Extreme Low Thermal Conductivity in Nanoscale 3D Si Phononic

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Extreme Low Thermal Conductivity in Nanoscale 3D Si Phononic Crystal with Spherical Pores Lina Yang,† Nuo Yang,*,‡ and Baowen Li*,†,‡,§,∥ †

Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore ‡ Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, 200092 Shanghai, People’s Republic of China § Graphene Research Center, National University of Singapore, Singapore 117542, Republic of Singapore ∥ NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 117456, Republic of Singapore S Supporting Information *

ABSTRACT: In this work, we propose a nanoscale threedimensional (3D) Si phononic crystal (PnC) with spherical pores, which can reduce the thermal conductivity of bulk Si by a factor up to 10,000 times at room temperature. Thermal conductivity of Si PnCs depends on the porosity, for example, the thermal conductivity of Si PnCs with porosity 50% is 300 times smaller than that of bulk Si. The phonon participation ratio spectra demonstrate that more phonons are localized as the porosity increases. The thermal conductivity is insensitive to the temperature changes from room temperature to 1100 K. The extreme-low thermal conductivity could lead to a larger value of ZT than unity as the periodic structure affects very little the electric conductivity. KEYWORDS: Phononic crystal, thermal conductivity, phonon localization, thermoelectrics, thermoelectric material, molecular dynamics

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produced. The measurement show that the thermal conductivity of 2D Si PnCs is as low as 2 W/m·K, meanwhile the values of electrical conductivity were comparable to those of bulk Si thin film.16 Moreover, Gillet et. al studied the thermal conductivity of an atomic 3D PnC where molecular-size selfassembled Ge quantum-dots (QDs) are distributed periodically in Si.18 From Boltzmann transport equation (BTE) method, they predicted that the thermal conductivity could be reduced by several orders of magnitude when compared with bulk Si. By lattice dynamics, Davis and Hussein demonstrated that the phonon band structure of a nanoscale Si PnC supercell is significantly different from that of bulk Si.19,20 Both boundary scattering and the dispersion of the PnC play an important role in reducing the thermal conductivity. In our recent numerical study on isotopic nanoscale 3D phononic crystals, where 28Si and isotopes MSi atoms are assembled periodically in the three directions,14 we found that the thermal conductivity of the structure with 6 nm period length and 2 in mass ratio is 2.14 W/m·K at 1000 K, which is only 4.3% of pure 28Si. The reduction of thermal conductivity is due to a reduction of group velocities, phonon localizations, and band gaps. In a most

hermoelectric material provides much hope for converting heat into electricity. It can also be used as solid-state Peltier coolers in integrated circuits, an outstanding challenge for electronic engineers. Although many works have been done, we are still far from having a recipe for thermoelectric materials. Nanostructuring provides an effective way to increase figure of merit (ZT) by reducing the thermal conductivity without affecting electronic property.1−8 Among many nanostructured materials, Si phononic crystals (PnCs) are considered promising candidates. On one hand, Si has the advantage of low-cost compared with other thermoelectric materials. It is also environmentally friendly and widely used in semiconductor industry.9−13 On the other hand, PnCs are constructed by a periodic array of scattering inclusions distributed in a host material, which can affect the transport of terahertz lattice vibrations, phonons, when the period length decreases to nanometers. Because of its periodic change of the density and/or elastic constants, PnCs exhibit phononic band gaps.14,15 This remarkable property is very different from those of traditional materials (like adding random/disordered impurities), and PnCs can be engineered to achieve new functionalities, such as modulating thermal conductivity. A successful methodology has been reported to implement two-dimensional (2D) PnC geometry in Si,16,17 which is compatible with standard CMOS fabrication and can be mass © 2014 American Chemical Society

Received: October 8, 2013 Revised: January 27, 2014 Published: February 21, 2014 1734

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K, which means that the thermal conductivity obtained from EMD is a little bit overestimated. This discrepancy is caused by both the inaccuracies of semiempirical potentials used in EMD and the impurity of the sample in experiments. Nevertheless, this discrepancy does not affect our comparison of EMD results of bulk Si and 3D Si PnC due to the same simulation methods and potential parameters. We calculate the thermal conductivity of 3D PnCs with different porosity at room temperature. In all of the following simulations of PnCs, we use 16 units as the side length of cell, which is large enough to overcome the finite size effect (finite size effect in Supporting Information I) and 8 units as period length. As shown in Figure 2a, the thermal conductivity rapidly decreases as the porosity increases. When the porosity is 30%, the thermal conductivity is 1.67 W/m·K. Moreover, when porosity increases to 90%, the thermal conductivity is decreased to 0.022 W/m·K, which is only 0.01% of bulk Si. We compare thermal conductivity of different Si nanostructures in Table 1 which shows that the thermal conductivity of 3D Si PnCs is even smaller than Si nanowires (NWs) and other Si nanostructures. Because PnCs do not have impurities and disorders, the significant reduction is due to the periodic spherical pores. In bulk porous materials, the thermal conductivity value may be predicted from the Fourier classic Eucken model,22,27 (κporous/κsolid)Eucken = (1 − ϕ)/(1 + ϕ/2), and Russell model,28 (κporous/κsolid)Russell = (1 − ϕ2/3)/(1 − ϕ2/3 + ϕ). When porosity is 90%, the predicted value of κporous is about 12 and 11.2 W/m· K, which is even one order larger than the thermal conductivity of PnC with porosity of 30%. The classical Eucken model and Russell model are only good in structures that are much larger than the phonon mean free path. The models are not valid in nanoscale PnC, due to the strong size effect in nanostructures.22,27 The thermal conductivity of polycrystalline Si is almost decreased 1 order of magnitude compared with bulk silicon. The thermal conductivity of random nanoporous polycrystalline Si, where holes are randomly arranged both in their positions and sizes, has been measured to be 0.04 W/m· K.29 Moreover, Alvarez et al. predicted the thermal conductivity of the random nanoporous polycrystalline Si as 0.02 W/m·K by a phonon hydrodynamic approach model.30 However, the value of ZT is small compared to Si nanowires because the electron mobility will be severely degraded at the same time.31 Besides porosity effect, we also investigate the temperature effect on the thermal conductivity of Si PnCs in the range from 300 to 1100 K (shown in Figure 2b). Both the thermal conductivity of bulk Si and the thermal conductivity of Si PnCs with three different porosities, 50%, 80%, and 90%, are calculated. Different from the bulk Si, it shows that the thermal conductivity of 3D PnCs change little as the increase of temperature. That is, the thermal conductivity of 3D PnCs is insensitive to temperature. A similar temperature dependence is also found in nanoporous SiGe23 and thin Si NWs.32 In pure bulk Si, there are normal phonon−phonon scatterings (crystal momentum conserved in three-phonon interactions) and Umklapp phonon−phonon scatterings (crystal momentum not conserved). The high-temperature thermal conductivity is dominated by Umklapp scatterings and decrease as ∼T−1. In PnCs, the very low frequency phonons contribute mostly to thermal conductivity (shown in Figure 2c), due to the localization of high frequency phonons (details in Figure 3). These low frequency phonons, whose wavelengths are larger than the unit cell of PnC, have large mean free path and weak

recent experiment, Ma et al. found that the thermal conductivity of nanoscale 3D Si PnC is below 10 W/m·K.21 Similar to 3D Si PnC structure, the Si thin film with periodic cylinder nanopores22,23 can also significantly reduce lattice thermal conductivity with specific choices of the pore size and spacing. However, Si PnCs and nanoporous Si could preserve their electrical properties with little degradation;16,22 consequently, the reduction of thermal conductivity in nanoscale Si PnCs and nanoporous Si could lead to a larger ZT than unity. The 3D Si PnC is constructed by periodic arrangement of nanoscale supercell constructed from a cubic cell with a spherical pore (shown in Figure 1). The spherical pore

Figure 1. Structures of nanoscale 3D Si PnCs. The period length of 3D PnCs is 8 units, and the side length of simulation cell is 16 units. The periodic boundary condition is applied in simulation. The lattice constant is 0.543 nm of Si, and 1 unit represents 0.543 nm. (a) The structure of PnC with one corner cutting off and its porosity is 50%. (b) The structure PnC with porosity of 90%. (c,d) Normalized energy distribution on the PnC at 300 K with porosity of 70% and 90%, respectively. The intensity of the energy is depicted according to the color bar.

corresponds to a set of missing atoms and is not perfectly spherical at the atomic scale. The center of the pore is located at the center of a cubic cell {1/2,1/2,1/2}. Together with the origin of the cubic cell, they could be looked on as Wyckoff 1b in space group # 221 (Pm3̅m). The period length of PnC is the distance between centers of two nearest supercells. The porosity is defined as the ratio of number of removed atoms in pore to the total number of atoms in a cubic Si cell. In this work, the thermal conductivity of 3D PnCs is calculated by the equilibrium molecular dynamics (EMD), namely, the Green−Kubo method, which is an effective method in calculating thermal conductivity in semiconductors.12,24 We focus on the thermal conductivity of 3D PnCs at high temperature (⩾300 K). (Simulation details are given in the Supporting Information III) To compare the thermal conductivity of 3D PnC with bulk Si, we calculate the thermal conductivity of bulk Si at 300 K by EMD method, which is found to be 170 ± 16 W/m·K. In the simulation of bulk Si, the volume of cell is 12 × 12 × 12 units3 (1 unit is 0.543 nm). The periodic boundary conditions are applied in three directions. Our result is comparable to Henry et al.’s EMD results, 160 W/m·K.25 The experimental value of thermal conductivity of bulk Si at 300 K26 is around 156 W/m· 1735

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Figure 2. (a) Thermal conductivity of PnCs versus porosity at 300 K. The side length of the simulation cell is 16 units and period length is 8 units. The thermal conductivity decreases rapidly as the increase of porosity. (b) Thermal conductivity of PnCs and bulk Si versus the temperature. Thermal conductivity of PnCs is insensitive to the temperature. The PnCs have the same period length (8 units) and side length (16 units). The cubic simulation cell of bulk Si has 12 units in side length. The error bar is standard deviation of 12 simulations with different initial conditions. (c) Cumulative thermal conductivity as a function of frequency for PnC with porosity of 90% and for Bulk Si at 300 K. Our calculated cumulative thermal conductivity for bulk silicon is compared with results from ref 15.

Table 1. Thermal Conductivity in Different Nanoscale Materials material

temperature (K)

3D Si PnC porosity: 90% porosity: 50% isotope-doped SiNW11 Si nanotube36 GeNWs with Si-coating37,38 SiNWs with Ge-coating39 Si/Ge superlattice40 nanoporous SiGe23 3D PnC (Ge QDs in Si)18 2D Si PnC17 nanomesh16 SiNW array16 bulk Si Bulk Si25 Bulk Si26

300

method

thermal conductivity (W/m·K)

MD

300 300 300 300 300 300 300 300 280 300 300 300 300

0.022 0.56 0.4 3.0 4.7 2.8 1.2 0.36 0.95 4.8 1.9 3.5 170 160 156

MD MD MD MD MD MD BTE experiment experiment experiment MD MD experiment

phonon−phonon scatterings in PnCs.15 Therefore, the thermal conductivity of PnCs is not sensitive to the change of temperature. Furthermore, we calculate the cumulative thermal conductivity as a function of frequency at 300 K, which is shown in Figure 2c. The details of calculation method can be found in Henry and Chen’s work.25 It is clearly seen that the heat is primarily carried by phonons with frequencies lower than 0.1 THz in PnCs, which contribute 90% to thermal conductivity. This is very different from in bulk Si in which more high frequency phonons contribute to thermal conductivity. This is why the thermal conductivity is insensitive to the change of temperature as shown in Figure 2b. To understand the reduction of thermal conductivity by porosity, we carry out a vibration eigen-mode analysis on 3D Si

PnCs. The mode localization can be quantitatively characterized by participation ratio P.14 The participation ratio (P) for phonon mode k is defined through the normalized eigenvector uiα,k Pk =

1 N 3 N ·∑i = 1 (∑α = 1 ui*α , kuiα , k)2

(1)

where N is the total number of atoms, ui,α,k is calculated by general utility lattice program (GULP).33 When there are less atoms participating in the motion, the phonon mode has a smaller P value. For example, P is 1/N when only one atom vibrates in the localized mode. When all atoms participate in the motion, P is calculated out as 1. That is, the smaller the value of P, the more localized the phonon mode. 1736

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decreases as the porosity increases. When the porosity is 50%, the thermal conductivity is 0.56 W/m·K. Moreover, when porosity increases to 90%, the thermal conductivity is decreased to 0.022 W/m·K, which is only 0.01% of bulk Si. Because of the reduction in thermal conductivity, the PnC with spherical pores could lead to a larger ZT than unity. Moreover, we find that the thermal conductivity changes little when temperature increases from 300 to 1100 K. The participation ratio spectra of both PnCs and bulk Si show that there are more phonons localized in PnCs at boundaries, which causes a lower thermal conductivity. Quantitatively, Si PnC with larger porosity has a larger LR value. As a consequence, the thermal conductivity of Si PnCs decreases as porosity increases. There are much advances in fabricating nanoscale 2D PnCs and 3D printing, which may help fabricating nanoscale 3D PnCs in the future.

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Figure 3. Participation ratio spectra of Si PnCs and bulk Si. A cell of 4 × 4 × 4 units3 is used in the calculation of participation ratio of bulk Si. Compared with bulk Si, the participation ratios of PnC have smaller value, which means phonon modes in PnC are likely localized. The participation ratio in bulk Si is almost in the range between 0.5 and 1.0, which characterizes the extended modes. LR values of PnCs are also shown in each channel. According to the definition of LR, there are more localized phonon modes in PnCs with larger porosity.

ASSOCIATED CONTENT

S Supporting Information *

Simulation and calculation details. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*(N.Y.) E-mail: [email protected]. *(B.L.) E-mail: [email protected].

Participation ratio spectra of 3D Si PnCs with different porosities are shown in Figure 3. We also calculate the participation ratio (P) of bulk Si, using a cell of 4 × 4 × 4 units3 with periodic boundary conditions. The values of P in bulk Si are almost in the range between 0.5 and 1, which represent extended phonon modes.34 Obviously, the values of P in 3D PnCs are much smaller than those of bulk Si, which means that phonon modes in PnCs are likely localized, which corresponds to the reduction of thermal conductivity in Si PnCs. In order to quantitatively analyze the phonon localization in PnCs, we further define the number of phonon modes whose value of P is less than 0.5 divided by the total number of phonon modes as localization ratio (LR). According to the definition, a larger LR value means there are more localized phonon modes. Also shown in Figure 3, LR values of PnCs with porosity of 70%, 80%, and 90% are 34%, 58%, and 91%, respectively, namely, there are more localized phonon modes with the increase of porosity. The extreme-low thermal conductivity of 3D PnCs mainly come from the localization of phonons. Generally, band gaps are observed in other phononic crystal structures.14,35 As shown in Figure 3c,d, there are only some ultranarrow bandgaps for porosity 80% and 90% at low frequencies (