In this work, we theoretically study steady state thermoelectric transport through a single-molecule junction with a long chain-like bridge. Electron transmission through the system is computed using a tight-binding model for the bridge. We analyze dependences of thermocurrent on the bridge length in unbiased and biased systems operating within and beyond the linear response regime. It is shown that the length-dependent thermocurrent is controlled by the lineshape of electron transmission in the interval corresponding to the HOMO/LUMO transport channel. Also, it is demonstrated that electron interactions with molecular vibrations may significantly affect the length-dependent thermocurrent.
Presently, charge transport through molecular systems is an important research field because of possible applications of these systems in molecular electronics.1–4 The key element and basic building block of molecular electronic devices is a single-molecule junction including a couple of metallic/semiconducting electrodes linked by a molecular bridge. Alongside other tailored nanoscale systems (such as carbon-based nanostructures and quantum dots), single-molecule junctions hold promise for enhanced efficiency of heat-to-electric energy conversion.5–12 Therefore, thermoelectric properties of single-molecule junctions are being explored both theoretically and experimentally.
In general, thermoelectric charge transport through molecular junctions is controlled by simultaneous driving by electric and thermal driving forces. The combined effect of these forces depends on several factors including the bridge geometry and the characteristics of its coupling to the leads13–24 and electron-electron interactions.25–34 Thermoelectric transport characteristics may be affected due to the interaction between transmitting electrons and environmental nuclear motions35–54 and the effects of quantum interference.55,56 Electron-photon interactions may also bring changes in thermoelectric properties of nanoscale systems.57 Under certain conditions (e.g., in single-molecule junctions with ferromagnetic electrodes and/or with a magnetic molecule used as a linker), spin polarization of electrons may significantly influence thermoelectric transport resulting in several new phenomena such as the spin Seebeck effect.58–69
It was repeatedly demonstrated that electron transport through molecules strongly depends on the molecular length. Length-dependent electronic conductance as well as the electronic contribution to heat conductance and thermoelectric response were observed and discussed in molecular junctions with repeating molecular units such as benzene or phenyl rings.13–23,70,71 These linkers provide a better opportunity to observe relationships between transport coefficients and the length of the linker. For other kinds of molecular bridges, these relationships are less distinct due to the diversity of specific properties of different parts of the bridge. Usually, these transport characteristics are measured assuming that thermal gradient Δθ applied across the system is much smaller than the average temperature θ characterizing the latter.72 In this case, thermal driving forces remain relatively weak and the response of the system is linear in Δθ. Correspondingly, transport characteristics such as electron conductance and thermopower appear to be independent on Δθ. However, as the temperature gradient Δθ increases, the system may switch to a regime of operation nonlinear in Δθ. In several recent studies, nonlinear thermoelectric transport through molecular junctions and other tailored nanoscale systems has been discussed.35,37,43,49,58,73–76 The nonlinear Seebeck effect was already observed in semiconducting quantum dots and single molecule junctions.19,77,78
An important characteristic of thermoelectric transport through molecular junctions and similar nanoscale systems is the thermocurrent, defined as a difference between the charge current flowing through a biased system in the presence of a temperature gradient Δθ and the current flowing through the same system when the temperature gradient is removed (Δθ = 0).77 According to this definition, Ith represents the contribution to the charge current which appears due to the thermally excited transport of charge carriers through a molecular junction. The thermocurrent is more convenient for measuring and modeling than some other characteristics of thermoelectric transport such as the Seebeck coefficient. Also, it may be studied within the same computational approach over a wide range of Δθ values both within and beyond the linear response regime. Despite these advantages, the thermocurrent properties in nanoscale systems were not so far thoroughly studied. In accord with the well-pronounced dependences of both electron tunnel conductance and thermopower on the length of a molecular bridge, we expect Ith to be length-dependent as well. Also, we may expect that molecular vibrations can significantly affect thermocurrent. In the present work, we focus on this observable and its dependences on the molecular linker length and on the effect of vibrational phonons.
II. MODEL AND MAIN EQUATIONS
In the following analysis, we assume coherent electron transmission to be the predominant transport mechanism. We simulate a linker in a single-molecule junction by a periodical chain of N sites. Each single site is assigned to an on-site energy Ei and coupled to its nearest neighbors with the coupling strengths βi−1,i and βi,i+1, respectively (2 ≤ i ≤ N − 1). Within the simplest version of this model, all on-site energies are assumed to be equal (Ei = E0) as well as all coupling parameters (βi−1,i = βi,i+1 = β). Such simple chain models are often used to represent molecular bridges comprising repeating units where π − π coupling dominates electron transport and the parameter β characterizes the coupling between adjacent π orbitals.79 The schematics of this model showing relevant parameters employed in further computations is presented in Fig. 1. This simple chain model may be modified by introducing gateway states different from the rest. This could be achieved by separating out two sites at the ends of the chain, setting on these terminal sites on-site energies Ei = EN = ϵ which differ from E0 and suggesting that these sites are coupled to their neighbors with different strength (β1,2 = βN−1,N = δ). It was shown that gateway states may significantly affect the length dependence of the Seebeck coefficient19,80 and similar effects may appear in the thermocurrent. Such effects can be investigated with the simple model advanced here.
The Hamiltonian of a single-molecule junction where the chain-like bridge interacts with a single vibrational mode may be written as
For a simple chain, the term HM that represents the molecular bridge coupled to the phonon mode has the form
Here, are creation and annihilation operators for electrons with the spin σ on the bridge site “i,” a†, a are creation and annihilation operators for the phonon mode, and δik is the Kronecker symbol. The last term in expression (2) describes the electron-vibron interaction. In the following analysis, we assume that the coupling parameters accept nonzero values only when i = j (Λii ≡ Λ) and when i = j ± 1 (Λi,i+1 = Λi,i−1 ≡ M). It is reasonable to postulate that M < Λ.
The terms Hγ (γ = L, R) correspond to noninteracting electrons on the leads with energies ϵrγσ,
where and crγσ create and annihilate electrons on the leads. Within the employed model, only terminal sites of the chain are coupled to electrodes, so the transfer Hamiltonian HT can be written as
This term describes electron tunneling between the bridge and the electrodes, where factors τrγσ characterize the coupling of relevant electron states on the bridge to those on the leads. Finally, the term Hph represents the vibrational mode with the frequency Ω,
We note that the last term in the expression (2) may be simplified by assuming that electron-phonon coupling parameters Λii take on the same value for all electron states on the bridge. However, this simplification is fitting well with the assumption concerning on-site energies, so we use it in following calculations. As follows from Eq. (2), interactions between electrons on the bridge are omitted from consideration. More advanced models can be used but the simple model (2) already shows the essential physics that needs to be addressed.
To eliminate the electron-phonon coupling term from the Hamiltonian (1), we employ the commonly used small polaron (Lang and Firsov) transformation which converts the Hamiltonian HM into where .81,82 As a result, we obtain
Here, the on-site energy acquires a polaronic shift originating from electron-vibron interaction: = E0 − Λ2/ℏΩ. Assuming a sufficiently weak electron-phonon coupling (Λ ≪ β), the coupling parameter β is renormalized in a similar way:
The transfer Hamiltonian (6) also undergoes a transformation. Within the accepted model, the second term in the expression for the operator “s” commutes with HT. Thus HT transformation reduces to the substitution of renormalized coupling parameters for τrβσ,
In addition to the Hamiltonian (1)–(5), we assume that the vibrational mode is coupled to a thermal phonon bath and that this coupling is strong enough so that the phonon maintains its thermal equilibrium state throughout the process. Consequently, the expectation value of the phonon operator X may be presented as follows:83
where Nph denotes the equilibrium phonon population. Here, we consider the low temperature regime where kθ ≪ Λ, ℏΩ, so may be approximated by which does not depend on temperature. By replacing X by , we decouple electron and phonon subsystems. Then the electron Green’s function on the Keldysh contour may be approximated as a product of the pure electronic term computed basing on the Hamiltonian ( being the transformed transfer Hamiltonian) and the Franck-Condon factor,37,38,84,85
This approximation is the most sensitive step in the accepted computational approach. It is inherent within the Born-Oppenheimer approximation and justified when a significant difference occurs between time scales characterizing electronic and vibrational dynamics. In the processes involving electron transport through molecules, this difference in time scales is by no means obvious. However, it was shown that the Born-Oppenheimer approximation in the diabatic representation may be employed in the analysis of such processes as well,86 and this is how the approximations (9) may be understood.
The Fourier transform of the retarded Green’s function for the electrons on the bridge may then be found in the form
In this expression (γ = L, R),
Thus the coupling between the bridge and the electrodes is reduced due to the effect of molecular vibrations, being a manifestation of the Franck-Condon physics typical to this model, which is further expressed by the dynamical term K(t, t′) discussed below. Within the wide band approximation, one may disregard dependences of ΓL,R on E and treat these parameters as constants. In the following calculations, we focus on a symmetrically coupled single-molecule junction: ΓL = ΓR = Γ. We remark that at the low temperatures considered here, this symmetry is not violated by electron-phonon interactions.
When gateway states are taken into consideration, the equation for the retarded electron Green’s function accepts the form
This equation is derived assuming that the bridge chain is symmetrically coupled to the electrodes. Note that the on-site energy and the coupling parameter are shifted due to electron-phonon interactions in the same way as E0 and β.
Next, consider the function K(t,t′). Since we assumed that the phonon subsystem maintains its thermal equilibrium, it can be evaluated to yield85
Expanding K(t, t′) in the terms of Bessel functions, one obtains the following expression for the spectral function matrix elements:
Here, the factors Lm have the form
and Im(Z) is the modified Bessel function of the mth order.
Specific manifestations of electron-vibron interactions in transport characteristics of molecular junctions are determined by three relevant energies. These are the coupling strengths of the electrodes to the molecular linker expressed by Γ0, the electron-phonon coupling parameter Λ, and the thermal energies being the temperatures associated with the left and right electrodes. Here, we consider the situation when the system is weakly coupled (Γ0 ≪ Λ, ℏΩ) and the characteristic temperatures are low (kθL,R ≪ Γ0). Also, we assume that electron-phonon coupling is rather weak (Λ < ℏΩ). Under these conditions, one may expect a pronounced vibrational structure to appear in the electron transmission function which may significantly affect characteristics of thermoelectric transport. Expanding the Bessel functions in power series and keeping first terms in these expansions, we get
so, the expression (15) may be reduced to
The charge current flowing through the symmetrically coupled junction then takes the form87
Here, fL,R(E) are Fermi distribution functions for electrodes. In the considered system where only first and last sites on the molecular bridge are coupled to electrodes, each N × N transfer matrix ΓL,R has a single nonzero element . Using this feature and expression (15) for spectral function, we may present the current I in Landauer form
where electron transmission function equals
The obtained expression for the electron transmission describes the situation when an electron on the bridge with the initial energy E may absorb and/or emit phonons thus moving to a state with the energy E ± mℏΩ. In weakly coupled junctions (Γ0, Λ ≪ ℏΩ), the broadening of these states is sufficiently small for them to serve as transport channels for traveling electrons and the electron transmission may be roughly estimated as a sum of contributions from all these channels.
When the electrodes are kept at different temperatures, the on-site energies acquire corrections proportional to the thermal energy kΔθ (k being the Boltzmann constant) making the Green’s function matrix elements temperature-dependent.75,77 These corrections may bring noticeable changes in the electron transmission provided that on-site energies take on values comparable with kΔθ. However, when E0 ≫ kΔθ which is typical for molecular junctions at sufficiently low temperatures, the effect of temperature on the Green’s function becomes negligible. In further analysis, we omit these corrections thus making the Green’s function and electron transmission temperature independent.
As follows from Eqs. (10) and (12), the retarded electron Green’s function is represented by the N × N matrix. Solving Eq. (10), one gets an explicit expression for the matrix element G1N in the case of a simple chain79,88,89
Here, the determinant ΔN(E, Γ) is given by
When we take into consideration the gateway states, the expression for accepts the form
and the determinant for N ≥ 3 equals80
In the following analysis, we assume that the temperature of the right electrode is kept constant whereas the temperature of the left electrode varies. Also, we assume that , so Δθ = θL − θR > 0. According to its definition, the thermocurrent is given by
where V is the bias voltage applied across the junction and the Fermi functions fL,R are computed for different leads temperatures and for different chemical potentials μL,R of the leads. The chemical potentials are shifted with respect to each other by the bias voltage. We suppose that V is symmetrically distributed between the electrodes, so μL,R = μ ∓ eV/2, μ being the chemical potential of electrodes in an unbiased system. As the electron charge is negative, a positive bias voltage shifts μL above μR. Within the accepted approximations, electron transmission τ(E) included in the integrand in Eq. (20) is temperature independent and is given by Eq. (21). We apply Eqs. (18)–(26) to analyze the dependence of the thermocurrent on the molecular bridge length.
III. RESULTS AND DISCUSSION
Consider first the case where the molecular vibrations do not affect electron transport through the system. Then the expression for the electron transmission takes the simple form
In the following computations concerning the bridge with gateway states, we use E0 = −4.47 eV, ϵ = −1.85 eV, Γ0 = 2.85 eV, δ = 1.27 eV, and β = 2.28 eV. These are the values that were derived for single-molecule junctions with gold leads and oligophenyl bridges.19 For a simple tight-binding chain, we put E0 = −4.6 eV, β = 2.2 eV, and Γ0 = 3 eV. We use two close sets of relevant parameters to better elucidate the effect of gateway states on the thermocurrent. In both cases, the HOMO appears to be located slightly below the chemical potential of electrodes in the unbiased system (μ = 0) and, therefore, serves as the primary channel for the thermally induced transport. Consequently, the charge carriers pushed through the system from the left (hot) to the right (cool) electrode by the thermal gradient Δθ are holes, and the thermocurrent Ith through an unbiased junction takes on positive values. Obviously, the thermocurrent increases with Δθ. This dependence is linear for small Δθ but it becomes superlinear at higher thermal bias, as presented in Fig. 2.
Turning now to the bridge-length dependence of the thermocurrent, Figs. 2 and 3 show that Ith decreases, for a given Δθ, when the bridge length N increases. This may be contrasted with the observation13,15–19 that the Seebeck coefficient increases with bridge length. This apparent contradiction may be resolved by turning to the length dependence of the bridge conductance G. The latter falls very rapidly with increasing bridge length, and this fall is typically much more pronounced than the corresponding increase in the thermopower S. An example of comparative behavior of these characteristics is displayed in Fig. 2 (see the right inset). The curves plotted in the inset are computed using the expressions90
These expressions satisfactory describe G and S at low temperatures provided that the transmission τ(E) smoothly varies in the close vicinity of the chemical potential μ: (|E − μ| < kθR). So, the weakening of Ith originates from the exponential fall of the molecular conductance, and it may appear simultaneously with the rise of the thermopower.
Electron transport driven by a thermal gradient through an unbiased single-molecule junction is dominated by a single HOMO/LUMO. The dependence of thermoelectric transport characteristics (including Ith) on the molecular bridge length results from the fact that the peak in τ(E) associated with the transmitting becomes sharper and narrower as the length of the molecular bridge increases. The accepted values of relevant energies result in asymmetry of the model for no energy levels appear above the Fermi level. This asymmetry was deliberately introduced to make changes in the HOMO/LUMO profile originating from variations in the bridge length more pronounced, as it happens with terminal transmission peaks within tight-binding models. However, we remark that the sharpening of the electron transmission peaks caused by the molecule lengthening is their inherent property. One may expect this effect to be manifested irrespectively of the specific model describing molecular bridge.
When gateway states are present, significant changes in the profile of the HOMO/LUMO transmission peak may occur causing variations in the length dependences of thermoelectric transport characteristics.19,80 In the chosen model, we may analyze changes in the thermocurrent originating from the effect of gateway states. As shown in Fig. 3 (right inset), in this case the HOMO peak in the plot of τ(E) versus E appears to be significantly broader than other resonance features. The peak broadening is especially pronounced for relatively short bridges (N ≤ 5). The distorted HOMO profile is supposed to be responsible for nonlinear temperature dependences of thermopower observed in experiments on molecular junctions.19 In the present case, the same HOMO distortion causes a rapid fall of Ith occurring when the number of sites increases from N = 3 to N = 5. For longer bridges, the distortion of the HOMO profile becomes less distinct, and the fall of Ith slows down. On the contrary, in the case of a simple chain, Ith decreases nearly uniformly as the chain lengthens.
To further elucidate the nature of electron transport controlled by a thermal bias, we assume that the gate voltage Vg is applied to the junction, which shifts molecular bridge energy levels upwards along the energy scale. As follows from τ(E) profiles shown in Figs. 2 and 3, at small Vg magnitudes, the electron conduction remains low, and Ith takes on low values (see the inset in Fig. 4). As Vg increases, transport channels associated with higher conductance come into play bringing a significant increase in the thermocurrent magnitude. When a certain molecular level approaches the electrodes chemical potential μ from below, this level starts to serve as a transport channel for holes pushed by the thermal bias from the left (hot) electrode to the right (cool) one. Accordingly, the thermocurrent takes on positive values. When this level crosses E = μ, the holes flow becomes counterbalanced by the electrons flow in the same direction, and Ith equals zero. When the molecular level is shifted slightly above E = μ, it serves as a transport channel for electrons, so Ith accepts negative values. As a result, a derivative-like feature appears in the Ith versus Vg plot. The electron transport induced by a thermal gradient applied across an electrically unbiased junction occurs solely on the condition that the molecular orbital serving as a transport channel is located in a close proximity of the electrodes chemical potential. Therefore, when the molecular energy level is shifted farther upwards, away from E = μ, it ceases to serve as a transport channel.
As the gate voltage increases, another molecular energy level may approach the chemical potential bringing another derivative-like feature into the Ith versus Vg plot. The resulting Ith(Vg) behavior for our model bridge is shown in Fig. 4. When the transmission peaks are well separated, the total number of derivative features is equal to the number of bridge states, namely, the number of bridge sites.
When Vg = 0 and a bias voltage V is applied across the system, electron transport is simultaneously driven by electric and thermal biases. The magnitude of Ith is determined by Δθ regardless of the electrical bias strength and polarity provided that the bias is sufficient for molecular orbitals associated with relatively high electron conductance to appear in the conduction window. Its sign, however, depends on the voltage bias. Each time a molecular transmission channel crosses μ, the sign of Ith changes, imparting an oscillatory contribution to the overall current. The thermocurrent defined by Eq. (27) represents a relatively small part of the total charge current I induced by the combined action of electrical and thermal driving forces even in weakly electrically biased molecular junctions. This is illustrated in Fig. 5.
Similar oscillations of electrical conductance and thermopower were studied for multilevel quantum dots weakly coupled to electrodes.59,91,92 Thermopower oscillations accompanying varying chemical potential of the electrodes were observed in single-molecule junctions.93 It was shown that Coulomb interactions between electrons on a multilevel dot significantly affect oscillations of electron transport characteristics in these systems, whereas in the present work, we did not take these interactions into account. Nevertheless, we believe that the main reason for these oscillations and quasi-oscillations of Ith described here is the same. These features appear when molecular orbitals (or quantum dot levels) cross the boundaries of the conduction window. The latter may be created in different ways and obviously different reasons may cause the shift of energy levels of the quantum dot and/or molecule.
As discussed before, in a weakly coupled system, the effect of vibronic interactions may be analyzed by considering the contribution of different vibronic levels to the transmission. This leads to the appearance of the vibronic structure in the transmission function (observed in inelastic tunneling spectroscopy) and to the corresponding effect on the thermal conductance. In Fig. 6, we display the transmission function τ(E) calculated using Eq. (21) for a 3-site bridge (N = 3). One observes that in the presence of electron-vibron interactions, each of the three electronic transmission peaks is replaced by a set of narrower peaks associated with the vibronic levels. The center of each set is shifted with respect to the original peak position. This happens due to the polaronic shift of energy levels on the bridge. These changes in the electron transmission profile take place over the whole range of relevant values of the tunnel energy E, including the interval around E = μ. As before, we assume that μ = 0. As shown in Fig. 6, the profile of τ(E) near E = μ may be significantly distorted. The τ(E) profile varies depending on the electron-phonon coupling strengths Λ and M as well as on the number of bridge sites and the phonon mode frequency.
These variations are expected to be most strongly pronounced in the characteristics of thermally induced electron transport through unbiased molecular junctions since in this case, the conduction window around E = μ is very narrow. One may expect alterations in molecular conduction and thermocurrent behavior to appear. For example, the length-dependent molecular conductance through a simple chain may be partly suppressed as Λ and M increase as displayed in Fig. 6. The thermocurrent itself shows changes in behavior reflecting the effect of electron-vibron coupling (see Fig. 7). One sees that Ith may display a minimum and/or a derivative-like feature at certain values of Λ provided that the molecular bridge includes five or more sites, that is, the bridge is sufficiently long. These features are signatures of vibronic levels appearing in the electron transmission. When the electron-phonon interaction becomes sufficiently large, these features disappear. This happens because with an increase in the coupling strengths Λ, M is accompanied by the increase of the polaronic shift (assuming that ℏΩ remains fixed) and by the decrease in the parameter β which narrows down the energy range where τ(E) values are not too small. As a result, all transmission peaks slide to the left, leaving solely smoothly falling tails within the vicinity of E = μ. Also, one may observe changes in the Ith dependences on the bridge length (see the inset) which appear due to the effect of the vibrational mode. We note that Eqs. (10), (21), and (22) were derived assuming a simple tight-binding model for the molecular linker. Nevertheless, these expressions give a reasonable approximation for the electron transmission which is suitable for the semiqualitative analysis of the response of thermocurrent to molecular vibrations.
In this work, we have studied some aspects of steady thermoelectric transport through a single-molecule junction with a chain-like linker of an arbitrary length. Specifically, we focus on the thermocurrent Ith, defined as the change in the charge current at a given bias voltage V due to an imposed temperature difference between the two leads. To compute Ith, we model the bridge as a tight binding chain of identical sites. We have highlighted several characteristics properties of the thermocurrent:
In unbiased systems characterized by tunneling conductance, Ith decreases with increasing bridge length. This behavior is caused by the exponential fall of the bridge conductance with increasing bridge length.
The length dependence of Ith may be significantly affected by the profile of the HOMO/LUMO peak in the electron transmission function. Specifically, the particular HOMO profile caused by the gateaway states associated with terminal sites on the molecular bridge may significantly change the behavior of the length-dependent thermocurrent. Similar effect of gateaway states on the thermopower of single molecule junctions was observed and analyzed in earlier studies.19,80 We have shown that Ith may experience the gateaway states influence within a wide range of temperature variations.
When gate or bias potentials bring the system closer to resonance transmission Ith, its bridge length dependence is affected by the energy dependent variations in the electron transmission profile near the electrodes Fermi energy. In a gated/biased molecular junction, the thermocurrent changes sign several times, as the voltage increases. The profile Ith(Vg) or Ith(V) looks like a sequence of peaks and dips. The number of peaks in this picture corresponds to the number of conducting bridge states, which in the considered model reflects the number of sites on the bridge chain. This change in the sign indicates the change of charge carriers (electron/holes) involved in the transport process. It happens when a certain molecular orbital crosses the boundary of the conduction window when either the bias or gate voltage increases. These peaks become more pronounced as the temperature gradient rises. However, at a sufficiently strongly gated/biased system when all molecular orbitals cross the boundary E = μ or are situated inside the conduction window, Ith becomes zero.
Within our model, we have analyzed the effect of molecular vibrations on the thermocurrent. In particular, we have found that in weakly coupled molecular junctions (Λ, Γ0 ≪ ℏΩ) at sufficiently low temperatures (Λ, Γ0 ≫ kθ), electron interactions with vibrational modes may qualitatively change the length dependences of Ith. Specifically, it was shown that Ith may display dips and/or derivative-like features at certain values of electron-phonon coupling parameters. These features may appear in systems with sufficiently long molecular bridges (N ≥ 5), at sufficiently weak electron-phonon coupling (Λ < ℏΩ). They are signatures of vibronic levels occurring in the electron transmission.
The system used in the present analysis is simple, and more detailed models, both regarding the molecular electronic structure and its coupling to the environment, are required for quantitative calculations of thermocurrent through long molecular bridges. Nevertheless, we believe that the results presented and discussed here capture some essential physics and may be helpful in studies of thermoelectric transport through tailored nanoscale systems.
We thank G. M. Zimbovsky for help with the manuscript preparation. This work was supported by the U.S. NSF-DMR-PREM 1523463 and the U.S. NSF Grant No. CHE1665291.