Determining factors of thermoelectric properties of semiconductor nanowires
- Denis O Demchenko^{1}Email author,
- Peter D Heinz^{2} and
- Byounghak Lee^{2}
DOI: 10.1186/1556-276X-6-502
© Demchenko et al; licensee Springer. 2011
Received: 14 June 2011
Accepted: 19 August 2011
Published: 19 August 2011
Abstract
It is widely accepted that low dimensionality of semiconductor heterostructures and nanostructures can significantly improve their thermoelectric efficiency. However, what is less well understood is the precise role of electronic and lattice transport coefficients in the improvement. We differentiate and analyze the electronic and lattice contributions to the enhancement by using a nearly parameter-free theory of the thermoelectric properties of semiconductor nanowires. By combining molecular dynamics, density functional theory, and Boltzmann transport theory methods, we provide a complete picture for the competing factors of thermoelectric figure of merit. As an example, we study the thermoelectric properties of ZnO and Si nanowires. We find that the figure of merit can be increased as much as 30 times in 8-Å-diameter ZnO nanowires and 20 times in 12-Å-diameter Si nanowires, compared with the bulk. Decoupling of thermoelectric contributions reveals that the reduction of lattice thermal conductivity is the predominant factor in the improvement of thermoelectric properties in nanowires. While the lattice contribution to the efficiency enhancement consistently becomes larger with decreasing size of nanowires, the electronic contribution is relatively small in ZnO and disadvantageous in Si.
Introduction
Using nanostructures for thermoelectric (TE) materials is a promising prospect as it opens up a possibility of controlling the TE properties by modifying the size and shape, in addition to the composition of the material. The TE properties of a material are characterized by a dimensionless figure of merit, ZT = TS ^{2} σ/(κ _{ e } + κ _{ l } ), where T, S, σ, κ _{ e } , and κ _{ l } are temperature, Seebeck coefficient (thermopower), electrical conductivity, electronic thermal conductivity, and lattice thermal conductivity, respectively. The main issue in finding large ZT materials is to balance the advantageous material properties, i.e., S and σ, with the detrimental κ _{ l } and κ _{ e } . The S, σ, and κ _{ e } are related to the electronic states and cannot be controlled independently in bulk materials. The κ _{ l } of bulk materials is a structural property that is also difficult to manipulate. It was suggested that reduction of the dimensionality should provide certain controllability over the individual transport coefficients, leading to a dramatic increase in ZT [1]. Bi_{2}Te_{3}/Sb_{2}Te_{3} epitaxial superlattices were reported to exhibit a very high ZT of 2.4 at room temperature [2], larger than the maximum ZT of 1.14 for a p-type (Bi_{2}Te_{3})_{0.25}(Sb_{2}Te_{3})_{0.72}(Sb_{2}Se_{3})_{0.03} alloy at room temperature [3], affirming the basic premise. In many cases, however, the decrease in phonon transport also leads to the reduction in power factor due to decrease in electron transport. The search for the balance between these factors is currently ongoing.
In this letter, we report the results of a theoretical investigation of TE properties of semiconductor nanowires to provide a microscopic picture for key contributing factors to improve the TE performance of nanostructures. The focus of this work is to distinguish electronic and lattice contributions to the enhancement of ZT in nanowires (NWs) by means of parameter-free calculations. Previous theoretical studies have provided proof-of-principle verification of ZT improvement in nanostructures [4, 5], but their analyses were based on the lattice thermal conductivity speculated from the bulk value. By combining molecular dynamics (MD), density functional theory (DFT), and Boltzmann transport equation (BTE) methods, we compute all of the transport coefficients without empirical parameters.
Methods of calculations
In this work, we used equilibrium MD simulations based on Green-Kubo formula, using LAMMPS [6, 7] code to compute lattice part of thermal conductivity. The interatomic interactions in ZnO are modeled using the Buckingham potential with long-range Coulomb interactions [8, 9]. This potential has been applied to describe elastic properties of ZnO nanostructures [10], their structural transformations [11, 12], and thermal properties [13–15]. In case of Si NW, we used Stillinger-Weber [16] potential, which has been applied to model thermoelectric properties of Si both in the bulk and on the nanoscale [17, 18]. The atomic and electronic structures of ZnO NW were calculated using planewave DFT method implemented in VASP code [19]. We used the generalized gradient approximation with Perdew-Burke-Ernzerhof exchange-correlation functional [20]. The k-point grid of 4 × 4 × 24 and an energy cutoff of 400 eV were used for all calculations. The TE coefficients of ZnO nanowires were calculated based on Boltzmann transport theory, using BoltzTraP [21]. The method calculates TE properties using the result of ab initio electronic structure method results. Within this semi-classical method, the group velocity and the mass tensor are calculated using the DFT band dispersion. Assuming a constant carrier relaxation time, one can calculate the TE coefficients from the transport tensors [22].
The only physical parameter that we do not seek to calculate from first principles is the relaxation time, τ. It is a complex function of atomic and electronic structure, temperature, impurities, and carrier concentrations. Experimentally, it has been reported that the carrier relaxation time in ZnO nanowires is close to the bulk relaxation time. The time-resolved terahertz spectroscopy measurements of electron conductivity yielded the relaxation time of 28 and 35 fs for 500-nm radius nanowires and films, respectively, at the electron density of 10^{17} cm^{-3}[23]. Throughout this paper, we adopt the ZnO bulk relaxation time τ = 30 fs for our calculations. The relaxation time in NW is generally larger than in the bulk [4], and therefore, our estimation of τ represents a correct quantitative picture. In the case of Si NW, we also used the carrier concentration-dependent τ fitted into the experimental data [24].
Results and discussion
The electronic structure and thermoelectric properties of Si [001] NW were computed for comparison. The diameter of the NW was 12 Å with single-bonded surface atoms removed and the surface atoms passivated by hydrogen to remove dangling bonds (inset to Figure 3). The computed band gap in the Si NW was 2.11 eV, in comparison with computed 0.56 eV in the bulk. The electronic properties of Si NWs have been studied in great detail in the past [28]. Electronic part of thermoelectric properties has also been addressed previously [4], suggesting improvement of ZT to as large as 1.55. However, the lattice thermal conductivity in these calculations was presumed instead of being calculated. The significance of the lattice conductivity is discussed below.
The heat transport through the phonon channels in nanowires is analyzed here via the classical MD simulations. The atomic structures for ZnO NWs used in MD calculations are the same as in the electronic part calculations, with their dimension in [0001] direction extended to 50 Å. Using periodic boundary conditions in this directions, our tests show that for these NWs, the effects of finite length are negligible. The value of the lattice thermal conductivity is the averaged integral of the heat current autocorrelation function (ACF). In bulk ZnO and ZnO NWs, the heat current ACF exhibits strong oscillations due to zero wave vector optical phonon modes. For example, ACF for the 8 Å ZnO NW shows a 56.7-meV phonon mode, which agrees well with the energy (approximately 53 meV) of the optical branch at the Γ-point for bulk wurtzite ZnO [29]. Although these optical phonon oscillations do not contribute to the heat current, they significantly complicate the numerical integration of the heat current ACF [30]. To avoid numerical instabilities due to this optical phonon mode, a very small MD step of 0.1 fs was taken in our MD calculations. κ _{ l } were averaged over the total MD run times of 3 ns. At room temperature, the experimental κ _{ l } of bulk ZnO varies from about 50 to 140 W/Km [31, 32], depending on growth conditions. The bulk values of κ _{ l } = 85 W/Km in [0001] direction were obtained by extrapolating to infinity the results for a series of simulation box sizes. For the NWs, a significant reduction in κ _{ l } is obtained. The converged values of the κ _{ l } at room temperature are 7.9, 9.2, and 11.7 W/Km for 8, 10, and 17 Å NWs, respectively. In case of Si NW, there are no optical phonon oscillations in the ACF and computing κ _{ l } is relatively straightforward. In the bulk Si, we obtain κ _{ l } = 254 W/Km at room temperature. This value is overestimated by about 70% in comparison with experiment, which is typical for Stillinger-Weber potential. In the 12 Å Si NWs, κ _{ l } is reduced to 2.8 W/Km. This value is close to the presumed lattice thermal conductivity in the previous study by Vo et al. [4]. This reduction of κ _{ l } can stem from several processes, such as three-phonon Umklapp, mass difference, and boundary/surface scattering. In ZnO and Si NWs considered here, the primary process is likely the increased surface phonon scattering, evident from the fact that κ _{ l } decreases with increasing surface to volume ratio.
The calculated TE properties of ZnO NWs at room temperature are shown in Figure 1 as a function of carrier concentration. Compared with the bulk TE coefficients, σ changes are unfavorable to the ZT while S and κ _{ e } change positively. Notice that the unfavorable change in σ is much larger than the favorable increase in the magnitude of S, but the quadratic dependence of ZT on S nearly cancels the effect of linear dependence on σ. The combination of these changes, together with the change in κ _{ l } , conspires a significantly improved ZT in NWs. The changes of individual TE transport coefficients are consistent with the tendencies found in previous calculations of Si NW [4], nanoporous Si [5], and Si NW computed here (not shown). The maximum ZT is more than 30 and 20 times larger than that of the bulk in ZnO and Si, respectively, confirming the basic proposition of ZT enhancement in nanostructures. However, both Si and ZnO nanowires considered here are ultra thin; therefore, these results represent the largest values for ZT enhancements possible solely due to scaling down the sizes and using these materials on the nanoscale.
Enhancement of ZT in ZnO and Si NW
NW diameter (Å) | ||||
---|---|---|---|---|
ZnO | Si | |||
8 | 10 | 17 | 12 | |
ζ = ZT*/ZT*_{bulk} | 31.5 | 18.9 | 8.17 | 20.5 |
ζ _{ e } = ZT*(κ _{ l } ^{bulk})/ZT*_{bulk} | 3.25 | 2.15 | 1.15 | 0.264 |
ζ _{ l } = ζ/ζ _{ e } | 9.69 | 8.79 | 7.10 | 77.7 |
Finally, we note that the maximum values of ZT we obtain for ZnO NW is approximately 0.05 and similarly it is approximately 0.06 in Si NW. This is much smaller than the previously calculated value in Si NW, approximately 1.55 [4]. The differences between these calculations appear to arise from the differences in computed electric conductivity and consequently power factors. The values of electric conductivity for bulk Si computed here are in agreement with that of the experiment [23] for electron concentrations between 10^{17} to 10^{20} cm^{-3} as well as previous calculations (Ref. [5]), while it appears to be overestimated in Ref. [4]. The values of ZT computed here suggest that reducing the scale of nanostructures alone does not improve the ZT to a practical degree. Our findings indicate that other mechanisms must be involved in the improvement of ZT, in addition to using nanocrystals. For example, in recent experiments, the enhanced ZT values in nanowires were obtained due to surface roughness and impurity scattering [33, 34].
Conclusion
In conclusion, we presented the results of BTE calculations of ZnO and Si NWs based on electronic and lattice properties computed using DFT and classical MD, respectively. Not all electronic and lattice thermal coefficients change favorably when size of the nanowire is reduced. By decoupling the effects of electronic and lattice thermal conductivity on ZT values of semiconductor nanowires, we found that the reduction in κ _{ l } plays a predominant role in the enhancement of ZT. For example, for the 8 Å ZnO and 12 Å Si nanowire, this enhancement is by a factor of 10 and 78, respectively. We conclude that for ZnO nanowires with diameter of 17 Å and larger, the lattice thermal conductivity contribution to the enhanced ZT is overwhelmingly more important than that of the electrons. Opposite to the common belief, the electronic contribution changes disadvantageously in Si NWs. In both materials, although the improvement in ZT is substantial in comparison with the bulk, the ZT of about approximately 0.05 are the maximal achievable values as a result of scaling down of the materials to the nanoscale level (down to about 10 Å).
Declarations
Acknowledgements
This work used computational facilities of the VCU Center for High Performance Computing and Texas State University High Performance Computing Center.
Authors’ Affiliations
References
- Hicks L, Dresselhaus M: Effect of quantum-well structures on the thermoelectric figure of merit. Phys Rev B 1993, 47: 12727–12731. 10.1103/PhysRevB.47.12727View Article
- Venkatasubramanian R, Siivola E, Colpitts V, O'Quinn B: Thin-film thermoelectric devices with high room-temperature figures of merit. Nature 2001, 413: 597–602. 10.1038/35098012View Article
- Ettenberg MH, Jesser WA, Rosi FD: A new n-type and improved p-type pseudo-ternary (Bi _{ 2 } Te _{ 3 } )(Sb _{ 2 } Te _{ 3 } )(Sb _{ 2 } Se _{ 3 } ) alloy for Peltier cooling. In Proceedings of 15th International Conference on Thermoelectrics. Edited by: Caillat T. Piscataway: IEEE; 1996:52–56.View Article
- Vo T, Williamson AJ, Lordi V, Galli G: Atomistic design of thermoelectric properties of silicon nanowires. Nano Letters 2008, 8: 1111–1114. 10.1021/nl073231dView Article
- Lee J-H, Galli GA, Grossman JC: Nanoporous Si as an efficient thermoelectric material. Nano Letters 2008, 8: 3750–3754. 10.1021/nl802045fView Article
- Plimpton SJ: Fast parallel algorithms for short-range molecular dynamics. J Comp Phys 1995, 117: 1–19. 10.1006/jcph.1995.1039View Article
- LAMMPS Molecular Dynamics Simulator[http://lammps.sandia.gov]
- Binks DJ, Grimes RW: Incorporation of monovalent Ions in ZnO and their influence on varistor degradation. J Am Ceram Soc 1993, 76: 2370–2372. 10.1111/j.1151-2916.1993.tb07779.xView Article
- Grimes RW, Binks DJ, Lidiard AB: The exent of zinc-oxide solution in zinc chromate spinel. Philos Mag A 1995, 72: 651–668. 10.1080/01418619508243791View Article
- Agrawal R, Peng B, Gdoutos EE, Espinosa HD: Elasticity size effects in ZnO nanowires - a combined experimental-computational approach. Nano Letters 2008, 8: 3668–3674. 10.1021/nl801724bView Article
- Dai L, Cheong WCD, Sow CH, Lim CT, Tan VBC: Molecular dynamics simulation of ZnO nanowires: size effects, defects, and super ductility. Langmuir 2010, 26: 1165–1171. 10.1021/la9022739View Article
- Kulkarni AJ, Zhou M, Sarasamak K, Limpijumnong S: Novel phase transformation in ZnO nanowires under tensile loading. Phys Rev Lett 2006, 97: 105502–105506.View Article
- Kulkarni AJ, Zhou M: Tunable thermal response of ZnO nanowires. Nanotechnology 2008, 18: 435706–435712.View Article
- Kulkarni AJ, Zhou M: Surface-effects-dominated thermal and mechanical responses of zinc oxide nanobelts. Acta Mech Mech Sinica 2006, 22: 217–224. 10.1007/s10409-006-0111-9View Article
- Kulkarni AJ, Zhou M: Size-dependent thermal conductivity of zinc oxide nanobelts. Appl Phys Lett 2006, 88: 141921–141924. 10.1063/1.2193794View Article
- Stillinger FH, Weber TA: Computer simulation of local order in condensed phases of silicon. Phys Rev B 1985, 31: 5262–5271. 10.1103/PhysRevB.31.5262View Article
- Volz SG, Chen G: Molecular dynamics simulation of thermal conductivity of silicon nanowires. Appl Phys Lett 1999, 75: 2056–2058. 10.1063/1.124914View Article
- Volz SG, Chen G: Molecular-dynamics simulation of thermal conductivity of silicon crystals. Phys Rev B 2000, 61: 2651–2656. 10.1103/PhysRevB.61.2651View Article
- Kresse G, Furthmüller J: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B 1996, 54: 11169–11186. 10.1103/PhysRevB.54.11169View Article
- Perdew JP, Burke K, Ernzerhof M: Generalized gradient approximation made simple. Phys Rev Lett 1996, 77: 3865–3868. 10.1103/PhysRevLett.77.3865View Article
- Madsen GKH, Singh DJ: BoltzTraP: a code for calculating band-structure dependent quantities. Comp Phys Comm 2006, 175: 67–71. 10.1016/j.cpc.2006.03.007View Article
- Ashcroft NW, Mermin ND: Solid State Physics. New York: Holt, Rinehart, and Winston; 1976.
- Baxter J, Schmuttenmaer CA: Conductivity of ZnO nanowires, nanoparticles, and thin films using time-resolved terahertz spectroscopy. J Phys Chem B 2006, 110: 25233–25239.
- Gaymann A, Geserich HP, Van Löhneysen H: Temperature dependence of the far-infrared reflectance spectra of Si: P near the metal-insulator transition. Phys Rev B 1995, 52: 16486–16493. 10.1103/PhysRevB.52.16486View Article
- Duke CB, Meyer RJ, Paton A, Mark P: Calculation of low-energy-electron-diffraction intensities from ZnO(1010). II. Influence of calculational procedure, model potential, and second-layer structural distortions. Phys Rev B 1978, 18: 4225–4240. 10.1103/PhysRevB.18.4225View Article
- Shen X, Pederson MR, Zheng J-C, Davenport JW, Muckerman JT, Allen PB[http://arxiv.org/pdf/cond-mat/0610002v1]
- Wang B, Zhao J, Jia J, Shi D, Wan J, Wang G: Structural, mechanical, and electronic properties of ultrathin ZnO nanowires. Appl Phys Lett 2008, 93: 021918–021921. 10.1063/1.2951617View Article
- Vo T, Williamson AJ, Galli G: First principles simulations of the structural and electronic properties of silicon nanowires. Phys Rev B 2006, 74: 045116–045128.View Article
- Serrano J, Manjón FJ, Romero AH, Ivanov A, Lauck R, Cardona M, Krisch M: The phonon dispersion of wurtzite-ZnO revisited. Physica Status Solidi (B) 2007, 244: 1478–1482. 10.1002/pssb.200675145View Article
- Landry ES, Hussein MI, McGaughey AJH: Complex superlattice unit cell designs for reduced thermal conductivity. Phys Rev B 2008, 77: 184302–184315.View Article
- Wolf MW, Martin JJ: Low temperature thermal conductivity of zinc oxide. Physica Status Solidi (A) 1973, 17: 215–220. 10.1002/pssa.2210170124View Article
- Özgür Ü, Gu X, Chevtchenko S, Spradlin J, Cho SJ, Morkoc H, Pollak FH, Everitt HO, Nemeth B, Nause JE: Thermal conductivity of bulk ZnO after different thermal treatments. J Electronic Materials 2006, 35: 550–555. 10.1007/s11664-006-0098-9View Article
- Boukai AI, Bunimovich Y, Tahir-Kheli J, Yu J-K, Goddard WA III, Heath JR: Silicon nanowires as efficient thermoelectric materials. Nature 2008, 451: 168–171. 10.1038/nature06458View Article
- Hochbaum AI, Chen R, Delgado RD, Liang W, Garnett EC, Najarian M, Majumdar A, Yang P: Enhanced thermoelectric performance of rough silicon nanowires. Nature 2008, 451: 163–167. 10.1038/nature06381View Article
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