Experimental Micro/Nanoscale Thermal Transport

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Library of Congress Cataloging-in-Publication Data:

Wang, Xinwei, 1948-

Experimental micro/nanoscale thermal transport / Xinwei Wang.

pages cm

Includes bibliographical references.

ISBN 978-1-118-00744-0 (hardback)

1. Nanostructured materials—Thermal properties. 2. Heat—Transmission. I. Title.

TA418.9.N35W365 2012

620.1'1596—dc23

2011047244

Preface

With the fast development of nanoscience and nanotechnology, it has become more and more important to understand various physical properties of nanoscale and nanostructured materials in order to evaluate their unique characteristics and apply them to different engineering applications. Nanoscale and nanostructured materials could have very different thermal conductivity, since the energy carriers (phonons or electrons) can be strongly scattered by the extremely constrained material feature size, and their dispersion relation can also be altered. Although tremendous effort has been dedicated to modeling the thermal transport in micro/nanoscale materials and exploring how and to what extent their unique material size and structure change their thermal conductivity, it ultimately requires experiments to validate these modeling predictions. Owing to the great complexity and variety of micro/nanoscale material structures, largely because of varying manufacturing/growth conditions, measurement is becoming critical for obtaining accurate information about the thermophysical properties of these materials, monitor their quality, and provide the knowledge base for device performance optimization. One example of thermal characterization application is the performance evaluation of thermoelectric materials, which can be used to convert thermal energy to electricity. The performance of thermoelectric materials can be described using the figure of merit Z = σS²/k, where σ is the electric conductivity; S, the Seebeck coefficient; and k, thermal conductivity. It can be seen that accuracy of k measurement directly affects the figure of merit. Many novel thermoelectric materials are in the form of thin films or nanowires, which make accurate thermal conductivity measurement more challenging. Although such measurement is critical to confirm the novel performance claims of these materials.

In the past, various books were published to introduce to micro/ nanoscale thermal transport. These books, together with some excellent journal reviews, cover comprehensive knowledge about micro/nanoscale thermal transport, from its unique feature, physics background, and material structure to theoretical analysis, numerical modeling, and experimental characterization. On the other hand, it is realized that this area is still under fast development, partly owing to the emergence of novel materials. Instead of an extended review to cover various technologies developed by researchers to characterize thermophysical properties and thermal phenomena, this book focuses on the novel technology development, material thermal characterization, and thermal transport study conducted by the author and his laboratory. From the perspective of materials, the thermal characterization study covers materials of films (micro- to nanometers thick); single one-dimensional materials, wire/tube bundles, and highly packed and highly aligned one-dimensional materials; and material interface thermal transport phenomena. In terms of technology development for thermal excitation, pulsed, step, and periodic photon and electric excitations have been employed. To measure the thermal response of the material, its electrical resistance, thermal radiation, acoustic vibration, and photon scattering have been used.

This book is designed to cover the details of the novel technology development, from experimental principle, physical model, and experiment conduction to data analysis, result uncertainty assessment, and result physical interpretation. It will help readers adopt the covered technologies, or design specific technologies to characterize their unique materials, and to realize high accuracy thermophysical properties measurement and thermal transport study. Chapter .1 provides a general introduction to thermal transport at micro/nanoscales, including the micro/nanoscale thermal transport constrained by the material dimension or internal structure feature size, thermal transport constrained by time, and thermal transport constrained by the size of physical process. Numerical techniques are discussed on how to predict the thermal conductivity or thermal transport phenomenon at micro/nanoscales, including the molecular dynamics simulation, lattice Boltzmann method (LBM), and direct energy carrier relaxation tracking. Chapter 2 discusses how to characterize thermal transport using thermal excitation and sensing in the frequency domain. The frequency domain photoacoustic technique, photothermal radiation technique, three-omega technique, and optical heating and electrical thermal sensing technique are discussed in detail. These techniques can be used to measure the thermophysical properties of films/coatings and conductive/nonconductive wires. Chapter 3 covers transient technologies in the time domain, involving photon and electric heating. The thermal response of the sample is tracked by observing its electrical resistance change. For nonconductive samples, a metallic coating (e.g., Au) is deposited on the surface of the sample to function as a heater and thermal sensor. In Chapter 4, the focus is on techniques in which the material is subjected to static heating (electric or photon heating), and its temperature is measured by evaluating its electrical resistance or Raman signal. Various materials are discussed for thermal characterization, including microwires, solid films, and films/bundles composed of nanoscale wires. Chapter 5 deals with steady-state thermal characterization but is more focused on temperature measurement, which has very broad applications in evaluating the thermal characteristics of micro/nanoscale structures. Also, in this chapter, several techniques used/developed by other researchers are introduced, in anticipation to broaden the knowledge in this area. A transient photo-heating and thermal sensing technique is proposed in order to eliminate the effect of electrical contact resistance and wire–base connection in technologies involving thermal excitation and sensing based on electrical resistance.

Chapter 1

Introduction

In the past decades, tremendous advancement in nanoscience and nanotechnology has prompted a wide spectrum of unique applications of nanoscale and nanostructured materials. Examples include nanowires and nanostructured materials as novel thermoelectric modules in waste heat recovery, nanowire-based sensors, and composites embedded with carbon nanotubes (CNTs) to achieve superior mechanical strength and significant thermal conductivity enhancement. The unique structure of nanomaterials (nanoscale and nanostructured) makes their physical properties (e.g., thermal conductivity and mechanical strength) differ significantly from the values of the bulk counterparts.

1.1 Unique Feature of Thermal Transport in Nanoscale and Nanostructured Materials

The thermal transport in solid materials is sustained by the transport (movement and collision) of phonons (dielectric and semiconductive materials) and free electrons (metals). A phonon is a quasi-particle representing the quantization of the modes of lattice vibrations of periodic, elastic crystal structures of solid. This is more like a photon represents the quantization of light, which is an electromagnetic wave. To picture the thermal transport by phonons, the movement of phonons in a solid can be thought like gas molecules/atoms filling a space. Phonons move around in the solid, and the hot phonons collide with the cold ones, and energy exchange between them takes place, leading to thermal transport. The thermal conductivity of a solid can be described in a simplified form:

1.1 1.1

where ρ is the density of the material (for phonons); cp, the specific heat; v, the group velocity of phonons or electrons; and l, the mean free path (MFP). The physical meaning of l can be regarded as the average distance a phonon travels between two successive collisions. Or it can be pictured in the statistical way like this: if a phonon has an excessive energy δE over those surrounding it, after it travels a distance x, its excessive energy will decay to δEl due to collision as δEl = exp( − l/λ)δE. For electrons, ρcp should take the volumetric specific heat of free electrons. It should be emphasized that Equation .1 represents a very simplified model in which only one energy carrier relaxation time (τ) is used: τ = l/v. Take phonon as an example, since the lattice can vibrate at different frequencies and directions, phonons have a wide band in terms of their frequency, velocity (dispersion relationship), and relaxation time. The velocity and MFP in Equation .1 take an average of the phonon velocity and relaxation time. In classical heat conduction, the heat flux (q′′) and temperature gradient is related to each other by the Fourier law of heat conduction (in one-dimensional heat conduction scenario) as q′′ = − kdT/dx.

1.1.1 Thermal Transport Constrained by Material Size

Heat transfer at micro/nanoscales takes place and deserves special attention and treatment due to the rise of several scenarios: the size of the material is comparable to the MFP of phonons in bulk materials, the existence of extensive nanograins and grain boundaries within the material significantly alters the movement and transport of energy carriers, the way heat conduction happens and its characteristic size is constrained by the physical process of interest, and the heat transfer characteristic length is limited to nanometers by the physical process even if the material itself is at macroscale.

Figure 1.1a shows the phonon scattering by the top and bottom boundary of thin films. The MFP l in Equation .1 for bulk materials usually is induced by phonon–phonon scattering, phonon–electron scattering, or scattering by defects in the material. When the material size is reduced to a very small scale, like the thin film shown in Figure 1.1, boundary scattering of phonons becomes more important in comparison with the phonon–phonon scattering inside the material. This is because boundary scattering roughly is proportional to the size of the surface area, while the inside phonon–phonon scattering is proportional to the material volume. When the film gets very thin, its surface-to-volume ratio: As/V − L−1 becomes very large (L, film thickness). As can be seen, when the film is becoming thinner, the surface boundary scattering becomes more dominant, which will significantly increase the scattering events of phonons, thereby reducing their MFP and thermal conductivity. The discussion provided here is intended to illustrate the general physical picture of energy carriers scattering by surfaces. For detailed derivation of surface scattering behavior and its effect on thermal conductivity, readers are encouraged to read the numerous papers published in this area in the past 20 years. When phonons reach the surface of the film, the scattering can be diffusive or reflective, sort of like the situation when a light beam reaches a surface, mirror reflection and diffusive reflection can take place. As a general rule, when the surface roughness is smaller than the phonon wavelength, the scattering tends to be specular, while diffusive scattering will dominate if the surface roughness is larger than the phonon wavelength.

Figure 1.1 Schematic demonstration of (a) phonon scattering by the top and bottom boundary in thin films and (b) phonon scattering at the grain boundary in nanocrystalline materials.

In nanocrystalline materials (Fig. .1b), when a phonon reaches the grain boundary/interface, it has the probability of transmitting the boundary to the other side or is reflected back by the interface. The interface phonon scattering is determined by several factors, including the acoustic impedance mismatch at the interface, crystographical orientation mismatch at the interface, interface roughness, and structural defect at the interface. The acoustic impedance (Z) is defined as Z = ρc, where ρ is the material density and c is the sound speed. If the acoustic impedance of the materials next to the grain boundary differs significantly from each other, the phonon will have more chance to be reflected at the interface. This is more like when a light beam reaches the interface of two materials (such as water and air), part of the light will transmit the interface and part of it will be reflected. Assuming the light coming inside a medium of refractive index n1 and transmitting to a medium of refractive index n2, the reflection at the interface for normal incidence is [(n1 − n2)/(n1 + n2)]². Similarly, when phonons reach a material interface in the normal direction, the interface reflection is [(Z1 − Z2)/(Z1 + Z2)]² (1). This is understandable since phonons are lattice vibrations. The traveling behavior of the vibration is the stress/acoustic wave, which shares great similarity with the acoustic wave behavior in materials. In the past, significant research has been conducted about phonon scattering by nanoscale and nanostructured materials, and numerous papers have been published by researchers worldwide.

To illustrate how the thermal conductivity becomes anisotropic in nanoscale materials and how it changes with the material size, Figure 1.2 shows the thermal conductivity of a freestanding argon crystal film (2) and nanocrystalline argon (3) (at 30 K) changing with the film thickness and grain size. Figure 1.2a shows that the thermal conductivities in the x, y, and z directions decrease with the decreasing thickness of the film. It reflects the fact that boundary scattering at the top and bottom surfaces introduces diffuse phonon scattering in the three directions. The thermal conductivity in the z direction is more affected by the thickness, which is caused by the smaller size and the free boundary condition applied in this direction. When the thickness is comparable to the MFP of phonons ( ∼ 1.5 nm as calculated later) in argon at 30 K, the thermal conductivity in the z direction varies with thickness significantly due to the strong boundary scattering. After the thickness becomes large, the ratio of boundary scattering events to the internal phonon scattering becomes much smaller. As a result, the thermal conductivity tends to be constant and becomes close to the values in the x and y directions. The thermal conductivity at large film thicknesses is around 0.55 W/m K, which is close to the measured thermal conductivity of argon crystal at 30 K, 0.78 W/m K.

Figure 1.2 (a) Variation of the thermal conductivity of an argon film (at 30 K) with its thickness in the thickness (z) and in-plane (x and y) directions (2) and (b) variation of the thermal conductivity of nanocrystalline argon against the grain size inside. Also shown in (b) is the thermal conductivity of single argon nanoparticles (3).

Figure 1.2b clearly indicates that with the increasing nanograin size, the thermal conductivity of the nanocrystalline material goes up. Compared with our previous result for freestanding nanoparticles consisting of single crystals (also shown in Figure 1.2b) (2), it is found that the thermal conductivity of nanocrystalline materials is a little larger than that of nanoparticles with the same characteristic size. The reason is that the nanograins in the nanocrystalline materials under study are not exactly spheres, but close to cubes, which could have less constraint on the movement of energy carriers than spheres. This will lead to a little longer MFP of phonons in nanocrystalline materials than that in single nanoparticles. Another reason is that for freestanding nanoparticles, the scattering at the boundary is total reflection, whereas for nanograins in nanocrystalline materials, some phonons can penetrate the boundary. As a result, there will be less reduction in thermal transport by boundary scattering in nanocrystalline materials. The thermal conductivity reduction in nanocrystalline argon observed in Figure 1.2b is not induced only by phonon scattering at grain boundaries. In comparison with the bulk counterpart, the nanocrystalline material has a density reduction due to the local disorder at grain boundaries. The density reduction becomes larger for nanocrystalline materials composed of smaller grains. Part of the thermal conductivity reduction observed in Figure 1.2b is caused by the low density of the nanocrystalline material. To rule out the effect of the density on the thermal conductivity reduction, the Maxwell method can be applied to calculate the effective thermal conductivity of nanocrystalline argon assumed full density of the single-crystal counterpart. The equation in the Maxwell method is written as follows (3):

1.2 1.2

where α is the ratio of thermal conductivity of cavities to the thermal conductivity of nanocrystalline argon (here, α = 0); φ, the volume fraction of the cavity; kcal, the calculated thermal conductivity, and keff, the effective thermal conductivity of nanocrystalline argon without cavities.

The result (effective thermal conductivity) is shown in Figure 1.2b as well. It is clear that after taking out the density effect, the effective thermal conductivity of nanocrystalline argon becomes slightly greater but still much less than that of the bulk counterpart. This indicates that the phonon scattering at grain boundaries is the most important factor in the thermal conductivity reduction observed in nanocrystalline material. The presence of phonon scattering at grain boundaries will give rise to a boundary thermal resistance, namely, Kapitza resistance. Assuming the nanograin itself has the same thermal conductivity as the bulk counterpart, the effective thermal conductivity of nanocrystalline materials is related to the Kapitza resistance as follows (3):

1.3 1.3

where R is the Kapitza resistance, d is the characteristic nanograin size, and k0 is the thermal conductivity of bulk argon. In this work, k0 takes the value of 0.55 W/m K based on our molecular dynamics (MD) work on thermal transport in nanoscale argon at 30 K (2). It needs to be pointed out that Equation .13 is derived from the Fourier law. When the nanograin size is extremely small, the uncertainty induced by the heat transfer deviation from the Fourier law could be significant. The Kapitza resistance discussed here includes the effect of the non-Fourier thermal transport in nanograins. The variation of the Kapitza resistance versus the grain size is calculated based on the effective thermal conductivity (shown in Figure 1.2b) and is plotted out in Figure 1.3. The result shows that the Kapitza resistance is not constant over the grain sizes studied in this work. It is in the order of 10−9 m² K/W. For smaller grain sizes, the calculated Kapitza resistance is smaller. On the other hand, the significantly increased boundary interface area in the material overshadows this reduction in the Kapitza resistance, making the overall thermal conductivity smaller.

Figure 1.3 (a) Estimated Kapitza resistance at the nanograin interface for the nanocrystalline material studied in Figure 1.2b (3) and (b) the structure configuration for the nanocrystalline material studied in Reference (3). The nanograin size is about 6.25 nm, and the picture shown here is only for a layer of 0.38 nm thickness in order to clearly show the atom configuration at the grain boundary.

1.1.2 Thermal Transport Constrained by Time

Another kind of micro/nanoscale thermal transport that differs significantly from the classical one is induced by ultrafast thermal excitation, like that in picosecond (10−12 s) and femtosecond (10−15 s) laser–material interaction. In the past, ultrafast laser–material processing has been studied extensively and intensively due to the great advantage of ultrafast lasers in material processing, such as cutting, drilling, welding, sintering, forming, and cleaning. In ultrafast laser–material interaction, the laser heating time is very short, even comparable or shorter than the relaxation time of energy carriers in the material. Under such situations, the Fourier law of heat conduction becomes insufficient and questionable to describe the related heat transfer, especially for the very early stage (including heating and sometime after laser heating) thermal transport. This is because the ultrafast laser heating can quickly bring the material temperature to an elevated level and establish a temperature gradient. On the other hand, heat transfer does not immediately arise in response to the temperature gradient because in order for heat to be conducted, the energy carriers need time to collide with their neighbors, and this time is the energy carrier's relaxation time mentioned above. To account for this special heat transfer, the non-Fourier law of heat conduction has been applied extensively, which usually takes the following form

or in a more straightforward way

A direct consequence of this non-Fourier law of heat conduction is that it eliminates the ambiguity of infinite heat transfer speed encountered within the limit of Fourier's law of heat conduction. In the past, extensive research (mostly theoretical and modeling) has been reported on thermal transport in ultrafast laser heating considering the effect of non-Fourier heat conduction. A thermal wave is usually found inside the material and it decays fast during its propagation. Figure 1.4 shows the temperature distribution predicted by solving the Boltzmann transportation equation (BTE) for phonons in silicon on ultrafast laser heating (4). Also shown in Figure 1.4 is the temperature prediction by the non-Fourier model (hyperbolic heat conduction equation, HHCE) and classical heat transfer model (parabolic heat conduction equation, PHCE). It is obvious that a temperature wave is observed in the prediction using both BTE and HHCE.

Figure 1.4 Temperature distributions along the laser-incident (x) direction in the target for femtosecond laser heating. Target material, silicon; laser wavelength, 266 nm; laser pulse, Gaussian distribution with full-width at half maximum of 50 fs, centered at 100 fs. The laser comes from the x direction, and x = 0 is the surface of the target. Source: From Reference 4.

1.1.3 Thermal Transport Constrained by the Size of Physical Process

As for the third scenario of micro/nanoscale thermal transport, the way heat conduction happens and its characteristic size is constrained by the physical process of interest. One typical example is the heat conduction in the substrate during surface nanostructuring using laser-assisted scanning probe microprobe (SPM) as shown in Figure 1.5. The SPM tip scans over the substrate surface with a distance of several angstroms (10−10 m) to a few nanometers. A pulsed laser beam is manipulated to be nearly parallel to the sample surface to irradiate the SPM tip. Taking a metallic SPM tip as an example, when the laser irradiates the tip, the tip will act like a receiving antenna to collect the laser (an electromagnetic wave). Such laser beam collection will induce an eddy current in the tip (oscillation of electrons). Then the tip will act like an emitting antenna (just like the emitting tower of a radio station) to emit an extremely focused light as illustrated in Figure 1.5. This near-field focused light exists in a very small region ( ≤ 10 nm), while it is extremely enhanced. Figure 1.6 shows the simulation result about the enhanced optical field when a laser (532 nm wave length) shines on a tungsten tip scanning over a silicon substrate. In the model, the distance between the tip apex and the substrate is 5 nm, the tip apex radius is 30 nm, the laser polarization direction follows the axis direction of the tip, and the incident angle of the laser is 10° with respect to the horizontal direction. It is clear that in a small region less than 10 nm below the SPM tip, the electrical field is enhanced significantly ( > 15 times). The local optical field intensity will be more than 200 times stronger than the original incident laser beam. This extremely focused optical field can heat up the substrate, leading to phase change and surface nanostructuring. Since the size of the heating region by the near-field focused optical field is usually less than the MFP of energy carriers in the substrate, the continuum approach becomes questionable for predicting the local thermal transport, phase change, stress, and structure evolution. Special treatment considering the noncontinuous effect at nanoscales must be taken into account when studying the underlying physical processes.

Figure 1.5 Illustration to demonstrate how the near-field focused optical field is formed when a laser is irradiating an SPM tip scanning over a substrate.

Figure 1.6 The electrical field distribution (a) outside the tip and (b) inside the tip when a 532-nm wavelength laser irradiates a tungsten tip. In (a), it is clear that an extremely enhanced optical field arises below the SPM tip in a very small region ( < 10 nm).

The scenarios discussed above for micro/nanoscale thermal transport that need special treatment are not complete to cover all situations but rather to provide typical senses on why micro/nanoscale thermal transport has come to the researchers' attention and what are their potential applications.

1.2 Molecular Dynamics Simulation of Thermal Transport at Micro/Nanoscales

When the characteristic size of thermal transport comes down to micro/nanoscales (comparable to the MFP of energy carriers), the classical equations describing thermal transport cannot be applied directly for a few reasons. First, the thermal transport phenomenon is becoming noncontinuous at micro/nanoscales and it challenges the classical approaches based on the continuum assumptions. Second, the strong size effect by the nanoscale material or nanoscale inside structures significantly reduces the thermal conductivity, making the bulk counterpart's values incapable of being used anymore to describe the thermal transport at micro/nanoscales. Third, at micro/nanoscales, the thermal conductivity is not an intrinsic property of the material anymore. It depends on how it is defined and the physical process it gets involved. To address various challenges and issues in micro/nanoscale thermal transport, MD simulation has arisen as a powerful approach to provide insight into the fundamental physics in micro/nanoscale thermal transport and predict the material thermal behavior at micro/nanoscales under various optical and mechanical excitations. This section focuses on MD simulations to calculate the thermal conductivity of materials and study the nanoscale transport and phase change in laser-assisted SPM-based surface nanostructuring. It is intended to give readers the first (not complete) impressions on how MD simulation is used in studying thermal transport.

1.2.1 Equilibrium MD Prediction of Thermal Conductivity

The basis of MD simulation is to solve the Newtonian equation to obtain the position, force, and velocity of each atom in the system. Each atom has the following movement equation:

1.4 1.4

where mi and ri are the mass and position, respectively, of atom i and N is the total number of atoms in the system. Fij is the interaction force between atoms i and j. For many materials of face-centered crystal (FCC) structure, the Lennard–Jones (LJ) 12-6 potential is a good choice to describe the atomic interaction (2):

1.5 1.5

1.6

where ϕij is the LJ potential between two atoms; ε, the LJ well depth; σ, the equilibrium separation parameter; and rij, the distance between two atoms (rij = ri − rj). When rij is much larger than σ, the two terms in