Physics and methods of altering thermal conductivity in nanostructures

Transcription

1 1 - Pearson Physics and methods of altering thermal conductivity in nanostructures Richard L. Pearson III University of Denver Graduate Comprehensive Exam: Written January 15, 2013 richard.pearson@du.edu Abstract: Nanostructures with particular thermal conductivities are needed in many applications; some are needed with high conductivities to aid in removing heat from devices; others require materials and structures with low thermal conductivities to prevent heat flow from reaching heat-sensitive equipment. The altering of a structure s operating temperature and size provides a means to adjust its thermal conductivity. Recent theoretical calculations have described ways to alter phonon band structures in thin films and superlattices, in order to both minimize and maximize a structure s thermal conductivity. This paper describes the mechanisms that provide that alteration. 1. Introduction Current limitations to common technologies such as computer processing (Hassan, Humaira, & Asghar, 2010), construction (T. W. Love, 1973), automobiles, batteries (Whittingham, 2012), and photovoltaics (Nayak, Garcia-Belmonte, Kahn, Bisquert, & Cahen, 2012) have been set by the materials used in their creation. Simply put, the properties of materials shape how they can be used. Properties of materials become extremely intriguing when fabricated devices are on the same size-scale as its material s crystalline structure. Nanotechnology research is interested in the properties of devices at those nanometer scales, particularly those properties concerning management and manipulation of heat flow and energy conversion. For instance, thermal management is crucial in integrated circuits (Goodson & Ju, 1999) and thermoelectric energy conversion (Chen & Shakouri, 2002). Thus, the bulk of this paper will focus on one particular property relating to the transfer of heat in nanostructures: thermal conductivity. More specifically, a focus on the physics of techniques that alter the thermal conductivity of thin film and superlattice structures will be described. Background information regarding this topic will be provided in 2. Discussion of phonons and their role in heat transport is found in 3. Section 4 derives

2 2 - Pearson the equations for thermal conductivity. Methods to optimize thermal conductivity for thin films and superlattices are presented in Background information Before discussing how structural interfaces affect phonons and the thermal conductivity of the structure, there are other things that need to be addressed. For example, the definition of a phonon must be conferred. Though the following topics are hefty subjects in-and-of-themselves, a small portion has been set aside at the beginning of this paper to describe facets of physics in connection with the overall topic in mind. Specifically, crystal structures, lattice vibrations, and phonons will be presented. 2.1 Crystal structures and lattice vibrations Crystals are created by atoms that are chemically bound together and create a form of periodicity and symmetry. The arrangements of the atoms at their various locations of minimum energy create a lattice. Since crystal lattices are bound together by potential energies, the atoms are able to vibrate/oscillate around the minimum energy point. In effect, a lattice can be modeled by attaching springs from one atom to another, with the force of the spring representing the slight perturbations of the atomic potentials. A simple, yet effective portrayal of a diatomic, linear (and periodic) lattice is shown in Figure 1. Two equations of motion, one for each atom, describes the unit cell s motion (the force constant between atoms is ): ( ) ( ) Figure 1. A cartoon illustrating a linear, periodic, diatomic lattice. Vibrations are illustrated and identified. Image obtained from Singh (2010).

3 3 - Pearson The solution to a second order differential is a plane wave defined for each particle, where is the wave number of the plane wave: ( ) ( ) These solutions can go back into the equations of motion. By doing so, we now have coupled eigenvalue equations that can be solved with the eigenvalue [ ]. The dispersion relation results from the determinant being set to zero: ( ) [( ) ] This frequency equation provides a dependence on the wave number and the distance between identical/periodic planes in the lattice ( ). This, in fact, holds true for all lattices (Ibach & Luth, 2009). The sine-squared function permits the periodicity of the waves, but also provides quantized frequencies. Classical physics provides quantized frequencies and associated energies; the quantization of the amplitudes comes from quantum mechanics. These quantized, atomic vibrations are called phonons (Mahan, 2011). 2.2 Optical and acoustic modes Solutions to this dispersion relation are plotted in Figure 2. There are two separate vibration branches: optical and acoustic. Along with the two branches, there are two types of modes in branch: longitudinal and transverse (not shown in figure). These two additional modes account for the direction that the atoms move relative to the propagation direction (either parallel or perpendicular). These extra modes contain different energies, which is important to have in calculations and modeling. Some aspects concerning the optical phonon branch are as follows: 1) atoms vibrate in opposition to one another; 2) the frequencies are higher than the acoustic branch; 3) the energies are higher than the acoustic branch; 4) near, energies are still quite large; and 5) at low temperatures, optical contribution is negligible (Singh, 2010).

4 4 - Pearson In contrast, here are some aspects concerning the acoustic branch: 1) atoms vibrate in phase; 2) lower frequencies (sound waves); 3) lower energies; 4) near, ; 5) at low temperatures, acoustic waves are still valid, due to the long wavelength. As for the transverse and longitudinal modes in each branch, the simple difference between them is that the longitudinal modes Figure 2. Solutions to the dispersion relation are plotted here, with frequency along the vertical and wave number on the horizontal. There are two separate branches that describe the atomic vibrations, or phonons. Image obtained from Singh (2010). generally have a higher energy. These are the modes that have displacement along the propagation direction (a compression wave), where the restoring forces are stronger than for the transverse waves (Ziman, 1960). 2.3 Phonon heat capacity Albert Einstein and Peter Debye provided additional ways to model and think of lattice vibrations. They provided a way to find the total energy in lattice structures, which led to phonon heat capacities Einstein s model In 1907 Albert Einstein first described a solid as a collection of identical oscillators that consisted of a quadratic potential energy, effectively a collection of harmonic oscillators (Schroeder, 2000). Based on that model, he derived an expression for the energy of a lattice and subsequently a volumetric heat capacity (by using, where an objects temperature by ), labeled as the Einstein model: number of atoms in the lattice, energy units for the oscillators. is the heat capacity and is the Boltzmann constant, is the amount of energy needed to raise ( ) (, and ), where is the total or the size of the

5 5 - Pearson The Einstein approach is consistent with observations when and. However, when, the model predicts an exponential decay as instead of the observed dependence. Fortunately, Peter Debye provided his own model five years later Debye s contribution Debye realized there was a maximum number of vibrational modes available in a solid. He associated the energy of a phonon as, where is the speed of sound in the solid, is the mode, and is the maximum phonon wavelength of twice the crystal length. Then using Bose-Einstein statistics, found the total energy in lattice vibrations as ( ) where the accounts for the three polarization states of the phonon in each state, and where is the Planck function that describes the average number of energy units the phonon has at a given. Debye also integrated over a sphere, rather than a cube, and arrived at a total energy for lattice vibrations of ( ) By definition, and ( ). The Debye temperature is defined as ( ) and which is associated with the highest allowed mode of vibration. It is from this total energy equation that we can look at the phonon contribution to the heat capacity. 3. Heat transport and phonon scattering A simple definition and description of a phonon was provided in the previous section. As was mentioned, a phonon is a quantization of the vibrational modes in solids. Heat is predominately transported in a solid or crystal by those vibrational modes. There is an electron portion of heat transport, but in insulators and on the scale and temperatures that we are discussing, we ignore the electrical part and move forward with

6 6 - Pearson the phonon analysis. Therefore, to analyze the effectiveness of the phonon transport, we will look at events that prohibit the phonon from its transporting. 3.1 Heat transport How efficient a structure is in transporting heat is displayed in the structure s thermal conductivity measurement. Thermal conductivity has units of, which is energy per time per distance per temperature. Therefore, higher the thermal conductivity, the more energy a structure is able to move per time and distance. Ballistic heat transport is the movement of phonons without inhibition. This limit is nearly reached at very low temperatures and small structure size, when the mean free path of the phonon is much larger than the characteristic size of the structure (Chen G., 2002). So then what makes a material or structure a less-than-perfect heat conductor? What interrupts the movement of phonons? The answer is simply this: a structure with a thermal conductivity has scattering events that prohibit phonons from moving through the lattice. 3.2 Phonon scattering There are many different types of scattering that could be discussed here. Current practice requires writing the scattering terms as either a mean free path ( ) or a relaxation time ( ). These two factors describe the distance a phonon moves between scattering events and the associated time between those events. Three scattering events will be presented here, along with how they are included together to impact the thermal conductivity Matthiessen s rule This is a simple rule that combines all of the relaxation time scattering events into one. It is of the same format as when combining resistors in parallel circuits:

7 7 - Pearson The is then input into a thermal conductivity equation. The scattering components from left to right are boundary, defects and dopants, normal, and Umklapp processes. The sum at the end is to include any other scattering process that could be included in this calculation. All relaxation time equations are taken from Chantrenne, Joulain, & Lacroix (2009) Boundary scattering The boundary scattering is when a phonon intersects with the edge of the structure. This relaxation time is solely associated with the characteristic size of the structure: the closer the characteristic size is to the mean free path of the phonon, the more this scattering will influence the thermal conductivity. The form of the equation is, where is the average group phonon velocity, is the characteristic size of the structure, and is a fitting factor from to (Chantrenne, Joulain, & Lacroix, 2009). Additional discussion is found in Ziman (1960) and Srivastava (1990) Anharmonic scattering Anharmonic scattering primarily refers to two scattering processes: normal and Umklapp. These refer to phonon-phonon collisions and the wave vectors associated with the collisions and the resulting phonon. The majority of the derivations and equations found in this paper are done in -space, or momentum space. In solid state physics, this is a reciprocal space called the Brillouin zone. The importance of this reciprocal space is that all of the physics necessary to describe properties of crystals and atomic arrangements can be mathematically completed in this space (Beiser, 2003). The anharmonic scattering refers to the addition of the momentum vectors, as shown in Figure 3. As can be seen, the Umklapp collisions produce a third phonon wave vector that opposes the heat conduction. The normal process does not and so contributes very little to the overall resistance of heat transfer. On the other hand, the Umklapp process provides a significant contribution to the decrease in thermal conductivity. Its relaxation time equation is presented by Joshi and Majumbar (1993):, where and are constants relating to mass and crystal structure, respectively.

8 8 - Pearson Figure 3. Normal (left) and Umklapp (right) phonon-phonon scattering depicted with two momentum vectors colliding and creating a third phonon. Note the difference between the two: all of the normal scattering processes have resulting momenta continuing along the same general as the others. The Umklapp process is a collision that does not conserve momentum and is a resistive factor in the thermal conductivity. Images obtained from Chang (Solid State Physics). 4. Thermal conductivity Thermal conductivity ( ) is an important material and structure property for applications involving the management and manipulation of heat, especially in nanostructures such as integrated circuits, data storage, as well as thermophotovoltaic and thermoelectric devices (Chen, Borca-Tasciuc, & Yang, 2004). An optimization of for various applications has fueled studies on the subject; both experimental and theoretical investigations as to how to increase or decrease manipulate have been published. A few of the ways to includes the following: increasing/decreasing a structure s operating temperature, alter the thickness of a thin film (Cahill, Bullen, & Lee, 1999), and change the periodicity of superlattices (Garg, Bonini, & Marzari, 2011). These will be discussed in 5. However, before delving into the ways to alter the thermal conductivity of a structure, numerical expressions must be presented. This will include a simple, but effective derivation from Fourier s law. The second expands on Fourier with the Boltzmann transfer equation. Note that very little step-by-step guides of the derivations could be located at the time of writing this paper, which is why they are included; hopefully, this will aid others attempting to put the pieces together. 4.1 Kinetic theory derivation Kinetic theory a theory that describes the microscopic behavior and interactions of particles in order to explain macroscopic relationships (Nave, 2013) provides an approach to derive an equation for thermal

9 9 - Pearson conductivity,. This approach displays the dependencies of, but fails to completely describe it on nanometer scales. Following Ziman (1960), we start by relating thermal conduction to a temperature gradient, according to the 2nd law of thermodynamics: a system in non-equilibrium, i.e. with a temperature gradient, will transfer (thermal) energy, e.g. conduction, to obtain equilibrium. Therefore, in a collection of particles (say, phonons) that each possess a heat capacity1,, we define a temperature gradient,. If a phonon travels with a velocity ( ), then its energy ( ) would have to change to keep it in thermal equilibrium with its local surroundings, as per the energy-per-time equation below: The total is then a summation of all of the phonons in a volume,. If the phonons contribute to this current in the distance they travel between scattering events, then one of the components of the volume can be described by the mean free path,, where is the relaxation time, or the time between scattering events; so, we can take where the volume. Therefore, if the total number of phonons is, then the total energy flow can be written as where ) ), then we can find the average velocity of the ) and the average mean free path ( or average velocity does not matter (denoted as ). By definition,. Because the motion is isotropic, the direction of the ). A thermal conductive flux can be found as 1 )(, or the total specific heat. If we assume isotropic movement throughout the volume (the phonon group ( ( velocity is the same in every direction, e.g. is the area of the other dimensions of We have chosen this heat capacity so that it requires an energy of the collection of particles by. per particle to change the temperature of

10 10 - Pearson This is the form of Fourier s heat conduction law, where the thermal conductivity is defined as Note that, or the specific heat at constant volume. From the derivation one can see that the thermal conductivity relies on the specific heat, average velocity, and mean free path of the phonons. Each is responsible for how materials react in different regimes, including low/high temperatures and macro/nano-scales. However, there are some limitations to this definition of the thermal conductivity. It is important to explain that Fourier s law is a classical approach, meaning it holds true for structures with characteristic length scales ( ) greater than the phonon mean free path (MFP):. However, it is still widely used to approximate thermal conductivity of nanostructures because other approaches require so much more derivation and computational effort. It is a convenient way to determine a number of properties of a structure. In nanostructures, this is commonly the case. Also, the length of the mean free path or relaxation time is affected by a number of different scattering events, which also must be considered when deriving thermal conductivity values. Additionally, the macroscopic definition of a temperature gradient begins to lose meaning as regions over which a local temperature can be defined shrinks (Joshi & Majumdar, 1993). Therefore, a quantum definition of temperature must be described (Cahill, et al., 2003). Though that will not be specifically discussed here, the next section demonstrates how the Boltzmann transport equation helps to expand the kinetic solution. 4.2 Boltzmann transport equation derivation The Boltzmann equation The Boltzmann transport equation (BTE) is another approach to solving for the thermal conductivity of phonons in a material. It permits the usage of phonon angular frequencies and Bose-Einstein statistics. The subsequent discussion follows details from Chantrenne, Joulain, & Lacroix (2009), Ibach & Luth (2009), and Sellan (2012).

11 11 - Pearson The general form of the BTE describes changes to a distribution function ( ), where is the average number of phonons at a time in the volume of and wave vector : ( ) ( ) ( ) ( ) The first term on the left is the time evolution of the distribution function. The second term describes the movement of phonons within the structure. Another way of writing the phonon angular frequency portion of that term is ( ) ( ), which is by definition the phonon group velocity. We can also write the second-half of that term as. The last term on the left side of the equation covers any external force ( ) acting on the phonon distribution (Chantrenne, Joulain, & Lacroix, 2009). This term is neglected, but is important when using the BTE to describe the motion of a distribution of electrons in an external electric or magnetic field. The phonon distribution scattering term could be considered the most vital part of the phonon BTE. It accounts for all of the scattering processes the phonon distribution participates in. Simplifications to this term are necessary to solve the equation. This is typically performed by using the relaxation time approximation, which is to say that a perturbed phonon distribution will return to an equilibrium state in a certain time ( ) and go through another disturbance. Therefore, we state that ( ) ( ) where is the phonon distribution function at thermodynamic equilibrium. The phonon scattering events are collected via the Matthiessen rule as discussed in Coulomb s law & Boltzmann statistics Before we can move forward with the BTE, Boltzmann statistics must be addressed in the context of crystal lattice structures and the associated forces. The bonds that make up a lattice structure depend on the attractive and repulsive forces between the lattice atoms/ions. Bonding occurs when the potential energies associated with those lattice forces ( )is minimized. The minimization in the potential

12 12 - Pearson energy profile is estimated by a harmonic oscillator. This estimation allows crystal dynamics to predict normal mode frequencies in oscillations, but also estimates probable energy states for phonon transport of thermal energy (Ibach & Luth, 2009). Specifically, the harmonic estimation predicts energies of ( ) with a probability given by the Boltzmann distribution, ( is the partition function and is the Boltzmann constant). The partition function is defined as ( ) ( ) ( ) Therefore, the probability is given as [ ( ) ] ( ) ( ) To find the average energy (which is what is needed for thermal energy transport), we have ( ) ( ) ( ) ( ) [ ] [ ( ) ] { [ ( ) ] } ( ) To get rid of the summation, the approximations of ( ) and ( ) were used. The Bose-Einstein distribution, ( ), was substituted in for the final equation. The derivation of the average energy will be used at the end of the next section.

13 13 - Pearson Fourier law expansion A thermal conductive flux can be obtained by creating a thermal current by deviating the phonon distribution from an equilibrium state. Therefore, we can write the flux as ( ) where is the volume of the region in interest. If we remain in an isotropic environment where thermal currents are in a steady-state (no acceleration or velocity change), then the first term in the phonon BTE now goes to zero ( ). Then, the BTE is ( ) ( ) ( ) and can be rearranged to input into the flux equation above. We are then left with ( ) ( ) ( ) ( ) ( ) ( ) This now has the same form as Fourier s conduction law, so that the thermal conductivity is ( ) ( ) ( ) ( ) ( ) The derivation with the BTE allows for computation models to include frequency and wave vector dependencies. The phonon distribution is simply given by the Bose-Einstein distribution. If we would like to write the thermal conductivity equation in terms of average phonon energy, we can include the from the end of the previous section after taking a temperature derivative: ( ), so that ( ) ( ). The importance of tracking the wave vectors and phonon frequencies becomes apparent as we look into how to effectively optimize the thermal conductivity of a nanostructure.

14 14 - Pearson In correlating this equation with the simpler kinetic version, a phonon specific heat can be identified as ( ) ( ), which is another material property that can be measured, modeled, and manipulated. 5. Optimizing thermal conductivity The optimization of thermal conductivity comes in twofold: improving maximum and minimum limits. Obviously, the development of various apparatus will require one or the other. Finding a structure that optimizes performance (say, in thermoelectronics or thermophotovoltaics) will determine the quality of the apparatus. There are many ways to enhance the thermal conductivity: material choice, temperature selection, nanostructure fabrication, size manipulation, and others. The scope of this paper prohibits an exhaustive look into each optimization technique. However, a brief summary regarding the effects of temperature and size specifically with thin film thickness and superlattice periods on thermal conductivity will be included in this section. 5.1 Temperature effects From the kinetic derivation of the thermal conductivity, it is not completely clear that there is any sort of temperature dependence. However, in different temperature regimes, a component of ( ) dominates as per each factor s own dependence on temperature. The temperature limits are based on the system s relation to the Debye temperature,, described in High temperature: To investigate the high temperature regime, we will look at the temperature dependence of the mean free path. The average group velocity and heat capacity will not be an overwhelming influence, as per the temperature dependence of the mean free path. However, an analysis of the heat capacity is presented for justification. We begin with applying the constraint of onto the Debye energy (from 2). It is reproduced here: ( ) In the high temperature regime, the

15 15 - Pearson upper limit of the integral becomes small, as does. This permits the exponential in the denominator to become, which allows us to write the energy as [ ] This is the same energy that can be defined by the equipartition theorem: the average energy of any quadratic degree of freedom at a temperature is. For a solid crystal made up of atoms, each with six degrees of freedom, the total energy is (Schroeder, 2000). To find the heat capacity, we simply use its definition,, which follows the Einstein approximation, too. Therefore, the heat capacity is a constant at high temperatures. Recall that the mean free path is defined as, where is the relaxation time or the time between scattering events. As described in 3, the factor accounts for all of the phonon scattering processes, which include, but are not limited to, boundary scattering, impurity scattering, Umklapp and normal processes. In the high temperature limit, the dominant process is the Umklapp process, which allows the mean free path to be approximated as (Mahan, 2011). This gives the thermal conductivity a dependence at high temperatures Low temperature: At low temperatures, the influence of anharmonic processes and the average group velocity are diminished and do not significantly contribute to the mean free path. The phonons at these temperatures have long wavelengths. This allows them to only scatter at the surface of the solid. The contribution of the boundary scattering at these lower temperatures can be described by the heat capacity. We will, therefore, begin again with our derived energy, ( ). At small s, the integral limit approaches, permitting the use of known definite integrals. We are left with the following energy:

16 16 - Pearson This provides us with a phonon heat capacity of. Therefore, the thermal conductivity has a dependence at low temperatures Summary If we compile the two previous limits, one will see that at high temperatures (and increasing temperatures), the thermal conductivity decreases; at low temperatures (and decreasing temperatures), the thermal conductivity also decreases. This presents a curve that includes a local maximum in its thermal conductivity. An example of this is shown by single crystal Si in Figure 4. The Si thermal conductivity peak is around, but is almost times less at and. There is a peak in the thermal conductivity as the effects of surface and anharmonic scattering reach a minimum. Obviously, systems can be altered to match the thermal conductivity specifications of materials, especially when seeking for a high thermally conductive material, as with heat sink devices to cool CPU s (Seshasayee, 2011). However, there are many systems that are in need of low thermally conductive materials, which inherently become an insulation source (Accuratus, 2005). Many of these applications are in temperature regions between the limiting cases presented previously. Recent research has focused on finding other ways to decrease thermal conductivity of materials in intermediate temperature regions (i.e. around room temperature). The remainder of this paper will discuss those additional techniques. Figure 4. This plots the temperature dependence of thermal conductivity in single crystal germanium. Clarke (2003) reproduced this figure from previous data obtained by Glassbrenner & Slack (1964) and included the different dependency regions presented in 5.1. The first region, where size effects dominate, has. In region II, the peak is associated with a combination of I and III effects. Region III has where anharmonic scattering (Umklapp) dominates. Finally, region IV is where, associated with the phonon mean free path reaching a minimum.

17 17 - Pearson 5.2 Size effects Because phonons can be described as quantum waves, a phonon-interface scattering event can be described similar to an electromagnetic wave hitting an interface (Brillouin, 1946). Phonon-interface scatterings are more important in thermal conductivity measurements as the structure s characteristic size decreases. When the structure sizes becomes smaller than the phonon mean free path ( ), many of the phonon scattering events will take place at the interfaces or boundaries of the structure. This is fairly intuitive. When the structure size decreases below the average distance a phonon travels between collisions (i.e. the mean free path), then the scattering events will occur more often at the boundaries of the structure. Additionally, the decrease in creates, effectively, more boundaries-per-area in the structure; therefore, more events are bound no pun intended to happen. The significance of is described for thin films and superlattice structures in the subsequent sections Thin film thickness The thermal conductivity dependence on the thickness of a thin film is demonstrated by the relaxation time describing boundary scatterings, decreases so that, as described in We know that as the thickness, the thermal conductivity becomes more dependent on. Therefore, one would expect a decrease in thermal conductivity as thickness decreased. Many experimental results have supported this prediction. One of these experiments was from Cahill, Bullen, & Lee (1999) who observed this effect on SiO2 film (Figure 5) in order to demonstrate the sensitivity of a newly developed method of measuring thermal conductivity, called the 3 method.2 This method permitted them to find no statistically significant decrease in the thermal conductivity until the film reached. Figure 5 shows thermal conductivity s dependence on temperature and thin film thickness. 2 Simply, the 3 method measures an oscillation of temperature and resistance at numerous frequencies. The measurements are taken of a heater metal wire on the thin film. The slope of a ( ) plot provides the thermal conductivity. Refer to Cahill, Katiya, & Ableson (1994) for a complete description of the measurement method.

18 18 - Pearson In a more recent, theoretical paper, McGaughey et al. (2011), outline a more complete way to model thin film conductivity. They begin with the definition of thermal conductivity based on the relaxation time approximation of the Boltzmann transport equation, as derived ( ) ( in 4.2: ( ) ). The importance of their work is the inclusion of more than one group velocity and relaxation time, unlike previous models: the phonon group velocity is a function of the wave vector dispersion branch and the Figure 5. This shows two dependencies of the thermal conductivity: temperature and film thickness in an SiO2 film. No conductivity decrease is seen until the film is 100 nm thick. Before that, the films followed the bulk value. Figure retrieved from Cahill, Bullen & Lee (1999). (or groups of modes); the relaxation time is a function of and the film thickness. The Matthiessen rule is still used to combine anharmonic and boundary scattering for the relaxation time calculation. Additionally, there are heat capacities associated with each phonon mode ( The ). part of the group velocity is directional: in a thin film, one can describe phonon properties in a cross-plane or in-plane orientation. The only difference between the orientations is in the group velocity definitions. Otherwise, their thermal conductivity would be the same. Though there could be large differences in the cross- and in-plane thermal conductivities (Sadeghi, Djilali, & Bahrami, 2011), the modeling efforts here only show significant difference at smaller thicknesses. Still, the in-plane thermal conductivity exhibits the same dependence on film thickness as the cross-plane term. This shows the importance of size effects at these small scales. Figure 6 displays the in-plane thermal conductivity model from McGaughey et al. (2011) and plots it against previous models and experimental results. By including a mode-dependent group velocity and relaxation time, the models were better able to fit the

19 19 - Pearson Figure 6. The New Model line displays the thermal conductivity solution from McGaugey et al. (2011), for a silicon thin film at 300 K. The other two lines are solutions to previous models that did not include mode dependent group velocity or relaxation time. Thermal conductivity is plotted on the vertical axis. The film thickness is along the horizontal axis: nanometers on top, unit-less on the bottom. Note the thermal conductivity dependence on L. Image retrieved from McGaugey et al. (2011). experimental results. However, all the models predicted higher conductivities than were obtained, including their treatment of a nanowire with the same diameter as the thickness of the thin films.3 The discrepancy between the models and experiments falls on the assumptions made during the model derivation, after obtaining the initial thermal conductivity from the BTE. The assumptions included the following: 1) only acoustic phonons contribute to the thermal conductivity; and 2) only one acoustic branch of phonons exists, i.e. there is no difference between longitudinal or transverse waves. One must be reminded that the overall goal of McGaughey et al. (2011) was to provide a simple model that required no fitting parameters or numerical calculations, but that better coordinated with experimental results. In that respect, they were successful. Ironically, Tian et al. (2011) published an article entitled On the importance of optical phonons to thermal conductivity in nanostructures one month prior to their publication. In their opening paragraph, they paraphrase Ward and Broido (2010) and state that optical phonons provide an important scattering channel [ meaning that the acoustic phonons use them for scattering purposes ] for 3 Nanowires and nanoscale thin films require similar treatment to calculating thermal conductivity. The difference arises in the boundary scattering relaxation time, or the average time a phonon moves through the nanostructure before impacting the sides of the structure. The values presented in McGaughey et al. (2011) are as follows: and.

20 20 - Pearson acoustic phonons, and therefore, if removed from the system, would lead to a dramatic increase in the thermal conductivity. This is a possible explanation for the McGaughey et al. (2011) results predicting higher thermal conductivities. Even though Tian et al. (2011) model silicon nanowires, they emphasize the fact that their results can be applied to many nanostructures they do not even mention nanowire in the abstract. As we saw previously in McGaughey et al. (2011), the numerical derivations are identical between nanowires and thin films. The significance of both Ward and Broido (2010) and Tian et al. (2011) was to introduce the necessity of including optical phonons in thermal conductivity modeling. The analysis of Tian et al. (2011) found that the optical phonon contribution to the thermal conductivity increases by more than 10% when the thickness of an Si nanowire decreases from to nm. Therefore, for Si structures with characteristic lengths of about nm, optical phonons must be considered to obtain accurate thermal conductivity predictions. The inclusion of optical phonons and their importance in regulating thermal conductivity applies directly to superlattices Superlattice periodicity Superlattices are simply thin films stacked on top of each other. Typically, the layers are made of alternating materials. These structures have a unique affect on thermal conductivity; rather, superlattices have some distinct properties that allow manipulation of thermal conductivity through the structure. It is its periodic arrangement that produces these unique properties. A pair of layers is called the period of the superlattice. The affect of the superlattice periodicity is examined here. Phonons and periodic structures Phonons moving through a structured lattice are analogous to electrons moving through a periodic potential. The periodic potentials are found along the crystal lattice at atomic bonding points. This, effectively, is the hypothetical situation described by Einstein and discussed in section The result of this periodic potential results in Bloch wave analysis, which places limits on allowed energies for waves with certain wave numbers. This general description is called the band theory of solids. See further discussion in Brillouin (1946), Ziman (1960), and Uman (1974). The analogous nature between electrons

21 21 - Pearson and phonons provides further details and descriptions of phonon dispersion plots, or figures, such as Figure 2. However, in superlattice structures, when the entire structure itself is periodic, additional constraints are found. Another analogy was made for phonons in periodic materials by Narayanamurti et al. (1979), this time to optical dielectric filters. These optical filters are designed so that the layer thickness permits and therefore, rejects specific wavelengths of light through structure. Narayanamurti et al. (1979) created a superlattice phonon filter with stacks of GaAs and AlGaAs with layer thicknesses at a quarterwavelength, similar to a quarter-wave stack in optics. In phonon acoustics, the analogy to an index of refraction is acoustical impedance. The index of refraction is the ratio of the speed of light in a material relative to the speed of light in a vacuum. Acoustical impedance is defined as mass density of the material and, where is the is the speed of sound in the material (Cahill, et al., 2003). The transmission coefficient from one layer to another is described as ( ). Figure 7 is a theoretical calculation of transmission through a superlattice phonon filter. The figure illustrates the gaps that the superlattice creates by its periodic nature. Figure 7. This plot shows the theoretical calculation of transmission through a GaAs/AlGaAs superlattice phonon filter. The derivation of phonon transmission follows the optical derivation of light transmitting through thin films. However, the index of refraction is replaced with acoustical impedance, which is a measure of the relative phonon speed in different materials. The plot describes the filtering process available to superlattice structures. The top plot is based on an ideal superlattice with no disorder along the interfaces. The bottom plot inclused a random 10% disorder as described by the trace at the base of the figure. Image obtained from Narayanamurti et al. (1979).

22 22 - Pearson A byproduct of the transmission filter and interface scattering is an adjustment of the phonon dispersion plots. Figure 8 shows the dispersion of bulk Si and Ge compared to the in- and cross- plane dispersions of a 2x2 layer superlattice of SiGe. The effect concerning the superlattice filtering is seen in the cross-plane dispersion, the lower-right plot. The significance of these findings are discussed in the next section. The lower-left plot describes the condition of phonons moving in the plane of the superlattice, rather than perpendicular to the superlattice surface in the cross-plane motion. The previously discussed size effects are responsible for the increased number of phonons and the overlapping nature of the modes. The immense scattering provides a reduction in the thermal conductivity along the in-plane direction as well. Thermal conductivity reduction The immediate discussion resulting in this superlattice feature is the fact that only a limited number of Figure 8. The two plots, (a) and (b) are phonon dispersions, or phonon band structures, for bulk Si and bulk Ge. Plots (c) and (d) are phonon dispersions for a double period Si/Ge superlattice. (c) represents the phonon bands present across the in-plane direction of the superlattice, while (d) shows the cross-plane dispersion. Note the increased number of phonons in (c) compared to (a) and (b), due to increased scatterings at the structure boundary and phonon-phonon collisions. The (d) plot represents the formation of mini band gaps due to the filtering process in the cross-plane direction. The thermal conductivity is heavily decreased by the decrease in group velocity. Image obtained from Chen et al. (2004).

23 23 - Pearson phonon frequencies are allowed to pass. Therefore, a limited amount of heat is able to be conducted through the structure, decreasing the thermal conductivity of the device. The reflections at the superlattice boundaries prevent the phonons from transferring their carried energy fluidly through the interfaces. Additionally, the creation of mini bandgaps flattens out the allowed phonon bands, thereby decreasing the group velocity of the phonons ( ), and therefore the thermal conductivity (see Figure 8d). Figure 9 demonstrates the effective decrease in thermal conductivity of two different superlattices as compared to their bulk equivalent. The reduction in the SiGe superlattice is about five times less than the bulk alloy. Thermal conductivity increase As it turns out, superlattice structures have the ability to increase the thermal conductivity of materials as well. This feature was experimentally observed by Venkatasubramanian & Colpitts (1997) in a Bi2Te3/Sb2Te3 superlattice. They measured a thermal conductivity decrease until the dimensions of the superlattice became approximately the size (and smaller) of the unit-cell dimensions of the material. Theoretical calculations by Simkin & Mahan (2000) explain that when the layer thickness is less than the mean free path of the phonons, wave theory must be used to accurately calculate the thermal conductivity. Figure 9. This chart displays results from Cahill, Bullen, & Lee (1999), who used their developed 3 method to measure the thermal conductivity of PbTe/PbSe and Si/Ge superlattices. Both superlattices are compared to bulk materials. They demonstrate that a significant drop can be seen by using a superlattice structure instead of bulk material.

24 24 - Pearson Wave theory includes phonon modes and interference when determining the thermal conductivity (see section 4.2 in this paper). By incorporating these modes, they were able to predict an increase in conductivity when the period of the hypothetical superlattice was less than the mean free path of the phonons. So, what creates the different characteristics of superlattices? The key to this distinction falls back on the topic of optical phonon inclusion in modeling efforts (Tian, Esfarjani, Shiomi, Henry, & Chen, 2011). A recent paper by Garg, Bonini, and Marzari (2011) explained that thermal conductivity increased for short-period superlattices because a lower number of anharmonic scattering events were occurring. Their modeling included all phonon modes available in an SiGe superlattice. Results of their computation are found in Figure 10. The upper figure shows the dependency of thermal conductivity on the superlattice period. The lower figure shows two phonon dispersion plots; one for an average material (described as a hypothetical material that had the average mass and potentials of Si 28 and Ge 70 ) and the other for a SiGe superlattice consisting of one layer of each material. The dispersion curve describes a number of important features of the material at hand. It shows the phonon frequencies outlined along a specific direction; here, the x-axis refers to a specific crystal orientation direction. Because frequency is directly related to energy, the dispersion plot provides a visual way to interpret energy conservation in phonon-phonon collisions. Garg, Bonini, and Marzari (2011) included points on the plots of various phonon modes, detailed as a ( ). Viewing the average material dispersion, one sees that the collision of two acoustic modes, ( ) ( ), results in an output optical mode, ( ). This scattering/collision event detracts from the heat conduction. However, if one tries to add two acoustic modes together in the SiGe superlattice, the resulting optical phonon is not allowed, i.e. the resulting phonon is found in a forbidden zone or phonon band gap. The large gap between optical and acoustic phonons removes many of the phonon-phonon scattering events that are present in the high temperature regions.

25 25 - Pearson Figure 10. (top) This plot describes the effect on thermal conductivity of minimizing the periodicity in a SiGe superlattice. Note that the superlattice data was collected at 300 K, which is not near any extreme. Both in- and cross-plane measurements result in a dramatic increase in thermal conductivity. (bottom) The plot on the left is the phonon dispersion for a hypothetical mass and potential energy mix of Si28 and Ge70. On the right is a superlattice structure of SiGe. The superlattice creates band gaps, or forbidden zones, so that there is no interaction between those phonons. Example phonon additions are shown in both plots (Garg, Bonini, & Marzari, 2011). In contrast to the thermal conductivity decrease seen previously, even though the short period superlattice filters many of the phonons, the short periodicity creates large band gaps between the phonon modes, instead of mini band gaps as previously described. The size of the gap, therefore, prohibits phonon-phonon scattering. Recall that was dependent upon those particular scattering events, especially in the higher temperature ranges. Though the decreased size of the superlattice layers increases the boundary scattering, the long wavelength (shorter frequency) phonons carry the heat through the superlattice much more effectively without the collisions with the optical phonons. Figure 11 shows calculations from Garg, Bonini, and Marzari (2011) that displays a thermal conductivity increase of 23% for short-period SiGe superlattice, as compared to pure Si.

26 26 - Pearson Thermoelectronics and other fields requiring high thermal conductivity structures will benefit greatly from this new understanding. Additional manipulation of phonon dispersion curves may result in even better material outputs. 6. Conclusion Two ways to effectively alter the thermal conductivity of a structure is to change its operating temperature and change its characteristic size. Figure 11. This chart displays the results from the calculations of Garg, Bonini, & Marzari (2011). The thermal conductivity calculations use the entirety of the phonon modes. They find that a period-two superlattice (consisting of one layer of Si and one layer of Ge) provides a thermal conductivity higher than its associated bulk values. The average substance is a hypothetical material that has been mass and potential energy averaged from the two constituent parts of the superlattice; it is just used for comparison purposes. See text for further details. Operating temperatures determine the types of phonon scattering that dominate the thermal conductivity calculation. Higher temperatures allow more energy into the structure and enable increased anharmonic scattering (phonon-phonon interactions), which decreases thermal conductivity. Lower temperatures tend to freeze out everything but the long mean free path phonons. Still, the dependency causes the thermal conductivity to drop quickly. Nanostructures also create a decreased thermal conductivity by having their size proportional to the mean free path of the phonons. As structure sizes decrease, the phonons are more likely to scatter off the surfaces. This drives the conductivity down as well. The size decrease also reconfigures a structure s phonon dispersion, which creates and also destroys certain scattering channels between phonons. In superlattices, it was found that the size of the superlattice period was small enough (less than the phonon mean free path), a substantial increase in thermal conductivity would occur. In the superlattice structure, the thermal conductivity increase was due to band gaps forming in the phonon dispersion,

27 27 - Pearson thereby decreasing the number of phonon-phonon scattering events and increasing the relaxation time. This increase drove the thermal conductivity above values of the superlattices pure materials. The manipulation of the phonon band structure is providing another avenue of thermal conductivity management. Depending on the application, a structure may be altered to increase or decrease scattering events. Even newly fabricated structures, like phononic nanomesh, have been developed to specifically alter the phonon band structures to optimize thermal conductivity (Yu, Mitrovic, Tham, Varghese, & Heath, 2010).

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