Length- and temperature-dependent crossover of charge transport across molecular junctions
Ya-Lin Lo (),1,2Shih-Jye Sun (),3,*and Ying-Jer Kao ()1,2,4,† 1Department of Physics, National Taiwan University, Taipei 106, Taiwan
2Center of Theoretical Sciences, National Taiwan University, Taipei 106, Taiwan 3Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan 4Center for Quantum Science and Engineering, National Taiwan University, Taipei 106, Taiwan (Received 18 March 2011; revised manuscript received 21 June 2011; published 4 August 2011) We study the electronic transport in a molecular junction, in which each unit is coupled to a local phonon bath, using the nonequilibrium Green’s function method. We observe that the conductance oscillates with the molecular chain length and that the oscillation period in odd-numbered chains depends strongly on the applied bias. This oscillatory behavior is smeared out at the bias voltage near the phonon energy. For the phonon-free case, we find a crossover from tunneling to thermally activated transport as the length of the molecule increases. In the presence of the electron-phonon interaction, the transport is thermally driven and a crossover from thermally suppressed conduction to assisted conduction is observed.
DOI:10.1103/PhysRevB.84.075106 PACS number(s): 72.60.+g, 73.23.−b, 73.63.−b, 81.07.Nb
I. INTRODUCTION
Research on molecular electronics is a fast-progressing field due to its great application potential. Molecular junctions with single molecules or molecular assemblies sandwiched between two electrodes are of particular interests since they are potential candidates as the building blocks for the next generation of electronics.1 One important issue associated with electron
transport in molecular devices is to understand the role that phonons play in the conduction mechanism. In general, the electron-phonon interaction is expected to modify transport properties when the phonon relaxation time is comparable to the typical transport time of the junction.2 Thanks to
the rapid development of new experimental techniques, it is now possible to experimentally study quantum transport phenomena in a single-molecule wire using break junctions,3
STM,4–6and crossed-wire tunnel junctions.7Vibronic contri-butions to the charge transport have been identified in the low-bias conductivity of molecular wires,8,9 step-like
fea-tures in current-voltage characteristics,10–14and
temperature-dependent conduction.15–17 It has been argued that the
transmission rate of the electron decays exponentially with length for short chains, which is a signature of tunneling transport.18 For long chains, however, thermal excitations
dominate at high temperature and a crossover to thermally activated diffusion with Ohmic behavior best describes elec-tron transfer in this regime.19,20 In molecular junctions with
fixed molecular length, a crossover from coherent to incoherent conduction, as temperature increases, has been observed.21–23
In addition, strong experimental evidence demonstrated that there is a crossover from tunneling to hopping of the charge-transport mechanism as the length of the molecule increases.24–26 What is not clear, however, is the underlying mechanism of this crossover and how it is modified in the presence of electron-phonon interactions.
In this paper, we study electrons interacting with local phonon baths in the molecular wires and we treat the onsite electron-phonon interaction perturbatively to second order in coupling strength. This may become important in molecules which couple to the vibration modes provided by the local functional group, and the charge transport characteristics
can change significantly. Using the standard nonequilibrium Green’s function (NEGF) method and a tight-binding model for the molecular chain, we study the conduction mechanism as a function of length and temperature. We observe a crossover of the electron conduction mechanism from tunneling to thermally activated transport as the molecular length increases in the absence of the electron-phonon interaction. In the presence of electron-phonon coupling, a crossover from thermally suppressed to assisted conduction is also observed.
The paper is organized as follows: In Sec.II, we describe the microscopic model of the molecular wire, and the theoretical framework to compute the transport properties in the presence of the electron-phonon interaction. In Sec.III, we show our results and interpret their physics based on our proposed model and approximations. Finally, we conclude in Sec.IV.
II. MODEL AND THEORETICAL METHOD
We model the molecule as a one-dimensional chain of N units with each unit being coupled locally to a phonon bath (Fig.1). The full Hamiltonian of the molecular system with the local electron-phonon interaction can be written as
H = Hl+ Hc+ Hel+ Hph+ He-ph. (1)
We assume two semi-infinite one-dimensional leads, and the lead Hamiltonian is given by
Hl= α∈L,R ∞ i=1 αc†i,αci,α− ηα i,j ci,α† cj,α, (2)
where ci,α† (ci,α) is the creation (annihilation) operator of the
carrier at the ith unit in lead α= R,L, ηα is the hopping
integral in the leads, and αis the onsite energy in the α lead.
The coupling between leads and the molecule is given by
Hc=
α∈L,R,i∈1,N
Vα(c†1,αdi+H.c.), (3)
where the L(R) lead only couples to the first (N th) unit of the molecule via the coupling constant VL(VR) and di†(di) is the
FIG. 1. One molecular chain, sandwiched between two leads, consists of N units with each unit coupled to a local phonon bath.
the middle wire. The Hamiltonian for the molecule is given by a single-orbital tight-binding model,
Hel= N i=1 idi†di− t i,j di†dj, (4)
where t is the hopping integral between the nearest-neighbor units and iis the orbital energy at the ith molecular unit.
For simplicity, we set the vibrational mode ω= ω0 of the phonon baths to be the same and the electron-phonon interaction is given by Hph= N i=1 ω0b†ibi, (5)
where i is unit index and bi (b†) is the phonon annihilation
(creation) operator, and
He-ph= γ N
i=1
di†di(b†i + bi), (6)
where γ is the strength of the electron-phonon coupling. The current through an interacting region driven by a bias voltage is obtained by the Keldysh NEGF:27
I = ie 2h dωTr{[fL(ω,μL)L− fR(ω,μR)R] × [Gr (ω)− Ga(ω)]+ [L− R]G<(ω)}, (7)
where α∈L,R are related to the imaginary parts of the
self-energies of leads, α∈L,R= [αa(ω)− αr(ω)]/(2i),28and fα(ω,μα)= 1/(1 + eβ(ω−μα)) are Fermi-Dirac distributions in
the reservoirs.
The full (retarded, advanced, and lesser) Green’s functions are obtained by solving the Dyson equation:
G= g + gVG, (8)
where G is the full Green’s function with the electron-phonon interaction, g is the noninteracting Green’s function, and V is the interaction part. Following the notations in Ref.29, we solve the Dyson equation by keeping terms up to second order in the electron-phonon coupling γ to obtain the full Green’s function:
G= g(I + γ GQ), (9)
where GQ contains all the information about the
electron-phonon interaction,
GQij = di(bi+b†i); dj†δi,j, (10)
where we use superscript Q to indicate the presence of the phonon operators b and b†. a; b† represents one of the
Keldysh Green’s functions with (composite) operators a and
b†, and their Fourier transforms are defined as
G<ab(ω)= i dt eiωtb†a(t), (11) G>ab(ω)= −i dt eiωta(t)b†,
for the lesser and greater Green’s functions, and
Grab(ω)= −i ∞ 0 dt ei(ω+i0+)t[a(t),b†]+, (12) Gaab(ω)= i 0 −∞ dt ei(ω−i0+)t[a(t),b†]+,
for the retarded and advanced Green’s functions, respectively. Moreover, in our model we consider only the onsite electron-phonon interaction; therefore, GQ is diagonal and can be written as GQij = gij bi+ b†i + γ G QQ ii δij, (13) where GQQii = di(bi+ b†i); (bi+ bi†)d † i. (14)
The retarded Green’s function GQQ,rcan be evaluated as
GQQ,rii (ω)= −i ∞ 0 dt eiωt[di(bi+ b†i); (bi+ b†i)di†]+t = i dω 2π g>i,i(ω) b ib†i ω− ω0− ω+ iδ + b†ibi ω+ ω0− ω+ iδ − g< i,i(ωt) bibi† ω+ ω0− ω+ iδ + b†ibi ω− ω0− ω+ iδ . (15)
Collecting all the terms, the inverse of the full Green’s function is given as
G−1 = g−1− ph, (16)
ph= γ bi + b†iδi,j+ γ2GQQ= ph1 + ph2 , (17)
where the bare Green’s function g is defined as the noninteracting-electron Green’s function together with the self-energy corrections from the leads:
g= (ω − Hel− lead)−1. (18)
The lead self-energy lead describes the interaction between the leads and the molecular chain and, for semi-infinite one-dimensional leads, is given by30,31
lead = α∈R,L α, (19) r,aα = |Vα| 2 ω−α 2 ∓ i η2 α− ω−α 2 2δi,jδi,n, (20) α<= iαfα(ω), (21) α>= −iα[1− fα(ω)], (22)
where n is 1 (N ) for α= L (R). One remaining quantity required to evaluate the current is the full lesser Green’s function G<, which can be calculated through the Gr
and Ga:
G<= Gr<Ga, (23)
<= lead< + ph<, (24) where the full lesser self-energy is broken up into contributions from the leads and the phonon, and the former is given by Eq. (21). For the phonon self-energy, 1ph in Eq. (17) gives a shift in energy. To evaluate the expectation value of the phonon operators, we perturb the phonon wave function to the first order in γ : | ni = |ni + n=n ni|He−ph|ni En− En |ni, = |n i + γ ω0 di+di n=n ni|bi+ bi+|ni n− n |ni, (25)
where|ni and |ni are phonon states and the energy difference
is proportional to the difference of their phonon quantum numbers: bi + b†i = i 2γ ω0 dω 2πg <. (26) The second term ph<,2= γ2GQQ,<, where GQQ,<is diagonal with matrix elements
GQQ,<ii (ω)= i dt eiωt(bi+ bi†)(bi(t)+ bi†(t))di†di(t), = bi†bigii<(ω− ω0)+ bib†ig < ii(ω+ ω0), (27)
whereb†b = (eβω0− 1)−1is the phonon occupation number.
III. RESULTS AND DISCUSSION
We consider the simplest case of the model, Eq. (2), where the two electrodes are identical (ηL= ηR ≡ η) and have the
same coupling to the molecule (VL= VR≡ Vc= 1). In the
following, unless otherwise noticed, the energy will be in units of Vc. The parameters used in this paper are L= R = i =
0, η= −5, t = −2. The symbol T indicates the temperature both in the reservoirs and phonon baths. The conductance is in units of G0= 2e2/ h, the current is in units of I0= 2e/h, and the resistance is in units of R0= 1/G0. The current in Eq. (7) is driven by a bias voltage Va. We assume that the left
lead is connected to the ground, with the chemical potentials
μL= 0 and μR = −e Va. We obtain the Green’s functions,
for example, in Eq. (15) by performing numerical integration in the real frequency domain, and the small positive δ is set to 0.01.
A. Even-odd effect
We start by considering short chains first. Figures 2 and 3 show the behavior of currents at small γ for chains with odd and even sites at T = 0.025, and the phonon mode is at ω0= 0.4. This case is in the quantum limit as
T /ω0 1. First we observe that odd-numbered chains shows significantly larger current than even-numbered chains, as
FIG. 2. (Color online) Current as a function of γ at ω0= 0.4 for different odd-numbered chains: N= 5 (circles), 7 (triangles), and 9 (squares) at Va= 0.1. The injection electron hits a resonance state
at the corresponding γ to produce the maximum.
found in Refs32and33. For even-numbered chains, in the absence of electron-phonon interaction (γ = 0) and at small bias, the chemical potential of the leads falls in the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), thus the tunneling current is small. On the other hand, for odd-numbered chains, there exists a state close to the chemical potential of the lead, and resonant tunneling is possible, which produces a finite current.32,33The electron-phonon interaction has a strong
effect on the conductance of the odd-numbered chains. The molecular state near the chemical potential of the lead is shifted by the interaction, and a resonance occurs at a finite γ (Fig.2). The resonance is not observed in the even-numbered chains, and the conductance increases monotonically as γ increases. One expects this even-odd effect should vanish when the number of sites becomes large as the HOMO-LUMO gap decreases with more atoms in the molecular wire. Moreover, this transport behavior depends strongly on these physical parameters, the Fermi level of the leads, and the HOMO and LUMO of the molecules. Thus, one expects that this
FIG. 3. (Color online) Current as a function of the electron-phonon interaction with ω0= 0.4 for different even-numbered chains: N= 4 (circles), 6 (triangles), and 8 (squares) at Va = 0.1.
FIG. 4. (Color online) Conductance as a function of length with
γ = 0 for different bias voltages Va= 0.3, 0.4, 0.5. Inset shows the
maxima of the transmission probability are at N = 31, 61 and N = 41 for E= 0.4 and E = 0.3, respectively.
even-odd effect would be tuned as these physical parameters are changed, and the effects are reversed.
B. Length dependence
We will focus on odd-numbered chains in the following discussion. Figure4shows conductance versus chain length at bias voltages Va= 0.3, 0.4, 0.5. It is clear that the conductance
also shows an oscillatory behavior as the chain length increases, consistent with published results.34,35 The period of the oscillation is roughly inversely proportional to the injection energy of the electron from the lead. This should be distinguished from the oscillatory behavior in the even-odd effect, as discussed in the previous section, since this behavior cannot be easily explained by the simple argument of whether a state is present near the Fermi energy. The conductance
maxima and minima can be attributed to the constructive and destructive interferences of the outgoing and reflected wave functions, which can be better understood by solving the cou-pled Schr¨odinger equations36,37for transmission probabilities
as a function of the electron injection energy. Details of the calculation are given in the appendix. The inset of Fig. 4 shows the transmission probability as a function of chain length with external energy E= 0.3, 0.4, and the resonances occur at roughly the same molecular length as in the Green’s function calculation, confirming the above statement. Figure5 shows the conductance as a function of length with the phonon frequency ω0= 0.4, and electron-phonon coupling γ = 0.1 at various biases. At low bias, the electron-phonon interaction has little effect on the length dependence. As the bias increases (Va= 0.31), the maximum is suppressed and the peak position
starts to shift. When the external bias is close to the phonon frequency (Va= 0.36, 0.41), signature of the oscillation is
completely washed out. Further increasing the bias above the phonon frequency, the oscillatory behavior reappears. Figure6 shows the total density of states for the molecules with different length N :
D(E)= −Tr(ImGr(E)), (28)
which roughly corresponds to the conductance at bias Va = E
[see Eq. (7)]. Comparing Figs. 6(a) and 6(b), it is clear that, near E= ω0 = 0.4, the density of states is significantly modified by the electron-phonon interaction. Far away from this energy, the density of states of the two cases are almost identical, and the oscillatory behavior reappears. Since we are in the quantum limit where T /ωo 1, there are few
thermally excited phonons present at low bias. Increasing the bias to larger than ω0, physical phonons can be excited and the inelastic transport process becomes available. However, these physically excited phonons act as impurities by adding an onsite potential,38 and the constructive and destructive
interference patterns reappear. On the other hand, close to the threshold of the phonon frequency, one expects large
(e) (a)
(d)
(b) (c)
(f)
FIG. 5. (Color online) Conductance as a function of length at (a) Va= 0.21, (b) Va = 0.31, (c) Va= 0.36, (d) Va= 0.41, (e) Va = 0.44,
FIG. 6. (Color online) Molecular density of states for N sites in (a) phonon-free and (b) γ = 0.1, ω0= 0.4 cases.
fluctuations of the phonon number, and the phase coherence is lost and the interference pattern disappears.
C. Crossover of the conduction mechanism
It has been observed in experiments that the conduction mechanism of conjugated molecular wires changes from tunneling to hopping as the length of the molecule increases in oligophenyleneimine (OPI),24oligonaphthalenefluoreneimine
(ONI),39 oligo(p-phenylene ethynylene)s (OPE),25 and
oligophenylenetriazole (OPT).26 At low bias voltage,
tunnel-ing transport dominates for short chains and the conductance
Gis described by Magoga’s law,40
G= G0e−βN, (29)
where G0= 2e2/ h, N is the molecular length, and β is the material-dependent inverse decay length. Experimentally, one way to distinguish the tunneling transport from the thermally activated transport is to compare the β values at different regimes.41 For example, in the conjugated OPI molecule,24
a larger β= 3.0 nm−1 is observed in the short wire, while in the long chain, a comparatively smaller value of β= 0.9 nm−1 is observed. It is argued that two different β values for molecular wires of different lengths based on the same oligomers imply that there is a change in the conduction mechanism. In contrast, in the thermally activated regime, the resistance increases only linearly with the length and the conductance has a strong temperature dependence. Therefore, we use the following criteria to distinguish tunneling and thermally activated transport in molecular wires:42 (1) Large β with weak temperature dependence of the conductance corresponds to tunneling, and small β with strong temperature dependence corresponds to thermally activated transport. (2) Linear length dependence of the resistance corresponds
FIG. 7. (Color online) Conductance vs length at Va= 0.1.
Triangles (squares) are for γ= 0.1, and ω0= 0.2 (0.7). Circles are for γ= 0. The slope corresponds to β in Eq. (29).
to thermally activated transport, and exponential decay of the conductance corresponds to tunneling.
Figure 7 shows the conductance versus length with fits of βs in different regimes. First we discuss the case where the electron-phonon interaction is absent. Tunneling transport dominates in short chains with a larger β= 0.11 (per unit), while thermally activated transport prevails in long chains (N 21) with a smaller β = 0.04 and a linear growth of resistance with length (Fig.8). In addition, in the long-chain molecule such as N= 31 [Fig. 9(a)], a crossover from tunneling to thermally activated transport is observed as the temperature increases. At high temperatures, conductance grows as the temperature increases, which is a characteristic of
FIG. 8. (Color online) Resistance as a function of length at
Va= 0.1 V. Circles corresponds to γ = 0, and triangles (squares)
corresponds to γ= 0.1 with ω0= 0.2 (0.7). Solid lines show the linear relation between the resistance and the chain length.
FIG. 9. (Color online) Conductance as a function of the inverse temperature for different lengths: N= 5 (circles), 11 (squares), and 31 (triangles) at Va= 0.1. ω0= 0.2 and (a) γ = 0 and (b) γ = 0.1.
the thermally activated transport conduction. The conductance is less sensitive to temperature at low temperature, indicating the dominant transport is due to tunneling. For a short chain N = 5, in contrast, the conductance is temperature independent in the whole temperature regime, suggesting tunneling conduction.
Figure 10 summarizes the conduction mechanism deter-mined as described above for different temperatures and chain lengths. It is clear that, at low temperature, tunneling dominates. At high temperature, a change in the transport mechanism from tunneling to thermally activated transport occurs as the length increases. Experiments show that the electron transport mechanism changes abruptly at a particular length.24–26,39 However, our calculation [Fig. 10 (a)] shows that a crossover region appears between N = 7 and N = 19, where the characteristics of tunneling and thermally activated transport coexist.18 Interestingly, for long-chain
molecules, there exists another transition from tunneling to thermally activated transport as the temperature increases. For short molecules, however, no such transition is observed and tunneling is still the main transport mechanism at high temperature.
The electron-phonon interaction can significantly change the transport characteristics. Figure 7 shows the conduc-tance versus length for γ = 0.1, with phonon frequencies
FIG. 10. (Color online) Charge transport mechanism for different length and inverse temperature at Va= 0.1 for (a) γ = 0. Circles
indicate tunneling transport, triangles correspond to thermally acti-vated transport, and squares represent the crossover regime with the mixture of tunneling and thermally activated transport characteristics (see text), and (b) for γ= 0.1 and ω0= 0.2. The up and down triangles indicate thermally assisted and suppressed conduction, respectively, and stars represent a transition between these two behaviors.
ω0= 0.2 (triangles), 0.7 (squares). The inverse decay length β decreases in the presence of the electron-phonon interaction for short chains, suggesting that the coherent electron tunneling becomes weaker as the interaction with phonons is turned on. Increasing ω0 decreases the probability of the phonon processes, and the behavior of the conductance reverts back to the noninteracting case (γ = 0 or ω0→ ∞). In addition, for
ω0= 0.2, the resistance is proportional to the length even in short chains (Fig.8), indicating thermally activated transport for all molecular sizes. The temperature dependence of the conductance [Fig.9(b)] shows an interesting crossover from a thermally assisted transport in the long molecule (N= 31) to a thermally suppressed transport in the short molecule (N= 5), and the transition occurs roughly at N = 11. This crossover of the temperature-dependent conduction as length increases replaces the change of the transport mechanism in the phonon-free case [Fig.10(b)].
To better understand the temperature dependence of the conductance at bias Va, formally defined as
g(Va)= lim +→0
I(Va+ +)− I(Va− +)
2+ , (30)
where +is an infinitesimal positive number, we perform fur-ther analysis as follows: When the electron-phonon coupling is weak, the first term in the current definition [Eq. (7)] dominates and the density of states is almost the same for the two bias voltages separated by the infinitesimal +, thus
g(Va)≈ lim +→0 e 4h+ dωTr{ ˜F(Va)D(ω)}, ˜ F(Va,T ,ω)= ˜f(Va+ +,T ,ω)− ˜f(Va− +,T ,ω), (31) ˜f(Va,T ,ω)= fL(ω,μL)L− fR(ω,μR)R, D(ω)= i[Gr(ω)− Ga(ω)],
where TrD(ω) gives the total density of states of the molecular chain, including the effects from electron-phonon coupling and ˜f(Va,T ,ω) is the net electron distribution from two leads
at temperature T and bias Va. The derivative of the electron
distribution at Va is given by lim+→0F(V˜ a,T ,ω)/(2+). We
define a parameter ϒ as ϒ(Va,T1,T2)= dωTr{ ˜F(Va,T1,T2,ω)D(ω)} = dω ˜F11(ω)D11(ω)+ ˜FN N(ω)DN N(ω) (32) where the difference between distributions at two different temperatures is ˜F(Va,T1,T2,ω)= ˜F(Va,T1,ω)− ˜F(Va,T2,ω) and has only the two nonzero matrix elements ˜F11and ˜FN N
due to the coupling to the left lead L
11 and right lead N NR ,
respectively.
The physics of a thermally activated or suppressed transport is thus dictated by ϒ. For T1> T2, ϒ(Va,T1,T2) is positive for thermally activated transport, and negative for thermally suppressed transport. Since the density of states has a small temperature dependence, ˜F(ω) would play the major role in the temperature behavior. Assuming the density of states at sites 1 and N are approximately equal due to symmetry, we can rewrite Eq. (32) as
ϒ(Va,T1,T2)≈ dω[ ˜F11(ω)+ ˜FN N(ω)]D11(ω) = dω ˜F(ω)D11(ω),
where ˜F = ˜F11+ ˜FN N. The inset of Fig. 11 shows the ˜F(Va,T1= 300 K, T2= 70 K, ω) as a function of energy, We notice ˜F is negative near ω= −Va and becomes
positive away from this region. Figure11shows the integrand of Eq. (32). For N= 5, there is more negative area than the positive one, which gives ϒ < 0, and the transport is thermally suppressed. On the other hand, for the N = 31 chain, there is more positive area than the negative one, resulting in a thermally activated transport. In summary, the temperature dependence of the conductance comes principally from ˜F(ω), and the characteristics of the molecules and the electron-phonon effects enter through the density of states
FIG. 11. (Color online) The integrand in Eq. (32) as a function of energy ω for N= 5 (circles) and N = 31 (triangles) at Va= 0.1, and
T1= 300 K, T2= 70 K. Inset: Temperature difference of the effective electron distributions of the leads ˜F(ω). See text for details. D(ω). For N = 5, the weak electron-phonon interaction results in a shift of the density of states toward ω= −Va. This picks
up more negative contribution in ˜F, resulting in thermally suppressed transport.
IV. CONCLUSION
In conclusion, using the nonequilibrium Green’s function method, we study the length and temperature dependencies of electron transport in molecular junctions. We model the molecular wire using a simple tight-binding model, with each site coupled locally to a phonon bath. We treat the electron-phonon interaction perturbatively up to second order in the electron-phonon coupling.
We find different behaviors of conductance in short chains. Larger conductance is observed in odd-numbered chains, and a resonance is present when the electron-phonon coupling increases. On the other hand, for even-numbered chains, low conductance is observed with a monotonic increase as the electron-phonon coupling increases. In the absence of the electron-phonon interaction, we observe a length-dependent crossover of the conduction mechanism from tunneling to thermally activated transport at high temperature. We find that, at high temperatures, tunneling transport dominates in short chains, while thermally activated transport dominates in long chains. For a long molecule of fixed length, there exists also a crossover from tunneling at low temperature to thermally activated transport at high temperature. In the presence of the electron-phonon interaction, the coherent tunneling is interrupted by the phonon and the length crossover disappears. The thermally activated transport becomes the main transport mechanism. At high temperature a crossover of temperature dependent conduction from the thermally suppressed to as-sisted transport as the length increases, in place of the crossover in the transport mechanism in the phonon-free case.
Without resorting to the ab initio machinery, we are able to extract the essential physics in these molecular junctions using a simple model, and the results qualitatively agree with
a wide range of experiments on molecular wires. The approach in this paper can be generalized to study molecules with site-dependent coupling strengths and phonon energies. This may be important to understand the mechanical roles of the functional groups. In addition, the results can be extended to the spin-dependent case to study the properties of the magnetic tunneling junctions. Further improvement can also be made by self-consistently calculating the charge density at each site. These questions will be addressed in a future study.
ACKNOWLEDGMENTS
We thank Serguei Brazovski and Natasha Kirova for useful discussions. We are grateful to the National Center for High-Performance Computing for computer time and facilities. This work was supported by the National Science Council in Taiwan through Grants No. NSC-98-2112-M-390-001-MY3, No. NSC-97-2120-M-006-010- (SJS), No. NSC-97-2628-M-002 -011-MY3, and No. 99-2120-M-NSC-97-2628-M-002-005- (YLL and YJK), and by NTU Grants No. 65 and No. 99R0066-68 (YJK). The hospitality of the National Center for Theoreti-cal Sciences, Taiwan, where the work was initiated, is greatly acknowledged.
APPENDIX
In this Appendix, we provide an alternative method to study the transport properties of molecular chains by solving coupled Sch¨odinger equations in the tight-binding limit for the onsite wave functions. For illustrative purposes, we consider the molecular transport without the electron-phonon interaction, and the full Hamiltonian can be written as
H= Hl+ Hc+ Hel, (A1)
where the Hamiltonian of two leads Hl, the molecular
electronic system Hel, and the coupling between them
Hc are described by Eqs. (2) to (4). We solve the
site-dependent Schr¨odinger equations Eφi =
jHijφj with
the tight-binding model to obtain the transmission and reflection amplitudes. In the leads, the dispersion relation
E− ¯α= 2ηαcos(kαaα) for α= L,R is employed, where ¯α
includes both the onsite energy and the chemical potential in the α lead, aα is the lattice constant, and kl (kr) is the
incoming (outgoing) wave vector. We use plane waves as the wave functions in leads:
φL i = a1e
iklxli + a2e−iklxil, (A2)
φRi = a3eikrxir, (A3)
where, in the left lead, the total wave function φiLconsists of
two components of the incoming and reflected waves, and the transmitted wave in the right lead. The transition and reflection probabilities are defined as
T = c a3 a1 2, (A4) R= c a2 a1 2, (A5)
where c is a normalization constant given by|T |2+ |R|2= 1. The current across the molecular junction is given by
I =
dET(E)[fL(E,μL)− fR(E,μR)]. (A6)
After solving the Schr¨odinger equations, the transition amplitude a3/a1 and the reflection amplitude a2/a1 are obtained by a3 a1 = 2ηLVLVRcos(klal)eiklx l 1e−ikrxr1 (E− R− ηReikrar) V2 L− (E − L− ηLeiklal)(E− 1− tD2) φNc φc 1 , (A7) a2 a1 = (E− L− ηLe−iklal)(E− 1− tD2)− VL2 V2 L− (E − L− ηLeiklal)(E− 1− tD2) e2iklxl1. (A8)
The ratio of wave functions at the N th and the first site is obtained by solving the site-dependent Schr¨odinger equations of the molecular chain, which gives
φNc φ1c = N n=2 Dn, (A9)
where the Green’s function on the nth site is given by
Dn= t E− n
, (A10)
which propagates the wave function from site n− 1 to site n as φc
n= Dnφnc−1. Moreover, the molecular chain is influenced
by leads through the weak coupling Vα, and these effects are
included in the self-energies. We replace the onsite energies at
the ends of the chain (n= 1,N) with effective onsite energies, which include the self-energies from the leads. For example, the Schr¨odinger equation at site N is given by
EφNc = tφNc−1+ NφNc + VRφ1L, (A11) and the Green’s function DN = t/(E − ¯N) is obtained with
the effective onsite energy ¯N= N+ VRφ1L/φNc, where the
ratio φ1L/φNc is obtained by solving the Schr¨odinger equation of the right lead:
φ1L φNc =
VR E− R− ηReikrar
. (A12)
On the other hand, the effective onsite energy ¯N can also
be described by the self-energy from the right lead. ¯N = N+ R, and Rcan be obtained from Eq. (20).
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