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High Thermoelectric Performance of Ge/Si Core-Shell Nanowires: First-Principles Prediction Xin Chen, Yanchao Wang, and Yanming Ma* State Key Lab of Superhard Materials, Jilin UniVersity, Changchun 130012, People’s Republic of China ReceiVed: February 05, 2010; ReVised Manuscript ReceiVed: April 12, 2010
In an effort to enhance the thermoelectric performance in Si nanowires, we have explored the thermoelectric figure of merit (ZT) of the Ge/Si core-shell structured nanowires by using ab initio electronic structure calculations in the framework of Boltzmann transport theory. Our results show that the ZT value of the Ge/Si core-shell nanowire with p-type doping can reach 0.85 at 300 K, significantly larger than the observed ZT value of 0.36 in pure Si nanowires. The underlying mechanism for this enhanced ZT is mainly attributed to the reduced lattice thermal conductivity in the Ge/Si core-shell structure. Moreover, we suggest that appropriate Ge content in the Ge/Si core-shell nanowire may further optimize its ZT value. The current research proposed a way to design high-performance thermoelectric materials through a proper construct of heterostructured materials. I. Introduction Thermoelectric performance involves the energy conversion between heat and electricity in waste heat recovery and environmentally friendly refrigeration, and thus thermoelectric materials are expected to play an increasingly important role in energy savings to meet future energy challenges. The conversion efficiency of a thermoelectric material is characterized by its figure of merit ZT: ZT ) S2σT/κ, where S is the Seebeck coefficient, σ is the electrical conductivity, T is the mean operating temperature, and κ is the thermal conductivity. A ZT value exceeding 3 is required for commercially competitive energy conversion systems, while commercially available thermoelectric materials today have a typical room temperature ZT of only about 1. Since the three physical properties S, σ, and κ are coupled with each other, it remains a major difficulty to further enhance the ZT value in bulk thermoelectric materials. Since Hicks and Dresselhaus in the early 1990s proposed the possibility to enhance ZT with nanostructures,1,2 theoretical3,4 and experimental studies5-11 have extensively explored and confirmed the feasibility. Very recently, Si nanowires (SiNWs) have been demonstrated to possess a 100-fold improved thermoelectric performance over bulk Si near room temperature12,13 owing to the reduced lattice thermal conductivity in nanowires. However, the increased ZT value in SiNWs is still very low. For example, ref 12 reported a ZT value of about 0.36 at 300 K in 10-nm-wide SiNWs with a p-type doping of 2 × 1020 cm-3. Thus improvement of the thermoelectric performance in SiNWs is greatly demanded for real application. In an effort to enhance the thermoelectric performance of SiNWs, we have extensively investigated the thermoelectric properties of Ge/Si core-shell nanowires (NWs) by a combination of density functional theory (DFT) with the Boltzmann transport equation. This is in line with the recent progress in experimental synthesis of core-shell heterostructure NWs.14 We aimed to provide a microscopic description of Ge/Si core-shell NWs and to predict the optimal ZT values for guidance in future experiments. Our results showed that the ZT value for the p-type Ge/Si core-shell NWs * To whom correspondence should be addressed. E-mail: mym@ jlu.edu.cn.
reaches a high of 0.85 at room temperature, much larger than that of pure SiNWs. It is thus proposed that the Ge/Si core-shell NWs could stand as a promising candidate for nanoscale thermoelectric applications. The underlying mechanism for the high thermoelectric performance is presented. II. Computational Details We consider Ge/Si core-shell NWs oriented along the [001] direction as shown in Figure 1. A typical model can be created on the basis of diamond-structured nanocrystals with cylindrical shape and H-terminated surface atoms.15-17 We construct Ge/ Si core-shell NWs with a fixed total number of atoms N ) 77 (57 Si/Ge atoms and 20 H atoms) and five different settings of core (Ncore) and shell atoms in the unit cell, where Ncore ) 0, 9, 21, 33, and 57, is denoted as SiNW, Ge/Si_#1, Ge/Si_#2, Ge/ Si_#3, and GeNW, respectively. Since the total number of atoms is fixed, the diameter of the wire remains nearly invariant (about 1.6 nm) at different core contents. The geometry optimization and the band structure calculations have been performed within the projector augmented wave method18 and the generalized gradient approximation19 within the PW91 functional20 through the VASP package.21 The plane-wave kinetic energy cutoffs of 330 eV and the Monkhorst-Pack k-point meshes of 1 × 1 × 12 are chosen to ensure that all the structures are well-converged to be better than 0.1 meV. The transport coefficients including electrical conductivity σ and the Seebeck coefficient S were obtained by using the semiclassical Boltzmann theory and the rigid band approach.22 The phonon calculations were performed by the force constant method based on the harmonic approximation through the PHONON package.23,24 A convergence test gave the use of 1 × 1 × 2 supercell in the phonon calculation. III. Results and Discussion The calculations of S and σ22,25-27 involve the explicit treatment of electronic relaxation time, τe, which is a complex functional of temperature, atomic structure, electron energy, and carrier concentrations and is typically treated as a constant for simplicity and convenience.22,28 Here, the τe value (1.4 × 10-14 s) obtained for bulk Si by fitting the calculated σ/τe into the
10.1021/jp101132u 2010 American Chemical Society Published on Web 04/23/2010
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Figure 1. Cross section and side views of optimized hydrogenated SiNW, GeNW, and Ge/Si core-shell NWs oriented along the [001] direction. Side views of the figures are given in four unit cells for clarity, with each unit cell separated by the dashed lines. The yellow, green, and gray spheres represent Si, Ge, and H atoms, respectively.
measured electrical conductivity29 was applied into Si wires since the studied NWs have similar covalent bonding behaviors with that of bulk as suggested by our calculated charge densities.
In fact, the application of bulk τe into NWs has been successfully employed in ref 30. It is noteworthy that ref 30 reported the ratio of electronic relaxation time of SiNWs over bulk Si as
Figure 2. Calculated Seebeck coefficients (a, b), electrical conductivities (c, d), and power factor (e, f) for SiNW, Ge/Si_#1, Ge/Si_#2, Ge/Si_#3, and GeNW as a function of carrier concentration at 300 K. The black and red lines in the uppermost two panels correspond to the xx and zz components of transport tensors, respectively. The left and right panels represent p- and n-type materials, respectively.
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Figure 3. Electronic band structures of SiNW, Ge/Si_#1, Ge/Si_#2, Ge/Si_#3, and GeNW. The red arrows are to show the energy gap directions. Zero energy is set at the Fermi level. Red ellipses illustrate the highest valence bands near the zone-center.
Figure 4. Phonon dispersion relations of SiNW, Ge/Si_#1, Ge/Si_#2, Ge/Si_#3, and GeNW.
τwire/τbulk ≈ 1-4, indicating larger τe in NWs. Their results suggest that the use of bulk τe in the constant relaxation time model possibly underestimates the ZT values (proportional to τe) of NWs.30 Our calculated transport coefficients (S and σ) at 300 K for various wires are shown in Figure 2. Since the wires adopt the tetragonal symmetry, the transport tensors are diagonal and the xx and yy components are equal. It is found that the electrical conductivities (Figure 2c,d) in the z direction are greatly larger than those in the x and y directions. This can be justified by the true nature of wires. For the [001]-oriented NWs, 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. Since carriers in NWs move only along the wire direction, we mainly focused on the zz component of the power factor S2σ as depicted in Figure 2e,f. It is found that p-doped Ge/ Si_#3 and SiNW have the highest S2σ values, which are mainly attributed to their favorable electronic band structures (Figure 3). One observes from Figure 3 that valence band maximum (VBM) and conduction band minimum (CBM) have similar dispersive behaviors in the [001] direction, indicating nearly
invariant effective mass. Besides, we find that the highest valence bands are 2-fold degenerate near the zone-center (depicted by ellipses). Transport properties are closely related to the electronic states near VBM for p-doped materials, where a higher degree of degeneracy contributes more bands to the electronic transport properties. This fact accounts for the high S2σ values of p-doped SiNW and Ge/Si_#3. Thermal conductivity κ includes electronic (κe) and lattice (κl) components. κe can be derived from the Wiedemann-Franz relation: κe ) L0σT, where L0 ) (πκB/e)2/3 ) (156 µV/K)2 is Lorentz number. κl can be determined by the kinetic formula31 κl ) 1/3cV2τl, where c is the lattice specific heat per unit volume and is usually assumed to be constant, V is the average phonon velocity defined as V ) ∇qω, and τl is the phonon relaxation (scattering) time. The phonon group velocity for each mode can be obtained through numerical differentiation of phonon frequencies. Figure 4 shows the calculated phonon spectra for various wires with four acoustic branches, different from bulk. The lowest two acoustic modes are degenerated, denoted as modes 1 and 2, proportional to q2, while the other two acoustic modes (modes 3 and 4) with higher frequency and larger slopes
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Figure 6. The calculated ZT values as a function of carrier concentrations for p-doped (left panel) and n-doped (right panel) wires at 300 K.
Figure 5. The calculated phonon group velocities of acoustic modes as a function of phonon frequency in SiNW, GeNW, and Ge/Si core-shell NWs. Since the two lowest modes (modes 1 and 2) are two-degenerated branches, we therefore only present the results for modes 1 and 3 in part a, ignoring the velocities of mode 2 for clarity. The phonon group velocities for mode 4 in these wires are illustrated in part b. The horizontal short-dot lines in parts a and b correspond to the group velocities for bulk transverse (TA) and longitudinal acoustic (LA) branches, respectively. Part c represents the average group velocities for acoustic modes in wires. The bulk value (horizontal shortdot lines in part c) is shown for comparison.
are linear in q near the zone-center. It is found that the phonon frequency decreases and the acoustic branches shift to low frequency as Ge content increases. This is as expected since the atomic mass of Ge is larger than that of Si. We further calculated the bond lengths for these wires and found that the average bond lengths (including Si-Si, Si-Ge, and Ge-Ge bonds) increase with increasing Ge concentration. This fact is responsible for the decreased phonon frequency since a longer bond distance generally gives a smaller force constant and hence a lower vibration frequency.32 The lattice thermal conductivity κl is mainly dominated by the contribution from acoustic phonons.33,34 Thus only the phonon group velocities for the four acoustic branches (Figure 5) are calculated to derive κl. It is evident that the phonon velocities decrease in the sequence of SiNW f Ge/Si_#1 f Ge/Si_#2 f Ge/Si_#3 f GeNW. We further calculated the average group velocity for acoustic modes as a function of j (pω) ) [∑nVn(pω)Nn(pω)]/[∑nNn(pω)], where phonon energy:35 V Nn(pω) is the number of oscillators with frequency ω on the nth mode, given by the Plank distribution Nn ) 1/[exp(pωn/ j (pω) for wires is presented in Figure kBT) - 1]. The resulting V 5c, much lower than that in the bulk, and the average phonon velocities are found to decrease with increasing Ge content, which is in good accordance with the phonon results. For j (pω) over phonon simplicity and convenience, the average V frequency is used to calculate κl. We adopt the least-squares j (pω) data. The obtained method to find the best fit lines on our V best fit lines are nearly horizontal. As a good approximation, we exploit the mean function values of fitted lines to the average velocities (here, 4.66 × 105, 4.18 × 105, 3.44 × 105, 2.98 × 105, 2.48 × 105 cm/s for SiNW, Ge/Si_#1, Ge/Si_#2, Ge/Si_#3, GeNW, respectively). This approach is similar to the fittedasymptote method used in ref 36. Our result for average phonon
velocity of SiNW is in good agreement with previous results,36 supporting the validity of our method. Using the known κl of SiNW (about 1 W/mK)12,13 and our calculated average velocities, we derive κl for Ge/Si_#1, Ge/Si_#2, Ge/Si_#3, and GeNW to be 0.80, 0.54, 0.41, and 0.29 W/mK, respectively. Figure 6 shows the calculated ZT values along the c axis with different carrier concentrations for p- and n-doped wires at 300 K. Remarkably, it is found that p-doped Ge/Si_#3 NW possesses the highest ZT value of 0.85, which is significantly larger than our calculated optimal ZT value of 0.42 and the experimental data of 0.3612 for pure SiNW. It should be noted that the optimal ZT value in p-doped Ge/Si_#3 is only slightly higher than that of GeNW. However, it is necessary to point out that in heterostructures, Ge/Si core-shell NWs may have much lower τl than those of pure Si/GeNWs for the additional phonon scattering at core-shell interfaces. Donadio et al.37 have proposed that vibrational lifetimes in core-shell NWs are about 10-fold lower than those of conventional NWs. The use of constant τl approximation in this work neglects those contributions from interfaces and thus results in an overestimation of τl and κl for Ge/Si core-shell NWs. Therefore, a larger ZT value might be expected for Ge/Si core-shell NWs. Nevertheless, the current calculations have demonstrated that Ge/Si core-shell NWs possess largely enhanced thermoelectric performance over the pure SiNWs and are potential candidates for thermoelectric applications. IV. Conclusion In conclusion, we have employed first-principles calculations and Boltzmann transport theory to explore the thermoelectric performance of Ge/Si core-shell NWs. Our calculations have demonstrated that the Ge/Si core-shell NWs possess much larger ZT values than the pure SiNWs. We suggest that the excellent thermoelectric performance in Ge/Si core-shell NWs is mainly attributed to their remarkable depression of the lattice thermal conductivity. In addition, we provide an optimal carrier concentration of 2.64 × 1025 m-3 to achieve the maximum ZT value in Ge/Si_#3. Moreover, it is possible to further improve thermoelectric performance of Ge/Si core-shell NWs by choosing appropriate core and shell contents, fine optimization of wire sizes, doping, orientation, or tuning the surface structure. The principle of the largely enhanced thermoelectric performance of Ge/Si core-shell NWs can be extended to other kinds of core-shell NWs. We believe that our work will stimulate future research on other core-shell NWs, and proposes an effective way to pursue novel thermoelectric materials at the nanoscale.
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Acknowledgment. We are thankful for financial support from the China 973 Program under Grant No. 2005CB724400, the National Natural Science Foundation of China (NSFC) under grant No. 10874054, the NSFC awarded Research Fellowship for International Young Scientists under grant No. 10910263, the research fund for Excellent young scientist in Jilin University (No. 200905003), and the 2007 Cheung Kong Scholars Program of China. References and Notes (1) Hicks, L. D.; Dresselhaus, M. S. Effect of Quantum-Well Structures on the Thermoelectric Figure of Merit. Phys. ReV. B 1993, 47, 12727– 12731. (2) Hicks, L. D.; Dresselhaus, M. S. Thermoelectric Figure of Merit of a One-Dimensional Conductor. Phys. ReV. B 1993, 47, 16631–16634. (3) Balandin, A.; Wang, K. L. Significant Decrease of the Lattice Thermal Conductivity due to Phonon Confinement in a Free-Standing Semiconductor Quantum Well. Phys. ReV. B 1998, 58, 1544–1549. (4) Faleev, S. V.; Le´onard, F. Theory of Enhancement of Thermoelectric Properties of Materials with Nanoinclusions. Phys. ReV. B 2008, 77, 214304–214312. (5) Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’Quinn, B. ThinFilm Thermoelectric Devices with High Room-Temperature Figures of Merit. Nature 2001, 413, 597–602. (6) Harman, T. C.; Taylor, P. J.; Walsh, M. P.; LaForge, B. E. Quantum Dot Superlattice Thermoelectric Materials and Devices. Science 2002, 297, 2229–2232. (7) Hsu, K. F.; Loo, S.; Guo, F.; Chen, W.; Dyck, J. S.; Uher, C.; Hogan, T.; Polychroniadis, E. K.; Kanatzidis, M. G. Cubic AgPbmSbTe2+ m: Bulk Thermoelectric Materials with High Figure of Merit. Science 2004, 303, 818–821. (8) Li, S.; Toprak, M. S.; Soliman, H. M. A.; Zhou, J.; Muhammed, M.; Platzek, D.; Muller, E. Fabrication of Nanostructured Thermoelectric Bismuth Telluride Thick Films by Electrochemical Deposition. Chem. Mater. 2006, 18, 3627–3633. (9) Sootsman, J. R.; Pcionek, R. J.; Kong, H.; Uher, C.; Kanatzidis, M. G. Strong Reduction of Thermal Conductivity in Nanostructured PbTe Prepared by Matrix Encapsulation. Chem. Mater. 2006, 18, 4993–4995. (10) Wang, W.; Zhang, G.; Li, X. Manipulating Growth of Thermoelectric Bi2Te3/Sb Multilayered Nanowire Arrays. J. Phys. Chem. C 2008, 112, 15190–15194. (11) Guo, A. T.; Zhou, B.; Guo, W. Structural Characterization and Thermoelectric Transport Properties of Uniform Single-Crystalline Lead Telluride Nanowires. J. Phys. Chem. C 2008, 112, 11314–11318. (12) Boukai, A. I.; Bunimovich, Y.; Tahir-Kheli, J.; Yu, J. K.; Goddard Iii, W. A.; Heath, J. R. Silicon Nanowires as Efficient Thermoelectric Materials. Nature 2008, 451, 168–171. (13) Hochbaum, A. I.; Chen, R.; Delgado, R. D.; Liang, W.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. Enhanced Thermoelectric Performance of Rough Silicon Nanowires. Nature 2008, 451, 163–167. (14) Lauhon, L. J.; Gudiksen, M. S.; Wang, D.; Lieber, C. M. Epitaxial Core-Shell and Core-Multishell Nanowire Heterostructures. Nature 2002, 420, 57–61. (15) Musin, R. N.; Wang, X. Q. Quantum Size Effect in Core-Shell Structured Silicon-Germanium Nanowires. Phys. ReV. B 2006, 74, 165308– 165312.
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