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Quantum interference independence of the heteroatom position
Accepted Manuscript Research paper Quantum interference independence of the heteroatom position Zainelabideen Yousif Mijbil PII: DOI: Reference:
S0009-2614(18)31006-6 https://doi.org/10.1016/j.cplett.2018.12.012 CPLETT 36146
To appear in:
Chemical Physics Letters
Received Date: Revised Date: Accepted Date:
18 September 2018 29 November 2018 6 December 2018
Please cite this article as: Z.Y. Mijbil, Quantum interference independence of the heteroatom position, Chemical Physics Letters (2018), doi: https://doi.org/10.1016/j.cplett.2018.12.012
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Quantum interference independence of the heteroatom position Zainelabideen Yousif Mijbil Chemistry and Physiology Department, Veterinary Medicine College, Al-Qasim Green University, Al-Qasim Town, Babylon, Iraq Email: [email protected], [email protected]
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ABSTRACT
We introduce a novel calculation to the transmission coefficient of molecules with single impurity. The study employs tight binding Green’s function method and new derived analytical formula to investigate chromene-like (even) and doped azulene-like (odd) model systems. Under certain circumstances, these specific even and odd models are respectively shown single and double critical points, at which the transmission becomes independent of the heteroatom position. Moreover, a counterintuitive correlation between the transmission and the coupling strength has analytically and numerically been approved: the transmission decreases with strengthening lead-nodes couplings.
1. INTRODUCTION
Over the recent decades, quantum interference (QI) in molecular junctions has been extensively investigated and modulated both experimentally [1-7] and theoretically [8-14]. QI is simply defined as the interactions between quantum states of electrons passing through a molecule which bridges the gap between generally two metallic electrodes. The pristine appearance of these quantum interactions is clearly seen in the observed spectra of the coherent transmission as destructive (dip) or constructive (peak) patterns. The practical importance of such patterns lies in the potential ability to properly control them and, thus, typically tune the electrical conductance of the molecular devices. Consequently, understanding these QI interactions and their defining parameters, like molecular structure, opens a wide window for engineering advanced molecular units [15] such as quantum interference effect transistor (QuIET) [3, 16, 17] and interferometers[18]. Therefore, future molecular electronics is generally shaped by the factors that control QI. For instance, it is found that electrical conductance of oligoynes will be independent of length at a specific energy [19], or the positive correlation between the electrical conductance and the length of the pi-conjugated molecular wires under certain conditions[20]. However, the length is seriously precluded by molecular conformation that gives rise to uncommon declination of the electrical conductance as the length of oligothiophenes is increased [21]. Furthermore, Ismael and co-workers have investigated the dependence of the molecular QI states on connectivity locations to the external leads[22]. They have found that the connectivity would have no effect on Fano resonances, which is localized states attached to a continuum[23]. But it, on the other hand, would highly define Mach-Zender resonances[22], which are multiple-path interacted states[24]. Furthermore, the reliance of QI on the junction symmetry was successfully delineated by for example Taniguchi’s work [1] and Yang et al. [25]. Similar work done by Garner et al. [26] focuses on the new QI rules that appear in aromatic molecules with substituted atoms. In addition, the insensitivity of QI to the substituents at the anchoring sites has been theoretically and experimentally approved by Tsuji et al. [27]. Experimentally, using azulene molecule as a modal system, Xia et al. [28] have found that QI illustrates patterns based on even-odd connection configurations. The literature is replete with papers which emphasized the vital role of QI[7, 9, 10, 29-31] and it relation to heteroatoms in molecular junction[32].
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In the present work, we aim to prove that QI is insensitive to the position of the heteroatom in the structurer of the backbone molecule as long as a particular bias is applied for a certain molecular length. In other words, the transmission coefficient is independent of the position of the heteroatom. Therefore, the fabrication of the such particular nanostructures would be much more flexible and affordable. Based on that, we expect that chromene-like systems, so-called model (a), would demonstrate identical electrical properties when the position of the heteroatom is changed. The same behaviour is presumed to appear in the transmission spectra of anthracenelike unit and doped-azulene model, so-called model (b).
2. COMPUTATIONAL METHODS
Green’s function code OLIFE [33] has been employed to calculate the electronic characteristics of the above model systems. The code combines tight binding Hamiltonians (TBH) with Green’s function (GF) method to produce the electronic transmission coefficient and electronic conductance. The incorporation of TB with GF ensures well formulated, fastly obtained, and simply analysed electronic properties owing to the elementary TBH. Mainly, the TBH principally consist of two parameters: the onsite energy of the orbital (main diagonal elements) and the coupling energy between two successive orbitals (off diagonal elements). Furthermore, energy units of the TBH are moderately convenient since they can be scaled by one parameter, such as the coupling energy in the leads, to gain plain numbers with arbitrary units. Despite its simplicity, tight binding approach is considered as a very successful method with impressive results [29, 34-38]. Therefore, OLIFE can be considered as a friendly tool to calculate the conductance, σ, by applying Landauer formula [39], 2e2 f dE ( E ) ( ), (1) h E where e is the electron charge, h is Plank’s constant, E is the energy, τ(E) is the transmission coefficient, f (E) is the Fermi-Dirac distribution function = [1 exp ( E EF ) ]1 , EF is the Fermi energy, β = kBT, kB is Boltzman’s constant, and T is the temperature. In addition, the transmission coefficient can be calculated from ( E ) | t |2 , (2)
The transmission amplitude, t, can be calculated from Fisher-Lee relation[40] for one allowed channel, because we are allocating a single energy for each site,
G ,
1 t eik i v
(3)
Gδ,μ is the Green’s function, which defines the susceptibility of a site δ to the perturbation on site μ, ħ = h/2π, v is the “group velocity” in the input contact, and k is the wave number of the output contact. For more details about the methodology, see Ref. [24, 39]. We recommend reference [24], for it worthily represents a thorough and simple tutorial.
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Figure 1. A schematic representation of the possible model molecules when a heteroatom (oxygen) occupies different positions in the structure. The molecules are chromene (a), 3HIsochromene (b), chromone (c), and coumarin (d).
Figure 2: (a) is the tight binding model (a) of molecule (a) in Figure 1, where the grey circles represent perfectly periodic leads, the white circles depict the molecular “atoms”, and finally the colored circles are for clear illustration of the positions that the heteroatom may take. (b) shows the transmission of the model above. The black, blue, and orange curves respectively illustrate the transmission of the model when the impurity occupies the sites 6, 7, and 8. The energies used to produce the graphs are εL=εM= 0, εI=-1, γL=γM=-0.5, and α=0.
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3. RESULTS
Molecules take different forms by chemically altering the position of a single atom in their structure. Hence, their chemical properties may severely or tenuously be modified. For example, chromene (benzopyran or 2H-Chromene) molecule becomes 3H-Isochromene without significance change when its single oxygen atom is at site number five instead of site number four, as illustrated in Figure 1. In contrast, the irritant chromone molecules change into toxic coumarin molecules, when the duoble-bonded oxygen atom is at site number seven in lieu of site number five, as shown in the same figure. Consequently, the electrical conductance of these molecules would differ as a result of the changes in the position of the oxygen atom. Therefore, efforts should also be increased to obtain a pure sample of one type of molecular conformation in order to have a nanodevice with consistent electrical properties. Baring this in mind, a selected sample of one type of molecules with multiple forms, which yields insignificant statistical variations in their electrical conductance, is an effective choice for conventional molecular devices. For this reason, model (a) of chromene molecule has been investigated and depicted it in Figure 2a. On top of that, chromene molecule possesses considerable potentials in molecular electronics since it works as a rectifier or switch [41, 42]. Figure 2 also shows the electronic transmission of model (a) as a function of energy and the dependence of the transmission on the heteroatom position, namely sites six, seven, and eight. It is worth mentioning that we have adopted physics convention for labeling the sites in the above figures. Moreover, the circuit in Figure 2a is considered closed when sites two and nine are attached to input and output external leads, respectively. In addition, Figure 2b gives rise to a main result, which is the intersection of all curves in one point, at the center of the energy band. In this particular case, zooming in the intersection point reveals that the curves do not overlap exactly at the same point. Such result does not crucially affect the findings, which are supported by the experimental data of Coen et al. [43]. Coen and the co-workers have measured the optical properties of chromene while it is attached to a terthiophene molecule. They have found that the optical spectra of chromene are invariant when the terthiophene molecule occupies different positions [43]. The independence of the transmission from the hetero-orbital position is also similar to the theoretical results of Spark et al. [19] where they have found that the transmission is independent of the length of the molecule. Spark et al. have calculated the electrical conductance for multibranch system and have similarly found an intersection point, which they have designated as a critical point. Based on that, the current work may be reasonably regarded as a possible generalization to their excellent work, since it shows that no matter where the position of the heteroatom is, the electrical conductance will almost be the same. For the sake of generality, we have checked whether the intersection point persists to appear in other model systems. Therefore, we have chosen anthracene, tetracene, pentacene, and 1,2 benzopyrene. The results of the latter three model molecules were presented in the ESI. The anthracene molecule exceeds the chromene in size, but acquires similar even-symmetry structure as shown in Figure 3a. Likewise, we have assumed it incorporates a heteroatom, which has been laid in various positions from three to seven each time alone. At each location, we have calculated the transmission curve for the associated structure, as illustrated in Figure 3b. Figure 3 clearly displays similar trend, where all curves intersect with each other at a unique point at the center of the band. The last part of this section includes the calculation of the transmission coefficient of model (b). Figure 3b shows model b, which is unlike the previous case in a sense that it has two asymmetric
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rings. The first ring consists of five sites while the second ring is defined by seven sites. In other words, each side of model b is determined by odd number of sites. This model exhibits two different ‘transmission bands’ with different critical points for each. The first band is associated with even-occupied position of the heteroatom, while the second distinct band is uniquely associated with odd-occupied positions, as shown in Figure 3d. The figure also illustrates that the intersection point of the even band stems from constructive QI in contrast to the odd band critical point, which is attributed to the destructive interactions of QI.
Figure 3: Molecular structures of the model systems and the corresponding transmission coefficients. (a) is the anthracene model and (b) is model (b) of the doped-azulene. (c) and (d) illustrate the transmission coefficient of anthracene and azulene respectively as a function of the location of the impurity. The zero in the legend refers to intact structure of the molecules. The parameters are zero for all onsite energies except the heteroatom, which has -1 value, the coupling elements were -0.5 for all except the one between the lead and the molecule, which is -0.3. The leads, yellow triangles, were attached to (2) and (13) orbitals in the case of the anthracene model and to (2) and (10) orbitals in the case of model (b).
4. DISCUSSION
Our results reveal the ineffective role of the heteroatom position on τ(E) at the center of the energy band. A result which is obtained by only changing the position of the heteroatom along one branch of the molecule and keeping all energies fixed. Thus, to explain the intersection point, we first need to analyze the transmission dependence on the position of the impurity along the specified branch. The specified doped branch is structurally similar to Breit-Wigner (BW) segment in a sense that it includes an input lead, scatterer, and output lead, as shown in Figure 4. Thus, our initial guess to explain the intersection point is BW. For conciseness, the details of the
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derivation and related parameters of BW are put in the ESI. The final equation of the transmission coefficient via BW derivation takes the form, m2 (4) (E) 2 2 2 n sin k where m 2 L R sin k , n ( R2 L2 ), and n cos k (1 E ) . E is the energy, γ is the coupling in the leads, γL(R) is the coupling between the impurity and left (right) lead, and ε1 is the onsite energy of the heteroatom. The major result of Eq.(4) is the independence of τ(E) from the position of the impurity when E=ε1. In addition to this, the intersection point follows the onsite energy of the impurity. In other words, the intersection point refers to the orbital energy of the impurity. These two facts are well illustrated in Figure 5. The success of BW is also supported by the results of the calculated transmission coefficient of a mono-doped one-dimensional six-site chain, which simulates the number of sites in the doped branch of model (a). These results properly describe the emergence of the critical point as the position of the heteroatom is changed along the chain, see Figure 6c. Furthermore, the chain tight binding findings verify the predictions of Eq.(4) by showing that the transmission coefficient decreases firstly with coupling energy in the molecule (γ), Figure 6b, and secondly with the onsite energy of the heteroatom (ε1), Figure 6c. One should be aware that for each panel of Figure 6 we fixed all parameters except the corresponding variable, more details can be found in the caption of the figure. Hence, BW formula reveals that increasing either ε1 or γ would further localize the molecular potential and thus develop an impedance to the flowing electrons which results in lower transmission coefficient.
Figure 4. An illustration for the analogy between a segment of a molecule with a single defect (blue part) and lead-scatterer-lead structure of Breit-Wigner. The arrows refer to the flowing direction of the electrons.
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Figure 5. Transmission coefficient of Breit-Wigner formula (4) as a function of the coupling between the impurity and leads (γL and γR). Panel (a) is obtained by using the parameters ε0=0.0, ε1=0.5, γ=0.5. Panel (b) is similarly obtained via adjusting the parameters to have the values ε0=0.0, ε1=0.1, γ=-1.0.
Figure 6. Transmission of a six-site chain with single impurity. (a) The effect of changing the position of the impurity on the transmission coefficient. The central panel is a cartoon illustration of the chain where the black circle represents the heteroatom and the light blue circles stand for the other five sites. The left and right yellow shaded sides depict the electrodes. The conditions to have the results are: onsite energy of both leads and scattering region was set to zero, while the impurity’s onsite energy was equal to -1, the coupling in the leads and chain was -0.5, and finally the coupling between the leads and chain was 0.2. (b) Transmission versus the coupling in the chain (γ), which contains an impurity at site 4. To produce the results in (b) we have only changed the coupling in the leads to be -5. (c) Transmission dependance on onsite energy of the heteroatom (ε1) at site 4. The parameter of panels (c) and (a) are the same.
However, BW provides no clue about why there are two odd-even discernible intersection points in the odd-symmetry segment. Therefore, we have derived a new analytical formula, which is especially derived for this work but published elsewhere[44]. The analytical model appropriately describes the transmission coefficient of a chain with single impurity of onsite energy (ε0) attached to the left branch (1) and right branch (2) of the molecule by coupling matrix elements, γs and γs′ respectively. The left and right branches of the system are defined by the onsite energies ε1 and ε2 and their interorbital couplings γ1, γ2, respectively. These left and right branches are also connected to left and right nodal sites via coupling elements α1 and α2. Then, the nodal sites, with onsite energies ε and έ, are linked to their corresponding electrodes by the hopping elements αL and αR. Finally, the left lead (L) and right lead (R) are described by the onsite energies εL and εR in addition to the coupling elements γL, and γR, respectively. The whole system is shown in
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Figure 7 with the molecule labelled 1, 2, …, m-1, m, m+1, … N-1, N, where N is the total number of the atomic sites in the molecule and m is the position label of the impurity. The impurity may occupy the positions from m=2 until m=N-1. It is worth mentioning that the impurity may allocate at the starting and ending positions but with a slight precaution. The left (right) leads have been labelled with -1(1), -2(2), …,-∞(∞), and the nodal sites have been labelled with zero site on both sides. Dimensionless wave numbers kμ=cos-1[(εμ -E)/(2γμ)], and group velocities aμvμ/ħ emerge for any given energy E, where aμ is segment μ spacing factor, vμ=dE/dkμ=2γμsinkμ, and the index μ is either L, 1, 2,or R. Applying Landauer definition of the transmission coefficient, ( E ) (vR vL ) | t ( E ) |2 , into our analytical formula, the final transmission coefficient takes the form
( E ) vL (
L 2 GRL 2 R 2 ) | | ( ) vR L aL R
(5)
where vL and vR respectively describe the rate of the flowing current in the left and right leads, (αL/γL) and (αR/γR) illustrate the transparency of the nodal sites to the flowing waves, GRL is the Green’s function which defines the sensitivity of site R to the perturbation in site L, and the last parameter is aL E ( L2 / L ) eik . The general form of the last mathematical expression is similar to equation (5) in Ref.[19], but they are different in details. It should be mentioned that we have applied almost the same notations used in Sparks et al. work [19] after the generous permission of Prof. C. J. Lambert. The legitimate reason for performing this is to reasonably facilitate the comparison between our derivation and their derivation, since the two teams have dealt with almost similar systems. However, the current work is a step further since it deals with systems incorporating single impurity. L
Explaining the mono-critical point of the even system and di-critical points of the odd system requires a number of simplifications matching the numerical input parameters. These approximations include calculating τ(E) at energy equals to zero, E=0, setting the onsite energies of all sites to be zero except the impurity, ε0=-1.0, and setting the coupling elements between any two adjacent sites to be -0.5 except the coupling elements between the leads and the relevant nodal sites, αL= αR=-0.3. These assumptions mean that we have a symmetric system, namely the details of the left lead, left nodal site, left molecular branch, and left coupling to the impurity are identical to their right counterparts. As a result, kμ = π/2 and vμ =1, and equation (5) takes the form 4 L2
1 (E) ( 2 ) L d 2
2
(6)
Section C in the ESI explicitly illustrates the details of obtaining equation (6) and its parameters, namely d and Δ. Significantly, it can be clearly seen from Eq. (6) that the transmission is inversely correlated to the lead-nodal couplings, αL and αR, Such behavior is also proven both numerically and analytically in Figure 7. Furthermore, the product of d with Δ can be abbreviate to have four formulae depending on the chain total number of sites (N) and the position of the impurity (m).
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Figure 7. The transmission coefficient dependence on the strength of the coupling (αL and αR) between the leads and nodes (anchor group). Generally, panel (a) shows the structure of a sixsite chain with single impurity at site number (4). The chain is attached to two external leads via nodal sites. Panel (b) illustrates the transmission variation with the couplings mentioned above. In (b), continuous lines represent the analytical solutions, whereas the dotted curves represent numerical data. The coupling between the nodal sites and leads depicted in the dark yellow, blue, and black curves correspond to the values of -0.5, -0.7, and -0.9 respectively. Our ansatz onsite energies are (εL(R)=0) in the left(right) lead, (ε, έ=0) for the left(right) nodal sites, (ε1(2)=0) for the left(right) branches of the system, and (ε0=-1.0) for the impurity. In the same manner, the ansatz coupling elements are (γL(R)=-0.5) in the left (right) lead, (αL(R)) between the left (right) lead and the left (right) nodal site, (α1(2) )=-0.2) between the left (right) nodal site and the left (right) branch of the molecule, (γs(s′)=-0.5) between the impurity and the left (right) branch of the molecule, and finally (γ1(2)=-0.5) in the left (right) branch of the molecule.
N 1 4 1 N even , meven ; d (1) 2 i(2 2 ) 0 1 2 aL aL N 1 4 1 N even , modd ; d (1) 2 i(2 2 ) 0 1 2 aL aL
N odd , meven ; d 0 N odd , modd ; d (1)
N 1 2
512 1 4 15 2 0 i 2 1 2 0 aL aL
(7) (8) (9) (10)
According to equations (7) and (8), τ(E) would have the same value at the center of the band. This means that the impurity position is irrelevant to the transmission if the total number of units in the chain is even. However, the transmission explodes to infinity at the center when the total
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number of sites in the chain is odd and the impurity occupies even-numbered positions, according to equation (9). When both N and m are odd-numbered, equation(10) shows that τ(E) would have a definite value which is always less than the case of odd-N and even-m. The last two cases represent the justifications for the two critical points appeared in Figure 3. The oscillation in the conductance as the length, in other words the number of atoms, of the branch/chain changes is attributed to the QI due to phase shifting[45]. Figure 8 and the equations from (7) to (10) together support the phase justification since the left panel of Figure 8 illustrates a single phase wherever is the position of the impurity. In consistent with the calculations, the left panel of Figure 8 shows two patterns, we named the first arrangement as the OFF-pattern due to the low conductance that emerges from the destructive interference. In contrast, the second arrangement is the ON-pattern that shows a high electrical conductance because of the constructive interference. The ON-OFF patterns, or in general language the oscillating conductance with the length of the molecular wire, is a generic feature for changing the length of the molecule [19, 45-48]. However, this is the first time, as far as we know, the dependence of the electrical conductance on the position of the impurity has been considered. Finally, the chain with even number of sites, N, and either even or odd impurity-occupied position, m, shows single significant arrangement, as shown in the right panel of Figure 8. The even-N system always shows single position occupied by the impurity, another single position occupied by the ordinary site of the chain and unit of coupled sites. Thus, the phase of the electronic wave persists to maintain one pattern, which typically produces only one interaction. The ultimate point to declare is that the latter four equations show that τ(E) is not only independent of the position of the impurity but also independent of the length of the chain. Consequently, equation (5) inclusively explains the results of Sparks et al.[19], and proves its own general applicability.
Figure 8. The effect of the impurity position on the arrangement patterns of the one-dimensional chains, namely nine site (odd) chain on left, and eight site (even) chain on right. The hollow circles represent the ordinary sites while the black circle stands for the impurity atom. The left odd chain exhibits two patterns depending on the position of the impurity. The first pattern is the OFF, low conductance, pattern of the odd occupied sites. The second pattern is the ON, high conductance, pattern with even occupied sites. The right even chain shows only one patter regardless of the position of the occupied site of the impurity.
In order to have a comprehensive idea about applying the our results to real systems, a number of factors should seriously be considered. A prominent factor is charge transfer [49] between the
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main molecule and anchor groups or leads. Charge transfer can completely modify standard results by shifting the transmission toward one of the frontier orbitals, the high occupied molecular orbital (HOMO) or the lowest unoccupied molecule orbital (LUMO)[19]. For instance, thiol (SH) anchor group is well known to create a HOMO dominated junction. Whereas, amine and pyridine give rise to LUMO dominated transmission[50]. Therefore, anchor groups with low charge transfer are preferable to obtain similar behaviour to what we have already elucidated earlier.
5. CONCLUSIONS
We have calculated the dependence of the transmission coefficient on the position of the impurity in two model systems: model (a) with even-symmetry shape (chromene-like) and model (b) with odd-symmetry shape (doped zulene-like). For our particular set of values of onsite and coupling energies, the transmission curves result from moving the heteroatom along one branch of these specific model systems show that the even-symmetry path systems exhibit one critical point associated with ON state due to high electronic conductance. At the same time, the oddsymmetry path system possesses two critical points, resembling the ON-OFF behaviour. Such results would help in lowering the fabrication cost of molecular junctions and supporting the efforts to tune the molecular conductivity by chemically changing the position of the impurity/heteroatom in the molecules. Furthermore, for such specific type of polyaromatic hydrocarbons, we have numerically and analytically proven a new counterintuitive behaviour; the transmission declines as the lead-node coupling strength grows.
ACKNOWLEDGEMENT
The author gratefully thanks Prof. C. J. Lambert for the consent to implement his notations in our work. The sincere gratitude should also be paid to Dr. Ahmed Baqer Sharba for linguistic and scientific revision.
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Goringe C.M., Bowler D.R., and Hernández E., Tight-binding modelling of materials. Reports on Progress in Physics, 1997. 60(12): p. 1447. Ramakrishnan R., A Simple Hückel Molecular Orbital Plotter. Journal of Chemical Education, 2013. 90(1): p. 132-133. Datta S., Electronic Transport in Mesoscopic Systems. Cambridge Studies in Semiconductor Physics and Microelectronic Engineering (Book 3)1995: Cambridge University Press. 396. Fisher D.S. and Lee P.A., Relation between conductivity and transmission matrix. Physical Review B, 1981. 23(12): p. 6851-6854. Yassar A., Rebière-Galy N., Frigoli M., Moustrou C., Samat A., Guglielmetti R., and Jaafari A., Molecular switch devices realised by photochromic oligothiophenes. Synthetic Metals, 2001. 124(1): p. 23-27. Zhang X., Hou L., and Samorì P., Coupling carbon nanomaterials with photochromic molecules for the generation of optically responsive materials. Nature Communications, 2016. 7: p. 11118. Coen S., Moustrou C., Frigoli M., Julliard M., Samat A., and Guglielmetti R., Spectroscopic properties of thiophene linked [2H]-chromenes. Journal of Photochemistry and Photobiology A: Chemistry, 2001. 139(1): p. 1-4. Mijbil Z.Y., Analytical formula for calculating transmission coefficient of onedimensional molecules with single impurity. Solid state communications, 2018. Smit R.H.M., Untiedt C., Rubio-Bollinger G., Segers R.C., and van Ruitenbeek J.M., Observation of a Parity Oscillation in the Conductance of Atomic Wires. Physical Review Letters, 2003. 91(7): p. 076805. Miao F., Ohlberg D., Stewart D.R., Williams R.S., and Lau C.N., Quantum Conductance Oscillations in Metal/Molecule/Metal Switches at Room Temperature. Physical Review Letters, 2008. 101(1): p. 016802. Tada T., Nozaki D., Kondo M., Hamayama S., and Yoshizawa K., Oscillation of conductance in molecular junctions of carbon ladder compounds. Journal of the American Chemical Society, 2004. 126(43): p. 14182-14189. Halbritter A., Csonka S., Mihály G., Shklyarevskii O.I., Speller S., and van Kempen H., Quantum interference structures in the conductance plateaus of gold nanojunctions. Physical Review B, 2004. 69(12): p. 121411. Mijbil Z.Y., Aboud H.I., and Abdul-Lettif A.M., Ab-initio Calculation for Intermolecular Charge Transfer of V-III Zinc Blende Clusters Due to Compression. American Journal of Condensed Matter Physics, 2014. 4(2): p. 25-31. Sun L., Diaz-Fernandez Y.A., Gschneidtner T.A., Westerlund F., Lara-Avila S., and Moth-Poulsen K., Single-molecule electronics: from chemical design to functional devices. Chemical Society Reviews, 2014. 43(21): p. 7378-7411.
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Manuscript title: Quantum interference independence of the heteroatom position
Highlights: 1. The transmission coefficient of doped molecules is insensitive to the possible
change of the position of the impurity as well as the length of the molecule branch containing the impurity. 2. A counterintuitive relation between the transmission and the coupling strength is demonstrated: an increasing coupling can lead to a decreasing transmission. 3. Single (double) critical point(s) is found for even (odd) symmetric system.
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Graphical abstract
17
S0009-2614(18)31006-6 https://doi.org/10.1016/j.cplett.2018.12.012 CPLETT 36146
To appear in:
Chemical Physics Letters
Received Date: Revised Date: Accepted Date:
18 September 2018 29 November 2018 6 December 2018
Please cite this article as: Z.Y. Mijbil, Quantum interference independence of the heteroatom position, Chemical Physics Letters (2018), doi: https://doi.org/10.1016/j.cplett.2018.12.012
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Quantum interference independence of the heteroatom position Zainelabideen Yousif Mijbil Chemistry and Physiology Department, Veterinary Medicine College, Al-Qasim Green University, Al-Qasim Town, Babylon, Iraq Email: [email protected], [email protected]
1
ABSTRACT
We introduce a novel calculation to the transmission coefficient of molecules with single impurity. The study employs tight binding Green’s function method and new derived analytical formula to investigate chromene-like (even) and doped azulene-like (odd) model systems. Under certain circumstances, these specific even and odd models are respectively shown single and double critical points, at which the transmission becomes independent of the heteroatom position. Moreover, a counterintuitive correlation between the transmission and the coupling strength has analytically and numerically been approved: the transmission decreases with strengthening lead-nodes couplings.
1. INTRODUCTION
Over the recent decades, quantum interference (QI) in molecular junctions has been extensively investigated and modulated both experimentally [1-7] and theoretically [8-14]. QI is simply defined as the interactions between quantum states of electrons passing through a molecule which bridges the gap between generally two metallic electrodes. The pristine appearance of these quantum interactions is clearly seen in the observed spectra of the coherent transmission as destructive (dip) or constructive (peak) patterns. The practical importance of such patterns lies in the potential ability to properly control them and, thus, typically tune the electrical conductance of the molecular devices. Consequently, understanding these QI interactions and their defining parameters, like molecular structure, opens a wide window for engineering advanced molecular units [15] such as quantum interference effect transistor (QuIET) [3, 16, 17] and interferometers[18]. Therefore, future molecular electronics is generally shaped by the factors that control QI. For instance, it is found that electrical conductance of oligoynes will be independent of length at a specific energy [19], or the positive correlation between the electrical conductance and the length of the pi-conjugated molecular wires under certain conditions[20]. However, the length is seriously precluded by molecular conformation that gives rise to uncommon declination of the electrical conductance as the length of oligothiophenes is increased [21]. Furthermore, Ismael and co-workers have investigated the dependence of the molecular QI states on connectivity locations to the external leads[22]. They have found that the connectivity would have no effect on Fano resonances, which is localized states attached to a continuum[23]. But it, on the other hand, would highly define Mach-Zender resonances[22], which are multiple-path interacted states[24]. Furthermore, the reliance of QI on the junction symmetry was successfully delineated by for example Taniguchi’s work [1] and Yang et al. [25]. Similar work done by Garner et al. [26] focuses on the new QI rules that appear in aromatic molecules with substituted atoms. In addition, the insensitivity of QI to the substituents at the anchoring sites has been theoretically and experimentally approved by Tsuji et al. [27]. Experimentally, using azulene molecule as a modal system, Xia et al. [28] have found that QI illustrates patterns based on even-odd connection configurations. The literature is replete with papers which emphasized the vital role of QI[7, 9, 10, 29-31] and it relation to heteroatoms in molecular junction[32].
2
In the present work, we aim to prove that QI is insensitive to the position of the heteroatom in the structurer of the backbone molecule as long as a particular bias is applied for a certain molecular length. In other words, the transmission coefficient is independent of the position of the heteroatom. Therefore, the fabrication of the such particular nanostructures would be much more flexible and affordable. Based on that, we expect that chromene-like systems, so-called model (a), would demonstrate identical electrical properties when the position of the heteroatom is changed. The same behaviour is presumed to appear in the transmission spectra of anthracenelike unit and doped-azulene model, so-called model (b).
2. COMPUTATIONAL METHODS
Green’s function code OLIFE [33] has been employed to calculate the electronic characteristics of the above model systems. The code combines tight binding Hamiltonians (TBH) with Green’s function (GF) method to produce the electronic transmission coefficient and electronic conductance. The incorporation of TB with GF ensures well formulated, fastly obtained, and simply analysed electronic properties owing to the elementary TBH. Mainly, the TBH principally consist of two parameters: the onsite energy of the orbital (main diagonal elements) and the coupling energy between two successive orbitals (off diagonal elements). Furthermore, energy units of the TBH are moderately convenient since they can be scaled by one parameter, such as the coupling energy in the leads, to gain plain numbers with arbitrary units. Despite its simplicity, tight binding approach is considered as a very successful method with impressive results [29, 34-38]. Therefore, OLIFE can be considered as a friendly tool to calculate the conductance, σ, by applying Landauer formula [39], 2e2 f dE ( E ) ( ), (1) h E where e is the electron charge, h is Plank’s constant, E is the energy, τ(E) is the transmission coefficient, f (E) is the Fermi-Dirac distribution function = [1 exp ( E EF ) ]1 , EF is the Fermi energy, β = kBT, kB is Boltzman’s constant, and T is the temperature. In addition, the transmission coefficient can be calculated from ( E ) | t |2 , (2)
The transmission amplitude, t, can be calculated from Fisher-Lee relation[40] for one allowed channel, because we are allocating a single energy for each site,
G ,
1 t eik i v
(3)
Gδ,μ is the Green’s function, which defines the susceptibility of a site δ to the perturbation on site μ, ħ = h/2π, v is the “group velocity” in the input contact, and k is the wave number of the output contact. For more details about the methodology, see Ref. [24, 39]. We recommend reference [24], for it worthily represents a thorough and simple tutorial.
3
Figure 1. A schematic representation of the possible model molecules when a heteroatom (oxygen) occupies different positions in the structure. The molecules are chromene (a), 3HIsochromene (b), chromone (c), and coumarin (d).
Figure 2: (a) is the tight binding model (a) of molecule (a) in Figure 1, where the grey circles represent perfectly periodic leads, the white circles depict the molecular “atoms”, and finally the colored circles are for clear illustration of the positions that the heteroatom may take. (b) shows the transmission of the model above. The black, blue, and orange curves respectively illustrate the transmission of the model when the impurity occupies the sites 6, 7, and 8. The energies used to produce the graphs are εL=εM= 0, εI=-1, γL=γM=-0.5, and α=0.
4
3. RESULTS
Molecules take different forms by chemically altering the position of a single atom in their structure. Hence, their chemical properties may severely or tenuously be modified. For example, chromene (benzopyran or 2H-Chromene) molecule becomes 3H-Isochromene without significance change when its single oxygen atom is at site number five instead of site number four, as illustrated in Figure 1. In contrast, the irritant chromone molecules change into toxic coumarin molecules, when the duoble-bonded oxygen atom is at site number seven in lieu of site number five, as shown in the same figure. Consequently, the electrical conductance of these molecules would differ as a result of the changes in the position of the oxygen atom. Therefore, efforts should also be increased to obtain a pure sample of one type of molecular conformation in order to have a nanodevice with consistent electrical properties. Baring this in mind, a selected sample of one type of molecules with multiple forms, which yields insignificant statistical variations in their electrical conductance, is an effective choice for conventional molecular devices. For this reason, model (a) of chromene molecule has been investigated and depicted it in Figure 2a. On top of that, chromene molecule possesses considerable potentials in molecular electronics since it works as a rectifier or switch [41, 42]. Figure 2 also shows the electronic transmission of model (a) as a function of energy and the dependence of the transmission on the heteroatom position, namely sites six, seven, and eight. It is worth mentioning that we have adopted physics convention for labeling the sites in the above figures. Moreover, the circuit in Figure 2a is considered closed when sites two and nine are attached to input and output external leads, respectively. In addition, Figure 2b gives rise to a main result, which is the intersection of all curves in one point, at the center of the energy band. In this particular case, zooming in the intersection point reveals that the curves do not overlap exactly at the same point. Such result does not crucially affect the findings, which are supported by the experimental data of Coen et al. [43]. Coen and the co-workers have measured the optical properties of chromene while it is attached to a terthiophene molecule. They have found that the optical spectra of chromene are invariant when the terthiophene molecule occupies different positions [43]. The independence of the transmission from the hetero-orbital position is also similar to the theoretical results of Spark et al. [19] where they have found that the transmission is independent of the length of the molecule. Spark et al. have calculated the electrical conductance for multibranch system and have similarly found an intersection point, which they have designated as a critical point. Based on that, the current work may be reasonably regarded as a possible generalization to their excellent work, since it shows that no matter where the position of the heteroatom is, the electrical conductance will almost be the same. For the sake of generality, we have checked whether the intersection point persists to appear in other model systems. Therefore, we have chosen anthracene, tetracene, pentacene, and 1,2 benzopyrene. The results of the latter three model molecules were presented in the ESI. The anthracene molecule exceeds the chromene in size, but acquires similar even-symmetry structure as shown in Figure 3a. Likewise, we have assumed it incorporates a heteroatom, which has been laid in various positions from three to seven each time alone. At each location, we have calculated the transmission curve for the associated structure, as illustrated in Figure 3b. Figure 3 clearly displays similar trend, where all curves intersect with each other at a unique point at the center of the band. The last part of this section includes the calculation of the transmission coefficient of model (b). Figure 3b shows model b, which is unlike the previous case in a sense that it has two asymmetric
5
rings. The first ring consists of five sites while the second ring is defined by seven sites. In other words, each side of model b is determined by odd number of sites. This model exhibits two different ‘transmission bands’ with different critical points for each. The first band is associated with even-occupied position of the heteroatom, while the second distinct band is uniquely associated with odd-occupied positions, as shown in Figure 3d. The figure also illustrates that the intersection point of the even band stems from constructive QI in contrast to the odd band critical point, which is attributed to the destructive interactions of QI.
Figure 3: Molecular structures of the model systems and the corresponding transmission coefficients. (a) is the anthracene model and (b) is model (b) of the doped-azulene. (c) and (d) illustrate the transmission coefficient of anthracene and azulene respectively as a function of the location of the impurity. The zero in the legend refers to intact structure of the molecules. The parameters are zero for all onsite energies except the heteroatom, which has -1 value, the coupling elements were -0.5 for all except the one between the lead and the molecule, which is -0.3. The leads, yellow triangles, were attached to (2) and (13) orbitals in the case of the anthracene model and to (2) and (10) orbitals in the case of model (b).
4. DISCUSSION
Our results reveal the ineffective role of the heteroatom position on τ(E) at the center of the energy band. A result which is obtained by only changing the position of the heteroatom along one branch of the molecule and keeping all energies fixed. Thus, to explain the intersection point, we first need to analyze the transmission dependence on the position of the impurity along the specified branch. The specified doped branch is structurally similar to Breit-Wigner (BW) segment in a sense that it includes an input lead, scatterer, and output lead, as shown in Figure 4. Thus, our initial guess to explain the intersection point is BW. For conciseness, the details of the
6
derivation and related parameters of BW are put in the ESI. The final equation of the transmission coefficient via BW derivation takes the form, m2 (4) (E) 2 2 2 n sin k where m 2 L R sin k , n ( R2 L2 ), and n cos k (1 E ) . E is the energy, γ is the coupling in the leads, γL(R) is the coupling between the impurity and left (right) lead, and ε1 is the onsite energy of the heteroatom. The major result of Eq.(4) is the independence of τ(E) from the position of the impurity when E=ε1. In addition to this, the intersection point follows the onsite energy of the impurity. In other words, the intersection point refers to the orbital energy of the impurity. These two facts are well illustrated in Figure 5. The success of BW is also supported by the results of the calculated transmission coefficient of a mono-doped one-dimensional six-site chain, which simulates the number of sites in the doped branch of model (a). These results properly describe the emergence of the critical point as the position of the heteroatom is changed along the chain, see Figure 6c. Furthermore, the chain tight binding findings verify the predictions of Eq.(4) by showing that the transmission coefficient decreases firstly with coupling energy in the molecule (γ), Figure 6b, and secondly with the onsite energy of the heteroatom (ε1), Figure 6c. One should be aware that for each panel of Figure 6 we fixed all parameters except the corresponding variable, more details can be found in the caption of the figure. Hence, BW formula reveals that increasing either ε1 or γ would further localize the molecular potential and thus develop an impedance to the flowing electrons which results in lower transmission coefficient.
Figure 4. An illustration for the analogy between a segment of a molecule with a single defect (blue part) and lead-scatterer-lead structure of Breit-Wigner. The arrows refer to the flowing direction of the electrons.
7
Figure 5. Transmission coefficient of Breit-Wigner formula (4) as a function of the coupling between the impurity and leads (γL and γR). Panel (a) is obtained by using the parameters ε0=0.0, ε1=0.5, γ=0.5. Panel (b) is similarly obtained via adjusting the parameters to have the values ε0=0.0, ε1=0.1, γ=-1.0.
Figure 6. Transmission of a six-site chain with single impurity. (a) The effect of changing the position of the impurity on the transmission coefficient. The central panel is a cartoon illustration of the chain where the black circle represents the heteroatom and the light blue circles stand for the other five sites. The left and right yellow shaded sides depict the electrodes. The conditions to have the results are: onsite energy of both leads and scattering region was set to zero, while the impurity’s onsite energy was equal to -1, the coupling in the leads and chain was -0.5, and finally the coupling between the leads and chain was 0.2. (b) Transmission versus the coupling in the chain (γ), which contains an impurity at site 4. To produce the results in (b) we have only changed the coupling in the leads to be -5. (c) Transmission dependance on onsite energy of the heteroatom (ε1) at site 4. The parameter of panels (c) and (a) are the same.
However, BW provides no clue about why there are two odd-even discernible intersection points in the odd-symmetry segment. Therefore, we have derived a new analytical formula, which is especially derived for this work but published elsewhere[44]. The analytical model appropriately describes the transmission coefficient of a chain with single impurity of onsite energy (ε0) attached to the left branch (1) and right branch (2) of the molecule by coupling matrix elements, γs and γs′ respectively. The left and right branches of the system are defined by the onsite energies ε1 and ε2 and their interorbital couplings γ1, γ2, respectively. These left and right branches are also connected to left and right nodal sites via coupling elements α1 and α2. Then, the nodal sites, with onsite energies ε and έ, are linked to their corresponding electrodes by the hopping elements αL and αR. Finally, the left lead (L) and right lead (R) are described by the onsite energies εL and εR in addition to the coupling elements γL, and γR, respectively. The whole system is shown in
8
Figure 7 with the molecule labelled 1, 2, …, m-1, m, m+1, … N-1, N, where N is the total number of the atomic sites in the molecule and m is the position label of the impurity. The impurity may occupy the positions from m=2 until m=N-1. It is worth mentioning that the impurity may allocate at the starting and ending positions but with a slight precaution. The left (right) leads have been labelled with -1(1), -2(2), …,-∞(∞), and the nodal sites have been labelled with zero site on both sides. Dimensionless wave numbers kμ=cos-1[(εμ -E)/(2γμ)], and group velocities aμvμ/ħ emerge for any given energy E, where aμ is segment μ spacing factor, vμ=dE/dkμ=2γμsinkμ, and the index μ is either L, 1, 2,or R. Applying Landauer definition of the transmission coefficient, ( E ) (vR vL ) | t ( E ) |2 , into our analytical formula, the final transmission coefficient takes the form
( E ) vL (
L 2 GRL 2 R 2 ) | | ( ) vR L aL R
(5)
where vL and vR respectively describe the rate of the flowing current in the left and right leads, (αL/γL) and (αR/γR) illustrate the transparency of the nodal sites to the flowing waves, GRL is the Green’s function which defines the sensitivity of site R to the perturbation in site L, and the last parameter is aL E ( L2 / L ) eik . The general form of the last mathematical expression is similar to equation (5) in Ref.[19], but they are different in details. It should be mentioned that we have applied almost the same notations used in Sparks et al. work [19] after the generous permission of Prof. C. J. Lambert. The legitimate reason for performing this is to reasonably facilitate the comparison between our derivation and their derivation, since the two teams have dealt with almost similar systems. However, the current work is a step further since it deals with systems incorporating single impurity. L
Explaining the mono-critical point of the even system and di-critical points of the odd system requires a number of simplifications matching the numerical input parameters. These approximations include calculating τ(E) at energy equals to zero, E=0, setting the onsite energies of all sites to be zero except the impurity, ε0=-1.0, and setting the coupling elements between any two adjacent sites to be -0.5 except the coupling elements between the leads and the relevant nodal sites, αL= αR=-0.3. These assumptions mean that we have a symmetric system, namely the details of the left lead, left nodal site, left molecular branch, and left coupling to the impurity are identical to their right counterparts. As a result, kμ = π/2 and vμ =1, and equation (5) takes the form 4 L2
1 (E) ( 2 ) L d 2
2
(6)
Section C in the ESI explicitly illustrates the details of obtaining equation (6) and its parameters, namely d and Δ. Significantly, it can be clearly seen from Eq. (6) that the transmission is inversely correlated to the lead-nodal couplings, αL and αR, Such behavior is also proven both numerically and analytically in Figure 7. Furthermore, the product of d with Δ can be abbreviate to have four formulae depending on the chain total number of sites (N) and the position of the impurity (m).
9
Figure 7. The transmission coefficient dependence on the strength of the coupling (αL and αR) between the leads and nodes (anchor group). Generally, panel (a) shows the structure of a sixsite chain with single impurity at site number (4). The chain is attached to two external leads via nodal sites. Panel (b) illustrates the transmission variation with the couplings mentioned above. In (b), continuous lines represent the analytical solutions, whereas the dotted curves represent numerical data. The coupling between the nodal sites and leads depicted in the dark yellow, blue, and black curves correspond to the values of -0.5, -0.7, and -0.9 respectively. Our ansatz onsite energies are (εL(R)=0) in the left(right) lead, (ε, έ=0) for the left(right) nodal sites, (ε1(2)=0) for the left(right) branches of the system, and (ε0=-1.0) for the impurity. In the same manner, the ansatz coupling elements are (γL(R)=-0.5) in the left (right) lead, (αL(R)) between the left (right) lead and the left (right) nodal site, (α1(2) )=-0.2) between the left (right) nodal site and the left (right) branch of the molecule, (γs(s′)=-0.5) between the impurity and the left (right) branch of the molecule, and finally (γ1(2)=-0.5) in the left (right) branch of the molecule.
N 1 4 1 N even , meven ; d (1) 2 i(2 2 ) 0 1 2 aL aL N 1 4 1 N even , modd ; d (1) 2 i(2 2 ) 0 1 2 aL aL
N odd , meven ; d 0 N odd , modd ; d (1)
N 1 2
512 1 4 15 2 0 i 2 1 2 0 aL aL
(7) (8) (9) (10)
According to equations (7) and (8), τ(E) would have the same value at the center of the band. This means that the impurity position is irrelevant to the transmission if the total number of units in the chain is even. However, the transmission explodes to infinity at the center when the total
10
number of sites in the chain is odd and the impurity occupies even-numbered positions, according to equation (9). When both N and m are odd-numbered, equation(10) shows that τ(E) would have a definite value which is always less than the case of odd-N and even-m. The last two cases represent the justifications for the two critical points appeared in Figure 3. The oscillation in the conductance as the length, in other words the number of atoms, of the branch/chain changes is attributed to the QI due to phase shifting[45]. Figure 8 and the equations from (7) to (10) together support the phase justification since the left panel of Figure 8 illustrates a single phase wherever is the position of the impurity. In consistent with the calculations, the left panel of Figure 8 shows two patterns, we named the first arrangement as the OFF-pattern due to the low conductance that emerges from the destructive interference. In contrast, the second arrangement is the ON-pattern that shows a high electrical conductance because of the constructive interference. The ON-OFF patterns, or in general language the oscillating conductance with the length of the molecular wire, is a generic feature for changing the length of the molecule [19, 45-48]. However, this is the first time, as far as we know, the dependence of the electrical conductance on the position of the impurity has been considered. Finally, the chain with even number of sites, N, and either even or odd impurity-occupied position, m, shows single significant arrangement, as shown in the right panel of Figure 8. The even-N system always shows single position occupied by the impurity, another single position occupied by the ordinary site of the chain and unit of coupled sites. Thus, the phase of the electronic wave persists to maintain one pattern, which typically produces only one interaction. The ultimate point to declare is that the latter four equations show that τ(E) is not only independent of the position of the impurity but also independent of the length of the chain. Consequently, equation (5) inclusively explains the results of Sparks et al.[19], and proves its own general applicability.
Figure 8. The effect of the impurity position on the arrangement patterns of the one-dimensional chains, namely nine site (odd) chain on left, and eight site (even) chain on right. The hollow circles represent the ordinary sites while the black circle stands for the impurity atom. The left odd chain exhibits two patterns depending on the position of the impurity. The first pattern is the OFF, low conductance, pattern of the odd occupied sites. The second pattern is the ON, high conductance, pattern with even occupied sites. The right even chain shows only one patter regardless of the position of the occupied site of the impurity.
In order to have a comprehensive idea about applying the our results to real systems, a number of factors should seriously be considered. A prominent factor is charge transfer [49] between the
11
main molecule and anchor groups or leads. Charge transfer can completely modify standard results by shifting the transmission toward one of the frontier orbitals, the high occupied molecular orbital (HOMO) or the lowest unoccupied molecule orbital (LUMO)[19]. For instance, thiol (SH) anchor group is well known to create a HOMO dominated junction. Whereas, amine and pyridine give rise to LUMO dominated transmission[50]. Therefore, anchor groups with low charge transfer are preferable to obtain similar behaviour to what we have already elucidated earlier.
5. CONCLUSIONS
We have calculated the dependence of the transmission coefficient on the position of the impurity in two model systems: model (a) with even-symmetry shape (chromene-like) and model (b) with odd-symmetry shape (doped zulene-like). For our particular set of values of onsite and coupling energies, the transmission curves result from moving the heteroatom along one branch of these specific model systems show that the even-symmetry path systems exhibit one critical point associated with ON state due to high electronic conductance. At the same time, the oddsymmetry path system possesses two critical points, resembling the ON-OFF behaviour. Such results would help in lowering the fabrication cost of molecular junctions and supporting the efforts to tune the molecular conductivity by chemically changing the position of the impurity/heteroatom in the molecules. Furthermore, for such specific type of polyaromatic hydrocarbons, we have numerically and analytically proven a new counterintuitive behaviour; the transmission declines as the lead-node coupling strength grows.
ACKNOWLEDGEMENT
The author gratefully thanks Prof. C. J. Lambert for the consent to implement his notations in our work. The sincere gratitude should also be paid to Dr. Ahmed Baqer Sharba for linguistic and scientific revision.
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Manuscript title: Quantum interference independence of the heteroatom position
Highlights: 1. The transmission coefficient of doped molecules is insensitive to the possible
change of the position of the impurity as well as the length of the molecule branch containing the impurity. 2. A counterintuitive relation between the transmission and the coupling strength is demonstrated: an increasing coupling can lead to a decreasing transmission. 3. Single (double) critical point(s) is found for even (odd) symmetric system.
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Graphical abstract
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