Toward the Design of Bithermoelectric Switches - ACS Publications

Article Cite This: J. Phys. Chem. C 2018, 122, 24436−24444

pubs.acs.org/JPCC

Toward the Design of Bithermoelectric Switches Thijs Stuyver,*,† Paul Geerlings,† Frank De Proft,† and Mercedes Alonso† †

Algemene Chemie, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

Downloaded via UNIV OF WINNIPEG on October 26, 2018 at 06:33:38 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

S Supporting Information *

ABSTRACT: In this work, we explore the design of so-called “reversible bithermoelectric switches”, molecules which can be switched reversibly between a positive and negative Seebeck coefficient through the application of external stimuli. We focus on heptaphyrins, a class of expanded porphyrins that can be shifted between a Hückel and Möbius topology, and demonstrate that these molecules are promising candidates for such an electronic function. Our calculations lead to the conclusion that the molecular switch between these two πconjugation topologies causes the thermopower to change considerably from +50 μV/K to −40 μV/K, respectively. We address some of the limitations of our approach and expect that this behavior will be retrieved experimentally as well, although chemical modifications of the molecules and/or electrodes might be required. Bithermoelectric switches could potentially constitute an entirely new class of molecular switches which, instead of switching between an ON and an OFF state (the modus operandi of most molecular switches so far), revert the direction of the heat and/or charge transport. This could potentially lead to a wide range of novel technological applications.

■

INTRODUCTION In the field of single-molecule electronics, a growing effort is being devoted to obtaining a better understanding of the thermoelectric properties of molecular electronic devices.1,2 The development of molecular thermoelectric devices is of great technological importance since it offers an efficient pathway toward the conversion of waste heat into electricity via the so-called Seebeck effect. Alternatively, these devices could also be employed in cooling applications by harnessing the Peltier effect.3 Initially, the interest in the thermoelectric properties of molecules was mainly fueled by the opportunity offered by the thermoelectric effect to determine straightforwardly the location of the Fermi level of the molecular junction and the nature of conduction (n- or p-type). In this context, we want to point to the pioneering work of Paulsson and Datta on the theoretical side4 and of Reddy and co-workers on the experimental side.5,6 The simultaneous measurement of conductance and thermopower of a molecular junction was first reported by the Venkataraman group.7 Later on, attention started to shift toward the design of efficient molecular thermoelectric devices.8−11 The quantity generally used to characterize the performance of a thermoelectric device (also called the figure of merit), can be expressed as follows ZT = S2Tσ /κ

thermopower S of a molecular junction, which can be defined as12 S = − lim

ΔT → 0

I=0

(2)

is generally the quantity of interest in the field of singlemolecule thermoelectronics. The following analytical expression can be derived for this quantity starting from the expression for the current in the Landauer formalism12 ÄÅ ∂f (ε) ÉÑ ∞ ÅÅ− Ñ d ετ ( ε )( ε − μ ) ∫ ÅÅ ∂ε ÑÑÑ 1 −∞ ÅÇ ÑÖ S(μ , T ) = − ÄÅ ∂f (ε) ÉÑ ∞ Å Ñ eT ∫−∞ dετ(ε)ÅÅÅÅ− ∂ε ÑÑÑÑ (3) Ç Ö with e = 1.6 × 10−19 C, τ(ε) the transmission function, μ the chemical potential within the device, and f(ε) the Fermi function of the contacts. Often, this expression is simplified by taking the low-temperature (first-order) approximation, also known as the Sommerfeld approximation:13,15,14 S(1)(μ , T ) = −

i ∂ ln[τ(ε)] yz 1 ∂τ(μ) π 2 kB π 2 kB zz =− kBT kBT jjjj z ∂ε 3 e 3 e τ(μ) ∂ε k {μ

(4)

For the reader’s convenience, a complete derivation of eqs 3 and 4 can be found in the Appendix. We note here that, next to the condition of low temperature, an additional condition for

(1)

where S corresponds to the thermopower (also known as the Seebeck coefficient), T is the absolute temperature, σ is the electrical conductivity, and κ is the thermal conductivity. The © 2018 American Chemical Society

ΔV ΔT

Received: August 9, 2018 Revised: September 27, 2018 Published: October 9, 2018 24436

DOI: 10.1021/acs.jpcc.8b07753 J. Phys. Chem. C 2018, 122, 24436−24444

Article

The Journal of Physical Chemistry C the validity of this approximation appears in this derivation, namely that the transmission function τ(ε), should be close to invariant on the scale of kBT.12 Considerable advances have been made in obtaining qualitative (chemical) insights into the thermoelectric properties of molecular junctions. The effect of an increasing length of molecular wires on the thermopower has been probed9,15 as well as the effect of different anchor units.16 Furthermore, the realization that the thermopower is proportional to the slope of the logarithm of the transmission function, as revealed by eq 4, has led to the proposal of highly efficient thermoelectric devices based on the occurrence of Quantum Interference (QI) features.3,9,11,17 QI is capable of sharply reducing the transmission probability around specific energy values due to a mutual cancellation of the different transport channels through the active molecule,18 thus leading to considerable slopes in the transmission spectrum on a logarithmic scale and, as a consequence, high Seebeck coefficients. The factors governing the occurrence of QI are reasonably well understood nowadays,19−24 leading to almost full chemical control over these features. For example, it has been demonstrated theoretically that the positioning of QI features relative to the Fermi level of the device can be controlled by heteroatom substitution,25−28 thus enabling the transformation of a common molecular framework from a heating (positive Seebeck coefficient) to a cooling function (negative Seebeck coefficient).3,11 Lambert and co-workers coined the term “bithermoelectricity” to describe such heteroatom-triggered sign inversions of the thermopower.3 Next to heteroatomic substitution, an alternative strategy exists to switch the sign of the thermopower, namely electrostatic gating.29 In this strategy, a voltage is applied to an electrostatic gate, which leads to a shift in the location of the Fermi level of the device. Since the coupling between the gate and the molecule is usually weak, the voltages needed to move the Fermi level from one side of a QI feature to the other can be rather large, reducing the practical applicability of this approach. Additionally, electrostatic gating is not very appealing from a technological perspective since one of the functions envisioned for thermoelectric devices is the generation of electrical power from heat, whereas gating consumes electricity in its own right.3 Up to this point, little attention has been devoted to the design of reversible thermoelectric switches, i.e., molecules capable of switching between a positive and a negative thermopower upon application of external stimuli.30 Such bithermoelectric switches could potentially constitute an entirely new class of molecular switches which, instead of switching between an ON and an OFF state (the modus operandi of most molecular switches so far), revert the direction of the heat and/or charge transport. This reversal could then be triggered in two different ways: either by directly applying external stimuli to switch the molecule from one state to the other and consequently changing the sign of the Seebeck coefficient S in the process or through the application of a voltage or temperature difference between the contacts respectively (see Scheme 1 for all the different configurations which can be envisioned for such a switch).31 Next to multidimensional switches, bithermoelectric switches could also potentially find applications as energy/heat converters connecting reservoirs with an unstable temperature gradient between them or even as molecular thermostats.

Scheme 1. Schematic Representation of the Different Configurations Possible in a Bithermoelectric Switcha

a

On the left-hand side of the scheme, the transport is temperature driven (initially, the voltage is equal on both sides, but a bias will emerge over time). On the right-hand side, the transport is voltage driven (initially, the temperature is equal on both sides, but a temperature gradient will emerge over time). The molecular junction is represented by two rectangular reservoirs on the side and an ellipsoid representing the active molecule. The color codes are the following: red corresponds to a (relatively) hot reservoir, blue to a cold reservoir, black to an elevated voltage, gray to a reduced voltage, green to the molecule in the state with a positive Seebeck coefficient, and purple to the state with a negative Seebeck coefficient.

One of the reasons for the limited interest in the exploration of bithermoelectric switches so far is presumably the challenge of designing suitable candidate molecules for such a function. For a molecule to be a promising candidate for a bithermoelectric switch, it should be able to switch between two distinct and stable configurations, both giving rise to a QI feature upon incorporation into a molecular electronic device. In addition, both states must be reversibly interconvertible upon the application of a desirable external stimulus. Furthermore, the QI features of the two (or more) stable configurations should be located at appreciably different energies, e.g., in one configuration close to the HOMO level and in the other configuration close to the LUMO level, so that the slope of the transmission spectrum around the Fermi level (and thus the Seebeck coefficient) will switch signs upon interconversion between the states. In this paper, a promising candidate bithermoelectric switch is proposed for the first time, based on 24437


Article

The Journal of Physical Chemistry C

Figure 1. Hückel (H), Möbius (M), and figure-eight (F) conformations of the heptaphyrins connected along their longitudinal axis and their aromaticity character. The different linker unit groups, denoted by R, are specified at the bottom (from left to right: unsubstituted, −OCH3 and −CN substituted thiolphenylethynyl groups).

a class of macrocycles which have recently been identified by some of the present authors as promising candidates for electronic switching functions, the so-called expanded porphyrins.32,33 Expanded porphyrins are a class of porphyrinoid macrocycles flexible enough to switch between different πconjugation topologies, namely Hückel and Möbius. These topology interconversions can be triggered by a variety of external stimuli.34−36 As shown by our recent research, the aromaticity of the π-conjugated system is closely determined by the molecular topology, in agreement with the well-known Hü ckel and Mö bius aromaticity rules.37−40 Accordingly, topologies with an even number of half-twists in the πstructure (i.e., with an even linking number Lk) follow the Hückel rule for aromaticity, whereas those with an odd number of half-twists follow the Möbius aromaticity rule.41 Protonation and deprotonation of the individual five-membered rings of [4n] π-electron macrocycles induce topology changes enabling the interconversion from antiaromatic to aromatic systems.42−44 Furthermore, the [4n + 2] and [4n] π-electron states can be easily interconverted by reversible two-electron redox reactions.35,45 In our previous computational works,32,33 penta-, hexa-, and heptaphryins were assembled into gold molecular junctions along their longitudinal axis through the inclusion of linker groups (Figure 1), and ratios of conductance up to 103 could be estimated theoretically for this configuration of contacts upon switching the aromaticity and/or topology of the macrocycle. The differences in conductance were mainly caused by the appearance/disappearance of QI upon the aromaticity switch. However, there was one striking exception. For the heptaphyrins, the difference in conductance between the Möbius and Hückel/twisted-Hückel conformations arose by a shift in the location of the QI feature (Figure 2). Indeed, this feature is exactly what is needed for a functional bithermoelectric switch. The unusual behavior of the transmission functions for the heptaphyrin macrocycles32 and the realization that this feature corresponds to the required behavior for an efficient thermoelectric switch constitutes the starting point of the present study. In a first step, the potential of heptaphyrins to act as bithermoelectric switches will be assessed for the first time. Second, strategies will be discussed to improve the performance of the resulting thermoelectric devices.

Figure 2. Dependence of the transmission function of [32]heptaphyrin with the π-conjugation topology. Red lines are linked to aromatic structures, whereas blue lines to antiaromatic configurations.

■

COMPUTATIONAL METHODS The initial structures of the different Hückel and Möbius topologies of [32]heptaphyrin were obtained from our previous works in which an exhaustive conformational analysis was performed for different protonation states.37 Linker groups, i.e., (substituted) thiolphenylethynyl groups, were attached to the different meso-positions of the macrocycles along the approximate longitudinal axis, as shown in Figure 1. The resulting geometries were fully optimized at the B3LYP46,47/6-31G(d) level of theory, as implemented in the Gaussian 09 software.48 The performance of the B3LYP hybrid functional on the geometries and relative conformational energies of expanded porphyrins was assessed in our previous benchmarks from comparison with the experiment.37,39,40 All transmission calculations were performed using the NonEquilibrium Green’s Function (NEGF) method combined with DFT as implemented in the Artaios code,49,50 a postprocessing tool for Gaussian 09. The electrode geometry used in the transmission calculations consists of nine gold atoms per electrode, which are arranged as a six-atom triangular fcc-gold (111) surface, with a second layer consisting of three gold atoms. The distance between gold atoms in the clusters was set to 2.88 Å. From the optimized molecular structures, the thiol’s hydrogen atoms were removed and the Au9 clusters were attached in accordance with the method24438


Article

The Journal of Physical Chemistry C ology presented in a recent study.51 The adsorption site is the middle fcc-hollow site of the first layer. The Au−S distance was set to 2.48 Å.52 For the resulting structures, single-point calculations were performed at the B3LYP/LanL2DZ level of theory using the Gaussian 09 software. In the final step, the Hamiltonian and overlap matrices were extracted to carry out the NEGF calculation within the wide-band-limit (WBL) approximation using the postprocessing tool Artaios.53 In the WBL approximation, we used a constant value of 0.036 eV−1 for the local density of states (LDOS) of the electrode surface. This value was taken from the literature.53 The thermoelectric calculations were performed with an inhouse developed code. This new code is a postprocessing tool, which, starting from a transmission spectrum obtained with, e.g., the Artaios code, calculates the thermopower as a function of the chemical potential within the device according to eq 3. We decided not to employ the Sommerfeld approximation (cf. eq 4) since the transmission spectra considered exhibit significant variation in the region around the Fermi level (cf. Figure 2). An analysis of the validity and accuracy of the implementation can be found in the Supporting Information. The temperature has been set consistently to 298 K.

Figure 4. Plot of the thermopower S as a function of the chemical potential of the system expressed in eV relative to the original Fermi level for the Möbius 32H-M.

addressed, namely the positioning of the Fermi level in the molecular junction. It is well-known that the Fermi level of molecular electronic devices, determined through transmission calculations, are generally ill-defined.11 In our approach, which has been applied extensively in electron-transport calculations,21,22,26,54,55 the Fermi level is estimated to be located halfway between the HOMO and LUMO energy levels of the entire molecular junction (i.e., the molecule combined with the linkers and gold clusters on both sides). This is a reasonable approximation and leads to results that appear to approach the experimental findings for the conductance in the literature.56−58 Nevertheless, this convention is hardly more than an educated guess, and computationally demanding calculations, such as GW, are needed in order to obtain a better estimate.59,60 For complex macrocycles like heptaphyrins, such an approach unfortunately becomes computationally intractable. Furthermore, the thermopower can be expected to be very sensitive to deviations in the value of the Fermi level. To give some perspective, Figure 5 shows the thermopower plots in Figures 3 and 4 with slightly shifted Fermi energies. As such, there is some uncertainty concerning the results obtained above. It is hypothetically even possible that experimentally, the Fermi level could be shifted to such a large extent that the Seebeck coefficient switches signs for one of the topologies discussed or that the value becomes much smaller than the one obtained through our calculations, destroying the expected bithermoelectric switching behavior. However, as mentioned before, it is well-known that the relative positioning of the Fermi level and QI features can be controlled through chemical modification, i.e., substitution with electron-donating and electron-withdrawing groups.25 This approach, first taken in the context of thermoelectronics by Solomon and co-workers to enhance the thermopower of quinoid molecules by aligning QI features with the Fermi level of the gold electrodes,11 could be used advantageously to design substituted heptaphyrins with the desired properties for an efficient bithermoelectric switch, i.e., the QI features for the two topologies each on one side of the actual Fermi level of the molecular junction.3,11,25 One caveat in this respect within the context of the present study is the possibility that a heptaphyrin, heavily substituted in its core region, may no longer exhibit the desired switching properties between Hückel and Möbius topology, which is central to its functioning as a bithermoelectric switch. As such, the exhaustive conforma-

■

RESULTS AND DISCUSSION The shape of the plot in Figure 2 already gives an indication that, upon switching between the Hückel and Möbius states, the heptaphyrin will exhibit an opposite sign of thermopower. In order to determine the actual difference in Seebeck coefficient between the two topologies, thermoelectric calculations were performed. The results of these calculations can be found in Figures 3 and 4.

Figure 3. Plot of the thermopower S as a function of the chemical potential of the system expressed in eV relative to the original Fermi level for Hückel 32H-H.

From the plots of the thermopower as a function of the chemical potential in the Hückel and Möbius topologies, we can conclude that at room temperature the Seebeck coefficient shifts from approximately +50 μV/K to −40 μV/K upon topology switching. These obtained values are quite elevated compared to the thermopower values usually reported in the literature.3 As such, heptaphyrin could potentially act as a functional bithermoelectric switch with a reasonable efficiency for both topologies. Even though the preliminary findings above are quite promising, an important issue ignored so far remains to be 24439


Article


Figure 5. Plot of the thermopower S as a function of the chemical potential of the system expressed in eV relative to the original Fermi level for: (a) Hückel 32H-H and (b) Möbius 32H-M with Fermi levels shifted by −0.1 eV.

tional analysis performed for the unsubstituted systems mentioned in the Computational Methods would have to be replicated for the resulting substituted analogues before meaningful conclusions could be drawn.37 As a result, we decided to limit ourselves in the present exploratory study only to the investigation of substitutions on the phenyl-groups in the linker units. Such substitution patterns retain the central expanded porphyrin in its entirety, and thus we can expect the topological switching properties of the system to not be influenced significantly. Indeed, similar relative energies between the different π-conjugation topologies are computed for the thiolphenylethynyl-substituted heptaphyrins as compared to the unsubstituted macrocycle.32 The substitution pattern for the linker units is shown in Figure 1. −CN and −OCH3 functional groups were selected because of their pronounced electron-withdrawing and electron-donating properties, respectively. The resulting transmission spectra are presented in Figures 6 and 7. The computed spectra demonstrate that tuning the linker groups enables a shift of the interference features by a few tenths of an electron volt in a uniform way for both topologies, and one can expect even more wiggle room once the porphyrinoid molecule itself is modified.11,25,26

Figure 7. Transmission spectra for the Möbius 32H-M with different substitution patterns for the linker units (cf. Figure 1 for the full structures).

These results illustrate the chemical control over the thermoelectric properties of the considered macrocycles, evidencing that the substitution approach could be used experimentally to tune the positions of the QI features in case that the actual Fermi levels deviate significantly from the computational estimates. We should note here that special consideration should also be given to the interplay between the chosen substitution patterns and the stability of the resulting expanded porphyrins. Often, expanded porphyrins require, e.g., pentafluorophenyl groups at the meso-positions, and such stabilizing groups will evidently influence the positions of the QI features in their own right.34,35 Finally, we also note that an alternative approach for the tuning of the QI features around the Fermi levels of the electrodes has very recently been developed by the Solomon group.61 In their approach, the traditional gold electrodes, used throughout this study, are replaced by transition-metal dichalcogenides. The work function (and, thus, the Fermi energy) of these alternative electrodes can be tuned across the entire molecular energy range (from HOMO to LUMO), making this a potentially suitable alternative to the substitution method discussed above. Further investigations will be needed to determine the necessity of tuning the alignment of the QI features and the

Figure 6. Transmission spectra for the Hückel 32H-H with different substitution patterns for the linker units (cf. Figure 1 for the full structures). 24440


Article

The Journal of Physical Chemistry C efficiency of the different methods at our disposal to this end. Nevertheless, the present study offers a clear proof-of-principle that reversible bithermoelectric switches based on expanded porphyrins could be feasible.

∂f (ε) (ε − μ) ∂f (ε) (μ − μi ) − (T − Ti ) T ∂ε ∂ε ∂f (ε) ∂f (ε) (T − Ti ) (μ − μi ) − (ε − μ) = f (ε) − T ∂ε ∂ε

fi (ε) ≈ f (ε) −

■

(A.6)

CONCLUSIONS In this work, we investigated heptaphyrins, a class of expanded porphyrins which can be reversibly shifted between Hückel and Möbius topologies, as potential candidates for bithermoelectric switches at the molecular scale. Our calculations lead to the conclusion that a topology switch between the Hückel and Mö bius form causes the Seebeck coefficient S or thermopower to change considerably from +50 μV/K to −40 μV/K. We addressed some of the limitations of our approach, and we expect that this behavior will be retrieved experimentally as well, although chemical modifications of the molecule and/or electrodes might be required. Bithermoelectric switches could potentially constitute an entirely new class of molecular switches which, instead of switching between an ON and an OFF state (the modus operandi of most molecular switches so far), revert the direction of the heat and/ or charge transport.

■

Substitution of this new expression for f L (ε) and f R (ε) into the expression for the current (eq A.1) gives ÄÅ 2e ∞ ÅÅ ∂f (ε) dετ(ε)ÅÅÅ− (μL − μR ) − (ε − μ) I=− ÅÅ ∂ε h −∞ ÅÇ É ∂f (ε) (TL − TR ) ÑÑÑÑ ÑÑ T ∂ε ÑÑÑÖ (A.7)

∫

Setting I = 0, we obtain an analytical expression for the thermopower S(μ,T) after some calculus ÄÅ É ∞ ÅÅ ∂f (ε) ∂f (ε) ΔT ÑÑÑÑ ÑÑ dετ(ε)ÅÅÅ− (μL − μR ) − (ε − μ) ÅÅÅ ∂ε −∞ ∂ε T ÑÑÑÖ Ç (A.8) =0 ÄÅ É ∞ ÅÅ ∂f (ε) ÑÑÑ ΔT Å ÑÑ + (μ L − μ R ) dετ(ε)ÅÅ− ÅÅÅ ∂ε ÑÑÑÑ −∞ T Ç Ö ÄÅ ÉÑ ∞ ÅÅ ∂f (ε) ÑÑ ÑÑ = 0 dετ(ε)(ε − μ)ÅÅÅ− ÅÅÅ ∂ε ÑÑÑÑ −∞ (A.9) Ç Ö

∫

∫

APPENDIX A

Derivation of the Analytical Expression of the Thermopower

∫

The starting point of this derivation is the expression for the current in the Landauer formalism I=−

2e h

∞

∫−∞ dετ(ε)[fL (ε) − fR (ε)]

− (A.1)

1 e ε − μ / kT + 1

with μ = the chemical potential inside the device. When a small bias ΔV = (μL−μR)/(−e) and temperature difference ΔT = TL−TR are applied, the distribution functions for the contacts can be expanded around μ and T as follows

ÅÄ Ç

∫−∞ dετ(ε)ÅÅÅÅÅ− ∞

É

∂f (ε) Ñ ÑÑ Ñ ∂ε Ñ ÑÖ

(A.10)

(A.12)

which corresponds exactly to eq 3.

■

(A.3)

APPENDIX B

Derivation of the Sommerfeld Approximation

with f(ε) representing the Fermi function with the equilibrium chemical potential μ and temperature T and i = L, R. With the help of the explicit expression for the Fermi functions (eq A.2), the derivatives to μ and T can be written in a more convenient form

In eq A.12, the ∂f(ε)/∂ε in the denominator can be approximated as −δ(ε−μ) under two conditions: (i) the temperature has to be low and (ii) the function τ(ε) should be close to invariant in the region of the peak in ∂f(ε)/∂ε; i.e., τ(ε) has to be approximately constant on the scale of kBT in that region. In that case, the denominator can be written as ÄÅ É ∞ ∞ ÅÅ ∂f (ε) ÑÑÑ ÑÑ ≈ dετ(ε)ÅÅÅ− dετ(ε)δ(ε − μ) = τ(μ) Ñ −∞ −∞ ÅÅÅÇ ∂ε ÑÑÑÖ

i yz ε − μ / kT ij 1 yz ∂f (μ) ∂f (ε) 1 zz e jj− zz = − = −jjjj ε − μ / kT z kT ∂μ ∂ε e 1 + k { k { −2

∫

(A.4) i ∂f (T ) 1 = −jjjj ε − μ / kT ∂T + ke

T

(μL − μR )

(A.2)

∂f (μ) ∂f (T ) fi (ε) ≈ f (ε) + (μ − μi ) + (T − Ti ) ∂μ ∂T

=

É

∂f (ε) Ñ ÑÑ Ñ ∂ε Ñ ÑÖ

ÅÄ ∂f (ε) ÑÉ ∞ ∫−∞ dετ(ε)(ε − μ)ÅÅÅÅÅ− ∂ε ÑÑÑÑÑ 1 Ç Ö − −e =− ÄÅ ∂f (ε) ÉÑ ∞ Å Ñ ΔT eT Å Ñ ∫−∞ dετ(ε)ÅÅÅ− ∂ε ÑÑÑ (A.11) Ç Ö ÄÅ ∂f (ε) ÉÑ ∞ ÅÅ− Ñ ετ ε ε − μ d ( )( ) ∫ ÅÅ ∂ε ÑÑÑ ΔV 1 −∞ ÅÇ ÑÖ − =− = S(μ , T ) ÅÄÅ ∂f (ε) ÑÉÑ ∞ ΔT eT ∫−∞ dετ(ε)ÅÅÅÅ− ∂ε ÑÑÑÑ Ç Ö

where τ(ε) corresponds to the transmission function and f i is the Fermi function for the right/left (i = R/L) lead. The general form of a Fermi function is as follows: f (ε) =

ΔT

ÄÅ

ÅÅ 1 ∫−∞ dετ(ε)(ε − μ)ÅÅÇÅ− ∞

(μ L − μ R )

yz ε − μ / kT ij ε − μ yz i ε − μ yz ∂f (ε) zz e jj− z = −jjj zz 2 z z kT 1{ k T { ∂ε k {

∫

(B.1)

−2

In the numerator, ∂f(ε)/∂ε cannot be approximated as a δ function due to the presence of (ε − μ), which is not an approximately constant function in the region around ε = μ. Instead, we start by expanding τ(ε) around μ as follows

(A.5)

As such, eq A.3 can be rewritten as follows 24441


Article


i ∂τ(ε) zy zz (ε − μ) + ... τ(ε) ≈ τ(μ) + jjjj z k ∂ε { μ

With the help of the L’Hopital’s rule, it can be demonstrated that the first term in eq B.8 will vanish, leaving only the second one, which cancels eq B.7 perfectly. Combining eq B.1 with what remains of eq B.6 then finally leads to the expression for the Sommerfeld approximation of the Seebeck coefficient or thermopower found in eq B.4 of the main text.

(B.2)

ÄÅ É ÅÅ ∂f (ε) ÑÑÑ Å ÑÑ Å dετ(ε)(ε − μ)Å− ÅÅÅ ∂ε ÑÑÑÑ −∞ Ç Ö ÅÄÅ ∂f (ε) ÑÉÑ i ∂τ(ε) y ∞ Å ÑÑ jj zz Ñ+ dε(ε − μ)ÅÅÅ− = τ(μ) ÅÅÅ ∂ε ÑÑÑÑ jjk ∂ε zz{ −∞ Ç Ö μ ÄÅ ÉÑ ∞ Å Ñ Å ∂f (ε) ÑÑ Ñ dε(ε − μ)2 ÅÅÅ− ÅÅÅ ∂ε ÑÑÑÑ −∞ Ç Ö

leading to

∫

∞

■

∫

∫

* Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b07753. Verification of the validity of the implemented code (PDF)

(B.3)

The first term in this sum will be zero, since the integrand is an antisymmetric function with respect to ε = μ. The second term can now be partially integrated as follows: ÄÅ É ∞ Å ∂f (ε) ÑÑÑ ij ∂τ(ε) yz 2Å Å Ñ jj zz j ∂ε z −∞ dε(ε − μ) ÅÅÅÅ− ∂ε ÑÑÑÑ k {μ ÅÇ ÑÖ

■

*E-mail: [email protected].

i ∂τ(ε) yz ij ∂τ(ε) yz zz [(ε − μ)2 f (ε)]∞ j z = −jjjj −∞ + j z j ∂ε zz k ∂ε { μ k {μ

ORCID

Thijs Stuyver: 0000-0002-8322-0572 Notes

The authors declare no competing financial interest.

■

∞

∫−∞

(B.4)

ACKNOWLEDGMENTS T.S. acknowledges the Research Foundation-Flanders (FWO) for a position as research assistant (11ZG615N). P.G. and F.D.P. acknowledge the Vrije Universiteit Brussel (VUB) for a Strategic Research Program. F.D.P. also acknowledges the Francqui foundation for a position as Francqui research professor. M.A. thanks the FWO for a postdoctoral fellowship (12F4416N) and the VUB for financial support. Computational resources and services were provided by the Shared ICT Services Centre funded by the Vrije Universiteit Brussel, the Flemish Supercomputer Center (VSC), and FWO.

We can now finally apply the Sommerfeld expansion (cf. refs 62 and 63) ∞

μ

∫−∞ dεf (ε)H(ε) = ∫−∞ dεH(ε) +

π2 (kT )2 H′(μ) + ... 6 (B.5)

where H(ε) is some well-behaved function of ε and where terms with higher order derivatives of H(ε) are neglected to the second term in eq B.4: i ∂τ(ε) yz zz 2jjjj z k ∂ε { μ

■

∞

∫−∞ dεf (ε)(ε − μ)

ÄÅ i ∂τ(ε) yz ÅÅÅÅ zz ÅÅ = 2jjjj z Å k ∂ε { μÅÅÇÅ

i ∂τ(ε) yz zz = 2jjjj z k ∂ε { μ

ÉÑ ÑÑ μ π ∂ ÑÑ ÑÑ dε(ε − μ) + (kT )2 (ε − μ) Ñ 6 ∂ε −∞ ε=μ Ñ ÑÖÑ i

y π2 (kT )2 {μ 3

∫−∞ dε(ε − μ) − jjjj ∂τ∂(εε) zzzz μ

k

(B.6)

It can now be demonstrated that the first term in eq B.6 and the first term in eq B.4 cancel each other out. Focusing first on the first term in eq B.6 and performing the substitution x = ε − μ in the integral gives i ∂τ(ε) yz zz = 2jjjj z k ∂ε { μ

i

y

∫−∞ dε(ε − μ) = 2jjjj ∂τ∂(εε) zzzz ∫−∞ dxx μ

k

{μ

0

i ∂τ(ε) yz 2 0 i ∂τ(ε) yz zz [x ]−∞ = −jjj z lim (ε − μ)2 = jjjj z j ∂ε zz ε →−∞ ∂ ε k {μ k {μ

(B.7)

The first term in eq B.4, on the other hand, can be written as i ∂τ(ε) yz zz [(ε − μ)2 f (ε)]∞ − jjjj −∞ z k ∂ε { μ

i ∂τ(ε) yz i ∂τ(ε) yz zz lim (ε − μ)2 f (ε) + jjj z lim = −jjjj z j ∂ε zz ε →−∞ k ∂ε { με →∞ k {μ (ε − μ)2 f (ε)

REFERENCES

(1) Aradhya, S. V.; Venkataraman, L. Single-molecule junctions beyond electronic transport. Nat. Nanotechnol. 2013, 8, 399−410. (2) Butcher, P. N. Thermal and electrical transport formalism for electronic microstructures with many terminals. J. Phys.: Condens. Matter 1990, 2, 4869−4878. (3) Sangtarash, S.; Sadeghi, H.; Lambert, C. J. Connectivity-driven bi-thermoelectricity in heteratom-substituted molecular junctions. Phys. Chem. Chem. Phys. 2018, 20, 9630−9637. (4) Paulsson, M.; Datta, S. Thermoelectric effect in molecular electronics. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 241403. (5) Reddy, P.; Jang, S. Y.; Segalman, R. A.; Majumdar, A. Thermoelectricity in molecular junctions. Science 2007, 315, 1568− 1571. (6) Tan, A.; Sadat, S.; Reddy, P. Measurement of thermopower and current-voltage characteristics of molecular junctions to identify orbital alignment. Appl. Phys. Lett. 2010, 96, 013110. (7) Widawsky, J. R.; Darancet, P.; Neaton, J. B.; Venkataraman, L. Simultaneous determination of conductance and thermopower of single molecule junctions. Nano Lett. 2012, 12, 354−358. (8) Ke, S. H.; Yang, W.; Curtarolo, S.; Baranger, H. U. Thermopower of molecular junctions: an ab initio study. Nano Lett. 2009, 9, 1011−1014. (9) Bergfield, J. P.; Solis, M. A.; Stafford, C. A. Giant thermoelectric effect from transmission supernodes. ACS Nano 2010, 4, 5314−5320. (10) Karlström, O.; Linke, H.; Karlström, G.; Wacker, A. Increasing thermoelectric performance using coherent transport. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 113415.

2

∫

AUTHOR INFORMATION

Corresponding Author

∫

dεf (ε)2(ε − μ)

ASSOCIATED CONTENT

S

(B.8) 24442


Article

The Journal of Physical Chemistry C (11) Strange, M.; Seldenthuis, J. S.; Verzijl, C. J. O.; Thijssen, J. M.; Solomon, G. C. Interference enhanced thermoelectricity in quinoid type structures. J. Chem. Phys. 2015, 142, 084703. (12) Lunde, A. M.; Flensberg, K. On the Mott formula for the thermopower of non-interacting electrons quantum point contacts. J. Phys.: Condens. Matter 2005, 17, 3879. (13) Paulsson, M.; Datta, S. Thermoelectric effect in molecular electronics. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 241403. (14) Malen, J. A.; Yee, S. K.; Majumdar, A.; Segalman, R. A. Fundamentals of energy transport, energy conversion, and thermal properties in organic-inorganic heterojunctions. Chem. Phys. Lett. 2010, 491, 109−122. (15) Hüser, F.; Solomon, G. C. From chemistry to functionalitiy: trends for the length dependence in molecular junctions. J. Phys. Chem. C 2015, 119, 14056−14062. (16) Su, T. A.; Neupane, M.; Steigerwald, M. L.; Venkataraman, L.; Nuckolls, C. Chemical principles of single-molecule electronics. Nat. Rev. Mater. 2016, 1, 16002. (17) Garcia-Suarez, V. M.; Lambert, C. J.; Manrique, D. Z.; Wandlowski, T. Redox control of thermopower and figure of merit in phase-coherent molecular wires. Nanotechnology 2014, 25, 205402. (18) Solomon, G. C.; Andrews, D. Q.; Hansen, T.; Goldsmith, R. H.; Wasielewski, M. R.; Van Duyne, R. P.; Ratner, M. A. Understanding quantum interference in coherent molecular conduction. J. Chem. Phys. 2008, 129, 054701. (19) Tsuji, Y.; Estrada, E.; Movassagh, R.; Hoffmann, R. Quantum interference, Graphs, Walks and Polynomials. Chem. Rev. 2018, 118, 4887−4911. (20) Markussen, T.; Stadler, R.; Thygesen, K. S. The relation between structure and quantum interference in single molecule junctions. Nano Lett. 2010, 10, 4260−4265. (21) Yoshizawa, K.; Tada, T.; Staykov, A. Orbital views of the electron transport in molecular devices. J. Am. Chem. Soc. 2008, 130, 9406−9413. (22) Tsuji, Y.; Hoffmann, R.; Movassagh, R.; Datta, S. Quantum interference in polyenes. J. Chem. Phys. 2014, 141, 224311. (23) Stuyver, T.; Fias, S.; De Proft, F.; Geerlings, P. Back of the envelope selection rule for molecular transmission: A curly arrow approach. J. Phys. Chem. C 2015, 119, 26390−26400. (24) Tsuji, Y.; Yoshizawa, K. Frontier orbital perspective for quantum interference in alternant and nonalternant hydrocarbons. J. Phys. Chem. C 2017, 121, 9621−9626. (25) Andrews, D. Q.; Solomon, G. C.; Van Duyne, R. P.; Ratner, M. A. Single molecule electronics: increasing dynamic range and switching speed using cross-conjugated species. J. Am. Chem. Soc. 2008, 130, 17309−17319. (26) Garner, M. H.; Solomon, G. C.; Strange, M. Tuning conductance in aromatic molecules: constructive and counteractive substituent effects. J. Phys. Chem. C 2016, 120, 9097−9103. (27) Li, X.; Staykov, A.; Yoshizawa, K. Orbital views of the electron transport through heterocyclic aromatic hydrocarbons. Theor. Chem. Acc. 2011, 130, 765−774. (28) Tsuji, Y.; Stuyver, T.; Gunasekaran, S.; Venkataraman, L. The influence of linkers on quantum interference: A linker theorem. J. Phys. Chem. C 2017, 121, 14451−14462. (29) Kim, Y.; Jeong, W.; Kim, K.; Lee, W.; Reddy, P. Electrostatic control of thermoelectricity in molecular junctions. Nat. Nanotechnol. 2014, 9, 881. (30) Rincon-Garcia, L.; Ismael, A. K.; Evangeli, C.; Grace, I.; RubioBollinger; Porfyrakis, K.; Agraït, N.; Lambert, C. J. Molecular design and control of fullerene-based bi-thermoelectric materials. Nat. Mater. 2016, 15, 289−295. (31) Datta, S. Lessons from nanoelectronics. A new perspective on transport − part A: basic concepts; World Scientific: Singapore, 2017. (32) Stuyver, T.; Perrin, M. L.; Geerlings, P.; De Proft, F.; Alonso, M. Conductance switching in expanded porphyrins through aromaticity and topology changes. J. Am. Chem. Soc. 2018, 140, 1313−1326.

(33) Stuyver, T.; Fias, S.; Geerlings, P.; De Proft, F.; Alonso, M. Qualitative insights into the transport properties of Hückel/Möbius (anti-)aromatic compounds: application to expanded porphyrins. J. Phys. Chem. C 2018, 122, 19842. (34) Tanaka, Y.; Saito, S.; Mori, S.; Aratani, N.; Shinokubo, H.; Shibata, N.; Higuchi, Y.; Yoon, Z. S.; Kim, K. S.; Noh, S. B.; Park, J. K.; Kim, D.; Osuka, A. Metalation of expanded porphyrins: a chemical trigger used to produce molecular twisting and Möbius aromaticity. Angew. Chem. 2008, 120, 693−696. (35) Osuka, A.; Saito, S. Expanded porphyrins and aromaticity. Chem. Commun. 2011, 47, 4330−4339. (36) Yoon, Z. S.; Osuka, A.; Kim, D. Möbius aromaticity and antiaromaticity in expanded porphyrins. Nat. Chem. 2009, 1, 113− 122. (37) Alonso, M.; Geerlings, P.; De Proft, F. Viability of Möbius topologies in [26]- and [28] Hexaphyrins. Chem. - Eur. J. 2012, 18, 10916. (38) Alonso, M.; Geerlings, P.; De Proft, F. Conformational control in [22]- and [24]-pentaphyrins (1.1.1.1.1) by meso substituents in their N-fusion reaction. J. Org. Chem. 2013, 78, 4419−4431. (39) Woller, T.; Contreras-Garcia, J.; Geerlings, P.; De Proft, F.; Alonso, M. Understanding the molecular switching properties of octaphyrins. Phys. Chem. Chem. Phys. 2016, 18, 11885. (40) Alonso, M.; Geerlings, P.; De Proft, F. Exploring the structurearomaticity relationship in Hückel and Möbius N-fused pentaphyrins using DFT. Phys. Chem. Chem. Phys. 2014, 16, 14396. (41) Rappaport, S. M.; Rzepa, R. S. Intrinsically chiral aromaticity. Rules incorporating linking number, twist, and write for higher-twist Möbius annulenes. J. Am. Chem. Soc. 2008, 130, 7613. (42) Alonso, M.; Geerlings, P.; De Proft, F. Topology switching in [32] heptaphyrins controlled by solvent, protonation, and meso substituents. Chem. - Eur. J. 2013, 19, 1617−1628. (43) Saito, S.; Shin, J. Y.; Lim, J. M.; Kim, K. S.; Kim, D.; Osuka, A. protonation-triggered conformational changes to Möbius aromatic [32] heptaphyrins (1.1. 1.1. 1.1. 1). Angew. Chem. 2008, 120, 9803− 9806. (44) Cha, W.-Y.; Yoneda, T.; Lee, S.; Lim, J. M.; Osuka, A.; Kim, D. Deprotonation induced formation of Mobius aromatic [32]heptaphyrins. Chem. Commun. 2014, 50, 548−550. (45) Sung, Y. M.; Oh, J.; Cha, W.-Y.; Kim, W.; Lim, J. M.; Yoon, M.C.; Kim, D. Control and switching of aromaticity in various all-azaexpanded porphyrins: spectroscopic and theoretical analyses. Chem. Rev. 2017, 117, 2257−2312. (46) Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098−3100. (47) Becke, A. D. Becke’s three parameter hybrid method using the LYP correlation functional. J. Chem. Phys. 1993, 98, 5648−5652. (48) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A. et al. Gaussian 09, Revision B.01; Gaussian, Inc.: Wallingford CT, 2009. (49) Deffner, M.; Gross, L.; Steenbock, T.; Voigt, B. A.; Solomon, G. C.; Herrmann, C. Artaios − a code for postprocessing quantum chemical electronic structure calculations; Universität Hamburg, 2010−2017. (50) Herrmann, C.; Solomon, G. C.; Subotnik, J. E.; Mujica, V.; Ratner, M. A. Ghost transmission: how large basis sets can make electron transport calculations worse. J. Chem. Phys. 2010, 132, 024103. (51) Schlicke, H.; Herrmann, C. Controlling molecular conductance: switching off π sites through protonation. ChemPhysChem 2014, 15, 4011−4018. (52) Bilić, A.; Reimers, J. R.; Hush, N. S. The structure, energetics, and nature of the chemical bonding of phenylthiol adsorbed on the Au (111) surface: Implications for density-functional calculations of molecular-electronic conduction. J. Chem. Phys. 2005, 122, 094708. (53) Herrmann, C.; Solomon, G. C.; Ratner, M. A. Designing organic spin filters in the coherent tunneling regime. J. Chem. Phys. 2011, 134, 224306. 24443


Article

The Journal of Physical Chemistry C (54) Stuyver, T.; Zeng, T.; Tsuji, Y.; Fias, S.; Geerlings, P.; De Proft, F. Captodative substitution: a strategy for enhancing the conductivity of molecular electronic devices. J. Phys. Chem. C 2018, 122, 3194− 3200. (55) Tsuji, Y.; Hoffmann, R. Frontier orbital control of molecular conductance and its switching. Angew. Chem., Int. Ed. 2014, 53, 4093−4097. (56) Taniguchi, M.; Tsutsui, M.; Mogi, R.; Sugawara, T.; Tsuji, Y.; Yoshizawa, K.; Kawai, T. Dependence of single-molecule conductance on molecule junction symmetry. J. Am. Chem. Soc. 2011, 133, 11426− 11429. (57) Tsuji, Y.; Hoffmann, R. Frontier orbital control of molecular conductance and its switching. Angew. Chem., Int. Ed. 2014, 53, 4093−4097. (58) Stuyver, T.; Blotwijk, N.; Fias, S.; Geerlings, P.; De Proft, F. Exploring electrical currents through nanographenes: visualization and tuning of the transmission paths. ChemPhysChem 2017, 18, 3012− 3022. (59) Jin, C.; Strange, M.; Markussen, T.; Solomon, G. C.; Thygesen, K. S. Energy level alignment and quantum conductance of functionalized metal-molecule junctions: density functional theory versus GW calculations. J. Chem. Phys. 2013, 139, 184307. (60) Strange, M.; Rostgaard, C.; Häkkinen, H.; Thygesen, K. S. Selfconsistent GW calculations of electronic transport in thiol- and amine-linked molecular junctions. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 115108. (61) Jin, C.; Solomon, G. C. Controlling band alignment in molecular junctions: utilizing two-dimensional transition-metal dichalcogenides as electrodes for thermoelectric devices. J. Phys. Chem. C 2018, 122, 14233−14239. (62) Sommerfeld, A. Zur elektronentheorie der metalle auf grund der Fermischen statistik. Z. Phys. 1927, 47, 1928. (63) Landau, L.; Lifshitz, E. Physique statistique; Editions Mir Moscou, 1967.

24444


Toward the Design of Bithermoelectric Switches - ACS Publications

Recommend Documents