Quantum phonon transport through channels and molecules—A Perspective

In this Perspective, we discuss thermal transport mediated by phonons in quantum channels, at the atomic and the molecular scale. To explore the richness of transport physics in general, it has proven useful to study transport on various length scales and explore low-dimensional geometries. Small contacts or nanowires, for example, have been used to explore charge, heat, or mass transport. Phonon transport through contacts is already a well-known way to explore fundamental transport properties (see, for example, Refs. 1–7).

A contact between two bodies, such as a mechanical contact between a tip and a surface, can be experimentally scaled down such that different characteristic length scales and, thereby, transport regimes can be tested. For example, the transition from diffusive to ballistic transport of charge or heat carriers has been studied by reaching contact diameters below the mean-free path of carriers.⁶ Similarly, transport in one-dimensional structures, such as wires, undergoes characteristic changes when transitioning between these two transport regimes.⁷ 

Reducing the sample dimensions even further leads to quantization effects, a transport regime well-studied for charge transport and much less explored for heat transport. In one-dimensional nanowires, for example, the energy spectrum of electronic wave functions is discrete for waves moving perpendicular to the wire, and only along the wire, a continuous dispersion relation can be observed. In such situations, the electrical conductance assumes discrete, quantized values, which are multiples of the conductance quantum $2 e 2 / h$⁠, including the spin degeneracy. Here, e is the electron charge and h is Planck's constant. The case of charge transport leading to quantized conductance values is well-known and studied for several decades now. In particular, for samples of relatively short length, the Landauer–Buttiker picture of transport is applicable, expressing transport in terms of such (quantized) transport channels, which can be described by a transmission coefficient (⁠ $τ ≤ 1$⁠).

The quantum of thermal conductance g₀ is an upper bound for the thermal conductance per channel, but it is saturated only for noninteracting, ballistic transport. In fact, at high temperatures, the ballistic bound g₀ typically is far from being saturated, but it can be reached and measured at low temperatures. A universal extension to this bound, which is valid also for interacting systems, was derived in Ref. 12 based on exact sum rules. This general bound, expressed in terms of the system's characteristic energy scales, is saturated at high temperatures, but still in the quantum regime and even when interactions are strong. As we discuss next, measuring phonon-controlled quantum of thermal conductance in atomic scale constrictions remains a challenge.

The observation of quantized heat transport based on phonons has only rarely been claimed. One important reason for this may be the small magnitude; g₀ is only $≈$300 pW/K at 300 K, and $≈$10 pW/K at 10 K, see Eq. (1). To estimate under what constriction diameter quantization can be expected, one can consider the characteristic dimension of a phonon wave packet through the thermal wavelength $λ t h = h c / ( k B T )$⁠,¹⁷ where c is the phonon velocity. At room temperature, this can nearly reach atomic dimensions. The highly original landmark paper of Schwab et al.¹⁷ has observed a minimum of g₀ for four channels in nano-constrictions of <200 nm in silicon nitride when T was lowered to below 1 K. These findings, however, were not experimentally reproduced. Quantized thermal transport was invoked in the so-called nano-asperities of contacts with atomic roughness to explain the large pressure-dependence of heat transport in nano-contacts.²⁰ However, the indications are somewhat indirect and remain controversial.^21,22

A recent and very careful study by Tavakoli et al.²³ tackled the problem of quantized phonon transport also by using micro-fabricated constrictions in silicon nitride. Similar to the study of Schwab et al.,¹⁷ the shape of the constriction was chosen to form an impedance matching,²⁴ a prerequisite of achieving ideal transmission (τ = 1) in microscale devices. (This impedance matching is similar and analogous to matching acoustic mismatch between sound in 3D air and the speaker of a hifi sound system.) However, the quantization effect through reaching a minimum constant conductance of $N g 0$ once lowering T was not observed by Tavakoli et al. Instead, the conductance reduced further, which was interpreted as imperfect transmission, in accordance with theoretical predictions.²⁵ It, therefore, remains to be shown to what extend the quantization of heat transport can be viewed as universal in phonon transport of nanowire systems and constrictions, or it can merely be viewed as a theoretical upper limit.

A somewhat special case for the measurement of quantized transport are so-called quantum point contacts (QPCs).^26,27 QPCs are narrow constrictions between two wide conducting regions. One example is the constriction between electrically conducting regions in two-dimensional conductors. For sufficiently narrow constrictions, transport through such constrictions is quantized in electrical and thermal conductance signals.²⁷ Experimentally, QPCs are easily accessible using the break junction technique. In this method, the narrow constriction is formed through the slow and gradual breaking of a mechanical contact of ductile metals. In charge transport, quantized conductance can be routinely observed in such QPCs. The ability to form single-atom junctions using break-junction techniques has led to the observation of electrical conductance quantization in gold and several other metals. A particular benefit in studying break-junction QPCs is the ease to make an atomically thin constriction, which exhibits quantization effects up to room temperature. It is conceptually simple and can be easily reproduced. Such QPC is also referred to as atomic junctions.

Quantized heat transport in atomic junctions of gold was recently measured by one of the authors and team,^8,28 see Fig. 1, and also independently studied by Cui et al.²² Both studies used micro-fabricated heater-sensors, operated in vacuum at room temperature. The approach is based on the well-established break-junction technique in which a mechanical contact between a gold tip and a gold surface is slowly broken such that conductance steps can be observed in the electrical signal.⁸ For this, a break-junction experiment was set up with one of the electrodes being situated on a suspended micro electro-mechanical system (MEMS). The MEMS features a heater/sensor element capable of reaching sub-nW/K sensitivity to thermal conductance, when biased to a few tens of K with respect to the other electrode, a tip remaining at room temperature, see Fig. 2. During the experiment, the gold tip was repeatedly pushed into and out of contact with a gold electrode on the MEMS sensor, which was heated to above ambient temperature. The resulting small change in temperature measured in the MEMS sensor at constant heating power was then detected and used to calculate the thermal conductance of the junction by subtracting the much larger conductance of the MEMS supporting beams. Figure 2 shows the experimental arrangement with tip and MEMS heater/sensor.

The results showed thermal conductance quantization in parallel with the electrical conductance quantization, see Fig. 1. Within the measurement uncertainty (of below 20%), the Wiedemann Franz (WF) relationship was confirmed. This implies that, in these junctions, the heat is almost entirely carried by electronic excitations and phonon transport plays a minor role. This is expected because the experiments were performed at room temperature, and therefore, far above the Debye temperature of gold and the realm of applicability of quantized phonon transport. Apart from having additional heat carriers increasing thermal conductivity, the Wiedemann Franz (WF) law can be violated if the electronic scattering processes involved affect heat and charge in different ways. Therefore, the confirmation of WF law enables insight into the nature of scattering processes. Numerical simulations and analytical estimations are in agreement with the experimental data. Similar conclusions were achieved also in Ref. 22, by comparing gold and platinum atomic wires.

An open question is whether quantized transport of the phonon contribution would be observable in QPCs at lower temperatures,²⁹ i.e., below the Debye temperature of gold, at which three or four phononic transport channels would have to be added to the electronic ones. An experimental study of this regime would require sensitivies on the order of 1 pW/K, which is certainly a challenge. It is not readily clear how much the phonons would contribute to heat transport in QPCs at low temperatures. In contrast to electronic QPCs, phonons are sensitive to an impedance mismatch formed by the spatially varying diameter of the constriction on the atomic scale, as was already predicted using classical molecular dynamics simulations,³⁰ compare also Panzer and Goodson.³¹ Atomistic quantum mechanical transport calculations in break-junction geometries predict negligible phonon contribution at room temperature^29,32 in agreement with the aforementioned experiments. At reduced temperatures, however, below the Debye temperature of the bulk metals, the phonon contribution is predicted to appear,²⁹ albeit staying well below the quantization value. The somewhat idealized system of an atomic QPC appears beneficial to relate experimental to theoretical transmission factors.

A natural next step from considering single, quantized channels of transporting heat via phonons is to modify that transport channel, add functionality, and link the transport channel to local degrees of freedom inside the channel. To this end, probably the most tangible approach is the use of molecular systems. Heat transport through organic molecules is a fascinating area of research, as it combines unresolved conceptual questions with slowly increasing availability of experimental data.³⁴ To begin with, one needs to distinguish between thermal transport in organic materials, such as amorphous polymers and fibers from molecular junctions, the latter considered here. In amorphous polymers,³⁵ heat transfer is mainly limited by energy transfer between the interlaced molecular groups through relatively weak van der Waals interactions or electrostatic bonds. Therefore, although thermal transport along a single chain, i.e., intra-molecular transport, can be large and ballistic,^35–37 in amorphous media we typically find diffusive transport and low thermal conductivity. In contrast, in a molecular junction setup, when one or more molecules are individually bound to two solid electrodes (or thermal reservoirs), heat transport can be quantum coherent, meaning that it can be described by coherent quantum mechanical waves.^33,38 (Many people find the notion useful in which the waves move as quasi-particles in a ballistic fashion, applying a Landauer approach based on a scattering picture). In either case, thermal transport in/along saturated organic molecules is typically dominated by nuclear-vibrational contribution, while heat transport due to electrons can be largely neglected.

Turning to the status of experimental methods to measure thermal transport across molecules, one should distinguish between ensemble-based methods^{36,39,42–45} and recent work using single molecules,^33,41 see Fig. 3. Ensemble-based methods [Figs. 3(a)–3(f)] typically use self-assembled monolayers (SAMs) sandwiched between two electrodes³⁹ or one solid electrode and a liquid counter electrode.^45,46 These samples can then be studied using optical techniques, of which time-domain thermoreflectance has been particularly successful.³⁹ However, the fabrication of such sandwiched structures is challenging, because defects and imperfections of the surfaces can influence the transport strongly. Ideally, atomically flat surfaces and rather perfect films are required over macroscopic areas. A typical uncertainty for transport measurements of SAMs is, therefore, the percentage of molecules of a film that are actually contacted (typically chemisorbed) to both electrodes. For example, a careful study by Majumdar et al.⁴³ estimated that by only about 50%. Despite these inherent difficulties, there are beautiful measurements with high reproducibility studying the effect of molecular length and bonding strength on the transport. To address the limitation of defects on the interpretation, a small ensemble (tens to hundreds of molecules) has been studied using the contact of a tip of a scanning thermal microscope to a SAM. This study,⁴⁰ however, while showing the potential of the method, reported systematic errors larger than reported for optical studies. Nevertheless, the community is awaiting with interest new tip-based studies currently being performed.

A second limitation of using SAMs, relevant to this study, is that only a small group of molecules actually form well-defined molecular layers, typically involving simple molecules, such as alkane chains, while relevant current questions are related to more complicated molecular designs. This motivates the use of single molecule junctions, Figs. 3(g)–3(i). For this however, the ability to contact individual molecules has to be combined with sensitive heat flux measurements to reach a sensitivity to detect the conductance of a single molecule on the order of a few tens of pW/K. Based on the aforementioned experiments on single-atom junctions, two groups have recently succeeded to measure heat transport across single molecules.^33,41 Both groups have used essentially the same approach, by combining a break-junction method based on a scanning tunneling microscope (STM) with a heater/sensor micro-fabricated using MEMS technology. The method combines charge and heat transport measurements, and the well-understood charge transport properties of single molecules serve to probe the integrity of single-molecule junctions.

The first demonstration of heat transport on the single molecule level utilized molecules that are well known from previous charge and thermal transport studies. Oligo(phenylene ethynylene) (OPE3) is probably the best characterized molecule in electrical break junctions to date,³³ and alkane chains^33,41 have been used in numerous experimental and theoretical studies on thermal transport through SAMs. The two studies^33,41 agree on the value of the thermal conductance for the latter, a first validation of the methodology. The results, shown in Fig. 3(f), compare well to simulation results,^33,47 if one assumes coherent phonon transport across the junction and confirms the negligible contribution of heat transport by charge carriers. Therefore, experiments could already confirm or falsify certain assumptions from theory, which are described in the following.

There is already a large number of studies offering theoretical predictions of heat transport across molecular junctions. Looking from a bird's eye view, we first notice that the problem of heat transfer is challenging, because it is inherently multiscale. Phonons contributing to room-temperature heat transport have phonon wavelengths and mean free paths ranging from atomic dimensions to several micrometers. On the other hand, the calculation of phonon properties on the atomic scale is much less prone than excited electronic states to systematic errors of first-principle methods like density functional theory (DFT), which result in difficulties in predicting electronic conductance of molecular junctions.

There are several methods available to study heat transport mediated by molecular vibrations along molecular junctions. It has been found in almost all experimental and computational studies that heat transport is coherent along linear molecules with repeating units of up to a few nanometers in length, such as alkane chains of up to ten carbon atoms. In this case, the simple picture of an (energy-dependent) transmission coefficient $τ ( ω )$ in a Landauer transport formalism is well justified. Fundamentally, Landauer transport is built on the assumption that it is sufficient to consider only the harmonic part of interatomic interactions. Such scattering theory calculations, which were first performed with simplified force fields^38,48 now employ density functional theory for the phonon transmission function and the language of nonequilibrium Green's functions.^{32,33,49–51}

Vibrational heat transport beyond the harmonic approximation can be studied using molecular dynamics simulations, which have been applied to molecular junctions.^43,52–54 However these typically are restricted to the classical limit, with notable exceptions at high computational costs.⁵⁵ Therefore, it is a significant challenge to study quantum heat transport beyond the harmonic approximation. Anharmonic terms in the force field can be treated perturbatively in the language of either Green's functions^52,56–58 or with quantum master equation methods.^59–61 Other theoretical approaches rely on the construction of renormalized phonons in nonlinear systems,⁶² the introduction of inelastic phonon–phonon scattering in a phenomenological manner using fictitious reservoirs,⁶³ or the application of quantum corrections to classical molecular dynamics simulations.^58,64

It should be emphasized that molecular junction experiments have not yet revealed signatures of molecular anharmonicity in transport, as focus has been placed on relatively simple junctions, such as alkane chains; these uniform molecules can display signatures of anharmonicity only at very high temperatures.³⁸ This is to be contrasted with pump–probe experiments of vibrational energy flow within organic and biomolecules in solution, which rely on anharmonic interactions.⁶⁵ In accord, so far methods developed to capture anharmonic effects in molecular junctions typically adopt empirical united-atom force fields, rather than ab initio all-atom potentials.

Some of the main conclusions on the mechanisms of phonon transport in molecular junctions, based on theoretical and experimental insights, are as follows:

1. Several studies indicate that it is dangerous to apply simple estimations that have proven useful for bulk solids. In particular, by applying kinetic theory to phonon gases, the thermal conductivity should be proportional to the phonon group velocity (as well as heat capacity and mean free path). This implies that stiffer bonds along a molecule, which should lead to higher phonon velocities, would lead to better thermal conductance of a junction. This is not the case, and to learn that one needs to take into account the phonon density of states in relevant energy ranges, which is rich for both stiff and soft chains.

2. Coherent transport dominates vibrational heat conduction in relatively short and uniform molecules. This applies, for example, to the aforementioned experimental studies of single molecule transport^33,41 and is well supported by simulations. One of the direct implications is the length-dependence of thermal transport, saturating (in accord with ballistic transport) for long chains.

3. Sensitivity to the coupling strength of the molecular–electrode bond has been observed using time-domain thermo-reflection (TDTR) experiments (see Ref. 39 as an example), and it was predicted early on using various methods.^38,53 In the studied examples of small and simple molecules like alkane chains, the effect of binding to the reservoirs appears to be stronger than variations of the molecular length, in accord with coherent-ballistic conduction mechanism. Thus, the contact resistance plays a major role in these systems.

4. Beyond coherent-ballistic transport, anomalous or diffusional heat transport was exemplified in pump–probe (donor–acceptor type) experiments as well as at the meso- and microscale, see, e.g., Ref. 66 yet not in molecular junctions of nanoscale length, see point (2).

5. Apart from strong differences between chemically versus van der Waals-bonded molecules, variations in thermal conductance between different molecular junctions are relatively small, in both experimental and simulation data. The phononic thermal conductance roughly stays within the range of a few to a few tens of pW/K for simple and short organic molecular junctions as studied in the context of molecular electronics. (Larger differences appear for buckyballs fullerenes or very long molecules, of course.) As a qualitative explanation, the thermal conductance G in linear uniform chains scales as $G ∝ γ · k B$ in the weak metal–molecule coupling limit and as $G ∝ Ω 2 · k B / γ$ in the opposite case. Here, γ represents the molecule–metal coupling energy (in units of frequency) and Ω is the representative interparticle vibrational frequency (related to the harmonic force constant). Interestingly, in either limits (say Ω = 20 meV and γ = 5 or γ = 500 meV), the thermal conductance is contained within the range $G ∼ 1 – 20$ pW/K. While the aforementioned scaling relations are based on the harmonic potential,⁴⁸ they can be justified approximately beyond that using the picture of renormalized phonons.⁶⁷ It is worth emphasizing that both expressions for G reflect the role of the solid-molecule contact on conduction, showing a Kramer-like turnover of G with γ.

In conjunction with ongoing developments, there are significant open questions and research opportunities in the field of heat transport along molecular junctions. These relate both to conceptual understanding and abstraction and to possibilities of phonon engineering to increase or reduce thermal conductance. In the following, we list questions already addressed using theory but not yet verified through experiments, which, given recent experimental progress, should be possible in the near future.

1. Similarly to what has been discussed in the molecular electronics community for charge transport, phonon transport could also display quantum interference effects.^49,51 Figure 4(a) gives an example, based on the popular backbone of oligo(phenylene ethynylene) (OPE3), in which phonon transmission around the central aromatic ring has two paths leading to destructive phonon interference in the meta configuration. In the para configuration, in contrast, there is no difference in path-length and, therefore, no destructive interference. Through destructive interferences, a reduction in conductance by about a factor of two is predicted.

2. Another proposed mean to reduce thermal conductance is the attachment of pendant side groups to molecules.^51,68–70 Local vibrational states can induce resonances in transport reminiscent of Fano resonances, which are prominent in different fields of transport. According to predictions, a single side-group can reduce the thermal conductance of a molecular junction by a factor of up to three, see [Figs. 4(d)–4(f)] for an example based on side-groups in benzenediamine. Another example for the predicted influence of pendant side groups is given in Figs. 4(g)–4(i), in which a single side group reduces strongly the conductance of bipyridine connected to thiobenzene anchor groups.⁶⁸ This example shows that extremely low thermal conductance at room temperature of only a few pW/K may be reached even for relatively short molecule junctions and with proper chemical binding to metal electrodes. Such a control and suppression of thermal conductance have potential applications in thermoelectric energy conversion or for the use of junctions as thermodynamic machines in general.

3. Harnessing anharmonic interactions toward functional devices anharmonic effects are expected to be important in flexible molecules made of weakly connected and distinct components, where raising the temperature induces, for example, bond weakening, torsional motion, or even conformational changes. Asymmetric molecular junctions with prominent anharmonic couplings can support an anisotropic heat flow (diode) effect.^59,71,72 The realization of molecular thermal rectification would be a valuable contribution for two reasons. First, there are some predictions or demonstrations of thermal diode behavior,^73–78 but mostly with only modest rectification, in particular when compared to electrical diodes. Therefore, current technological applications are mostly limited to opening and closing mechanical contacts and thereby switching thermal conductance. Truly passive and scalable thermal diodes could be an asset for thermal management. Second, a thermal diode would be an excellent example of true phonon engineering.

4. Identifying—and creating—molecular junctions that display quantum thermal transport beyond coherent behavior remains an open challenge. While meso- and micro-scale junctions show the onset of anomalous and normal diffusive transport, it is not clear yet what range of transport mechanisms can molecular junctions realize, beyond coherent transport. While we do not expect diffusive transport to develop at the scale of few nanometers, complex mechanisms may arise due to atomic heterogeneity and disorder, allowing phonon localization thus the suppression (and more generally control) of thermal conductance.

5. Effect of the environment: In most experimental situations and certainly in anticipated technological applications, a molecule will not be isolated but surrounded by solvent, residues, or other molecules. There is a significant experimental challenge to create a clean and defined experimental environment, even when using nominally perfect molecular SAMs.⁷⁹ Molecular dynamics simulations indicate a clear influence of the molecular neighbors⁴³ when using alkane SAMs. Conceptually, there is a question whether lateral coherence is large enough to cover two neighboring molecules. More importantly, however, this is closely related to defects and technological efforts around the integration of molecular films.

6. Beyond passive control of thermal conductance by modifying the molecular structure, active control by means of mechanical forces, electric fields, and light excitation is yet to be explored. Particularly exciting is the prospect of building on advances in molecular electronics, and using electric current to induce mechanical forces and control vibrational heat flow at wish.

Atomic contacts, quantum point contacts (QPCs), and single molecular junctions in general may appear far away from any technological relevance at first sight. However, the research on tribology⁸⁰ has shown clearly that interfaces between solids with roughness oftentimes employ distributed atomic contacts, see Fig. 5. That this has profound implications on the pressure dependence of thermal contacts has already been hypothesized.⁸¹ Research on thermal QPCs, therefore, will contribute to a bottom-up picture of thermal boundary conductance or thermal contact resistance. Such interface effects are predominant on the sub-micron scales already for reasons of classical scaling. For the construction and verification of models of thermal contacts, the independent measurement of heat transport in point contacts is needed. An improved understanding of phonon scattering in atomic-scale structures will also feed into the engineering of novel devices. For example, memristive devices as used today for neuromorphic computing can exhibit current-carrying filaments with atomic diameters.

Research on nanoscale phonon transport is anticipated to have technological impact in the quest for thermal processes at high efficiency, particularly for reducing self-heating of densely packed nanodevices on modern computer chips. In general, the contrast between good and bad thermal conductors (of about four orders of magnitude) is rather small compared to the control and variability of electrical conductance, spanning over more than 30 orders of magnitude. Thermal engineering is, therefore, an increasingly important topic. On the molecular level, this is already being done empirically by comparing, e.g., different polymeric materials. The technological community would welcome novel ideas that help with the trade-off between mechanical processing and thermal properties. Whether junction geometries will be applied widely is less clear, given the few examples used today, e.g., in some sensor implementations.

An exciting prospect for the field of thermal transport is the growing cross fertilization between different physical platforms, nanostructures, and molecular junctions, optomechanical devices and superconducting circuit quantum electrodynamics (with studies of photonic heat transport for the latter). Open question and predictions listed above, e.g., on the realization of a thermal diode by utilizing the vibrational anharmonicity, can be tested in such highly controlled devices.^83,84 Figure 6 presents a quantum heat diode based on a superconducting circuit⁸² utilizing principles proposed in Ref. 71, albeit in the context of phonon heat transport through molecular devices. Once theoretical predictions on the passive and active control of quantum heat transport are tested and established in such engineered and well-controlled devices, an exciting prospect is to utilize the enormous diversity of materials, nanostructures, and molecules to build and enrich the functionality of quantum phonon transport systems.

In conclusion, studies of phonon transport through quantum conductance channels are a lively field of research. Pioneering original experiments now open research questions of fundamental interest and significant technological potential.

A.G. and B.G. gratefully acknowledge support from Hatef Sadeghi for the insightful discussions. D.S. acknowledges support from an NSERC Discovery Grant and the Canada Research Chair program. This work also received funding from the Swiss National Science Foundation with Project No. 200660 and from the European Commission H2020-FETOPEN projects “EFINED,” Grant Agreement No. 766853, and “QuIET,” Grant Agreement No. 767187.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.