DESIGNING THERMAL TRANSPORT IN ENERGY MATERIALS: A FIRST PRINCIPLES APPROACH Derek A. Stewart1, Anupam Kundu2, Wu Li2, Lucas Lindsay3, Natalio Mingo2, and David Broido4 1
Cornell Nanoscale Facility, Cornell University, Ithaca, NY 2 LITEN, CEA-Grenoble, Grenoble, France 3 Naval Research Laboratory, Washington, DC 4 Department of Physics, Boston College, Chestnut Hill, MA
[email protected] Introduction A clear understanding of heat transfer due to lattice vibrations or phonons is crucial for a number of industrial energy applications including thermal barrier coatings for aeronautics, thermoelectric materials, nanoscale heat conduits for next generation electronics, and even heat transfer deep in the Earth. A theoretical approach that could provide robust predictive capabilities and insight into phonon processes in materials would be very beneficial for energy research. However, due to the numerical complexity of this challenge, the heat transfer community has been forced to rely on approximate solutions with limited predictive capability. In this overview of our recent work[1-4], I will present an effective first principles approach for predicting lattice thermal conductivity that demonstrates excellent agreement with the measured thermal conductivity of key industrial semiconductors (i.e. diamond, silicon, and germanium)[1,2]. This approach uses second and third order interatomic force constants calculated from density functional perturbation theory. These force constants are incorporated into an iterative solution of the phonon Boltzmann transport equation where the phonon-phonon scattering processes are calculated explicitly. We have also expanded this framework to consider thermal transport in thermoelectric alloys (SiGe) with embedded nanoparticles[3] and size effects on thermal transport in diamond nanowire heat conduits[4]. Phonon Boltzmann Transport Equation. The general expression of the phonon Boltzmann transport equation (BTE) in materials was developed by Peierls in 1929. In the bulk limit, it consists of a term representing phonon diffusion due to a thermal gradient and a collision term that denotes all possible scattering mechanisms that can impede thermal transport. In a steady state system, these two terms are equal and we can relate the thermal conductivity to the temperature gradient in the system. In the technological applications discussed above, the key thermal transport takes place at or above room temperature where phonon-phonon scattering is the dominant form of thermal resistance. Phonon-phonon interaction is due to the intrinsic anharmonicity of the crystal lattice (a perfect infinite harmonic crystal would have infinite thermal conductivity). Three phonon scattering events can be divided into two categories: Normal scattering events which conserve momentum and Umklapp scattering events which flip the phonon direction and lead to thermal resistance. The phonon-phonon scattering process is very difficult since it is inelastic, requires both accurate harmonic and anharmonic force constants, and also requires a precise knowledge of the phase space and matrix elements for three-phonon scattering. These difficulties forced researchers for almost 80 years to rely on fitted relaxation time approximations to solve the BTE. This unfortunately limited the predictive capabilities of this approach and also obscured how specific phonon scattering events affect a material’s thermal conductivity. In the last twenty years, molecular dynamics based on empirical potentials has provided another way to predict thermal transport in materials.
However, since this is a classical approach, it is best suited for materials at high temperatures. In addition, since empirical potentials must be carefully developed for each material, it is difficult to add impurities or introduce new combinations of materials into a system. In order to improve on this situation, two key improvements are necessary. The first is a method to explicitly calculate the phonon collision term and accurately solve the BTE. The second is an approach that can accurately predict the harmonic and anharmonic interatomic force constants for a wide range of crystalline materials To address these issues, we developed an iterative BTE approach that incorporates second order and third order interatomic force constants determined using density functional theory. The BTE collision term is calculated explicitly and phonon-phonon scattering terms can also be sorted based on type (normal versus Umklapp) and participating phonon branch. Approach Iterative Solution of the Boltzmann Transport Equation: The Boltzmann transport equation can be written in terms of a nonequilibrium phonon distribution, nλ=n0λ+n1λ, where n0λ is the equilibrium Bose distribution and n1λ is the deviation due to the thermal gradient. In this expression, λ, is a shorthand used to denote both the phonon q vector and the phonon branch index, α. The deviation from the equilibrium phonon distribution for a given λ can be related to the self-consistent lifetimes, .
n1 n0 n0 1 where
vz
and
are
vz dT k BT 2 dz
(1)
respectively the z-component of the group
velocity in mode λ and the phonon frequency. This allows us to rewrite the linearized BTE in terms of the phonon lifetimes.
0 0 (2) where 0 can be related to the scattering rates due to anharmonic (phonon-phonon) and harmonic (isotopes, impurities) interactions for a given λ and Δλ is a linear function of the ensemble of phonon lifetimes that takes into account interactions between different modes. The 0 is defined in terms of known quantities and this can
be calculated directly and used as an initial guess for . This is basically equivalent to the relaxation time approximation which assumes all scattering events will bring phonons back to the equilibrium distribution. Due to this assumption, the thermal conductivity calculated using the relaxation time approximation is always less than the actual thermal conductivity. The iterative approach corrects for this and allows us to determine the selfconsistent phonon lifetimes. Further details of the calculations can be found in the following works [1,2,5,6]. Once the self consistent phonon lifetimes are determined, the thermal conductivity can be calculated by integrating contributions from the entire Brillouin zone.
dq 1 2 z 2 n n v 1 0 0 2 k BT 2 3
(3)
Ab-Initio Force Constants. To determine the initial guess for the phonon lifetimes, we need the interatomic force constants. These terms can be calculated using density functional perturbation theory[7,8]. In this approach, the Kohn-Sham equations for the charge density, self-consistent potential, and orbitals are linearized
Prepr. Pap.-Am. Chem. Soc., Div. Energy Fuels Chem. 2012, 57(2), 558 Proceedings Published 2012 by the American Chemical Society
with respect to changes in wave function, density, and potential variations. For phonon calculations, the perturbation in the ionic potential, δV, is periodic with the phonon wavevector, q. This perturbation leads to a corresponding change in the charge density, δn(r). Since the perturbation generated by a phonon is periodic with respect to the crystal lattice, we can Fourier transform the selfconsistent equations for the first order correction, and consider the Fourier components, δn(q+K) of the change in electron density, where K is a reciprocal lattice vector. This problem can be addressed on a sufficiently dense grid in the unit cell Brillouin zone and fast Fourier transforms can be used to convert back to real space. Interatomic force constants can also be calculated using a real space approach (see Esfarjani et al. [9]) To achieve a high figure of merit in a thermoelectric, we want to minimize a material’s thermal conductivity while maximizing the power factor. Recent work has shown that embedding nanoparticles in alloys is an effective route to improve the figure of merit[10]. We consider SiGe alloys with embedded Ge nanoparticles that have applications in on-chip electronics cooling. We use the virtual crystal approximation to determine the effective interatomic force constants and thermal conductivity of the alloy. Phonon scattering due to embedded nanoparticles can be calculated using a rigorous Tmatrix approach or a Born approximation (see details in Ref. [3]). Results and Discussion The lattice thermal conductivity for isotopically pure diamond and silicon are compared with experiment in Fig. 1. The agreement is good over a wide temperature range. Although not shown here, we can also predict the thermal conductivity of a semiconductor with a natural isotope distribution by adding a scattering term for isotopes[2]. A key benefit of our approach is that we have direct information on all possible phonon-phonon scattering events. This allows us to explore the role of optical branches in thermal transport. Since these branches have very small group velocities, they are often ignored in thermal transport models. However, our results show that optical branches play an important role in phonon scattering and thermal resistance[2]. In Fig. 1, the thermal conductivity is also plotted for the case where all phonon scattering involving optical branches has been removed. This leads to a substantial increase in the thermal conductivity, demonstrating the importance of optical branches of thermal transport. Manipulating the position of optical branches relative to acoustic branches could provide a novel way to modify a material’s thermal conductivity.
Figure 1. Ab-initio calculations of isotopically pure lattice thermal conductivity of Si and diamond (solid green & purple lines) from full calculations and for the case where acoustic-optic phonon-phonon scattering channels are omitted (dashed green and purple curves)[2]. Diamonds denoted measured values for diamond and Si. Figure 2 shows that embedding nanoparticles in an alloy provides an effective means to reduce the thermal conductivity of a
material. The calculated thermal conductivity for Si0.5Ge0.5 at 800K is 6.07 W/m-K which is higher than experimentally measured values 4.5-5 W/m-K. A recent ab-initio study of SiGe alloys found that the virtual crystal approximation slightly overestimates the thermal conductivity[11]. Adding 5% volume fraction of Ge nanoparticles reduces the thermal conductivity to 2 W/m-K, a 66% reduction. The graph shows the Born approximation provides a good estimate for the thermal conductivity of the nanostructured material.
Figure 2. Calculated thermal conductivity of Si0.5Ge0.5 with embedded Ge nanoparticles, (T=800K), for different nanoparticle volume fractions, as a function of nanoparticle diameter. Symbols denote T-matrix results and lines denote Born approximation[3]. We have also expanded our first principle Boltzmann transport approach to consider the thermal conductivity of diamond nanowires[4]. Here, the phonon distribution is space dependent and we must account for scattering at the nanowire surface. We find the largest thermal conductivity is always found in diamond nanowires with a [001] growth direction. Experimentally, the effect of boundary scattering on the temperature dependence of the thermal conductivity should be evident for diamond nanowires with diameters as large as 400 nm. Conclusions By combining an iterative solution of the Boltzmann transport equation for phonons with interatomic force constants determined using density functional theory, we have developed an effective approach to predict the thermal conductivity of materials, alloys with embedded nanoparticles, and nanowires. This new technique is an important asset in the search for new energy materials. Acknowledgement. This research has been supported in part by NSF CBET Grants 065130, 0651427, 0651381, 1066634, 1066404, and an IRG grant from the European Union. Some simulations were done at the Cornell Nanoscale Facility, a NNIN site funded by NSF. References (1) Broido, D. A.; Malorny, M.; Birner, G.; Mingo, N.; and Stewart, D. A. Appl. Phys. Lett., 2007, 91, 231922. (2) Ward, A.; Broido, D. A.; Stewart, D. A.; and Deinzer, G. Phys. Rev. B, 2009, 80, 125203. (3) Kundu, A.; Mingo, N.; Broido, D. A.; and Stewart, D. A. Phys. Rev. B, 2011, 84, 125426. (4) Li., W.; Mingo, N.; Lindsay, L.; Broido, D. A.; Stewart, D. A.; and Katacho, N. A. 2012, submitted to Phys. Rev. B (5) Omini, M.; and Sparavigna, A. Phys. Rev. B, 1996, 53, 9064. (6) Broido, D. A.; Ward, A.; and Mingo, N. Phys. Rev. B, 2005, 72, 014308. (7) Baroni, S.; de Gironcoli, S.; Dal Corso, A.; and Giannozzi, P. Rev. Mod. Phys., 2001, 73, 515. (8) Deinzer, G.; Birner, G.; and Strauch, D. Phys. Rev. B, 2003, 67, 144304.
Prepr. Pap.-Am. Chem. Soc., Div. Energy Fuels Chem. 2012, 57(2), 559 Proceedings Published 2012 by the American Chemical Society
(9) Esfarjani, K.; Chen, G.; and Stokes, H. T.; Phys. Rev. B., 2011, 84, 085204. (10) Kim, W.; Zide, J.; Gossard, A.; Klenov, D.; Stemmer, S.; Shakouri, A.; and Majumdar, A. Phys. Rev. Lett., 2006, 96, 045901. (11) Garg, J.; Bonini, N.; Kozinsky, B.; and Marzari, M. Phys. Rev. Lett., 2011, 106, 045901.
Prepr. Pap.-Am. Chem. Soc., Div. Energy Fuels Chem. 2012, 57(2), 560 Proceedings Published 2012 by the American Chemical Society
