Surface Disordered Ge–Si Core–Shell Nanowires as Efficient

Aug 13, 2012 - Independent Research/Technology and Production Sciences through Grant No. 11-104592 and the Sapere Aude program. □ REFERENCES...
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Surface Disordered Ge−Si Core−Shell Nanowires as Efficient Thermoelectric Materials Troels Markussen* Center for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark S Supporting Information *

ABSTRACT: Ge−Si core−shell nanowires with surface disorder are shown to be very promising candidates for thermoelectric applications. In atomistic calculations we find that surface roughness decreases the phonon thermal conductance significantly. On the contrary, the hole states are confined to the Ge core and are thereby shielded from the surface disorder, resulting in large electronic conductance values even in the presence of surface disorder. This decoupling of the electronic and phonon transport is very favorable for thermoelectric purposes, giving rise to promising room temperature figure of merits ZT > 2. It is also found that the Ge−Si core−shell wires perform better than pure Si nanowires. KEYWORDS: GeSi core−shell nanowires, transport, thermoelectrics

T

degree of control of both diameter, length, and doping profile.12,13 In Ge-core Si-shell NWs the band offset naturally leads to hole accumulation in the core region without the need for impurity doping. Ballistic hole transport and very long mean free path were reported,14 indicating a smooth interface between the Ge core and Si shell that does not scatter the holes, which are confined to the core region. Moreover Ge−Si core shell wires have been demonstrated to have potential applications within high-performance electronics15,16 and quantum computing.17 A number of theoretical works have studied the electronic properties of Ge−Si NWs with density functional theory (DFT) methods focusing on quantum confinement effects on the band structure,18−23 strain effects,21,24−27 phonon band structures,28,29 multishell structures,30 and doping properties.31 In qualitative agreement with experiments14,32 several works have shown that the highest valence band states (the hole states) are spatially localized to the Ge core region.21,27,31 In addition to this, the phonon thermal conductance has been studied with continuum methods33 and in a number recent works with classical molecular dynamics (MD) approaches.34−36 These studies have found a large reduction of the thermal conductance of Ge−Si core−shell wires, in agreement with recent experimental findings.37 Initial studies of thermoelectric properties of Ge−Si NWs were presented by Chen et al.28 based on Boltzmann theory but not taking the spatial confinement explicitly into account. A large number of theoretical works have studied the electronic and thermal properties of pure SiNWs.38

hermoelectric materials can convert thermal energy to electrical energy or vise versa and are applied in solid-state generators or refrigerators. The efficiency of a thermoelectric material is characterized by the dimensionless figure of merit ZT, given by1

ZT =

S2GT κ ph + κe

(1)

where S is the Seebeck coefficient, G the electronic conductance, T the temperature, and κph and κe are the phonon- and electron contributions to the thermal conductance, respectively. Materials with ZT ∼ 1 are regarded as good thermoelectrics, while ZT > 3 is required to compete with conventional refrigerators or generators.2 An ideal thermoelectric material is a so-called phonon-glass electron-crystal (PGEC) where the phonons are scattered by a disordered glasslike (amorphous) structure, giving a low κph, while the electrons see a perfect crystal with no scattering resulting in a high power factor, S2G.3,4 In this Letter we propose that Ge−Si core−shell nanowires could act as a PGEC material with a high thermoelectric efficiency. Semiconductor nanowires are promising candidates as efficient thermoelectric materials.5 Initial interest in nanowires was motivated by the large density of states in confined onedimensional systems leading to an increased power factor.6 Another advantage of nanowires is a potentially very low thermal conductance due to surface scattering by the phonons.7,8 Indeed, recent experiments have shown that silicon nanowires (SiNWs) can be efficient thermoelectric materials although bulk silicon is not.9,10 The nanowires conduct charge well, but the thermal conductance is dramatically reduced due to phonon scattering at the rough surfaces. Core−shell nanowires of Ge and Si can be synthesized with the vapor−liquid−solid (VLS) growth method11 with a high © 2012 American Chemical Society

Received: May 31, 2012 Revised: August 8, 2012 Published: August 13, 2012 4698

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varying radius R(θ,z), which we generate following refs 41 and 42 as detailed described in the Supporting Information. The average radius fluctuation is 1 Å, and the length of the disordered region is 8 nm and limited by computational costs (the disordered wires contain up to 3000 atoms). The interface between the Ge core and Si shell is assumed to be disorder-free. While these structures certainly represent idealized structures which will be demanding to produce experimentally, the high degree of control in the VLS growth does make it possible to make nanometer sized NWs43 and core−shell heterostructures with epitaxial and sharp boundaries12,13 and atomically smooth edges.44 The outer surface may be roughness disordered from the VLS growth45 (under certain growth conditions) or from a subsequent etching process.10 All of the nanowire structures are initially relaxed using interatomic potentials parametrized by the Tersoff empirical potential (TEP) model46,47 as implemented in the “general utility lattice program” (GULP).48 The atomic structure is relaxed using periodic boundary conditions. For this reason each surface disordered wire piece is terminated in both ends with two disorder-free unit cells indicated with dashed boxes in Figure 1. For use in phonon transmission calculations we also calculate the force constant matrix, K, using GULP. Our description of the phonon thermal conductance is limited to the harmonic approximation thus neglecting phonon−phonon scattering. The harmonic approximation is always valid at low temperatures. Experimental studies of silicon films49 showed a room temperature mean free path of ∼300 nm for the dominant phonons. For relatively short wires with lengths L ≳ 100 nm we thus estimate the harmonic approximation to be fairly accurate, although MD simulations have shown that anharmonic effects at the Ge−Si interface significantly reduce the thermal conductivity.34−36 Including such anharmonic effects would presumably lead to lower thermal conductance values than those reported here. The electronic system is describe by a 10 orbital sp3d5s* nearest neighbor tight-binding model recently developed for Si, Ge, and their alloys50 with an accurate description of strain effects. All relevant electronic properties are derived from the Hamiltonian matrix H. Ideally, we would like to describe the electronic system with first-principles DFT methods. However, modeling a disordered wire with ∼3000 atoms is out of the range with DFT methods. For the pristine wires we have compared the TB calculations with results from DFT (see Supporting Information) showing a quantitative agreement between band structures as well as cross sectional shape of the Bloch states. This comparison supports the quality of the TB parametrization as well as the implementation of it. The electronic conductance is calculated from the electronic transmission function Te(ε). This we obtain from H using a recursive Green’s function method,51 which scales linearly with the wire length and thus allows for an efficient calculation of the transmission function for even long wires. The scattering region, shown in Figure 1e is coupled to two semi-infinite perfect wires (leads), which are included through selfenergies.51 The scattering region shown in Figure 1e includes two unit cells of the disorder-free leads. To simulate ensemble averaged properties of even longer wires we use an incoherent addition method. In previous works39,52 we have shown that the sample averaged electron as well as phonon resistances for long wires with length L can be calculated as

From previous studies it appears that Ge−Si core−shell NWs may be an even better candidate for thermoelectric purposes than pure SiNWs: Assuming that NWs can be synthesized with a smooth core−shell interface that does not scatter the confined holes, good electronic properties can be achieved.14 Since the holes are confined to the core region, they will presumably not be affected by surface roughness disorder (SRD) at the outer Si surface. However, the phonon thermal conductance will still be strongly reduced by the SRD, thus rendering the Ge−Si NWs as phonon-glass-electron-crystal materials and ideal for thermoelectric applications. In agreement with this hypothesis, it is shown in this Letter by means of numerical simulations that in Ge−Si core−shell NWs with SRD the phonon thermal conductance is strongly suppressed, while the hole transport is much less affected, leading to very promising thermoelectric figure of merits ZT > 2. While similar concepts of decoupling the electron and phonon transport properties have been suggested theoretically for SiNWs,39,40 we here find that Ge−Si core−shell wires have significantly larger ZT than pure SiNWs. In Figure 1 we show cross section views of the considered pristine NWs (a−d). We consider Ge−Si core−shell NWs (a,c)

Figure 1. Wires oriented along the ⟨110⟩ direction (a,b) and ⟨111⟩ direction (c,d). For both directions we compare Ge−Si core−shell wires (a,c) with pure SiNWs (b,d). In all cases, the outer diameter of the pristine wires are 2.1 nm, and the diameter of the Ge core is 1.1 nm. Silicon atoms are shown as yellow balls, germanium as green, and hydrogen as white. Panel e illustrates a full scattering region (cut through the center of the wire). A disordered region is centered between two disorder free unit cells on each side, marked with dashed boxes. The whole structure is repeated periodically in the wire direction for the relaxation and calculation of the force constant matrix.

and pure SiNWs (b,d) oriented along the ⟨110⟩ and ⟨111⟩ directions. The outer diameter is ∼2.1 nm, while the diameter of the Ge core region is ∼1.1 nm. The outer surface is passivated with hydrogen atoms. Panel e shows a side view of a wire with SRD. The SRD profile corresponds to a randomly

R(L , E) = R c(E) + ⟨R s(E)⟩L /Ls 4699

(2)

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where Re(E) = 1/; 0(E) is the energy dependent contact resistance, that is, the resistance of the infinite, disorder-free 2 wire measured in units of G−1 0 = h/(2e ) = 12.9 kΩ. The sample-averaged scattering resistance, ⟨Rs(E)⟩ = 1/⟨; (E)⟩ −1/ ; 0(E) is calculated from the average of the individual transmission functions ⟨; (E)⟩ of typically 5−10 relatively short (length Ls = 8 nm) wires with different realizations of the random SRD. The transmission of a long wire of length L is then simply ; (L,E) = 1/R(L,E). In the case of Ge−Si ⟨110⟩ wires we have explicitly tested that the computationally easy incoherent addition, eq 2, agrees with much more timeconsuming fully coherent, sample averaged results (see the Supporting Information). From the scattering- and contact resistance we may also calculate the energy-dependent elastic mean free path (MFP) as le(E) = LsRc(E)/⟨Rs(E)⟩.52 Defining the function Lm(μ): Lm(μ) =

2 h

⎛ ∂f (E , μ) ⎞ ⎟ ∂E ⎠

∞

∫−∞ dE ;(E)(E − μ)m ⎜⎝−

and second valence bands is observed for the Ge−Si wire. Not only the band structure of the two wires differ but also the corresponding Bloch states. The insets in Figure 2 show the cross sectional distribution of the electron density of the Bloch state in the highest lying valence bands.55 Importantly, the valence band state in the Ge−Si wire is confined to the Ge core region (indicated with red ellipse) with 70% of the density on the Ge atoms. In comparison, the SiNW valence band state is more delocalized with only 40% of its density in the core region (dashed ellipse). In the Supporting Information we show that hole confinement in the Ge core is even more pronounced in larger diameter wires. The confinement of the Ge−Si valence band to the core region is in agreement with previous experimental and theoretical studies.14,21,25,31,32 From the difference in cross sectional shape of the Bloch states we anticipate that valence band transport in the Ge−Si NWs is less sensitive to surface disorder than in the SiNWs, simply because the holes in the Ge−Si wires do not “feel” the surface as much as the holes in SiNWs. Figure 3 fully confirms this expectation. Here we show the electron transmission function for SRD Ge−Si NWs (a and b) and pure SiNWs (c and d), oriented along the ⟨110⟩ and ⟨111⟩ directions. The different solid curves are obtained for different realizations of the random SRD. The dashed curves show the pristine wire transmission functions, ; 0(E), in the absence of disorder. For the Ge−Si NWs the transmissions quickly approach unity for energies in the valence band showing that there is only a small scattering probability. On the contrary, in the SiNWs the holes are strongly scattered, and for the ⟨110⟩ SiNW the sample-tosample variations are significant. Although the holes in the Ge−Si NWs are clearly less scattered by the SRD than in the SiNWs, this difference is not clearly visible in the room temperature conductance for such short wires. In Table 1 we compare the sample averaged conductance for the different wires. Since we do not explicitly model any dopant atoms, we do not know the chemical potential. In all results, we will thus assume that the chemical potential μ = Evb, where Evb is the valence band edge, that is, the highest energy in the valence band. The leftmost values in Table 1 correspond to wires with length L = 8 nm of the disordered region, while the rightmost values (*) correspond to wires with length L = 160 nm. The former values are obtained from the transmission functions shown in Figure 3, and the latter values are calculated using eq 2. Notice that conductance values for the short wires (L = 8 nm) are relatively similar with no clear differences between Ge−Si and pure Si wires. However, for the longer wires (numbers in parentheses), there are very clear differences with the Ge−Si NWs having almost an order of magnitude larger conductances. Similar differences between Ge−Si and Si-NWs are seen in the calculated MFPs, which represent an average over energies close to the valence band edge. The Ge−Si NWs have ∼30 times longer MFP than than the Si-NWs, which explains the large difference in the long-wire conductance values. From Table 1 we see that κe follow the same trends as for G with the Ge−Si wires having rather high values even for the long wires, while for the SiNW κe is reduced by a factor of 10 when going from short to long wires. The thermopower values, ⟨S⟩, increase by ∼25% in the long Ge−Si wires, whereas it remains more constant for the SiNWs. We now turn to the phonon contribution to the thermal conductance. Figure 4a show the phonon transmission function through a pristine Ge−Si ⟨110⟩ NW and through five different

(3)

where f(E,μ) is the Fermi−Dirac distribution function at the chemical potential μ, the electronic properties entering eq 1 are given by Ge = e2L0(μ), S = L1(μ)/[eTL0(μ)], and κe = {L2(μ) − [L1(μ)]2/L0(μ)}/T, where T is the temperature and e is the electron charge. The phonon transmission function ; ph(ω) at frequency ω is obtained in a similar way as the electronic transmission with the substitutions H→K and εI→ω2M, where I is the identity matrix and M is a matrix with the atomic masses in the diagonal and zeros elsewhere. From the phonon transmission function, the phonon thermal conductance is calculated as:53,54 κ ph(T ) =

ℏ2 2πkBT 2

∫0

∞

dωω 2 ;ph(ω)

eℏω / kBT (eℏω / kBT − 1)2

(4)

We begin our numerical analysis by comparing the electronic band structure of disorder free Ge−Si NWs and pure SiNWs, which are shown in Figure 2 for ⟨110⟩ oriented wires. While the conduction band structures are quite similar for the two wires, the valence bands differ significantly from each other. The valence bands of the Ge−Si wire are higher in energy than for the SiNW, and a larger energy separation between the first

Figure 2. Electronic band structures of ⟨110⟩ Ge−Si core−shell (left) and Si-NW (right). The insets show the spatial profile of the electron density of the Bloch state in the highest lying valence bands (hole state) calculated at the Γ-point (k = 0). In the case of Ge−Si NW, the hole state is localized within the Ge core region, while it is more delocalized in the SiNW. 4700

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Figure 3. Electronic transmission functions through wires with a L = 8 nm long region of SRD. Different solid lines correspond to different realizations of the random SRD. The transmission of the pristine wires are shown by dashed lines. In the Ge−Si core−shell wires (a,b) there is a high transmission probability and weak scattering. In the pure SiNWs (c,d) the holes are significantly more scattered. All energies are shown relative to the valence band edges, Evb, of the pristine wires. The wire orientations are marked in the figures.

Table 1. Sample Averaged Electronic and Phonon Propertiesa ⟨G⟩ (G0) Ge−Si ⟨110⟩ Ge−Si ⟨111⟩ Si ⟨110⟩ Si ⟨111⟩

0.44 0.35 0.47 0.19

(0.21) (0.20) (0.03) (0.06)

⟨κe⟩ (nW/K) 0.11 0.08 0.13 0.06

(0.06) (0.05) (0.008) (0.006)

⟨S⟩ (μV/K) 127 156 170 232

⟨MFP⟩ (nm)

κ0ph (nW/K)

241 284 8.4 6.4

2.69 1.37 3.16 1.89

(163) (192) (186) (222)

⟨κph⟩ (nW/K) 0.66 0.38 0.85 0.62

(0.07) (0.07) (0.09) (0.09)

⟨κph⟩/κ0ph 0.25 0.28 0.27 0.32

(0.03) (0.05) (0.03) (0.05)

⟨ZT⟩ 0.21 0.43 0.39 0.36

(0.95) (1.50) (0.26) (0.70)

⟨G⟩, ⟨κe⟩, and ⟨S⟩ are calculated at μ = Evb, where Evb is the valence band edge. The temperature is in all cases T = 300 K. Properties which depend on the wire length are calculated at L = 8 nm and L = 160 nm with the latter values (*). The MFP is a mean value in the energy interval E−Evb = [−0.05; 0.0] eV close to the valence band edge. κ0ph is the phonon thermal conductance of the pristine wires, and ⟨κph⟩ is the sample averaged value of the disordered wires. a

Ge−Si wire. It is noteworthy that the reduction factors are very similar for the different wires as seen in Table 1. Figure 5 shows calculated values of ZT versus chemical potential for ⟨111⟩ (a) and ⟨110⟩ oriented wires (b). The solid lines correspond to L = 160 nm long disordered wires, while the dashed lines are obtained for the pristine, disorder-free wires. The energy axis is in all cases shown relative to the valence band edge Evb. In all cases ZT is seen to peak around the band edge, and in the case of Ge−Si NWs very high values of ZT ∼ 1.5 are obtained. The values at μ = Evb are shown in Table 1. Panels c and d show how ZT evaluated at μ = Evb change with wire length in the limit of very long wires. Even though the short wire results are similar for Ge−Si and Si-NWs the long-wire results give significantly different results. The Ge−Si wires show a monotonously increasing ZT approaching 1.5 and 2.4 for ⟨110⟩ and ⟨111⟩ oriented wires, respectively. The pure SiNWs, on the other hand reach a maximum ZT at wire lengths 24 nm for ⟨110⟩ wires and 700 nm for the ⟨111⟩ wires. As already seen in Table 1 this difference is due to the longer valence band mean free paths in the Ge−Si wires where the holes are confined to the core region and thus shielded from the disordered outer surface. Even though the results for the very long wires with length of several micrometers shown in Figure 5c and d represent a very idealized situation and the

wires with SRD (the corresponding electronic transmission functions for the same wires are shown in Figure 3a). We first notice that the transmission through the disordered wires is almost independent of the actual realization of the SRD. Second, the transmission is greatly reduced as compared to the pristine wire. This is further visualized in panel b showing the phonon thermal conductance κph versus temperature. The average phonon thermal conductance values at T = 300 K are shown in Table 1 for the different wires at different lengths. In similarity with SiNWs56 the Ge−Si NWs show a pronounced anisotropy with the ⟨110⟩ wires having almost twice as large conductance as the ⟨111⟩ wires. Comparing Ge−Si and SiNWs with the same crystallographic orientations the Ge−Si wires have lower conductance values due to the larger mass of Ge leading to lower phonon energies and velocitiessee phonon band structures in the Supporting Information. Notice that for the short disordered wires, and in particular for the pristine wires, the phonon thermal conductance κph is significantly larger than the electronic contribution, κe. However, for the long Ge−Si wires κph and κe are almost equal. This implies that further reduction of κph will not lead to significant increases in the ZT values. The inset in Figure 4b shows the reduction factor κph/κ0ph compared to the pristine 4701

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core−shell wires.34−36 However, experiments indicate that surface scattering is the dominant mechanism in thin NWs as witnessed by the absence of a clear peak in the thermal conductivity versus temperature plot, which for bulk samples is due to anharmonic (umklapp) scattering.7,37 The inclusion of anharmonic effects as well as isotope scattering would decrease κph and tend to increase the computed ZT values. For the electronic calculations electron−phonon scattering would lead to reduced conductance values thus decreasing ZT. In the above calculations we have passivated the outer Si surface with hydrogen atoms. While real NWs indeed can be Hpassivated after HF etch, the typical experimental situation is that the surface is oxidized forming SiO2. Modeling the amorphous SiO2 is computationally much more difficult than using the H-passivation. It is likely that a SiO2 passivation would somehow affect the phonon transmission calculation. However, since the holes in the Ge−Si core shell wires are confined to the core, we do not think that a different surface passivation or surface reconstruction would have a strong effect on the electronic transport, and we believe our main conclusions are independent of the specific surface. In a real NW the core−shell interface will not be completely perfect, and imperfections would influence both the electronic and phononic transport properties. Preliminary calculations shown in the Supporting Information show that interface disorder mainly affects the phonon transmission while the electronic transmission is almost unaffected. While more work is required to draw final conclusions, these results do indicate that a certain amount of interface disorder would improve the thermoelectric properties. We also note that it has been experimentally shown that Ge−Si core−shell wires can have hole transport with mean free path larger than 100 nm,14 indicating relatively weak Ge−Si boundary scattering as well as weak electron−phonon interactions, which further justifies the simplified calculations presented in this work. Recent theoretical works have shown that surface disordered graphene nanoribbons can have high ZT values,57 but it was

Figure 4. Phonon transmission function vs phonon energy (a) and thermal conductance κph vs temperature (b) of pristine and disordered ⟨110⟩ oriented Ge−Si NWs. Inset in b shows the reduction factor κph/ κ0ph of the disordered wires compared to the pristine wire conductance, κ0ph. It is noteworthy that wires with different realizations of SRD have very similar transport properties.

values should not be taken literally, they do indicate that Ge−Si wires with SRD are even more promising candidates as efficient thermoelectric materials than pure SiNWs with SRD, which has experimentally been shown to have ZT ∼ 0.6−1.9,10 The presented results are obtained for idealized structures focusing only on the effect of SRD scattering while neglecting all other scattering mechanisms. For the phonon transport, anharmonic phonon−phonon scattering could be important especially for the core−shell wires, as recent MD calculations have shown a significant thermal conductance reduction in

Figure 5. ZT vs chemical potential for ⟨111⟩ (a) and ⟨110⟩ oriented wires (b) shown for Ge−Si (blue) and SiNWs (red). Solid curves correspond to 160 nm long wires with SRD while dashed curves represent the pristine, disorder-free wires. Panels (c) and (d) show ZT vs wire length calculated at the valence band edge, μ = Evb. 4702

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(10) Hochbaum, A.; Chen, R.; Delgado, R. D.; Liang, W.; Garnett, E. C.; Najarian, M.; Marumdar, A.; Yang, P. Nature 2008, 451, 163. (11) Wagner, R. S.; Ellis, W. C. Appl. Phys. Lett. 1964, 4, 89. (12) Lauhon, L.; Gudiksen, M.; Wang, C.; Lieber, C. Nature 2002, 420, 57−61. (13) Zhao, Y.; Smith, J. T.; Appenzeller, J.; Yang, C. Nano Lett. 2011, 11, 1406−1411. (14) Lu, W.; Xiang, J.; Timko, B. P.; Wu, Y.; Lieber, C. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10046−10051. (15) Xiang, J.; Lu, W.; Hu, Y.; Wu, Y.; Yan, H.; Lieber, C. Nature 2006, 441, 489−493. (16) Liang, G.; Xiang, J.; Kharche, N.; Klimeck, G.; Lieber, C. M.; Lundstrom, M. Nano Lett. 2007, 7, 642−646. (17) Hu, Y.; Churchill, H. O. H.; Reilly, D. J.; Xiang, J.; Lieber, C. M.; Marcus, C. M. Nat. Nanotechnol. 2007, 2, 622−625. (18) Musin, R. N.; Wang, X.-Q. Phys. Rev. B 2005, 71, 155318. (19) Musin, R. N.; Wang, X.-Q. Phys. Rev. B 2006, 74, 165308. (20) Migas, D. B.; Borisenko, V. E. Phys. Rev. B 2007, 76, 035440. (21) Amato, M.; Palummo, M.; Ossicini, S. Phys. Rev. B 2009, 80, 235333. (22) Peköz, R.; Raty, J.-Y. Phys. Rev. B 2009, 80, 155432. (23) Yang, L.; Musin, R. N.; Wang, X.-Q.; Chou, M. Y. Phys. Rev. B 2008, 77, 195325. (24) Liu, N.; Lu, N.; Yao, Y.-X.; Li, Y.-R.; Wang, C.-Z.; Ho, K.-M. J. Phys. Chem. C 2011, 115, 15739−15742. (25) Peng, X.; Logan, P. Appl. Phys. Lett. 2010, 96, 143119. (26) Huang, S.; Yang, L. Appl. Phys. Lett. 2011, 98, 093114. (27) Peng, X.; Tang, F.; Logan, P. J. Phys.: Condens. Matter 2011, 23, 115502. (28) Chen, X.; Wang, Y.; Ma, Y. J. Phys. Chem. C 2010, 114, 9096− 9100. (29) Peelaers, H.; Partoens, B.; Peeters, F. M. Phys. Rev. B 2010, 82, 113411. (30) Zhang, L.; d’Avezac, M.; Luo, J.-W.; Zunger, A. Nano Lett. 2012, 12, 984−991. (31) Amato, M.; Ossicini, S.; Rurali, R. Nano Lett. 2011, 11, 594− 598. (32) Li, L.; Smith, D. J.; Dailey, E.; Madras, P.; Drucker, J.; McCartney, M. R. Nano Lett. 2011, 11, 493−497. (33) Yang, R.; Chen, G.; Dresselhaus, M. S. Nano Lett. 2005, 5, 1111−1115. (34) Chen, J.; Zhang, G.; Li, B. J. Chem. Phys. 2011, 135, 104508. (35) Hu, M.; Zhang, X.; Giapis, K. P.; Poulikakos, D. Phys. Rev. B 2011, 84, 085442. (36) Hu, M.; Giapis, K. P.; Goicochea, J. V.; Zhang, X.; Poulikakos, D. Nano Lett. 2011, 11, 618−623. (37) Wingert, M. C.; Chen, Z. C. Y.; Dechaumphai, E.; Moon, J.; Kim, J.-H.; Xiang, J.; Chen, R. Nano Lett. 2011, 11, 5507−5513. (38) Rurali, R. Rev. Mod. Phys. 2010, 82, 427−449. (39) Markussen, T.; Jauho, A.-P.; Brandbyge, M. Phys. Rev. B 2009, 79, 035415. (40) Markussen, T.; Jauho, A.-P.; Brandbyge, M. Phys. Rev. Lett. 2009, 103, 055502. (41) Lherbier, A.; Persson, M. P.; Niquet, Y.-M.; Triozon, F.; Roche, S. Phys. Rev. B 2008, 77, 085301. (42) Persson, M. P.; Lherbier, A.; Niquet, Y.-M.; Triozon, F.; Roche, S. Nano Lett. 2008, 8, 4146−4150. (43) Wu, Y.; Cui, Y.; Huynh, L.; Barrelet, C.; Bell, D.; Lieber, C. Nano Lett. 2004, 4, 433−436. (44) Moutanabbir, O.; Senz, S.; Scholz, R.; Alexe, M.; Kim, Y.; Pippel, E.; Wang, Y.; Wiethoff, C.; Nabbefeld, T.; Meyer zu Heringdorf, F.; Horn-von Hoegen, M. ACS Nano 2011, 5, 1313−1320. (45) Goldthorpe, I. A.; Marshall, A. F.; McIntyre, P. C. Nano Lett. 2008, 8, 4081−4086. (46) Tersoff, J. Phys. Rev. B 1988, 38, 9902. (47) Tersoff, J. Phys. Rev. B 1989, 39, 5566. (48) Gale, J. D. J. Chem. Soc., Faraday Trans. 1997, 93, 629. (49) Ju, Y.; Goodson, K. Appl. Phys. Lett. 1999, 74, 3005−3007.

also shown that the thermoelectric performance is reduced for wider ribbons.58 Since the results shown above are obtained for very thin wires, one may suspect that a similar ZT reduction would be seen for larger diameter wires. However, studying thicker and more realistic sized wires becomes computationally very demanding. Preliminary calculations shown in the Supporting Information for ∼4 nm diameter Ge−Si wires indicate that, for a given degree of SRD, the ZT decreases with increasing diameter. However, by increasing the degree of SRD, it is possible to retain or even improve the ZT in larger diameter Ge−Si core shell wires. However, further studies are needed to draw final conclusions. Future works should focus on optimizing the core- and shell diameters and the degree of disorder. In conclusion we have shown that Ge−Si core−shell NWs with SRD act as a phonon-glass electron-crystal material with very promising thermoelectric figure of merit ZT > 2 for long disordered wires. The SRD strongly reduce the phonon thermal conductance, while the hole transport in the Ge−Si wires is affected by the disorder to a much smaller degree due to the spatial localization of the hole states to the Ge core region.

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ASSOCIATED CONTENT

S Supporting Information *

Details on the construction of nanowires with surface roughness disorder (SRD) and phonon band structures of ⟨110⟩ wires as well as electronic band structures for larger diameter wires, a comparison of the tight-binding model used in the main text and with results from DFT calculations, an explicit comparison between the in-coherent approximate method applied in the main text and fully coherent calculations, results for Ge−Si wires with interface disorder, and transport calculations for larger (3.9 nm) diameter wires. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author thanks Riccardo Rurali for inspiring discussions and acknowledges support from The Danish Council for Independent Research/Technology and Production Sciences through Grant No. 11-104592 and the Sapere Aude program.

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REFERENCES

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