Thermoelectric Transport from First-Principles—Biphenyl-Based Single-Molecule Junctions

Chapter

Abstract

Using first-principles electronic structure methods in conjunction with nonequilibrium Green function (NEGF) techniques, we study the thermoelectric transport through biphenyl-based single-molecule junctions. We show, based on our recently published works and their present extension to include also the electron energy current, that the single-molecule conductance, junction thermopower, and electron thermal conductance strongly depend on the choice of the molecular anchor group and on the geometry of the investigated gold-biphenyl-gold contacts. We compare two different anchor groups, sulfur and cyano. The electron-donating S anchor group gives rise to a positive thermopower, while the electron-withdrawing cyano anchor results in a negative thermopower. For the S-terminated biphenyl a strong variation of the transport coefficients with respect to the binding motif is observed, for CN-terminated biphenyl such variations remain small.

1 Introduction

Since the first realization of single-molecule devices [1, 2, 3], controlling their transport properties has become a major theme in the research in molecular electronics. Using single molecules as conducting system provides thereby a great flexibility for tailoring the transport properties by changing the geometrical and chemical structure of the molecule [4, 5, 6, 7].

Due to the strong dependence of the transport on the atomistic details, an accurate theoretical description requires a first-principles treatment of the electronic and geometrical properties of the molecular junctions. Such a first-principles approach is usually based on a combination of density functional theory (DFT) with nonequilibrium Green function (NEGF) techniques. Despite the well-known shortcomings of DFT when it comes to predict quasiparticle excitations [8], the DFT + NEGF scheme has demonstrated to be able to capture and explain important trends observed in experiment [4, 7, 9]. Moreover it provides the basis for higher levels of theory like, e.g., quasiparticle methods [8].

In this work we are going to give a short overview on how the thermoelectric transport properties can be obtained from the Landauer-Büttiker approach formulated within the NEGF theory. After presenting the theoretical framework, we illustrate the application of the DFT + NEGF scheme for a selection of our previously obtained results on the conductance [10] and thermopower [11] of biphenyl-based molecular junctions. Additionally, we include in this work also the electron thermal conductance. We discuss the role of the metal-molecule configuration on the thermoelectric transport properties and the possibility to tune the sign of the thermopower, and hence the type of charge carriers, by the choice of the used anchor group. Here we consider S- and CN-terminated biphenyl molecules, with the chemical structure given in Fig. 1a. We observe that, depending on the binding site of the sulfur anchor on the gold surface, the molecular conductance, thermopower and electron thermal conductance can vary by one order of magnitude for the same molecule. We show that the anchor group determines the type of charge carriers and hence the sign of the thermopower. For sulfur we observe a positive thermopower, characteristic for hole-like transport through the highest occupied molecular orbital (HOMO), while for cyano a negative thermopower is obtained, corresponding to electron-like transport through the lowest unoccupied molecular orbital (LUMO).
Fig. 1

a The studied biphenyl molecule is either terminated by X = S or X = CN anchors. b Partitioning of the contact geometry into L, C and R, here displayed for the S-terminated biphenyl

2 Thermoelectric Transport Coefficients from NEGF Theory

In this section we will shortly introduce the theoretical framework used to calculate the transport properties of the molecular junctions. An extensive discussion of our transport approach can be found in reference [12].

The system we have in mind is depicted in Fig. 1b and consists of a nanoscale device part C, here a single molecule, connected to two macroscopic electrodes L and R. The phase-coherent transport through such a molecular junction, and nanoscale conductors in general, can be conveniently described by means of the Landauer-Büttiker approach. This formalism relates the charge current \( I \) and energy current \( Q \) to the energy-dependent transmission probability \( \tau (E) \) of an electron from one electrode, through the device part C, into the other electrode [13, 14],
$$ I = \frac{2e}{h}\int {\text{d}} E\tau (E)\left[ {f(E,\mu_{\text{L}} ,T_{\text{L}} )} \right.\left. { - f(E,\mu_{\text{R}} ,T_{\text{R}} )} \right], $$
(1)
$$ Q = \frac{2}{h}\int {\text{d}} E\tau (E)\left[ {f(E,\mu_{\text{L}} ,T_{\text{L}} )} \right.\left. { - f(E,\mu_{\text{R}} ,T_{\text{R}} )} \right](E - \mu ). $$
(2)
Here, \( f(E,\mu ,T) = \left\{ {\exp [(E - \mu )/k_{\text{B}} T] + 1} \right\}^{ - 1} \) is the Fermi function describing the left and right electrodes, which are assumed to be infinitely large reservoirs of non-interacting electrons at the chemical potentials \( \mu_{L} \) and \( \mu_{R} \), and at the temperatures \( T_{L} \) and \( T_{R} \), \( e = |e| \) is the absolute value of the electron charge, \( k_{\text{B}} \) the Boltzmann constant, and \( h \) the Planck constant. In linear response the thermoelectric transport coefficients, namely the conductance \( G \), thermopower \( S \), and electron thermal conductance \( \kappa \), can be obtained from  [14, 15, 16]
$$ G = \frac{{2e^{2} }}{h}K_{0} , $$
(3)
$$ S = - \frac{{K_{1} }}{{eTK_{0} }}, $$
(4)
$$ \kappa = \frac{2}{hT}\left( {K_{2} - \frac{{K_{1}^{2} }}{{K_{0} }}} \right). $$
(5)
The integral coefficients above are given by
$$ K_{n} = \int {\text{d}} E\tau (E)\left( { - \tfrac{\partial f(E)}{\partial E}} \right)(E - \mu )^{n} . $$
(6)
In the low-bias limit the chemical potential is approximately given by the electrode Fermi energy, \( \mu = E_{F} \). For low temperatures, compared to \( E_{F} \), we can perform a Sommerfeld expansion of Eq. (6) and, to leading order in \( k_{B}^{{}} T \), Eqs. (3)–(5) are solely determined by the transmission and its energy derivative at \( E_{F} \)
$$ G = \frac{{2e^{2} }}{h}\tau (E_{F} ), $$
(7)
$$ S = - \left. {\frac{{\pi^{2} k_{\text{B}}^{2} T}}{3e}\left( {\frac{{\partial_{E} \tau (E)}}{\tau (E)}} \right)} \right|_{{E = E_{\text{F}} }} , $$
(8)
$$ \kappa = \frac{\begin{aligned} \hfill \\ 2k_{B}^{2} \pi^{2} T \hfill \\ \end{aligned} }{3h}\tau (E_{F} ). $$
(9)
Eventually, the transport coefficients are determined by the energy-dependent transmission probability \( \tau (E) \), which contains all information about the system.Using a nonorthogonal, local basis allows partitioning of the basis states into L, C, and R regions (Fig. 1b). Within this basis the system Hamiltonian \( \varvec{H} \) and analogously the overlap matrix \( \varvec{S} \) can be written in a block form
$$ \varvec{H} = \left( {\begin{array}{*{20}c} {\varvec{H}_{LL} } & {\varvec{H}_{LC} } & 0 \\ {\varvec{H}_{CL} } & {\varvec{H}_{CC} } & {\varvec{H}_{CR} } \\ 0 & {\varvec{H}_{RC} } & {\varvec{H}_{RR} } \\ \end{array} } \right). $$
(10)
To calculate \( \tau (E) \), we take advantage of the NEGF formalism, which allows to express the transmission function in terms of Green’s functions
$$ \tau (E) = {\text{Tr}}[\varGamma_{L} G_{CC}^{r} \varGamma_{R} G_{CC}^{a} ], $$
(11)
with the retarded Green function
$$ \varvec{G}_{CC}^{r} = [E\varvec{S} - \varvec{H}_{CC} -\varvec{\varSigma}_{L}^{r} (E) -\varvec{\varSigma}_{R}^{r} (E)]^{ - 1} $$
(12)
and the advanced Green function \( \varvec{G}_{CC}^{a} = [\varvec{G}_{CC}^{r} ]^{\dag } \), the contact self-energies accounting for the semi-infinite leads
$$ \varvec{\varSigma}_{X}^{r} = (\varvec{H}_{CX} - E\varvec{S}_{CX} )\varvec{g}_{XX}^{r} (\varvec{H}_{XC} - E\varvec{S}_{XC} ), $$
(13)
the spectral density of the leads \( \varvec{\varGamma}_{X} (E) = i(\varvec{\varSigma}_{X}^{r} -\varvec{\varSigma}_{X}^{a} ) \), and the electrode surface Green function \( \varvec{g}_{XX}^{r} (E) \), with \( X = \{ L,R\} \). Here we have assumed that by choosing C large enough, we can neglect the direct coupling between the electrodes. The system Hamiltonian \( \varvec{H} \) is approximated by the Kohn-Sham matrix, obtained from DFT. In that sense we construct the Green functions by using the Kohn-Sham eigenvalues and wavefunctions as approximation for the corresponding quasiparticle quantitiesenergies entering the Green functions. For further discussion we refer to our previous work in reference  [12].

3 Results

3.1 Contact Geometries and Electronic Structure

The electronic structure and contact geometries are determined within DFT, as implemented in the quantum chemistry package TURBOMOLE 6.4 [17] We use the gradient-corrected BP86 exchange-correlation functional [18, 19] and employ the def2-SV(P) basis set and the corresponding Coulomb fitting basis [20, 21, 22]. The procedure, how the junction geometries are constructed, can be found in reference [10] for the S-terminated biphenyl and in reference [4] for the CN-terminated biphenyl. The contact geometries are displayed in Fig. 2.
Fig. 2

For S anchors, we consider hollow, bridge, and top binding sites to Au with the corresponding contact geometries called S-HH, S-BB, and S-TT1, respectively. For CN, we consider binding to single Au atoms in two different top positions, labeled CN-TT1 and CN-TT2. In all cases the binding sites are the same on both sides

3.2 Thermoelectric Transport Properties

We consider two different anchor groups, connecting the biphenyl molecule to the Au electrodes, namely S and CN (Fig. 1a). Sulfur forms a covalent bond with the Au atoms of the electrode, and it is known that several metastable binding motifs exist. Therefore we select three representative geometries [10, 23] where S binds symmetrically on each side either to three Au atoms in the hollow position (S-HH), to two Au atoms in the bridge position (S-BB), or to a single one in the top position (S-TT1) (Fig. 2). CN-anchored biphenyl binds selectively via the nitrogen lone pair to a single Au atom [5, 24]. Here we consider binding to a single, highly coordinated Au atom (CN-TT1) and to a low-coordinated one (CN-TT2), respectively (Fig. 2).

Neglecting phonons, the charge and heat current through the biphenyl is carried by the π-electron system and depends strongly on the degree of π-conjugation, the broadening of the molecular orbitals due to interactions with the metal electrodes, and the alignment of the molecular frontier orbitals with respect to the electrode Fermi energy [10, 11].

The calculated dihedral angle φ for the connected molecule and the thermoelectric transport coefficients, calculated at T = 300 K, are summarized in Table 1. The transmission spectra are displayed in Fig. 3. All the transport coefficients have been calculated by means of Eqs. (3)–(5), taking the broadening of the Fermi function into account. However, it has to be noted that even at room temperature T = 300 K, the deviations from the lowest-order Sommerfeld expansion, as given by Eqs. (7)–(9), remain small.
Table 1

Dihedral angle φ, molecular conductance G, thermopower S, and electron thermal conductance κ for the various contact geometries, evaluated at T = 300 K

 

φ [°]

G [2e2/h]

S [μV/K]

κ [pW/K]

S-HH

35.1

0.01

1.81

8.08

S-BB

12.9

0.13

39.14

77.52

S-TT1

17.9

0.09

28.08

54.17

CN-TT1

33.1

0.002

-87.83

1.86

CN-TT2

35.3

0.003

-53.02

1.90

Fig. 3

Transmission spectra for all considered junction geometries

The degree of π-conjugation depends on the dihedral angle φ between the two phenyl rings [10]. For the H-terminated biphenyl, the gas-phase dihedral angle is given by φ = 35.9°. Comparing this to the angle which is adopted, if the molecule is connected to the Au electrodes (Table 1), we see that for S-HH, CN-TT1, and CN-TT2 φ remains almost at its gas-phase value, while for S-BB and S-TT1 φ decreases to 12.9° and 17.9°, respectively. The change in φ can be related to the mechanical strain, which is exerted if the molecule is connected in S-BB or S-TT1 positions. For the S-connected molecules, we find a strong dependence of the transport coefficients on the configuration of the molecule with respect to the Au surface. If connected in the HH position G, S, and κ are largely decreased, as compared to BB and TT1. The reason for this is two-fold. First, the degree of π-conjugation is larger for S-BB and S-TT1 due to the reduced φ as compared to S-HH. Second, for S-HH the overlap between the molecular orbitals and the states in the electrodes is small, which results in a relatively weak hybridization between them. On the other hand, for S-BB and S-TT1, the molecular orbitals and electrode states are rather strongly hybridized due to the tilted molecular geometry, which in turn increases the level broadening and moves the HOMO resonance closer to EF [10, 11]. For the CN-linked molecules, the thermoelectric transport coefficients are roughly comparable, which is consistent with the experimentally observed conductance histograms [5].

From Eq. (9) we can see that the sign of the thermopower is determined by the slope of the transmission function at the Fermi energy and can be related to the type of charge carriers present in the system [25]. If the HOMO is closer to EF, the sign of the thermopower is positive, S > 0, and hole-like excitations are transported through the HOMO. Conversely, for S < 0, the LUMO is closer to EF and thus electron-like excitations are transported through the LUMO.

From Table 1. we can see that the sign of the thermopower is determined by the anchor group. For S-linked biphenyl we obtain a positive thermopower, and for CN-linked biphenyl it is negative. This behavior can be related to the Hammett constant of the molecules. Sulfur is electron donating and transfers negative partial charge into the molecular π-electron system [26], hence increasing the Coulomb repulsion. This will in turn increase the orbital energies, moving the HOMO resonance closer to EF (Fig. 3). Thus, as discussed above, we observe a positive thermopower. In contrast, for the electron withdrawing CN group [26] the Coulomb repulsion is decreased, which moves the orbital energies down (Fig. 3), resulting in LUMO-dominated transport and hence a negative thermopower.

4 Conclusions

Combining first-principles electronic structure, obtained within the framework of density functional theory, and the Landauer Büttiker approach, formulated with nonequilibrium Green functions, we studied the thermoelectric transport through biphenyl-based molecular junctions with Au electrodes. We investigated two different anchor groups, connecting the molecule to the Au electrodes, namely sulfur and cyano. For S-terminated molecules we observed a strong dependence of the thermoelectric transport properties on the binding motif. In hollow position the molecular conductance, the junction thermopower and the electron thermal conductance are strongly reduced as compared to geometries, where the molecule is either in bridge or top position. This behavior can be traced back to the different degree of hybridization between the molecular orbitals and states in the Au electrodes as well as to the molecular geometry itself. CN on the other hand binds selectively to single Au atoms, and we observed for the two studied top geometries just a moderate variation of the transport coefficients. Moreover, we showed that the anchor determines the type of charge carriers present in the system, which is ultimately related to the sign of the thermopower. The electron-donating S anchor gives rise to hole-like conduction through the HOMO, resulting in a positive thermopower. Oppositely, for the electron-withdrawing CN anchor we observe a negative thermopower, which is characteristic for electron-like conduction through the LUMO.

Notes

Acknowledgments

This work was partly supported by a FY2012 (P12501) Postdoctoral Fellowship for Foreign Researchers from the Japan Society for Promotion of Science (JSPS) and by a JSPS KAKENHI, i.e. ‘Grant-in-Aid for JSPS Fellows’, grant no. 24·02501.

F.P. gratefully acknowledges financial support from the Carl Zeiss Foundation as well as the collaborative research center of the German science foundation, SFB 767, through project C13.

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Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Nanosystem Research Institute (NRI) ‘RICS’National Institute of Advanced Industrial Science and Technology (AIST)TsukubaJapan
  2. 2.Department of PhysicsUniversity of KonstanzConstanceGermany

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