We design thermal emitters based on gold micrograting structures with VO_{2}-filled slots for enhanced far-field thermal rectification. We numerically calculate the thermal rectification ratio for two different approaches, peak extinction and peak shift. In peak extinction, the VO_{2} phase transition switches the radiative coupling of the surface plasmon on and off. In peak shift, the phase transition shifts the wavelength of the radiatively-coupled surface plasmon. We vary the extinction coefficient of VO_{2} to determine the effect on rectification for each approach. In both cases, the rectification ratio can be increased by increasing the VO_{2} loss in the metallic state. This suggests that highly efficient, microstructured thermal rectifiers can be achieved via suitable manipulation of the optical properties of VO_{2}.

## INTRODUCTION

While electronic devices form the foundation of modern computing systems, their operation is limited in high temperature environments.^{1} As a result, there has been considerable interest in the development of thermal analogs of electronic devices, which would operate by controlling heat flow rather than electron transport. These include thermal memories, logic gates, and thermal rectifiers.^{2–9} A key figure of merit for thermal rectifiers is the rectification ratio R, which describes the asymmetry in forward and backward heat flow in the device. A variety of mechanisms have been proposed to achieve asymmetric flow, based on either conductive^{10–15} or radiative^{16} approaches. Radiative approaches consider the heat flow between two bodies separated by a gap and generally use a phase-change material such as VO_{2}, which exhibits a significant change in optical and electrical properties around the metal-insulator transition temperature. In recent years, several VO_{2}-based tunable absorbers have been proposed^{17–22} and incorporated in thermal rectifier designs.^{23–25}

Designs for near-field devices, where the gap is small compared to the thermal wavelength, have yielded high rectification ratios^{26,27} due to the participation of evanescently-decaying surface waves in radiative transport.^{28,29} However, experimental implementation is likely to require tight tolerances on alignment and fabrication. Far-field devices can relax these constraints, if strategies can be found to achieve high rectification ratios. A recent design has achieved an *R* > 11 for far-field devices based on Fabry-Perot cavities.^{30} In this case, a phase transition in VO_{2} toggles the system between one that supports a well-defined Fabry-Perot resonance and one that does not.

In this work, we explore an alternative approach to far-field rectifiers based on surface plasmon modes. Using grating structures composed of gold, VO_{2}, and silicon, we can selectively couple evanescently-decaying surface plasmon modes to far-field radiation. This geometry allows us to study two different approaches to thermal rectification. In the peak extinction approach, we exploit the metal-insulator transition in VO_{2} to switch the radiative coupling from the surface plasmon mode on and off. In the peak shift approach, we shift the resonant wavelength of the surface plasmon mode. In each case, we show that the highest rectification ratios are achieved by tuning the loss of the VO_{2} material in the metallic state to best match the properties of gold. Several studies have demonstrated the effect of doping and changes in annealing parameters on the electrical and optical properties of VO_{2} thin films.^{31–35} This work suggests a new direction for the improvement of thermal rectifier performance through material property optimization. Moreover, it demonstrates that with appropriate design strategies for near- to far-field coupling, evanescently-decaying surface waves can be leveraged for far-field rectifier designs.

## MATHEMATICAL FORMALISM FOR FAR-FIELD RADIATIVE THERMAL RECTIFIERS

Consider emitters 1 and 2 maintained at two different temperatures *T _{1}* =

*T*and

_{c}*T*=

_{2}*T*(

_{h}*T*>

_{h}*T*) as shown in Fig. 1(a). In accordance with the second law of thermodynamics, heat will flow from emitter 2 to emitter 1 in the absence of any external work on the system. We refer to this as the forward direction. On the other hand, the direction of heat flow will be reversed if emitter 1 is maintained at

_{c}*T*and emitter 2 at

_{h}*T*. However, it is possible to construct a device which allows heat flow in the forward direction while suppressing it in the reverse direction. Such a device that enables unidirectional heat flow between two bodies maintained at different temperatures is referred to as a thermal rectifier.

_{c}^{36}

Thermal rectification is quantified by the rectification ratio *R* defined as^{37}

where $Pf$ and $Pr$ represent radiative power exchanged in the forward and reverse directions, respectively. For two planar bodies maintained at temperatures *T _{h}* and

*T*, the forward and reverse radiative powers exchanged

_{c}^{38}in the wavelength range $[\lambda 1,\lambda 2]$ are defined as

Here, $\Gamma f,r(\lambda ,Th,Tc)$ represent the forward and reverse channeled emissivities, respectively. For two bodies at temperatures *T _{h}* and

*T*with emissivities

_{c}*ɛ*

_{1}and

*ɛ*

_{2}, respectively, the channeled emissivity is given by

## APPROACHES TO ACHIEVE THERMAL RECTIFICATION

We will discuss two approaches of achieving thermal rectification: peak extinction and peak shift. In the more common peak extinction approach, emitter 1 is temperature-dependent, while emitter 2 is a temperature-independent black body. Therefore, $\epsilon 2\mu (\lambda )=1$, and the expression for channeled emissivity simplifies to

As a result, *P _{f}* and

*P*are directly proportional to the emissivities of emitter 1 in the low and high temperature states, respectively. Rectification can, thus, be enhanced by increasing the difference in emissivities between the two states. Figure 1(b) shows an example emission spectrum suitable for the peak extinction approach. In its insulating state, emitter 1 has a well-defined peak at 10

_{r}*μ*m. The peak is considerably suppressed in the metallic state. This leads to the increased heat transfer in the forward direction relative to the backward direction, causing thermal rectification. This approach has previously been used for the construction of VO

_{2}-based far-field thermal rectifiers.

^{39}

Here, we introduce an alternative approach based on peak shift. In this approach, the emissivity of emitter 1 is temperature-dependent, while that of emitter 2 is temperature-independent. Hence, the channeled emissivity can be simply defined as

It is important to note that $\Gamma $ is no longer dependent on the temperature of emitter 2. Consider the example spectra shown in Fig. 1(c). At a particular wavelength $\lambda 0=15\mu m$, for example, we have $\epsilon 2\mu (\lambda 0)$ and $\epsilon 1\mu (\lambda 0,Tc)$ equal to 1. Therefore, from Eq. (5), $\Gamma f(\lambda 0,Tc)=1$. On the other hand, $\epsilon 1\mu (\lambda 0,Th)=0$, and hence the channeled emissivity for the reverse direction $\Gamma r(\lambda 0,Th)=0$. The difference in integrated channeled emissivities for the forward and reverse directions results in thermal rectification.

Intuitively, for this approach to yield high rectification, it is essential for the emissivity of emitter 2 to be strongly peaked, rather than constant. It is the temperature-dependent change in the spectral overlap between emitter 1 and 2 that results in rectification. From Eq. (5), the channeled emissivity is only nonzero over wavelength ranges for which both *ɛ*_{1} and *ɛ*_{2} are nonzero. For the forward case, the channeled emissivity is nonzero where the two peaks overlap; in the backward case, it is small everywhere.

In this study, we exploit the ability of metal gratings to exhibit surface plasmon polariton (SPP) modes to achieve rectification using both the peak extinction and peak shift approaches. For a planar metal-dielectric interface, SPPs propagate parallel to the interface and decay exponentially in the perpendicular direction.^{40} A photon incident on the interface can excite an SPP if it has the same wavevector component parallel to the interface (k_{||}) as the SPP. Figure 2(a) shows the dispersion relation of an SPP at a metal-air interface as a function of k_{||}. Since the curve does not intersect the k_{||} = 0 axis at any point, a normally incident plane wave cannot excite an SPP. However, if we introduce a grating on the metal surface, the wavevector space becomes periodic.^{40} As a result, the SPP dispersion relation gets folded at the first Brillouin zone boundaries and intersects the k_{||} = 0 axis. This makes it possible for normally incident photons at certain frequencies to excite an SPP on the metal-air interface.

VO_{2} has a phase transition from insulating to the metallic state near 340 K. By designing a metal/vanadium dioxide grating that supports SPP in the insulating state, several new opportunities for far-field thermal rectifier designs become possible. In the section below, we first present a peak extinction design. In this case, the phase transition effectively toggles the structure between a planar metal slab and a metal/dielectric grating, switching the radiative coupling of the SPP on and off. We then present a peak shift design, where the phase transition changes the effective slot depth of the grating. This shifts the wavelength of the SPP, reducing the spectral overlap with a second, temperature-independent emitter.

## RESULTS AND DISCUSSION

To illustrate the peak extinction approach, we take emitter 1 to be a gold micrograting structure with VO_{2}-filled slots. One unit cell of the grating is shown in Fig. 3(a). Emitter 2 is taken to be a black body. Normal-incidence emissivity spectra are computed using Lumerical FDTD solutions, assuming that emissivity is equal to absorptivity from Kirchoff's law. The optical constants for gold and silicon are obtained from the built-in material library, while those for VO_{2} are taken from Ref. 41. For all our calculations, we define *T _{h}*

_{ }= 340 K + Δ

*T*and

*T*

_{c}_{ }= 340 K − Δ

*T*, where Δ

*T*= 30 K. We optimized the slot dimensions to yield the highest rectification ratio

*R*, as shown in Fig. 3(b). A slot depth of 950 nm and a slot width of 400 nm give an optimal value of 4.86.

The emissivity spectra for the optimized grating are shown in Fig. 3(c). For the insulating state of VO_{2}, we observe a strong emissivity peak close to 15 *μ*m. This is consistent with coupling to a normal-incidence SPP mode of the metal-dielectric grating. For the metallic state of VO_{2}, the emissivity is strongly suppressed. This behavior is similar to that expected for a solid block of metal, for which the SPP cannot couple to normal-incidence light. This result suggests that strong rectification is achieved due to the resemblance of the metallic state of VO_{2} to gold. However, the optical constants of metallic VO_{2} and gold are not identical. We, thus, explore the effect of increasing the loss in the metallic state with the goal of increasing the rectification value. We note that the qualitative behavior exhibited in Fig. 3(c) is similar to that observed in Ref. 30; the smaller value of *R* we obtain largely results from a narrower emission peak.

Figure 4(a) shows emission spectra for the metallic state of emitter 1 with the imaginary part of the refractive index (extinction coefficient) scaled by a factor C_{M}. As C_{M} increases, the metallic state becomes increasingly lossy. This causes the emissivity to reduce, ultimately mimicking the spectrum obtained for a hypothetical case of VO_{2} replaced with gold (that is, a solid, gold slab). Figure 4(b) shows a scatter plot of *R* vs C_{M}. As C_{M} increases and the metallic state emissivity reduces, *R* increases to a value of around 16.6.

In order to increase the rectification ratio further, we scale the extinction coefficient of insulating VO_{2}. Figure 4(c) shows emission spectra for the insulating state of emitter 1 with extinction coefficients scaled by a factor C_{I}. An increase in C_{I} causes the peak emissivity to reduce and the prominent emission peak to broaden. Figure 4(d) shows a scatter plot of *R* vs the scaling factor C_{I} with C_{M} fixed at 4. The rectification ratio is directly influenced by the overlap integral between the black body spectrum and the insulating state emissivity. In an ideal case, one would expect a sufficiently broad peak with near-perfect emissivity to yield the highest rectification. However, for the grating structure under study, the peak width and height are inversely related [Fig. 4(c)]. Hence, there is a trade-off between peak broadening and peak height reduction. An initial increase in the value of *R* is caused by peak broadening, which dominates over peak height reduction to give a maximum rectification value of 20.67 at a scaling factor of 6. Beyond this point, the peak width remains nearly constant, while the peak height continues to decrease. As a result, the rectification ratio reduces.

To illustrate the peak shift approach, we study a gold micrograting structure with slots partly filled with VO_{2} and Si. The unit cell is shown in Fig. 5(a). For emitter 2, we choose a gold grating similar to emitter 1 but with 0.9 *μ*m deep slots completely filled with Si. Figure 5(b) shows the emission spectra for the metallic and insulating states of emitter 1 along with that of emitter 2. For *T*_{1} = *T _{h}*, VO

_{2}is in the metallic state, and emitter 1 behaves as a metal-dielectric grating with a slot depth given by the height of the Si region. The SPP mode is close to 12

*μ*m and couples with normally incident light, giving a broad emission peak. However, for

*T*

_{1}=

*T*, VO

_{c}_{2}is in its insulating state and emitter 1 acts as a metal-dielectric grating with a slot depth given by the sum of the Si and VO

_{2}heights; Si was chosen for the dielectric material due to the similarity between the values of n for Si and insulating VO

_{2}. As a result of the change in slot depth, the SPP mode shifts to 15

*μ*m and gives a sharp emission peak at that wavelength. This peak overlaps with the emission peak of emitter 2, which is designed to have an SPP mode at 15

*μ*m.

In order to increase the value of *R* for the peak shift approach, it is desirable to tune the emission peak of emitter 1 in the metallic state to increase peak emissivity and reduce overlap with the insulating state. This is achieved by scaling the extinction coefficient of metallic VO_{2} by a factor of C_{M} greater than 1. Figure 6(a) shows the metallic state emission spectra for a few values of C_{M}. As the value of C_{M} increases, metallic VO_{2} becomes increasingly lossy, ultimately causing the emission spectrum to look similar to that for the hypothetical case of VO_{2} replaced with gold. Figure 6(b) shows a scatter plot of *R* vs the metallic state scaling factor C_{M}. As the value of C_{M} increases and the metallic state emission peak becomes sharper, *R* increases to a value of around 7.3. In order to increase the value of *R* further, we scale the extinction coefficient of insulating VO_{2}. Figure 6(c) shows the insulating state emission spectra for a few values of the scaling factor C_{I} with C_{M} fixed at 10. It can be seen that scaling does not have a significant impact on the emission spectrum. This claim is further substantiated by Fig. 6(d), which shows a scatter plot of *R* vs the insulating state scaling factor C_{I}. The rectification ratio increases from 7.3 to 8.4 with an increase in the value of C_{I}. However, the change is not as significant as in the peak extinction approach.

It is possible to have a third approach by using a temperature independent emitter 2 instead of a black body in the peak extinction approach. If we use the grating design for emitter 1 described in the peak extinction approach along with that for emitter 2 in the peak shift approach, we get a rectification ratio of 6.58. This is higher than the values of 4.86 and 1.38 obtained for peak extinction and peak shift, respectively, for unscaled VO_{2}. By using a scaling factor of C_{M} = 10, we obtain a value of 14.79 for the rectification ratio. Combining this with a scaling factor of C_{I} = 2, we get *R* = 15.15.

## CONCLUSION

In conclusion, we have used gold micrograting structures with VO_{2}-filled slots to achieve enhanced, far-field thermal rectification. We exploited the ability of the gratings to exhibit tunable SPP modes and investigated two approaches to far-field rectification. In the peak extinction approach, the coupling of the SPP mode to the far field is switched on and off with temperature. In the peak shift mode, the resonant wavelength of the SPP mode is shifted to change the spectral overlap with a second, resonant emitter. For both approaches, we studied the dependence of the rectification ratio *R* on the extinction coefficients of the metallic and insulating states of VO_{2}. We observed that rectification increases as the extinction coefficient of metallic VO_{2} approaches that of gold. By scaling the insulating state extinction coefficient of VO_{2} in addition to the metallic state, we were able to achieve a rectification ratio of 20.67 for the peak extinction approach. These observations suggest that materials research focused on manipulation of the optical properties of VO_{2} could yield more efficient thermal rectifier devices, as the general strategy of tuning the optical loss could potentially yield higher rectification ratios for a variety of device geometries. Moreover, we have shown that by appropriately considering mechanisms for coupling evanescently-decaying surface modes to the far field, such modes can play a critical role in far-field rectifier design.

## ACKNOWLEDGMENTS

This work was funded in part by the Defense Advanced Research Projects Agency under Agreement No. HR00111820046. The views, opinions, and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. R.A. was funded in part by the USC Viterbi Graduate School Merit Fellowship.

## REFERENCES

*Introduction to Heat Transfer*(Wiley, 1990).