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DOI: 10.1039/D0NH00164C
(Communication)
Nanoscale Horiz., 2020, Advance Article

Ali K. Ismael^{ab} and
Colin J. Lambert*^{a}
^{a}Department of Physics, Lancaster University, Lancaster LA1 4YB, UK. E-mail: c.lambert@lancaster.ac.uk
^{b}Department of Physics, College of Education for Pure Science, Tikrit University, Tikrit, Iraq

Received
19th March 2020
, Accepted 13th May 2020

First published on 13th May 2020

This article highlights a novel strategy for designing molecules with high thermoelectric performance, which are resilient to fluctuations. In laboratory measurements of thermoelectric properties of single-molecule junctions and self-assembled monolayers, fluctuations in frontier orbital energies relative to the Fermi energy E_{F} of electrodes are an important factor, which determine average values of transport coefficients, such as the average Seebeck coefficient 〈S〉. In a worst-case scenario, where the relative value of E_{F} fluctuates uniformly over the HOMO–LUMO gap, a “worst-case scenario theorem” tells us that the average Seebeck coefficient will vanish unless the transmission coefficient at the LUMO and HOMO resonances take different values. This implies that junction asymmetry is a necessary condition for obtaining non-zero values of 〈S〉 in the presence of large fluctuations. This conclusion that asymmetry can drive high thermoelectric performance is supported by detailed simulations on 17 molecules using density functional theory. Importantly, junction asymmetry does not imply that the molecules themselves should be asymmetric. We demonstrate that symmetric molecules possessing a localised frontier orbital can achieve even higher thermoelectric performance than asymmetric molecules, because under laboratory conditions of slight symmetry breaking, such orbitals are ‘silent’ and do not contribute to transport. Consequently, transport is biased towards the nearest “non-silent” frontier orbital and leads to a high ensemble averaged Seebeck coefficient. This effect is demonstrated for a spatially-symmetric 1,2,3-triazole-based molecule, a rotaxane-hexayne macrocycle and a phthalocyanine.

## New conceptsThe design of new thermoelectric materials for converting waste heat directly into electricity is a global challenge. To avoid the cost and toxicity of available inorganic materials, there is a need to identify organic materials, whose thermoelectric performance can be optimised through chemical synthesis, guided by principles of molecular-scale quantum transport. However single-molecule junctions and self-assembled monolayers suffer for atomic-scale variability at the molecule–electrode interfaces and in the surroundings, which lead to a decrease in their thermoelectric performance. To overcome this problem, we introduce a new concept of utilising organic molecules with ‘silent orbitals’. Silent orbitals are frontier orbitals, which are localised within the core of a molecule and reduce the effect of fluctuations, leading to an increase in thermoelectric performance. This general principle is illustrated by examining the properties of 17 different molecules, including a 1,2,3-triazole-based molecule, a rotaxane-hexayne macrocycle and a phthalocyanine, all of which possess silent orbitals. |

(1) |

Despite intense efforts to improve the thermoelectric performance of single molecules, reported values of their Seebeck coefficients remain disappointingly low, because in experimental measurements of single-molecule thermoelectric properties, the location of the Fermi energy E_{F} relative to the energies of frontier orbitals is sensitive to environmental effects, the shape and spacing of electrodes and the manner in which a molecule binds to the electrodes. Consequently measured single-molecule transport properties such as the Seebeck coefficient S exhibit large measurement-to-measurement fluctuations. To accommodate these fluctuations, it is standard practice to make many thousands of measurements of S, and then to quote their average value 〈S〉. In many cases, individual measurements of S could be large and positive or large and negative, but 〈S〉 is small due a cancellation of opposite signs in the average. Therefore, it is of interest to develop strategies for avoiding this sign cancellation and improving the ensemble-averaged thermoelectric performance, even in the presence of large fluctuations.

The aim of this paper is to confront this issue by considering a worst-case scenario, in which the relative value of E_{F} varies randomly between the energy E_{H} of the transmission resonance associated with highest occupied molecular orbital (HOMO) and the energy E_{L} of the resonance associated with the lowest unoccupied molecular orbital (LUMO). We choose this range of E_{F}, because most molecules measured to date are not redox active. For non-redox active molecules, E_{F} is constrained to lie between the HOMO and LUMO, because if E_{F} lies above E_{L} or below E_{H}, the molecule would gain or lose an electron respectively. Our aim is to identify molecular design strategies, which would lead to a large value of 〈S〉, even in a worst-case scenario, where E_{F} is a random number, uniformly distributed between E_{H} and E_{L}.

Fig. 1 (a) Three molecules containing ‘silent’ orbitals, based on: 1 phthalocyanine, 2a rotaxane-hexayne macrocycle and 3 a spatially-symmetric 1,2,3-triazole-based molecule (b) a localised HOMO and (c) a delocalised LUMO (for frontier orbitals of 1 and 2 see Fig. S26–S28 of the ESI†). |

Fig. 2 Three examples of transmission functions of the 1,2,3-triazole based molecule 3 of Fig. 1. The grey curve (labelled 3a) corresponds to an improbable symmetric situation, in which each terminal group binds with precisely the same geometry to atomically identical electrodes. The yellow and brown curves (labelled 3b, 3c) show the transmission functions of slightly asymmetric junctions, for which the transmission resonance at energy E_{H} falls silent. |

The grey transmission curve in Fig. 2 corresponds to an improbable symmetric junction, in which each terminal group binds with precisely the same geometry to atomically identical electrodes. In this case, there is a narrow transmission resonance associated with the HOMO, at energy E_{H}, with a maximum value of T(E_{H}) = 1 and a much broader resonance associated with the LUMO, at energy E_{L}, which also has a maximum value of T(E_{L}) = 1. As discussed below, these high values of T(E_{H}) and T(E_{L}) are a consequence of spatial symmetry. However, experimentally, in a real junction, where molecules bind randomly to electrodes, such a precise symmetry is highly improbable. In contrast, the yellow and brown curves in Fig. 2 show the transmission functions of slightly asymmetric junctions, which arise when the distances between the terminal groups and the nearest gold electrode atoms at opposite ends of the molecule differ by a mere 0.1 Å, or their angles differ by <5°. This tiny symmetry breaking causes the HOMO to fall ‘silent’ and the HOMO resonance to disappear, whereas the LUMO resonance is unaffected and maintains a high value of T(E_{L}) = 1.

This destruction of narrow resonances due to symmetry breaking is advantageous, because as demonstrated below, in a worst-case scenario, where the Fermi energy fluctuates between the HOMO and LUMO, the average room-temperature Seebeck coefficient of the improbable symmetric junction **3a** (grey curve of Fig. 2) is almost zero. In contrast, for the slightly asymmetric junctions **3b** and **3c** (yellow/brown curves of Fig. 2) the average room-temperature Seebeck coefficients are large and can exceed 140 μV K^{−1}. Fig. S6 and S7 (ESI†) show that this is a generic feature of such molecules, because as shown in Fig. S26 and S27 (ESI†), **1** possesses a localised LUMO, whereas **2** possess a localised HOMO. Consequently, as demonstrated by Fig. S6 and S7 (ESI†), in the presence of small symmetry-breaking fluctuations in their junction geometries, their LUMO and HOMO resonances respectively fall silent and their average Seebeck coefficients are enhanced.

To demonstrate why non-redox-active, symmetric molecules possessing a silent frontier orbital are the molecules of choice in a worst-case-scenario, we now present a detailed analysis of four typical examples of transmission coefficients, which could be encountered in a spatially-symmetric single-molecule junction. Each transmission coefficient in Fig. 3 possesses HOMO and LUMO resonances and passes through a minimum at some energy E_{0} within the HOMO–LUMO gap. Fig. 3 demonstrates that even if a molecular junction is spatially symmetric, plots of the associated transmission function versus energy are not necessarily symmetric about the middle of the HOMO–LUMO gap. Fig. 3a and b show transmission coefficients, which are symmetric about the gap centre E_{HL} = (E_{H} + E_{L})/2, whereas Fig. 3c (Fig. 3d) shows a transmission coefficient, which is asymmetric, with a predominantly positive (negative) slope over a large energy range E_{0} < E < E_{L} (E_{H} < E < E_{0}).

Fig. 3a shows a transmission curve, which exhibits constructive quantum interference (CQI) within the HOMO–LUMO gap, signalled by a rather smooth parabolic minimum at an energy E_{0} between the HOMO and LUMO. Fig. 3b is a transmission function, which exhibits destructive quantum interference (DQI), signalled by a sharp dip in T(E) at an energy E_{0}. For these transmission coefficients, E_{0} is located in the middle of the HOMO–LUMO gap, whereas for those Fig. 3c and d, the DQI dip is either below or above the gap centre. (For more details of the model leading to these transmission coefficients, see Section 1 of the ESI.†)

If E_{F} is uniformly distributed between E_{H} and E_{L}, then molecules exhibiting symmetric transmission curves such as Fig. 3a and b are undesirable for thermoelectric applications, because the slope of T(E) and therefore the sign of S (from eqn (1)) is equally likely to be positive or negative and the average Seebeck coefficient 〈S〉 would be zero. On the other hand, for the transmission coefficient shown in Fig. 3c, the slope is positive and S is negative over a wide range of energy and therefore at first sight, such asymmetric transmission coefficients would appear to be more desirable. However, care must be taken, because in a worst-case scenario, when E_{F} is uniformly distributed between the HOMO and LUMO energies, E_{H} and E_{L}, we now show that for the transmission coefficients of Fig. 3c and d, 〈S〉 = 0. Indeed when eqn (1) is valid, the average Seebeck coefficient is

(2) |

This theorem is easily proved, because if E_{F} is a random variable with a probability distribution P(E_{F}), then by definition

(3) |

Since integration is the reverse of differentiation, the WCS theorem (2) is obtained.

The WCS theorem tells us that if the HOMO and LUMO transmission resonances are equal (i.e. if T(E_{L}) = T(E_{H})), then 〈S〉 = 0, even if the transmission function is asymmetric. Consequently in a worst-case scenario, the average Seebeck coefficients of the junctions of Fig. 3 vanish, because for these junctions T(E_{L}) = T(E_{H}).

An alternative way of viewing this vanishing of 〈S〉 is to note that the slopes of the plots of T(E) versus E in the upper panels of Fig. 3 change sign at some energy E = E_{0} and consequently, the corresponding plots of S(E_{F}) versus E_{F} in the lower panel change sign at E_{F} = E_{0}. In the worst-case scenario, where P(E_{F}) = 1/Δ, for E_{L} > E_{F} > E_{H} and zero outside this range, eqn (3) becomes

(4) |

(5) |

To avoid the equality T(E_{L}) = T(E_{H}), we note that the Breit Wigner formula^{39} tells us that for weakly coupled molecules with non-degenerate orbitals, a molecular junction (but not necessarily the molecule) must be spatially asymmetric. To illustrate this point, we note that when the transmission function T(E) possesses a resonance at an energy E_{a} (where in our case E_{a} is either E_{H} or E_{L}), the Breit–Wigner formula takes the form

(6) |

(7) |

(8) |

Fig. 4 Upper panel: Examples of transmission functions associated with spatially asymmetric junctions, obtained from the tight-bonding model of Fig. S1 (ESI†). Lower panel: Four the corresponding Seebeck coefficient S obtained using eqn (1). Note that the area under each curve of S(E_{F}) between E_{H} and E_{0} (shaded grey) is not equal to the area under the curve between E_{0} and E_{L} (shaded yellow), because for each of the four transmission coefficients in the upper panel, T(E_{H}) ≠ T(E_{L}). |

In contrast with the Seebeck plots of Fig. 3, which correspond to spatially-symmetric junctions, for the spatially-asymmetric junctions of Fig. 4 the (positive) area A_{H} of the grey shaded region is not equal and opposite to the (negative) area A_{L} of the yellow shaded region, and therefore from eqn (4) 〈S〉 ≠ 0.

We now argue that best way to achieve such asymmetric junctions is to utilise molecules with silent frontier orbitals, such as **1**, **2** and **3**. To demonstrate this result, we first consider a more obvious strategy of using spatially asymmetric molecules, such as **8**, **9**, **10**, **11**, **13**, **14**, **15**, **16**, **17** in Fig. 5. These contrast with the symmetric molecules **2**, **3**, **4**, **6**, **12**, which possess a σ_{v} mirror plane and **5**, **7**, which possess a C_{2} axis. As expected, Fig. 6 shows that in a worst-case scenario, the room-temperature values of 〈S〉 of these asymmetric molecules (obtained by evaluating the finite temperature version of eqn (1), shown in eqn (S1) of Section 1 of the ESI†) are non-zero, whereas symmetric junctions formed from the more symmetric molecules (**1–7** and **12**) possess negligibly small values of 〈S〉. The transmission curves of each of these molecules are shown in the ESI,† along with their values of E_{H}, E_{0} and E_{L}. In Fig. 6, results for their room-temperature 〈S〉 are plotted against the transmission asymmetry parameter . The latter is close to 0 or 1 when T(E_{H}) ≪ T(E_{L}) or T(E_{H}) ≫ T(E_{L}) respectively and equals 0.5 when the HOMO and LUMO transmission resonances have equal values. Fig. 6 shows that negative values of 〈S〉 tend to arise when δ_{H} is small, whereas positive values of 〈S〉 tend to arise when δ_{H} is close to unity. In other words, the value of room-temperature 〈S〉 is closely correlated with δ_{H}, in agreement with the WCS theorem and eqn (2).

Fig. 5 Structures of the molecules 4–17 (see Fig. S4 and S5 for more details and Fig. S26–S43 for their frontier orbitals, ESI†). |

Fig. 6 Results for the room temperature values of 〈S〉 versus the parameter δ_{H}, which characterises the relative values of the HOMO and LUMO transmission resonances. The points labelled 1a, 2a, 3a indicate the value of 〈S〉 obtained when 1, 2, 3 are placed symmetrically in a junction. The points 1b, 2b, 3b and 1c, 2c, 3c the values of 〈S〉 obtained when 1, 2, 3 are placed asymmetrically in a junction in two slightly different geometries. Table S1 (ESI†) shows the actual values of these average Seebeck coefficients. |

Although the asymmetric molecules possess non-zero values of 〈S〉, Fig. 6 reveals that a more advantageous strategy is to use molecules, such as the phthalocyanine **1**, the rotaxane **2**,^{40} or the 1,2,3-triazole based molecule **3** (see Fig. 1), which possesses a ‘silent’ frontier orbital. To illustrate this feature, Fig. 6 shows the values of 〈S〉 associated with **1**, **2** and **3**, in three junction geometries, denoted **1a**, **1b** and **1c** (similarly **2a**, **3a**, **2b**, **3b** and **2c**, **3c**). The points labelled **1a**, **2a**, **3a** show the values of 〈S〉 for the highly improbable symmetric junction corresponding to the grey curves of Fig. 2 and Fig. S6 and S7 (ESI†), whereas the points **1b**, **2b**, **3b** and **1c**, **2c**, **3c** show the values of 〈S〉 obtained in the more-probable slightly asymmetric junctions, with silent orbitals, corresponding to yellow and brown curves of Fig. 2 and Fig. S6 and S7 (ESI†). Clearly, the values of 〈S〉 for the junctions **1c**, **2c**, **3c**, with silent orbitals, are higher in magnitude than those of any of the other molecules.

From the WCS theorem and the Breit–Wigner formula, it is clear that the above behaviour is generic. For asymmetric junctions, T(E_{L}) ≠ T(E_{H}) and from eqn (2) 〈S〉 is non-zero. To understand why molecules with a silent frontier orbital, such as **3** are advantageous over spatially-asymmetric molecules, it is interesting to examine the consequences of the WCS theorem (2) and a modified version of (7). At energy E_{a}, the latter should be modified when x_{a} is small, because then the contribution to T(E_{a}) from all other orbitals cannot be ignored. (This is clear in the brown curve of Fig. 2, because the feature associated with the silent HOMO sits on a smooth background curve due to contribution from all other orbitals.) Therefore we replace eqn (7) by

T(E_{a}) = x_{a} + c_{a}
| (9) |

(10) |

Of course in a real experiment, a worst-case scenario may not be encountered and E_{F} may have a non-uniform probability distribution peaked at a particular value of E_{F}. However we are not aware of any experimental measurement, which can determine the distribution of E_{F} relative to frontier orbitals. The extreme opposite of a worst-case scenario is to assume a ‘best-case scenario’ in which E_{F} does not fluctuate and has a fixed value. However in practice this is completely unrealistic and does not occur in a real experiment. In the ESI,† (Fig. S44), we explore some examples of distributions, which lie between these extremes. Interestingly, the molecules **1b**, **2b**, **3b**, **1c**, **2c**, **3c** with silent orbitals deliver large Seebeck coefficients, which are relatively insensitive to the changes in the distribution of E_{F}, which again shows that these molecules are advantageous for thermoelectricity under realistic laboratory conditions. In this study, we deliberately chose unbiased distributions, in which the fluctuations of E_{F} are distributed symmetrically about the gap centre. Clearly the thermoelectric performance of these molecules could be improved by biasing the distributions of E_{F} using a judicious choice of anchor group. If the worst-case-scenario value of 〈S〉 is negative, then this could be improved by utilising pyridyl anchors, which bias E_{F} towards the LUMO, whereas if the worst-case-scenario value of 〈S〉 is positive, then this could be improved by utilising thiol anchors, which bias E_{F} towards the HOMO.

So far, we have discussed the average Seebeck coefficients of single-molecule junctions. In Fig. S44 (ESI†), we also show results for the conductance-weighted average 〈S_{SAM}〉 (defined in eqn (S9), ESI†), which as discussed in ref. 34, is the average Seebeck coefficient of a self-assembled monolayer (SAM) of a parallel array of non-interacting molecules. This reveals that molecules with silent frontier orbitals also lead to high values of 〈S_{SAM}〉 and to high values of the power factor 〈S_{SAM}〉^{2}〈G〉.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0nh00164c |

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