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Transporte de calor en nanoestructuras


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Transporte de calor en nanoestructuras

  1. 1. Journal of Nanoparticle Research 2: 199–204, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. Brief communication Particularities of heat conduction in nanostructures Gang Chen Nanoscale Heat Transfer and Thermoelectrics Laboratory, Department of Mechanical and Aerospace Engineering, University of California at Los Angeles, Los Angeles, CA 90095-1597, USA (Tel.: 310-206-7044; Fax: 310-206-2302; E-mail: Received 29 December 1999; accepted in revised form 4 January 2000 Key words: nanoscale heat transfer, nanoparticles, nanowires, phonons, superlattices, thermal conductivity, thin films, microscale effects Abstract Heat conduction in nanostructures differs significantly from that in macrostructures because the characteristic length scales associated with heat carriers, i.e., the mean free path and the wavelength, are comparable to the character- istic length of nanostructures. In this communication, particularities associated with phonon heat conduction in nanostructures, the applicability of the Fourier law, and the implications of nanoscale heat transfer effects on nan- otechnology are discussed. Introduction Heat conduction is typically treated as a diffusion pro- cess that is governed by the Fourier law, q = −k T (1) where q is the local heat flux and T the temperature gradient, k is the thermal conductivity of the material that is temperature dependent. In nanostructures such as nanoparticles, nanowires, and nanoscale thin films, the applicability of the Fourier law is highly ques- tionable because the length scales associated with the energy carriers become comparable to or larger than the characteristic length of the nanostructures (Tien & Chen, 1994). Discussion in this communication will emphasize phonons that are major heat carriers in dielectrics and semiconductors, although an analogy to electrons is straightforward. According to quantum mechanics principles, energy carriers in solids such as phonons and electrons have both wave and particle characteristics. The length scales associated with the wave nature of phonons are their wavelength and the coherence, and the length scale with the particle nature is the mean free path length. The phonon mean free path is the average distance that a phonon particle travels before each collision. The internal collision may be caused through scattering by phonons, impurities, and defects. Boundary scattering is often grouped together with the internal scattering processes, but one should con- sider that boundary scattering is a surface process while internal scattering is a volumetric process. The phonon scattering is highly frequency dependent, which makes the estimation of the mean free path difficult and prone to uncertainties. For example, although a simple kinetic theory leads to an estimation of phonon mean free path in silicon at room temperature to be ∼40 nm, more detailed studies show that the mean free path of these phonons that carry heat is ∼250–300 nm (Ju & Goodson, 1999). The phonon wavelength in a crystal ranges from the inter-atomic spaces to the crystal size. The spectral distribution of the phonon energy density is wide as determined by the density of states and the Bose–Einstein statistics. It is similar to the spectrum of the thermal radiation from a blackbody truncated in the short wavelength side since the shortest phonon wavelength in a crystal is twice the lattice constant.
  2. 2. 200 The central wavelength is an average of all the wave- lengths weighted against the phonon energy density, and is representative of the wavelengths of most pop- ulated phonons (Chen, 1997). The coherence length is inversely proportional to the effective bandwidth of phonons. It is an indicator whether the wave nature is importantornot.Ifthecharacteristiclengthofthestruc- ture is much larger than the coherence length, phonons can be treated as particles. In Figure 1, the estimated coherence length, the cen- tral wavelength, and the mean free path of phonons in bulk GaAs as a function of temperature are shown. As the characteristic lengths of nanostructures, i.e., the diameter of nanoparticles and nanowires, and the thick- ness of thin films, become comparable to the mean free path or even the coherence length, the diffusion approximation underlying the Fourier law is no longer valid. Size and interface effects must be taken into consideration. Depending on the relative magnitude of the characteristic length of the nanostructures and the characteristic lengths of the energy carriers, one may treat the energy carriers as particles or as waves. The latter requires consideration of the phase informa- tion of the energy carriers. Different transport regimes and the governing principles for photons, electrons, and phonons are summarized in Table 1. The basic Figure 1. Characteristic length scales of phonons in bulk gallium arsenide. principles governing the transport of energy carriers are very similar and analogy can often be made to under- stand the particularities of phonon transport by exam- ining the photon and electron transport. Applicability of the Fourier law In general, the Fourier theory applies only to the diffu- sion regime as indicated in Table 1. The failure of the Fourier law in nanostructures may occur to all variables in the constitutive relation, namely the temperature and its gradient, and the thermal conductivity. First, temperature is an equilibrium concept. Although heat transfer is intrinsically a nonequilib- rium process, the deviation from equilibrium is usually small and a local thermal equilibrium is assumed. The establishment of equilibrium requires enough scatter- ing among phonons to thermalize their energy. When the structure is small compared to the mean free path, the equilibrium cannot be established and temperature cannot be defined in the conventional sense. A simple example is illustrated in Figure 2(a), which shows a thin film sandwiched between two thermal reservoirs. Consider now the limit that no internal scattering exists inside a thin film. There will be two groups of phonons
  3. 3. 201 Table 1. Transport regimes for three common energy carriers: Device characteristic length (such as thickness) is h; O denotes the order-of-magnitude of a length scale; the listed MFP and coherence lengths are typical values but these values are strongly material and temperature dependent Length scale Regimes Photon Electron Phonon Wave regimes Coherence length, L photon: 1 µm–1 km electron: 1–100 nm phonon: ∼ 1–10 nm h < O(L) wave regime Maxwell EM theory Quantum mechanics Quantum mechanics h ∼ O(L) Optical coherence Electron coherence Phonon coherence partial coherence regime theory theory theory Particle regimes Mean free path, photon: 10 nm–1 km electron: ∼100 nm phonon: ∼100 nm h < O( ) ballistic regime Ray tracing Ballistic transport Ray tracing h ∼ O( ) Radiative transfer Boltzmann transport Boltzmann transport quasi-diffusive regime equation equation equation h > O( ) Diffusion Ohm’s and Fourier’s law diffusion regime approximation Fourier’s laws Figure 2. Illustration of nonequilibrium nature of ballistic phonon transport across a thin film in the absence of internal scat- tering, (a) counter-propagating phonons do not interact with each other due to the absence of scattering and (b) equivalent local tem- perature if those phonons interact and were thermalized locally (temperature independent specific heat assumed). propagating inside the film: one is at temperatureT1 and the other at temperature T2. The local energy density spectrum deviates significantly from the equilibrium distribution. If these phonons were to interact and become thermalized, an equivalent local temperature can be defined. The temperature distribution accord- ing to such a definition inside the film is shown in Figure 2(b). At the boundaries, temperature jump hap- pens due to the nonequilibrium nature of phonons and the artificial definition of local temperature. Clearly, in the above example, the concept of temper- ature gradient is useless. The proportionality between the local heat flux and local temperature gradient as seen in the Fourier law is indicative of diffusion pro- cesses, while the above-example is a purely ballistic transport process. In such a process, the origin of phonons, which determines their spectral energy den- sity, is most important, while in the diffusive trans- port regime, the origin of the heat carrier is of no significance. For the intermediate cases between the totally ballistic transport and totally diffusive transport, the heat flux depends on the trajectory of phonons. Mathematically, it is a path integral, as dictated by the Boltzmann transport equation. The failure of the Fourier law is reflected in the ther- mal conductivity of nanostructures. Strictly speaking, unless a local temperature gradient can be established such that the Fourier law is applicable, the concept of thermal conductivity is meaningless. In reality, how- ever, it is still a convenient parameter to measure and has been used to gauge the size effects in nanostruc- tures. The thermal conductivity is no longer an intrinsic material property but a structure property, and it may depend on how the heat source is applied. Generally, because the interface imposes additional resistance to the heat flow compared to bulk materials, the effective thermal conductivity of nanostructures is smaller than those of their corresponding bulk materials. Clearly, the Fourier theory is not valid if one is interested in the local temperature and their gradients. It may, however, be applied to certain heat conduc- tion configurations in nanostructured materials with a modified structural-dependent thermal conductivity, as long as the domain of interests is much larger than the phonon mean free path inside the nanostructure.
  4. 4. 202 Examples are heat conduction along long nanowires or thin film planes, in which local temperature gradient can be established in the heat flux direction, and heat conduction in macroscopic structures made of nano- materials such as nanoparticles. Heat transfer in nanostructures Studies under the name ‘microscale heat transfer’ have surged in the heat transfer community over the last decade (Tien, 1997), although the length scale at which deviation from the classical Fourier law in solids typically occurs at nanoscale. Some of the research is illustrated below. Heat transfer in thin films has received most attention. Experimental and theoretical studies have been carried out to investigate thermal conductivity of various thin films used in microelectronics, photonics, and thermoelectrics (Goodson & Ju, 1999; Chen, 2000). In amorphous materials with short phonon mean free path, the size effect is typically less important than that of those caused by microstructural variations. The most dramatic size effects are observed in differ- ent crystalline superlattices with nanometer periodicity (Yao, 1987), which show orders of magnitude reduc- tion in thermal conductivity compared to the prediction of the Fourier diffusion theory. Studies of thermal transport in nanowires are few at this stage but will become increasingly important as new technologies such as carbon nanotubes are being implemented. A few experimental studies have been reported on the electron–phonon interaction in and the thermal conductance of nanowires at low temperatures (Potts et al., 1991; Seyler & Wybourne, 1992; Tighe et al., 1997), including the observation of quantized phonon conductance (Roukes, 1999). Thermal con- ductivity of carbon nanotube bundles have also been reported (Hone et al., 1999; Yi et al., 1999). Boltzmann transport equation and molecular dynamics simulation of thermal conductivity of nanowires was carried out, but the modeling results await experimental confirma- tion (Walkauskas et al., 1999; Volz & Chen, 1999). Heat conduction away from nanoparticles and between interconnected nanoparticles is representative of many applications such as nanoparticles or nano- devices embedded in a host medium, thermal insula- tion, and opal structures for light manipulation. The thermal pathway from a heated nanoparticle to the ambient depends on the coupling of the nanoparticle phonon spectrum with the surrounding medium, and depends on the ratio of the nanoparticle diameter to the phonon mean free path in the surrounding medium. The former determines the interface thermal resistance and the later resembles the rarefied gas conduction external to an object. Due to the phonon rarefaction, the effective thermal conductivity that the nanoparticle ‘feels’ about the surrounding medium is significantly reduced (Chen, 1996), as shown in Figure 3. Intercon- nected nanoparticles may form regular porous struc- tures such as opals and random porous structures such as nanoporous Si. There have been several studies of heat transfer in these nanostructures (Arutyunyan et al., 1997; Gesele et al., 1997; Chung & Kaviany, 2000). Technology implications Since energy dissipation is a fundamental process, the particularities of heat transfer in nanostructures have important implications to the rapidly developing nan- otechnology. Size effects on phonons often lead to reduced thermal pathways from the hot region to the cold region. This can be detrimental to microelectronic and photonic devices and the lessons learnt from the theoretical and experimental studies could be used to optimize the design of those devices. On the other hand, many other applications require low thermal conductivity materials, such as in thermoelectrics, thermal protective coatings, and microelectromechani- cal systems. Various nanofabrication and characteriza- tion techniques, data storage devices, also call for the understanding of nanoscale heat transfer phenomena to control the fabrication, writing, and interpretation of experimental data. Taking thermoelectrics as an example, it is well known that the Peltier effect and the Seebeck effect of electrons and holes can be used for refrigeration and power generation due to the energy associated with the motion of these charged carriers (Goldsmid, 1964). The solid-state Peltier coolers are commercially avail- able but their applications have been limited by their low efficiency. A good thermoelectric material requires several attributes such as a large Seebeck coefficient for maximum cooling capability, a high electrical con- ductivity to reduce the Joule heating, and a low thermal conductivity to minimize thermal leakage from the hot to the cold side. Satisfying those requirements have been proven to be a difficult task (DiSalvo, 1999) in bulk materials. Nanostructures offer new pathways by manipulating electron and phonon transport through size effects (Dresselhaus et al., 1999; Chen, 2000).
  5. 5. 203 Figure 3. Effective thermal conductivity as a function of the particle size parameter (particle radius normalized by the phonon mean free path in the surrounding medium) for heat conduction in the vicinity of a nanoparticle embedded in a host material. As the particle size becomes smaller than the phonon mean free path in the host material, the effective thermal conductivity that the particle feels about the surrounding is reduced due to rarefied gas effect (after Chen, 1996). Another interesting possibility is to utilize size effects between nanoscale heterogeneous materials for the design of low thermal conductivity protective coatings. It has been shown that phonon reflection at interfaces can drastically reduce thermal conductivity of superlattices and we can infer from those studies that coatings made of heterogeneous nanoparticles may have superior thermal insulating characteristics and thermal stability, compared to single phase nanoparti- cle coatings. Acknowledgments This work is supported in part by a NSF Young Investigator Award and a DOD MURI grant on low- dimensional thermoelectrics. References Arutyunyan L.I., V.N. Bogomolov, N.F. Kartenko, D.A. Kurdyukov, V.V. Popov, A.V. Prokof’ev, I.A. Smirnov & N.V. Sharenkova, 1997. Thermal conductivity of a new type of regular-structure nanocomposites: PbSe in opal pores. Phys. Solid State 39, 510–514. Chen G., 1996. Nonlocal and nonequilibrium heat conduction in the vicinity of nanoparticles. J. Heat Transf. 118, 539–545. Chen G., 1997. Size and interface effects on thermal conductivity of superlattices and periodic thin-film structures. J. Heat Transf. 119, 220–229. Chen G., 2000. Phonon heat conduction in superlattices and nanostructures. In: Semimetals and Semiconductors (in press). Chung J.D. & M. Kaviany, 2000. Effects of phonon pore scatter- ing and pore randomness on effective conductivity of porous silicon. Int. J. Heat and Mass Transf. 42, 521–538. DiSalvo F.J., 1999. Thermoelectric cooling and power generation. Science 285, 703–706. Dresselhaus M.S., G. Dresselhaus, X. Sun, Z. Zhang, S.B. Cronin, T. Koga, J.Y. Ying & G. Chen, 1999. The promise of low-dimensional thermoelectric materials. Microscale Thermophysical Eng. 3, 89–100. Gesele G., J. Linsmeier, V. Drach, J. Fricke & R. Arens-Fischer, 1997. Temperature-dependent thermal conductivity of porous silicon. J. Phys. D: Appl. Phys. 30, 2911–2916. Goldsmid H.J., 1964. Thermoelectric Refrigeration. Plenum Press, New York. Goodson K.E. & Y.S. Ju, 1999. Heat conduction in novel elec- tronic films. Ann. Rev. Mat. 29, 261–293. Hone J., M. Whitney, C. Piskoti & A. Zettl, 1999. Thermal con- ductivity of single-walled carbon nanotubes. Phys. Rev. B 59, R2514–R2516. Ju Y.S. & K.E. Goodson, 1999. Phonon scattering in silicon films with thickness of order 100 nm. Appl. Phys. Lett. 74, 3005–3007. Potts A., M.J. Kelly, D.G. Hasko, C.G. Smith, D.B. Hasko, J.R.A. Cleaver, H. Ahmed, D.C. Peacock, D.A. Ritchie, J.E.F. Frost & G.A.C. Jones, 1991. Thermal transport in free- standing semiconductor fine wires. Superlattices & Microstruc- tures 9, 315–318.
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