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Origin of the High Thermoelectric Performance in Si Nanowires: A First-Principle Study Xin Chen, Yanchao Wang, Yanming Ma,* Tian Cui, and Guangtian Zou State Key Lab of Superhard Materials, Jilin UniVersity, Changchun 130012, P.R. China ReceiVed: April 2, 2009; ReVised Manuscript ReceiVed: June 4, 2009
First-principles calculations on the electronic, transport, and lattice vibrational behaviors of Si nanowires (SiNWs) are performed to reveal the origin of the high thermoelectric performance in SiNWs. Our results show that the power factor (S2σ) of SiNWs deviate only slightly from those of bulk Si originating from the similar bonding characters between SiNWs and bulk Si. In contrast, lattice heat transport of SiNWs has been largely revised by the spatial confinement, leading to a significant reduction of group velocity and phonon scattering time, thus reducing dramatically the lattice thermal conductivity. The extremely low lattice thermal conductivities together with the invariant power factor are responsible for the greatly enhanced figure of merit (ZT) of SiNWs, in excellent agreement with experiments. We suggest that our exploration on the physical principle of the high thermoelectric efficiency in SiNWs can be helpful for the deep understanding of SiNWs and sheds strong light on the search or even design of new low dimensional thermoelectric materials through phonon engineering. Introduction The suitability of a material for thermoelectric applications is characterized by its figure of merit ZT: ZT ) S2σT/κ, where S is the Seebeck coefficient, σ is the electronic conductivity, and κ is the thermal conductivity. It is believed that a ZT value exceeding 3 is necessary for a desired goal. In fact, so far ZT in bulk materials has remained stuck in the range of ∼0-1 at low temperature (e300 K). It is a big challenge to increase ZT in bulk thermoelectric materials, because the three properties (S, σ, and κ) are coupled with each other, and optimizing one parameter often adversely affects another. For example, the increase of S normally leads to a decrease in σ, and an increase in σ implies an enhanced κ, as given by the Wiedemann-Franz law. Only until recent years have both theoretical1-3 and experimental studies4-8 shown that large improvements in ZT could be achieved in nanostructured systems due to band structures and phonon engineering. Very recently, this enhancement was further confirmed by the 100-fold improvement in ZT values of Si nanowires (SiNWs) over bulk Si below or around room temperature.9,10 Moreover, it was proposed in refs 9 and 10 that the improved efficiency in SiNWs was attributed to a large reduction of κ as compared to bulk Si while a high power factor is maintained. The much lower thermal conductivity of SiNWs over bulk Si with an invariant of power factor is of fundamental interest and significant importance for thermoelectric applications. However, the exact physical mechanism remains unknown. We here have extensively investigated the electronic, transport, and lattice vibrational behaviors of SiNWs by highly precise first-principles calculations to probe the physical origin of the high thermoelectric performance. Computational Approach The calculations were performed using density functional theory (DFT) within projector augmented wave pseudopotentials11 and the generalized gradient approximation12 expressed by PW91 functional13 using the VASP program package.14 The * Author to whom any correspondence should be addressed. E-mail:
[email protected].
Figure 1. Cross section and side views of optimized SiNW(25) (left panels) and SiNW(57) (right panels). Side views of the figures are given in four unit cells for clarity, separated by the blue dashed lines. The yellow (light) and gray (dark) spheres represent Si and H, respectively. The Si atoms denoted as 1, 2, 3, and 4 are shown for the discussions in the phonon density of states. Atom 1 remains central. Atoms 2, 3, and 4 are surface atoms.
high kinetic energy cutoffs in the range of 320-350 eV are chosen by the convergence check. The Monkhorst-Pack k-point meshes of 1 × 1 × 10 to 1 × 1 × 14 are found to provide sufficient accuracy in the calculation of total energies and forces. For electronic transport calculations, we used the semiclassical Boltzmann theory15 and the rigid band approach.16,17 The rigid band approach to conductivity is based on the transport distribution, σRβ(ε) ) 1/N∑i,ke2τi,kVR(i,k) Vβ(i,k) δ(ε-εi,k)/dε, where VR(i,k) is a component of the group velocities. Once the transport distribution is known, all transport coefficients necessary to determine ZT can be directly obtained. A more detailed method and discussion on the calculation of transport coefficients is found in ref 18. This ab initio approach has been successful in rationalizing and predicting the optimal doping level of known compounds.19-24
10.1021/jp903061m CCC: $40.75 2009 American Chemical Society Published on Web 07/02/2009
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Figure 2. Electronic band structures of the bulk Si (left panel), SiNW(25) (middle panel), and SiNW(57) (right panel). The red arrows positioned on the top of the valence band and bottom of the conduction band are to make the energy gap clearer. Zero energy is set at the Fermi level.
The phonon calculations were performed by the force constant method based on the harmonic approximation through PHONON package.25,26 Convergence test gave the use of a 1 × 1 × 2 supercell in the phonon calculation. Results and Discussion We consider SiNWs oriented along the [001] direction as adopted in earlier first-principles studies.27,28 SiNWs are initially cut from the ideal bulk Si crystal in rodlike forms, and subsequently their atomic structures and lattice parameters are fully relaxed.27,29,30 The distance between neighboring nanowires is chosen to be greater than 10 Å in order to avoid cell to cell interaction. The bare SiNWs are metallic due to the existence of dangling bonds on the surface. This is in contrast to the experimental observation of insulating SiNWs. The dangling bonds are, therefore, passivated with H atoms, and the systems are then reoptimized. Figure 1 shows the optimized H-saturated SiNWs for the system sizes containing 25 and 57 Si atoms, denoted as SiNW(25) and SiNW(57), respectively, whose average diameters are 0.93 and 1.63 nm. It is found from the electronic band structures (Figure 2) that the SiNW(25) and SiNW(57) are semiconductors with band gaps of 1.1 and 1.6 eV, respectively, which are in good accordance with the results in ref 27. We also presented the electronic band structure of bulk Si in Figure 2 for comparison. It is clear that the DFT band gap of 0.6 eV for bulk is much smaller than those of nanowires, induced by quantum confinement.29-31 The calculated transport coefficients (S and σ) at 300 K for bulk Si and SiNWs are shown in Figure 3. Since the SiNWs adopt the tetragonal symmetry, the transport tensors are diagonal and the xx and yy components are equal. The electrical conductivities for both p- and n-doped materials are illustrated in parts c and d of Figure 3, respectively. It is shown that the electrical conductivities in the z direction are greatly larger than those in the x and y directions for SiNWs. This is confirmed by the true nature of nanowires. For the [001] SiNWs, carriers (electrons and holes) are confined to move only along the z direction and are bounded in the x and y directions with infinite potential barriers. Furthermore, it is found that neither S nor σ for both SiNWs and bulk Si shows any significant changes. This is in good agreement with the experimental results.9,10 To understand these similarities, we have presented the calculated charge densities for SiNWs and bulk Si on the left panels of Figure 4. It is evident that for both SiNWs and bulk Si, similar
significant charge accumulations along the Si-Si bonding directions are revealed to characterize the nearly identical covalent bonding in both materials. It is also noteworthy that we did not find any charge distribution between neighboring wires. This indicates negligible wire-wire interaction to validate our theoretical model. We have calculated the Si-Si pair distribution function of bulk Si and SiNWs up to the third nearest neighbors, as depicted in Figure 4d. One observes that the first peak positions of SiNWs are nearly identical to that of bulk Si (2.36 Å). This further implies that the bonding situation in SiNWs remains largely unchanged over bulk Si. Further inspection on the crystal structures of SiNWs reveals that nearly all Si atoms (including surface Si atoms with the consideration of H-passivation) maintain in a closed shell configuration with four nearest neighbors, similar to the bulk structure. We thus conclude that the preserved tetrahedral bonding character in SiNWs is responsible for the negligible changes of electronic transport properties compared with those in bulk Si. We now turn to the discussion of thermal conductivity, including electronic (κe) and lattice (κL) contribution. Since phonons are the primary carriers of thermal energy in semiconductors, a smaller value of κL would lead to a larger ZT. κL can be determined by the well-known kinetic formula32 κL ) 1/3cVl ) 1/3cV2τ, where c is the lattice specific heat per unit volume, V is the average phonon velocity defined as V ) ∇qω, l is the phonon mean free path, and τ is the phonon relaxation (scattering) time. If the lattice specific heat is assumed to be constant, then we only need to consider the size dependence of V and τ to derive κL. It is known that the lattice thermal conductivity is mainly dominated by the contribution of the acoustic phonons.33,34 We therefore focus on the low-frequency phonon spectra of SiNW(25) and SiNW(57) as depicted in parts a and b of Figure 5, respectively. The acoustic modes of bulk Si (red pentacle lines) are also presented for comparison. We note that our calculated zone-center optical phonon frequency (15.3 THz) of bulk Si is in excellent agreement with the experimental Raman data of 15.3 ( 0.3 THz.35 The phonon spectrum of SiNWs shows four acoustic branches characterized by limqf0Vi(q) ) 0, where Vi(q) is the frequency of mode i for the wave vector q, dramatically different from bulk. Two of these branches are linear in q and can therefore be identified as the longitudinal and transverse acoustic modes, while the two-degenerated branches proportional to q2 are characteristic for wires.36 It is
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Figure 3. Calculated p-type (a) and n-type (b) Seebeck coefficients and p-type (c) and n-type (d) electrical conductivities for bulk Si, SiNW(25), and SiNW(57) as a function of carrier concentration at 300 K.
Figure 4. The charge densities of bulk Si (a), SiNW(57) (b), and SiNW(25) (c). The Si-Si pair distribution up to the third nearest neighbors for bulk Si, SiNW(57), and SiNW(25) is shown in part d. The dotted lines in part d represent the Si-H pair distribution.
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Figure 5. The low-frequency phonon spectra of SiNW(25) (a) and SiNW(57) (b). The two acoustic modes with higher frequency near zone center and larger slopes are longitudinal (LA) and transverse (TA) acoustic modes for SiNWs, respectively, linear in q. The lowest acoustic modes are two-degenerated branches for wires, proportional to q2. The red pentacle lines in parts a and b represent acoustic modes of bulk Si, including longitudinal and transverse acoustic modes.
Figure 6. (a) The calculated PDOS for bulk Si and central Si atoms of SiNWs. The PDOS for surface Si atoms of SiNW(57) and SiNW(25) are shown in parts b and c, respectively. The atoms denoted as 1, 2, 3, and 4 are illustrated in Figure 1. For the surface Si atoms of SiNWs, peaks exist also at about 64 THz (not shown), due to the Si-H bond stretching vibrations.
evident from Figure 5 that the slopes of linear acoustical modes for nanomaterials are lower than those of bulk Si, especially for SiNW(25) with a smaller diameter (Figure 5a). According to the formula of V, the smaller slope of phonon spectrum represents a lower phonon velocity. Therefore, the resulting acoustic phonon velocity V of nanowires is evidently reduced. Besides, it is found (Figure 5) that the low optic modes are clearly mixed with the acoustic phonons in SiNWs, indicating a strong coupling of acoustic and optical phonons and that energy transfer between these modes is easy. This is in apparent contrast to the phonons in bulk Si.35 Phonon relaxation time τ is an indication of phonon modes scattering or a measure of temperature-dependent anharmonic effects and is commonly given by Matthiessen’s rule, expressing the total inverse lifetime as the sum of the inverse lifetimes corresponding to each scattering mechanism. In SiNWs, the acoustic modes carrying the heat flow will be strongly scattered by these low-frequency optic modes, leading to the decrease of τ. The reduced V and τ in SiNWs are jointly responsible for its extremely low lattice thermal conductivity κL. Generally, a longer Si-Si distance implies a smaller force constant and hence a lower vibration frequency.37 Therefore,
the Si-Si pair distribution (Figure 4) could help us to understand the predicted low frequencies in SiNWs. One observes that there are longer Si-Si distances close to the major peaks of SiNWs, which are not present in bulk Si. The particular structure of SiNWs under a spatially confined constraint attributes to these different Si-Si bond interactions. This is further confirmed by the calculations of phonon density of states (PDOS) for surface and central Si atoms (Figure 6). It is found that while for the central atoms, the PDOS are quite similar to that of the bulk (Figure 6a), PDOS of the surface Si atoms in SiNWs are dramatically different from that of bulk Si atoms. Thus, the spatial confinement is the key to revising the PDOS of nanowires and leads to the large phonon spectrum modification. Conclusion In conclusion, we have explored the origin of the high thermoelectric capability in SiNWs by a highly precise firstprinciple method. Our calculations show that the power factors of SiNWs are not significantly affected by the low dimensional structures, i.e., possessing the similar transport coefficients (S and σ) to the bulk values. This originates from the similar
High Thermoelectric Performance in Si Nanowires bonding behavior in SiNWs over bulk Si. However, spatial confinement has an important effect to revise the lattice heat transport. The reduced size and dimension of the SiNWs increases the influences of the surfaces, leading to a significant modification of the vibrational properties (dispersion curves and phonon group velocities). Meanwhile, optic modes of nanomaterials are low in frequency, which strongly scatter the heatcarrying acoustic modes to reduce phonon scattering time. The reduced V and τ in SiNWs jointly result in a decrease of the lattice thermal conductivity. Thus, the major contribution to the 100-fold ZT enhancement of SiNWs comes from the lattice thermal conductivity reduction. Our calculations have verified the exciting prospect that SiNWs can be applied as novel nanoscale thermoelectric materials. Since very recently a theoretical work has found that the room temperature thermal transport of SiNWs is highly anisotropic with a strong diameter and wire directions dependence,38 it is possible that further improvements on thermoelectric performance can be made through fine optimization of nanowire sizes, doping, and orientation. We believe that the discovery of the physical origin of the low thermal conductivity in SiNWs is greatly helpful for searching for or even designing the new low dimensional thermoelectric materials through phonon engineering. Acknowledgment. We are thankful for financial support from the NSAF of China under Grant No. 10676011, the National Natural Science Foundation of China under Grant No. 10874054, the China 973 Program under Grant No. 2005CB724400, the Program for 2005 New Century Excellent Talents in University, and the 2007 Cheung Kong Scholars Program of China. References and Notes (1) Balandin, A.; Wang, K. L. Phys. ReV. B 1998, 58, 1544. (2) Hicks, L. D.; Dresselhaus, M. S. Phys. ReV. B 1993, 47, 12727. (3) Faleev, S. V.; Le´onard, F. Phys. ReV. B 2008, 77, 214304. (4) Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’Quinn, B. Nature 2001, 413, 597. (5) Harman, T. C.; Taylor, P. J.; Walsh, M. P.; LaForge, B. E. Science 2002, 297, 2229. (6) Hsu, K. F.; Loo, S.; Guo, F.; Chen, W.; Dyck, J. S.; Uher, C.; Hogan, T.; Polychroniadis, E. K.; Kanatzidis, M. G. Science 2004, 303, 818.
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