We give a review of the theoretical approaches for predicting spectral phonon mean free path and thermal conductivity of solids. The methods can be summarized into two categories: anharmonic lattice dynamics calculation and molecular dynamics simulation. In the anharmonic lattice dynamics calculation, the anharmonic force constants are used first to calculate the phonon scattering rates, and then the Boltzmann transport equations are solved using either standard single mode relaxation time approximation or the Iterative Scheme method for the thermal conductivity. The MD method involves the time domain or frequency domain normal mode analysis. We present the theoretical frameworks of the methods for the prediction of phonon dispersion, spectral phonon relaxation time, and thermal conductivity of pure bulk materials, layer and tube structures, nanowires, defective materials, and superlattices. Several examples of their applications in thermal management and thermoelectric materials are given. The strength and limitations of these methods are compared in several different aspects. For more efficient and accurate predictions, the improvements of those methods are still needed.
In recent years, increasing attention has been focused on seeking novel structures and materials with desired thermal properties, especially thermal conductivity. High thermal conductivity can help remove heat rapidly and reduce device temperatures so as to improve performance of nanoelectronics and optoelectronics, while low thermal conductivity is desired in thermoelectrics for improving the figures of merit
Gaining a deeper physical insight into the spectral phonon properties, for example, the spectral phonon relaxation time and mean free path, is necessary to correctly explain experimental results and accurately predict and guide the further designs and applications. Analytical models have been used by Balandin and Wang to estimate frequency-dependent phonon group velocity and various phonon scattering rates including phonon-phonon, phonon-impurity, and phonon-boundary scattering processes. They used this approach to observe the strong modification of acoustic phonon group velocity and enhanced phonon scattering rate due to boundary scattering in semiconductor quantum wells, so as to successfully explain their significantly reduced lattice thermal conductivity [
The methods of predicting spectral phonon relaxation times and mean free paths become increasingly important for predicting the thermal properties of numerous novel materials. For instance, superlattice structure is found to be an effective way to suppress the thermal conductivity because of the interface mass mismatch scattering [
Many methods have been proposed and applied to predict spectral phonon relaxation time in the last half century. At the earliest, Klemens and other researchers obtained the frequency-dependent phonon relaxation time mostly by long-wave approximation (LWA) and Deybe model: Klemens obtained the phonon relaxation times by Umklapp (
In this work, we present a review of the methods of predicting spectral phonon properties, discuss the applications to each method, and compare them in different aspects. Section
Spectral phonon mean free path (MFP), determined by phonon scattering rate, dominates the behavior of thermal properties, especially the thermal conductivity
If isotropic heat transport is assumed, the integration of
The early theoretical predictions of phonon relaxation times for different scattering processes are briefly summarized in Table
Analytical models of inverse relaxation time for different scattering processes.
Scattering process | Inverse relaxation time |
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Herringa |
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Callawayb |
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Klemensc |
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Klemensd |
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Callawayb |
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Hollande |
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Asen-Palmer et al.f |
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References: a[
Boundary scattering
The last equation in Table
For the system that contains several scattering mechanisms, the Matthiessen rule is often used to evaluate the total scattering rate,
These frequency dependent relaxation time expressions in Table
In perturbation theory, the steady-state phonon BTE [
The RTA assumes that deviation of single phonon mode population decays exponentially with time:
Considering only three-phonon scattering, (
The Standard SMRTA assumes that the system is in its complete thermal equilibrium, except that one phonon mode
Different from the Standard SMRTA, the other method to solve the phonon BTE allows all the modes to be in their thermal nonequilibrium states at the same time. By replacing the occupation numbers
Equation (
ALD methods can be divided into classical method and
In (
To calculate the relaxation time, one can use Standard SMRTA scheme [
One way to predict thermal conductivity
In addition to intrinsic phonon scattering
Without any fitting parameters, Standard SMRTA with
Percent error (color online) in
One flaw of the Standard SMRTA is that it does not grasp the interplay between the
For Si and Ge at room temperature where the
In contrast, the
The calculated intrinsic lattice thermal conductivity of diamond for the Standard SMRTA (dashed line) and the Iterative Scheme (solid line), both by
One important application of ALD calculation is to predict and understand the thermal conductivity of thermoelectric materials and help to design higher thermoelectric performance structures. Based on first principle calculation, Shiga et al. [
(a) Spectral phonon relaxation times of pristine PbTe bulk at 300 K by Standard SMRTA scheme with first principle IFCs. Relaxation times of (b) normal and (c) Umklapp processes, respectively. The solid lines plot (a)
Spectral phonon relaxation times of PbSe bulk (squares) and PbTe bulk (crosses) at 300 K by Standard SMRTA scheme with IFCs from first principle calculation: (a) TA, (b) LA, (c) TO, and (d) LO. Reprinted with permission from [
The accumulated thermal conductivity of different bulk materials as a function of phonon mean free path at room temperature calculated from ALD by first principle approach. (Diamond [
For single- and multilayer 2D materials, the boundary scattering from the sides perpendicular to the transport direction is much weaker than for 3D systems [
Vibrations in 2D lattices are characterized by two types of phonons: those vibrating in the plane of layer (TA and LA) and those vibrating out of plane, so called flexural phonons (ZA and ZO). Lindsay et al. [
The selection rule mentioned above does not hold for multilayer graphene, twisted graphene, graphite (because of the interlayer coupling), CNT (due to the curvature), graphene nanoribbon (GNR) (due to boundary scattering), substrate-supported graphene (due to scattering with the substrate), and defective graphene (due to defective scattering). Therefore, the thermal conductivity of these structures is typically lower than that of single layer graphene, and the contribution of each phonon mode changes [
Thermal conductivity
For 2D materials and nanotube structures, the
The ratio (color online) between thermal conductivity predicted from Iterative Scheme (
For nanowires, the Casimir model (Table
From second-order perturbation theory [
Equation (
Superlattices (SLs), composed of periodically arranged layers of two or more materials, have been extensively investigated in the aspect of thermal transport. Because of the heat transport suppression by interfaces and mass mismatch, superlattice has been designed to have a lower thermal conductivity than pure bulk. SLs are classified into two categories: diffuse and specular interfaces. The phonons in the first case are diffusively scattered by interfaces, while the phonons in the latter one propagate through the whole structure as if in one material, so-called coherent phonon transport [
Generally,
Garg et al. [
Scattering rate (color online) of TA mode at 300 K due to (a) absorption of an acoustic phonon to yield another acoustic phonon, (b) absorption of an acoustic phonon to yield an optical phonon, (c) absorption of an optical phonon to yield another optical phonon, and (d) total scattering rate, along (
More generally, Broido and Reinecke [
Thermal conductivities calculated by Standard SMRTA method (dashed lines) and Iterative Scheme (solid lines) of
The time domain normal mode analysis based on MD simulation was first proposed by Ladd et al. [
The calculation of normal mode coordinate
Here, the frequency domain normal mode analysis is demonstrated by a simplified version; for detailed derivation, see [
In some works, the total SED function for a given wave vector
According to Ong et al. [
Figures
Autocorrelation functions of potential and total energies of time-dependent normal modes of TA mode at
SED functions (color online) of (a) empty CNT and (b) water-filled CNT along
The relaxation times predicted from MD simulation includes the effects of three-, four-, and higher-order phonon scattering processes; in contrast, ALD calculation only considers the lowest one. Thus, the ALD calculation may lose its accuracy when temperature increases, since the higher-order anharmonicity of lattice becomes greater for higher temperature due to thermal expansion. For instance, Turney et al. [
The inverse relaxation times (color online) of LA and TA mode of argon as functions of wave vector in [
Compared to ALD calculation, MD simulation is a better tool for predicting the phonon properties of complex systems, such as the CNT filled with water and the graphene supported by substrate. So far, it is hard for ALD method to handle the extrinsic phonon scattering processes other than the Umklapp scattering without fitting parameters. However, in the MD simulation, the surrounding influence is reflected by the atomic vibrating trajectory of the studied system. Qiu and Ruan [
The relaxation time (color online) of suspended (“s”) and
(a) Contributions (color online) and (b) the corresponding percentages of thermal conductivity from ZA, TA, LA, and TO modes in suspended (“s”) and supported (“p”) SLG at different temperatures. Reprinted with permission from [
Due to the low computational complexity, the NMA methods have been applied to many cases. Time-domain NMA was used for Ar [
Spectral phonon relaxation time bulk PbTe (color online) at 300 K and 600 K. Reprinted with permission from [
Contribution (color online) of each phonon mode to total thermal conductivity of PbTe bulk at 300 K and 600 K. Reprinted with permission from [
The phonon properties of Bi2Te3 are studied by time-domain NMA [
(a) Phonon relaxation times (color online) of of Bi2Te3 along the Γ-
The normalized accumulated thermal conductivity (color online) of several bulk materials at room temperature as a function of phonon MFP. (Si (MD) [
The three methods, anharmonic lattice dynamics based on Standard SMRTA, iterative anharmonic lattice dynamics, and normal mode analysis, can all predict thermal conductivity by calculating the velocities, relaxation times, and specific heats of all phonon modes. The applications are listed in Table
Applications of the numerical methods in predicting spectral phonon properties and thermal conductivity.
Materials | Methods* | FP† | Reference | Year |
---|---|---|---|---|
Ar | 1 and 3 | [ |
1986 | |
Ar | 2 | [ |
1995 | |
Ar and Kr | 2 | [ |
1996 | |
Ar | 3 | [ |
2004 | |
Ar | 1 and 3 | [ |
2009 | |
Ar, Si thin films | 1 | [ |
2010 | |
C, Si, and Ge | 1 |
|
[ |
1995 |
C, isotope-doped C, Si, and Ge | 1 and 2 |
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[ |
2009 |
C (Pure and natural) (extreme pressure) | 1 and 2 | [ |
2012 | |
C nanowire | 1 and 2 |
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[ |
2012 |
Si (isotope-doped) | 1 | [ |
1999 | |
Si (isotope-doped) | 1 | [ |
2001 | |
SiC | 2 | [ |
2002 | |
Si and Ge | 1 |
|
[ |
2003 |
Si | 1 and 2 | [ |
2005 | |
Si and Ge | 2 |
|
[ |
2007 |
Si | 3 | [ |
2008 | |
Si (isotope-doped) | 2 | [ |
2009 | |
Si | 1 |
|
[ |
2011 |
Si | 1 | [ |
2012 | |
Si Nanowire | 3 | [ |
2009 | |
Si, Ge | 1 |
|
[ |
2010 |
Si/Ge, |
1 and 2 | [ |
2004 | |
Si/Ge SLs | 1 |
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[ |
2011 |
Si/Ge SLs | 1 | [ |
2013 | |
Si/Ge SLs | 1 |
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[ |
2013 |
SiGe alloys with embedded nanoparticles | 2 |
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[ |
2011 |
SiGe alloys | 1 |
|
[ |
2011 |
Si, Ge, and Si0.5Ge0.5 | 1 | [ |
2012 | |
Si/Ge, GaAs/AlAs, and |
1 and 2 |
|
[ |
2008 |
Ge | 3 | [ |
2010 | |
Ge | 4 | [ |
2013 | |
Semiconductors (Groups IV, III–V, and II–VI) | 1 | [ |
2008 | |
Graphene | 1 and 2 | [ |
2010 | |
Graphene and graphite | 1 and 2 | [ |
2011 | |
Graphene (supported and suspended) | 4 | [ |
2012 | |
Graphene (free-standing and strained) | 1 |
|
[ |
2012 |
Graphene and graphite | 1 |
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[ |
2013 |
CNT to graphene (diameter dependence) | 1 and 2 | [ |
2010 | |
CNT | 4 | [ |
2006 | |
CNT | 1 and 2 | [ |
2009 | |
CNT (empty and water-filled) | 4 | [ |
2010 | |
CNT (on amorphous silica) | 4 | [ |
2011 | |
BN (pristine and isotope-doped) | 1 and 2 | [ |
2011 | |
BN (multilayer and nanotubes) (pristine and isotope-doped) | 2 | [ |
2012 | |
Mg2 |
2 | [ |
2012 | |
Compound semiconductors (Si, Ge, GaAs, Al-V, Ga-V, In-V, SiC, AlN, etc.) | 1 and 2 |
|
[ |
2013 |
Ionic solids (MgO, |
2 | [ |
2011 | |
GaN (GaAs, GaSb, and GaP) (pristine and isotope-doped) | 1 and 2 |
|
[ |
2012 |
PbTe | 1 |
|
[ |
2012 |
PbTe | 4 | [ |
2011 | |
PbTe, PbSe, and |
1 |
|
[ |
2012 |
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4 | [ |
2013 | |
Heusler | 3 | [ |
2011 | |
MgO | 4 |
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[ |
2009 |
Polyethylene | 3 | [ |
2009 | |
Polyethylene | 3 | [ |
2009 | |
GaAs | 1 |
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[ |
2013 |
GaAs/AlAs SLs | 1 |
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[ |
2012 |
Comparison of different methods for predicting spectral phonon relaxation time and thermal conductivity.
Methods | Analytical model | ALD calculation | MD simulation | |||
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Standard SMRTA | Iterative Scheme | Time NMA | Frequency NMA | |||
Equations | Table |
Equations ( |
Equations ( |
Equations ( |
Equations ( |
Equations ( |
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Characteristics | Lots of approximations, need fitting parameters |
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Eigenvectors needed | Eigenvectors not needed | |
Need 2nd- and 3rd-order IFCs | Need interatomic potential (or | |||||
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Suitable for | Long wavelength, Debye model | Low temperature higher-order anharmonicity not large | Higher than Debye temperature | |||
Temperature not too low, quantum effect negligible | ||||||
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Accuracy | Low | Medium | Higher | Higher | ||
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Computational complexity | High | Higher | Low | |||
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Applications so far | Some thermal conductivity analysis and prediction | Pure and isotope-doped bulk, alloy superlattice, nanostructures | Pure lattice Materials with surrounding influences | |||
Further research | Temperature dependent IFCs, 4th- and higher-order phonon scattering | Accurate interatomic potential, large domain first principle MD, defects, boundaries |
All the three methods are based on phonon Boltzmann Transport Equation and relaxation time approximation. To obtain the spectral phonon relaxation time, the first two methods calculate three-phonon scattering rates from anharmonic interatomic force constants, while the last method calculate the linewidth of spectral energy in frequency domain or the decay rate of spectral energy in time domain from molecular dynamics. Since the first two methods ignore the 4th- and higher-order phonon scattering processes, they are only valid at low temperature. The first two methods differ with each other at solving the phonon BTE: the first method assumes single mode RTA, while the second one solves the linearized BTE iteratively instead. As a result, the first method treats
These numerical methods have been applied to numerous materials and structures and revealed lots of physical nature that has never been reached before. The acoustic phonons are verified to have the ~
The mathematic preparations are
Relaxation time approximation assumes
In Standard SMRTA, only
Substituting (
The Iterative Scheme solves phonon BTE (
Substituting (
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the National Science Foundation, Air Force Office of Scientific Research, and the Purdue Network for Computational Nanotechnology (NCN) for the partial support.