A one-way street for phonon transport: past, present and future of solid-state thermal rectification

Thermal rectification is the dependence of the thermal conductivity on the direction of heat propagation. This fascinating effect could be exploited for the design of thermal devices, such as the thermal diode, and could find application in energy harvesting and thermal management. Where did we start from, what has been achieved and what does the future hold?

least to some extent, as thermal diodes because of the difference between accumulating defects next to the cold or to the hot side: in the former case they reduce considerably the conductance by adding additional scattering; in the latter case phonon scattering is already dominated by high temperature anharmonic effects and adding structural imperfections has little effect.One of the highest experimental thermal rectification was obtained in LaCoO 3 /La 0.7 Sr 0.3 CoO 3 junctions [12] and amounted to 47% (see figure 1).
Despite the undeniable progresses made in the last couple of decades, experimental demonstrations are still scarce and rectification factors are in most of the cases modest.What is the reason?Is this just as-good-as-it-gets -and thus a reminder of why we do not live in a phononic world-or are we missing something?If one looks at the results of Dames for junctions between existing bulk materials [24] it seems, indeed, that there is no qualitative leap forward ahead of us.In these systems thermal rectification derives from the different temperature dependence of the thermal conductivity, κ(T), of the constituent materials.However, κ(T) of many semiconductors of practical interest are awfully similar and consistently lead to small rectifications [25].Additionally, thermal rectification depends on the difference between the temperature of the hot and cold reservoirs and hence vanishes for small thermal gradients.And if this was not enough, there are also significant hurdles arising when it comes to the practical implementation of a thermal diode.The most important one is perhaps the need for a highly (thermally) insulating substrate.Otherwise, a large fraction of the phonons would not travel through the main device, but rather via the substrate, washing out the rectification behavior.Indeed, sizeable rectification values are often obtained in suspended nanostructures [13,14].
So, do we still have some cards up our sleeves?There are indeed some avenues that are worth exploring -and that indeed are being explored-and that in principle can alleviate some of the limitations discussed above.
It would be tempting to leave conduction aside and turn to other heat transfer mechanisms.Good values of thermal rectification can be achieved with convection [26] or radiation [27,28].However, these technologies are not suited for micro-and nano-electronics applications due to their limited reliability (fluid leakage, exact separation between radiating materials) and/or scalability.We should bear in mind, though, that sticking to conduction does not necessarily mean being stuck with phonons.In materials with a high electrical conductivity, e.g.metals or degenerately doped semiconductors, a substantial part of heat can be carried by electrons and under certain circumstances, electronic thermal rectification can outperform the phononic counterpart.The most successful demonstrations on this effect, however, rely on superconductor-metal junctions and the operation is thus restricted to temperatures in the sub-kelvin regime [29].
Bringing all the ideas outlined above to the nanoscale offer additional design parameters that might be exploited to engineer thermal rectification.On the one hand, the nanostructuring itself can provide the spatial dependence of κ in an otherwise homogeneous system.An example is a nanowire with an abrupt variation of diameter [30], combining a narrow and a thicker segments, and thus a rather flat with a 1/T-dependence of κ (see e.g.[31]).Other instances are carbon nanocones [32] or graphene step junctions [33].At the same time, interface thermal resistances (ITR) may not be negligible at the nanoscale, if compared to the overall thermal resistance, and can play a role in determining the thermal rectification [34,35].In particular, if the materials on the two sides of a junction have different thermal resistances, the temperature of the interface changes upon reversing of the thermal bias [35] and so does the ITR due to its dependence on temperature [36].
Phase-change materials [37] are a very appealing alternative for the design of efficient thermal diodes.They normally exist in two states that can exhibit very dissimilar κ(T) behaviors and thus permit to create junctions between materials with opposing thermal conductivity trends [38,39], the ideal situation to obtain high rectifications in a heterojunction (with conventional materials one can at most aim at combining a decreasing κ(T), where phonon scattering is ruled by anharmonic effects, and a roughly temperature independent κ, such as in alloys where mass disorder is the dominant scattering mechanism [24]).Although thermal diodes based on phase-change materials have now been around for a while [37,40], it looks like we have barely scratched the surface of the possibilities they offer.As an example, Swoboda and coworkers [41] recently proposed a multilayer structure with a theoretical rectification factor of 119%, whereas a junction between Ag 2 S 0.6 Se 0.4 and Ag 2 S 0.1 Te 0.9 exhibited the highest experimental value reported to date for a solid-state thermal diode [42], amounting to 170%.
A final mention goes to organic semiconductors, which are very popular for thermoelectric applications due to their low thermal conductivities, but are normally overlooked for thermal management and thermal logic [43].While as a general rule the rectifications are indeed small [44], theoretical predictions [45] and experimental observation [46] indicate that counterexamples exist and that more efficient thermal diodes can be engineered.As for now, no fundamentally new rectification mechanisms have emerged and organic semiconductors have been dealt the same cards as their inorganic counterparts (e.g., temperature dependence of the thermal conductivity, nanostructuring, or molecular orientation, which plays a similar role to the degree of crystallinity in an inorganic solid).Yet, materials properties differ and it is in principle possible that significantly larger rectification degrees can be obtained.This is another area where most likely there is still much room for improvement.
It is difficult to say what does the future hold for thermal rectification.On the one hand it might seem we have run out of ideas and that graded systems or heterojunctions have already given everything they had to give (yes, rectification could be perhaps increased a little, but no game changer seems to be waiting around the corner).On the other hand, there are different approaches and/or material systems that received less attention and that can still potentially provide a breakthrough.A good example is in that of two-dimensional materials, which revolutionized many aspects of nanoscience and are now making their way in the field of thermal rectification [47].A remarkable case is the recent report of thermal rectification as high as 96% in MoSe 2 -WSe 2 lateral junctions [48].What is undeniable is that all these attempts at devising efficient thermal diodes played an important role in increasing our understanding of heat transfer and phonon manipulation.Indeed, this fundamental knowledge has far reaching consequences that go beyond the specific challenge of thermal rectification, and is going to be useful in other contexts, e.g.thermoelectricity or other phonon-based logic devices.

Figure 1 .
Figure 1.(a) Schematic of an oxide thermal rectifier; reprinted with permission from [12].Copyright 2009 American Institute of Physics.(b) Scanning electron microscopy image of the graphene thermal diode of [13]; Creative Commons CC-BY license.(c) Sketch of the thermal diode based on a holey Si membrane of [14]; Creative Commons CC-BY license.Thermal rectification is driven (a) by the temperature dependence of the thermal conductivity of materials on the two side of the heterojunctions and (b,c) by an engineered distribution of nanoscale defects.