Department of Electronics, Peking University Zhang Gang

1. Violation of Fourier’s law and Anomalous Heat Diffusion in Low-Dimensional Nano Materials Department of Electronics, Peking University Zhang Gang

2. Outline • Introduction • Thermal Conductivity of Carbon Nanotubes • Thermal Conductivity of Silicon Nanowires • Heat diffusion in Silicon Nanowires • Conclusion

3. I. Introduction Dependence of mean time to failure on temperature 80C 90C 108C Steve Kang et al. Electrothermal analysis of VLSI Systems, Kluwer 2000 110C 1 cm On chip temperature contour

4. I. Introduction CNT Thermal ConductivityW/m-K 1 10 100 1000 SiNW cSi Alumina diamond copper Most of the variation comes from the mean free path of the carrier = 1/3 c Lv Thermal conductivity Phonon velocity Mean free path Heat capacity

5. I. Introduction Fourier’s law (1768 - 1830) French mathematicians and Egyptologist Local heat flux, [W·m−2] Temperature gradient, [K.m−1] Thermal Conductivity, [W·m−1·K−1]

6. I. Introduction Thermal Conductivity in 1-D lattices jN vs N on double-logarithmic

7. II. Thermal Conductivity in Carbon Nanotubes S. Maruyama / Physica B 323 (2002) 193–195

8. II. Thermal Conductivity in Carbon Nanotubes The thermal conductivity k vs tube length in log-log scale for (5,5) and (10,10) tubes at 300 and 800 K. In all cases, k~Lβ with β changes from case to case. The solid line, whose slope is the value of β, is the best-fit one.

9. II. Thermal Conductivity in Carbon Nanotubes Normalized resistance vs normalized sample length for different CNT and BNNT samples

10. III. Thermal Conductivity in Silicon Nanowires Measured thermal conductivity of different diameter Si nanowires. The number beside each curve denotes the corresponding wire diameter.

11. III. Thermal Conductivity in Silicon Nanowires Nano Letters, 8, 276 (2008) Diameter(nm) PRB 73, 153303 (2006) Nano Letters, 2003, 3, 1713 APL, 75, 2056 (1999) APL, 95, 063102 (2009) Nano Letters, 2007, 7, 1155. Nano Today 4, 393 (2009) PRL 102, 195901 (2009)

12. III. Thermal Conductivity in Silicon Nanowires

13. Molecular Dynamics Stillinger -Weber Potential III. Thermal Conductivity in Silicon Nanowires Heat flux J is the thermal flux through unit cross section in unit time, dT/dx is the temperature gradient. F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985). Nose, S., J. Chem. Phys. 81, 511 (1984) Hoover, W. G., Phys. Rev. A 31, 1695 (1985)

14. III. Thermal Conductivity in Silicon Nanowires

15. III. Thermal Conductivity in Silicon Nanowires The thermal conductivity of SiNWs vs longitude length Lz. Group velocity 6400 m/s Relaxation time 10ps Mean Free Path 60nm

16. III. Thermal Conductivity in Silicon Nanowires The phonon density of states along the longitude direction of SiNWs with different lengths. The phonon density of states of bulk Si is also shown for reference.

17. III. Thermal Conductivity in Silicon Nanowires The thermal conductivity of SiNWs (with fixed transverse boundary condition) vs longitude length Lz. Inset is the length dependent thermal conductivity of SiNWs with free (transverse) boundary condition.

18. IV. Heat diffusion in Silicon Nanowires

19. IV. Heat diffusion in Silicon Nanowires The spread of the pulse

20. IV. Heat diffusion in Silicon Nanowires 0.4 0.27 0.15 The behavior of energy diffusion in SiNW at room temperature and at 1000 K. The length of SiNW is 140 nm. The inset show the discussion for SiNW with free boundary condition.

21. Conclusions • Thermal Conductivity of SiNWs diverges with the length k~ Lβ • The value of β depends on temperature and length • Anomalous heat diffusion is responsible for the length dependent thermal conductivity Fourier’s law is brokendown in Nano Structures

22. Acknowledgements Prof. LI Baowen YANG Nuo YAO Donglai CHEN Jie SHI Lihong ZHANG Kaiwen 2009年国家优秀自费留学生奖学金

23. Thanks！