Light-driven electron transport through a molecular junction based on cross-conjugated systems

Light-driven electron transport through a molecular junction based on cross-conjugated systemsLiang-Yan Hsu, Dan Xie, and Herschel Rabitz

Citation: The Journal of Chemical Physics 141, 124703 (2014); doi: 10.1063/1.4895963 View online: http://dx.doi.org/10.1063/1.4895963 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electron transport through nano-MOSFET in presence of electron-electron interaction AIP Advances 3, 032124 (2013); 10.1063/1.4795736 Guidelines for choosing molecular “alligator clip” binding motifs in electron transport devices J. Chem. Phys. 134, 154708 (2011); 10.1063/1.3581097 Theoretical investigation of electron transport modulation through benzenedithiol by substituent groups J. Chem. Phys. 129, 034707 (2008); 10.1063/1.2955463 Electron Transport through TripleQuantumDot Systems AIP Conf. Proc. 893, 799 (2007); 10.1063/1.2730132 Formalism, analytical model, and a priori Green’s-function-based calculations of the current–voltagecharacteristics of molecular wires J. Chem. Phys. 112, 1510 (2000); 10.1063/1.480696

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THE JOURNAL OF CHEMICAL PHYSICS 141, 124703 (2014)

Light-driven electron transport through a molecular junction basedon cross-conjugated systems

Liang-Yan Hsu,a),b) Dan Xie,b) and Herschel Rabitzc)

Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

(Received 16 August 2014; accepted 7 September 2014; published online 23 September 2014)

This work explores light-driven electron transport through cross-conjugated molecules with differ-ent numbers of alkenyl groups. In the framework of coherent quantum transport, the analysis usessingle-particle Green’s functions together with non-Hermitian Floquet theory. With realistic param-eters stemming from spectroscopy, the simulations show that measurable current (∼10−11 A) causedby photon-assisted tunneling should be observed in a weak driving field (∼2 × 105 V/cm). Current-field intensity characteristics give one-photon and two-photon field amplitude power laws. The gapbetween the molecular orbital and the Fermi level of the electrodes is revealed by current-field fre-quency characteristics. Due to generalized parity symmetry, the cross-conjugated molecules with oddand even numbers of alkenyl groups exhibit completely different current-polarization characteristics,which may provide an advantageous feature in nanoelectronic applications. © 2014 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4895963]

I. INTRODUCTION

Light-driven electron transport (LDET) has attractedbroad interest due to the experimental observation of photon-assisted tunneling (PAT) in mesoscopic systems1–3 sincethe 1960s. As a key feature of LDET, PAT has been ex-tensively studied in superconductor-insulator-superconductortunnel junctions1, 4, 5 and semiconductor nanostructures6, 7 ex-posed to microwave radiation.2, 3 Recently, LDET8–10 inmolecular junctions has become an active field in the do-main of molecular electronics.11–17 Theoretical studies us-ing a tight-binding model have shown that photon-assistedtunneling,18, 19 coherent quantum ratchets,20 and coherent de-struction of tunneling21 could be observed in systems withweak hopping integrals, e.g., � � 0.1 eV,22 or in the presenceof strong fields, e.g., at |E| � 2 × 107 V/cm.23, 24 However,for molecules the former regime is not reasonable and thelatter domain could cause possibly undesirable physical orchemical processes. This issue was recently addressed witha proposal to use phenyl-acetylene macrocycles25, 26 basedon the concept of destructive quantum interference originat-ing from the nature of the molecular structure.27–32 In thisstudy we show that, in addition to phenyl-acetylene macro-cycles, cross-conjugated systems are also potential candi-dates for the observation of photon-assisted tunneling, andthis family of molecules may be applied to single-moleculeoptoelectronics.25, 33

Cross-conjugated systems have drawn attention dueto their natural abundance and interesting electronicstructure.34–37 Recently, destructive quantum interferencewith tunneling has been theoretically predicted38–41 and ex-perimentally observed42 in the cross-conjugated systems.However, these studies focus on electron transport in the

a)Electronic mail: [email protected])L.-Y. Hsu and D. Xie contributed equally to this work.c)Electronic mail: [email protected]

presence or absence of a static field, while LDET in cross-conjugated systems has not received such attention. To ex-plore the feasibility of LDET in molecular junctions, we pro-pose a family of cross-conjugated molecules (see Fig. 1(a))and analyze their transport properties by employing single-particle Greens functions along with the non-Hermitian Flo-quet theory.21, 25, 43 Our analysis is within the framework ofcoherent single electron transport, which is suitable for shortmolecules at low bias and at low temperature.44–46 The inelas-tic current caused by molecular vibrations is neglected sincethe conductance changes caused by molecular vibrations arevery small under off-resonant conditions.47, 48

II. THEORETICAL MODELING

Molecules 1–7 (see Fig. 1(a)) and two electrodes forminga single-molecule junction within a time-periodic driving field(a laser field) are illustrated in Fig. 1(b). Molecules 1–6 arecross-conjugated systems with different numbers of alkenylgroups and Molecule 7 is a linear polyene. The system inFig. 1(b) can be modeled as a time-dependent Hamilto-nian H(t) comprised of light-driven molecular HamiltonianHmol(t), electrode Hamiltonian Helec, and molecule-electrodecoupling Hcontact. To facilitate the theoretical analysis ofLDET in the cross-conjugated systems (i.e., appropriately re-duce the complexity of Hmol(t)), first, we neglect the effectof anchoring groups (thiol groups in Fig. 1(a)). Second, weassume that each cross-conjugated molecule is located in thex-y plane, and Hmol(t) is treated in the framework of a Hückelmodel and the electric dipole approximation, i.e.,

Hmol(t) =∑

n

(E0 − ern · E(t))|n〉〈n| +∑nn′

�n′n|n′〉〈n|,

(1)

where |n〉 stands for the pz-orbital on the nth carbon at po-sition rn, E0 is the on-site energy, �nn′ is the resonance

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124703-2 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

FIG. 1. (a) Cross-conjugated systems (Molecules 1–6) and a linear conjugated system (Molecule 7). Molecules 1–6 have alkenyl groups, indicated by the reddashed lines. All molecules have anchoring groups, two thiol groups, which can chemically bond to the electrodes. (b) A junction consists of two electrodes (Land R electrodes), a single molecule, and a laser field propagating in the z direction (optical gate) with frequency ω and polarization angle β. Molecules 1–7are placed in the xy-plane and the blue double headed arrow denotes laser polarization.

integral, and E(t) is a time-dependent electric field with aperiod T and angular frequency ω = 2π /T, i.e., E(t) = E(t+ T ) = E(t + 2π/ω). As a result, Eq. (1) has a time-periodicsymmetry Hmol(t) = Hmol(t + 2π /ω). In addition, E(t) is a lin-early polarized monochromatic electric field with amplitude Aand polarization angle β in x-y plane, i.e.,

E(t) = A(cos β ex + sin β ey) cos ωt, (2)

where ex and ey denote unit vectors in x and y directions,respectively.40 The coupling strength between two carbonsites �nn′ depends on the carbon bond length, and the reso-nance integral-bond length relationship is

�nn′ = −67.28 exp

(− dnn′

0.424

)eV, (3)

which is determined from experimental optical transitions inpolydiacetylenes.49 The bond length dnn′ = |rn − rn′ | is ob-tained from the geometry optimizations of isolated cross-conjugated molecules at the B3LYP/6-31 G(d) level usingthe Gaussian 09 program.50 The Hückel model captures themain electronic structure of the cross-conjugated systems, asfirst-principle calculations show that the corresponding HO-MOs (highest occupied molecular orbital) and LUMOs (low-est unoccupied molecular orbital) are linear combinations ofpz-orbitals. We use hydrogens instead of thiol groups in ourgeometry optimizations and take 4 digits after the decimalpoint for each optimized bond length and bond angle in or-der to capture the main geometry characteristics of the cross-conjugated systems.

The two electrodes are described by a non-interactingelectron gas model, i.e.,

Helec =∑lq

εlq |lq〉〈lq|, (4)

where |lq〉 stands for the orbital with energy εlq in the lead lwith the mode q, and l = L and R stand for the left and theright leads. The molecule-electrode couplings are modeled as

Hcontact =∑

q

VLq,u|Lq〉〈u| + VRq,v|Rq〉〈v| + H.c., (5)

where |u〉 and |v〉, respectively, represent the pz-orbital onthe contact carbon atoms u and v, and the matrix elementof the coupling function can be expressed as [�l(ε)]n′n = 2π∑

q V ∗lq,n′Vlq,nδ(ε − εlq). Moreover, by using Eq. (5), the cou-

pling functions reduce to [�(ε)L]n′n = [�(ε)L]n′nδunδun′ and[�R(ε)]n′n = [�R(ε)]n′nδvnδvn′ .

By invoking Floquet theory, the time-average currentwith spin degeneracy is computed using the Landauer-typeformula21, 25, 26, 51

IR =2e

h

∞∑k=−∞

∫ ∞

−∞dε

[T

(k)LR (ε)fR(ε) − T

(k)RL (ε)fL(ε)

], (6)

where fL(R)(ε) = (1 + e(ε−μL(R))/kBθ )−1 is the Fermi function

of electrode L(R) with chemical potential μL(R) at temperatureθ .

T(k)

LR(RL)(ε) denotes the kth component of the transmissionfunction for an incident electron from the right (left) electrodeto the left (right) electrode, which is computed via26

T(k)

LR (ε) = Tr[�L(ε + k¯ω)GR(k)(ε)�R(ε)GA(k)(ε)], (7)

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124703-3 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

T(k)

RL (ε) = Tr[�R(ε + k¯ω)GR(k)(ε)�L(ε)GA(k)(ε)], (8)

where T(k)

LR (ε) (T (k)RL (ε)) corresponds to the tunneling of an

electron from the electrode R(L) to the electrode L(R) withenergy ε accompanied by k-photon absorption (k > 0) oremission (k < 0). Note that in general cases T

(k)LR (ε) = T

(k)RL (ε)

for molecules without generalized parity (GP) symmetry.51

To facilitate the theoretical analysis, we adopt the wide-band limit approximation and assume symmetric couplings[�(ε)R]vv = [�(ε)L]uu = �. As a result, the kth componentof the transmission function reduces to T

(k)RL (ε) = �2|G(k)

vu (ε)|2and T

(k)LR (ε) = �2|G(k)

uv (ε)|2, where � = 0.4 eV is a reasonablevalue for molecular junctions.28, 52

The kth component of the retarded Green function is26

GR(k)nn′ (ε) =

∑λ∈1st BZ

+∞∑ζ=−∞

φn,−kλ,ζ φ

n′,0λ,ζ

ε − qλζ

, (9)

where “1st BZ” denotes the first Brillouin zone,53 and φn,kλ,ζ as

well as qλζ can be derived from the time-independent infinite-dimensional eigenvalue matrix equation∑

nk

[HF ]n′k′,nkφn,kλ,ζ = qλζ φ

n′,k′λ,ζ . (10)

Here, [HF ]n′k′,nk denotes the matrix elements of the time-averaged Floquet Hamiltonian over a period T, which can becast as

[HF ]n′k′,nk = [H

(k′−k)mol

]n′n − i�

2δk′k(δunδun′ + δvnδvn′ )

+ δn′nδk′kk¯ω, (11)

where H(k′−k)mol is the Fourier coefficient of Hmol(t),

H(k′−k)mol = 1

T

∫ T

0dtHmol(t)e

−i(k′−k)ωt . (12)

Note that if we define TLR(ε) = ∑∞k=−∞ T

(k)LR (ε) and

TRL(ε) = ∑∞k=−∞ T

(k)RL (ε), Eq. (6) can be expressed as

IR =2e

h

∫ ∞

−∞dε[TLR(ε)fR(ε) − TRL(ε)fL(ε)]. (13)

Moreover, when the generalized parity symmetry is held,TLR(ε) = TRL(ε) is derived and the transmission function canbe defined as T(ε) = TLR(ε). As a result, Eq. (13) reduces tothe Landauer formula,

IR = 2e

h

∫ ∞

−∞dεT (ε)[fR(ε) − fL(ε)], (14)

but the transmission function T (ε) = ∑∞k=−∞ T

(k)LR (ε) in-

cludes the description of both light-driven (k = 0) and un-driven (k = 0) electron transport.

III. RESULTS AND DISCUSSION

A. Transmission spectra

Under the field-free condition, T(k)

LR (ε) = T(k)

RL (ε)= T

(k)LR (ε)δk0, Eq. (6) reduces to the Landauer formula

IR = 2eh

∫ ∞−∞ dεT (ε)[fR(ε) − fL(ε)], where T (ε) = T

(0)LR (ε).

The peaks and dips in Fig. 2 reflect the resonance (con-structive quantum interference) and antiresonance (de-structive quantum interference) of the tunneling electron,respectively.30, 54 In the weak molecule-electrode couplinglimit, the positions of peaks in each plot correspond tothe molecular orbital energies derived from the Hückelmodel.54 Fig. 2 shows that T(ε) of Molecules 1–6 dra-matically decreases in close proximity of ε = E0, whileMolecule 7 does not show a dip. Compared with Molecule7, Molecules 1–6 exhibit extremely small transmissionnear ε = E0. The transmission suppression originates fromantiresonance caused by the alkenyl groups.38–40 Generally,the antiresonance dip widens and deepens when the length ofthe molecular chain increases (the number of alkenyl groupsincreases). Note that within such a dip window where T(ε)is below 10−10, the current I ≈ eVSDT (ε)/π¯ ∼ 10−16 Ais experimentally undetectable under a small source-drainvoltage (VSD ∼ 0.05 V). That is, a tunneling electron cannotreadily pass through Molecules 1–6, indicating that theantiresonance condition can be utilized in the design of theoff-state of a single-molecule optoelectronic switch.

When the cross-conjugated molecules are exposed to aweak laser field, the dip (antiresonance) close to the Fermilevel vanishes. Considering the junction described in Fig. 1(b)and the polarization angle β = 0◦, the molecular dipole-field

−5 0 510

−8

10−6

10−4

10−2

100

ε−εF (eV)

T(ε

)

Molecule 1Molecule 3Molecule 5Molecule 7

−5 0 510

−8

10−6

10−4

10−2

100

ε−εF (eV)

T(ε

)

Molecule 2Molecule 4Molecule 6Molecule 7

(a) (b)

FIG. 2. Transmission functions of cross-conjugated molecules with (a) odd and (b) even number of alkenyl groups in the absence of the laser field for � = 0.40eV and εF = E0 (the on-site energy of pz-orbital on the carbon atoms). As a comparison, the transmission function of the polydiacetylene molecule (molecule7 in Fig. 1) is plotted in cyan.

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124703-4 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

hω (eV)

ε−ε F

(eV

)

0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

−10

−8

−6

−4

−2

0

hω (eV)

ε−ε F

(eV

)

0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

−10

−8

−6

−4

−2

0

hω (eV)

ε−ε F

(eV

)

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

−10

−8

−6

−4

−2

0

hω (eV)

ε−ε F

(eV

)

0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

−10

−8

−6

−4

−2

0

hω (eV)

ε−ε F

(eV

)

0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

−10

−8

−6

−4

−2

0

hω (eV)

ε−ε F

(eV

)

0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

−10

−8

−6

−4

−2

0

(a) (b)

(c) (d)

(e) (f)

FIG. 3. Transmission of cross-conjugated molecules with 1–6 alkenyl groups for εF = E0, A = 0.005 V/Å, and � = 0.4 eV. Here, ω is the laser frequency, ε

is the electron energy, and the colors denote the magnitude of transmission on a logarithmic scale. (a)–(f) correspond to the transmission spectra of Molecules1–6, respectively.

interaction reduces to

ern · E(t) = exnA cos(ωt), (15)

where xn is the x-coordinate of the nth carbon atom. Underthis condition, generalized parity symmetry is held, suchthat TLR(ε) = TRL(ε) = T(ε). By substituting Eq. (15) intoEq. (1) and using Eqs. (7)–(10), we can determine thetransmission function T(ε).

Fig. 3 shows T(ε) of Molecules 1–6 as a function oflaser frequency ω and electron energy ε, where the colordenotes the magnitude of the transmission on a logarithmic

scale. In each transmission plot, the straight lines correspondto tunneling processes that involve different photon absorp-tion or emission. For example, in Fig. 3(c), the horizon-tal lines C1 on top and C2 on bottom correspond to zero-photon assisted tunneling, and thus the lines have slope 0(i.e., the transmission lines do not vary with the laser fre-quency); the vertical positions of C1 and C2 correspond tothe HOMO and LUMO energies (∼±1.5 eV), respectively.Within the HOMO-LUMO gap, there are two groups oflines in mid-high frequency region (>1.2 eV): C3–C6 hav-ing slope −1 while C7–C10 having slope +1. These lines

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124703-5 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

with slopes −1 (+1), corresponding to tunneling processesfacilitated by emitting (absorbing) a photon, cross and formdiamond-shape areas. In the Hückel model, a longer cross-conjugated molecule (with more pz-orbitals) has more molec-ular orbitals, providing additional conduction channels. Thissituation is reflected by the number of the lines in the trans-mission spectra in Figs. 3(a)–3(f). Furthermore, in the lowfrequency region (¯ω < 1.2 eV, dark green area), we canalso see some bright lines with slope ±2 (C11–C14). Theselines correspond to two-photon assisted tunneling. Generally,in weak fields the transmission caused by one-photon assistedtunneling is much larger than that caused by two-photon as-

sisted tunneling, which can be understood from a perturbationanalysis.

B. Photon-assisted tunneling current

In order to reduce the effect of inelastic current andthe complexity of the modeling, we assume that the source-drain voltage is small (0.05 V), the chemical potentials of thetwo electrodes are symmetric, i.e., μL,R = εF ± VSD/2, andthe Fermi level εF is located halfway between the HOMOand LUMO (εF = E0). The current-frequency characteris-tics of Molecules 1–6 are depicted in Fig. 4 on linear and

1 2 3 4 5 6 7

2

4

6

8

10

x 10−12

hω (eV)

I (am

pere

)

1 2 3 4 5 6 7

10−12

10−11

1 2 3 4 5 6 7

2

4

6

x 10−11

hω (eV)

I (am

pere

)

1 2 3 4 5 6 7

10−16

10−14

10−12

1 2 3 4 5 6 7

5

10

15

x 10−11

hω (eV)

I (am

pere

)

1 2 3 4 5 6 7

10−16

10−14

10−12

10−10

1 2 3 4 5 6 7

1

2

3

x 10−10

hω (eV)

I (am

pere

)

1 2 3 4 5 6 7

10−18

10−16

10−14

10−12

10−10

1 2 3 4 5 6 7

2

4

6x 10

−10

hω (eV)

I (am

pere

)

1 2 3 4 5 6 7

10−20

10−17

10−14

10−11

1 2 3 4 5 6 7

2

4

6

8

x 10−10

hω (eV)

I (am

pere

)

1 2 3 4 5 6 7

10−20

10−17

10−14

10−11

(a) (b)

(c) (d)

(e) (f)

FIG. 4. Current-frequency characteristics of cross-conjugated molecules for εF = E0, VSD = 0.05 V, kBθ = 1.0 × 10−4 eV, � = 0.4 eV, and A = 0.005 V/Å.(a)–(f) correspond to the current-frequency characteristics of Molecules 1–6, respectively. The blue lines show the magnitude of the currents on a linear scale,while the green lines show the same on a logarithmic scale.

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124703-6 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

TABLE I. Molecular orbital (MO) energies of Molecule 3 (3 alkenylgroups) with respect to E0 based on the Hückel model calculation.

MO Energy (eV) MO Energy (eV)

1 −5.880 14 5.8802 −5.398 13 5.3983 −4.656 12 4.6564 −3.863 11 3.8635 −2.765 10 2.7656 −1.982 9 1.9827 (HOMO) −1.506 8 (LUMO) 1.506

logarithmic scales; the two scales permit focusing on dis-tinct photon-assisted tunneling features. As shown in eachlinear scale plot, Molecule 1, 2, 3, 4, 5, and 6 exhibit 3, 5,7, 9, 11, and 13 current peaks, respectively. That is, a cross-conjugated molecule with n alkenyl groups shows 2n + 1current peaks. Note that when the Fermi level εF is not lo-cated halfway between the HOMO and LUMO, the number ofcurrent peaks doubles. The positions of current peaks corre-spond to the intersections of εF and one-photon quasistates inthe transmission spectra25 (see Fig. 3). Moreover, the currentpeaks also reflect the energy difference between the Fermilevel of the electrodes and the resonant-state energies of thesingle-molecule junction. For instance, εF − E3, 7(HOMO)= E3, 8(LUMO) − εF = 1.506 eV corresponds to the firstcurrent peak in the linear scale in Fig. 4(c), where En1,n2

de-notes the energy of the n2th molecular orbital in Molecule n1,e.g., E2, 7 denotes the seventh orbital of Molecule 2. The otherpeaks on the linear scale in Figs. 4(a)–4(f) can be explainedin the similar way. Note that in the Hückel model, Molecule3 (with 3 alkenyl groups) has 14 molecular orbital energies,E3, 1–14, which appear in pairs with opposite energies with re-spect to εF (see Table I).

From Figs. 4(a) to 4(f), the position of the first one-photon assisted tunneling peak redshifts with the increasingmolecular chain length, indicating the narrowing gap betweenthe HOMO and LUMO. In addition, the overall magnitude

of tunneling current also rises. This occurs because a largerdipole is induced by the laser field in a longer molecule. Theheight of each peak on the linear scale in Fig. 4 can be under-stood by the overlap of the wavefunction of two quasistates,φ

n,∓1λ,ζ φ

n′,0λ,ζ . Note that a weak field of 0.005 V/Å can result in

detectable current peaks up to 10−11–10−10 A, indicating thatphoton-assisted tunneling can be applied to the design of theon-state of a single-molecule optoelectronic switch.

The current peaks on the logarithmic scale in the fre-quency range ¯ω <1.5 eV caused by the two-photon assistedtunneling are shown in Fig. 4. The peaks become readily dis-cernible for molecules containing more alkenyl groups. Forexample, in Fig. 4(c), we observe the current peaks caused bytwo-photon assisted tunneling at 0.75 and 0.99 eV, which areequal to (εF − E3, 7)/2 and (εF − E3, 6)/2, respectively. Notethat some current peaks caused by two-photon assisted tun-neling are obscured. For example, the peak at (εF − E3, 5)/2 ismasked by the effects of one-photon assisted tunneling.

Fig. 5 shows the current-field intensity characteristics ofMolecule 4 and 5 in the presence of a laser field for sev-eral frequencies. By varying the field amplitude A between5 × 10−5 V/Å and 0.2 V/Å, we can observe either quadratic orquartic dependence on the field amplitude. For the solid lines,one-photon assisted tunneling (the quadratic dependence) isdominant over the range of A < 0.1 V/Å. On the other hand,for the dashed lines, the current-field intensity characteristicsshow a one-photon feature in the weak field range and two-photon feature in the stronger field regime. The two results areconsistent with a previous study.25 Note that the field ampli-tude range of one-photon and two-photon assisted tunnelingdepends on the system and the frequency. Take the dashedlines in Fig. 5 as an example, for Molecule 4 two-photonassisted tunneling is dominant for A > 10−3 V/Å, while forMolecule 5 two-photon assisted tunneling is dominant for A> 10−4 V/Å. On the other hand, when the field amplitude islarger than 0.1 V/Å, the power law no longer holds due toStark shifting of the quasistates. The quadratic and quarticdependence on the field amplitude originates from the one-photon Green’s function G

(±1)vu (ε) ∝ E and the two-photon

10−4

10−3

10−2

10−1

10−20

10−15

10−10

A (V/angstrom)

I (am

pere

)

1.451 eV1.748 eV2.260 eV0.725 eV0.874 eV1.130 eV

10−4

10−3

10−2

10−1

10−20

10−15

10−10

A (V/angstrom)

I (am

pere

)

1.421 eV1.623 eV1.976 eV0.711 eV0.812 eV0.988 eV

(a) (b)

FIG. 5. Current-field intensity characteristics of (a) Molecule 4 and (b) Molecule 5 in a laser field at particular frequencies for εF = E0, VSD = 0.05V, � = 0.4eV, and kBθ = 1.0 × 10−4 eV. Solid (dashed) lines correspond to driving frequencies that are equal to the energy differences (half of the energy differences)between the MO energies and the Fermi level.

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124703-7 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

0 30 60 90 120 150 180

−5

0

5

x 10−10

β (degree)

I (am

pere

)

Molecule 1 (2.000 eV)Molecule 3 (1.506 eV)Molecule 5 (1.421 eV)

0 30 60 90 120 150 180

−5

0

5

x 10−10

β (degree)

I (am

pere

)

Molecule 2 (1.629 eV)Molecule 4 (1.451 eV)Molecule 6 (1.404 eV)

(a) (b)

FIG. 6. Current-polarization characteristics of cross-conjugated molecules with (a) odd and (b) even numbers of alkenyl groups at zero bias for εF = E0,A = 0.005 V/Å, � = 0.4 eV, and β ∈ [0◦, 180◦].

Green’s function G(±2)vu (ε) ∝ E2, respectively.25 Further dis-

cussion about field-amplitude power laws can be found in aprevious work.26 Our simulations indicate that, in addition tophenyl-acetylene macrocycles, the field amplitude power lawsshould be observed in the cross-conjugated systems.

C. Polarization effects

Besides amplitude and frequency, the field polarizationalso plays a key role in the optical control of electron dynam-ics. For example, laser polarization enables the manipulationof the aromaticity of benzene55, 56 as well as aromatic ring cur-rents in molecules.57–60 These studies motivate examining thelinear polarization effects on LDET through molecular junc-tions in the weak field regime.

The currents caused by one-photon assisted tunnelingin Molecules 1–6 are computed for a fixed field amplitudeA = 0.005 V/Å with the polarization angle β from 0◦ to180◦. In order to focus on the polarization effect, we setVSD = 0, which intuitively should result in a zero-currentwithout the driving field. The chosen field frequencies corre-spond to the gaps between the Fermi level and the HOMOof Molecules 1–6. The results show that molecules withodd and even numbers of alkenyl groups exhibit completelydifferent current-polarization characteristics, as shown inFigs. 6(a) and 6(b). For molecules with an odd number ofalkenyl groups (Molecules 1, 3, 5 in Fig. 1), the tunnelingcurrents are zero only at β = 0◦, 90◦, and 180◦; in betweenthese values, the magnitude and direction of the currentscan be fully modulated by the laser polarization. However,molecules with an even number of alkenyl groups (Molecules2, 4, 6 in Fig. 1) exhibit zero current over the entire polar-ization range. This can be explained by the generalized paritysymmetry SGP = (r, t) → (−r, t + π/ω).53 As discussed inSec. II, the symmetric transmission TLR(ε) = TRL(ε) holdsonly if GP is fulfilled. Since the molecules with odd numberof alkenyl groups are axially symmetric and the moleculeswith even number of alkenyl groups are centrosymmetric,the GP condition is met only at particular polarization angles(0◦, 90◦, 180◦) for the former, but at any polarization angle for

the latter. Consequently, the non-zero light-driven current atzero bias is allowed in Molecules 1, 3, 5, but not in Molecules2, 4, 6.

This symmetry-based property can be exploited in the de-sign of molecular optoelectronic devices. The field configura-tion aided by the molecular symmetry can serve as a type ofdevice “remote control,” which flexibly modulates the char-acteristics of the current.

IV. CONCLUSION

To summarize, we have presented a general principle foroperating a single-molecule switch based on cross-conjugatedsystems by utilizing destructive quantum interference andphoton-assisted tunneling. LDET through cross-conjugatedmolecules with odd and even numbers of alkenyl groupsare investigated over a broad range of field frequency andamplitude. In addition, generalized parity symmetry can beexploited to devise remote control via the modulation offield configuration together with the consideration of molec-ular symmetry. As the analysis is within the weak-fieldlimit, a perturbation treatment should yield similar resultsto those given by the Floquet method. One-photon and two-photon assisted tunneling can be observed in the transmissionplots, current-frequency characteristics, and current-field in-tensity characteristics. Moreover, in current-frequency char-acteristics, the current peaks resulting from one-photon as-sisted tunneling reveal the relation between the electronicstructure of the molecule and the electrode Fermi level.Multi-photon assisted tunneling allows for operation of amolecular device over a wide range of laser frequencies. Suchflexibility may enable new applications in molecular elec-tronics. Note that some physical processes such as photon-induced resonant tunneling61 cannot be described because ourapproach does not consider laser-induced electron excitationin the electrodes (Eq. (4) is time-independent). This issue willbe explored in a future study. We hope that this study mo-tivates additional experimental and theoretical investigationsinto single-molecule optoelectronic devices.

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124703-8 Hsu, Xie, and Rabitz J. Chem. Phys. 141, 124703 (2014)

ACKNOWLEDGMENTS

The authors acknowledge the financial support fromthe National Science Foundation (NSF) (Grant No. CHE-1058644), (U.S.) Army Research Office (USARO) (Grant No.W911NF-13-1-0237), and PPST of the United States.

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