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 parameterfree 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 8Ådiameter ZnO nanowires and 20 times in 12Ådiameter 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 ptype (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 parameterfree calculations. Previous theoretical studies have provided proofofprinciple 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 GreenKubo 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 longrange 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 [1315]. In case of Si NW, we used StillingerWeber [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 PerdewBurkeErnzerhof exchangecorrelation functional [20]. The kpoint 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 semiclassical 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 timeresolved terahertz spectroscopy measurements of electron conductivity yielded the relaxation time of 28 and 35 fs for 500nm 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 concentrationdependent τ fitted into the experimental data [24].
Results and discussion
When the ZnO NW is grown in [0001] direction, the surface oxygen atoms move outward away from the nanowire center while the zinc atoms moving inward toward the center (Figure 1 (ad)). This "buckling" of the surface atoms has been reported in elastic lowenergy diffraction experiments, where ZnO bonding on the surface of ZnO (1010) is tilted by an angle of 12° from the [0001] axis [25]. In our calculations, the tilting angle is 9.6°, in agreement with previous DFT results [26]. We also find that the bond length of the OZn surface dimer is reduced from 2.07 to 1.9 Å (similar to Ref. [27]). This surface reconstruction removes the surface states from the nanowire band gap. The computed electronic band gaps are 1.65, 1.47, and 1.13 eV for 8, 10, and 17 Å NWs, showing gap opening due to the quantum confinement. In comparison, the bulk ZnO band gap in our calculations is 0.73 eV.
Figure 1. Relaxed atomic structure and roomtemperature transport coefficients and ZT of ZnO NWs. (a), (b), and (c) shows the [0001] cross section of 8, 10, and 17Ådiameter NWs, respectively. Grey spheres denote Zn atoms and red spheres indicate O atoms. (d) shows the side view of 8Ådiameter NW. In the transport coefficients plots, the red, blue, and green lines denote the results for NW of diameter 8, 10, and 17 Å, respectively. The bulk values (black dashed lines) are shown for comparison. (Color online).
The electronic structure and thermoelectric properties of Si [001] NW were computed for comparison. The diameter of the NW was 12 Å with singlebonded 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 56.7meV phonon mode, which agrees well with the energy (approximately 53 meV) of the optical branch at the Γpoint 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 StillingerWeber 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 threephonon 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.
To distinguish the electronic and lattice contribution to the improvement of ZT separately, we computed the ZT of NWs using the lattice thermal conductivity of bulk ZnO, κ_{l}^{bulk}, and that of NWs, κ_{l}^{MD}. The top panel of Figure 2 shows the ZT of ZnO NW using κ_{l}^{MD }of each NW and the bottom panel shows the ZT using κ_{l}^{bulk}. While a large enhancement of maximum ZT in NW has contributions from both electronic and lattice part (top panel), the increase in ZT at the bottom panel stems solely from the changes in the electronic structure. In Table 1 we show the ratio of the maximum figureofmerit of nanowire, ZT*, to the bulk counterpart, ZT*_{bulk}: ζ = ZT*/ZT*_{bulk}. The enhancement of ZT due to the changes in electronic structure can be parameterized by ζ_{e }= ZT*(κ_{l}^{bulk})/ZT*_{bulk}, where ZT*(κ_{l}^{bulk}) is the ZT of NWs calculated using the bulk lattice thermal conductivity. It appears that the quantum confinement has only modest effects, i.e., ζ_{e }= 3.25, 2.15, and 1.15 for 8, 10, and 17 Å nanowires, respectively. The remaining contribution of ζ comes from the lattice part, ζ_{l }= ζ/ζ_{e}. We see a much larger effect of lattice contribution, i.e., ζ_{l }= 9.69, 8.79, and 7.10 for the same set of NWs. We also observe that both contributions of the electronic and the lattice part to the enhancement of ZT decrease as the diameter of the NW increases. In particular, the electronic contribution to the ZT enhancement becomes negligible when the NW diameter is as large as 17 Å.
Figure 2. ZT of ZnO NWs of 8 Å (red), 10 Å (blue), and 17 Å (green) in diameter. The bulk values (black dashed lines) are shown for comparison. Top panel: The lattice contribution of thermal conductivity obtained from the molecular dynamics calculations is used. Bottom panel: The lattice contribution of thermal conductivity of bulk ZnO (85 W/Km) is used for the calculation of the NW ZT. (Color online).
Table 1. Enhancement of ZT in ZnO and Si NW
This enhancement of ZT due to the reduction of the lattice thermal conductivity is even more pronounced in Si NWs (Figure 3). As presented in Table 1 the changes in the electronic structure on the nanoscale lead to the reduction of ZT by nearly a factor of 4, with ζ_{e }= 0.264, in contrast to those of ZnO. This reduction stems from the decreased electric conductivity in comparison with the bulk. The thermopower, on the other hand, remains relatively unchanged. As a result, the power factor in Si nanowires is about 4.6 times smaller than that in the bulk. Nevertheless, the reduction in the lattice thermal conductivity is by almost a factor of 90 leads to the more than 70fold lattice contribution to the enhancement of ZT. (Note that the lattice thermal conductivity of Si nanowires is highly anisotropic and very sensitive to the geometry of the surface.)
Figure 3. ZT of 12Ådiameter Si NW's (red) and bulk Si (black). The dashed (red) line is the ZT of the same Si NW calculated with bulk lattice thermal conductivity. The inset shows the crosssection atomic configuration of the NW. The NW axis direction is [001]. The lattice contributions of thermal conductivity of bulk Si and the NW, 254 and 2.8 W/Km, respectively, were derived from MD calculations. (Color online).
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).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
DOD carried out molecular dynamics calculations and some of the density functional theory (DFT) calculations. PDH and BL performed DFT calculations of the electronic part of thermal conductivity. All authors read and approved the final manuscript.
Acknowledgements
This work used computational facilities of the VCU Center for High Performance Computing and Texas State University High Performance Computing Center.
References

Hicks L, Dresselhaus M: Effect of quantumwell structures on the thermoelectric figure of merit.
Phys Rev B 1993, 47:1272712731. Publisher Full Text

Venkatasubramanian R, Siivola E, Colpitts V, O'Quinn B: Thinfilm thermoelectric devices with high roomtemperature figures of merit.
Nature 2001, 413:597602. PubMed Abstract  Publisher Full Text

Ettenberg MH, Jesser WA, Rosi FD: A new ntype and improved ptype pseudoternary (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; 1996:5256.

Vo T, Williamson AJ, Lordi V, Galli G: Atomistic design of thermoelectric properties of silicon nanowires.
Nano Letters 2008, 8:11111114. PubMed Abstract  Publisher Full Text

Lee JH, Galli GA, Grossman JC: Nanoporous Si as an efficient thermoelectric material.
Nano Letters 2008, 8:37503754. PubMed Abstract  Publisher Full Text

Plimpton SJ: Fast parallel algorithms for shortrange molecular dynamics.
J Comp Phys 1995, 117:119. Publisher Full Text

LAMMPS Molecular Dynamics Simulator [http://lammps.sandia.gov] webcite

Binks DJ, Grimes RW: Incorporation of monovalent Ions in ZnO and their influence on varistor degradation.
J Am Ceram Soc 1993, 76:23702372. Publisher Full Text

Grimes RW, Binks DJ, Lidiard AB: The exent of zincoxide solution in zinc chromate spinel.
Philos Mag A 1995, 72:651668. Publisher Full Text

Agrawal R, Peng B, Gdoutos EE, Espinosa HD: Elasticity size effects in ZnO nanowires  a combined experimentalcomputational approach.
Nano Letters 2008, 8:36683674. PubMed Abstract  Publisher Full Text

Dai L, Cheong WCD, Sow CH, Lim CT, Tan VBC: Molecular dynamics simulation of ZnO nanowires: size effects, defects, and super ductility.
Langmuir 2010, 26:11651171. PubMed Abstract  Publisher Full Text

Kulkarni AJ, Zhou M, Sarasamak K, Limpijumnong S: Novel phase transformation in ZnO nanowires under tensile loading.
Phys Rev Lett 2006, 97:105502105506. PubMed Abstract  Publisher Full Text

Kulkarni AJ, Zhou M: Tunable thermal response of ZnO nanowires.

Kulkarni AJ, Zhou M: Surfaceeffectsdominated thermal and mechanical responses of zinc oxide nanobelts.
Acta Mech Mech Sinica 2006, 22:217224. Publisher Full Text

Kulkarni AJ, Zhou M: Sizedependent thermal conductivity of zinc oxide nanobelts.
Appl Phys Lett 2006, 88:141921141924. Publisher Full Text

Stillinger FH, Weber TA: Computer simulation of local order in condensed phases of silicon.
Phys Rev B 1985, 31:52625271. Publisher Full Text

Volz SG, Chen G: Molecular dynamics simulation of thermal conductivity of silicon nanowires.
Appl Phys Lett 1999, 75:20562058. Publisher Full Text

Volz SG, Chen G: Moleculardynamics simulation of thermal conductivity of silicon crystals.
Phys Rev B 2000, 61:26512656. Publisher Full Text

Kresse G, Furthmüller J: Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set.
Phys Rev B 1996, 54:1116911186. Publisher Full Text

Perdew JP, Burke K, Ernzerhof M: Generalized gradient approximation made simple.
Phys Rev Lett 1996, 77:38653868. PubMed Abstract  Publisher Full Text

Madsen GKH, Singh DJ: BoltzTraP: a code for calculating bandstructure dependent quantities.
Comp Phys Comm 2006, 175:6771. Publisher Full Text

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 timeresolved terahertz spectroscopy.

Gaymann A, Geserich HP, Van Löhneysen H: Temperature dependence of the farinfrared reflectance spectra of Si: P near the metalinsulator transition.
Phys Rev B 1995, 52:1648616493. Publisher Full Text

Duke CB, Meyer RJ, Paton A, Mark P: Calculation of lowenergyelectrondiffraction intensities from ZnO(1010). II. Influence of calculational procedure, model potential, and secondlayer structural distortions.
Phys Rev B 1978, 18:42254240. Publisher Full Text

Shen X, Pederson MR, Zheng JC, Davenport JW, Muckerman JT, Allen PB[http://arxiv.org/pdf/condmat/0610002v1] webcite

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, 93:021918021921. Publisher Full Text

Vo T, Williamson AJ, Galli G: First principles simulations of the structural and electronic properties of silicon nanowires.

Serrano J, Manjón FJ, Romero AH, Ivanov A, Lauck R, Cardona M, Krisch M: The phonon dispersion of wurtziteZnO revisited.
Physica Status Solidi (B) 2007, 244:14781482. Publisher Full Text

Landry ES, Hussein MI, McGaughey AJH: Complex superlattice unit cell designs for reduced thermal conductivity.

Wolf MW, Martin JJ: Low temperature thermal conductivity of zinc oxide.
Physica Status Solidi (A) 1973, 17:215220. Publisher Full Text

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, 35:550555. Publisher Full Text

Boukai AI, Bunimovich Y, TahirKheli J, Yu JK, Goddard WA III, Heath JR: Silicon nanowires as efficient thermoelectric materials.
Nature 2008, 451:168171. PubMed Abstract  Publisher Full Text

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, 451:163167. PubMed Abstract  Publisher Full Text