A Density Functional Theory Based Electron Transport Study of Coherent Tunneling Through Cyclic Molecules Containing Ru and Os as Redox Active Centers
AA Density Functional Theory Based Electron Transport Study of Coherent TunnelingThrough Cyclic Molecules Containing Ru and Os as Redox Active Centers
Xin Zhao and Robert Stadler ∗ Institute for Theoretical Physics, TU Wien - Vienna University of Technology,Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria (Dated: December 24, 2018)In our theoretical study in which we combine a nonequilibrium Green’s function (NEGF) approachwith density functional theory (DFT) we investigate branched compounds containing Ru or Os metalcomplexes in two branches, which due to their identical or different metal centers are symmetricor asymmetric. In these compounds the metal atoms are connected to pyridyl anchor groups viaacetylenic and phenyl spacer groups in a meta-connection. We find there is no destructive quantuminterference (DQI) feature in the transmission function near the Fermi level for the investigatedmolecules regardless of their symmetry, neither in their neutral states nor in their charged states.We map the structural characteristics of the range of molecules onto a simplified tight-bindingmodel in order to identify the main differences between the molecules in this study and previouslyinvestigated ferrocene compounds in order to clarify the structural sources for DQI, which we foundfor the latter but not for the former. We also find that local charging on one of the branches onlychanges the conductance by about one order of magnitude which we explain in terms of the spatialdistributions and charge-induced energy shifts of the relevant molecular orbitals for the branchedcompounds.
A. Introduction
Under the restrictions of low bias current flowingthrough small molecules absorbed to metal electrodes inultrahigh vacuum at very low temperature, the field ofsingle-molecule electronics has become accessible for anonequilibrium Green’s function (NEGF) approach com-bined with density functional theory (DFT) , which al-lows for an atomic interpretation of experimental resultsin a mechanical break-junction (MCBJ) or scanning tun-neling microscope (STM) setup . Destructive quan-tum interference (DQI) effects allow for the designof logical gates and memory cells in single moleculeelectronics as well as the implementation of thermoelec-tric devices , since DQI is supposed to significantlyreduce the conductance in conjugated π systems wheresuch effects were even observed at room temperature .Experimentally, the design and synthesis of branchedcompounds containing ferrocene moieties in each branchhas been proposed for the purpose of creating sin-gle molecule junctions, where the combination of quan-tum interference effects with redox gating for coherentelectron tunneling as well as the electrostatic correlationbetween spatially distinct redox centers for electron hop-ping can be explored. A detailed theoretical analysisof branched compounds containing ferrocene centers hasbeen published in our previous work . In order to assessthe generality of this analysis we apply the same modelsand methods to cyclic Me (P(CH ) ) (C H ) (C H ) bis(pyridyl-diacetylide) molecules (Fig. 1), where thetwo metal atoms Me are Ru or Os and we use the nota-tion of Ru/Os, Os/Os, Ru/Ru for complexes containingsymmetrical and asymmetrical branches.There are experimental and theoretical studies on co-herent tunneling and electron hopping through single-branched molecules containing Ru atoms as redox-active centers , where Ref. focuses on the comparisonbetween a coherent tunneling and a hopping mechanismin dependence on the molecular length for rutheniumbis(pyridylacetylide) wires, and the work in Ref. inves-tigated redox-switches with coherent tunneling for theelectron transport through the junction and the switch-ing induced by hopping.For the work on branched ferrocene compounds weused a simple but effective model, i.e. a combined atomicorbital (AO)/fragment orbital (FO)-tight-binding (TB)model where we keep only the p z AOs for each atomin both anchoring groups and one relevant bridge FO.For the molecules in Ref. we found that the through-space coupling between the two anchor groups is the de-cisive parameter causing DQI effects in the interestingenergy region, i.e. in the gap between the highest occu-pied molecular orbital (HOMO) and the lowest unoccu-pied molecular orbital (LUMO) and close to the Fermilevel E F . For the molecules which are the focus of thisstudy (Fig. 1) asymmetry can be introduced by ex-changing one of the Ru redox centers with Os, and dueto the longer organic bridges, the redox centers are moredecoupled from the pyridyl anchors for these moleculesthan for the ferrocene compounds we studied in . Inthis article, we want to address two issues: i) How do theTB models we used for the branched ferrocene moleculesin Ref. interpret coherent electron transport for thisnew type of molecules? ii) Will the decreased through-space couplings due to the longer molecular length havean impact on the occurrence of DQI?The paper is organized as follows: Section I gives thecomputational details for all NEGF-DFT calculations. Insection II we present transmission functions from thesecalculations for all junction geometries covered in thisstudy and discuss their characteristics features, wherecomparisons of single and double branched molecules are a r X i v : . [ phy s i c s . c h e m - ph ] D ec FIG. 1:
Chemical structure of a cyclicRu (P(CH ) ) (C H ) (C H ) bis(pyridyl-diacetylide)molecule. made. In section III we derive topological tight-bindingmodels from the DFT calculations based on the schemein our previous study where we found DQI for Ferrocenecontaining compounds and identify the structural sourcesfor the absence of DQI in all Ru, Os containing com-pounds in the current study. In section IV we com-pare the conductance from NEGF-DFT calculations forcharged systems with the corresponding neutral ones forassessing the usefulness of these double-branched systemsas molecular switches. We conclude with a summary insection V. I. COMPUTATIONAL DETAILS FOR THENEGF-DFT CALCULATIONS
For the computation of transmission probabilities T ( E ), we performed DFT calculations with a PBE XC-functional within a NEGF framework using a linearcombination of atomic orbitals (LCAO) as basis seton a double zeta level with polarization functions (DZP)using the GPAW code , where a grid spacing of 0.2˚A for the sampling of the potential in the Hamiltonian on a real space grid is used. In our transport calcula-tions, the scattering region is formed by the respectivemetal organic compounds and three and four layers forthe upper and lower fcc gold electrodes, respectively, ina (111) orientation and with 6 ×
10 gold atoms in theunit cell within the surface plane. The distance betweenthe Au ad-atom attached to the electrodes surface andthe N atom of the pyridyl anchor groups is 2.12 ˚A andfor the k points only the Γ point is used in the scatteringregion for evaluating T ( E ) due to the rather large cellsizes in our simulations. II. RESULTS AND DISCUSSION OF THENEGF-DFT CALCULATIONS
In Fig. 2 we illustrate the molecular junctions de-rived from the compound in Fig. 1, where we vary thetwo metal atoms acting as redox-centers to be Ru/Os,Os/Os and Ru/Ru. In the resulting transmission func-tions T ( E ) for all three combinations and for comparisonalso compounds with single-branched cases of Ru and Os,which we show in Fig. 3, the HOMO peaks are close tothe Fermi level for all investigated systems. Hence we ex-pect the conductance to be dominated by the MOs below E F .Our definition of DQI is that the transmission througha system with more than one MO around E F is lower thanthe sum of the individual contributions of these MOs to T ( E ) . The exact energetic position of the Fermienergy within the HOMO-LUMO gap, which is also af-fected by the underestimation of this gap in our DFTcalculations with a semi-local parametrization of the XCfunctional, will have a crucial impact on the quantita-tive value obtained for the conductance but qualitativelyDQI will always result in a significant conductance low-ering for the structures where it occurs regardless of thedetails of the Fermi level alignment .We note that for molecule Ru/Os (green curve, Fig.3) the peak splitting in the occupied region is distinctdue to the intrinsic asymmetry caused by the differentmetal atoms in the two branches. In order to clarifywhether there are DQI effects in the region of the HOMO-LUMO gap, we employ a simple TB model with a MObasis, where the eigenenergies of the molecular orbitalsand their individual coupling values to the electrodes canbe obtained by diagonalizing the subspace of the trans-port Hamiltonian with the basis functions centered on themolecule . We use Larsson’s formula to calculatean approximation for the transmission function T ( E ),where only the frontier orbitals, namely the HOMO andthe LUMO are included. As one can see from Fig. 4 thecontributions from only the frontier molecular orbitalsreproduce the characteristic features (blue curve) of theDFT result (red curve) for molecule Ru/Ru. From Fig.4 we also see that when the HOMO and LUMO bothare coupled to the electrodes ( (HOMO + LUMO) , bluecurve) we achieve identical result as from the individualFIG. 2: Junction setups containing the molecule in Fig. 1,where for the double-branched molecule (left panel) the twometal atoms are varied as Ru/Ru, Ru/Os and Os/Os andfor the single branched molecule (right panel) the singlemetal atom is either Ru or Os.
FIG. 3:
Transmission functions from NEGF-DFTcalculations for the double-branched molecules Ru/Ru (solidred line), Os/Os (solid black line), Ru/Os (green solid line)as well as for single-branched Ru (dashed red line) and Os(dashed black line) in the junction setup shown in Fig. 2 intheir neutral states, respectively. contributions (HOMO + LUMO , cyan curve) aroundE F , which means no DQI of electron transport throughthese two MOs is occurring for the system Ru/Ru. For allother systems, namely Ru/Os, Os/Os, single-branchedRu and Os, we obtain the same results (not shown here).We also compare the transmission functions for single(red and black dashed lines) and double-branched (redand black solid lines) molecules to illustrate the impactof the number of branches in Fig. 3. As we can seethe transmission functions for the Ru (dashed red line)and Ru/Ru (solid red line) molecules are rather similar,and the zero-bias conductances differ only by a factor FIG. 4: Transmission functions calculated from NEGF-DFT(red curve) and Larsson’s formula for the system Ru/Ru,where for the latter the blue curve denotes (HOMO +LUMO) and the cyan curve HOMO + LUMO . of about 1.5. The amount of peaks in the occupied re-gion for the double-branched molecule is higher than theone in the single-branched molecule as there are moremolecular states coupled to the electrodes for the former.Comparing Os (dashed black line) and Os/Os (solid blackline), we find the same qualitative differences but the con-ductance for the double-branched molecule is now onlyhigher by about a factor of 0.6 for the single-branchedmolecule.The next question is then what is the reason forthe absence of DQI in the HOMO-LUMO gap whenthese molecules share the design ideas with the ferrocenemolecules we studied before . In order to address thisquestion we apply the AO-FO model we previously usedfor the ferrocene systems in the next section and takethe single-branched Ru molecule as an example due toits representative transmission function, which is verysimilar to those of the double-branched Ru/Ru, Os/Os,Ru/Os and single-branched Os molecules. III. COMPARISON OF SIMPLIFIED 3 × Based on the scheme we developed in Ref. we usethe simplified 3 × p z AOs of the anchor groups containing pyridyland the attached acetylene moieties by diagonalizing thesubspace of the transport Hamiltonian corresponding toeach carbon or nitrogen atom on these groups and pickingFIG. 5:
Transmission functions for a single-branched Rumolecule, where the red curve is from a NEGF-DFTcalculation. The magenta curve is obtained from aNEGF-TB calculation with one FO anchor state on eachside and one bridge FO, where the anchor FO is obtainedfrom a subdiagonalization of the transport Hamiltoniancontaining the p z state of each carbon or nitrogen atom. Forthe electrodes a chain of single AOs with onsite energies of0.83 eV and coupling values of -5.67 eV within theelectrodes is used. the p z states, which can be identified by their onsite en-ergies and symmetry. For the bridge group two relevantbridge FOs (again obtained by a subdiagonalisation) inthe occupied region are considered where we define Ruplus Phosphine ligands and conjugated spacers includingthe acetylene and benzene groups as part of the bridge.Then we diagonalize the two anchor group subspaces nowdefined only by p z orbitals in order to get the relevantanchor FOs on both sides. We then minimize the Hamil-tonian to the most simple one, which only contains thethree most relevant states, i.e. one FO on each anchorgroup and one bridge FO. We show T ( E ) obtained fromNEGF-TB calculations for such a 3 FO model in Fig. 5,and find that if qualitatively reproduces the characteris-tic features found in NEGF-DFT calculations.The most distinct differences between the molecules inFig. 1 and those studied in Ref. are the molecularlength and the type of metal centers. In our previouswork we found that in order to observe a DQI featureclose to E F , the through-space coupling t D needs to beneither too small nor too big in size so that the DQIfeature will not be pushed outside the relevant energyregion around E F .For a direct comparison of the molecules in our presentstudy with those from our previous study we define the 3 × H mol = ε L t L t D t L ε B t R t D t R ε R TABLE I: Parameters entering the 3 × . Ru Os Fct L -0.026 -0.023 0.27t R D ε -1.5 -1.5 0.6 for the single-branched Ru, Os and ferrocene (Fc) sys-tems in Table I, where ε L,B,R are the respective onsiteenergies of the three FOs, t L,R the electronic couplingsbetween the two anchor FOs and the bridge FO and t D the direct coupling between the anchor FOs.We can see that the coupling values of t L , t R and t D connecting the three FOs for the three systems Ru, Osand Fc differ in: i) the size of the couplings t L/R betweenanchor and bridge, which are one magnitude smaller forRu and Os compared with the ones in Fc; ii) the through-spacing coupling t D , which are three magnitudes smallerfor the Ru and Os molecules, and iii) the onsite energy ε B of the bridge FO we used for the 3 × ε are larger in size for Ruand Os and the sign differs compared with the ferrocenemolecule.The differences in couplings can be interpreted by visu-alizing the FOs in Fig. 6, where the symmetry of the FOson the anchors in Fig. 6(a) for the Ru molecule is equiv-alent to what is found for the ferrocene molecule (Fig. 6b)), but t L/R is decreased markedly, because in betweenthe metal complex and the pyridyl group there is nowa benzene unit separating the two for the Ru molecule.The benzene groups are defined as part of the bridge inour subdiagonalization but the resulting bridge FOs closeto E F show no localization on them. In addition, the in-creased length also strongly reduces the direct couplingt D between the two anchor groups. The bridge state onthe Ru molecule is also localized on the adjacent triplebonds, while for Fc the state is confined to the ferrocenemoiety.Having established above that the parameters t L/R , t D and ∆ ε distinguish the ferrocene molecule Fc with a DQIfeature close to the LUMO from the single-branched sys-tems Ru and Os, we want to further explore the relativeimportance of these parameters, where we focus on thecomparison of Ru and Fc regarding the three parametersentering the Hamiltonian. If we mark the three parame-ters for ferrocene molecule Fc as F , F , F and for Ru (a)(b) FIG. 6:
Spatial distributions of the three FOs in thesimplified model for a) single-branched Ru with the anchorFO on each side at 1.3 eV and the bridge FO at -0.2 eV; b)single-branched Fc with the anchor FO on each side at 1.05eV and the bridge FO at 1.66 eV . TABLE II: The three parameters defining the 3 × Ru Fc∆ ε (parameter 1) -1.5 0.6t L/R (parameter 2) 0.025 0.25t D (parameter 3) 5.0E-05 -0.023t D /t L/R -2.0E-03 -0.092 as R , R , R , where the numbers refer to those given inTable II, there are six possible combinations of them forforming a 3 × L/R with 0.025 eV and 0.25 eV for Ru and Fc,respectively, and plot the transmission functions for thesix resulting Hamiltonians based on the different combi-nations of the three parameters in Fig. 7.As one can see modifying any one of the three parame-ters in the Hamiltonian of Fc leads to the disappearanceof the DQI feature in the interesting region, which indi-cates that all three parameters have a fundamental influ- FIG. 7:
Transmission functions obtained from NEGF-TBcalculations using the 3FO-model of the single-branched a)Fc molecule (black solid line) and b) Ru molecule (blacksolid line), where the symbols F and R represent Fc and Ru,respectively, and the indices 1-3 refer to the numbers inTable II. In panel a) we start from the parametrization ofFc in Table II (F F F ) and replace each of the parametersindividually by the one corresponding to Ru, and calculate T ( E ) for R F F (green line), F R F (blue line) andF F R (cyan line). In panel b) we permute the parametersin the opposite direction starting from the parametrisationof Ru in Table II (R R R ) and obtaning T ( E ) for F R R (green line), R F R (blue line) and R R F (cyan line). ence on the DQI absence for Ru in the relevant energyregion and its their interplay which is decisive for the ab-sence or occurrence of DQI features within the HOMO-LUMO gap. The through-space coupling t D i, however isstill special in the sense that taking the value from Fcin the Ru Hamiltonian (R R F in Fig. 7b), induces aDQI minimum slightly below the HOMO peak.In the following we keep two parameters fixed and varyone in a systematic way to further investigate the roleeach parameter plays. In Fig. 8 we illustrate the relationbetween each of these three parameters and the energyposition of the DQI-induced minimum E , which we ob-tain from the eigenenergies and amplitudes at the contactsites of the three MOs resulting from a diagonalizationof the parametrized 3 × using the procedure described in detail in Ref. .The such obtained single parameter dependencies canbe summarized as linear for E (∆ ε ) (Fig 8a), as quadraticfor E (t L/R ) (Fig 8b), and as multiplicative inverse forE (t D ) (Fig 8c). The black curve in Fig. 8a illustratesthat for the single-branched Ru system with t D and t L/R fixed to the values in Table II, E lies always above ∼
12 eV regardless of the variation of ∆ ε . For the Fcsystem (red curve in Fig. 8) E is around -1.1 eV, i.e.close to E F , for ∆ ε chosen as in the real system but ispushed below -3 eV when the value is replaced by theone corresponding to the Ru molecule.While E shows a significant dependence on t L/R inFig 8b for both systems with a maximum at t
L/R =0,they differ strongly in the sense that this maximum whichis defined by ε B corresponds to the LUMO peak for Fcand the HOMO peak for Ru, meaning that a variation oft L/R only allows to introduce a DQI minimum into theHOMO-LUMO gap for the former but not for the latter.A variation of the parameter t D on the other side allowsfor E to cross the HOMO-LUMO gap for both systemsas ca be seen from Fig 8c albeit for values about an orderof magnitude smaller for Ru when compared with Fc andfor different signs.We put our findings on the t D dependence of E to atest by performing NEGF-TB calculations for a range ofvalues of t D where ∆ ε and t L/R have been kept fixed tothe values of the Ru Hamiltonian in Table II. In Fig. 9awe show that the real value for Ru in Table II (0.00005eV) is too small to cause DQI anywhere near the gap,a value largeer by about one order of magnitude (0.0004eV) makes a DQI feature appear above the LUMO andif the value is too large (0.09 eV), the feature mergeswith the HOMO peak. An intermediate value (0.0002eV) places the DQI induced minimum optimally withinthe gap and close to the Fermi level, where its influenceon the conductance will be most pronounced. In Fig. 9bwe confirm the findings of Fig 8c, namely that the signof t D matters significantly for the location of E .For a physical interpretation of our results we note thatthe ratio t D /t L/R we obtain from the optimal t D valuein Fig. 9a (0.002 eV) is 0.08 meaning that its magnitudeis very close to that for the Fc system and about two or-ders of magnitude larger than the real value of t D for theRu molecule in Table II. It is intuitively plausible thatthis ratio plays such a significant role for the occurrenceof DQI, where since it is a wave phenomenon destruc-tive interference needs two different path ways which arehighly asymmetrical with respect to each other but notmore than one order of magnitude apart in their respec-tive couplings. In our model these two path ways are FIG. 8: E versus a) the energy difference ∆ ε between theanchor and bridge states; b) the coupling value t L/R and c)t D for Ru (black curve) and Fc (red curve), respectively. Inpanel a) the E values resulting from ∆ ε (Ru) and ∆ ε (Fc)at -1.5 and 0.6 eV are also marked as black and red dots,respectively. FIG. 9:
Transmission functions calculated with NEGF-TBfor the single-branched Ru system for a) t D = 0.00005 eV(black line), 0.0004 eV (red line), 0.002 eV (green) and 0.09eV (blue curve), and for b) t D = -0.00005 eV (black line),-0.0001 eV (red line), -0.008 eV (green line) and -0.2 eV(blue line), respectively. represented by the transport through the metal centervia t L/R and the transport from anchor to anchor medi-ated by t D as also illustrated in Fig. 6. This seems to be ageneral rule regardless of the detailed quantitative valuesfor ∆ ε , t L/R and t D , which will facilitate the chemicaldesign of molecules enabling DQI in their electron trans-port characteristics in the future. IV. EFFECT OF CHARGING
In order to ensure the charge neutrality in the unitcell of the system, which is necessary also for a junctionwith a charged molecule when applying periodic bound-ary conditions for electronic-structure calculations, thecountercharge to the complex has to be an explicit partof the cell, where we use Cl − as a counterion (Fig. 10). FIG. 10: Junction geometry for two neighbouring cells inthe periodic setup for the scattering region of doublebranched Ru/Os containing a chlorine atom for achievinglocal charging with a distance of 5.2 ˚A between Cl and thecloser-lying metal atom.
We used the generalized ∆ SCF method for calculatingthe charging effect in this section, where one additionalelectron on the chlorine p shell is subtracted from themolecule. In this way the molecule is charged and wekeep the neutrality of the unit cell in our calculations.This approach makes use of the flexibility of the gener-alized ∆ SCF method to define the spatial expansion ofan orbital which is forced to contain an electron as anarbitrary linear combination of Bloch states and inour calculations is localized on a single Cl atom only asis predefined at the beginning of each iteration step. Theself-consistency cycle then progresses as usual, but withthe electron density of this particular orbital as a con-tribution to the external potential. In this way we canfix the electron occupation for the Cl manually, whichsolves the self-interaction problem implicitly and makesthis method ideal for introducing localized charges intoa junction .The redox centers with localized d states on the metalatom in the investigated systems are stabilized by fourdonor ligands, which suggests a tetragonal ligand field .According to Ligand field theory, for an octahedral fieldwith Jahn Teller distortion, the orbitals d x − y and d z are in higher lying and the orbitals d xz , d yz and d xy inlower lying energy levels (Fig. 11a). We find thatthe peaks in the occupied region close to the Fermi levelhave contributions mostly from d orbitals with d xz or d yz symmetries (Fig. 11b), where the d xy lies further belowin energy (Fig. 11c). For the systems investigated herethe d -orbitals d xz and d yz are fully occupied for all sys-tems when in their neutral state, while for the chargedstate when Ru(II) is ixidized to Ru(III) (Fig. 11a), ei-ther d xz or d yz is singly occupied according to Hund’srule. This single occupation of a localized state leadsFIG. 11: a) Electron occupation of the Ru d orbitals n a low-spin configuration for neutral and charged molecules accordingto ligand field theory, b) spatial localization of the four relevant occupied orbitals and c) their corresponding energies withinthe junction and with respect to the Fermi level for three systems, namely Ru/Ru, Ru/Os and Os/Os in their neutral andcharged states, respectively. to the necessity of spin polarisation in our DFT calcu-lations because spin-up and spin-down orbitals are thennot equivalent in energy anymore.From the respective MO eigenergies within the junc-tion in Fig. 11c, which we obtained from a subdiagonal-ization of the transport Hamiltonian, it can be seen thatfor the asymmetric system Ru/Os the amount of the shiftinduced by charging is smaller than for the two symmet- ric systems Ru/Ru and Os/Os. For Ru/Os also differsfrom the other two molecules in the energetic sequenceof the orbitals, where there are no degeneracies for thiscase and no changes in sequence when moving from theneutral to the charged system. We also show the differ-ence for the energies of the MOs on the branch closerto Cl and on the other branch with respective d xz and d yz symmetries in Table III, which confirms the sameFIG. 12: Transmission functions for a) Ru/Ru, b) Ru/Osand c) Os/Os in their respective neutral (black solid line)and charged states (red solid line), where for the latter theaverage of spin-up and spin-down contributions has beenused. trends.Fig. 12 shows the transmission functions for eachbranched molecule in this study (Ru/Ru, Ru/Os, Os/Os)in their respective neutral and charged states, where weconducted spin-polarized NEGF-DFT calculations for allcharged systems. It can be seen that the peak splittingfor the two symmetrically built molecules Ru/Ru and Os/Os after charging is pronounced, where MOs on thebranch closer to the chlorine atom shifted more with re-spect to the Fermi level, and the MOs on the other branchalmost have not been affected by the chlorine charging ef-fect. For the asymmetrically built molecule Ru/Os thepeak splitting caused by charging is less distinct, becauseof the peak splitting already occuring in the neutral casedue to the in-built asymmetry of the molecule, wherecharging does not seem to increase this splitting muchfurther.In Table IV we list the conductance of the neutraland charged states for each system as well as the par-tial charge on each molecule in order to investigate thesources of the asymmetry in MO energies as induced bycharging and their effect on the coherent electron trans-port through the junction. While there is a marked dif-ference between the partial charges on the two branchesfor the charged versions of all three molecules, the con-ductance of the charged systems changes only slightlywhen compared with their neutral counter parts, sincethe energy shifts of the peaks in the occupied regionare relatively small and the conductance is dominatedby those rather narrow peaks. Therefore, we concludethat these molecules are not suitable for redox switches,since although one of the two branches can be selectivelyoxidized this does not result in a significant DQI inducedreduction of the conductance.
V. SUMMARY
In this study we investigated the potential use ofbranched molecules containing different metal centers intwo branches as molecular transistors where the switch-ing would be achieved by a redox process allowing to al-ternate between an ON and an OFF state, which woulddiffer in their conductance by the occurrence of DQI ef-fects in only one of these two redox states. We did,however, not find a DQI effect in the coherent electrontransport through the branched molecules in our study,neither in their neutral nor in their charged states.By comparing our results with previously studied fer-rocene compounds, we further developed a scheme for theanalysis of the structural conditions for the occurence orabsence of DQI in branched metal complexes with redoxactive groups in each branch. We found that the ratioof the through-space coupling t D and the couplings be-tween anchor and bridge states t L/R play a decisive rolein this context, which is signficant for chemical designpurposes. These parameters, however, are barely alteredby the oxidation of one of the two branches. As a conse-quence, the charging effect on the conductance of thesecyclic molecules is not pronounced where only a rathermoderate upward shift in energy of the narrow peaks inthe transmission curves corresponding to occupied MOson one branch is found.Our findings and the analysis scheme we developed arelikely to facilitate the design of DQI-based redox switches0TABLE III: Energy differences in eV between the occupied d orbitals close to E F with symmetries d xz and d yz localized on the two branches M and M , where M is closer to the Cl atom and contains Os for the mixed caseRu/Os. For M we mark the respective d orbitals as d xz or d yz and for M as d (cid:48) xz or d (cid:48) yz . We use the same notationas in Fig. 11 for all double branched molecules in their respective neutral and charged states. For all systems wedefine ∆ ε ( d xz ) = d (cid:48) xz − d xz and ∆ ε ( d yz ) = d (cid:48) yz − d yz . ∆ ε ( d xz ) ∆ ε ( d yz )Ru/Os Os/Os Ru/Ru Ru/Os Os/Os Ru/Runeutral 0.120 0.001 0.002 0.090 0.001 0.003charged/spin up 0.121 0.064 0.098 0.096 0.094 0.090charged/spin down 0.184 0.159 0.157 0.123 0.131 0.115 TABLE IV: Partial charges as obtained from a Baderanalysis for each of the two branches (metal complexplus all penyl and acetylene spacers but without thepyridyl anchors) of the double branched neutral andcharged molecules, where M and M denote the branchcontaining the metal center further away from andcloser to the chloride ion, respectively. All values for thecharges are given in fractions of electrons. Theconductance G for all molecules as defined by T ( E F ) isgiven in units of G . M M GRu/Ru (neutral) -0.038 -0.037 4.94 × − Ru/Ru (Cl) -0.208 -0.662 4.48 × − Ru/Os (neutral) -0.0345 -0.0378 1.26 × − Ru/Os (Cl) -0.173 -0.710 5.58 × − Os/Os (neutral) -0.036 -0.036 9.19 × − Os/Os (Cl) -0.230 -0.653 4.46 × − and to interpret experimental observations on such com-plex molecules in the future. ACKNOWLEDGMENTS
XZ and RS both have been supported by the Aus-trian Science Fund FWF (project number No. P27272).We are indebted to the Vienna Scientific Cluster VSC,whose computing facilities were used to perform all cal-culations presented in this paper (project No. 70671).We gratefully acknowledge helpful discussions with GeorgKastlunger. ∗ Email:[email protected] M. Ratner,
Nature Nanotech. , 378 (2013). E. L¨ortscher,
Nature Nanotech. , 381 (2013). Y. Meir and N. S. Wingreen,
Phys. Rev. Lett. , , 2512(1992). M. Brandbyge, J. L. Mozos, P. Ordejon, J. Taylor and K.Stokbro, Phys. Rev. B , 165401 (2002). Y. Xue, S. Datta and M. A. Ratner, Chem. Phys. , 151(2002). A. R. Rocha, V. M. Garcia-Suarez, S. W. Baily, C. J. Lam-bert, J. Ferrer and S. Sanvito, Nature Materials , 335(2005). K. S. Thygesen and K. W. Jacobsen, Chem. Phys. ,111 (2005). C. Joachim, J.K. Gimzewski, R.R. Schlittler and C. Chavy,
Phys. Rev. Lett. , 2102 (1995). M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin and J.M.Tour,
Science , 252 (1997). J. Reichert, R. Ochs, D. Beckman, H.B. Weber, M. Mayorand H.v. L¨ohneysen,
Phys. Rev. Lett. , 176804 (2002). R.H.M. Smit, Y. Noat, C. Untiedt, N.D. Lang, M.C. vanHemert, and J.M. van Ruitenbeek,
Nature (London) ,906 (2002). M. Mayor, H. B. Weber, J. Reichert, M. Elbing, C. vonH¨anisch, D. Beckmann and M. Fischer,
Angew. Chem. Int.Ed. , 5834 (2003). C. J. Lambert,
Chem. Soc. Rev. , 875 (2015). R. Stadler, S. Ami, M. Forshaw and C. Joachim,
Nanotech-nology , S115 (2004). R. Stadler, M. Forshaw and C. Joachim,
Nanotechnology , 138 (2003). R. Stadler and T. Markussen,
J. Chem. Phys. , 154109(2011). C. M. Finch, V. M. Garcia-Suarez and C. J. Lambert,
Phys. Rev. B , 033405 (2009). C. M. Guedon, H. Valkenier, T. Markussen, K. S. Thyge-sen, J. C. Hummelen and S. J. van der Molen,
Nat. Nan-otechnol. , 305 (2012). M. S. Inkpen, T. Albrecht and N. J. Long,
Organometallics , 6053 (2013). G. Kastlunger and R. Stadler,
Phys. Rev. B , 125410(2015). F. Schwarz, G.Kastlunger, F. Lissel, H. Riel, K. Venkate-san, H. Berke, R. Stadler and E. L¨ortscher,
Nat Nanotech-nol , , 170176 (2016). X. Zhao,G. Kastlunger, and R. Stadler,
Phys. Rev. B ,085421 (2017). J. P. Perdew, K. Burke and M. Ernzerhof,
Phys. Rev. Lett. , 3865 (1996). A.H. Larsen, M. Vanin, J.J. Mortensen, K.S. Thygesen andK.W. Jacobsen,
Phys. Rev. B , , 195112 (2009). J.J. Mortensen,L. B. Hansen and K.W. Jacobsen,
Phys.Rev. B , 035109 (2005). J. Enkovaara et al.
Phys.: Conf. Ser. , 253202 (2010). R. Stadler, K. S. Thygesen and K. W. Jacobsen,
Phys. Rev.B , 241401(R) (2005). R. Stadler,
Phys. Rev. B , 125401 (2009). X. Zhao, V. Geskin and R. Stadler,
J. Chem. Phys. ,092308 (2017). G. Kastlunger and R. Stadler,
Phys. Rev. B , , 035418 (2013). S. Larsson,
J. Am. Chem. Soc. , 4034 (1981). M. A. Ratner,
J. Phys. Chem. , 4877 (1990). G. Kastlunger and R. Stadler,
Phys. Rev. B , 115412(2014). P. Sautet and M.-L. Bocquet,
Phys. Rev. B , 4910(1996). J. Gavnholt, T. Olsen, M. Engelund and J. Schiøtz,
Phys.Rev. B , , 075441 (2008). T. Olsen, J. Gavnholt and J. Schiøtz,
Phys. Rev. B , ,035403 (2009). J. Bendix, T. Birk, T. Weyhermuller,
Dalton Trans. , 2737(2005). C. E. Sch¨affer, C. Anthon and J. Bendix.
CoordinationChemistry Reviews, Theory and Computing in Contempo-rary Coordination Chemistry , 575 (2009). F. Schwarz, G. Kastlunger, F. Lissel, C. Egler-Lucas, S.N.Semenov, K. Venkatesan, H. Berke, R. Stadler and E.L¨ortscher,
Nat. Nanotech. ,170 (2016). R. S. Mulliken,
J. Chem. Phys. ,23