Molecular-scale thermoelectricity : a worst-case scenario

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 EF 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 EF 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.


Tight binding model
Supplementary Figure  The ground state Hamiltonian and optimized geometry of each molecule was obtained using the density functional theory (DFT) code [41]. The local density approximation (LDA) exchange correlation functional was used along with double zeta polarized (DZP) basis sets and the norm conserving pseudo potentials. The real space grid was defined by a plane wave cut-off of 250 Ry. The geometry optimization was carried out to a force tolerance of 0.01 eV/Å. This process was repeated for a unit cell with the molecule between gold electrodes where the optimized distance between Au and the anchor group was obtained. From the ground state Hamiltonian, the transmission coefficient, the room temperature electrical conductance and Seebeck coefficient was obtained.
The above series of molecules were then connected to gold electrodes and the resulting junction geometries were relaxed. As examples, junctions formed from 3 and 6 are shown in Figure S5.

DFT results for the transmission functions and Seebeck coefficients of structures 1-17.
For each of the above molecules, Supplementary Figures 6-18 show the transmission coefficients as a function of electron energy and Seebeck coefficients evaluated at room temperature using equation S1 for various Fermi energies , relative to the DFT-predicted Fermi energy . The grey plot is obtained when the molecule is located symmetrically within the junction. 1b and 1c: The yellow and brown plots are obtained from slightly asymmetric junctions, in which the distance (or angle) to one electrode differs slightly from the other. (Right panel). Corresponding Seebeck coefficients S E for junctions 1a-c. Note the negative yellow/brown area is much smaller than positive grey area for the asymmetric junctions 1b and 1c, which possess silent LUMOs. The black solid curves are plots of equation S1 (see section 1 above), whereas the red solid lines are obtained using the low-temperature formula equ. 1 of the main text.

Supplementary Figure 9: Transmission coefficient T(E) for 4 (Left panel).
Seebeck coefficient S of the same molecule (Right panel). Note the positive grey area equals negative grey area due to symmetry. The black solid curves are plots of equation S1 (see section 1 above), whereas the red solid lines are obtained using the low-temperature formula equ. 1 of the main text. Note the negative yellow/brown area is slightely bigger than positive grey area due to asymmetry. The black solid curves are plots of equation S1 (see section 1 above), whereas the red solid lines are obtained using the low-temperature formula equ. 1 of the main text.

Summary of Seebeck calculations
For each of the above molecules, Table S1, shows the average Seebeck coefficients and the values of . We also show values of the following quantities, which capture different aspects of asymmetry Δ and . These are defined by can be regarded as an alternative to Δ as an indicator of transmission-coefficient asymmetry. is the average of all Seebeck coefficients obtained for values of between the HOMO and , whereas is the average of all Seebeck coefficients obtained for values of between the LUMO and . ≈ for 1a, 2a, 3a, 4-9 and 12 whereas, > or vice versa for the rest (1b, 1c, 2b, 2c, 3b, 3c, 8-11 and 13-17

Wave function plots for isolated molecules with their optimised geometries
The plots below (Supplementary Figures 26-43 shows plots of and for various values of . Interestingly, the molecules 1b, 2b, 3b, 1c, 2c, 3c with silent orbitals deliver large Seebeck coefficients, which are relatively insensitive to the value of , which shows that these molecules are advantageous for thermoelectricity. These distributions are unbiased, in the sense that they are symmetric about the gap centre. Clearly the thermoelectric performance of these molecules could be improved by biasing the distributions of with a judicious choice of anchor group. If the worst-case-scenario value of is negative, then this could be improved by utilising pyridyl anchors, which bias towards the LUMO, whereas if the worst-case-scenario value of is positive, then this could be improved by utilising thiol anchors, which bias towards the HOMO. Figure S45 shows the power factor SG / G plotted against for both symmetric and asymmetric molecules. It demonstrates that symmetric molecules have quite low power factors (grey circles), whereas asymmetric molecules have higher power factor and the asymmetric junctions containing molecules with silent orbitals consistently deliver high values.

Power factor
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