Dephasing in a Molecular Junction Viewed from a Time-Dependent

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Dephasing in a Molecular Junction Viewed from a Time-Dependent and a Time-Independent Perspective Hasan Rahman, Patrick Karasch, Dmitry A. Ryndyk, Thomas Frauenheim, and Ulrich Kleinekathöfer J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00955 • Publication Date (Web): 12 Mar 2019 Downloaded from http://pubs.acs.org on March 21, 2019

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The Journal of Physical Chemistry

Dephasing in a Molecular Junction Viewed from a Time-Dependent and a Time-Independent Perspective Hasan Rahman,† Patrick Karasch,‡ Dmitry A. Ryndyk,∗,‡ Thomas Frauenheim,‡ and Ulrich Kleinekath¨ofer∗,† †Department of Physics and Earth Sciences, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany ‡Bremen Center for Computational Materials Science, Department of Physics, University of Bremen, Am Fallturm 1, 28359 Bremen, Germany E-mail: [email protected]; [email protected]

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The Journal of Physical Chemistry 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

Abstract The current through a molecular junction can be determined numerically in a multitude of ways. Some of these methods like the Landauer scheme are only valid for coherent transport and the steady state regime while other schemes are able to treat time-dependent electronic currents across molecular junctions subject to fluctuating environments. In time-independent formalisms, the effect of thermal environments can be introduced in several ways. Here we focus on the vibronic dephasing model within a Green’s function approach. This time-independent scheme is contrasted with the time-dependent non-equilibrium Green’s function (TD-NEGF) approach in which the effect of the thermal environment manifests itself in fluctuating site energies. It is found that the time-averaged results of the TD-NEGF approach agree excellently with those from the vibronic model. This numerical comparison helps to better understand the connection between these different pure dephasing schemes which both have their advantages and specific applications.

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Introduction

In recent years the scientific interest in the charge transport characteristics of molecular junctions has grown enormously, both theoretically 1–5 as well as experimentally. 6–11 Molecular junctions form integral elements in molecular electronics and their theoretical understanding is essential for the development of functional molecular systems such as nanoscale electronics. 12–14 The molecules utilized in such junctions for the formation of molecular wires often consist of conjugated structures. 15–18 Another class of systems contains biological molecules like DNA due to their important role in the life sciences as well in as in electronic applications. 19–21 Experimentally, molecular junctions may be realized, e.g., as mechanically controlled break junctions or as scanning tunneling microscopy break junctions allowing a direct current measurement. 4 Moreover, light-field-driven currents have been of considerable interest recently. 22–24 2


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To model the current flow through a single molecule, several theoretical approaches have been developed. The coherent time-independent charge transport can be calculated by means of the Landauer-B¨ uttiker approach (also known as the scattering method), which is practically formulated in terms of the Green’s function technique. 25–28 Apart from scattering approaches, nonequilibrium Green’s function (NEGF) based approaches and quantum master equations (QME) schemes 29 are among the alternative approaches which may be utilized for the determination of the charge dynamics through molecular junctions. 30,31 When charge transport is influenced by environmental effects and temperature-dependent vibrations, quantum coherence is not preserved and scattering approaches are not sufficient as they are often limited to coherent scenarios. To overcome this drawback, the LandauerB¨ uttiker method can be extended by the B¨ uttiker probe (BP) model 32–35 which allows taking decoherence effects into account empirically. Note that in the present study all vibrational modes associated with any nuclear vibration are denoted as phonons. 1 The coupling to these phonons induces decoherence effects, 36–39 influences conduction 40,41 and also has functional applications, e.g., in molecular switches. 42 In theoretical simulations the conducting molecule is in many cases assumed to be in a static conformation, i.e., rigid. Under the influence of fluctuations in the surrounding environment 43 or due to external time-dependent fields such as laser pulses, 6,7,44,45 the molecule in the junction needs, however, to be treated time-dependently. Time-dependent effects can be taken into account using quantum master equations (QMEs), 29,46–48 which are usually based on a perturbative treatment of the molecule-lead coupling. Therefore, the validity range of QME approaches is limited to the case of weak coupling between molecular wire and leads. It is possible, though, to calculate higher-orders of this perturbative treatment systematically in a hierarchical fashion 49 and in this manner to deal with strong system-lead coupling strengths. Furthermore, electron interaction effects on the current dynamics can easily be incorporated in the QME formalism. 46,50 Another well-known method to calculate charge transport is the many-body nonequilibrium Green’s function formalism 27,28 which is not lim-

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ited to coherent motion of electrons. It allows taking into account decoherence phenomena caused by physical sources like electron-electron interaction or electron-phonon coupling. Approaches based on time-dependent nonequilibrium Green’s functions (TD-NEGFs) 51–54 facilitate the treatment of time-dependent effects in charge transport through quantum systems. An efficient version 55 of the TD-NEGF formalism that builds upon the decomposition of the Fermi function and the spectral densities 51 is utilized in the present study. This TDNEGF formalism is known to be exact for describing the non-interacting case, i.e., when electron interactions are neglected. Moreover, this omission of correlation between particles consequently leads to the fact that all involved matrices scales as N × N with N being the number sites in the system. This enables the TD-NEGF scheme to be more efficient than QME approaches for the calculation of the dynamics of larger systems.

a)

b)

ΓL(ε) ΓR(ε)

μL E11

E2

EN

ΓL(ε) ΓR(ε)

μL

μR

E1

E2

Env.

Env.

EN

μR

Env.

Figure 1: Schematic representation of the dephasing models investigated here. In the timedependent case a) dephasing is caused by site fluctuations while in the time-independent case b) dephasing is the result of static couplings to phonon environments. There are several options how to include decoherence effects in time-independent charge transport calculations. One way is to use the B¨ uttiker probe model which relies on the idea of virtual probes coupled to the system. There are two main variants of this probe technique, the “dephasing” probes which allow only elastic scattering and the “voltage” probes also allowing for inelastic-dissipative scattering. 35,56 Here we focus on the former variant in which the strength of the dephasing is chosen empirically. On the other hand, a more physical approach is to go beyond the coherent theory and to use an NEGF method which allows in4


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cluding electron-phonon interactions directly in the so-called vibronic dephasing model, 57–61 which will be called VD-NEGF approach in the following. In the elastic approximation, the electron-phonon coupling can be taken into account by an additional self-energy, which depends on the Green’s function at the same energy. A comparison of the dephasing B¨ uttiker probe and the VD-NEGF approach has been reported earlier. 61 As mentioned already above, environmental effects on the charge transport through a molecular junction can also be treated in a time-dependent manner using, e.g., the TDNEGF formalism. The vibrations of the environment but also of internal vibrations lead to time-dependent fluctuations of the site energies which in turn influence the conduction. The TD-NEGF scheme has been compared in detail to an ensemble of Landauer calculations. 62,63 The latter scheme has been found to be reasonable when delocalization of the molecular eigenstates responsible for transport is not changed drastically and when the charge transfer between molecular wire and the respective contacts is fast with respect to the molecular correlation time. 63 The VD-NEGF scheme is an improvement of the original Landauer formalism and naturally the question arises how the VD-NEGF scheme compares to the TD-NEGF approach. Although both methods describe physical decoherence phenomena induced by phonons, the way the electron-phonon interaction is realized is quite different. The aim of this paper is to compare both models and to understand the similarities as well as the differences. A schematic diagram of the two models is given in Fig. 1. In the next Section the two scheme are described in more detail while Section 3 describes how the two models can be compared. Thereafter, the results are shown and discussed. Finally, the findings of the comparative study are summarized in the last Section.

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2 2.1

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Description of the model and methods Description of the Hamiltonian and the couplings

As usual, the molecular junction is modeled as a chain of N sites in contact with fermionic reservoirs. The total Hamiltonian of the molecular wire sandwiched between two electrodes, which are assumed to be in equilibrium, can be written as

ˆ ˆ S (t) + H ˆR + H ˆ SR + H ˆB + H ˆ SB H(t) =H

(1)

ˆ S (t) describes the time-dependent electronic properties of the relevant system part, where H namely the molecular wire which actually can consist of a single molecule only. It can be represented as a quantum wire by using a nearest-neighbor tight-binding model

ˆ S (t) = H

N X

Ei (t)c†i ci

− ∆(t)

i=1

N X

c†i ci+1

+

c†i+1 ci

(2)

i=1

where c†i (ci ) denotes the creation (annihilation) operator at site i, Ei (t) the time-dependent onsite energies and ∆(t) the possibly time-dependent charge transfer integral between to adjacent sites. The leads are defined as fermionic (electronic) baths

ˆR = H

X X α=L,R

εα,k b†α,k bα,k

(3)

k

where the operator b†α,k (bα,k ) creates (annihilates) a particle with energy εα,k in the reservoir state k in lead α = L/R. The wire-lead coupling is characterized by the tunneling Hamiltonian ˆ SR = H

XX α,k

α∗ † α † ci bαk Tk,i bα,k ci + Tk,i

(4)

i

α where Tk,i describes the coupling between reservoir state k and system state i. The details

of the coupling between molecular wire and fermionic reservoirs can be assembled in the

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reservoir spectral density also termed broadening function. In general, non-vanishing couα can be present between any site i of the wire and any state k of each pling amplitudes Tk,i

reservoir α leading to

Γα,ij () = 2π

X

α α∗ Tk,i ()Tk,j ()δ( − αk ) .

(5)

k

This broadening function is often given in a matrix form with respect to the site indices, i.e., Γα (). In the numerical examples below, we assume for simplicity that only the first (site 1) or left-most and the last (site N) or right-most site of the wire are coupled to the left and right electrode, respectively. Thus, only one element for each of the matrices ΓL () and ΓR () is non-zero in this case and will be denoted ΓL () and ΓR () . The main objective of this study is to investigate the effect of additional phonon baths on the conduction through a molecular junction. This is described by the last two terms in Eq. (1). The coupling of the phonon bath to the electronic system is described by

ˆ SB = H

XX i,j

ξij,q (dq + d†q )c†i cj ,

(6)

q

where ξij,q represents the coupling strength and the operator d†q (dq ) creates (annihilates) a phonon with the respective energy. In the applications below, we assume that the electronphonon interaction occurs only site-locally, i.e., ξij,q = δij ξii,q . Then the coupling Hamiltonian reads

ˆ SB = H

XX i

ξii,q (dq + d†q )c†i ci =

q

X

Φi c†i ci ,

(7)

i

where we have denoted the bath part of the system-bath interaction as Φi for later reference. Thereby all information regarding the coupling between system and phonon bath are

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characterized by the so called phonon spectral density Ji (ω) at site i

Ji (ω) =

2 X ξii,q δ(ω − ωq ) 2m ω q q q

(8)

with mq and ωq denoting the mass and frequency of the bath oscillators, respectively. It can be determined as 64,65 βω Jj (ω) = π

Z∞ dt Cj (t) cos(ωt)

(9)

0

where the inverse temperature is given by β = 1/(kB T ). This expression shows that the spectral density can be obtained as a half-sided Fourier transform of the energy autocorrelation function Ci (t) = h∆Ei (t)∆Ei (0)i with the energy fluctuations ∆Ei = Ei − hEi i. At this point we need to state that for a consistent use of the high-temperature limit, the prefactor βω/π is being employed instead of 2 tanh(β~ω/2)/(π~) as used earlier. 66,67 Moreover, the spectral density of the present Holstein model is connected to that of the similar CaldeiraLeggett model JCL,j (ω) by a factor, i.e., JCL,j (ω) = π~ Jj (ω). To describe the coupling to the environment by a single number, on usually defines the reorganization energy of the bath λi as

Z∞ λi =

Ji (ω) dω . ω

(10)

0

The results below will also be discussed in terms of the reorganization energy which sometimes is also termed polaron shift.

2.2

Time-dependent Green’s function method

As mentioned earlier, the TD-NEGF formalism utilized herein is introduced in Ref. 51 where the authors combine the TD-NEGF formalism with a spectral decomposition of the spectral density belonging to the electronic baths. This decomposition results in an expression of the self-energies composed of a weighted sum of exponentials, forming the basis of the derivation of differentials equations for reduced-system density operator in the single-particle basis 8


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ρs.p. S (t). This operator is related to the multi-particle operator of the density matrix by † ρs.p. S (t) = T rS {ci cj ρS (t)}.

The self-energies of the leads are given by 51

Σ< α (t1 , t2 )

=

i 2π

i Σ> α (t1 , t2 ) = − 2π

R∞ −∞

R∞ −∞

dε fα (ε − µα )e−iε(t1 −t2 )/~ Γα (ε)

dε fα (−(ε − µα ))e−iε(t1 −t2 )/~ Γα (ε) ,

(11) (12)

where fα (ε) denotes the Fermi distribution of lead α and µα the corresponding chemical potential. Different decomposition schemes can be used for the Fermi function such as the partial fractional decomposition 68 or the Ozaki decomposition scheme. 69 The broadening function, Γα (ε), for the coupling to the leads, i.e., the density of states of the leads weighted by coupling factors, needs to be given in a specific form. As discussed above, if only the first and last site of the molecular wire are connected to the respective lead, the matrix Γα (ε) has only one non-vanishing element which is assumed to be of the form

Γα (ε) =

NL X l=1

2 Wαl Γαl 2 (ε − εαl )2 + Wαl

(13)

with the real parameters Wαl , Γαl , and εαl determined, in general, during a fitting procedure. 70 In the TD-NEGF approach the coupling through the phonon bath is treated indirectly, i.e., through the fluctuation of the site energies induced by the environment. For a tightbinding system without leads this approach has been detailed in Ref. 67. Under the assumption that the phonon bath always stays in equilibrium if initially in equilibrium, the effect of the bosonic bath can be transformed into an additional time-dependent site energy ˆ i |ΨB (t)i with the bath term of the system-bath term at site i, namely ∆Ei (t) = hΨB (t)| Φ ˆ as defined in Eq. 7. Therefore, the site energy at site i, i.e., Ei (t), is replaced by coupling Φ Ei (t) + ∆Ei (t) to account for the environmental effects. How these site energy fluctuations can be generated for a specific example is delineated below. Thus for the description of the 9



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TD-NEGF only the molecular wire and its coupling to the fermionic reservoirs needs to be taken into account. The assumption that the coupling to the leads can be written as a sum of Lorentzian functions, leads to the fact that the self-energies are given by a weighted sum of exponential functions, i.e.,

Σ≷ α =

X k,σ

Σ≷,σ α,k (t) =

X

σ

−iχα,k t Γ≷,σ . α,k e

(14)

k,σ

29,51 σ The time evoThe explicit expressions for Γ≷,σ α,k and χα,k have been reported elsewhere.

lution of the single particle reduced density matrix of the wire is then given by 29,51 i 1X ∂ρs.p. S (t) = − [HS (t), ρs.p. [Παk (t) + Π†αk (t)] S (t)] + ∂t ~ ~ α,k

(15)

∂Παk (t) i = − [HS (t) − χ+ αk ]Παk (t) ∂t ~

(16)

>,+ (t, t2 )Σ< (t , t) − G (t, t )Σ (t , t) . 2 2 2 α α

(18)

t0

Moreover, G> and G< denote the greater and lesser Green’s functions, respectively, and Σ≷ are the corresponding self-energies. The Green’s functions are defined in terms of system

10


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creation and annihilation operators i † hc (t1 ), ci (t2 )i , ~ j i † G> ij (t1 , t2 ) = − hci (t1 ), cj (t2 )i . ~ G< ij (t1 , t2 ) =

(19) (20)

In terms of Green’s functions and reservoir self-energies the time-dependent current flowing from or to the lead α is given by 71 2e Iα (t) = Re Tr ~

Z

∞

α (ε) = −i (1 − fα (ε)) Γα (ε) ,

(27)

where Γα denotes the above defined broadening functions and fα (ε) the equilibrium Fermi functions of the leads. As in Sec. 2.2, we consider decoherence effects by electron-phonon interactions, Eq. 6, and also assume a bath of independent oscillators. The corresponding lesser self-energy then reads 28

Σ< vib (ε)

X i Z 0 0 = Mq G< (ε − ε0 )Mq D< 0,q (ε )dε , 2π q

(28)

where (Mq )ij ≡ ξij,q are the coupling matrix elements between electrons and phonons. We use the quasi-elastic Vibronic Dephasing (VD) model, 57–61 i.e., we assume that the energy of the phonons ~ωq is small compared to all other energy scales. Furthermore it is assumed that the electron-phonon interaction is site-local, i.e., only i = j terms are nonzero. With these approximations we can study the influence of pure dephasing which is comparable to

12


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the model described in Sec. 2.2. The corresponding self-energy reads 61 h

i R(

Dephasing in a Molecular Junction Viewed from a Time-Dependent

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