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Prediction of high zT in thermoelectric silicon nanowires with axial germanium heterostructures

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a Letters Journal Exploring the: Frontiers or Pnrsics

June 2011

EPL, 94 (2011) 67001 www.epljournal.org

doi: 10.1209/0295-5075/94/67001

Prediction of high zT in thermoelectric silicon nanowires with axial germanium heterostructures

M. Shelley and A. A. MosTOFi(a)

The Thomas Young Centre for Theory and Simulation of Materials, Imperial College London London SW7 2AZ, UK

received 11 March 2011; accepted in final form 5 May 2011 published online 30 May 2011

PACS 73.63.Nm - Quantum wires

PACS 72.20.Pa - Thermoelectric and thermomagnetic effects PACS 63.22.Gh - Nanotubes and nanowires

Abstract - We calculate the thermoelectric figure of merit, zT = S2GT/(k + Ke), for p-type Si nanowires with axial Ge heterostructures using a combination of first-principles density-functional theory, interatomic potentials, and Landauer-Buttiker transport theory. We consider nanowires with up to 8400 atoms and twelve Ge axial heterostructures along their length. We find that introducing heterostructures always reduces S2G, and that our calculated increases in zT are predominantly driven by associated decreases in kj. Of the systems considered, (111) nanowires with a regular distribution of Ge heterostructures have the highest figure of merit: zT = 3, an order of magnitude larger than the equivalent pristine nanowire. Even in the presence of realistic structural disorder, in the form of small variations in length of the heterostructures, zT remains several times larger than that of the pristine case, suggesting that axial heterostructuring is a promising route to high-zT thermoelectric nanowires.

Copyright © EPLA, 2011

In recent years, silicon nanowires (SiNWs) have been proposed for use as chemical sensors [1], photovoltaics [2] and thermoelectrics [3,4]. Hicks and Dresselhaus [5] first identified that NWs could be used to improve the thermoelectric figure of merit, zT, over bulk and two-dimensional superlattice (2DSL) values. While the dramatic increases that were predicted have not been yet realised, much progress has been made: refs. [3,4] report zT ~ 1 for SiNWs, a 100-fold increase over the value for bulk Si.

The measure of the performance of a thermoelectric material is given by its figure of merit zT = S2GT/(k; + Ke), where S, G and T are the Seebeck co-efficient, electronic conductance and average temperature of the two contacts, respectively, and k; and Ke are the lattice and electronic contributions to the thermal conductance, respectively. zT may, therefore, be increased by designing materials that have either higher thermoelectric power factor S2G, or lower thermal conductivity. The relatively high zT seen in recent experiments on SiNWs has been attributed to both of these effects: an increase in S

(a) E-mail: a.mostofi8imperial.ac.uk

resulting from enhanced phonon drag [4], and a decrease in k resulting from surface scattering of phonons [3,6].

By analogy with 2DSLs, which show a reduction in K compared to its value both in the bulk and in the alloy limit [7], superlatticed NWs have been proposed as a possible route toward high-zT thermoelectrics [8], through both enhancement of the power factor S2G and reduction of k;, as compared to pristine NWs. Experimental evidence for Si-SiGe NWs [9] supports the idea that superlatticing results in a reduction in k; , although direct experimental evidence for increased S2G is still missing.

In this letter, we use a combination of first-principles density-functional theory simulations and calculations with interatomic potentials in order to compute zT for axially heterostructured Si-Ge NWs within the coherent transport regime using the Landauer-Buttiker approach [10,11]. More specifically, we calculate zT at 300K for thin (<2nm diameter), p-type (110), (111) and (211) H-passivated SiNWs (fig. 1, top) containing: a) single axial Ge heterostructures with lengths ranging from 0.4 nm to 4.3 nm (fig. 1, middle); b) multiple Ge heterostructures of uniform length, distributed along the length of the NW either randomly, periodically, or as

Fig. 1: (Colour on-line) Top: Cross-sections of SiNWs. Labels indicate the crystal direction of the longitudinal (z) axis (pointing into the page). Diameters (left-to-right): 0.78nm, 1.02 nm, 1.44 nm, 1.14 nm and 1.06 nm. Middle: A single Ge heterostructure in a SiNW (Ge and Si atoms in magenta and blue, respectively). Bottom: A multiple heterostructure nanowire (MHNW) with an arbitrary distribution of Ge hetero-structures.

a Fibonacci chain (fig. 1, bottom); and c) multiple Ge heterostructures whose lengths approximately follow a Gaussian distribution.

Our calculations demonstrate that: i) the introduction of a single Ge heterostructure in a SiNW can lead to a 3.5-fold increase in zT as compared to the equivalent pristine SiNW; ii) this can be further enhanced to a 7.4-fold increase by introducing multiple Ge heterostruc-tures and by controlling their spatial distribution along the length of the SiNW; iii) this observed enhancement in zT is almost entirely due to a reduction in thermal conductivity rather than an increase in the power factor S2G —indeed, we find that introducing Ge heterostructures in an SiNW always results in a decrease in S2G; and iv) introducing further disorder, in the form of a variability in the length of the Ge heterostructures within a SiNW, results in a decrease in zT as compared to the case in which the heterostructures are all identical, highlighting the importance of precise atomic-scale control that may be required in order to fabricate high-zT NWs.

Our method (described in detail elsewhere [12]), in which accurate yet compact model Hamiltonians of large-scale systems are constructed from first-principles calculations, enables us to study transport through meso-scale systems with modest computational cost: our largest simulations consist of a conductor region of length 116 nm (8432 atoms) coupled to semi-infinite leads. Our procedure is largely automated, which has made it possible to perform high-throughput calculations and undertake a comprehensive study of a large structural parameter space. With little modification, our general approach may be easily used to calculate transport properties in other quasi-one-dimensional systems.

We note that analogous approaches have been used recently for calculating electronic and/or thermal transport in large-scale 1D systems, e.g., in SiNWs [13,14] and carbon nanotubes [15-17].

Electronic transport properties. — Starting from a plane-wave density-functional theory (PW-DFT) calculation1 , a unitary transformation is applied to the extended ground-state eigenfunctions in order to obtain maximally localized Wannier functions (MLWFs) [19] and, hence, the Hamiltonian matrix in the basis of MLWFs. Due to the localized nature of MLWFs in real-space, the Hamil-tonian matrix can be spatially partitioned and used in so-called "lead-conductor-lead" Landauer-Buttiker [10,11] transport calculations, using standard Green function techniques [12,20-23]. In our case, the semi-infinite "leads" are pristine (H-passivated) SiNWs and the central "conductor" region comprises of SiNW with some axial distribution of Ge heterostructures. An example of such a system exhibiting a single Ge heterostructure is shown in fig. 1 (middle panel). Although we begin with PW-DFT calculations with periodic boundary conditions, we determine electronic transport properties under open boundary conditions. Once the electronic density of states of the conductor and the transmission function T(e) are calculated, one can write [24,25] G = e2L0(^), S = Li(^)/eTLo(^) and Ke = 1 (L2(m) - [Li(^)]2/Lo(^)j, where

= 2 £ T (e)(e - ,)-(

and f (e, = 1 /{exp[(e — /kBT] + 1} is the Fermi-Dirac function at chemical potential In this letter, we focus on hole transport, so that we can associate ^ with a carrier concentration that is driven by p-doping2.

As full DFT structural relaxation of our large heterostructured NWs would have been computationally intractable, the atomic configurations used for the electronic calculations were built by piecing together unit cells of pristine Si and Ge NWs whose equilibrium lattice parameters3 were calculated separately with DFT. For the smallest NWs considered, the results obtained from our approach had only small quantitative and no qualitative difference as compared to those from the equivalent fully relaxed structure [26].

Figure 2 shows the maximum4 thermoelectric power factor S2Gmax for a range of Ge heterostructure lengths in (110) (black solid lines), (111) (red dashed line) and (211) (blue dot-dashed line) SiNWs. We note first that in

xWe use the Quantum-Espresso package [18], the local-density approximation for exchange and correlation, norm-conserving pseudopotentials, a 400 eV energy cut-off for the PW basis set, and r-point sampling of the Brillouin zone.

2This doping is "artificial" in the sense that we do not directly include dopant atoms in our calculation.

3These were 3.775 A (3.910 A), 9.224A (9.497A), and 6.470 A (6.692 A), for our largest diameter Si (Ge) NWs in the (110), (111) and (211) growth directions, respectively.

4As can be seen from eq. (1), the electronic transport coefficients are functions of chemical potential Throughout this work, S2Gmax and zTmax are defined to be the maximum values of the power factor and the figure of merit, respectively, as a function of The maximal power factors shown in fig. 2 are obtained, in all cases, for values of ^ within 30 meV of the Si valence band edge.

й 2 % 2

А^Д<110> 0.78nm 0-0<110> 1.02nm B-O<110> 1.44nm 0-0<111> 1.14nm Q-Q<211> 1.06nm

Pristine Si

Heterostructure Length (nm)

Pristine Ge

Fig. 2: (Colour on-line) Maximum thermoelectric power factor S2Gmax at 300 K for single Ge heterostructures in (110) (black solid lines), (111) (red dashed lines) and (211) (blue dot-dashed lines) SiNWs. Three diameters 0.78nm (triangles), 1.02 nm (circles) and 1.44 nm (squares) are shown for the (110) direction. Pristine SiNWs are shown plotted as zero heterostructure length and pristine Ge NWs values are shown on the right.

A-A<110> 0.78 nm 0-©<110> 1.02 nm B-H<110> 1.44 nm ®0<111> 1.14 nm Q-Q<211> 1.06 nm

"s-g—e—e--

e~e—o—G—e—в—e— о

-0' ri ~ e-5

Pristine 1

Heterostructure Length (nm)

4 Pristine Ge

Fig. 3: (Colour on-line) Dependance of kj (top panel) and zTmax (bottom panel) at 300 K as a function of Ge heterostruc-ture length for (110), (111) and (211) SiNWs. Labelling is equivalent to fig. 2, again plotting pristine SiNWs as zero heterostructure length and pristine Ge NWs also shown on the right.

no case does the introduction of an axial heterostructure result in an increase of S2Gmax, the value of which is, at best, approximately the same as that of a pristine SiNW. The similarity of the results between pristine SiNWs and GeNWs is also interesting to note. Since all the NWs investigated only have a single channel that is available for conduction at the top of the valence manifold, we confirm that the most important factor for S2Gmax in quasi-one-dimensional systems is the number of conducting channels at this edge [27].

The oscillations in fig. 2 can be explained by a model in which the heterostructure is considered as a 1D quantum potential well of width L, corresponding to the length of the heterostructure, and depth V0, corresponding to the band offset between Si and Ge. The reflection amplitude of a wave incident on such a well vanishes when the well-known Fabry-Perot resonance condition is satisfied, qL = nn, where q is the wave vector inside the well and n is an integer. For holes entering the heterostructure, this condition gives En(L) = — ^m L + V0, where m* is the effective mass of holes and En (L) are energies at which resonances occur in the transmission. For the (111) NWs, which show the strongest oscillations in S2Gmax, plotting the resonance energies against 1 /L2 produces an excellent linear fit (not shown), with correlation coefficient r2 = 0.998, giving m* =0.28me and V0 =0.32eV. Models such as this may be a useful additional tool for the optimization of heterostructure lengths in thermoelectric devices [28].

Phononic transport properties. — We determine the lattice thermal conductance k; in an analogous way to the electronic conductance, by segmenting the system into a lead-conductor-lead geometry. Instead of finding the Green function of a Schrodinger-type eigenvalue problem, we determine the Green function that solves the eigenvalue problem relating nuclear displacements u to the dynamical matrix K and phonon frequency w: Ku = w2u. Applying the thermal equivalent of the Caroli formula [20,29], we obtain the phonon transmission function, T(w), in the

limit of non-interacting phonons and coherent phonon transport. One can then write [30]

ft2 Гж 2 T

Kl = 2ПкВТ2 J0 T(eW*Bt - 1)2

dw. (2)

Determination of the dynamical matrix using first-principles methods is computationally intractable for the large NW supercells with heterostructures that are considered here. To structurally relax the NWs and obtain their dynamical matrices, therefore, we use Tersoff potentials [31], which have been shown to give accurate values for lattice thermal conductivities for thin pristine SiNWs, as compared to DFT calculations [32]. It is worth noting that the approach outlined above neglects Umklapp scattering, which would further decrease k; at the temperature with which we are concerned (300 K).

Results for K; are shown in fig. 3 (top panel) for the same single Ge heterostructure SiNWs discussed earlier. We find that, by introducing a single Ge heterostructure, the lattice thermal conductivity can be reduced by a factor of five for the (111) growth direction, as compared to the corresponding pristine SiNW, giving a value of 0.1nWK-1. Reductions in the (110) and (211) direction are also significant (approximately a factor of four). We tentatively suggest that the longer unit cell in the (111) direction may account for the greater reduction seen in that growth direction. It may also be seen that, in the (110) direction, K; increases with diameter as more phonon modes become available. We also observe this trend in larger diameter pristine SiNWs in the (111) and (211) directions (results not shown).

zT for SiNWs with a single Ge heterostructure.

— Figure 3 (bottom panel) combines our results for electronic and phononic transport coefficients for the single Ge heterostructure systems discussed above and shows our calculated values of zTmax as a function of hetero-structure length. It can be seen that heterostructured

Left Lead

- Conductor ■

Right Lead

Table 1: Structural details of the MHNWs in the (111) and (211) growth directions. The different MHNWs (periodic, Fibonacci and random) are built by placing ten single heterostructures along the length of the wire.

Fig. 4: (Colour on-line) Schematic illustration of MHNWs studied. Top: periodic conductor; middle: a typical random arrangement; bottom: Fibonacci chain pattern, with units A and B (see text). Si and Ge sections are in blue and magenta, respectively.

SiNWs in the (111) direction display the greatest values of the figure of merit zTmax ~ 1.4. Such high values, however, are not found consistently across the range of heterostruc-tures studied. Such variations may limit the values of zT observed in realistic SiNWs since experimental control over heterostructure length is, currently at least, limited to length scales comparable to, or greater than, the differences in length that are investigated here [33]. In the (110) and (211) directions, we find zTmax < 1, mainly due to the higher lattice thermal conductivities found in these systems. We note that, across the range of systems studied, the ratio of lattice and electronic thermal conductances, K;/Ke, lies between 3 and 10, therefore, k; is the dominant contribution to the denominator of zT. This emphasizes the importance of reducing the lattice thermal conductivity for high-zT NWs.

zT for SiNWs with multiple Ge heterostruc-tures. — Next, we consider much longer SiNWs with many Ge heterostructures along their length. Such multiple heterostructure nanowires (MHNWs) are shown schematically in fig. 4. These systems are too large for brute-force PW-DFT calculations. Instead, we use the Hamiltonian matrices of single heterostructure calculations as "building-blocks" for constructing model Hamiltonians of much larger (up to ~ 8400 atom) MHNWs, with negligible loss of accuracy. Our approach, which relies on exploiting the nearsightedness of electronic structure that becomes manifest when Hamiltonian matrices are represented in a basis of MLWFs, is described in detail in ref. [12].

Once the model Hamiltonian for the MHNW is constructed, the electronic transport properties under open boundary conditions are calculated in exactly the same way as described above for SiNWs with single Ge heterostructures. For the lattice thermal conductivity, an analogous "building-block" scheme is used in which the (short-ranged) dynamical matrices of the single heterostructures that comprise the MHNWs are combined to construct dynamical matrices for the MHNWs. Under the assumption that phonons remain phase coherent across the length of the MHNW, this dynamical matrix is then used to calculate the phonon transmission function T(w) and, hence, the coherent lattice thermal conductance KC°h according to eq. (2).

(111) (211)

Single heterostructure length (nm) 3.80 1.34

Periodic Total MHNW length (nm) Total number of atoms 93.3 7208 50.9 3608

Random Total MHNW length (nm) Total number of atoms 93.3 7208 49.6 3520

Fibonacci Total MHNW length (nm) Total number of atoms 93.3 7208 49.6 3520

It is unclear whether the phase coherence length of phonons is comparable to the lengths of the MHNW systems that we consider (up to ~ 116nm). Therefore, we also calculate the lattice thermal conductance in an ohmic regime, K°hm, in which the total resistance of a given MHNW is the sum of the thermal resistances of each individual heterostructure that constitutes the MHNW [34]. For a NW with N heterostructures, each of which in isolation gives a transmission 7i(E), we compute the transmission function as [35]

T0hm(E) :

N - (1 - N)7/70'

where T =^2i=i 7i/N is the average transmission of the isolated heterostructures, and 70 is the transmission for the pristine NW. Having obtained Tohm(E), K°hm is calculated from eq. (2). KÇ?oh and K0hm are used to estimate upper and lower bounds for zT.

We consider MHNWs in the (111) and (211) growth directions, with diameters of 1.14nm and 1.06nm, respectively, and Ge heterostructure lengths of 3.80 nm and 1.34 nm, respectively. For each of these two growth directions, three qualitatively distinct heterostructure distributions are considered (shown in fig. 4): i) random, ii) periodic, and iii) Fibonacci chain5, each with ten heterostructures along the length of the NW. Structural details are given in table 1. For MHNWs with a random distribution of heterostructures, T(e) and T(w) are ensemble averaged over 250 independent configurations.

Figure 5 (top panel) shows the thermal conductances of the MHNWs detailed in table 1 and compares them to the pristine values. A prominent feature is the large

5A Fibonacci chain is an example of a 1D quasicrystal [36]: it displays local translational symmetries, yet remains aperiodic in toto. Exceptionally low kj values have been reported experimentally for 3D quasicrystals [37], thus the introduction a Fibonacci chain distribution of heterostructures could be a systematic method to reduce kj. The Fibonacci chain MHNWs are designed such that the length ratio of structural units A and B that comprise them is as close as possible to the golden ratio (1 + \/5)/2. These structural units each contain a Ge heterostructure between lengths of SiNW and the total chain is built with three iterations (n = 0, 1, 2) of the sequence: An+i = AnBn, Bn+1 = An, with Aq = A and Bo = B.

K; (nWK )

Pristine Periodic Fibonacci Random

Pristine Periodic Fibonacci Random

Pristine Periodic Fibonacci Random

Pristine Periodic Fibonacci Random

K (nWK )

<111> Ohmic <111> Coherent <211> Ohmic Ta <211> Coherent

on KCoh, we see that, in the coherent regime, random patterning results in values of the figure of merit as high as zT = 2. Conversely, (111) MHNWs display significant reductions in S2G as the disorder increases, which tend to counteract similar decreases in K^oh, leaving zTmax approximately constant at ~ 1.7. In both (111) and (211) MHNWs the calculated zTmax increases if the thermal transport is assumed to be ohmic, since K0hm < K^oh. In this regime, K0hm is invariant with respect to the

6The electronic properties are always calculated within a fully coherent model but, depending on whether

Kohm or Kcoh is used,

the value of ß at which zTmax occurs changes slightly and, hence, the value of S2G, which is dependent on ß.

0.5 • 3

:V^ | 1 S2 h0

y 0.1 1

QQ Near-periodic, Coh. QQ Near-periodic, Ohm Periodic, Coh. Periodic, Ohm. 0.05 0 1

Number of hets.

Number of hets.

Number of hets.

S G (x10 WK )

Fig. 5: (Colour on-line) Transport properties at 300 K of (111) (red, solid bars) and (211) (blue, striped bars) periodic, Fibonacci and random patterned MHNWs, and comparison to the pristine cases. Results from fully coherent phononic transport (111) ((211)) calculations have a solid shading (bold stripe), while results in the ohmic regime have a lighter shading (fine stripe). Top panel: the lattice (left) and electronic (right) contributions to the thermal conductance, respectively. Bottom panel: zTmax (left) and S2G (right) at the value of j that maximizes zT for each system.

reduction of k; due to heterostructuring, with (111) MHNWs displaying smaller values than (211). Using the coherent model (red solid bars/blue bold striped bars), K; is reduced by factors of between five and eight when compared to the pristine results and reduces as the disorder is increased (periodic to Fibonacci to random patterning). The ohmic model (red shading/fine blue stripes), results in reductions of k; by factors of ~12 and 8.5 in (111) and (211) MHNWs, respectively. We note that KCoh/Ke and K0hm/Ke are found to be between 2.5 and 7 —only marginally smaller than those values we obtained for single heterostructure NWs.

zTmax and S2G are shown in fig. 5 (bottom panel)6. It is striking that in no case does S2G increase due to heterostructuring, and the (211) direction performs best over the range of MHNWs considered, showing only small decreases (with respect to pristine) as the disorder increases from periodic to random. Together with the pronounced effect that increased disorder has

Fig. 6: (Colour on-line) Comparison of S2G (left panel), kj (middle panel) and zTmax (right panel) at 300 K in periodic (green triangles) and near-periodic (red squares) (111) MHNWs as a function of the number of Ge heterostructures (hets.). Results using the coherent (coh.) and ohmic (ohm.) phonon transport models are shown with solid and dashed lines, respectively.

distribution of heterostructures, therefore, zTmax in this regime will follow the behaviour seen in S2G with a value of zT = 2.3 for the (211) direction (almost independent of the distribution of heterostructures), and up to zT = 3 in the (111) direction, with a periodic arrangement of heterostructures.

The effect of variability of heterostructure length. — The MHNWs that we have discussed thus far consist of multiple instances of identical heterostructures. Experimental synthesis techniques do not have this level of atomic precision, therefore, we have investigated the effect of introducing some variability of the lengths of the heterostructures that comprise the MHNW. In particular, we compare periodic patterned MHNWs with identical heterostructures with "near-periodic" MHNWs that are comprised of heterostructures whose lengths are Gaussian distributed about a mean length that is given by the heterostructure length used in the "true" periodic case, with a standard deviation a that corresponds to approximately 1/3 (1/2) a unit cell in the (111) ((211)) direction. For the near-periodic MHNWs, the transmission functions are ensemble averaged over 250 independent configurations of the disorder to model a "typical" MHNW of this type.

We consider MHNWs with up to twelve heterostructures along their length. Figure 6 shows a comparison between near-periodic (red squares) and periodic (green triangles) MHNWs in the (111) growth direction. The resulting values for zTmax are shown in the right panel. In the ohmic phonon transport regime (dashed lines), the near-periodic system displays a dramatic reduction in zTmax as compared to the periodic case, which arises from the sharp reduction that is found in S2G (left panel), combined with the fact that K; does not decrease very much (middle panel). When considering the coherent regime (solid lines), the reduction in S2G for the near-periodic MHNW is also large (left panel), as compared to the periodic MHNW, but associated decreases in K; are also observed (middle panel) so that the resultant drop

in zTmax (right panel), as compared to the periodic case, is much less pronounced. For the near-periodic MHNWs, both phonon transport regimes display a maximum in zTmax with respect to heterostructure length after the introduction of approximately four heterostructures.

We note that the significant decreases in S2G due to the variability in the heterostructure length is consistent with our earlier conclusion that increased disorder tends to reduce the power factor, as was seen when comparing periodic, Fibonacci and random distributions. We also note that in the ohmic phonon transport regime, there is little difference in k; between periodic and near-periodic MHNWs, which follows the earlier observation that there is almost no dependence of k; on heterostructure length in single heterostructure NWs (fig. 3, top panel).

Finally, comparing near-periodic MHNWs in the (111) and (211) growth directions, we find that S2G;111^ < S2G<211\ which may have been expected from the stronger dependence of S2Gmax on heterostructure length in (111) SiNWs with a single heterostructure (fig. 2). We also find that K;111^ < K;211\ which also could have been predicted from the trends observed for single heterostructure SiNWs (fig. 3). However, the delicate balance between S2G and k; make it difficult to use calculations on SiNWs with a single heterostructure to predict trends in zTmax for our near-periodic MHNWs, highlighting the need for accurate first-principles approaches. Among the near-periodic MHNWs studied, the (111) direction with four heterostructures performed best, with zT ~ 1.5-1.6.

In conclusion, we have performed first-principles calculations on thin, p-type (110), (111) and (211) SiNWs with Ge heterostructures. In all cases studied, a decrease of thermoelectric power factor S2G is observed when a heterostructure is introduced, and any increase in the figure of merit zT is due to a corresponding reduction in Kl. We have built model Hamiltonians for MHNWs with over 8400 atoms while retaining first-principles accuracy. A similar method was applied to the dynamical matrices of MHNWs to obtain the thermal conductance Kl for such structures. In such MHNWs we again find that S2G is always reduced and that increases in zT are driven predominantly by significant decreases in k;. We find values as high as zT = 3 in (111) MHNWs with periodic arrangements of Ge heterostructures. The intricate balance between S2G and k;, however, makes zT strongly dependent on the details of the system at the atomic level: in structures that model the kind of disorder that may be present in realistic MHNWs, more modest values of zT = 1.6 are found, which is still a factor of four greater than the pristine SiNW case. Our calculations suggest axial heterostructuring to be a promising route to high-zT nanowire thermoelectrics.

We are grateful to the High Performance Computing Facility at Imperial College London, and to the EPSRC

and E.ON's International Research Initiative. We thank N. Poilvert, N. Marzari and Y.-S. Lee for discussions.

REFERENCES

[1] Cui Y. et al., Science, 293 (2001) 1289.

[2] Garnett E. and Yang P., Nano Lett., 10 (2010) 1082.

[3] Hochbaum A. I. et al., Nature, 451 (2008) 163.

[4] Boukai A. I. et al., Nature, 451 (2008) 168.

[5] Hicks L. D. and Dresselhaus M. S., Phys. Rev. B, 47 (1993) 16631.

[6] Li D. et al., Appl. Phys. Lett., 83 (2003) 2934.

[7] Borca-Tasciuc T. et al., Superlattices Microstruct., 28 (2000) 199.

[8] Lin Y.-M. and Dresselhaus M. S., Phys. Rev. B, 68 (2003) 075304.

[9] Li D. et al., Appl. Phys. Lett., 83 (2003) 3186.

10] Landauer R., Philos. Mag., 21 (1970) 853.

11] Buttiker M. et al., Phys. Rev. B, 31 (1985) 6207.

12] Shelley M., Poilvert N., Mostofi A. A. and Marzari N., http://arxiv.org/abs/1101.3754 (2011).

13] Markussen T. et al., Phys. Rev. B, 74 (2006) 245313.

14] Markussen T. et al., Phys. Rev. Lett., 103 (2009) 055502.

15] Lee Y.-S. et al., Phys. Rev. Lett., 95 (2005) 076804.

16] Rocha A. R. et al., Phys. Rev. Lett., 100 (2008) 176803.

17] Savic I. et al., Phys. Rev. Lett., 101 (2008) 165502.

18] Giannozzi P. et al., J. Phys.: Condens. Matter, 21 (2009) 395502.

19] Marzari N. and Vanderbilt D., Phys. Rev. B, 56 (1997) 12847.

20] Caroli C. et al., J. Phys. C: Solid State Phys., 4 (1971) 916.

21] Lee D. H. and Joannopoulos J. D., Phys. Rev. B, 23 (1981) 4988; 4997.

22] Meir Y. and Wingreen N. S., Phys. Rev. Lett., 68 (1992) 2512.

23] Nardelli M. B. et al., Phys. Rev. B, 64 (2001) 245423.

24] Sivan U. and Imry Y., Phys. Rev. B, 33 (1986) 551.

25] Esfarjani K. et al., Phys. Rev. B, 73 (2006) 085406.

26] Shelley M., Theory and Simulation of Semiconducting Nanowires for Thermoelectric Applications, PhD Thesis, Imperial College London (2011).

27] Kim R. et al., J. Appl. Phys., 105 (2009) 034506.

28] Wang S. and Mingo N., Phys. Rev. Lett., 79 (2009) 115316.

29] Wang J.-S. et al., Eur. Phys. J. B, 62 (2008) 381.

30] Yamamoto T. and Watanabe K., Phys. Rev. Lett., 96 (2006) 255503.

31] Tersoff J., Phys. Rev. B, 37 (1988) 6991; 39 (1989) 5566.

32] Markussen T. et al., Nano Lett., 8 (2008) 3771.

33] Wen C. et al., Science, 326 (2009) 1247.

34] Markussen T. et al., Phys. Rev. Lett., 99 (2007) 076803.

35] Markussen T. et al., Phys. Rev. B, 79 (2009) 035415.

36] Levine D. and Steinhardt P. J., Phys. Rev. Lett., 53 (1984) 2477.

[37] Pope A. L. and Tritt T. M., Thermal conductivity of quasicrystalline materials, in Thermal Conductivity, edited by Tritt T. M. (Springer) 2004, pp. 255-259.