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Ali
Ismael
*^{ab},
Alaa
Al-Jobory
^{ac},
Xintai
Wang
^{a},
Abdullah
Alshehab
^{a},
Ahmad
Almutlg
^{a},
Majed
Alshammari
^{a},
Iain
Grace
^{a},
Troy L. R.
Benett
^{d},
Luke A.
Wilkinson
^{d},
Benjamin J.
Robinson
^{a},
Nicholas J.
Long
^{d} and
Colin
Lambert
*^{a}
^{a}Department of Physics, Lancaster University, Lancaster LA1 4YB, UK. E-mail: k.ismael@lancaster.ac.uk
^{b}Department of Physics, College of Education for Pure Science, Tikrit University, Tikrit, Iraq
^{c}Department of Physics, College of Science, University of Anbar, Anbar, Iraq
^{d}Department of Chemistry, Imperial College London, MSRH, White City, London, W12 0BZ, UK

Received
7th July 2020
, Accepted 6th October 2020

First published on 19th October 2020

If the Seebeck coefficient of single molecules or self-assembled monolayers (SAMs) could be predicted from measurements of their conductance–voltage (G–V) characteristics alone, then the experimentally more difficult task of creating a set-up to measure their thermoelectric properties could be avoided. This article highlights a novel strategy for predicting an upper bound to the Seebeck coefficient of single molecules or SAMs, from measurements of their G–V characteristics. The theory begins by making a fit to measured G–V curves using three fitting parameters, denoted a, b, c. This ‘ABC’ theory then predicts a maximum value for the magnitude of the corresponding Seebeck coefficient. This is a useful material parameter, because if the predicted upper bound is large, then the material would warrant further investigation using a full Seebeck-measurement setup. On the other hand, if the upper bound is small, then the material would not be promising and this much more technically demanding set of measurements would be avoided. Histograms of predicted Seebeck coefficients are compared with histograms of measured Seebeck coefficients for six different SAMs, formed from anthracene-based molecules with different anchor groups and are shown to be in excellent agreement.

Herein we propose a simple and straightforward method of estimating an upper bound for the Seebeck coefficient of single molecules and self-assembled molecular layers (SAMs), based on measuring their conductance–voltage characteristics alone.

Since the latter are available to many experimental groups, this should speed up the screening of potential molecules for thermoelectric applications, without the need for direct measurement of their Seebeck coefficients. On the other hand, if the latter is also measured, then the proposed method enables a consistency check between measurements of complementary transport properties.

lnT(E) = a + b(E − E_{F}) + c(E − E_{F})^{2} | (1) |

In what follows, the coefficients a, b, c of this ‘ABC’ theory will be determined by fitting the above expression to measured low-voltage conductance–voltage curves, under the assumption that a, b and c do not change with voltage. Information about T(E) has been extracted from experimental measurements previously.^{37,38} ABC theory is aimed at describing off-resonance transport, since this is the most common case in molecular junctions and self-assembled monolayers. Of course, by applying an electrostatic of electrochemical gate, one could move transport towards resonance, but this is not relevant from the point of view of identifying thermoelectric materials. Our approach also applies to non-symmetric junctions, as demonstrated by molecule **1**, and is not limited to the wide band approximation. In fact eqn (1) can describe many molecular junctions, but it could fail at high bias, because the proposed I–V fitting assumes that I–V curves are symmetric and therefore it should not be applied to junctions exhibiting strong rectification. However, it should be noted that the Seebeck effect is a low bias phenomenon, because typical values of the Seebeck coefficient are in the range of microvolts per Kelvin.

To acquire this fitting, we measure the current versus voltage at M different locations (labelled j) across a SAM. At each location, the current I^{j}_{exp}(V_{i}) is measured at a series of N voltages labelled V_{i} between −1 V and +1 V, where N is typically several hundred. The corresponding conductance is defined to be G^{j}_{exp}(V_{i}) = I^{j}_{exp}(V_{i})/V_{i}. For each location j, we then computed the mean square deviations

(2) |

In this expression, G(V_{i}, a, b, c) = I(V_{i}, a, b, c)/V_{i} where I(V_{i}, a, b, c) is the theoretical current obtained from the Landauer formula:

(3) |

(4) |

(5) |

In fact, we found that in all cases, c was small and in many cases setting c = 0 yielded an acceptable fit. In what follows, we show results obtained by setting c = 0 and fitting a and b only. In Table S1 of the ESI,† we also show results obtained by allowing c to be non-zero.

It should be noted (see Section 3 of the ESI†) that ABC theory cannot predict the sign of the Seebeck coefficient, because it can only predict the magnitude of the coefficient b. To illustrate this point, note that at low-enough temperatures, the current I due to a source-drain bias voltage V, and the Seebeck coefficient S are given by

(6) |

(7) |

This yields for the low-bias electrical conductance G,

G = G_{0}e^{a} | (8) |

Assuming that an adequate fit can be obtained with c = 0, integration of eqn (6) yields

(9) |

(10) |

In Section 3 of the ESI,† it is demonstrated that the current I(V, a, b, c) in eqn (3) is an even function of b. This is also evident in the low-temperature eqn (9) and (10), since and coshy are even functions of y. Therefore, a fit to these formulae cannot determine the sign of b, because in eqn (2), χ_{j}^{2}(a, b, c) = χ_{j}^{2}(a, −b, c). In other words, if a minimum of χ_{j}^{2}(a, b, c) is found for a particular value of b, then there will also be a minimum at −b.

From eqn (7) and (1), this fitting yields the modulus of the Seebeck coefficient via the relation

|S| = −α|e||b|T | (11) |

If S is a random variable, then the average of |S| is greater than or equal to the average of S. Therefore, from the average of |S|, ABC theory yields an upper bound for the average Seebeck coefficient.

Eqn (6) to (10) are valid at low temperatures only. At finite temperatures, the exact formula (3) is used to perform the fitting. In what follows, by simultaneously measuring both current–voltage relations and Seebeck coefficients, we demonstrate that ‘ABC’ theory indeed predicts an upper bound for the Seebeck coefficient from I–V curves.

We measured several hundred I–V curves for SAMs formed from the six anthracene-based molecules shown in Fig. 1, whose synthesis was reported previously,^{39,40} and then applied the above procedure to calculate the modulus of the Seebeck coefficient from each curve. Starting from a single raw I–V curve such as that shown in Fig. S8,† the ratio G = I/V was obtained (left panel of Fig. S9†) and the spike at zero volts was eliminated, as shown in the right-hand panel of Fig. S9.† Finally, we obtained a fitted curve from I–V data, as shown in Fig. S10.† The same procedure was repeated for each I–V curve of a given molecule. Fig. 2 shows a comparison between a fitted curve and the corresponding experimental data.

Fig. 2 An example of the fitting process, experiment data (blue-circles) and fitted curve (red-solid line), also see curve fitting process in the ESI.† |

To demonstrate the validity of ABC theory, we also made many measurements of Seebeck coefficients at different locations across the SAM, and for each molecule, constructed a histogram of the resulting values. These are the green histograms of Fig. 3. These histograms were then ‘folded’ to yield the yellow histograms of experimental |S| values shown in Fig. 3 for each of the molecules **1–6**.

Fig. 3 shows that the experimental and predicted histograms are in qualitative agreement. To make a quantitative comparison, we first computed the average |S| (denoted |S_{ABC}|) from the red histograms and compared this with the average |S| (denoted |S_{exp}| from the yellow histograms. These values are shown in Fig. 4, for each of the 6 molecules. This plot demonstrated strong overlap between experimental and ABC-theory values, clearly demonstrating the predictive ability of ABC theory. Our aim is to compare theory with experiment and since in the experimental histograms are fitted to a single Gaussian, we follow the same approach for the theoretical histograms. There are two peaks in the histograms of Fig. 3 (molecules **5** and **4**). For molecule **5** these occur at |S| = 32.7 and |S| = 41.8 taking the average of these yields |S| = 37.2 which is very close to our quoted value for the most-probable |S| (i.e. |S| = 37.3). Similarly, for molecule **4** these occur at |S| = 11.4 and |S| = 24.3 taking the average of these yields |S| = 17.8, which is close to our quoted value (|S| = 17.5). Therefore fitting to a single Gaussian provides an adequate prediction for |S| for the studied molecules.

Fig. 4 Experimental and ABC-theory predictions for average of the magnitudes of Seebeck coefficients 〈|S|〉 (yellow- and red-circles respectively). |

The averages in Fig. 4 were obtained by making a Gaussian fit to the experimentally-measured (green) histograms, as is common practice in the literature. If each of the green histograms of measured values of S is assumed to approximate a Gaussian distribution of the form

(12) |

This means that measured values of |S| possess a folded Gaussian distribution of the form f(|S|) = p(|S|) + p(−|S|). i.e.

(13) |

(14) |

For |S_{0}| < σ, f(|S|) has a maximum at |S| = 0, whereas for |S_{0}| > σ, the maximum occurs at |S| ≠ 0. The blue curves in Fig. 3 show a fit of this function to each of the red histograms. The black curves show plots of the corresponding Gaussian distributions. For the experimental averages corresponding to the yellow points in Fig. 4 and for the ABC-predicted averages corresponding to the red points in Fig. 4, the average was computed by fitting a folded Gaussian f(|S|) to the histogram of predicted values of |S| and then using the formula

Fig. 5 shows a comparison between the resulting σ values for each of the molecules, obtained by fitting eqn (13) to the red histograms and by fitting eqn (12) to the green histograms. This shows that qualitative information about the widths of the distributions can also be obtained from ABC theory. Fig. 5 shows similar results for the ABC standard deviations σ_{ABC} and the experimental σ_{Exp.} for most molecules, whereas there is a larger difference for molecules **1** and **2**. To address this point, the distributions of the root mean square deviations χ_{i} (see eqn (2)) from each individual G–V fit (labelled i), for the 6 molecules, are shown in Fig. S24.† The mean values 〈χ〉 of these values of χ_{i} are shown in Table S3† for each molecule. This shows that molecule **2** has the largest root mean square deviations 〈χ〉 = 1.5 × 10^{−2} and this corresponds to the largest difference Δσ = σ_{ABC} − σ_{Exp.} between standard deviations of the theory and experiment. Similarly, molecule **1** has the next highest value of 〈χ〉 and the next highest value of Δσ. Molecule **3** has the lowest value of Δσ and the lowest value of 〈χ〉. This correlation between 〈χ〉 and Δσ is shown more clearly in Fig. S25† and demonstrates that the fitting parameter 〈χ〉 is an indicator of the accuracy of the predicted value of |S| made by ABC theory.

Fig. 5 Standard deviations σ obtained from experiment and predicted ABC theory data (yellow- and red-circles). |

It is worth mentioning that in the above analysis, Seebeck coefficients have been calculated by fitting to G–V curves rather than I–V curves. Table S2† shows a comparison between the results obtained from I–V fits and G–V fits for twelve different sets of I–V measurements and show that the results are comparable.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0na00772b |

This journal is © The Royal Society of Chemistry 2020 |