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Letter
Molecular Orbital Rule for Quantum Interference in Weakly Coupled Dimers: Low-Energy Giant Conductance Switching Induced by Orbital Level Crossing Daijiro Nozaki, Andreas Lücke, and Wolf Gero Schmidt J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b02989 • Publication Date (Web): 20 Jan 2017 Downloaded from http://pubs.acs.org on January 24, 2017
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The Journal of Physical Chemistry Letters
Molecular Orbital Rule for Quantum Interference in Weakly Coupled Dimers: Low-Energy Giant Conductivity Switching Induced By Orbital Level Crossing Daijiro Nozaki* Andreas Lücke, and Wolf Gero Schmidt †
Lehrstuhl für Theoretische Materialphysik, Universität Paderborn, 33095 Paderborn, Germany
ABSTRACT: Destructive Quantum interference (QI) in
molecular junctions has attracted much attention in recent years. It can tune the conductance of molecular devices dramatically, which implies numerous potential applications in thermoelectric and switching applications. There are several schemes that address and rationalize QI in single molecular devices. Dimers play a particular role in this respect, since the QI signal may disappear, depending on the dislocation of monomers. In this Letter, we derive a simple rule that governs the occurrence of QI in weakly coupled dimer stacks of both alternant and non-alternant polyaromatic hydrocarbons (PAHs) and extends the Tada-Yoshizawa scheme. Starting from the Green’s function formalism combined with the molecular orbital (MO) expansion approach it is shown that QI-induced antiresonances and their energies can be predicted from the amplitudes of the respective monomer terminal molecular orbitals. The condition is illustrated for a toy model consisting of two hydrogen molecules and applied within density-functional calculations to alternant dimers of oligo(phenylene-ethynylene) and non-alternant PAHs. Minimal dimer structure modifications that require only a few meV and lead to an energy crossing of the essentially preserved monomer orbitals are shown to result in giant conductance switching ratios.
Charge transfer across non-covalently bonded molecular bridges is ubiquitous in nature, e.g. in long-range signaling1 or in photosynthesis,2 but it also controls the functionality of man-made materials like organic semiconductors.3,4 It depends sensitively on the distance and respective orientation of the molecules. Particularly pronounced conductance drops may result from socalled quantum interferences (QI), i.e., destructive superpositions of broadened electronic eigenstates.5–10 This quantum effect at the molecular scale leads to a large on/off conductance ratio11–23 that can be exploited in molecular switches and sensors.24,25 It can also serve to filter hole or electron transmission and thus enhance the thermoelectric performance of molecular junctions.26,27 Many molecular junctions have been designed and several theoretical models have been developed in order to understand the occurrence of quantum inferences. Metasubstituted phenyl rings,12,18,20,22–37 T-shaped,35-39 and cross-conjugated molecules8,11,12,33,40 are intensively explored in this context. For instance, the TadaYoshizawa or MO approach can explain the QI at the center of HOMO-LUMO (H-L) by the cancellation of the contributions from frontier MOs.10,15-23 The local molecular orbital (LMO) method allows for explaining the QI in cross-conjugated systems.8,33,34,40 Conductance measurements on dimer stacks9,41-45 and theoretical studies on QI effect in π-stacked systems9,4547 have been performed to explore the influence of dislocations and rotations. Li et al. have reported the QI in covalently-binded cyclophane dimers using the MO approach.22 Solomon and co-workers46,47 have shown that QI in tidily aligned stacks of alternant polyaromatic hydrocarbons (PAHs) can be understood in terms of the Coulson-Rushbrooke-McLachlan (CRM) pairing theorem.48-51 A general condition applicable to arbitrary dimer stacking configurations covering both alternant and nonalternant PAHs still needs to be established, however: The CRM theory cannot be applied to non-alternant PAHs including odd-membered rings50,51 such as az-
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ulene-derivatives. Also, the applicability of the graphical Markussen-Stadler-Thygesen (MST) scheme and the MO approach has been under debate for some systems.12,52-59 A detailed comparison of the predictions from the CRM theory, the MO approach, and the MST scheme for a variety of compounds and an analysis of their relationship to each other are summarized in Ref. [59].
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We start from the contacted molecular dimer Hamiltonian H = HM+ΣL+ΣR, where HM is the Hamiltonian for the dimer and ΣL/R are the self-energy terms for left/right contacts. The retarded/advanced Green’s function is then given by Gr/a(E) = [(E±iδ)I−HM−ΣL−ΣR]−1 and allows for calculating the transmission TLR(E) = Tr[ΓLGr(E)ΓRGa(E)], where ΓL/R are the spectral densities of the left/right contacts defined as ΓL/R ≡ i[ΣL/R− Σ†L/R]. Since the present focus is on the dimer QI, the contacts are treated in the wide-band limit (WBL) ΣL/R ≡ −iγ. Also, we assume the coupling to the contacts to be sufficiently strong to rule out Coulomb blockades and allow for coherent tunneling. If ( ), are the atomic orbitals (AOs) that contact the dimer from the left (right) hand side, the electron transmission is | , (1) LR = 4 |LR with the retarded Green’s function in Lehmann representation15-23,32,35,51 LR = ∑
L | |R
.
(2)
Here Ψm is the m-th eigenvector of the contacted dimer obtained within WBL. L/R
Figure 1. (a) MO diagram of weakly interacting H2 molecules in stacked (left) and linear (right) configurations. (b) Transmission spectra of the stacked and (c) the linear dimer calculated within the WBL (ΣL/R = −i0.025|β|) for different intermolecular couplings β (intramolecular transfer integral). The resonant peaks of the original monomers split at E = ±β due to the weak intermolecular interaction. Stronger intermolecular coupling leads to larger splitting. The QI is seen in stacked dimers as antiresonance in the middle of the gap. The electron-hole symmetry frequently breaks in real molecules, i.e, the Fermi energy does not coincide with the middle of the H-L gap. This happens, e.g, if external fields are applied50,60,61 or one of monomers in dimers is dislocated.9,46,47 A scheme to predict the QI energy would be very helpful in such cases. In this Letter, a condition for QI in weakly coupled molecular dimers is derived from the Green’s function formalism.62 We mention that a similar expression, starting from a MO explansion of the Green’s function, was derived earlier15-23. The present scheme is first illustrated for the simple case of interacting H2 molecules and then shown to correctly predict transmission antiresonances and their energies for dimers of both alternant and non-alternant PAHs. For this purpose, comparison is made to density-functional theory (DFT) based nonequilibrium Green’s function (NEGF) and scattering approaches.
Let HO/LU be the HOMO/LUMO of the isolated lhs/rhs monomers and εH/L the corresponding energies. Upon coupling, the energy levels split and the HOMO/HOMO-1 and LUMO/LUMO+1 energies are approximately given by εH±∆H and εL∓∆L, respectively. The splitting is small compared to the monomer gap, Eg = εL−εH ≫ ∆L/H, and the amplitudes of the monomer molecular orbitals are nearly preserved for sufficiently weak coupling. The frontier orbitals of the dimer stack (see also Fig. 1(a) left) can then be approximated by Brédas’s MO expansion as63 L
R
ΨLU+1 = ( LU ) ± ( LU )/√2,
(3)
ΨLU = ( LU ) ∓ ( LU )/√2,
L
R
(4)
ΨHO = ( HO ) ∓ ( HO )/√2,
L
R
(5)
ΨHO-1 = ( HO ) ± ( HO )/√2.
(6)
L
R
Note that the molecular orbitals of the dimer stack are obtained here by linear combinations of the monomer MOs, in contrast to the LMO approach.8,33,34 Assuming that the Green’s function around the Fermi energy can be approximated from the four levels HOMO−1 to LUMO+1 and that the contributions from other states are negligible because of the large denominator in eq. (2), one finds LR ≈ ∑ .HO-1,HO,LU,LU+1
L | |R
.
(7)
Here and in the following E contains a small imaginary part. The definitions of the MOs together with the relation
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/L ( LU/HO ) = /R ( LU/HO ) = 0, allow for expanding (7) as 1
2LR ≈ ± 3
H ∆H
,
(8)
1
∓
H ∆H
3
∓
L ∆L
± (9)
L ∆L
4 = /L( HO ) / HO (R )
with
and
5 = /L( LU ) / LU (R ) . Introducing a normalized
energy in the H-L gap as 6 = − 89 /: ; 0≤c≤1 one finds ±1
6 2LR ≈ ;
< ∆H
∓
1
;< ∆H
3
∓ ;=
< ∆L
3
± ;=
< ∆L
,
(10) For mid-gap energies assuming ∆ = ∆L/R, one obtains 13
6 LR = 1/2 ≈ ±
< ∆
13
∓
,
(11)
< ∆
which yields the condition a = b for mid-gap QI, i.e.,
/L ( HO ) / HO (R ) = /L ( LU ) / LU (R ),
(12)
Using eqs. (3)-(6) and (8) this can be written as L |ΨHOΨHO|R = L |ΨLU ΨLU |R .
(13)
Equation (13) allows for the prediction of mid-gap antiresonances from the signs and amplitudes of the dimer HOMO and LUMO at the left and right terminal contacts. If Eq. (13) holds exactly, QI occurs at the exact middle of the H-L gap, as predicted by the MO approach.15-23 As long as eq. (7) is valid, however, the antiresonance energy is obtained from the MO terminal amplitudes by 6=
1±?13∆@ 13∆@ 13/∆@ AB 13∆@
; D =
∆A
