^{1}and Sanjiv Sinha

^{1}

### Abstract

Nanostructured single-crystalsilicon exhibits a remarkable increase in the figure of merit for thermoelectric energy conversion. Here we theoretically investigate a similar enhancement for polycrystalline silicon inverse opals. An inverse opal provides nanoscale grains and a thin-film like geometry to scatterphonons preferentially over electrons. Using solutions to the Boltzmann transport equation for electrons and phonons, we show that the figure of merit at 300 K is fifteen times that of bulk single-crystalsilicon. Our models predict that grain boundaries are more effective than surfaces in enhancing the figure of merit. We provide insight into this effect and show that preserving a grain size smaller than the shell thickness of the inverse opal increases the figure of merit by as much as 50% when the ratio between the two features is a third. At 600 K, the figure of merit is as high as 0.6 for a shell thickness of 10 nm. This work advances the fundamental understanding of charge and heat transport in nanostructured inverse opals.

Support for this work is in part from the Department of Energy (ARPA-E) under Contract No. DOE-DE-AR-0000041PF-ARRA, from the Department of Defense (DARPA) under Grant No. N66001-11-1-4154 and in part from the National Science Foundation under the Grant No. NSF-CBET-09-54696-CAREER. The authors thank Paul Braun and his research group in the Materials Science Department at the University of Illinois for insightful discussions.

I. INTRODUCTION

II. RELATION BETWEEN GEOMETRY AND THE FIGURE OF MERIT

III. MODELING OF THERMOELECTRIC PROPERTIES

A. Electrical conductivity

B. Thermal conductivity

C. Seebeck coefficient

IV. RESULTS AND DISCUSSION

V. CONCLUSION

##### B82B1/00

## Figures

The schematic of an inverse opal unit cell.

The schematic of an inverse opal unit cell.

Scattering rates of electrons for different processes in silicon at 300 K. Solid lines represent scattering mechanisms insensitive to doping. Dashed and dotted-dashed lines represent doping levels of and , respectively. The shell thickness and average grain size are both 25 nm.

Scattering rates of electrons for different processes in silicon at 300 K. Solid lines represent scattering mechanisms insensitive to doping. Dashed and dotted-dashed lines represent doping levels of and , respectively. The shell thickness and average grain size are both 25 nm.

Thermal conductivity as a function of doping concentration at 300 K. The open circles, triangles, and squares are measurement values from Refs. 38, 60, and 61. The red, green, and blue curves are calculations for P-doped 3m thick, As-doped 174 nm thick, and As-doped 74 nm thick single-crystal silicon films, respectively.

Thermal conductivity as a function of doping concentration at 300 K. The open circles, triangles, and squares are measurement values from Refs. 38, 60, and 61. The red, green, and blue curves are calculations for P-doped 3m thick, As-doped 174 nm thick, and As-doped 74 nm thick single-crystal silicon films, respectively.

Electron mobility in bulk and thin-film single-crystal silicon, bulk polysilicon, and polysilicon inverse opals at 300 K. The shell thickness and grain size are both 25 nm. The data for bulk silicon are from Ref. 62 (open circles) and Ref. 63 (crosses).

Electron mobility in bulk and thin-film single-crystal silicon, bulk polysilicon, and polysilicon inverse opals at 300 K. The shell thickness and grain size are both 25 nm. The data for bulk silicon are from Ref. 62 (open circles) and Ref. 63 (crosses).

The calculated effective thermal conductivity of polysilicon inverse opals as a function of shell thickness. We assume the average grain size to be equal to the shell thickness in these calculations.

The calculated effective thermal conductivity of polysilicon inverse opals as a function of shell thickness. We assume the average grain size to be equal to the shell thickness in these calculations.

The Seebeck coefficient at 300 K as a function of doping. The solid lines represent phonon-drag, carrier diffusion, and total Seebeck coefficients, respectively, in single-crystal bulk silicon. The dashed line is the Seebeck coefficient in polysilicon inverse opals. The open circles are data from Ref.45.

The Seebeck coefficient at 300 K as a function of doping. The solid lines represent phonon-drag, carrier diffusion, and total Seebeck coefficients, respectively, in single-crystal bulk silicon. The dashed line is the Seebeck coefficient in polysilicon inverse opals. The open circles are data from Ref.45.

The power factor for different silicon structures at 300 K as a function of doping.

The power factor for different silicon structures at 300 K as a function of doping.

The ratio between the electrical and thermal conductivities in relation to the ratio for bulk silicon. The doping level is and the temperature is 300 K for all curves. Grain boundaries contribute more to the enhancement than surfaces as evident from the curves for thin film and polycrystalline silicon with similar feature size. Inverse opals combine both effects and are slightly better than bulk polysilicon when the grain size equals the shell thickness.

The ratio between the electrical and thermal conductivities in relation to the ratio for bulk silicon. The doping level is and the temperature is 300 K for all curves. Grain boundaries contribute more to the enhancement than surfaces as evident from the curves for thin film and polycrystalline silicon with similar feature size. Inverse opals combine both effects and are slightly better than bulk polysilicon when the grain size equals the shell thickness.

The mean free path of electrons and phonons due to grain boundary scattering as a function of energy. The grain size is 25 nm. Mean free paths for electrons are consistently larger than those for phonons. The Casimir limit is shown for comparison with boundary scattering. The discontinuity in the phonon curves arises from the segmented dispersion assumed in Holland's model.^{35}

The mean free path of electrons and phonons due to grain boundary scattering as a function of energy. The grain size is 25 nm. Mean free paths for electrons are consistently larger than those for phonons. The Casimir limit is shown for comparison with boundary scattering. The discontinuity in the phonon curves arises from the segmented dispersion assumed in Holland's model.^{35}

The projected *ZT* as a function of shell thickness. The solid curves represent different ratios, *r*, of the average grain size to the shell thickness. Choosing grain sizes smaller than the shell thickness enhances the figure of merit for feature sizes 10 nm.

The projected *ZT* as a function of shell thickness. The solid curves represent different ratios, *r*, of the average grain size to the shell thickness. Choosing grain sizes smaller than the shell thickness enhances the figure of merit for feature sizes 10 nm.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content