Advances in Thermal Conductivity

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Annu. Rev. Mater. Res. 2012.42:179-209. Downloaded from www.annualreviews.org by Colorado School of Mines on 10/18/12. For personal use only.

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Advances in Thermal Conductivity Eric S. Toberer,1 Lauryn L. Baranowski,1 and Chris Dames2 1 Physics Department, Colorado School of Mines, Golden, Colorado 80401; email: [email protected] 2 Department of Mechanical Engineering, University of California, Berkeley, California 94720; email: [email protected]

Annu. Rev. Mater. Res. 2012. 42:179–209

Keywords

The Annual Review of Materials Research is online at matsci.annualreviews.org

nanostructures, nanowires, transport, thermoelectrics, thermal barrier coatings, thin films

This article’s doi: 10.1146/annurev-matsci-070511-155040 c 2012 by Annual Reviews. Copyright All rights reserved 1531-7331/12/0804-0179$20.00

Abstract This review discusses recent advances in materials engineering to control thermal conductivity. We begin by presenting theories of heat conduction for general material classes, focusing on common approximations and trends. Next, we discuss characterization techniques for measuring thermal conductivity and the underlying transport properties. Advanced materials at the frontiers of thermal transport, such as rattlers, complex unit cells, nanowires, and nanocomposites, are treated in depth using experimental data and theoretical predictions. The review closes by highlighting several promising areas for further development.

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1. INTRODUCTION


Fundamental to all applications are the release and absorption of energy. Managing the ensuing heat flow involves selecting a material that possesses an appropriate thermal conductivity, either low or high. Obtaining such thermal conductivity in the face of other design requirements can be exceedingly difficult and often limits device performance. In the past decade, we have seen major advances in materials with tailored thermal conductivity and enhanced understanding of the underlying transport behavior in such materials. These advances have directly translated to improvements in energy and electronic applications. Many electronic applications require high thermal conductivity materials to facilitate heat extraction. Excessive heating can lead to poor device performance and decreased lifetimes. Heat generation in electronics can arise from a variety of sources, including Joule heating, solar flux, and even exothermic reactions. Localized Joule heating is prevalent in high power density electronics such as integrated circuits, supercapacitors, LEDs, and lasers. As these devices move to the nanoscale, both the power density rises and the ability to extract heat decreases. A similar challenge is faced by nanostructured and concentrated solar cells, in which increased temperature reduces device efficiency due to dark current. A final example in which heat extraction is critical for performance is that of batteries, where Joule heating and exothermic reactions can lead to undesired chemical reactions and device failure. At the opposite extreme are materials that must minimize heat transfer. Thermoelectric devices require low thermal conductivity materials to reduce the parasitic transfer of heat across each leg. At the same time, thermoelectric materials must be good electrical conductors, creating a design conflict. Thermal insulation is designed to prevent heat from reaching critical components. For example, the thermal barrier coating (TBC) of turbine blades enables the turbine surface temperature to be higher than the melting point of the structural blade material. This review highlights materials at the extremes (low and high) of thermal conductivity and the modeling and characterization techniques that have been critical to achieving these recent advances. We begin with an overview of classical heat conduction phenomena. From this basis, we investigate how nanostructuring affects phonon transport. In bulk materials, we focus on structurally complex materials for thermoelectrics and thermal barrier coatings. In both applications, we see the rise of rattling compounds that alter the acoustic branch of the phonon dispersion. We finish with a discussion of current opportunities to design materials with tailored thermal conductivity.

2. CONDUCTION HEAT TRANSFER 2.1. Core Concepts For a classical, isotropic material, the defining equation for heat conduction is Fourier’s law, q = −k∇T ,

1.

where q is the heat flux vector, ∇T is the temperature gradient vector, and k is the scalar thermal conductivity, expressed in the International System of Units (SI) as W m−1 K−1 . This isotropic scalar form is applicable to cubic crystal structures, polycrystalline materials, and truly anisotropic atomic arrangements such as amorphous solids. Often a single type of energy carrier dominates k; for multiple carriers, the conductivities are added as follows: k = kL + ke + kr + kg + . . . .

2.

to account for the lattice (kL ), electrons (ke ), photon radiation (kr ), gas molecules (kg ), etc. Prominent systems in which multiple carriers are important include thermoelectric materials (kL and ke ), thermal barrier coatings at high temperature (kL and kr ), and aerogels (kL , kr , and kg ). 180

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a

b 105


Silicon Si

103

T0

102 101

NaCl T3

Si-Ge

Cu alloy

100

Natural

103 102

Nanowire

101

Doped

Si-Ge

PMMA Air

100

10

100

1,000

kMin

Amorphous

T1 1

Nanograined

a-SiO2

10–1 10–2

Isotopically purified

104

k L (W m–1 K–1)

Thermal conductivity (W m–1 K–1)

Cu 104 T1

10 –1

Temperature (K)

1

10

100

1,000

Temperature (K)

Figure 1 (a) Thermal conductivity survey with representatives of various materials classes. The temperature power laws indicated for certain materials are representative of well-known theoretical results. The Cu alloy is Cu0.84 Mn0.12 Ni0.04 . (b) Thermal conductivity of various types of bulk and nanostructured silicon. Sources: References 8, 9, 20–23, 32, 59, 86, 182, 183.

Figure 1 shows a sampling of the great diversity of k values and temperature trends for various material classes. These trends can be understood using kinetic theory, by which a collection of particles with volumetric heat capacity [C ( J m−3 K−1 )] and group velocity [v (ms−1 )] undergo random walks with mean free paths (MFPs) [l (m)] in a temperature gradient. For an isotropic three-dimensional material, 1 3. k = Cvl, 3 where multiple scattering mechanisms i may be combined using Matthiessen’s rule (1, 2): l i−1 . 4. l −1 = In general C, v, and l each depend on polarization and frequency, requiring a summation and integration, respectively, in Equation 3. It is interesting to recognize that C, which is primarily of interest for transient processes (e.g., phase transitions, heating, and cooling), is also deeply coupled with k, even at steady state. The key to understanding k for any material is to understand its C, v, and l. In the remainder of this section, we briefly outline the conventional wisdom for several material classes, emphasizing behavior around room temperature. This discussion of C, v, and l will provide a framework to understand the magnitude and T dependence of experimental k data, as well as rationalize the effects on k of changes in material composition and microstructure. Further details about the materials physics of k may be found in References 3–8, and Reference 9 is a helpful compilation of the experimental k(T ) for numerous bulk materials.

2.2. Electrons in Metals and Semiconductors The thermal conductivity of metals is dominated by free electrons, whereas in semimetals and heavily-doped semiconductors, both kL and ke are important. kL will be discussed at length in the next section. Focusing on ke , for metals and degenerate semiconductors with parabolic bands near www.annualreviews.org • Advances in Thermal Conductivity

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the Fermi energy, the electrons (or holes, as appropriate) can be understood as a free electron gas. To an excellent approximation, the electrons that contribute to k travel at a constant Fermi velocity (v F ) and obey C = bT, where b is a constant, and v F and b are widely tabulated (1). For pure, high-quality metals around room temperature and above, the dominant scatterer of electrons is phonons, with an electron MFP given roughly by l∼

50T melt 1/3

ηPUC T

,

5.


where ηPUC is the number density of primitive unit cells and T melt is the melting point (2). The T −1 dependence of l reflects the phonon population above the Debye temperature (θ D ). The C of electrons goes as T 1 ; thus, in this regime ke is expected to be independent of T. Equation 5 is a useful guideline for T as low as θ D /2. For example, θ D for Cu is 315 K (1), and as shown in Figure 1, copper’s k(T ) is roughly constant above 150 K. Because free electrons carry both heat and charge, there is an important relationship between the electrical conductivity (σ ) and the electronic contribution to the thermal conductivity (ke ): ke = L. σT

6.

Equation 6 is known as the Wiedemann-Franz law, where L is the Lorenz number. For a freeelectron gas with parabolic dispersion near the Fermi energy (such as most metals and degenerate semiconductors) with purely elastic scattering, L = π 2 k2B /3e 2 = 2.44 × 10−8 WK−2 . This elastic approximation is very good for scattering of electrons by impurities and point defects at all temperatures; it is also a good approximation for scattering of electrons by phonons at temperatures near and above θ D because the energy exchanged with a phonon is sufficiently small compared to kB T, where kB is Boltzmann’s constant (1, 2). For example, in pure metals like Cu and Pt this free-electron L is within 10% of the experimental value for T from 1,000 K down to ∼250 K. However, the experimental L decreases by a factor of two or more in these metals at lower cryogenic temperatures, before finally returning back to the free-electron value as T → 0 K. For semiconductors, it is common practice to use Equation 6 to estimate ke from the measured σ and an approximate L. L can be readily estimated from the Boltzmann transport equation. For the special case of nondegenerate semiconductors limited by phonon scattering, L approaches 1.5 × 10−8 WK−2 . If both n- and p-type carriers are present, ke exceeds the simple summation of the electron and hole terms. This “ambipolar effect” can be estimated using ke = ke, p + ke,n + σ p σn (Sp − Sn )2 T , where Sn and Sp are the Seebeck coefficients of the n- and p-type carriers, σ p +σn respectively (10, 11).

2.3. Lattice Conduction in Semiconductors and Dielectrics: Phonon-Phonon Scattering Heat conduction in dielectric and light-to-moderately doped semiconductor crystals is dominated by kL . For many crystalline semiconductors, heat is conducted primarily by the acoustic phonons, and Debye theory is helpful for understanding the major trends (optical phonons are discussed separately below in the context of complex and amorphous materials). The Debye dispersion relation is ω = v s q for 0 ≤ q ≤ qD , where ω is the phonon frequency, v s is the sound velocity, q is the magnitude of the wavevector, and qD is the Debye cutoff wavevector given by q D = (6π 2 ηPUC )1/3 . Note that when qD is defined in this way, the acoustic branch is linear and the optical modes are ignored. An alternative representation is to define qD in terms of the atomic volume, which introduces a 1/3 factor of N basis into qD and θ D , where Nbasis is the number of atoms in a primitive unit cell. 182

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To understand the acoustic phonon contribution to kL , we must also consider the heat capacity (Equation 3). At T well below θ D , the heat capacity of acoustic phonons follows the Debye T 3 4 law: C = 12π ηPUC k B (T /θ D )3 . At higher temperatures, C for acoustic phonons approaches the 5 constant Dulong-Petit value of 3ηPUC k B , which can be used as low as T = 12 θ D with errors less than 20% as compared to the exact Debye calculation. A convenient interpolation formula (3) for all T that is never in error by more than 12% compared to the full Debye calculation is C = 3ηPUC k B /(1 + 4π5 4 ( θTD )3 ). In pure materials, the dominant phonon scattering mechanism transitions from boundary scattering to phonon-phonon Umklapp scattering with increasing temperature. Even highly disordered materials can be limited by Umklapp scattering at sufficiently high temperatures. Phonons interact through anharmonic interatomic potentials; thus, understanding the extent of anharmonicity in a material is critical to quantifying Umklapp scattering. It is interesting that we utilize simple harmonic crystal models to understand the phonon dispersion but require anharmonic potentials to obtain reasonable values for l. The Gruneisen parameter (γ ) is a scalar quantity used ¨ to quantify the extent of this anharmonicity in a material. For a completely harmonic material, γ is zero, and any nonzero value (positive or negative) denotes anharmonic interactions (12). For most materials, γ is positive and equal to γ = 3α BT /C, where α is the linear thermal expansion coefficient, BT is the isothermal bulk modulus, and here C is lattice heat capacity (4, 13, 14). The phonon MFP associated with Umklapp scattering at high T is 4

l umkl =

βMvs 1/3

k B V PUC ω2 γ 2 T

,

7.

where β is a constant of order unity [typically ( 43 π 2 )1/3 ≈ 2.0] (4), M is the average mass per atom, −1 . For T near and above θ D , applying Equation 7 to the Debye model gives (4, and V PUC = ηPUC 13, 14) kL =

(6π 2 )2/3 M vs3 . 2/3 2 4π 2 V PUC γ T

8.

This expression emphasizes the tendency of materials with low sound velocity and high anharmonicity to have low thermal conductivity, and also the k ∝ T −1 behavior at high T (Figure 2). Another striking feature depicted in Figure 2 is the prominent T 3 trend in k at very low T, arising from the Debye T 3 specific heat and the fact that the only remaining phonon scattering mechanism at low T is boundary scattering, to be described further below.

2.4. Point Imperfections: Alloys, Isotope Scattering, and Point Defects Point inhomogeneities—such as alloy atoms, dopants, isotope variations, and point defects— also scatter electrons and phonons and reduce k. Starting with electrons, Figure 1a shows one example for metals, where the presence of Mn and Ni atoms in the copper alloy manganin (Cu0.84 Mn0.12 Ni0.04 ) reduces copper’s ke by a factor of 20 around room temperature, and by over four orders of magnitude at very low T. Phonons are also scattered by point defects, which for wavelengths larger than the defect size results in the Rayleigh scattering law, 2 −4 ∝ φc 2 ω4 , l −1 p.d . ∝ φc λ

9.

where φ is the fractional concentration of the point defects, λ is the phonon wavelength, and c is a contrast parameter proportional to the relative deviations in mass and/or bonding strength (13, 15, 16), which can be even stronger due to the local strain field (17, 18). Equation 9 is only www.annualreviews.org • Advances in Thermal Conductivity

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100

k L /k L peak

10 –1

10 –2 Ge d-C Si NaCl MgO

10 –3


10 –4 0.01

0.1

1

10

100

T/ Tpeak Figure 2 Homologous plot of the thermal conductivity for several simple dielectrics, showing the universality of T 3 and T −1 laws at low and high T, respectively. Sources: References 9, 183, 184.

appropriate for defects smaller than the phonon wavelength, which is generally believed to be a good approximation for atomic-scale defects such as vacancies and atomic substitutions. For defects much larger than the wavelength, such as large voids and inclusions of a different phase, analysis using ray optics is more appropriate; and the intermediate regime including the effects of polydispersity of defect size has also been considered (110). Figure 1b illustrates several examples for silicon. First, the natural composition is only 92.2% 28 Si, with the balance 4.7% 29 Si and 3.1% 30 Si (19). Compared with isotopically purified 28 Si (99.98%), the mass contrast in natural Si is important from ∼10 K to 100 K, the regime where both Umklapp and macroscopic boundary scattering are very weak (20). Figure 1b also shows the strong effect of scattering by dopant atoms (21). Even though the P concentration is only 0.040%, their scattering efficiency is much stronger than that from the 7.8% of Si isotopes because the P atoms have much greater contrast in both mass and bonding stiffness with the host 28 Si, and the scattering goes as the square of such contrast (Equation 9). Note that this doping concentration of 2.0 × 1019 cm−3 corresponds to σ ∼ 3 × 104 S/m; so, from Equation 6, ke ∼ 0.2 W m−1 K−1 and is negligible compared with kL . Finally, Figure 1b also shows Si1−x Gex alloys with x = ∼20% (22) to ∼30% (23), which gives even further reduction in kL due to the higher concentration and mass contrast of the Ge atoms.

2.5. Amorphous Solids and the Minimum Thermal Conductivity A limiting case of point defect scattering is when every atom is disordered, as in an amorphous solid. Indeed, the form of Si in Figure 1b with the lowest k is the amorphous phase (24). In this limit, the kinetic theory framework of Equation 3 is no longer strictly appropriate because it is no longer possible to speak of the phonons as particles with MFPs much longer than their wavelengths. Nevertheless, Equation 3 is still useful for defining a conceptual lower bound on kL , by postulating that the MFP for all phonons is half of their wavelength: l ≈ λ/2 (14, 25). This approach is roughly consistent with a more sophisticated analysis that identifies diffusons as the most important vibrational entity in amorphous solids (26). An example of this first approximation is shown in Figure 1b for a Debye model of silicon. The comparison between experimental data 184

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and the calculated kL,Min is typical of many amorphous and disordered materials (27): kL,Min appears to be a reasonable lower bound for the amorphous phase in the plateau region (here T > ∼200 K, although experimental data were not available for the plateau of amorphous Si in Figure 1b). At lower T, the kL,Min calculation’s assumed MFPs for long-wavelength phonons must be shorter than what is realized experimentally. Similar kL,Min analyses can also be helpful to estimate the lower-bound kL of amorphous polymers (28) and the contribution of optical phonons (4, 29). Materials with k values below kL,Min have been reported in disordered, layered crystals which are randomly stacked (29a). This unusually low thermal conductivity normal to the layers likely arises from the structural disorder and weak inter-layer bonding. The minimum thermal conductivity in the high temperature limit is given by


2/3 (vs ,L + 2vs ,T ), kL,Min,highT = 0.40k B ηatom

10.

with the group velocity approximated by an isotropic Debye model (v s,L and v s,T are the longitudinal and transverse sound velocities, respectively). Equation 10 can be used as low as T/θ D = 0.4 with no more than 20% error compared with the exact calculation.

2.6. Other Energy Carriers: Photons and Gas Molecules When materials with low kL and low ke are at high T (e.g., aerogels and thermal barrier coatings), the heat transported by photons is also important. This radiative transport is diffusive and can be viewed as heat conduction by the photon gas (kr ). For very low k materials at high T, an important strategy to minimize kr is to add photon scatterers to reduce l, with sizes that should be comparable to or larger than the characteristic infrared wavelengths (30). In porous low-k materials such as aerogels, the heat conduction through the interstitial gases (kg ) can be an important parasitic shunt path for heat flow. For a monatomic gas modeled as elastic k

3/2

spheres, kg = ( 5√6 π ) d 2 mB 1/2 T 1/2 , where d is the effective scattering diameter, which is weakly T dependent (3). Thus, ideal gases have a kg independent of pressure, and smaller, lighter gases, such as He, have higher kg .

3. MEASUREMENT TECHNIQUES Here we consider both steady-state and dynamic measurement techniques to determine k, as well as techniques that reveal the phonon dispersion, v g , l, and anharmonicity.

3.1. Thermal Conductivity and Diffusivity Accurate measurements of k are difficult, as reflected by variation in the literature for nominally identical samples (9). Such variation arises from both the inherent difficulty in the measurement as well as subtle chemical and microstructural differences (porosity, grain orientation and size, defect and impurity levels) between materials. Determining k for bulk materials has traditionally involved steady-state measurements. This approach extracts k from the rate of heat flow, cross-sectional area, and temperature gradient across a flat sample inserted between two temperature-controlled blocks (31). Although widely used, this method is critically sensitive to thermal contact resistances (Rc ). If no special care is taken, typical Rc are in the range ∼10−3 to 10−4 m2 K W−1 , and might be reduced to ∼10−5 m2 K W−1 using polished surfaces, clamping pressure, and interstitial greases and/or foils (32). For accurate measurements of k, this Rc should be much less than the sample’s area-specific thermal resistance www.annualreviews.org • Advances in Thermal Conductivity

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t/k, where t is the sample thickness. This criterion invariably fails for thin, conductive samples: for example, 1-mm-thick Al has t/k ∼ 4 × 10−6 m2 K W−1 . This issue can be partially overcome by incorporating additional temperature measurements on the sample itself or using a comparative method with a standard material (33). Another common technique for bulk samples is the laser flash thermal diffusivity measurement, which monitors the transient temperature on the back side of a planar sample in response to an optical heat pulse on the front side. Compared with the steady-state method, the flash method requires more sophisticated hardware but has easier sample preparation and, being noncontact, is much less sensitive to Rc issues. The flash method fundamentally gives the thermal diffusivity (D ≡ k/C), so determining k also requires knowledge of C. For bulk samples, C is typically determined with differential scanning calorimetry and a density measurement. Recent round-robin tests led by H. Wang and co-workers at Oak Ridge National Laboratory have, however, exposed a serious issue in experimental heat capacity measurements: variation in measured C is far greater than the variation in thermal diffusivity among laboratories. For thermal conductivity measurements above the Debye temperature, use of the Dulong-Petit value for C may indeed be preferable. This approach facilitates comparison of data between laboratories and does not introduce spurious effects into the thermal conductivity values calculated from D. In practically all solids of interest, from metals to dielectrics, the total heat capacity CL +Ce +Cr +Cg +· · · is dominated by phonons from far above room T down to cryogenic T. Note that this also implies that it is difficult to make an independent measurement of the electronic heat capacity (Ce ), the relevant quantity for ke , except at cryogenic temperatures (1).


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3.2. Supporting Measurements To develop advanced thermal materials, we must look beyond measurements of kL and ke . If we look to Equations 7 and 9, a broad range of phonon frequencies contribute to kL . As the phonon group velocity and scattering mechanisms are strongly frequency dependent, a complete understanding of kL requires advanced methods to probe the underlying physical phenomena of C, v, and l. 3.2.1. Phonon dispersion. The classic approaches to understanding thermal transport have employed a Debye model (linear dispersion). Such approaches may prove successful at low temperatures, where the activated phonon modes only populate the acoustic branches. However, at temperatures where optical modes are populated, this simple model is expected to break down, and a detailed understanding of the actual phonon dispersion is needed to successfully model thermal transport. With the development of advanced X-ray and neutron sources, direct measurements of the phonon dispersion (discussed below) are becoming more common and offer an unprecedented opportunity to understand phonon dynamics. These techniques are enabling the measurement of frequency-dependent group velocities and phonon MFPs. Concurrently, we are seeing the rise of new computational methods to understand the detailed lattice dynamics and phonon scattering from first principles (34). Experimentally validating such results remains an ongoing challenge. These approaches will not necessarily capture all of the complexities of the phonon dispersion, as seen in the mode-mode repulsion observed experimentally in PbTe (38). Nevertheless, computational estimates of v g are a significant improvement over the commonly used Debye model. 3.2.2. Inelastic scattering. Inelastic scattering techniques yield information on the relative positions and motions of atoms in solid or liquid samples and are thus critical in understanding phonon transport. Inelastic scattering can be conducted with photons, electrons, or neutrons, and all such interactions involve the exchange of energy and momentum between the scattering particle and 186

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a phonon. From the changes in wavevector and energy of the scattered particle, one can infer information about the associated phonon involved in this inelastic process. Raman scattering (visible or near-visible photons), inelastic X-ray scattering (IXS), and inelastic neutron scattering (INS) are the three most common inelastic techniques used to characterize phonons. In Raman scattering, the momentum of the incident light is so low that only phonons near the center of the Brillouin zone can be involved. Nevertheless, the high photon fluxes and relatively similar energies between photons and phonons make Raman scattering useful for investigating the optical phonon frequencies at the zone center. Raman scattering can also be used to investigate the role of point defects, interfaces, and intermolecular forces in reducing the thermal conductivity, by measuring phonon peak frequency shifts and Raman line width (41–43). By moving to photons with higher energy through IXS, one can likewise map out the whole Brillouin zone. Such measurements are challenging, however, as the change in photon energy is on the order of meV, while the energy of X-rays is on the order of keV. One thus requires extremely fine energy resolution of at least E/E = 10−7 (44). The high fluxes of IXS allow study of very small samples (on the order of 10−6 mm3 ) alleviating the need for large single-crystal specimens. IXS can also probe samples at very high pressures (44), thereby enabling measurement of the mode-dependent Gruneisen parameter (discussed below). ¨ The similarities of neutron and phonon momenta and energy allow much of the Brillouin zone to be mapped out with INS (45). When powder samples are employed, INS data yield the phonon density of states. These curves are naturally weighted towards the edge of the Brillouin zone and thus complement Raman measurements. INS measurements on single crystals yield sections of the phonon dispersion. From such measurements, one can directly obtain the frequency dependence of v g and infer the lifetimes [τ (ω) ≡ l(ω)/v g (ω)] from the dispersion curve linewidths. However, single-crystal INS measurements can be challenging due to the need for macroscopic single crystals. The results of inelastic scattering experiments have been used to understand the phonon dispersions of many low thermal conductivity materials, such as PbTe, La4 Te3 , Zn4 Sb3 , AgPbm SbTe2+m , clathrates, and skutterudites (35–40). INS and IXS data from these materials reveal an array of complex behavior in the phonon dispersion, including coupling between modes, rattling modes, and mode broadening. In materials with rattler modes, Raman scattering can determine the frequencies of the rattling guest atoms (4, 40, 46–49). 3.2.3. Anharmonic behavior: measuring the Gruneisen parameter. When phonon-phonon ¨ Umklapp scattering dominates conduction, the Gruneisen parameter is the critical metric of an¨ harmonic phonon interactions. Experimentally, the Gruneisen parameter can be determined in ¨ a variety of ways. If only the bulk γ is desired, then α and C can be measured directly and BT can be determined from speed-of-sound measurements (51). In practice, a large bulk Gruneisen ¨ parameter is often used to explain low values of measured thermal conductivity. However, a more complete picture of the anharmonicity in a material is obtained by considering mode-specific ωi for a mode of frequency ωi . Figure 3 shows the mode Gruneisen parameters, given as γi = ∂∂ ln ¨ ln V Gruneisen parameter of silicon ranges from −2 to +1.5, information which would be lost were ¨ only the bulk value (0.56) to be considered (14, 52). Experimentally, frequency-dependent characterization techniques must be used, most commonly Raman spectroscopy, infrared spectroscopy, and inelastic neutron scattering (53–55). In these methods, temperature or pressure is used to induce frequency shifts in the measured phonon dispersion, and the magnitudes of the frequency shifts are then used to calculate the mode Gruneisen parameters. ¨ Mode Gruneisen parameters appear to be underutilized in the literature. In many cases, when ¨ the numerical values of the mode Gruneisen parameter are calculated, these data are not used for ¨ www.annualreviews.org • Advances in Thermal Conductivity

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2

Grüneisen ratio

1

0


–1

–2

Γ

Κ

Χ

Γ

L

Figure 3 Mode-specific Gruneisen parameters reveal the anharmonicity of specific phonon modes and thus the ¨ tendency for phonon-phonon scattering. Here, the γ associated with the Si acoustic (red ) and optical (blue) modes has been calculated. The anharmonicity is determined by the magnitude, not the sign, of γ . Reprinted with permission from Reference 52.

any further analysis, or even connected with the thermal conductivity. Although some studies (12, 56) do discuss the connection between the mode Gruneisen parameter and the resultant thermal ¨ conductivity, much more could be done to use the γ values explicitly for making other predictions or to inform future research.

4. ADVANCED MATERIALS Dramatic advances have been made in the development of new materials with tailored thermal conductivity. These include nanostructured materials, structurally complex materials, and materials with highly unusual phonon modes such as rattling.

4.1. Nanostructured Materials Recent decades have seen an explosion in research on the k of nanostructured materials, including thin films, superlattices, nanowires, and nanostructured bulk materials (Figure 4). These materials typically have characteristic lengths, Lchar , in the range of 10 to 100 nm, and single-nm periods are common in superlattices. Graphene and carbon nanotubes may be considered as the limiting case in which Lchar can be a single atom (57, 58). Nanostructured materials have several distinct features that can enable hitherto unprecedented applications: their smaller size allows more devices to be packed in a given space (e.g., transistors); they have large specific surface area (advantageous for catalysis, sensing, and nanostructured thermoelectrics); and in the smallest structures, quantum confinement effects change the states of photons and electrons (e.g., quantum well lasers). In this section, we discuss the often dramatic reductions in k caused by the increased scattering at boundaries and interfaces, which is a critical aspect of performance for many of these applications. As an example of the reduced k in nanostructured materials, consider phonons in nanostructured Si. As illustrated in Figure 5, reductions in k by a factor of 5 or 10 compared with bulk single-crystal 188

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Nanowires

b

a

d

c

100 nm

20 nm

Vapor-liquidsolid Si

e

100 nm

Electrochemically etched Si

100 nm

100 nm

~0.3% mismatch (Si0.76Ge0.24/Si0.84Ge0.16)

Lithographically patterned Si

~3.1% mismatch (Si0.9Ge0.1/Si0.1Ge0.9)

Nanobulk/nanocomposites Growth direction

g

f

h

x y

i

iii

iv

i

j

ii 500 nm

3 nm

10 nm

4 nm

ErAs in In0.53Ga0.47As

Si

Nanocrystalline

1 µm

2 µm Widmanstätten ppts.

SrTe in PbTe

Nanodots endotaxially embedded in a matrix

Lamellae

Sb2Te3 nanostructures in PbTe by spontaneous decomposition

Figure 4 A selection of nanostructured materials (nanowires, superlattices, and nanocomposite/nanocrystalline materials). In all cases, the additional scattering by boundaries and interfaces reduces the thermal conductivity significantly. Figures used with permission from References 59, 92, 100, 101, 103, 185, 186. 100

Film

Si, 300 K

k L (W m–1 K–1)


Superlattices

Ju & Goodson (1999)

80

Liu et al. (2000)

60

Wang et al. (2011)

40

Li et al. (2003)

Nanocrystalline

Nanowire Boukai et al. (2008) 20

0 10

Hochbaum et al. (2008)

100

1,000

Critical thickness (nm) Figure 5 Measured thermal conductivity of various Si wires, films (in-plane), and nanocrystalline materials at room temperature as a function of diameter, film thickness, and average grain size, respectively. For comparison, the thermal conductivity of bulk single-crystal Si at 300 K is approximately 148 W m−1 K−1 . Sources: References 59, 84, 86, 93, 187, 188. www.annualreviews.org • Advances in Thermal Conductivity

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Si are common even at 300 K. The effect is even more impressive at lower T, with nanocrystalline Si and Si nanowires both showing k more than 1,000× smaller than that of bulk Si below 30 K (Figure 1b). Although harmful for many applications such as transistors and optoelectronics, this reduced k can be exploited in thermoelectrics and thermal barrier coatings. In most cases, the tremendous k reductions by boundary scattering can be readily understood with simple models based on classical or particle size effects. Such models are valid only when the energy carriers can be treated as incoherent particles, which is generally a good approximation when Lchar λ (coherent wave effects are discussed below in Section 4.3). For simple geometries including nanowires, thin films, and nanocrystalline and nanoporous bulk materials, the boundary −1 −1 + l bdy , where lbulk is the scattering can be well described by adding a term to Equation 4: l −1 = l bulk effective MFP accounting for all bulk scattering mechanisms and lbdy accounts for the additional interfacial/boundary scattering. Often l bdy = a Lchar , where a depends on the structure and is commonly of order unity if the nanostructure has non-negligible interfacial roughness (3, 59). When l bdy l bulk , the limiting behavior is simply k k = a lLchar , where bulk bulk Cvl bulk d ω 3k = 11. l bulk ≡ Cvd ω Cvd ω


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is an average bulk MFP. A simpler expression l bulk = 3k/Cvs is also sometimes used, but, because it ignores the flattening of the dispersion relation at high ω, it is only exact well below θ D , whereas at higher T it can severely underestimate l bulk by a factor of 6 or more (60, 61). 4.1.1. Typical mean free paths. Because boundary scattering is only significant when Lchar is comparable to or smaller than lbulk , it is very helpful to have good estimates of the bulk MFPs. If we focus on phonons, it is common to approximate all the MFPs as l bulk , known as a lumped or gray approach (Equation 11). For example, in bulk Si at 300 K, lbulk ∼ 225 nm is a typical gray estimate (for comparison, the cruder estimate from 3k/Cvs is only ∼45 nm) (60, 61). However, more detailed calculations that account for the ω dependence of the phonon scattering (Equations 7 and 9) show that there is actually a broad spread in the bulk MFPs (61–64). Figure 6 shows the accumulation functions, defined as the fraction of the total kL carried by MFPs less than the indicated value. Thus, 10% of the heat in bulk Si at 300 K is carried by MFPs below ∼90 nm, whereas 90% is carried by MFPs less than ∼15 μm. The long tail of this distribution clearly reaches far beyond the common gray estimate of ∼225 nm. Figure 6a further shows that the MFP distribution for a SiGe alloy is even broader than Si, with the 90% cutoff MFP now in the neighborhood of 100 μm at 300 K (61, 63). Such a result is initially surprising, because a conventional estimate for the lumped effective MFP for the SiGe alloy is l bulk ≈ 40 nm. Indeed, one expects the additional alloy scattering should always tend to reduce the MFPs. However, further consideration shows that by strongly suppressing the short-λ (high ω) phonons in the alloy (recall Equation 9), the phonons remaining to contribute to kL are now biased towards longer λ (lower ω), thus skewing the distribution towards longer MFPs (61–63). The longer tail in the alloy’s MFP distribution also means that SiGe is more sensitive than Si to further reductions in kL by boundary scattering (Figure 6b), which again is opposite of expectations from simpler analysis using the lumped MFPs of ∼225 nm for Si and ∼40 nm for SiGe. There have been exciting recent developments in evaluating MFP distributions from both theory and experiment. Modern first-principles calculations of kL with no free parameters (34) allow much more accurate calculations of the MFP distributions. These calculations do not rely on any simplifying assumptions about the phonon dispersion or Umklapp scattering rules, allowing them to also critically evaluate scattering laws such as Equation 7. Researchers have also begun 190

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Figure 6 (a) Calculated MFP accumulation functions for bulk Si and SiGe alloy at 300 K from three different references, showing the broad MFP distribution in Si and even broader distribution in SiGe. Conventional estimates of the lumped MFP are also included for comparison. (b) Calculated (lines) and measured ( points) thermal conductivity of Si and SiGe nanowires at 300 K. Sources: DC = Dames & Chen (61); BMV = Bera, Mingo & Volz (63); HC = Henry & Chen (64); Li et al. (86).

developing new experimental methods to probe the MFP spectrum (65, 66) and to present their results on plots like Figure 6a. These exciting measurements would be further aided by rigorous solutions of the relevant frequency-dependent Boltzmann transport equations used to interpret the data. 4.1.2. Transport normal to boundaries: contact resistance, films, and superlattices. Thermal transport across heterostructure interfaces is important for thermal management of many micro- and nanostructured devices. This is especially true when Lchar is comparable to or smaller than lbulk ; for example, optoelectronic and thermoelectric superlattices often have periods on the ˚ order of Angstroms. Modeling of the cross-plane boundary scattering lengths (lbdy ) and the thermal contact resistance (Rc ) for films and superlattices is notoriously difficult and almost invariably involves the phonon transmissivity and/or absorbtivity as a free parameter (60, 67). Nevertheless, several helpful trends are clear. First, the strength of interfacial reflection increases with the contrast in acoustic impedance ρv s between two adjacent layers, where ρ is the mass density (68, 69). In real materials, increasing the acoustic contrast is often correlated with increasing the lattice mismatch, causing defects which also reduce kL (70) (Figure 4d–e). Second, in the limit of a very rough or defective interface, it is sometimes imagined that every incident phonon is absorbed and reemitted (69), equivalent to perfect thermalization and giving a relatively large value of Rc . At the other extreme, even the best quality interfaces cannot have Rc lower than certain fundamental thermodynamic bounds (71, 72). Empirically, Rc for very high quality interfaces are typically in the range of 10−8 to 10−7 m2 K W−1 (see, for example, figure 1 in Reference 73 and figure 10 in Reference 189). As an example of the cross-plane kL of superlattices, four Si/Si0.7 Ge0.3 samples were measured by Huxtable et al. (70). At 300 K, the effective kL increased by a factor of ∼1.7 as the period increased by a factor of 4 (from 4.5 nm to 30 nm), the expected trend for mixed boundary and bulk phonon scattering. References 24, 74, and 75 review additional results for films and superlattices: as expected, kL generally increases monotonically with Lchar . However, for very short period superlattices, there have been some intriguing reports of nonmonotonic trends of kL with period (76–78), for which it has been suggested that partially coherent wave effects may be important (79, 80). www.annualreviews.org • Advances in Thermal Conductivity

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4.1.3. Transport parallel to boundaries: films, superlattices, and nanowires. Thermal transport parallel to boundaries is important for nanowires (potential applications include thermoelectrics, optoelectronics, transistors, and memory) and in-plane conduction in superlattices and thin films (applications include microelectronics and microelectromechanical systems). For both films and superlattices, the in-plane kL is somewhat less sensitive to boundary scattering than is the cross-plane kL . This follows from simple line-of-sight arguments, and the fact that in-plane phonons can be specularly reflected from smooth surfaces/interfaces (81). A surface is generally considered smooth if its roughness is smaller than one-tenth of the wavelength of the energy carrier, resulting in highly specular reflections (82). As an example of the in-plane kL of thin films, Asheghi, Goodson, and coworkers (83, 84) measured high-quality Si films with thicknesses ranging from 3 μm down to 20 nm (Figure 5). They found that the measurements were described well by the classical size effect (85) assuming fully diffuse boundaries for T down to ∼20 K. At this temperature, in the regime of strong boundary scattering, the dominant thermal wavelengths in Si are ∼2 nm to ∼7 nm (61). This indicates that even high-quality silicon-on-insulator films have sufficient roughness to cause fully diffuse scattering. At lower T, the specularity and kL were found to increase somewhat, and this trend was attributed to the further increase in the thermal average wavelengths, which at low T scale as λ ∝ T −1 . For nanowires, kL has also been measured in various materials for diameters (D) down to around 20 nm. For D of ∼50 nm and larger, many measurements (86, 87) (Figure 5) are readily understood (61, 82, 87–89) through the classical size effect with fully diffuse boundaries, corresponding simply to lbdy = D, the so-called Casimir limit. lbdy > D has also been observed in Si wires at T < 10 K, corresponding to increasingly specular reflections (90, 91). However, for smaller diameter and rougher nanowires the body of experimental results is more complicated and currently defies simple understanding. A major challenge is that the measured k can be strongly sensitive to the method used to synthesize the wires (92). For example, Figure 5 shows k for various Si nanowire morphologies, some of which were also shown in Figure 4a–c. The electrochemically etched (93) Si nanowires were found to have k ∼ 5–10 times smaller than comparable vapor-liquid-solid grown wires (86), with lithographically-patterned nanowires somewhere in between. The conventional boundary scattering models (61, 82, 87– 89) have proven inadequate to explain many of these results. Thus modelers have been inspired to explore alternative explanations, including perturbation theory (94) and a mixture of coherent and incoherent transport (95, 96), although there is not yet a widely accepted predictive understanding.


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4.1.4. Nanostructured bulk materials. Nanocrystalline, nanoporous, and nanocomposite bulk materials (Figure 4f–j) can possess extraordinarily low thermal conductivity due to boundary scattering. Such materials are often more suitable than nanowires and superlattices for low cost, mass production applications. One method to make such materials is based on hot pressing of nanopowders, resulting in final grain sizes typically in the range of 10 to 100s of nm (59, 97, 98). Another common synthesis route uses thermal treatment to induce the precipitation of internal nanostructures from an initially homogeneous ingot. This route has been used to create dots a few nm in diameter (99, 100), as well as pseudosuperlattices with characteristic layer thicknesses typically ∼100 nm (101, 102) and other shapes (103, 104). These nanostructured bulk materials are of practical interest because they can exhibit remarkable magnetic, optical, and mechanical properties (105, 106). In particular, the internal nanostructuring can reduce kL by an order of magnitude or more compared with a single crystal, making these materials very promising for thermoelectrics (97–100, 102, 104, 107). 192

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To understand the kL reduction in nanocrystalline materials, the most common approach (98, 108) is to set the boundary scattering length equal to the average grain size: l Bdy = Davg . However, this simple expression ignores possible variations in grain boundary transmissivity and specularity, which must depend on the detailed processing conditions, intergranular phases, etc. A recent study of nanocrystalline silicon with high-quality grain boundaries found that a more appropriate expression is l Bdy = α Davg where α ≈ 0.5 for models using a Debye dispersion and α ≈ 0.9 for a Born-von Karman dispersion (59). The same study also identified a frequency dependence (l Bdy ∝ ω−1 ) due to wavelength-dependent grain boundary transmission, consistent with previous theoretical analysis (13), although in practice this is likely only important below ∼80 K. Composites with inclusions much larger than the MFPs are fruitfully analyzed with effective medium theories like the Maxwell-Garnett result, keff 1 + 2rφ , = kmatrix 1 − rφ

12.

where r ≡ (kparticle − kmatrix )/(kparticle + 2kmatrix ) and φ is the volume fraction of the particles (109). For smaller inclusions, the boundary scattering physics may include both Rayleigh (Equation 9) and ray optics regimes (110), though for inclusions with a lattice mismatch of ∼10% or greater, the additional scattering by the surrounding strain field may also need to be considered (17, 18). In materials where the inclusions are comparable to or smaller in size than a substantial part of the MFP distribution, the reduction in k can be stronger than effective medium theories predict due to subcontinuum effects (59, 104). Porous materials with large pores can be analyzed using effective medium theories like Equation 12 with kparticle = 0, but, as the pore sizes become smaller than the bulk MFPs (59, 111–113), such continuum theories break down. For example, the porosity effect measured in References 59 and 113 was more than twice as strong as the prediction of Equation 12, for φ in the range from 3% to 26%. The deviations from Equation 12 should become stronger as the pore sizes become ever smaller compared with the bulk MFPs, until the pores become even smaller than the phonon wavelengths (110), in which case a Rayleigh scattering law like Equation 9 is appropriate.

4.2. Measurements of Nanostructures The classical measurement techniques described above in the section on thermal conductivity and diffusivity are not appropriate for nanostructures. Alternative techniques have been developed, which face major challenges in sample preparation and identification, minimization of parasitic heat losses, and ensuring good thermal contacts for accurate temperature readings (114, 115). 4.2.1. Films and superlattices. As an example of the tremendous challenges involved in measuring k across planar structures, as noted above the conventional steady-state method for macroscopic samples uses grease or metal foil contacts with typical contact resistance of Rc ∼ 10−5 –10−4 m2 K W−1 . But a typical film of thickness t = 1 μm and k = 10 W m−1 K−1 has t/k of only ∼10−7 m2 K W−1 , making the signal less than ∼1% of the background contact resistance. Thus, macroscopic pressure contacts are completely inappropriate for measuring k across microscopic planar structures. Instead, it is necessary to use atomically intimate contacts, usually evaporated metal, which may have Rc in the range of 10−8 to 10−7 m2 K W−1 (73). One very common technique is the 3ω method, which monitors the sinusoidal temperature response of a Joule-heated metallic line microfabricated on top of the film of interest, which itself should be on a higher k substrate (116–118). The thermal contact and substrate spreading resistances are best www.annualreviews.org • Advances in Thermal Conductivity

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subtracted through control experiments using different film thicknesses and/or omitting the film completely (116, 119), although these methods have also been adapted to measure the thermal contact resistance itself (120, 121). The other common approach is pump-probe laser thermoreflectance (122, 123). These methods again typically rely on a thin evaporated metal layer for intimate thermal contact, and monitor the transient temperature of the metal as the heat from an ultrafast laser pulse conducts into the underlying film of interest and substrate. Compared with 3ω methods, thermoreflectance methods generally have simpler sample preparation and the potential to study ultrafast (∼ps) phenomena such as electron-phonon nonequilibrium. Challenges include the hardware complexity, relatively weak thermoreflectance coefficient, and more complicated data analysis (which can however give the thermal contact resistance). Sample preparation for in-plane measurements is usually more demanding than for crossplane measurements because of the risk of major parasitic heat losses through the film-substrate interface. Although in some cases these lateral losses may be manageable (114, 124–126), the best solution is to eliminate them by suspending the film. Creating the heater and temperature sensors usually requires additional microfabrication steps (127, 128); however, if the film is electrically conducting or can be coated with a thin metal, it is more convenient to Joule heat the film itself, which also acts as its own temperature sensor (84, 129, 130). All-optical methods to measure the in-plane k of suspended films include the transient grating method (131) and Raman spot method (132). The latter has also been used for graphene (133, 134), requiring attention to the thermal boundary conditions and the power deposited (134). Other measurements of graphene have used microfabricated platforms (135) and a heat spreader method (136).


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4.2.2. Individual nanowires and nanotubes. Measurements of k for individual nanowires and nanotubes are even more challenging than for films. The most common approach, pioneered by Shi and coworkers (137, 138), is to suspend the wire between microfabricated platforms for heating and sensing. For electrically conducting wires, a self-heating method (139, 140) is arguably the most convenient but requires careful consideration of both thermal and electrical contact resistances. In a related technique, the nanowire is suspended with one tip on a heat sink and the other tip at the midpoint of a suspended metal line, forming a “T” (141, 142). The metal line is Joule heated, and its average temperature is reduced by the thermal conductance through the nanowire. Although this T-bridge method has easier fabrication than the two-platform method (137), there are thermal matching considerations for good sensitivity (142). If the heater line is patterned from a thin metal film (143), care should be taken in the electrode design to minimize the radially-logarithmic electrical spreading resistance between the V+/V− leads and the narrow heater line. Optical methods are less common for nanowire thermal measurements, though a recent development uses noncontact Raman thermometry (144–146). As with films (132–134), the biggest challenge is determining the optical power deposited.

4.3. Phononic Structures The focus on nanomaterials thus far in this review has been on scattering from interfaces or boundaries. In low dimensional and periodically ordered structures, the potential for profound changes in the phonon dispersion due to coherent wave interference effects must also be considered. In the last two decades, there has been a slowly developing effort in phononic materials, which are analogous to photonic structures (147). Here, the interest is in controlling the group velocity of acoustic phonons and even developing metamaterials (148). Periodic composites with elastic contrast can produce large stop-bands within the phonon dispersion (149) at acoustic 194

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frequencies. It is essential to recognize that these structures interact most strongly with phonons whose wavelengths are on the order of the structure’s periodicity (150). When the phononic structure is in the nanometer regime, the structures are termed hypersonic phononic crystals. This is an odd term, as their dominant effect is actually to slow down v g well below the speed of sound (151, 152). Some of these structures can be topologically complex—e.g., chiral and core-shell systems—and yield equally complex phonon dispersions (153–155). Such dispersions could make these materials ideal for acoustic waveguides, mechanical filters, and vibration isolators. Much of the interest in the kL of phononic structures has focused on reducing kL for thermoelectric applications (156). At room temperature, the thermally relevant wavelengths in Si are typically ∼1–10 nm (61, 64); thus, for these phononic structures to have a measureable impact on kL , their periodicity should also be in the nanometer regime and with comparable smoothness. Furthermore, at elevated temperatures, scattering by normal and Umklapp processes may cause the phonon waves to lose coherence before experiencing the requisite multiple reflections from the periodic phononic crystal. We expect phononic structures to have the greatest potential to impact kL at low temperature, at longer wavelengths, and in materials with low anharmonicity. Experimentally, it can be quite difficult to distinguish between incoherent interfacial scattering and coherent group velocity reductions as the primary mechanism for the reduced kL observed in nominally phononic structures. Accurate measurements of the frequency-dependent lifetimes of the thermally-relevant phonons may eventually allow these effects to be separated.

4.4. Complexity Within the Unit Cell When one is considering the lattice thermal conductivity of crystalline solids, the Debye model can be effective for simple solids. However, for complex materials with Nbasis 1 atoms in the primitive cell (and thus 3Nbasis − 3 optical modes), the Debye model breaks down. First, the group velocity in such solids is far from constant, with much of the heat trapped in low velocity optical modes. Second, a Debye model fails to capture the Umklapp scattering options available to phonons in such a complex dispersion. Finally, the Debye model entirely misses complex structural interactions such as rattling modes. One approach to modeling materials with complex unit cells is to treat the acoustic branch within the Debye framework and to treat the low velocity optical phonon modes as minimally conductive. Here, we use the kMin framework developed above for glasses, with loptical = λ/2. Although this calculation approach is only minimally more complex than the Debye model, it is surprisingly successful (within a factor of 2) for predicting experimental thermal conductivity for a broad range of compounds (4). Ab initio calculations of such a complex cell are bound to be extremely challenging, and simple models like this allow separation of the acoustic and optical contributions to the thermal conductivity. Such a separation is valuable when considering the potential impact of disorder (nanostructures, point defects) on phonon scattering. This property of complex materials to trap their heat in low velocity optical modes has led to some of the lowest kL for crystalline solids to date. The following two sections discuss how such materials are revolutionizing thermoelectrics and thermal barrier coatings. 4.4.1. Thermoelectric materials. Of the modern thermoelectric materials with figure of merit (zT) values near unity, several achieve their extremely low thermal conductivity without nanostructures, rattling (discussed below), or point defect scattering. Compounds such as Yb14 MnSb11 and Ca3 AlSb3 reach their minimum thermal conductivity values at high temperature (157–159) (Figure 7a). The minimum thermal conductivity (Equation 10) is compound-specific and determined from the measured speed of sound. In the cases listed above the primitive unit cells are quite www.annualreviews.org • Advances in Thermal Conductivity

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Temperature (K) Figure 7 Si0.8 Ge0.2 and YSZ, the classic high temperature materials for (a) thermoelectrics and (b) thermal barrier coatings, respectively, have much higher kL than the complex advanced materials under development. Sources: SiGe (190), Ca3 AlSb3 (159), Ba8 Ga16 Ge30 (191), Yb14 MnSb11 (157), YSZ (164), Gd8 Ca2 (SiO4 )6 O2 (166), Yb2 Zr2 O7 (171), Bi4 Ti3 O12 (192), Ba2 AlDyO5 (193).

complex, containing between 26 and 104 atoms. Such an observation is consistent with the results on the empty clathrate Si136 by Nolas et al. (160), who observe glass-like thermal conductivity even without rattling atoms. To illustrate the effect of complexity within the unit cell, consider the case of Yb14 MnSb11 , which has 104 atoms in the primitive cell and an extremely low kL of 0.6 W m−1 K−1 at 300 K (157). Separating the acoustic and optical modes into Debye and glass-like contributions, respectively, yields approximately equivalent contributions from the sum of the 3 acoustic modes and the sum of the 309 optical modes. As these materials intrinsically possess low kL , they are particularly attractive for thermoelectrics. Most other thermoelectric materials require structural disorder within and beyond the unit cell to achieve similarly low thermal conductivity values (104, 161, 162). However, structural disorder can lower the electronic mobility, add additional processing challenges, and be annealed away during operation of the device. Inspired by the success of Yb14 MnSb11 and related compounds, we see many opportunities for new high efficiency thermoelectric materials based on structurally complex materials with soft bonding and heavy atoms (so as to reduce v s and minimize the minimum thermal conductivity of Equation 10).

4.4.2. Thermal barrier coatings. Crystalline oxide materials do not typically possess low thermal conductivity, as the low densities and fairly stiff bonding lead to high group velocities. However, in the classic thermal barrier coating (TBC) material yttria-stabilized zirconia (YSZ), 196

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low thermal conductivity (∼2 W m−1 K−1 ) is found at high temperatures (∼1,400 K) (163). Here, oxygen vacancies are introduced by alloying scatter high frequency phonons. Additionally, the high mass contrast in ZrO2 (Zr: 91.2 amu, O: 16 amu) leads to large stop-bands between phonon modes. These stop-bands flatten the phonon dispersion, further reducing the contribution from the optical modes. Beyond the unit cell, the growth techniques used to form TBCs inherently generate porosity. Such porosity can decrease the phonon MFP of the longer wavelength phonons that are not readily scattered by point defects. Additionally, these pores can reduce kr by scattering photons (30), which become a significant source of heat transfer at high temperature. Further improvement in TBCs requires the development of alternative materials that satisfy a broad range of requirements, including exceptionally low thermal conductivity (Figure 7b). Alternative TBC material exploration initially concentrated on the pyrochlore compounds (e.g., Gd2 Zr2 O7 and La2 Zr2 O7 ) and has since branched out to include many other oxides [e.g., Bi4 Ti3 O12 , ReBa2 AlO5 , ReSrAl2 O7 , Gd8 Ca2 (SiO4 )6 O2 (Re = rare earth element)] (164–168, 173, 192). Like many of the new thermoelectric materials, the primitive unit cells are generally large and have high mass contrast. In many of these compounds, oxygen vacancies play a major role in the low thermal conductivity that is observed (166, 169, 170). Alloying on the cationic sites is likewise prevalent (170, 174). In contrast to the nanostructured thermoelectric materials, most new TBC materials are not intentionally nanostructured. This comes from an understanding that (a) the phonon MFPs in TBC materials are already reduced to near-minimum values at high temperatures and (b) nanostructures may not survive under such extreme conditions. Nevertheless, variation in microstructure may account for some of the variation in kL observed in the literature for nominally identical samples. If we look beyond the usual point defect and boundary scattering sources, some of these materials have rather interesting lattice dynamics. Lattice softening is found at the fluorite-pyrochlore phase transition in (Sm1−x Ybx )2 Zr2 O7 , as observed by a decrease in sound velocity and Young’s modulus (171). At this composition, kL is suppressed and increases with increasing temperature like a glass. A second example is the related compound Ybx La2−x Zr2 O7 , which upon alloying exhibits a significantly suppressed kL (172). The authors of Ref. (172) suggest the Yb may be rattling (discussed below), as the atomic displacement parameter of Yb on the La site is quite large and the lanthanide contraction leads to a significant difference in diameter between La and Yb cations. Measurements of the Ybx La2−x Zr2 O7 phonon dispersion are required to confirm this rattling behavior.

4.5. Rattling When an atom is only loosely bound within a comparatively stiff framework, the frequency of the associated optical modes can drop into the acoustic regime. Such behavior is termed rattling and is found in several thermoelectric materials. In the most general definition, rattling behavior occurs when a guest atom within a lattice is underconstrained and only weakly bound to the host lattice (4). Rattling behavior can be explored using a simple ball and spring model as shown in Figure 8a. The host framework can be modeled as atoms of mass m1 connected by springs c1 . The guest atoms, of mass m2 , are bound to the host lattice by springs c2 . When c2 c1 , the contribution of c1 is negligible and the material behaves as a simple two-atom solid. Decreasing the value of c2 leads to the emergence of rattling behavior, causing the frequency of the associated optical mode to drop into the acoustic frequency range. In the phonon dispersion, this interaction is seen as the avoided crossing between the optical and acoustic branches. The strength of this interaction, or the extent of the avoided crossing, increases as c2 stiffens. This avoided crossing behavior is observed experimentally when the phonon dispersion is measured using INS, as seen in Figure 8b (40). www.annualreviews.org • Advances in Thermal Conductivity

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Rattling behavior can be directly observed by INS and IXS as flat modes which cross the acoustic branch in the phonon dispersion (39, 40). Rattling can also be seen from the Raman spectrum, in which low frequency peaks appear when rattler modes are present (47, 175) (Figure 8c). The intensity of the rattler mode peak is also correlated with the amplitude of the rattler vibration (49). The increase in low frequency vibrational states caused by rattling also can be readily detected in the low temperature heat capacity (176). When rattling modes are present, low lattice thermal conductivity can be anticipated due to a decrease in both v g and l. Originally, resonant scattering was proposed to explain the low 198

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Lattice constant (Å)

Figure 9 (a) For filled skutterudite compounds XFe4 Sb12 , a clear increase in lattice parameter is observed with increasing filler size. Such behavior suggests the framework is being stretched by the guest atom. (b) A strong correlation between lattice constant and kL is observed, consistent with the understanding that the rattling frequency depends on the bonding between the framework and the guest (180).

experimental kL in skutterudites and clathrates. In the resonant scattering theory, the vibrations of the guest atoms are considered to be independent of the host lattice (177). Such ideas are consistent with modeling of skutterudites, which indicates that the mode Gruneisen parameters ¨ are at a maximum near the frequency of the guest atoms (178). However, the transport models associated with resonant scattering have tended to assume a constant group velocity, which does not fully capture the details of the rattling interaction (4). Further reduction in l arises from the increase in the number of allowed Umklapp processes when rattling modes are present. Figure 8 shows that in the frequency regime near the avoided crossing, the group velocity is much lower than the speed of sound. When Umklapp scattering dominates, the spectral kL scales with [v g (ω)]2 . Thus, even a minor reduction in v g leads to significant decrease in kL . This effect is somewhat counterintuitive, as the affected frequency range where v g is reduced scales with c2 (4). Thus, increased rattler-framework coupling may affect a broader range of acoustic phonons. Low thermal conductivity in skutterudite and clathrate rattlers has been experimentally observed since the mid-1990s. The frequency of the rattling mode appears to play a major role in the magnitude of kL . In the skutterudites, the positive correlation between atomic displacement parameter (due to the guest atom) and lattice thermal conductivity has been known for over a decade (179). This trend has been shown with varied guest atoms X in the fixed framework of the XFe4 Sb12 compounds, for which the speed of sound could be expected to stay relatively constant (180). The strong correlation between guest atom ionic radius, lattice constant, and the resulting lattice thermal conductivity can be seen in Figure 9. These two trends suggest that the larger guest ion stretches the cage and increases the lattice constant and the bond strength between the guest and host framework. Increasing this coupling should increase the rattler frequency, potentially driving the rattler mode out of the acoustic regime. Conversely, the underconstrained guest atoms are expected to have rattling modes within the acoustic frequency regime. Although guest-free Fe4 Sb12 is not stable, the cobalt analog (CoSb3 ) has a much higher kL of ∼4 W m−1 K−1 at similar carrier concentrations (190). www.annualreviews.org • Advances in Thermal Conductivity

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4.6. High k Bulk Materials


High thermal conductivity materials are required for thermal management in a broad range of applications, ranging from integrated circuits and optoelectronics to power plants. Although lower k materials can achieve equivalent thermal conductance values, there are trade-offs in the corresponding geometric parameters. High k is found in both good metals and dielectrics, arising from ke and kL , respectively. The Wiedemann-Franz law suggests the highest ke will be found in good metals such as silver and copper. (In superconductors, the Wiedemann-Franz law does not apply, and despite their perfect σ , superconductors are poor thermal conductors because only the normal electrons can carry heat.) Electron scattering from defects and grain boundaries can, however, lead to variation between different samples. Advanced intermetallic or carbon-based compounds which have electronic (and thus thermal) conductivities in excess of silver and copper could eventually herald a new generation of high ke materials. Developing new nonmetallic crystals with high thermal conductivity is quite challenging. Following the discussion above, we would expect high kL in high-quality crystals with high group velocity and low anharmonicity to minimize phonon-phonon scattering. High group velocity is found in materials with low atomic mass, stiff bonding, and simple crystal structures. Complex crystal structures have much of the heat trapped in low velocity optical modes. These requirements prove to be accurate, with high kL in compounds such as diamond, SiC, AlN, BeO, and BP (18). Some structurally complex materials (e.g., Si3 N4 , α-B) also possess high thermal conductivity (181). Here, small Gruneisen parameters and stiff bonding overshadow the reduced group velocity ¨ of the optical modes. In these high thermal conductivity materials, phonon scattering sources that are typically neglected can begin to limit transport. Isotopic variation as a source of point defect scattering is one such example (Figure 1b). We see this in the dramatic rise in thermal conductivity of diamond, Si, and Ge when the material is isotopically purified. This impact is most prominent at low temperatures near the peak in kL (T ), before Umklapp scattering dominates. Empirically, the atomic coordination geometry is found to correlate with the magnitude of γ — increasing coordination number leads to a rise in anharmonicity (14). For example, four-coordinate diamond-like compounds have γ ∼ 0.7, whereas rock salt compounds (6 nearest neighbors and much less directional bonding) have much larger Gruneisen parameters (∼1.5). ¨

4.7. Phonon Freezeout at Low T: Diamond as a Thermal Insulator Although this review has emphasized properties around 300 K, for applications including bolometers and potential low-T thermoelectrics, there is also interest in nanostructured materials at cryogenic T, where some of the conventional wisdom about kL gets turned on its head. As an example, Figure 10a shows calculations (61) for three nanostructured materials with a boundary scattering length of 115 nm (chosen to allow comparison with the Si nanowire measurements from Reference 86). At 300 K, kL of nanostructured diamond exceeds that of nanostructured PbTe by a factor of 170, an unsurprising result in light of Equation 3 because PbTe’s heavy atoms give it a low v s , compared with diamond’s stiff, light structure. However, at low T the trend is completely opposite, with the kL of nanostructured diamond now dramatically lower than that of nanostructured PbTe, e.g., by a factor of 43 at 10 K. This crossover is readily understood through the freezeout of the phonons, depicted in Figures 10b and c. For T well below θ D , the number of occupied phonon modes scales as vs−3 , related to the thermally populated volume in reciprocal space. The same scaling applies to 200

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a

b 3

k L (W m–1 K–1)

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Nanowire, d = 115 nm PbTe Si, Li et al. Si d-C

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Wavevector, q (nm–1) –0.5

0

0.5

1

2 kBT

1 0

c 1


0.1 qx 0.01

1

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100

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Temperature, T (K) qy

Figure 10 (a) Phonon modeling predicts an inversion in kL with cooling between PbTe, Si, and d-C. (b) This behavior can be understood in terms of phonon freezeout, where the lower group velocity of PbTe leads to more occupied states for a given T (e.g., here for 10 K). This population of states in the reciprocal lattice is shown schematically in (c) for the three materials considered here (61, 86).

the heat capacity (the classic Debye T 3 law), and thus kL = 13 Cvl bdy scales as v S−2 in the low-T boundary scattering limit: in this regime, materials with high v s have low thermal conductivity. In contrast, in the more familiar high-T boundary scattering limit, kL scales as ηPUC v S .

5. REMAINING CHALLENGES AND FUTURE DIRECTIONS In this review, we have explored how new materials and morphologies have been developed with unprecedented thermal transport properties. Extraordinarily low thermal conductivity has been observed in nanostructured materials and materials with complex unit cells. Concurrent with these emerging materials, experimental and computational techniques have been developed to probe the underlying transport phenomena. However, major opportunities still remain. One such area includes determining frequency-resolved phonon MFPs and relating these values to experimental mode-dependent Gruneisen measurements and general design rules for anharmonic materials. In ¨ the realm of nanostructured materials, scalable synthesis routes that yield controlled morphologies are needed, as well as further advances in characterization. Good progress has been made with materials possessing rattling atoms or complex unit cells; much potential remains to extend these approaches in advanced materials. As the field progresses, our growing understanding of phonon transport will further enable the predictive design of materials with desired thermal properties.

DISCLOSURE STATEMENT The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review. www.annualreviews.org • Advances in Thermal Conductivity

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ACKNOWLEDGMENTS C.D. gratefully acknowledges financial support from the DARPA/DSO NMP program (W911NF-08-C-0058) and the US National Science Foundation (NSF) (CBET 0854554 & 1055317). L.L.B. was supported by the Department of Defense through the National Defense Science and Engineering Graduate Fellowship Program. E.S.T. and L.L.B. acknowledge support from the NSF MRSEC program, REMRSEC Center, Grant No. DMR 0820518. The authors acknowledge G. Jeffrey Snyder, Glen Slack, and Alexandra Zevalkink for their insightful discussions. LITERATURE CITED


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Contents

Annual Review of Materials Research Volume 42, 2012

Three-Dimensional Tomography of Materials (Manfred Ruhle and David N. Seidman, Guest Editors) ¨ Atom Probe Tomography 2012 Thomas F. Kelly and David J. Larson p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 Electron Microscopy of Biological Materials at the Nanometer Scale Lena Fitting Kourkoutis, Jurgen ¨ M. Plitzko, and Wolfgang Baumeister p p p p p p p p p p p p p p p p p33 Electron Tomography in the (S)TEM: From Nanoscale Morphological Analysis to 3D Atomic Imaging Zineb Saghi and Paul A. Midgley p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p59 Fatigue and Damage in Structural Materials Studied by X-Ray Tomography Philip J. Withers and Michael Preuss p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p81 Measurement of Interfacial Evolution in Three Dimensions D.J. Rowenhorst and P.W. Voorhees p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 105 Optical Sectioning and Confocal Imaging and Analysis in the Transmission Electron Microscope Peter D. Nellist and Peng Wang p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 125 Three-Dimensional Architecture of Engineering Multiphase Metals Guillermo Requena and H. Peter Degischer p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 145 X-Ray Tomography Applied to the Characterization of Highly Porous Materials Eric Maire p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 163 Current Interest Advances in Thermal Conductivity Eric S. Toberer, Lauryn L. Baranowski, and Chris Dames p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 179 Bio-Inspired Antifouling Strategies Chelsea M. Kirschner and Anthony B. Brennan p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 211

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Bio-Inspired Self-Cleaning Surfaces Kesong Liu and Lei Jiang p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 231 Ferroelastic Materials Ekhard K.H. Salje p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 265 High-Strain-Rate Deformation: Mechanical Behavior and Deformation Substructures Induced George T. (Rusty) Gray III p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 285


The Magnetocaloric Effect and Magnetic Refrigeration Near Room Temperature: Materials and Models V. Franco, J.S. Blázquez, B. Ingale, and A. Conde p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 305 Responsive Surfaces for Life Science Applications Hidenori Kuroki, Ihor Tokarev, and Sergiy Minko p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 343 Second-Generation High-Temperature Superconductor Wires for the Electric Power Grid A.P. Malozemoff p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 373 Solid-State Dewetting of Thin Films Carl V. Thompson p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 399 Surface-Bound Gradients for Studies of Soft Materials Behavior Jan Genzer p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 435 Index Cumulative Index of Contributing Authors, Volumes 38–42 p p p p p p p p p p p p p p p p p p p p p p p p p p p 469 Errata An online log of corrections to Annual Review of Materials Research articles may be found at http://matsci.annualreviews.org/errata.shtml

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