆T AD for a 1:1 ratio of the two parts) by field change in the other (second) part and after a sufficient settling time larger than the characteristic time constant. Then this initial temperature change in the first part (from heat flow from the other part) is superimposed by an additional ∆T AD by field change in the first part. To form closed cycles a similar, but reversed, field control is repeated, and results in a total (peak-to-peak) temperature change of more than ∆T AD (possibly even higher than 2∆T AD as will be explained) in a part of one EC element (for example on parts of the surface). For proper configurations of distributed geometry and distributed electrical field excitation, the thermal low-pass behavior of the distributed thermal capacitance is used to emulate the functionality of a thermal switch. Obviously, using the low-pass property as thermal switch only results in an alternating current-like (AC, high frequency component) switching behavior, in contrast to mechanical thermal switches which realize a true direct current-like (DC, low-frequency component) thermal switching behavior. Since the electrocaloric element is operated with a continuous cyclic excitation signal, the AC thermal switch is sufficient to establish a temporary temperature difference spacially alternating between at least two parts within all-solid-state EC elements during continuous heat pump operation. Figure 1(a) shows a conventional electrocaloric cascade exemplary for N = 2 independent electrocaloric elements which are connected by three thermal switches to the heat sink, source or to each other. The conventional cascade with N stages without temperature generation (under not thermal load conditions) allows a maximum temperature span of N∆T AD , and requires N + 1 thermal switches. Figure 1(b) shows an example of a spacial-temporal configuration, which is an example of this work's approach, which now enables to reach up to 2∆T AD in at least one part (for example on one surface of the first part) of one EC element. The two parts of the one all solid-state EC element are assumed to be thermally connected with the same (or better) thermal conductivity compared to the bulk of the electrocaloric element, for example by soldering of two EC parts to one element. This ensures good heat flux between the two parts. Figure 1(c) shows another implementation of a the spacial-temporal configuration, where the cascading is realized using both surfaces of the two parts of one element. The number or required thermal switches for a heat pump with the same overall temperature span is now reduced, resulting from the halved number of all-solid-state EC elements compared to figure 1(a). The increased temperature change results from heat regeneration between two parts within an all-solid-state EC element, which is indicated by arrows in figures 1(b) and (c).
For simplicity, all cycles analyzed in this work are based on Brayton-cycles. By replacement of the instantaneous (step-like) electric field excitation with a slower, possibly two-step field change, also Carnot-like cycles could be realized, where heat is transferred (slower) to the sink and source in almost isothermal conditions. Exemplary, one complete thermal cycle for figure 1(b) is explained in detail, by analyzing the heat flow and temperature changes between the eight time instances (labeled in figure 1(b) as 1-8): • 1-2) A rapid electrical field change (increase) in the first part of the all-solid-state EC element causes an electrocaloric temperature increase ∆T AD of the first part's temperature T 1 . Since the field in the second part is not changed yet, the temperature T 2 initially does not change. • 2-3) Since the two parts are permanently attached, heat is transferred from the first to the second element.
This heat regeneration causes the temperature of the first part to decrease and of the second part to increase (indicated by arrows in the T-S diagram). • 3-4) A rapid electrical field change (increase) in the second part causes an electrocaloric temperature increase of ∆T AD in the second part. Since the temperature was already increased in the previous step, the The electrocaloric temperature change is modeled by pulse sources, and the electrocaloric elements as discrete thermal RC-lines. The figure shows simulation results. The temperatures T on the (electrocaloric) elements are shown as differences from a reference temperature T0, for example room-temperature or an internal heat storage element T0 such as a large passive block of material. total temperature change, and peak temperature change (observed for example at a surface of the second part which is not part of the permanent connection of the two parts) is now above ∆T AD . • 4-5) At the beginning of this phase, the (surface) temperature of the second part T 2 is now higher than T 1 and immediately a heat regeneration between both parts is initiated. If at the same time the surface of the second part is connected (by a thermal control device, for example a thermal diode) to a heat sink, then heat can be dumped to the hot site. However, as soon as because of the regeneration between both electrocaloric parts the surface temperature of the second part reaches below the hot site temperature, the thermal control device has to break the thermal connection to the hot site. Otherwise, heat would start to be also absorbed from the hot site, reducing the thermal power of the system. • 5-6-7-8-1) The same steps are repeated, however with reversed field changes, and absorbing heat from the cold site by another thermal control device. Since the overall (peak-to-peak) temperature change is up to 2∆T AD (no load condition), this all-solid-state arrangement enables to exceed 1∆T AD per cascade stage, simplifying the system and reducing the number of required thermal control devices.
While figure 1 showed complete cascade systems, where multiple EC stages can be cascaded and which could be used for example in electrocaloric heat pumps, in the following only one EC stage is further analyzed, and for simplicity only the case for maximum temperature span (no thermal loading) is considered. Figure 2 shows exemplary implementations of the proposed approach to realize a spatio-temporal solid-state electrocaloric effect with increasing maximum temperature difference (in each cycle) of surface temperature up to 1∆T AD (figure 2(a)) temperature change of one component with no thermal load; reaching 2∆T AD (figure 2(b)) by slow regeneration to a reference temperature T 0 (from a permanently attached reference temperature element); up to 3∆T AD (figure 2(c)) by using a permanently attached and large (here with an active electrocaloric mass ratio of 10:1) electrocaloric element to pre-heat/pre-cool another electrocaloric element at the surface, such that a delayed electrocaloric effect on the smaller element is superimposed on the preconditioned temperature; and up to 4∆T AD (figure 2(d)) by permanently adding an even slower heat regeneration to a reference temperature to the previous configuration. The electrocaloric element used in figure 1(a) is based on figure 2(a), and in the example in figures 1(b) and (c) based on figure 2(c), while in figure 1(b) the heat sink and source are connected by thermal switches to the same side of the EC element, whereas in figure 1(c) the heat sink and source are connected alternately to both sides of the EC element. All simulations and shown waveforms are based on distributed RC-line models, where the total thermal capacitance and resistance (of each EC element) is modeled with a network of series connected lumped resistors and capacitors (10 resistors and capacitors with 1/10-times the total thermal resistance or capacitance are used in the actual simulations as shown in figure 2(a), which models the distributed RC-line already sufficiently accurate). The adiabatic temperature change of the electrocaloric elements is normalized as ∆T AD = 1 K (without restriction of generality), and modeled as a common-mode rectangular source to the thermal elements. Simulation of these thermal circuits is done in this work using the electrical equivalent components in an electrical circuit simulator (LTSpice). One exemplary full simulation is provided in the supplementary materials of this publication. For configurations with at least two independently and actively controlled electrocaloric parts of an EC element, a phase shift between the electric field source signals (and thus phase shifted electrocaloric effects) is indicated by the phase shifted rectangular waveforms close to the sources. For a simplified visualization of the temperature span build-up, the reference temperature (typically around room temperature) is offset in the figures to zero. For simplified analysis of the relevant fundamental effects, the first-principle calculations assume only one fixed electrocaloric temperature change for one given voltage (electrical field) step, and no non-linear properties such as temperature or voltage/field-dependent electrical or thermal capacitances or resistances. Also, an ideal electrocaloric material with zero dissipative losses is assumed. The electrocaloric and other thermal parts are simplified with fixed (but distributed) thermal capacitances and resistances.

Experimental verification of over 2∆T AD surface temperature change in all-solid-state electrocaloric element
A minimal experiment is constructed to demonstrate that more than 2∆T AD (surface) temperature change can be actually realized in a real all-solid-state element consisting of several permanently connected (electrocaloric) parts. Typically, to quantify the adiabatic temperature change of electrocaloric materials, the material to be measured is intentionally attached to a much larger passive element (for example a large cooling and heating plate). This allows for example to quickly measure material properties with a temperature sweep, or heat flux to a base plate. Often, very thin layers of electrocaloric element are fabricated on top of other carrier substrates, which is also similar to the setup in figure 2(b). The previously described setups are thus already often used for material characterization (but not for heat pump implementations) and already give a maximum temperature change of up to 2∆T AD , either because of heat regeneration to a baseplate or to the environment at slow cycle frequencies. Since up to twice the adiabatic temperature change is already state-of-the-art, the experiment here aims for the range 2 . . . 3∆T AD , and implements the circuit from figure 2(c) with an asymmetric thermal capacitance ratio. While the experiment is under no thermal loading (except heat flow to the environment), it could be applied to a heat pump with a thermal cycle as shown in figure 1(b). Figure 3 shows an all-solid-state arrangement, which consists of one electrocaloric capacitor (forming the second part of the one EC element, C EC,2 in figure 2(c)) on top of twelve electrocaloric capacitors (forming the first part of the one EC element, C EC,1 in figure 2(c)). This work uses commercial Y5V SMD capacitors (as in [8,35], Multicomp MC1210F476Z6R3CT) which show a small electrocaloric effect at room temperature (≈23 • C), since they are based on a modified BaTiO 3 dielectric which is known to have an electrocaloric effect [36,37]. The setup was built by first soldering 12 EC MLCCs together at both electrodes. Then, the electrode of one additional EC MLCC was soldered on the middle of the larger surface of the combined 12 EC MLCCs. The soldering of the 1 to the 12 EC MLCCs ensures good heat flux between those two parts. The electrical voltage (field) of both groups is separately provided by a continuous square voltage source. The surface temperature T on top of the one electrocaloric capacitor (surface of the second part of the EC element, T 2 in figure 2(c) is measured with a thin thermocouple (type K with low mass and thin wires, Omega CHAL-003). Also, thin electrical wires are used to connect the capacitors, and the arrangement is kept above the base plate, which helps to thermally isolate the arrangement from the surroundings. The arrangement was measured in still ambient air at room temperature under adiabatic conditions, and the small surface heat convection between the samples and the air, as well as the parasitic heat flux from the samples through the electrical wires and thermocouple wires does not yet significantly influence the temperature measurement results (surface temperatures).

Measurement results
First, for comparison, all electrocaloric parts are driven by the same rectangular voltage (0 V/32 V driving voltages, both voltages (V 1/2 ) applied to the capacitors are equal and thus in phase) to mimic the conventional arrangement as in figure 2(a), where no spatio-temporal control is used. This triggers the electrocaloric in all parts of the arrangement at the same time with the same direction. Assuming ideal thermal isolation from the surrounding, all parts of the EC element undergo the same temperature change simultaneously, and no heat flow occurs between parts of the EC element. As expected, the measured surface temperature is the same as everywhere else in the arrangement, and equal to one times the adiabatic electrocaloric temperature change 1∆T AD . With the in-phase measurement, the adiabatic temperature change ∆T AD = 0.22 K is measured. Figure 4 shows the temperature measurement and driving fields during the first 80 s (conventional operation, for cycles shown). This operation strategy is typically used in cascaded electrocaloric heat pump prototypes and limits the maximum temperature difference per stage to below one times ∆T AD (see also figure 1(a)).
Then, a phase shift is introduced between the two voltage pulses: The rising voltage edge for charging and adiabatic temperature increase of the smaller electrocaloric capacitor (V 2 ) is delayed with respect to the rising voltage edge of the larger electrocaloric capacitor (V 1 ), to realize the spatio-temporal configuration as in figure 2(c). Within one cycle, now four characteristic time regions result. After an initial slow temperature increase of slightly below ∆T AD (since only part of the solid-state element is driven), an additional temperature change is superimposed as soon as the remaining part of the arrangement is driven. Initially, the larger part of the assembly is driven, such that within that part a temperature change of ∆T AD occurs. Since the larger part is thermally (permanently) connected to the smaller part which was not yet driven, the temperatures within both parts slowly equalize. Since the temperature on top surface of the smaller part (and not on or within the larger part) is measured in figure 4, a slower temperature change is observed in this phase. The reason for this slower temperature change is that the distributed RC-line between the larger and smaller part is a thermal low pass. Since the smaller part is now pre-heated by the larger part, as soon as the smaller part is driven, the full adiabatic temperature change of the smaller part is superimposed on the previous initial value, resulting in an observed surface temperature increase during the first two phases below but close to two times the adiabatic electrocaloric temperature change. After this two phases, the remaining two additional phases then complete the cycle in the reverse direction.
In total, a total peak-to-peak maximum surface temperature change of 0.49 K is measured. The theoretical possible calculated value of 0.63 K (around 2.8-times the adiabatic change, not 3-times; calculated from the 12:1 ratio of the thermal capacitances of the two parts which is not infinite, discussed later) is not completely reached here. The thermal mass of the wires and thermocouples, as well as the rather fast pulse-train both contribute that the measured value stayed below the theoretical maximum. Nevertheless, the measured value of 0.49 K exceeds twice the adiabatic temperature change (2∆T AD = 0.44 K) clearly, by 23% of the adiabatic temperature change (0.49 K is 223% of the adiabatic temperature change 0.22 K). The measurement thus experimentally verifies this work's approach for a spatio-temporal solid-state electro-caloric effect exceeding twice the adiabatic temperature change. Compared to a conventional cascading approach where in each stage the complete electrocaloric element is driven with just one voltage, this work's approach, or the experiment in particular, has a significantly increased maximum surface temperature change per solid-state element. If the surface which is measured in the experiment is contacted alternately between a heat sink and heat source, it allows to realize a temperature change significantly above that of conventional cascaded electrocaloric heat pump systems (see figures 1(b) and (c) with elements as in figures 2(b)-(d). In the following, the mechanisms is further analyzed for different configurations and up to even higher temperature changes per all-solid-state element using simulations.  Figure 5, an implementation of figure 2(a) in a heat pump, shows the conventional setup of a heat switch (or diode) based electrocaloric heat pump with one active electrocaloric element between a heat sink and source. Starting from a zeroed initial condition, the temperature span between the hot and cold side raises up to 1∆T AD under thermal no-load conditions. With increased thermal loading of the heat sink/source, the temperature span is reduced from this value. This paper only analyzes the no-load condition to investigate the maximum achievable temperature span. However, all discussed configurations can be operated with thermal loading, and then in all cases the achievable temperature span is reduced from the maximum values.

Example up to 1∆T AD temperature span per stage (conventional case)
The configuration in figure 5, operated with Brayton cycles, is the basis of many electrocaloric heat pump demonstrators in literature. Due to the low ∆T AD of typical electrocaloric materials, cascading of many stages is often used. In that case, each additional level requires additional thermal switches (or diodes), since the electrocaloric elements have to be temporal and spatially separated, which is thus not an all-solid-state solution to increase the temperature span beyond ∆T AD in each stage.

Example up to 2∆T AD temperature span per stage
Two different implementations to realize up to 2∆T AD are discussed, the first using one electrocaloric element and a much larger passive element for regeneration, and the second using two similar sized active electrocaloric elements. Figure 6, a modified implementation of figure 2(b) in a heat pump, is an extension of figure 5 by an additional much larger thermal element which is permanently thermally connected to the electrocaloric element. The modification to figure 2(b) is that the second thermal element is not electrocaloric active, but used only as passive regenerator. This additional (passive) element together with the electrocaloric element is an all-solid state element (permanently physically connected). The additional passive element serves as a regenerator to the electrocaloric element, such that on a slower time scale the temperature of the electrocaloric element decays to around the average of the hot and cold site temperatures. If the thermal capacitance of the regenerator is much larger than that of the electrocaloric part, it adds half of ∆T H−C as passive regeneration just before the electrocaloric effect of ∆T AD is triggered. In total, the maximum achievable temperature span is now (2 − CEC,1 CEC,1+CEC,0 )∆T AD , which for C C,0 ≫ C EC,0 is limited by 2∆T AD (it is assumed that the mass and thermal capacitance of the multiple parts are proportional to their electrical capacitances). Figure 7 shows the temperatures along the positions from the heat sink/source through the one to the second electrocaloric element. The time evolution (increasing time indicated by the direction of the arrows) of the temperatures is shown around the two switching instances which happen due to the two edges of the electric field pulse which triggers the electrocaloric effect. The different colors indicate different time instances within one cycle.  figure 2(a) of one active electrocaloric element between a heat-sink and source, to realize up to 1∆TAD. The heat sink/source temperature span buildup is also shown.. Figure 6. An heat pump implementation of figure 2(b), using one active electrocaloric part and a permanently attached passive thermal regenerator (10:1 mass ratio) alternately connected to a heat sink/source, to realize up to 2∆TAD. The heat sink/source temperature span buildup is also shown. A practical implementation is as follows: In some heat switch based electrocaloric prototypes the electrocaloric element is mechanically moved between the heat sink and source. The actuator (often a servo) is mechanically attached to the electrocaloric element for this purpose. In many prototypes it is attempted and suggested to realize this mechanical fixture with as little thermal capacitance and as high as possible thermal resistance to avoid thermal leakage. For example, a thin wooden stick is used in [8] (permanently attached to the active electrocaloric element) as mechanical fixture, and thin electrical wires are used to electrically interface the electrocaloric element. Theoretically, the fixture holder has the same functionality as the passive regenerator in figure 7 if it is thermally isolated from the environment, or at the midpoint temperature between the heat sink and source temperatures. Proper dimensioning of the thermal impedance (by adjusting the geometry of the components), such that the thermal time constant to the heat sink or source is much faster than to the fixture (regenerator), and selecting a system operation frequency just between the two related characteristic thermal frequencies, should allow to increase the temperature span in a heat pump demonstrator compared to the system where the mechanical fixture is intentionally optimized away. Of course, again, the introduction of a second characteristic frequency, which has to be separated from the one between the heat sink/source and electrocaloric element, will reduce the achievable system frequency and thus power density. For the sake of increased temperature span, and at the same time not increasing the  figure 2(b), using two independently-controlled but permanently attached active electrocaloric parts (1:1 mass ratio) between a heat sink and source, to realize up to 2∆TAD. The heat sink/source temperature span buildup is also shown.. required electrical charging energy, such a passive regenerator arrangement might be justified and could theoretically even improve the relative COP, since the temperature span of the system is slightly increased. Figure 8, an implementation of figure 2(b) in a heat pump, now uses two active electrocaloric parts which are permanently attached as an all solid-state element. In the example, both elements have the same thermal capacitance, and they are driven with a phase-shifted electric field. With this ratio of 1:1, if the electrocaloric element is triggered in one part while the all solid-state combined element is neither in contact with the heat sink or source (temperature below the hot and above the cold site temperature), its initial temperature change of ∆T AD will cause heat flow to the other attached part, such that after a thermal settling time both parts will have the same temperature. If the thermal settling is finished, in both parts a temperature change of 1 2 ∆T AD occurred. If now the electrocaloric effect is triggered in the second part, the full adiabatic temperature change ∆T AD is now added to the pre-cooled or pre-heated values of 1 2 ∆T AD . In total, for the 1:1 capacitance ratio, a total temperature span up to 2∆T AD occurs at least at the surface (or part of the element) of the all-solid-state element. If this combination of two permanently connected (all solid-state) electrocaloric parts as one element is used in the typical heat switch (or diode) based heat pump system, a no-load temperature span up to 2∆T AD could be realized without the need for more thermal switches than the conventional heat pump system from figure 5. Figure 9 shows the temperatures along the positions from the heat sink/source through the one to the second electrocaloric part. The time evolution of the temperatures is shown around the four switching instances which happen due to the phase-shifted electric field pulses. Figure 10. An heat pump implementation of figure 2(c) with two active electrocaloric parts (10:1 mass ratio) between a heat sink and source to realize up to 3∆TAD. The heat sink/source temperature span buildup is also shown.. Figure 11. An heat pump implementation of figure 2(d) with two active electrocaloric parts (10:1 mass ratio) and an additional permanently attached passive material part between a heat sink and source to realize up to 4∆TAD. The heat sink/source temperature span buildup is also shown. Figure 10 shows an implementation of figure 2(c) to realize up to 3∆T AD . The setup is similar to the previously discussed case using two active electrocaloric elements. However, instead of a 1:1 ratio, which just reached 2∆T AD now a significantly larger second element is used, which increases the pre-cooling/heating temperature change to CEC,1 CEC,1+CEC,2 ∆T AD , on top of that then the full ∆T AD of the smaller element is added. This two phases of a full cycle are followed by two additional phases, where first in the larger element the reversal of the electrocaloric effect −∆T AD is triggered, which after a thermal settling time brings both parts to the common average temperature of CEC,2 CEC,1+CEC,2 ∆T AD , below that then the full −∆T AD of the smaller element is subtracted. The total maximum temperature span thus is 1∆T AD + CEC,1+CEC,2 )∆T AD . This gives 2∆T AD for a capacitance ratio of 1:1 (as the previous case), ≈ 2.8∆T AD for the 10:1 ratio (see also the experiment in section 2.2), and is bound by the limit of 3∆T AD for C EC,1 ≫ C EC,2 .

Example up to 3∆T AD temperature span per stage
The position-dependence of the temporal evolution is similar to the previous discussed case and figure 9, different only in the ratio of EC material and the possible temperature span. So no additional diagrams are presented here.
Of course, the additional large active electrocaloric part has to be also driven by an (external) voltage source, and causes both additional dielectric losses in the electrocaloric material as well as external losses from the charging circuit. The achievable relative system COP of such a heat pump thus is expected to be lower compared to a conventional system (see e.g. [17] for system COP calculations). However, this work focuses on an approach to increase the temperature difference in on solid-state element, which obviously comes at the expense of other system parameters. A possible use-case could be a heat pump systems which allows either the conventional operation (both active electrocaloric elements controlled in phase) with high efficiency, but also an operation with increased temperature span (beyond what is possible with the conventional approach but otherwise similar arrangement) but reduced performance.
This simulated heat pump system uses a all-solid-state arrangement similar to the one which was experimentally demonstrated in section 2.2.

Example up to 4∆T AD temperature span per stage
To further increase the temperature span, another (third) part is added to the all-solid-state element. Figure 11 shows an implementation of figure 2(d) to realize up to 4∆T AD . This third part is passive, and very slowly brings the average temperature of the complete element to a reference temperature (the average of the hot and cold temperatures during continuous operation). To maintain the spatial temperature effects, a third time-scale is required, further reducing the power density of the complete system for the sake of higher temperature span. Compared to the previous case, the two previously middle temperature levels which were separated by CEC,1 CEC,1+CEC,2 ∆T AD are now slowly brought to one common average temperature. This additional regeneration adds another CEC,1 CEC,1+CEC,2 ∆T AD to the achievable temperature span, such that a maximum of 4∆T AD can be realized. However, again, the required additional time-scale and additional thermal mass will further reduce the power density of the system. This case is discussed here only to show that a spatio-temporal all-solid-state electrocaloric effect can be realized which significantly exceeds twice the adiabatic temperature change, and the approach could be further refined for higher temperature changes or improved power density.

Beyond 4∆T AD temperature span per stage
Theoretically, the presented approach can be extended by an arbitrary number of additional levels, increasing the spatio-temporal temperature effect. However, to maintain the relation between each two consecutive parts of the complete element requires an more than proportionally increased additional material effort, and a significantly slower system operation.
Even though there might be applications where this additional effort to realize an all-solid-state electrocaloric effect significantly higher than twice the adiabatic temperature change might be justified, the authors suspect that most likely only the case for up to 2T AD with the discussed rather simple system implementations might be actually beneficial for heat pump systems, since heat pumps require competitive performance and power density. There might be other applications (e.g. in industry or manufacturing) with other requirements, for example applications that require only a small amount of heat, but a fast cyclic temperature source of an increased temperature change.

Conclusion
It was shown and experimentally verified that in an all-solid-state electrocaloric element, locally a temperature change exceeding twice the adiabatic temperature change can be realized using only a spatio-temporal low-pass filter effect which results from the distributed thermal capacitance and resistance. Electrocaloric elements are a key component of emerging electrocaloric heat pumps. Since the electrocaloric temperature change in most known materials is too small to realize heat pumps with a sufficiently high temperature span, most prototypes today use a cascading approach to multiply the temperature effect by using many individual and physically separated electrocaloric elements, which requires an increased number of thermal switches (or diodes). This work's alternative approach allows to reduce the number of thermal switches or diodes in heat pump systems by spatio-temporal doubling (or more) of the electrocaloric temperature effect within all-solid-state electrocaloric elements. A reduced number of thermal switches and stages contributes to simplify the system, and fewer (mechanical) thermal switches could improve the reliability of such systems. Furthermore, the power density of such systems can be increased, for example in the two-element arrangement, where the attachment of two parts to one does not additional volume such as for example an additional thermal switch element. While it was theoretically shown that even four times the adiabatic temperature change could be realized in an solid-state element, practically a trade-off between temperature effect and power density or performance has to be found, since this work's approach for increased temperature effect is enabled at the expense of increased complexity and additional partly inactive and passive elements. The effect of this work's approach on the thermal power density and system frequency should be further investigated, and actual cascaded heat pump system demonstrators could be realized using this work's approach of spatio-temporal all-solid-state EC elements. Multi-material combinations could also benefit from this work's theory. For example, larger multi-layer electrocaloric ceramics could be extended by separately controlled, but physically permanently attached additional electrocaloric layers of the same or other electrocaloric materials (such as polymers) at the surfaces which form the interfaces in cascaded electrocaloric heat pumps.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.