## Abstract

We demonstrated nanoscale solid-state thermal rectification. High-thermal-conductivity carbon and boron nitride nanotubes were mass-loaded externally and inhomogeneously with heavy molecules. The resulting nanoscale system yields asymmetric axial thermal conductance with greater heat flow in the direction of decreasing mass density. The effect cannot be explained by ordinary perturbative wave theories, and instead we suggest that solitons may be responsible for the phenomenon. Considering the important role of electrical rectifiers (diodes) in electronics, thermal rectifiers have substantial implications for diverse thermal management problems, ranging from nanoscale calorimeters to microelectronic processors to macroscopic refrigerators and energy-saving buildings.

The invention of nonlinear solid-state devices, such as diodes and transistors, that control electrical conduction marked the emergence of modern electronics. It is apparent that counterpart devices for heat conduction, if they could be fabricated, would have deep implications for thermal circuits, thermal management, and the field of phononics in general. In recent years, some theoretical proposals for thermal rectifiers have been put forward (*1*–*4*), but these usually require complex coupling between individual atoms and substrates that are difficult to achieve experimentally. However, as noted by Peierls (*5*), heat transport in one dimension can be anomalous, and the breakdown of Fourier's law in one-dimensional (1D) systems may be coupled with extraordinary nonlinear thermal effects (*6*), including rectification.

Nanotubes are nearly 1D and thus are ideal materials for exploring thermal rectification effects. Previous studies have demonstrated that the thermal conductivity of 1D carbon nanotubes (CNTs) and boron nitride nanotubes (BNNTs) is high and dominated by phonons (*7*, *8*). For unmodified nanotubes with uniform mass distribution, the thermal conductance is symmetric (i.e., independent of the direction of axial heat flow). To investigate asymmetric thermal propagation in a suitable 1D inhomogeneous medium, we modified CNTs and BNNTs so that they assumed a non-uniform axial mass distribution (Fig. 1).

Pristine multiwalled BNNTs were first synthesized by means of an adaptation of a previously reported method (*9*), yielding samples with a typical outer diameter of ∼30 to 40 nm and a length of ∼10 μm. High-quality CNTs with diameters ranging from 10 to 33 nm were prepared by means of conventional arc methods (*10*). Individual tubes were placed on a custom-designed microscale thermal conductivity test fixture (*11*), with the use of a piezo-driven manipulator operated inside a scanning electron microscope (SEM). In brief, the fixture incorporates independently suspended SiN_{x} pads, with symmetrically fabricated Pt film resistors serving as either heaters or sensors. One end of the nanotube was bonded to the heater, the other end was bonded to the sensor, and the body of the nanotube was suspended in the vacuum in between.

Figure 2A shows an SEM image of a multiwalled CNT mounted to the test fixture and B the corresponding low-magnification transmission electron microscopic (TEM) image of the same CNT. For thermal conductance measurements, a known power *P* was supplied to the heater while resistance changes of the heater and sensor were used to determine the resulting temperature changes of the heater (Δ*T*_{h}) and sensor (Δ*T*_{s}) pads. The thermal conductance *K* of the nanotube was determined from Δ*T*_{h} and Δ*T*_{s} with the use of the relation (1) Because of unavoidable non-uniformities in the construction of the test fixture itself, the system with the attached pristine nanotube was first calibrated to establish residual asymmetry by switching the roles of the heater and sensor. All the measurements were done at room temperature.

Nanotubes were engineered in situ while mounted to the test fixture in the SEM. Trimethyl-cyclopentadienyl platinum (C_{9}H_{16}Pt) was deposited non-uniformly along the length of the nanotube in an attempt to achieve the non-uniform mass-loading geometry depicted in Fig. 1. Figure 2C shows a TEM image of the same CNT as in Fig. 2, A and B, after mass loading. The deposited C_{9}H_{16}Pt was found to be amorphous and tightly bound to the CNT. The sample mass near the right contact has clearly been increased (Fig. 2C). Indeed, the mass loading is even more effective than Fig. 2C might suggest: The molecular weight of C_{9}H_{16}Pt (∼319 g/mol) is much larger than that of (C–C)_{5} or (BN)_{5} (∼120 g/mol), and because the molecular volumes are similar, the mass density is correspondingly higher as well.

Depositing C_{9}H_{16}Pt on a nanotube has several possible effects on the sample thermal conductance. The most obvious is that the fused C_{9}H_{16}Pt forms an additional thermal conductance channel on parts of the sample. To test for the magnitude of this symmetrical enhancement, we suspended the fused C_{9}H_{16}Pt across the test fixture pads and measured its thermal conductance. At room temperature, the thermal conductivity of C_{9}H_{16}Pt was found to be temperature-independent and less than 1% of that of the nanotube. Hence, its direct thermal contribution can be neglected.

After mass loading, the thermal conductance of the nanotube was again tested in both directions. Thermal rectification of the nanotube is defined as (2) where *K*_{L→H} and *K*_{H→L} are the thermal conductances of the nanotube when heat flows from low-mass to high-mass ends or from high-mass to low-mass ends, respectively. For the CNT in Fig. 2, the measured thermal conductivity was 305 W/(m·K), and the rectification effect at room temperature was 2%.

Figure 3, A to C, shows three BNNTs that were also mass-engineered with C_{9}H_{16}Pt. The respective thermal rectifications were found to be 7, 4, and 3%. The arrows in Fig. 3 denote the direction of heat flow in which a higher thermal conductance was observed. All measurements showed that a higher thermal conductance was observed when heat flowed from the high-mass region (where more C_{9}H_{16}Pt was deposited) to the low-mass region. Because electrons do not contribute to the thermal transport for BNNTs, the observed rectification effects can be attributed to nonelectronic excitations.

Figure 3D shows, in detail, the relation of Δ*T*_{h} versus Δ*T*_{s} for the BNNT of Fig. 3A before and after the deposition of C_{9}H_{16}Pt. Equation 1 can be expressed as *Ps*/Δ*T*_{h}(1–*s*^{2}), which, for *s* ≡ Δ*T*_{s}/Δ*T*_{h} ≪ 1, reduces to *Ps*/Δ*T*_{h}. Thus, the slope of the Δ*T*_{h} versus Δ*T*_{s} curve is proportional to absolute thermal conductance. *K*_{L→H} and *K*_{H→L} of the pristine nanotube are symmetric. After mass loading, *K*_{H→L} and *K*_{L→H} differ by 7%, well above the measurement uncertainty (∼1%).

We now examine the origin of the observed thermal rectification. An asymmetric geometrical shape can, in principle, introduce asymmetric boundary scattering of phonons, whereby the thermal conductance can be reduced in one direction while it is increased in the other direction. In this scenario, thermal conductance is higher when heat flows from the narrow region to a wide region. Using the definition of Eq. 2, this always leads to a negative rectification coefficient, even for models where the boundary-scattering coefficient is mass-dependent. However, the thermal rectification observed in our experiments was always positive, and therefore any effect due to asymmetric shape was not dominant. Indeed, the *sp*^{2} bonds in nanotubes are qualitatively much stronger than the bonds between fused C_{9}H_{16}Pt molecules; thus, phonons should be mainly confined within the nanotubes, with relatively minor geometrical boundary-scattering effects.

A worthwhile analogy can also be made to photon wave propagation. The reflectivity *R* and the reflection coefficient *r* of a wave propagating across different media follow (3) where *k*_{i} and *k*_{t} are the wave numbers of incident waves and transmitted waves, respectively. The squaring of the expression in Eq. 3 ensures that *R* is independent of the direction of incident waves. Because phonons are quanta of waves, the above result demonstrates that thermal rectification is not expected for ordinary wave transport. Similarly, impedance mismatching due to contact resistance will not lead to thermal rectification. In addition, nonlinear perturbative effects such as umklapp processes only decrease the total thermal conductance of the nanotube, without rectification.

Theoretical work has suggested the presence of stable solitons in nanotubes (*12*, *13*). Solitons are nonperturbative solutions of nonlinear systems. They are localized particle-like entities that can collide with each other without changing shape. Within a general class of soliton models, asymmetry of heat flow for an inhomogeneous medium is a common feature (*14*–*16*). As an example, the reflection amplitude *r* for the Korteweg–de Veries equation is (*14*) (4) where *m*_{1} and *m*_{2} are the mass of atoms whose displacement constitutes the incident and transmitted waves, respectively. The most important result of Eq. 4 is the asymmetry with respect to *m*_{2}/*m*_{1}. The direction of the thermal rectification is positive (better heat flow from high- to low-mass regions), which is consistent with the engineered nanotube rectification results presented above. For a crude estimate of the magnitude of the rectification effect, Eq. 4 yields a rectification of ∼7% for a *m*_{2}/*m*_{1} of ∼5 [close to the molecular weight ratio of C_{9}H_{16}Pt to (C–C)_{5} or (BN)_{5}], consistent with the 2 to 7% rectification effects observed for mass-loaded nanotubes. Obviously, more-refined models of soliton transport in mass-loaded nanotubes, taking into account details of geometry, elastic constants, and mass distributions, are needed; but the key point is that linear or nonlinear perturbative systems do not lead to thermal rectification, whereas nonperturbative soliton models naturally do. The stronger ionic nature of BNNTs over CNTs also favors the nonlinearity. This may be the reason why BNNTs show a larger thermal rectification effect than CNTs.

With the availability of nonlinear thermal control, phonons should no longer be considered the unwanted by-products of electronics. Phonons, like electrons and photons, are information carriers and should be processed accordingly. Historically, semiconductor- or superconductor-based devices have been used to access thermal signals as soon as they are generated. Thermal rectifiers should make it possible to process thermal currents independently and convert them into electronic signals only when it is most efficient to do so.