Nanoscale thermal interface rectification in the quantum regime

Controlling heat flow is important for energy conversion and storage, thermal insulation, additive manufacturing, and electronics cooling. Functional thermal components that can control heat flow on demand will enable the next era of dynamic thermal control for energy saving and contribute to the blueprint of carbon neutrality. In analogue to modern electronics, the concepts of thermal diodes,1,21. M. Y. Wong, C. Y. Tso, T. C. Ho, and H. H. Lee, “ A review of state of the art thermal diodes and their potential applications,” Int. J. Heat Mass Transfer 164, 120607 (2021). https://doi.org/10.1016/j.ijheatmasstransfer.2020.1206072. C. Y. Tso and C. Y. H. Chao, “ Solid-state thermal diode with shape memory alloys,” Int. J. Heat Mass Transfer 93, 605–611 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2015.10.045 thermal transistors,3–53. D. Jou and L. Restuccia, “ Nonlinear heat transport in superlattices with mobile defects,” Entropy 21, 1200 (2019). https://doi.org/10.3390/e211212004. L. Wang and B. Li, “ Thermal logic gates: Computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007). https://doi.org/10.1103/PhysRevLett.99.1772085. B. Q. Guo, T. Liu, and C. S. Yu, “ Quantum thermal transistor based on qubit-qutrit coupling,” Phys. Rev. E 98, 022118 (2018). https://doi.org/10.1103/PhysRevE.98.022118 and thermal switches2,62. C. Y. Tso and C. Y. H. Chao, “ Solid-state thermal diode with shape memory alloys,” Int. J. Heat Mass Transfer 93, 605–611 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2015.10.0456. J. Ma, R. Zhu, D. Lu et al., “ Experimental and theoretical studies of a thermal switch based on shape-memory alloy cladded with graphene paper,” Energy Sources, Part A 42, 898–908 (2020). https://doi.org/10.1080/15567036.2019.1602202 have been proposed. Our understanding is still far from a comprehensive picture despite studies on the underlying mechanisms that drive these thermal functional components.The thermal rectification effect, which allows heat to flow preferentially in one direction, is one of the cornerstones to achieving tunable thermal materials and devices. Thermal rectification requires spatial asymmetry and a nonlinearity77. C. Dames, “ Solid-state thermal rectification with existing bulk materials,” J. Heat Transfer 131, 061301 (2009). https://doi.org/10.1115/1.3089552 in lattice or its thermal transport properties. Asymmetric nanostructures8–138. H. Ma and Z. Tian, “ Significantly high thermal rectification in an asymmetric polymer molecule driven by diffusive versus ballistic transport,” Nano Lett. 18, 43–48 (2018). https://doi.org/10.1021/acs.nanolett.7b028679. B. Li, L. Wang, and G. Casati, “ Thermal diode: Rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004). https://doi.org/10.1103/PhysRevLett.93.18430110. B. Li, J. Lan, and L. Wang, “ Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005). https://doi.org/10.1103/PhysRevLett.95.10430211. N. Yang, N. Li, L. Wang, and B. Li, “ Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007). https://doi.org/10.1103/PhysRevB.76.02030112. N. Yang, G. Zhang, and B. Li, “ Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009). https://doi.org/10.1063/1.318358713. C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “ Solid-state thermal rectifier,” Science 314, 1121 (2006). https://doi.org/10.1126/science.1132898 were found to give thermal rectification in a single material without heterojunctions by providing the non-separable thermal conductivity dependence on space and temperature. In bulk materials, thermal rectification typically happens at bi-material junctions7,14–177. C. Dames, “ Solid-state thermal rectification with existing bulk materials,” J. Heat Transfer 131, 061301 (2009). https://doi.org/10.1115/1.308955214. N. A. Roberts and D. G. Walker, “ A review of thermal rectification observations and models in solid materials,” Int. J. Therm. Sci. 50, 648–662 (2011). https://doi.org/10.1016/j.ijthermalsci.2010.12.00415. W. Kobayashi, Y. Teraoka, and I. Terasaki, “ An oxide thermal rectifier,” Appl. Phys. Lett. 95, 171905 (2009). https://doi.org/10.1063/1.325371216. H. Kang and F. Yang, “ Thermal rectification via heterojunctions of solid-state phase-change materials,” Phys. Rev. Appl. 10, 024034 (2018). https://doi.org/10.1103/PhysRevApplied.10.02403417. M. Hu, P. Keblinski, and B. Li, “ Thermal rectification at silicon-amorphous polyethylene interface,” Appl. Phys. Lett. 92, 211908 (2008). https://doi.org/10.1063/1.2937834 where two materials have different temperature-dependent thermal conductivities providing nonlinearity. The rectification of thermal interface conductance reported so far was all caused by the thermal strain/warping at the interface.1414. N. A. Roberts and D. G. Walker, “ A review of thermal rectification observations and models in solid materials,” Int. J. Therm. Sci. 50, 648–662 (2011). https://doi.org/10.1016/j.ijthermalsci.2010.12.004 Is there an intrinsic rectification of thermal interface conductance without mechanical deformation or contact area changes at the interfaces?To answer this question, we need to first isolate the contribution of the two bulk materials and focus on an interface region of a few nanometers thick. If bulk leads or heat reservoirs remain harmonic, their thermal resistance will stay zero no matter whether the heat flow direction is flipped or not. By adding the anharmonicity to the interface region, the thermal rectification effect should come from the interface region alone. Then we need to consider the temperature regime. In the classical regime above the Debye temperature, all the phonon modes are excited, and, hence, the phonon populations remain the same regardless of the heat flow direction. Even with anharmonicity, the inelastic scatterings cannot excite extra phonons. As a result, one cannot observe the thermal interface rectification in the classical regime,1818. T. Feng, Y. Zhong, J. Shi, and X. Ruan, “ Unexpected high inelastic phonon transport across solid-solid interface: Modal nonequilibrium molecular dynamics simulations and Landauer analysis,” Phys. Rev. B 99, 045301 (2019). https://doi.org/10.1103/PhysRevB.99.045301 and we need to look into the quantum regime where the phonon populations follow the Bose–Einstein distribution and change with the heat flow direction. Last but not the least, we need anharmonicity at the interface. When the interface is harmonic, there is no nonlinear effect to activate the blocked states, and only the overlapped phonon populations between the two materials allow phonon propagation. Correspondingly, we cannot observe the rectification either.Therefore, we need a way to calculate the heat flow across the nanometer-thick three-dimensional (3D) anharmonic interface with harmonic leads in the fully quantum regime. This was a big challenge due to the lack of a proper tool. Our recent development of the anharmonic atomistic Green's function (AGF) method for 3D interfaces1919. J. Dai and Z. Tian, “ Rigorous formalism of anharmonic atomistic Green's function for three-dimensional interfaces,” Phys. Rev. B 101, 41301 (2020). https://doi.org/10.1103/PhysRevB.101.041301 filled the gap and allowed us to look into the intrinsic thermal interface rectification in the quantum regime. In this work, we demonstrated the thermal interface rectification at a perfect 3D interface of 1.592 nm thick and uncovered a thermal rectification mechanism for solid-state interfaces in the quantum regime. The anharmonic scatterings across the interface act on the temperature-dependent phonon populations on both sides of the interface and generate the necessary nonlinearity to achieve thermal rectification. This is an intrinsic and universal phenomenon for nanoscale interfaces. We also presented and explained how the temperature and anharmonic strengths affect the thermal rectification ratios. These physical insights are essential to advancing our fundamental understanding of the nanoscale thermal rectification effects in the quantum regime and facilitating the thermal diode to create functional thermal devices and control heat flow on demand.The 3D anharmonic AGF formalism was developed in our previous work.1919. J. Dai and Z. Tian, “ Rigorous formalism of anharmonic atomistic Green's function for three-dimensional interfaces,” Phys. Rev. B 101, 41301 (2020). https://doi.org/10.1103/PhysRevB.101.041301 In general, the traditional AGF was formulated within the harmonic regime. The 3D anharmonic AGF overcomes this limitation and can treat interfacial thermal transport in the practical temperature ranges. On the other hand, the 3D anharmonic AGF is derived in the fully quantum-mechanical regime and, therefore, is applicable to low temperature ranges or the quantum regime. It allows us to investigate the heat flow of a 3D interface of a few nanometers thick.The system is composed of two semi-infinite leads and one central region. The system periodically repeats in the lateral direction to represent a 3D interface. The system of interest is depicted in Fig. 1. Retarded Green's function, Gr, describes the dynamics of phonons in the center region, taking the effect of the leads and the anharmonic phonon scattering in the central region into account through self-energies, Gr=ω2I−HC−ΣLr−ΣRr−ΣMr−1,(1)where HC is the harmonic force constant matrix of central region and ΣLRr≡HCLCRgLRrHLCRC. HCLCR and HLCRC are harmonic force constant matrices connecting the left or right lead to the center, and gLRr is the uncoupled retarded Green's function for the semi-infinite leads. The many-body self-energy ΣMr includes the anharmonic phonon scatterings of the central region into the Green's function Gr. Here, the traditional harmonic AGF is in the wave picture to give the phonon transmission, but the anharmonic scattering is in a particle picture. The kinetic equation connects scattering terms (Σ>/<) with correlation terms (G>/<). Therefore, in the central region, it is a combined wave and particle picture. The computation of ΣMr is complex and requires the third-order force constants for the central region. Calculations of first-principles lattice dynamics were performed on pure aluminum (Al) to extract the second- and third-order interatomic force constants (IFCs) from density functional theory (DFT). Calculations of the harmonic IFCs were performed using the QUANTUM ESPRESSO2020. P. Giannozzi, S. Baroni, N. Bonini et al., “ QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials,” J. Phys.: Condens. Matter 21, 395502 (2009). https://doi.org/10.1088/0953-8984/21/39/395502 and Phonopy,2121. A. Togo and I. Tanaka, “ First principles phonon calculations in materials science,” Scr. Mater. 108, 1–5 (2015). https://doi.org/10.1016/j.scriptamat.2015.07.021 and the third-order IFCs were calculated using thirdorder.py.2222. W. Li, J. Carrete, N. A. Katcho, and N. Mingo, “ ShengBTE: A solver of the Boltzmann transport equation for phonons,” Comput. Phys. Commun. 185, 1747–1758 (2014). https://doi.org/10.1016/j.cpc.2014.02.015 All calculations are based on generalized gradient approximation (GGA) with projector augmented wave (PAW) pseudopotentials. We used the lattice constant (a = 3.98 Å) and force constants of Al on both sides, the mass of Al on the left side, and a heavier mass on the right side, where the mass of heavy-Al is the mass of Al multiplied by mass ratio.As introduced in the previous work,1919. J. Dai and Z. Tian, “ Rigorous formalism of anharmonic atomistic Green's function for three-dimensional interfaces,” Phys. Rev. B 101, 41301 (2020). https://doi.org/10.1103/PhysRevB.101.041301 the heat current can be written in the following way: JL(R)ω=1N2As∑Q→Tr[Σ̃LR>(<)ω,Q→G̃<(>)ω,Q→−Σ̃LR<(>)ω,Q→G̃>(<)ω,Q→]ℏω2π,(2)where the L and R represent the heat current entering/leaving the anharmonic central region from the harmonic left or right lead, respectively, the G̃<(>) and Σ̃LR<(>) are the coupled non-equilibrium Green's functions and self-energies in reciprocal space, N is the number of transverse wavevectors and As is the cross-plane area of Al unit cell.The interfacial thermal conductance was calculated by integrating the heat current, σ=1ΔT∫0∞dω JLRω=1ΔT1N2As∫0∞dω∑Q→Tr[Σ̃LR>ω,Q→G̃<ω,Q→−Σ̃LR<ω,Q→G̃>ω,Q→]ℏω2π,(3)where TL and TR represent the equilibrium temperature of the left and right leads (reservoirs), respectively. The temperature difference ΔT is set to be ΔT=TL−TR. And we define an average temperature of the system as T=(TL+TR)/2, which will be the x-axis in Fig. 2(a).To clearly observe the thermal rectification effects, we set a large temperature difference between the two leads. To be consistent, the ΔT was set to be 100 K, and the lowest T is 50 K. We defined the positive direction to be from Al to heavy-Al side, as marked in Fig. 1. To quantify the difference in the conductance for the positive and negative directions, we defined the rectification ratio as η=(σpos−σneg)/σpos×100%.(4)The central region length was set to be four-cell long, and, hence, there are enough atoms to participate in the anharmonic scatterings to establish a stable trend. Meanwhile, the central region is small enough to be considered a nanoscale interface region. The semi-infinite leads are harmonic, and their thermal resistance is 0. Hence, the rectification effect we observe is from the variation of the thermal interface conductance itself, not from the variation of the bulk thermal conductivity. This is fundamentally different from previous interface rectifications that were observed due to the change of lead bulk thermal conductivities.There are two necessary conditions to generate such thermal rectification for a 3D interface of a few nanometers thick: (1) Quantum regime gives temperature-dependent phonon population. As heat flow direction is flipped, the phonon population changes. In the quantum regime, as temperature increases, more phonon modes are excited following the Bose–Einstein distribution. The phonon population, i.e., the number of phonons that are excited, can be defined as DoSω×fB.E.ω,TL(R) for the leads. Phonon density of states (DOS) is the number of states that can potentially be occupied at a certain frequency, and the Bose–Einstein distribution gives how many phonons occupy a state at a certain frequency and temperature. The phonon population of the left or right lead describes the number of phonon modes that are excited at a certain frequency and a given lead temperature. By switching the heat flow direction, the DoS remains the same for a certain lead, but the Bose–Einstein distribution changes with temperature, leading to varying phonon populations of the two leads. (2) Anharmonic interactions at the interface open up phonon channels to access the temperature-dependent phonon population. In the harmonic regime, there must be corresponding states on both sides to accommodate phonons traveling across the interface. As shown in Fig. 3, the phonon population of the Al lead above the maximum frequency of the heavy-Al lead will not contribute to the thermal interface transport for both positive and negative directions, or this part of the phonon population is blocked. Furthermore, below the maximum frequency of heavy-Al, the phonon populations of Al and heavy-Al are not fully overlapped, and the un-overlapped part is also prevented from interfacial transport in the harmonic picture. In other words, this part of phonon population is blocked, too. The phonon population from the un-overlapped region, or the blocked phonon population, will be partially activated by anharmonic scatterings and facilitate phonon transport across the interface, as shown in previous works.19,2319. J. Dai and Z. Tian, “ Rigorous formalism of anharmonic atomistic Green's function for three-dimensional interfaces,” Phys. Rev. B 101, 41301 (2020). https://doi.org/10.1103/PhysRevB.101.04130123. Y. Guo, M. Bescond, Z. Zhang et al., “ Quantum mechanical modeling of anharmonic phonon-phonon scattering in nanostructures,” Phys. Rev. B 102, 195412 (2020). https://doi.org/10.1103/PhysRevB.102.195412 As heat flow changes the direction, the number of anharmonicity-excited phonon populations differs. Moreover, the phonon scattering rates vary with the temperature non-linearly, following a 1/T dependence for Umklapp scattering, adding another layer of nonlinearity. Thus, the number of blocked populations that become possible to participate in interface transmission due to inelastic scatterings varies non-linearly with temperature and, thus, the heat flow direction. This essentially leads to asymmetric heat flow. Therefore, the anharmonicity, acting on the temperature-dependent phonon population, is crucial to the existence of quantum thermal rectification at the interface.The thermal rectification ratio, η, decreases monotonically with the temperature for different mass ratios, as shown in Fig. 2(a). As shown in Figs. 3(a)–3(d), when the average temperature increases for Al/3Al, the phonon populations of the lower temperature side increase more significantly, and the phonon population overlap increases. In other words, as the average temperature increases, the portion of the blocked population decreases, and the relative importance of anharmonicity in creating thermal rectification reduces. This explains why the thermal rectification ratio decreases with increasing temperature for Al/3Al, and the same is true for Al/1.5Al and Al/5Al [Fig. 2(a)]. If the average temperature is further reduced, there may be a peak value of the rectification ratio as the anharmonicity gets weaker.Clearly, Al/1.5Al, Al/3Al, and Al/5Al have different possible phonon populations, as shown in Figs. 3(e), 3(a), and 3(f). If we simply compare the size of the blocked population, Al/5Al has the largest, and Al/1.5Al has the smallest. However, the thermal rectification ratio does not follow this trend [Fig. 2(a)]. Although we are confident of the universal existence of the intrinsic thermal interface rectification, it is yet difficult to present a general theory for different material systems because the interplay between different modes and the anharmonic scattering rates is intricate. Here, we provide a possible limiting factor of such quantum rectification effect in the Al-based system where only acoustic modes are present. In general, we do not expect all the phonons emitted from the high-temperature lead to be accepted by the blocked part in the other lead. The smaller phonon population area essentially limits the final thermal conductance in a certain direction. For example, in the case of Al/3Al, at 100 and 150 K, the smaller phonon populations would be 3Al lead in the positive direction and Al lead in the negative direction, while at 200 and 250 K, the smaller phonon population changes to Al lead in the positive direction and stays as Al lead in the negative direction, as marked in Figs. 3(a)–3(d) with shades. We defined the ratio between the smaller area of the phonon population in the positive direction over the smaller area of the phonon population in the negative direction and then plotted the thermal rectification vs this ratio of pos/neg. There is a monotonically increasing trend, as shown in Fig. 2(b). In other words, the thermal rectification ratio in an Al-based interface is probably limited by the ratio between the smaller phonon populations in both directions. We choose Al as a model system because of its simple structure. Although electronic thermal conductivity is dominant in bulk Al, phonon–phonon scattering is usually dominant for metal/dielectric interfaces.2424. S. Sadasivam, N. Ye, J. Feser et al., “ Thermal transport across metal silicide-silicon interfaces: First-principles calculations and Green's function transport simulations,” Phys. Rev. B 95, 085310 (2017). https://doi.org/10.1103/PhysRevB.95.085310 Therefore, we expect the observation of the proposed rectification at metal/dielectric and dielectric/dielectric interfaces.The strength of anharmonicity, which determines how likely phonons can participate in an inelastic scattering, also governs the thermal rectification ratio. To show the role of anharmonicity in the rectification and quantify it correspondingly, we multiplied a parameter, χ, onto the third-order force constants in the anharmonic AGF calculations. Obviously, when χ equals 1, it reduces to the anharmonic cases we discussed above. At the very beginning, the thermal rectification ratio, η, increases with χ since there are more blocked states being activated by anharmonicity. Then, the thermal rectification ratio reaches a maximum, and the excitation process becomes saturated. Finally, the thermal rectification ratio decreases because strong anharmonicity χ introduces larger thermal resistance, suppressing the rectification (Fig. 4).

Because of the computational limitations, we did not calculate the thermal rectification ratio for thicker interface regions. However, we can analyze the possible trend for longer interface within our study in this work. On one hand, with a fixed ΔT, if we have a nanometer-thick interface, we can have a sharp temperature jump across the atomic junction at the interface, which will be weakened if the central region is too long, or the rectification will be too weak to observe. On the other hand, as the thickness increases, the thermal rectification reduces because of increasing thermal resistance. At a very large thickness, the thermal resistance will dominate, and the thermal rectification should approach 0. This essentially reduces to the case for bulk interfaces. In other words, the intrinsic thermal interface rectification can only be observed in a nanometer-thick interface.

In summary, we presented the observation of the intrinsic thermal rectification of nanometer-thick three-dimensional (3D) solid interfaces in the quantum regime, enabled by our recent development of the 3D anharmonic AGF. The combination of anharmonic phonon scattering and the temperature-dependent phonon population results in such a rectification effect. The rectification ratio is positively related to the ratio between the smaller phonon populations in both heat flow directions of Al-based interfaces. Additionally, the thermal rectification first increases with the strength of anharmonicity and then decreases due to increasing thermal resistance. The nanoscale thermal rectification effect of the 3D interface offers complete insight into thermal rectification mechanisms in a quantum regime and will help guide the future design of nanoscale thermal diodes.

This work was sponsored by the Department of the Navy, Office of Naval Research, under ONR Award Nos. N00014-18-1-2724 and N00014-22-1-2357. The calculations of the force constants was performed using the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation (Grant No. ACI-1053575).

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Jinghang Dai: Conceptualization (supporting); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Project administration (equal); Resources (equal); Software (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Zhiting Tian: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Supervision (lead); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

REFERENCES

1. M. Y. Wong, C. Y. Tso, T. C. Ho, and H. H. Lee, “ A review of state of the art thermal diodes and their potential applications,” Int. J. Heat Mass Transfer 164, 120607 (2021). https://doi.org/10.1016/j.ijheatmasstransfer.2020.120607, Google ScholarCrossref2. C. Y. Tso and C. Y. H. Chao, “ Solid-state thermal diode with shape memory alloys,” Int. J. Heat Mass Transfer 93, 605–611 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2015.10.045, Google ScholarCrossref3. D. Jou and L. Restuccia, “ Nonlinear heat transport in superlattices with mobile defects,” Entropy 21, 1200 (2019). https://doi.org/10.3390/e21121200, Google ScholarCrossref4. L. Wang and B. Li, “ Thermal logic gates: Computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007). https://doi.org/10.1103/PhysRevLett.99.177208, Google ScholarCrossref, ISI5. B. Q. Guo, T. Liu, and C. S. Yu, “ Quantum thermal transistor based on qubit-qutrit coupling,” Phys. Rev. E 98, 022118 (2018). https://doi.org/10.1103/PhysRevE.98.022118, Google ScholarCrossref6. J. Ma, R. Zhu, D. Lu et al., “ Experimental and theoretical studies of a thermal switch based on shape-memory alloy cladded with graphene paper,” Energy Sources, Part A 42, 898–908 (2020). https://doi.org/10.1080/15567036.2019.1602202, Google ScholarCrossref7. C. Dames, “ Solid-state thermal rectification with existing bulk materials,” J. Heat Transfer 131, 061301 (2009). https://doi.org/10.1115/1.3089552, Google ScholarCrossref8. H. Ma and Z. Tian, “ Significantly high thermal rectification in an asymmetric polymer molecule driven by diffusive versus ballistic transport,” Nano Lett. 18, 43–48 (2018). https://doi.org/10.1021/acs.nanolett.7b02867, Google ScholarCrossref9. B. Li, L. Wang, and G. Casati, “ Thermal diode: Rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004). https://doi.org/10.1103/PhysRevLett.93.184301, Google ScholarCrossref, ISI10. B. Li, J. Lan, and L. Wang, “ Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005). https://doi.org/10.1103/PhysRevLett.95.104302, Google ScholarCrossref, ISI11. N. Yang, N. Li, L. Wang, and B. Li, “ Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007). https://doi.org/10.1103/PhysRevB.76.020301, Google ScholarCrossref12. N. Yang, G. Zhang, and B. Li, “ Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009). https://doi.org/10.1063/1.3183587, Google ScholarScitation, ISI13. C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “ Solid-state thermal rectifier,” Science 314, 1121 (2006). https://doi.org/10.1126/science.1132898, Google ScholarCrossref, ISI14. N. A. Roberts and D. G. Walker, “ A review of thermal rectification observations and models in solid materials,” Int. J. Therm. Sci. 50, 648–662 (2011). https://doi.org/10.1016/j.ijthermalsci.2010.12.004, Google ScholarCrossref15. W. Kobayashi, Y. Teraoka, and I. Terasaki, “ An oxide thermal rectifier,” Appl. Phys. Lett. 95, 171905 (2009). https://doi.org/10.1063/1.3253712, Google ScholarScitation,

留言 (0)

沒有登入
gif