## Phonon thermal conductivity suppression of bulk silicon nanowire composites for efficient thermoelectric conversion |

Optics Express, Vol. 18, Issue S3, pp. A467-A476 (2010)

http://dx.doi.org/10.1364/OE.18.00A467

Acrobat PDF (7066 KB)

### Abstract

Vertically-aligned silicon nanowires (SiNWs) that demonstrate reductions of phonon thermal conductivities are ideal components for thermoelectric devices. In this paper, we present large-area silicon nanowire arrays in various lengths using a silver-induced, electroless-etching method that is applicable to both n- and p-type substrates. The measured thermal conductivities of nanowire composites are significantly reduced by up to 43%, compared to that of bulk silicon. Detailed calculations based on the series thermal resistance and phonon radiative transfer models confirm the reduction of thermal conductivity not only due to the increased air fraction, but also the nanowire size effect, suggesting the soundness of employing bulk silicon nanowire composites as efficient thermoelectric materials.

© 2010 OSA

## 1. Introduction

1. F. J. DiSalvo, “Thermoelectric cooling and power generation, ” Science **285**(5428), 703–706 (1999). [CrossRef] [PubMed]

6. R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, “Thin-film thermoelectric devices with high room-temperature figures of merit,” Nature **413**(6856), 597–602 (2001). [CrossRef] [PubMed]

7. S. M. Lee, D. G. Cahill, and R. Venkatasubramanian, “Thermal conductivity of Si-Ge superlattices,” Appl. Phys. Lett. **70**(22), 2957–2959 (1997). [CrossRef]

8. A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of rough silicon nanowires,” Nature **451**(7175), 163–167 (2008). [CrossRef] [PubMed]

9. A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard 3rd, and J. R. Heath, “Silicon nanowires as efficient thermoelectric materials,” Nature **451**(7175), 168–171 (2008). [CrossRef] [PubMed]

10. K. Q. Peng, Y. J. Yan, S. P. Gao, and J. Zhu, “Synthesis of Large-Area Silicon Nanowire Arrays via Self-Assembling Nanoelectrochemistry,” Adv. Mater. **14**(16), 1164–1167 (2002). [CrossRef]

## 2. Experimental and measurement

_{3}and HF solution to initiate the electroless metal deposition. During the process, the silver ions captured electrons from the silicon valence band, forming silver dendrite and meanwhile oxidized the silicon. The silicon dioxide was then etched away by the HF solution, causing silver particles to sink into bulk silicon. The remaining silicon therefore formed nanostructures such as holes or wires that depend on the distribution and density of silver particles. At the end of etching process, the samples were dipped in a HNO

_{3}solution to remove the silver residues.

_{3}/HF solution with a concentration of 0.02M/5M at room temperature for 2.5 hours and 5.5 hours, respectively. The rest of the samples were etched with AgNO

_{3}/HF concentrations of 0.03M/6.5M, 0.04M/6.5M and 0.05M/6.5M for 24 hours. Here, different etching treatments were adapted to obtain a large span of nanowire lengths. In particular, the concentration of the AgNO

_{3}solution was increased to assure the supply of silver and to avoid the lateral etching due to the long etching time. Figure 2(a) and 2(b) show the tilted, 45° scanning electron microscopic (SEM) images for the 35- and 215-μm-long nanowire samples, respectively. The nanowires with high aspect ratios appear to be bundled in groups due to the electrostatic force. The resulting surface morphology depends on the etching conditions, as discussed previously by Peng

*et al.*[11

11. K. Q. Peng, Y. Yan, S. P. Gao, and J. Zhu, “Dendrite-Assisted Growth of Silicon Nanowires in Electroless Metal Deposition,” Adv. Funct. Mater. **13**(2), 127–132 (2003). [CrossRef]

12. Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. **2**(12), 770–774 (2007). [CrossRef]

13. W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,” J. Appl. Phys. **32**(9), 1679–1684 (1961). [CrossRef]

14. D. G. Cahill, “Thermal conductivity measurement from 30~750K: the 3ω method,” Rev. Sci. Instrum. **61**(2), 802–808 (1990). [CrossRef]

15. S. Mo, P. Hu, J. Cao, Z. Chen, H. Fan, and F. Yu, “Effective Thermal Conductivity of Moist Porous Sintered Nickel Material,” Int. J. Thermophys. **27**(1), 304–313 (2006). [CrossRef]

17. H. Wang, J. Y. Feng, X. J. Hu, and K. M. Ng, “Reducing thermal contact resistance using a bilayer aligned CNT thermal interface material,” Chem. Eng. Sci. **65**(3), 1101–1108 (2010). [CrossRef]

18. Y. He, “Rapid thermal conductivity measurement with a hot disk sensor Part 1. Theoretical considerations,” Thermochim. Acta **436**(1-2), 122–129 (2005). [CrossRef]

*κ*is the thermal diffusivity and

*t*is the measuring time. As this condition is satisfied, the thermal wave does not reach the horizontal boundaries of the sample during the transient recording. The sample dimensions in the xy-plane therefore could be assumed infinite. In this work, the sensor number was 4922 with a radius of 1.46 cm, and the sample size was one-quarter of a 6-inch silicon wafer. Therefore, Δp is ~2.35 cm. Since

*κ*is 0.9 cm

^{2}/s and the measuring time is 1.2 sec, the calculated probing depth is ~2.1 cm which satisfies Eq. (1). Nevertheless, the thickness of a thin slab sample cannot be ignored. Since the introduced heat could be reflected multiple times inside the sample, a concept of image heat sources is introduced [18

18. Y. He, “Rapid thermal conductivity measurement with a hot disk sensor Part 1. Theoretical considerations,” Thermochim. Acta **436**(1-2), 122–129 (2005). [CrossRef]

18. Y. He, “Rapid thermal conductivity measurement with a hot disk sensor Part 1. Theoretical considerations,” Thermochim. Acta **436**(1-2), 122–129 (2005). [CrossRef]

## 3. Calculation methods

*d*,

_{1}*k*, and

_{1}*A*are the thickness, thermal conductivity, and cross-sectional area of the silicon substrate, respectively;

_{1}*d*,

_{2}*k*, and

_{2}*A*are those of a nanowire in a unit cell; and

_{2}*k*is the effective thermal conductivity of the composite including substrate. By setting

_{eff}*k*

_{1}=

*k*

_{2}, the reduction in

*k*is solely due to the air fraction arising between two different cross-sectional areas. Here, the air volume ratio is defined as the etched silicon volume (air gaps) to the combined nanowire volume including air gaps, which is estimated using the following expression:where

_{eff}*w*and

*l*are the total weight and sample thickness before etching,

*Δw*is the weight difference before and after the etching process, and

*Δl*is the etching depth. The air volume ratio is estimated to be 30% for the 35-μm long and slightly increased to be 45% for the 215-μm long samples due to the lateral etching. Therefore, the ratio is set to be 40% in our simulation. Although the thermal conductivity of the nanowire composites reduces with the increase of air fraction, it is not desirable for thermoelectric materials, since the electrical conductivity also reduces concurrently. The nanowire size effect, on the other hand, can reduce heat transfer without compromising electron conduction.

19. G. Chen, “Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices,” Phys. Rev. B **57**(23), 14958–14973 (1998). [CrossRef]

*f*is a time-dependent probability distribution function of phonons, which depends on particle position, time

*t*, and phonon group velocity

*v.*The right hand side of Eq. (4) denotes a scattering term, which arises from the collision between particles. As the collisions tend to relax the system to equilibrium, the scattering term can be rewritten using a relaxation time approximation:where

*f*is the Bose–Einstein distribution of phonons, which depends on the local equilibrium temperature, and τ is the relaxation time.

_{0}*θ*is the azimuthal angle and

*ϕ*is the polar angle in a spherical coordinate system, and

*D*(

*ω*) is the phonon density of states as a function of the phonon frequency

*ω*. The product of

*D*(

*ω*) and the distribution function

*f*on the right hand side of Eq. (6) is the number of phonons per unit volume. We note that the phonon dispersion relation is not taken into account here due to the incoherent nature of phonon transport in nanostructures. Several studies have shown that diffusive interface scattering is the dominant factor of phonon transport in low-dimensional systems, where the coherence of phonons may be destroyed after a few scattering events [22

22. B. Yang and G. Chen, “Lattice Dynamics Study Of Anisotropic Heat Conduction in Superlattices,” Microscale Thermophys. Eng. **5**(2), 107–116 (2001). [CrossRef]

23. G. Chen and M. Neagu, “Thermal Conductivity and Heat Transfer in Superlattices,” Appl. Phys. Lett. **71**(19), 2761–2763 (1997). [CrossRef]

*I*is the equilibrium phonon intensity which has a spatial dependence. The EPRT treats phonon transport as radiation during ballistic heat transport and assumes that phonons, as the wave packets of energy, behave identically to photons. The integrals in (7) can then be approximated by a discrete ordinate method with appropriate weighting functions. Here, an S

_{0}_{4}method with twelve direction cosines and corresponding quadrature weights is employed. Therefore, the EPRT becomes a partial differential equation. The air is assumed to be adiabatic and isothermal boundary conditions are set along the propagation direction. where I

^{+}and I

^{-}represents the forward and backward phonon intensity, respectively. By assuming a constant specific heat

*C*, the effective temperature is obtained as:where I represents the phonon intensity at local equilibrium at each position solved by Eq. (7). In the calculations, we set

*C*= 0.93 × 10

^{6}J/m

^{3}

_{·}K,

*v*= 1804 m/s, and Λ = 260 nm. The average heat flux q is obtained by:where

*L*is the propagation length and

*q*being the local heat flux obtained from the phonon intensity.

_{x}*q*can be expressed as:where

_{x}*w*and

_{m}*w*are weighting functions of the direction cosines in the S

_{n}_{4}method. In the end, the effective thermal conductivity is obtained by:

## 4. Results and discussion

*ρ*is defined as:where

*δ*is the characteristic interface roughness. The characteristic phonon wavelength defined as λ =

*hν*is about 1 nm at room temperature, where

_{s}/k_{B}T*h*is the Planck constant,

*ν*is the sound speed of the material and

_{s}*k*the Boltzmann constant.

_{B}*ρ*is a number between 0 and 1, where

*ρ*= 1 represents the specular reflection of phonon transport at smooth interfaces, and

*ρ*= 0 a totally diffuse boundary condition. Any number in between represents a partially specular and partially diffuse scattering condition. Figure 4(a) and 4(b) illustrate the specular and diffuse reflections of phonon transport occurring at smooth and rough interfaces, respectively. The specular boundary condition (

*ρ*= 1) means that all phonons are reflected following the Fresnel law of reflection, while a totally diffuse boundary condition (

*ρ*= 0) means that all the phonons are reflected to random directions with equal probabilities when encountering a boundary. For silicon nanowires obtained by the electroless etching method, the interface roughness is about 1-5 nm [8

8. A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of rough silicon nanowires,” Nature **451**(7175), 163–167 (2008). [CrossRef] [PubMed]

*ρ*is nearly zero, justifying the use of a totally diffuse boundary condition. Studies have also shown that the totally diffuse boundary condition can well describe the heat conduction of a nanowire [25

25. D. Terris, K. Joulain, D. Lacroix, and D. Lemonnier, “Numerical simulation of transient phonon heat transfer in silicon nanowires and nanofilms,” J. Phys.: Conf. Ser. **92**, 012077 (2007). [CrossRef]

26. H. Y. Chen, H. W. Lin, C. Y. Wu, W. C. Chen, J. S. Chen, and S. Gwo, “Gallium nitride nanorod arrays as low-refractive-index transparent media in the entire visible spectral region,” Opt. Express **16**(11), 8106–8116 (2008). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | F. J. DiSalvo, “Thermoelectric cooling and power generation, ” Science |

2. | G. S. Nolas, J. Sharp, and H. Goldsmid, Thermoelectrics: Basic Principles and New Materials Developments, Springer, New York (2001). |

3. | G. A. Slack, CRC Handbook of Thermoelectrics, D. M. Rowe Ed., Boca Raton, Florida, (1995). |

4. | R. Venkatasubramanian, “Recent Trends in Thermoelectric Materials Research III, in Semiconductors and Semimetals,” Academic Press |

5. | G. Chen, “Recent Trends in Thermoelectric Materials Research III, in Semiconductors and Semimetals,” Academic Press |

6. | R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, “Thin-film thermoelectric devices with high room-temperature figures of merit,” Nature |

7. | S. M. Lee, D. G. Cahill, and R. Venkatasubramanian, “Thermal conductivity of Si-Ge superlattices,” Appl. Phys. Lett. |

8. | A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of rough silicon nanowires,” Nature |

9. | A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard 3rd, and J. R. Heath, “Silicon nanowires as efficient thermoelectric materials,” Nature |

10. | K. Q. Peng, Y. J. Yan, S. P. Gao, and J. Zhu, “Synthesis of Large-Area Silicon Nanowire Arrays via Self-Assembling Nanoelectrochemistry,” Adv. Mater. |

11. | K. Q. Peng, Y. Yan, S. P. Gao, and J. Zhu, “Dendrite-Assisted Growth of Silicon Nanowires in Electroless Metal Deposition,” Adv. Funct. Mater. |

12. | Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. |

13. | W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,” J. Appl. Phys. |

14. | D. G. Cahill, “Thermal conductivity measurement from 30~750K: the 3ω method,” Rev. Sci. Instrum. |

15. | S. Mo, P. Hu, J. Cao, Z. Chen, H. Fan, and F. Yu, “Effective Thermal Conductivity of Moist Porous Sintered Nickel Material,” Int. J. Thermophys. |

16. | J. L. Zeng, Z. Cao, D. W. Yang, L. X. Sun, and L. Zhang, “Thermal conductivity enhancement of Ag nanowires on an organic phase change material,” J. Therm. Anal. Calorim. (to be published). |

17. | H. Wang, J. Y. Feng, X. J. Hu, and K. M. Ng, “Reducing thermal contact resistance using a bilayer aligned CNT thermal interface material,” Chem. Eng. Sci. |

18. | Y. He, “Rapid thermal conductivity measurement with a hot disk sensor Part 1. Theoretical considerations,” Thermochim. Acta |

19. | G. Chen, “Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices,” Phys. Rev. B |

20. | K. Miyazaki, T. Arashi, D. Makino, and H. Tsukamoto, “Heat Conduction in Microstructured Materials,” IEEE Trans. Compon. Packag. Tech. |

21. | S. Sihn, and K. Ajit, Roy, “Nanoscale Heat Transfer using Phonon Boltzmann Transport Equation,” COMSOL Conference (2009). |

22. | B. Yang and G. Chen, “Lattice Dynamics Study Of Anisotropic Heat Conduction in Superlattices,” Microscale Thermophys. Eng. |

23. | G. Chen and M. Neagu, “Thermal Conductivity and Heat Transfer in Superlattices,” Appl. Phys. Lett. |

24. | J. M. Ziman, “Electrons and Phonons,” Oxford University Press, London, (1985). |

25. | D. Terris, K. Joulain, D. Lacroix, and D. Lemonnier, “Numerical simulation of transient phonon heat transfer in silicon nanowires and nanofilms,” J. Phys.: Conf. Ser. |

26. | H. Y. Chen, H. W. Lin, C. Y. Wu, W. C. Chen, J. S. Chen, and S. Gwo, “Gallium nitride nanorod arrays as low-refractive-index transparent media in the entire visible spectral region,” Opt. Express |

**OCIS Codes**

(030.5620) Coherence and statistical optics : Radiative transfer

(120.6810) Instrumentation, measurement, and metrology : Thermal effects

(130.1750) Integrated optics : Components

(160.4236) Materials : Nanomaterials

**ToC Category:**

Radiative Transfer

**History**

Original Manuscript: July 13, 2010

Revised Manuscript: September 3, 2010

Manuscript Accepted: September 3, 2010

Published: September 9, 2010

**Citation**

Ting-Gang Chen, Peichen Yu, Rone-Hwa Chou, and Ci-Ling Pan, "Phonon thermal conductivity suppression of bulk silicon nanowire composites for efficient thermoelectric conversion," Opt. Express **18**, A467-A476 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-S3-A467

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### References

- F. J. DiSalvo, “Thermoelectric cooling and power generation, ” Science 285(5428), 703–706 (1999). [CrossRef] [PubMed]
- G. S. Nolas, J. Sharp, and H. Goldsmid, Thermoelectrics: Basic Principles and New Materials Developments, Springer, New York (2001).
- G. A. Slack, CRC Handbook of Thermoelectrics, D. M. Rowe Ed., Boca Raton, Florida, (1995).
- R. Venkatasubramanian, “Recent Trends in Thermoelectric Materials Research III, in Semiconductors and Semimetals,” Academic Press 71, 175–201 (2001).
- G. Chen, “Recent Trends in Thermoelectric Materials Research III, in Semiconductors and Semimetals,” Academic Press 71, 203–259 (2001).
- R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, “Thin-film thermoelectric devices with high room-temperature figures of merit,” Nature 413(6856), 597–602 (2001). [CrossRef] [PubMed]
- S. M. Lee, D. G. Cahill, and R. Venkatasubramanian, “Thermal conductivity of Si-Ge superlattices,” Appl. Phys. Lett. 70(22), 2957–2959 (1997). [CrossRef]
- A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of rough silicon nanowires,” Nature 451(7175), 163–167 (2008). [CrossRef] [PubMed]
- A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard, and J. R. Heath, “Silicon nanowires as efficient thermoelectric materials,” Nature 451(7175), 168–171 (2008). [CrossRef] [PubMed]
- K. Q. Peng, Y. J. Yan, S. P. Gao, and J. Zhu, “Synthesis of Large-Area Silicon Nanowire Arrays via Self-Assembling Nanoelectrochemistry,” Adv. Mater. 14(16), 1164–1167 (2002). [CrossRef]
- K. Q. Peng, Y. Yan, S. P. Gao, and J. Zhu, “Dendrite-Assisted Growth of Silicon Nanowires in Electroless Metal Deposition,” Adv. Funct. Mater. 13(2), 127–132 (2003). [CrossRef]
- Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2(12), 770–774 (2007). [CrossRef]
- W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,” J. Appl. Phys. 32(9), 1679–1684 (1961). [CrossRef]
- D. G. Cahill, “Thermal conductivity measurement from 30~750K: the 3ω method,” Rev. Sci. Instrum. 61(2), 802–808 (1990). [CrossRef]
- S. Mo, P. Hu, J. Cao, Z. Chen, H. Fan, and F. Yu, “Effective Thermal Conductivity of Moist Porous Sintered Nickel Material,” Int. J. Thermophys. 27(1), 304–313 (2006). [CrossRef]
- J. L. Zeng, Z. Cao, D. W. Yang, L. X. Sun, and L. Zhang, “Thermal conductivity enhancement of Ag nanowires on an organic phase change material,” J. Therm. Anal. Calorim. (to be published).
- H. Wang, J. Y. Feng, X. J. Hu, and K. M. Ng, “Reducing thermal contact resistance using a bilayer aligned CNT thermal interface material,” Chem. Eng. Sci. 65(3), 1101–1108 (2010). [CrossRef]
- Y. He, “Rapid thermal conductivity measurement with a hot disk sensor Part 1. Theoretical considerations,” Thermochim. Acta 436(1-2), 122–129 (2005). [CrossRef]
- G. Chen, “Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices,” Phys. Rev. B 57(23), 14958–14973 (1998). [CrossRef]
- K. Miyazaki, T. Arashi, D. Makino, and H. Tsukamoto, “Heat Conduction in Microstructured Materials,” IEEE Trans. Compon. Packag. Tech. 29(2), 247–253 (2006). [CrossRef]
- S. Sihn, and K. Ajit, Roy, “Nanoscale Heat Transfer using Phonon Boltzmann Transport Equation,” COMSOL Conference (2009).
- B. Yang and G. Chen, “Lattice Dynamics Study Of Anisotropic Heat Conduction in Superlattices,” Microscale Thermophys. Eng. 5(2), 107–116 (2001). [CrossRef]
- G. Chen and M. Neagu, “Thermal Conductivity and Heat Transfer in Superlattices,” Appl. Phys. Lett. 71(19), 2761–2763 (1997). [CrossRef]
- J. M. Ziman, “Electrons and Phonons,” Oxford University Press, London, (1985).
- D. Terris, K. Joulain, D. Lacroix, and D. Lemonnier, “Numerical simulation of transient phonon heat transfer in silicon nanowires and nanofilms,” J. Phys.: Conf. Ser. 92, 012077 (2007). [CrossRef]
- H. Y. Chen, H. W. Lin, C. Y. Wu, W. C. Chen, J. S. Chen, and S. Gwo, “Gallium nitride nanorod arrays as low-refractive-index transparent media in the entire visible spectral region,” Opt. Express 16(11), 8106–8116 (2008). [CrossRef] [PubMed]

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