Perspective: Thermal and thermoelectric transport in molecular junctions

The intriguing idea¹ of building electrical components and interconnects using single molecules has greatly inspired researchers over the past several decades and led to the emergence of the field of “molecular electronics.” Transport characteristics of junctions of molecules between electrodes (Fig. 1(a)), called molecular junctions (MJs), have been extensively explored to investigate the feasibility of creating molecular devices with desirable transport characteristics. Both experimental and theoretical work has shown that MJs can indeed be employed to achieve unique and interesting charge transport characteristics.^2–9 Owing to technical and computational advancements over the last decade, energy (thermal) transport and conversion properties of MJs have attracted considerable attention.^10–16 Fundamentally, probing thermal transport in MJs is useful to understand the limits to the applicability of classical theories at the nanoscale. Moreover, such studies are also essential for future practical applications of MJs in electronic devices and for developing high-efficiency energy conversion devices. For example, it has been computationally proposed that by carefully designing MJs, it is possible to achieve very efficient thermoelectric energy conversion.^17–20 Furthermore, single molecular chains are also expected to feature ultrahigh thermal conductivities, in strong contrast to the relatively poor thermal conductivity of polymer materials,^21,22 indicating that a more detailed understanding of thermal transport in such molecules may enable the creation of polymer based materials with high thermal conductivity.

In this article, we review current understanding of thermal and thermoelectric properties of MJs. Insights obtained from both theoretical and experimental investigations of both self-assembled monolayer (SAM) based and single molecule based MJs will be described. This paper is organized as follows: in Section II, we briefly introduce the Landauer formalism that is used to describe transport in MJs; subsequently in Section III, we review the theoretical and experimental advancements in studying thermoelectric effects in MJs; in Section IV, recent work on heat conduction, heat dissipation, and local heating/cooling of MJs will be discussed. Finally, we conclude by highlighting some open questions in the field.

It is clear from the above expression, which is valid if the transmission function is smooth and slowly varying in an energy range of ∼k_BT around E_F, that the slope of the transmission function at E_F determines the sign and magnitude of the Seebeck coefficient.

In Sec. III, we describe thermoelectric phenomena in MJs. We begin by describing the reasons why thermoelectric properties of MJs are interesting. Subsequently, we describe theoretical and computational studies that highlight the potential of MJs for thermoelectric energy conversion. Finally, we describe the currently available experimental techniques and the progress to date in probing thermoelectric effects of MJs.

One of the first works to explore thermoelectric properties in MJs was by Paulsson and Datta,²⁸ where they analyzed the thermoelectric current and voltage of benzenedithiol (BDT) based MJs. Their findings suggested that the thermoelectric current and voltage output of BDT MJs were large enough to be measured and insensitive to the molecule-electrode coupling details. Further, their work suggested that the relative position of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) with respect to the Fermi energy plays an important role in determining the sign of the Seebeck coefficient, indicating that thermoelectric measurements of MJs can reveal important information about the electronic structures of MJs.

Subsequently, several groups theoretically studied the thermoelectric properties of MJs to explore the feasibility of achieving high ZT.^{17,18,28,32–45} For example, Finch et al.¹⁷ presented a computational study for CSW-479-bipyridine MJs with gold electrodes (Fig. 2(a)). They showed that by tuning the orientation of a side group with respect to the molecular backbone, the thermopower could be dramatically enhanced, resulting in a large ZT value. They attributed this enhancement to Fano resonances,⁴⁶ which arise from the presence of degenerate energy levels in the molecular backbone and the side groups. Specifically, they found that the transmission function showed a peak, which was shifted towards the Fermi energy by controlling the angular orientation of the side group.

Bergfield et al.¹⁸ investigated polyphenyl ether (PPE) molecules bridged between gold electrodes (Fig. 2(b)). Their work suggested that a sharp peak in the transmission function near the Fermi energy could arise from quantum interference effects,^41,47,48 and thus could lead to a large Seebeck coefficient. Further, they showed that by increasing the number of phenyl rings in the molecule, very high values of ZT (>4) were achievable.

Besides modifying the shape of transmission function by means of quantum interferences, recent work has also found that spin-crossover could be used to tune the charge transport^49–51 and thermoelectric properties⁵² of molecule-ferromagnetic metal junctions. For example, Ghosh et al.⁵² studied a spin-crossover molecule Fe(2-(1H-pyrazol-1-yl)-6-(1H-tetrazole-5-yl)pyridine)₂ ([Fe(L)₂]) which has a sharptransition from a low-spin (LS) state to a high-spin (HS) state upon elevating the temperature of the junction, and both spin and electron contributed to the thermoelectric current. In this work, ZT for the HS state (at 350 K) was found to be 4 times higher than the ZT for the LS state (at 250 K).

Vacek et al.²⁰ studied thermoelectric properties in heli-cene MJs when the helical shaped junctions are mechanically stretched or compressed (Fig. 2(c)). Their calculations showed that when the molecular junction is compressed, a significant enhancement of electrical conductance and a sign-change in thermopower, originating from the change in the overlap of the wave functions of nearest atoms, is expected along with a concomitant increase in ZT. Upon stretching, the ZT value was found to first decrease and subsequently increase. They suggested that the idea of mechanically controlling electric conductance and thermopower could also be applied to other non-planar shaped molecules including cycloparaphenylenes, cyclacenes, ball-like molecules, carbon cages, tailored fullerenes, fullerene cages, and short DNA molecules.

As described above several computational studies have suggested that it may be feasible to perform thermoelectric energy conversion at efficiencies close to the Carnot limit (i.e., very high values of ZT). However, it is well know that operation close to the Carnot limit implies operation and the reversible limit and therefore, the power output is negligible. In practice, as has been highlighted in recent works,^44,53,54 it is essential to understand the relationship between efficiency and power output under irreversible conditions. The Curzon-Ahlborn (CA) limit, which describes the thermodynamically maximum efficiency of a heat engine operating under conditions where the power output is maximized, is therefore of great interest. The maximum efficiency of a heat engine operating at maximum output power is given by $η C A = η C 2 + η C 2 8 + O ( η C 3 ) + ⋯$⁠, where $η C A ( η C )$ is the Curzon-Ahlborn (Carnot) efficiency. To achieve this maximum output power, the transmission function of MJs has to be rationally designed.

Taken together, computational work on the thermoelectric efficiencies of MJs suggests various strategies to achieve energy conversion at very high efficiencies (close to the Carnot limit) and large power outputs (close to the CA-limit). Key to achieving this goal is to develop strategies for achieving transmission functions that show rapid variations at the Fermi energy (e.g., delta/step shaped functions or an asymmetric Lorentzian-like transmission function⁵⁵). Other approaches to tune the transmission function include redox control of the quantum interference within phase coherent molecular wires,⁵⁶ variation of the transition metal-center in porphyrin-based conjugated molecules,⁵⁷ and creating π-π overlap between planar aromatic anchor groups and electro-burnt graphene electrodes.⁵⁸ In the following, we will discuss past experimental work on probing the thermoelectric properties of MJs.

Several groups have experimentally quantified the thermoelectric properties of MJs. In Table I, we summarize the results obtained so far (to the best of our knowledge). Below, we first provide a brief discussion of experimental techniques that were developed to measure the thermoelectric properties of MJs in two-terminal configurations, i.e., techniques that enable the measurement of a voltage output from junctions created by trapping molecules between a hot and a cold electrode. Subsequently, we review experimental work that elucidated the dependence of thermoelectric properties of MJs on molecular length, end groups, molecular structures and conformation, electrode materials, and temperature. Finally we discuss three-terminal techniques that enable tuning of the thermoelectric properties of MJs.

Some of the initial experimental studies of thermoelectric properties of MJs were performed by Reddy et al.⁵⁹ In performing these studies the authors adapted a STM based technique called the STM break-junction (STMBJ) technique that was originally developed by Xu and Tao⁶⁰ for probing electrical transport in MJs (Fig. 3(a)). Since the electrical and thermoelectric properties of MJs are pivotal to thermoelectric energy conversion, we first introduce the STMBJ technique and then describe how it was modified⁵⁹ for single-molecule thermoelectric measurements.

In the STMBJ technique a voltage bias is applied between a sharp tip of an STM (typically made of Au) and a Au substrate covered with a monolayer of molecules that are chemically bound to the substrate via endgroups such as thiols (–SH).The tip is displaced towards the substrate while the electrical conductance between the tip and the sample is being monitored. This process continues until an electrical contact is established and an electrical conductance of ∼5 G₀ or greater is observed (signifying the formation of Au–Au atomic contacts). During this process, in addition to the formation of Au–Au contacts, some molecules stochastically bridge the electrodes (see Fig. 3(b)). Subsequently, the tip is withdrawn from the substrate, while monitoring the electrical conductance. This process results in a scenario where a few molecules bridge the tip and substrate. By further withdrawing the tip, the molecular bridges break successively until there is only one molecule bridging the tip and the substrate. During this withdrawal process the electrical conductance of the tip-molecule-substrate junctions is measured. Typical conductance traces obtained in such an experiment⁶¹ are shown in Fig. 3(c). A substantial number of conductance traces can then be used to analyze and obtain statistically significant information about the electrical properties of MJs.

In order to investigate thermoelectric effects of MJs, the STMBJ techniques are modified so as to enable the application of a temperature difference across the MJs. In Fig. 4(a), a single MJ formed by using the STMBJ technique is depicted. The Au substrate is heated while the STM tip is kept at room temperature by placing it in contact with a large thermal reservoir. Because thermal conductance of the MJ is much smaller (quantified later in this article) than the thermal conductance of the gold STM tip, a tip-substrate temperature difference, $Δ T$⁠, is readily established.^55,57

The procedure to measure the Seebeck coefficient involves trapping of multiple molecules between the electrodes by following the procedure described above. Subsequently, the voltage bias and the current amplifier that are used to monitor the current are disconnected and a voltage amplifier is connected to measure the thermoelectric voltage induced by $Δ T$ (see Fig. 4(a)). The tip is slowly withdrawn until all the trapped molecules break off. During this process, the output voltage, $Δ V$, is continuously monitored. When the last molecule breaks off, the thermoelelctric voltage vanishes. This output voltage $Δ V$⁠, due to $Δ T$⁠, is a measure of the Seebeck coefficient and is obtained by $S = − Δ V / Δ T$.

Reddy et al.⁵⁹ employed the above-described STMBJ technique to probe the thermoelectric properties of MJs created from benzenedithiol, dibenzenedithiol, and tribenzenedithiol molecules and Au electrodes (Fig. 5(a)). These experiments revealed that the Seebeck coefficient of these aromatic MJs is positive and increases monotonically (and approximately linearly) with molecular length.

The STMBJ technique as applied to thermoelectric studies was further improved by Widawsky et al.⁶² who succeeded in simultaneously recording the electrical conductance and thermopower of single MJs in STMBJ technique by measuring both the zero-bias thermocurrent and the electrical conductance under a small applied bias. Using this method the authors explored several amine–Au linked and pyridine–Au linked MJs. From these measurements they observed a positive Seebeck coefficient for amine–Au linked MJs and a negative Seebeck coefficient for pyridine–Au linked MJs, confirming that transport in amine–Au junctions is HOMO dominated whereas that in pyridine–Au junctions is LUMO dominated.

In addition to the STMBJ technique, a different technique that is based on an atomic force microscope (AFM) was developed by Tan et al.⁶³ to probe thermoelectric effects in SAM based MJs. Briefly, a Au-coated AFM probe made of Si, that is at ambient temperature, is placed in contact with a heated Au substrate that is covered with a monolayer of molecules, while the deflection of the cantilever is continuously monitored andmaintained at a constant value using feedback control (Fig. 4(b)). The electrical conductance and the Seebeck coefficient of the tip-SAM-metal junctions are directly measured by monitoring the current output under a small bias and by measuring the voltage output under a temperature difference.

Mechanically controlled break junction (MCBJ) technique is another widely used experimental approach for probing the electrical and thermoelectric transport properties of MJs. The first mechanical “break junction” experiment was conducted by Moreland and Ekin⁶⁴ for electron tunneling in superconductors. Later Muller and van Ruitenbeek⁶⁵ developed a MCBJ technique in which a suspended metallic wire with a notch at the center is attached to a thin substrate (Fig. 4(c)). Further, the flexible device is mounted in a three-point contact configuration. The center of the device is pushed by a piezoactuator whose motion can be controlled precisely by external bias. This process results in an elongation of the wire and eventually leads to the fracture of the wire at the notch, exposing clean metal surfaces. By controlling the motion of the actuator, the gap size between the electrodes can be systematically controlled allowing one to close and reopen the gap. Specifically, the displacement between the electrodes is a small fraction (also called reduction factor) of the displacement of the actuator due to the geometry of the MCBJ platform. This reduction factor enables very precise control of the separation of the electrodes with picometer stability. Given the excellent stability of the MCBJ technique, it has been employed extensively in studies of transport properties of MJs and atomic junctions.^66–74 

Reed et al.⁷³ applied the MCBJ technique to perform charge transport measurements in organic molecule based MJs. Briefly, they exposed the Au electrodes in an MCBJ setup to benzenedithiol (BDT) molecules. This enabled them to successfully trap BDT molecules between the Au electrodes. Given the stability of the MCBJ platform, they were able to trap MJs and perform electrical transport experiments in them.

Tsutsui et al.⁷⁴ adapted the MCBJ technique for probing thermoelectric properties of MJs by integrating a Pt serpentine heater into one of the electrodes, which enabled them to apply temperature differentials across MJs (Fig. 4(c)). Using this approach, the authors probed both the electrical conductance and the Seebeck coefficient of Au–BDT–Au junctions and reported that the transport properties were sensitive to the geometric configuration of BDT molecules. They showed that upon mechanical stretching, in some cases, the molecule tilts into an upright direction and the Au–S bonds are elongated, resulting in a weaker coupling to the contact and a gradual shift of the HOMO level, accompanied by a decrease in conductance and a change (slight increase or decrease) in the Seebeck coefficient. In other cases they found that the configuration of the contact was dramatically changed, so that E_F shifted towards HOMO level, leading to an increased conductance and Seebeck coefficient.

Researchers have explored the dependence of the thermoelectric properties of MJs on molecular length. For Au–alkanedithiol (ADT)–Au junctions, the thermopower was found (Fig. 5(a)) to decrease linearly with increasing N (N is the number of carbon atoms in the backbone).⁷⁵ In contrast, for aromatic MJs, e.g., Au–phenylenediamine–Au⁷⁵ and Au–phenylenedithiol–Au^59,75,76 junctions, the thermopower was found to increase linearly with length (Fig. 5(a)). The authors⁷⁵ attributed the opposite trend in alkanedithiols to the presence of metal induced gap states. For aromatic junctions similar results were found by Widawsky et al.⁷⁷ for trimethylstannylmethyl-terminated polyphenyls with 1−4 phenyl rings (P1− P4) (Fig. 5(b)). However, they also found that the thermopower of trimethylstannyl-terminated alkanes with N = 6, 8, and 10 (C6, C8, and C10) has little length dependence (Fig. 5(b)).

Length dependence of thermopower was also studied in junctions made from DNA molecules of different sequences (Fig. 5(c)).^78,79 For GC sequence double-stranded DNA molecules, Li et al.⁷⁹ reported a linear increase in resistance with molecular length whereas the Seebeck coefficient was small and weakly dependent on the length. This is because hopping of holes along the molecules dominates the charge transport in GC sequences where Gs act as hopping sites. When AT pairs, which are expected to act as a tunneling barrier, were inserted in the middle of GC sequence, both the conductance and thermopower changed. It was found that when the number of inserted AT block base pairs m is smaller than 4, the resistance increased exponentially with the length of the block, and the Seebeck coefficient increased linearly and is large compared with the size of the GC molecule. When m ≥ 4, the transport mechanism was reported to transition from tunneling to hopping, in which regime the resistance shows weak dependence on length, and the Seebeck coefficient drops to small values.

In addition to the above described measurements of length dependence, Evangeli et al.⁸⁰ created a C₆₀ dimer by contacting a tip that had one C₆₀ molecule adhered to it with another C₆₀ molecule which adhered to a Au substrate (Fig. 5(d)). They found that the thermopower of the C₆₀ dimer is approximately doubled in comparison to single C₆₀ junctions, resulting in a relatively high Seebeck coefficient of −33 $μ V / K$⁠. Computational results suggested that the thermopower and ZT increased with the number of C₆₀ monomers in the molecular chain due to the intermolecular interaction, suggesting that it may be possible to improve thermoelectric properties by manipulating C₆₀ junctions.

The end groups bridging molecules to electrodes play an important role in charge transport and thermoelectric properties of MJs.^81–84 Experimental measurements of thiol-terminated aromatic MJs showed a positive thermopower^34,74,75 while measurements of isocyanide-terminated junctions showed a negative thermopower.⁸⁵ In addition, experimental work on trimethyltin-terminated aromatic molecules⁷⁷ demonstrated slightly higher thermopower compared to thiol groups.⁸⁵ Balachandran et al.⁸³ provided a theoretical explanation for end-group effects on thermopower of MJs. Specifically, they investigated five different end groups (isocyanide, nitrile, amine, thiol, and hydroxyl) which couple triphenyl molecule to Au electrodes (Fig. 6) and studied the influence of end groups. They found that isocyanide, nitrile, and amine end-groups led to charge transfer out of the molecule upon coupling with the electrodes, resulting in a downward shift of the energy levels, which positioned the HOMO peak closer to the Fermi energy. In contrast, thiol and hydroxyl end-groups resulted in charge transfer from the electrodes into the molecule, which led to a slight downward shift in the energy levels thus placing the LUMO peak closer to the Fermi energy.

The effect of chemical structure on thermoelectricity of MJs has been probed by adding electron-withdrawing/donating groups (fluorine, chlorine, and methyl) to BDT molecules (Fig. 7(a)).⁸⁶ It was suggested that the presence of electron-withdrawing groups (fluorine and chlorine) on a BDT molecule shifts the energy levels of MJs downwards, leading to a decreased thermopower, while the presence of electron-donating group (methyl) groups shift the energy levels upwards, leading to an increased thermopower. Since BDT has HOMO dominated charge transport properties, shifting theenergy levels up (down) gives rise to a larger (smaller) slope in the transmission function at the Fermi level.

The effect of molecular conformation on the electrical conductance has also been studied by several groups.^87–89 For example, it was shown that in aromatic molecules, when the planes associated with each of phenyl rings are twisted from being “flat” (⁠$θ = 0 °$⁠, $θ$ indicates the twist angle between two benzene plates) to being “perpendicular” (⁠$θ = 90 °$⁠), the conductance decreases by a factor of 30.⁸⁷ The thermoelectric properties of MJs created from fullerene based molecules were investigated by Lee et al.⁹⁰ They experimentally investigated three fullerene derivatives (C₈₂, Gd@C₈₂, and Ce@C₈₂) and found enhanced thermopower of Gd@C₈₂ and Ce@C₈₂ (Fig. 7(b)). Computational analysis of the same system suggested that this enhancement was due to the encapsulated metal atoms that induced changes in the geometric and electronic structure of the fullerene molecule.

Researchers have also explored the effect of mechanical deformations of MJs on their thermoelectric properties.^74,91 For example, by studying Au–Sc₃N@C₈₀–Au junctions, where an Sc₃N molecule is imbedded into a C₈₀ cage (Fig. 7(c)), Rincon-Garcia et al.⁹¹ found that the magnitude and the sign of thermopower both depend on the orientation of the molecule and force applied to the molecule. Further, computational analysis by the authors also suggested that the introduced Sc₃N creates a sharp resonance near the Fermi energy, and the location of the peak in transmission function depends strongly on the molecular orientation and applied pressure, so the thermopower can exhibit both a positive and a negative sign, depending on the relative position of the transmission peak with respect to the Fermi level. They emphasized that the transport resonance plays an import role in thermoelectric performance and that Sc₃N@C₈₀ can act as a bi-thermoelectric material.

Although gold is the prototypical electrode material in most MJ experiments, other metals have also been studied to identify potentially interesting thermoelectric properties.^11,39,92,93 The advantage of other electrode materials lies in the possibility of bringing the Fermi energy of electrodes closer to the orbital that dominates the charge transport properties of molecules.^94,95 Past work on fullerene based molecules (C₆₀, PCBM ([6,6]-Phenyl-C₆₁-butyric acid methyl ester) and C₇₀) (Fig. 8(a), showing C₆₀ only) showed that the Seebeck coefficient is suppressed when one of the Au electrodes is replaced by Pt. In contrast, it was found that the Seebeck coefficient is doubled when one of the Au electrodes was replaced with Ag.⁹⁶ Enhancements in the Seebeck coefficient were also reported in experiments involving Ni electrodes. Computations revealed that these enhancements arise from the spin-split hybridized states that are generated when the HOMO orbital of the BDT molecule couples with Ni electrodes (Fig. 8(b)).⁹⁷ In addition to these studies, computational work has pointed out that semiconducting electrodes⁴⁵ and carbon nanotubes²⁹ could suppress the “electron-like” contributions to the thermopower and cutoff the lower energy tails of HOMO transmission.⁴⁵ In these cases the output power of the systems was found to be greatly boosted.

According to Landauer formula, thermopower is expected to be linearly dependent on the average temperature of the molecular junction (see Eq. (2)). Kim et al.⁹⁸ varied the average temperature of Au–BDT–Au MJs and experimentally verified this prediction (Fig. 9). Since the figure of merit ZT is proportional to the power factor (G_eS²) and temperature T, and inversely proportional to the thermal conductance G_th, this work indicates that ZT can potentially increase as T³ if the thermal conductance is independent of temperature, as is expected when the operating temperature is higher the Debye temperature of the electrodes.

Key to improving the thermopower of a MJ is to modify the transmission function such that the slope and magnitude of the transmission function at the Fermi energy are maximized. However, most measurements of thermoelectric properties of MJs have relied on two-terminal measurements, which do not offer control on the electronic structure of the junction. Past research^99–109 has shown that it is possible to create three-terminal devices that can tune the electronic structure of junctions via a gate electrode. These devices are often created by a process calledgap by exposing the electrodes to molecules in a solution. The broken portions of the Au nanowire create two terminals (source and drain electrodes), while the Si back gate serves as a gate electrode. While several groups have used three terminal devices for tuning electrical transport, such devices could not be readily used for thermoelectric measurements due to the challenges in establishing temperature differences across the nano-gap between electrodes.

electromigration which creates three-terminal MJs by inserting molecules in nanometer-sized gaps formed during electromigration.^103,110 In creating these devices, a gold nanowire is first fabricated using electron-beam lithography on a doped Si layer which is coated with a thin dielectric layer. Subsequently, a large electric current is applied to the nanowire, causing the movement of metal atoms, which creates a nanometer or sub-nanometer sized gap in the nanowire. Molecules are deposited into the nanoscale

In order to overcome the limitations of two-terminal thermoelectric measurements, Kim et al.¹¹¹ created novel three-terminal devices based on EBJ and integrated an electric heater into one electrode (source or drain) that allowed them to apply temperature differences across MJs while electrostatically gating their electronic structure. Using such devices (Fig. 4(d)), they probed the thermoelectric properties of Au–biphenyldithiol (BPDT)–Au and Au–C₆₀–Au junctions. These measurements revealed that the thermoelectric properties can be significantly improved when the dominant transport orbital is located closely to Fermi energy. In analyzing these experimental results, the transmission function of the junctions was approximated by a Lorentzian function. For both Au–BPDT–Au junction (Fig. 10(a)) and Au–C₆₀–Au junctions (Fig. 10(b)), the thermoelectric properties were found to be tunable when a gate voltage was applied. This work demonstrated the feasibility of improving thermoelectric properties by tuning the electronic structure of MJs.

In this section we provide a review of the studies of thermal transport in MJs and polymer chains. We begin by first describing the computational and theoretical work on thermal transport in MJs. Subsequently, we will describe the status of current experimental work and compare the results from experiments with those from computational predictions.

Thermal transport in macroscopic systems is well described by Fourier’s law, $J = − κ ∇ T$⁠, where J is the heat current density, $κ$ is the thermal conductivity, and T is the absolute temperature. However, numerous computational studies have suggested that Fourier’s law is inapplicable to modeling thermal transport in single molecule chains and junctions. Historically, this important characteristic was identified in the pioneering work of Fermi, Pasta, and Ulam (the FPU problem)¹¹² where they studied the dynamics of a 1D lattice chain of N monoatomic particles. They found that a small nonlinearity in the harmonic 1D chain can give rise to a surprising divergence in thermal conductivity. Specifically, they found that for a chain of N atoms, the thermal conductivity varies as $κ ∝ N α , α > 0$⁠. More recently, several other groups have explored thermal transport in 1D-chains based on the FPU chain model.^113,114 Specifically, simulations performed using non-equilibrium molecular dynamics (NEMD) and equilibrium Green-Kubo methods confirmed that a divergence in thermal conductivity is to be expected for various configurations of the interatomic potential as well as for chains with a diatomic basis. Several of these studies reported that the value of the exponent (α) is ∼0.4, suggesting that such divergences are a general feature of 1D chains.

Despite the early theoretical findings leading to important insights into thermal transport in 1D atomic chains, the originally considered systems were relatively simple and did not include the effects of electrodes and the coupling between the chains and the electrodes. In order to capture these additional effects, more realistic 1D structures such as MJs and atomic junctions have also been recently explored. For example, Segal et al.¹¹⁵ studied thermal conduction through alkane chains bridging two thermal reservoirs. They derived a Landauer-like expression for heat flux from a generalized Langevin equation. By considering the phonon modes of a given molecule and the thermal reservoir, they calculated the thermal conductance and revealed interesting dependencies of the thermal conductance on the length and temperature of the junction (Figs. 11(a) and 11(b)). Specifically, they found that in the presence of weak reservoir-molecule coupling, the molecule chain achieved a maximum thermal conductance when the number (N) of the back-bone carbon atoms equals 4. Further, they found that a size-independent conductance is established when N > 10, indicating a strong divergence in the thermal conductivity. However, they found that when the reservoir-molecule coupling was strengthened, a Fourier-like thermal conductivity was recovered.

Ab initio methods were also employed by Sadeghi et al.¹¹⁶ to study the thermal conductance of alkanes (N = 2, 4, 8, and 16) and oligoynes (N = 2, 4, 8). They identified the contributions to the thermal conductance from electrons and phonons and found that the phononic contribution for alkanes is ∼700 times larger than the electronic contribution. Similar calculations for oligoynes suggested that the phononic contribution is only 30 times larger than the electronic contribution. Further, similar to the findings of Segal et al.,¹¹⁵ the thermal conductance of both alkanes and oligoynes in this work was found to first increase and then decrease with the molecule length (Figs. 11(c) and 11(d)).

More recently, Kl$o ¨$ckner et al.¹¹⁷ combined DFT and NEGF approaches to calculate the thermal conductance of alkanes (N = 2 to 30) and PTFE molecule (where H atoms in alkanes are replaced by F atoms) with different end anchoring groups (thiol and amine) (Figs. 11(e) and 11(f)). They reported that the phononic thermal conductances in all scenarios are relatively insensitive to the molecular length. Further, they explored the effect of the anchoring group by calculating the thermal conductance for both thiol- (–SH) and amine– (–NH₂) terminated MJs. They found that the thiol-terminated MJs had a higher conductance than the amine-terminated ones.

Markussen¹¹⁸ performed first-principle simulations of phonon transport through both linear-conjugated and cross-conjugated molecules. He showed that in cross-conjugated molecules, such as meta-connected benzene, the phonon transmission function exhibits destructive quantum interference. This observation is analogous to the destructive quantuminterference effects for electron transport in similar molecules (Figs. 12(a) and 12(b)).¹¹⁹ Further, he found that the cross-conjugated MJs reduce the thermal conductance by a factor of 2-5 compared to the linear-conjugated counterparts (Figs. 12(c) and 12(d)).

In addition to studies in single MJs, heat conduction through self-assembled monolayer (SAM) of molecules covalently bonded to the substrates has also been extensively investigated. Select computational results for different combinations of solid-SAM-solid/liquid junctions are summarized in Fig. 13. Specifically, Luo and Lloyd^120–122 performed non-equilibrium and equilibrium molecular dynamics (NEMD) calculations in Au–SAM–Au junctions. The thermal conductance of such junctions was found to be independent of length and the applied external pressure. These results were also confirmed by Duda et al.¹²³ who calculated the interface thermal conductance across the Au–SAM junctions. They found that the thermal conductance is nearly constant upon varying the molecular length. Further, Goicochea et al.¹²⁴ studied thermal transport across silica-SAM-water junctions. They found that compared to the silica-water junctions, the thermal conductance is enhanced by 1.8–3.2 fold when the silica surface is chemically functionalized by alkysilanol (–Si(OH)₂–(CH₂)_n–OH) to become hydrophilic. In addition, Hu et al.¹²⁵ studied thermal transport in Au–SAM–Si junctions using molecular dynamics simulations and found that the phonontransmission function shows distinct oscillatory behavior with strong frequency dependence, which indicates phonon interference effects. Further, they showed that this interference effect diminishes with increasing SAM thickness.

The observation of unusual phonon transport phenomena in MJs have led to several proposals for control and manipulation of thermal properties.¹²⁶ As shown in Fig. 14,it has been proposed theoretically that MJs can be used to create thermal rectifiers^127–130 (heat flow enhanced in one direction but suppressed in the other direction due to the asymmetrical molecule-reservoir coupling), field-effect phononic thermal transistors¹³¹ (by reversibly switching between the acoustic torsion and optical phonon modes), mechanically tunable thermal modulators,¹³² heat pumps¹³³ (flowing heat flux in the absence of temperature difference due to the phononic analogy of berry-phase effect), and quantum heat ratchets¹³⁴ (directed flow of current even when the net average temperature difference across the junction $Δ T = 0$⁠).

In addition to the studies of thermal transport in short MJs (N < 30), another interesting topic is thermal transport in long polymer molecular chains. Here, computational/theoretical work was mainly inspired by the experimental observation¹³⁵ of high thermal conductivity in highly drawn polyethylene (PE) fibers. In contrast to the explorations of thermal transport in 1D chains, long polymer molecules represent realistic 1D systems that can test the limit of Fourier’s law. Manyresearchers^{21,22,136–138} have simulated thermal conduction in single polymer chains of various lengths and found that the thermal conductivity is several orders of magnitude higher than that in bulk materials. Similar to the results from the simple 1D lattice model, thermal conductivity increases with molecular length, and even shows divergence in some cases. Particularly, Henry and Chen found that the thermal conductivity of a single PE chain can reach 350 (W/m)/K when the molecular length is longer than 1 $μ m$ (Fig. 15(a)).²¹ Further, Sasikumar and Keblinski¹³⁸ studied the role of chain conformation on phonon transport. They found that by introducing gauche conformations (kinks), the curved chain was shown to have much smaller thermal conductance than the straight chain because of the increase of phonon scattering centers (Fig. 15(b)). Further, Liu and Yang²² found that the polymer chains with aromatic monomers had a thermal conductivity ∼5 times larger than that of a PE chain (Figs. 15(c) and 15(d)) and concluded that thermal transport in these long molecular chains was determined by the competition between ballistic and diffusive phonons.

Another topic of interest in the study of thermal transport in molecular junctions and chains is the development of strategies to suppress phononic thermal conductance. Such strategies can positively affect thermoelectric energy conversion efficiency of molecular devices by reducing phononic thermal conductance. Proposals in this direction include, for example, taking advantage of the degree of mismatch in the vibrational (phonon) density of states of the electrodes (which is effectively continuous) and the molecules (which are discrete)^116,139 as well as designing π-stacked molecular structures with weak coupling between different parts of the molecules.¹⁴⁰ 

While computational studies of thermal transport have been pursued for several years, corresponding experimental investigations have begun only recently. The strategy to quantify molecular thermal conductance is conceptually straightforward. An applied temperature difference (analogous to a voltage bias applied during electrical measurements) results in a relatively small heat current (analogous to electrical current) across the MJs, which need to be measured to quantify thermal conductances. Given the challenges associated with measuring the small heat currents in that arise in MJs (especially single molecule junctions), experimental studies have been somewhat limited.

The primary platform employed for probing thermal transport in MJs are SAM based junctions as they provide a well characterized junction structure with a relatively large area over which experiments can be performed.¹⁴¹ Some of the first transport studies were performed across solid-SAM-liquid junctions. For example, Ge et al.¹⁴² studied the thermal conductance in solid-SAM-water junctions. In this experiment the SAM (OTS, C₁₈, PEG-silane, C₁₁OH) was bonded to a solid surface (Au or Al) to chemically modify the surface. The measurement of thermal conductance of the junction was performed using the time domain thermoreflectance¹⁴³ (TDTR) technique, which characterizes the thermal conductivity of a thin film by measuring the temperature-dependent change of surface reflectance under transient laser heating. They found that the thermal conductance between hydrophilic surfaces and water was several times larger than that between hydrophobic surfaces and water, indicating the important role of the SAM on modulating the interface thermal transport. These findings were also confirmed by Tian et al.¹⁴⁴ who studied the thermal transport across the solid-liquid interface. They found that by adding a SAM (thiolated alkane molecules) between Au and ethanol the thermal conductance is enhanced 2-5 times, depending on the length of the MJs.

In addition to the studies of Solid-SAM-liquid configuration, MJs have also been studied in solid-SAM-solid configurations. Specifically, Wang et al.¹³⁹ measured the thermal conductance of Au–ADT–GaAs junctions (N = 8, 9 and 10). The molecular junction sample was prepared by a nanotransfer printing technique, where a gold thin film, patterned on a silicon stamp, was transferred onto a SAM chemically bonded to a GaAs substrate. Subsequently, the patterned Au film was employed as a thermometer to measure the thermal conductivity of the SAM junctions using the $3 ω$ method.¹⁴⁵ In this work, the measured thermal conductance was found to be insensitive to the length of the MJs (Fig. 16(a)).

In another work, Wang et al.⁴² studied thermal transport through long-chain alkanethiols of various lengths that were bonded to a Au substrate (Fig. 16(b)). A nonlinear coherent vibrational spectroscopy technique, called the time-resolved sum-frequency generation (SFG) technique, was employed to detect the ultrafast thermal induced orientational disorder when heat is transported to the terminal –CH₃ head groups. They found that phonon transport in the alkane chains is ballistic and the thermal conductance, which is estimated to be ∼50 (pW/K)/molecule, is limited by the Au–SAM interface.

These original experiments of thermal transport in solid-SAM-solid junctions not only demonstrated how thermal transport could be systematically probed but also identified several questions that needed further elucidation; for example, what is the influence of the SAM’s molecular structure, the chemical composition of the electrodes, and the interactions between the molecule and the electrode on their thermal transport properties? How can the geometry of MJs be controlled to optimize (maximize or minimize) the thermal conductance? Several studies have been performed to answer these questions and are described below briefly.

O’Brien et al.¹⁴⁶ probed the thermal conductance of metal-SAM-dielectric interfaces and compared them to the thermal conductance of metal-dielectric interfaces. The samples were prepared by assembling different organosilane molecules with identical molecular structure except for the end groups (one with –SH, another with –CH₃) on two different dielectric structures (SiO₂ or TiO₂). By employing the TDTR technique, they demonstrated that the thermal conductance across the interfaces can be significantly increased (more than a factor of 4) by introducing strong bonding organic monolayers to facilitate phonon transport through both metal-molecule and dielectric-molecule interfaces (Fig. 16(c)). Specifically, they found that the –SH end-group functionalization is more effective in boosting thermal transport than the –CH₃ end-group functionalization and demonstrated that interfacial thermal conductance can be tuned over an order of magnitude by altering the interfacial bonding strength.

In addition to the study of O’Brien et al., Losego et al.¹⁴⁷ systematically measured the thermal conductance of Au–SAM-quartz junctions. All the organic molecules studied had a silane group (–SiH₃) at one end, while the other end was varied to be methyl (–CH₃), thiol (–SH), amine (–NH₂), or bromine (–Br). Their measurements of thermal transport revealed that the binding strength had a strong impact on the conductance. Specifically, they found that covalent-like bonds at the interface such as thiol-gold bonds yielded a thermal conductance that was ∼2 times higher than that given by van der Waals interactions, such as the cases with a –CH₃ moiety (Fig. 16(d)) interacting with Au. Further, they observed a linear dependence of interfacial thermal conductance on the covalent bond density, suggesting that intermolecular interactions do not significantly contribute to thermal transport.

More recently, Majumdar et al.¹⁴⁸ studied the impact of vibrational mismatch of electrodes on the thermal transport characteristics of SAM junctions. This measurement was conducted by employing a frequency-domain thermo-reflectance¹⁴⁹ (FDTR) technique. Different from the TDTR technique, which records the time domain response of the surface reflectance due to temperature change, the FDTR technique records the frequency domain response. They found that Au–ADT–Au junctions yield a thermal conductance that is ∼2 times higher than Au–ADT–Pd junctions (Fig. 16(e)). The ratio of the Debye temperatures of the electrodes that sandwich the MJs was used as a measure of the vibrational mismatch between the electrodes to explain the differences in the thermal conductance. These studies show that since thermal transport in MJs is dominated by phonons, the vibrational modes inside the electrodes matter significantly even though the thermal carriers inside the electrodes are electrons.

In addition to the above conclusions, Meier et al.¹⁵⁰ performed experiments where they probed thermal transport using scanning probes with integrated heater thermometers in a vacuum environment. Their measurements were performed on SAM layers assembled between a Au surface and a silica-coated tip of their scanning probe. Specifically, they created Au–ADT–SiO₂ junctions, where one end group (–SH) was chemically bonded to the Au substrate while the other (–CH₃) was weakly interacting with the tip and studied heat flow from a heated substrate into their tip as the length of the molecules was varied. They found that the thermal conductance of Au–ADT–SiO₂ junctions showed a maxima when the number of carbon atoms (N) was increased from 2 to 4. Upon further increasing N from 8 to 14, the thermal conductance was found to be invariant within measurement uncertainty (Fig. 16(f)). This observed length-dependent thermal conductance matched well with the predictions by Segal et al.¹¹⁵ We note that the advantage of the SThM over the former techniques is the capability to probe thermal transport in a highly localized region (<10 nm).

To summarize, current experimental work has successfully examined thermal transport properties of MJs in monolayers. However, experimental elucidation of thermal transport in single molecule junctions and single polymer chains has remained inaccessible. Specifically, the prediction of ultrahigh thermal conductivity in single polymer chains remains unverified. However, noteworthy efforts have succeeded in probing the thermal conductivity of nanofibers of polymer materials.^151–153 Such polymer nanofibers, which contain thousands or more single polymer chains, have provided an interesting preview of the thermal transport phenomena that arise in single-polymer chain based predictions. However, directly connecting the results from such nanofiber measurements to those of single polymer chains is complicated due to an incomplete knowledge of the crystallinity, orientation of the polymer fibers and the cross-linking between the chains.

Electric current flow and heat dissipation are often concomitant and involve complicated interactions among energy carriers. When a current I is driven by a constant bias V applied across a nanoscale junction, the total power dissipation is Q = IV. In MJs created from short molecules, if the transport through the molecular region is elastic, the Landauer approach can be used to describe heat dissipation at the electrodes of the MJs as described by Eqs. (5) and (6).

Inelastic processes, such as those arising from electron-phonon interactions in the molecular region pose significant challenges to modeling and result in a fraction of the power being dissipated in the molecular region. Given the small heat capacity of the molecules, even a small amount of heat dissipation leads to a significant increase in the local kinetic energy of atoms in the molecular region. Such scenarios raise important questions regarding the validity of the notion of temperature in this strongly non-equilibrium state. It is well known that the classical definition of temperature is applicable to only systems in equilibrium and is given by $T = ∂ U ∂ S$⁠, where U is the internal energy of the system and S is its entropy. Therefore, the description of non-equilibrium scenarios that arise during dissipation requires the definition of an operational temperature (also called the “effective temperature”¹¹). Below, we provide a description of the computational and experimental work that was performed to understand heat dissipation in MJs.

Various computational works have investigated heat dissipation and local heating effects in MJs and the microscopic mechanisms behind these phenomena. For example, heat dissipation in MJs was studied by Zotti et al.³⁸ based on the Landauer picture (Fig. 17). They performed ab initio transport calculations and showed that heat dissipation in short molecular junctions, where transport is expected to be elastic, is intimately related to the electronic structure of MJs. Specifically, they found that the heat dissipated in the two electrodes of a MJ is in general asymmetric and the degree of asymmetry is determined by the sign and magnitude of the thermopower of the MJs.

Heat dissipation in systems featuring inelastic effects (non-Landauer picture, Fig. 18(a)) was studied by Segal and Nitzan¹⁵⁴ where they estimated the fraction of heat dissipated in the molecular region and its dependence on the molecular length and coupling strength of the molecule with the electrodes. By comparing the predictions from different models (Fig. 18(b)), they found that the classical heat conduction theory underestimates the temperature rise in the molecular region, suggesting the need to apply microscopic models to understand the phonon energy transport in MJs.

Based on first principle approaches, Pecchia et al.¹⁵⁵ presented a theory of heat dissipation and local temperature of MJs. They found that local temperature is strongly influenced by electron-phonon scattering. By applying this theory to metal-styrene–Si junctions, they found that the local temperature sharply increases after a critical bias, which indicates the strong excitation of the lowest vibrational mode in the styrene molecules. Along these lines, Galperin et al.¹⁵⁶ also studied heating in current-carrying MJs and obtained a unified model to describe both heat dissipation and the electronic and vibrational contributions to heat conduction in MJs.

In addition to the contribution of electron-phonon scatterings to local heating, the effect of electron-electron scatterings has also been studied by D’Agosta et al.¹⁵⁷ who derived a model to describe the local temperature based on an electron liquid based hydrodynamic approach. They found that in MJs the electron-electron interactions, which could have comparable scattering rate as the electron-phonon interactions, may play an important role in determining the local temperature.

The effect of molecular length on local heating was also studied by Yang et al.¹⁵⁸ who calculated the local temperature increase in MJs containing 1 to 4 Al atoms. Specifically, they found that at a given bias the local temperature increase is larger in longer chains (Fig. 18(c)). Further, Chen et al.¹⁵⁹ estimated the local heating of various ADT MJs (Fig. 18(d)). They found that in the presence of good molecule-electrode coupling, the local temperature increase was smaller when the molecular length increases. Further, local temperature increase was predicted to be insensitive to the molecular length when the molecule-electrode coupling was weak. These works suggest that inelastic interactions inside the molecule and heat dissipation in the electrodes play important roles in determining the local temperature.

In addition to studies of local heating effects, local cooling in MJs has also attracted much attention.^160–164 For example, Galperin et al.¹⁶⁰ proposed two cooling mechanisms: one based on the depletion of hot electrons inside the high potential electrode; the other based on inelastic electron transport. The former achieved a cooling effect by shifting the non-equilibrium electron distribution, which defines the effective temperature. The latter leveraged the inelastic tunneling in which the electron extracts energy from vibrational modes while transmitting through the junction. More recently, Lykkebo et al.¹⁶⁴ studied local cooling effects in realistic MJs. They presented cooling schemes based on inelastic electron tunneling and found that under voltage biases across the MJs the local temperature can potentially reach values below the ambient temperature. This study pointed out the possibility of realizing local molecular cooling by rationally designing the MJs.

Several experimental studies have been performed to quantify heat dissipation in MJs and obtain insights into the inelastic processes in the molecular region and the electrodes. One of the first studies in this area was conducted by Huang et al.¹⁶⁵ who studied current induced local heating in MJs by measuring the force required to break the molecule-electrode covalent bonds. By taking advantage of the fact that the chemical bond breakdown process is thermally activated, the authors were able to define an effective temperature of the junction as a function of the applied bias. Based on their analysis they suggested that, under ambient conditions, an ∼30 K temperature increase is generated when a bias of 1 V is applied across Au–octanedithol–Au junctions (Fig. 19(a)).

The instability of MJs induced by applied voltage biases has been studied^166,167 by quantifying the frequency of two-level fluctuations (TLF) in conductance of MJs.^166,167 It was found that the frequency of the conductance fluctuations increase with increasing bias and the effective temperature of the MJ is proportional to $V$ (Fig. 19(b)). Along the same lines, Tsutsui et al.¹⁶⁸ reported an estimation of the local temperature by fitting the molecular junction lifetime to a model that takes into consideration both the applied bias and theeffective temperature. From these studies it was found that Au–BDT–Au molecule junctions were heated up by ∼200 K at room temperature under an applied bias of ∼1 V (Fig. 19(c)). Further, inelastic electron tunneling spectroscopy¹⁶⁹ (IETS) studies were used to study the onset of heat generation in MJs due to the excitation of the lowest energy phonon mode.

Huang et al.¹⁷⁰ also studied the effective temperature of a series of ADT MJs by using the STMBJ technique (Fig. 19(d)). In this work, the stretching lengths of a MJ at breakdown were measured and fitted to a model that describes the dependence of stretching length on effective temperature. They found that the local temperature increases with the applied bias until ∼0.8 V and then decreased in the presence of higher biases, suggesting that strong electron-electron scattering led to local ionic cooling. Further, they studied the dependence of effective temperature on the molecular length and found that at a given bias the effective temperature was higher for shorter MJs, indicating that electron-phonon scattering is reduced when MJs were longer.

Besides the measurements of local temperatures using break-junction techniques, Ioffe et al.¹⁷¹ and Ward et al.¹⁷² also leveraged surface-enhanced Raman spectroscopy to study this problem (Fig. 20). In their works, the Stokes (S) and anti-Stokes (AS) components of the Raman scattering were measured and the AS/S ratio, which is temperature dependent, was used to estimate the local heating effects. Specifically, Ward et al.¹⁷² characterized both the effective vibrational and electronic temperatures of OPV3 molecular junctions and observed a pronounced heating effect with ∼200–300 K local temperature increase in the presence of applied bias ∼0.4 V. In contrast, Ioffe et al.¹⁷¹ reported a much smaller temperature increase (30–50 K) for the same bias in DBDT MJs. These studies indicate that local Raman spectroscopy may serve as a valuable technique to probe local temperatures at the atomic scale.

In addition to these studies, Schulze et al.¹⁷³ used a STM to measure heating and heat dissipation processes in single C₆₀ molecules. Specifically, they observed the thermally induced decomposition of molecular structures. In the tunneling regime, they found that dissipation of ∼20 $μ W$ of power was sufficient to cause strong molecule heating and a breakdown of the fullerene cage. In contrast they found that when the molecule was in good contact with both the electrodes, a larger amount of power was necessary for the breakdown of the fullerene cage. Given these observations, they concluded that in the case of junctions with good contact with electrodes either the heat generation in the molecule was less effective or the heat dissipation through phonon-electrode coupling was stronger.

We note that in the experiments discussed above, some were aimed at resolving the inelastic scattering mechanisms in the molecular regions of MJs, while others focused on resolving heat dissipation in the electrodes and assumed that there is negligible dissipation in the molecular regions. These goals do not necessarily contradict each other: as discussed above, the power supplied to sustain the electron flow in MJs must be dissipated in either the electrodes or the molecular region, or both. The heat dissipation in the molecular region of short MJs is usually a very small portion of the total heat dissipation. But even such minute levels of dissipation cause large increases in the local temperature.

Experimental interrogation of thermoelectric energy conversion, heat transport, and heat dissipation in MJs poses several challenges overcoming which is critical to testing current predictions and stimulating future theoretical and computational work. We conclude this review by discussing some outstanding challenges.

A key challenge in thermoelectric energy conversion is to experimentally demonstrate large thermoelectric efficiencies (i.e., ZT values >1) in MJs. While several computations have suggested the feasibility of achieving efficient and/or high power output, thermoelectric energy conversion experiments have so far failed to achieve this goal. Further development of thermoelectric measurements schemes, especially three terminal techniques, may be required to successfully explore the limits of thermoelectric energy conversion in MJs. Moreover, it is also important to explore strategies for developing thermoelectric devices/materials where MJs form the functional subunits.

Probing thermal transport in single MJs and polymer chains is critical for achieving efficient thermoelectric energy conversion and for understanding the physics of heat transfer in these fundamentally different structures. While ensemble based experiments (such as in SAM and polymer nanofibers) have made impressive progress in understanding thermal transport properties, they fail to reveal heterogeneities and can only reveal averaged properties. Single-molecule thermal transport measurement will enable direct comparisons among multiple theories in this field and help build guiding principles for designing MJs for optimal thermal transport.

Recent work by Cui et al.^184,185 has shown that it is possible to systematically study heat transfer due to both thermal radiation and conduction at the nanoscale. Specifically, very recent advances by Cui et al.¹⁸⁵ have shown that novel picowatt-resolution scanning calorimetric probes can be employed to measure thermal transport in metallic single-atom junctions. Further, they observed the phenomena of thermal conductance quantization at room temperature. This recent work offers new opportunities to explore thermal transport in molecular junctions, one-dimensional chains of atoms and individual polymer chains, all of which have been studied extensively for over half a century but have not been experimentally accessible.

Another topic of interest is to probe heat dissipation and measuring local temperatures. Existing experimental approaches are not completely capable of resolving the validity of competing theoretical models. To overcome this challenge, high resolution heat dissipation measurements at the single-molecule level are required. Such measurements will enable establishing a precise limit for the fraction of heat dissipation in the molecular region and will play a critical role in verifying the proposed models for heating/cooling of molecules directly.

We acknowledge support from the U.S. Department of Energy Basic Energy Sciences through a grant from the Scanning Probe Microscopy Division under Award No. DE-SC0004871.