Thermal diodes, regulators, and switches: Physical mechanisms and potential applications

Controlling thermal transport is a key challenge of modern technology, energy conversion systems, heating and refrigeration, manufacturing and materials processing, data storage, and electronics thermal management. However, engineers seeking improved thermal control are limited by the traditional toolkit of linear, static, and passive thermal components, such as thermal resistors and thermal capacitors. This paucity of thermal options pales in comparison to the rich selection of highly nonlinear, switchable, and active components in the electrical domain. For example, integrated electrical circuits for information processing rely on nonlinear building blocks such as transistors and diodes, and effectively transmitting electrical power without switches would be a tall order. Since switchable and nonlinear thermal components are not nearly as mature as their electrical counterparts, contemporary thermal research explores many different mechanisms which could provide these new capabilities.

In this article, we review experimental demonstrations of switchable and nonlinear thermal components—thermal diodes, switches, and regulators. Each of these components is defined by a characteristic transfer function connecting the heat flow (⁠ $Q$⁠) to the driving temperature difference (⁠ $ΔT)$⁠. Thermal regulators have a nonlinear transfer function, meaning that $Q$ is not linearly proportional to $ΔT$⁠. The transfer function of a thermal diode depends on the polarity of $ΔT$⁠: $Q ( Δ T ) ≠ Q ( − Δ T )$⁠. This defining asymmetry implies that the diode's transfer function must also be nonlinear, since the ratio $Q/ΔT$ cannot be a constant everywhere. Finally, the transfer function of a thermal switch can be actively controlled by a non-thermal parameter, such as pressure or voltage. After establishing these definitions in Sec. II, Sec. III surveys the physical mechanisms across the fields of heat conduction, convection, and radiation that lead to thermal rectification, regulation, and switching. With an eye towards potential applications, we focus our attention primarily on mechanisms which can operate near room temperature and atmospheric pressure. In Secs. IV A and IV B, we then examine examples of nonlinear or active thermal circuits which have been developed to improve the efficiency of solid-state refrigeration and energy harvesting cycles. In Sec. IV C, we present nonlinear and active thermal circuits which are analogous to well-known electrical circuits. We summarize the current status and future opportunities for thermal diodes, regulators, and switches in Sec. IV D.

This article complements several other recent reviews. Li et al.¹ summarized predominantly computational predictions for nonlinear nanoscale phononic devices. Ben-Abdallah and Biehs² presented theoretical predictions for nonlinear thermal radiation components in the far- and near-field. Roberts and Walker³ summarized rectification mechanisms in bulk solids and theoretical predictions of rectification in nanostructures. DiPirro and Shirron⁴ surveyed thermal switches for cryogenic refrigeration applications, and Shahriari et al.⁵ recently discussed boiling and condensation heat transfer switching using electric fields. Compared with these works, this review focuses on experimental demonstrations of thermal diodes, switches, and regulators across all fields of heat transfer, with an emphasis on the changes in thermal properties across phase transitions and on switching heat transfer using electric and magnetic fields. Though concepts of thermal computing applications have also been proposed,^1,6 we focus here on possible applications in thermal control and energy conversion.

Throughout this review, we will draw upon analogies between thermal and electrical components. This serves the dual goals of understanding thermal systems using intuition from electrical engineering and providing inspiration for adapting useful electrical circuits into the thermal domain. To motivate this approach, Fig. 1 compares the linear response theory of thermal and electrical resistors and capacitors. (There are no thermal elements that are analogous to electromagnetic inductors.⁷) The analogy between current flow in electrostatic resistor networks and heat flow in thermal resistor networks is exact at steady-state, because the governing equation, constitutive relations, and boundary conditions are all identical for electrical and thermal transport.

Analogies involving capacitance, however, must be treated with more caution. There is a strict analogy between the self-capacitance in the electrical domain, which is the voltage change required to increase the charge stored by an isolated conductor, and the thermal capacitance, which is the temperature change required to increase the energy stored by an isolated thermal conductor. However, there is no thermal analogy for the mutual capacitance of the more common parallel-plate capacitor, which characterizes the electrical potential difference between two conductors. Therefore, every capacitor in a thermal circuit must have one terminal grounded, which unfortunately prohibits thermal analogies to certain interesting electrical circuits such as voltage triplers.

Adding nonlinear and switchable thermal components to this traditional toolkit of linear resistors and grounded capacitors leads to qualitatively new thermal capabilities, including tunable thermal control and other examples discussed in Sec. IV such as rectification, heterodyning, and amplification. We now more thoroughly introduce the three primary components discussed in this review: thermal diodes, switches, and regulators.

Components are characterized by a transfer function (constitutive relation) relating flow rates to their driving potentials. For example, electrical transistors are characterized by their steady-state current-voltage diagrams. In the thermal domain, a convenient representation of the steady-state transfer function is a plot of the heat flow (⁠ $Q$⁠) as a function of the driving temperature difference (⁠ $ΔT)$ across the device. Although the absolute magnitude of the applied voltage need not be specified for electrical transfer functions, material properties often do depend on the absolute temperature. Therefore, when presenting the thermal transfer functions, it is most proper to fix the average temperature $T a v g =( T h + T c )/2,$ where $T h$ is the hot-side temperature and $T c$ is the cold-side temperature.

In this review, we will focus on three fundamental thermal components: thermal diodes, thermal regulators, and thermal switches. Figure 2 shows the symbols, transfer functions, and an illustrative example of a heat conduction mechanism for each component. We now discuss each of these components in detail.

A thermal diode is a two-terminal element that rectifies heat currents. The left column of Fig. 2 shows a typical transfer function of a thermal diode. If the polarity of the temperature difference $ΔT$ is switched at fixed $T a v g$⁠, the magnitude of the heat flow changes. The figure of merit for a thermal diode is the rectification ratio, here defined as

where the magnitude of the forward heat flow $Q f w d$ is greater than the magnitude of the reverse heat flow $Q r e v$ at the same $Δ T$⁠, and it is also understood that $T a v g$ is held constant. In general, $γ$ will be a function of $ΔT$ and $T a v g$⁠.

Rectification requires both a spatial asymmetry and a nonlinearity. For example, in an electrical p–n diode, the asymmetry is imposed by the levels of p-type and n-type doping, and the nonlinearity arises from the exponential dependence of the minority carrier concentrations on the voltage difference across the space-charge region.⁸ The second law of thermodynamics poses constraints on rectification which may not be obvious on first glance; for example, the second law implies that an electrical diode cannot rectify the Johnson noise of a load resistor, as long as the resistor and the diode are at the same temperature.⁹ Similar second law restrictions of course also apply for thermal diodes, as we shall see throughout this review.

An example of a thermal diode mechanism is shown in the bottom left panel of Fig. 2 and discussed further below in Sec. III A 2. This “junction thermal diode” consists of two materials with different temperature-dependent thermal conductivities $k T$ providing the nonlinearity. In the forward mode, when the left material is hot and the right material is cold, both materials have high $k$ and the heat flux is large; in the reverse mode, the materials both have low $k$ and the heat flux is smaller.¹⁰ 

An example application of thermal diodes is a solid-state refrigeration cycle utilizing the magnetocaloric¹¹ or electrocaloric¹² effect. Thermal diodes allow for strong thermal coupling when heat is pumped out of the refrigerator, while blocking undesirable heat backflow into the refrigerator during other portions of the cycle. Several solid-state cooling cycles using thermal switches to mimic thermal diode behavior have been demonstrated,^13–16 and in Sec. IV A, we will analyze an electrocaloric refrigeration cycle using thermal diodes. Other applications of thermal diodes are also suggested in Sec. IV C 1.

A thermal regulator is a two terminal component with a strongly nonlinear $Q$ −  $ΔT$ transfer function at fixed $T a v g$⁠, as shown in the center column of Fig. 2. One common function of a thermal regulator is to maintain a desired critical temperature $T c$ by switching between a high conductance and a low conductance state. The thermal regulator is analogous to an electrical varistor (portmanteau of “variable resistor”). All thermal diodes are also technically thermal regulators since the transfer function of a diode is nonlinear, but in contrast with a thermal diode, the transfer function of a regulator can be symmetric in $ΔT$⁠. The essential feature of the diode's transfer function is the asymmetry, while the essence of a regulator's transfer function lies in the sharp nonlinear response.

A figure of merit for a thermal regulator is the on/off switching ratio

where $G= d Q d ( Δ T )$ is the differential thermal conductance, which is the local slope of the $Q$ −  $ΔT$ transfer function, as shown in the center column of Fig. 2. This figure of merit $r$ represents the ratio of the “on” state, or largest differential conductance $G on$⁠, to the “off” state, or smallest differential conductance $G off$⁠.

An example of a thermal regulator shown in Fig. 2 is a phase change material with a sharp change in k at a critical temperature $T c$⁠. This regulator example displays a nonlinear but symmetric $Q$ −  $ΔT$ curve with a crossover from a low $G$ to a high $G$ state at $Δ T = T c − T a v g$⁠. Quantitative data for two examples of these phase-change thermal regulators are shown below in Figs. 3 and 4. For most thermal regulator applications, phase transition hysteresis would be ideally minimized, such that the temperature is always regulated at the same $T c$⁠. Finally, in addition to the substantial change in k at $T c$⁠, materials undergoing first-order phase transitions also have a latent heat, which can assist in regulating the temperature at $T c$ for transient heat loads.

In an example application of regulation, a thermal regulator using bimetallic buckling¹⁷ reduced the power required to maintain the temperature of a chip-scale atomic clock at 70±0.1 °C as the external temperature was varied from −40 to +50 °C. Having different $G$ at different temperatures can also be useful to improve the transient thermal response without sacrificing steady-state performance. A familiar example is a vehicle's thermostat, which traps heat to quickly raise the engine temperature until the desired temperature $T c$ is reached. Once the system is at $T c$⁠, the wax in the thermostat melts, allowing coolant to flow freely to prevent overheating. In a related application, thermal regulators have been tested on catalytic converters^18,19 to reduce cold-start emissions by keeping the converter hot enough between drive times, while switching to a high- $G$ state at high temperatures to keep the converter from overheating. Several additional applications of regulators are presented in Sec. IV C 3.

As indicated in the right column of Fig. 2, we define a thermal switch as a two-terminal component with a $Q$ −  $ΔT$ constitutive relation that depends on a non-thermal control parameter, such as an electric field, magnetic field, or applied pressure. Changing this control parameter changes the thermal conductance. The on/off switching ratio $r$ introduced for thermal regulators in Eq. (2) is also a natural figure of merit for a thermal switch, where now $G o n$ is the highest differential conductance that can be achieved by tuning the control parameter, and $G o f f$ is the smallest achievable differential conductance.

An example of a thermal switch is a giant magnetoresistive (GMR) device,²⁰ illustrated in the bottom right of Fig. 2. Since k depends on the magnetic field (⁠ $B$⁠), the heat flux can be tuned by changing $B$⁠. As shown in Fig. 7, thermal switch ratios of $r=$ 2 have been achieved at room temperature using this mechanism in GMR multilayers.²⁰ 

Spacecraft thermal management²¹ is one motivating example for thermal switches. Radiators which dump the excess heat generated in the spacecraft out to deep space must be sized for the peak heat rejection loads. However, when the spacecraft is generating much less heat, the same radiators may be too effective in rejecting heat, potentially causing the working fluid to freeze. Thermal switches can be used to actively control the heat rejection rates and spacecraft temperature. In Sec. III, we will discuss many different thermal switches, some of which were developed for spacecraft applications using the mechanisms of conduction (Sec. III A 4), convection (Sec. III B 2), and radiation (Sec. III C 4). Developing lighter and faster thermal switches with larger $r$ would reduce costs and improve performance of these spacecraft thermal management systems, and many other potential applications, including the two examples of periodically switched circuits discussed in Sec. IV C 2.

We now proceed to the heart of this article, which is a review of experimentally demonstrated mechanisms of switchable and nonlinear heat transfer, with a focus on mechanisms which function near room temperature. As a non-exhaustive introduction to Secs. III A–III C, Table I summarizes measured rectification ratios and switching ratios for a variety of physical mechanisms that have been used to make thermal switches, regulators, and diodes.

We organize the following discussion by the dominant mode of heat transfer: conduction, convection, and radiation.

Heat in solids is primarily carried by electrons in metals and by phonons in semiconductors and insulators. The kinetic theory model for the thermal conductivity is $k=CvΛ/3$⁠, where $C$ is the volumetric specific heat, $v$ is the group velocity, and $Λ$ is the mean free path. Modifications to the microstructure or environment which change $C,v,$ or $Λ$ can therefore impact k, resulting in thermal diode, regulator, or switch behavior. Here, we discuss changes in $k$ across phase transitions and review recent diode implementations which utilize $k(T)$ nonlinearities. We also review thermal switches and regulators which function by bringing materials into and out of contact, by intercalating ions into layered structures, and by utilizing magnetic and electric fields to control $k$⁠.

Since steady-state heat transfer depends only on the material's thermal conductivity and not on the specific heat or the latent heat of a phase transition, we focus here on changes in k occurring across phase transitions. For transient thermal regulation, the large latent heats of phase change materials such as waxes, salts, or water can also be useful.

Metal-insulator transitions: Metal-insulator transitions (MITs) dramatically change the electrical conductivity $σ$⁠. In the best-known example, $σ$ can increase by 5 orders of magnitude when vanadium dioxide crosses a MIT around $T M I T$ = 340 K. It is natural to suspect that the thermal conductivity of VO₂ may also show a large change across the MIT, since the standard single-electron Wiedemann-Franz (WF) law (Lorenz number $L= L 0$ = 2.44 × 10⁻⁸WΩ/K²) predicts that both electrons and phonons contribute significantly to k in the metallic state, while in the insulating state only the phonon $k$ remains. Indeed, several experiments^22,23 have measured switch ratios as large as $r=$ 1.6 in polycrystalline VO₂ films. However, these experiments measured $k$ in the cross-plane direction and $σ$ in the in-plane direction, so the WF law could not be quantitatively tested.

A recent experiment²⁴ using VO₂ single-crystal nanobeams [Fig. 3(a)] measured $k$ and $σ$ both in the in-plane direction and surprisingly found that the in-plane total $k$ changes by <5% across the MIT, even with high $σ$ in the metallic state. This small change in the total thermal conductivity across the MIT is consistent with early $k$ measurements of bulk VO₂.^25,26 Two possible explanations for this behavior have been historically suggested: either the phonon thermal conductivity $k p h$ significantly decreases across the transition to a metallic state as the electron thermal conductivity $k e l$ increases in accordance with the WF relation,²⁶ or the change in $k p h$ is small and the typical WF relation (⁠ $L= L 0$⁠) is not valid in metallic VO₂.²⁵ Based on calculations of $k p h$ in the metallic state using first-principles phonon dispersions and a fit to experimental measurements of energy-dependent phonon linewidths, Lee et al.²⁴ argue that $k p h$ is nearly unaffected by the phase transition, and conclude the correlated electrons in the metallic state of VO₂ carry much less heat than predicted by traditional single-electron theories (⁠ $L< L 0$⁠).

Interestingly, however, when the VO₂ nanobeams are doped with only a few percent of W, thermal regulator behavior is observed.²⁴ As shown in Fig. 3(b), the phase transition temperature of VO₂ is lowered to $T M I T =$ 310 K at 2.1 at. % doping of W, and $k$ switches from 3.4 W m⁻¹K⁻¹ in the insulating phase at low $T$ to 5.2 W m⁻¹K⁻¹ in the metallic phase at high $T$⁠, resulting in $r=1.5$⁠. Lee et al.²⁴ interpret the data as indicating that the electrons in the metallic state of VO₂ tend to recover the WF behavior as the W doping increases. Other materials displaying metal-insulator²⁷ or metal-metal²⁸ transitions may also be promising thermal regulators; for example, a ferromagnetic magnetite material²⁹ has shown $r$ = 1.5 across a MIT at 200 K.

Phase change memory: Chalcogenides such as Ge₂Sb₂Te₅ (GST), which is used in phase-change memory devices, can be switched between metastable structural states at room temperature.^30,31 The crystal structure of GST is determined by the thermal history. Rapidly quenching from a high temperature results in an amorphous state at room temperature, but crystalline structures can be recovered by reheating the material above the crystallization temperature and slowly cooling back to room temperature. Since crystalline materials have larger $k$ than their amorphous counterparts, a thermal switch can be implemented by controlling the crystallinity. Lyeo et al.³² showed that the room temperature $k$ of GST can range from 0.2 to 1.5 W m⁻¹K⁻¹ depending on the thermal history, though reversible switching was not demonstrated. The authors also note that the changes in $k$ achieved using rapid laser crystallization are smaller (⁠ $r=$ 1.2) than when the global temperature was slowly raised above the crystallization point. Reifenberg et al.³³ and Lee et al.³⁴ have measured increases in k of a factor of 3 for GST films when the temperature is slowly ramped, but also did not report reversible switching.

Miscellaneous solid-solid transitions: Modest switching ratios have been observed around other solid-solid phase transitions near room temperature. Huesler alloys³⁵ have shown $r$ as large as 1.6 across a magnetostructural phase transition, and $r$ = 1.8 has been measured^36,37 across a structural phase transition near $T$ = 400 K in Cu_2-xSe. At high pressures (>10 GPa), $r$ =3 has been observed in semiconductor-metal transitions³⁸ due to the enhanced electronic contribution to $k$⁠. Charge-density wave (CDW) phase transitions can impact both the electron and the phonon contributions to k. For example, a 25% increase in $k$ of octahedral TaS₂ was observed³⁹ at the CDW transition at 350 K, and similar $r$ has been measured^40,41 in other CDW systems.

Melting solids: Typically, a liquid will have a lower thermal conductivity than its corresponding solid phase near $T melt$⁠, so melting a solid decreases k. Several familiar examples in the vicinity of room temperature include H₂O (⁠ $r=3.5$ at $T melt =$ 0 °C),⁴² waxes (⁠ $r=2$⁠, with $T melt$ ranging from 50 to 80 °C),⁴³ or liquid metals such as mercury (⁠ $r=4$ at $T melt =−$ 38 °C).⁴⁴ In a comprehensive recent study, Kim and Kaviany⁴⁵ analyzed different solid-liquid transitions to find the most promising candidates for achieving high $r$⁠. They identified an interesting class of materials which are semiconducting in the solid state but metallic in the liquid state (all with $T melt$>700 K). Since the electrons in the liquid metal carry the heat effectively, $k$ actually increases in the liquid state. First principles calculations of switching ratios as large as 15 for CdSb at $T melt =$ 850 K were in good agreement with previous measurements⁴⁶ of k and $σ$⁠.

Composites: Composite materials undergoing liquid-solid transitions can also show thermal regulator behavior. If the liquid matrix freezes into a crystalline structure, the solid composite particles are pushed towards the grain boundaries and into better thermal contact, improving the heat transfer at low temperatures. As shown in Fig. 4, Zheng et al.⁴⁷ demonstrated this mechanism using graphite flakes suspended in hexadecane, and obtained $r$ as large as 3. They also showed that this switching was reversible over 7 thawing/freezing cycles. Schiffres et al.⁴⁸ showed that the switching ratio can be tuned by controlling the freezing rate. Additional measurements for a variety of liquids and composite particles^49–54 also resulted in $r$ in the range of 2–3.

This section emphasizes new advances in solid-state rectification since the 2011 review of Roberts and Walker.³ 

Ballistic phonon rectification: The idea of ballistic thermal rectification is to use asymmetric scattering elements, such as internal pyramids or sawtooth shaped surfaces, to preferentially scatter the phonons more strongly in one direction than another. However, such concepts require further careful thinking to ensure the vision remains compatible with the second law of thermodynamics. Maznev et al.⁵⁵ showed theoretically that to obtain a ballistic phonon thermal diode, a nonlinearity must be present in the system in addition to an asymmetric scattering structure. In accordance with this theory, Chen et al.⁵⁶ and Kage et al.⁵⁷ measured no measurable phonon rectification in asymmetrically etched silicon structures which lacked a significant nonlinearity. Although Schmotz et al.⁵⁸ inferred thermal rectification from measurements in an asymmetrically etched silicon membrane, Maznev et al.⁵⁵ showed that the experimental design lacked the proper symmetry reversals to conclusively claim rectification.

Rectification measurement technique: Chiu et al.⁵⁹ recently demonstrated a new experimental technique to test for nonlinearities. Since nonlinear systems generate higher order harmonics, time-periodic heating of a thermal diode should cause higher harmonics in the temperature response. These higher harmonics can be easily measured using a lock-in amplifier, although electrical nonlinearities also need to be considered. Using this technique, Chiu et al.⁵⁹ observed no rectification above the measurement threshold of 0.5% for either pristine nanowires or asymmetrically loaded mass-loaded BN nanotubes with diameter $d=$ 100 nm. This null result contradicts the previously reported 7% rectification reported for asymmetrically mass-loaded BN nanotubes with $d≈$ 30 nm.⁶⁰ Chiu et al.⁵⁹ attribute this difference to the improved experimental design of the higher-harmonics technique compared with the previous steady-state measurement, but also note that the different nanotube diameters might also play a role.

Junction diodes: A simple mechanism for a solid-state thermal diode arises in composite bars made of two materials with different temperature-dependent thermal conductivities $k(T)$⁠. If the thermal conductivities of materials 1 and 2 scale like $k 1 ∝ T n 1$ and $k 2 ∝ T n 2 ,$ then for small temperature differences after optimizing geometry,¹⁰ the rectification is

Many demonstrations^61,62 of this “junction diode” rectification mechanism were performed at low temperatures, where $( T h − T c ) / T avg$ is large and $k$ depends strongly on temperature. Above room temperature, Takeuchi et al.⁶³ achieved $γ=0.1,$ and Nakayama and Takeuchi⁶⁴ measured $γ=0.6$ with $ΔT=240$ K. Recently, Wang et al.⁶⁵ used a focused ion beam to create nanopores in one region of suspended graphene, causing the irradiated and pristine sections to have different $k(T)$ trends near room temperature. Unexpectedly, the reported measurement of $γ=0.26$ for $ΔT=35$ K exceeds the maximum rectification predicted by Eq. (3) by a factor of 3, warranting closer examination.

New junction diode theory: Interestingly, although measurements of $γ$ = 0.2 were initially reported for a single material with an asymmetric shape,⁶⁶ further theoretical investigations by Wang et al.⁶⁷ showed that, in fact, rectification should not occur in asymmetric single materials with position-independent $k(T)$⁠. Similarly, Go and Sen⁶⁸ showed that rectification is forbidden for steady state one-dimensional heat flow if the thermal conductivity can be separated as $k T , x =X x Θ(T)$⁠, where $X x$ is a function which depends only on position and $Θ(T)$ is a function which depends only on temperature. In the junction diode mechanism, the composite bar's thermal conductivity depends on (x,T) in a nonseparable manner, and rectification is observed. In another recent theoretical advance, Herrera, Luo and Go⁶⁹ numerically examined transient thermal transport in junction diodes, and predicted that the transient rectification ratios can be even larger than the steady-state values.

Phase transition junction diodes: Junction thermal diodes utilizing the temperature dependence of k across phase transitions have also been recently investigated. Both Zhang and Lou⁷⁰ and Cottrill and Strano⁷¹ constructed analytical models to describe rectification in junction diodes where one material undergoes a phase change. These models can be used to optimize $γ$ by selecting the dimensions and the temperature range of the diode. Experimental demonstrations of such phase-transition-exploiting junction diodes include a diode constructed of a Nitinol shape memory alloy and graphite which displayed⁷²  $γ=0.5$ for $ΔT=160$ K, a diode made of gadolinium and silicon showed⁷³  $γ=0.6$ for $ΔT=30$ K, and a diode constructed from a phase-change polymer PNIMAP and PDMS resulted in⁷⁴  $γ=0.5$ for $ΔT=20$ K. Kobayashi et al.⁷⁵ measured $γ=$ 0.1 with small temperature differences of 2 K across an oxide structural phase transition. Wang et al.⁷⁶ developed a foam-paraffin composite to utilize the solid-liquid phase transition of paraffin while maintaining mechanical stability. They measured $γ=0.35$ with $ΔT=30$ K for this paraffin composite and PMMA junction diode. Excitingly, Wang et al.⁷⁶ also used these junction thermal diodes to build a thermal diode bridge which rectified an oscillating temperature profile (see Sec. IV B for a related rectification application). Finally, asymmetric VO₂ nanobeams displayed⁷⁷  $γ=0.3$ for $ΔT=1$ K, which was attributed to the existence of metal-insulator domains along the length of the nanobeam.

Intercalating atoms between the atomic planes of layered materials changes the bonding strength and microstructure. Ex situ measurements indicate that permanent intercalation affects the thermal conductivity,^78,79 and recent in situ experiments^80,81 have gone even further to show that electrochemistry can be used to reversibly switch k. Delithiating the battery cathode material LiCoO₂ decreased k by up to 40%,⁸⁰ and the changes were reversible upon re-lithiation. The switching timescales were on the order of 1 hour, limited by the ion diffusion time. It was also recently shown that intercalating Li ions between black phosphorous layers⁸¹ decreases the thermal conductivity in all three of the distinct crystallographic directions. For example, in the cross-plane direction, reversible switching ratios of $r=$ 1.6 were demonstrated over 100 cycles.

An efficient form of thermal switching, with some of the highest demonstrated switching ratios, relies on the simple mechanism of bringing materials in and out of contact. Achieving the largest $r$ requires reducing the parasitic heat pathways in the off state due to convection or radiation, and minimizing the thermal contact resistance at the conducting interface in the on state by using polished surfaces, a thermal interface material, and moderate to high contact pressures (at least 10 kPa recommended to obtain $r$ =100).^82,83 For example, McKay and Wang⁸⁴ used a bistable latching solenoid to make and break thermal contact in their thermal switch circuit discussed in Sec. IV B, and Wang et al.¹⁵ fabricated a silicon-based contact switch integrated with a linear actuator to achieve a measured $r=27$(see also Sec. IV A). Westwood⁸⁵ constructed a switch with an estimated $r=2000$ using a silicone interface and 10 kPa contact pressures to increase the on-state conductance $G o n$ between two metals in contact, and also reduced $G o f f$ by inserting a low- $k$ aerogel between the metals in the off state.

Utilizing liquids to make thermal contact at the interface can overcome the need for high contact pressures in solid-solid interfaces. A liquid metal droplet switch⁸⁶ showed $r$ > 200 due to the high liquid metal $k.$ This switch achieved 100 Hz switching frequencies, and was used in a MEMS piezoelectric heat engine demonstration.⁸⁷ Thermal switches using dielectric fluid droplets^88–90 have also been implemented to obtain high $G o n$⁠, though $G o f f$ was not reported.

Electrostatic actuation: Another technique to improve thermal contact uses electrostatic forces to snap plates together. These electrostatic switches have been used to control the heat flux reaching spacecraft radiators,^91–93 but typically require large snap-in and snap-out voltages to overcome stiction. Supino and Talghader⁹⁴ achieved $r$ = 2 using an electrostatic MEMS thermal switch with an applied voltage of 26 V. Interestingly, since the electrostatic actuation time was much shorter than the characteristic thermal time constants in this application, pulse-width electrostatic modulation was used to achieve continuous sweeping of the effective thermal conductance $G e f f$⁠. Kim et al.⁹⁵ built a MEMS electrostatic thermal switch which used an applied voltage of 100 V to snap suspended gold beams into contact with a hot source, decreasing the thermal resistance to the heat sink. The switching ratio was $r$ = 5–6, and the switching was shown to be repeatable over > 200 cycles.

Nanoscale contact switching: The thermal conductivity of boron nanoribbon bundles⁹⁶ has been reversibly switched by 20% by wetting the nanoribbons using different liquids, which changes the bonding strength between nanoribbons. Thermal switching ratios of 5 were achieved by “telescoping” multi-wall carbon nanotubes.⁹⁷ By cutting the outer nanotube walls and pulling on the nanotube, the outer walls slide past the rigid inner walls, reducing the pathways for heat transfer.

Thermal expansion regulators: A common example of a thermal regulator utilizing temperature-dependent mechanical contact is the thermal expansion valves in HVAC systems. These valves use the thermal expansion of a gas or liquid to control the cross-sectional area and resistance to fluid flow, passively maintaining the temperature at an optimal setpoint. Similarly, the expansion of paraffin waxes in the liquid state has been used to create regulators⁹⁸ with $r=$ 60 for applications such as thermal control of a Mars rover. Chapter 10 in the Gilmore textbook⁹⁹ includes several designs of paraffin regulators that achieved $r=$ 100 at room temperature. Copic and Hart¹⁰⁰ combined paraffin and carbon nanotubes forest to obtain large-stroke thermal switching actuators with a small footprint. Thermal expansion of metal droplets upon melting¹⁰¹ has been used to create thermal regulators with estimated $r$ = 25. As an example of solid-state thermal expansion regulators, a bimetallic disc scavenged from a commercial thermostat has been used as a thermal regulator¹⁷ with $r$ = 1.2 in air and $r$ = 1.8 in vacuum. Thermal diodes utilizing the effects of asymmetric bimaterial thermal expansion have also been developed, with measured $γ=0.07$ in solid composites¹⁰² and $γ$ =1 for a mercury-air interface.¹⁰³ 

Very large $r$ can be obtained if small gaps between objects are pressed closed due to thermal expansion. To achieve good thermal contact between objects for high $G o n$⁠, small gap sizes and high coefficient of thermal expansion (CTE) materials are desirable; however, manufacturing costs and cycling performance are concerns associated with small gap sizes, and high- $k$ and machinable metals typically do not have large CTE. Innovative geometries and composite systems have been developed to address these problems. Guo et al.¹⁰⁴ reported that the thermal expansion of a flexible rod bringing metal pieces into contact at room temperature yield switching ratios $r$ = 400, and Marland et al.¹⁰⁵ reported $r$ > 1000 at $T=230$ K using high CTE polymers to open and close a 100 $μ$ m gap between metal pieces.

At the nanoscale, Choe et al.¹⁰⁶ recently demonstrated a thin film VO₂ nanogap thermal regulator with $r=2–2.5$⁠. The VO₂ film expands when the temperature is increased above the metal-insulator transition temperature $T M I T$⁠, pressing together two polysilicon layers which are separated by a nanogap. The nanogap was created by etching a 20 nm sacrificial layer deposited between the poly-silicon. The switching was reversible over 100 cycles, and the fastest switching speeds were estimated to be $∼$1 ms.

Rather than using thermal expansion as the mechanism for making/breaking thermal contact, Bulgren et al.¹⁰⁷ used the temperature dependent magnetization of permanent magnets to reversibly actuate a thermally conducting disc. At low $T$⁠, the strong magnetic force keeps the disc thermally coupled to the magnet, but at high $T$⁠, the magnetization decreases and springs pull the disc out of contact. Further investigation of $r$ for this mechanism would be of interest.

Shape memory alloys: SMAs are promising thermal regulators due to the large mechanical deformation induced when the SMA is heated above the solid-solid phase transition. Benafan et al.¹⁰⁸ used SMA springs to create a thermosiphon-based regulator with $r$ = 4, and SMA springs have also been used to regulate fluid flowrate and control the convective heat transfer.¹⁰⁹ SMAs have been combined with bimetallic strips in a thermal diode demonstration.¹¹⁰ Tso and Chao¹¹¹ designed and fabricated a SMA spring-based thermal diode with $γ$ > 90, showing that very large switching and rectification can be achieved near room temperature using SMAs.

Electrons transport both charge and heat. The well-known Wiedemann-Franz (WF) relation, which is strictly applicable for elastic scattering in metals, relates the electronic conductivity (⁠ $σ$⁠) and the electronic contribution to thermal conductivity $( k e l )$ as $k e l =σTL$⁠, where $L$ is the Lorenz number. For many metals, $L$ is within 10% of the free electron (or Sommerfeld) value¹¹²  $L 0 =2.44× 10 − 8 WΩ K − 2$⁠.

Gating thermal conductivity: It would be quite appealing to re-appropriate electronic transistor technologies such as MOSFETs to control heat currents via electrostatic gating, as illustrated in Figs. 5(a) and 5(b). However, it turns out that the phononic contribution to the thermal conductivity $k p h$ often swamps the electronic contribution $k e l$ in the regime where the electronic concentration $n$ can be readily gated. Figure 5(c) displays this fundamental tradeoff for bulk materials.

For small $n$ typical of moderately doped semiconductors, gating can effectively change $k e l$ by controlling the local carrier concentration, but the total conductivity $k t o t = k e l + k p h$ is negligibly affected, because $k p h ≫ k e l$⁠. A typical estimate of the minimum possible $k p h$ for a fully dense amorphous material at room temperature ranges from $k p h , m i n ≈$ 0.3–3 W m⁻¹K⁻¹; for a specific material, $k p h , m i n$ can be more accurately estimated using the model of Cahill, Watson, and Pohl.¹¹³ For many single- or poly-crystalline materials, $k p h$ can be up to several orders of magnitude larger than $k p h , m i n$⁠; the purpose of using the optimistic (low) value $k p h ≈ k p h , m i n$ here is to illustrate that if $k e l$ is already much smaller than $k p h , m i n$⁠, gating $k t o t$ in the real material would be quite challenging.

For large $n$ found in heavily doped semiconductors or metals, $k e l$ is a significant (or even dominant) portion of $k t o t$⁠; however, the challenge now is that it is difficult to gate large $n$⁠. The depth of the gated channel, where $n$ and thus $k e l$ are substantially altered from their far-field bulk values, can be estimated by the characteristic Thomas-Fermi screening length $( l T F )$⁠.¹¹⁴ Representative values of $l T F$ at 300 K are given along the top axis of Fig. 5(c) assuming a free electron dispersion relation and using the dielectric constant (⁠ $ϵ$⁠) of silicon. For semiconductors $n < 10 20 cm − 3$ at room temperature, $l T F$ reduces to the simpler Debye screening length $l d = ϵ k b T e 2 n$⁠, where $ϵ$ is the dielectric constant, $k b$ is the Boltzmann constant, and $e$ is the electron charge. Note crucially that $l d$ is proportional to $1/ n$⁠, indicating that free carriers are very effective in screening electric fields. Due to this screening restriction, the maximum $n$ for which significant gating effects on $k t o t$ can occur in a channel at least ∼2 nm thick is on the order of $n∼ 10 18$ cm⁻³.

Considering these competing effects, the potential sweet spot for seeking gatable $k t o t$ in 3D materials at 300 K lies in the regime of low $k p h$⁠, moderate $n$⁠, and large mobility $μ$⁠, as indicated by the un-shaded regime in the upper-left of Fig. 5(c). It will be necessary to achieve $σ$ at the very least larger than 1400 S/cm (⁠ $ρ<0.7 mΩ cm)$ to obtain $k e l >1$ W m⁻¹K⁻¹ (comparable to $k p h , m i n$⁠), yet with n no more than ∼10¹⁸ cm⁻³ to maintain a channel of at least ∼2 nm thickness. Interestingly, the requirements of high $σ$ and low $k p h$ at intermediate $n$ are also shared by thermoelectric materials,¹¹⁵ so some thermoelectric systems may be promising avenues for electrostatic gating of $k$⁠.

Recent developments in fabricating two-dimensional (2D) materials with high $σ$ may open new opportunities for gating $k e l$⁠, as illustrated in Fig. 5(d). 2D semimetals or semiconductors such as graphene and black phosphorene have enticingly large mobilities, but also typically have large $k p h$⁠. A different approach is to gate 2D metals, which compensate for their small $μ$ with large $n$ to achieve $k e l$ comparable to the minimum $k p h$⁠. 2D systems also have a unique advantage because their carriers are confined to the 2D plane, which means they cannot efficiently screen electric fields applied in the out-of-plane direction. Thus, the carrier concentration even in metals can be controlled by electrostatic gating without strong screening effects.¹¹⁶ Recently, high- $σ$ 2D metals and semimetals such as Ti₂Se,¹¹⁶ encapsulated NbSe_2,¹¹⁷ and Mo₂C¹¹⁸ have been fabricated, and metallic 2D electron gases have been observed at oxide interfaces.¹¹⁹ The conclusion of this analysis [Fig. 5(d)] is that $k t o t$ switching may be possible in 2D materials, though large electric fields would be required to make significant changes to $n$ for 2D metals. Another critical requirement will be to keep any parasitic shunt heat transfer pathways through encapsulating layers and substrates to an absolute minimum.

Wiedemann-Franz breakdown: Another avenue for gating $k t o t$ can be found in the regimes where the WF relation breaks down. If the electrons carry much more heat than predicted by the WF law (⁠ $L≫ L 0$⁠), $k e l$ could be larger than $k p h$⁠, even in the low $σ$ materials which are easily gated. One significant breakdown of the WF law occurs when the carriers are strongly interacting and the classic Fermi liquid (single-electron) picture no longer accurately describes the transport.^120,121 Though many of the deviations from the WF law occur at low temperatures,^122,123 Wakeham et al.¹²⁴ used measurements of the thermal Hall effect to infer a large $L/ L 0 >10$ at room temperature in Li_0.9Mo₆O₁₇. However, Cohn et al.¹²⁵ conducted additional transport measurements for this material and proposed an alternative bipolar transport model with L ≈ L₀.

Non-Fermi liquid behavior has been recently observed in graphene at liquid nitrogen temperatures.¹²⁶ When the electron-electron scattering is strong, the carriers come to their own local equilibrium and $L≫ L 0$⁠.¹²¹ Figure 6, reproduced from Crossno et al.,¹²⁶ shows that $k e l$ of h-BN encased graphene (measured using a novel Johnson noise thermometry technique¹²⁷) can be switched by a factor of 5 using electrostatic gating at 75 K. In this strongly interacting regime near zero gate voltage (Dirac point), $k e l$ is much larger than predicted by the WF relation (⁠ $L/ L 0 ∼22)$⁠, while at larger gate voltages, the transport is well-described by the Fermi liquid picture and the WF law holds. In addition, at higher temperatures, the increased electron-phonon scattering rate breaks the strongly interacting regime near zero gate voltage, decreasing $k e l$ and reducing the efficiency of thermal switching. A previous study¹²⁸ using resistance thermometry also showed that $k e l$ of suspended graphene can be gated from 2–4 Wm⁻¹K⁻¹ at 100 K. It must be noted that these exciting studies^126,128 only measured $k e l ,$ so future work measuring $k t o t$ as a function of gate voltage and temperature would be very useful to understand the fundamental limits for thermal switching using graphene.

Although phonon transport in most materials is unaffected by electric fields (⁠ $E$⁠), materials such as ferroelectrics, which display a spontaneous electric polarization below the Curie temperature, offer promise for tunable $k p h$ using $E$ fields.

Domain walls: The ferroelectric domain wall density is thought to impact phonon transport.¹²⁹ In the absence of a poling $E$ field, large samples have domains with different polarizations. By applying an $E$ field, the domain wall density decreases as the dipoles align with the field. Room temperature $k$ switching by reconfiguring domain walls was recently demonstrated¹³⁰ in Pb(Zr,Ti)O₃, resulting in an ∼8% change in $k$ with 475 kV/cm $E$ fields and sub-second switching times. The proposed mechanism envisions that the domain walls scatter phonons in a manner analogous to grain boundary scattering, so controlling the domain wall density with an $E$ field can switch $k p h .$

Ferroelectric transitions: Ferroelectric phase transitions involve atomic displacements. This atomic re-arrangement near the phase transition can be influenced by $E$ fields, as indicated by heat capacity measurements, $C E .$¹³¹ Though field-dependent $k$ measurements at high temperatures are sparse, $E$ fields of 4.2 kV/cm were used to increase $k p h$ by 20% in the ferroelectric material triglycine sulfate¹³² near the phase transition at 48 °C. $k(T)$¹³³ and $C(T)$¹³¹ measurements indicate that BaTiO₃ may be another promising candidate for achieving $k p h (E)$⁠.

Ferroelectric strain coupling with metals: If $E$ fields induce strain in a dielectric substrate, and that induced strain affects the electrical conductivity of a metallic thin film, then $k e l$ of the metallic film could be tuned. Though the thermal conductivity was not measured, 20% $σ$ modulation in metallic FeRh alloy films on ferroelectric substrates has been observed¹³⁴ using 1 kV/cm electric fields near the FeRh magnetic phase transition at 375 K. Using the WF relation, this $σ$ modulation would correspond to a thermal conductivity change of $Δ k e l ∼5$ Wm⁻¹K⁻¹.

Thermomagnetic phenomena arise from the transport of heat by charged particles in a magnetic field $(B)$⁠. The WF relation still holds in the presence of $B$ fields,¹³⁵ so the $k e l (B)$ tensor can in principle be determined from the dependence of the $σ$ tensor and the thermoelectric tensors on the magnetic field $B$⁠. Given the numerous tensorial quantities in play, care must be taken to account for the various thermoelectric and thermomagnetic phenomena when reconstructing the $k$ tensor from electrical measurements.

Magnetoresistance: Though classical longitudinal and transverse magnetoresistance models for bulk materials¹³⁵ predict that the $B$ field increases the electrical resistivity and decreases $k e l$⁠, the magnitude of this effect is negligible (<1%) for most metals at room temperature. For ferromagnetic metals, the anisotropic magnetoresistance effect utilized in magnetic read heads can change $k$ by several percent.¹³⁶ Interestingly, larger classical magnetoresistance effects have been observed in semimetals such as bismuth which have atypical Fermi surfaces. Early $B$-dependent measurements of bismuth¹³⁷ and bismuth-antimonide alloys¹³⁸ measured $k$ switching ratios around $r$ = 1.2 near room temperature. More recent measurements of room temperature $σ$ have shown electrical switching ratios up to 2.5 at 4 T in bismuth thin films¹³⁹ and 2.5 at 9 T in single crystals of high- $σ$ NbP.¹⁴⁰ (Here, and throughout this review, we report the free-space magnetic field $B= μ 0 H$⁠, where μ₀ is the vacuum magnetic permeability.) Further investigation of the maximum achievable $k$ switching ratios in these semimetals could be quite interesting.

Giant magnetoresistance (GMR) occurs when nanoscale metallic ferromagnets are separated by thin nonmagnetic metallic domains.¹⁴¹ The resistivity in both the cross-plane and in-plane directions depends on the relative magnetizations of the ferromagnetic domains; if the magnetic moments are anti-parallel, then the resistivity is high, while aligning the moments with an external field decreases the resistivity. In these metallic systems, the electrons dominate the heat transfer, and magnetic switching of $k$ has been demonstrated both at cryogenic^142–145 and room temperatures.^20,146,147 Figure 7(a) shows an experimental setup used by Kimling et al.²⁰ to measure the cross-plane $k$ of a Co/Cu multilayer. Figure 7(b) shows that $k$ switching ratios of $r=$ 2 can be achieved with 0.2 T fields at room temperature. The in-plane $k$ of GMR multilayers also depends on the $B$ field,^146,147 with measured $r$ = 1.2.

Though other forms of magnetoresistance have displayed large $σ$ switching, for the purposes of $k$ switching, it is essential to consider whether electrons or phonons carry the heat. For example, tunnel magnetoresistance (TMR)¹⁴⁸ occurs when metallic ferromagnets are separated by nonmagnetic insulating domains. Unlike in the GMR metal-metal systems, the thermal transport at these TMR junctions is likely dominated by phonons, since applying the interfacial form of the WF relation¹⁴⁹ to measured contact resistance values in Fe/MgO/Fe interfaces¹⁴⁸ yields electronic thermal boundary conductances at least three orders of magnitude smaller than typical phonon boundary conductances.¹⁵⁰ Likewise, the colossal magnetoresistance (CMR)^151–154 effect associated with the paramagnetic-ferromagnetic phase transition in some oxides can produce electronic switching ratios as large as 10³ near the phase transition. However, $k p h$ is often much larger than the WF prediction for $k e l$ in these materials due to the small $n$⁠. Interestingly, Visser et al.¹⁵⁵ reported a $k$ switching ratio around $r=$ 1.3 in CMR systems with low $σ$⁠, indicating that the phonon transport may also be impacted by the $B$ field near the phase transition. Euler et al.¹⁵⁶ also measured $r$ = 1.2 near the CMR phase transition and attributed the switching to a combination of electron and phonon contributions.

$B$ field microstructure alignment: Magnetic forces can change the microstructure of some materials, which is another route to obtain $k p h (B)$⁠. One example is the magnetic alignment of liquid crystals,¹⁵⁷ which are assemblies of molecules with a preferred alignment in their low-temperature nematic phase. Since heat is more efficiently carried along the backbone of the molecule than between molecules, aligning the molecules increases $k$ in the aligned direction and reduces $k$ in the perpendicular directions.¹⁵⁸ Shin et al.¹⁵⁹ achieved $r$ = 1.5 by heating the liquid crystal to a high temperature disordered state, applying a $B$ field to align the molecules, and then cooling to the high-viscosity aligned state to freeze in the orientation before removing the field. Sahraoui et al.¹⁶⁰ measured similar $r$ using $E$ fields to align polymer-dispersed liquid crystals. In this measurement, the liquid crystal recovered its isotropic orientation once the $E$ field was removed. Thermal devices such as concentrators¹⁶¹ and diodes¹⁶² have been proposed using the dependence of $k$ on liquid crystal alignment.

Other low-dimensional materials with anisotropic magnetic susceptibilities such as nanotubes,¹⁶³ graphene oxide,¹⁶⁴ and graphene¹⁶⁵ can also be oriented in aqueous suspensions using $B$ fields. Sun et al.¹⁶⁶ showed that the thermal conductivity of suspended graphite flakes with attached magnetic nanoparticles can be switched by a factor of 3 in 0.05 T fields, though mild hysteresis was observed over 10 cycles.

Joule heating is a nonlinear coupling between the electrical and thermal domains. If $σ$ depends on temperature (or other external parameters), the heat dissipated via Joule heating can be controlled, even though the thermal properties themselves (e.g., $k$⁠) are not varying. This electrothermal coupling was recently utilized¹⁶⁷ to passively prevent battery thermal runaway. If the battery overheats, thermal expansion breaks a percolation pathway, which increases the electrical resistance and reduces the Joule heating. Transition-edge bolometers¹⁶⁸ use electrothermal feedback to quickly reset the bolometer at the desired temperature after the electrical resistance is perturbed by the absorption of light. The electrothermal coupling near the phase transition in a VO₂ nanobeam was also exploited¹⁶⁹ to create a thermal Schmitt trigger, where a small increase in the input temperature results in a large increase in output temperature, due to the VO₂ metal-insulator transition and the electrothermal feedback.

The key engineering parameter in convection is the average heat transfer coefficient $h≡ Q / ( A Δ T ) ,$ where A is the surface area. $h$ depends on the geometry, fluid properties, flow velocity and state (laminar or turbulent), and whether the fluid is driven by an external pressure gradient or induced by buoyancy (forced vs. free convection), any of which represents a potential mechanism to control the heat transfer.

Switchable and nonlinear convection have been widely investigated for many years. Clearly, highly efficient thermal switching is regularly achieved using valves or pumps to control the fluid flowrates, and therefore $h$⁠. From a fundamental perspective, nonlinear convective heat transfer has also been extensively studied because the coupling between the hydrodynamic and thermal domains leads to rich physics. Indeed, Lorenz's nonlinear model for natural convection led to the discovery of chaotic behavior in deterministic systems.¹⁷⁰ 

Our coverage of switchable and nonlinear heat convection here focuses on recent experimental demonstrations of thermal rectification, thermal switching with thin gas gaps, and heat flux control with electric and magnetic fields. In addition, we illustrate several examples of mature thermal diode and regulator technologies utilizing liquid-vapor phase change in heat pipes.

Natural convection: Consider a fluid which is confined between two horizontal plates that are separated by a distance $L$⁠. Since this configuration has no free surfaces (i.e., no liquid-vapor interfaces), surface tension phenomena such as the Marangoni effect do not come into play. Most fluids have positive thermal expansion coefficients (β), so fluids in a gravitational field (g) which are heated from below will expand and rise. This bulk fluid motion results in efficient heat transfer from the hot bottom plate to the cold top plate. However, when this same fluid is heated from above while the bottom plate is cold, the fluid is stable against natural convection and the heat is transferred only by conduction, a broken symmetry which results in rectification. The rectification ratio increases with increasing Rayleigh numbers $R a L$⁠, a dimensionless group defined as $R a L = g β Δ T L 3 / α ν$⁠, where $L$ is the confining fluid gap distance, $α$ is the thermal diffusivity, and $ν$ is the kinematic viscosity. Using an experimental correlation¹⁷¹ valid over the range $3 × 10 5 <R a L <7× 10 9$⁠, and neglecting a weak dependence on the Prandtl number $Pr=ν/α$⁠, the predicted rectification is $γ≈0.07 R a L 1 / 3$⁠. For liquid water at room temperature with $ΔT=$ 50 K and $L$ =1 cm, this correlation gives $γ≈7.$ In this $R a L$ regime, $γ$ increases linearly with $L$⁠, so very large rectification (⁠ $γ>100$⁠) is accessible for large gaps $(L>15$ cm) and temperature differences (⁠ $ΔT=$ 50 K).

From the $R a L$ definition, we see that increasing the material property figure of merit $β / α ν$ will increase $γ.$ Interestingly, although gases have larger $β$⁠, liquids have much smaller $α$ and $ν$⁠, and correspondingly much larger $γ$ can be achieved using liquids (such as water or ethylene glycol) than using gases.³ However, this natural convection mechanism may be restricted to use in only specialized applications, since $γ$ depends critically on the orientation with respect to gravity, and relatively large temperature differences are required to obtain high forward-mode $h$ (for example, $ΔT=50$ K is required to obtain $h$ = 500 Wm⁻²K⁻¹ for water).

Heat pipes: Larger $γ$ with higher forward-mode conductances at smaller $ΔT$ can be achieved using the liquid-vapor phase change in heat pipe thermal diodes. Heat pipe diodes are a relatively mature technology, and they have been developed for diverse applications ranging from temperature control of spacecraft sensors and electrical devices,¹⁷² thermal energy storage,¹⁷³ nighttime radiative cooling,¹⁷⁴ and temperature control of buildings for reduced energy consumption.¹⁷⁵ In a large-scale infrastructure application, gravity-driven heat pipe thermal diodes known as thermosiphons were installed on the Trans-Alaska Pipeline System¹⁷⁶ and the Qinghai-Tibet Railway¹⁷⁷ to prevent permafrost melting. Many different heat pipe thermal diode designs with asymmetric shapes have been demonstrated (including the liquid trap diode, the liquid blockage diode, and the gas blockage diode), and these designs have been discussed in textbooks.^175,178

One example that provides a sense of heat pipe thermal diode performance is the liquid trap diode developed by Groll et al.¹⁷⁹ This diode has an asymmetric shape, with a liquid trap appended on one end of a traditional heat pipe. In the forward mode, the end of the heat pipe with the liquid trap is heated, the liquid in the liquid trap evaporates, and the thermal diode functions as a typical heat pipe. When the other end of the heat pipe is heated, however, the liquid condenses in the liquid trap and cannot return to the hot section. Evaporation ceases, and heat is carried only by conduction. This liquid-trap diode has a large forward-mode conductance (⁠ $G f w d =$ 13.8 W/K, in agreement with an estimated evaporation heat transfer coefficient $h$ = 7000 Wm⁻²K⁻¹) and shows large rectification (⁠ $γ=$ 310) with a small temperature difference (⁠ $ΔT=10$ K). However, like other heat pipe thermal diodes, the liquid-trap diode only works for certain gravitational orientations, since the liquid trap floods the main heat pipe if tilted.

Jumping droplets: The jumping droplet thermal diode^180,181 illustrated in Fig. 8 is a phase change convective thermal diode which does not depend on the gravitational orientation. The jumping droplet mechanism occurs when multiple small droplets coalesce into a large droplet and use the energy gained from reducing the surface area to jump off a superhydrophobic surface.¹⁸² 

Figure 8(a) shows a schematic of the jumping droplet thermal diode. In the forward mode, the vapor evaporated from the hot superhydrophilic surface condenses on the cold superhydrophobic surface. The liquid is efficiently returned to the hot side by droplet jumping from the cold side. In the reverse mode, droplet jumping does not occur on the cold superhydrophilic surface, and fluid circulation in the diode ceases. As a result, the heat is mainly transferred by conduction through the spacer, and the diode has a low $G r e v$⁠. As shown in Fig. 8(b), Boreyko, Zhao, and Chen¹⁸¹ constructed a jumping droplet thermal diode with $γ=$ 23, and showed that there was no dependence on gravitational orientation. Increasing the vapor temperature increased the rectification, with measured values as large as $γ=$ 150. Zhou et al.¹⁸³ demonstrated a similar jumping droplet thermal diode with $γ=$ 5. They reported partial surface degradation after use, which will be an important consideration in applications.

Tsukamoto et al.¹⁸⁴ also utilized superhydrophobic and hydrophilic surfaces in a thermal diode. However, instead of using the jumping droplet mechanism to return the condensed water to the hot hydrophillic surface in the forward mode, their MEMS device uses surface tension forces to propel the droplets back to the hot surface by wicking through microchannels. This device achieved $γ=2$ with very small temperature differences $(ΔT$ =1 °C).

Liquid-vapor phase change: The strong nonlinearities of $h$ during liquid-vapor phase change can be used for thermal regulation. Here, we give only a few illustrative examples of nonlinearities in pool boiling¹⁷¹ and refer the reader to the phase change heat transfer literature for more details.^185,186

Boiling heat transfer coefficients can increase by more than a factor of 100 with only small (10 °C) increases in superheat above the saturation temperature $T s a t$⁠. The temperature dependence of the heat flux $q″$ also evolves through the phase change. In the crossover from laminar natural convection to nucleate boiling, for example, the temperature scaling of $q″$ changes from $(ΔT) 5 / 4$ to $( Δ T ) 3$⁠. This strong $h(T)$ dependence could be used to regulate the temperature 5–10 °C above $T s a t$ where the onset of nucleate boiling occurs. Finally, and most dramatically, fluid dryout at the critical heat flux (⁠ $q ″ C H F )$ can even lead to negative differential thermal resistance (NDR), in which $q″$ decreases as the controlled $ΔT$ increases until the heat flux reaches a minimum value $q ″ m i n$ at the Leidenfrost point. In Sec. IV C 3, we discuss a circuit which could utilize this NDR in a thermal amplifier. If the heat flux $q″$ is controlled instead of the temperature, the boiling curve displays hysteresis (where $ΔT$ at a given $q″$ value depends on the thermal history) and instability (where large jumps in $ΔT$ result from small changes in $q″$ near $q ″ C H F$ or $q ″ m i n$⁠).

Variable conductance heat pipe (VCHP): The VCHP^99,175,187 is a widely studied thermal regulator. Figure 9(a) shows a gas-blockage VCHP, which has a low $G$ at low temperature and a high $G$ at high temperature. The working fluid vapor travelling from the evaporator towards the condenser sweeps the non-condensable gas (NCG) along with the vapor, until the NCG is concentrated at the condenser end of the heat pipe. The vapor then condenses and is refluxed towards the evaporator using a wick. For a low evaporator temperature (⁠ $T evap )$⁠, there is relatively little working fluid vapor, so the NCG blankets the cold condensation section at temperature $T cond$ and the effective heat transfer area on the cold side is small. At higher temperatures, the working fluid's vapor pressure increases and the vapor compresses the NCG, increasing the effective area for condensation, and correspondingly $G$⁠. The switching temperature can be varied by controlling the amount of the NCG. In addition, if an external temperature control unit is set around the gas reservoir to control the NCG volume and pressure, the VCHP can act as an active thermal switch.¹⁸⁸ The gas blockage VCHP has been used for temperature control in spacecraft¹⁸⁸ and vehicle¹⁸⁹ components.

Figure 9(b) compares experimental data from Leriche et al.¹⁸⁹ for an ordinary heat pipe and a VCHP. When the temperature rise $ΔT= T evap − T cond$ reaches 50 °C (corresponding to $T evap =80 °$C) the VCHP switches to a high conductance state, with switch ratios around 200. This sharp change in the $Q−ΔT$ curve means that the VHCP can regulate the temperature effectively. However, this VCHP did not work for all gravitational inclinations, which was thought to be due to the gravitational forces limiting the rate at which the condenser liquid was wicked back to the evaporator.¹⁸⁹ 

A gas gap thermal switch tunes the heat transfer by changing the pressure P of an encapsulated gas. Gas gap switches operate in the boundary scattering (Knudsen flow) regime, where the gas molecule mean free path $Λ$ is set by the confining gap width $d$ rather than by the intrinsic molecule-molecule collisions. In this boundary scattering regime, $k ∝P$ at fixed temperature, since the volumetric heat capacity of the gas scales linearly with the gas density. Gas gap switches do not work at higher pressures or for larger gaps because in the bulk regime where molecule-molecule scattering dominates, $Λ∝ P − 1$ and $k$ becomes independent of $P$⁠. Therefore, the gas gap switch requires pressures $P crit < 760 Torr Λ RP / d$⁠, where $Λ RP$ is the intrinsic mean free path in the gas at room pressure. Since impractically small gaps of $d<100$ nm would be required for air at STP to be in the boundary scattering regime, gas gap switches operate at lower pressures. Gas gap switches with measured room temperature $r$ > 500 have been fabricated¹⁹⁰ using helium under high vacuum $(5× 10 − 7$ Torr) and $=250 μm$⁠.

More recently,¹⁹ room-temperature switching ratios of $r=$ 10 have been measured with $d=20$ mm using hydrogen gas. Here, a thermal regulator was created by pulling a high vacuum and controlling the hydrogen pressure via the temperature-dependent hydrogen adsorption onto an adjacent metal hydride. In another recent example, freeze-casted polymer foams¹⁹¹ with small pore sizes of $d$ = 20  $μm$ showed $r$ > 10 using helium at 0.5 Torr and room temperature. From an applications perspective, cryocoolers using gas gap switches may cool down faster than traditional insulation systems,¹⁹² since the large gas $k$ in the bulk regime at high $T$ allows heat to be quickly removed from the cryocooler, until the boundary scattering regime is reached at low $T$ to provide steady-state insulation.

This section supplements the recent review by Shahriari et al.,⁵ which summarized the use of electric fields to control boiling and condensation heat transfer rates.

Electrowetting: The contact angle (wettability) between a droplet and a dielectric substrate can be changed with an electric field.¹⁹³ Cha et al.¹⁹⁴ developed an electrowetting thermal switch with two parallel plates and a liquid droplet on the dielectric bottom plate. The droplet height depends on the contact angle, so a DC voltage can switch between droplet contact and noncontact with the upper plate using the electrowetting effect, showing $r=$ 2.4 for water droplets. McLanahan et al.¹⁹⁵ made a thermal switch using electrowetting to control the lateral droplet motion. In the on state, heat is transferred by convection and conduction through a glycerin droplet bridging two plates, while in the off state, the droplet is shuttled aside using an electric field, and heat is transferred only through the air gap. Switching ratios of $r=$ 4 at ambient pressure and $r=$ 15 at 80 Pa were obtained.

Boiling control with  $E$ fields: Cho, Mizerak, and Wang¹⁹⁶ showed that E fields can be used to enhance nucleate boiling by attracting charged surfactants to the electrodes. The adsorbed surfactants make the surface more hydrophobic, so the E field enhances the bubble nucleation rate and increases the boiling h. Similarly, if the E field repels the surfactants, the nucleation rate decreases as the surfactants desorb, and the boiling h decreases. Switch ratios as large as $r$ = 10 were measured using relatively small voltages (0.1 to 2 V) in boiling of de-ionized water with sodium dodecyl sulfate surfactants at superheats of $ΔT=8$K. Images of the bubble nucleation showed that the switching times were <1 s, and the $h$ measurements showed good repeatability.

Larger $E$ fields can also be used to suppress the Leidenfrost effect, in which liquid droplets are levitated by a thin vapor film over the superheated substrate. If a voltage is applied between the substrate and the liquid, then an electrostatic force pulls the liquid droplet into contact with the substrate.¹⁹⁷ Using measurements of the droplet vaporization lifetime, Shahriari, Wurz, and Bahadur¹⁹⁸ estimated that the film boiling heat transfer rate of isopropyl alcohol superheated by 120 °C could be increased by a factor of 20 with an applied voltage of 1 kV to break the Leidenfrost state. Shahriari, Hermes, and Bahadur¹⁹⁹ also showed that the film-boiling heat flux from a hot metal sphere quenched in isopropanol can be increased by a factor of 5 using a DC bias of 2 kV.

$E$ field effect on jumping droplets: The jumping droplet mechanism utilized in the thermal diodes discussed in Sec. III B 1 has also been used in thermal switches. After the droplets jump off a superhydrophobic surface, the droplet motion can be controlled with electric fields because the droplets have a remnant surface charge.²⁰⁰ Miljkovic et al.²⁰¹ showed that applying electric fields of 75 V/cm can enhance $h$ in water condensation by a factor of 2. The $E$ field prevents the droplets from returning to the surface after they have jumped off, which makes more room for new droplets to condense, and results in larger condensation rates. In one application of a jumping droplet thermal switch, Oh et al.²⁰² used 200–300 V/cm $E$ fields to direct jumping droplets towards electronics hotspots, increasing the heat removal rate by 20% compared with the no-field measurements.

Ferrofluids: Magnetic fields affect heat transfer in ferrofluids,²⁰³ which are fluids containing suspensions of nanoscale magnetic particles such as magnetite. It is energetically favorable for the magnetic moments of the particles to align with the $B$ field, but the average magnetization (⁠ $M$⁠) decreases at higher $T$ due to the increased thermal fluctuations. This $M(T)$ dependence induces natural convection when magnetic fields are applied, leading to measured switch ratios of $r$ = 2.^204,205 Magnetic enhancements of forced convection also have been reported,²⁰⁶ and $B$-field induced particle agglomeration is another possible mechanism for increased heat transfer.^{203,207–209} Applying non-uniform $B$ fields to ferrofluids causes forced convection resulting from the magnetic body force, leading to $r$ = 16 switching.²¹⁰ In one recent application, Puga et al.²¹¹ developed a ferrofluid thermal switch with $r$ = 2.6, investigated the frequency response from 0.1 to 30 Hz and demonstrated that the switch could be used to cool an LED (see also Sec. IV C 2).

Magnetorheological fluids: As compared with the ferrofluids, magnetorheological fluids (MRFs) have larger magnetic particles which do not remain suspended in solution. Cha et al.²¹² made a thermal switch using MRFs of 1–3  $μ$m carbonyl iron particles dispersed in silicone oil. As illustrated in Fig. 10(a), the MRF partially fills the chamber, but in a $B$ field, the MRF aligns in columns which bridge the gap with high- $k$ heat conduction pathways²¹³ between hot and cold plates. Figure 10(b) shows how the thermal conductance of the MRF can be switched between the on and off states using a 0.1 Hz magnetic field. The MRF $k$ switching ratio was measured to be as large as 12, but the switching ratios shown in Fig. 10(b) was reduced to $r$ = 1.3 due to the additional series resistance of heat dissipation from the cold side of the switch to the ambient air.

Single-component fluids: The framework of magnetoconvection^214,215 describes heat transfer in electrically conducting fluids in the presence of external fields. Switching ratios of $r$ = 2 have been observed in natural convection^216–218 or forced convection²¹⁹ of liquid metals with $B$ fields up to 0.5 T. Controlling the heat transferred from a surface to an ionized gas or an electrolyte using B fields would be appealing for applications such as orbital re-entry or hypersonic flows,^215,220 but prohibitively large B fields are required to achieve significant heat transfer modulation. Electric fields have been shown to enhance convection in gases by up to a factor of 5 using the “corona wind” effect,^221,222 where large electric fields ionize the gas and the Coulomb force repels the charged electrons. Electrophoretic effects of electric fields on free charges in gases and liquids also increase the heat transfer rates, though the enhancement depends on a number of experimental variables.^221,223 Finally, natural convection augmentations as large as a factor of 3 have been reported in low- $σ$ fluids,^223,224 and it is also known that $E$ or $B$ fields can influence natural convection in ordered liquid crystals.^225,226

Thermal radiation is a broadband phenomenon, with most of the heat at room temperature carried by photons in the near infrared wavelengths. The spectral hemispherical emissivity (⁠ $ε λ$⁠) describes how well a surface emits over all angles, compared with a blackbody emitting at the same wavelength. The total emissivity $ϵ t o t$ is obtained by integrating over all wavelengths, weighted by the Planck spectrum at a given $T$⁠.

In this section, we discuss how $ϵ t o t$ (and closely associated properties such as the absorptivity, reflectivity, or transmissivity) depends on $T$⁠, particularly near phase transitions. We review photon diode mechanisms and emissivity switching with electric fields. The heat transferred between objects depends on the geometrical view factors, which can be controlled using louvers. Finally, we discuss recent experiments showing that thermal radiation in the near field is highly dependent on the distance between objects.

To construct radiative thermal regulators, materials with highly temperature dependent optical properties are required. Many materials have weakly temperature dependent $ϵ t o t (T)$⁠, which can originate either from fundamentally $T$-dependent $ε λ$ (e.g., the Hagen-Rubens relation predicts that $ε λ (T)$ scales as $T 1 / 2$ for metals²²⁷), or from the $T$-dependent Planck distribution weighting (e.g., $ϵ t o t (T)$ of Al₂O₃ decreases²²⁸ roughly as $T − 1 / 2$ due to the increased Planck weighting of the smaller $ε λ$ for $λ<2 μm$ at high $T$⁠). However, to achieve high switching ratios, more dramatic changes in $ϵ t o t$ are needed.

Metal-insulator transitions: The broadband emissivity of vanadium dioxide^229,230 changes across the metal-insulator transition (MIT) at 68 °C, indicating that VO₂ can be used as a thermal radiative regulator. The low-temperature insulating state has a larger emissivity than the high-temperature metallic state, with recently reported^231–233  $ϵ t o t$ switching ratios of $r≈2–3$ across the MIT. These large changes in $ϵ t o t$ have been utilized in temperature-tunable infrared absorbers using VO₂ thin films,^234–236 and VO₂ has also been investigated as a coating to modulate the solar transmittance through windows for building climate control.²³⁷ Ito et al.²³⁸ recently demonstrated that the near-field radiative transport (see also Sec. III C 5) across a 370 nm gap between VO₂ and fused quartz also decreases as the VO₂ temperature increases above $T MIT$⁠. Other materials displaying metal-insulator transitions can also be thermal regulators; for example, $ϵ t o t$ of doped manganese oxides^239,240 increases by a factor of 3 as the temperature is increased from −50 to +50 °C. In Sec. III C 2, we will discuss an implementation of a thermal diode using the VO₂ MIT.

Interestingly, since $ϵ t o t$ dramatically decreases across the phase transition, the total power radiated from VO₂ is non-monotonic, displaying negative differential thermal resistance (NDR). This non-monotonic dependence of emitted power on temperature difference has been utilized to demonstrate thermal camouflaging,²⁴¹ where VO₂ films at different temperatures emit the same radiative power.

Phase-change memory: In the visible wavelengths, the different reflectivities of amorphous and crystalline phase-change materials such as GST are utilized in data storage applications. However, amorphous and crystalline GST also have quite different properties in the near infrared wavelengths relevant for thermal radiation.²⁴² For example, $ε λ$ of GST films on a metal substrate were switched²⁴³ by a factor of 10 over the wavelength range of 8–14  $μ$m, and similar switching was also observed in the transmissivity over 1.4 –2  $μ$m in a gold grating filled with germanium telluride.²⁴⁴ 

As a side note, the term “thermochromic” is sometimes used to indicate T-dependent optical properties in the visible (rather than infrared) wavelengths. Although there are many thermochromic polymers and liquid crystals that are used for temperature sensing,^245,246 the broadband infrared properties relevant for thermal emission are typically much less affected by temperature.²⁴⁷ Likewise, though metamaterials displaying temperature-dependent resonance or narrowband absorption phenomena have also been developed,^248,249 broadband phenomena are required to switch $ϵ t o t$ and affect the heat transfer.

Fundamentals of photon rectification: The Lorentz reciprocity relations²⁵⁰ state that in any optical system which is linear and obeys time-reversal symmetry (TRS), the total transmitted power is invariant under TRS, even though individual modes may have asymmetric transmission. The second law of thermodynamics provides additional constraints on radiative diodes, stating that no rectification can be observed in linear systems, even in the presence of magnetic fields²⁵¹ which break TRS. To achieve rectification, both a nonlinearity (such as temperature dependent $ϵ t o t$⁠) and an asymmetry are required.⁵⁵ The necessity for both nonlinearity and asymmetry was illustrated experimentally by some control measurements on a ballistic photon diode,⁵⁶ which observed no rectification when an asymmetric pyramidal scatterer was placed between blackbody reservoirs. That study also highlights the importance of paying close attention to experimental symmetries when exploring rectification, since the other experiments⁵⁶ were subsequently found to have a fundamental symmetry flaw^252–254 which invalidated their attempts to demonstrate rectification.

Interestingly, $ϵ t o t (T)$ and an asymmetry are necessary, but not sufficient, conditions for rectification. For example, consider two diffuse parallel plates (1 and 2), each of which have arbitrary spectral emissivity, $ϵ λ , 1 ≠ ϵ λ , 2$⁠. Though the spectral emissivities do not depend on $T$⁠, the Planck weighting still gives each plate $(i)$ a different temperature-dependent $ϵ t o t , i (T)= ∫ 0 ∞ ϵ λ , i E b λ , T d λ /σ T 4$⁠, where $E b λ , T$ is the Planck blackbody emissive power. Perhaps unexpectedly, however, it can be shown²⁵⁵ that no rectification occurs in this scenario, even with the asymmetry and nonlinearity of the two plates with different $ϵ t o t , i (T)$⁠. Furthermore, Chen et al.⁵⁶ showed that rectification is, in fact, strictly forbidden between any two surfaces with temperature-independent but otherwise arbitrary $ϵ λ$⁠, as long as any scattering phenomena in the volume between the two surfaces are elastic. However, if photons are scattered inelastically between the two plates, rectification can occur, even if $ϵ λ$ does not depend on T. For example, if a thin, opaque, diffuse, black plate is placed between plates 1 and 2 to inelastically absorb and re-emit photons, rectification now becomes possible.³ 

Demonstration of a radiative diode: As shown in Fig. 11(a), Ito et al.²⁵⁶ fabricated a radiative diode consisting of a VO₂ film on a silicon wafer, exchanging radiation with quartz. This diode utilizes the highly nonlinear emissivity of VO₂ around the MIT, as discussed in Sec. III C 1. Figures 11(b) and 11(c) show that this demonstration achieved rectification ratios around 2 with $ΔT=$ 70 K. In addition, doping the VO₂ with W decreased the MIT temperature $T M I T$ by 25 K while maintaining $γ$⁠. The rectification ratio is independent of $ΔT$ once the hot side temperature is increased above $T M I T$⁠. From a diode design perspective, it was desirable to use a high- $ϵ$ second material such as quartz, since the total resistance to radiation in the reverse mode should be dominated by the low $ϵ$ of the metallic VO₂. In addition to rectification, this demonstration also displays negative differential thermal resistance, since the decrease in $ϵ$ across the MIT reduces the heat flux, even though $ΔT$ is increasing.

Electrochromic materials have optical properties which depend on the electric field. To influence the radiative emission, broadband changes in $ϵ λ$ over the infrared wavelengths are required. Since solar energy lies mostly in the visible wavelengths, changing the absorptivity or transmissivity in the visible range can also modify the heat transfer if the material is receiving solar irradiation.

Electrochromic chemical reactions: Commercial electrochromic systems use reduction/oxidation (redox) processes to modify the optical properties of electrochromic solids or liquids.^257,258 If a solid-state electrochromic material such as WO₃ is used, a voltage at the electrode drives lithium cations from an electrolyte into the WO₃. The resulting redox reactions in the WO₃ cause large changes in the broadband optical properties. Figure 12(a) shows transmittance measurements²⁵⁹ from an electrochromic glass manufactured by the company Gesimat. This device uses complimentary WO₃ and Prussian Blue (ferrite ferrocyanide) electrochromic films separated by a polymer electrolyte. Changing the polarity of the applied DC voltage (0.5–2 V) switches the integrated transmittance over the incident solar irradiation by a factor of 9. The switching timescales in this demonstration were around 10 min.

Similar reflectance switching ratios have been measured using polymers in the near infrared,²⁶⁰ and redox processes can also modify the reflectivity in the far infrared.²⁶¹ Figure 12(b) shows reflectivity measurements of conducting polymers by Topart and Hourquebie²⁶² with $r$ = 2.5 between 5 and 20 $μ$m. Switch ratios of $r=$ 2 across the wavelength band of 2–40 $μ$m have also been reported in an all-solid state electrochromic system.²⁶³ 

Liquid crystals: Ordered liquid crystals display birefringence, leading to optical phase shifts that depend on the liquid crystal orientation.²⁶⁴ With the assistance of polarizing filters, the total transmissivity can be switched by orienting the liquid crystals;^265,266 however, strong absorption losses in the infrared²⁶⁷ coupled with polarization filtering losses limits the achievable $r$⁠. Polymer dispersed liquid crystals have been used as broadband unpolarized infrared switches,²⁶⁸ with $r$ = 5 in the 2–5 $μ$m range and $r$ = 2 in the 8–14 $μ$m range. The transient microstructure reorientation in ferroelectric liquid crystals²⁶⁹ has also been used to decrease the transmissivity by a factor of 2 in the 8–12 $μ$m band, and even larger switching ratios²⁷⁰ on the order of $r$ = 10 have been reported over 0.5–1.5 $μ$m wavelengths.

Electrostatically tuned resonances:  $E$ fields can shift the resonant absorption peaks in metamaterials or thin films, though this has only a small effect on the heat transfer. Electrostatic gating reduced the reflectance of graphene on a thin silica film²⁷¹ by 3% for $λ$ between 3 and 6 $μ$m, and increased $ϵ λ$ of a patterned graphene resonator²⁷² by 1% in the mid-IR. Narrowband $ϵ λ (E)$ has also been observed in quantum well structures^273,274 with photonic crystal patterning. A MEMS device²⁷⁵ displayed a 50% decrease in the normal reflectivity at 6 $μ$m, but the integrated reflectivity between 5 and 10 $μ$m only decreased by 5%. Likewise, electrostatic control of the gap size in a MEMS plasmonic structure²⁷⁶ caused a factor of 2 increase in the reflectivity at 1.6 $μ$m, but only a 10% decrease in the integrated reflectivity from 1 to 2 $μ$m. Generally, it appears difficult to achieve large $r$ by controlling resonance phenomena due to the broadband, hemispherical, and unpolarized nature of far-field thermal radiation. Although narrow bandpass filters placed in series could be used to emphasize the importance of thermal radiation within the small wavelength range which can be efficiently switched, this filter will dramatically reduce the total radiative heat transfer, and parasitic conduction or convection pathways will become more important.

The radiative heat transfer between surfaces depends on the view factor (⁠ $F 12$⁠), defined as the fraction of the total energy leaving surface 1 which is intercepted by surface 2. By controlling the view factors using mechanical actuation, the heat transfer can be efficiently switched. A common implementation uses low- $ϵ$ metallic mechanical louvers^21,99 covering a high- $ϵ$ emitter. When the louvers are closed, there is a higher resistance to radiation, and the emitter cannot efficiently dump the heat to the environment. Switching ratios of $r$ = 6 have been achieved in spacecraft applications.⁹⁹ Thermal regulators can also be constructed by actuating the louvers using a bimetallic switch, or using a shape-memory alloy actuator to unfold a stowed radiator with a high $ϵ$⁠.^277,278

Thermal radiation is typically a far-field phenomena, since the distance between radiating objects $L$ is almost always large compared with the thermal photon wavelengths $λ$⁠. However, recent experiments near room temperature reviewed by Song et al.²⁷⁹ have probed heat transfer in the near-field, where $L≪λ$⁠. In this regime, evanescent electromagnetic waves tunnel between materials, increasing the heat flux above the far-field blackbody limit. Since the tunneling currents depend critically on $L$⁠, near-field heat transfer can lead to impressive switching. Recent room temperature measurements²⁸⁰ showed $r$ as large as 100–1000 for SiO₂-SiO₂ surfaces or Au-Au surfaces as the gap was reduced to $L$ < 100 nm. For larger values of $L=3 μ$m, Elzouka and Ndoa²⁸¹ reported thermal rectifications of 10% at $ΔT=150$ K using the asymmetric thermal expansion of a MEMS structure to control $L$⁠. Overall, for the purpose of thermal switching it is likely more practical (and effective²⁸²) to bring materials fully in and out of contact (see Sec. III A 4) than to rely on near field mechanisms, since it is difficult to maintain and control such small yet finite $L$ over macroscopic distances.

Interestingly, theory predicts that near-field radiation between semiconductors can be tuned with electric²⁸³ or magnetic²⁸⁴ fields. Demonstrating this switching using external fields would highlight the unique capabilities of near-field radiation, since the heat conduction through interfaces in contact is insensitive to external fields. Theory also predicts that near-field thermal diodes could be constructed using materials which have temperature-dependent electromagnetic resonances. Otey, Tung, and Fan²⁸⁵ first proposed this concept and analyzed the rectification between SiC polytypes, and Song et al.²⁷⁹ reviewed subsequent theoretical predictions of near-field thermal rectification. In a similar exploitation of temperature-dependent properties, Ito et al.²³⁸ recently demonstrated a near-field thermal regulator which used the metal-insulator transition (MIT) of VO₂ to reduce the heat transfer across a gap of $L=370$ nm by a factor of 1.6.

Now that we have examined the mechanisms of switchable and nonlinear heat transfer in great detail, we consider several potential applications of thermal diodes, switches, and regulators in simple thermal circuits. First, we examine applications to solid-state refrigeration cycles. We then consider energy scavenging schemes. Finally, we present eight additional thermal circuits that can be implemented using nonlinear and switchable thermal devices. These thermal circuits are directly analogous to commonly utilized electrical circuits, and offer new thermal capabilities.

Thermal diodes or thermal switches can be used to implement both the adiabatic and isothermal portions of solid state thermodynamic cycles at a single physical location. Here, we examine a solid state refrigeration cycle in detail to see why thermal diodes are useful in this application, and to quantify the rectification ratios γ required to obtain good cycle performance. Though we focus here on an electrocaloric refrigeration example,^14–16,286 many of the same principles will also apply to adiabatic demagnetization refrigeration (magnetocaloric) cycles.^4,13,287

Electrocaloric: Figure 13 summarizes an electrocaloric (EC) Brayton refrigeration cycle described by the temperature-entropy diagram in Fig. 13(a). Adiabatically applying an electric field (⁠ $E$⁠) during process 1 → 2 increases the EC temperature by the amount $θ$ (which can be as large as $∼$10 K in ferroelectric polymers,²⁸⁸ but is more typically <1 K). Heat then flows from the EC material to the hot reservoir at $T h$ (2 → 3). After the $E$ field is adiabatically removed (3 → 4) the EC temperature $T EC$ decreases below the cold reservoir temperature $T c$⁠, and finally (4 → 1) heat flows from the cold reservoir to the EC material to complete the cycle. Clearly, if at any point in the cycle, heat can flow back from the hot reservoir into the EC material, or from the EC material into the cold reservoir, the cycle performance is degraded. One way to block this undesirable heat flow, while still allowing for strong thermal coupling to pump heat out of the cold reservoir, is to use two thermal diodes. The thermal circuit describing the EC system is shown in Fig. 13(b), where the thermal capacitance of the EC material is $C t$⁠. Each diode is reverse-biased for 3 of the 4 stages of the cycle: the hot-side diode is only in the forward mode during (2 → 3) when heat flows from $T EC$ to $T h$⁠, and the cold-side diode is only in the forward mode during (4 → 1) when heat flows from $T c$ to $T EC$⁠. At all other times, the diodes block heat from flowing back into the refrigerator.

Thermal switches are also used to mimic thermal diode behavior during the different stages of the cycle; in fact, all experimental implementations of EC cycles that we are aware of have utilized contact thermal switches, since solid state thermal diodes are less common than contact switches. We present this new implementation because using passive thermal diodes rather than active switches could substantially reduce the system complexities associated with active thermal switching, and the performance of the diode circuit and switch circuit for refrigeration will be identical if the diode's forward/reverse conductances are equal to the switch's on/off conductances.