1. 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

2. 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

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

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

5. 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

6. 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 ….

7. 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:

8. 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

9. 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

10. 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