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Negative differential conductance and hysteretic current switching of benzene molecular junction in a transverse electric field
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 465202 (http://iopscience.iop.org/0957-4484/25/46/465202) View the table of contents for this issue, or go to the journal homepage for more
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Nanotechnology Nanotechnology 25 (2014) 465202 (7pp)
doi:10.1088/0957-4484/25/46/465202
Negative differential conductance and hysteretic current switching of benzene molecular junction in a transverse electric field Wen-Huan Zhu, Guo-Hui Ding and Bing Dong Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, Peopleʼs Republic of China E-mail: [email protected] Received 15 May 2014, revised 22 September 2014 Accepted for publication 24 September 2014 Published 30 October 2014 Abstract
We study charge transport through single benzene molecular junction (BMJ) directly sandwiched between two platinum electrodes by using a tight-binding model and the nonequilibrium Greenʼs function approach. Pronounced negative differential conductance is observed at finite bias voltage, resulting from charge redistribution in BMJ and a Coulomb blockade effect at the interface of molecule-electrode contacts. In the presence of a transverse electric field, hysteretic switching behavior and large spin-polarization of current are obtained, indicating the potential application of BMJ for acting as a nanoscale current modulator or spintronic molecular device. Keywords: negative differential conductance, molecular junction, switches 1. Introduction
investigation of transport properties of molecular junction devices. In fact, the conductance through BMJ reported by the pioneering experimental work [4] is much lower than theoretically calculated values [18]. However, recent experiments [17] achieved significant progress regarding this problem with the geometry of direct binding of a benzene molecule between two platinum (Pt) electrodes, and highly conductive molecular junctions were formed. In addition, using a break-junction technique, Meisner et al [23] found that direct metal-π interaction can cause significant variability in molecular junction conductance. In the present work, by numerical simulations based on a tight-binding model with the non-equilibrium Greenʼs function (NEGF) approach, we study the charge transport properties of a π electron molecular junction by directly connecting a benzene molecule as a bridge between two Pt electrodes (see figure 1). It is widely acknowledged that negative differential conductance (NDC) is a very useful property in traditional semiconductor device applications, and recently NDC has also been found in some organic molecular junctions [24–28]. In this work, we show that pronounced NDC phenomena can be observed in BMJ, and its intrinsic mechanism originates
The miniaturization of electron circuits leads to the goal of using single-molecule junctions as components of nanoscale electronic devices [1], which is currently under active research in the field of molecular electronics [2, 3]. The study of charge transport through single-molecule junctions is also a powerful means to probe intrinsic physical, chemical, and even biological processes in molecules. Since the pioneering experimental results of the conductance of the single benzene molecular junction (BMJ) were measured by Reed et al [4], charge transport through BMJ has attracted great interest, and has become one of the most intensively studied nanoscale systems [5–19]. In particular, based on BMJ, molecular singleelectron devices operating in the Coulomb blockade regime [5], quantum interference effect molecular transistors[6, 8, 9, 20], resonant-tunneling effects [10, 11], nonlinear optical effects [12], and thermoelectric effects [13, 15] were investigated. To obtain efficient charge transportation, one critical issue is the development of ideal molecule–electrode contacts [16, 17, 21–23], which has become a challenge for the 0957-4484/14/465202+07$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
W-H Zhu et al
Nanotechnology 25 (2014) 465202
operator with spin σ = ↑ , ↓. t = 2.6 eV is the nearestneighbor hopping matrix element [6], and ϵ0 is assumed to be a constant −U0 2 below the equilibrium Fermi energy of the Pt electrode (E F = −7.67 eV ) to ensure particle-hole symmetry at half-filling [35]. U0 is the on-site Coulomb repulsive interaction. By considering the screening effect of substrates, we take a relatively small value of U0 (U0 = 3.0 eV ). As shown in figure 1, Vg is the back-gate voltage. Vti is the on-site potential induced by the transverse voltage Vt perpendicular to the transport direction (longitudinal direction), where a linear potential drop along the transverse direction is assumed. The transverse voltage Vt can be generated by a capacitor field of two charged disks at a certain distance from each other. The axis of the capacitor is perpendicular to the transport direction [10, 11]. Although in our following calculation a large transverse electric field (≈2 V/Å) is considered, it is noted that the transverse electric field only drops linearly with increasing the distance of the capacitor disks, and the tunneling amplitude will decrease exponentially with the increasing between distances of the molecular atoms and the capacitors. Therefore, the electron current in the transverse direction can be suppressed in experiments. ni is the electron number operator on the ith lattice site. Vij = U0 (1 + 0.6117rij2 )−1 2 is the intersite Coulomb interaction potential given by the Ohno parametrization [6, 9], and rij is the distance between the atomic sites i and jin units of Å. The expectation values of the spin-resolved occupation number 〈niσ 〉 and Fock exchange terms 〈ci†σ c jσ 〉 (i ≠ j ) can be calculated by self-consistent equations. Within the NEGF approach, the current passing through the benzene molecule is obtained by the Landauer formula [36, 37],
Figure 1. Schematics of the single BMJ system. Along the transport
direction (x direction), the bias voltage Vb is directly applied to the benzene molecule on the SiO2 substrate via source and drain platinum electrodes. The transverse electric field introduced by the transverse voltage Vt lies in the BMJ plane (x–y plane) and is perpendicular to the transport direction. The back-gate voltage Vg (z direction) is modulated by a back gate under the SiO2 substrate.
from the charge redistribution in BMJ with increasing bias voltages and the underlying Coulomb blockade effect at the interface of molecule–electrode contacts. Moreover, molecular switches, in which the conductance can be switched between two states with an applied voltage, can play a key role in logic and memory devices [29], and we find that BMJ can exhibit current switching and hysteresis behavior [26, 27, 30, 31] with large on–off ratios of currents [24] in the presence of a transverse electric field. Another observation is that partial spin-polarized current can be achieved in BMJ, resulting from the splitting of spin-resolved electron occupation numbers on carbon atoms of the benzene molecule by the transverse voltage. In fact, both from fundamental and technological points of view, molecular spintronics [30, 32, 33] is a new and interesting link between spintronics and molecular electronics, which will enable the manipulation of spins and charges in molecular electronic devices. We propose that molecular junctions with transverse electric fields can be a potential component in current switching or spintronic devices.
(
We describe electron states of the benzene molecule by means of a tight-binding model for the π orbital of carbon atoms, which takes into account the on-site and long-range Coulomb interactions. In the self-consistent mean-field (MF) theory, the system will be modeled by the following Hamiltonian within the Hartree–Fock approximation [6, 9, 26, 34]:
∑
(c
† iσ c jσ
)
+ h .c . +
(
)
i
+ ∑ Vij ( ni − 1)
(
nj − 1
)
i≠j
−
∑ Vij (
)
c j†σ ciσ ci†σ c jσ + h .c . ,
(
dωT ω , Vb, Vg, Vt R
E F + eVb 2
∫E
F
− eVb 2
(
) )
dωT ω , Vb, Vg, Vt ,
(2)
(3)
which is an essential quantity that determines the nonlinear transport properties of this molecular electronic system. Here ΓL (R) is the tunneling-width matrix characterizing the coupling strength of the molecule to the left (right) metallic electrode, and induces the broadening of the molecular orbital energy levels. ΓL (R) is a 6 × 6 matrix, with 〈1| ΓL |1〉 and 〈4| ΓR |4〉 being only nonzero elements. In order to model the interface between the molecule and the electrodes more realistically, we take into account the finite band width effects and assume the density of states of electrodes around the Fermi energy to be a Lorentzian form. Therefore the
i, σ
+ U0 ∑ ni ↑ ni ↓ + ni ↓ ni ↑
EL
∫E
T = Tr [ GrΓR GaΓL ],
∑ ( ϵ0 + Vg + Vti ) niσ
< i, j >, σ
2e h 2e = h
where e is the electron charge, ω the incident electron energy, and h Planckʼs constant. A bias voltage Vb is applied symmetrically between the source and drain electrodes. EL = EF + eVb 2 (ER = EF − eVb 2) is the EF of the left (right) electrode after the bias is applied. The transmission probability T (ω , Vb, Vg, Vt ) through the junction is given by the general formula [37, 38],
2. Methods
H MF = −t
)
I Vb, Vg, Vt =
(1)
i ≠ j, σ
where ci†σ (ciσ ) denotes the creation (annihilation) electron 2
W-H Zhu et al
Nanotechnology 25 (2014) 465202
L
I (e ħ-1)
0.8 0.6 0.4
0.4
0.2
0.3
0 -3
Vg
occurrence of the NDC. (2) Nonzero magnetic moments on site 2 and 3 atoms (figures 3(d) and (e)) are induced in the region with about 7.4 V < Vb < 9.3 V, due to the splitting of electron occupation numbers Nup and Ndown. (3) The increasing of occupation numbers of site 1 and 4 atoms (figures 3(b) and (c)) at the voltage Vb ≈ 9.3 V is attributed to a new molecular orbital (LUMO+1) participating in the electron tunneling process. The corresponding transmission probabilities T (ω , Vb ) (figures 4(a) and (b)) at Vg = 0 and Vt = 0 show four peaks because of the degeneracy of energy levels of benzene molecular orbitals, which agrees with the results of the previous calculation with a Hückel Hamiltonian for the π system of benzene [40]. In fact, in this parallel and symmetric configuration, there are two quasi-localized π orbitals of benzene molecule, which donʼt participate in the electron transport through this molecular junction. One unexpected phenomenon is that the T curves are not continuous, and they suddenly split at the voltage (Vb ≈ 7.4 V) corresponding to the energy difference between LUMO and HOMO, indicating the occurrence of NDC at that voltage. Since the abrupt change of the transmission probability at Vb ≈ 7.4 V gives rise to its asymmetry with respect to the reference Fermi energy ω = EF (EF = −7.67 eV), it leads to the sudden decrease of the current according to the Landauer formula [36, 37] in the Methods section. Actually, explanations of NDC in molecular systems still remain controversial in the literature [3]. The charging and conformational changes of molecule were recognized as a possible mechanism for explaining NDC in molecular junctions reported by the first experiment [24]. It was shown that the asymmetry between LUMO and HOMO levels can cause asymmetry of coupling of the molecule to leads at high bias voltage, resulting in dramatic differences of potential profiles and electron occupation number distributions in different parts of this molecular junction along the transport direction, which are responsible for the Coulomb blockade effect appearing at the interface of molecule-lead contacts [38]; on the other hand, Ratner and his co-workers [26, 27] showed that a polaron model with self-consistent Hartree approximation can successfully predict NDC, switching, and also hysteresis I-V characteristics of molecular junctions. Our calculation suggests NDC of molecular junctions can be explained based solely on a tight-binding model of π electrons with the on-site and long-range Coulomb interactions. In addition to NDC, the spin-resolved components Tup and Tdown near the NDC region (7.4 V < Vb < 9.3 V) and in the large bias region (Vb > 11 V) are distinct, which induces a little spin-polarization of the current in these two regions as shown in figure 3(a). Next, we study the effects of the transverse electric field based on the I-V characteristic of this molecular junction. The current image as a function of Vb and Vt with fixed Vg = 3 V is shown in figure 5. For low values of Vb and Vt, there is a region where the resistance is very high, and similar results have been found by Di Ventra et alʼs first-principle calculations [10, 11]. Moreover, we find a zero current region (indicated by an arrow) is achieved at a finite transverse
0.5
Vt=0
0.2 0.1
NDC
-2 -1
) (V
2
0 0
6
4
8
10
12
14
16
Vb (V)
Figure 2. Current image as a function of Vg and Vb without Vt. The NDC region is indicated by an arrow.
hybridization strengths are 〈1| ΓL |1〉 =
〈4| ΓR |4〉 =
2)2
Γ (D . (ω − E R )2 + (D 2)2
Γ (D 2)2 (ω − E L )2 + (D 2)2
and
Γ is equal to 1.0 eV, and
D = 16 eV characterizes the band width of Pt electrodes [39]. G r (a) represents the retarded (advanced) Greenʼs function matrix of molecular states.
3. Results and discussion We first consider the system without external transverse voltages (Vt = 0). The image of the current I through the benzene molecule as a function of the back-gate voltage Vg and the bias voltage Vb is shown in figure 2. At the small absolute value of Vg, a pronounced NDC (indicated by an arrow) is observed when Vb exceeds approximately a threshold value (Vb ≈ 7.4 V), which corresponds to the energy difference between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) near the Fermi level. In the large bias region (Vb > 11 V), the current starts to decrease because of the mismatching of conduction bands of the left and right leads due to the finite band width effect. With the used parameter Γ = 1 eV in our calculation, the unit of current eΓ −1 ≈ 243μA and the magnitude of the corresponding current agree well with the previous first-principle calculations [18]. In order to understand the mechanism of the above NDC phenomena, we examine the I-V characteristic and its corresponding electron occupation number distribution in the molecule as a function of Vb in figure 3. One can see that the spin-resolved current components Iup and Idown begin to split in the NDC region (Vb ≈ 7.4 V) with corresponding splitting of the spin-up Nup and spin-down Ndown electron number distributions of the benzene molecule (figures 3(b)–(e)). There are some interesting features on these electron number distributions: (1) When the NDC happens (Vb ≈ 7.4 V), the total electron numbers Ntotal on the carbon atoms located at the interface of molecule–electrode contacts (site 1 and 4 carbon atoms as shown by the inset of figure 3(c)) have an abrupt increase; nevertheless, electron numbers on other carbon atoms decrease simultaneously. It is easy to see that the charge accumulation on the two interface carbon atoms increases the Coulomb repulsive potential for the tunneling electron, and therefore a blockade effect for charge transport through the molecule emerges, which results in the 3
W-H Zhu et al
Nanotechnology 25 (2014) 465202
a) Vg=0, Vt=0
b)
c)
1.2
L
I (e ħ-1)
0.5
0.4
Ntotal
1
Nup
2
1.0
5 4
0.8
3
Site 4
Ndown
Site 1
6
0.6 0.4
0.3
0.2
1.0
d)
e)
Site 2
Site 3
0.8 Itotal
0.6
Iup
0.1
0.4
Idown
0.0 0
2
4
6
8
10
12
Electron Number N
0.6
14 16 0
4
8
12
0
4
8
0.2 16
12
Vb (V) Figure 3. Current (a) and electron numbers on carbon atoms of benzene molecule (b–e) vs Vb without Vg and Vt, whose spin-resolved components are also plotted. Since Vb is applied to the benzene symmetrically, electron number distributions of site 6 and 5 carbon atoms (not shown here) have the same results as those of site 2 and 3 atoms, respectively. The inset of (c) shows the actual positions of site numbers of carbon atoms.
L
I (e ħ-1)
0.1 0.08
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
Vg=3V
0.06 0.04 0.02
)
(V Vb
0 1 0.5 0 0
1
2
3
4
5
Vt (V)
6
7
8
Figure 5. Current image as a function of Vb and Vt with Vg = 3 V. The zero current region is indicated by an arrow.
with Vb = 0.5 V of the current image of figure 5 and its corresponding spin-resolved current components Iup and Idown are plotted in figure 6(a). One can see that, in the presence of Vt, Itotal shows valley and peak structures as observed experimentally [4, 24] and theoretically [10, 11, 18] in some similar molecular systems, and the current approaches zero at Vt ≈ 4.7 V. Therefore, this BMJ can act as a molecular switching device [26, 27, 29, 30] with large on–off ratios [24] of current upon the application of a transverse electric field, which is in accordance with the results based on first-principle calculations [10, 11]. Interestingly, except for the large on–off ratios of currents, we also observe bistability and hysteresis phenomena of current switching in this BMJ by sweeping the transverse voltage from low value to high value, and then back to low value (figure 6(e)). This reversible switching characteristic of molecular junctions between two conducting states in response to the bias voltage has been proposed by recent experimental [31] and theoretical [26, 27] studies. However, it should be noted that here the hysteretic switching is controlled by a transverse electric field instead of the bias voltage. Furthermore, we find that at a large value of Vt, large splitting of current components Iup and Idown is observed
Figure 4. Spin-up Tup (a) and spin-down Tdown (b) components of transmission probabilities T (ω, Vb ) as a function of energy ω and bias voltage Vb without Vg and Vt.
voltage Vt, and the exact value of this critical voltage Vt depends on how the electron number is redistributed in BMJ within the self-consistent calculation. In this case, the molecular orbital, by which the electron could tunnel previously, is totally decoupled from the leads because of Vt. To detail the underlying features of I-V characteristics with an applied transverse electric field, a sectional drawing 4
W-H Zhu et al
Nanotechnology 25 (2014) 465202
Figure 6. Current (a) and electron numbers on carbon atoms of the benzene molecule (b–d) vs Vt with Vg = 3 V and Vb = 0.5 V, whose spinresolved components are also plotted. Due to the symmetry, similar results for the electron numbers on site 3, 4, and 5 atoms are not shown here. (e) Bistable and hysteretic current switching behaviors for three different values of Vb while Vt is swept from 4.0 V to 6.5 V, and then back to 4.0 V. One of the routes (Vb = 0.3 V) is marked by arrows.
figure 6(a). The difference is due to the electron number splitting (figures 6(b) and (c)) coming from Vt.
(figure 6(a)), indicating this BMJ can be useful in the molecular spintronic applications [30, 32, 33]. The large peak of Itotal can be mainly attributed to the Iup component of this total current. Using first-principles density functional and NEGF methods, Liu et al [30] proposed to obtain spin-polarized current by taking advantage of the energy difference between the antiparallel and parallel states of the molecular spins. However, we show that the large spin-polarization of the current can be realized by the splitting of electron occupation numbers Nup and Ndown of site 1 and 2 atoms induced by the transverse electric field (figures 6(b) and (c)). Moreover, there is a significant increase of electron occupation number on site 6 atom (figure 6(d)), further suggesting that Vt greatly influences the electron distributions in BMJ [10, 11]. Therefore, modulating Vt is a powerful means to get the best performance on current switching and spin-polarization properties of this molecular junction. We also observe that the corresponding transmission probabilities T (ω , Vt ) in the presence of transverse voltage Vt (figures 7(a) and (b)) have two more ridges than the degenerated molecular orbital energy level case (figure 4), because the degeneracy is broken by Vt, which causes strong modulation of the potential profiles and the density of states of this molecular system [10, 11]. Hence, the two previous quasilocalized molecular orbitals in the parallel and symmetric configuration begin to have tunnel coupling with the leads. It is interesting to note that the peak positions of transmission probabilities show strong dependence on the transverse voltage Vt. Furthermore, the branches of Tup and Tdown near the Fermi level ω = −7.67 eV in the relevant energy window (ω = [−7.92, −7.42]eV when EF = −7.67 eV and Vb = 0.5 V, according to the Landauer formula in the Methods section) are enlarged in figures 7(c) and (d), respectively. One can see the distinct difference between Tup and Tdown, which results in the corresponding spin-polarized current peak as shown in
4. Conclusions In summary, I-V characteristics through BMJ with direct metal and carbon atoms of molecule couplings are studied, based on a tight-binding model and NEGF approach. Due to the on-site and long-range Coulomb interactions in the benzene molecule, pronounced NDC at the bias voltage corresponding to the energy difference between LUMO and HOMO is predicted, with its intrinsic mechanism coming from the charge redistribution in BMJ and the Coulomb blockade effect induced by charge accumulations on carbon atoms at the interface of molecule–electrode contacts. The valley-peak structure of the I-V curve with a high resistance region at the low transverse voltage makes BMJ behave as a switching device with large on–off ratios. Hysteretic behavior of current is also found in the switching voltage region, which indicates the bistability of this system in the presence of the transverse electric field. In addition, by applying the transverse voltage, spin-polarized current can be obtained owing to splitting of spin-resolved electron occupation numbers on carbon atoms. Therefore, our calculation suggests that BMJ with transverse electric fields can be a potential candidate in current switching and spintronics applications. However, it should be noted that for BMJ some interesting features, e.g., NDC and the bistability, are obtained in the large bias regime or with large transverse electric field, which might cause difficulties in experimental realization. We notice that the exact values of the bias voltage or transverse electric field for the occurrence of NDC or the bistability are largely determined by the energy gap between the HOMO and LUMO in the molecule. Therefore, one may expect that the rich 5
W-H Zhu et al
Nanotechnology 25 (2014) 465202
Figure 7. Images of spin-up Tup (a) and spin-down Tdown (b) components of transmission probabilities T (ω, Vt ) as a function of ω and Vt with Vg = 3 V and Vb = 0.5 V, whose frequency parts near the Fermi energy (ω = [−7.92, −7.42]eV) are enlarged in (c) and (d), respectively.
transport phenomena can be much more easily observed in molecular junctions with a relatively small energy gap or artificial semiconductor quantum dot systems.
[10] di Ventra M, Pantelides S T and Lang N D 2000 Appl. Phys. Lett. 76 3448–50 [11] Rashkeev S N, di Ventra M and Pantelides S T 2002 Phys. Rev. B 66 033301 [12] Hansen T, Hansen T, Arcisauskaite V, Mikkelsen K V, Kongsted J and Mujica V 2010 J. Phys. Chem. C 114 20870–6 [13] Reddy P, Jang S Y, Segalman R A and Majumdar A 2007 Science 315 1568–71 [14] Taniguchi M, Tsutsui M, Yokota K and Kawai T 2009 Nanotechnology 20 434008 [15] Bergfield J P, Solis M A and Stafford C A 2010 ACS Nano 4 5314–20 [16] Seminario J M, de la Cruz C E and Derosa P A 2001 J. Am. Chem. Soc. 123 5616–7 [17] Kiguchi M, Tal O, Wohlthat S, Pauly F, Krieger M, Djukic D, Cuevas J C and van Ruitenbeek J M 2008 Phys. Rev. Lett. 101 046801 [18] di Ventra M, Pantelides S T and Lang N D 2000 Phys. Rev. Lett. 84 979–82 [19] Zhu W H, Ding G H and Dong B 2014 IEEE Trans. Electron Devices 61 1168–74 [20] Stafford C A, Cardamone D M and Mazumdar S 2007 Nanotechnology 18 424014 [21] Dadosh T, Gordin Y, Krahne R, Khivrich I, Mahalu D, Frydman V, Sperling J, Yacoby A and Bar-Joseph I 2005 Nature 436 677–80 [22] Smit R H M, Noat Y, Untiedt C, Lang N D, van Hemert M C and van Ruitenbeek J M 2002 Nature 419 906–9 [23] Meisner J S, Ahn S, Aradhya S V, Krikorian M, Parameswaran R, Steigerwald M, Venkataraman L and Nuckolls C 2012 J. Am. Chem. Soc. 134 20440–5 [24] Chen J, Reed M A, Rawlett A M and Tour J M 1999 Science 286 1550–2 [25] Reed M A 1999 Proc. IEEE 87 652–8
Acknowledgments This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2011CB925603, the National Natural Science Foundation of China (Grant Nos. 91121021 and 11074166), and Shanghai Natural Science Foundation (Grant No. 12ZR1413300). Our computations were supported by the center for HPC, Shanghai Jiao Tong University.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Aviram A and Ratner M A 1974 Chem. Phys. Lett. 29 277–83 Nitzan A and Ratner M A 2003 Science 300 1384–9 Tao N J 2006 Nat. Nanotechnol. 1 173–81 Reed M A, Zhou C, Muller C J, Burgin T P and Tour J M 1997 Science 278 252–4 Stokbro K 2010 J. Phys. Chem. C 114 20461–5 Cardamone D M, Stafford C A and Mazumdar S 2006 Nano Lett. 6 2422–6 Muller C J, Vleeming B J, Reed M A, Lamba J J S, Hara R, Jones L II and Tour J M 1996 Nanotechnology 7 409 Ke S H, Yang W and Baranger H U 2008 Nano Lett. 8 3257–61 Rincón J, Hallberg K, Aligia A A and Ramasesha S 2009 Phys. Rev. Lett. 103 266807 6
W-H Zhu et al
Nanotechnology 25 (2014) 465202
[33] Bogani L and Wernsdorfer W 2008 Nat. Mater. 7 179–86 [34] Yeganeh S, Ratner M A, Galperin M and Nitzan A 2009 Nano Lett. 9 1770–4 [35] Wang X F, Chakraborty T and Berashevich J 2010 Nanotechnology 21 485101 [36] Meir Y and Wingreen N S 1992 Phys. Rev. Lett. 68 2512–5 [37] Datta S 1995 Electronic Transport in Mesoscopic Systems (Cambridge: Cambridge University Press) [38] Cheraghchi H and Esfarjani K 2008 Phys. Rev. B 78 085123 [39] Smith N V, Wertheim G K, Hüfner S and Traum M M 1974 Phys. Rev. B 10 3197–206 [40] Hansen T, Solomon G C, Andrews D Q and Ratner M A 2009 J. Chem. Phys. 131 194704
[26] Galperin M, Ratner M A and Nitzan A 2005 Nano Lett. 5 125–30 [27] Yeganeh S, Galperin M and Ratner M A 2007 J. Am. Chem. Soc. 129 13313–20 [28] García-Suárez V M and Lambert C J 2008 Nanotechnology 19 455203 [29] Phoa K, Neaton J B and Subramanian V 2009 Nano Lett. 9 3225–9 [30] Liu R, Ke S H, Baranger H U and Yang W 2005 Nano Lett. 5 1959–62 [31] Blum A S, Kushmerick J G, Long D P, Patterson C H, Yang J C, Henderson J C, Yao Y, Tour J M, Shashidhar R and Ratna B R 2005 Nat. Mater. 4 167–72 [32] Rocha A R, García-Suárez V M, Bailey S W, Lambert C J, Ferrer J and Sanvito S 2005 Nat. Mater. 4 335–9
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Negative differential conductance and hysteretic current switching of benzene molecular junction in a transverse electric field
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 465202 (http://iopscience.iop.org/0957-4484/25/46/465202) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 139.80.123.36 This content was downloaded on 23/04/2015 at 13:07
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 25 (2014) 465202 (7pp)
doi:10.1088/0957-4484/25/46/465202
Negative differential conductance and hysteretic current switching of benzene molecular junction in a transverse electric field Wen-Huan Zhu, Guo-Hui Ding and Bing Dong Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, Peopleʼs Republic of China E-mail: [email protected] Received 15 May 2014, revised 22 September 2014 Accepted for publication 24 September 2014 Published 30 October 2014 Abstract
We study charge transport through single benzene molecular junction (BMJ) directly sandwiched between two platinum electrodes by using a tight-binding model and the nonequilibrium Greenʼs function approach. Pronounced negative differential conductance is observed at finite bias voltage, resulting from charge redistribution in BMJ and a Coulomb blockade effect at the interface of molecule-electrode contacts. In the presence of a transverse electric field, hysteretic switching behavior and large spin-polarization of current are obtained, indicating the potential application of BMJ for acting as a nanoscale current modulator or spintronic molecular device. Keywords: negative differential conductance, molecular junction, switches 1. Introduction
investigation of transport properties of molecular junction devices. In fact, the conductance through BMJ reported by the pioneering experimental work [4] is much lower than theoretically calculated values [18]. However, recent experiments [17] achieved significant progress regarding this problem with the geometry of direct binding of a benzene molecule between two platinum (Pt) electrodes, and highly conductive molecular junctions were formed. In addition, using a break-junction technique, Meisner et al [23] found that direct metal-π interaction can cause significant variability in molecular junction conductance. In the present work, by numerical simulations based on a tight-binding model with the non-equilibrium Greenʼs function (NEGF) approach, we study the charge transport properties of a π electron molecular junction by directly connecting a benzene molecule as a bridge between two Pt electrodes (see figure 1). It is widely acknowledged that negative differential conductance (NDC) is a very useful property in traditional semiconductor device applications, and recently NDC has also been found in some organic molecular junctions [24–28]. In this work, we show that pronounced NDC phenomena can be observed in BMJ, and its intrinsic mechanism originates
The miniaturization of electron circuits leads to the goal of using single-molecule junctions as components of nanoscale electronic devices [1], which is currently under active research in the field of molecular electronics [2, 3]. The study of charge transport through single-molecule junctions is also a powerful means to probe intrinsic physical, chemical, and even biological processes in molecules. Since the pioneering experimental results of the conductance of the single benzene molecular junction (BMJ) were measured by Reed et al [4], charge transport through BMJ has attracted great interest, and has become one of the most intensively studied nanoscale systems [5–19]. In particular, based on BMJ, molecular singleelectron devices operating in the Coulomb blockade regime [5], quantum interference effect molecular transistors[6, 8, 9, 20], resonant-tunneling effects [10, 11], nonlinear optical effects [12], and thermoelectric effects [13, 15] were investigated. To obtain efficient charge transportation, one critical issue is the development of ideal molecule–electrode contacts [16, 17, 21–23], which has become a challenge for the 0957-4484/14/465202+07$33.00
1
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W-H Zhu et al
Nanotechnology 25 (2014) 465202
operator with spin σ = ↑ , ↓. t = 2.6 eV is the nearestneighbor hopping matrix element [6], and ϵ0 is assumed to be a constant −U0 2 below the equilibrium Fermi energy of the Pt electrode (E F = −7.67 eV ) to ensure particle-hole symmetry at half-filling [35]. U0 is the on-site Coulomb repulsive interaction. By considering the screening effect of substrates, we take a relatively small value of U0 (U0 = 3.0 eV ). As shown in figure 1, Vg is the back-gate voltage. Vti is the on-site potential induced by the transverse voltage Vt perpendicular to the transport direction (longitudinal direction), where a linear potential drop along the transverse direction is assumed. The transverse voltage Vt can be generated by a capacitor field of two charged disks at a certain distance from each other. The axis of the capacitor is perpendicular to the transport direction [10, 11]. Although in our following calculation a large transverse electric field (≈2 V/Å) is considered, it is noted that the transverse electric field only drops linearly with increasing the distance of the capacitor disks, and the tunneling amplitude will decrease exponentially with the increasing between distances of the molecular atoms and the capacitors. Therefore, the electron current in the transverse direction can be suppressed in experiments. ni is the electron number operator on the ith lattice site. Vij = U0 (1 + 0.6117rij2 )−1 2 is the intersite Coulomb interaction potential given by the Ohno parametrization [6, 9], and rij is the distance between the atomic sites i and jin units of Å. The expectation values of the spin-resolved occupation number 〈niσ 〉 and Fock exchange terms 〈ci†σ c jσ 〉 (i ≠ j ) can be calculated by self-consistent equations. Within the NEGF approach, the current passing through the benzene molecule is obtained by the Landauer formula [36, 37],
Figure 1. Schematics of the single BMJ system. Along the transport
direction (x direction), the bias voltage Vb is directly applied to the benzene molecule on the SiO2 substrate via source and drain platinum electrodes. The transverse electric field introduced by the transverse voltage Vt lies in the BMJ plane (x–y plane) and is perpendicular to the transport direction. The back-gate voltage Vg (z direction) is modulated by a back gate under the SiO2 substrate.
from the charge redistribution in BMJ with increasing bias voltages and the underlying Coulomb blockade effect at the interface of molecule–electrode contacts. Moreover, molecular switches, in which the conductance can be switched between two states with an applied voltage, can play a key role in logic and memory devices [29], and we find that BMJ can exhibit current switching and hysteresis behavior [26, 27, 30, 31] with large on–off ratios of currents [24] in the presence of a transverse electric field. Another observation is that partial spin-polarized current can be achieved in BMJ, resulting from the splitting of spin-resolved electron occupation numbers on carbon atoms of the benzene molecule by the transverse voltage. In fact, both from fundamental and technological points of view, molecular spintronics [30, 32, 33] is a new and interesting link between spintronics and molecular electronics, which will enable the manipulation of spins and charges in molecular electronic devices. We propose that molecular junctions with transverse electric fields can be a potential component in current switching or spintronic devices.
(
We describe electron states of the benzene molecule by means of a tight-binding model for the π orbital of carbon atoms, which takes into account the on-site and long-range Coulomb interactions. In the self-consistent mean-field (MF) theory, the system will be modeled by the following Hamiltonian within the Hartree–Fock approximation [6, 9, 26, 34]:
∑
(c
† iσ c jσ
)
+ h .c . +
(
)
i
+ ∑ Vij ( ni − 1)
(
nj − 1
)
i≠j
−
∑ Vij (
)
c j†σ ciσ ci†σ c jσ + h .c . ,
(
dωT ω , Vb, Vg, Vt R
E F + eVb 2
∫E
F
− eVb 2
(
) )
dωT ω , Vb, Vg, Vt ,
(2)
(3)
which is an essential quantity that determines the nonlinear transport properties of this molecular electronic system. Here ΓL (R) is the tunneling-width matrix characterizing the coupling strength of the molecule to the left (right) metallic electrode, and induces the broadening of the molecular orbital energy levels. ΓL (R) is a 6 × 6 matrix, with 〈1| ΓL |1〉 and 〈4| ΓR |4〉 being only nonzero elements. In order to model the interface between the molecule and the electrodes more realistically, we take into account the finite band width effects and assume the density of states of electrodes around the Fermi energy to be a Lorentzian form. Therefore the
i, σ
+ U0 ∑ ni ↑ ni ↓ + ni ↓ ni ↑
EL
∫E
T = Tr [ GrΓR GaΓL ],
∑ ( ϵ0 + Vg + Vti ) niσ
< i, j >, σ
2e h 2e = h
where e is the electron charge, ω the incident electron energy, and h Planckʼs constant. A bias voltage Vb is applied symmetrically between the source and drain electrodes. EL = EF + eVb 2 (ER = EF − eVb 2) is the EF of the left (right) electrode after the bias is applied. The transmission probability T (ω , Vb, Vg, Vt ) through the junction is given by the general formula [37, 38],
2. Methods
H MF = −t
)
I Vb, Vg, Vt =
(1)
i ≠ j, σ
where ci†σ (ciσ ) denotes the creation (annihilation) electron 2
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Nanotechnology 25 (2014) 465202
L
I (e ħ-1)
0.8 0.6 0.4
0.4
0.2
0.3
0 -3
Vg
occurrence of the NDC. (2) Nonzero magnetic moments on site 2 and 3 atoms (figures 3(d) and (e)) are induced in the region with about 7.4 V < Vb < 9.3 V, due to the splitting of electron occupation numbers Nup and Ndown. (3) The increasing of occupation numbers of site 1 and 4 atoms (figures 3(b) and (c)) at the voltage Vb ≈ 9.3 V is attributed to a new molecular orbital (LUMO+1) participating in the electron tunneling process. The corresponding transmission probabilities T (ω , Vb ) (figures 4(a) and (b)) at Vg = 0 and Vt = 0 show four peaks because of the degeneracy of energy levels of benzene molecular orbitals, which agrees with the results of the previous calculation with a Hückel Hamiltonian for the π system of benzene [40]. In fact, in this parallel and symmetric configuration, there are two quasi-localized π orbitals of benzene molecule, which donʼt participate in the electron transport through this molecular junction. One unexpected phenomenon is that the T curves are not continuous, and they suddenly split at the voltage (Vb ≈ 7.4 V) corresponding to the energy difference between LUMO and HOMO, indicating the occurrence of NDC at that voltage. Since the abrupt change of the transmission probability at Vb ≈ 7.4 V gives rise to its asymmetry with respect to the reference Fermi energy ω = EF (EF = −7.67 eV), it leads to the sudden decrease of the current according to the Landauer formula [36, 37] in the Methods section. Actually, explanations of NDC in molecular systems still remain controversial in the literature [3]. The charging and conformational changes of molecule were recognized as a possible mechanism for explaining NDC in molecular junctions reported by the first experiment [24]. It was shown that the asymmetry between LUMO and HOMO levels can cause asymmetry of coupling of the molecule to leads at high bias voltage, resulting in dramatic differences of potential profiles and electron occupation number distributions in different parts of this molecular junction along the transport direction, which are responsible for the Coulomb blockade effect appearing at the interface of molecule-lead contacts [38]; on the other hand, Ratner and his co-workers [26, 27] showed that a polaron model with self-consistent Hartree approximation can successfully predict NDC, switching, and also hysteresis I-V characteristics of molecular junctions. Our calculation suggests NDC of molecular junctions can be explained based solely on a tight-binding model of π electrons with the on-site and long-range Coulomb interactions. In addition to NDC, the spin-resolved components Tup and Tdown near the NDC region (7.4 V < Vb < 9.3 V) and in the large bias region (Vb > 11 V) are distinct, which induces a little spin-polarization of the current in these two regions as shown in figure 3(a). Next, we study the effects of the transverse electric field based on the I-V characteristic of this molecular junction. The current image as a function of Vb and Vt with fixed Vg = 3 V is shown in figure 5. For low values of Vb and Vt, there is a region where the resistance is very high, and similar results have been found by Di Ventra et alʼs first-principle calculations [10, 11]. Moreover, we find a zero current region (indicated by an arrow) is achieved at a finite transverse
0.5
Vt=0
0.2 0.1
NDC
-2 -1
) (V
2
0 0
6
4
8
10
12
14
16
Vb (V)
Figure 2. Current image as a function of Vg and Vb without Vt. The NDC region is indicated by an arrow.
hybridization strengths are 〈1| ΓL |1〉 =
〈4| ΓR |4〉 =
2)2
Γ (D . (ω − E R )2 + (D 2)2
Γ (D 2)2 (ω − E L )2 + (D 2)2
and
Γ is equal to 1.0 eV, and
D = 16 eV characterizes the band width of Pt electrodes [39]. G r (a) represents the retarded (advanced) Greenʼs function matrix of molecular states.
3. Results and discussion We first consider the system without external transverse voltages (Vt = 0). The image of the current I through the benzene molecule as a function of the back-gate voltage Vg and the bias voltage Vb is shown in figure 2. At the small absolute value of Vg, a pronounced NDC (indicated by an arrow) is observed when Vb exceeds approximately a threshold value (Vb ≈ 7.4 V), which corresponds to the energy difference between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) near the Fermi level. In the large bias region (Vb > 11 V), the current starts to decrease because of the mismatching of conduction bands of the left and right leads due to the finite band width effect. With the used parameter Γ = 1 eV in our calculation, the unit of current eΓ −1 ≈ 243μA and the magnitude of the corresponding current agree well with the previous first-principle calculations [18]. In order to understand the mechanism of the above NDC phenomena, we examine the I-V characteristic and its corresponding electron occupation number distribution in the molecule as a function of Vb in figure 3. One can see that the spin-resolved current components Iup and Idown begin to split in the NDC region (Vb ≈ 7.4 V) with corresponding splitting of the spin-up Nup and spin-down Ndown electron number distributions of the benzene molecule (figures 3(b)–(e)). There are some interesting features on these electron number distributions: (1) When the NDC happens (Vb ≈ 7.4 V), the total electron numbers Ntotal on the carbon atoms located at the interface of molecule–electrode contacts (site 1 and 4 carbon atoms as shown by the inset of figure 3(c)) have an abrupt increase; nevertheless, electron numbers on other carbon atoms decrease simultaneously. It is easy to see that the charge accumulation on the two interface carbon atoms increases the Coulomb repulsive potential for the tunneling electron, and therefore a blockade effect for charge transport through the molecule emerges, which results in the 3
W-H Zhu et al
Nanotechnology 25 (2014) 465202
a) Vg=0, Vt=0
b)
c)
1.2
L
I (e ħ-1)
0.5
0.4
Ntotal
1
Nup
2
1.0
5 4
0.8
3
Site 4
Ndown
Site 1
6
0.6 0.4
0.3
0.2
1.0
d)
e)
Site 2
Site 3
0.8 Itotal
0.6
Iup
0.1
0.4
Idown
0.0 0
2
4
6
8
10
12
Electron Number N
0.6
14 16 0
4
8
12
0
4
8
0.2 16
12
Vb (V) Figure 3. Current (a) and electron numbers on carbon atoms of benzene molecule (b–e) vs Vb without Vg and Vt, whose spin-resolved components are also plotted. Since Vb is applied to the benzene symmetrically, electron number distributions of site 6 and 5 carbon atoms (not shown here) have the same results as those of site 2 and 3 atoms, respectively. The inset of (c) shows the actual positions of site numbers of carbon atoms.
L
I (e ħ-1)
0.1 0.08
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
Vg=3V
0.06 0.04 0.02
)
(V Vb
0 1 0.5 0 0
1
2
3
4
5
Vt (V)
6
7
8
Figure 5. Current image as a function of Vb and Vt with Vg = 3 V. The zero current region is indicated by an arrow.
with Vb = 0.5 V of the current image of figure 5 and its corresponding spin-resolved current components Iup and Idown are plotted in figure 6(a). One can see that, in the presence of Vt, Itotal shows valley and peak structures as observed experimentally [4, 24] and theoretically [10, 11, 18] in some similar molecular systems, and the current approaches zero at Vt ≈ 4.7 V. Therefore, this BMJ can act as a molecular switching device [26, 27, 29, 30] with large on–off ratios [24] of current upon the application of a transverse electric field, which is in accordance with the results based on first-principle calculations [10, 11]. Interestingly, except for the large on–off ratios of currents, we also observe bistability and hysteresis phenomena of current switching in this BMJ by sweeping the transverse voltage from low value to high value, and then back to low value (figure 6(e)). This reversible switching characteristic of molecular junctions between two conducting states in response to the bias voltage has been proposed by recent experimental [31] and theoretical [26, 27] studies. However, it should be noted that here the hysteretic switching is controlled by a transverse electric field instead of the bias voltage. Furthermore, we find that at a large value of Vt, large splitting of current components Iup and Idown is observed
Figure 4. Spin-up Tup (a) and spin-down Tdown (b) components of transmission probabilities T (ω, Vb ) as a function of energy ω and bias voltage Vb without Vg and Vt.
voltage Vt, and the exact value of this critical voltage Vt depends on how the electron number is redistributed in BMJ within the self-consistent calculation. In this case, the molecular orbital, by which the electron could tunnel previously, is totally decoupled from the leads because of Vt. To detail the underlying features of I-V characteristics with an applied transverse electric field, a sectional drawing 4
W-H Zhu et al
Nanotechnology 25 (2014) 465202
Figure 6. Current (a) and electron numbers on carbon atoms of the benzene molecule (b–d) vs Vt with Vg = 3 V and Vb = 0.5 V, whose spinresolved components are also plotted. Due to the symmetry, similar results for the electron numbers on site 3, 4, and 5 atoms are not shown here. (e) Bistable and hysteretic current switching behaviors for three different values of Vb while Vt is swept from 4.0 V to 6.5 V, and then back to 4.0 V. One of the routes (Vb = 0.3 V) is marked by arrows.
figure 6(a). The difference is due to the electron number splitting (figures 6(b) and (c)) coming from Vt.
(figure 6(a)), indicating this BMJ can be useful in the molecular spintronic applications [30, 32, 33]. The large peak of Itotal can be mainly attributed to the Iup component of this total current. Using first-principles density functional and NEGF methods, Liu et al [30] proposed to obtain spin-polarized current by taking advantage of the energy difference between the antiparallel and parallel states of the molecular spins. However, we show that the large spin-polarization of the current can be realized by the splitting of electron occupation numbers Nup and Ndown of site 1 and 2 atoms induced by the transverse electric field (figures 6(b) and (c)). Moreover, there is a significant increase of electron occupation number on site 6 atom (figure 6(d)), further suggesting that Vt greatly influences the electron distributions in BMJ [10, 11]. Therefore, modulating Vt is a powerful means to get the best performance on current switching and spin-polarization properties of this molecular junction. We also observe that the corresponding transmission probabilities T (ω , Vt ) in the presence of transverse voltage Vt (figures 7(a) and (b)) have two more ridges than the degenerated molecular orbital energy level case (figure 4), because the degeneracy is broken by Vt, which causes strong modulation of the potential profiles and the density of states of this molecular system [10, 11]. Hence, the two previous quasilocalized molecular orbitals in the parallel and symmetric configuration begin to have tunnel coupling with the leads. It is interesting to note that the peak positions of transmission probabilities show strong dependence on the transverse voltage Vt. Furthermore, the branches of Tup and Tdown near the Fermi level ω = −7.67 eV in the relevant energy window (ω = [−7.92, −7.42]eV when EF = −7.67 eV and Vb = 0.5 V, according to the Landauer formula in the Methods section) are enlarged in figures 7(c) and (d), respectively. One can see the distinct difference between Tup and Tdown, which results in the corresponding spin-polarized current peak as shown in
4. Conclusions In summary, I-V characteristics through BMJ with direct metal and carbon atoms of molecule couplings are studied, based on a tight-binding model and NEGF approach. Due to the on-site and long-range Coulomb interactions in the benzene molecule, pronounced NDC at the bias voltage corresponding to the energy difference between LUMO and HOMO is predicted, with its intrinsic mechanism coming from the charge redistribution in BMJ and the Coulomb blockade effect induced by charge accumulations on carbon atoms at the interface of molecule–electrode contacts. The valley-peak structure of the I-V curve with a high resistance region at the low transverse voltage makes BMJ behave as a switching device with large on–off ratios. Hysteretic behavior of current is also found in the switching voltage region, which indicates the bistability of this system in the presence of the transverse electric field. In addition, by applying the transverse voltage, spin-polarized current can be obtained owing to splitting of spin-resolved electron occupation numbers on carbon atoms. Therefore, our calculation suggests that BMJ with transverse electric fields can be a potential candidate in current switching and spintronics applications. However, it should be noted that for BMJ some interesting features, e.g., NDC and the bistability, are obtained in the large bias regime or with large transverse electric field, which might cause difficulties in experimental realization. We notice that the exact values of the bias voltage or transverse electric field for the occurrence of NDC or the bistability are largely determined by the energy gap between the HOMO and LUMO in the molecule. Therefore, one may expect that the rich 5
W-H Zhu et al
Nanotechnology 25 (2014) 465202
Figure 7. Images of spin-up Tup (a) and spin-down Tdown (b) components of transmission probabilities T (ω, Vt ) as a function of ω and Vt with Vg = 3 V and Vb = 0.5 V, whose frequency parts near the Fermi energy (ω = [−7.92, −7.42]eV) are enlarged in (c) and (d), respectively.
transport phenomena can be much more easily observed in molecular junctions with a relatively small energy gap or artificial semiconductor quantum dot systems.
[10] di Ventra M, Pantelides S T and Lang N D 2000 Appl. Phys. Lett. 76 3448–50 [11] Rashkeev S N, di Ventra M and Pantelides S T 2002 Phys. Rev. B 66 033301 [12] Hansen T, Hansen T, Arcisauskaite V, Mikkelsen K V, Kongsted J and Mujica V 2010 J. Phys. Chem. C 114 20870–6 [13] Reddy P, Jang S Y, Segalman R A and Majumdar A 2007 Science 315 1568–71 [14] Taniguchi M, Tsutsui M, Yokota K and Kawai T 2009 Nanotechnology 20 434008 [15] Bergfield J P, Solis M A and Stafford C A 2010 ACS Nano 4 5314–20 [16] Seminario J M, de la Cruz C E and Derosa P A 2001 J. Am. Chem. Soc. 123 5616–7 [17] Kiguchi M, Tal O, Wohlthat S, Pauly F, Krieger M, Djukic D, Cuevas J C and van Ruitenbeek J M 2008 Phys. Rev. Lett. 101 046801 [18] di Ventra M, Pantelides S T and Lang N D 2000 Phys. Rev. Lett. 84 979–82 [19] Zhu W H, Ding G H and Dong B 2014 IEEE Trans. Electron Devices 61 1168–74 [20] Stafford C A, Cardamone D M and Mazumdar S 2007 Nanotechnology 18 424014 [21] Dadosh T, Gordin Y, Krahne R, Khivrich I, Mahalu D, Frydman V, Sperling J, Yacoby A and Bar-Joseph I 2005 Nature 436 677–80 [22] Smit R H M, Noat Y, Untiedt C, Lang N D, van Hemert M C and van Ruitenbeek J M 2002 Nature 419 906–9 [23] Meisner J S, Ahn S, Aradhya S V, Krikorian M, Parameswaran R, Steigerwald M, Venkataraman L and Nuckolls C 2012 J. Am. Chem. Soc. 134 20440–5 [24] Chen J, Reed M A, Rawlett A M and Tour J M 1999 Science 286 1550–2 [25] Reed M A 1999 Proc. IEEE 87 652–8
Acknowledgments This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2011CB925603, the National Natural Science Foundation of China (Grant Nos. 91121021 and 11074166), and Shanghai Natural Science Foundation (Grant No. 12ZR1413300). Our computations were supported by the center for HPC, Shanghai Jiao Tong University.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Aviram A and Ratner M A 1974 Chem. Phys. Lett. 29 277–83 Nitzan A and Ratner M A 2003 Science 300 1384–9 Tao N J 2006 Nat. Nanotechnol. 1 173–81 Reed M A, Zhou C, Muller C J, Burgin T P and Tour J M 1997 Science 278 252–4 Stokbro K 2010 J. Phys. Chem. C 114 20461–5 Cardamone D M, Stafford C A and Mazumdar S 2006 Nano Lett. 6 2422–6 Muller C J, Vleeming B J, Reed M A, Lamba J J S, Hara R, Jones L II and Tour J M 1996 Nanotechnology 7 409 Ke S H, Yang W and Baranger H U 2008 Nano Lett. 8 3257–61 Rincón J, Hallberg K, Aligia A A and Ramasesha S 2009 Phys. Rev. Lett. 103 266807 6
W-H Zhu et al
Nanotechnology 25 (2014) 465202
[33] Bogani L and Wernsdorfer W 2008 Nat. Mater. 7 179–86 [34] Yeganeh S, Ratner M A, Galperin M and Nitzan A 2009 Nano Lett. 9 1770–4 [35] Wang X F, Chakraborty T and Berashevich J 2010 Nanotechnology 21 485101 [36] Meir Y and Wingreen N S 1992 Phys. Rev. Lett. 68 2512–5 [37] Datta S 1995 Electronic Transport in Mesoscopic Systems (Cambridge: Cambridge University Press) [38] Cheraghchi H and Esfarjani K 2008 Phys. Rev. B 78 085123 [39] Smith N V, Wertheim G K, Hüfner S and Traum M M 1974 Phys. Rev. B 10 3197–206 [40] Hansen T, Solomon G C, Andrews D Q and Ratner M A 2009 J. Chem. Phys. 131 194704
[26] Galperin M, Ratner M A and Nitzan A 2005 Nano Lett. 5 125–30 [27] Yeganeh S, Galperin M and Ratner M A 2007 J. Am. Chem. Soc. 129 13313–20 [28] García-Suárez V M and Lambert C J 2008 Nanotechnology 19 455203 [29] Phoa K, Neaton J B and Subramanian V 2009 Nano Lett. 9 3225–9 [30] Liu R, Ke S H, Baranger H U and Yang W 2005 Nano Lett. 5 1959–62 [31] Blum A S, Kushmerick J G, Long D P, Patterson C H, Yang J C, Henderson J C, Yao Y, Tour J M, Shashidhar R and Ratna B R 2005 Nat. Mater. 4 167–72 [32] Rocha A R, García-Suárez V M, Bailey S W, Lambert C J, Ferrer J and Sanvito S 2005 Nat. Mater. 4 335–9
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