Selective Transmission of Phonons in Molecular Junctions with

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Selective Transmission of Phonons in Molecular Junctions with Nanoscopic Thermal Baths Leonardo Rafael Medrano Sandonas, Alvaro Rodriguez Mendez, Rafael Gutierrez, Jesus M. Ugalde, Vladimiro Mujica, and Gianaurelio Cuniberti J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 26 Mar 2019 Downloaded from http://pubs.acs.org on March 26, 2019

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The Journal of Physical Chemistry

Selective Transmission of Phonons in Molecular Junctions with Nanoscopic Thermal Baths Leonardo Medrano Sandonas,†,‡ Álvaro Rodríguez Méndez,† Rafael Gutierrez,∗,† Jesus M. Ugalde,¶,§ Vladimiro Mujica,k,§ and Gianaurelio Cuniberti†,‡,⊥ †Institute for Materials Science and Max Bergmann Center of Biomaterials, TU Dresden, 01062 Dresden, Germany ‡Center for Advancing Electronics Dresden, TU Dresden, 01062 Dresden, Germany ¶Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), P.K. 1072, 20080 Donostia, Euskadi, Spain §Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain kArizona State University, School of Molecular Sciences, Tempe, AZ 85287, USA ⊥Dresden Center for Computational Materials Science (DCMS), TU Dresden, 01062 Dresden, Germany E-mail: [email protected]

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Abstract A fundamental problem for thermal energy harvesting is the development of atomistic design strategies for smart nano-devices and nano-materials that can be used to selectively transmit heat. We carry out here an extensive computational study demonstrating that heterogeneous molecular junctions, consisting of molecular wires bridging two different nano-contacts, can act as a selective phonon filter. The most important finding is the appearance of gaps on the phonon transmittance spectrum, which are strongly correlated to the properties of the vibrational spectrum of the specific molecular species in the junction. The filtering effect results from a delicate interplay between the intrinsic vibrational structure of the molecular chains and the different Debye cutoffs of the nanoscopic electrodes used as thermal baths.

INTRODUCTION Heat transport in most materials is a consequence of two interrelated processes: electron and phonon transport. Phonon transport is the dominant mechanism whenever electronic excitations and electron transport can be neglected. 1–4 In metals heat flow is dominated by electrons, while in insulators heat is transmitted solely by phonons. This study focuses on the phononic mode of heat transfer in molecular junctions consisting of a molecule attached to two nano-contacts. Addressing phonon transport requires the efficient calculation of the corresponding force constant matrices, which can be obtained using first-principles 5–8 or semi-empirical 9–13 electronic structure approaches. A quantum description of phonon transport can then be achieved by using propagation techniques based on e.g. Green’s functions. 14 For elastic transport, the result is analogous to the Landauer theory 15 of electric conductance for mesoscopic systems that is based on the idea of counting discrete conductance channels for electronic transport through a constraint. In this picture, conductance is interpreted as a quantum scattering process and the total conductance is the sum of the partial conductances through individual channels. Important pioneering work on this field has been carried out 2


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by Nitzan and co-workers, 16–19 who have considered, in addition to the phononic regime, the explicit interplay between vibrational and electronic degrees of freedom and their role in heat dissipation in molecular junctions. 20 An important breakthrough in the field was the theoretical prediction 21 and subsequent observation in a mesoscopic nanostructure 22,23 of the quantization of the phononic thermal conductance at low temperatures, in analogy to the conductance quantization for charge transport. In contrast to the electrical conductance quantization (with conductance quantum=e2 /h), the quantum of thermal conductance κ0 is not a constant, but depends on the temperature T : κ0 = π 2 kB2 T /3h, with kB being the Boltzmann constant and h the Planck constant. This result already demonstrates a fundamental difference between charge and phonon transport. Another basic difference relies on the different energy ranges determining the corresponding transport properties: in the case of electrons the relevant energy window lies around the Fermi level, while for phonons the thermal conductivity is formally an integral result of the whole vibrational spectrum. The difficulty of working with a broad spectrum of excitations naturally possesses major challenges in designing thermal devices such as cloaks and rectifiers, 3,4 phononic wires and phononic crystals, 24,25 or processing information based on phononic computing. 26 However, this feature can also be of advantage for designing phonon filters, since mismatches in the vibrational spectral densities in a composite nanostructure can be exploited to selectively tune the phonon transmission channels. The idea of phonon filtering has been previously considered for interfaces between different mesoscopic materials. 27–35 However, if a molecular system is used to bridge the two contacts −this situation can be viewed as an extended interface situation−, some questions arise that are connected to specific molecular features: Can phonon filtering be further engineered by modifying the chemical composition and structure of the molecular bridge? How can the phenomenon be related to selection rules based on the interplay between the phonon modes in the two electrodes of the junction and the discrete vibrational spectrum of the molecule? Addressing both issues entails the possibility of designing devices and materials specifically 3



tailored to control interfacial heat and energy distribution. 36–43 A macroscopic analogue of such a device, although based on an entirely different physical mechanism, namely non-linear flows, is the Ranque-Hilsch vortex tube, 44,45 where a gas flowing at a given temperature is split into two streams at different temperatures. That it is indeed possible to fabricate such a device without violating the laws of thermodynamics, was established during the last century when it was used for train refrigeration. 46 The nano-junction phonon filter described below does not, of course, operate by splitting molecular flows in gas phase, but it could be the seed for a true energy splitting nano-device by exploiting selective filtering at different interfaces.

COMPUTATIONAL METHOD Phonon filter model In this study, we proposed nano-junctions consisting of two colinear (6,6)-nanotubes (NT) joined by a central molecular structure as selective molecular-scale phonon filter, see Figure 1 for a schematic representation. Here, the nanotubes act as two heat baths kept at the same temperature. For the sake of specificity, we consider the left NT as a reference bath with a broad phonon frequency spectrum that is kept fixed for all studied junctions, thus playing the role of a "source" of phonon modes. The molecular system in the central part acts as a mode selector, and selected modes are then propagated to the right contact. The Debye cut-off ωD in the frequency spectrum of the right bath can be additionally varied by modifying the material the NT is made of, see Figure 1. For this we use, besides carbon NTs, boron-nitride (BN) and Silicon carbide (SiC) nanotubes. As we will show below, the filtering capability sensitively depends on the existence of a mode-specific propagation that results from the combined effect of molecular vibrations selection rules and the overlap of the contact spectral densities with the molecular region. We remark that our goal is to stress the possibility of chemically engineering the interface between two contacts to achieve selective 4


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filtering of vibrational modes. Hence, although any nanostructure bridging our nanoscopic thermal baths will in general terms reduce the overall phonon transmission function due to additional scattering, the selective suppression of transmission can only be achieved by specific molecular species. In this sense, we do not consider as a filter such a "trivial" global reduction of transmission. Further below we propose an effective measure of the filtering quality of a molecular junction based on a modification of the well-known KullbackLeibler divergence in information theory. We also remark that the issue of the vibrational mismatch between thermal baths and molecules has already been experimentally studied in the context of self-assembled alkane-based monolayers formed between metallic substrates made of different elements. 47 Our investigation addresses, however, features at the level of single or few molecules.

Transport calculation method The phonon transport problem is treated using atomistic Green’s function techniques (AGFT) as implemented in an in-house version of the DFTB+ code. 12,13,48 For calculating the electronic energies, from where the force constants will be computed, we have considered different Slater-Koster parametrizations, depending on the molecular junction under study. We use for the case of the CNT-X-CNT junctions the mio-0-1 parametrization, 49 and for the CNT-X-BNNT and CNT-X-SiCNT junctions the matsci parametrization. 50 Next, a partition of the whole system into left bath, device region, and right bath is carried out. The mass-weighted force constant matrix, K, for each subsystem is obtained by numerically differentiating the forces calculated using the DFTB method. 51 In our study, we neglect phonon-phonon interactions, since we are mostly interested in the filtering effect in the region of low up to room temperatures. At these temperatures high-frequency modes, which are expected to be stronger affected by phonon-phonon coupling due to the larger available scattering space, are not contributing much to the thermal conductance. Moreover, these modes are also those which experience the strongest transmission suppres5



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sion. Even though, a more quantitative statement about the influence of phonon-phonon interactions can only be achieved by a many-body formulation explicitly including third and fourth order anharmonicities. 52,53 Using the previously obtained DFTB-based atomistic information, the retarded harmonic phonon Green’s function (Gr ) for the central region (including the coupling to the thermal baths through self-energies) can be calculated, Gr (ω) = (ω 2 I − K − ΣrL (ω) − ΣrR (ω))−1 . Here, I is the identity matrix, and ω the vibrational frequency. ΣL/R are the self-energies for the left (L) and right (R) baths, respectively, which are calculated as described in Refs. 14,54 Having obtained the GF and self-energy matrices, the phonon transmission spectrum, τph (ω), can be computed as τph (ω) = Trace (Gr (ω)ΓL (ω)Ga (ω)ΓR (ω)) .

(1)

h i ΓL/R (ω) = i ΣrL/R (ω) − ΣaL/R (ω) are the spectral densities and Ga (ω) = [Gr (ω)]† is the advanced Green’s function. As far as only elastic scattering processes are considered (no anharmonic interactions or coupling to electronic degrees of freedom), this formalism is mathematically similar to the Landauer approach for charge transport. For systems in which elastic scattering is dominant, AGFT describes phonon transport very accurately. The corresponding thermal conductance can then be calculated as

κph

1 = 2πkB T 2

Z 0

∞

(~ω)2

e~ω/kB T τph (ω)dω, (e~ω/kB T − 1)2

(2)

where kB and h are Boltzmann and Planck constants, respectively. The previous expression is obtained by a linear expansion in the applied temperature difference ∆T of the quantity NB (T + ∆T ) − NB (T ), where NB is the Bose-Einstein distribution. The phonon density of 2ω states per atomic site is defined as, ηi (ω) = − (Im Gr [ω])ii . In our calculations, geometπ rical optimizations of the setups shown in Figure 1 were performed until the absolute value of the inter-atomic forces was below 10−6 atomic units. The monomers of ethylene, benzene, and azobenzene have a length of around 2.7 Å, 4.6 Å, and 10.7 Å, respectively. 6


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RESULTS AND DISCUSSION As a first step, we address the influence of the interconnecting molecular chains on the filtering effect (see Figure 2(a)). This is shown in Figure 2(b) for a junction where both thermal baths consist of the same material, (6,6)-CNTs, with a Debye frequency of ∼ 1800 cm−1 . This has the advantage that any selective filtering effects can be traced back to the molecular system, since the spectral densities of both thermal baths are identical. As bridging molecular systems we have chosen four parallel chains of ethylene, benzene, and azobenzene (see Figure S1 in Supporting Information (SI) for the ethylene case). Figure 2(b) shows the phonon transmission functions τph for both benzene and azobenzene chains, where the emergence of spectral gaps in the phonon transmission, induced by the presence of the molecular chains, can be clearly seen. The interplay between chain length, chain composition, and the filtering effect becomes evident from the figure: while in both systems the bridging molecular chains induce local phonon gaps (highlighted with dashed-line circles in Figure 2(b)) in the transmission spectrum, there is a clear difference between the influence of benzene and azobenzene on the overall transmission spectrum. The emergence of phonon gaps can be best seen by comparing the transmission functions of the nanotube-molecule junctions with the corresponding transmissions of the infinite carbon NT (as a grey background, providing the reference spectral range for phonon selectivity) as well as with the transmission of an infinite, periodic molecular chain (brown background), where both baths and the scattering region are composed of the same molecule (benzene or azobenzene). First notice that due to phonon scattering in the molecular region and at the contact interfaces, the overall transmission of the molecular junctions is reduced by roughly a factor of four when compared with the infinite CNT. In the case of benzene (upper panel of Figure 2(b)), various interesting features are found: first, in the frequency regions centered around 500 cm−1 and 1000 cm−1 a progressive suppression of the transmission with increasing length is observed, already developing into a full phonon gap for chains composed of 12 benzene monomers. Additionally, the high-frequency sector of the transmission function 7



at frequencies above 1200 cm−1 is also strongly affected by the molecules, showing transmission suppression within various frequency windows. Much less affected are, as expected, the low frequency modes below 500 cm−1 , where the phonon channels are mostly dominated by out-of-plane modes (XY-plane), see Figure S2 in the SI. As shown in Figure S3 in the SI, in junctions with shorter lengths low-frequency vibrations are mostly determined by the CNTs, while molecular modes in this range become increasingly dominant, as expected, with increasing chain length, since long-wave length modes can then better propagate. 38 For the azobenzene junction, the influence of the molecular wires become even stronger. As shown in the lower panel of Figure 2(b), chains with only two monomers are already enough to open phonon gaps. Moreover, azobenzene chains are also able to filter out roughly half of the spectral range in the phonon transmission, i.e. almost all phonon channels above 1000 cm−1 are blocked or have a very low transmission. In contrast to the benzene case, the transmission at low frequencies is strongly reduced; this may be related to the lower number of modes as quantified by the cumulative phonon transmission A(ω) (computed for the three types of molecular chains in Figure S4 in the SI) and a larger structural distortion of the azobenzene chains (when compared with the benzene chains), which induces additional scattering for lower frequency modes. 42,55 This effect is also reflected in the thermal conductance κph values plotted in Figure S5 and S6 in the SI. Indeed, besides the considerable reduction of the thermal conductance compared to the pure CNT (by a factor of ∼4), κph (T ) values for a CNT-junction with two benzene monomers are larger than those in the ethylene and azobenzene junctions. Also, the thermal conductance for all monomers displays a linear dependence with the number of parallel molecular chains j bridging the nanotubes, with the benzene-based junctions displaying the highest slope. Although the detailed nature of the molecule-induced selection rules on the phonon transport is not obvious, it is clear from the previous discussion that channel selection and, hence, the phonon filtering effect, can be largely controlled by the molecule type. The complex nature of the phonon filtering effect translates thus into a rich phonon channel structure. 8


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In Figure 2(c) we have attempted to quantify the degree of filtering induced by a specific molecular bridge, a key design magnitude, by defining the quantity τKL (j, N ):

τKL (j, N ) =

1 CNT ωD

Z

CNT ωD

dω τCNT (ω) ln

0

τCNT (ω) , τMJ,j (ω)

(3)

where the index j refers to the number of molecular chains in parallel interconnecting the two thermal baths and N the number of monomers in one chain. τCNT (ω) and τMJ,j (ω) are the corresponding transmission functions for an infinite CNT and for the CNT-molecule junction containing j molecular chains in parallel. τKL (j, N ) describes the deviation from the "source of modes" distribution, quantified in our case by the transmission spectrum of the carbon nanotube. The integral is divided by the Debye frequency of the carbon nanotube to make it dimensionless. A trivial perfect filter (zero transmission) would give τKL (j, N ) → ∞, while no filtering at all yields τKL (j, N ) = 0. This quantity has been chosen in close analogy with the Kullback-Leibler divergence used in probability theory and in information theory to quantify the "distance" or similarity between two probability distributions. 56 Clearly, other combinations of transmission functions are conceivable, but they will only lead in general to a redefinition of the condition for filtering. Figure 2(c) shows τKL (j, N ) for the case of four interconnecting chains (j = 4) with varying number of monomers per chain (see Figure S7 in SI for the case of τKL as a function of the length of the molecular chain L). Here, we see that the azobenzene-based junctions display the highest efficiency for filtering due to the efficient blocking of high-frequency modes above 1000 cm−1 already for a small number of monomers (compare with Figure 2(b)), i.e., for larger number of monomers the distortions on the phonon transmission will become weaker. As a result, their corresponding τKL have the highest values and show a tendency to saturate for N > 6. The ethylene-based chains are less efficient, but still display a larger effect than the benzene chains; this is mostly related to the complete filtering of frequencies larger than 1500 cm−1 (see Figure S1 and S4 in the SI) as well as to the presence of a relatively large phonon gap in the region between 500 cm−1

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and 750 cm−1 and the resonances arising in the molecular junctions (which are reduced for larger chains). Finally, τKL for benzene-based junctions also increases with the length of the molecular chain and shows a tendency to saturate for N > 12 monomers. Additionally, we have found a reduction of τKL (j, 4) after increasing the number of parallel chains j, which just reflects the overall increase in the transmission function of the molecular junctions with increasing number of parallel chains (see Figure S8 in the SI). Figure 3 highlights now the additional effect of changing the material of one of the thermal baths on the phonon filtering effect. For this, we have studied junctions with BN nanotubes (BNNT) and with SiC nanotubes (SiCNT). We only show for the sake of simplicity benzenebased junctions (see Figure S9 in SI for the other cases). In these calculations, all chains are composed by four monomers of the respective molecule under study. For reference, the upper panel of Figure 3 displays the transmission function of an infinite CNT, an infinite BNNT, as well as the transmission of a CNT-BNNT junction. As expected, the phonon transmission of the junction is reduced when compared to the perfect tubes due to interface scattering effects (interface thermal resistance) and, consequently, its thermal conductance is also affected (see Figure 4(a)). 31,34,57,58 Additionally, due to the difference in Debye frequencies ωD of both materials, the BN tube acts as a (non-selective) low-pass filter, since all phonon channels with frequencies above the BNNT Debye frequency (ωD ∼ 1400cm−1 ) are blocked. This behavior is also seen in the lower panel of Figure 3, where the results for the nanotubemolecule junctions are shown for the case of benzene chains with four monomers. Due to the reduction of the Debye frequency when going from CNT (ωD ∼ 1800 cm−1 ) to BNNT (ωD ∼ 1400 cm−1 ) to SiCNT (ωD ∼ 880 cm−1 ), an increasing blocking of the high frequency channels is found, related to the low-pass filter effect previously mentioned. However, the presence of the molecules in the junction can lead to an additional, more selective filtering. This effect is clearly seen for the BNNT case, where the benzene chains induce spectral gaps in the frequency regions between the vertical dashed lines. For SiCNT based junctions, on the other hand, this effect cannot be clearly seen anymore, due to the relatively small ωD of 10


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the SiCNT, so that the influence of the molecules is strongly masked by the filtering imposed by the SiCNT (see Figure S10 in the SI). The evolution of the thermal conductance as a function of temperature for each studied molecular junction is displayed in Figure 4. Here, again for reference, Figure 4(a) shows the thermal conductance of pure CNT, BNNT, and of the CNT-BNNT junction. All molecular junctions lead to a dramatic reduction of the thermal conductance by roughly a factor of 3-4, the effect being more pronounced for the azobenzene junction. Notice also that in this latter case the thermal conductance barely changes, when going from CNT-CNT to CNT-BNNT junctions (see Figure 4(d)). This reflects the previously mentioned fact that azobenzene junctions almost completely suppress vibrational transport channels with frequencies above 1000 cm−1 . As a result, the narrower spectral density of the BNNT does not serve to cut off any additional transmission channels (see also Figure S6(b) in this respect). The saturation of the thermal conductance is determined by the nanotube with the smaller Debye cutoff frequency, the value of the saturation point is, however, strongly influenced by the specific composition of the molecular chains. Finally, to correlate the emergence of phonon gaps in the transmission functions of the junctions, we carried out a real space study of the projected phonon density of states (PDOS) for the CNT-azobenzene-SiCNT junction with four azobenzene monomers per molecular chain. The results are shown in Figure 5, focusing on two spectral intervals highlighted as I and II in the top left panel. In the bottom left panel, we show the quantity RDOS that we define as the ratio of the vibrational density of states, projected on a certain spatial region of the junction, to the total vibrational density of states of the scattering region. Notice that in the frequency region 600-800 cm−1 the phonon transmission probability vanishes, despite the fact that the RDOS on the CNT and SiCNT surface rings as well as on the molecular region do not vanish. This is related to the fact that the low-DOS states appearing for the SiCNT (red curve) correspond to largely localized vibrations at the molecule-tube interface, while the spectral density of the semi-infinite SiCNT bath is exactly zero in that frequency region. 11



As a result, the transmission is fully suppressed. The right panel of Figure 5 displays the spatial resolution of the PDOS for both regions I and II. As it can be clearly seen, in region I there is a very low weight on the azobenzene chains, which explains the pseudo-phonon gap in that region (notice that with only four azobenzene units the full phonon gap has not yet developed, compare with Figure 2). In region II, on the contrary, a larger contribution from the molecular region is found. However, although the SiCNT has a large spectral weight in that region (see the bottom left panel), the corresponding spectral weight of the CNT is relatively small, thus resulting in a relatively small number of transmitting phonon channels (∼ 1).

CONCLUSIONS To conclude, we have shown, using relatively simple structural models supported by an extensive computational effort, that single or few molecule junctions provide the possibility of acting as selective vibrational filters. The selectivity of the junction is mediated to a large extent by transport channels determined by the vibrational structure of the molecule. 59 In the case of asymmetric junctions, with thermal baths consisting of different materials, a natural (low-pass) filtering can be additionally achieved due to the differences in Debye frequency of the baths. However, molecular chains based on azobenzene can already develop a low-pass filter behavior even if both nanotubes are made of the same material. It is further conceivable to additionally obtain high-pass filtering effects, i.e. supressing low-frequency modes, by e.g. depositing the molecular devices on a substrate. 60 Work in this direction is currently in progress. We have further quantified the quality of the filtering effect by defining a parameter resembling the Kullbak-Leibler distance known from information theory. Although this quantity may not provide a universal fingerprint to characterize the filtering capability of a molecular junction, it still affords a compact way to quantify it and describe its dependence on junction parameters such as the length, structure and chemical nature

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of the molecular bridge. In brief, the combination of contact-mediated low-pass filtering in one of the thermal baths and molecule-induced selective filtering, suggests a promising design route to engineer nanoscale phononic filters by exploiting the rich molecular chemical space and the nature of the nano-contacts, e.g., the use of 2D-materials. An issue, which will require further inquiry in future studies is the influence of interaction effects, such as electronphonon coupling (EPC), on the filtering effect. A rough estimate of EPC contributions to the phonon self-energy within a simple model Hamiltonian (see Supplementary Information) yields a broadening of the vibrational modes in the order of 0.3-1 meV, depending on the specific vibrational frequency and strength of the EPC. These values may be enough to partly wash out small gaps, such as those found for azobenzene chains below 500 cm−1 (see. Fig. 2). However, larger gaps at higher frequencies will not be considerably affected by the EPC and hence, the filtering effect is expected to be robust against such interaction effects. The current study has contributed to shed light into the underlying physical mechanisms mediating nanoscale heat flow, thus opening the possibility for developing more efficient energy harvesting technologies such as quantum cooling devices. 61

Acknowledgement L.M.S. thanks to the Deutscher Akademischer Austauschdienst (DAAD) for the financial support. A.R.M. thanks to the National Council of Science and Technology (CONACYT) for the scholarship granted towards his master studies. This work has also been partly supported by the German Research Foundation (DFG) within the Cluster of Excellence "Center for Advancing Electronics Dresden". V.M. acknowledges financial support from the DRESDEN Fellowship Programme. We acknowledge the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for providing computational resources.

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Supporting Information Available The following files are available free of charge. • sup-info.pdf: Additional results related to the nanoscale phonon filters proposed in this study. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Figure 1: Schematic representation of the molecule-based phonon filter. A two-terminal junction is considered, where the role of the thermal baths is played by two semi-infinite nanotubes. We consider symmetric junctions, where both contacts are (6,6) carbon nanotubes (CNT) as well as asymmetric junctions, where one contact is a (6,6) CNT and the other can be a (6,6) Boron nitride NT or a (6,6) Silicon carbide NT. The thermal baths are bridged in all cases by four molecular chains in parallel. We use chains consisting of ethylene (∼2.6 Å), benzene (∼4.4 Å), and azobenzene (∼10.1 Å). The spectral density of the CNT serves as a "source" of phonon modes. In the described setup, two main effects lead to phonon filtering: the difference in Debye frequency ωD for asymmetric junctions filters high-frequency modes (low-pass filter); the molecular chains selectively induce additional phonon gaps in specific spectral windows (mode selector) in dependence of the specific chain chemical composition.

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Figure 2: Length dependence of the phonon filtering effect in a symmetric CNT-moleculeCNT junction. (a) Atomistic structure of benzene- and azobenzene-based molecular junctions, highlighting the scattering region through which the phonon transmission is computed, and the two thermal baths, which consist of semi-infinite CNTs. (b) Phonon transmission τ (ω) as a function of the frequency ω for benzene (top panel) and azobenzene (bottom panel) junctions. Each junction consists of four molecular chains in parallel. Each chain contains 2 (blue) and 12 (red) monomers for benzene and 2 (blue) and 8 (red) monomers for azobenzene. In the plots we also include for reference two additional cases: transmission of an infinite CNT (grey background) and a single infinite molecular chain (brown). Highlighted with dashed-line circles are the regions, where phonon gaps clearly develop by increasing the chain length. Notice that clear phonon gaps develop for azobenzene already for the shorter chain with two monomers. Additionally, azobenzene displays a very strong low-pass filter behavior for frequencies above ∼ 1000 cm−1 . (c) The function τKL (j = 4, N ), defined by Equation (1) in the text, as a function of the number of monomers (N ) in the different studied molecular chains for the case j = 4 chains in parallel. This quantity, which ressembles a similarity with the Kullback-Leibler divergence, serves as a measure of the phonon filtering ability of the molecular junctions.

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Figure 3: Top panel: Phonon transmission function of an infinite CNT (grey), an infinite BNNT (brown), and of a CNT-BNNT junction. The junction displays the lowest transmission due to phonon scattering at the interface. Notice also the low-pass filter behavior induced by the smaller Debye frequency of the BNNT. Lower panel: Phonon transmission function for CNT-(four monomers)benzene-X junctions, with X= CNT, BNNT, and SiCNT. The vertical dashed lines highlight the spectral regions, where the benzene molecular wires in the CNT-benzene-BNNT junction (red line) are acting as a mode selector. In the junction with SiCNT (green line) the influence of the molecular wire is masked by the rather small Debye frequency ωD of the SiC nanotube, which fully blocks all phonon transport channels above ωD .

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CNT BNNT CNT-BNNT

8.0

CNT-X-SiCNT CNT-X-CNT CNT-X-BNNT 1.2 1.2 (c) (d)

1.2 (b)

(a)

6.0

κph [nW/K]

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0.8

0.8

0.8

0.4

0.4

0.4

4.0

2.0 Benzene

Ethylene

0.0

0

400

800

0

0

400

800

Temperature [K]

0

0

400

Azobenzene

800

0

0

400

800

Temperature [K]

Figure 4: Thermal conductance as a function of the temperature. Left panel: Thermal conductance of the infinite CNT and BNNT as well as of the CNT-BNNT junctions. The remaining panels show the thermal conductance for (b) ethylene, (c) benzene, and (d) azobenzene in the different junction types with a lenght of chain corresponding to four monomers. Notice the strong suppression of the thermal conductance in all cases when comparing with the CNT-BNNT junction.

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Figure 5: (a) Transmission function of a CNT-(four monomers)azobenzene-SiCNT junction. (b) Ratio RDOS of the vibrational density of states, projected on a certain spatial region of the junction, to the total vibrational density of states in the scattering region. The blue line corresponds to the projection on the CNT surface, similarly the red line refers to the SiCNT surface, and the grey background is the projection on the azobenezene chains. (c) Spatial resolution of the the projected DOS for the frequency regions I and II, highlighted in yellow in the transmission plot. On region I, there is almost no contribution arising from the molecular region, so that a phonon gap starts to emerge (it only becomes a full gap for longer molecular chains, see Figure 2). In region II, on the contrary, there is a larger contribution from the molecular region; however, the CNT has a weaker contribution there and, hence, the transmission is rather small.

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Graphical TOC Entry

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Selective Transmission of Phonons in Molecular Junctions with

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