MODE RESOLVED THERMAL TRANSPORT IN AMORPHOUS SILICON: A MOLECULAR DYNAMICS STUDY

Transcription

1 MODE RESOLVED THERMAL TRANSPORT IN AMORPHOUS SILICON: A MOLECULAR DYNAMICS STUDY Christopher H. Baker and Pamela M. Norris Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia 2294 Amorphous materials have found application in all manner of consumer products and continue to show promise for next generation technologies. Yet thermal transport in this class of materials remains a point of dispute. We studied the mechanisms of thermal transport in amorphous silicon using molecular dynamics simulation and wavelet transform techniques. Perturbation of a single atom in the equilibrated structure introduced broadband vibrational energy, which propagated outward. Analyzing the system dynamics with the wavelet transform provided information on the spectral wave velocity and diffusivity. We compare initial results to the assumptions underlying two current theories of thermal transport in amorphous materials. Introduction Amorphous materials have applications in many technologies where heat transfer is of concern. Examples include common window glass, semiconductor alloys of silicon, 1 and phase-change memory, 2 among others. Just as for crystalline materials, the small length scales found in developing technologies requires a microscopic theory of thermal transport in amorphous materials. A successful theory must not only explain the bulk conductivity based on the microscopic structure, but it must also correctly describe the length dependent conductivity at small length scales, when device dimensions are on the order of the mean-free-path of energy carriers. Despite the theoretical and practical interest in amorphous materials over the last century or so, there remains controversy over how to model thermal transport through the highly disordered atomic structure found in amorphous materials. Theories of thermal transport in solid systems revolve around the vibrational modes of the system. In crystals, the normal modes of vibration arising from interatomic forces and the symmetry of the crystalline lattice are specifically referred to as phonons. Vibrations also no doubt play a role in the amorphous phase, where there is no lattice, and we will examine two theories based on this principle, those pioneered by 1) Cahill and Pohl 3,4 and 2) Feldman, Allen, and Bickham. 5 It should be noted that the models presented by these authors are by no means isolated works and rely on concepts developed by the physics and microscale thermal transport communities over decades and that this does not constitute an exhaustive list of amorphous transport models, e.g. the fracton model of transport. 6,7 The Cahill-Pohl model 3,4 approaches the issue from the crystalline state. The thermal conductivity is calculated by assuming a crystalline dispersion, but with the key modification that these phonons have a mean-free-path equal to one half their wavelength, instead of on the order of tens of nanometers to microns as they would in the crystalline phase. 8 Feldman et al. begin with the eigenstates of a simulated amorphous system itself, which give the allowed vibrational states. 5 By comparing the eigenvectors with the wavevectors of a crystal, the authors define three types of vibrons (analogous to phonons) that transmit energy through different mechanisms: propagons, diffusons, and locons, which transmit energy ballistically, diffusively, and not at all. We present a simulation methodology which enables the direct observation of microscopic heat conduction in amorphous materials, where the properties of heat carriers can be discriminated by frequency. A pure 28 Si system is used for this study because of the abundance of experimental and computational literature on the same system. In this way, the assumptions behind current theories of thermal transport in amorphous materials can be directly tested. Methods The methodology consists of two stages. In the first stage, an equilibrium amorphous structure at zero kelvin is generated. In the second stage, kinetic energy is injected into the system by giving one atom an instantaneous non-zero velocity. The deposited energy then radiates outward and its spatial evolution measured with respect to time. Since the Fourier spectrum of a step function contains energy at all frequencies, this method introduces kinetic energy at all frequencies into the system (but not with equal weight). The time and frequency component of the energy at a certain distance from the epicenter can be discerned using the wavelet transform. 9,1 Baker 1

2 The open source molecular dynamics program LAMMPS 11 was used for the simulation. To begin, a 15 3 unit cell cube of 28 Si was constructed. The Stillinger-Weber potential 12 was used to describe the atomic interactions with the parameter set formulated by Vink et al.. 13 We chose this parameter set since it is optimized for amorphous silicon (a- Si) specifically. A time step of.5 femtoseconds was used throughout the simulation. To produce the amorphous structure, the atomic velocities were set to a distribution corresponding to 5 K and run in the NPT ensemble (constant number, pressure, and temperature) for 4 picoseconds at 1 bar and 3 K, melting the crystal. The thermostat was then modified to cool the system down to about 1 K over the course of 7.5 ps, which is a cooling rate of about 4 x 1 14 K/s, a little faster than rates used in the literature. 14 At this point, an NPH ensemble (constant enthalpy) was employed in conjunction with a fictitious damping coefficient of.1 ev ps/å 2 to drain the remaining thermal energy over the course of 2 ps, leaving the system at a temperature on the order of 1 8 K. The atomic coordinates of the equilibrated structure were printed. Finally, all atomic velocities were set to zero in preparation for the dynamic portion of the simulation. Broadband vibrational energy was introduced to the system by setting the velocity of a single atom to 1. Å/ps (kinetic energy of about 3 mev) with a random directional orientation. The system was then allowed to evolve for 3 ps, printing the velocity components of each atom every 3 time steps. This generated 21 values for velocity as a function of time, where the velocity oscillates as a function of time. The time series of each atom was analyzed using the wavelet transform 9,1 to extract the kinetic energy content of the atom as a function of time and frequency. The resulting description of the energy was summed over atoms with the same radial distance (bin size of 1.95 Å) from the atom at the epicenter. The wavelet transform is essentially a compromise between analysis of a signal in the time domain versus the frequency domain; it contains both temporal and spectral information. For analysis of the data, a mother wavelet frequency of THz was used to give equal resolution in time and frequency at about 5 THz. Other parameters used in the transform were the same as used in a previous work, which should be consulted for details of the transform. 1 g(r) amorph. cryst r(å) Figure 1: Radial distribution function of the equilibrated structure. Triangles denote neighbor shells in a crystal of equivalent density (46.8 nm 3 ). The third nearest neighbor spike has disappeared which is characteristic of amorphous silicon. 15,16 Over a length scale on the order of about 8 Å, the atoms are essentially randomly distributed. n(r) (atoms/å) Sim. Expt r(å) Figure 2: Number distribution function of the equilibrated structure and experimental measurements of pure a-si. 17,18 The simulated system captures the amorphous structure with slight quantitative differences. Baker 2

3 g(θ) (atoms/θ) θ (deg.) Figure 3: Angular distribution function of the equilibrated structure. The peak corresponds to the crystalline tetrahedral angle, but has some finite spread because of the disorder in the system. Results A large amount of information can be derived from the equilibrated structure. The most important fingerprint of an amorphous material is its radial distribution function (RDF), which gives the number of atoms as a function of distance from an arbitrary central atom. The RDF is visualized in two ways: normalized with respect to the RDF of an ideal gas of equivalent density, or normalized such that integration yields the total number of atoms within a sphere of radius r. Both versions have been referred to as the RDF in the literature; here we use RDF for the former and number distribution function (NDF) for the latter. The RDF is plotted in Fig. 1. A short range order is apparent due to the spikes in g(r) within the first several angstroms which quickly decay in magnitude, yielding a completely random structure on the length scale of about 8 angstroms and greater. The third nearest neighbor shell in the crystal structure has disappeared, which is characteristic of a-si. The NDF, n(r), is compared to the experimentally determined function for a-si 17,18 in Fig. 2. The simulated a-si system shows good agreement with experiment. One notable difference is in the first neighbor peak, where the experimental distribution is slightly narrower. The angular distribution function, g(θ), also displays properties expected for an amorphous system, the distribution of angles between nearest neighbors is centered at the tetrahedral value of 19.5 and the width indicates the disorder in the system (Fig. 3). A notable feature is the tail at larger angles which is also present in another simulation of a-si. 14 Despite the good agreement with experimental probability (%) % 5.2% neighbors (atoms) Figure 4: Distribution of coordination number. Despite the close correspondence between n(r) with experimental data, the coordination number is overestimated in the simulated system; it is predominantly 4 in real a-si with a small percentage of 3 and 5 coordinated atoms. results for the number distribution function and the expected shape of g(θ), other metrics indicate that the obtained amorphous system does not completely reproduce experimental a-si. Counting the number of nearest neighbors for each atom, defined as those that lie within the first minimum in n(r) results in a histogram of the coordination number of the system (Fig. 4). Whereas atoms are predominantly 5- fold coordinated in the simulated system, the experimental value is an average of 3.88 coordination, where most atoms are four-fold coordinated. 17 Even though amorphous systems are completely disordered, defects are typically defined in this context as atoms that are under- or over-coordinated with respect to the crystalline system. Hence the simulated system is heavily defected (over-abundance of 5-fold coordinated atoms) with respect to experimental a-si. This over coordination, perhaps arising from the interatomic potential overestimating the Si-Si bond strength, may also be the cause of a difference in the vibrational properties between the simulated system and the experimental system. The vibrational density of states was calculated using the harmonic lattice dynamics program GULP 19 and compared with experimental values found using inelastic neutron scattering 2,21 (Fig. 5). Agreement with the simulated DOS is obtained only after scaling the experimental DOS by a factor of 1.55 in frequency. After shifting, there is good qualitative agreement between the two curves, indicating that, while not exactly reproducing the amorphous state of a-si, the simulated system does show all the characteristics of an amorphous system. Baker 3

4 DOS (arb. units) Sim. Expt f (THz) Figure 5: Density of states of the simulated and experimental systems. 2,21 The experimental DOS has been shifted by a factor of 1.55 in frequency and both distributions are normalized to have unit area. The shift causes good qualitative agreement between the simulated and experimental systems. Figure 6: Energy distribution versus time for 2.9 THz frequency. The color is proportional to the wavelet energy density (ev/ps THz); absolute magnitudes should not be compared between figures. The analysis of the dynamic portion of the simulation is shown in Figs The figures present the radial distribution of energy as a function of time for increasing frequencies. The wavelet transform introduces some uncertainty into the results, having a lower temporal resolution at low frequencies compared to higher frequencies. This manifests itself in the width of the energy ridge in Fig. 6. At higher frequencies, the temporal resolution increases (with the tradeoff occurring in the frequency resolution), leading to a narrower stem near the origin (e.g. Figs. 7 and 8). As the frequency increases, the energy goes from pure wave transport (Fig. 6), to diffusive-wave transport (Fig. 7), to almost purely diffusive transport (Fig. 8). Linearly fitting the wave propagation in Fig. 6 results in a velocity of about 2.4 km/s. Energy distribution versus time for 8.21 THz fre- Figure 7: quency. Discussion We now discuss the implications of our data with respect to two theories of thermal transport in amorphous materials. In the Cahill-Pohl model, 3,4 which is often used to estimate the thermal conductivity of amorphous solids, the key assumptions are that heat carriers travel at the crystalline sound speed (for longitudinal and transverse acoustic branches) and that the mean-free-path of these modes is one-half their wavelength. This results in an expression for the conductivity: Energy distribution versus time for THz fre- Figure 8: quency. Baker 4

5 f (THz) LO LA TO TA k (A 1 ) Figure 9: Dispersion of crystalline phase in the (1) direction. Acoustic, optical, longitudinal, and transverse modes are noted. Dashed line is the zone center extrapolation, equivalent to Debye type dispersion. π Λ = 6 1/3 T 2 Θi kb n 2/3 /T v i i Θ i x 3 e x (e x 1) 2 dx, hf (1) where k B is Boltzmann s constant, n is the atomic number density, v i is the Debye velocity of mode i, T is temperature, Θ i is the mode specific Debye temperature (proportional to the frequency at maximum wavevector), and x is the reduced phonon energy: k B T. The strength of this theory is that it has no adjustable parameters and requires only basic knowledge of the crystalline phase of the material in question. Yet, from Fig. 6, the measured velocity in the amorphous phase is only 2.4 km/s compared to Debye velocities of 11. km/s (longitudinal) and 7.1 km/s (transverse) in the crystalline phase at the same density (Fig. 9). This suggests two possible explanations for why the Cahill-Pohl model still yields accurate estimates of Λ. The first is that it overestimates group velocities while underestimating meanfree-paths. Referring to Figs. 9 and 6 the halfwavelength of Debye modes with a frequency of 2.9 THz is about 26.4 Å (LA) or 17. Å (TA). The energy in Fig. 6 clearly travels farther than this distance without showing diffusive energy loss. Furthermore, experimental evidence has shown that the sound speed of a-si is less than that of the crystalline phase, 22 although their value is about 8% of the weighted crystalline sound speed versus our value of 29%. Our much lower value of sound speed may be due to the over-coordinated, highly defected structure. It is unclear how this would affect comparisons made to the Cahill-Pohl model as our highly defected structure would surely lead to increased scattering of phonons. Further study with simulated amorphous structures with more accurate coordination will be necessary. In the model of Feldman et al., 5 energy carriers in amorphous structures are divided into three characteristic types. Propagons travel ballistically with long mean-free-paths, diffusons transmit energy through an intrinsic diffusivity, and locons do not contribute to transport. The authors estimate the transitions between carrier types to occur at about 4.8 THz for the propagon-diffuson transition and about 17.4 THz for the diffuson-locon transition in a-si. These frequencies are in reasonable agreement with our findings, that the transitions occur at about 3.95 THz and THz, respectively. The agreement is not entirely surprising since Feldman et al. used the Stillinger-Weber potential in their calculations. The diffuson-locon transition appears to correlate with the acoustic to optical transition in the dispersion of about 17.4 THz for the (1) direction (Fig. 9). Disagreement between our results and those of experiment, especially with regard to the amorphous structure, result from two sources. The first is the preparation procedure used. It is possible that reducing the cooling rate will allow the structure to equilibrate to a lower energy state with four-fold coordination. The second is in the use of the interatomic potential. The Stillinger-Weber potential may not be able to recreate experimental a-si with the desired degree of fidelity, no matter what parameter set it used, and in this case, a different potential like the Tersoff 23,24 or density functional theory based Car-Parinello molecular dynamics 25,26,27 would more closely reproduce the desired system. The accuracy of the results may also be improved by adjusting the parameters used in the wavelet transform and performing the same study on independently generated structures or on the same structure for different orientations of the initial kinetic energy. Ultimately, fitting the results to a model of ballistic diffusive transport will allow the frequency resolved velocity and diffusivity (related to the mean-free-path) to be measured quantitatively. Summary In summary, we have presented a method by which the velocity and diffusivity of different vi- Baker 5

6 brational modes can be directly measured in simulated amorphous systems. Initial results indicate that the Cahill-Pohl model may achieve its accuracy by overestimating mode velocities and underestimating mean-free-paths. The data agreed qualitatively with the theory and findings of Feldman et al.. Further improvements to the study, through a more refined structure generation procedure and a more accurate description of interatomic forces, will enable quantitative comparison to theoretical treatment and experimental findings for transport in amorphous structures. Acknowledgements The authors would like to acknowledge financial support from the Air Force Office of Scientific Research (Grant No. FA ). C. H. B. is grateful for support from the Virginia Space Grant Consortium through a Graduate STEM Research Fellowship and the Department of Defense through a NDSEG Fellowship. References [1] P. G. Le Comber, Journal of Non-Crystalline Solids 115, 1 (1989). [2] M. Wuttig and N. Yamada, Naure Materials 6, 824 (27). [3] D. G. Cahill and R. O. Pohl, Solid State Communications 7, 927 (1989). [4] D. G. Cahill, S. K. Watson, and R. O. Pohl, Physical Review B 46, 6131 (1992). [5] J. L. Feldman, P. B. Allen, and S. R. Bickham, Physical Review B 59, 3551 (1999). [6] A. Aharony, S. Alexander, O. Entin-Wohlman, and R. Orbach, Physical Review Letters 58, 132 (1987). [7] A. Jagannathan, R. Orbach, and O. Entin- Wohlman, Physical Review B 39, (1989). [8] W. Kim, R. Wang, and A. Majumdar, Nano Today 2, 4 (27). [9] G. Kaiser, A Friendly Guide to Wavelets (Birkhauser, Boston, 1994). [1] C. H. Baker, D. A. Jordan, and P. M. Norris, Physical Review B 86, 1436 (212). [11] S. Plimpton, Journal of Computational Physics 117, 1 (1995), lammps.sandia.gov. [12] F. H. Stillinger and T. A. Weber, Physical Review B 31, 5262 (1985). [13] R. L. C. Vink, G. T. Barkema, W. F. van der Weg, and N. Mousseau, Journal of Non- Crystalline Solids 282, 248 (21). [14] M. Ishimaru, S. Munetoh, and T. Motooka, Physical Review B 56, (1997). [15] J. Fortner and J. S. Lannin, Physical Review B 39, 5527 (1989). [16] D. A. Drabold, P. A. Fedders, O. F. Sankey, and J. D. Dow, Physical Review B 42, 5135 (199). [17] K. Laaziri, S. Krycia, S. Roorda, M. Chicoine, J. L. Robertson, J. Wang, and S. C. Moss, Physical Review Letters 82, 346 (1999). [18] K. Laaziri, S. Krycia, S. Roorda, M. Chicoine, J. L. Robertson, J. Wang, and S. C. Moss, Physical Review B 6, 1352 (1999). [19] J. D. Gale and A. L. Rohl, Molecular Simulation 29, 291 (23). [2] W. A. Kamitakahara, H. R. Shanks, J. F. McClelland, U. Buchenau, F. Gompf, and L. Pintschovius, Physical Review Letters 52, 644 (1984). [21] W. A. Kamitakahara, C. M. Soukoulis, H. R. Shanks, U. Buchenau, and G. S. Grest, Physical Review B 36, 6539 (1987). [22] I. R. Cox-Smith, H. C. Liang, and R. O. Dillon, Journal of Vacuum Science and Technology A 3, 674 (1985). [23] J. Tersoff, Physical Review B 37, 6991 (1988). [24] J. Tersoff, Physical Review B 39, 5566 (1989). [25] R. Car and M. Parrinello, Physical Review Letters 55, 2471 (1985). [26] R. Car and M. Parrinello, Physical Review Letters 6, 24 (1988). [27] I. Štich, R. Car, and M. Parrinello, Physical Review B 44, 1192 (1991). Baker 6