Diffusive Conductors Research Articles

Anomalous crossover between thermal and shot noise in macroscopic diffusive conductors

We predict the existence of an anomalous crossover between thermal and shot noise in macroscopic diffusive conductors. We first show that, besides thermal noise, these systems may also exhibit shot noise due to fluctuations of the total number of carriers in the system. Then we show that at increasing currents the crossover between the two noise behaviors is anomalous, in the sense that the low-frequency current spectral density displays a region with a superlinear dependence on the current up to a cubic law. The anomaly is due to the nontrivial coupling in the presence of the long-range Coulomb interaction among the three time scales relevant to the phenomenon, namely, diffusion, transit, and dielectric relaxation time.

Long-range Coulomb interaction and the crossover between quantum and shot noise in diffusive conductors

Frequency-dependent nonequilibrium noise in quantum-coherent diffusive conductors is calculated with account taken of long-range Coulomb interaction. For long and narrow contacts with strong external screening the crossover between quantum and shot noise takes place at frequencies much smaller than the voltage drop across the contact. We also show that under certain frequency limitations, the semiclassical and quantum-coherent approaches to shot noise are mathematically equivalent.

The role of boundary conditions in shot noise in elasticdiffusive conductors

The role played by boundary conditions at a kineticlevel in determining the noise properties of mesoscopicdiffusive conductors is evaluated by decomposing the currentfluctuations, into fluctuations of boundaries,self-consistent electric potential and the intrinsic Langevinsource. We show that the self-consistent contribution is essentiallycontrolled by the range of energies determined by thedistribution function at the contacts together with thedifferential conductivity in energy space. The intrinsiccontribution is completely determined by the boundary conditionsand is expressed in closed analytical form in these terms. Thepresent formulation provides a unifying microscopic approach ofthe noise properties of mesoscopic diffusive conductors showingthat (i) in the absence of Coulomb interaction the Fano factorγ takes values bounded in the range (1/3)⩽γ⩽1 and (ii) in the presence of Coulomb interaction the valuesof γ are no longer bounded in the above range andenhanced shot noise becomes possible.

Scanned probe microscopy of electronic transport in carbon nanotubes.

We use electrostatic force microscopy and scanned gate microscopy to probe the conducting properties of carbon nanotubes at room temperature. Multiwalled carbon nanotubes are shown to be diffusive conductors, while metallic single-walled carbon nanotubes are ballistic conductors over micron lengths. Semiconducting single-walled carbon nanotubes are shown to have a series of large barriers to conduction along their length. These measurements are also used to probe the contact resistance and locate breaks in carbon nanotube circuits.

High-fieldnoise in metallic diffusive conductors

We analyse high-field current fluctuations in degenerate conductorsby mapping the electronic Fermi-liquid correlations at equilibriumto their semi-classical non-equilibrium form. Our resultingBoltzmann description is applicable to diffusive mesoscopic wires.We derive a non-equilibrium connection between thermal fluctuationsof the current and resistive dissipation. In the weak-field limitthis is the canonical fluctuation-dissipation theorem. Away fromequilibrium, the connection enables explicit calculation of theexcess `hot-electron' contribution to the thermal spectrum. Weshow that excess thermal noise is strongly inhibited by Pauliexclusion. This behaviour is generic to the semi-classical metallicregime.

Observation of photon-assisted noise in a diffusive normal metal-superconductor junction

We report measurements of nonequilibrium noise in a diffusive normal metal-superconductor (N-S) junction in the presence of both dc bias and high-frequency ac excitation. We find that the shot noise of a diffusive N-S junction is doubled compared to a normal diffusive conductor. Under ac excitation of frequency nu the shot noise develops features at bias voltages |V| = hnu/(2e), which bear all the hallmarks of a photon-assisted process. Observation of these features provides clear evidence that the effective charge of the current carriers is 2e, due to Andreev reflection.

Energy Redistribution Between Quasiparticles in Mesoscopic Silver Wires

We have measured with a tunnel probe the energy distributionfunction of quasiparticles in silver diffusive wires connectedto two large pads (“reservoirs”), between which a biasvoltage was applied. From the dependence in energy and biasvoltage of the distribution function we have inferred theenergy exchange rate between quasiparticles. In contrast withpreviously obtained results on copper and gold wires, thesedata on silver wires can be well interpreted with the theoryof diffusive conductors either solely, or associated withanother mechanism, possibly the coupling to two-level systems.

Shot-noise suppression in nondegenerate semiconductors: the role of an energy-dependent scattering time

We present a Monte Carlo analysis of shot-noise suppression associated with long-range Coulomb interaction in nondegenerate diffusive conductors. Different energy dependencies of the elastic scattering time τ( ε)∝ ε α are considered. The shot-noise suppression factor γ is found to increase from about 1 3 to 0.9 when α decreases from 1 to −1.5. The simulations evidence dramatic changes in the profiles of electric field, carrier concentration and velocity in the above range of α values.

Local non-equilibrium distribution of charge carriers in a phase-coherent conductor

We use the scattering matrix approach to derive generalized Bardeen-like formulae for the conductances between the contacts of a phase-coherent multiprobe conductor and a tunneling tip which probes its surface. These conductances are proportional to local partial densities of states, called injectivities and emissivities. The current and the current fluctuations measured at the tip are related to and effective local non-equilibrium distribution function. This distribution function contains the quantum-mechanical phase-coherence of the charge carriers in the conductor and is given as products of injectivities and the Fermi distribution functions in the electron reservoirs. The results are illustrated for measurements on ballistic conductors with barriers and for diffusive conductors.

Kinetic theory of shot noise in nondegenerate diffusive conductors

We investigate current fluctuations in nondegenerate semiconductors, on length scales intermediate between the elastic and inelastic mean free paths. We present an exact solution of the nonlinear kinetic equations in the regime of space-charge limited conduction, without resorting to the drift approximation of previous work. By including the effects of a finite voltage and carrier density in the contact region, a quantitative agreement is obtained with Monte Carlo simulations by GonzAilez et al., for a model of an energy-independent elastic scattering rate. The shot-noise power P is suppressed below the Poisson value PPoisson=2eAa (at mean current Aa) by the Coulomb repulsion of the carriers. The exact suppression factor is close to 1/3 in a three-dimensional system, in agreement with the simulations and with the drift approximation. Including an energy dependence of the scattering rate has a small effect on the suppression factor for the case of short-range scattering by uncharged impurities or quasielastic scattering by acoustic phonons. Long-range scattering by charged impurities remains an open problem.

Local densities, distribution functions, and wave-function correlations for spatially resolved shot noise at nanocontacts

We consider a current-carrying, phase-coherent multi-probe conductor to which a small tunneling contact is attached. We treat the conductor and the tunneling contact as a phase-coherent entity and use a Green's function formulation of the scattering approach. We show that the average current and the current fluctuations at the tunneling contact are determined by an effective local non-equilibrium distribution function. This function characterizes the distribution of charge-carriers (or quasi-particles) inside the conductor. It is an exact quantum-mechanical expression and contains the phase-coherence of the particles via local partial densities of states, called injectivities. The distribution function is analyzed for different systems in the zero-temperature limit as well as at finite temperature. Furthermore, we investigate in detail the correlations of the currents measured at two different contacts of a four-probe sample, where two of the probes are only weakly coupled contacts. In particular, we show that the correlations of the currents are at zero-temperature given by spatially non-diagonal injectivities and emissivities. These non-diagonal densities are sensitive to correlations of wave functions and the phase of the wave functions. We consider ballistic conductors and metallic diffusive conductors. We also analyze the Aharonov-Bohm oscillations in the shot noise correlations of a conductor which in the absence of the nano-contacts exhibits no flux-sensitivity in the conductance.

Microscopic analysis of shot-noise suppression in nondegenerate diffusive conductors

We present a theoretical investigation of shot-noise suppression due to long-range Coulomb interaction in nondegenerate diffusive conductors. Calculations make use of an ensemble Monte Carlo simulator self-consistently coupled with a one-dimensional (1D) Poisson solver. We analyze the noise in a lightly doped active region surrounded by two contacts acting as thermal reservoirs. By taking the doping of the injecting contacts and the applied voltage as variable parameters, the influence of elastic and inelastic scattering in the active region is investigated. The transition from ballistic to diffusive transport regimes under different contact injecting statistics is analyzed and discussed. Provided significant space-charge effects take place inside the active region, long-range Coulomb interaction is found to play an essential role in suppressing the shot noise at $\mathrm{qU}\ensuremath{\gg}{k}_{B}T.$ In the elastic diffusive regime, momentum space dimensionality is found to modify the suppression factor \ensuremath{\gamma}, which within numerical uncertainty takes values respectively of about 1/3, 1/2, and 0.7 in the 3D, 2D, and 1D cases. In the inelastic diffusive regime, shot noise is suppressed to the thermal value.

Observing the Berry phase in diffusive conductors: Necessary conditions for adiabaticity

We investigate Berry phase effects in the magnetoconductance of diffusive systems, and determine the precise criterion for adiabaticity within the weak-localization formalism. We show that the exact solution of the Cooperon propagator for the special case of a cylindrically symmetric texture agrees with the adiabatic approximation in the adiabatic limit characterized by $\ensuremath{\tau}\ensuremath{\gg}{(1/d)(l}^{2}{/L}^{2}).$ We point out that orientational inhomogeneities in the magnetic field induce dephasing effects that can mask the Berry phase (and any other phase-coherent phenomena) for certain parameter values of system and field.

Noise in multiterminal diffusive conductors: Universality, nonlocality, and exchange effects

We study noise and transport in multiterminal diffusive conductors. Using a Boltzmann-Langevin equation approach we reduce the calculation of shot-noise correlators to the solution of diffusion equations. Within this approach we prove the universality of shot-noise in multiterminal diffusive conductors of arbitrary shape and dimension for purely elastic scattering as well as for hot electrons. We show that shot-noise in multiterminal conductors is a non-local quantity and that exchange effects can occur in the absence of quantum phase coherence even at zero electron temperature. It is also shown that the exchange effect measured in one contact is always negative -- in agreement with the Pauli principle. We discuss a new phenomenon in which current noise is induced by thermal transport. We propose a possible experiment to measure locally the effective noise temperature. Concrete numbers for shot-noise are given that can be tested experimentally.

PHOTON SHOT NOISE

A recent theory is reviewed for the shot noise of coherent radiation propagating through a random medium. The Fano factor [Formula: see text] (the ratio of the noise power and the mean transmitted current) is related to the scattering matrix of the medium. This is the optical analogue of Büttiker's formula for electronic shot noise. Scattering by itself has no effect on the Fano factor, which remains equal to 1 (as for a Poisson process). Absorption and amplification both increase the Fano factor above the Poisson value. For strong absorption [Formula: see text] has the universal limit [Formula: see text], with f the Bose–Einstein function at the frequency of the incident radiation. This is the optical analogue of the one-third reduction factor of electronic shot noise in diffusive conductors. In the amplifying case the Fano factor diverges at the laser threshold, while the signal-to-noise ratio [Formula: see text] reaches a finite, universal limit.

Noise properties and ac conductance of mesoscopic diffusive conductors with screening

A theory of non-equilibrium (``shot'') noise and high frequency conductance in diffusive mesoscopic conductors with screening is presented. Detailed results are obtained for two simple geometries, for both large and short electron-electron scattering length $l_{ee}$, at frequencies of the order of the inverse Thouless time $1/\tau_T$. The conductance and the noise are found to exhibit significant frequency dependence. For $L \ll l_{ee}$, the high-frequency ($\omega\tau_T \gg 1$) shot noise spectral density $S_I(\omega)$ approaches a finite value between $2eI/3$ and $2eI$, depending on the screening properties of the system, with temperature corrections to $S_I(\omega)$ being linear in $T$. However, when $L \gg l_{ee}$, $S_I(\omega)$ grows as $\omega^{1/4}$ (at T=0), is not upper-bound by $2eI$, and has a temperature-dependent component quadratic in $T$. As a result, measurements of $S_I(\omega, T)$ can be utilized as a probe of the strength of electron-electron scattering.

Shot noise in diffusive conductors: A quantitative analysis of electron-phonon interaction effects

Using the 'drift-diffusion-Langevin' equation, we have quantitatively analyzed the effects of electron energy relaxation via their interaction with phonons, generally in presence of electron-electron interaction, on shot noise in diffusive conductors. We have found that the noise power $ S_I(\omega )$ (both at low and high observation frequencies $\omega $) drops to half of its 'mesoscopic' value only at $\beta \gtrsim 100,$ where $\beta $ is the ratio of the sample length $L$ to the energy relaxation length $l_{% {\rm ph}}$ (the latter may be much larger then the dephasing length). It means in particular that at low temperatures the shot noise may be substantial even when $L\sim 10^{-2}$ -- $10^{-1}$ cm, and the conductor is 'macroscopic' in any other respect.

Direct determination of the electron-electron mean free path in diffusive mesoscopic samples using shot noise

Using the 'drift-diffusion-Langevin' equation, we have recently shown that finite-frequency shot noise in diffusive mesoscopic conductors is very sensitive to the ratio $\gamma \equiv L/\lee$ between the sample length $L$ and the electron-electron mean free path $\lee$. In this work we present numerical calculations of the noise at arbitrary value of $\gamma$. If coupled with accurate noise measurements, the results presented here could serve as a new and independent way of determining $\lee$ in a given sample.

Quantum Shot Noise at Local Tunneling Contacts on Mesoscopic Multiprobe Conductors

New experiments which measure the low-frequency shot-noise spectrum at local tunneling contacts on mesoscopic structures are proposed. The current fluctuation spectrum at a single tunneling tip is determined by local partial densities of states. The current-correlation spectrum between two tunneling tips is sensitive to the nondiagonal density of states elements which are expressed in terms of products of scattering states of the conductor. Such an experiment permits the investigation of correlations of electronic wave functions. We present specific results for a clean wire with a single barrier and for metallic diffusive conductors.

Universality of Shot Noise in Multiterminal Diffusive Conductors

We prove the universality of shot noise in multiterminal diffusive conductors of arbitrary shape and dimension for purely elastic scattering as well as for hot electrons. Using a Boltzmann-Langevin approach we reduce the calculation of shot-noise correlators to the solution of a diffusion equation. We show that shot noise in multiterminal conductors is a nonlocal quantity and that exchange effects can occur without quantum phase coherence even at zero electron temperature. Concrete numbers for shot noise are given that can be tested experimentally.