Regime ballistic

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

Transport of electrons along conducting wires surrounded by insulators have been studied for several decades mechanisms of the transport phenomena involved are nowadays well understood (see [1, 2, 3] for review). In the ballistic regime where the mean free path is much longer than the wire lengths, l 3> d, the conductance is given by the Sharvin expression, G = (e2/-jrh)N, where N (kpa)2 is the number of transverse modes, a, is the wire radius, a Fermi wave vector. For a shorter mean free path diffusion controlled transport is obtained with the ohmic behavior of the conductance, G (e2/ph)N /d, neglecting the weak localization interference between scattered electronic waves. With a further decrease in the ratio /d, the ohmic behavior breaks down due to the localization effects when /d < N-1 the conductance appears to decay exponentially [4]. 

One should be cautioned to ascribe the case Rq/ f — 1 to the ballistic regime with the tunnel barrier having Z parameter of about 0.5. In most cases this value of Z in the BTK [9] fitting corresponds to the diffusive regime of current flow. This happens in many cases, where the same Z value have many junctions with quite different materials as dissimilar electrodes. 

The question may arise whether the self-energy effects are important in the normal state. These are known to be smaller than the inelastic backscattering nonlinearities in the ballistic regime [18]. If we decrease the contact size d or the elastic mean free path li in order to make the inelastic contribution negligible, the latter parameters become comparable to the Fermi wave length of charge carriers and the strong nonlinearities connected with localization occur, which masks the desired phonon structure [19]. 

One should also invoke Fermi statistics. A typical tunnel curve is shown in Fig. 12 for SET model with D = 14 a.u., a = 1 a.u., the work function of electrodes W = 0.4 a.u., the Fermi energy Ee = 0.2 a.u., and the polarizability a = 200 a.u. (of Na atom). The potential drops near the interface of the source-drain electrodes, as it should for the ballistic regime. The tunnel curve has a single shallow well at a small bias voltage. When the latter increases, the well becomes deeper, and the dot is attracted to the inter-electrode gap center... 

At present, advances in nanotechnology allow to create semiconductor nanostructures in which linear dimensions of the conductive channel in the direction of propagation of the electron wave are smaller than the mean free path of the electron. In such a channel particles move in a ballistic regime that allows to study experimentally the effects of ballistic transport in such structures, in particular, various electron interference effects [1]. A large number of theoretical works were devoted to the investigation of electron quantum ballistic transport in ID and 2D nanostructures whose common feature is the presence in quantum channels of regions of a sharp (nonadiabatic) variation either of the channel s geometry or a potential relief in it [2-4]. 

Reasoning along the same lines as for the DIRW case, we obtain the following result In the diffusive as well as the ballistic regime, the stable state (1/2, 1/2) invades the unstable state (0, 0) in the Fisher DDRW in the form of a propagating front. The front travels at constant velocity v with v e [wddrw> K)-... 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

At short timescales (t < 500 ps) the MSD exhibits a quadratic time dependence. This is known as the ballistic regime, where particle collisions are infrequent. In nanoporous materials, an intermediate regime starts when particles are colliding but only with a subset of the other particles due to the confinement. When particles are able to escape the local environment and explore the full macroscopic network, the diffusive regime is reached. In the diffusive regime, the MSD becomes linear with time, exhibiting a slope of one on a log-log scale. 

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