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Advance progress in the processing of materials with structure on nanometer length scales has created a demand for greater scientific understanding of thermal transport in nanoscale devices, individual nanostructures, and nanostructured materials. Classical molecular dynamics simulations are promising as a powerful tool for calculations of thermal conductance and phonon scattering, and will provide the correlation with experiment and theory in the near future. Silicon microelectronics is now in the nanoscale regime. Experiments have demonstrated that the proximity of interfaces and the extremely small volume of heat dissipation strongly modify thermal transport, causing problems of thermal management. Microelectronic devices are too large to handle in the atomic-level simulation at the moment; therefore, calculations of thermal transport must rely on solutions of the Boltzmann transport equation. Advances in measurement methods, such as 3ω method, time-domain thermo reflectance, micro fabricated test structures, and the scanning thermal microscope, are enabling new capabilities for nanoscale thermal metrology.
Key words: molecular dynamics simulations, Boltzmann transport equation, phonon scattering, mean free path.
The communal impact of research on thermal transport in materials is clearly enormous. The history of thermal transport goes back to primitive humankind who combined empirical observations with applications (specific heat of stone for warmth, straw for insulation, wooden tools for manipulating fire, etc.). Recent research on thermal transport in nanostructures has expanded its extremes.
Many of these nanoscale structures already have important commercial applications, while others are studied scientifically. The most important devices, in terms of worldwide sales, are the integrated circuits used in computer circuits and memories. Thermal management of integrated circuits is the biggest challenge in microelectronics today (Chu, R. C., et al2004). Ultra low thermal conductivity materials can be used for thermal barrier coatings in jet engines (Padture, n. p., et al, 2001) whereas extremely high thermal conductivities can be used for heat sinks (Schelling, P. K., et al, 2005)
There are two types of heat carriers in solids: electrons and crystal vibrations (or phonons). This review focuses on nonmetallic materials where phonons are the dominant carrier. A phonon is a quantum of crystal vibrational energy and is analogous to the photon. Just as a solid radiates out a Planck-like distribution of photons, it also contains a Planck-like distribution of phonons within itself. Phonons have two fundamental lengths: wavelength and mean free path. At room temperature, the dominant heat-carrying phonons typically have wavelengths of 1-3 nm and mean free paths of 10-100 nm. By using nanostructures comparable in size to these length scales, one can then manipulate thermal transport in solids (Cahill, D. G., et al, 2003). This clearly necessitates an understanding of heat transport beyond that achievable at the continuum level. So far, no analytical theories have adequately treated the wave nature of phonons that will be manifest at these length scales.
This review will summarize the recent researches done on thermal transport in nanostructures, dividing this broad topic into few subtopics namely, Molecular dynamics simulations, Advances in theoretical knowledge, Boltzmann transport equation (BTE), Advances in metrological techniques, Heat transfer in Nano-material interfaces and Heat transport in nanoscale structures.
Molecular dynamics simulations
Molecular dynamics simulations are continuing as an effective tool for calculations and understandings of thermal conductance and phonon scatterings. C.S. Wang, et al, 2007 have done molecular dynamics simulations, aim to gain understanding in the heat transfer through the channel including the influence of the contact resistances which become important in small-scale systems. The heat transfer properties of the finite-space system were measured at a quasi-steady non-equilibrium state achieved by imposing a longitudinal temperature gradient to the channel. The results indicate that the total thermal resistance is characterized not only by the thermal boundary resistances of the solid-liquid interfaces but also by the thermal resistance in the interior region of the channel. The overall thermal resistance is determined by the balance of the thermal boundary resistances at the solid-liquid interfaces and the thermal resistance attributed to argon adsorption on the lateral walls. As a consequence, the overall thermal resistance was found to take a minimum value for a certain surface potential energy.
John W. et al, 2006 simulated isoenergy molecular dynamics combined with the Green-Kubo approach to calculate the heat current time-dependent autocorrelation function and determine the lattice thermal conductivity. They showed that the inverse temperature dependence of the lattice thermal conductivity fail at low temperatures when the atomic diameters of the two species differ. The thermal conductivity was nearly a constant across temperatures for species with different atomic diameters. Overall, it is shown that there is a dramatic decrease of the lattice thermal conductivity with increasing atomic radii ratio between species and a moderate decrease due to mass disorder.
Molecular dynamics simulation and the Boltzmann transport equation were used to analyze the phonon transport in nanometer films (Wang Zenghui, et al, 2006). Although the measurement of nanoscale film thermal conductivity has been made in the thickness range of about 100 nm, the direct measurement tools face problems due to the difficulties in preparing the samples and test apparatus for the thickness range of less than 100 nm. It was showed in this paper from the simulation results that there exists the obvious size effect on the thermal conductivity when the film thickness reduces to nanoscale from microscale, and the feasibility of the molecular dynamics method is a particularly good alternative method to investigate the physical characteristics of nanometer film when direct measurement are difficult to make.
Hydrodynamics simulations are powerful tool for understanding the particle behavior. J.H. Jeong, et al, 2003 have modified the numerical steps involved in a smoothed particle hydrodynamics (SPH) simulation. Specifically, the second order partial differential equation (PDE) is decomposed into two first order PDEs. Using the ghost particle method, consistent estimation of near-boundary corrections for system variables is also accomplished. Also the SPH equations for heat conduction to verify the numerical scheme were focused. Each particle carries a physical entity (here, this entity is temperature) and transfers it to neighboring particles, thus exhibiting the mesh-less nature of the SPH framework. This methodology can be applied to complex geometries in nanoscale heat transfer.
Advances in theoretical knowledge
Chunhai Wang, 2004 has developed a mathematical model for micro thermal analysis based on heat conduction equation. It interprets the principle of micro thermal analysis by relating its signals directly to established parameters of materials, such as thermal conductivity and specific heat. Classical equations for transport processes predict infinite speed of the perturbations propagation. From molecular point of view, this feature is rather doubtful, because, for instance in gas or plasma, any perturbation propagates as a result of molecules or charged particles (ions, electrons) interaction. Isaac Shnaid 2007 has presented an exact statistical description of the perturbations propagation in gas and plasma based on the Boltzmann equation written in the most general form.
Three heat transfer modes in tip-sample thermal interaction were analyzed with experimental data and modeling (Ste´phane Lefe`vre. Et al, 2006) the tip-sample thermal interaction involves conduction at solid-solid contact as well as conduction through the ambient gas and through the water meniscus. It is concluded that the three modes contribute in a similar manner to the thermal contact conductance but they have distinct contact radii. Li Zhang et al, 2006 has applied a model of 2-D heat transfer in a multilayered film structure and the mark size difference was explained by the thermal conductivity of the substrate. Here a field emission current from a scanning tunneling microscope (STM) tip is used as the heating source and pulse voltages of 3-7V with a duration of 500 ns were applied to a CoNi/Pt multilayered film which is fabricated on a bare silicon and oxidized silicon, respectively.
Xing Xiaokai, 2007 has developed an innovative electromagnetic anti-fouling (EAF) technology for heat transfer enhancement. A series of experiments with and without an EAF device were performed to find out the mechanism of heat transfer enhancement of the proposed technology. In these experiments, the variation of the fouling thermal resistance vs. time was measured and the scanning electron microscope (SEM) micrographs of the fouling layers were taken to prove the results.
Boltzmann transport equation -BTE
Most theories of transport in solids employ the Boltzmann transport equation (BTE), for both electron and phonon transport. The forms of the equation, and the form of the various scattering mechanisms, are very well known. This theory can explain, in bulk homogeneous materials, the dependence of the electrical conductivity and the thermal conductivity. One erroneous of the Boltzmann equation in nanoscale is its treatment of the electrons and phonons as classical particles. Few modifications have done recently Rodrigo A. Escobar, et al, 2006, Jorge E. Fernandez, et al, 2007 and A. Bulusu, et al, 2007, on this theory to suit the specific circumstances.
According to Jorge E. Fernandez, et al, 2007, The Boltzmann-Chandrasekhar vector equation is the best model known for describing the diffusion of incoherent photon beams with arbitrary polarization state. They have presented brief comparison of the transport equations (scalar and vector). Then they described the state-of-the-art of the transport codes developed at Bologna based on these models.
Escobar, et al, 2006, have used the lattice Boltzmann method (LBM) to investigate one-dimensional, multi-length and -time scale transient heat conduction in crystalline semiconductor solids, in which sub-continuum effects are important. The implementation of this method and its application to electronic devices are described in their paper. A silicon-on-insulator transistor subject to Joule heating conditions is used as a case study to illustrate the essence of the LBM. LBM results were compared for the diffusive to the ballistic transport regimes, with various hierarchical methodologies of heat transport such as the Fourier, Cattaneo, and ballistic-diffusive transport equations.
The Boltzmann transport equation is often used for non-continuum transport when the mean free path of phonons is of the order of device sizes. One particular application involves heat generation in electronic devices. The size of this generation region is often smaller than the mean free path of phonons, which suggests the generation Knudsen number is large and non-continuum models are appropriate. A. Bulusu, et al, 2007 have made a comparison between the continuum and non-continuum models using a one-dimensional BTE and diffusion equation, The focus of this comparative study is the behavior of each model for various Knudsen numbers for the device size and generation region. Results suggest that non-continuum distributions are similar to continuum distributions except at boundaries where the jump condition results in deviations from continuum distributions.
Advances in metrological techniques
The goal of successful commercial implementation of nanoscale materials places a burden on the metrology of these materials and devices that extend beyond the characterization of the fundamental materials properties. This burden involves not only the small physical dimension of the material but also the general accuracy of the measurement.
The possibility to use the scanning thermal microscope for a quantitative determination of the local heat conductivity λ at material surfaces is evaluated and critically discussed (H. Fischer 2005). Here two different methods of operation have been applied for the determination of the probe to sample heat flux in the local thermal analysis (LTA) mode and the analysis of heat flow data derived from thermal maps in scanning experiments (SThM). Both methods lead to a comparable accuracy in the determination of λ. The SThM shows the highest sensitivity for small λ, and is useful in a λ range between 0.05 and 20 W/m K. Also new multiwire calibration standard is introduced in this paper.
S. Lef`ever, et al, 2004 have used the 3ω method to enhance the sensitivity of the devices to a larger range. They have used both a thermal model and experimental results from the calibration procedure to study the thermal behavior of standalone probes. The two approaches provide data in very good agreement on the full measured frequency domain. Several geometric and thermal parameters are deduced from the comparison. Those quantities will be key inputs for future heat transfer modeling of the tip-sample contact.
Yu-Ching Yang, et al, 2007 have proposed a general methodology for determining the thermal conductance between the probe tip and the workpiece, during microthermal machining, using Scanning Thermal Microscopy (SThM). The processing system was considered as inverse heat conduction problem with an unknown thermal conductance. Temperature dependence for the material properties and thermal conductance in the analysis of heat conduction is taken into account. The conjugate gradient method is used to solve the inverse problem. Furthermore, this methodology can also be applied to estimate the thermal contact conductance in other transient heat conduction problems.
Heat transfer in Nano-material interfaces
Here the consideration is on thermal conductance of an isolated interface i.e., an interface that is separated from other interfaces by a distance that is large compared to the mean-freepath of the lattice vibrations that dominate heat transport in the material. In this limit, coherent superposition of lattice waves reflected or transmitted by adjacent interfaces can be ignored. Some advanced experiments have been conducted in last few years as well.
J. Amrit, 2004 has defined two possible heat conduction regimes, namely a classical surface effect regime and a scattering effect regime. The distinction between these regimes depends upon the (l_/λ) ratio, where the surface roughness length is l_ and the phonon wavelength is λ. In a preliminary experiment it is shown that heat conduction across a silicon-superfluid helium interface at ~2 K can be entirely explained by the scattering effect regime where diffuse phonon scattering from surface irregularities of nanometric scale lengths play a dominant role.
Andrei G. Fedorov, et al, 2007 illustrates the transport of precursor molecules to the substrate surface. Depending on the operating pressure either continuous advection-diffusion mass conservation equation or the kinetic Boltzmann Transport Equation (BTE) describes the transport of precursor molecules to the substrate surface. At the surface, some of the precursor molecules are adsorbed, spatially redistributed by surface diffusion and, finally, a fraction of the adsorbed molecules become converted into a solid deposit. This occurs upon interaction with back-scattered primary electrons and secondary electrons, yielded by the substrate and deposit upon impingement of the high-energy primary electron beam. The interactions of the primary electrons with the substrate and nanoscale-confined deposit possibly induce significant localized heating. Such energy transfer process is complex, involves non-classical heat conduction, and may greatly influence the deposition process.
The work of A. Flores Renteria, et al, 2006 demonstrates that variation of the EB-PVD process parameters alters the resulting columnar morphology and porosity of the coatings. The physical properties and, most importantly, thermal conductivity, are greatly affected by these morphological alterations. Correlation of shape and surface-area changes in all porosity types of the analyzed coatings revealed that the thermal conductivity of these coatings is influenced primarily by size and shape distribution of the pores and secondarily by the pore surface-area available at the cross section perpendicular to the heat flux.
D. Majcherczak, et al, 2006 has presented an experimental thermal study of contact with third body. The thermal study of sliding contact is complex due to numerous physical aspects highly coupled. Heat generation mechanisms are still badly known due to the complex interactions between mechanical, thermal and physico-chemical behaviors and surface degradations. In the thermal field, the literature generally classifies the sliding contact as perfect or imperfect contact, depending if the asperities are taken into account. In the both cases thermal continuity at the interface of the two sliding bodies is assumed. Few studies consider a contact with third body. It is usually neglected because of its weak thermal conductivity. In the goal to better appreciate the third body role on the thermal aspect, an experimental set-up has been realized. Comparison between the thermal scene and the surface observations has allowed connecting the third body accumulation with local surface heating.
Heat transport in nanoscale structures
While phonon dynamics and transport in a single unit is important and must be studied, the collective behavior of a large number of units can add to more complexity, i.e., there may be some collective modes which are not found when considering only single units. For example, when the phonon coherence length scales are larger than the size of the unit, phonon interference effects lead to modified dispersion relations.
Hence, the appearance of both wave effects at nanoscales and diffusive heat flow at bulk scales poses challenges in predicting thermal transport in these materials. Such predictions are important because nanostructured materials find many applications.
Hongwei Liu, et al, 2007 have studied the flow and heat transfer in vacuum packaged MEMS devices numerically by using the direct simulation Monte Carlo method. This research can improve the understanding of gas flow and heat transfer in vacuum packaged MEMS devices. E. Ziambaras, et al, 2005 have shown that both material-interface scattering and total internal reflection significantly limit the SiC-nanostructure phonon transport and hence the heat dissipation in a typical device. The phonon-interface scattering produces a heterostructure thermal conductivity significantly smaller than what is predicted in a traditional heat transport calculation. They have showed that the high temperature heat flow across the metal/SiC interface is limited by total internal reflection effects and maximizes with a small difference in the metal/SiC sound velocities.
Various size effects on carrier transport in nanostructures can be utilized to engineer new structures with improved energy conversion efficiency. G. Chen, et al, 2004 have used the thermal conductivity of superlattices to illustrate the phonon transport characteristics in nanostructures. Nanoscale thermal radiation phenomena, such as interference, tunneling, and surface waves, that can potentially be exploited to improve the efficiency of thermo photovoltaic power generation devices were also discussed.
Complicated fundamental issues continuing in advances in theory, computation, and experimental techniques promise the nanoscale thermal transport will remain a dynamic field of investigation for many years to come. Large-scale molecular dynamics should be used to evaluate the sensitivity of the thermal conductance to the structural and chemical disorder of the interface. A challenge is to include the electronic and quantum mechanical effects in atomic-level simulations of thermal transport.
Techniques such as Monte Carlo solutions of the classical or quantum Boltzmann transport equation would find effective in higher level computations. Improvements in scanning thermal microscopy and other nanoscale metrology methods are needed to make quantitative comparisons between experimental observations of nanoscale transport and the predictions of theory.