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Density of States and Transmission in Molecular Transport Junctions
Density of States and Transmission in Molecular Transport Junctions Zsolt Bihary and Mark A. Ratner Department of Chemistry, Northwestern University, Evanston, IL 60208, USA Abstract Electron transport through molecular junctions (a molecule coordinated to two electrodes) is a nonequilibrium phenomenon, and corresponds to a current/voltage spectroscopy. We discuss such transport in two different limits. In the scanning tunneling microscope limit, where the coupling to one electrode is very much stronger than that to the other, the density of states (DOS) along the molecule effectively dominates the transport. However, when the couplings to the two electrodes are comparable, then the DOS itself is inadequate to determine either mechanism or magnitude of transport. The DOS still describes the quantum interference effects and the statistical aspects of transport, but the actual mechanism is described by another factor, that we call the transmittance. This transmittance function modulates the DOS, due to the effects of electronic structure changes (and, though not explored here, other couplings through interelectronic correlations or vibronic coupling). In the limit of transport outside of the band, the superexchange-type exponential decay with length enters not through the DOS, but through the transmittance function. Contents 1. Introduction 2. Formalism and model 3. Transmission in STM experiments 4. Transmission through a molecular junction 5. Conclusions Acknowledgements References
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1. INTRODUCTION The electronic spectrum and the current/voltage spectrum of molecular systems are related, since transmission through a molecular junction is greatly enhanced if the conduction electrons are injected at energies close to the molecular resonances. This relationship is even more pronounced in Scanning Tunneling Microscopy (STM). Indeed the modeling and the interpretation of such experiments is most simply expressed by the Tersoff–Hamann formula which gives the STM current as proportional to the density of states (DOS) at the tip location and at the Fermi energy [1–6]. ADVANCES IN QUANTUM CHEMISTRY, VOLUME 48 ISSN: 0065-3276 DOI: 10.1016/S0065-3276(05)48003-X
q 2005 Elsevier Inc. All rights reserved
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Z. Bihary and M. A. Ratner
Molecular transmission, the measure of the probability that an electron with a given energy transmits current through the molecular junction, is clearly related to the electronic structure of the molecule. At energies close to resonances, transmission (differential conductance) shows peaks, reminiscent of the electronic spectra of the molecule. In this chapter, we explore the relationship between the (local) DOS of the molecule and the transmission function that characterizes its transport properties. To keep the discussion as simple and transparent as possible, we shall describe the molecular system with a single-band tight-binding Hamiltonian and treat the contacts as locally coupled, site-wise incoherent point sources, within the wide-band limit. We will use the standard non-equilibrium Green function formalism in the energy representation. Only elastic currents will be discussed, within the single-electron picture, therefore we neglect any internal interactions, such as electron–phonon coupling and electron–electron Coulomb repulsion. We find that while the current is indeed proportional to a scaled DOS, an additional factor, called the transmittance and describing the mechanism of the transport, enters as a proportionality factor, and can dominate the behavior in some situations. 2. FORMALISM AND MODEL In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle–hole propagators, which have been at the heart of Jens’ electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9–15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions: Gr ðEÞ Z ðE K Hel K Sr ðEÞÞK1
(1)
Ga ðEÞ Z ðE K Hel K Sa ðEÞÞK1 Z ½Gr ðEÞC
(2)
G!ðEÞ Z Gr ðEÞS!ðEÞGa ðEÞ
(3)
GOðEÞ Z Gr ðEÞSOðEÞGa ðEÞ: r
a
!
(4) GOðEÞ r
are the retarded, Hel is the Hamiltonian, G (E), G (E), G ðEÞ and advanced, lesser and greater Green functions, and S (E), Sa(E), S!ðEÞ and SOðEÞ are the corresponding self-energies. We use the Huckel (tight-binding) model to describe the molecular system. The basis for electronic states is a set of spatially localized orbitals that may be considered atomic orbitals, or orbitals associated with different groups of atoms, ‘sites’ within the molecule.
Density of States and Transmission in Molecular Transport Junctions
25
The Hamiltonian in second quantized notation is X C tij a^i a^j ; H^ el Z
(5)
i; j
^i Þ describes the creation (annihilation) of an electron at site i in the where a^C i ða molecule. In this chapter, we will investigate homogeneous, one-dimensional, single-band models defined by the Hamiltonian H^ 1D Z 3
n X iZ1
^i K t a^C i a
nK1 X ^iC1 C a^C ^i Þ: ða^C i a iC1 a
(6)
iZ1
Here, n is the number of sites, 3 is the energy of the sites, t is the hopping parameter. The conventional negative sign is used in the Hamiltonian to energetically favor long wavelength molecular states. The Hamiltonian, the Green functions, and the self-energies are all represented by matrices, using the atomic (site) basis (this corresponds to real-space representation). Within the elastic, non-interacting model we are considering here, the self-energy stems purely from the coupling to the leads (contacts). We shall study two-terminal arrangements, so the self-energies !;O !;O contain two terms: Sr;a;!;OZ Sr;a; C Sr;a; , referring to the source and the 1 2 emitter. As we are concerned about the effect of the molecular energy structure on the transmission, and not about effects due to the band structure of the leads, we simply take the contacts into account with self-energy terms in the wide band limit: i Sr1ð2Þ Z K G1ð2Þ 2
(7)
i Sa1ð2Þ Z C G1ð2Þ 2
(8)
S! 1ð2Þ Z CiG1ð2Þ f1ð2Þ ðEÞ
(9)
SO 1ð2Þ Z KiG1ð2Þ ½1 K f1ð2Þ ðEÞ
(10)
where G1(2) is the escape rate matrix and f1(2)(E) is the Fermi–Dirac distribution characterized by the chemical potential in the corresponding leads. Certain combinations of the self-energies yield the escape rate matrices: ! iðSr1ð2Þ K Sa1ð2Þ ÞZ iðSO 1ð2Þ K S1ð2Þ ÞZ G1ð2Þ , as can be easily verified using equations (7)–(10). We define the total escape rate matrix as the sum GZ G1 C G2 Z iðSr K Sa ÞZ iðSOK S!Þ. The self-energy terms due to the contacts do not depend on the Green functions themselves, therefore equations (1)–(4) can be solved in a straightforward manner; no self-consistent calculation is needed. Once the Green functions have been obtained, together with the self-energies, they allow calculation of the quantities of interest. In particular, the spectral function is given as AðEÞ Z iðGr ðEÞ K Ga ðEÞÞ Z iðGOðEÞ K G!ðEÞÞ;
(11)
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Z. Bihary and M. A. Ratner
or, combining the expressions (3) and (4) and noting the definition of G, we can also write: AðEÞ Z Gr ðEÞGGa ðEÞ:
(12)
The diagonal elements of the spectral function yield the local DOS at the corresponding site, while their sum, the trace, yields the DOS: 1 ½AðEÞii ; 2p
(13)
1 Tr½AðEÞ: 2p
(14)
Nði; EÞ Z NðEÞ Z
The current transmitted through the junction is given as an integral of the flux of electrons at the source (or, equivalently, at the emitter) over different energies: e O O ! (15) iðEÞ Z Tr½S! 1 ðEÞG ðEÞ K S1 ðEÞG ðEÞ h Ð I Z dEiðEÞ: (16) When internal interactions are neglected, the current is purely coherent (elastic), and the formalism yields expressions that are consistent with the Landauer formula. The current can be rewritten as ð e IZ (17) dEðf1 ðEÞ K f2 ðEÞÞTðEÞ; h where T is the transmission function. The expression f1(E)Kf2(E) in equation (17) can be viewed as the window function for transmission; it assumes non-zero (unity) value only at energies where one of the contacts is occupied, but the other has a hole. The transmission function describes the probability for such an electron to indeed transfer through the molecular junction. It is related to the differential conductance as GZ ð2e2 =hÞT, and hence is a primary observable in experiments. In what follows, we will characterize the (energy-dependent) transport of the molecular system with the function T(E) and compare it with the (local) DOS. From the equations introduced in this section, the transmission function can be derived as [16–25]: T Z Tr½G1 Gr G2 Ga :
(18)
3. TRANSMISSION IN STM EXPERIMENTS Let us consider a molecule absorbed on a metallic surface with an STM tip positioned at site k. This arrangement is schematically shown in Fig. 1a. One of the ‘leads’ in this case is the tip, the other one is the metal itself. The escape rate matrix for the tip can be modeled as a local contact, only coupling the molecule at site k: ½G1 ij Z G1 dij dik ;
(19)
Density of States and Transmission in Molecular Transport Junctions
27
Fig. 1. Two limiting cases for transport. The upper figure represents the scanning tunneling microscope limit, where the molecular structure coupling to the electrode is much stronger than that to the scanning tip. The lower figure shows the molecular wire junction, where the interactions with the two electrodes are comparable in magnitude.
where G1 now is simply the escape rate through the tip junction. The metallic surface interacts with each site, along the entire molecular chain, and we model its rate matrix as ½G2 ij Z G2 dij ;
(20)
where G2 is the escape rate to the metal at any given site. Figure 1a illustrates this coupling scheme. Plugging the assumed forms for the G matrices (equations (19) and (20)) into the general formula equation (18), we obtain the transmission function for the STM arrangement: X T Z G1 G2 Qkj ; (21) j
where the Q matrix is defined as [26] ½Qij Z jGrij j2 :
(22)
Using equation (12) we can express the diagonal elements of the spectral function and calculate the local DOS at site i: X 2pNði; EÞ Z G1 Qik C G2 Qij : (23) j
If we make the very reasonable assumption that the (overall) coupling strength to the metal is much stronger than that to the tip, the RHS of equation (23) is dominated by the second term. In this limit, expressing the local DOS at site k, where the tip is coupled, we obtain X 2pNðk; EÞ Z G2 Qkj : (24) j
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Z. Bihary and M. A. Ratner
Comparing this result with equation (21), we can express T with the local DOS: T Z G1 2pNðk; EÞ:
(25)
Our simple calculation thus reflects the well-known result [1–6] that the differential resistance measured in the tunneling current through an STM tip is proportional to the local DOS of the molecule at the tip. In this section, we did not use any concrete model for the Hamiltonian, the only assumption we needed to make was that the molecule is much more strongly coupled to the metal surface than to the tip, which, at least for metallic surfaces, is always the case. 4. TRANSMISSION THROUGH A MOLECULAR JUNCTION In this section, we consider a molecular junction [9–15]. The molecular chain is sandwiched between two metallic leads, and it is only coupled to them at its terminal sites. This arrangement is illustrated in Fig. 1b. The escape rate matrices now can be modeled as ½G1 ij Z G1 dij di1 ;
(26)
½G2 ij Z G2 dij din ;
(27)
where once again G1 and G2 are now the escape rates to the source and emitter contacts. Using equation (18), we can calculate the transmission function: T Z G1 G2 Q1n Z G1 G2 jGr1n j2 :
(28)
This is a well-known result [16–25], the transmission of a molecular chain is related to the 1Kn element of the Green function. Using equation (12), we can also calculate the local DOS and the DOS: 2pNði; EÞ Z G1 Q1i C G2 Qni ; 2pNðEÞ Z ðG1 C G2 Þ
X
Q1i :
(29) (30)
i
In equation (30) we used the fact that our molecular Hamiltonian is mirror symmetric. Again, in the limit of very different couplings G2 [ G1 , the transmission function can be related to the local DOS at the weak coupling terminal: TZG12pN(1,E). This is a natural and rather general result noted by many authors: Due to the strong coupling on one end, the molecule essentially becomes part of the corresponding lead, the transmission is limited and determined by the junction at the other terminal [27–29]. In the case when the couplings are comparable, however, no such simple result is available. We shall discuss this case next. For a single-site model (and, more generally, in the case of proportionate couplings, even for interacting central regions), a very general relationship exists
Density of States and Transmission in Molecular Transport Junctions
29
between the transmission and the spectral function [30]: TZ
G1 G2 A: G1 C G2
(31)
To generalize this relation for the multi-site model, we define the transmittance function t, whose physical meaning we will shortly discuss, as TZ
G1 G2 tA; G1 C G2
(32)
where A is the averaged spectral function, basically the size-scaled DOS: 1 NðEÞ A Z Tr½A Z : n 2pn
(33)
The transmittance is a dimensionless function of energy, just like the transmission. By definition, for a single-site model it equals unity, and is independent of energy. Using our simple model, we shall now calculate it for arbitrarily long chains. To keep the calculation tractable, we will only consider the weak-coupling limit ðG1 ; G2 / tÞ, in this case the advanced (and retarded) Green functions do not depend on the Gs, and for the model defined by equation (6) they are given as ½Gr ðEÞij Z K
1 sin iq sinðn C 1 K jÞq ; t sin q sinðn C 1Þq
cos q Z K
ðE K 3Þ ; 2t
(34)
(35)
if i% j, and Gr , just like H1D, is symmetric. These formulae yield real q values for energies such that jEK 3j! 2t, i.e., for energies within the molecular band. Outside of the band the formulae are still valid, but q becomes purely imaginary. In this parameter regime, substituting the trigonometric functions with their hyperbolic counterparts retains the above forms with real q. Using equations (12) and (18) once again, we can readily calculate the transmission function and the averaged spectral function: G1 G2 sin2 q ; t2 sin2 ðn C 1Þq
(36)
G1 C G2 sinð2n C 1Þq 2n C 1 K : sin q 4nt2 sin2 ðn C 1Þq
(37)
TZ A Z
The rapidly changing trigonometric denominator gives rise to divergences that correspond to the molecular resonances [16,27,28]. We can see that the DOS contains the same divergent term as the transmission, therefore, at least close to the molecular resonances, the transmission seems to be proportional to the scaled DOS, as suggested by the equation defining t (equation (32)). The spectral function (see equation (37)) contains the term G1CG2, which refers to the contacts
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Z. Bihary and M. A. Ratner
rather than to the molecule itself. This is already apparent in equation (12). Indeed, the spectral function in the sense of quantum transport should and does depend on the arrangement of the leads. In the case when the contact self-energies are purely imaginary (such as in the case studied here, see equations (7)–(10)), the effect of the contacts is to broaden the molecular resonant lines, without shifting them. When the weak coupling limit is taken, all the lines remain very narrow, and the spectral function is essentially that of a free molecule. This feature is somewhat disguised in equation (37), a more careful analysis around the poles of the Green function (that takes into account the small but non-zero self-energies) shows the proper limiting behavior. Comparing equations (36) and (37), and taking into consideration equation (32), we can write the transmittance function as tZ
4n sin2 q : 2n C 1 K sinð2n C 1Þq=sin q
(38)
No divergences and dependence on the contact parameters G1,2 remain in the form for t. It shows the transmittance function (at least in the weak-coupling limit) is indeed a well-defined molecular quantity. We can rewrite equation (38), taking into account the definition of q (see equation (35)) and the definition of the Chebyshev polynomials of the second kind Un(cos q)Z sin[(nC1)q]/sin q as tZ
4nð1 K ððE K 3Þ=2tÞ2 Þ : 2n C 1 K U2n ððE K 3Þ=2tÞ
(39)
For even indices, Un(x) is an even function, therefore the transmittance is a symmetric function of energy. For nZ1, t is constant and equals unity by definition, as we have mentioned. When nZ2, we obtain: tðn Z 2Þ Z
2t2 : ðE K 3Þ2 C t2
(40)
The transmittance function is now a Lorentzian, having its maximum at the middle of the molecular band (at EZ3). The DOS for the two-site model is also a combination of two Lorentzians, peaking at the resonances of the molecule (at 3Gt), but it is important to realize that the width of these peaks is determined by the coupling to the leads which can be very small (indeed, in our derivations we have taken such limit), while the range of the transmittance function is determined by t, the band-width parameter. We can already see a behavior that is quite general (as we shall demonstrate later): The sharp peaks in the transmission function are due to its being proportional to the DOS, but the proportionality factor, the transmittance, is a slowly varying function that assumes large values within the entire molecular band. On a more physical note we may say that the transmittance function characterizes the mechanism of the transition (ballistic within the band vs. tunneling out of the band), while the resonance structure, characterized by the DOS, reflects instances of quantum interference of the transferring electron which can greatly enhance or inhibit the transmission.
Density of States and Transmission in Molecular Transport Junctions
31
Figure 2 shows the scaled DOS (solid black line), the transmission function (dashed line) and the transmittance function (solid gray line) for a six-site chain. The functions were calculated numerically, directly applying equations (1), (2), (12), (18) and (32). The hopping parameter t was taken as the unit of energy, and the on-site energy was chosen as the energy reference (3Z0). To avoid divergence in the DOS and transmission function, a small value was chosen for the leakage rates to the electrodes ðG1 =tZ G2 =tZ 0:1Þ. In order to show the large changes of the functions, the plot is semi-logarithmic. Inspection of Fig. 2 confirms our conclusions: The DOS and the transmission function show the six resonance peaks within the molecular band. The transmittance function does not change significantly within the band, but it drops sharply at the band edge. The diminishing transmission function outside of the band is mainly a consequence of the drop in the transmittance, not in the DOS. We now investigate the long chain limit. In this case (n/N), within the molecular band ðjEK 3j! 2tÞ, the denominator in equation (39) is dominated by the term proportional to n, and we obtain: tinband ðn/NÞ Z 2ð1 K ððE K 3Þ=2tÞ2 Þ:
(41)
Transmittance within the band is a simple quadratic function of energy. Its maximum is at the middle of the band, and it goes to zero at the edges of the band. In the long chain limit, transmittance (like the band width) becomes independent of system size. Within the molecular band, sudden changes in the conductance
Fig. 2. Density of states, transmission and transmittance for the six site chain, as described in the text. The scaled state density (solid black line) exhibits resonances arising from eigenstates on the bridge. The transmittance (solid gray line) drops beyond the limits of the band, and shows only minimal oscillations within the band itself. The behavior of the overall resulting transmission function (dashed line) is determined by the scaled DOS within the band, and by the transmittance outside of the band.
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Z. Bihary and M. A. Ratner
(either due to size changes or by changing the Fermi energy) are dominated by the instances of quantum interference, while the mechanism is always ballistic transmission, signified by the slowly changing transmittance function. Far outside of the molecular band ðjðEK 3Þj[ 2tÞ, the situation is very different. The DOS is proportional to (t/EK3)2 and is independent of system size. The denominator in equation (39) is now dominated by the highest order term of the Chebyshev polynomial 22n((EK3)/2t)2n, and the numerator by ((EK3)/2t)2. Taking the limit, we obtain toffband ðn/NÞ Z n
t 2ðnK1Þ ; E K3
(42)
and the transmission function is proportional to ðt=ðEK 3 )) 2n . This is the standard McConnell super-exchange result [10,31,32]. Transmittance now falls exponentially with system size, and the decrease is stronger when the energy is farther from the molecular band. We can see, that in the off-resonant case it is the transmittance and not the DOS that is mostly responsible for the exponential size-dependence of conduction. Indeed, it is the tunneling mechanism, and not the (missing) instances of quantum interference that limits conductance in this regime. We defined the transmittance as the proportionality factor between transmission and the scaled DOS, and indeed, within the molecular band it was a natural choice. In the off-band regime, the DOS is independent of system size, therefore the transmittance contains an additional n factor.
Fig. 3. The transmittance function and its variation with length for the tight binding model. For the two-site case, the exact result demonstrates the transmittance as a Lorentzian. However, for longer chains the transmittance (as in Fig. 2) varies weakly within the band, and drops quite sharply outside the band – this latter dependence dominates the overall transport in this region.
Density of States and Transmission in Molecular Transport Junctions
33
Figure 3 shows the transmittance function for molecular chains with various lengths using the same parameters as in Fig. 2. The functions are shifted for better visibility. For nZ2, it is a Lorentzian as suggested by the exact result (equation (40)). With growing chain length, the transmittance function shows weak modulations within the band, and drops sharply outside of the band. We also show the exact result obtained in the limit of infinite chain length (equation (41)). 5. CONCLUSIONS It was realized long ago that conductance properties of a molecule are strongly related to the (local) DOS of the system [1,2]. We rederived the known relationship for an STM arrangement within the non-equilibrium Green function (Keldysh) formalism, using a simple tight-binding model for the molecule, and an empirical description for the contacts. We tried to generalize and explore the conductance-energy level relation also for a molecular junction arrangement. We found it to be both useful and productive of insight to dissect the transmission function into the DOS and a proportionality factor we called transmittance. This way, two very different but equally important aspects of molecular conductance can be individually considered. Transmission can be enhanced or suppressed by instances of quantum interference of the transmitting electron, and is also greatly influenced by the conduction mechanism. We argue that the former is characterized by the DOS, while the latter is determined by the transmittance. A rough but instructive analogy exists between our formulation of the transmittance t and the introduction of the transmission coefficient, k, in the activated complex theory of chemical reactions [33]. There one writes the rate constant as kZ kKn‡ , with Kn‡ the quasi-equilibrium constant relating concentration at the barrier top to that of the reactants. The analogy arises because in both situations a time rate of change (current or rate constant) is the product of a ¯ or Kn‡ ) and a dynamical, mechanistic factor statistical factor ((G1G2/G1CG2)A (t or k). In both cases limits exist in which the dynamical correction is unimportant, so that the statistics (G1G2/G1CG2 (scaledDOS) or Kn‡ ) completely determines the rate. The essential point of the present chapter is that this limit is appropriate for the STM experiment of Fig. 1a, but not, in general, for the transport junction experiment in Fig. 1b. While we have limited our discussion to a tight-binding chain, very similar considerations are relevant to interacting electron models as well as electron/ environment interactions such as vibronic coupling or dephasing. The general factorization into a scaled state density (describing the energetics of the molecule) and a transmittance that characterizes the transmission mechanism should remain valid and constructive for such more general, more realistic models. ACKNOWLEDGEMENTS We are grateful to the DOD MURI/DURINT program, to the DARPA Mole Apps program, and to the chemistry divisions of the NSF and ONR for support of
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this research. This paper is dedicated to Jens Oddershede with love, with thanks and with respect for 35 years of science and friendship. REFERENCES [1] J. Tersoff and D. R. Hamann, Phys. Rev. Lett., 1983, 50, 1998. [2] J. Tersoff and D. R. Hamann, Phys. Rev. B, 1985, 31, 805. [3] H.-J. Guntherodt and R. Wiesendanger (eds) Scanning Tunneling Microscopy I: General Principles and Applications to Clean and Adsorbate-Covered Surfaces, Springer, Berlin, 1992. [4] R. Wiesendanger and H.-J. Gu¨ntherodt (eds) Scanning Tunneling Microscopy II: Further Applications and Related Scanning Techniques, Springer, Berlin, 1992. [5] R. Wiesendanger and H.-J. Gu¨ntherodt (eds) Scanning Tunneling Microscopy III: Theory of STM and Related Scanning Probe Methods, Springer, Berlin, 1993. [6] C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press, New York, 1993. [7] J. Linderberg and Y. Ohrn, Propagators in Quantum Chemistry, Academic Press, New York, 1973. [8] (a) J. Oddershede, Adv. Quantum Chem., 1978, 11, 275; (b) J. Oddershede, Adv. Chem. Phys., 1987, 69, 201. [9] C. Joachim, J. K. Gimzewski and A. Aviram, Nature, 2000, 408, 541. [10] A. Nitzan, Annu. Rev. Phys. Chem., 2001, 52, 681. [11] A. Nitzan and M. A. Ratner, Science, 2003, 300, 1384. [12] M. Di Ventra, S. Evoy and J. R. Heflin (eds) Introduction to Nanoscale Science and Technology, Kluwer Academic Publishers, Norwell, 2004. [13] M. A. Reed and T. Lee (eds) Molecular Nanoelectronics, American Scientific Publishers, Valencia, CA, 2003. [14] (a) See in Mol. Electron.: Sci. Technol. (Ann. NY Acad. Sci.), 852; (b) Mol. Electron. II (Ann. NY Acad. Sci.), 960; (c) Mol. Electron. III (Ann. NY Acad. Sci.), 1006. [15] S. Datta, Quantum Transport, Atom to Transistor, Cambridge University Press, Cambridge, in press. [16] V. Mujica, M. Kemp and M. A. Ratner, J. Chem. Phys., 1994, 101, 6849. [17] Y. Q. Xue, S. Datta and M. A. Ratner, Chem. Phys., 2002, 281, 151. [18] A. W. Ghosh, P. Damle, S. Datta and A. Nitzan, MRS Bull., 2004, 29, 391. [19] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [20] T. Seideman and W. H. Miller, J. Chem. Phys., 1992, 97, 2499. [21] E. G. Emberly and G. Kirczenow, Mol. Electron.: Sci. Technol. (Ann. NY Acad. Sci.), 1998, 852, 54. [22] H. Ness and A. J. Fisher, Phys. Rev. Lett., 1999, 83, 452. [23] N. S. Hush, Mol. Electron. III (Ann. NY Acad. Sci.), 2003, 1006, 1. [24] K. Stokbro, J. Taylor, M. Brandbyge and P. Ordejon, Mol. Electron. III (Ann. NY Acad. Sci.), 2003, 1006, 212. [25] M. Buttiker, IBM J. Res. Dev., 1988, 32, 63. [26] Z. Bihary and M. A. Ratner, Phys. Rev. B, submitted. [27] V. Mujica, M. Kemp, A. Roitberg and M. A. Ratner, J. Chem. Phys., 1996, 104, 7296. [28] M. Kemp, A. Roitberg, V. Mujica, T. Wanta and M. A. Ratner, J. Phys. Chem., 1996, 100, 8349. [29] S. Datta, W. D. Tian, S. H. Hong, R. Reifenberger, J. I. Henderson and C. P. Kubiak, Phys. Rev. Lett., 1997, 79, 2530. [30] Y. Meir and N. S. Wingreen, Phys. Rev. Lett., 1992, 68, 2512. [31] H. McConnell, J. Chem. Phys., 1961, 35, 508. [32] M. A. Ratner, J. Phys. Chem., 1990, 94, 4877. [33] An elementary but enlightening discussion is given by D. G. Truhlar and the authors in R. S. Berry, S. A. Rice, J. Ross, Physical Chemistry, 2nd edn., Oxford, 2000, sec. 30.6.
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1. INTRODUCTION The electronic spectrum and the current/voltage spectrum of molecular systems are related, since transmission through a molecular junction is greatly enhanced if the conduction electrons are injected at energies close to the molecular resonances. This relationship is even more pronounced in Scanning Tunneling Microscopy (STM). Indeed the modeling and the interpretation of such experiments is most simply expressed by the Tersoff–Hamann formula which gives the STM current as proportional to the density of states (DOS) at the tip location and at the Fermi energy [1–6]. ADVANCES IN QUANTUM CHEMISTRY, VOLUME 48 ISSN: 0065-3276 DOI: 10.1016/S0065-3276(05)48003-X
q 2005 Elsevier Inc. All rights reserved
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Z. Bihary and M. A. Ratner
Molecular transmission, the measure of the probability that an electron with a given energy transmits current through the molecular junction, is clearly related to the electronic structure of the molecule. At energies close to resonances, transmission (differential conductance) shows peaks, reminiscent of the electronic spectra of the molecule. In this chapter, we explore the relationship between the (local) DOS of the molecule and the transmission function that characterizes its transport properties. To keep the discussion as simple and transparent as possible, we shall describe the molecular system with a single-band tight-binding Hamiltonian and treat the contacts as locally coupled, site-wise incoherent point sources, within the wide-band limit. We will use the standard non-equilibrium Green function formalism in the energy representation. Only elastic currents will be discussed, within the single-electron picture, therefore we neglect any internal interactions, such as electron–phonon coupling and electron–electron Coulomb repulsion. We find that while the current is indeed proportional to a scaled DOS, an additional factor, called the transmittance and describing the mechanism of the transport, enters as a proportionality factor, and can dominate the behavior in some situations. 2. FORMALISM AND MODEL In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle–hole propagators, which have been at the heart of Jens’ electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9–15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions: Gr ðEÞ Z ðE K Hel K Sr ðEÞÞK1
(1)
Ga ðEÞ Z ðE K Hel K Sa ðEÞÞK1 Z ½Gr ðEÞC
(2)
G!ðEÞ Z Gr ðEÞS!ðEÞGa ðEÞ
(3)
GOðEÞ Z Gr ðEÞSOðEÞGa ðEÞ: r
a
!
(4) GOðEÞ r
are the retarded, Hel is the Hamiltonian, G (E), G (E), G ðEÞ and advanced, lesser and greater Green functions, and S (E), Sa(E), S!ðEÞ and SOðEÞ are the corresponding self-energies. We use the Huckel (tight-binding) model to describe the molecular system. The basis for electronic states is a set of spatially localized orbitals that may be considered atomic orbitals, or orbitals associated with different groups of atoms, ‘sites’ within the molecule.
Density of States and Transmission in Molecular Transport Junctions
25
The Hamiltonian in second quantized notation is X C tij a^i a^j ; H^ el Z
(5)
i; j
^i Þ describes the creation (annihilation) of an electron at site i in the where a^C i ða molecule. In this chapter, we will investigate homogeneous, one-dimensional, single-band models defined by the Hamiltonian H^ 1D Z 3
n X iZ1
^i K t a^C i a
nK1 X ^iC1 C a^C ^i Þ: ða^C i a iC1 a
(6)
iZ1
Here, n is the number of sites, 3 is the energy of the sites, t is the hopping parameter. The conventional negative sign is used in the Hamiltonian to energetically favor long wavelength molecular states. The Hamiltonian, the Green functions, and the self-energies are all represented by matrices, using the atomic (site) basis (this corresponds to real-space representation). Within the elastic, non-interacting model we are considering here, the self-energy stems purely from the coupling to the leads (contacts). We shall study two-terminal arrangements, so the self-energies !;O !;O contain two terms: Sr;a;!;OZ Sr;a; C Sr;a; , referring to the source and the 1 2 emitter. As we are concerned about the effect of the molecular energy structure on the transmission, and not about effects due to the band structure of the leads, we simply take the contacts into account with self-energy terms in the wide band limit: i Sr1ð2Þ Z K G1ð2Þ 2
(7)
i Sa1ð2Þ Z C G1ð2Þ 2
(8)
S! 1ð2Þ Z CiG1ð2Þ f1ð2Þ ðEÞ
(9)
SO 1ð2Þ Z KiG1ð2Þ ½1 K f1ð2Þ ðEÞ
(10)
where G1(2) is the escape rate matrix and f1(2)(E) is the Fermi–Dirac distribution characterized by the chemical potential in the corresponding leads. Certain combinations of the self-energies yield the escape rate matrices: ! iðSr1ð2Þ K Sa1ð2Þ ÞZ iðSO 1ð2Þ K S1ð2Þ ÞZ G1ð2Þ , as can be easily verified using equations (7)–(10). We define the total escape rate matrix as the sum GZ G1 C G2 Z iðSr K Sa ÞZ iðSOK S!Þ. The self-energy terms due to the contacts do not depend on the Green functions themselves, therefore equations (1)–(4) can be solved in a straightforward manner; no self-consistent calculation is needed. Once the Green functions have been obtained, together with the self-energies, they allow calculation of the quantities of interest. In particular, the spectral function is given as AðEÞ Z iðGr ðEÞ K Ga ðEÞÞ Z iðGOðEÞ K G!ðEÞÞ;
(11)
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Z. Bihary and M. A. Ratner
or, combining the expressions (3) and (4) and noting the definition of G, we can also write: AðEÞ Z Gr ðEÞGGa ðEÞ:
(12)
The diagonal elements of the spectral function yield the local DOS at the corresponding site, while their sum, the trace, yields the DOS: 1 ½AðEÞii ; 2p
(13)
1 Tr½AðEÞ: 2p
(14)
Nði; EÞ Z NðEÞ Z
The current transmitted through the junction is given as an integral of the flux of electrons at the source (or, equivalently, at the emitter) over different energies: e O O ! (15) iðEÞ Z Tr½S! 1 ðEÞG ðEÞ K S1 ðEÞG ðEÞ h Ð I Z dEiðEÞ: (16) When internal interactions are neglected, the current is purely coherent (elastic), and the formalism yields expressions that are consistent with the Landauer formula. The current can be rewritten as ð e IZ (17) dEðf1 ðEÞ K f2 ðEÞÞTðEÞ; h where T is the transmission function. The expression f1(E)Kf2(E) in equation (17) can be viewed as the window function for transmission; it assumes non-zero (unity) value only at energies where one of the contacts is occupied, but the other has a hole. The transmission function describes the probability for such an electron to indeed transfer through the molecular junction. It is related to the differential conductance as GZ ð2e2 =hÞT, and hence is a primary observable in experiments. In what follows, we will characterize the (energy-dependent) transport of the molecular system with the function T(E) and compare it with the (local) DOS. From the equations introduced in this section, the transmission function can be derived as [16–25]: T Z Tr½G1 Gr G2 Ga :
(18)
3. TRANSMISSION IN STM EXPERIMENTS Let us consider a molecule absorbed on a metallic surface with an STM tip positioned at site k. This arrangement is schematically shown in Fig. 1a. One of the ‘leads’ in this case is the tip, the other one is the metal itself. The escape rate matrix for the tip can be modeled as a local contact, only coupling the molecule at site k: ½G1 ij Z G1 dij dik ;
(19)
Density of States and Transmission in Molecular Transport Junctions
27
Fig. 1. Two limiting cases for transport. The upper figure represents the scanning tunneling microscope limit, where the molecular structure coupling to the electrode is much stronger than that to the scanning tip. The lower figure shows the molecular wire junction, where the interactions with the two electrodes are comparable in magnitude.
where G1 now is simply the escape rate through the tip junction. The metallic surface interacts with each site, along the entire molecular chain, and we model its rate matrix as ½G2 ij Z G2 dij ;
(20)
where G2 is the escape rate to the metal at any given site. Figure 1a illustrates this coupling scheme. Plugging the assumed forms for the G matrices (equations (19) and (20)) into the general formula equation (18), we obtain the transmission function for the STM arrangement: X T Z G1 G2 Qkj ; (21) j
where the Q matrix is defined as [26] ½Qij Z jGrij j2 :
(22)
Using equation (12) we can express the diagonal elements of the spectral function and calculate the local DOS at site i: X 2pNði; EÞ Z G1 Qik C G2 Qij : (23) j
If we make the very reasonable assumption that the (overall) coupling strength to the metal is much stronger than that to the tip, the RHS of equation (23) is dominated by the second term. In this limit, expressing the local DOS at site k, where the tip is coupled, we obtain X 2pNðk; EÞ Z G2 Qkj : (24) j
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Z. Bihary and M. A. Ratner
Comparing this result with equation (21), we can express T with the local DOS: T Z G1 2pNðk; EÞ:
(25)
Our simple calculation thus reflects the well-known result [1–6] that the differential resistance measured in the tunneling current through an STM tip is proportional to the local DOS of the molecule at the tip. In this section, we did not use any concrete model for the Hamiltonian, the only assumption we needed to make was that the molecule is much more strongly coupled to the metal surface than to the tip, which, at least for metallic surfaces, is always the case. 4. TRANSMISSION THROUGH A MOLECULAR JUNCTION In this section, we consider a molecular junction [9–15]. The molecular chain is sandwiched between two metallic leads, and it is only coupled to them at its terminal sites. This arrangement is illustrated in Fig. 1b. The escape rate matrices now can be modeled as ½G1 ij Z G1 dij di1 ;
(26)
½G2 ij Z G2 dij din ;
(27)
where once again G1 and G2 are now the escape rates to the source and emitter contacts. Using equation (18), we can calculate the transmission function: T Z G1 G2 Q1n Z G1 G2 jGr1n j2 :
(28)
This is a well-known result [16–25], the transmission of a molecular chain is related to the 1Kn element of the Green function. Using equation (12), we can also calculate the local DOS and the DOS: 2pNði; EÞ Z G1 Q1i C G2 Qni ; 2pNðEÞ Z ðG1 C G2 Þ
X
Q1i :
(29) (30)
i
In equation (30) we used the fact that our molecular Hamiltonian is mirror symmetric. Again, in the limit of very different couplings G2 [ G1 , the transmission function can be related to the local DOS at the weak coupling terminal: TZG12pN(1,E). This is a natural and rather general result noted by many authors: Due to the strong coupling on one end, the molecule essentially becomes part of the corresponding lead, the transmission is limited and determined by the junction at the other terminal [27–29]. In the case when the couplings are comparable, however, no such simple result is available. We shall discuss this case next. For a single-site model (and, more generally, in the case of proportionate couplings, even for interacting central regions), a very general relationship exists
Density of States and Transmission in Molecular Transport Junctions
29
between the transmission and the spectral function [30]: TZ
G1 G2 A: G1 C G2
(31)
To generalize this relation for the multi-site model, we define the transmittance function t, whose physical meaning we will shortly discuss, as TZ
G1 G2 tA; G1 C G2
(32)
where A is the averaged spectral function, basically the size-scaled DOS: 1 NðEÞ A Z Tr½A Z : n 2pn
(33)
The transmittance is a dimensionless function of energy, just like the transmission. By definition, for a single-site model it equals unity, and is independent of energy. Using our simple model, we shall now calculate it for arbitrarily long chains. To keep the calculation tractable, we will only consider the weak-coupling limit ðG1 ; G2 / tÞ, in this case the advanced (and retarded) Green functions do not depend on the Gs, and for the model defined by equation (6) they are given as ½Gr ðEÞij Z K
1 sin iq sinðn C 1 K jÞq ; t sin q sinðn C 1Þq
cos q Z K
ðE K 3Þ ; 2t
(34)
(35)
if i% j, and Gr , just like H1D, is symmetric. These formulae yield real q values for energies such that jEK 3j! 2t, i.e., for energies within the molecular band. Outside of the band the formulae are still valid, but q becomes purely imaginary. In this parameter regime, substituting the trigonometric functions with their hyperbolic counterparts retains the above forms with real q. Using equations (12) and (18) once again, we can readily calculate the transmission function and the averaged spectral function: G1 G2 sin2 q ; t2 sin2 ðn C 1Þq
(36)
G1 C G2 sinð2n C 1Þq 2n C 1 K : sin q 4nt2 sin2 ðn C 1Þq
(37)
TZ A Z
The rapidly changing trigonometric denominator gives rise to divergences that correspond to the molecular resonances [16,27,28]. We can see that the DOS contains the same divergent term as the transmission, therefore, at least close to the molecular resonances, the transmission seems to be proportional to the scaled DOS, as suggested by the equation defining t (equation (32)). The spectral function (see equation (37)) contains the term G1CG2, which refers to the contacts
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Z. Bihary and M. A. Ratner
rather than to the molecule itself. This is already apparent in equation (12). Indeed, the spectral function in the sense of quantum transport should and does depend on the arrangement of the leads. In the case when the contact self-energies are purely imaginary (such as in the case studied here, see equations (7)–(10)), the effect of the contacts is to broaden the molecular resonant lines, without shifting them. When the weak coupling limit is taken, all the lines remain very narrow, and the spectral function is essentially that of a free molecule. This feature is somewhat disguised in equation (37), a more careful analysis around the poles of the Green function (that takes into account the small but non-zero self-energies) shows the proper limiting behavior. Comparing equations (36) and (37), and taking into consideration equation (32), we can write the transmittance function as tZ
4n sin2 q : 2n C 1 K sinð2n C 1Þq=sin q
(38)
No divergences and dependence on the contact parameters G1,2 remain in the form for t. It shows the transmittance function (at least in the weak-coupling limit) is indeed a well-defined molecular quantity. We can rewrite equation (38), taking into account the definition of q (see equation (35)) and the definition of the Chebyshev polynomials of the second kind Un(cos q)Z sin[(nC1)q]/sin q as tZ
4nð1 K ððE K 3Þ=2tÞ2 Þ : 2n C 1 K U2n ððE K 3Þ=2tÞ
(39)
For even indices, Un(x) is an even function, therefore the transmittance is a symmetric function of energy. For nZ1, t is constant and equals unity by definition, as we have mentioned. When nZ2, we obtain: tðn Z 2Þ Z
2t2 : ðE K 3Þ2 C t2
(40)
The transmittance function is now a Lorentzian, having its maximum at the middle of the molecular band (at EZ3). The DOS for the two-site model is also a combination of two Lorentzians, peaking at the resonances of the molecule (at 3Gt), but it is important to realize that the width of these peaks is determined by the coupling to the leads which can be very small (indeed, in our derivations we have taken such limit), while the range of the transmittance function is determined by t, the band-width parameter. We can already see a behavior that is quite general (as we shall demonstrate later): The sharp peaks in the transmission function are due to its being proportional to the DOS, but the proportionality factor, the transmittance, is a slowly varying function that assumes large values within the entire molecular band. On a more physical note we may say that the transmittance function characterizes the mechanism of the transition (ballistic within the band vs. tunneling out of the band), while the resonance structure, characterized by the DOS, reflects instances of quantum interference of the transferring electron which can greatly enhance or inhibit the transmission.
Density of States and Transmission in Molecular Transport Junctions
31
Figure 2 shows the scaled DOS (solid black line), the transmission function (dashed line) and the transmittance function (solid gray line) for a six-site chain. The functions were calculated numerically, directly applying equations (1), (2), (12), (18) and (32). The hopping parameter t was taken as the unit of energy, and the on-site energy was chosen as the energy reference (3Z0). To avoid divergence in the DOS and transmission function, a small value was chosen for the leakage rates to the electrodes ðG1 =tZ G2 =tZ 0:1Þ. In order to show the large changes of the functions, the plot is semi-logarithmic. Inspection of Fig. 2 confirms our conclusions: The DOS and the transmission function show the six resonance peaks within the molecular band. The transmittance function does not change significantly within the band, but it drops sharply at the band edge. The diminishing transmission function outside of the band is mainly a consequence of the drop in the transmittance, not in the DOS. We now investigate the long chain limit. In this case (n/N), within the molecular band ðjEK 3j! 2tÞ, the denominator in equation (39) is dominated by the term proportional to n, and we obtain: tinband ðn/NÞ Z 2ð1 K ððE K 3Þ=2tÞ2 Þ:
(41)
Transmittance within the band is a simple quadratic function of energy. Its maximum is at the middle of the band, and it goes to zero at the edges of the band. In the long chain limit, transmittance (like the band width) becomes independent of system size. Within the molecular band, sudden changes in the conductance
Fig. 2. Density of states, transmission and transmittance for the six site chain, as described in the text. The scaled state density (solid black line) exhibits resonances arising from eigenstates on the bridge. The transmittance (solid gray line) drops beyond the limits of the band, and shows only minimal oscillations within the band itself. The behavior of the overall resulting transmission function (dashed line) is determined by the scaled DOS within the band, and by the transmittance outside of the band.
32
Z. Bihary and M. A. Ratner
(either due to size changes or by changing the Fermi energy) are dominated by the instances of quantum interference, while the mechanism is always ballistic transmission, signified by the slowly changing transmittance function. Far outside of the molecular band ðjðEK 3Þj[ 2tÞ, the situation is very different. The DOS is proportional to (t/EK3)2 and is independent of system size. The denominator in equation (39) is now dominated by the highest order term of the Chebyshev polynomial 22n((EK3)/2t)2n, and the numerator by ((EK3)/2t)2. Taking the limit, we obtain toffband ðn/NÞ Z n
t 2ðnK1Þ ; E K3
(42)
and the transmission function is proportional to ðt=ðEK 3 )) 2n . This is the standard McConnell super-exchange result [10,31,32]. Transmittance now falls exponentially with system size, and the decrease is stronger when the energy is farther from the molecular band. We can see, that in the off-resonant case it is the transmittance and not the DOS that is mostly responsible for the exponential size-dependence of conduction. Indeed, it is the tunneling mechanism, and not the (missing) instances of quantum interference that limits conductance in this regime. We defined the transmittance as the proportionality factor between transmission and the scaled DOS, and indeed, within the molecular band it was a natural choice. In the off-band regime, the DOS is independent of system size, therefore the transmittance contains an additional n factor.
Fig. 3. The transmittance function and its variation with length for the tight binding model. For the two-site case, the exact result demonstrates the transmittance as a Lorentzian. However, for longer chains the transmittance (as in Fig. 2) varies weakly within the band, and drops quite sharply outside the band – this latter dependence dominates the overall transport in this region.
Density of States and Transmission in Molecular Transport Junctions
33
Figure 3 shows the transmittance function for molecular chains with various lengths using the same parameters as in Fig. 2. The functions are shifted for better visibility. For nZ2, it is a Lorentzian as suggested by the exact result (equation (40)). With growing chain length, the transmittance function shows weak modulations within the band, and drops sharply outside of the band. We also show the exact result obtained in the limit of infinite chain length (equation (41)). 5. CONCLUSIONS It was realized long ago that conductance properties of a molecule are strongly related to the (local) DOS of the system [1,2]. We rederived the known relationship for an STM arrangement within the non-equilibrium Green function (Keldysh) formalism, using a simple tight-binding model for the molecule, and an empirical description for the contacts. We tried to generalize and explore the conductance-energy level relation also for a molecular junction arrangement. We found it to be both useful and productive of insight to dissect the transmission function into the DOS and a proportionality factor we called transmittance. This way, two very different but equally important aspects of molecular conductance can be individually considered. Transmission can be enhanced or suppressed by instances of quantum interference of the transmitting electron, and is also greatly influenced by the conduction mechanism. We argue that the former is characterized by the DOS, while the latter is determined by the transmittance. A rough but instructive analogy exists between our formulation of the transmittance t and the introduction of the transmission coefficient, k, in the activated complex theory of chemical reactions [33]. There one writes the rate constant as kZ kKn‡ , with Kn‡ the quasi-equilibrium constant relating concentration at the barrier top to that of the reactants. The analogy arises because in both situations a time rate of change (current or rate constant) is the product of a ¯ or Kn‡ ) and a dynamical, mechanistic factor statistical factor ((G1G2/G1CG2)A (t or k). In both cases limits exist in which the dynamical correction is unimportant, so that the statistics (G1G2/G1CG2 (scaledDOS) or Kn‡ ) completely determines the rate. The essential point of the present chapter is that this limit is appropriate for the STM experiment of Fig. 1a, but not, in general, for the transport junction experiment in Fig. 1b. While we have limited our discussion to a tight-binding chain, very similar considerations are relevant to interacting electron models as well as electron/ environment interactions such as vibronic coupling or dephasing. The general factorization into a scaled state density (describing the energetics of the molecule) and a transmittance that characterizes the transmission mechanism should remain valid and constructive for such more general, more realistic models. ACKNOWLEDGEMENTS We are grateful to the DOD MURI/DURINT program, to the DARPA Mole Apps program, and to the chemistry divisions of the NSF and ONR for support of
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this research. This paper is dedicated to Jens Oddershede with love, with thanks and with respect for 35 years of science and friendship. REFERENCES [1] J. Tersoff and D. R. Hamann, Phys. Rev. Lett., 1983, 50, 1998. [2] J. Tersoff and D. R. Hamann, Phys. Rev. B, 1985, 31, 805. [3] H.-J. Guntherodt and R. Wiesendanger (eds) Scanning Tunneling Microscopy I: General Principles and Applications to Clean and Adsorbate-Covered Surfaces, Springer, Berlin, 1992. [4] R. Wiesendanger and H.-J. Gu¨ntherodt (eds) Scanning Tunneling Microscopy II: Further Applications and Related Scanning Techniques, Springer, Berlin, 1992. [5] R. Wiesendanger and H.-J. Gu¨ntherodt (eds) Scanning Tunneling Microscopy III: Theory of STM and Related Scanning Probe Methods, Springer, Berlin, 1993. [6] C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press, New York, 1993. [7] J. Linderberg and Y. Ohrn, Propagators in Quantum Chemistry, Academic Press, New York, 1973. [8] (a) J. Oddershede, Adv. Quantum Chem., 1978, 11, 275; (b) J. Oddershede, Adv. Chem. Phys., 1987, 69, 201. [9] C. Joachim, J. K. Gimzewski and A. Aviram, Nature, 2000, 408, 541. [10] A. Nitzan, Annu. Rev. Phys. Chem., 2001, 52, 681. [11] A. Nitzan and M. A. Ratner, Science, 2003, 300, 1384. [12] M. Di Ventra, S. Evoy and J. R. Heflin (eds) Introduction to Nanoscale Science and Technology, Kluwer Academic Publishers, Norwell, 2004. [13] M. A. Reed and T. Lee (eds) Molecular Nanoelectronics, American Scientific Publishers, Valencia, CA, 2003. [14] (a) See in Mol. Electron.: Sci. Technol. (Ann. NY Acad. Sci.), 852; (b) Mol. Electron. II (Ann. NY Acad. Sci.), 960; (c) Mol. Electron. III (Ann. NY Acad. Sci.), 1006. [15] S. Datta, Quantum Transport, Atom to Transistor, Cambridge University Press, Cambridge, in press. [16] V. Mujica, M. Kemp and M. A. Ratner, J. Chem. Phys., 1994, 101, 6849. [17] Y. Q. Xue, S. Datta and M. A. Ratner, Chem. Phys., 2002, 281, 151. [18] A. W. Ghosh, P. Damle, S. Datta and A. Nitzan, MRS Bull., 2004, 29, 391. [19] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [20] T. Seideman and W. H. Miller, J. Chem. Phys., 1992, 97, 2499. [21] E. G. Emberly and G. Kirczenow, Mol. Electron.: Sci. Technol. (Ann. NY Acad. Sci.), 1998, 852, 54. [22] H. Ness and A. J. Fisher, Phys. Rev. Lett., 1999, 83, 452. [23] N. S. Hush, Mol. Electron. III (Ann. NY Acad. Sci.), 2003, 1006, 1. [24] K. Stokbro, J. Taylor, M. Brandbyge and P. Ordejon, Mol. Electron. III (Ann. NY Acad. Sci.), 2003, 1006, 212. [25] M. Buttiker, IBM J. Res. Dev., 1988, 32, 63. [26] Z. Bihary and M. A. Ratner, Phys. Rev. B, submitted. [27] V. Mujica, M. Kemp, A. Roitberg and M. A. Ratner, J. Chem. Phys., 1996, 104, 7296. [28] M. Kemp, A. Roitberg, V. Mujica, T. Wanta and M. A. Ratner, J. Phys. Chem., 1996, 100, 8349. [29] S. Datta, W. D. Tian, S. H. Hong, R. Reifenberger, J. I. Henderson and C. P. Kubiak, Phys. Rev. Lett., 1997, 79, 2530. [30] Y. Meir and N. S. Wingreen, Phys. Rev. Lett., 1992, 68, 2512. [31] H. McConnell, J. Chem. Phys., 1961, 35, 508. [32] M. A. Ratner, J. Phys. Chem., 1990, 94, 4877. [33] An elementary but enlightening discussion is given by D. G. Truhlar and the authors in R. S. Berry, S. A. Rice, J. Ross, Physical Chemistry, 2nd edn., Oxford, 2000, sec. 30.6.