Charge Transport and Entropy Production Rate in Magnetically Active Molecular Dimer
CCharge Transport and Entropy Production Rate in Magnetically Active Molecular Dimer
J. D. Vasquez Jaramillo and J. Fransson ∗ Department of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden (Dated: October 23, 2018)We consider charge and thermal transport properties of magnetically active paramagnetic molecular dimer.Generic properties for both transport quantities are reduced currents in the ferro- and anti-ferromagnetic regimescompared to the paramagnetic and e ffi cient current blockade in the anti-ferromagnetic regime. In contrast, whilethe charge current is about an order of magnitude larger in the ferromagnetic regime, compared to the anti-ferromagnetic, the thermal current is e ffi ciently blockaded there as well. This disparate behavior of the thermalcurrent is attributed to current resonances in the ferromagnetic regime which counteract the thermal flow. Thetemperature di ff erence strongly reduces the exchange interaction and tends to destroy the magnetic control of thetransport properties. The weakened exchange interaction opens up a possibility to tune the system into thermalrectification, for both the charge and thermal currents. I. INTRODUCTION
Thermal transport properties in molecular junctions can beof electronic origin or mediated through lattice vibrations [1].There has been an increasing interest in the study of ther-mal properties in molecular junctions [2–5], stimulated byexperimental observations [2]. Several realizations of tun-neling junctions comprising noble metal electrodes and poly-mers that absorb, emit, and transmit thermal have been re-ported [4, 6]. The interest is, moreover, driven from theperspective of information science and technology with re-spect to entropy production rate [3, 7] and the meaning of thethermodynamics in low dimensional systems [3, 8, 9]. Thiswas motivated by the discovery of conducting polymers andsolitonic electronic transport mechanisms discovered by Shi-rakawa, see [10] and references therein. By this discoverypolyacetylene became the test bench, bridging the gap be-tween organic and inorganic chemistry regarding electronictransport [11–13]. Since then, charge transport has been ex-tensively studied theoretically and experimentally in molec-ular junctions [15–25]. From the same standpoint, thermaltransport studies were conceived and came to be conclusivein the upcoming years [2]. Subsequently, several theoreticalstudies demonstrated the possibility to conduct both thermaland charge through tunneling junctions [26–34], both in pres-ence and absence of lattice vibrations, although there is nogeneric framework that successfully is capable of describingthe thermodynamic properties of nano junctions [8, 14, 35].Theoretical predictions suggest all electrical control forboth reading and writing spin states in molecular dimers ∗ Electronic address: [email protected] [51, 52]. This prediction is based on the (electronically me-diated) indirect exchange interaction between the localizedspins which controls the charge transport properties [51, 52].Similar e ff ects were reported in Ref. [6], where the spinground state of a single metal complex is electrically con-trolled, imposing transition between high ( S = /
2) and low( S = /
2) spin configurations in a three terminal device.Here we build on the previous predictions made in Ref.[51, 53] for dimers of magnetic molecules in which the ef-fective spin-spin interactions are mediated by the propertiesof the delocalized electrons and extend to thermally inducedmagnetic and transport properties. In particular we studythermal transport and its response to changes of the mag-netic configurations. Our set-up pertains to, for instance,M-phthalocyanine (MPc), M-porphyrins, where M denotes atransition metal atom [61–64], e.g., Cr, Mn, Fe, Co, Ni, Cu,and also to bis(phthalocyaninato)R (TPc ) [65, 66], where Rdenotes a rare earth element, e.g., Tb. Such molecules can beinvestigated in, for example, mechanically controlled break-junctions [66, 67], in carbon nanotube assemblies [65] andscanning tunneling microscope [62, 63].We consider thermal and charge transport as the result ofthe electrothermal control of the junction. Accordingly, we in-vestigate the charge and thermal conductance with respect tothe bias voltage and thermal gradient across the junction. Withthis background we, furthermore, consider a non-equilibriumanalogue to the Seebeck coe ffi cient defined as the ratio be-tween the di ff erential conductances with respect to the ther-mal gradient and bias voltage, introduced in [68]. By thesame token, we consider the ratio of the energy di ff erentialswith respect to the thermal gradient and the bias voltage. Ourpredictions and results are based on non-equilibrium Greenfunctions defined on the Keldysh contour.Throughout this study we consider spin degenerate con- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n ditions in that we assume non-magnetic metals in the leadsas well as the absence of externally applied magnetic fields.The advantage with this set-up, compared to designs basedon ferromagnetic leads is that the dipolar and quadrupolarfields considered in Ref. [69] here becomes vanishinglysmall. Therefore the e ff ective isotropic electron mediatedspin-spin interactions dominates the properties and control ofthe molecular dimer. In this sense, our system would serve asa representation of paramagnetic spintronics, or paratronics . II. METHODSA. Magnetic molecular dimer in a junction
The specific set-up we address in this article comprises adimer of paramagnetic molecules which are embedded in se-ries in the junction between normal metallic leads, see Figure1 (a). Each paramagnetic molecule comprises a localized spinmoment which is embedded in a ligand structure, where the sp -orbitals define the spin-degenerate HOMO or LUMO or-bitals. We assume that the d -, or, f -orbitals that constitute themolecular spins, hybridize only weakly with the sp -orbitals,allowing to consider the spin moment in the localized momentpicture. As such, the localized moment interacts with the de-localized electrons only via local exchange. We also neglectspin-orbit coupling in the molecular orbitals as well as consid-ering them in single electron form, which is typically justifiedfor the sp -electrons. The molecular orbitals couple via tun-neling both to one another and to the adjacent lead.We model this set-up using the Hamiltonian H = H M + H int + H L + H R + H T . (1)Here, the molecular HOMO or LUMO levels are defined by H M = (cid:88) σ (cid:88) m = L , R ε m d † m σ d m σ + T c ( d † L σ d R σ + H . c . ) , (2)where d † m σ ( d m σ ) creates (annihilates) an electron in theleft ( L ) or right ( R ) molecule at the energy ε m = ε andspin σ = ↑ , ↓ , whereas T c defines the tunneling rate be-tween the molecules. Internally in molecule m , the local-ized spin moment S m interacts with the electron spin s m = (cid:80) σσ (cid:48) d † m σ σ σσ (cid:48) d m σ (cid:48) /
2, where σ is the vector of Pauli matri-ces, via exchange H int = (cid:88) m = L , R v m s m · S m , (3)where v m is the exchange integral, and we assume that v m = v .We focus on the case with non-magnetic leads, H L / R = (cid:88) k σ ∈ L / R ε k c † k σ c k σ , (4)where c † k σ creates an electron in the left ( L ; k = p ) or right ( R ; k = q ) lead at the energy ε k and spin σ . Tunneling between FIG. 1: (Color online) (a) Molecular dimer of paramagneticmolecules. An electron (at energy ε ) in each molecule interactswith the localized spin moment ( S m , m = L , R ) via exchange ( v m )with the electron in the adjacent molecule (tunneling rate T c ) andwith electron sin the left / right electrode (coupling Γ ). The left / rightnonmagnetic electrode is characterized by its electrochemical poten-tial ( µ L / R ). E ff ective molecular orbutals ( ε ± T c ) emerge from in-termolecular tunneling. (b) E ff ective exchange interaction betweenthe localized spin moments as function of the voltage bias V . (c)Occupation of the states in the spin dimer. The green curve repre-sents the occupation of the lowest energy eigenstate of the spin dimerwhich changes character between spin singlet and spin triplet statesas function of the voltage bias. Other colors analogously representthe occupation of the consecutively higher energy eigenstates. In theregion indicated by the red arrow three states are degenerate and forma spin triplet. Calculations are made at T L = T R = ε − µ = T c = v m = Γ =
1, meV. In panels (b)and (c), the ferromagnetic and antiferromagnetic regimes of the spindimer are indicated with red and black arrows, respectively. the leads and molecules is described by H T = (cid:88) p σ T L c † p σ d L σ + (cid:88) q σ T R c † q σ d R σ + H . c . (5)and we define the voltage bias V across the junction by eV = µ L − µ R , where µ χ , χ = L , R , denotes the electrochemical po-tential of the lead χ .In this way H = H L + H R + H T + H M provides a spin-degenerate background electronic structure which mediatesthe exchange interactions between the localized spin momentsin H int . The spin-degeneracy implies that these interactionsare purely isotropic [51, 53], such that we retain the Heisen-berg model only for the spins.
1. Exchange interactions
It has been shown that the Hamiltonian in Eq. (1) gives riseto the e ff ective spin model [53] H S = (cid:88) mn (cid:16) J mn S m · S n + D mn · [ S m × S n ] + S m · I mn · S n (cid:17) . (6)Here, the parameter J mn denotes the isotropic Heisenberginteraction, whereas D mn and I mn respectively denotes theDzyaloshinskii-Moriya and Ising interactions which both in-troduce anisotropic interactions into the system. In the present FIG. 2: (Color online) The exchange interaction J (units: meV) be-tween the molecular spins as function of voltage bias V and temper-ature di ff erence ∆ T = T R − T L for di ff erent gating conditions, suchthat ε − µ = , , , system, where the there is no spin-dependence imposed, eitherby external or internal forces, the anisotropic interactions van-ish, D mn = I mn =
0, and we retain the isotropic Heisenberginteraction only.In this paper, we treat the spin dimer as a closed systemin the sense that we require a conserved number of particlesfor which the occupations of the states are given by the Gibbsdistribution. This is a valid approximation for localized spinswhere the hybridization with the surrounding itinerant elec-trons is small such that nature of the localized electrons can bedescribed in terms of a Kondo-like model rather than a fluc-tuating spin model in the sense of the Hubbard or Andersonmodels.The spin-spin interactions are, nevertheless, influenced bythe tunneling current that flows through the molecular com-plex. In such set-ups, the e ff ective magnetic interaction pa-rameter J between the two spins can be calculated using theexpression, see Refs. [51, 53], J = − T c π v (cid:88) χ (cid:90) Γ χ f χ ( ω )( ω − ε ) ( ω − ε ) − T c − ( Γ / | ( ω − ε + i Γ / − T c | d ω, (7)where Γ = (cid:80) χ = L , R Γ χ , with Γ χ = π (cid:80) k σ ∈ χ T χ δ ( ω − ε k ), is thecoupling to the leads, and f χ ( ω ) is the Fermi function at thechemical potential µ χ . Here, we have assumed that the molec-ular level ε m = ε , m = L , R , and that the local exchange cou-pling v m = v , which is reasonable for equivalent molecules.We remark here that for a more general molecular assembly,that is, non-equivalent molecules and asymmetric couplingsto the leads, the above expression for the exchange interac-tion should be replaced by the general formulas given in Ref.[53]. While this article is focused on dimers with equivalentmolecules coupled symmetrically to the leads, we briefly dis-cuss deviations from this case below.Notice that while the exchange interaction is definedin terms of the two-electron propagator ( − i ) (cid:104) T s ( t ) s ( t (cid:48) ) (cid:105) [53], we employ the de-coupling approximation( − i )sp σ G ( t , t (cid:48) ) σ G ( t (cid:48) , t ). Here, G mn ( t , t (cid:48) ) is a single-electron Green function for the molecular orbitals projected onto the sites m and n , whereas sp defines the trace over spin1 / ff ects, this is farbeyond the scope of the present article.
2. Non-equilibrium variations
The salient features of the voltage bias dependence of J andthe corresponding occupation of the spin states in the dimerare plotted in Figure 1 (b), (c) for the case with ε − µ = ff erence of 5 K (for other parameters, see thefigure caption). For the details about the interdependence be-tween the electronic structure in the molecular orbitals and thespin-spin exchange we refer to Refs. [51, 53]. The tempera-tures of the leads are included in the respective Fermi functionon which the exchange interaction within the spin dimer de-pends, see Eq. (7). In addition to the dependencies on thetemperature of the leads, the exchange also depends on theenergy of the molecular levels. These dependencies are il-lustrated in Figure 2, which shows the exchange interactionenergy as function of the voltage bias and temperature di ff er-ence for di ff erent energies ε − µ . There is a clear distinctionbetween the situation in panel (a) compared to panels (b) –(d), which originates in the level position relative to µ .First, whenever the localized level ε − µ =
0, the molecu-lar bonding (at the energy: ε − T c ) and anti-bonding (at theenergy: ε + T c ) orbitals are centered symmetrically around µ which leads to that the influences from the left and rightleads is symmetric with respect to voltage bias and indepen-dent of whether the source or drain electrode is warmer thanthe other. The situation is schematically depicted in Figure 3(a), (b). The first thing to notice is a weakening of the ex-change interaction caused by the increased thermal spread ofthe electrons in the hot reservoir, compared to the cold reser-voir. In equilibrium, then, in comparison to the cold reservoirthe hot reservoir contains a larger number of electrons aboveand below both molecular orbitals which contribute to an anti-ferromagnetic interaction and simultaneously a smaller num-ber of electrons that contribute to the ferromagnetic exchange.This results in a weaker ferromagnetic interaction, which isverified in the simulations, see Figure 2 (a). The analo-gous argument holds for finite biases in the anti-ferromagneticregime, however, utilized in the opposite way. Due to thethermal distribution of the electrons, there are electrons inthe hot reservoir contributing to both the ferromagnetic andanti-ferromagnetic exchange which results in an e ff ectivelyweaker anti-ferromagnetic exchange, see Figure 2 (a). More-over, since the thermal distribution of the electrons is symmet-ric around the pertinent chemical potential, the e ff ect is equalregardless of the polarity of the voltage bias across the junc-tion, which basically indicates voltage bias symmetric trans-port properties of the junction under these conditions.Next, whenever the molecule is gated such that the local-ized level ε is o ff -set from the chemical potential µ , a fi-nite temperature di ff erence do generate an asymmetry withrespect to the voltage bias, see Figure 2 (b) – (d). One canconvince oneself that the exchange is symmetric with respectto the voltage bias at vanishing temperature di ff erence, how-ever, in general the asymmetry is conspicuous. Under thegiven conditions, ε − µ > T R with T L = ff ected by changes in the temperature di ff erence.This can be schematically understood from the sketch in Fig-ure 3 (c), showing µ L lying between the molecular orbitals.Then, the narrow thermal spread of the electrons in the leftlead maintains a strong contribution to the ferromagnetic ex-change while the contribution from the right lead is negligiblemore or less independently of its temperature. By the sametoken one can understand that the ferromagnetic regime fornegative voltages is extremely sensitive to the temperature ofthe right lead, see Figure 3 (d). Under these conditions, theexchange interaction is dominated by the contribution fromthe right lead while the contribution from the left lead is neg-ligible. Hence, at elevated temperatures there are competingcontributions to the exchange of both ferromagnetic and anti-ferromagnetic nature which leads to a very weak ferromag-netic exchange that weakens with increasing temperature dif-ference.
3. Deviations from perfectly equivalent and symmetric dimers
In realistic situations one can expect that the local parame-ters vary from molecule to molecule, although they are meantto be considered as equivalent. Here, we briefly discuss someimplications on the indirect exchange interaction under di ff er-ent local exchange integrals v m , finite level o ff -set ∆ = ε L − ε R ,between the molecular levels ε m , and asymmetric couplings tothe leads.For instance, for non-equivalent molecules such that ε (cid:44) ε , the exchange interaction J tends to become strongly asym-metric with respect to the voltage bias, see for instance Ref.[51]. It may even lead to situations where J is ferromag-netic (negative) for one polarity of the voltage and anti-ferromagnetic (positive) for the opposite. In those situations,one would expect the resulting transport properties to be sig-nificantly di ff erent for the two voltage polarities, somethingthat was also verified by a strong rectification property [51].Except for the strong dependence of the exchange inter-action on the level o ff -set between the two molecules, thedimer structure is remarkably robust against small variationsand asymmetries in the couplings Γ χ to the leads as well as toinequalities in the local Kondo exchange couplings v m . Dif-ferences of up to 20 % between the couplings of the twomolecules do not leads to any essential variations in the ex-change interaction and are, therefore, not expected to be detri-mental to the transport properties either. eVμ R μ L eVμ R μ L (a) (b)V>0 V<0T L T L T R T R eVμ R μ L eVμ R μ L (c) (d)V>0 V<0T L T L T R T R FIG. 3: (Color online) The combined role of the voltage bias andtemperature di ff erence on the tunneling conditions for (a), (c) pos-itive and (b), (d) negative bias, and (a), (b) symmetric and (c), (d)asymmetric molecular orbitals around µ (dashed). The left (right)lead is defined at the temperature T L ( T R ) and chemical potential µ L ( µ R ); in the figure T L < T R and µ L − µ R = eV . A low (high) tem-perature sustains a sharp (fuzzy) boundary between the occupied andunoccupied electron states in the metal.
4. Spin expectation values
For later purpose we introduce the notation (cid:104) S z χ (cid:105) for theexpectation values which are projections onto the molecule χ of the total spin expectation value (cid:104) S (cid:105) of the dimer. Theexpectation value (cid:104) S (cid:105) is calculated with respect to the spinHamiltonian H S = J S L · S R , with J obtained from Eq. (7). Aswe are considering the individual moments to remain in theirground states under all conditions, the restricted Hilbert spacecorresponding to this set-up is four dimensional labeled bythe states, say, {| ↑↑(cid:105) , | ↑↓(cid:105) , | ↓↑(cid:105) , | ↓↓(cid:105)} , where the first (second)entry corresponds to the left (right) spin. Then, the projectedexpectation value is calculated by the expression (cid:104) S z χ (cid:105) ≡(cid:104) σσ (cid:48) | exp {− β av E σσ (cid:48) } S z χ | σσ (cid:48) (cid:105) / (cid:104) σσ (cid:48) | exp {− β av E σσ (cid:48) }| σσ (cid:48) (cid:105) ,where the operator S z χ is expressed in the basis of the dimer.Here, also 1 /β av = k B ( T L + T R ) /
2, represents the averagetemperature of the two leads. Although this appears to bea severe simplification, it turns out that the spin occupationnumbers vary slowly as function of the temperature in thecurrent set-up, which justifies this simplistic treatment.We remark here that any temperature mediated by the tun-neling electrons vanish under the present condition in the em-ployed approximation. To clarify this point, we notice that thee ff ective spin model introduced in Eq. (6) in principle alsocontains the contribution H τ = − g µ B B τ · (cid:88) m S m , (8)where the magnetic field B τ is proportional to the currentthrough the junction, see, e.g., Refs. [53–55, 74–76]. Thisfield provides a torque on the loclized spin, however, onlywhen the spin-degeneracy is broken in the current. In otherwords, this current induced magnetic field vanishes wheneverthe tunneling current is spin-degenerate. Hence, since ourset-up is spin-degenerate by construction, the current inducedmagnetic field does not generate any renormalization of thelocalized spin and, in particular, the temperatures of the leadsare not transmitted via the charge current to the spins. B. Transport formulas
Here, we will briefly go through and summarize the ap-proach we employ to study the transport properties governedby the molecular spin dimer. First, the currents under interestare the charge and thermal currents, which are defined for theflows through interface χ as I χ c = − eI χ N , (9a) I χ Q = I χ E − µ I χ N , (9b)respectively, where I χ E ( N ) denotes the energy (particle) currentthrough the junction. Second, we notice that the energy andparticle currents through the system are defined as the rate ofchange of the energy and the particle number, respectively.Concerning the particle flux, conservation of particles ensuresthat the rate of particles passing through one of the interfacesequals the corresponding rate at the other interface. However,the component in the thermal current that is generated by theparticle flux is not a conserved quantity [70]. This problemcan be avoided by considering the local entropy productionrate in the molecular dimer, expressed through I Q = I LQ − I RQ = ( I LE − µ L I LN ) / − ( I RE − µ R I RN ) /
2. Then, since the particle contri-bution can be written as − ( µ L + µ R ) I LN / = − µ I LN / µ ,which is e ff ectively zero, the particle contribution to the ther-mal current vanishes.We remark here that the terminology entropy productionrate for the quantity I Q is justified in the stationary regimesince then the heat current I χ Q , χ = L , R , flowing through theleft or right interface equals the corresponding entropy flow I χ S multiplied by the temperature, T χ I χ S . This can be un-derstood by considering that the rate of change of the non-equilibrium grand potential vanishes in the stationary regime, ∂ t Ω =
0, which yields T ∂ t S = ∂ t (cid:104)H(cid:105) − µ∂ t (cid:104) N (cid:105) . On the onehand, the entropy production rate is given by the di ff erenceof the entropies flowing through the left and right interface, ∂ t S L − ∂ t S R , with dimensions energy divided by temperatureand time. On the other hand, the well defined and calculablequantity is here defined by the thermal currents I χ Q and, dueto the strong intimacy between the two quantities ∂ t S χ and I χ Q , we shall use the term energy production rate for I Q withdimensions energy divided by time.We express the fluxes at the interface χ in the generic forms I χ E = ddt (cid:104)H χ (cid:105) = − T χ Im (cid:88) k σ ε k ( − i ) F < k χσ ( t , t ) , (10a) I χ N = ddt (cid:104) N χ (cid:105) = − T χ Im (cid:88) k σ ( − i ) F < k χσ ( t , t ) , (10b)where we have defined the lesser form F < k χσ ( t , t (cid:48) ) of the trans-fer Green function F k χσ ( t , t (cid:48) ) = ( − i ) (cid:104) T c k σ ( t ) d † χσ ( t (cid:48) ) (cid:105) . Usingstandard theory [71] and restoring ¯ h , the currents can be com-pactly written on the form I χ E = ( − i ) Γ χ ¯ h sp (cid:90) ω (cid:16) f χ ( ω ) G >χ ( ω ) + f χ ( − ω ) G <χ ( ω ) (cid:17) d ω π , (11a) I χ N = ( − i ) Γ χ ¯ h sp (cid:90) (cid:16) f χ ( ω ) G >χ ( ω ) + f χ ( − ω ) G <χ ( ω ) (cid:17) d ω π . (11b)Here, sp denotes the trace over spin 1 / G >χ , χ = L , R , denotes the 2 × χ .
1. Derivation of the molecular Green function
We make further analytical progress by constructing an ex-plicit expression for the Green function G χ . To this end weinclude the broadening e ff ects from the couplings to the leadsas well as the inter-molecular tunneling and intra-molecularexchange interactions with the localized spin moments. Thesepresumptions lead to that we can write the retarded / advancedGreen functions G χ weighted on molecule χ as [51] G r / aL ( R ) ( ω ) =
12 ˜ T c (cid:88) s = ± ˜ T c σ − sv (cid:104) S zR ( L ) (cid:105) σ z ω − E s ± i Γ / , (12)where the excitation energies E ± = ε ± ˜ T c / T c = v (cid:104) S zL − S zR (cid:105) + T c .The Green function G for the full dimer system is a 4 × G = { G χχ (cid:48) } χ,χ (cid:48) = L , R , in which each entry is a 2 × G χχ (cid:48) = { G χσχσ (cid:48) } σ,σ (cid:48) = ↑ , ↓ . Here, the subscripts χ, χ (cid:48) refer to theleft (right) molecule if χχ (cid:48) = LL ( χχ (cid:48) = RR ) and coupling be-tween the molecules for χχ (cid:48) = LR or χχ (cid:48) = RL . For brevity,we write G χχ = G χ . Each entry is defined by G χσχ (cid:48) σ (cid:48) ( z ) = (cid:104)(cid:104) d χσ | d † χ (cid:48) σ (cid:48) (cid:105)(cid:105) ( z ) = (cid:90) ( − i ) (cid:104) T d χσ ( t ) d † χ (cid:48) σ (cid:48) ( t (cid:48) ) (cid:105) e − iz ( t − t (cid:48) ) dt (cid:48) . (13)The equation of motion for G can be summarized in the Dysonequation G = G + G Σ G , (14)where G is the bare Green function for the coupledmolecules, however, without couplings to the leads, whereas FIG. 4: (Color online) (a) Molecule ( L , R ) and spin projected (indi-cated by white arrows) densities of electron states of the left and rightnon-gated ( µ =
0) molecules, calculated using G LL ( RR ) σ given in Eq.(16), respectively, as function of the voltage bias V and energy ω .(b) Molecule projected eq-fullGreenFunction of the localized spins (cid:104) S z χ (cid:105) and the total magnetic moment (cid:104) S zL + S zR (cid:105) . Parameters are as inFigure 1. Σ defines the self-energy generated by the couplings tothe leads. It can be noticed that since molecule 1 (2)only couples to the left (right) lead, the retarded form ofthis self-energy can be written as the diagonal matrix Σ = ( − i )diag { Γ L ↑ , Γ L ↓ , Γ R ↑ , Γ R ↓ } /
2. Considering spin-degenerateconditions, we can set Γ χσ = Γ χ /
2. As an e ff ect of the Dysonequation for G , the corresponding lesser / greater forms aregiven by G > = G r Σ > G a , where the lesser / greater formsof the self-energy is given by Σ > ( ω ) = ( ± i ) 14 (cid:32) f L ( ± ω ) Γ L σ f R ( ± ω ) Γ R σ (cid:33) . (15)In order to find the analytical forms of the local Green func-tion, we notice that since the spin degrees of freedom areuncoupled, we can write G in block diagonal form. In thisrepresentation, the blocks are distinguished by the spin indexwhereas each block can be written on the form (cid:110) G r / a χχ (cid:48) σ ( ω ) (cid:111) χ,χ (cid:48) = L , R = (cid:34)(cid:32) ω − ε ± i Γ L / −T c −T c ω − ε ± i Γ R / (cid:33) − v σ z σσ (cid:32) (cid:104) S zL (cid:105) (cid:104) S zR (cid:105) (cid:33)(cid:35) − = (cid:32) ω − ε − v (cid:104) S zR (cid:105) ± i Γ R / T c T c ω − ε − v (cid:104) S zL (cid:105) ± i Γ L / (cid:33) ( ω − E r / a + )( ω − E r / a − ) , (16)where E r / as = ε + s ˜ T c / ∓ i ( Γ L +Γ R ) / = ε + s ˜ T c / ∓ i Γ / Γ χ = Γ / σ is, therefore, given by the entry G r / aLL σ ( ω ), which can also bewritten on the form G r / aL σ ( ω ) =
12 ˜ T c (cid:88) s = ± ˜ T c − sv (cid:104) S zR (cid:105) σ z σσ ω − E s ± i Γ / , (17)with E ± = ε ± ˜ T c / f χ ( ω ) G >χ ( ω ) − f χ ( − ω ) G <χ ( ω ) = ( − i ) Γ R (cid:16) f χ ( ω ) − f χ (cid:48) ( ω ) (cid:17) G r χχ (cid:48) ( ω ) G a χ (cid:48) χ ( ω ) , (18)out of which the only finite terms are the ones that couple thetwo molecules to one another and it is, therefore, necessary tostudy the forms of the site o ff -diagonal Green functions G LR and G RL . By a straight forward calculation one finds that these Green functions can be written as G r / a χχ (cid:48) ( ω ) = T c (cid:88) s = ± σ − s σ z ω − ε − sv (cid:104) S z χ (cid:105) ± i Γ / G r / a χ (cid:48) ( ω ) = T c T c (cid:88) ss (cid:48) = ± [ σ − s σ z ][ ˜ T c σ − s (cid:48) v (cid:104) S z χ (cid:105) σ z ][ ω − ε − sv (cid:104) S z χ (cid:105) ± i Γ / ω − E s (cid:48) ± i Γ / . (19) III. RESULTS AND DISCUSSION
Regarding the charge transport across the junction, most ofits properties can be understood in terms of the degree of de-localization of the electronic density in the molecular dimer.As have been discussed in a previous publication [51], themagnetic states and configurations lead to qualitatively dis-tinct regimes of the electronic properties of the dimer, some-thing that is conveyed over to the inherit transport propertiesof the molecular dimer. Accordingly, the conductance in theferromagnetic regime is expected to better, or, higher than inthe anti-ferromagnetic regime. This conjecture is based on thefact that in the anti-ferromagnetic regime the spin projectedelectronic density in the bonding state is strongly localized toone or the other molecule, see Figure 4 (a), and in the opposite -6 -3 voltage bias (mV) I Q ( M e V / s ) -6 -3 voltage bias (mV) ×10 -1 (c) (d) -303 I C ( n A ) -6 -3 voltage bias (mV) voltage bias (mV) (a) (b) T R -T L =0 KT R -T L =10 K FIG. 5: (Color online) (a), (b), The charge current I C (units: nA)and (c), (d), entropy production rate I Q (units: MeV / s) as functionof voltage bias V for T R − T L = T R − T L =
10K (red – bold) for (a), (c), ε − µ = ε − µ = fashion for the anti-bonding orbital. Therefore, an electron re-siding in one of the molecules has a nearly vanishing probabil-ity to tunnel over the other molecule which leads to a stronglysuppressed conductance. In the ferromagnetic regime, how-ever, the electronic density is more delocalized throughout themolecular dimer which allows for resonant tunneling betweenthe molecules and, in turn, leads to an increased conductance.These features and disparities of the two magnetic regimescan be observed in the current, see Figure 5 (a), (b), showingthe charge current I c as function of the voltage bias at zerotemperature di ff erence (black – faint) and T R − T L =
10 K (red– bold). Especially for vanishing temperature di ff erence, theanti-ferromagnetic and ferromagnetic regimes (indicated byblue and green arrows, respectively) are strikingly separatedby sharp current resonances which originates from a vanish-ing exchange between the spin. The absence of the exchangeinteraction leads to uncorrelated spins and a completely delo-calized charge density in the molecular dimer, which allowsfor a large current flow in the same way as in the high voltageregime where J → ff erence across the junction, thecurrent-voltage characteristics is necessarily symmetric when-ever then electronic structure of the molecular dimer are sym-metrically distributed, as is the case we consider here. In-troduction of a finite temperature di ff erence changes this sce-nario, however, a necessary condition for breaking the sym-metric current-voltage characteristics is that the bonding andanti-bonding orbitals do not surround µ symmetrically. Thiscan be seen in the red – bold traces in Figure 5. In panel (a),the molecular level ε = µ which accordingly leads to a sym-metric current as function of the voltage bias. In panel (b), incontrast, the molecular level satisfies ε − µ = FIG. 6: (Color online) (a), (c), The charge current I C (units: nA)and (b), (d), entropy conductance I Q (units: MeV / s) as function ofvoltage bias V and temperature di ff erence T R − T L for (a), (b), ε − µ = ε − µ = then generates a striking asymmetry in the current for finitetemperature di ff erences. This result can be traced back tothe dramatically changed, weakened, exchange interaction be-tween the spins which causes an increased degree of delocal-ization of the electronic density in the molecular dimer. In ef-fect, therefore, both the ferromagnetic and anti-ferromagneticregimes are essentially destroyed for negative voltages, seeFigure 5 (b) (red – bold). Hence, under the temperature di ff er-ence between the leads the molecular dimer functions partiallyas a rectifying device where the magnetically active regimescan be employed in the forward direction while the systembehaves like a normal conductor in the backward. In Fig-ure 6 (a), (c), we show contour plots of the charge currentas function of the voltage bias and temperature di ff erence for(a) ε − µ = ε − µ = ε − µ = ff erences,closely following the behavior of the exchange interactions,c.f., Figure 2.Next, in the discussion of the entropy production rate, weagain notice that most of the expected features can be ex-plained in terms of the properties of the electronic structurein the molecular dimer analogous as with the charge current.However, for vanishing temperature di ff erence between theleads a clear distinction compared to the charge current is thatthe finiteness of the entropy production rate strongly dependson the energy of the localized level in the molecular assem-bly. This can be traced back to the product of the energy andthe distribution functions ( G >χ ) of the molecular dimer in,for instance, Eq. (11a). One can make a simple compari-son between the qualitative properties of the two currents byassuming a Lorentzian model, 1 / [( ω + µ − ε ) + ( Γ / ], forthe device embedded between the leads at low temperatures.Then, under the voltage bias µ L / R = µ ± eV /
2, the two currentsbehave as I c ∼ Γ / (cid:18) arctan eV / − ε Γ / + arctan eV / + ε Γ / (cid:19) , (20a) I Q ∼
12 ln ( eV / − ε ) + ( Γ / ( eV / + ε ) + ( Γ / + ε − µ Γ / (cid:18) arctan eV / − ε Γ / + arctan eV / + ε Γ / (cid:19) . (20b)Hence, while both the charge current and entropy productionrate have normal on-sets associated with the energy ε , de-scribed by the arctan -component, the entropy production rateis also logarithmically resonant at ε . In addition, for the en-tropy production rate the on-set at ε is weighted by the posi-tion of ε relative to µ and, therefore, this contribution to theentropy production rate is expected to the small whenever ε lies in the vicinity of µ . The logarithmic contribution tends tobe small for large voltage biases, since it leads to an increas-ingly symmetric argument, and is accordingly only significantfor voltages such that either eV / − ε or eV / + ε is small.Extrapolating this discussion to our present case with themolecular dimer, we can verify these expected features underthe conditions of vanishing temperature di ff erence. This canbe be seen in Figure 5 (c), (d) (black – faint). At ε − µ = ff -set from µ , panel (b), yields a finite entropy pro-duction rate. In the latter case, the entropy production rate issmall in the magnetically active regime and grows large onlyat large voltages where the exchange interaction vanishes. Theentropy production rate tends to be e ffi ciently transmitted onlywhen the molecular dimer is in its fully conjugated state, thatis, when the exchange interaction is small. This is contrast tothe charge current where the less localized electronic density,in the molecular dimer, in the ferromagnetic regime yields asignificant di ff erence compared to the anti-ferromagnetic.The reason for this qualitative di ff erence between thecharge and entropy production rate, see Eq. (20), can befound in the properties of the two currents in the ferromag-netic regimes. In particular, the logarithmically resonant con-tribution to the entropy production rate, which is absent in thecharge current, peaks near the energies of the molecular bond-ing and anti-bonding orbitals, something which occurs withinthe ferromagnetic but not in the anti-ferromagnetic regime.Thereto, these resonant peaks are oppositely directed com-pared to the contributions varying like arctan x . As for theentropy production rate under finite temperature di ff erence be-tween the leads, see Figure 5 (c), (d) (red – bold) and Figure6 (b), (d), this behavior is verified. This feature can be tracedback to an increased degree of delocalization in the ferromag-netic regime due to the increased thermal broadening from thehotter reservoir.In the case where the localized level is o ff -set from µ , thereis an interesting observation to be made in the temperature andvoltage dependence of both the charge and entropy productionrate. At finite temperature di ff erence and voltage bias, thereis a strong asymmetry with respect to zero voltage which isclear signature that the magnetically active dimer potentiallycan be used for current rectification. Previously it was shown T (K) c u rr en t ( n A ) -4 40 T (K) -4 4001020 en t r op y p r od . ( M e V / s ) T (K) -4 40012 e xc hange ( m e V ) (a) (b) (c) V=3 mVV=2.5 mVV=2 mVV=1.5 mVV=1 mVV=0.5 mV
FIG. 7: (Color online) Thermal rectification. (a) Charge current, (b)entropy production rate, and (c) exchange interaction, as function ofthe temperature di ff erence for di ff erent voltage biases. Here, ε − µ = T c = in Ref. [51] that the charge current can be rectified by intro-ducing a level o ff -set between the localized levels in the twomolecules, something which possibly can be obtained by us-ing di ff erent types of molecules. The asymmetry induced bythe temperature di ff erence provides a di ff erent mechanism torectification which is also viable for the entropy productionrate. Indeed, the plots in Figure 6 (c), (d), illustrate that boththe charge and the entropy production rate is rectified at fi-nite temperature di ff erences upon changing the polarity of thevoltage bias. For this to be successful, the dimer has to beset-up with a finite level o ff -set from µ and a finite voltagebias. Moreover, the parameters of the molecular dimer haveto be tuned such that the o ff -set between the bonding and anti-bonding orbitals is larger than the thermal energy fed into thedimer from the hotter electrode. This enables a crossover fromthe anti-ferromagnetic regime to either the ferromagnetic orparamagnetic regime, under changes in the temperature dif-ference, which leads to strong variations in the currents. Inprinciple, the stronger anti-ferromagnetic exchange one canobtain for one polarity of the temperature di ff erence and theweaker the exchange can be in the opposite, regardless of sign,the larger the ratio between the currents for the two polarities.In Figure 7 we plot the (a) charge and (b) entropy produc-tion rate as function of the temperature di ff erence for di ff erentvoltage biases. We apply half the di ff erence to the tempera-ture in each lead such that T L = T − ∆ / T R = T + ∆ / T = V ≈ ff erential conductances,both charge conductance and di ff erential entropy productionrate and with respect to both voltage bias and temperature dif-ference. Hence, we calculate ∂ I x /∂ V and ∂ I x /∂ ∆ T , where x = c , Q , see Figs. 8 (a) – (d) and 9 (a) – (d). In Figs. 8and 9 we plot 1 / (1 + exp { σ F } ), where F is one of ∂ I x /∂ V , ∂ I x /∂ ∆ T , and S x , in order to lower the values of the high am-plitudes. First we notice that both currents have ranges withfairly rapid variations with the voltage bias and with the tem-perature di ff erence. While these properties can be traced backto the corresponding variations of the exchange interaction,we can predict a few consequences of these features on the ra- FIG. 8: (Color online) (a) – (d) Di ff erential charge conductance (a),(c), and entropy production rate (b), (d), with respect to the voltagebias (a), (b), and temperature di ff erence (c), (d). (e) S c and (f) S Q .Here, ε − µ =
0, while other parameters are as in Figure 1. All con-tours show (1 + exp { σ F } ) − , where F is one of ∂ I x /∂ V , ∂ I x /∂ ∆ T ,and S x , whereas σ is a scaling parameter. tios of the di ff erential conductance with respect to the temper-ature di ff erence and the di ff erential conductance with respectto the voltage bias, that is, S c = − ∂ I c / d ∆ T ∂ I c / dV , (21)where the notation S c is used since the ratio coincides with theSeebeck coe ffi cient in the limit V → ∆ T → S c is a non-equilibrium analogue of the See-beck coe ffi cient, however, we shall refrain from that nomen-clature for sake of not causing confusion with the concepts.Nontheless, as rapid variations in the charge current yielda large corresponding conductance and, oppositely, for slowvariations lead to a small conductance, we would expect that S c typically is large in the regions with weak variations of thecharge current. One is therefore led to think that S c mightbe large in the anti-ferromagnetic regime since the current isboth small and slowly varying with the voltage bias in thoseregimes, see Figs. 8 (a) and 9 (a). In addition, within the anti-ferromagnetic regimes, ∂ I c /∂ ∆ T varies rapidly, including pos-sible sign changes, near zero temperature di ff erence betweenthe leads. This conjecture is fairly well corroborated in theplots of the S c shown in Figs. 8 (e) and 9 (e). In particular, itcan be noticed in Figure 8 (e) that S c acquires large values inthe anti-ferromagnetic regime (voltages roughly in the ranges( − , −
2) and (2 ,
4) mV) for temperature di ff erences between0 K and 5 K. Except for these small regions of large S c , ittends to be small in the remainder of the magnetically activeregimes although not vanishing. In the case with a finite levelo ff -set, Figure 9 (e), however, S c tends to be large in the re-gions where the anti-ferromagnetic coupling is destroyed by FIG. 9: (Color online) (a) – (d) Di ff erential charge conductance (a),(c), and entropy production rate (b), (d), with respect to the voltagebias (a), (b), and temperature di ff erence (c), (d). (e), (f), Seebeckcoe ffi cients associated with the (e) charge, S c , and (f) heat, S Q , cur-rents. Here, ε − µ =
1, while other parameters are as in Figure 1. Allcontours show (1 + exp { σ F } ) − , where F is one of ∂ I x /∂ V , ∂ I x /∂ ∆ T ,and S x , whereas σ is a scaling parameter. the increased temperature di ff erence. There are clearly visi-ble domains at negative voltages which can be correlated withthe cross over between the anti-ferromagnetic and paramag-netic regimes. At positive voltages where the dimer remainsmagnetically active in a larger temperature range, S c is againsmall. We conclude that S c in both the ferromagnetic and anti-ferromagnetic regimes is small and typically becomes finite atthe crossovers to the paramagnetic regime. Hence, the spin-spin interaction provides a way to not only control the chargecurrent but also S c in the system.In analogy with the definition of S c associated with thecharge current, we can define a coe ffi cient S Q for the entropyproduction rate through S Q = − ∂ I Q / d ∆ T ∂ I Q / dV . (22)Although we cannot give this coe ffi cient an as simple interpre-tation as with the thermopower, we nevertheless find it usefulin the analysis of the influence of the magnetic configurationson the thermal properties. It can be seen in Figs. 8 (b) and9 (b) that the di ff erential entropy conductance, with respectto the voltage bias, has a non-trivial dependence on the volt-age bias and temperature bias. Moreover, the dependence onthe temperature di ff erence is intriguing. We can, however,notice regarding the generic features of S Q that its more orless vanishing in the magnetically active regimes except inthe crossover between the di ff erent regimes, where the en-tropy conductance varies slowly with the voltage bias but notnecessarily with the temperature di ff erence. The qualitativedi ff erence of the entropy conductance compared to the chargecurrent in the ferromagnetic regime, leads to that S Q provides0a complimentary piece of information about the system underinvestigation in addition to S c . IV. CONCLUSIONS
In summary we have theoretically studied the transportproperties of a dimer of paramagnetic molecules with local-ized spins embedded in a junction between metallic leads. Inparticular, we have addressed the charge and entropy conduc-tance flowing through the junction. It is demonstrated thatthe indirect exchange interaction between the localized spins,which previously has been shown to depend on the voltagebias and temperature di ff erence across the junction, acquiresa strongly asymmetric voltage bias dependence under finitetemperature di ff erence between the leads. This property wassubsequently is predicted to have implications on, for exam-ple, thermal rectification for both the charge and entropy con-ductance. Simultaneously, our calculations suggest that thetemperature of the source electrode has a stronger influenceon the properties of the indirect exchange than the drain. Itwas found, for instance, that while a voltage drop from thehotter to the colder reservoir tends to e ff ectively destroy thetunable properties of the indirect exchange, these propertiesare stable under the opposite voltage polarity.The transport properties of the dimer are intimately relatedto the indirect exchange, where the charge current is nearlyblockaded for anti-ferromagnetic exchange, whereas it is fi-nite for ferromagnetic exchange, and maximal whenever theexchange is negligible. These three regimes can be explainedby di ff erent characteristics of the electronic density. In theanti-ferromagnetic regimes, the spin projections of the den-sity is strongly localized to one molecule such that transportbetween the molecules is suppressed. When in the ferro-magnetic regime, the electron density is partially delocalizedwhich leads to an enhanced conductance, whereas the cur-rent can flow freely in the paramagnetic regime due to a com-pletely delocalized density. The entropy conductance followsthe same characteristics, as the charge current, in the anti-ferromagnetic and paramagnetic regimes. In the ferromag-netic regimes, however, the entropy conductance is stronglysuppressed by a contribution which is large, and nearly can-cels the regular entropy conductance contribution, near volt-age biases corresponding to the energy of the resonant states inthe molecular structure. By necessity, this resonant behavioroccurs in the ferromagnetic regimes which leads to a stronglysuppressed entropy conductance there as well.We, further, demonstrated that the non-equilibrium ther-mopower, in general is finite at the cross-over betweenregimes of di ff erent indirect exchange associated with smalldi ff erential conductance. Typically, this behavior can be sum-marized in that the thermopower is low within both the ferro-and anti-ferromagnetic regimes. Analogously, we introduceda thermal coe ffi cient as the ratio between the di ff erential en-tropy conductance with respect to the temperature di ff erenceand voltage bias. While some of its properties closely fol-low those ratios, additional features can be extracted, espe-cially at the cross over between the ferro- and antiferromag- netic regimes. In this sense, this ratio provides a complemen-tary sensitivity to the transport measurement which may showuseful in the analysis of the internal properties of the system.We remark that while our results presented in this arti-cle are quite qualitative, they are realistic from the follow-ing point of view. The results for the exchange interaction J presented in the previous section, are obtained using thebare single electron Green functions for the molecular lev-els. This means that the back-action e ff ect from the localspin moment is not included. In the calculations of the trans-port properties, on the other hand, the presence of the localspin moments are included, something which is crucial inorder to investigate possible signatures in the transport datathat originates from the spin configurations. As for the trans-port calculations we could have chosen to simply demonstratehow the charge and thermal transport depend on the natureof the exchange interaction, whether it is ferromagnetic (neg-ative), anti-ferromagnetic (positive), or paramagnetic (zero).Given the approximation in which we calculate the local elec-tronic Green function of the molecular dimer, we would obtainthe transport characteristics that are presented in the Results.However, instead of making assumptions about the values ofthe exchange interaction, we use the values as calculated bythe formulas provided in Exchange. In this way we incorpo-rate the voltage bias and thermal gradient dependence of theexchange also on the transport properties. While this approachcertainly has its limitations, we notice that a more thoroughstudy of the correspondence between the regimes with dif-ferent spin couplings and the associated transport propertiesrequires self-consistent calculations. Such calculations are,however, beyond the scope of the present investigation.Considering the limitations of the method, we yet believethat our findings are realistic and relevant to existing molecu-lar structures. The values of the local exchange interactionbetween localized spin and delocalized electrons can varybetween 0.5 – 20 meV [72, 73], which leaves a large win-dow of tuning freedom with respect to couplings to the leadsand HOMO / LUMO level o ff -set of the molecules. Moreover,since our predictions are stable with respect to di ff erences inthe local exchanges as well as the couplings to the leads, thisalso allows for flexibility in the design of potential experi-ments.The predictions discussed in this paper opens the possibil-ity to design nanoscale structures, in particular using mag-netic molecules, that have a strong sensitivity on the localspin states of the system, which can be measured through thecharge and thermal transport characteristics. In ways, this sug-gests an alternative utilization of the spin degrees of freedom,compared to the conventionally implemented spintronics, inwhich external magnetic fields are absent. The absence ofsuch fields, in turn, leads to that the spin-spin interactions areisotropic which implies a truly magnetically isotropic (param-agnetic) device. Experimental confirmation of our predictionsare feasible using state-of-the-art technology.1 V. ACKNOWLEDGEMENT
We thank S. Borlenghi Garoia, A. Bouhon, K. Björnson, H.Hammar, D. Kuzmanovksi and J. Schmidt for fruitful discus- sions. Financial support from Colciencias (Colombian Ad-ministrative Department of Science, Technology and Innova-tion) and Vetenskapsrådet is acknowledged. [1] Huang, K.
Statistical Mechanics , , 2nd ed. (Wiley).[2] Nitzan, A. Chemistry. Molecules Take the Heat. Science , , 759–760.[3] Pekola, J. P. Towards Quantum Thermodynamics in ElectronicCircuits. Nature Phys. , , 118–123.[4] Reddy, P.; Jang, S.-Y.; Segalman, R. A.; Majumdar, A. Ther-moelectricity in molecular junctions. Science , , 1568–1571.[5] Garrigues, A. R.; Wang, L.; del Barco, E.; Nijhuis, C. A.Electrostatic Control over Temperature-Dependent TunnellingAcross a Single-Molecule Junction. Nat. Comm. , ,11595.[6] Osorio, E. A.; Moth-Poulsen, K.; Van Der Zant, H. S. J.;Paaske, J.; Hedegård, P.; Flensberg, K.; Bendix, J.; Björnholm,T. Electrical Manipulation of Spin States in a Single Electro-statically Gated Transition-Metal Complex. Nano Lett. , , 105–110.[7] Parrondo, J. M. R.; Horowitz, J. M.; Sagawa, T. Thermodynam-ics of Information. Nat. Phys. , , 131–139.[8] Esposito, M.; Ochoa, M. A.; Galperin, M. Quantum Thermody-namics: A Nonequilibrium Green’s Function Approach. Phys.Rev. Lett. , , 080602.[9] Shastry, A.; Sta ff ord, C. A. Temperature and Voltage Measure-ment in Quantum Systems Far From Equilibrium. Phys. Rev. B:Condens. Matter , , 155433.[10] Dauxois, T.; Peyrard, M. Physics of Solitons (CambridgeUniversity Press, Cambridge).[11] Chiang, C.K.; Fincher, C.R.; Park, Y.W.; Heeger, A.J.; Shi-rakawa, H.; Louis, E. J.; Gau, S. C.; MacDiarmid, A. G. Electri-cal Conductivity in Doped Polyacetylene.
Phys. Rev. Lett. , , 1098–1101.[12] Fincher, C. J.; Peebles, D.; Heeger, A.; Druy, M.; Matsumura,Y.; MacDiarmid, A.; Shirakawa, H.; Ikeda, S. Anisotropic Op-tical Properties of Pure and Doped Polyacetylene. Sol. StateComm. , , 489–494.[13] Su, W. P.; Schrie ff er, J. R.; Hegger, A. J. Soliton Excitations inPolyacetylene. Phys. Rev. B , , 2099–2111.[14] Ochoa, M. A.; Bruch, A.; Nitzan, A. Energy distribution and lo-cal fluctuations in strongly coupled open quantum systems: Theextended resonant level model, Phys. Rev. B: Condens. MatterPhys. , 035420.[15] Rammer, J.; Smith, H. Quantum Field-Theoretical Methods inTransport Theory of Metals. Rev. Mod. Phys. , , 323–359.[16] Heeger, A. J.; Kivelson, S.; Schrie ff er, J. R.; Su, W. P. Solitonsin Conducting Polymers. Rev. Mod. Phys. , , 781–850.[17] Roth, S.; Bleier, H.; Pukacki, W. Charge Transport in Conduct-ing Polymers. Faraday Discussions of the Chem. Soc. , ,223–233.[18] Nakata, M.; Taga, M.; Kise, H. Synthesis of Electrical Conduc-tive Polypyrrole Films by Interphase Oxidative Polymerization– E ff ects of Polymerization Temperature and Oxidizing Agents. Polym. Journ. , , 437–441.[19] Meir, Y.; Wingreen, N. S. Landauer Formula for the CurrentThrough an Interacting Electron Region. Phys. Rev. Lett. , , 2512–2515.[20] Bao, Z.; Lovinger, A. J.; Dodabalapur, A. Organic Field-E ff ectTransistors with High Mobility Based on Copper Phthalocya-nine. Appl. Phys. Lett. , , 3066–3068.[21] Parthasarathy, G.; Burrows, P. E.; Khalfin, V.; Kozlov, V. G.;Forrest, S. R. A Metal-Free Cathode for Organic Semiconduc-tor Devices. Appl. Phys. Lett. , , 2138–2140.[22] Chen, G.; Shakouri, A. Heat Transfer in Nanostructures forSolid-State Energy Conversion. J. of Therm. Transf. , ,242.[23] Chen, G.; Chen, G. Nanoscale Heat Transfer and InformationTechnology. Appl. Phys. Lett. , , 1–3.[24] Haibo, M.; Schollwock, U. Dynamical Simulations of Po-laron Transport in Conjugated Polymers with the Inclusion ofElectron-Electron Interactions. J. Phys. Chem. A , ,1360–1367.[25] Haug, H. J.; Jauho, A.-P. Quantum Kinetics in Transport andOptics of Semiconductors , , 2nd ed. (Springer, Berlin).[26] Giazotto, F.; Heikkilä, T.T.; Luukanen, A.; Savin, A.M.;Pekola, J. P. Opportunities for Mesoscopics in Thermometryand Refrigeration: Physics and Applications. Rev. Mod. Phys. , , 217–274.[27] Dubi, Y.; Di Ventra, M. Colloquium: Heat Flow and Thermo-electricity in Atomic and Molecular Junctions. Rev. Mod. Phys. , , 131-155.[28] Liu, Y. S.; Chen, Y. C. Seebeck Coe ffi cient of ThermoelectricMolecular Junctions: First-Principles Calculations. Phys. Rev.B: Condens. Matter Phys. , , 193101.[29] Ludovico, M. F.; Lim, J. S.; Moskalets, M.; Arrachea, L.;Sánchez, D. Dynamical Energy Transfer in ac-Driven QuantumSystems. Phys. Rev. B , , 161306.[30] Esposito, M. Ochoa, M. A. and Galperin, M., Nature of Ther-mal in Strongly Coupled Open Quantum Systems. Phys. Rev.B: Condens. Matter Phys. , , , 235440.[31] Daré, A.M.; Lombardo, P. Time-Dependent ThermoelectricTransport for Nanoscale Thermal Machines. Phys. Rev. B , , 035303.[32] Arrachea, L.; Bode, N.; von Oppen, F. Vibrational Cooling andThermoelectric Response of Nanoelectromechanical Systems. Phys. Rev. B , , 125450.[33] Zhou, H.; Thingna, J.; Hänggi, P.; Wang, J.-S.; Li, B. BoostingThermoelectric E ffi ciency Using Time-Dependent Control. Sci.Rep. , , 14870.[34] Segal, D.; Agarwalla, B. K. Vibrational Heat Transport inMolecular Junctions. Ann. Rev. Phys. Chem. , , 185–209.[35] Ye, L.; Zheng, X.; Yan, Y.; Di Ventra, M. ThermodynamicMeaning of Local Temperature of Nonequilibrium Open Quan-tum Systems Phys. Rev. B , , 245105.[36] White, A. J.; Peskin, U.; Galperin, M. Coherence in Charge andEnergy Transfer in Molecular Junctions. Phys. Rev. B , ,205424.[37] Eich, F. G.; Di Ventra, M.; Vignale, G. Temperature-DrivenTransient Charge and Heat Currents in Nanoscale Conductors. Phys. Rev. B , , 134309. [38] Wang, R. Q.; Sheng, L.; Shen, R.; Wang, B.; Xing, D. Y. Ther-moelectric E ff ect in Single-Molecule-Magnet Junctions. Phys.Rev. Lett. , , 057202.[39] Ramos-Andrade, J.P.; Ávalos-Ovando, O.; Orellana, P.A.; Ul-loa, S. E. Thermoelectric Transport Through Majorana BoundStates and Violation of Wiedemann-Franz Law. Phys. Rev. B , , 155436.[40] Kim, H. S.; Gibbs, Z. M.; Tang, Y.; Wang, H.; Snyder, G. J.Characterization of Lorenz Number with Seebeck Coe ffi cientMeasurement. APL Mater. , , 041506.[41] Crépieux, A.; Šimkovic, F.; Cambon, B.; Michelini, F. En-hanced Thermopower Under a Time-Dependent Gate Voltage. Phys. Rev. B , , 153417.[42] Bergfield, J. P.; Sta ff ord, C. A. Thermoelectric Signatures ofCoherent Transport in Single-Molecule Heterojunctions. NanoLett. , , 3072–3076.[43] Koole, M.; Thijssen, J. M.; Valkenier, H.; Hummelen, J. C.; VanDer Zant, H. S. J. Electric-Field Control of Interfering Trans-port Pathways in a Single-Molecule Anthraquinone Transistor. Nano Lett. , , 5569–5573.[44] Valkenier, H.; Guédon, C.M.; Markussen, T.; Thygesen, K.S.;van der Molen, S. J.; Hummelen, J. C. Cross-Conjugation andQuantum Interference: A General Correlation? Phys. Chem.Chem. Phys. , , 653–662.[45] Fracasso, D.; Valkenier, H.; Hummelen, J. C.; Solomon, G. C.;Chiechi, R. C. Evidence for Quantum Interference in Sams ofArylethynylene Thiolates in Tunneling Junctions with EutecticGa-In (EGaIn) Top-Contacts. J. Amer. Chem. Soc. , ,9556–9563.[46] Aradhya, S. V.; Meisner, J. S.; Krikorian, M.; Ahn,S.; Parameswaran, R.; Steigerwald, M. L.; Nuckolls, C.;Venkataraman, L. Dissecting Contact Mechanics from Quan-tum Interference in Single-Molecule Junctions of StilbeneDerivatives. Nano Lett. , , 1643–1647.[47] Guédon, C.; Valkenier, H.; Markussen, T.; Thygesen, K.; Hum-melen, J.; van der Molen, S. Observation of Quantum Interfer-ence in Molecular Charge Transport. Nat. Nano. , , 305–309.[48] Markussen, T.; Thygesen, K. S. Temperature E ff ects on Quan-tum Interference in Molecular Junctions. Phys. Rev. B , ,085420.[49] Bessis, C.; Rocca, M. L. D.; Barraud, C.; Martin, P.; Lacroix,J. C.; Markussen, T. Probing Electron-Phonon Excitations inMolecular Junctions by Quantum Interference. Sci. Rep. , , 20899.[50] Darwish, N.; Díez-Pérez, I.; Da Silva, P.; Tao, N.; Gooding, J.J.; Paddon-Row, M. N. Observation of Electrochemically Con-trolled Quantum Interference in a Single Anthraquinone-BasedNorbornylogous Bridge Molecule. Ang. Chem., Int. Ed. , , 3203–3206.[51] Saygun, T.; Bylin, J.; Hammar, H.; Fransson, J. Voltage-Induced Switching Dynamics of a Coupled Spin Pair in aMolecular Junction. Nano Lett. , , 2824-2829.[52] Díazand, S.; Núñez, Á. S. Current-Induced Exchange Interac-tions and E ff ective Temperature in Localized Moment Systems. J. Phys: Condens. Matt. , , 116001.[53] Fransson, J.; Ren, J.; Zhu, J.-X. Electrical and Thermal Controlof Magnetic Exchange Interactions. Phys. Rev. Lett. , ,257201.[54] Fransson, J.; Zhu, J. X. Spin Dynamics in a Tunnel JunctionBetween Ferromagnets. New J. Phys. , , 013017.[55] Hammar, H.; Fransson, J. Time-Dependent Spin and TransportProperties of a Single Molecule Magnet in a Tunnel Junction. Phys. Rev. B. , , 054311. [56] Zimbovskaya, N. A. Transport Properties of Molecular Junc-tions , Vol. 254 (Springer, New York).[57] Moskalets, M.; Haack, G. Heat and Charge Transport Mea-surements to Access Single-Electron Quantum Characteristics. phys. stat. sol. , , 1600616.[58] Gaury, B.; Weston, J.; Santin, M.; Houzet, M.; Groth, C.;Waintal, X. Numerical Simulations of Time-Resolved QuantumElectronics. Phys. Rep. , , 1–37.[59] Zimbovskaya, N. A. Seebeck E ff ect in Molecular Junctions. J.Phys: Condens. Matt. , , 183002.[60] Simmonds, R. W. Thermal Physics: Quantum InterferenceHeats Up. Nature , , 358–359.[61] Heinrich, B. W.; Braun, L.; Pascual, J. I.; Franke, K. J. Protec-tion of Excited Spin States by a Superconducting Energy Gap. Nat. Phys. , , 765–768.[62] Heinrich, B. W.; Ahmadi, G.; Müller, V. L.; Braun, L.; Pas-cual, J. I.; Franke, K. J. Change of the Magnetic Coupling of aMetal-Organic Complex with the Substrate by a Stepwise Lig-and Reaction. Nano Lett. , , 4840–4843.[63] Heinrich, B. W.; Braun, L.; Pascual, J. I.; Franke, K. J. Tuningthe Magnetic Anisotropy of Single Molecules. Nano Lett. , , 4024–4028.[64] Brumboiu, I. E.; Haldar, S.; Lüder, J.; Eriksson, O.; Herper,H. C.; Brena, B.; Sanyal, B. Influence of Electron Correlationon the Electronic Structure and Magnetism of Transition-MetalPhthalocyanines. J. Chem. Theor. Comp. , , 1772–1785.[65] Urdampilleta, M.; Klyatskaya, S.; Cleuziou, J.-P.; Ruben, M.;Wernsdorfer, W. Supramolecular Spin Valves. Nat. Mater. , , 502–506.[66] Vincent, R.; Klyatskaya, S.; Ruben, M.; Wernsdorfer, W.; Bale-stro, F. Electronic Read-Out of a Single Nuclear Spin Using aMolecular Spin Transistor. Nature , , 357–360.[67] Liu, Z. F.; Wei, S.; Yoon, H.; Adak, O.; Ponce, I.; Jiang, Y.;Jang, W. D.; Campos, L. M.; Venkataraman, L.; Neaton, J.B. Control of Single-Molecule Junction Conductance of Por-phyrins via a Transition-Metal Center. Nano Lett. , ,5365–5370.[68] Fransson, J.; Galperin, M. Spin Seebeck Coe ffi cient of a Molec-ular Spin Pump. Phys. Chem. Chem. Phys. , , 14350-14357.[69] Misiorny, M.; Hell, M.; Wegewijs, M. R. Spintronic MagneticAnisotropy. Nat. Phys. , , 801–805.[70] Galperin, M.; Nitzan, A.; Ratner, M. A. Heat Conduction inMolecular Transport Junctions. Phys. Rev. B. , , 155312.[71] Jauho, A. P.; Wingreen, N. S.; Meir, Y. Time-dependent Trans-port in Interacting and Noninteracting Resonant-Tunneling Sys-tems. Phys. Rev. B. , , 5528–5544.[72] Coronado, E.; Day, P. Magnetic Molecular Conductors. Chem.Rev. , , 5419–5448.[73] Chen, X.; Fu, Y.-S.; Ji, S.-H.; Zhang, T.; Cheng, P.; Ma,X.-C.; Zou, X.-L.; Duan, W.-H.; Jia, J.-F.; Xue, Q.-K. Prob-ing Superexchange Interaction in Molecular Magnets by Spin-Flip Spectroscopy and Microscopy. Phys. Rev. Lett. , ,197208.[74] Chudnovskiy, A. L.; Swiebodzinski, J.; Kamenev, A. Spin-Torque Shot Noise in Magnetic Tunnel Junctions. Phys. Rev.Lett. , , 066601.[75] Ludwig, T.; Burmistrov, I. S.; Gefen, Y.; Shnirman, A. StrongNonequilibrium E ff ects in Spin-Torque Systems. Phys. Rev. B , , 075425.[76] Hammar, H.; Fransson, J. Transient Spin Dynamics in a Single-Molecule Magnet. Phys. Rev. B ,96