Molecular Dynamics - Driven Liouville von

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Electric Current Fluctuations Induced by Molecular Vibrations in the Adiabatic Limit: Molecular Dynamics - Driven Liouville von Neumann Approach Tse-Min Chiang, Qian-Rui Huang, and Liang-Yan Hsu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b12555 • Publication Date (Web): 09 Apr 2019 Downloaded from http://pubs.acs.org on April 9, 2019

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The Journal of Physical Chemistry

Electric Current Fluctuations Induced by Molecular Vibrations in the Adiabatic Limit: Molecular Dynamics - Driven Liouville von Neumann Approach Tse-Min Chiang,†,‡ Qian-Rui Huang,† and Liang-Yan Hsu∗,† †Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan ‡Department of Physics, National Taiwan University, Taipei 10617, Taiwan E-mail: [email protected]

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Abstract We investigate time-dependent electron transport through a molecular junction in the adiabatic limit at the density-functional tight-binding level using the molecular dynamics - driven Liouville von Neumann (MD-DLvN) approach. When electron transport involves nuclear dynamics at finite temperature (∼ 70 K) within the NVE ensemble, we find that the steady-state current cannot be achieved even for a very short molecule (trans-fumaronitrile). Furthermore, to establish a relationship between electric current fluctuations and molecular vibrations, we analyze the similarities and differences between the current noise spectra and the MD power spectra. Our simulations show that not all normal modes can bring about current fluctuations. Furthermore, when a normal mode satisfies a particular symmetry, the normal mode can lead to frequency doubling of current fluctuations. This investigation offers new directions for studying electronic dynamics in a non-equilibrium open quantum system.

I. Introduction Molecular electronics has become an active field of research in physical chemistry and nanotechnology during the past two decades. 1–6 Recently, due to advances in experimental techniques, the conductance of a variety of molecules has been successively reported, including DNA, 7 peptides, 8 conjugated oligomers, 9 porphyrins, 10 and extended metal-atom chains. 11 In order to quantitatively analyze molecular conductance, the Landauer approach and its extensions (including scattering formulations and the non-equilibrium Green’s function method) have been commonly employed to explore time-independent electron transport in a molecular junction. 12–16 In the state-of-the-art simulations, molecular motions are usually assumed to be stationary. 12–15 Under this condition, one can compute steady-state current and the corresponding shot noise. 16 However, electric current fluctuations caused by factors other than shot noise, e.g., molecular vibrations and thermal reservoirs, 17 cannot be described by time-independent electron transport with a fixed molecular geometry. In fact, 2


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molecular vibrations play a crucial role in nearly all fields of physical chemistry. To better understand the effect of molecular vibrations on electric current fluctuations, we study time-dependent electron transport through a molecular junction by incorporating molecular dynamics (MD), simulate the electronic dynamics and the transient current, and analyze a relationship between the current noise and the MD power spectra. Electronic dynamics in an open quantum system arouses broad interest in physics and chemistry, and it is of significant importance when investigating electron transport under time-dependent fields 18–21 or coupled with nuclear dynamics. The original Landauer approach is extensively used to study steady-state current, but it cannot describe the behavior of transient electronic dynamics. To address this issue, several cutting-edge methods have been developed to treat time-dependent electron transport through a molecular junction, including time-dependent Green’s function approaches, 22–28 real-time path-integral Monte Carlo methods, 29 multilayer multiconfiguration time-dependent Hartree simulations, 30,31 and constrained density functional theory methods. 32–34 These methods greatly advance our understanding of electronic dynamics in an open quantum system, but the practical applicability of these methods are restricted by the heavy computational demands due to the sophisticated algorithms or the modeling of the semi-infinite leads. The modeling of semi-infinite leads can be circumvented by adopting complex absorbing potentials 35,36 or ring boundary conditions, 37,38 but these methods cannot adequately describe the long-time behavior of electronic dynamics. To capture the correct long-time behavior, Sánchez, 39 Subotnik, 40 Hod, 41 and Franco 42 proposed simple but accurate methods to deal with the relaxation of electronic dynamics in the semi-infinite leads. Particularly, the pioneering works of Sánchez 39 et al. and Subotnik 40 et al. provided the first form of the equation of motion. Nevertheless, the formulation did not consider the proper damping of the coherences, resulting in severe unphysical behavior of the electronic dynamics. Hod 41,43 et al. presented a heuristic derivation of the correct equation of motion (termed the driven Liouville von Neumann equation of motion) and the site-to-state transformation, which extended the methodology with realis-

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tic atomistic model systems. Subsequently, the implementation for non-orthogonal basis-set representations, 44 the first proof of correspondence to Lindblad equation 45 and a parameterfree version of the equation of motion were established. 46 Franco 42 et al. presented the first non-heuristic derivation of the equation of motion and applied it to the dynamic perturbations. Inspired by these pioneering works, we extend the driven Liouville von Neumann (DLvN) method and apply it to explore the effect of nuclear motions on electric current fluctuation. When nuclear motions are not stationary, the nuclear kinetic energy leads to vibronic coupling and the interaction of electrons and nuclei alters the molecular electronic structure. The two factors are crucial in molecular conductance, and they are associated with the origin of inelastic electron tunneling 47–53 and the geometry dependence of conductance. 54–56 However, most analyses of these two factors are still in the framework of time-independent electron transport. The electronic dynamics, e.g., transient current and its fluctuation, caused by molecular vibrations does not receive much attention. As a result, in this study, we focus on electronic dynamics induced by molecular vibrations from a time-dependent view. For the practical applicability, first, we limit our study within the Born-Oppenheimer approximation, i.e., no vibronic coupling. Second, we treat nuclear motions using classical mechanics, i.e., no nuclear quantum effect. The two simplifications allow us to explore electronic dynamics coupled to a number of normal modes in complicated molecular systems. We organize this article as follows. In section II, we introduce the model Hamiltonian, our simulation method, and how to compute the time-dependent electric current, the currentcurrent correlation function, and the mass-weighted velocity autocorrelation function. In section III, we demonstrate the molecular dynamics - driven Liouville von Neumann (MDDLvN) simulation on fumaronitrile (FN, trans-1,2-dicyanoethylene, shown in Figure 1(a)) and analyze the effect of molecular vibrations on time-dependent electron transport. In section IV, we briefly summarize and provide an outlook.

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II. Method Model Hamiltonian The total Hamiltonian of a molecular junction is divided into three regions of interest, which are the left lead, the central molecule, and the right lead, as shown in Figure 1(b). Hence, ˆ the total Hamiltonian H(t) can be written as 

 ˆ ˆ ˆ VL,M 0   HL   ˆ ˆ M (t) VˆM,R  H(t) = VˆM,L H ,   ˆ ˆ ˆ0 VR,M HR X q q ˆ L(R) = εL(R) cq† H L(R) cL(R) ,

(1)

(2)

q

ˆ M (t) = H

X

n εnM (t)cn† M cM +

n

VˆL(R),M =

X

X

0 n0 VMn,n (t)cn† M cM + h.c. ,

(3)

n,n0 n,q n VL(R),M cq† L(R) cM ,

(4)

q,n

ˆ L, H ˆ M (t), and H ˆ R correspond to the block matrix representation of the left lead, the where H central molecule, and the right lead, respectively. VˆL(R),M represents the coupling between † the left (right) lead and the molecule and VˆM,L(R) = VˆL(R),M . The left (right) lead is modeled

by numbers of states with energies εqL(R) and with equal energy spacing ∆εqL(R) in the interval [Emin , Emax ], 42 and cqL(R) (cq† L(R) ) annihilates (creates) an electron with a quantum number q in the left (right) lead. cnM (cn† M ) is the annihilation (creation) operator for an electron with 0

energy εnM on the atomic orbital n of the central molecule. VMn,n (t) denotes the coupling ˆ M (t) is time-dependent because between different atomic orbitals in the molecule. Note that H ˆ M (t)) changes with the positions the molecular electronic structure (the matrix elements of H of the nuclei generated by the MD simulations. The matrix elements of the molecule-lead coupling (VˆL(R),M ) are set up according to the previous study 42 and we will discuss this in the subsection of Electronic Dynamics (Eqs. (7) and (8)).

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(a)

(b)

Figure 1: (a) Molecular structure of fumaronitrile (FN, trans-1,2-dicyanoethylene). (b) The structure of a molecular junction: the left electrode, the FN molecule, and the right electrode. In the simulation of electronic dynamics, we attach the FN molecule to two phenomenological driven lead models directly. In the simulation of molecular dynamics, we only consider the dynamics of the vinyl group in the FN molecule and fix its cyano groups. ˆ M (t) Moreover, we perform the unitary transformation for the molecular Hamiltonian H ˆ˜ (t) = Uˆ † (t)H ˆ M (t)UˆM (t). Due to from the atomic basis set to the energy basis set, i.e., H M M the non-orthogonality of the atomic orbitals, the unitary matrix for the molecule UˆM (t) has † ˆ 44 Hence, when the molecular Hamiltonian is diagonalized a relation UˆM (t)SˆM (t)UˆM (t) = I.

(energy basis), the total Hamiltonian and its unitary matrix are expressed as 

Vˆ˜L,M (t)



ˆ ˆ0   HL   ˆ † ˆ ˆ ˆ ˜ ˆ ˆ ˆ  ˜ M (t) V˜M,R (t) H(t) = U (t)H(t)U (t) = V˜M,L (t) H ;   ˆ ˜ ˆ ˆ0 VR,M (t) HR





ˆ ˆ0 ˆ0 I    ˆ  U (t) = ˆ0 UˆM (t) ˆ0 .   ˆ ˆ0 ˆ0 I

(5)

Electronic Dynamics To focus on the electronic dynamics of the central molecule and reduce the complexity of the two leads, we assume the same relaxation rate γ for each state in the two leads, which maintains the lead (reservoir) in thermal equilibrium. Under this condition, the electronic

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dynamics of the total system can be described by the DLvN equation as follows, 41,42,44 

i ˆ d ˜ ρˆ(t) = − H(t), ρˆ(t) − dt ~   ρˆL (t) ρˆL,M (t)  ρˆ(t) =  ρˆM,L (t) ρˆM (t)  ρˆR,L (t) ρˆR,M (t)

ρˆ0L

ρˆL (t) − γ  1 ρˆ (t) M,L ~  2 ρˆR,L (t)  ρˆL,R (t)   ρˆM,R (t) .  ρˆR (t)



1 ρˆ (t) 2 L,M

ρˆL,R (t)   , 1 ˆ0 ρ ˆ (t)  2 M,R  1 0 ρ ˆ (t) ρ ˆ (t) − ρ ˆ R R 2 R,M

(6)

Here, ρˆ0L(R) represents the density matrix of the left (right) lead in thermal equilibrium h q i−1 q according to the Fermi-Dirac distribution, fL(R) (εL(R) ) = 1 + exp (εL(R) − µL(R) )/kB T . Here, εqL(R) is the eigenstate of the finite lead model, kB is the Boltzmann constant, µL(R) is the chemical potential, and T is the temperature. The relaxation rate in the DLvN approach is related to the imaginary part of the selfenergy Γn,q L(R) contributed by the leads in the Landauer approach, and they can be expressed as: 42

n,q 2 Γn,q L(R) ≈ 2πη|VL(R) | , X X δ(ε − εqL(R) ) = η= q

q

(7) γ/π (ε −

εqL(R) )2

+ γ2

,

(8)

where η is the density of state of the left (right) lead, and the Lorentzian broadening is associated with the relaxation rate of each state in the left (right) lead. Based on the wide-band limit (WBL) approximation, Γn,q L(R) is independent of q (a constant), and one n,q can determine the value of VL(R) from Γn,q L(R) and η. Note that in this study we consider n,q the molecule-lead coupling VL(R) 6= 0 only when n belongs to the pz orbitals of the atoms

attached to the leads. The density matrix ρˆ(t) at time t (t ≥ t0 ) could be calculated numerically by direct integration of Eq. (6) with a given initial condition ρˆ(t0 ). The initial condition ρˆ(t0 ) is the 7



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steady-state density matrix derived by solving Eq. (6) based on an initial molecular geometry. The current is calculated according to 44 ( ) X |e| X X ˜ k,q k,q I(t) = VM,L (t)Im[ρk,q V˜M,R (t)Im[ρk,q M,L (t)] − M,R (t)] , ~ k q q

(9)

where e is an elementary charge and k is the index of the molecular orbital (the eigenstate ˆ M (t)). of H We analyze the electric current fluctuation by computing the current-current correlation function SI (t0 ) and its Fourier transform SI (ω) 16 SI (t0 ) = h∆I(t)∆I(t + t0 )i, Z ∞ 0 SI (ω) = 2 dt0 eiωt SI (t0 ).

(10) (11)

−∞

Here, the subscript I stands for current, the angle bracket h...i denotes the time average, and ∆I(t) = I(t) − hIi is the fluctuation in respect to the average current at every timestep.

Nuclear Dynamics We neglect the vibronic coupling and consider the electronic dynamics in the framework of the Born-Oppenheimer (BO) approximation. The nuclear motion and the electronic wavefunction are treated separately. The effect of the nuclear motion is implicitly included into ˆ M (t) in Eq. (6) via the MD simulation. For convenience, we refer the matrix elements of H to our approach as molecular dynamics-Driven Lioville von Neumann (MD-DLvN) method because the method incorporates MD into the DLvN equation. The MD-DLvN method is performed as follows. At the outset, the molecular geometry at each timestep is obtained by performing the molecular dynamics (MD) simulations; meanwhile, the molecular Hamiltoˆ M (t) and the corresponding overlap matrix SˆM (t) are constructed at every timestep. nian H ˆ M (t) derived from the MD simulaSubsequently, we incorporate the matrix elements of H

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tions into Eq. (6). Note that our approach is not a self-consistent approach, i.e., the density matrix derived from Eq. (6) does not alter molecular dynamics. To analyze the correlation between the current fluctuation and the nuclear dynamics, we compute the mass-weighted velocity autocorrelation function Sv (t0 ) and its power spectrum Sv (ω) according to 57,58

Sv (t0 ) =

N X

mα

α=1

Z

X hvαi (t) · vαi (t + t0 )i , hv (t) · v (t)i αi αi i=x,y,z

∞

Sv (ω) = 2

0

dt0 Sv (t0 ) · eiωt ,

(12) (13)

−∞

where the subscript v stands for velocity, mα denotes the mass of the α-th atom, vαi denotes the velocity of the atom, and Sv (ω) denotes the power spectrum, i.e., the Fourier transform of Sv (t0 ).

Computational Details We perform the MD simulations of the central FN molecule in a microcanonical (NVE) ensemble with the DFTB+ package. 59,60 The initial molecular geometry of the FN molecule is optimized first; in the MD simulation, we fix the cyano groups in the FN molecule, and we only allow the central vinyl groups to move. In principle, the cyano groups can have various molecule-lead configurations which can considerably influence the properties of electron transport. However, the emphasis of the molecule-lead configurations would obscure the main purpose of the work. In order to clearly demonstrate the relationship between the current noise spectra and the molecular vibrations, we reduce the number of normal modes by considering only the dynamics of vinyl group. The MD timestep is 0.01 fs, and the initial velocities of the atoms are set according to the Maxwell-Boltzmann distribution at 70 K, with a random seed 11044. 59 We use the mio-1-1 60 Slater-Koster files in the DFTB3 61 calculations, and the Hubbard derivatives of H, C and N atoms are set as -0.16 au. The 9



ˆ M (t) and SˆM (t), are molecular Hamiltonian and the corresponding overlap matrix, i.e., H obtained at every MD timestep. The electronic dynamics is performed using the fourth order Runge-Kutta method (RK4). The timestep of the numerical integration is 0.01 fs, and the total simulation time is 5000 fs. We attach the FN molecule to two phenomenological driven lead models directly, as shown in Figure 1(b). For the FN molecule, the HOMO-LUMO band gap is 4.359 eV. The large band gap indicates that the vibronic coupling should be small, and it is appropriate to neglect the coupling as the first order approximation. The chemical potential is set as µL = EF + Vb /2 and µR = EF − Vb /2, in which the bias voltage Vb is set as 0.3 V and we assume that the location of the Fermi level EF is at the midpoint between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). The conditions of the large band gap and the small bias indicate that the non-equilibrium effect should not be significant in our system, and it is reasonable to adopt a non-self-consistent approach to model nuclear motions, i.e., the molecular dynamics is not influenced by the density matrix derived from Eq. (6). The electron temperature is also set as 70 K. The imaginary part of the 62 self-energy Γn,q which corresponds to the sulfur-gold link in a gold-PDTL(R) is set as 0.4 eV,

god system (PDT: phenyl dithiol). For the cyano-gold link, Γn,q L(R) is smaller than 0.4 eV, but the choice of 0.4 eV does not affect any conclusion in this work. The relaxation rate γ is set as 0.005 eV. The energy interval [Emin , Emax ] is chosen as [EF −0.6 eVb , EF +0.6 eVb ] and the energy spacing ∆εqL(R) is equal to 0.001 eV (Recall the subsection of Molecular Hamiltonian). Details of the choice of the parameters 40,63–65 γ, [Emin , Emax ], and ∆εqL(R) can be found in the Supporting Information (SI).

III. Results and Discussions In this study, we use three methods to calculate electric current: 1. Landauer method: we follow the standard scattering approach without the calculation 10


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of the non-equilibrium density matrix. 16 It is appropriate to neglect the non-equilibrium effect when the simulation is under the low-bias condition and the frontier molecular orbitals are far away from the Fermi levels of the two leads. 2. DLvN (steady state): we assume that the current reaches a steady state at every timestep in the MD simulation, so one can obtain the steady-state current at each fixed molecular geometry by solving the Sylvester equation 40 with the condition

d ρˆ dt

= 0 (See

Supporting Information). 3. MD-DLvN method: it corresponds to the transient current computed via Eq. (6) ˆ M (t) is time-dependent. coupled to nuclear dynamics, i.e., H

Benchmark Evaluations The main purpose of this work is to explore the effect of molecular vibrations on electric current fluctuations in a molecular junction. Before studying the transient current and its fluctuations, we have to first verify whether the DLvN method and the Landauer method can give the same characteristics of the steady-state currents when molecular geometry changes. In the previous study, Hod et al. have already demonstrated that the DLvN method and the Landauer approach give the same steady-state current for a particular fixed molecular geometry based on a chosen fixed parameter γ. 41 In fact, when the molecular geometry changes, the trends of the steady-state currents given by the two methods may be different under a fixed parameter γ. In order to eliminate this concern, we start by a comparison of the steady-state electric current calculated using the DLvN approach and the Landauer current. Figure 2 shows that the characteristics of the currents derived from the two methods are almost identical at each timestep. The difference of the fluctuations of the two currents is minute and nearly the same (around 7.5%) at all times, indicating that the DLvN (steady-state) method captures the characteristics of Landauer current when the current fluc-

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tuations are induced by molecular vibrations. Note that the currents derived from the DLvN (steady-state) method and the Landauer method is under the assumption of the steady-state condition, which relies on the condition that the nuclear dynamics is much slower than the electronics dynamics so the steady state can be reached in every timestep. In fact, the steady-state condition may not be satisfied when nuclear dynamics is involved in electron transport. We will discuss this issue in the next section.

100

Landauer DLvN(steady state)

Difference(%)

30 80

I(nA)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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60

20 10 0

40

0

250 500 750 1000

t(fs)

20 0

0

200

400

t(fs)

600

800

1000

Figure 2: Comparison of Landauer current and the steady-state current calculated using the DLvN method in the wide-band limit approximation (Γn,q L(R) = 0.4 eV used in the Landauer method corresponds to γ = 0.005 eV used in the DLvN (steady-state) method). The difference of the current values given by the two approaches does not change significantly (around 7.5%), indicating that the DLvN method almost captures the characteristics of Landauer current.

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Transient Current Influenced by Nuclear Dynamics When nuclear dynamics is involved in electronic dynamics, the steady-state current and the transient current exhibit different behaviors. The comparison of two currents calculated using the DLvN (steady-state) method and the MD-DLvN method are shown in Figure 3. The transient current indeed resembles the result in Figure 2, where the electric current is assumed to reach a steady state at every timestep. The average current is nearly the same, which are 30.064 (nA) for the DLvN (steady-state) approach and 30.062 (nA) for the MDDLvN approach. Apparently, both currents oscillate at a frequency of 54.6 THz (∼ 1820 cm−1 ). In fact, this frequency corresponds to a C=C stretching mode, which is a bit higher than ordinary C=C bonds since we fixed the surrounding functional groups. Nevertheless, the amplitudes of the two currents at each timestep are somewhat different; more explicitly, there is a phase shift (∼ 3 fs) between the two currents, indicating that the electric current does not always reach a steady state in every timestep even for a very short molecule when molecular vibrations are involved in electron transport. With these issues in mind, we will now take a look at whether the transient current could match the steady-state current better if the nuclear motion is slower. To slow down the molecular vibrations, we replace the central vinyl group with their hypothetical isotopes which are ten times heavier (mC = 120 amu, mH = 10 amu), that makes the nuclear motion √ 10 times slower. Figure 3(b) and (d) show a comparison of electric currents through the FN molecule derived from the MD-DLvN method and the DLvN (steady-state) method, and both currents oscillate at a frequency of 17.4 THz (∼ 580 cm−1 ), which is

√1 10

of those in

Figure 3(a) and (c). Moreover, we find that the transient current nearly coincides with the steady-state current when the molecular vibrations are much slower. For real molecules, there are usually many vibration modes with frequencies higher than this hypothetical case (580 cm−1 ), implying that transient current cannot reach a steady state above 70 K in most cases.

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(a)

40.0

MD-DLvN DLvN(steady state)

35.0


I(nA)

35.0

I(nA)

37.5

32.5

32.5

30.0

30.0

27.5

27.5

25.0

25.0

22.5

(b)

40.0

37.5

0

40.0

1000

2000

t(fs)

(c)

3000

4000

5000


22.5

35.0

35.0

1000

2000

t(fs)

3000

(d)

4000

5000


I(nA)

37.5

32.5

32.5

30.0

30.0

27.5

27.5

25.0

25.0

22.5 1800

0

40.0

37.5

I(nA)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1825

1850

1875

1900

t(fs)

1925

1950

1975

2000

22.5 1500

1600

1700

t(fs)

1800

1900

2000

Figure 3: Comparison of electric currents calculated using the MD-DLvN approach and the DLvN (steady-state) method. (a)(c) Electric current through the original FN molecule. (b)(d) Electric current through the FN molecule with the atoms in the central vinyl group replaced with their hypothetical heavier isotopes (mC = 120 amu, mH = 10 amu).

Current Fluctuation Influenced by Nuclear Dynamics Next we consider the relationship between molecular vibrations and current fluctuations. To find their correlation, we analyze the characteristics of the current noise spectra (current fluctuations) and the MD power spectra (molecular vibrations) as shown in Figure 4.

14


0 10−18

(a)

1000

cm−12000

3000

0 10−18

(b)

cm−12000

3000 MD-DLvN MD

10−21 10−22

DLvN(steady state) MD-DLvN

0.1

0.2

eV

0.3

0.4

101 100

10−19

10−20

10−23 0.0

1000

S(A 2/Hz)

10−19

S(A 2/Hz)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


MD power spec rum (arb. uni )

Page 15 of 30

10−20

10−1

10−21

10−2

10−22

10−3

10−23 0.0

0.1

0.2

eV

0.3

0.4

Figure 4: (a) Current noise spectra based on the MD-DLvN approach and the DLvN (steadystate) method. The characteristics of these two spectra are nearly the same, with slight deviation in the amplitudes. (b) Comparison of the current noise spectrum based on the MD-DLvN approach (red line) and the MD power spectrum (green line). Note that most of the peaks of the red line have counterparts in the green line, except for the peak of the lowest frequency; the first peak of the red line is located at 0.02316 eV, exactly two times of the first peak at 0.01158 eV in the green line. It should be noted that not all vibration modes affect the electric current.

Figure 4(a) shows the current noise spectra of the two currents in Figure 3(a). The primary peaks in the two spectra are nearly overlapped except for the slight deviation in the amplitudes. Figure 4(b) is the comparison of the MD power spectrum and the current noise spectrum based on the MD-DLvN approach. The peaks of the current noise spectrum have counterparts in the MD power spectrum, but not the other way around; in the MD power spectrum, we can see that some vibrational modes affect the current fluctuation significantly (for example, the peaks at 0.1191 eV and 0.2258 eV), but some of them only produce a tiny influence (for example, the peaks at 0.01158 eV and 0.09016 eV). Figure 5 illustrates four selected normal modes (0.01159 eV, 0.09021 eV, 0.1190 eV and 0.2266 eV), which corresponds to the four strongest peaks (0.01158 eV, 0.09016 eV, 0.1191 eV and 0.2258 eV) in the MD power spectrum in Figure 4(b). The details of the normal mode analysis of the FN molecule are summarized in Supporting Information. 15



We accentuate the fact that the frequency of the first peak (0.02316 eV) in the current noise spectrum is exactly twice of the frequency of the first peak in the MD power spectrum (0.01158 eV). The frequency doubling can be attributed to the anti-symmetric nature (with respect to C2 -rotation and reflection in the C2h point group) of the corresponding normal mode, mode 1, which has a symmetric potential energy surface. Consequently, both the positive and negative displacements along this normal mode vector result in the equivalent molecular structures, leading to the same current response. Therefore, the frequency of current response is twice of the frequency of this normal mode. It is noteworthy that the frequency doubling here is due to the anti-symmetric normal mode, and has nothing to do with second-harmonic generation or non-linear processes.

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(a) mode 1 (93.47 cm−1 , 0.01159 eV)

(b) mode 4 (727.62 cm−1 , 0.09021 eV)

(c) mode 6 (959.95 cm−1 , 0.1190 eV)

(d) mode 10 (1827.41 cm−1 , 0.2266 eV)

Figure 5: Normal mode displacement vectors of the four selected vibrational modes.

As the aforementioned, the current fluctuations caused by the normal modes can be quite different. To understand the effect of the normal modes, we modify the geometry of the FN molecule along the normal mode vectors, and then use the Landauer method to calculate their transmission functions. Figure 6 shows the variations of the transmissions due to the four selected normal modes. The red line, the blue line, and the dashed line in each subfigure

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correspond to the total displacement as −0.1˚ A, 0 and +0.1˚ A, respectively. For a normal mode associated with a larger current response, its variation in the transmission function is also more significant. In addition, as shown in Figure 6(a) and (c), since mode 1 and mode 4 are anti-symmetric modes (as shown in Figure 5), their molecular geometries with the total displacements ∆q and −∆q are equivalent. The equivalent molecular geometries lead to the same transmission, which also explains the frequency doubling of mode 1 in the current noise spectra. In Figure 4(b), a primary peak (0.09021 eV) in the MD power spectrum is contributed by mode 4. Mode 4 should have a frequency doubling (0.18041 eV) in the current noise spectrum. However, the variation of the transmission contributed by mode 4 is too small as shown in Figure 6(b), so it is difficult to distinguish that peak in the current fluctuations in Figure 4(b).

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1.28 1e−3

1.28 1e 3

1.27

1.27

(a) mode-1 Δq

1.25 1.24 1.23

1.23

1.21

1.21

1.8

−5.5

1e 3

−5.4

−5.3

E(eV)

−5.2

1.20

−5.1

(c) mode-6

5

q

Δ

−Δq 0 Fermi level

Transmission

1.5

5.6

5.5

1e 3

1.7 1.6


1.24

1.22

−5.6

q

Δ

1.25

1.22

1.20

(b) mode-4

1.26


Transmission

Transmission

1.26

Transmission

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.4 1.3

4 3

5.4

5.3

5.2

5.1

5.4

5.3

5.2

5.1

E(eV)

(d) mode-10 q

Δ


2

1.2 1

1.1 1.0

5.6

5.5

5.4

5.3

E(eV)

5.2

5.1

5.6

5.5

E(eV)

Figure 6: The variations of the transmissions due to the four selected normal modes, with the normal mode displacement equal to 0 (blue), ∆q (red) and −∆q (dashed), where ∆q = 0.1˚ A. In (a) and (b), ∆q and −∆q lead to a similar change in the transmission functions due to the anti-symmetric normal mode vectors, which explains the frequency doubling of the current fluctuation due to mode 1 and mode 4. In (c) and (d), the differences of the transmission functions among the three the normal mode displacement are more significant, which indicates that mode 6 and mode 10 dominate the change in current fluctuations.

IV. Conclusion Electric current fluctuation in a molecular junction is a fundamental topic in statistical mechanics and physical chemistry, but the role of molecular vibrations has not been elucidated in detail. In the framework of the Born-Oppenheimer approximation, we have demonstrated several effects of molecular vibrations on transient currents through the FN molecule (a very short molecule). First, although our study is limited within the NVE ensemble and the 19



adiabatic limit, but it still gives an important physical insight: when a specific molecule has a weak vibronic coupling in vacuum, steady-state current cannot be achieved even for a very short molecular length. Moreover, for the FN molecule, the mean value of the current given by the DLvN steady-state approach and the mean value of the MD-DLvN transient current are nearly the same, but their fluctuations at each timestep are different. In addition, the peaks in the MD power spectra do not always correspond to the peaks in the current noise spectra. It means that a normal mode with larger contribution to the MD power spectrum does not guarantee a larger response in current fluctuations. Besides, we find the correspondence between the anti-symmetric normal modes and the frequency doubling in current fluctuations. Finally, we would like to emphasize that our conclusions are not limited to a specific molecule. The MD-DLvN simulation on a trans,trans-Mucononitrile system (see the Supporting Information) also exhibits apparent differences between the DLvN steady-state current and the MD-DLvN transient current. Although we have clearly demonstrated the role of molecular vibrations in time-dependent electron transport, several issues remain to be considered and further explored. First, in the ˆ M (t) is obtained from the DFTB+ package. present approach, the molecular Hamiltonian H ˆ M (t) is constructed from the density matrix of a closed system, not from the That is, H non-equilibrium density matrix ρˆM (t) in Eq. (6). To explore the non-equilibrium effect, it ˆ M (t). is needed to incorporate the non-equilibrium density matrix into the construction of H Second, the MD simulations are performed in the NVE ensemble and it is reasonable for an experimental setup in vacuum. In fact, for self-assembled monolayers or a molecule junction in solvent, the thermal reservoirs may lead to different results. To study the effect of thermal reservoirs, the consideration of the MD simulations in the NVT ensemble is required. Third, we neglect vibronic coupling in the present study and it is the origin of the nonadiabatic effect. How to simply include vibronic coupling into the MD-DLvN method is a challenging task. Fourth, the present MD-DLvN method is not limited to the wide-band limit approximation, and it is straightforward to include the effect of leads and extend the present method to a

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parameter-free version. 46 Fifth, in this work, we focus on the relationship between electric current fluctuations and the nuclear vibrations of the central molecule, i.e., we assume that the molecule-lead coupling is time-independent. In fact, the molecule-lead coupling should fluctuate and depend on time. When a few atoms of electrodes are included, one can expect that the current noise spectra should become more complicated. It deserves further study on analyzing the effect of the molecule-lead coupling on the current fluctuation. Sixth, we assume that the molecular vibrational modes are decoupled from the phonon baths of the leads. The phonon baths can give and take the energy of molecular vibrational modes, leading to the changes of electronic dynamics and current fluctuations. The observed inability to reach steady-state may be removed by including the effect of the phonon baths, but the impact of the phonon baths of the leads on current dynamics needs to be further clarified. Time-dependent electric transport coupled to nuclear motions is an emerging research area. We hope that the present study will motivate further investigations on electronic dynamics in a non-equilibrium open quantum system and the behaviors of current fluctuations in molecular electronics.

Supporting Information Available The Supporting Information is available free of charge via the Internet at http://pubs.acs.org. Choice of parameters; Sylvester equation; Cartesian coordinate of the fumaronitrile molecule; vibrational analysis; MD-DLvN simulation on trans,trans-Mucononitrile.

Acknowledgement The authors thank Prof. Oded Hod and Dr. Tamar Zelovich for useful discussions (the parameter-free DLvN method and the Sylvester equation), Mr. Chuan-Bin Huang for the preliminary study, and Dr. Yae-Lin Sheu for constructive advice. This research was supported by Academia Sinica and the Ministry of Science and Technology of Taiwan (MOST 21



106-2113-M-001-036-MY3).

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