Zhiliang
Pan‡
^{a},
Lin
Yang‡
^{a},
Yi
Tao
^{ab},
Yanglin
Zhu
^{c},
Ya-Qiong
Xu
^{de},
Zhiqiang
Mao
^{c} and
Deyu
Li
*^{a}
^{a}Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA. E-mail: deyu.li@vanderbilt.edu
^{b}School of Mechanical Engineering and Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, Southeast University, Nanjing, 210096, P. R. China
^{c}Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
^{d}Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235, USA
^{e}Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA

Received
29th June 2020
, Accepted 14th September 2020

First published on 14th September 2020

Understanding transport mechanisms of electrons and phonons, two major energy carriers in solids, are crucial for various engineering applications. It is widely believed that more free electrons in a material should correspond to a higher thermal conductivity; however, free electrons also scatter phonons to lower the lattice thermal conductivity. The net contribution of free electrons has been rarely studied because the effects of electron–phonon (e–ph) interactions on lattice thermal conductivity have not been well investigated. Here an experimental study of e–ph scattering in quasi-one-dimensional NbSe_{3} nanowires is reported, taking advantage of the spontaneous free carrier concentration change during charge density wave (CDW) phase transition. Contrary to the common wisdom that more free electrons would lead to a higher thermal conductivity, results show that during the depinning process of the condensed electrons, while the released electrons enhance the electronic thermal conductivity, the overall thermal conductivity decreases due to the escalated e–ph scattering. This study discloses how competing effects of free electrons result in unexpected trends and provides solid experimental data to dissect the contribution of e–ph scattering on lattice thermal conductivity. Lastly, an active thermal switch design is demonstrated based on tuning electron concentration through electric field.

While the positive contribution of free electrons to thermal transport can usually be evaluated using the Wiedemann–Franz law as a good approximation, the negative contribution through electron–phonon (e–ph) scattering has received relatively less attention and in fact, is often neglected in modeling lattice thermal conductivity of semiconductors. In contrast, extensive studies of other phonon scattering mechanisms, including phonon–phonon,^{2–5} phonon-boundary,^{6–9} and phonon-defects,^{10,11} have been carried out intensively in the past two decades in the efforts of understanding thermal transport through nanostructures and interfaces for better thermal management of electronic devices and construction of novel energy converters. This is partly due to the lack of experimental data clearly demonstrating the effects of e–ph scattering on phonon transport because in semiconductors, altering free electron concentrations through doping inevitably introduces defect scattering and it is difficult to distinguish the effects of e–ph scattering from those of defects.

Recently, using first principles calculations, it has been shown that e–ph scattering could have significant effects on the lattice thermal conductivity of heavily doped Si, metals, and SiGe alloys. For example, Liao et al.^{12} reported that e–ph scattering could lead to up to ∼45% reduction in the lattice thermal conductivity of heavily-doped silicon, which was overlooked in most previous studies. In addition, the influence of e–ph scattering on the lattice thermal conductivity of various metals has been examined by Wang et al.,^{13} as well as Jain and McGaughey;^{14} and both studies suggested significant impacts of e–ph scattering, especially in the lower temperature regime. The most remarkable result was reported by Xu et al.,^{15} which suggested a 60% reduction in lattice thermal conductivity of SiGe upon introducing the e–ph interactions.

Experimental attempts to clarify the effects of e–ph scattering on lattice thermal conductivity include some early measurements on heavily doped semiconductors;^{16} however, the complex phonon scattering process renders the analysis to be only qualitative. More recently, Liao et al.,^{17} measured the scattering rate between 250 GHz phonons and dynamically pumped electron–hole pairs in Si membrane to quantify the influence of e–ph scattering. In addition, we reported more direct data showing distinct signatures of e–ph scattering in the lattice thermal conductivity of NbSe_{3} nanowires as free electrons condensed due to charge density wave (CDW) phase transitions. In this paper, through reactivating the condensed electrons in NbSe_{3} nanowires in the temperature range between 41 and 59 K by electric field induced depinning, we further demonstrate that free electrons do not always contribute positively to thermal conductivity, which provides insights into the competing roles of free electrons in terms of thermal transport.

One interesting trait of CDW phase transitions is that the condensed electrons can be reactivated readily by an applied electric field.^{20,21} As CDW develops, the condensed free electrons are pinned to defects and surfaces, as a result of Fermi surface nesting.^{22} However, a rather small electric field can depin the condensed electrons to become free electrons again, which would contribute to both electrical and thermal transport. To examine the net effect of these electrons on thermal transport, we measure the thermal conductivity of NbSe_{3} nanowires without and with depinning the condensed electrons through modifying a well-established experimental scheme.^{23,24}

NbSe_{3} nanowires were prepared via liquid phase ultrasonic exfoliation from bulk crystals, which leads to small wires of irregular cross-sections with aligned molecular chains.^{25} The nanowires were then drop-casted onto a piece of polydimethylsiloxane (PDMS) and transferred to a measurement device and aligned between two side-by-side suspended membranes with integrated platinum resistance heaters/thermometers and electrodes, as shown in Fig. 1c. Electron beam induced deposition (EBID) was done using a dual-beam focused ion beam (FIB, FEI Helios NanoLab G3 CX) to locally deposit Pt/C mixture at the wire-electrode contacts to minimize the contact electrical and thermal resistance. The cross-section of the wire (inset of Fig. 1c) was obtained through direct observation using scanning electron microscopy (SEM) after it was cut open with the FIB following a procedure that we have reported before.^{25} We calculate the hydraulic diameter (D_{h}), 4 times the reciprocal of the surface-area-to-volume ratio (S/V), as the characteristic size of the nanowire for transport properties as it better represents the surface effects.^{26}

This depinning effect leads to a nanowire resistance change, and we define a resistance ratio as r = R_{0}/R_{d}, where R_{0} and R_{d} denote the electrical resistance without and with the depinning electric field, respectively. Given that the nanowire dimension remains the same during the depinning process, the resistance ratio can be written as r = σ_{d}/σ_{0} = (n_{d}eμ)/(n_{0}eμ). Here σ_{d} and σ_{0} are the electrical conductivity; n_{d} and n_{0} the corresponding carrier concentration under the depinned and pinned conditions. e is the elementary charge; and μ the electron mobility. Bardekn et al.,^{31} pointed out that the electron mobility is related to elastic properties of the material; and it has been shown that the measured Young's modulus of NbSe_{3} remains nearly the same (ΔE/E_{0} < 0.01%) with the application of the electric field.^{32} Moreover, Ong et al., directly measured the electron mobility of NbSe_{3} near the 2nd CDW phase transition temperature, which shows no carrier-concentration-dependence as CDW develops, confirming the negligible effects of the condensed electrons on charge carrier mobility.^{33} Therefore, it is reasonable to assume that μ remains the same at a given temperature without and with the depining electric field. In this case, r = n_{d}/n_{0}, that is, the resistance ratio can be regarded as the ratio of free carrier concentrations under depinned and pinned conditions.

Fig. 3a plots the extracted r for three different diameter wires, which indicates that as the nanowire size increases from 94 nm to 135 nm, r becomes larger. This means for thicker nanowires, a relatively larger portion of electrons can be released by the depinning electric field. Two main CDW pinining mechanisms need to be considered in NbSe_{3} nanowires: surface and impurity pinning, and for nanowires with smaller diameters, a relatively larger portion of electrons are pinned at surfaces due to the larger surface-area-to-volume ratio. Meanwhile, it has been suggested that electrons pinned by the surface are more difficult to be released compared to those pinned by impurities.^{34–40} As such, the relatively low electric field applied in our measurements (1.1–1.4 V cm^{−1}), which is not strong enough to fully active surface-pinned^{27} electrons renders a lower r for smaller hydraulic diameter wires as a result of their larger surface-area-to-volume ratio.

The enhancement of the free electron concentration upon depinning corresponds to an increase of the electronic thermal conductivity (κ_{e}), which is also shown in Fig. 3a. The most significant enhancement is observed for the largest wire with D_{h} = 135 nm, which demonstrates an r of ∼2.6, leading to a ∼160% increment in κ_{e}. This change contributes positively to the nanowire thermal conductivity.

The release of more free electrons, however, also leads to enhanced e–ph scattering, which poses resistance to phonon transport, and contributes negatively to the wire thermal conductivity. Indeed, the measured κ_{t} of the nanowires does not show enhancement in response to the increased κ_{e} upon depinning, but decreases as shown in Fig. 3b. The reduction in κ_{t} is more significant for thicker wires, corroborating with the higher r for larger wires. Depinning releases condensed free electrons, or it can be regarded as that CDW phase transition does not occur at 59 K. In this case, the distinct signatures in κ_{ph} and κ_{e} corresponding to the CDW disappear, as shown in Fig. 2a.

The lower κ_{t} indicates that the lattice contribution, κ_{ph}, decreases to a greater level than the enhancement in κ_{e}, as shown in Fig. 3c, which is due to the higher e–ph scattering rate as more free electrons are released. Based on the kinetic theory, the lattice thermal conductivity can be estimated as , where C, v, and l are the phonon heat capacity per unit volume, group velocity and mean free path (MFP), respectively. It has been reported that the heat capacity of NbSe_{3} only changes by about 1% during the CDW phase transition at 59 K.^{41} In addition, recent inelastic X-ray scattering studies show no sign of softening in phonon dispersion,^{42} consistent with the observation of marginal change in the Young's modulus across both CDW transitions.^{43} These results indicate that the phonon group velocity remains approximately the same through the CDW phase transitions. As such, the reduction in κ_{ph} must come from the change in phonon MFP, primarily due to the enhanced e–ph scattering as electrons are depinned.

When measuring thermal transport under depinning conditions, an electric field is applied to the nanowire. While the effect of Joule heating from this electric field has been considered in the derivation of the nanowire thermal conductivity (Section I in the ESI†), one might question whether the electric field could influence thermal transport in other ways. First, the applied electric field will accelerate electrons, which may potentially affect the electronic thermal conductivity. However, the drift velocity is estimated to be < 0.02% of the Fermi velocity in NbSe_{3} (Section III in the ESI†), which should not significantly affect the measured electronic thermal conductivity.

Another important factor when measuring the thermal conductance with a DC current through the nanowire sample is whether the Peltier effect alters the measured thermal conductivity. As discussed in the ESI,† we have carefully considered the Peltier effect and show that it can be eliminated in the calculation. This is because our approach measures the relative temperature increases on both heating and sensing membranes when we apply Joule heating to the heating membrane. As such, both Joule heating and Peltier effect from the nanowire sample simply present a background signal that can be canceled out in the thermal conductance calculation. Recently, Dong et al.,^{44} suggested an electric field dependent thermal conductivity in ferroelectric P(VDF-TrFE) nanofibers. The electric field they used (∼10^{5} V cm^{−1}) is 5 order of magnitude higher than the value (<1.5 V cm^{−1}) in our depinning measurements, and the effect of the small electric field on κ_{ph} in our study should be negligible.

To further show that the change in κ_{ph} is due to e–ph scattering with released electrons, we modeled κ_{ph} through combined first-principles calculations and phenomenological model. The Vienna ab initio simulation package (VASP)^{45} is used to derive the force constants with the same parameters as reported in our previous study.^{18} Then, the phonopy package^{46} is used to determine the frequency and velocity of each phonon by calculating phonon dispersion of NbSe_{3}. Under the framework of the Boltzmann transport equation, κ_{ph} along the molecular chain direction can be calculated by

(1) |

(2) |

The modelling results also show that comparing to other phonon scattering mechanisms, e–ph scattering indeed plays an important role in the relaxation time of phonons and it is the change in the e–ph scattering rate that leads to the reduced phonon MFP (Section IV in the ESI†). Moreover, the derived lattice thermal conductivity as a function of the depinning current for a 94 nm diameter wire at 47 K (Section V in the ESI†) indicates that the lattice thermal conductivity reduces as the depinning current escalates before it reaches 5.5 μA, beyond which no additional electrons are depinned. This observation suggests that the reduced lattice thermal conductivity is indeed due to the enhanced e–ph scattering as the free electron density increases upon depinning.

While the reduced κ_{ph} upon CDW depinning can be well-explained based on stronger e–ph scattering, in agreement with previous experiments,^{20,28} the overall reduction in κ_{t} is unexpected, which is different from the well-known plot illustrating the monotonically increasing trend of thermal conductivity as a function of the carrier concentration.^{1} As the ratio of the carrier concentration change is less than 2.6 in our experiment, to explore the effects in a wider range, as shown in Fig. 4, we calculated the κ_{ph} for the 135 nm nanowire at 45 K with r increasing to 8. We also estimated κ_{e} using the Wiedemann–Franz law, and the relative change in κ_{t} is plotted on the right axis in Fig. 4. Indeed, for r < 4.3, the increase of κ_{e} cannot compensate for the reduction in κ_{ph}, which results in a net reduction in κ_{t}, consistent with our experimental data. However as r further increases, κ_{ph} gradually saturates and the enhancement in κ_{e} becomes more significant, which consequently leads to the increasing trend in κ_{t}.

Fig. 4 Simulation results of thermal conductivity (κ_{ph}, κ_{e}, and κ_{t}) changes with carrier concentration ratio for D_{h} = 135 nm NbSe_{3} nanowire at T = 45 K. Dots in the plot are the experimental data showing good match with the simulation results. Data for Bi_{2}Te_{3} is from literature.^{1} |

It is important to note that the charge carrier concentration of the 135 nm NbSe_{3} nanowire is estimated to be 3.5 × 10^{19} cm^{−3} at 45 K. At this relatively low electron concentration regime, the effects of e–ph scattering on κ_{ph} is not saturated, as shown in Fig. 4. Moreover, owing to the strong e–ph scattering among other scattering mechanisms in NbSe_{3},^{18} the reduction of κ_{ph} caused by carrier concentration increase is larger than the κ_{e} increment, which explains a net negative change in κ_{t}. However, this could be different for other materials, for which even though κ_{ph} reduces as electron concentration increases, the relatively weak e–ph scattering would lead to a κ_{ph} reduction smaller than the κ_{e} enhancement. As shown in Fig. 4, we also plot the modeled relative κ_{t} change of Bi_{2}Te_{3} in the same carrier concentration regime,^{1} and different from our NbSe_{3} nanowires, it exhibits a monotonically increasing trend as the charge carrier concentration increases.

Notably, unlike extrinsic doping, the unique advantage of depinning induced carrier concentration change in NbSe_{3} nanowires is that it does not involve the effects from impurity scattering. Through careful comparison between the experimental data and modeling results, we exclusively show the lattice thermal conductivity reduction caused by e–ph scattering, and demonstrate a regime that is previously overlooked for total thermal conductivity change as carrier concentration increases.

The change in κ_{t} upon depinning also provides a potential mechanism of tuning the materials thermal conductivity. While for NbSe_{3}, the maximum tuning level is only ∼6%, similar mechanism of other CDW materials might provide a high on–off switch ratio. Control and modulation of material thermal properties is challenging but could impact a wide variety of applications and is being actively pursued by researchers.^{52} Different mechanisms have been explored to modulate thermal transport, such as asymmetric nanostructures,^{53–57} interface engineering,^{41–43,58} chemical composition modification,^{59} magnetic or electric fields^{60,61} or structure modulation.^{62,63} Cartoixa et al.,^{53} numerically studied thermal transport in telescopic Si nanowires. They observed a maximum thermal rectification of 50%. The modulation is achieved by the different temperature dependence for Si nanowires of different sizes. Ihlefeld et al.,^{61} claimed a maximum modulation in thermal conductivity of 11% with a giant electric field across nanoscale ferroelastic domain structure. Here we tested the modulation cycle using the depinning effect. A maximum modulation of ∼1.8 W (m K)^{−1} in κ_{ph} is achieved for the 135 nm sample, as shown in Fig. 5a. Repeated modulation in κ_{t} is confirmed in Fig. 5b as we switched the depinning voltage on and off. Collectively, our results demonstrate a new avenue to enable dynamic control of thermal transport in solid state systems through utilizing the CDW phase transition.

Fig. 5 Thermal switch behavior. (a) Measured lattice thermal conductivity variation during depinning test. (D_{h} = 135 nm). (b) Repeatability demonstration (D_{h} = 135 nm, T = 41 K). |

- G. Snyder and E. Toberer, Nat. Mater., 2008, 7, 105–114 CrossRef CAS.
- M. G. Holland, Phys. Rev., 1964, 134, A471–A480 CrossRef.
- L. Lindsay, D. A. Broido and N. Mingo, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 2–7 Search PubMed.
- J. S. Kang, M. Li, H. Wu, H. Nguyen and Y. Hu, Science, 2018, 361, 575–578 CrossRef CAS.
- T. Feng, L. Lindsay and X. Ruan, Phys. Rev. B, 2017, 96, 1–6 Search PubMed.
- D. Li, Y. Wu, P. Kim, L. Shi, P. Yang and A. Majumdar, Appl. Phys. Lett., 2003, 83, 2934–2936 CrossRef CAS.
- M. Asheghi, Y. K. Leung, S. S. Wong and K. E. Goodson, Appl. Phys. Lett., 1997, 71, 1798–1800 CrossRef CAS.
- W. Liu and M. Asheghi, Appl. Phys. Lett., 2004, 84, 3819–3821 CrossRef CAS.
- Y. S. Ju and K. E. Goodson, Appl. Phys. Lett., 1999, 74, 3005–3007 CrossRef CAS.
- C. T. Walker and R. O. Pohl, Phys. Rev., 1963, 131, 1433–1442 CrossRef CAS.
- J. H. Chen, W. G. Cullen, C. Jang, M. S. Fuhrer and E. D. Williams, Phys. Rev. Lett., 2009, 102, 1–4 Search PubMed.
- B. Liao, B. Qiu, J. Zhou, S. Huberman, K. Esfarjani and G. Chen, Phys. Rev. Lett., 2015, 114, 1–6 Search PubMed.
- Y. Wang, Z. Lu and X. Ruan, J. Appl. Phys., 2016, 119, 1–10 Search PubMed.
- A. Jain and A. J. H. McGaughey, Phys. Rev. B, 2016, 93, 1–5 CrossRef.
- Q. Xu, J. Zhou, T. H. Liu and G. Chen, Appl. Phys. Lett., 2019, 115, 1–4 Search PubMed.
- D. T. Morelli, J. P. Heremans, C. P. Beetz, W. S. Yoo and H. Matsunami, Appl. Phys. Lett., 1993, 63, 3143–3145 CrossRef CAS.
- B. Liao, A. A. Maznev, K. A. Nelson and G. Chen, Nat. Commun., 2016, 7, 1–7 Search PubMed.
- L. Yang, Y. Tao, J. Liu, C. Liu, Q. Zhang, M. Akter, Y. Zhao, T. T. Xu, Y. Xu, Z. Mao, Y. Chen and D. Li, Nano Lett., 2019, 19, 415–421 CrossRef CAS.
- H. Liu, C. Yang, B. Wei, L. Jin, A. Alatas, A. Said, S. Tongay, F. Yang, A. Javey, J. Hong and J. Wu, Adv. Sci., 2020, 7, 1–7 Search PubMed.
- R. M. Fleming, Phys. Rev. B: Condens. Matter Mater. Phys., 1979, 19, 3970–3980 CrossRef.
- P. Monçeau, N. P. Ong, A. M. Portis, A. Meerschaut and J. Rouxel, Phys. Rev. Lett., 1976, 37, 602–606 CrossRef.
- G. Grüner, Rev. Mod. Phys., 1988, 60, 1129–1181 CrossRef.
- L. Shi, D. Li, C. Yu, W. Jang, D. Kim, Z. Yao, P. Kim and A. Majumdar, J. Heat Transfer, 2003, 125, 881–888 CrossRef CAS.
- M. C. Wingert, Z. C. Y. Chen, S. Kwon, J. Xiang and R. Chen, Rev. Sci. Instrum., 2012, 83, 1–7 CrossRef.
- Q. Zhang, C. Liu, X. Liu, J. Liu, Z. Cui, Y. Zhang, L. Yang, Y. Zhao, T. T. Xu, Y. Chen, J. Wei, Z. Mao and D. Li, ACS Nano, 2018, 12, 2634–2642 CrossRef CAS.
- L. Yang, Y. Yang, Q. Zhang, Y. Zhang, Y. Jiang, Z. Guan, M. Gerboth, J. Yang, Y. Chen, D. G. Walker, T. T. Xu and D. Li, Nanoscale, 2016, 8, 17895–17901 RSC.
- S. Onishi, M. Jamei and A. Zettl, New J. Phys., 2017, 19, 023001 CrossRef.
- G. Grüner, A. Zawadowski and P. M. Chaikin, Phys. Rev. Lett., 1981, 46, 511–515 CrossRef.
- N. P. Ong and P. Monceau, Phys. Rev. B: Condens. Matter Mater. Phys., 1977, 16, 3443–3455 CrossRef CAS.
- R. M. Fleming, D. E. Moncton and D. B. McWhan, Phys. Rev. B: Condens. Matter Mater. Phys., 1978, 18, 5560–5563 CrossRef CAS.
- J. Bardeen and W. Shockley, Phys. Rev., 1950, 80, 72–80 CrossRef CAS.
- X. D. Xiang and J. W. Bril, Phys. Rev. B: Condens. Matter Mater. Phys., 1989, 39, 1290–1297 CrossRef CAS.
- N. P. Ong, Phys. Rev. B: Condens. Matter Mater. Phys., 1978, 18, 5272–5279 CrossRef CAS.
- J. C. Gill, EPL, 1990, 11, 175–180 CrossRef CAS.
- C. Brun, Z. Z. Wang and P. Monceau, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 1–11 CrossRef.
- S. Brazovskii, C. Brun, Z. Z. Wang and P. Monceau, Phys. Rev. Lett., 2012, 108, 1–4 CrossRef.
- P. J. Yetman and J. C. Gill, Solid State Commun., 1987, 62, 201–206 CrossRef CAS.
- B. M. Murphy, J. Stettner, M. Traving, M. Sprung, I. Grotkopp, M. Müller, C. S. Oglesby, M. Tolan and W. Press, Phys. B, 2003, 336, 103–108 CrossRef CAS.
- C. Brun, Z. Z. Wang, P. Monceau and S. Brazovskii, Phys. Rev. Lett., 2010, 104, 5–8 CrossRef.
- S. V. Zaitsev-Zotov, V. Y. Pokrovskii and P. Monceau, JETP Lett., 2001, 73, 25–27 CrossRef CAS.
- S. Tomić, K. Biljaković, D. Djurek, J. R. Cooper, P. Monceau and A. Meerschaut, Solid State Commun., 1981, 38, 109–112 CrossRef.
- H. Requardt, J. E. Lorenzo, P. Monceau, R. Currat and M. Krisch, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 1–4 CrossRef.
- J. W. Brill and N. P. Ong, Solid State Commun., 1978, 25, 1075–1078 CrossRef CAS.
- L. Dong, Q. Xi, J. Zhou, X. Xu and B. Li, arXiv, 2019.
- G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50 CrossRef CAS.
- A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 1–9 CrossRef.
- M. Asheghi, K. Kurabayashi, R. Kasnavi and K. E. Goodson, J. Appl. Phys., 2002, 91, 5079–5088 CrossRef CAS.
- J. P. Sorbier, H. Tortel, P. Monceau and F. Levy, Phys. Rev. Lett., 1996, 76, 676 CrossRef CAS.
- D. Reagor, S. Sridhar and G. Gruner, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 34, 2212 CrossRef CAS.
- J. L. Hodeau, M. Marezio, C. Roucau, R. Ayroles, A. Meerschaut, J. Rouxel and P. Monceau, J. Phys. C, 1978, 11, 4134 CrossRef.
- H. Requardt, J. E. Lorenzo, P. Monceau, R. Currat and M. Krisch, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 214303 CrossRef.
- S. Chu and A. Majumdar, Nature, 2012, 488, 294–303 CrossRef CAS.
- X. Cartoixà, L. Colombo and R. Rurali, Nano Lett., 2015, 15, 8255–8259 CrossRef.
- Y. Wang, A. Vallabhaneni, J. Hu, B. Qiu, Y. P. Chen and X. Ruan, Nano Lett., 2014, 14, 592–596 CrossRef CAS.
- C. W. Chang, D. Okawa, A. Majumdar and A. Zettl, Science, 2008, 314, 1121–1124 CrossRef.
- H. Wang, S. Hu, K. Takahashi, X. Zhang, H. Takamatsu and J. Chen, Nat. Commun., 2017, 8, 1–8 CrossRef.
- J. Lee, V. Varshney, A. K. Roy, J. B. Ferguson and B. L. Farmer, Nano Lett., 2012, 12, 3491–3496 CrossRef CAS.
- J. Yang, Y. Yang, S. W. Waltermire, X. Wu, H. Zhang, T. Gutu, Y. Jiang, Y. Chen, A. A. Zinn, R. Prasher, T. T. Xu and D. Li, Nat. Nanotechnol., 2012, 7, 91–95 CrossRef CAS.
- J. Cho, M. D. Losego, H. G. Zhang, H. Kim, J. Zuo, I. Petrov, D. G. Cahill and P. V. Braun, Nat. Commun., 2014, 5, 2–7 Search PubMed.
- N. H. Thomas, M. C. Sherrott, J. Broulliet, H. A. Atwater and A. J. Minnich, Nano Lett., 2019, 19, 3898–3904 CrossRef CAS.
- J. F. Ihlefeld, B. M. Foley, D. A. Scrymgeour, J. R. Michael, B. B. McKenzie, D. L. Medlin, M. Wallace, S. Trolier-Mckinstry and P. E. Hopkins, Nano Lett., 2015, 15, 1791–1795 CrossRef CAS.
- A. J. H. Mante and J. Volger, Physica, 1971, 52, 577–604 CrossRef CAS.
- J. Zhu, K. Hippalgaonkar, S. Shen, K. Wang, Y. Abate, S. Lee, J. Wu, X. Yin, A. Majumdar and X. Zhang, Nano Lett., 2014, 14, 4867–4872 CrossRef CAS.

## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp03484c |

‡ These authors contribute to the paper equally. |

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