Control of Phonon Transport by Phononic Crystals and Application to Thermoelectric Materials*

Masahiro Nomura1,2,3

1Institute of Industrial Science, The University of Tokyo, Tokyo 153–8505, Japan2Institute for Nano Quantum Information Electronics, The University of Tokyo, Tokyo 153–8505, Japan3JST, PRESTO, Kawaguchi 332–0012, Japan

Phonon transport and thermodynamic properties of nanostructured materials have been investigated and utilized to improve thermoelectric performance for various materials. In nanostructures, phonon transport is completely different from that in bulk materials and results in dramat-ic enhancement in the thermoelectric performance. This article reviews the impact of nanostructuring on the phonon transport and mainly focus-es on phononic crystal nanostructures, in which the wave nature of phonons also plays an important role. We demonstrate that it is important to ef�ciently scatter thermal phonons, which distribute to wide range of frequencies, with different phonon scattering mechanisms in the spatial domain. We also demonstrate an enhancement of thermoelectric property of polysilicon thin �lms by phononic crystal patterning. [doi:10.2320/matertrans.MF201606]

(Received December 7, 2015; Accepted February 5, 2016; Published June 25, 2016)

Keywords:　 phonon transport, phononics, phononic crystal, thermoelectrics

1.　 Introduction

Generally, in the case of solids the material is assumed large and homogenous enough that the thermal conductivity can be treated as simple heat diffusion. These heat carriers are either electrons or quantized lattice vibrations called pho-nons. In the case of metals, the majority carrier is the electron while, for semiconductors, the majority carrier is the phonon. In nanostructures, since the size of the system is equivalent to the mean free path (MFP) of the carrier, in order to more clearly understand the phenomenon of thermal conductivity we must consider the ballistic transport of phonons without describing their behavior as a diffusion process. Ballistic pho-non transport, as well as characteristic phonon transport due to scattering caused by the structure, has been reported in var-ious structures including atomic scale carbon nanotubes1), graphene2), semiconductor super lattices3,4), structures con-taining nano particles5,6), porous structures7,8), nanowires9) and phononic crystal (PnC) structures10–16). Since the thermal conductivity in nanostructures is system-dependent, using certain materials it is possible to fabricate monolithic devices with various thermal conductivities. Also, since it is now pos-sible to remarkably decrease the characteristic thermal con-ductivity of the materials by nanofabrication, new choices are available for developing thermoelectric conversion materi-als17,18). Using the difference in length of the MFP of elec-trons and phonons, it is possible to dramatically decrease the thermal conductivity of a material without negatively affect-ing the electrical conductivity. This approach was meant to emphasize the difference between electric charge and heat carriers, which means that the system is no longer restricted by the Wiedmann-Franz Law, which governs the temperature dependent ratio of thermal conductivity to electrical conduc-tivity in metals19). Until now, many have reported on a variety of materials with a high �gure of merit for thermoelectric

conversion by implementing nanostructuring in harmless and inexpensive materials with extremely low ZT values and in bulk materials that already possessed a high ZT value. Fig-ure 1 shows scanning electron microscope (SEM) and tunnel-ing electron microscope (TEM) images of several systems reporting such �ndings. Components that change the lattice thermal conductivity can be, as shown in Fig. 2, either classi-�ed as incoherent scattering processes such as impurity, sur-face, interface and phonon-phonon scattering under the parti-cle description, or as coherent scattering processes based on the wave description resulting from periodic PnC for acoustic waves. In order to analyze these phenomena, the former uses the relaxation time τ which becomes necessary when approx-imating scattering processes and is taken from the Boltzmann transport equation which serves as the foundation for classi-cal transport theory (Fig. 2). In this equation, f0 and f are the phonon distribution functions in equilibrium and non-equilib-rium states, and v is the velocity vector. The latter analyzes

* This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 79 (2015) 555–561.

Fig. 1　(a) ZT values for various thermoelectric materials17). (b) SEM image of nanoporous BiSbTe,8) (c) TEM image of Si nanowire,9) and (d) SEM image of two-dimensional Si phononic crystal nanostructure.

Materials Transactions, Vol. 57, No. 7 (2016) pp. 1022 to 1028 Special Issue on Recent Progress in Thermoelectrics -New Analyses and New Materials- (Thermoelectric Conversion Materials IX) ©2016 The Japan Institute of Metals and Materials OVERVIEW

phonon transport and calculates the phononic band diagram using �nite element method under Floquet conditions for the shape, density and young modulus of the periodic structure. The governing equation in the coherent scattering regime is given by the elastodynamic wave equation in Fig. 2: u as the displacement vector, ρ  =  2329 kg m−3 as the mass density, and λ =  84.5 and μ =  66.4 GPa as the Lame parameters for the case of silicon. Much like with the Knudsen number used in �uid dynamics, we must keep in mind the characteristic length of the system as well as the MFP when discussing heat transport. Thermal conduction in the incoherent and coherent regimes is based on different concepts: reduction of phonon relaxation time and reduction of the group velocity are the key factors in respective regime.

In this paper, we provide a great example of thermal con-ductivity in silicon PnC nanostructures exhibiting both semi-ballistic transport and wave nature of photons and which are described by both the particle and wave descriptions. In addition, we will discuss our current progress in thermoelec-tric conversion and the control of heat conductivity in multi-scale polysilicon designed while considering the multiscale nature of phonons.

2.　 Experimental Method

2.1　 Micro-thermore�ectance methodTypically, the thermal conductivity of mesoscale materials

is taken using electrical methods. With our measurement technique, we are able to detect small changes in temperature and measure thermal conductivity more precisely; however, it takes a long time to fabricate the samples, the environmental footprint is high and the through put is quite low. Since we will need to conduct measurements on various PnC struc-tures, we have developed an optical measurement technique called micro-thermore�ectance which has much higher throughput than the existing electrical methods. Time domain thermore�ectance (TDTR) is a method for measuring the thermal conductivity of thin �lm materials on which metal has been deposited by observing the change in temperature over time after using a pulsed laser to heat the �lm. Then, the experimental temporal evolution can be compared with simu-lation results20). Our custom-made micro-TDTR setup can measure in-plane thermal conductivity in micro/nano struc-tures using focused pump and probe laser beams21). The met-al layer converts the light to thermal energy, effectively acting as a transducer to detect temperature change by observing

changes in re�ectance.In our experiment, as shown schematically in Fig. 3(a), we

have devised a sample structure and focused light that we can apply to a micro-scale system. By making a suspended struc-ture from the active silicon layer of an SOI wafer, we form a central island which holds the aluminum pad and supports our structure on all sides. With such an arrangement, we can ensure that heat dissipation will occur through only the struc-ture to be measured, thus allowing us to compare with simu-lation data and obtain thermal conductivity values at high precision. The wavelength of the excitation source and probe laser source were 642 nm and 785 nm respectively, and the dot of focused light had a diameter of approximately 0.7 μm. We chose the wavelength of the probe laser so that the ther-mal re�ectance coef�cient, measured in percent change in re�ectance per unit change in temperature of the aluminum pad, would be large enough. The coef�cient value in our ex-periment was 0.015%/K. Since thermal conductivity is tem-perature dependent, it is desirable to heat the PnC structures as little as possible. In our experiment, we adjusted the pump laser power so that the increase in temperature remained be-low 2 K and con�rmed that the thermal conductivity was con-sistent within that range of temperature increase. In addition, the measurement took place in a vacuum so that air would not affect the heat dissipation in the channel which we are trying to measure.

The dimensions of the nanostructure were precisely veri-�ed using SEM and the simulation using �nite element meth-od was designed based on these dimensions. Using the ther-mal conductivity κ as the only variable, we compared with the values obtained from the TDTR signals using the technique

Fig. 2　Summary of phonon transport phenomena in bulk and nanostructures.

Fig. 3　(a) Schematic of the μTDTR measurement and cross-section of the sample structure. (b) Temporal evolution of the TDTR signals for an un-patterned membrane and PnCs with r =  73, 115, and 126 nm.

1023Control of Phonon Transport by Phononic Crystals and Application to Thermoelectric Materials

described above, and derived the most appropriate thermal conductivity coef�cient for our simulation. The physical model, as also seen in eq. 1, expresses an effective thermal conductivity for the entire device which only considers the decrease in thermal conductivity by surface scattering as a result of heat transfer.

ρCp∂T∂t− κ∆T = Q(t) (1)

The material used was single crystal Silicon with a density ρ of 2329 kg/m3, a speci�c heat at constant pressure Cp of 700 J/kg･K and a time dependent heat quantity per unit vol-ume given by the heating pulsed laser source Q(t) measured in W/m3. Consistent with the experiment, energy was sup-plied to the aluminum pad in a Gaussian distribution from time t =  0 to 500 ns. We can determine the thermal conductiv-ity via the least squares method by comparing the aluminum pad surface temperature simulation curve with our experi-mental results using thermal conductivity as the parameter. The TDTR signals (dots) from the 2D PnC nanostructure, which we will introduce in the next section, and the simula-tion curve (solid line) are shown in Fig. 3(b) and have pro-duced extremely reproducible results.

2.2　 Sample fabricationBeginning with either an SOI wafer on the market or a wa-

fer fabricated using low pressure chemical vapor deposition system, we fabricated the suspended structure necessary to measure the thermal conductivity by micro-thermore�ec-tance technique in PnC nanodevices like in Fig. 4(a). Then, using electron beam lithography and dry etching, the single crystal silicon nanostructures contained a 145 nm active layer above a 1 μm buried oxide layer, while the polysilicon nano-structures were fabricated on a silicon board with a 143 nm polysilicon crystal thin �lm grown by low pressure chemical vapor deposition system on top, and a 1.5 μm SiO2 layer above that. A 125 nm thick, 4 ×  4 μm aluminum square is deposited onto the remaining silicon island in the center so that there is a 1 μm margin around the perimeter of the pad. The island is completely suspended by the two PnC suspend-ed structures attached on either side.

3.　 Results and Discussion

3.1　 Control of thermal conductivity by multiscale archi-tecture

Following the Bose-Einstein distribution, the frequency distribution of phonons at room temperature can span from the MHz regime up to several THz. Among this spectrum, it is not clear which frequency band of phonons most contrib-utes to thermal conductivity. If the aim is to reduce thermal conductivity for thermoelectric conversion material develop-ment, it is effective to introduce phonon scattering mecha-nisms that �t to the frequency distribution. Biswas et al. have reported a rapid improvement in ZT with ef�cient scattering of various phonon wavelengths over a wide frequency do-main by creating three different scaled PbTe structures which they call an all-scale hierarchical structure22). Though we may also apply this approach for Si, since the distribution of thermal phonons in Si is quite different than in PbTe, we can follow a design based on the idea of a cumulative thermal conductivity. Figure 5 shows the cumulative thermal conduc-tivity plotted against the distribution of thermal phonon MFP in Si at 300 K, which is calculated based on Debye-Klemens

Fig. 4　(a) SEM image of the whole suspended Si structure with 2D PnC nanostructures. (b) Calculated phononic band diagram for the square lattice (a =  300 nm, r =  135 nm) and heat �ux spectra for an unpatterned thin �lm and the 2D PnC nanostructure.

Fig. 5　Cumulative thermal conductivity plotted as a function of phonon MFP. Vertical broken lines represent thermal phonon density at each MFP.

1024 M. Nomura

model23). The contribution to thermal conductivity by pho-nons is represented by the broken blue lines below. For exam-ple, phonons with MFP =  1 μm have a cumulative thermal conductivity of 0.5, but this means that half of the heat is be-ing carried by phonons with a MFP of less than 1 μm. So, the thermal phonon density (broken lines) is mostly distributed in the region between 100 nm to 10 μm where the cumulative thermal conductivity (solid line) rises up drastically. This is the target region where introducing scattering mechanisms will be most effective. This relatively long and broad thermal phonon MFP range in Si at room temperature agrees with the calculations based on �rst-principles24).

By PnC nanostructures, it is possible to scatter phonon modes in bulk with a MFP greater than 100 nm, but those with a MFP below 100 nm will remain largely unaffected. Consequently, when using polysilicon thin �lms with grain boundaries distributed in the 10’s of nm region, the belief is that this will enable the ef�cient scattering of thermal pho-nons with a short MFP. As a qualitative argument, based on the blue line representing MFP =  100 nm in Fig. 5, the wide range of thermal phonons distributed is considered to be cov-ered in multiscale by the various scales of boundary scatter-ing and surface scattering by PnC nanostructures.

Figure 6(a) shows the grain boundary distribution as ob-tained by analyzing TEM images of a non-doped Si surface grown by low pressure chemical vapor deposition system. As predicted, we were able to grow a polysilicon thin �lm with a grain size distribution spanning widely below 100 nm. By comparing the decrease in thermal conductivity by PnC nano-structures in both single crystal and polysilicon Si thin �lms, we can discuss whether or not the multiscale structures in sil-icon are effective. The thermal conductivity of PnC nano-structures with varying hole radii is shown in Fig. 6(b) for both single crystal and polysilicon thin �lms. The data for r =  0 nm corresponds to the thin �lm without holes. The thermal conductivity decreases as hole radius increases, but this is due to the drastic increase in occurrences of phonon surface scat-tering as the minimum distance between the inner walls of two adjacent holes, de�ned as the neck size (a −  2r), becomes narrow7). Comparing the in�uence of PnC nanostructure pat-terning on thermal conductivity for r =  100 nm with the val-ues obtained for the thin �lm without holes, we observed a decrease from 75 Wm−1K−1 to 38 Wm−1K−1 for single crys-tal, and 10.5 Wm−1K−1 to 4.3 Wm−1K−1 for polysilicon struc-

tures21). In Fig. 6(b), the thermal conductivity values for the thin �lm are clearly larger than the values obtained when ex-trapolating the values for PnC nanostructures from 50 nm to 0 nm. This discontinuous decrease in thermal conductivity is understood as the difference in phonon surface scattering. The surfaces causing phonon scattering in the membrane are parallel to the direction of thermal conductivity, but because the PnC nanostructures introduce holes with inner walls which are perpendicular to the direction of thermal conduc-tivity, this is thought to be a result of strong backscattering. Also, for hole radii of r >  115 nm, polysilicon Si showed a constant linear decrease while the single-crystalline Si showed a more drastic decrease in thermal conductivity as hole radius increased. Since the neck size becomes notably narrower compared to the 145 nm �lm thickness, it appears that the thermal conductivity in single-crystalline silicon is determined by neck size in this regime. On the other hand, since the boundary scattering at the interfaces of the polysili-con grain boundaries seem to determine the thermal conduc-tivity, it is quite possible that a similar drastic increase may begin to appear if the neck size were to narrow even more.

In this paper, in order to limit the discussion to room tem-perature measurements with thermoelectric conversion appli-cations in mind, though it is not plain to see the control of phonon transport based on the wave nature of phonons by the periodic structures that distinguish the �eld of phononics, it is possible to observe this phenomenon under suf�ciently low temperature environments. Figure 4(b) shows results for the calculated phononic band diagram and heat �ux spectrum in square lattice PnC nanostructures with a period of a =  300 nm, and a hole radius of r =  135 nm. The details of the simulation can be found in another work25). The band diagram for our arti�cial periodic structure differs much from that of bulk sil-icon. Since the momentum k within the Brillouin zone pro-vides a point of symmetry in small places, the band-folding effect occurs and multiple vibrational modes are being com-pletely �lled due to the �nite geometry. With an appropriate design, it is possible to open a complete band gap near the frequency vicinity of the fundamental vibrational mode, where the modes have become sparse, and with our structure this exists near 10 GHz. The propagation has been complete-ly prohibited for all phonons with frequencies within the band gap. A drop in the group velocity appears near the point of symmetry on the upper edge of the band gap, which is re�ect-

Fig. 6　(a) Histogram of the grain size measured on the surface and TEM image of the sample (inset). (b) Thermal conductivities for single-crystalline and polycrystalline Si PnC with a variety of radii. The broken horizontal lines represent the value for each crystal type membrane.

1025Control of Phonon Transport by Phononic Crystals and Application to Thermoelectric Materials

ed in the heat �ux. As shown in the right side of Fig. 4(b), we clearly observed lower heat �ux in the PnC nanostructures as compared to the membrane without holes (gray line). Since the difference in heat �ux becomes more signi�cant at higher frequencies, we expect that it is possible to achieve a large decrease in thermal conductivity at high temperatures, but the shortening of the phonon MFP and thermal phonon wave-length due to more frequent incoherent scattering events in-duced by a temperature increase will actually weaken the �uctuation caused by sensitivity to surface roughness, and the band-folding effect will vanish. The concept of heat transport control based on the wave nature of phonons has been known for quite some time, however studying this phenomenon has always been a challenge due to the exceedingly dif�cult pro-cess of detecting the extremely small wavelengths of thermal phonons (around 1 nm at room temperature). Therefore, we do not think that the signi�cant reduction of the thermal con-ductivities in PnCs shown in Fig. 6 stems from the wave na-ture of thermal phonons. In recent years, in low temperature environments in the range of 100 mK, there have been re-ports12) that suggest the possibility of making such observa-tions, and in our research we continue to �nd success in tun-ing the thermal conductivity of silicon at 4 K by manipulating its crystallinity using PnC nanostructures26).

3.2　 Improvement of thermoelectric conversion by nano-structuring

While bulk silicon is typically not attractive as a thermo-electric conversion material, since it is widely abundant, of-fers a low environmental load and integrates well with exist-ing microelectronics, there have been many attempts to develop silicon-based applications by dramatically improv-ing its capability as a thermoelectric material through nano-structuring. Thermoelectric conversion characteristics have been reported in polysilicon thin �lms produced using MEMS and CMOS fabrication techniques27–29).

The unitless Figure of Merit (ZT) of thermoelectric materi-als is expressed as S 2σT/κ, where S is the Seebeck coef�-cient, σ is the thermal conductivity, T is the temperature and κ is the thermal conductivity of the material. The thermoelec-tric ef�ciency η can be simply expressed in terms of ZT.

η =TH − Tc

TH·√

1 + ZT − 1√

1 + ZT + TC/TH

(2)

Here, TH is the temperature of the high temperature source, TC is the temperature of the low temperature source, and T is the average temperature. The �rst term is the Carnot ef�cien-cy which determines the upper limit. We have seen a number of works in the past few decades using a variety of material systems and structures which employ electrical enhance-ments that aim to increase the power factor S 2σ by consider-ing molecular structure and density of states near the Fermi surface, as well as structural enhancements which aim to de-crease lattice thermal conductivity by phonon transport con-trol, as an approach to realize a large ZT value. There also appear to be a large number of comprehensive explanations concerning the design guidelines of high ef�ciency thermo-electric conversion materials at the nanoscale30).

By patterning the PnC nanostructures on the doped polysil-

icon membrane, we were able to determine the degree of pos-sible improvement in our device. Figure 7(a–d) shows the pro�les and SEM images of 2D PnC nanostructures which were fabricated in order to make both electrical and thermal conductivity measurements. Boron and phosphorous were used as the p-type and n-type dopants at carrier concentra-tions of approximately 2 ×  1020 cm−3 and 8 ×  1020 cm−3, and �lm thicknesses of 149 nm and 145 nm respectively. The de-tails of the growth conditions are explained in Ref. 31. Sam-ples used for measuring electrical properties have 2D PnC nanostructures grown on SiO2 and Al electrical contacts evap-orated onto the structure to allow for four terminal electrical measurements. The pro�le of this structure is shown in Fig. 7(b). Samples used for measuring thermal properties have the same construction as described above.

Both electrical and thermal conductivity for p-type and n-type nano polysilicon 2D PnC nanostructures were ac-quired using the four terminal technique and micro-thermore-�ectance technique, and these results are shown in Fig. 8. Similar to thermal conductivity, electrical conductivity also shows a tendency to decrease as the hole radius increases, but unlike thermal conductivity the rate of decrease appears to be much slower and shows no discontinuous decrease from the values for the membrane. This is due to the in�uence of sur-face scattering by PnC nanostructure and the fact that the MFP of electric charge carriers is one order of magnitude shorter than that of a phonon. The Seebeck coef�cients used in calculating ZT for the results shown in Fig. 8(c) for p-type and n-type samples were 240 μV/K and −81 μV/K respec-tively32). It is clear that the value of ZT rises along with the increase in hole radius, and when r =  110 nm the ZT value for the p-type sample increased approximately 2 times, and n-type approximately 4 times, compared to the bulk mem-brane. This indicates that the dimensions of our structure em-phasize the difference in MFP between electric charge carri-ers and phonons as a result of the remarkable decrease in thermal conductivity compared to the change in electrical conductivity. By narrowing the neck width we might expect a further rise in the ZT value, but due to the increase in electri-cal resistance at hole radii r >  140 nm, in the region where the MFPs of electric charge carriers and phonons are compara-ble, there may exist an optimum hole diameter.

Fig. 7　SEM image and schematic of the cross section of (a, b) Si PnC nano-structures (a =  300 nm and r =  110 nm) for the four probe measurement, and (c, d) entire suspended Si microstructures supported by Si PnC nano-structures with circular holes aligned as a square lattice (a =  300 nm and r =  111 nm) for thermal conductivity measurement.

1026 M. Nomura

Thermoelectric conversion devices are often used by plac-ing the object against various surfaces with differing tempera-tures, so electric current and heat �ux is made to �ow verti-cally along this surface. However, the structures we have introduced were constructed so that the electric current and heat �ux �ow in-plane, so this aspect of the device will need to be assessed in future designs. Until now, electrical power per unit area on the range of 1 μWcm−2 with a 10 K tempera-ture difference has been reported using CMOS and MEMS technology to fabricate structures that can produce a tempera-ture difference along the in-plane direction26–28). As shown in Fig. 1(a), at 900 K silicon demonstrates the highest perfor-mance compared to other materials. At room temperature, silicon devices perform at only 10% of this level, which is relatively low when compared to materials such as BiTe which exhibit very high performance at room temperature.

However, when considering energy harvesting applications which are needed to build the wireless network that will con-tribute to a smart society, it is enough to drive the microsen-sors and when combined with a capacitor could potentially operate several times in one day which is suf�cient for a vari-ety of uses. A device that can output 1 μWcm−2 will be able to ful�ll this energy requirement and since it is cheap and has a low environmental load, silicon can be considered as one possible choice for thermoelectric conversion material devel-opment.

4.　 Conclusion

Since the 1990’s, improvements in nanostructure forming technology have taken thermoelectric conversion material de-velopment in a new direction. Since heat transport in nano-structures is a complex issue, even today many phenomena remain unclear and we hope to see development in this �eld in the near future. Certainly, if achievable, the development of a thermoelectric conversion material with both a high conver-sion ef�ciency and low environmental load will contribute to solving the current energy issues and building a smart society. In this paper, we have investigated the physics of thermal conductivity in polysilicon PnC nanostructures and its con-version characteristics in order to evaluate such a device as an approach to these modern day issues. We have shown that in order to effectively control thermal transport it is important to have control over thermal phonons dispersed over a large area, and we also demonstrated that an improvement in the

ZT value of polysilicon by three times that of bulk material is possible through PnC nanostructuring and multiscale layered structures with various scale phonon scattering mechanisms.

Acknowledgements

The contents of this work were obtained by the research activity with Jeremie Maire, Yuta Kage, Roman Anufriev at the University of Tokyo, and Oliver Paul and Dominik Moser at Freiburg University. The author acknowledge Kazuhiko Hirakawa and Junichiro Shiomi for fruitful discussion, and Yasuhiko Arakawa for technical support. This work was sup-ported by the Project for Developing Innovation Systems of MEXT, Japan, by KAKENHI (25709090).

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