The effect of coherent optical phonon on thermal transport

Transcription

1 Appl. Phys. A DOI 1.17/s The effect of coherent optical phonon on thermal transport Y. Zhang Y. Wang Received: 8 May 214 / Accepted: 22 July 214 Ó Springer-Verlag Berlin Heidelberg 214 Abstract Phonons are quantized lattice vibrations and the major heat carriers in most crystalline materials. We have been utilizing femtosecond phonon spectroscopy to excite and detect optical coherent phonons (CPs) in various materials. However, the impact of CPs to overall thermal transport is still unknown. In this study, we developed a small perturbation model in MD to simulate CPs and investigate the effects of CPs on thermal transport in Bi 2- Te 3 at temperatures ranging from 2 to 325 K. The phonon frequency and lifetime predicted by our model agree very well with experimental results. It is found that the effective thermal transport estimated with the heat current autocorrelation function shows a great enhancement upon CP generation and extension, especially at low temperatures. Our results suggest that it is possible to manipulate thermal transport effectively with CPs. 1 Introduction Controlling thermal transport is of great interest in various applications such as thermal management and energy harvesting. Alloys [1], interfaces [2] and isotopes [3] have been introduced to scatter phonons more effectively and to reduce lattice thermal conductivity. Even though negligible in bulk materials, optical phonons can play an important role in nanostructures where the structure size is comparable to optical phonon mean free path [4]. Phonons are generally incoherent, meaning no well-defined phase among atoms movements. Under extreme circumstances, Y. Zhang Y. Wang (&) Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA yaguo.wang@austin.utexas.edu such as illuminated by high-intensity femtosecond laser pulses, coherent phonons (CPs), of which the atoms vibrate at the same frequency and with well-defined phase [5, 6], can be generated, and the control of overall thermal transport by manipulating CPs in nanostructures might be feasible [7]. Experimentally, coherent optical phonons can be generated with femtosecond laser pulses. There are two well-accepted phenomenological theories to explain the generation of coherent optical phonons with femtosecond laser pulses, namely impulsively stimulated Raman scattering (ISRS) [8] and displacive excitation of coherent phonons (DECP) [9, 1]. In transparent materials, where the laser photon energy is well below the band gap, the underlying mechanism is ISRS. In ISRS theory, laser field E pushes the atoms as an impulse force and drives the oscillations, which are then damped by phonon scattering [8, 11]. In opaque materials, where usually the laser photon energy is above the band gap, DECP is assumed to be the mechanism. In DECP, the persistent electron-hole excitations temporarily shifts the equilibrium position of ion cores, which start to oscillate between its original and shifted equilibrium positions [9, 1]. Even though CPs have been studied intensively with ultrafast phonon spectroscopy [12 14], the effect of CPs on overall thermal transport has not been investigated. CP generation and relaxation are nonequilibrium processes and only exist for several picoseconds at room temperature, while transient approaches measuring thermal conductivity, such as timedomain thermo-reflectance (TDTR) [15 17], require data acquisition time over several nanoseconds. To make the effect of CPs prominent enough to be reflected in the overall thermal transport, CP lifetime has to be extended. There are two feasible ways to reach this goal. First, excite CPs at very low temperature, where Umklapp phonon scattering is not important and CP lifetime is much longer.

2 Y. Zhang, Y. Wang The second way is to excite the CPs periodically with multiple laser pulses separated by several periods of CP lifetime. Conducting either of the experiments suggested above is nontrivial. As the first attempt, we design MD simulations to mimic these two experiments and estimate the effective thermal transport with heat current autocorrelation function. CPs in bismuth telluride (Bi 2 Te 3 ) can be excited with femtosecond pulses through DECP and have been studied extensively in our group with ultrafast phonon spectroscopy [14]. In this study, we developed a small perturbation model in MD to simulate CPs and investigate the effects of CPs on lattice thermal transport in Bi 2 Te 3 at temperatures ranging from 2 to 325 K. As for simulations, Qiu and Ruan have developed a set of two-body potentials for Bi 2 Te 3 [18]. The predicted thermal conductivities with their potentials agree well with experiments [19]. Studying Bi 2 Te 3 will allow us to compare the simulation results with experiments directly. A similar approach has been applied to study the nonthermal melting induced by CP generation [2]. Instead of the influence of acoustic phonons [21 24], this study focuses on the effects of coherent optical phonon excitation on the lattice thermal transport. In the numerical method section, we describe the processes to generate CPs with our small perturbation model under single- and multi-pulse excitations, as well as to estimate effective thermal transport with heat current autocorrelation function. In the results and discussion section, we compare the effective thermal transport without CP, with single-pulse excited CP, as well as with multi-pulse excited CP, which will be denoted as, and multi-pulse CP, respectively, in the following discussion. 2 Numerical method 2.1 Coherent phonon generation with single and multiple pulses A small perturbation model was developed to simulate CPs in Bi 2 Te 3.Bi 2 Te 3 has five atoms in each rhombohedral unit cell and possesses 12 optical phonon branches, among which eight branches are nondegenerate. All optical phonons associate with the movements of this five-atom dipole. The spring ball model for A 1 1g phonon modes demonstrated in Fig. 1a. The two-body potentials for Bi 2- Te 3 developed by Qiu and Ruan [18], which has been used to predict various phonon properties in Bi 2 Te 3 accurately, are used in this paper to calculate interatomic forces. To excite A 1 1g phonons in MD, two pairs of Bi and TeI atoms are displaced slightly away from equilibrium positions, for example 1 % of nearest-neighbor distance 3 Å, as shown in Fig. 1a. This imitates the atomic displacements in the DECP process. Then, atoms are released to vibrate normally under interatomic forces. The transient atomic displacements are recorded and shown in Fig. 1b, which display similar features as the CP oscillations observed in ultrafast phonon spectroscopy (Fig. 1c). The transient displacements can be fitted in the same way as phonon oscillations with a chirped damping harmonic oscillator: Q ¼ expð t=sþ cos½ðx þ btþt þ uš ð1þ where s, X, b and u are CP lifetime, angular frequency, chirping coefficient and initial phase of phonon vibration, respectively. The phonon dephasing time and frequency obtained by fitting the transient displacements in MD are 3.2 ps and 1.83 THz, which agree well with 4. ps and 1.86 THz from ultrafast phonon spectroscopy at 3 K [14]. It has been shown that the CP properties predicted with our small perturbation model agree well with the experimental results, even though phonon relaxation process simulated in this model is purely classical and no electron effect is considered. Frequencies and lifetimes of A 1 1g CPs at different temperatures have been calculated and plotted in Fig. 2. Decrease in phonon frequencies, as shown in Fig. 2a, indicates a red-shift of phonon wavelength, due to thermal expansion and anharmonic interactions at higher temperatures [25]. As shown in Fig. 2b, when the temperature rises, CP lifetime is reduced due to the Umklapp phonon scattering. The Debye temperature T D of Bi 2 Te 3 is about 155 K [26]. CP lifetimes for both single-pulse and multipulse cases stay almost constant above Debye temperature. At T D, maximum number of phonon modes has been excited and phonon-phonon scattering reaches its maximum strength. As a result, the CP lifetime does not further decrease above Debye temperature. Both phenomena have been observed in experiments [27, 28]. Multi-pulse excitation of CPs is applied to extend the CP lifetime. As shown in Fig. 2c, the first pulse generates the CPs. The second pulse, which is weaker and delayed after five phonon periods, is used to compensate the loss in CP amplitude and recover the initial phonon amplitude generated by the first pulse. Pulses with the same power as the second pulse are given every five phonon periods afterward to compensate the loss and keep phonon amplitude at the same level, preventing the decay of CP as time develops. Via applying these secondary pulses periodically, CP lifetimes can be greatly extended. The multi-pulse CP lifetimes with 12 pulses have been shown in Fig. 2b. Comparing with at the same temperature (blue curve in Fig. 2c), CP lifetimes can be greatly extended via applying these secondary pulses periodically.

3 Effect of coherent optical phonon on thermal transport Displacement of Bi Atoms(a.u.) Fitting MD Simulation Coherent Phonon Amplitude (a.u.) (c) Fitting CP measured in experiments Delay Fig. 1 a Spring ball model for A 1 1g optical phonon; b transient atomic displacements of Bi atoms from MD; c transient atomic displacements of Bi 2 Te 3 in ultrafast phonon spectroscopy Phonon Frequency (THz) Single-Pulse CP Coherent Phonon Life Single-Pulse CP Multi-Pulse CP Displacement of Bi atoms (a.u.) (c) Single-Pulse CP Multi-Pulse CP Fig. 2 a Coherent phonon frequencies at different temperatures; b coherent phonon lifetimes at different temperatures excited by single and multiple pulses; c coherent phonons modeled in MD with single-pulse and multi-pulse excitations. The arrows mark the times applying pulses. The blue arrow and the first black arrow have been displaced vertically for clarity Since displacing atoms from equilibrium positions will inject addition potential energy into the system, the system temperature will increase until new equilibrium is reached. For example, the 1 % displacement will increase the final system temperature by 2 K, while giving 12 pulses leads to an increase of 25 K in the final system temperature. Thus, the final system temperatures are used when the results are demonstrated. 2.2 Effective thermal transport Heat flux * q is defined by: q * ¼ X e * i v i þ 1 X F * ij * * v i r ij 2 i i;j ð2þ e i ¼ 1 2 m i * v 2 i þ 1 X / * r ij 2 ð3þ j where * r is the atom position, * v is the velocity, F * is the interatomic force, e is the site energy and / is interatomic potential energy. Heat flux * q along both cross-plane and inplane direction is collected during simulation and heat current autocorrelation function (HCACF) is calculated accordingly in post-analysis. As seen in Fig. 3a, the HCACF has long-decay components and oscillatory components, representing contributions from acoustic and optical phonons. This phenomenon is common for crystal with multiple atoms per primitive cell [29]. The full HCACF can be expressed as: Z 1 D E * qðtþq ðþ dt ¼ Z 1 Z 1 þ A expð t=sþdt X B i expð t=s O i Þ cosðx itþdt i ð4þ where A stands for the acoustic phonon population, B i stands for the optical phonon population, s denotes the phonon dephasing time (lifetime) and x i is the oscillation frequency of optical phonons. When time goes to infinity, the first term of Eq. 4 will converge to a constant value and

4 Y. Zhang, Y. Wang HCACF (a.u.) Integral of HCACF (a.u.) Integral of HCACF Exponential fitting Lattice (c) Fig. 3 a Heat current autocorrelation function (HCACF); b integral of HCACF; and c temperature change in multi-pulse CP generation the net value of the second term will be zero. To reduce the computational cost, an exponential function is adopted to fit the integral of HCACF at finite time: Z t D E * qðuþq * ðþ du ¼ C s C s expð t=s Þ ð5þ in which C and s represent effective phonon population and phonon lifetime consisting of effects from all phonon modes, respectively. A typical fitting is demonstrated in Fig. 3b. The integral time is taken to be 15 ps for all cases, at which the integral of HCACF already converges and the system has been stabilized at a new temperature (Fig. 3c). We define a quantity K to describe the effective thermal transport as: K ¼ 1 k B VT 2 C s ð6þ where k B is Boltzmann constant, V is system volume, T is temperature. For the equilibrium case without CP generation, K is the same as lattice thermal conductivity defined in Green Kubo formula. However, because thermal conductivity is an intrinsic material property and the phenomena we observed here is temporary, the concept of thermal conductivity is no longer valid in this situation. The effect of CPs on thermal transport may last for several 1 ps or much longer time, depending how they are excited. Home-developed MD programs have been used to study phonon properties in Bi 2 Te 3. The MD simulation domain contains hexagonal lattices (39 Å 9 39 Å Å) with periodic boundary conditions along all three directions. With a time step of 2 fs, the simulations firstly run in NVT ensemble for 2 ps, then switch to NVE ensemble and run for another 4 ps to obtain equilibrium structures. After that, A 1 1g CPs are generated, and in the meanwhile, heat flux is collected for another 1nsperiod. 3 Results and discussion Using the method describe in Sect. 2, the K of Bi 2 Te 3 is investigated for cases of, and multipulse CP. Since the displacements of atoms are given in the cross-plane direction when A 1 1g CP is generated, there are negligible changes in the in-plane effective thermal transport K==, and thus, this section will focus on the cross- plane effective thermal transport K?, as shown in Fig. 4a. For the case, the K? decreases with temperature, which agrees well with the experimental data measured in poly crystalline Bi 2 Te 3 [19]. For the case of, the trend is similar over the entire temperature range. At very low temperatures, the single-pulse CP case shows much higher K? comparing with case. However, this difference decreases with temperature and eventually disappears around the Debye temperature. The predicted K? s of multi-pulse CP (12 pulses in our calculation) are even much higher. Similar to the singlepulse CP case, the difference decreases with temperature, but vanishes at a higher temperature, around 25 K. Here, we do not show the thermal conductivities below 75 K for multi-pulse CP. When CPs are excited, energy is injected into the sample and raises up the temperature. With 12 pulses used in our calculation, the injected energy boosts up the temperature by 25 K, which will shade the effect of CPs at low temperatures. A natural question to ask here is why and how the CP excitation affects the effective thermal transport. So we conduct more analysis to dig out the deeper reasons. According to Eq. 6, K? is affected by the effective phonon dephasing time s and effective phonon population C at a certain temperature. Thus, to find out the reason behind the enhancement in effective thermal transport when CPs are generated and extended, we plot the effective phonon dephasing times s and effective phonon populations C in Fig. 4b.

5 Effect of coherent optical phonon on thermal transport K* (W/mK) multi-pulse CP Effective Dephasing multi-pulse CP Callaway model multi-pulse CP Effective Phonon Population C * (a.u.) Fig. 4 a Thermal transport at different temperatures; b predicted effective phonon population and effective dephasing time at different temperatures As shown in Fig. 4b, the effective populations increase with temperature with similar trends for all three cases. The effective phonon populations C s of are slightly larger than that of over the entire temperature range, while the C s of multi-pulse CP are much greater. Exciting CP in the sample brings out extra A 1 1g optical phonons on top of the equilibrium phonons at each temperature [3]. When multiple pulses are applied, A 1 1g CPs are generated repeatedly, which gives even larger effective phonon populations. At low temperatures, the effective phonon dephasing times of both the and cases decrease with temperature rapidly, following very similar trends. At very low temperatures, phonons are scattered primarily by defects and grain boundaries [31, 32]. As temperature rises, the phonon-phonon Umklapp scattering becomes more important and reduces the phonon lifetime. According to Callaway s model [33], the Umklapp process at low temperatures produces a phonon relaxation time proportional to 1 ½e TD=bT T 3 Š, where b is a constant determined by the phonon spectrum of the material [34]. The trend of our predicted effective phonon relaxation times with temperature agrees well with that from Callaway s model. The absolute values of phonon lifetime for the case are much longer than that of case. The difference decreases with temperature and vanishes around Debye temperature. Similarly, the multi-pulse CP case shows much longer effective phonon lifetime until around 25 K. From the discussion above, one can see that excitation of coherent A 1 1g phonons can substantially boost the effective phonon population, as well as prolong the effective dephasing time, both of which contribute to the enhancement in K? observed in Fig. 4a. There are two competing factors that determine the K? : (1) phonon lifetime decreases with temperature and (2) phonon population increases with temperature. At the very lowest temperatures, increase in phonon population prevails over the decreasing phonon lifetime; hence, the effective thermal transport increases. At the intermediate and high temperatures, the number of phonons available for thermal excitation becomes less and the effect from decreasing phonon lifetime dominates; hence, the thermal transport changes downwardly. Exciting CPs introduces a third factor into this process by changing both the phonon population and lifetime. The effects on phonon population are prominent at low temperatures, especially for the case of multi-pulse CP. However, at intermediate and higher temperatures, increase in phonon population becomes less important and phonon lifetime is the governing factor. For both the and multi-pulse CP cases, enhancements in effective thermal transport are obvious when the effective phonon lifetimes are longer than that of case. The enhanced thermal transport shown in Fig. 4a could come from the CP mode directly or indirectly. Even only one CP is excited, the effective phonon population and the lifetime of all the phonon modes have been changed. The CPs can couple effectively with other phonon modes, which eventually enhance the overall thermal transport. As shown in Fig. 4b, Umklapp process scatters CPs in a similar way to incoherent phonons. With single-pulse excitation, the effect of CP on s is overwhelmed by phonon-phonon scatterings above Debye temperature. With multi-pulse excitation, this effect is quenched above 25 K. In this study, we have only applied 12 pulses to excite CPs periodically. However, our results suggest that the enhancement in K? could be observed above room temperature if more pulses are introduced. Moreover, in this particular case with Bi 2 Te 3, the Debye temperature is only about 155 K [26]. For materials with much higher Debye temperatures, it is possible to manipulate effective thermal transport via CP generation more effectively at

6 Y. Zhang, Y. Wang higher temperatures. Experimental investigation on this problem is necessary to verify this proposed approach. 4 Summary In summary, small perturbation model in MD has been developed to generate CPs in Bi 2 Te 3 and the effective thermal transport with CPs has been estimated. It has been found that CPs can substantially boost the effective phonon population and prolong the effective dephasing time, both of which contribute to the enhancement of thermal transport. Our results suggest that it is possible to manipulate lattice thermal transport effectively with CPs. Acknowledgments This work is supported both by National Science Foundation Grant No. CBET and by faculty start-up funds from Department of Mechanical Engineering and Cockrell School of Engineering at University of Texas at Austin, both of which are gratefully acknowledged. References 1. B. Poudel, Q. Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yan et al., Science 32(5876), (28) 2. R. Venkatasubramanian, E. Siivola, T.S. Colpitts, B. O quinn, Nature 413(6856), (21) 3. L. Yang, N. Yang, B. Li, Sci. Rep. 3, 1143 (213) 4. Z. Tian, K. Esfarjani, J. Shiomi, A.S. Henry, G. Chen, Appl. Phys. Lett. 99(5), (211) 5. M.V. Lebedev, O.V. Misochko, Phys. Solid State 51(9), 1843 (29) 6. L. Dhar, J.A. Rogers, K.A. Nelson, Chem. Rev. 94(1), 157 (1994) 7. M.N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M.T. Bulsara, A.J. Schmidt, A.J. Minnich, S. Chen, M.S. Dresselhaus, Z. Ren, E.A. Fitzgerald, G. Chen, Science 338(619), 936 (212) 8. G.C. Cho, W. Kütt, H. Kurz, Phys. Rev. Lett. 65(6), 764 (199) 9. H.J. Zeiger, J. Vidal, T.K. Cheng, E.P. Ippen, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 45(2), 768 (1992) 1. T.K. Cheng, J. Vidal, H.J. Zeiger, G. Dresselhaus, M.S. Dresselhaus, E.P. Ippen, Appl. Phys. Lett. 59(16), 1923 (1991) 11. Y. Shinohara, K. Yabana, Y. Kawashita, J.I. Iwata, T. Otobe, G.F. Bertsch, Phys. Rev. B 82(15), (21) 12. D. Boschetto, E.G. Gamaly, A.V. Rode, B. Luther-Davies, D. Glijer, T. Garl, O. Albert, A. Rousse, J. Etchepare, Phys. Rev. Lett. 1(2), 2744 (28) 13. Y. Wang, X. Xu, J. Yang, Phys. Rev. Lett. 12(17), (29) 14. Y. Wang, X. Xu, R. Venkatasubramanian, Appl. Phys. Lett. 93(11), (28) 15. M. Highland, B.C. Gundrum, Y.K. Koh, R.S. Averback, D.G. Cahill, V.C. Elarde, J.J. Coleman, D.A. Walko, E.C. Landahl, Phys. Rev. B 76(7), (27) 16. G.A. Antonelli, B. Perrin, B.C. Daly, D.G. Cahill, MRS Bull. 31(8), 67 (26) 17. Y.K. Koh, S.L. Singer, W. Kim, J.M.O. Zide, L. Hong, D.G. Cahill, A. Majumdar, A.C. Gossard, J. Appl. Phys. 15(5), 5433 (29) 18. B. Qiu, X. Ruan, Phys. Rev. B 8(16), (29) 19. Y. Wang, B. Qiu, A.J.H. McGaughey, X. Ruan, X. Xu, J. Heat Transf. 135(9), 9112 (213) 2. Y. Wang, X. Xu, Appl. Phys. A 11(3), 617 (213) 21. A.J.H. McGaughey, M. Kaviany, Phys. Rev. B 69(9), 9433 (24) 22. A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C.N. Lau, Nano Lett. 8(3), 92 (28) 23. D.R. Clarke, Surf. Coat. Technol , 67 (23) 24. J. Hone, M. Whitney, A. Zettl, Synth. Met. 13(1 3), 2498 (1999) 25. O. Blaschko, G. Ernst, G. Quittner, W. Kress, R.E. Lechner, Phys. Rev. B 11(1), 396 (1975) 26. G.E. Shoemake, J.A. Rayne, R.W. Ure Jr., Phys. Rev. 185(3), 146 (1969) 27. F. Vallée, F. Bogani, Phys. Rev. B 43(14), 1249 (1991) 28. M. Hase, K. Mizoguchi, H. Harima, S.-I. Nakashima, K. Sakai, Phys. Rev. B 58(9), 5448 (1998) 29. A.J.H. McGaughey, M.I. Hussein, E.S. Landry, M. Kaviany, G.M. Hulbert, Phys. Rev. B 74(1), 1434 (26) 3. A.Q. Wu, X. Xu, R. Venkatasubramanian, Appl. Phys. Lett. 92(1), 1118 (28) 31. G.S. Nolas, G. Fowler, J. Yang, J. Appl. Phys. 1(4), 4375 (26) 32. J. Yang, D.T. Morelli, G.P. Meisner, W. Chen, J.S. Dyck, C. Uher, Phys. Rev. B 67(16), (23) 33. J. Callaway, Phys. Rev. 113(4), 146 (1959) 34. P.G. Klemens, Encyclopedia of Physics (Springer, Berlin, 1956)