Lecture 12. Electron Transport in Molecular Wires Possible Mechanisms

Transcription

1 Lecture 12. Electron Transport in Molecular Wires Possible Mechanisms In Lecture 11, we have discussed energy diagrams of one-dimensional molecular wires. Here we will focus on electron transport mechanisms in a molecular wire connected between two electrodes. One step coherent tunneling We note that even without the molecular wire, electron may still tunnel through the vacuum between the two electrodes and lead to a finite conductance. However, the efficiency of such through-space process is rather poor and the tunneling probability decreases sharply over distance (separation between the two electrodes). For example, if the vacuum gap is greater than ~2 nm, the tunneling current is too small to be measured experimentally. When a molecular wire bridges across the two electrodes, the tunneling barrier lowers and the tunneling probability can be orders of magnitude greater than the through-space situation. At low bias voltages, the conductance, G, of the molecular wire (together with the two electrode contacts) is given by the well-known expression (Ref. 1) G( E F ) β ( EF ) L = GCe, (1) where G c is the contact conductance, β is the tunneling decay constant, and L is the length of the molecular wire. In most cases, the Fermi level of the electrodes falls in the HOMO LUMO gap of the molecular wire, and the tunneling barrier is determined by the LUMO level (HOMO in 2 m * φ the case of hole tunneling) relative to the Fermi level and β = 0, where φ is the h tunneling barrier.

2 If the molecular wire has an energy level perfectly aligned to the Fermi level of the electrodes, the tunneling barrier becomes zero and one might expect that the electron transport is ballistic or resonant tunneling and β=0, and the corresponding transmission probability is 100%. This situation rarely occurs in reality, and one important reason is the polarization of the molecule upon charging (internal structure or surrounding environment of the molecule). In other words, when an electron traverses through the molecular wire, it may polarize the molecule or its surrounding environment which will shift the energy level of the molecular wire away from the Fermi level and destroy the ballistic or resonance condition. Whether the polarization will take place or not depends on the relative time scales of the electron traversal and polarization processes. Obviously we need to take a look at the timescales for tunneling and structural relaxation in the molecule (or environment). Tunneling traversal times The question of how long an electron tunnel through an energy barrier has attracted much attention. One way to define the tunneling time was proposed by Büttiker and Landauer, which is based on imposing an internal clock, e.g., a sinusoidal modulation of the barrier height, on the tunneling system (Ref. 2). If the modulation frequency is very low, the tunneling electron sees a slowly varying barrier that goes up and down according to the modulation. On the other hand, if the modulation frequency is very high, the electron sees only an average perturbation to the tunneling barrier. The inverse of the crossover frequency separating the two regimes is defined as the tunneling time. For tunneling through a one-dimensional rectangular barrier,

3 (2) tunneling time is (3) if assuming that d = x 2 = x 1 is not too small, the tunneling energy E is sufficiently below U B, and energy, defined by Equation 3, is the imaginary velocity for the under-barrier motion. Using the same internal clock for electron transfer via a superexchange mechanism (Fig. 1) in which the donor and acceptor energy levels, E A = E D, coupled to opposite ends of a molecular bridge described by an N-state tight-binding model with nearest-neighbor coupling V B, with an energy gap ) yields (Ref. 3) (4). Fig. 1. Parameters used in the expressions for tunneling traversal times. Left: Tunneling through a rectangular barrier. Right: Bridge-mediated transfer, where the gray area denotes the band associated with the tight-binding level structure of the bridge.

4 Nitzan et al. (Ref. 3) have shown that the results given by Equations 3 and 4 are two limiting cases (wide-and narrow-band limits), and a more general expression for the tunneling time is (5). where U B E B - 2V B - E D is the difference between the initial energy E D and the bottom of the conduction band, E B - 2V B (see Fig. 1). When V B 0, U B E B, Equation 5 becomes Equation 4. In the opposite limit, V B with U B kept constant, Equation 5 becomes (6). Expressing V B in terms of the effective mass for the band motion,, using a = d/n, Equation 6 yields the Büttiker-Landauer result. For tunneling through a molecular spacer modeled as a barrier of width 10 Å (N = 2 3) and height U B - E E 1 ev, Equations 3 and 4 yield 0.2 fs and 2 fs, respectively, both considerably shorter than the vibrational period of molecular vibrations. When the barrier is lower or when tunneling is affected or dominated by barrier resonances, the traversal time becomes longer and the competing relaxation and dephasing processes in the barrier may become effective. This is expected to be the rule for resonance transmission through molecular bridges, because the bandwidth associated with the bridge states (i.e. the electronic coupling between them; see Fig. 1) is considerably smaller than that in metals. As a consequence, thermal relaxation and dephasing are expected to dominate electron transport at and near resonance.

5 Thermal relaxation during electron tunneling It has long been recognized that tunneling electrons interact and may exchange energy with the nuclear degrees of freedom of the molecule (or local environment such as solvent molecules). One realization of such processes is inelastic electron-tunneling spectroscopy (Ref. 4), where the opening of inelastic channels upon increasing the electrostatic potential difference between the source and sink metals is manifested as a peak in the second derivative of the tunneling current with respect to this potential drop. Recent applications of this phenomenon within scanningtunneling spectroscopy hold great promise for making the STM a molecular analytical tool (Ref. 5). Inelastic electron tunneling may also cause chemical bond breaking and chemical rearrangement in the tunneling medium, either by electron-induced consecutive excitation or via transient formation of a negative ion (Ref. 6). The Hamiltonian describing the inelastic tunneling is (Ref. 3) H = H + H + H (7) el ph el ph where H el is the electronic Hamiltonian (8). In Equation 8 the states (k) are taken to be different manifolds of continuous-scattering states, denoted by a continuous index k. The electronic Hamiltonian can describe a scattering process in which the electron starts in one electrode and ends in another, and the states {n} belong to the molecular wire that causes the scattering process. These states may be the eigenstates of the

6 target Hamiltonian, in which case V n,m in Equation 8 vanishes, or some zero-order representation in which the basis states are mutually coupled by the exact-target Hamiltonian. H ph is the Hamiltonian of the phonon bath (9). which represents the thermal environment as a harmonic-phonon bath. H is the electron-phonon interaction, usually written in the form of el ph (10). Here c j and c j (j = n, n', k) create and annihilate an electron in electronic state j, while b and b similarly create and annihilate a phonon of mode, of frequency.. The coupling between the electronic system described by Eq. 8 and the phonon bath by Eq. 9 is assumed in Equation 10 to originate from a molecular state ( n>) dependent shift in the equilibrium position of each phonon mode. An exact solution to this scattering problem can be obtained for the particular case where the target is represented by a single state n = 1 and the phonon bath contains one oscillator of frequency,. In this case, it is convenient to consider the oscillator as part of the target that is therefore represented by a set of states m with energies (the zero-point energy can be set to 0). If the oscillator is initially in the ground state (m = 0), the cross-section for electron tunneling (or scattering) from left to right is given by (Refs. 7-8) (11).

7 where are states of the shifted harmonic oscillator that corresponds to the temporary negative ion (electron residing on the target) and. and are the shifts and widths of the dressed-target states associated with their coupling to the continuous manifolds and (12). The exact solution shown in Equation 11 can be obtained because of the simplicity of the system, which was characterized by a single-intermediate electronic state and a single-phonon mode. In more realistic situations characterized by many-bridge electronic states and many-phonon modes, one needs to resort to approximations or to numerical simulations. References 1. Minimal attenuation for tunneling through a molecular wire, M. Magoga and C. Joachim, Phys. Rev. B, 57, (1998). 2. Büttiker M, Landauer R Phys. Rev. Lett. 49: Nitzan A, Jortner J, Wilkie J, Burin AL, Ratner MA J. Phys. Chem. B 104: Wolf EL Principles of Electron Tunneling Spectroscopy. New York: Oxford Univ. Press. 5. Stipe BC, Rezaei MA, Ho W Science 280: Foley ET, Kam AF, Lyding JW, Avouris P Phys. Rev. Lett. 80: Domcke W, Cederbaum LS J. Phys. B 13: Wingreen NS, Jacobsen KW, Wilkins JW Phys. Rev. Lett. 61:1396 HOMEWORK: 12-1 Read Refs. 3 and For tunneling through a molecular spacer modeled as a barrier of width 10 Å (N = 2 3) and height U B - E E 1 ev, Calculate the tunneling time using Equations 3 and 4.