Introduction to phonon transport

Transcription

1 Introduction to phonon transport Ivana Savić Tyndall National Institute, Cork, Ireland Materials for a Sustainable Energy Future Tutorials, Institute for Pure & Applied Mathematics, UCLA September 12, 2013 Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

2 Outline Thermal transport and sustainable energy. Overview of atomistic approaches to study thermal transport in materials: Boltzmann transport equation. Non-equilibrium molecular dynamics. Equilibrium molecular dynamics. Critical assessment of these approaches, recent developments and future challenges. Examples of their applications and insights gained from them. Ivana Savić, Introduction to phonon transport September 12, / 52

3 Basic problem How is energy (heat) transported in a semiconducting material due to the temperature gradient? T+ T heat flow T Fourier law: J = κ T; J - heat current, κ - thermal conductivity. κ = κ lattice +κ electronic +κ radiative. κ κ lattice in semiconductors. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

4 Why do we study thermal transport in semiconductors? Sustainable energy: convert waste heat into electricity. Ivana Savić, Introduction to phonon transport September 12, / 52

5 Thermoelectric energy conversion Maximum efficiency of the thermoelectric couple: η max = T 1+ZT 1 T + T 1+ZT +T/(T + T). transforming electricity into heat T cooling transforming heat into electricity T+ T heat source p type + e e n type p type e + e n type dissipation T+ T heat sink T V I + V Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

6 Thermoelectric energy conversion Thermoelectric figure of merit: ZT = σα 2 T/κ σ - electrical conductivity. α - Seebeck coefficient (α = V/ T). κ - thermal conductivity. transforming electricity into heat T cooling transforming heat into electricity T+ T heat source p type + e e n type p type e+ e n type dissipation T+ T heat sink T V I + V Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

7 Thermoelectric energy conversion If ZT, Carnot limit: η max T/(T + T). State-of-the-art bulk thermoelectric materials: ZT 1, η max 0.17 η Carnot. transforming electricity into heat T cooling transforming heat into electricity T+ T heat source p type + e e n type p type e + e n type dissipation T+ T heat sink T V I + V Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

8 Why is it difficult to increase ZT in bulk materials? ZT = σα 2 T/(κ el +κ latt ) Conflicting effects on ZT: α 1/σ m, κ el σ, κ latt σ. 1 zt zt Carrier concentration (cm 3 ) C. J. Snyder and E. Toberer, Nature Mater. 7, 105 (2008) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

9 Nanostructured thermoelectric materials ZT = σα 2 T/κ. Nanostructuring can improve ZT of bulk materials: M. S. Dresselhaus et al., Adv. Mater. 19, 1043 (2007) Reduce the thermal conductivity κ. The power factor σα 2 is reduced less than κ. incoming phonon outcoming phonon Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

10 High ZT nanostructured materials: Si nanowires with rough surface ZT 0.6, 60 times larger than that of bulk Si due to the κ reduction. a b c d A. I. Hochbaum et al., Nature 451, 163 (2008) Ivana Savic, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

11 Extremely anharmonic bulk materials Low ω optical branches and strong phonon-phonon interaction very low κ. D. T. Morelli et al., PRL 101, (2008) Ivana Savić, Introduction to phonon transport September 12, / 52

12 Thermal management in nano- and opto-electronics Solid state cooling via thermoelectric materials. E. Pop and K. E. Goodson, J. Electron. Packag. 128, 102 (2006) Ivana Savić, Introduction to phonon transport September 12, / 52

13 Lattice vibrations in solids Atom = ion cores + valence electrons. Ion cores move in a potential field generated by the average motion of valence electrons. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

14 Lattice vibrations in solids Displacements from equilibrium positions can be described as a linear combination of normal modes. Energy of a normal mode is quantized: (n+1/2) ω, n = 0,1,... Phonons are quanta of energy of normal modes. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

15 Lattice Hamiltonian of crystalline solids Total potential energy of the crystal in the harmonic approximation: V = V 0 + V u α (lb) u α (lb)+ lb,α V 2! u α (lb) u α (l b ) u α (lb)u α (l b ) lb,l b,α,α u α (lb) - deviation of atom lb from its equilibrium position in the α direction. Equations of motion: m b d 2 u α (lb) dt 2 = b,α 2 V u α (lb) u α (l b ) u α (l b ). Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

16 Lattice Hamiltonian of crystalline solids Reciprocal space: u α (lb) = 1 U α (q;b)exp[i(q x(l) ωt)]. m b q Equations of motion become ω 2 U α (q;b) = D αα (bb q)u α (q;b ). b α Dynamical matrix is defined as D αα (bb q) = G. P. Srivastava, The physics of phonons 1 2 V mb m b u l α (0b) u α (l b ) exp(iql ). Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

17 How do we obtain the interatomic potential? µs Time ns ps fs Empirical DFT A nm µm mm Length Density functional theory (DFT): No fitting parameters. Computationally expensive. Short length and time scales. Empirical potentials: Fitted to experimental data. Computationally less expensive. Longer length and time scales. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

18 Phonon band structure: Example of Si 15 Frequency (THz) 10 5 Experiment DFT Tersoff 0 Γ X K Γ L DFT reproduces well experimental phonon frequencies. Empirical potentials do not describe phonon frequencies as accurately as DFT. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

19 Lattice dynamics: beyond the harmonic approximation Harmonic approximation infinite phonon lifetimes and κ. Finite phonon lifetimes (but still infinite κ) due to boundaries, defects, impurities, alloying... C. J. Vineis et al., Adv. Mater. 22, 3970 (2010) Ivana Savić, Introduction to phonon transport September 12, / 52

20 Lattice dynamics: beyond the harmonic approximation Finite phonon lifetimes and finite κ due to anharmonicity: V = V 0 + V u α (lb) u α (lb)+ lb,α 0 1 2! b,b,b,α,α,α b,b,α,α 2 V u α (b) u α (b ) u α (b)u α (b ) ! 3 V u α (b) u α (b ) u α (b ) u α (b)u α (b )u α (b ) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

21 Three-phonon scattering q s q s q s q s q s q s (a) Annihilation of one phonon and creation of two phonons: ω(q,s) = ω(q,s )+ω(q,s ), q+g = q +q. (b) Annihilation of two phonons and creation of a third phonon: ω(q,s)+ω(q,s ) = ω(q,s ), q+q = q +G. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

22 Thermal conductivity in the kinetic theory of gases T+ T J L J T R l 0 l z Heat current: J z = nv z ǫ, n - concentration, v z - speed of gas atoms, ǫ - energy. [( ) ( )] J z = J L J R nv z k B T l dt dz T +l dt dz, l - mean free path. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

23 Thermal conductivity in the kinetic theory of gases T+ T J L J T R l 0 l z Thermal conductivity: κ z = c v v z l = c v v 2 zτ, c v - heat capacity, τ = l/v z - average collision time. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

24 More advanced theory: Boltzmann transport equation Time evolution of the distribution function n(r,v,t): n(r+dr,v+dv,t +dt) n(r,v,t) = dt n t r position, v velocity, t time. scattering Boltzmann transport equation (BTE): n t +v rn+ F m vn = dn dt F driving force. scattering Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

25 More advanced theory: Boltzmann transport equation Phonon Boltzmann transport equation in the steady state: v q,s T dn q,s dt = dn q,s dt scattering n q,s occupation ( = Bose-Einstein distribution in equilibrium), v q,s = dω q,s /dq group velocity. dn dt q s scattering v n (T+dT) v n q s q s q s q s (T) T+dT T Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

26 Scattering rates due to three-phonon processes Fermi s golden rule: P f i (3ph) = 2π f V 3 i 2 δ(ω f ω i ), Scattering term in BTE: dn q,s dt = ( P q s qs,q s 1 s,q s 3ph 2 Pq qs +P qs,q s q s + 1 ) 2 Pqs q s,q s q,s q,s ω q s ω q s q s q s q s q s Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

27 Scattering rates due to three-phonon processes Fermi s golden rule: P f i (3ph) = 2π f V 3 i 2 δ(ω f ω i ), Scattering term in BTE: dn q,s dt = ( P q s qs,q s 1 s,q s 3ph 2 Pq qs +P qs,q s q s + 1 ) 2 Pqs q s,q s q,s q,s ω ω q s q s q s q s q s q s Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

28 Thermal conductivity in the BTE framework Solve linearized BTE self-consistently to determine occupations. Heat current: J = 1 N q V ω q,s n q,s v q,s. q,s Thermal conductivity from Fourier law: κ αβ = J α T β 1 T 2 = N q V T 2 ω q,s n q,s v q,s T β. G. P. Srivastava, The physics of phonons q,s Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

29 Thermal conductivity from linearized BTE: Example of Si and Ge theory Si exp. Empirical potentials: D. A. Broido et al., PRB 72, (2005) First principles (DFT): D. A. Broido et al., APL 91, (2007) Very good agreement between first principles results and experiment. Empirical potentials give good trends, but are not very accurate. Ivana Savić, Introduction to phonon transport September 12, / 52

30 BTE in the relaxation time approximation approach (BTE-RTA) for thermal transport Relaxation time (τ q,s ) approximation: Assume equilibrium phonon distribution in all states except (q,s): τq,s 1 = 1 ( P q s qs,q n q,s ( n q,s +1) s + 2 P 1 q s,q s q,s q,s dn q,s dt scattering = n q,s n q,s. τ q,s qs ), Thermal conductivity: κ = 1 c q,s v 2 N q V q,sτ q,s, c q,s heat capacity. q,s G. P. Srivastava, The physics of phonons Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

31 Self-consistent BTE or BTE-RTA? Example of carbon diamond q q q q G q q Q q+q =q normal Q=q+q =q +G umklapp A. Ward et al., PRB 80, (2009) If umklapp scattering is weak, self-consistent BTE solution differs from BTE-RTA. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

32 Insights gained from BTE simulations of κ Mechanisms for low κ in bulk materials can be uncovered. Provide guidance in the search for more efficient materials. Z. Tian et al., PRB 85, (2012) Ivana Savić, Introduction to phonon transport September 12, / 52

33 Insights gained from BTE simulations of κ κ vs mean free path in bulk materials. Estimate nanostructure dimensions to achieve a desired κ reduction. Si A. J. Minich et al., PRL 107, (2009) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

34 Challenges for BTE approach applied to nanostructured and disordered materials How to solve BTE for large scale systems? Si Ge SL NW SL ND SL How to calculate the first principles input to BTE for large scale systems? How to extend BTE to treat disordered materials without resorting to the virtual crystal + effective lifetimes approximation? Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

35 Non-equilibrium MD (NEMD) approach to thermal transport Impose T in MD simulation by exchanging the energy of the atoms in the central and end slabs. Critical feature: existence of boundaries at the edges of the heat source and sink. F. Muller-Plathe, JCP 106, 6082 (1997) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

36 Extracting κ from the temperature profile Instantaneous local temperature in slab k: T k i k m iv 2 i K/Å K/Å T (K) Si z (Å) Heat flux J z determined from the transferred kinetic energy. κ = J z (t) /( T/ z). Y. He, I. Savić, D. Donadio, and G. Galli, PCCP 14, (2012) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

37 Size effects in NEMD simulations of κ 1/κ (W/mK) Lz (nm) Si T=1000K T=300K /Lz (nm) -1 Boundary scattering: l 1 L = l 1 +4L 1 z κ 1 L = κ 1 +CL 1. Difficult to ensure both the cross-section and length convergence: Is NEMD good enough for disordered materials? Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

38 Insights gained from NEMD simulations of κ Complex materials can be straightforwardly simulated. Si nanowires I. Ponomareva at al., Nano Lett. 7, 1155 (2007) Ivana Savić, Introduction to phonon transport September 12, / 52

39 Insights gained from NEMD simulations of κ Predict κ trends for complex materials. Si nanowires I. Ponomareva at al., Nano Lett. 7, 1155 (2007) Ivana Savić, Introduction to phonon transport September 12, / 52

40 NEMD approach to calculate κ from first principles Straightforward coupling of ab-initio and NEMD heat transport methods. Good agreement with experiments for MgO, predictions for temperatures and pressures in the Earth mantle. S. Stackhouse at al., PRL 104, (2010) Ivana Savić, Introduction to phonon transport September 12, / 52

41 Size effects in NEMD approach from first principles 0.08 Lz (nm) /κ (W/mK) T=1000K T=300K /Lz (nm) -1 First principles: S. Stackhouse at al., PRL 104, (2010) Empirical potentials: Y. He et al., PCCP 14, (2012) Much smaller sizes accessible than with empirical potentials. Much shorter simulation time than with empirical potentials: 10 ps versus 1 ns. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

42 Equilibrium MD (EMD) approach to thermal transport Green-Kubo formalism (fluctuation-dissipation theorem): κ = 1 Vk B T 2 Heat current obtained from EMD: 0 J(t)J(0) dt, J = d r i (t)ǫ i (t); i - atomic site, r i - coordinate, ǫ i - energy. dt i ǫ i can be defined for empirical potentials, but not in a unique way. Open problem: how to define the local energy from first principles? Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

43 Time scale for EMD simulations of κ κ = 1 ttr Vk B T 2 J(t)J(0) dt 0 <J(t)J(0)> CJ(t) Si κ (W/mK) t tr (ps) t corr (ps) t J(t)J(0) indirect measure of longest phonon lifetimes ( 100 ps). Total simulation time 10 ns. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

44 Size effects in EMD simulations of κ κ (W/m K) Si Number of atoms κ(w/mk) N (x1000) Si 0.5 Ge L cell (nm) Convergence for large samples ( atoms). Samples with atoms can be simulated with EMD. Disordered samples can be simulated with EMD. Y. He, I. Savić, D. Donadio, and G. Galli, PCCP 14, (2012) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

45 Insights gained from EMD simulations of κ Materials with very complex morphologies can be simulated. Closest to mimicking realistic materials so far! Si nanowires with surface roughness Y. He and G. Galli, PRL 108, (2012) Ivana Savić, Introduction to phonon transport September 12, / 52

46 Insights gained from EMD simulations of κ Powerful probe to determine which types of disorder cause strong scattering and κ reduction. Si nanowires with surface roughness Y. He and G. Galli, PRL 108, (2012) Ivana Savić, Introduction to phonon transport September 12, / 52

47 Calculating κ with lifetimes obtained from EMD EMD lifetimes from the exponential decay of the normal mode potential energy autocorrelation function. Use EMD lifetimes for κ: κ τemd = 1 N qv q,s c q,svq,s 2 τemd q,s. Typically κ EMD κ τemd for small system sizes ( 1000 atoms). A. J. H. McGaughey and M. Kaviani, PRB 69, (2004) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

48 κ with EMD lifetimes from first principles Can be calculated for small system sizes ( 100 atoms) and short simulation times ( 40 ps). Good agreement with experiments for MgO. N. de Koker, PRL 103, (2009) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

49 Can we use EMD lifetimes to study heat transport in disordered materials? Example of rough Si nanowires κ EMD 1 N qv q,s c q,svq,s 2 τemd q,s. Non-propagating and non-localized vibrations give additional contribution to κ. τ EMD +extra extra τ EMD D. Donadio and G. Galli, PRL 102, (2009) Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

50 Summary of atomistic approaches to heat transport Method EMD NEMD BTE Disorder Yes Yes Approximate Anharmonicity Yes Yes Approximate Transport regimes Any Any Propagating f BE?? Yes Ab-initio? Yes Yes f BE - the Bose-Einstein distribution. No perfect approach, they complement each other. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

51 Summary We need to understand the thermal conductivity to design new thermoelectric materials. Atomistic methods to calculate the thermal conductivity could be helpful. Lots of recent methodological developments in the field, especially in the domain of first principles calculations. Lots of insight gained from applications of these methods, but more insight is needed. Quite a few challenges ahead for these methods, especially to model disordered and nanostructured materials. Ivana Savić, ivana.savic@tyndall.ie Introduction to phonon transport September 12, / 52

52 Acknowledgments Prof. Giulia Galli, Department of Chemistry and Department of Physics, UC Davis, Davis, California, USA Dr. Yuping He, Department of Chemistry, UC Davis, Davis, California, USA Dr. Davide Donadio, Max Planck Institute for Polymer Research, Mainz, Germany Dr. Éamonn Murray, Tyndall National Institute, Cork, Ireland Prof. François Gygi, Department of Computer Science, UC Davis, Davis, California, USA Ivana Savić, Introduction to phonon transport September 12, / 52