Thermoelectric Driven Ring Currents in Single Molecules and

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Thermoelectric Driven Ring Currents in Single Molecules and Graphene Nanoribbons Jens Broe Rix, and Per Hedegard J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09626 • Publication Date (Web): 14 Jan 2019 Downloaded from http://pubs.acs.org on January 21, 2019

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The Journal of Physical Chemistry

Thermoelectric Driven Ring Currents in Single Molecules and Graphene Nanoribbons Jens Broe Rix†,‡ and Per Hedeg˚ ard∗,† †Nano-Science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, Denmark ‡Sino-Danish Center for Education and Research (SDC), Eastern Yanqihu Campus, University of Chinese Academy of Sciences, Huaibeizhen, Huairou Qu, 101408 Beijing, China E-mail: [email protected] Phone: +45 3532 0435

Abstract In a Seebeck coefficient measurement with vanishing transport current, it is usually assumed that local currents cancel as well. Here we present the existence of local electric currents in single molecule junctions under such experimental conditions. In contrast to ring currents induced by a magnetic field, the thermoelectric driven ring currents can show up in a variety of different patterns depending on how the molecule is connected to the leads. We derive an analytic expression to calculate local currents in tight binding models and use it to qualitatively predict the current patterns and to show that in alternant molecules, thermoelectric driven ring currents are odd as a function of a gate voltage. Finally, we show that local currents exist in gated graphene nanoribbons as well.

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Introduction A major focus in the field of thermoelectric transport in nanoscopic junctions has been on properties such as the Seebeck coefficient and the thermoelectric figure of merit which are properties related to the transport through junctions. In this paper, we will focus on transport within nanojunctions rather than through them and we demonstrate the existence of internal electric currents, which have not yet been described in the literature. The Seebeck coefficient S is usually measured by applying both ∆T and ∆V so that the electric net current cancels. From this zero-net-current condition it is often assumed that local current densities within a sample vanish but this assumption is not necessarily true. A simple example is two macroscopic components connected in parallel as shown in Figure 1. If both a temperature and potential difference are present so that there is no through-current, a ring current Iring =

S1 −S2 ∆T R1 +R2

law and Fourier’s law, I =

will exist, which can be shown from the combined Ohm’s

1 ∆V R

+

S ∆T . R

This might seem surprising at first sight, but

actually the concept was observed at the very beginning of thermoelectricity. In the 1820s Thomas J. Seebeck formed a loop of two different materials, heated one of the junctions and he observed the deflection of a compass needle. No current was flowing through Seebeck’s setup and it can therefore be thought of as the system illustrated in Figure 1.

R2, S2 TH

I=0

Iring

TC

R1, S1

Figure 1: Schematic illustration of a parallel setup subject to temperature and potential differences. Even though no electric current flows through the setup, a ring current exist due to the temperature difference and the difference in Seebeck coefficients of the two paths.

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Inside inhomogeneous or anisotropic samples, similar current loops can exist under the same experimental condutions. 1–3 In the inhomogeneous case, internal electric currents exist if ∇S ×∇T 6= 0 anywhere within a sample which can be shown by taking the curl of the linear transport equation j/σ = −S∇T − ∇V . One has to keep in mind that the usual textbook statement ∇V = −S∇T is not necessarily valid for inhomogeneous and anisotropic setups. Ring currents in single molecules have been studied in different contexts. An example is magnetically induced currents which play an important role in nuclear magnetic resonance spectroscopy. 4,5 Another example is in single molecule junctions subject to electric potential differences. 6,7 In this paper we will describe the interesting effect of internal thermoelectric driven currents in ring structured molecular junctions. These currents are similar to the one shown in Figure 1 but they appear in a fully quantum mechanical situation.

Theory We will start by reviewing the quantum theory of currents in single molecule junctions. In the single particle picture, the electric current from right to left is given by the Landauer formula 2e I= h

Z

dE T (E) nLF (E) − nR F (E)

(1)

where nαF is the Fermi function for lead α with chemical potential µα and temperature Tα . Using standard non-equilibrium Green function (NEGF) theory, the transmission function can be written as T (E) = Tr[ΓL G†M ΓR GM ](E)

(2)

where the full Greens function on the molecule is

GM =

E − HM

3

1 . − ΣL − ΣR


(3)


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Here Σα is the self-energy that appears because of the connection to lead α while Γα = −2 Im Σα . We now want to examine the local electric currents and we therefore consider the local current operator ie Iîj = tij (|iihj| − |jihi|) h ¯

(4)

which describes the electric current between the local orbitals i and j and where the hat indicates that it is an operator. tij is the real and positive hopping matrix element, tij = −hi|H|ji. To get the actual local current, we need to take into account the contributions D E from all the scattering states. Adding up all these contributions Iij = Iîj we get (see supporting information) 2e Iij = h

Z

dE Tij (E) nLF (E) − nR (E) F

(5)

where the bond transmission is given by

Tij (E) =

h ¯ Tr[ΓL G†M Iîj GM ](E). e

(6)

With Equations (5) and (6) it is possible to calculate the local electric currents for any junction under the applied temperature and potential differences ∆T = TL − TR and ∆V = −(µL − µR )/e. In this paper we are interested in the local currents in a Seebeck-typeexperiment where the electric through-current cancels. For a temperature difference ∆T we therefore use the condition I = 0 in Equation (1) to obtain ∆V . Inserting these ∆T and ∆V into Equation (5), we can obtain our results. When a single molecule is coupled to the leads via a single local orbital to both the left and right, the above equations simplify. If the left lead couples to site 1 and the right lead couples to site N , it can be shown analytically that the bond transmission is

Tij (E) = γij (E)T (E)

4


(7)

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where the function in front of the transmission function only depends on the isolated molecule in the following way (see supporting material)

γij (E) = tij

G01i G0jN − G01j G0iN G01N

(8)

where G0ij = hi|(E − HM )−1 |ji. This expression for the bond transmission is interesting since all the information about the leads is contained in the transmission T . The above simplification is only true in the single channel case. When multiple transport channels are available, the local currents will be the sum of contributions from the different channels, but they cannot be written on the simple form presented above. Since most single molecule junctions use binding groups, it is reasonable to assume that the molecules couple to the leads via single sites and we therefore use this assumption in the following analysis.

23 pA

1.4 pA

6.6 pA

1.5 pA

1.4 pA

6.1 pA

1.5 pA

0.9 pA

6.6 pA

1.2 pA

0.9 pA

Figure 2: Electric bond currents for different single molecular junctions at µL = 0.8εLUMO . The arrows together with the colors indicates the directions of the ring currents, while the blue and red dots indicate the cold and hot contacts, respectively. The single molecule calculations in this paper are primarily done on simple hydrocarbons such as benzene, naphthalene, and anthracene which are so-called alternant molecules. Alternant molecules follow the Coulson-Rushbrooke pairing theorem which e.g. states that 5



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the molecular orbital energies are symmetrically spread around a zero-energy level and that there is a sign change of the molecular orbitals at every other site when comparing a symmetry pair as e.g. the HOMO and LUMO. 8 In the supporting information we use the theorem to show that γij (E) in eq. (8) is an even function in energy γij (E) = γij (−E) for alternant molecules. Under the zero-net-current condition, local currents are related to the slope of γij . To show this, we expand the Fermi functions in eq. (5) to linear order in the temperatures and potentials around the averages T =

TR +TL 2

and µ =

µR +µL . 2

For a smooth function

T (E)γij (E) on the scale of kB T we can do a Sommerfeld expansion as shown in the supporting information to get the expression dγij (E) 2eπ 2 ˜ (kB T )(kB ∆T )T (µ) Iij ≈ 3h dE E=µ

(9)

where ∆T = TL − TR and the tilde in I˜ indicates the zero-through-current condition. The above approximation is good at low temperatures away from interference points and resonances. To obtain strong internal currents, Equation (9) indicates that γij (E) has to be steep at the chemical potential and that the transmission through the molecule has to be large. For alternant molecules where γij is even in energy, the thermoelectric driven bond currents are odd I˜ij (−µ) = −I˜ij (µ). We can now study the relation between destructive interference in the transport current and cancellation of the thermoelectric driven ring currents. Destructive interference occurs at energies E0 where G01N (E0 ) = 0 and if there is no special symmetry (such as E = 0 in alternant molecules) γij has a pole. We therefore need to be careful with the approximation and we show in the supporting information that close to destructive interference, Equation (9) needs to be multiplied by the function [(µ − E0 )2 −

π2 (kB T )2 ]/[(µ 3

− E0 )2 +

π2 (kB T )2 ]. 3

Consequently, destructive interference with no special symmetry leads to two nearby cancellations of the thermoelectric driven ring currents.

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As a side note, Equations (5) and (7) can be used to study the connection between local currents and destructive interference as done by Solomon et al. 7 When only a voltage difference is applied, the local currents are IijV ∝ γij (µ)T (µ)∆V ∝ G01N ∆V . This means that when G01N passes through zero, there will be destructive interference and the local currents will reverse.

Results and Discussion We will now present the results of thermoelectric driven bond currents in ring structured hydrocarbons. To obtain the results the model parameters are chosen as shown in Table 1. We use tij = tα when both i and j belong to region α while the amplitude tc is the one describing the coupling between the leads and the molecule. Especially the coupling tc and the position of µ relative to the molecular orbitals are difficult to estimate and they vary from junction to junction. Our choice of parameters results in a conductance G ≈ 10−3 G0 for para-connected benzene at µ = 0.7εLUMO . Table 1: The parameters for the single molecule calculations. Parameter tM tL = tR tc εM = εL = εR TL TR

Value 2.5 eV 5 eV 0.6 eV 0 295 K 300 K

The results for different benzene, naphthalene, and anthracene junctions are shown in Figure 2. We see the interesting effect of local electric ring currents even though there is no electric current through the junctions. Additionally we observe that very different current patterns can be obtained depending on how the molecules are connected to the leads. The benzene results demonstrate that the paths of the ring have to differ for a current to exist. The absence of a ring current in para-connected benzene can be predicted by symmetry. In 7



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naphthalene, ring currents of either opposite or same directions can be generated depending on how the molecule is connected to the contacts. As a result of two adjacent ring currents with the same intensity and direction, the current pattern is a single ring current along the perimeter of the naphthalene molecule. In anthracene, we see that two separated ring currents of opposite directions can be generated and that ring currents of different intensities can exist. The thermoelectric driven local currents are small but they are very sensitive to the choice of parameters. As an example if we set tc = 0.5tM , the resulting ring current in meta-coupled benzene is 0.34 nA at µL = 0.8εLUMO . Thermoelectric driven ring currents can be observed by measuring the induced magnetic field just as Thomas J. Seebeck did 200 years ago. As the system gets smaller, this becomes more difficult. If we approximate a benzene unit as a ring with radius r = 1.4 ˚ A, a ring current of 1 pA would induce a magnetic field of B = 4.5 nT in the center of the ring according to the Biot-Savart law. The variety of current patterns in Figure 2 are distinct from ring currents induced by a magnetic field. Benzene, naphthalene, and anthracene have diamagnetic responses and therefore the currents have the same direction in all rings. 4 For the particular examples to the right in Figure 2 the thermoelectric driven currents have similar patterns as if they were magnetically induced, but the two effect are not related in general. In principle a magnetic field could be used in combination with the thermoelectric currents to turn off some of ring currents while increasing others. In this way one could obtain a larger variety of patterns, but this requires very strong magnetic fields for our choice of parameters. 9 Rotate molecule and reverse current

Compare

Figure 3: For the anthracene junction we draw an initial guess for the ring currents. The colors indicate that the currents can have different amplitudes and directions. After a rotation which brings site 1 into site N and a reversal of the currents, the currents have to be unchanged. The symmetry argument therefore predicts the result in Figure 2. The current patterns shown in Figure 2 can be predicted qualitatively from a symmetry 8


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argument which follows from eqs. (5) and (8). The prediction rule is illustrated graphically in Figure 3 and it is based on the following argument: Consider a molecule that is invariant † under a rotation OR HM OR = HM that interchanges site 1 and site N . The rotation will take

site i into site i0 , OR |ii = |i0 i. Using this symmetry we can rewrite the numerator in Equation † † (8), G01j G0iN = h1|G0 |ji hi|G0 |N i = h1|OR G0 OR |ji hi|OR G0 OR |N i = hN |G0 |j 0 i hi0 |G0 |1i =

G01i0 G0j 0 N . By doing this on both terms in Equation (8) we get γij = −γi0 j 0 and consequently Iij = −Ii0 j 0 . This fact can be used to graphically predict the current pattern in a molecular junction with a symmetry: Draw a current in each ring of the molecule, rotate one contact point into the other and reverse the currents. The current patterns before and after the operations have to be the same. This is illustrated for an antracene junction in Figure 3. Except from the substituted benzene, all the patterns in Figure 2 can be predicted from this argument. Notice that the result Iij = −Ii0 j 0 is true for any applied bias or temperature difference and can therefore also be used to qualitatively predict the through-current. One could naively think that the current patterns can be understood by looking at the HOMO and LUMO since these are closest to the chemical potentials of the contacts. We have examined this by decomposing the molecular part of the scattering states into the molecular orbitals but we found that not only the HOMO and LUMO contribute to the currents. The weights of the molecular orbitals were found to be distributed in an unsystematic manner. The thermoelectric driven ring currents for six different junction are shown in Figure 4 as a function of the chemical potential of the lead relative to the molecular orbital energies. For the three alternant molecules to the left we see that the ring currents are odd as a function of µ as predicted analytically. The naphthalene junction has destructive interference at ∼ ±1.27 eV and as predicted in Equation (S34) the ring currents cancel twice close to the interference point. The azulene junction in the buttom right of Figure 4 has no destructive interference between the HOMO and LUMO but one of the ring currents turns off at µ = 0. This can again be understood by using Equation (9). Gated graphene ribbons is another example of nanoscopic junctions in which we expect

9



internal electric currents to exist. Consider the system shown in Figure 5 where a zigzag graphene ribbon is gated at roughly one half of the ribbon width in a short region. The gating modifies the charge density and therefore the Seebeck coefficient 10 and classical field theory indicates that a local currents have to exist in a zero-net-current measurement since ∇ × σj = ∇T × ∇S 6= 0. We model the junction by assuming that the ungated regions to the left and right are semi-infinite ribbons. We use a tight binding Hamiltonian and again consider only nearest neighbor hopping since this is sufficient to show the effect. The gated region is modeled by modifying the onsite energies. The self-energies are calculated from an iterative procedure 11 and the currents then follow from Equation (5). We use the same hopping amplitude tM and temperatures as shown in Table 1. The chemical potential for the left lead is set to µL = 0.5 eV while the right lead is found so that no current flows through the junction. The onsite energies in the gated region are set to εG = 0.6 eV and in the ungated region ε = 0. The resulting bond currents are shown in Figure 5 where the large black arrows are averages of the six bond currents in each honeycomb unit. The bond currents are shown with small arrows on top of the bonds, the longest of which has the current intensity I˜max = 0.6 nA. We do not show currents smaller than the cutoff 0.02I˜max . Just as for the single molecule calculations the results verify that ring currents exist even though no net electric current is flowing through the junction. To get a rough estimate of the generated magnetic field in the graphene ribbon, we consider a current of 1 nA that flows in a ring with radius 5 ˚ A. This generates a magnetic field of 1 µT in the center of the ring. A stronger gate voltage would increase the resulting magnetic field.

Conclusions From a tight binding model, we have shown the existence of internal electric current patterns in both single molecule junctions and gated graphene nanoribbons in Seebeck measurements

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where currents are usually assumed to vanish. The ring currents are driven partly by a temperature difference and partly by a potential difference, which allow the ring currents to exist without the flow of an electric net current through the junction. We showed that different current patterns can be obtained depending on how single molecules are connected to the contacts and demonstrated a symmetry argument to predict the patterns in symmetric molecules. We demonstrated that the bond transmission can be put on the simple form γT , where γ only depends on the properties of the isolated molecule. Finally, we showed the exixstence of isolated ring currents in graphene nanoribbons.

Acknowledgement J.B.R. thanks Professor Xiaohui Qiu from the National Center for Nanoscience and Technology in Beijing for hospitality, while part of this work was performed. We thank the Sino-Danish Center for Education and Research for funding.

Supporting Information Available The following files are available free of charge. Derivation of expression for local currents, Coulson-Rushbrooke pairing theorem, properties of γ for alternant molecules, and Sommerfeld expansion.

References (1) Silk, T. W.; Schofield, A. J. Thermoelectric Effects in Anisotropic Systems: Measurement and Applications. arXiv e-prints 2008, arXiv:0808.3526. (2) Fu, D.; Levander, A. X.; Zhang, R.; Ager, J. W.; Wu, J. Electrothermally Driven Current Vortices in Inhomogeneous Bipolar Semiconductors. Phys. Rev. B 2011, 84, 045205. 11



(3) Apertet, Y.; Ouerdane, H.; Goupil, C.; Lecoeur, P. Thermoelectric Internal Current Loops inside Inhomogeneous Systems. Phys. Rev. B 2012, 85, 1–3. (4) Steiner, E.; Fowler, P. W. Ring Currents in Aromatic Hydrocarbons. Int. J. Quantum Chem. 1996, 60, 609–616. (5) Sundholm, D.; Fliegl, H.; Berger, R. J. Calculations of Magnetically Induced Current Densities: Theory and Applications. WIREs Comput. Mol. Sci. 2016, 6, 639–678. (6) Rai, D.; Hod, O.; Nitzan, A. Circular Currents in Molecular Wires. J. Phys. Chem. C 2010, 114, 20583–20594. (7) Solomon, G. C.; Herrmann, C.; Hansen, T.; Mujica, V.; Ratner, M. A. Exploring Local Currents in Molecular Junctions. Nat. Chem. 2010, 2, 223–228. (8) Coulson, C. A.; Rushbrooke, G. S. Note on the Method of Molecular Orbitals. Math. Proc. Cambridge Philos. Soc. 1940, 36, 193. (9) Rai, D.; Hod, O.; Nitzan, A. Magnetic Fields Effects on the Electronic Conduction Properties of Molecular Ring Structures. Phys. Rev. B 2012, 85, 155440. (10) Zuev, Y. M.; Chang, W.; Kim, P. Thermoelectric and Magnetothermoelectric Transport Measurements of Graphene. Phys. Rev. Lett. 2009, 102, 1–4. (11) Sancho, M. P. L.; Sancho, J. M. L.; Sancho, J. M. L.; Rubio, J. Highly Convergent Schemes for the Calculation of Bulk and Surface Green functions. J. Phys. F: Met. Phys. 1985, 15, 851–858.

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Figure 4: Ring currents plotted as a function of the chemical potential of the left lead relative to the molecular orbital energies. The chemical potential of the right lead is calculated so that there is no through current. The currents are calculated using Equation (5) with the parameters in Table 1. Solid lines represent clockwise currents while dashed lines represent counterclockwise currents. The HOMO and LUMO are shown as vertical lines.

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T L, μL

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VG

T R, μR

Figure 5: Local electric currents in graphene under a zero-net-current Seebeck measurement. The gate is modeled by the onsite-energies εg = 0.6 eV in the grey region and the chemical potential of the left lead is chosen as µL = 0.5 eV. The small arrows in the bonds are obtained from calculation and the longest of these has the current intensity 0.6 nA. The big black arrows show the average of the six currents in the particular benzene unit. The leads are modeled as semi-infinite graphene nanoribbons with the same width as the main region.

Iring Cold

S

S

I=0 Figure 6: TOC Graphic.

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Thermoelectric Driven Ring Currents in Single Molecules and

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