A Thermal Switch for Coherent Phonons Based on a Molecular

Article pubs.acs.org/JPCC

A Thermal Switch for Coherent Phonons Based on a Molecular Junction Miquel Royo,† Aleandro Antidormi,†,‡ and Riccardo Rurali*,† †

Institut de Ciència de Materials de Barcelona (ICMAB−CSIC), Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (Ca), Italy

‡

S Supporting Information *

ABSTRACT: We present thermal and electron transport calculations in the ballistic regime of a graphene nanoribbon with a nanogap bridged by a stilbene molecular switch. We show that by commuting the stilbene molecule back and forth between its two stable isomers, the thermal conductance can be reversibly switched from a low to a high conduction state, providing a proofof-concept of a molecular junction-based thermal switch.

isomerization, where irradiation promotes a trans−cis conformational change, or stabilized mechanically, as discussed in this work. Recently, a configuration consisting of another diarylethene molecule connected to graphene contacts has been reported experimentally and shown to operate as a two-mode, single-molecule electrical switch.14 Ultraviolet and visible light were used to drive reversible isomerization of the diarylethene, commuting between conformations where a different degree of conjugation results in different electrical conductance values. Here we explore a very similar configuration consisting of a stilbene molecule sandwiched between two graphene nanoribbon (GNR) contacts and show that also thermal conductance can be tuned by switching the molecule between its two stable isomers. For comparison, we also study ballistic electron transport and critically address the differences.

A molecular switch is a molecule that can reversibly commute between two stable configurations in response to external stimuli.1 Photochromic molecular switches,2−4 where light induces a cis−trans isomerization, have attracted much of the interest in this kind of systems, but the conformational change between the two stable atomic configurations can also be triggered in other ways, ranging from tunneling electron current5−8 to temperature9 to pH.10,11 The physical, chemical, and mechanical properties of the two isomers in general differ. Therefore, these systems can operate as molecular scale transducers, where a conformational change can mediate the conversion of an external signal into a variation of a transport property.12−14 For these reasons, molecular switches are an interesting test-bed to study the interaction of external stimuli with molecular degrees of freedom and the coupling of the latter with physical−chemical properties and transport coefficients of the molecular system itself. Phononics aims at encoding and transmitting information with phonons. An obstacle that has considerably hindered its development is that phonons have no mass and no charge; thus, an external field cannot be used to control their propagation.15 Molecular switches, however, can mitigate this problem, providing an indirect control on phonon motion: an external field can trigger a specific isomerization which, in turn, modifies the thermal conductance of the molecule. Here we give a proof-of-concept of this effect with a stilbene molecular junction. Stilbene is a diarylethene with two phenyl groups connected by an ethene double bond and has two stable isomers: trans1,2-diphenylethylene, also called (E)-stilbene, and cis-1,2diphenylethylene, also called (Z)-stilbene. The latter is less stable because of the steric interactions force between the outof-plane aromatic rings, but it can be obtained by photo© XXXX American Chemical Society

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COMPUTATIONAL METHODS We study thermal and electron transport in the lowtemperature ballistic regime of a 1,2-diphenylethene molecule covalently bonded to two graphene nanoribbon (GNR) contacts. We consider (E)-stilbene and (Z)-stilbene, respectively the trans- and cis-isomer and both armchair and zigzag GNRs. The thermal and electron conductance are calculated using the Landauer formalism within the nonequilibrium Green’s function (NEGF) approach.16,17 As usual in this kind of calculations, we divide the system in three parts: a left and right contact and a central scattering region where the molecule is Received: March 10, 2017 Revised: April 27, 2017 Published: April 28, 2017 A

DOI: 10.1021/acs.jpcc.7b02284 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C placed. The GNRs in the contacts are assumed to be in their bulk state, i.e., sufficiently far from the perturbation induced by the molecular junction; thus, the scattering region must include long enough segments of the GNRs. A sketch of the computational setup is shown in Figure 1. For the electron

constant matrix in eq 4. ; ph(ω) is then obtained from the phonon equivalent of eq 3. We refer the interested reader to ref 16 for a more detailed description of the NEGF formalism.

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THERMAL TRANSPORT We begin our analysis by studying thermal transport through a molecular junction consisting of a stilbene molecule placed within a nanogap of a GNR and covalently bonded to its edges. At first we study the energetics of the system as a function of the nanogap width, Δz. When the GNR contacts are sufficiently separated, the molecule favors a trans-isomer and is under tensile strain for Δz > 3.95 Å. As the nanogap is reduced, the trans-isomer enters first the compression regime but then favors a conformational change by commuting to the cisisomer, as clearly indicated by the abrupt discontinuity of the E(Δz) curve of Figure 2 at Δz ∼3.6 (zigzag) and ∼3.8 Å

Figure 1. A (E)-stilbene and a (Z)-stilbene molecule within a nanogap in a zigzag GNR. The region in light yellow indicates the parts of the nanoribbon that were constrained during the structural relaxation and that were used to calculate the contact Green’s function and thus implement the open boundary conditions required for the transport calculations.

transport calculations, the scattering region and the contacts have been fully relaxed within density-functional theory (DFT) as implemented in the SIESTA code18 using an optimized19 single-ζ polarized basis set20 to expand the one-electron wave function, norm-conserving pseudopotentials to account for the core−electrons and the generalized gradient approximation (GGA) for the exchange-correlation energy.21 For the thermal transport calculations we relax the geometry and compute the force-constant matrix using the classical bond-order potential due to Brenner22 as implemented in the GULP code.23 In the limit of small applied bias the Landauer electron conductance at chemical potential μ reads Ge(μ) =

2e 2 h

⎛ ∂f (E , μ) ⎞ 0 ⎟d E ∂E ⎠

∫ ; e(E)⎜⎝−

Figure 2. Potential energy of the stilbene molecular junction as a function of the nanogap width Δz for both zigzag and armchair contacts. Energies are referred to the most stable configuration, i.e., (E)-stilbene for Δz ∼ 4 Å.

(armchair; in this case the isomerization is also associated with a partial release of the compressive strain). Notice that at a given value of Δz only one of the two isomers is stable. This also means that photoisomerization would only be possible if at least one of the contacts was mobile or if the bond with the molecule was sof ter than the C−C interaction of the simple configuration considered here: it is the light that triggers the conformational change, but the molecule must be able to slightly stretch/contract to allow it. Incidentally, we note that in this study we use mechanical compression as a practical approach to isomerization within our computational modeling. Yet, the ideal goal that our calculations suggest to pursue is photoisomerization and thus a light-actuated thermal switch. We computed the phonon transmission, ; ph(ω), and the thermal conductance, Gph(T), for the minimum energy configurations of both isomers. Obviously, the transmission of the pristine GNR undergoes a large decrease when a nanogap is opened and the two halves are only bridged by a stilbene molecule. This bottleneck effect is well-known and stems from the fact that the nanoconstriction, the stilbene molecule in this case, can only sustain the propagation of a subset of the incoming phonon modes (see ref 24 for a review of the analogous case of electron transport). What we are more interested in, however, is how ; ph(ω) changes when the molecule switches from the trans- to the cis-isomer. Our calculations indicate that, as a general rule, the transmission of (E)-stilbene is systematically larger than the one of (Z)-stilbene

(1)

where f 0 is the equilibrium Fermi−Dirac distribution, e is the electron charge, and h is the Planck constant. Similarly, the Landauer phonon conductance at temperature T for small temperature gradient is G ph(T ) =

ℏ 2π

⎛ ∂n0(ω , T ) ⎞ ⎟d ω ⎠ ∂T

∫ ω; ph(ω)⎜⎝

(2)

where n0 is the equilibrium Bose−Einstein distribution and ℏ is the reduced Planck constant. The electron transmission is calculated as ; e(E) = Tr[ΓL(E)GCr (E)ΓR (E)GCa (E)]

(3)

where ΓL/R(E) are broadening functions that account for the coupling between the contacts and the scattering region, while the retarded (advanced) Green’s function is computed as GCr , a(E) = [ESC − HC − ΣrL, a(E) − ΣrR, a(E)]

(4)

where HC and SC are the Hamiltonian and the overlap matrix and ΣL,R(E) is the self-energy of the left (right) contact. Analogously, one can calculate the phononic Green’s function 2 Gr,a C (ω) by replacing ESC with ω I and HC with the forceB

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The Journal of Physical Chemistry C for low- and midenergy phonons, while differences are not that clear-cut at high-frequency, where at some energy the transmission of (Z)-stilbene can be larger (see Figure 3a).

Figure 4. Thermal conductance of a (E)-stilbene (blue line) and a (Z)-stilbene molecule (red line) with zigzag GNR contacts. The conductance of a pristine zigzag GNR is also shown for comparison (black line). Inset: reduction of the thermal conductance upon ph isomerization, 1 − Gph cis/Gtrans.

diarylethene molecules more attractive is that such a conformational change can be induced with light. Therefore, it is interesting to question to which extent the change in conductance that we report in Figure 4 is an effect of the trans−cis isomerization or if it is just a mechanical effect similar to the one reported by Li and co-workers;26 the (Z)isomer can be seen as a compressed form of trans-stilbene, where compression induces a partial rotation of the phenyl groups, regardless of the fact that they are two individually stable isomers. To rule out this scenario, we have calculated Gph at a few selected temperatures as a function of the nanogap width. Our results are shown in Figure 5. As can be seen there, we indeed reproduce a strain dependence of the thermal conductance similar to the one previously reported.26 Additionally, however, we observe a clear signature, in the form of a discontinuity of Gph, of the molecular switching. This observation confirms that isomerization does produce a measurable change in Gph that goes beyond the effect of a bare mechanical compression.

Figure 3. (a) ; ph(ω) for a molecular junction featuring a (E)-stilbene (yellow curve) and a (Z)-stilbene molecule (red curve) with zigzag GNR contacts. The transmission for a pristine, continuous zigzag GNR is also shown for comparison (black curve). (b) Phonon dispersion of the GNR and single-mode analysis (see text) for the (E)stilbene and (c) for the (Z)-stilbene molecular junction. k* = k az/2 π, where az is the lattice parameter of the GNR along the transport direction, is the reduced wave vector. The case of armchair contacts is similar and reported in Supporting Information.

These qualitative observations can be rationalized in terms of the probability for individual phonon modes to be transmitted across the molecular junction. To this end, we calculate the Bloch matrix as described in ref 25, which in turn gives access to the transmission between individual phonon channels in the left and right contact (Figure 3b,c). The colormap in panels b and c illustrates the probability of transmission of each individual phonon across the molecular junction: black indicates full reflection while light yellow (red) indicates full transmission across a trans-(cis-)isomer. This analysis shows that acoustic phonons are largely reflected in the case of (Z)stilbene, while a good fraction of them is transmitted by the (E)-stilbene isomer. Midenergy optical phonons behave similarly, with a higher transmission across the trans-isomer. As a result, the thermal conductance suffers a reduction between 50% and 70%, in the temperature range studied, when the stilbene molecule switches from the trans- to the cis-isomer (see Figure 4). The reduction is stronger at low temperature where the phonons that are most affected by the cis−trans isomerization dominate heat transport. Recently, Li et al.26 have shown that the thermal conductance of an alkane chain can be tuned by mechanically compressing it. Here, in principle, we go one step further and consider the case in which the compression triggers a conformational switch between two stable isomers. What makes stilbene and similar

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ELECTRON TRANSPORT We now move to the study of electron transport. This analysis is useful to highlight the differences between the two switching processes and to study the effect of the conformational change on the thermoelectric properties of the molecular junction. We considered geometries similar to those of Figure 1, though for consistency we carried out structural relaxations within the same density-functional computational framework used to calculate the electronic structure and the transport properties. The zero-bias Landauer conductances for a zigzag GNR bridged by a (E)-stilbene and a (Z)-stilbene molecule are displayed in Figure 6. As it can be seen, Gel(μ) is fully suppressed in the μ < |1| eV energy interval, where the continuous GNR featured two conductance quanta. An exception is a small peak at zero energy, reminiscent of the peak due to the edge states of the pristine GNR. To elucidate the nature of this peak, we have computed the eigenchannel that carries most of the current at three selected energies, corresponding to the conductance features indicated by small arrows in Figure 6. The peak at zero is indeed due to an edge C

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molecule. The active role of the molecule in assisting the tunneling is further highlighted by the fact that, due to its spatial orientation, the (E)-stilbene isomer connects efficiently the upper edge state of the GNR on the left of the junction with the one of the lower edge state on the right-hand side of the junction, where the coupling of the molecule with the GNR is stronger. On the other hand, at μ ∼ ±1.2 eV the (E)-stilbene isomer has no longer an accessory role, as shown by the transmission eigenchannels that reveal a strong localization on the molecule: a common behavior in nanoconstrictions or nanowires connecting two bulk leads. The conductances of the transand the cis-isomer are substantially similar for μ < 0, while for μ > 0 the onset of the current for the (E)-stilbene occurs slightly before than in the case of the (Z)-stilbene. The comparison between the electron and phonon conductances and their variation upon isomerization indicates that the thermal switch effect is much more robust than the electronic switch effect. The phonon conductance of (E)stilbene is consistently larger than that of (Z)-stilbene throughout all the temperature range considered, though the ph exact value of 1 − Gph cis/Gtrans depends on the temperature: it has the highest values at low temperature, but it is still of the order of 0.5 at room temperature. In the case of electron transport, on the other hand, Getrans/Gecis depends critically on the chemical potential: much larger values can be obtained, but at certain values of μ it can even occur that the cis-isomer is more conductive than the trans-isomer. A clear example of this behavior is the electron conductance at μ ∼ −1.25 eV, where a small misalignment in the conductance peaks is such that Getrans/ Gecis goes from ≪1 to ≫1 within a very narrow energy range. This behavior derives directly from the different statistics that rule electrons and phonons. For electrons to travel from one contact to the other, Pauli exclusion principle requires the existence of accessible states in the channel over an energy window of few kBT around the chemical potential. In other words, only those states energetically close to the chemical potential contribute to the electronic conductance. In practice, this is incorporated in the Landauer formula (eq 1) through the energy derivative of the Fermi−Dirac distribution function, thus whenever ; e(E) is characterized by narrow resonant peaks, the same will happen to Ge(E). On the other hand, more than one phonon can occupy the same energy state which enables transport between occupied states in contacts and channels. Therefore, at a given temperature the phononic conductance accumulates contributions from all occupied phonon states weighted by the temperature derivative of the Bose−Einstein distribution in eq 2. In the present harmonic approximation this effect also means that whenever T2 > T1, Gph(T2) ≥ Gph(T1). As a final remark, we observe that a conceptually similar case to the one considered here was tackled by Ranganathan and coworkers.27 They concluded, however, that tuning of thermal conductance upon photoisomerization was not straightforward. An important distinctive feature of our setup, however, is that the molecule backbone resembles closely that of the contacts, i.e., sp2 bonded C atoms; thus, the transmission across the nanogap is non-negligible. In the system studied by Ranganathan et al.,27 where a polymer is sandwiched between bulk Si contacts, phonon scattering is already very strong due to the chemical and structural heterogeneity of the molecular junction, thus masking possible differences arising from the different configuration of the isomer. Additionally, in our

Figure 5. Gph at five selected temperatures as a function of the separation between the contacts for (a) a zigzag and (b) an armchair GNR. The dashed line indicates in each case the trans−cis switching onset.

Figure 6. Gel(μ) at T = 100 K for a zigzag GNR with a nanogap bridged by a (E)-stilbene (blue curve) and a (Z)-stilbene molecule (red curve). The conductance for a pristine, continuous zigzag GNR is also shown for comparison (black curve). The insets show the spatially resolved eigenchannels at μ = 0, and ±1.2 eV.

state of the GNR contact, which transmits throughout the molecular junction. Notice that it is not direct tunneling of the edge state through the nanogap and the molecule does play an active role. We have verified that either removing the molecule or considering wider GNR contacts leads to the disappearance of the peak at zero energy, indicating that (a) the nanogap is sufficiently big to suppress tunneling and that (b) only a narrow enough GNR allows coupling of the edge states with the D

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calculation we address the low-temperature regime where phonons are coherent and only elastic scattering due to the molecule is accounted for.28 This also contributes to amplify the effect of thermal transport deriving from different structural configurations of the molecular junction. In summary we have shown that not only can the (photo)isomerism of a molecular switch be used to tune the thermal conductance but the performance of such a thermal switch is even more robust than its electronic counterpart. In the coherent low-temperature regime the phononic conductance can undergo variations of up to the 70% upon a trans−cis conformational change that can be induced mechanically or triggered by photons of suitable wavelength, as demonstrated recently in another molecular switch of the same family. Single mode analysis reveals that the lower thermal conductance of the cis-isomer is due to a considerable reflection of acoustic and midenergy optical phonons.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b02284. Thermal conductances of the molcular junction with armchair contacts. Details for thermoelectric figure of merit (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Aleandro Antidormi: 0000-0002-5266-8147 Riccardo Rurali: 0000-0002-4086-4191 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support by the Ministerio de Economı ́a, Industria y Competitividad (MINECO) under grant FEDER-MAT2013-40581-P and the Severo Ochoa Centres of Excellence Program under grant SEV-2015-0496 and by the Generalitat de Catalunya under grant no. 2014 SGR 301 and through the Beatriu de Pinós fellowship program, 2014 BP_B00101. We thank the Centro de Supercomputación de Galicia (CESGA) for the use of their computational resources.

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REFERENCES

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The Journal of Physical Chemistry C (28) We note that the validity of the coherent (ballistic) transport regime extends up to room temperature as the graphene intrinsic phonon mean free path at such temperatures is ∼600 nm (∼100 nm) in suspended (supported) samples,29 i.e., much larger than the simulated molecular junction device. (29) Pop, E.; Varshney, V.; Roy, A. K. Thermal Properties of Graphene: Fundamentals and Applications. MRS Bull. 2012, 37, 1273−1281.

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