Heat Conduction by Molecular Dynamics Technique Sebastian Volz National Engineering School of Mechanics and Aerotechnics Laboratory of Thermal Studies UMR CNRS 668 Poitiers, France Denis Lemonnier - Lab. of Thermal Studies - Poitiers Jean-Bernard Saulnier - Lab. of Thermal Studies - Poitiers Gang Chen - NanoHeat Transfer and Thermoelectrics Lab. - UCLA Pierre Beauchamp - Laboratoire de Métallurgie Physique - Poitiers
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UNDERSTANDING AND MONITORING MATERIALS PROPERTIES NEW MATERIALS contain nano-micro architectured structures o NANOFIBERS: multiscale complex materials ultra insulating (.7 W/mK) Si Nanoparticles chain Vacuum Absorbing MicroParticle o SUPERLATTICES monitoring thermal conductivity Bulk /1<< Bulk 5 Å to 1m o NANOWIRES templates: 1 nm - 1m monitoring anisotropy G. Chen - Ni nanowires template
LOW-DIMENSIONAL PHYSICS FOR HEAT CONDUCTION o Ballistic Transport of Phonons BALLISTIC DIFFUSIVE Size L< Mean Free Path L Boundary Scattering More resistive Diffuse Specular o Phonon Confinement v G and Reduction (rad.thz) 9 8 7 6 5 4 3 1 u t v. u t Bulk Si nanowire.5 1 q x (nm -1 ) v v ( u) l t q BULK.grad eff (size)< BULK T
SITUATION of MD Small Scales Phonon Transport: Ballistic Diffusive Phonon Particle: Boltzmann TE No interference, Interface transfer Small and Large scales Phonon Wave No p-p scattering, limited by wl Interface transfer easier Molecular Dynamics - MD Classical Heat Conduction q BULK.gradT Atomic Period.1ps Lattice Constant.5nm Relaxation time 1ps Mean Free Path 1nm
MOLECULAR DYNAMICS TECHNIQUE nd NEWTON LAW COMPUTE ALL ATOMIC TRAJECTORIES STILLINGER-WEBER POTENTIAL M d ri dt N j1 j i F ij u /. -1. -BODY 3-BODY r ij jik r ik r ij / a=1.8-1.84 Coordonnée Y (Angstroem) -1.85-1.86-1.87-1.88-1.89 t=.3 ps -1.9 8.76 8.78 8.8 8.8 8.84 8.86 8.88 Coordonnée X (Angstroem)
ADVANTAGES OF MD TECHNIQUE o Phonon Scattering is Difficult to Model: Phonon Particle Approach: Relaxation Time?? Phonon Wave Approach: No scattering. MD PROVIDES A COMPLETE DESCRIPTION OF PHONON SCATTERING EXAMPLE: NANOWIRE o Phonon Transport Approach Assumes Fully Periodic Lattices MD ALLOWS TO INCLUDE ATOMIC DEFAULTS and STRAINS EXAMPLE: SUPERLATTICE o Non-Equilibrium Short Time Heat Conduction MD DESCRIBE HT BEHAVIOUR AT GigaHTz FREQUENCIES EXAMPLE: IN BULK SI Corrected T(K) 6 5 4 3 1 Quantum Effects BULK Si correction No correction T Debye =65K 1 3 4 5 6 TMD (MD (K) U)
HEAT FLUX by MD o Kinetic and Work terms q e i Kinetic Term v i solids N N 1 1 ( t) e i v i v i. Fij r V i1 j 1 j i. ij ij Work Term v i F ij r ij solids W K
THERMAL CONDUCTIVITY by MD o Fluctuation-Dissipation Theorem =, Thermal Conductivity gradt X The Flux q q 3k V B T Autocorrelation e i d X gradt The Force Silicon at K and 5K
SILICON NANOWIRE MD MODEL RIGID BOUNDARY PERIODIC BOUNDARY Phonon CONDITIONS Energy Conserved MD BOX RIGID BOUNDARY Free standing Bi Nanowire, M.S.Dresselhaus
BOLTZMANN TRANSPORT EQUATION o 1D solution to BTE: Boundary Scattering ONLY g t v. gradg g Bulk T g g( r) Bulk.cos...1 G( r,p) x T (Ziman -Electrons and Phonons) M(r) T Infinite Length S p x D 1 q S v.. g. D. d. d r r 4 4 Bulk nw 1 S. Volz and G. Chen, Heat and Technology, 18, 37,. Size Bulk Function of G only
COMPARISON BETWEEN MD&BTE RESULTS Is boundary scattering the only cause forthermal conductivity reduction?.8 S. Volz and G. Chen, Applied Physics Letters, 57, 56, 1999.
PHONON CONFINEMENT EFFECT ON HEAT CONDUCTION o Ballistic Transport of Phonons BALLISTIC DIFFUSIVE Size L< Mean Free Path L Boundary Scattering More resistive Diffuse Specular o Phonon Confinement v G and Reduction (rad.thz) 9 8 7 6 5 4 3 1 u t v. u t Bulk Si nanowire.5 1 q x (nm -1 ) v v ( u) l t q BULK.grad effective < BULK T
RPTE - THE DISCRETE ORDINATE METHOD Ω.gradL L L o S 8, discretizing in 48 directions (, w m ) for finite length wire L p. g. p Confinement effect p r v m. L k, m k L k, m L k, m RIGID -PERFECTLY REFLECTING BOUNDARY 1 =33K =3K L r Heat Flux q RIGID -PERFECTLY REFLECTING BOUNDARY 1-8 nm Confinement effect di q r w. L i, j m k, mi,, j m k 15, m148, ql. T T 1 T ij? (nm -1 ),6,5,4,3,,1, =1/ =1/(v. Bulk 1 3 4 5 6 7 circular frequency, (rad.thz)
PHONON CONFINEMENT vs BOUNDARY SCATTERING 8, thermal conductivity (W/mK) 7, 6, 5, 4, 3,, 1,, Phonon Confinement Effect Only Bulk =15 W/mK at 3K Boundary Scattering+Ph. C. Effects 4 6 8 p = p =.5 p = 1 nanowire length (nm) PHONON CONFINEMENT: 5% REDUCTION BOUNDARY SCATTERING: 7% REDUCTION S. Volz and D. Lemonnier, Physics of Low-Dimensional Structures, 5/6, 91,.
Si/Ge SUPERLATTICE MD MODELING x 38.9A y 8 Ge Si Ge 8 16 8 8 z Periodic Boundary Conditions Ge Si Ge.A
STRAIN EFFECT ON SUPERLATTICE STRUCTURE o Starting with mean lattice constant o Implementing Conjugate Gradient Method.5.4.3 DISPLACEMENT (A)..1 -.1 -. 5 1 15 5 3 3 -.3 -.4 Ge Si Ge -.5 [1] ATOMIC PLANE NUMBER
SUPERLATTICE THERMAL CONDUCTIVITY CROSS-PLANE THERMAL CONDUCTIVITY (W/mK) 1 1 1.1 With Minimisation Procedure Without Minimisation Procedure Trend for Experimental Results RBTE Solution 15 5 3 35 LAYER THICKNESS (A) S. Volz, J.B Saulnier, G. Chen, P. Beauchamp, Microelectronics Journal 31 (9-1) 815,.
EFFECTIVE THERMAL CONDUCTIVITY AT GIGAHERTZ FREQUENCIES o Fluctuation Dissipation Theorem V i t qbg z q bg q bg t e dt gradt... 3. k. T q B q t q e t e qbg gradt V 3. k. T B. q 1 ( ) -1 dependence at Giga frequencies Ge - GHtz BULK =15W/mK SiO - 9GHtz
CONCLUSION MOLECULAR DYNAMICS TECHNIQUE: o COMPLETELY DESCRIBES PHONON SCATTERING o ALLOWS THE SIMULATION OF DEFAULTS/STRAINS o GIVES ACCESS TO NON-EQUILIBRIUM REGIMES - IS VERY HEAVY IN TERMS OF COMPUTATION TIME - RELIES ON THE INTERACTION POTENTIAL VALIDITY - DOES NOT INCLUDE QUANTUM EFFECTS
ULTRA SHORT TIME HEAT CONDUCTION
-1 LAW FOR Si THERMAL CONDUCTIVITY AT GIGAHTZ FREQUENCIES