Ge Superlattice Nanowires with Ultralow Thermal Conductivity

Oct 29, 2012 - system, a constant heat flux JQ = (dQ/dt)/A in the longitudinal direction is ..... 28 unit cells, while a Ge slice has 27 × 27 unit ce...
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Si/Ge Superlattice Nanowires with Ultralow Thermal Conductivity Ming Hu* and Dimos Poulikakos† Laboratory of Thermodynamics in Emerging Technologies, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland ABSTRACT: The engineering of nanostructured materials with very low thermal conductivity is a necessary step toward the realization of efficient thermoelectric devices. We report here the main results of an investigation with nonequilibrium molecular dynamics simulations on thermal transport in Si/Ge superlattice nanowires aiming at taking advantage of the inherent one dimensionality and the combined presence of surface and interfacial phonon scattering to yield ultralow values for their thermal conductivity. Our calculations revealed that the thermal conductivity of a Si/Ge superlattice nanowire varies nonmonotonically with both the Si/Ge lattice periodic length and the nanowire cross-sectional width. The optimal periodic length corresponds to an order of magnitude (92%) decrease in thermal conductivity at room temperature, compared to pristine single-crystalline Si nanowires. We also identified two competing mechanisms governing the thermal transport in superlattice nanowires, responsible for this nonmonotonic behavior: interface modulation in the longitudinal direction significantly depressing the phonon group velocities and hindering heat conduction, and coherent phonons occurring at extremely short periodic lengths counteracting the interface effect and facilitating thermal transport. Our results show trends for superlattice nanowire design for efficient thermoelectrics. KEYWORDS: Si/Ge superlattice nanowire, thermal conductivity, coherent phonon, thermoelectrics hermoelectric materials find important applications in the direct conversion of thermal energy to electric power and in solid-state cooling.1 Although thermoelectric devices possess unique advantages such as high reliability, lack of moving parts, and the ability to be scaled down to small sizes, the energy conversion efficiencies of these devices remain a generally poor factor that severely limits their competitiveness and range of employment.2 To materialize widespread use of thermoelectrics at levels that would impact global energy issues the material and device efficiencies existing currently will need to be improved significantly. The energy conversion efficiency of thermoelectric devices is characterized by the figure-of-merit: ZT = S2σT/κ, where S, σ, and κ are the Seebeck coefficient, electrical conductivity, and thermal conductivity of the material, respectively, and T is the absolute temperature.2 In general, development schemes to improve thermoelectric conversion efficiency are driven by the need to maximize the Seebeck coefficient, S, and to balance the competing requirements of high electrical conductivity, σ, and low thermal conductivity, κ. To this end, significant scientific effort is dedicated to reducing the thermal conductivity. Nanostructuring of existing thermoelectrics (nanowires, superlattices, composite matrices, etc.) has emerged as a promising pathway to improve thermoelectric performance by manipulating phononic transport.3 In particular, control of interfaces in thermoelectric materials can play a critical role in meeting this challenge. Interfaces affect each of the above properties involved in the ZT coefficient and can have profound effects when present at the high densities typical of nanomaterials. Advances over the past decade show that it is possible to enhance the ZT coefficient in nanoscale systems by using

T

© 2012 American Chemical Society

phonon scattering at interfaces to reduce the thermal conductivity and quantum confinement and carrier scattering effects to enhance the power factor, S2σ. Many improvements in thermoelectric performance have been demonstrated in bulk materials with embedded nanostructures4 or multilayer thinfilm geometries.5 Along the pathway of employing individual nanostructures, semiconductor nanowires (NWs) receive exceptional attention, due to their inherent one-dimensionality resulting in low thermal conductivity and simultaneous high carrier mobility, leading to a behavior that is drastically different from that of their corresponding bulk material. Recent progress in this direction, either by experimental fabrication and measurements or through theoretical modeling, includes investigations of smooth and rough Si nanowires,6,7 Si/Ge core−shell nanowires,8−12 amorphous layers on the surface of Si nanowires,13 and Si nanotubes by conceptually drilling a small hole at the center of Si nanowires.14 Both numerical simulations8−10 and experiments12 have suggested that modification of a nanowire surface with coatings can cause significant reduction in room-temperature thermal conductivity, as compared with pristine nanowire. Using molecular dynamics (MD) simulations,8−10 it was shown that a Ge layer of only 1− 2 unit cell thickness conformal to a Si nanowire can lead to a 75% decrease in thermal conductivity at room temperature compared to uncoated Si nanowires. In this Letter, we investigate a concept that combines the thermal conductivity reduction advantages of nanowires with Received: May 25, 2012 Revised: September 13, 2012 Published: October 29, 2012 5487

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conditions were used in the transverse directions. We set 2.2 nm long regions at both ends as “rigid walls”. During the first stage of MD simulations, we relaxed the system at 300 K for 8 ns with the walls moving freely in the longitudinal direction, corresponding to zero pressure, using a Nosé-Hoover thermostat.34,35 After free boundary relaxation, we immobilized the walls and continued to relax the system with NVT (constant particles, volume, and temperature) and NVE (constant particles and volume without thermostat) ensembles each for 1 ns. Following equilibration, we computed the thermal conductivity of the system using nonequilibrium MD. A heat source and a heat sink were placed at the left and right end of the superlattice nanowire, respectively, to mimic possible future plausible experiments. The constant heat flux method with the Muller-Plathe algorithm36,37 was adopted, in which the heat flux is determined by an input parameter and the resulting temperature gradient is calculated. Once the steady state is reached, which typically takes 10−20 ns depending on the system, a constant heat flux JQ = (dQ/dt)/A in the longitudinal direction is established, where A is the cross-sectional area of the superlattice nanowire. After that, a time averaging of the temperature profile is performed for an additional 15 ns. The nonlinear effects on the temperature gradient arisen from the heat source/sink are avoided by fitting only the middle region of the time-averaged temperature profile with a linear function to obtain the temperature gradient. Finally, the thermal conductivity is calculated with Fourier’s law

those of superlattices and present the results of molecular dynamics simulations of phonon transport in Si/Ge superlattice nanowires (SLNWs). Although in recent years a considerable amount of interest has been dedicated to the investigation of heat conduction in two-dimensional superlattice films15−19 and one-dimensional superlattice nanowires,20−23 to the best of our knowledge, the mechanism of thermal transport in superlattice nanowires, along with the effects of important parameters, such as the individual layer thickness in the superlattice (periodic length), the wire diameter (or cross-sectional width) and the temperature, on the thermal transport properties of the nanowire are not thoroughly understood. As confirmed by experiments,24−26 thin Si/Ge slices are periodically stacked one upon the other to form a singlecrystalline Si/Ge superlattice NW, which serves as our model system (see Figure 1). The longitudinal direction is set along

Figure 1. Snapshot of a typical Si/Ge superlattice nanowire used as model system in the MD simulations. The total length of the wire (L), cross-sectional width (D), and periodic length (Lp = LSi + LGe) are indicated. Color code: yellow: Si, blue: Ge.

κ=−

JQ ∂T /∂z

(1)

where JQ is the heat flux in the longitudinal direction and ∂T∂z is the temperature gradient. The dependence of the thermal conductivity of Si/Ge superlattice nanowires on the periodic length is shown in Figure 2a. All of the superlattice nanowires shown in Figure 2a have the same cross section of 3.07 × 3.07 nm2, but their lengths range from 139 to 556 nm. We also show the thermal conductivity of pure smooth Si nanowires and Si0.5Ge0.5 alloy nanowires with same length for comparison. The alloy nanowires were generated by randomly replacing 50% of Si atoms in pristine Si NWs with Ge atoms. We first noticed that the thermal conductivity of Si/Ge superlattice nanowires is remarkably reduced as compared to that of pure smooth Si nanowires. For the three lengths considered, the reduction percentage in thermal conductivity reaches values as high as 92%. In other words, the thermal conductivity of a Si/Ge superlattice nanowire can amount to only 8% of that of an equivalent pristine Si nanowire, which is a markedly stronger reduction than that found in previous studies8,14 exploring various concepts of thermal conductivity reduction in Si-based nanowires. This can be translated into almost an order of magnitude enhancement in the ZT coefficient, considering that the electrical transport properties of a very thin superlattice nanowire does not change noticeably compared to a pure smooth nanowire, as evidenced by recent ab initio calculations.38,39 Specifically, only a 25% decrease in S2σ was found in Si nanowires with periodic Ge heterostructures.38 Thus, the significant reduction in thermal conductivity makes Si/Ge SLNWs a very promising nanomaterial for high efficiency thermoelectrics.

the z-axis, so that atoms in the same layer have the same zcoordinate. The initial Si nanowire is constructed from diamond-structured bulk silicon with n × n × m unit cells in the x-, y-, and z-directions, where n ranges from 2 to 28, corresponding to cross-sectional widths in the range from 0.77 to 10.8 nm, and m ranges from 256 to 1024, corresponding to a wire lengths in the range from 139 to 556 nm, respectively. After constructing the Si NW, the Ge segments were built by periodically replacing Si atoms with Ge, resulting in epitaxial Si/Ge interfaces. The structure of a typical Si/Ge superlattice nanowire is shown in Figure 1. For a fixed total length of a nanowire (L), the lengths of the Si and Ge segments (LSi and LGe, respectively) vary from 1 unit cell, that is, 0.543 nm (lattice constant of Si at room temperature), to half of the total length, that is, L/2. In all cases, both the Si and the Ge layer axes lie along the [001] crystal axis of the diamond lattice. The surfaces in the transverse directions (x- and y-axes) for both Si and Ge are of the (110) type. Nanowires made with this surface orientation were found to be stable with respect to disordering or surface reconstruction.27 In all MD simulations performed herein, the Tersoff potential28 with original parameters optimized for Si−Ge systems was used to describe the Si−Si, Ge−Ge, and Si−Ge interactions in Si/Ge superlattice NWs. This potential has been used to predict the thermal transport properties of Si nanowire with amorphous layer,29 nanoporous Si−Ge,30 Si−Ge quantum dot superlattices,31 and Si−Ge nanocomposites.32 All MD calculations were performed using the LAMMPS33 package with a time step of 1 fs throughout. Free boundary 5488

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⎞ 1 1 ⎛l = NW ⎜ NW + 1⎟ κ κ∞ ⎝ Lz ⎠

(2)

κNW ∞

where is the thermal conductivity of an infinitely long NW and lNW is the effective phonon mean free path in the NW. The κNW ∞ values for pristine Si NW, Si0.5Ge0.5 alloy nanowire, Si/Ge SLNW, and fully amorphous Si NW are obtained to be 45.3, 6.03, 4.44, and 1.35 W/mK, respectively, indicating that as the structure becomes more heterogeneous the thermal conductivity is reduced further. For an infinitely long NW the reduction percentage in the thermal conductivity of a Si/Ge SLNW as compared with a pristine Si NW still remains very high at 90%. For very large lengths, the thermal conductivity of Si/Ge SLNW is even lower than the alloy limit. It is worth pointing out that our simulation result of 45.3 ± 1.9 W/mK, extrapolated from Figure 2b for an infinitely long pristine Si NW with 3.07 × 3.07 nm2 cross-section, is consistent with Donadio’s results,13,29 who reported a thermal conductivity of ∼40 ± 10 W/mK (read from their graph) for the same cross-sectional width and orientation of longitudinal direction using equilibrium MD simulation and the same Tersoff potential. We note that the thermal conductivity of a pristine Si NW obtained with the Tersoff potential is expected to be somewhat higher than that obtained with the Stillinger− Weber (SW) potential.8,9,41 This can be explained by the tendency of the Tersoff potential to yield stiffer Si bonds. Separate simulations using the Tersoff potential yield a thermal conductivity of 243 ± 11 W/mK for bulk Si. Although markedly higher than the value of 132 ± 7 W/mK obtained with the SW potential, the result is consistent with previous studies42 and confirms the stiffer Si−Si bonds in the Tersoff potential. The major prohibiting disadvantage to the employment of the SW potential for the studied superlattice nanowires is its lack of stability especially at short periodic lengths. We also note the discrepancy between our MD results (as well as those from others from MD) for a pristine Si NW and experimental values.43 This is plausibly attributed to defects, impurities, surface roughness, and oxidation, which are always present in the experimental samples and impact adversely on the thermal conductivity of the nanowire. Theoretical calculations assume that the Si NWs are perfect crystals with atomically smooth surfaces, which facilitate heat conduction. Another noticeable feature in Figure 2a is the nonmonotonic dependence of the thermal conductivity of Si/Ge SLNWs on the periodic length. The same trend that the thermal conductivity increases with decreasing periodic length was observed for superlattice f ilms in experiments,44 in theoretical analysis,45 and in MD simulations for other systems46,47 as well. However, the nonmonotonic dependence shown in Figure 2a was not observed in previous MD simulation for an idealized Kr/Ar superlattice nanowire.22 The minima in thermal conductivity can be attributed to the two competing mechanisms governing heat conduction in SLNWs, to be discussed in sequence. First, we explain why the thermal conductivity of Si/Ge SLNWs is drastically reduced as compared to pristine NWs. To this end, we computed the phonon dispersion curve of bulk Si, pristine Si NW, and Si/Ge SLNW with cross section 3.07 × 3.07 nm2 and periodic length of 1.09 nm using the PHONOPY software.48 In the results presented in Figure 3a it is clearly shown that the dispersion curve of the four acoustic phonons (longitudinal acoustic, transverse acoustic, and twist acoustic, two transverse modes are degenerated) in SLNW is significantly depressed, compared

Figure 2. (a) Periodic length dependence of the thermal conductivity of Si/Ge superlattice nanowires for different total lengths. The superlattice nanowires with epitaxial and nonepitaxial growth have cross-sectional areas of 3.07 × 3.07 and 10.8 × 10.8 nm2, respectively. The solid and dashed lines represent the thermal conductivity of pure smooth Si nanowires and Si0.5Ge0.5 alloy nanowires with same length and cross-section width, respectively. Color code: red: 139 nm, green: 278 nm with cross-sectional area of 3.07 × 3.07 nm2, blue: 556 nm, magenta: 278 nm with cross-sectional area of 10.8 × 10.8 nm2. (b) Length dependence of 1/κ on 1/L for four different systems: pure smooth Si nanowire, Si/Ge superlattice nanowire, Si0.5Ge0.5 alloy nanowire, and fully amorphous Si nanowire.

In Figure 2b, we present the reciprocal of the thermal conductivity of Si/Ge SLNWs versus the reciprocal of the nanowire length for Si/Ge periodic length of 4.34 nm (8 unit cells thick) and compare it with fully amorphous Si NWs, Si0.5Ge0.5 alloy NWs, and pure smooth Si NWs. The amorphous NW was produced by melting a rather long pristine nanowire in a confined channel at elevated temperature (e.g., 3000 K) and then rapidly quenching the melt to room temperature. The channel has fixed xy dimensions of 3.07 × 3.07 nm2 and free boundaries in the z-direction; thus the amorphous structure can be fully relaxed. The as-formed amorphous NW was then cut to the desired length for further calculations. When calculating the thermal conductivity, the fixed boundary in the x- and ydirections was removed to mimic the free surface of a freestanding NW. We fitted the obtained data points using a linear 40 function to obtain κNW ∞ and lNW 5489

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to bulk Si and pristine Si NW. As a result, the phonon group velocity in SLNW is drastically reduced (Figure 3b). Note that in Figure 3a we only show the acoustic phonons, which are anticipated to dominate the thermal transport in SLNW. However, in the full range of dispersion curves (not shown for brevity) the phonon frequency of the optical modes can go up to 18 THz. The phonon group velocity was obtained from the relation Vg = (∂ω/∂q), where q is the wave vector. Using the single mode relaxation time approximation of the Boltzmann equation, the contribution of each phonon mode (κi) to the total thermal conductivity can be expressed as κi(q) = Ci(q)vi2(q)τi(q)

(3)

where Ci is the specific heat, vi is the group velocity, and τi is the phonon relaxation time. If the phonon group velocity is significantly reduced, the thermal conductivity is anticipated to be very low. Moreover, phonon boundary scattering (due to the small cross-sectional width of SLNW) and phonon scattering occurring at Si/Ge interfaces will largely reduce the phonon relaxation time, which is an additional negative effect on heat conduction in SLNW. Therefore, we conclude that the large reduction in thermal conductivity of Si/Ge SLNW originates from the large depression in phonon group velocity aided by the phonon interfacial and boundary scattering. We now explain why the thermal conductivity of Si/Ge SLNW is expected to increase as periodic length becomes very shorter, that is, less than 4.34 nm for the case of Figure 2a. To this end, we performed a standard real-space mode analysis, by assuming that the normal-mode solution has the form uiαλ = (1/mi1/2)εiα,λeiωλt. The normal mode eigenfrequencies ωλ and their corresponding eigenvectors components εiα,λ are obtained by solving the lattice dynamical equation ωλ2εiα , λ =

Figure 3. (a) Phonon dispersion curves of bulk Si (solid lines), pristine Si nanowires (dashed lines), and Si/Ge superlattice nanowires (dotted lines) with cross section 3.07 × 3.07 nm2 and periodic length 1.09 nm. For all cases, only the acoustic branches are shown for brevity. Color code: red: longitudinal, blue: transverse, pink: twist acoustic. Note that the two transverse acoustic modes are degenerated and the label "a" should be interpreted differently for different systems, that is, "a" is the lattice constant for bulk Si and pristine Si nanowire, while the same "a" denotes the periodic length for Si/Ge superlattice nanowire. (b) Corresponding phonon group velocity vs frequency. The same group lines and color code are used.

Φiα , jβ =

∑ Φiα ,jβ εjβ ,λ jβ

(4a)

1 ∂ 2V mimj ∂uiα ∂ujβ

(4b)

Here i and j denote the atom, α and β (= x, y, or z) are the Cartesian directions, λ denotes a branch of phonon modes, uiα is the displacement of atom i in the αth direction, m is the atomic mass, Φ is the dynamical matrix, and V is the total

Figure 4. Regional mode weight factor for Si/Ge superlattice nanowires with different Si/Ge periodic lengths. (a) 1.09 nm, (b) 2.17 nm, (c) 4.34 nm. All superlattice nanowires have cross-sectional area of 3.07 × 3.07 nm2. Color code: red: interface, blue: interior, black: surface. 5490

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epitaxial SLNWs and the position of the minimum shifts to an even shorter periodic length (2.172 nm). To the best of our knowledge, no experimental results on Si/Ge SLNWs with such small periodic length exist (we could not find any in the literature); hence, no direct comparison can be made between our MD simulation and experiments. The shortest periodic length we can find in literature for a superlattice f ilm54 is about 3 nm, where the thermal conductivity still follows the trend of reduction with decreasing periodic length. It is, however, not studied/shown experimentally whether the conductivity continues to decrease when the periodic length is reduced further below 2 nm. The nanowire cross section can significantly influence thermal transport. As the lateral dimension is reduced to the nanometer scale, which is comparable to or is even smaller than the phonon mean free path in a solid, phonon boundary scattering at the free surface of the nanowire becomes important, hindering phonon propagation. Therefore, it is necessary to investigate how the thermal conductivity of SLNWs changes with cross-sectional width and compare this result to the behavior of a pristine NW. Figure 5a depicts the cross-sectional width dependence of thermal conductivity of Si/

potential energy. After obtaining the eigenvectors, we calculated the mode weight factor defined as8−10,49 fj′,λ =

∑ ′ ∑ (εjα ,λ)2 j

α

(5)

where the prime denotes that the sum over j (atoms) is alternatively restricted to the interior of the superlattice nanowire, the Si/Ge interface, and the free surface. The mode weight factor provides information on the relative contributions from individual modes from different regions. The Si/Ge interfacial region is defined by the Si and Ge atoms, which have at least one neighbor from the other element within the cutoff distance of the force field. The Si or Ge atoms in the interior region have exactly four nearest neighbors from the same element determined by the diamond structure, while those in the surface region have only three nearest neighbors. Consequently, the sum of the mode weight factors in the superlattice nanowire is always equal to unity, that is, f interior,λ + f interface,λ + fsurface,λ ≡ 1. Any change in one component leads to an adverse change in other component(s). In Figure 4 we demonstrate the regional mode weight factor dependence on the periodic length for the case of Si/Ge SLNW with cross section of 3.07 × 3.07 nm2. The periodic length of 4.34 nm corresponding to the minimum of thermal conductivity and two shorter periodic lengths of 2.17 and 1.09 nm were considered. We first notice that the mode weight factor from the surface region does not change appreciably with the periodic length. However, from Figure 4 it is evident that, as the periodic length is decreased, the phonon mode weight factors of the interior and the interface regions approach each other, and finally, at the shortest periodic length (1.09 nm) they are practically equal. That is to say, for each branch of phonon modes, all of the atoms from the interior and interface region participate and vibrate with the same frequency. They contribute equally to the total vibration and the phonon modes are nonlocalized. Same as presented in two-dimensional superlattice films,17,50,51 such coherent phonon behavior is believed to eliminate the effect of phonon interfacial scattering and thus facilitate thermal transport. Combining this conduction enhancement with the previously explained mechanism hindering conduction in SLNW explains the nonmonotonic dependence of the thermal conductivity of Si/ Ge SLNW on the periodic length. It is worth emphasizing that our full phonon dispersion curves (not shown for brevity) are similar with those of previous studies.52 Clearly, there is miniband formation in our superlattice nanowire as well, as discussed for other systems in the literature.18,45,52,53 We also checked the mini-band changes for different periodic boundary lengths. Compared to the mechanism of mini-band formation and change presented before, the two competing mechanisms we proposed in this paper (interface modulation and coherent phonons) explain the phenomenon from a different perspective, which we find more straightforward to understand. We also considered the lattice mismatch between Si and Ge in Si/Ge SLNW with cross-sectional width 10.8 nm and total length 278 nm. In this case the Ge slices are nonepitaxially grown on Si. Specifically a Si slice has lateral dimension of 28 × 28 unit cells, while a Ge slice has 27 × 27 unit cells, accounting for the fact that that the lattice constant of Ge is about 4% larger than Si. The results are included in Figure 2a. It is found that the thermal conductivity of nonepitaxial SLNW still has a nonmonotonic dependence on the periodic length, except that the minimum is less evident than that occurring in the cases of

Figure 5. (a) Thermal conductivity of Si/Ge superlattice nanowires with different cross-sectional widths as a function of the Si/Ge periodic length at 300 K. The solid and dashed lines represent the thermal conductivity of pure smooth Si nanowires and Si0.5Ge0.5 alloy nanowires with the same length, respectively. (b) Thermal conductivity of Si/Ge superlattice nanowires with fixed periodic length of 4.34 nm as a function of wire cross-sectional width. The right axis shows the percent reduction in thermal conductivity of Si/Ge superlattice nanowires relative to that of pure smooth Si nanowires. All nanowires in a and b have the same length of 278 nm. 5491

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Ge SLNWs. The thermal conductivities of pristine Si NW, Si0.5Ge0.5 alloy NW, and a two-dimensional superlattice film are also shown for comparison. All superlattice nanowires along with the superlattice film have the common feature that they reach the lowest thermal conductivity at the Si/Ge periodic length of 4.34 nm and that superlattice nanowires generally have much lower thermal conductivity than their counterpart pristine nanowires and also the superlattice film. More importantly, for the fixed “optimal” Si/Ge periodic length of 4.34 nm we increased the cross-sectional width of Si/Ge SLNWs up to 10.8 nm. The results are shown in Figure 5b. Similar to the periodic length dependence in Figure 2a, the thermal conductivity of Si/Ge SLNWs exhibits a nonmonotonic dependence on cross-sectional width. As the nanowire cross-sectional width is reduced down to 3.07 nm, the thermal conductivity decreases. Below that cross-sectional width, the thermal conductivity increases again. This means that 3.07 nm is the optimal cross-sectional width corresponding to the lowest thermal conductivity in SLNW. A reduction percentage of 91.2% was found in this case, as compared with a pristine Si NW. Note that the reduction percentage remains around 93% even for cross-sectional widths up to 10.8 nm. Our MD simulation result is consistent with previous MD studies from other groups,13,29 in that in thin Si nanowires (especially for cross-sectional width less than 3 nm) phonon confinement through dimensionality reduction does not necessarily lead to low values of thermal conductivity. To explain the unexpected increase in thermal conductivity with nanowire cross-sectional width below 3.07 nm, we first calculated the heat capacity of the SLNWs with cross-sectional widths ranging from 0.77 to 3.07 nm. The computed heat capacities of Si/Ge SLNWs are 15.1, 19.8, and 23.7 J/(mol·K) for cross-sectional widths of 0.77 nm, 1.54 nm, and 3.07 nm, respectively, indicating that the capacity decreases with decreasing cross-sectional width. Obviously, this cannot explain the increase in the thermal conductivity as cross-sectional width becomes smaller, since the conductivity is linearly proportional to heat capacity according to the Boltzmann equation (see eq 3). We are aware that work on the diameter dependence of the thermal conductivity of Si nanowires has been published, which incorporates a diameter-dependent phonon mean free path.55,56 Such results have shown a monotonic decrease in the thermal conductivity with decreasing diameter. However, the diameters considered in these studies (at least 20 nm) are much larger than the cases of our MD simulations. Therefore, it is questionable if such theory can be predictive of the behavior of SLNW in the size range considered in our MD simulations. So far in the literature there are few studies focusing on the diameter dependence of the thermal conductivity of thin Si nanowires. For the first time here, we believe, we tried to plausibly explain the unexpected nonmonotonic diameter dependence in terms of the long wavelength surface modes propagating along the surface in extremely small cross-sectional width nanowires. Further experimental confirmation will be sought in future studies. Figure 6a clearly shows that, as the SLNW becomes thinner, the phonon transport is dominated by the surface modes, as evidenced by the significant increase in the mode weight factor, especially for low frequency phonons (less than 5 THz). This is understandable considering that the thinnest nanowire has the strongest surface effect and is more flexible to vibrate at low frequencies. More evidence is provided by the vibrational density of states (VDOS) of surface atoms in Figure 6b, showing enhancement of the power density of low

Figure 6. Comparison of surface mode weight factor (a) and vibrational density of states of surface atoms (b) in Si/Ge superlattice nanowires with different cross-sectional widths. All nanowires have the same Si/Ge periodic length of 4.34 nm.

frequency phonons in the thinnest SLNW. Since low frequency phonons conduct heat more efficiently, the enhancement of low frequency phonon modes can explain the high thermal conductivity for very thin SLNW. We also investigated the temperature dependence of the thermal conductivity of Si/Ge SLNWs. The results are reported in Figure 7. Due to the quantum nature of thermal vibrations in the low-temperature range, classical molecular dynamics results below 50 K are not reliable. The thermal conductivity of a pure Si NW exhibits a sharp peak at low temperatures then decays rapidly as the temperature increases. This 1/Tα trend originates from the enhanced phonon−phonon Umklapp scattering, as pointed out in ref 43. Surprisingly, the thermal conductivity of a Si/Ge superlattice nanowire shows a slow reduction rate throughout the entire temperature range studied. Experiments20 have shown a similar temperature independence of the thermal conductivity of Si/Ge SLNWs between 200 and 300 K, as identified in our MD simulation. This behavior is believed to originate from the presence of extremely low phonon group velocities, that is, nearly nonpropagating vibrational modes, as identified in Figure 3 and a previous study.9 In contrast, the thermal conductivity of fully amorphous Si NW remains constant from 50 to 600 K. This temperature independence is due to the presence of fully nonpropagating vibrational modes in the amorphous structure. The thermal 5492

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ACKNOWLEDGMENTS M.H. would like to thank Xiaoliang Zhang of the Chinese Academy of Sciences for his help in computing the phonon dispersion curve. Computational support from the Brutus Cluster at ETH Zurich is gratefully acknowledged. This work was supported by a grant from the Swiss National Supercomputing Centre-CSCS under project ID s243 and s359.

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Figure 7. Temperature dependence of the thermal conductivity of Si/ Ge superlattice nanowires in comparison with pure smooth Si nanowires, Si0.5Ge0.5 alloy nanowires, and fully amorphous Si nanowires. All wires have the same cross section of 3.07 × 3.07 nm2 and the same length of 278 nm.

conductivity of Si0.5Ge0.5 alloy NW resides between that for Si/ Ge SLNW and amorphous Si NW. It is close to the amorphous Si NW at low temperatures and gradually approaches the Si/Ge SLNW as the temperature increases. The temperature dependence results suggest that Si/Ge SLNWs may have a considerably larger ZT coefficient than pristine Si NWs at temperatures below 300 K. In conclusion, employing nonequilibrium molecular dynamics simulations we have studied thermal transport in Si/Ge superlattice nanowires. An order of magnitude reduction (up to 92%) in the thermal conductivity of Si/Ge superlattice nanowires relative to pristine nanowires was found. The thermal conductivity of Si/Ge superlattice nanowires depends nonmonotonically on both the periodic length and the crosssectional width. By calculating the phonon dispersion curve and performing normal-mode analysis, we explained that the nonmonotonic length dependence stems from two competing mechanisms governing heat conduction in Si/Ge superlattice nanowires, that is, interface modulation in the longitudinal direction significantly depressing the phonon group velocities and hindering heat conduction, and coherent phonons occurring at extremely short periodic lengths counteracting the interface effect and facilitating thermal transport. The nonmonotonic dependence on the cross-sectional width originates from the enhanced long wavelength surface modes dominating the phonon transport in ultrathin superlattice nanowires. The obtained results suggest that, by optimizing the periodic length and the wire cross-sectional width, superlattice nanowires can be very good candidates as materials for highperformance, efficient thermoelectrics.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.H.). Notes

The authors declare no competing financial interest. † E-mail: [email protected] (D.P.). 5493

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