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Meet the Author
ADRIAN BEJAN is the J. A. Jones Professor of Mechanical Engineering at Duke University in Durham, North Carolina.
ALLAN D. KRAUS is a retired professor and practicing consulting engineer in Beechwood, Ohio. He is a frequent adjunct to the engineering faculty at The University of Akron in Ohio.
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Heat Transfer Handbook
By Adrian Bejan Allan D. Kraus
John Wiley & Sons
ISBN: 0471390151Chapter One
Microscale Heat TransferANDREW N. SMITH
Department of Mechanical Engineering United States Naval Academy Annapolis, Maryland
PAMELA M. NORRIS
Department of Mechanical and Aerospace Engineering University of Virginia Charlottesville, Virginia
18.1 Introduction
18.2 Microscopic description of solids 18.2.1 Crystalline structure 18.2.2 Energy carriers 18.2.3 Free electron gas 18.2.4 Vibrational modes of a crystal 18.2.5 Heat capacity Electron heat capacity Phonon heat capacity 18.2.6 Thermal conductivity Electron thermal conductivity in metals Lattice thermal conductivity
18.3 Modeling 18.3.1 Continuum models 18.3.2 Boltzmann transport equation Phonons Electrons 18.3.3 Molecular approach
18.4 Observation 18.4.1 Scanning thermal microscopy 18.4.2 3[omega] technique 18.4.3 Transient thermoreflectance technique
18.5 Applications 18.5.1 Microelectronics applications 18.5.2 Multilayer thinfilm structures
18.6 Conclusions Nomenclature References
18.1 INTRODUCTION
The microelectronics industry has been driving home the idea of miniaturization for the past several decades. Smaller devices equate to faster operational speeds and more transportable and compact systems. This trend toward miniaturization has an infectious quality, and advances in nanotechnology and thinfilm processing have spread to a wide range of technologicalareas. A few examples of areas that have been affected significantly by these technological advances include diode lasers, photovoltaic cells, thermoelectric materials, and microelectromechanical systems (MEMSs). Improvements in the design of these devices have come mainly through experimentation and macroscale measurements of quantities such as overall device performance. Most studies of the microscale properties of these devices and materials have focused on either electrical and/or microstructural properties. Numerous thermal issues, which have been largely overlooked, currently limit the performance of modern devices. Hence the thermal properties of these materials and devices are of critical importance For the continued development of hightech systems.
The need for increased understanding of the energy transport mechanisms of thin films has given rise to a new field of study called microscale heat transfer. Microscale heat transfer is simply the study of thermal energy transfer when the individual carriers must be considered or when the continuum model breaks down. The continuum model for heat transfer has classically been the conservation of energy coupled with Fourier's law for thermal conduction. In an analogous manner, the study of "gas dynamics" arose when the continuum fluid mechanics models were insufficient to explain certain phenomena. The field of microscale heat transfer bears some striking similarities. One area of similarity is in the methodology. Usually, the first attempt at modeling is to modify the continuum model in such a way that the microscale considerations are taken into account. The more common and slightly more difficult method is application of the Boltzmann transport equation. Finally, when both of these methods fail, the computationally exhaustive molecular dynamics approach is typically adopted. These three methods and specific applications will be discussed in more detail.
Figure 18.1 demonstrates four different mechanisms by which electrons, the primary heat carriers in metallic films, can be scattered. All of these scattering mechanisms are important in the study of microscale heat transfer. The mean free path of an electron in a bulk metal is typically on the order of 10 to 30 nm, where electron lattice scattering is dominant. However, when the film thickness is on the order of the mean free path, boundary scattering comes important. This is referred to as a size effect because the physical size of the film influences the transport properties. Thin films are manufactured using a number of methods and under a wide variety of conditions. This can have a serious influence on the microstructure of the film, which influences defect and grain boundary scattering. Finally, when heated by ultrashort pulses, the electron system becomes so hot that electronelectron scattering can become significant. Thus, microscale heat transfer requires consideration of the microscopic energy carriers and the full range of possible scattering mechanisms.
In the first section of this chapter we focus on defining and describing the microscopic heat carriers. The free electrons are typically responsible for thermal transport in metals. The governing statistical distribution is presented and discussed, along with equations for thermal conduction and the electron heat capacity. In an insulating material, thermal transport is accomplished through the motion of lattice vibrations called phonons. These lattice vibrations or phonons are discussed in detail. The primary heat carriers in semiconductor materials are also phonons, and therefore the thermal transport properties of semiconductors are determined in the same manner as for insulating materials. The formulations for these energy carriers are then used to explain and calculate the phonon thermal conductivity and lattice heat capacity of crystalline materials.
Experimental observation and measurement of microscale thermophysical properties is the subject of the next section. These techniques can be either steadystate, modulated, or pulsed transient techniques. Steadystate techniques typically focus on measuring the surface temperature with high spatial resolution, while the transient techniques are better suited for measuring transport properties on microscopic length scales. The majority of these techniques utilize either one or more of the following methods for determining thermal effects; nanoscale thermocouples, the temperature dependence of the electrical resistance of a microbridge, or thermal effects on the refractive index monitored using optical techniques. Three common methods for measuring microscale thermal phenomena are discussed in more detail.
In the final section we focus on specific applications where consideration of microscale heat transfer is important. For example, the microelectronics industry is perpetually looking for materials with lower dielectric constants to keep pace with the miniaturization trend. Unfortunately, materials that are good electrical insulators are typically also good thermal insulators. As another example, highpower diode lasers and, particularly, vertical cavity surfaceemitting laser diodes are often limited by the dissipation of thermal energy. These devices are an example of the increased trend toward multilayer thinfilm structures. Recently, developers of thermoelectric materials have been using multilayer superlattice structures to reduce thermal transport normal to the material. This could significantly increase the efficiency of thermoelectric coolers. These examples represent just a few areas in which advancements in nanotechnology will have a dramatic impact on our lives.
18.2 MICROSCOPIC DESCRIPTION OF SOLIDS
To proceed with a discussion of microscale heat transfer, it is necessary first to examine the microscopic energy carriers and the basic heat transfer mechanisms. In metals, thermal transport occurs primarily from the motion of free electrons, while in semiconductors and insulators, thermal transport occurs due to lattice vibrations that travel about the material much like acoustic waves. In this chapter a conscious decision was made to minimize the presentation of quantum mechanical derivations and focus on a more physical presentation. More detailed descriptions of the material presented in this section can be found in most basic solidstate physics textbooks such as those of Ashcroft and Mermin (1976) and Kittel (1996). The theoretical descriptions of electrons and phonons usually include an assumption that the material has a crystalline structure. Therefore, this section begins with the basic relevant concepts of crystalline structures.
18.2.1 Crystalline Structure
The atoms within a solid structure arrange themselves in an organized manner such that the potential energy stored within the lattice is minimized. If the structure has longrange order, the material is referred to as crystalline. Once this structure is formed, smaller individual pieces of the crystalline can usually be identified that, when repeated in each direction, comprise the entire solid material. This type of material is then referred to as single crystalline. Most real materials contain grains, which are single crystalline; however, when the grains meet, a grain boundary is formed and the material is described as polycrystalline. In this section the assumption is made that the materials are single crystalline. However, the issues of grain size and boundaries, which arise in polycrystalline materials, are very important to the study of microscale heat transfer since the grain boundaries can scatter energy carriers and impede thermal transport.
The smallest of the individual structures that make up the entire crystal are called unit cells. Once the crystal has been broken down into unit cells, it must be determined whether the unit cells make up a Bravais lattice. Several criteria must be satisfied before a Bravais lattice can be identified. First, it must be possible to define a set of vectors, R, which can describe the location of all points within the lattice,
(18.1) R = [n.sub.i]a.sub.i] = [n.sub.i]a.sub.1] + [n.sub.2][a.sub.2] + [n.sub.3][a.sub.3]
where [n.sub.1], [n.sub.2], and [n.sub.3] are integers. The set of primitive vectors [a.sub.i] are defined in the same manner. The three independent vectors [a.sub.i] can be used to translate between any of the lattice points using a linear combination of these vectors. Second, the structure of the lattice must appear exactly the same regardless of the point from which the array is viewed. Described in another way, if the lattice is observed from the perspective of an individual atom, all the surrounding atoms should appear to be identical, independent of which atom is chosen as the observation point.
There are 14 threedimensional lattice types (Kittel, 1996). However, the most important are the simple cubic (SC), facecentered cubic (FCC), and bodycentered cubic (BCC). These structures are Bravais lattices only when all the atoms are identical, as is the case with any element. When the atoms are different, these structures are not Bravais lattices. The NaCl structure is an example of a simple cubic structure, as shown in Fig. 18.2a, where the sodium and chloride atoms occupy alternating positions. For this structure to meet the criteria of a Bravais lattice, to be seen as identical regardless of the viewing point, the Na and Cl atoms must be grouped. Whenever two atoms are grouped, the lattice is said to have a twopoint basis. In a Bravais lattice each unit cell contains only one atom, while each unit cell of a lattice with a twopoint basis will contain the two grouped atoms. When each sodium atom is grouped with a chlorine atom, the result is a Bravais lattice with a twopoint basis, shown in Fig. 18.2b by dark solid lines.
It is also possible to have a lattice with a basis even if all the atoms are identical; the most important crystal structure that falls into this category is the diamond structure. The group IV elements C, Si, Ge, and Sn can all have this structure. In addition, many IIIV semiconductors, such as GaAs, also have the diamond structure. The diamond structure is a FCC Bravais lattice with a twopoint basis, or equivalently, the diamond structure is composed of two offset FCC lattices.
For the purposes of understanding microscale heat transfer, there are two important concepts regarding crystalline structures that must be understood. The first is the concept of a Bravais lattice, which has just been presented. It is important in the study of energy transport on a microscale basis to know the Bravais lattice structure of the material of interest and whether or not the crystal is a lattice with a basis. The second important concept is the idea of the recriprocal lattice.
The structure of a crystal has an intrinsic periodicity that begins with the Bravais lattice unit cell. Certain properties, such as the electron density of the material, will vary between lattice sites but will vary periodically with the lattice. It is also common to be dealing with waves or particles with wavelike properties traveling within the crystal. In both cases, it is advantageous to define a recriprocal lattice. The set of all wave vectors k, which represent plane waves with the periodicity of a given Bravais lattice, is described by the recriprocal lattice vectors. Given the Bravais lattice vector R, the reciprocal lattice vectors can be defined as the set of vectors that satisfy the equation
(18.2) [e.sup.iK·(r+R) = [e.sup.ik·r]
where r is any vector within the lattice. It can be shown that the recriprocal lattice of a Bravais lattice is also a Bravais lattice, which also has a set of primitive vectors b. It turns out that the recriprocal lattice of a FCC lattice is a BCC lattice, the reciprocal lattice of a BCC lattice is FCC, and the reciprocal lattice of a SC lattice is still simple cubic.
Once the reciprocal lattice vectors have been defined, the Brillouin zone can be found. The Brillouin zone is a unit cell of the reciprocal lattice centered on a particular lattice site and containing all points that are closerto that lattice site than to any other lattice site. According to the definition of a Bravais lattice, if the Brillouin zone is drawn around each lattice point, the entire volume will be filled and each Brillouin zone will be identical. The manner in which the Brillouin zone is constructed geometrically is: (1) Draw lines from one reciprocal lattice site to all neighboring sites, (2) draw planes normal to each line that bisect the line, and (3) end each plane once it has intersected with another plane. The result is a choppy sphere that contains all the points closer to the central reciprocal lattice point than any other reciprocal lattice point. Threedimensional representation of the Brilloiun zone can be found in most solidstate physics texts (Kittel, 1996).
18.2.2 Energy Carriers
Thermal conduction through solid materials takes place both by the transport of vibrational energy within the lattice and by the motion of free electrons in a metal.
Continues...
Table of Contents
Preface.
Contributors.
1. Basic Concepts (Allan D. Kraus).
2. Thermophysical Properties of Fluids and Materials (R. T Jacobsen, E. W. Lemmon, S. G. Penoncello, Z. Shan, and N. T. Wright).
3. Conduction Heat Transfer (A. Aziz).
4. Thermal Spreading and Contact Resistances (M. M. Yovanovich and E. E. Marotta).
5. Forced Convection: Internal Flows (Adrian Bejan).
6. Forced Convection: External Flows (Yogendra Joshi and Wataru Nakayama).
7. Natural Convection (Yogesh Jaluria).
8. Thermal Radiation (Michael F. Modest).
9. Boiling (John R. Thome).
10. Condensation (M. A. Kedzierski, J. C. Chato, and T. J. Rabas).
11. Heat Exchangers (Allan D. Kraus).
12. Experimental Methods (José L. Lage).
13. Heat Transfer in Electronic Equipment (Avram BarCohen, Abhay A. Watwe, and Ravi S. Prasher).
14. Heat Transfer Enhancement (R. M. Manglik).
15. Porous Media (Adrian Bejan).
16. Heat Pipes (Jay M. Ochterbeck).
17. Heat Transfer in Manufacturing and Materials Processing (Richard N. Smith, C. Haris Doumanidis, and Ranga Pitchumani).
18. Microscale Heat Transfer (Andrew N. Smith and Pamela M. Norris).
19. Direct Contact Heat Transfer (Robert F. Boehm).
Author Index.
Subject Index.
About the CDROM.