Electron-Transfer-Induced Thermal and Thermoelectric Rectification

TL;DR

This paper demonstrates that electron transfer across temperature gradients in molecular structures can induce both thermal and thermoelectric rectification, expanding the understanding beyond phononic heat transport.

Contribution

It introduces electron transfer as a new mechanism for thermal and thermoelectric rectification in molecular junctions, previously dominated by phononic effects.

Findings

Electron transfer can generate thermal rectification in molecular systems.

Electron hopping causes asymmetric Seebeck effects, enabling thermoelectric rectification.

Electron transfer effects are significant in designing molecular thermal and thermoelectric devices.

Abstract

Controlling the direction and magnitude of both heat and electronic currents using rectifiers has significant implications for the advancement of molecular circuit design. In order to facilitate the implementation of new transport phenomena in such molecular structures, we examine thermal and thermoelectric rectification effects that are induced by an electron transfer process that occurs across a temperature gradient between molecules. Historically, the only known heat conduction mechanism able to generate thermal rectification in purely molecular environments is phononic heat transport. Here, we show that electron transfer between molecular sites with different local temperatures can also generate a thermal rectification effect and that electron hopping through molecular bridges connecting metal leads at different temperatures gives rise to asymmetric Seebeck effects, that is, thermoelectric rectification, in molecular junctions.

Equations

$$E_m\left(x_1,x_2,\dots ,x_N\right)=\frac{1}{2}\sum _{j\in A\cup B}{\kappa }_j{\left(x_j-{\lambda }_j^{(m)}\right)}^2+E_m^{\prime },$$

$$k_{m\to n}^{\pm }$$

$$\times \exp \left[-\frac{{\left(\Delta E_{nm}+E_{\text{R}}\right)}^2}{4k_{\text{B}}\left(T_A{E}_{\text{R}A}+T_B{E}_{\text{R}B}\right)}\right],$$

$$\alpha =\frac{E_{\text{R}A}}{E_{\text{R}B}}.$$

$${\mathcal{J}}_{\mathcal{Q}}^{\pm }={\mathcal{J}}_{\text{el}}^{\pm }\frac{2(T_B-T_A)E_{\text{R}A}{E}_{\text{R}B}}{T_A{E}_{\text{R}A}+T_B{E}_{\text{R}B}},$$

$$R=\left|\frac{{\mathcal{J}}_{\mathcal{Q}}^{+}}{{\mathcal{J}}_{\mathcal{Q}}^{-}}\right|.$$

$$M\underset{k_{a\to M}^{\pm }}{\overset{k_{M\to a}^{\pm }}{\rightleftharpoons }}a\underset{k_{b\to a}^{\pm }}{\overset{k_{a\to b}^{\pm }}{\rightleftharpoons }}b\underset{k_{M\to b}^{\pm }}{\overset{k_{b\to M}^{\pm }}{\rightleftharpoons }}M$$

$$\Phi =-\sum _{j=1}^{\infty }{S}_j^{\pm }\Delta T^j,$$

Full text

Cite as

G. T. Craven, D. He, and A. Nitzan, Phys. Rev. Lett. 121, 247704 (2018)

DOI

10.1103/PhysRevLett.121.247704

Rectifiers are devices that promote preferential current flow in one direction. The development and application of single-molecule and solid-state electronic rectifiers has been integral in the advancement of both nanoscale and macroscale electrical circuit designs Aviram and Ratner (1974); Streetman and Banerjee (2005); Coropceanu et al. (2007); Xiang et al. (2016); Plett et al. (2017). Thermal rectifiers, which facilitate unidirectional heat flow, have also been developed and these devices have potential applications in diverse technologies including thermal management systems for nanoscale electronics, energy-harvesting devices, and thermal circuits which use heat instead of electricity to perform operations Li et al. (2012); Narayana and Sato (2012); Maldovan (2013). In molecular environments where heat conduction is dominated by vibrational (phononic) energy transfer, thermal rectification may result from nonlinear coupling, that is, anharmonicity, and the breaking of symmetry. Such phenomena have been the focus of significant theoretical and experimental examination Terraneo et al. (2002); Li et al. (2004); Segal and Nitzan (2005); Chang et al. (2006); Segal (2008); Cottrill and Strano (2015); Wang et al. (2017). Phononic heat conduction has been discussed as a possible mechanism for the operation of thermal components such as thermal transistors Li et al. (2006); Ben-Abdallah and Biehs (2014); Joulain et al. (2016), thermal memory Wang and Li (2008), and thermal logic gates Wang and Li (2007). Thermoelectric rectification effects have also been predicted in solid-state devices where they are applied to induce and control electrical currents using thermal sources Kuo and Chang (2010); Zhang et al. (2012).

Recently, a new mechanism for heat transfer was identified in molecular systems undergoing electron transfer (ET) events Craven and Nitzan (2016). It has been shown that electron hopping between molecular sites of different local vibrational temperatures is accompanied by heat transfer even when the net electronic current vanishes. Such electron-transfer-induced heat transport (ETIHT) can make a substantial contribution to the heat transfer between sites that are characterized by large reorganization energies (i.e., strong electron-phonon couplings) Chen et al. (2017) and was also shown to lead to a thermal transistor effect in a properly engineered donor-acceptor site geometry Craven and Nitzan (2017).

In this Letter, we show that the transfer of electrons between molecular donor-acceptor sites of different temperatures can induce thermal rectification when the two sites are characterized by different reorganization energies, and that the electronic conduction intrinsic to this process results in thermoelectric rectification in molecular nanostructures. The underlying ET process between charge transfer sites is described by a Marcus formalism Marcus (1956, 1993) (see also Levich Kuznetsov and Ulstrup (1999) and Hush Hush (1958, 1961)) that is augmented to treat the case of ET across a thermal gradient Craven and Nitzan (2016). This theory is based on a linear response (harmonic) model of the thermal environment. Such a harmonic environment cannot, by itself, cause rectification of heat transport. Here, the coupling of vibrational modes to the electronic subsystem is shown to provide the nonlinear component needed for rectification behavior to be present. Thus, there are two major findings in this Letter: (a) observation that electron transfer events across thermal gradients in purely molecular environments can generate significant thermal rectification and (b) demonstration that such systems can show thermoelectric rectification in molecular junctions operating under zero current conditions.

The general formalism we apply follows the results of Ref. 23, adapted to show the rectification effects examined here. The first system we consider comprises two molecular sites which have respective local solvent environments $A$ and $B$ (see Fig. 1). Electron transfer occurs between these two sites via a hopping mechanism. The system has two electronic states: $a$ and $b$, which respectively correspond to excess electron localization on the molecular charge transfer site in environments $A$ and $B$, i.e., $a\equiv\{A^{-},B\}$ and $b\equiv\{A,B^{-}\}$. Each environment contains a set of nuclear vibrational modes that are affected by electron localization on the charge transfer site in that environment. Note that, herein, we use $A$ and $B$ to denote the respective environment and the set of modes that belong to that environment. The polaron response of the nuclear vibrational modes to the electron localization on each of the sites is described using the Marcus formalism for ET in which the energy surface for electronic state $m\in\left\{a,b\right\}$ is

[TABLE]

where $x_{j}$ and $\kappa_{j}$ are, respectively, the coordinate and force constant of the $j$th mode, $E^{\prime}_{m}$ is the electronic origin of state $m$, and $\lambda^{(m)}_{j}$ is a polaron-induced shift from equilibrium of mode $j$ due to the electron localization. The reorganization energy of each mode, which is a measure of the magnitude of the electron-phonon interaction, is $E_{\text{R}j}=\tfrac{1}{2}\kappa_{j}(\lambda^{(a)}_{j}-\lambda^{(b)}_{j})^{2}$ not (a); Galperin et al. (2007). Finally, $\Delta E_{ab}=-\Delta E_{ba}=E^{\prime}_{a}-E^{\prime}_{b}$ is the difference in the energy origins of the states, that is, the reaction free energy.

Our model considers ET between two sites embedded in environments with different local temperatures $T_{A}=T-\Delta T/2$ and $T_{B}=T+\Delta T/2$ (with $\Delta T=T_{B}-T_{A}$). The phonons that respond strongly to the difference between the charge distributions in the two electronic states are localized near sites $A$ and $B$ and are assumed to be equilibrated at the temperature of the local environment not (b). To examine thermal rectification effects in this system we consider two thermal bias states: $\Delta T>0$ (denoted by “$+$”) and $\Delta T<0$ (denoted by “$-$”).

The ET rate between states can be derived in the strong electron-phonon coupling limit using a bithermal ($T_{A}$ and $T_{B}$) formulation of the Marcus semiclassical transition state theory Craven and Nitzan (2016), or, equivalently, Fermi’s golden rule evaluated in the strong coupling/high temperature limit not (c); Lin (1966); Lin et al. (2002); Nitzan (2006). In this limit, nuclear tunneling is ignored and the bithermal ET rate in the corresponding thermal bias state ($\pm$) takes the form

[TABLE]

where $V_{mn}$ is the coupling between energy surfaces, $k_{\text{B}}$ is Boltzmann’s constant, $E_{\text{R}A}=\sum_{j\in A}E_{\text{R}j}$ and $E_{\text{R}B}=\sum_{j\in B}E_{\text{R}j}$ are the partial reorganization energies associated with the modes in environments $A$ and $B$, respectively, and $E_{\text{R}}=E_{\text{R}A}+E_{\text{R}B}$ is the total reorganization energy for the electronic transition.

The structure of Eq. (Electron-Transfer-Induced Thermal and Thermoelectric Rectification) implies that the electron transfer rate is not symmetric in the two thermal bias states for $E_{\text{R}A}\neq E_{\text{R}B}$. The asymmetry between the partial reorganization energies of environments $A$ and $B$ can be expressed in terms of the parameter

[TABLE]

When $\alpha=1$, the system is symmetric and the total reorganization energies of each environment are equal. When $0\leq\alpha\neq 1$ the system is asymmetric and in this case we expect to observe thermal rectification effects. The asymmetrical property can arise from several sources: (a) the presence of more modes that participate in the ET process in one environment than the other, (b) different electronic characteristics of the solvent about each charge transfer site, (c) the extent of electron localization on the two sites, and/or (d) the temperature dependence of the reorganization energies Vath et al. (1999); Derr and Elliott (1999).

Now, consider the energy change in each environment that is induced by the ET process. To examine this heat current $\mathcal{J}_{\mathcal{Q}}$ we first consider the probabilities $p^{\pm}_{a}$ and $p^{\pm}_{b}$ for the system to be found in the respective electronic states $a$ or $b$ under the corresponding positive or negative thermal bias. The time-evolution of the occupancy probabilities obey the kinetic equations $\dot{p}^{\pm}_{a}=-\dot{p}^{\pm}_{b}=-k^{\pm}_{a\to b}p^{\pm}_{a}+k^{\pm}_{b\to a}p^{\pm}_{b}$. At steady state, the net electronic current vanishes and the unidirectional electron flux is $\mathcal{J}^{\pm}_{\text{el}}=k^{\pm}_{a\to b}p^{\pm}_{a}=k^{\pm}_{b\to a}p^{\pm}_{b}$. The steady state heat current for the corresponding thermal bias state:

[TABLE]

is a product of the electron flux $\mathcal{J}_{\text{el}}^{\pm}$ and the energy change generated by each ET event Craven and Nitzan (2016, 2017); not (d). Equation (4) implies that in a system with nonvanishing reorganization energies, the heat current generated by the transfer of electrons is nonzero for $T_{A}\neq T_{B}$. In the case of $\alpha\neq 1$, $|\mathcal{J}_{\mathcal{Q}}^{\pm}(\Delta T)|\neq|\mathcal{J}_{\mathcal{Q}}^{\pm}(-\Delta T)|\,\forall\,\Delta T\neq 0$ which implies that in asymmetric electron transfer reactions between molecules in environments with different local temperatures, thermal rectification effects can be induced solely from the transfer of electrons.

It is clear from Eq. (4) that in general $\mathcal{J}^{+}_{\mathcal{Q}}\neq\mathcal{J}^{-}_{\mathcal{Q}}$. This asymmetry is analogous to that reported for an anharmonic vibrational mode, specifically a two-level system, bridging two bosonic reservoirs with different temperatures Segal and Nitzan (2005). The magnitude of the rectification can be quantified through the thermal rectification ratio

[TABLE]

Figure 2 illustrates rectification ratios generated by electron-transfer-induced heat transport between molecular charge transfer sites over variation of $\Delta T$. For $\alpha=1$, Eq. (5) gives the expected result that $R=1$. For $\alpha=2$, $R>1$ for $\Delta T>0$ illustrating that thermal rectification is present in asymmetric systems with temperature gradients. Note that for $\alpha\neq 1$, $R$ varies nonlinearly in $\Delta T$. Comparing the results for $\alpha=2$ and $\alpha=5$, it can be observed that $R$ is also nonlinear in $\alpha$ and, thus, for constant $\Delta T$, small variations in the asymmetry of the system can result in large changes in the observed rectification ratio. It is of interest that the observed thermal rectification effects are generated using the harmonic surfaces given in Eq. (1). This is in contradiction with the traditional posit, arising from the theories that describe phononic thermal diodes, that thermal rectifiers must be anharmonic.

The thermal rectification ratios depend strongly on the reaction free energy $\Delta E_{ab}$ which appears implicitly in Eq. (4) through the $\mathcal{J}^{\pm}_{\text{el}}$ term. Shown in Fig. 3 are rectification ratios as a function of $\Delta E_{ab}$ with $\Delta T$ held constant. In the $\alpha=2$ curve, note that $R$ is nonlinear over variation of $\Delta E_{ab}$ and also symmetric with respect to the sign of $\Delta E_{ab}$. This implies that the magnitude of the free energy difference $|\Delta E_{ab}|$ is the pertinent quantity in the rectification ratio and not its direction ($\Delta E_{ab}$ or $-\Delta E_{ab}$) with respect to the applied temperature difference $\Delta T$.

While the observation just made is not easily amenable to experimental observation, its implication in a junction environment suggests a possible experimental demonstration as described below. Consider the model molecular junction shown in Fig. 4(a) in which the previous model is augmented by placing the two molecular sites as a bridge between two metal electrodes, $M_{A}$ and $M_{B}$, which are characterized by respective equilibrium temperatures $T_{A}=T-\Delta T/2$ and $T_{B}=T+\Delta T/2$ and chemical potentials $\mu_{A}=\mu+e\Phi/2$ and $\mu_{B}=\mu-e\Phi/2$, with $\Phi$ being the voltage bias across the junction not (c). Each molecule in the bridge is taken to be in thermal equilibrium at the temperature of the corresponding metal to which it is coupled. The system has three electronic states: $a\equiv\{A^{-},B\}$, $b\equiv\{A,B^{-}\}$, and $M\equiv\{A,B\}$, where $M$ denotes the state in which an excess electron is not found in the molecular bridge. The reaction free energies for electron insertion into electrodes $M_{A}$ and $M_{B}$ are $\Delta E_{M_{A}}=E^{\prime}_{M}-E^{\prime}_{a}$ and $\Delta E_{M_{B}}=E^{\prime}_{M}-E^{\prime}_{b}$, respectively, where $E^{\prime}_{M}=\mu$ is the energy origin of state $M$ and $\Delta E_{M_{A}}-\Delta E_{M_{B}}=\Delta E_{ba}$. The kinetic scheme for this system is Lehmann et al. (2002); Polizzi et al. (2012)

[TABLE]

where the interfacial molecule-metal ET rate constants are calculated from Marcus theory Marcus (1965); Nitzan (2006); not (c) and the bithermal rate constants for ET between the molecular sites are taken from Eq. (Electron-Transfer-Induced Thermal and Thermoelectric Rectification).

The unidirectional electron flux between molecules in the forward $A\to B$ and reverse $B\to A$ directions for the corresponding thermal bias states are $\mathcal{J}_{a\to b}^{\pm}$ and $\mathcal{J}_{b\to a}^{\pm}$. In the zero current case, the electron-transfer-induced heat current in the junction takes a functional form analogous to (4) where now $\mathcal{J}^{\pm}_{\text{el}}=\mathcal{J}_{a\to b}^{\pm}=\mathcal{J}_{b\to a}^{\pm}$. The condition of the electronic current $I$ to be zero while the bias is maintained is given by the condition $\mathcal{J}_{a\to b}^{\pm}(\Phi)-\mathcal{J}_{b\to a}^{\pm}(\Phi)=0$. It is important to note that, unlike the Landauer case, zero electric current does not imply zero heat current. The voltage bias leading to $I=0$ for a given temperature difference can be expressed as Çipiloğlu et al. (2004); Sánchez and López (2013)

[TABLE]

where $S^{\pm}_{k}$ is the $k$th order Seebeck coefficient in the corresponding thermal bias state, with $S_{1}$ being the standard linear Seebeck coefficient. In contrast to coherent transport, here, $S_{k}^{+}$ can be different from $S_{k}^{-}$. Next, we show numerical examples of the thermal and thermoelectric rectification phenomena associated with this model.

Electron-transfer-induced thermal rectification ratios in the junction with $I^{\pm}=0$ are shown in the Fig. 4(b) as a function of $\Delta E_{ab}$ for various values of $\alpha$ and $\Delta E_{M_{A}}$ not (c). The observed rectification effects arise from relations between the reorganization energies, the reaction free energies of both the interfacial and molecule-to-molecule ET events, and the zero-current voltage bias. For $\alpha=1$, the rectification ratio $R$ can be different from unity, which is observed prominently in the case $\Delta E_{M_{A}}=-0.25\,\text{eV}$ but is almost absent for $\Delta E_{M_{A}}=0.25\,\text{eV}$ not (e). This implies that the nonlinearity and asymmetry required to induce thermal rectification can arise solely from the electronic structure of the molecular bridge (expressed through $\mathcal{J}^{\pm}_{\text{el}}$), and that rectification can occur even when the reorganization energies are the same in each environment, which is not the case for the molecule-molecule system considered earlier. Also note that when $\Delta E_{M_{A}}<0$ ($\Rightarrow E^{\prime}_{a}>\mu$), $R(\Delta E_{ab})\neq R(-\Delta E_{ab})$ for all shown values of $\alpha$, which differs from the molecule-molecule thermal rectification ratios shown in Fig. 3 which depend only on the magnitude of $\Delta E_{ab}$ not (f). For $\Delta E_{M_{A}}>0$ ($\Rightarrow E^{\prime}_{a}<\mu$) which is shown using solid curves, $R(\Delta E_{ab})\approx R(-\Delta E_{ab})$ for both symmetric ($\alpha=1$) and asymmetric ($\alpha=5$) environments.

The rectification of thermoelectric response in the molecular junction model is demonstrated in Fig. 5. Shown are the values of $\Phi$ that lead to $I^{\pm}=0$ as a function of $\Delta T$ for various values of $\alpha$ with $\Delta E_{ab}$ held constant. Each panel of the figure shows the results for different values of $\Delta E_{M_{A}}$ and $\Delta E_{M_{B}}$. It is of significance that for incoherent transport $\Phi(\Delta T)\neq-\Phi(-\Delta T)\,\forall\,\Delta T\neq 0$, which states that when the temperature bias is reversed, the voltage bias leading to zero current does not simply change sign. This is in contrast to the coherent limit of transport calculated using the Landauer formalism (shown by a dashed black line) where $\Phi(\Delta T)=-\Phi(-\Delta T)$ as long as the transmission coefficient remains constant not (c); Nitzan (2006). This asymmetry with respect to $\Delta T$ implies that $S^{+}_{k}\neq S^{-}_{k}$ for even values of $k$. Moreover, in Fig. 5(a) for $\alpha=1$ and Fig. 5(b) for $\alpha=2$, $\Phi(\Delta T)\approx\Phi(-\Delta T)$ which expresses the counter-intuitive result that when the thermal bias state is reversed, the voltage bias required to generate zero electronic current can remain unchanged for bridges with certain electronic properties. It can also be observed in Fig. 5(b) that the sign of the first-order Seebeck coefficient $S_{1}$ for incoherent transport can be different than for coherent transport. This implies that the direction of thermoelectric current can be controlled by altering the electronic properties of the junction environment, i.e., the reorganization energies.

In conclusion, we have shown that thermal rectification effects can be induced by the transfer of electrons between molecular sites that are seated in environments with different local temperatures. The electron-transfer-induced thermal rectification mechanism differs from the standard phononic heat transport mechanism which has historically been assumed to be the sole heat conduction channel in purely molecular systems. We have also demonstrated that the intrinsic electronic conduction that accompanies this heat transfer can give rise to thermoelectric rectification in a model molecular junction. The treatment of electron-transfer-induced thermal and thermoelectric rectification effects in molecular junctions and at molecule-electrode interfaces will provide insight into how thermal gradients affect electron scattering and charge currents in molecular electronic devices.

Electron-transfer-induced thermal rectification effects could be observed experimentally by controlling the electrical currents in molecular electronic diodes Aviram and Ratner (1974) using thermal gradients, and also by examining systems with suppressed phononic heat currents such as $\pi$-stacked molecular junctions Kirs̆anskas et al. (2014); Li et al. (2017) and molecular donor-acceptor dyads Hedström et al. (2017) with weak intramolecular vibrational couplings. In the case of molecule-to-molecule electron transfer (see Fig. 1), measuring heat changes in each molecule’s environment while controlling the local temperature will require the implementation of recent advances in nanoscale thermometry Sadat et al. (2010); Menges et al. (2012, 2016); Mecklenburg et al. (2015). For molecular junctions (see Fig. 4(a)), the phenomena predicted in this Letter could be measured by employing similar experimental techniques to those that have been previously used to probe the electronic Capozzi et al. (2015), thermoelectric Reddy et al. (2007); Tan et al. (2011), and thermal Lee et al. (2013); Kim et al. (2014); Cui et al. (2017) properties of such systems.

The research of A.N. is supported by the Israel-U.S. Binational Science Foundation, the German Research Foundation (DFG TH 820/11-1), the U.S. National Science Foundation (Grant No. CHE1665291), and the University of Pennsylvania. D.H. is supported by NSFC of China (Grants No. 11675133 and No. 11335006), NSF of Fujian Province (Grant No. 2016J01036).