Elastic Mean Free Path Research Articles

Simultaneous observation of small- and large-energy-transfer electron-electron scattering in three-dimensional indium oxide thick films

In three-dimensional (3D) disordered metals, the electron-phonon (e-ph) scattering is the sole significant inelastic process. Thus the theoretical prediction concerning the electron-electron scattering rate as a function of temperature T in 3D disordered metal has not been fully tested thus far, though it was proposed 40 years ago (Schmid A., Z. Phys., 271 (1974) 251). We report here the simultaneous observation of small- and large-energy-transfer scattering in 3D indium oxide thick films. In the temperature region , the temperature dependence of resistivity of each film obeys Bloch-Grüneisen law, indicating that the films possess degenerate-semiconductor characteristics in electrical transport property. In the low-temperature regime, as a function of T for each film can not be ascribed to e-ph scattering. To quantitatively describe the temperature behavior of , both the 3D small- and large-energy-transfer scattering processes should be considered. (The small- and large-energy-transfer scattering rates are proportional to and T2, respectively.) In addition, the experimental prefactors of and T2 are proportional to and (kF is the Fermi wave number, ℓ is the electron elastic mean free path, and EF is the Fermi energy), respectively, which are completely consistent with the theoretical predictions. Our experimental results fully demonstrate the validity of the theoretical predictions concerning both small- and large-energy-transfer scattering rates.

A universal equation for computing the beam broadening of incident electrons in thin films

A universal equation for computing the beam broadening of incident electrons in thin films is presented. This equation is based on the concepts of anomalous diffusion with the Hurst exponent H. When the thickness to elastic mean free path ratio, t/λ, is greater than 1, the Hurst exponent goes to 0.5 and this random walk behavior leads to the Goldstein et al. [1] beam broadening equation when non-relativistic screened Rutherford elastic cross-sections are used. When t/λ≪1, the lack of elastic collisions for the electron trajectories gives an H exponent of 1 and a different beam broadening equation is obtained. A general equation to compute the beam broadening that takes into account the variation of H with t/λ is presented and this equation was fitted and validated with Monte Carlo simulations of electron trajectories in thin films.

Anomalous dephasing scattering time of Zr80Sn20−xFex alloys at low temperature

We report the results of a comprehensive study of weak electron localization (WEL) and electron–electron interaction (EEI) effects in disordered Zr80Sn20−xFex alloys. The resistivity in absence of magnetic field shows a minimum at temperature T=Tm and follows ρo5/2T1/2 law within the temperature range 5K≤T≤Tm, which suggests predominant EEI effect. Magnetoresistivity is positive due to strong spin–orbit interaction. The dephasing scattering time is dominated by the electron–phonon scattering. The electron–phonon scattering rate shows anomalous behavior and obeys the relation τe−ph−1∝T2le, where le is the electron elastic mean free path. The zero temperature scattering time (τ0) strongly depends on the disorder and its magnitude decreases with increasing disorder resistivity ρ0. Such anomalous behavior cannot be explained in terms of existing theories.

Electron–Phonon Energy Relaxation Time in Thin Strongly Disordered Titanium Nitride Films

We have measured the energy relaxation times from the electron bath to the phonon bath in strongly disordered TiN films grown by atomic layer deposition. The measured values of τ eph vary from 12 to 91 ns. Over a temperature range from 3.4 to 1.7 K, they follow T -3 temperature dependence, which are consistent with values of τ eph reported previously for sputtered TiN films. For the most disordered film, with an effective elastic mean free path of 0.35 nm, we find a faster relaxation and a stronger temperature dependence, which may be an additional indication of the influence of strong disorder on a superconductor.

Linear mean free path and quadratic temperature dependence of electron-phonon scattering rate in V82Al18-xFex alloys at low temperature

We have reported a comprehensive study on temperature and disorder dependence of inelastic electron dephasing scattering rate in disordered V82Al18-xFex alloys. The dephasing scattering time has been measured by analysis of low field magnetoresistance using the weak localization theory. In absence of magnetic field the variation of low temperature resistivity rise follows the relation Δρ(T)∝−ρ05/2T, which is well described by three-dimensional electron-electron interactions. The temperature-independent dephasing rate strongly depends on disorder and follows the relation τ0−1∝le, where le is the electron elastic mean free path. The inelastic electron-phonon scattering rate obeying the anomalous relation τe−ph−1∝T2le. This anomalous behavior of τe−ph−1 cannot be explained in terms of current theories for electron-phonon scattering in impure dirty conductors.

Enhancement of thermoelectric effect in diffusive superconducting bilayers with magnetic interfaces

We demonstrate that thermoelectric currents in superconducting bilayers with a spin-active interface are controlled by the two competing processes. On one hand, spin-sensitive quasiparticle scattering at such an interface generates an electron-hole imbalance and yields an orders-of-magnitude enhancement of the thermoelectric effect in the system. On the other hand, this electron-hole imbalance gets suppressed in the superconductor bulk due to electron scattering on nonmagnetic impurities. As a result, large thermoelectric currents can only flow in the vicinity of the spin-active interface and decay away from this interface at a distance exceeding the electron elastic mean free path $\ensuremath{\ell}$. The magnitude of the thermoelectric effect reaches its maximum provided $\ensuremath{\ell}$ becomes of order of the total bilayer thickness.

Photoelectron transport in the surface region of solids: universal analytical formalism for quantitative applications of electron spectroscopies

An advanced analytical theory describing electron transport in the surface region of solids may have accuracy comparable to Monte Carlo simulations of electron trajectories, however such an approach requires knowledge of a parameter called the single scattering albedo. This parameter is material dependent and can be calculated from the elastic mean free path and transport mean free path for signal electrons. An attempt is made to derive a simple expression that accurately describes the energy dependence of single scattering albedo in a wide energy range from 50 eV to 30 keV for 78 elemental solids. For these solids and the considered energy range, the mean percentage deviations between the reference values and values calculated from the fitted function were found to be generally well below 1%; the largest value of this deviation was equal to 0.86% (europium). Calculation of the single scattering albedo with high accuracy requires only five fitted coefficients for a given element. Recommendations are also given for calculations of this parameter for compounds. Different predictive formulas expressed in terms of the single scattering albedo are briefly discussed.

Effect of atomic structure on excitation of the electronic subsystem of a solid by a swift heavy ion

An effect of different phase states of a solid on excitation of its electronic subsystem due to penetration of a swift heavy ion (SHI) is examined on example of silicon dioxide (crystalline quartz vs. amorphous glass). The complex dielectric function formalism describing collective response of the electronic and ionic subsystems of a condensed target to excitation is used to calculate scattering cross sections of a penetrating ion and electrons generated due to target ionizations. A Monte Carlo model based on these cross sections is applied for tracing electron kinetics in the nanometric vicinity of the trajectory of a swift heavy ion. It is demonstrated that differences of the maximal values of the SHI energy losses and the electron inelastic mean free paths calculated for two phase states of SiO2 do not exceed 10–15%, whereas the elastic mean free paths differ more significantly.

Nanoscale three-dimensional reconstruction of elastic and inelastic mean free path lengths by electron holographic tomography

Electron holography at medium resolution simultaneously probes projected electrostatic and magnetostatic potentials as well as elastic and inelastic attenuation coefficients with a spatial resolution of a few nanometers. In this work, we derive how the elastic and inelastic attenuation can be disentangled. Using that result, we perform the first three dimensional tomographic reconstruction of potential and (in)elastic attenuation in parallel. The technique can be applied to distinguish between functional potentials and composition changes in nanostructures, as demonstrated using the example of a GaAs—Al0.33Ga0.67As core-shell nanowire.

Anderson localization in two-dimensional graphene with short-range disorder: One-parameter scaling and finite-size effects

We study Anderson localization in graphene with short-range disorder using the real-space Kubo-Greenwood method implemented on graphics processing units. Two models of short-range disorder, namely, the Anderson on-site disorder model and the vacancy defect model, are considered. For graphene with Anderson disorder, localization lengths of quasi-one-dimensional systems with various disorder strengths, edge symmetries, and boundary conditions are calculated using the real-space Kubo-Greenwood formalism, showing excellent agreement with independent transfer matrix calculations and superior computational efficiency. Using these data, we demonstrate the applicability of the one-parameter scaling theory of localization length and propose an analytical expression for the scaling function, which provides a reliable method of computing the two-dimensional localization length. This method is found to be consistent with another widely used method which relates the two-dimensional localization length to the elastic mean free path and the semiclassical conductivity. Abnormal behavior at the charge neutrality point is identified and interpreted to be caused by finite-size effects when the system width is comparable to or smaller than the elastic mean free path. We also demonstrate the finite-size effect when calculating the two-dimensional conductivity in the localized regime and show that a renormalization group beta function consistent with the one-parameter scaling theory can be extracted numerically. For graphene with vacancy disorder, we show that the proposed scaling function of localization length also applies. Lastly, we discuss some ambiguities in calculating the semiclassical conductivity around the charge neutrality point due to the presence of resonant states.

Experimental measurements of the coherent field resulting from the interaction of an ultrasonic shock wave with a multiple scattering medium

Whereas multiple scattering and shock wave formation are known to be antagonistic phenomena, this work concentrates on the interaction of an ultrasonic shock wave with a random multiple scattering medium. The shock wave is generated by long distance propagation of a short pulse (4 periods at a 3.5 MHz central frequency) in water before it encounters the scattering medium (a slab-shaped random set of parallel metallic rods). Transmitted waves are recorded over hundreds of positions along the lateral dimension of the slab to estimate the ensemble-averaged transmitted field ⟨ϕ(t)⟩, also known as the coherent wave. Experiments are repeated for different thicknesses L of the slab and different emission amplitudes. The elastic mean free path le (i.e the typical distance for the decreasing of the coherent intensity |⟨ϕ(t)⟩|2 due to scattering) is determined as well as the harmonic rate of the averaged transmitted wave. Experimental results are discussed and compared to the linear case.

On the validity of trajectory methods for calculating the transport of very low energy (

It is easily demonstrated that a trajectory picture of low energy electron transport in condensed matter is not compatible with the Heisenberg uncertainty principle. The uncertainty in the position of a low energy electron is large and may in fact be larger than an entire simulated trajectory. This might be interpreted to mean that trajectory methods are not applicable. However, this conclusion is not correct. In the present paper, the evidence for the validity of low energy electron trajectory simulation is discussed, as well as the wave aspects and quantum nature of low energy electron transport in liquids and amorphous solids.It is pointed out that the validity of a trajectory approach to low energy electron transport in a liquid or amorphous solid partly is due to its ability to reproduce the average results of coherent elastic multiple wave scattering in a randomlike medium, and moreover that this ability may be further enhanced by the presence of inelastic scattering. The resulting validity of the trajectory method may be referred to as circumstancial validity, which is of a nature different from the explicit validity of trajectory methods which are compatible with the uncertainty principle.A previous systematic analysis of the limits of circumstancial validity is revisited and discussed for the basic case of multiple elastic scattering of a particle in a random medium of point scatterers. The detailed limits of circumstancial validity are graphically demonstrated in terms of particle wavelength, average distance between scatterers and elastic mean free path. Their immediate applicability to neutron transport is noted. The approximate nature of the point scatterer model as regards electron transport is adressed. In order to obtain an extrapolation of the result of the point scatterer model, it is observed that an increasing error of the trajectory method appears together with an increased amplitude of the multiple wave scattering taking place within the medium. On the basis of this observation, an extrapolation is proposed which provides a rough estimate of the relative error of the trajectory method when applied to multiple elastic scattering of low energy electrons in real liquids or amorphous solids.

Imaging the Conductance of Integer and Fractional Quantum Hall Edge States

We measure the conductance of a quantum point contact (QPC) while the biased tip of a scanning probe microscope induces a depleted region in the electron gas underneath. At finite magnetic field we find plateaus in the real-space maps of the conductance as a function of tip position at integer (\nu=1,2,3,4,6,8) and fractional (\nu=1/3,2/3,5/3,4/5) values of transmission. They resemble theoretically predicted compressible and incompressible stripes of quantum Hall edge states. The scanning tip allows us to shift the constriction limiting the conductance in real space over distances of many microns. The resulting stripes of integer and fractional filling factors are rugged on the micron scale, i.e. on a scale much smaller than the zero-field elastic mean free path of the electrons. Our experiments demonstrate that microscopic inhomogeneities are relevant even in high-quality samples and lead to locally strongly fluctuating widths of incompressible regions even down to their complete suppression for certain tip positions. The macroscopic quantization of the Hall resistance measured experimentally in a non-local contact configuration survives in the presence of these inhomogeneities, and the relevant local energy scale for the \nu=2 state turns out to be independent of tip position.

Coherent transmission of an ultrasonic shock wave through a multiple scattering medium

We report measurements of the transmitted coherent (ensemble-averaged) wave resulting from the interaction of an ultrasonic shock wave with a two-dimensional random medium. Despite multiple scattering, the coherent waveform clearly shows the steepening that is typical of nonlinear harmonic generation. This is taken advantage of to measure the elastic mean free path and group velocity over a broad frequency range (2-15 MHz) in only one experiment. Experimental results are found to be in good agreement with a linear theoretical model taking into account spatial correlations between scatterers. These results show that nonlinearity and multiple scattering are both present, yet uncoupled.

Characterisation of ultrasonic structural noise in multiple scattering media using phased arrays

The ultrasonic inspection of multiple scattering media gives rise to structural noise which makes it difficult to detect potential damage or crack inside the component. In order to predict the performances of ultrasonic inspection over such complex media, scattering models can be used. Such models rely on specific key parameters describing the multiple scattering process, which can be determined by specific measurements and post-processing techniques. Such experiments were carried out on stainless steel plates using linear phased-arrays. They consist in recording the response matrix constituted by impulse responses between all the elements of the array. By conducting post-processing on this matrix, we measure the elastic mean free path le and the correlation distance dc of the recorded noise. Additionally, the dynamic behaviour of the coherent backscattering effect was studied in order to measure the diffusion constant D. Plane-wave beamforming has been applied to the response matrix to improve the angular resolution and the signal-to-noise ratio of the backscattered intensity. Details of postprocessing techniques will be shown.

Electron-electron scattering in three-dimensional highly degenerate semiconductors

We have measured the low-field magnetoresistances of a series of Sn-doped indium oxide thick films in the temperature T range of . The electron dephasing rate as a function of T for each film was extracted by comparing the magnetoresistance data with the three-dimensional weak-localization theoretical predictions. We found that the extracted varies linearly with . Furthermore, at a given T, varies linearly with , where kF is the Fermi wave number, and l is the electron elastic mean free path. These features are well explained in terms of the small-energy-transfer electron-electron scattering time in three-dimensional disordered conductors. This electron dephasing mechanism dominates over the electron-phonon scattering process because the carrier concentrations in our films are ∼ 3 orders of magnitude lower than those in typical metals, which resulted in a greatly suppressed electron-phonon relaxation rate. Our result is the first quantitative demonstration of this unique three-dimensional electron-electron scattering time in experiments.

Suppression of surfacep-wave superconductivity in disordered topological insulators

The paper proposes a self-consistent Green function description of the induced surface superconductivity in a disordered three-dimensional topological insulator (TI) coupled to an s-wave superconductor. We recover earlier results regarding the induced spin-triplet p-wave pairing, showing that a mixture of p- and s-wave pair correlations appears as a result of broken spin-rotation symmetry on the helical surface of the TI. Unlike the s-wave pairing, the p-wave component is found to be suppressed in dirty TIs in which the elastic mean-free path is much smaller than the superconducting coherence length. The suppression is due to the generic nonlocality of the spin-triplet correlations, which makes them strongly dependent on the mean-free path in a disordered system. In dirty TIs the induced superconductivity is predicted to be predominantly s-wave like. In cleaner TIs, however, the p-wave component may reach a magnitude comparable with (but not larger than) the s-wave pairing.

Faraday rotation in a disordered medium

The Faraday rotation angle $\ensuremath{\Theta}$ is calculated in a diffusive regime on a three-dimensional disordered slab. It is shown that $\mathrm{tan}\phantom{\rule{0.16em}{0ex}}\ensuremath{\Theta}$ is (i) an oscillating function of the magnetic field or the medium's internal properties, and (ii) proportional to the ratio of the inelastic mean-free path ${l}_{\mathrm{in}}$ to the mean-free path $l$, that is, to the average number of photon scatterings. The maximum rotation is achieved at frequencies when the photon's elastic mean-free path is minimal. We have obtained the rotation angle of backscattered light taking into account the maximally crossed diagrams. The latter leads to an ellipticity in the backscattered wave that can serve as a precursor of weak localization. The critical strength of the magnetic field is ${B}_{c}\ensuremath{\sim}{g}_{c}\ensuremath{\sim}\ensuremath{\lambda}/l$ beyond which rotation in the backscattered wave disappears. The traversal time of an electromagnetic wave through the slab is estimated in a diffusive regime. The disorder enhanced the traversal time by an additional factor ${l}_{\mathrm{in}}/l$ in comparison with a free light propagation time. Comparison with the experimental data is carried out.

Experimental measurements of the coherent field resulting from the interaction of an ultrasonic shock wave with a multiple scattering medium

Whereas multiple scattering and shock wave formation are known to be antagonistic phenomena, this work concentrates on the interaction of an ultrasonic shock wave with a random multiple scattering medium. The shock wave is generated by long distance propagation of a short pulse (four periods at a 3.5 MHz central frequency) in water before it encounters the scattering medium (a slab-shaped random set of parallel metallic rods). Transmitted waves are recorded over hundreds of positions along the lateral dimension of the slab to estimate the ensemble-averaged transmitted field 〈Φ(t)〉, also known as the coherent wave. Experiments are repeated for different thicknesses L of the slab and different emission amplitudes. The elastic mean free path le (i.e., the typical distance for the decreasing of the coherent intensity ∣〈Φ(t)〉∣2 due to scattering) is determined as well as the harmonic rate of the averaged transmitted wave. Experimental results are discussed and compared to the linear case.

The DMPK equation for mesoscopic quantum transport revisited

A recent Drude model description of the metallic regime and of a channel-averaged elastic mean free path (mfp), ℓ0, in an N-channel tight-binding wire identifies the Thouless localization length, Nℓ0, as a proper lower bound of macroscopic length scales (“mean free path”) for the DMPK equation describing the localized regime of the wire. The mfp ℓ0 leads to a metallic regime which is consistent with Dorokhov's microscopic transmission analysis in terms of a nominal elastic mfp. On the other hand, the validity of Mello's derivation of universal conductance fluctuations in the metallic regime based on the DMPK equation is restored if the mfp ℓ′, of order Nℓ0, in that equation is replaced by the correct mean free path ℓ0.