Title:

Timedependent quantum transport and fluctuations in molecular junctions

This thesis contains a theoretical treatment of the timedependent quantum transport of electrons through molecular junctions. In particular, the nonequilibrium Green's function (NEGF) method is used to obtain the timedependent current and current fluctuations in response to an arbitrarily timedependent spatially homogeneous bias in the leads and molecular region. All subsystems in the nanojunction are assumed to be noninteracting throughout, and in addition the wideband limit approximation (WBLA) is made in order to simplify the treatment of the leadmolecule coupling. The theoretical work involves a complete analytical solution in the twotime plane of the KadanoffBaym equations for all components of the molecular region Green's function defined on the imaginary time KonstantinovPerel' contour. This enables the derivation of expressions for timedependent quantities such as the particle number in the molecular region and a generalized timedependent LandauerBüttiker (LB) formula for the current through an arbitrarily large conducting molecule. The partitionfree approach is assumed for the switchon of a bias in the leads, which enables us to study the effects on the transient transport of the initial coupling between the subsystems of the nanojunction. The analytical expression for the full twotime Green's function is then used to study the quantum correlations between currents in the different leads. Using manybody methods, the twotime current correlation function is expressed in terms of a matrix product of Green's functions which are all known exactly. Therefore, in similar fashion to the generalized LB current, the current correlations generalize the LB theory of shot and thermal noise so that it is (i) partitionfree and (ii) valid for an arbitrary timedependent bias. To check its generality, this formula is shown to reproduce results for the zero and finitefrequency current noise that are known from the literature on this topic. The formulas for the current and current fluctuations are originally obtained in terms of integrals over the frequency and time planes. To render these formulas useful for calculations on large molecules they are then reexpressed with all frequency integrals replaced with special functions, which may be computed via the Padé expansion of the Fermi function. This results in a formalism which is ideally multiscale in time, i.e. which does not get more expensive with increasing number of time steps, and which can therefore be used to study the shorttime transient and steadystate transport quantities with identical computational expense. Some numerical calculations of the current and noise response to the switchon of a timedependent bias are then performed. The first kind of timedependent driving considered in this thesis is periodic. Some preliminary calculations are performed on periodicallydriven quantum dots, and the effect on the current of varying the driving frequency, temperature, leadmolecule coupling and driving amplitude is studied. Next, the problem of quantum pumping is considered, and sufficient conditions are found for a periodically driven system with zero timeaverage bias to generate a net timeaveraged current through the system. The effects of finite size on the current response to static and periodic biases are investigated, again using the molecular wire as an illustrative model. Next, calculations of the quantum pump current are presented for a particular biharmonic bias model, and in particular the effect on this current of breaking the sinusoidal shape of the driving field. Finally, the discussion turns to calculations of the crosslead current correlations in extended systems in response to static and periodic biases. By investigating molecular wires of different lengths and internal hopping parameters, a signature of the traversal time for electrons crossing the sample is found in both the transient and steadystate crosslead current correlations. As there are no restrictions on the nature of the timedependent bias driving the system out of equilibrium, one can use this formalism to study the electronic response to a classically fluctuating bias in the leads. The second kind of timedependent driving studied in this thesis is therefore nondeterministic, and exact formulas for the biasaveraged current response to a stochastic bias are derived for the first time. This problem is formally analogous to the use of the Langevin equation approach to obtain a relation between the conductance in a classical circuit and the strength of fluctuations in an external driving bias: the socalled Nyquist theorem. An exact quantum version of the Nyquist theorem is derived for a driving bias with colour noise which relates the strength of fluctuations, the correlation time of fluctuations, the temperature and conductance in a molecular junction. Due to the nonlinear relation between current and driving bias in these junctions, it is not always true that the conductance will decrease with increasing fluctuation strength. Instead, there can be particular intermediate values of the fluctuation strength at which peaks in the conductance are observed, in a stochastic resonance effect.
