Molecular Dynamics Simulation of Thermal Conductivity of Si/SiGe Nanowire Superlattice Yongxing Shen1, Annica Heyman2, and Ningdong Huang2 Departments of Materials Science and Engineering1 and Applied Physics2 Stanford University, CA 94305

Introduction Si / SiGe Nanowire Superlattices Si SixGe1-x (x~0.75) (other names: bamboo structures, axial heterostructures) Si SixGe1-x (x~0.75) 16~23 Å ~38 Å Potential application as thermoelectric materials: & high electrical conductivity low thermal conductivity

Why Low Thermal Conductivity? Definition: Si / SiGe nanowire superlattices have phonon scattering centers like: Imperfections (Ge) Surface Interfaces

Earlier Experimental Results (Li, et al, Appl. Phys. Lett. 2001) kSi bulk = 241 W/mK at 200 K

Our Primary Goal To study the interfacial effect on thermal conductivity (k) of Si / SiGe nanowire superlattice by comparing k’s of and Si SixGe1-x

But this is all in non-equilibrium …. Thermal Conductivity J  T = T1 T = T2 < T1  T Temperature gradient [K/m] J = - k  T Heat current [W/m2] = [J/m2s] Thermal conductivity [W/mK] But this is all in non-equilibrium ….

Fluctuation-Dissipation Theorem Fully developed in the 1950s by R. Kubo. Provides general relationship between Internal fluctuations in the absence of disturbance Response of system to external disturbance & Non-equilibrium Characterized by a response function Equilibrium Characterized by a correlation function (of relevant quantity) Transport coefficient Correlation function For transport problems: or for large t:

Calculating k using the FDT Thermal conductivity: (a = x, y, or z) or with and Ei = (Ekin)i + (Epot)i Compare with diffusion coefficient D is given by: or Using the standard MD average substitution: < > = ensemble average  < >time we can calculate

Technical Details Code to use: MDPackage developed in Prof. KJ Cho’s group (NVE, Langevin dynamics, conjugate gradient) Interatomic potential: Tersoff potential (Tersoff, 1989) (can handle Si, Ge, and C and alloys of Si-C and Si-Ge) For Tersoff, usually still adopts How does one define the potential energy per atom, (Epot)i , when a non-pairwise interatomic potential is used? The formulas given so far have been classical limits of corresponding quantum mechanical relations. (For T >> TD  640 K in Si (Debye temperature) this is a naturally a good approximation.) Long NVE simulation times necessary (correlation lengths of 100 ps not uncommon) The length of the supercell puts a limit on the maximum wavelength of the phonons permitted

Simulations Steps Create initial structure Relax at T = 0 K (conjugate gradient method) Equilibrate at T = 300 K using Langevin dynamics Run at constant energy (NVE) to extract qa(t) Calculate k from q Repeat for different supercell sizes (for convergence) and different compositions, diameters, block lengths and structures (for real data) ~ 23 Å 4 x ~19 Å