First-Principles Study of Multiterminal Quantum Interference Controlled

Article pubs.acs.org/JPCC

First-Principles Study of Multiterminal Quantum Interference Controlled Molecular Devices Yukihiro Okuno*,† and Taisuke Ozaki‡ †

Research and Development Management Headquarters, FUJIFILM Corporation, Minamiashigara, Kanagawa 250-0193, Japan Research Center for Simulation Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa 923-1292, Japan

‡

S Supporting Information *

ABSTRACT: Multiterminal molecular devices controlled with the quantum interference effect (QIE) are studied by using a nonequiblium Green’s function (NEGF) method with density functional theory (DFT). An antiresonance quantum interference connection of the source and drain leads, which gives almost zero transmission in the absence of the gate lead, is considered for benzenedithiol (BDT) and 18-annulene molecules. We investigate how the gate lead approaching the molecule affects the quantum interference effect. The results show that the gate lead strongly influences the transmission spectra, and the behaviors of the transmission spectra in the QIE-controlled molecular devices depend on the detailed molecular orbital structure and the coupling strength of the leads and molecules. In particular, the σtype orbital between the gate lead and the molecule has a strong effect on the transmission spectra at the Fermi level, and it masks the QIE control of the gate lead in some cases. will not flow through the benzene ring when using a metaconnected position. In such a linkage position which influences the transmission through the molecule, the quantum interference effect (QIE) of the electron was introduced to interpret the specific electron transport behavior in the conjugated molecules.8−12 The current is influenced by the destructive (antiresonance) or constructive (resonance) quantum interference between the two paths around symmetric molecules. Typically, QIE appears in the aromatic systems, and the large variation in transmission is expected by means of a structure change from para- to meta-connection to the leads.8,13−15 Furthermore, the antiresonance QIE also appears in the case that the electron paths are not spatially separated, as pointed out by Solomon et al.,16,17 and it may not necessarily be an aromatic ring molecule as a system exhibiting QIE. Such an interference effect can be controlled by a third terminal that provides the elastic scattering or dephasing, and one can switch the current through the molecule by using the third terminal.8,9 QIE originates from the structure of a molecule and may not be affected by the details of the molecule−lead contact. For this reason, QIE-controlled molecular devices are considered to have a strong tolerance to thermal effects and the subtleties of the structure in a molecule−lead contact. This is an attractive property of a QIE-controlled molecular device because the use of single molecules as a functional device is generally very

1. INTRODUCTION Recent developments in nanofabrication techniques have enabled us to measure currents through molecular wires1 using a single isolated molecule attached to measuring leads.2−6 Among these developments, devices using single molecules as nanoscale conductors have attracted great attention. Many theoretical and experimental studies have been reported concerning the molecular devices1 because of expected exotic properties particular to small molecules, which are considered to be an ultimate goal in the miniaturization of electronic devices. Various π-conjugated molecular junctions, such as benzenedithiol (BDT) and polycyclic aromatic hydrocarbons, are investigated for a variety of potential applications in molecular devices such as switches, rectifiers, and memories.3,7 In such single molecular devices, the conductance is not simply proportional to the applied voltage; rather, it generally shows the step-like features corresponding to resonances with molecular eigenstates. Such features of the transport properties are the key factor for the use of single molecules as a functional device. Therefore, an understanding of the electron transport properties through a single molecule is indispensable to develop a single molecular device. Among such single molecular devices, a novel mechanism to control electron transport through a single molecule has been proposed.8,9 The conductivity of the molecular ring strongly depends on the linkage position between the molecule and lead terminals, and the current through the molecule is controlled by the linkage position. For example, a current can flow through a benzene ring with a para-connected position to the leads, but the current © 2012 American Chemical Society

Received: September 24, 2012 Revised: December 13, 2012 Published: December 14, 2012 100

dx.doi.org/10.1021/jp309455n | J. Phys. Chem. C 2013, 117, 100−109

The Journal of Physical Chemistry C

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current through the molecule by the third lead. This is an important point in the realization of a QIE-controlled molecular device because the controllable variable of the effect of decoherence due to the third lead is the distance between the molecule and the third lead. The distance is assumed to be controlled by piezo-voltage, but precise control of the distance will not be easy. The realistic treatment of such a lead− molecule contact in electric calculation is difficult for a semiempirical treatment, and an ab initio calculation is indispensable. We will show that a σ-type orbital between the third lead and molecule masks the QIE control in some cases of lead−molecule systems, which is the missing effect in a semiempirical treatment.8 Furthermore, we investigate the effect of the voltage on the third lead to its transmission properties. Using the MPSH analysis, we can understand the physical origin of the behavior in the transmission due to the applied voltage. We will show the peculiar behavior of the σtype orbital around the third lead due to the gate voltage, which is generally missed in the treatment by semiempirical methods.

sensitive to the contact between leads and molecules, as well as the thermal effect.18 Therefore, it is important to investigate the validity of a QIE-controlled molecular device in realistic systems, in which the treatment of a multiterminal structure is essential to investigate a QIE-controlled molecular device, unlike the usual single molecular device.7 The study by Cardamone et al.8 shows the basic concept of a QIE-controlled molecular device and concretely suggests two molecules, benzene and 18-annulene, in which QIE can be expected. Their original concept of QIE-controlled devices is as follows: If two separated paths between leads along which electrons move have a difference in the length d of the condition kFd = (2n + 1)π, where kF is a Fermi wave vector, then the destructive QIE is realized at the Fermi energy level. As a result, this condition makes the transmission node at the Fermi level. In such a situation, a third lead approaching one of the electron paths introduces the decoherence and breaks the QIE condition, kFd = (2n + 1)π. Basically, their idea is correct, but the physical picture of QIE is obscure from a molecular orbital (MO) point of view. What is changed in the electric state when the QIE condition is broken by the third lead is not explicit. Furthermore, their argument is based on a discussion of the semiempirical π-electron model, and they neglect the σtype orbitals in their argument. An intuitive understanding based on realistic MOs is indispensable to investigate the feasibility of the QIE-controlled molecular device; for this reason, self-consistent (SC) treatment of the electric potential within an ab initio treatment is needed. This is our motivation for this study. An ab initio calculation study can show the correct wave function of the QIE-controlled molecular device, and it can give a physical picture of the breaking of the QIE condition. Ke et al.15 studied QIE in molecular rings by firstprinciples calculation, from the point of view that is somewhat similar to ours. However, they used two-terminal systems and did not show the influence of the third terminal, which is the key instrument of QIE-controlled molecular devices. From the theoretical side, some multiterminal transport calculation studies are presented, but many of them are based on a semiempirical tight-binding method.19−21 As far as we know, few works are available on ab initio calculations of multiterminal quantum systems except Saha’s studies.22,23 In this study, we present the mechanism of the QIE for benzenedithiol (BDT) and 18-annulene molecules, which are treated in the work of Cardamone et al.,8 with realistic leads utilizing firstprinciples calculations, based on a nonequiblium Green’s function (NEGF) method for multiterminal systems, with density functional theory (DFT). In the following section, we will show the MO picture of the QIE control in the third lead by utlizing the molecular projected self-consistent Hamiltonian (MPSH) analysis, which is appropriate for the transport problem in lead−molecule systems. We examine the control possibility of the QIE device by using the third lead. We also investigate the influence of the contact between the leads and molecules by comparing two types of leads consisting of silver and aluminum atoms, which have a strong and weak binding to the molecule, respectively. As described above, QIE-controlled molecular devices are considered to be insensitive to the electric structure of a molecule−lead contact. However, it is not explicit that this insensitivity is also applicable to the contact between the third lead and a molecule. If the coupling of the lead and the molecule is strong enough to destroy the QIE in the molecule in a wide range of distances between the third lead and the molecule, then it becomes difficult to control the

2. COMPUTATIONAL METHOD We will first describe the detailed treatment of the multiterminal system implemented within the first-principles calculation methods and QIE on transmission of a molecular device. We employ the nonequilibrium Green’s function (NEGF) method based on the density functional theory (DFT).24,25 Our implementation of the NEGF method, coupled with DFT, is based on localized pseudoatomic orbitals (PAOs)26 and norm-conserving pseudopotentials,27 and most of the details follow our implementation of the two-terminal system.25 We implemented the three- and four-terminal NEGFDFT method as an extension of the two-terminal transport calculation part of OpenMX.28 Details of the implementation specific to the multiterminal systems are discussed below. We separate the system into four and five parts for the three- and four-terminal systems, respectively: a central region, which includes a scattering region denoted by C0, and semi-infinite left, right, up, and down lead regions denoted by Li, Ri, Ui (third lead), and Di (fourth lead), where i = 0,1,···, as shown in Figure 1. Note that lead Di is not introduced for the threeterminal system. We extend the scattering region by including the regions L0, R0, U0, and D0 into C0 to take account of the

Figure 1. Configuration of a system, treated by the multiterminal NEGF-DFT method. We treat the C0, L0, R0, U0 (third lead), and D0 (fourth lead, if needed) as an extended central scattering region. 101



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almost the same transmission spectra. Once the potential is selfconsistently determined, the transmission spectra from the lead i and j is given by

relaxation of the electronic structure around the interfaces between the leads (L0, R0, U0, and D0) and the central region C0. All cells Li, Ri, Ui, and Di, arranged semi-infinitely, contain the same number of atoms with the same structural configuration, respectively. The lead regions Li, Ri, Ui, and Di are set up in such a way that localized basis functions belonging to each region do not have any overlap with those belonging mutually to the other regions, and these semi-infinite leads are treated by a surface Green’s function method.29 To make this study self-contained, we give a necessarily minimum explanation of the multiterminal NEGF-DFT treatment. The density matrix for the multiterminal case is given by ρν , ν ′ =

∑∫

−∞

l

R 1/2 † 1/2 R 1/2 = Tr[(Γ1/2 i G Γ j ) Γi G Γ j ]

1 [G(ε)Γl(ε)G(ε)]ν , ν ′ π

(1)

A B (0) Gba(ε) = Gbb(ε)(Σba (ε) + Σba (ε))Gaa (ε )

Gbb(ε) =

with i R [Σl − ΣlA ] 2

(3)

Σl =

† Vg l l Vl

(4)

1 A B ε − β − Σbb − Σbb −

A B A B (Σba )(Σab ) + Σba + Σab A B ε − α − Σaa − Σaa

(9)

and (0) Gaa ( ε) =

1 A B ε − α − Σaa − Σaa

(10)

G(0) aa (ε)

where Gbb(ε) is the full Green’s function of site b and is the uncoupled Green’s function of site a. α and β are on-site energies of sites a and b, respectively. ∑Abb and ∑Bbb are the virtual excursions into paths A and B. The self-energy that describes the excitation going to site a and back to site b is given by (∑Aba + ∑Bba)(∑Aab + ∑Bab)/(ε − α − ∑Aaa − ∑Bbb). Then, it turns out that zeroes of Green’s function Gba originate from two factors, ∑Bba(ε) + ∑Bba(ε) and Gbb(ε)G(0) aa (ε). One can interpret the zeroes of the former as a result of the quantum interference of the multipath. For example, in the free electron model, if the phase difference of the two paths is π, then cancellation occurs between the self-energy of paths A and B. The latter, the zeroes of Gbb(ε)G(0) aa (ε), is the resonance zeroes,17 as a result from the poles of the self-energy ΣA or ΣB. The expression, eq 8, is convenient for clarifying the origin of the zero transmission but rather cumbersome to treat. Tada and Yoshizawa12 used the site’s Green’s function Gba(ε) to interpret the transmission properties of QIE in molecular devices. They first interpreted the antiresonance transmission from the MO structure of the scattering region. Thier arguments are based on the site’s Green function with the MO. The site’s Green’s function is given in the following expression12,13

where gl is the surface Green’s function of the semi-infinite lth lead, Vl the interaction between the central scattering region and lth lead, Σl the self-energy from the lth lead, and Γl representing the broadening function for the lth lead. By these quantities, we can treat the open system with semi-infinite leads. In the evaluation of the density matrix, we assume that the electron population is in equilibrium and nonequilibrium in the energy regime below μm and the energy regime between μm and μl, respectively, in the same treatment as for the twoterminal case.30 We determined the Hartree potential through Poisson’s equation. The solution of Poisson’s equation ϕ(r) is obtained by using a fast Fourier transform (FFT) technique and a neutral atom potential method.31 The Hartree potential, VH, is determined up to the linear term as VH(r) = ϕ(r) + a r + b

(8)

with

(2)

Γl =

(7)

if the energy dependence of the coupling (Γi) between the molecule and the lead can be neglected. The expression implies that one of the origins of the QIE on the suppression of the current flow (antiresonance QIE) corresponds to the zeros of Green’s function, GR(ε) = 0, at the Fermi energy level. As shown by Hansen et al.,17 if we denote atoms as sites a and b, which are connected to left and right leads, and if we can separate the electron path between sites a and b into two spatially separated paths, A and B, Green’s function from the site a to b, Gba, can be represented as

where l denotes the index of the lead, ν and ν′ the indexes of localized atomic orbitals, μm the lowest electrochemical potential of the corresponding mth lead in all the electrochemical potentials, and η a positive infinitesimal. nlν,ν′(ε) is defined as nνl , ν ′(ε) =

(6)

A 1/2 1/2 R 1/2 Tij = Tr[Γ1/2 j G Γi Γi G Γ j ]

dεnνl , ν ′(ε)f (ε − μl )

∫−∞ dεG(ε + iη)f (ε − μm) ∞ ∫−∞ dεnνl ,ν′(ε)(f (ε − μl ) − f (ε − μm))

2e 2 Tr[Γi(ε)GR (ε)Γj(ε)G A (ε)] h

Here, we will discuss the QIE of a single molecular device within the NEGF formalism. It is important to note that the transmission, eq 6, can be rewritten as

∞

1 = − Im π +∑ l≠m

∞

Tij =

(5)

where a and b are determined from the boundary conditions of the Poisson equation. The use of the FFT technique for the solution of Poisson’s equation is approximation when the periodic boundary condition violates by the semi-infinite leads. However, when the scattering region is not a strongly polarized system and there is a sufficient distance between the boundary of the system and the scattering region, the solution given by FFT is a reasonably good result. We compared the results by FFT and a more sophisticated finite difference method25 for the two-terminal single BDT molecule device, and we obtained

Gba(ε) ≃

∑ k

* Cak Cbk ε − Ek ± iη

(11)

by directly diagonalizing the molecule wave function in the scattering region and neglecting the self-energy correction of 102



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Figure 2. Total DOS of Al (left) and Ag (right) nanowire leads.

the coupling to the lead, where Cbk is the kth MO coefficient at the bth atomic site in an orthogonal basis and Ek is the kth MO energy. Although the expression, eq 11, cannot distinguish the multipath or resonance zeroes, it is easy to handle within the approximation of neglecting the self-energy effect of the lead and gives a simple rule in estimating the value of the site’s Green’s function Gba(ε). If only the transmission near the Fermi energy EF is of interest, it is enough to know MOs near the Fermi energy, EF, and the simplest expression that takes account of the contribution from only the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) levels to the transmission is derived as Gba(E F) ≃

Cb*HOMOCaHOMO Cb*LUMOCaLUMO + E F − εHOMO ± iη E F − εLUMO ± iη

described in the previous section. To investigate the dependence on the conductance of the metallic species for the lead, we select two atomic species, silver and aluminum, and construct the lead structure with a FCC [100] surface consisting of five atoms in one unit cell of the lead. In Figure 2, we show the total density of states (TDOS) of these leads of nanowire because these quantities strongly affect the transmission spectra through the coupling term of the lead and the scattering region, Γl, in eq 3. We first discuss the results of a benzenedithiol (BDT) device as an example of a simple QIE device. Figure 3 shows the para-

(12)

The expression implies that a nearly zero transmission is realized by the antiresonance QIE if the MO coefficients of HOMO and LUMO at sites a or b are small or a cancellation between the Green’s function components of HOMO and LUMO occurs,12−14 where the cancellation occurs when the sign of the product, Cb × Ca, of the MO coefficient for HOMO is the same as that for the LUMO and the Fermi energy is located at the middle of the two energy levels of HOMO and LUMO. Furthermore, if there are two degenerate states near the Fermi level, εHOMO1 and εHOMO2, for example, then the site’s Green’s function is approximately given as Gba(E F) ≃

Figure 3. Structure of a (a) para- and (b) meta-connected BDT molecular device.

Cb*HOMO1CaHOMO1 C* C + bHOMO2 aHOMO2 E F − εHOMO1 ± iη E F − εHOMO2 ± iη

and meta-connected BDT configuration. BDT is one of the most theoretically14,34,35 and experimentally2 well-investigated molecules, due to its simple structure. The meta-connected device is expected to show the destructive (antiresonance) QIE14,15,17 in π-conjugated molecular orbitals, and it will show the low transmission spectra at the Fermi level. Sulfur atoms in the BDT molecule are situated on the top site of the lead atoms. The distance between the sulfur atom and atom in the lead was fixed at 1.8 Å. In the calculation, we determine the optimized molecular structure by the first-principles calculation and fixed the molecular structure in the lead−molecule system. We do not consider the relaxation of atomic positions due to the coupling to the leads. Figure 4 shows the transmission spectra of the BDT molecule connected at the para- and meta-position with the Ag and Al leads. We see that the transmission of the paraconnected BDT device has a finite weight at the Fermi level, while that of the meta-connected BDT device almost vanishes at the Fermi level. In addition to this, it is found that the

(13)

In this case, the cancellation occurs between two degenerate states when the sign of the product, Cb × Ca, is different. These MO rules for transmission were confirmed by Taniguchi et al.32 and are quite predictive. Therefore, we will refer to eqs 12 and 13 for interpretation of the calculated results later on. In the calculations, the Perdew−Burke−Ernzerhof (PBE) exchange-correlation (XC) functional,33 which is derived within the generalized gradient approximation (GGA), was adopted. The optimized PAO basis functions denoted by C6.0-s2p2d1, H5.0-s2p2, S7.0-s2p2, Ag7.0-s1p1d1, and Al7.0-s1p1d1,26 were applied.

3. RESULTS AND DISCUSSION 3.1. Single BDT Molecular Device. In this section, we show the results of the QIE-controlled molecular device with three terminal leads calculated by the NEGF-DFT method 103



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Figure 4. Computed transmission spectra of two-terminal para- and meta-structure BDT molecular devices. Left and right figures show the transmissions of the Al and Ag leads, respectively.

and drain as the gate lead approaches the BDT molecule, which is a reversal behavior, when compared to the case of the Al lead, and an unexpected effect by the gate lead. To understand the results of these calculated properties in the transmission spectra of the three-terminal BDT device, we now analyze which orbitals on the BDT molecule are reflected in the transmission. To analyze the relation between the transmission spectra peaks and the molecular orbitals affected by the leads, we use a molecular-projected self-consistent Hamiltonian (MPSH) matrix.35 Since the spatial distribution of molecular orbitals is influenced by leads, molecular orbitals of an isolated molecule are not suited to analyzing the transmission of the molecular device. The MPSH states are eigenstates of the molecular orbital projected Hamiltonian that originally includes the selfenergies of the leads. Though the MPSH energies do not always coincide with the transmission peaks, the MPSH analysis helps us to qualitatively understand the origin of the transmission peaks with the corresponding MPSH molecular orbitals. Hereafter, we use the notations “HOMO” and “LUMO” for the eigenstates of the MPSH, and we set the chemical potential of the lead as a zero reference energy of MPSH. In Figure 7, we show the MPSH eigenstates of the three-terminal BDT device with Ag leads, where the distance between the gate lead and the BDT molecule is 1.0 or 2.0 Å. Figure 7 shows that the σ-type MPSH orbital already appears at a distance of 2.0 Å (the LUMO orbital in distance 2.0 Å with the Ag leads). Approaching the gate lead to the BDT molecule, the weight of the wave function of the HOMO eigenstate moves toward the Ag atom in the gate lead due to the strong coupling between the Ag lead and the BDT molecule, and the weight of the wave function on the S atom that is connected to the source lead decreases as a result. Furthermore, at a distance of 1.0 Å with the Ag leads, the energy level of the HOMO eigenstate is shifted down from the Fermi level, when compared to the case of distance of 2.0 Å. These facts are the reason why the transmission spectral weight near the Fermi level decreases when the gate Ag lead approaches the BDT molecule. In the Al lead case, the MPSH analysis shows that the σ-type orbital also contributes to a finite transmission spectra weight near the Fermi level as in the Ag lead case, but the σtype orbital only appears in the case of distance of 1.0 Å. We show the MPSH states with the Al lead in the Supporting Information, where detailed analyses are given. On the transmission spectra other than the source−drain terminal, we see a large transmission between the gate and other terminals at the Fermi level. In particular, the

absolute value of the transmission at the Fermi level for the Al lead is much smaller than that of the Ag lead in the paraconnected device. This is because the coupling between Al−S is weaker than that of Ag−S. The reason that the antiresonance QIE occurs in the meta-connected BDT device can be interpreted, by eq 13, as that cancellation between degenerate states (HOMO orbitals, in this case) is occurring. It is also noted that the structures of the transmission spectra for the Ag and Al leads are very different from each other, reflecting the differences in the TDOS of the nanowire lead (see Figure 2). By these calculations, we can confirm that the antiresonance QIE works in the meta-connected BDT single molecular devices with both the Ag and Al nanowire leads, and the lead atom species strongly influence the transmission spectra. Next, we introduce the third lead in the meta-connected BDT device by approaching the third lead to one of the hydrogen atoms in BDT, as in Figure 5. In this paper, we name

Figure 5. Structure of the three-terminal BDT molecular device. S, D, and G denote the source, drain, and gate leads, respectively.

the first and the second lead as the source and drain leads, respectively, and the third lead as the gate lead. In Figure 6, we show transmission spectra between every pair, source−drain, source−gate, drain−gate, of the terminal leads for the two cases with distances of 1.0 and 2.0 Å between the gate lead and the BDT molecule. We can find that the transmission at the Fermi level between the source and drain in the device with the Al leads increases as the gate approaches the BDT molecule. This is an expected result from the effect of the gate lead, which introduces the breaking of the antiresonance QIE in the metaconnected molecule. On the other hand, the result of the Ag leads shows a decrease in the transmission between the source 104



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Figure 6. Computed transmission spectra of the three-terminal BDT QIE device with Al (top panel) and Ag (bottom panel) leads. We set the distance between the BDT and the gate lead (third lead) at 1.0 and 2.0 Å. We also show the source−drain two-terminal BDT transmission spectra.

switching mechanism by approaching the gate lead in the threeterminal BDT molecule is different from the destructive effect of the quantum interference within the π orbitals; rather, it arises from the appearance of the σ-type wave function between the gate lead and the BDT molecule and the change in the energy level of the σ-type MO. These results are because of the strong π−σ orbital coupling between the molecule and gate lead, and thereby the expected QIE control within the π orbitals in the BDT molecule is masked by the π−σ hybridization. 3.2. Single 18-Annulene Molecular Device. We will now provide the results of the three-terminal 18-annulene device. The 18-annulene molecule is proposed as the QIE-controlled device because the size of the molecule can avoid the tunnel current from the σ-type orbital that masks the QIE.15 Figure 8

Figure 7. Eigenstates of MPSH in the three-terminal BDT molecular device with Ag leads. The left side and the right side are the eigenstates of the distance between the gate lead and the BDT molecule at 1.0 and 2.0 Å, respectively. We set the chemical potential of the lead at zero reference energy and show the eigenvalues of each MPSH state.

Figure 8. Configuration of the (a) 1−10 and (b) the 1−9 connected 18-annulene molecular device. The 1−9 connected device is expected to show low transmission at the Fermi level.

transmission between the gate and the drain is the largest at the Fermi level because it is the shortest distance between the terminals. The transmission between the gate and the drain shows that the tunnel current from the gate lead atom to the hydrogen atom of the BDT molecule is remarkable. Therefore, if we block the tunneling current from the gate lead, we must set the third gate lead as an insulator. Our results for the three-terminal BDT molecule device show that the expected switching of transmission by the gate lead does not necessarily occur and depends on the atomic species of the gate lead. Furthermore, we emphasize that the

shows configurations of the 1−10 (a) and the 1−9 (b) connected 18-annulene device, where the 1−10 (1−9) configuration corresponds to the constructive (destructive) interference,15 and Figure 9 shows the corresponding transmission spectra of the two-terminal devices with the Al and Ag leads, respectively. We can confirm from Figure 9 that the transmission spectra of the 1−9 connected devices are almost zero at the Fermi level, while the 1−10 connected transmission spectra have a finite weight. From analysis of the MPSH eigenstates (not shown here), the cancellation occurs between 105



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Figure 9. Computed transmission spectra of the 1−9 and the 1−10 connected, 2-terminal, and 18-annulene molecule devices. The left and right figures show the transmissions with Al and Ag leads, respectively.

for the Al and Ag leads, as shown in Figure 11. In the calculations, the distance between the gate lead and the 18annulene molecule is changed from 0.5 to 2.5 Å. We can observe that the transmission spectra at the Fermi level increase as the distance between the gate lead and the molecule decreases in both the Al and Ag lead cases. It seems that the gate lead controls the transmission, as Cardamone et al. suggested.8,9 In the Al lead case, by approaching the gate lead to the molecule, the energy level splittings of the degenerate LUMOs and HOMOs occur by approaching the gate lead, and this effect breaks the cancellation of the transmission between the degenerate MOs, as we can see in eq 13. These effects are the primary reasons that approaching the gate lead makes a finite contribution to the transmission spectra at the Fermi level. The breaking of the cancellation of the MO’s contribution to the transmission spectra is the expected result of the destruction of the antiresonance quantum interference. This is exactly the realization of the QIE control by the gate lead that Cardamone et al. proposed.8,9 Detailed MPSH analysis of the Al lead case is given in the Supporting Information. In the Ag lead case, we show the MPSH eigenstates of the 18-annulene molecule in Figure 12 in the case of the distances of 1.0 and 2.5 Å between the gate lead and the molecule. From

the degenerate HOMO’s and degenerate LUMO’s eigenstates in the Green function (eq 13) of the 1−9 configuration, and its antiresonance QIE can be interpreted by the analyses of Yoshizawa et al.12−14 Just like the BDT device, we introduce the gate lead into the 1−9 configuration of the antiresonance QIE, as shown in Figure 10, and we obtain the transmission spectra

Figure 10. Structure of the 3-terminal 18-annulene molecular device. S, D, and G denote the source, drain, and gate leads.

Figure 11. Computed transmission spectra of the 3-terminal 18-annulene device with Al (top panel) and Ag (bottom panel) leads. The left, center, and right figures in each panel show the source−drain, source−gate, and drain−gate terminal transmission spectra, respectively. We fixed the distance between the gate lead and the 18-annulene molecule at 0.5, 1.0, and 2.5 Å. 106



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is originally located at 1.3 eV higher than the Fermi level in the absence of a gate lead, is shifted down toward the Fermi level, and its energy level becomes 0.2 eV. The effect of the downshift of the LUMO energy level produces finite transmission spectra at the Fermi level in the case of distances of 1.0 and 0.5 Å. The peak of the S−D transmission spectra around −0.3 eV in the two-terminal and the distance of 2.5 Å case stems from the HOMO and the second HOMO eigenstates. These eigenstates are also shifted down by the gate lead, and they make the peak structure of the transmission spectra around −0.5 eV in the cases of distances 1.0 and 0.5 Å. Thus, in the 18annulene molecule QIE device, we see that the finite transmission spectra at the Fermi level due to the gate lead mainly originate from the fact that the degenerate energy levels are split by the hybridization of the gate lead and molecule, and the energy level of the split state approaches the Fermi energy. Energy level splitting of the degenerate states results in the breaking of the cancellation of the contributions to the transmission spectrum from these states. The shift of the MPSH eigenenergy toward the Fermi level increases the transmission spectra at the Fermi level. This is the expected result that the gate lead breaks the antiresonance QIE and is the manifestation of the breaking of QIE from the MO point of view. Furthermore, the σ-type orbitals appear near the Fermi level in the Ag lead case, and it also brings a finite contribution to the transmission spectra at the Fermi level. Such σ-type orbitals are not observed in the Al lead case, and the property of the orbital which contributes the finite transmission at the Fermi level depends on the lead species. On the transmission spectra other than the source−drain, the gate−drain transmission has the largest weight at the Fermi level. The tunneling current through the gate lead and the hydrogen atom in the 18annulene molecule is significant, as in the BDT molecule case. 3.3. Gate Voltage Effect. Finally, we investigate the effect of the gate lead voltage to the transmission spectra between the source and drain terminals. The distance between the 18annulene molecule and the gate lead is fixed to be 1.0 Å, and the gate voltage is changed from −0.4 to 0.4 V. We show the calculated transmission spectra of the source−drain (S−D), source−gate (S−G), and drain−gate (D−G) terminals in

Figure 12. Eigenstates of the MPSH in the three-terminal-18-annulene molecular device with Ag leads. The left and right figures correspond to the distance of 1.0 and 2.5 Å, respectively.

the MPSH eigenstates in Figure 12, one can observe that the MOs are strongly modified by approaching the gate lead to the 18-annulene molecule. Due to the strong hybridization between orbitals in the Ag lead and the 18-annulene, the σ-type MOs located on the Ag atom in the gate lead appear in the second LUMO and the second HOMO in the case of distance of 1.0 Å. The degeneracy of LUMO states is broken by the gate lead, and its orbital structure has sizable weight near the gate lead (see Figure 12). Furthermore, the energy level of the LUMO, which

Figure 13. Computed transmission spectra of the three-terminal 18-annulene device with Al (top panel) and Ag lead (bottom panel). Three figures in each panel show the transmission between the source−drain (left), source−gate (center), and drain−gate (right) terminals. We also show S−D transmission of gate voltage at V = −0.8 V in the Ag lead case. 107



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transmission with a gate voltage of −0.8 V in Figure 13 of the Ag lead case. Therefore, we can expect a strong nonlinear dependence of the S−D conductance to the gate voltage. In the Al lead case, the HOMO energy level is deeper than that of the Ag lead case and is not the σ-type orbital; therefore, we cannot see this effect within the range of the voltage that we changed. Further analysis of the Al lead case is given in the Supporting Information. The change in MPSH energy levels by the gate voltage is not uniform in the Ag lead case. Due to the strong hybridization by the Ag 5s orbitals, some of the MPSH eigenstates near the Fermi level (the second HOMO and the LUMO, in this case) have substantial weight on the atoms around the gate lead, and these states are subject to a stronger influence by the gate voltage than in other states. Therefore, if there are σ-type orbitals weighted around the gate lead near the Fermi level, we can selectively control these orbitals and change the transmission by the gate voltage effectively. The LUMO and second HOMO in the Ag lead are exactly this case. On the transmission of the S−G (the source and gate) and D−G (the drain and gate) terminals, the gate voltage dependence at the Fermi level for the Al lead shows that positive voltage increases the spectra for both the S−G and the D−G cases, due to the energy shift of the LUMO and second LUMO energy levels, and the behaviors are the same as that of the S−D case. On the other hand, in the Ag lead case, the behaviors of the S−G and D−G transmissions at the Fermi level are different from that of the Al lead. The spectra at the Fermi level in the S−G and D−G transmissions are increased by the negative voltage, and they show the reverse behavior of the S−D transmission. In the S−G and D−G transmissions, the main contribution to the spectra at the Fermi level stems from the HOMO and the second HOMO states in the Ag lead case. On the other hand, in the S−D transmission, it stems from the LUMO state. Accordingly, the energy level shift of the HOMO and second HOMO by the gate voltage causes a shift in the transmission peak at the Fermi level in the S−G and D−G transmission, and the negative gate voltage shifts these energy levels toward the Fermi level. This is why the negative gate voltage increases the transmission spectra at the Fermi level in the S−G and the D−G transmission in the Ag lead case. The disappearance of the LUMO peak in the transmission spectra of the S−G and the D−G terminal of the Ag lead case may be attributed to the structure of the lead DOS at the applied gate voltage. Thus, the behavior of the transmission spectra with gate-terminal voltage depends on the detailed structures of the molecular wave functions and the lead species.

Figure 13 for the Al and Ag leads. In these figures, we set zero reference energy as the chemical potential of the source and the drain leads. We can see from Figure 13 that transmission at the Fermi level between the source and drain leads increases (decreases) by applying positive (negative) gate voltage for both the Al and Ag lead cases within a range of voltage from −0.4 to 0.4 V. For the Ag lead case, we also show the MPSH eigenstates for the gate voltages of −0.4 and 0.4 V in Figure 14.

Figure 14. Eigenstates of the MPSH three-terminal 18-annulene molecular device with finite gate voltage in the case of the Ag lead. The left side and the right side are the eigenstates of the case of gate voltage at V = 0.4 and −0.4 V, respectively.

We can see from Figure 14 that the structures of the MPSH eigenstates are not largely modified by the gate voltage, but the energy levels of eigenstates are influenced. The same is true of the Al lead case (not shown here). The LUMO in the Ag lead case contributes to the transmission spectra at the Fermi level in the range of gate voltage of −0.4 to 0.4 V. The positive gate voltage shifts down the energy level of the LUMO, and this is the reason for the increase of transmision at the Fermi level by the positive gate voltage. The reason that the absolute value of the S−D transmission spectra peak near the Fermi level is rather small in the Ag lead case is that the σ-type LUMO eigenstate has little weight on the atom connected to the source or drain leads. As we can see from Figure 14, the HOMO and LUMO energy levels are shifted down to −0.67 and 0.01 eV, respectively, in the case of gate voltage of 0.4 V. These eigenstates of HOMO and LUMO correspond to the peaks of transmission spectra around −0.7 and −0.25 eV, respectively, in the case of gate voltage of 0.4 V with Ag lead. The differences in the transmission peaks and the MPSH eigenenergies may be attributed to the approximation of MPSH. By applying a negative gate voltage (−0.4 V), the LUMO energy level is shifted up, and the transmission at the Fermi level decreases. Furthermore, the energy levels of HOMO and the second HOMO are interchanged by the negative gate voltage because the wave function of the second HOMO is weighted around atoms near the gate lead and is strongly influenced by the gate voltage. The negative gate voltage also shifts up the energy levels of the HOMO and the second HOMO. This causes the upper shift of the transmission peak around −0.5 eV by the negative gate voltage in the Ag lead case. Further negative gate voltage shifts the HOMO energy level up toward the Fermi level, and it increases the transmission spectra at the Fermi level. We can see this behavior of the S−D

4. SUMMARY We have investigated the three-terminal QIE-controlled molecular device by first-principles methods with two different types of leads, Ag and Al nanowires. As functional molecules in which the QIE works, we used BDT and 18-annulene molecules. The results of the three-terminal QIE device with BDT molecules indicate that the expected QIE control for meta-connected molecules does not necessarily occur; the behavior of the transmission spectra depends on the lead species. Depending on the strength of the coupling between the lead atom and the molecule, the σ-type MO appears near the Fermi level and causes complicated transmission behavior, as in the Ag lead case. The influence of the gate lead on a small molecule, such as BDT, is so large that the σ-type MO suppresses the QIE control. On the 18-annulene molecular device, we have confirmed that the antiresonance QIE can be 108



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(13) Yoshizawa, K.; Tada, T.; Staykov, A. J. Am. Chem. Soc. 2008, 130, 9406−9413. (14) Tsuji, Y.; Staykov, A.; Yoshizawa, K. J. Am. Chem. Soc. 2011, 133, 5955−5965. (15) Ke, S. H.; Yang, W.; Baranger, H. U. Nano Lett. 2008, 8, 3257− 3261. (16) Solomon, G. C.; Andrew, D. Q.; Hansen, T.; Goldsmith, R. H.; Wasielewski, M. R.; Van Duyne, R. P.; Ratner, M. A. J. Chem. Phys. 2008, 129, 054701. (17) Hansen, T.; Solomon, G. C.; Andrews, D. Q.; Ratner, M. A. J. Chem. Phys. 2009, 131, 194704. (18) Venkataraman, L.; Klare, J. E.; Tam, I. W.; Nuckolls, C.; Hybertsen, M. S.; Steigerwald, M. L. Nano Lett. 2006, 6, 458. (19) Terasawa, A.; Tada, T.; Watanabe, S. Phys. Rev. B 2009, 79, 195436. (20) Dutta, P.; Maiti, S. K.; Karmakar, S. N. Org. Electron. 2010, 11, 1120−1128. (21) Cook, B. G.; Dignard, P.; Varga, K. Phys. Rev. B 2011, 83, 205105. (22) Saha, K. K.; Lu, W.; Bernholc, J.; Meunier, V. J. Chem. Phys. 2009, 131, 164105. (23) Saha, K. K.; Branislav, B. K.; Nikolic, K.; Meunier, V.; Lu, W.; Bernholc, J. Phys. Rev. Lett. 2010, 105, 236803. (24) Taylor, J.; Guo, H.; Wang, J. Phys. Rev. B 2001, 63, 245407. (25) Ozaki, T.; Nishio, K.; Kino, H. Phys. Rev. B 2010, 81, 035116. (26) Ozaki, T. Phys. Rev. B 2003, 67, 155108. Ozaki, T.; Kino, H. Phys. Rev. B 2004, 69, 195113. (27) Morrison, I.; Bylander, D. M.; Kleinman, L. Phys. Rev. B 1993, 47, 6728−6731. (28) OpenMX uses the pseudoatomic basis functions and the normconservative pseudo potentials (http://www.openmx-square.org/). (29) Lopez Sancho, M. P.; Lopez Sancho, J. M.; Rubio, J. J. Phys. F: Met. Phys. 1985, 15, 851−858. (30) Ozaki, T. Phys. Rev. B 2007, 75, 035123. (31) Ozaki, T.; Kino, H. Phys. Rev. B 2005, 72, 045121. (32) Taniguchi, M.; Tsutsui, M.; Mogi, R.; Sugawara, T.; Tsuji, Y.; Yoshizawa, K.; Kawai, T. J. Am. Chem. Soc. 2011, 133, 11426−11429. (33) Perdew, J. P.; Burke, K.; Ernzerhof., M. Phys. Rev. Lett. 1996, 77, 3865−3868. (34) Xue, Y.; Datta, S.; Ratner, M. A. J. Chem. Phys. 2001, 115, 4292. (35) Stokbo, K.; Taylor, J.; Brandbyge, M.; Mozos, J. L.; Ordejon, P. Comput. Mater. Sci. 2003, 27, 151−160.

controlled by the gate lead. By the MPSH analysis, we have shown that the breaking of the antiresonance QIE in an 18annulene molecule is basically caused by the energy level splitting and shift of the degenerate LUMO states when approaching the gate lead to the molecule. The appearance of the σ-type MO near the Fermi level with bonding to the gate lead atom also affects the transmission spectra, as in the Ag lead case of our study. Therefore, in addition to the change in the energy level of the π-type MO around the Fermi level, which is expected from the initial guess of a QIE controlled device,8,9 we found that the new σ-type MO appears and strongly influences the transmission spectra in the strong lead-molecule coupling condition, such as in the case of the Ag lead. Moreover, we showed the finite gate voltage can also control the antiresonating state. The applied gate voltage causes the energy level shift of the MO which is the main reason for the change of the transmission spectra by the gate voltage. The strong gate voltage changes the MO, which contributes to the transmission at the Fermi level. In such a situation, we expect nonlinear behavior of the source−drain conductance due to the gate lead voltage. The gate voltage affects the σ-type MO in the bonding to the gate lead atom, in particular, and we can selectively control the energy level of this type of orbital by utilizing gate voltage. These results suggest new possibilities in controlling the transmission properties of a single molecular device.

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ASSOCIATED CONTENT

S Supporting Information *

Additional figures of MPSH states of the Al leads are shown. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: yukihiro_okuno@fujifilm.co.jp. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. H. Watanabe and Dr. K. Furuya of FUJIFILM Corp. for their support.

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REFERENCES

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