Time-dependent quantum transport and fluctuations in molecular junctions

This thesis contains a theoretical treatment of the time-dependent quantum transport of electrons through molecular junctions. In particular, the nonequilibrium Green's function (NEGF) method is used to obtain the time-dependent current and current fluctuations in response to an arbitrarily time-dependent spatially homogeneous bias in the leads and molecular region. All subsystems in the nanojunction are assumed to be noninteracting throughout, and in addition the wide-band limit approximation (WBLA) is made in order to simplify the treatment of the lead-molecule coupling. The theoretical work involves a complete analytical solution in the two-time plane of the Kadanoff-Baym equations for all components of the molecular region Green's function defined on the imaginary time Konstantinov-Perel' contour. This enables the derivation of expressions for time-dependent quantities such as the particle number in the molecular region and a generalized time-dependent Landauer-Büttiker (LB) formula for the current through an arbitrarily large conducting molecule. The partition-free approach is assumed for the switch-on 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 two-time Green's function is then used to study the quantum correlations between currents in the different leads. Using many-body methods, the two-time 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) partition-free and (ii) valid for an arbitrary time-dependent bias. To check its generality, this formula is shown to reproduce results for the zero- and finite-frequency 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 re-expressed 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 short-time transient and steady-state transport quantities with identical computational expense. Some numerical calculations of the current and noise response to the switch-on of a time-dependent bias are then performed. The first kind of time-dependent driving considered in this thesis is periodic. Some preliminary calculations are performed on periodically-driven quantum dots, and the effect on the current of varying the driving frequency, temperature, lead-molecule 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 time-average bias to generate a net time-averaged 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 cross-lead 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 steady-state cross-lead current correlations. As there are no restrictions on the nature of the time-dependent 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 time-dependent driving studied in this thesis is therefore non-deterministic, and exact formulas for the bias-averaged 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 so-called 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.Open Acces