Weak Localization Correction Research Articles

Low magnetic field anomaly of the Hall effect in disordered two-dimensional systems: Interplay between weak localization and electron-electron interaction

The nonlinear behavior of the Hall resistivity at low magnetic fields in single quantum well GaAs/In$_x$Ga$_{1-x}$As/GaAs heterostructures with degenerated electron gas is studied. It has been found that this anomaly is accompanied by the weaker temperature dependence of the conductivity as compared with that predicted by the first-order theory of the quantum corrections to the conductivity. We show that both effects in strongly disordered systems stem from the second order quantum correction caused by the effect of weak localization on the interaction correction and vice versa. This correction contributes mainly to the diagonal component of the conductivity tensor, it depends on the magnetic field like the weak localization correction and on the temperature like the interaction contribution.

Properties of graphene: a theoretical perspective

In this review, we provide an in-depth description of the physics of monolayer and bilayer graphene from a theorist's perspective. We discuss the physical properties of graphene in an external magnetic field, reflecting the chiral nature of the quasiparticles near the Dirac point with a Landau level at zero energy. We address the unique integer quantum Hall effects, the role of electron correlations, and the recent observation of the fractional quantum Hall effect in the monolayer graphene. The quantum Hall effect in bilayer graphene is fundamentally different from that of a monolayer, reflecting the unique band structure of this system. The theory of transport in the absence of an external magnetic field is discussed in detail, along with the role of disorder studied in various theoretical models. We highlight the differences and similarities between monolayer and bilayer graphene, and focus on thermodynamic properties such as the compressibility, the plasmon spectra, the weak localization correction, quantum Hall effect, and optical properties. Confinement of electrons in graphene is nontrivial due to Klein tunneling. We review various theoretical and experimental studies of quantum confined structures made from graphene. The band structure of graphene nanoribbons and the role of the sublattice symmetry, edge geometry and the size of the nanoribbon on the electronic and magnetic properties are very active areas of research, and a detailed review of these topics is presented. Also, the effects of substrate interactions, adsorbed atoms, lattice defects and doping on the band structure of finite-sized graphene systems are discussed. We also include a brief description of graphane -- gapped material obtained from graphene by attaching hydrogen atoms to each carbon atom in the lattice.

Dimensional dependence of weak localization corrections and spin relaxation in quantum wires with Rashba spin-orbit coupling

The quantum correction to the conductivity in disordered quantum wires with linear Rashba spin-orbit coupling is obtained. For quantum wires with spin-conserving boundary conditions, we find a crossover from weak anti- to weak localization as the wire width W is reduced using exact diagonalization of the Cooperon equation. This crossover is due to the dimensional dependence of the spin relaxation rate of conduction electrons, which becomes diminished, when the wire width is smaller than the bulk spin precession length $L_{SO}$. We thus confirm previous results for small wire width, $W/L_{SO}<= 1$ [PRL98,176808(2007)], where only the transverse 0-modes of the Cooperon equation had been taken into account. We find that spin helix solutions become stable for arbitrary ratios of linear Rashba and Dresselhaus coupling in narrow wires. For wider wires, the spin relaxation rate is found to be not monotonous as function of wire width: it becomes first enhanced for W on the order of the bulk $L_{SO}$ before it becomes diminished for smaller wire widths. In addition, we find that the spin relaxation is smallest at the edge of the wire for wide wires. The effect of the Zeeman coupling to the magnetic field perpendicular to the 2D electron system is studied and found that it shifts the crossover from weak anti- to weak localization to larger wire widths $W_c$. When the transverse confinement potential of the quantum wire is smooth (adiabatic), the spin relaxation rate is found to be enhanced as W is reduced. We find that only a spin polarized state retains a finite spin relaxation rate in such narrow wires. Thus, we conclude that the injection of polarized spins into nonmagnetic quantum wires should be favorable in wires with smooth confinement potential. Finally, in wires with tubular shape, corresponding to transverse periodic boundary conditions, we find no reduction of the spin relaxation rate.

Quantum coherence and magnetic scattering

The time τϕ over which an electron can maintain its phase coherence at low temperatures is of fundamental importance in mesoscopic systems. The observability of many phenomena, such as the Aharonov Bohm effect, the universal conductance fluctuations, the weak localisation correction to the conductance, persistent current in ringstructures and many more rely on a long enough phase coherence time. In disordered conductors and within the standard Fermi liquid picture, the phase coherence time is expected to diverge at zero temperature. However, most experiments show a saturating phase coherence time at low temperatures. This saturation has often been attributed to the presence of a small amount of magnetic impurities giving rise to the so-called Kondo effect. In this paper, we present a brief review of recent advances, both experimental and theoretical, in the understanding of dephasing by magnetic impurities in the framework of the Kondo effect.

Semiclassical transport in nearly symmetric quantum dots. II. Symmetry breaking due to asymmetric leads

In this work-the second of a pair of articles-we consider transport through spatially symmetric quantum dots with leads whose widths or positions do not obey the spatial symmetry. We use the semiclassical theory of transport to find the symmetry-induced contributions to weak localization corrections and universal conductance fluctuations for dots with left-right, up-down, inversion, and fourfold symmetries. We show that all these contributions are suppressed by asymmetric leads; however, they remain finite whenever leads intersect with their images under the symmetry operation. For an up-down symmetric dot, this means that the contributions can be finite even if one of the leads is completely asymmetric. We find that the suppression of the contributions to universal conductance fluctuations is the square of the suppression of contributions to weak localization. Finally, we develop a random-matrix theory model which enables us to numerically confirm these results.

Semiclassical transport in nearly symmetric quantum dots. I. Symmetry breaking in the dot

We apply the semiclassical theory of transport to quantum dots with exact and approximate spatial symmetries; left-right mirror symmetry, up-down mirror symmetry, inversion symmetry, or fourfold symmetry. In this work-the first of a pair of articles-we consider (a) perfectly symmetric dots and (b) nearly symmetric dots in which the symmetry is broken by the dot's internal dynamics. The second article addresses symmetry-breaking by displacement of the leads. Using semiclassics, we identify the origin of the symmetry-induced interference effects that contribute to weak localization corrections and universal conductance fluctuations. For perfect spatial symmetry, we recover results previously found using the random-matrix theory conjecture. We then go on to show how the results are affected by asymmetries in the dot, magnetic fields, and decoherence. In particular, the symmetry-asymmetry crossover is found to be described by a universal dependence on an asymmetry parameter gamma_{asym} . However, the form of this parameter is very different depending on how the dot is deformed away from spatial symmetry. Symmetry-induced interference effects are completely destroyed when the dot's boundary is globally deformed by less than an electron wavelength. In contrast, these effects are only reduced by a finite amount when a part of the dot's boundary smaller than a lead-width is deformed an arbitrarily large distance.

Dc Conductivity of an Array of Josephson Junctions in the Insulating State

We consider microscopically low-temperature transport in weakly disordered arrays of Josephson junctions in the Coulomb blockade regime. We demonstrate that at sufficiently low temperatures the main contribution to the dc conductivity comes from the motion of single-Cooper-pair excitations, scattered by irregularities in the array. Being proportional to the concentration of the excitations, the conductivity is exponentially small in temperature with the activation energy close to the charging energy of a Cooper pair on a superconductive island. Applying a diagrammatic approach to treat the disorder potential we calculate the Drude-like conductivity and obtain weak localization corrections. At sufficiently low temperatures or strong disorder the Anderson localization of Cooper pairs ensues.

Semiclassical approach to the ac conductance of chaotic cavities

We address frequency-dependent quantum transport through mesoscopic conductors in the semiclassical limit. By generalizing the trajectory-based semiclassical theory of dc quantum transport to the ac case, we derive the average screened conductance as well as ac weak-localization corrections for chaotic conductors. Thereby we confirm respective random matrix results and generalize them by accounting for Ehrenfest time effects. We consider the case of a cavity connected through many leads to a macroscopic circuit which contains ac-sources. In addition to the reservoir the cavity itself is capacitively coupled to a gate. By incorporating tunnel barriers between cavity and leads we obtain results for arbitrary tunnel rates. Finally, based on our findings we investigate the effect of dephasing on the charge relaxation resistance of a mesoscopic capacitor in the linear low-frequency regime.

Anomalous Hall effect in granular ferromagnetic metals and effects of weak localization

We theoretically investigate the anomalous Hall effect in a system of dense-packed ferromagnetic grains in the metallic regime. Using the formalism recently developed for the conventional Hall effect in granular metals, we calculate the residual anomalous Hall conductivity $\sigma_{xy}$ and resistivity $\rho_{xy}$ and weak localization corrections to them for both skew-scattering and side-jump mechanisms. We find that, unlike for homogeneously disordered metals, the scaling relation between $\rho_{xy}$ and the longitudinal resistivity $\rho_{xx}$ does not hold. The weak localization corrections, however, are found to be in agreement with those for homogeneous metals. We discuss recent experimental data on the anomalous Hall effect in polycrystalline iron films in view of the obtained results.

Magnetotransport properties of individual InAs nanowires

We probe the magnetotransport properties of individual InAs nanowires in a field-effect transistor geometry. In the low magnetic field regime we observe magnetoresistance that is well described by the weak localization description in diffusive conductors. The weak localization correction is modified to weak antilocalization as the gate voltage is increased. We show that the gate voltage can be used to tune the phase coherence length $({l}_{\ensuremath{\phi}})$ and spin-orbit length $({l}_{\text{so}})$ by a factor of 2. In the high field and low-temperature regime we observe that the mobility of devices can be modified significantly as a function of magnetic field. We argue that the role of skipping orbits and the nature of surface scattering is essential in understanding high-field magnetotransport in nanowires.

Effects of disorder in three-dimensionalZ2quantum spin Hall systems

In this paper, we address ourselves to the nonmagnetic disorder effects onto the quantum critical point, which intervenes the three-dimensional ${Z}_{2}$ quantum spin Hall insulator (topological insulator) and an ordinary insulator. The minimal model describing this type of the quantum critical point is the single copy of the $3+1$ Dirac fermion, whose topological mass $m$ induces the phase transition between the topological insulator and an ordinary one. We first derive the phase diagram spanned by the mass term $m$, chemical potential $\ensuremath{\mu}$, and strength of the disorder within the self-consistent Born approximation. By way of this, we find a finite density of state appears even at zero energy and at the phase-transition point, i.e., $m=\ensuremath{\mu}=0$, if the strength of the disorder potential exceeds some critical value. To infer the structure of the low-energy effective theory around these zero-energy states, we further calculated the weak-localization correction to the conductivity. To be more specific, we have found that the diffuson is dominated by the charge diffusion mode and parity diffusion mode. While the charge diffusion mode always carries the diffusion pole, the parity diffusion mode becomes massless only at $m=0$, but suffers from the infrared cutoff for nonzero $m$. Corresponding to this feature of the diffuson, the Cooperon is also composed of two quasidegenerate contributions. We found that these two give rise to the same magnitude of the anti-weak-localization (AWL) correction with each other at $m=0$. As a result, when the topological mass $m$ is fine tuned to be zero (but for generic $\ensuremath{\mu}$), the AWL correction becomes doubled (quantum correction doubling). Based on this observation, we will discuss the possible microscopic picture of the ``levitation and pair-annihilation'' phenomena, recently discovered by Onoda et al. [Phys. Rev. Lett. 98, 076802 (2007)].

Weak localization inAlxGa1−xAs/GaAs/AlxGa1−xAsheterostructures with electrostatically induced random antidot array

main concern was with the investigations of the ballistic systems, in whichl > d, D, wherel is the mean free path, d andD are the size of antidots and period of the antidots array, respectively. The rich diversity of the transport phenomena, such as commensurable oscillations, the peculiarities due to the trajectories rolling along the array of antidots was observed. The antidots array structures in the diffusion regime (l < d,D) were not studied essentially. Such structures are interesting in some aspects. First of all, the quantum corrections to the conductivity due to weak localization (WL) and electron-electron (ee) interaction have to reveal the specific features when the phase breaking length, L� = p D��, where D is the diffusion coefficient and�� is phase breaking time, or the temperature length, LT = p D/T, become larger than d,D at decreasing temperature (hereafter we set kB = 1, ~ = 1). Secondly, the large enough negative gate voltage has to deplete the channels between the antidots (the channel widthw0 is aboutD−d) and as a result to lead to crossover to the hopping conductivity. Besides, from the transport properties standpoint the antidots arrays are the fine model of the granular media. In contrast to the granular metallic film, the parameters of the “granules” and “barriers” are reliably known and can be changed continuously within wide range. In this paper we report the results of the experimental study of the weak localization correction to the conductivity in the structure with the random array of antidots. We show that the change of the magnetoresistance at arising of the antidots and increase of the antidots size results from the change of statistics of closed paths. Namely, the contribution of the trajectories with the large enclosed area is strongly suppressed. The experimental area distribution function is in a reasonable agreement with the function obtained from the computer simulation. The random antidots array was made on the single quantum well heterostructure with the electron density n = 1.5 × 10 12 cm −2 and mobility µ = 19000 cm 2 /(Vs) δ - Si t 2 GaAs

Electron dephasing scattering rate in two-dimensional GaAs/InGaAs heterostructures with embedded InAs quantum dots

We report a comprehensive study of weak-localization and electron-electron interaction effects in a GaAs/InGaAs two-dimensional electron system with nearby InAs quantum dots, using measurements of the electrical conductivity with and without magnetic field. Although both the effects introduce temperature dependent corrections to the zero magnetic field conductivity at low temperatures, the magnetic field dependence of conductivity is dominated by the weak-localization correction. We observed that the electron dephasing scattering rate τφ−1, obtained from the magnetoconductivity data, is enhanced by introducing quantum dots in the structure, as expected, and obeys a linear dependence on the temperature and elastic mean free path, which is against the Fermi-liquid model.

Temperature and magnetic-field dependence of the quantum corrections to the conductance of a network of quantum dots

We calculate the magnetic-field and temperature dependence of all quantum corrections to the ensemble-averaged conductance of a network of quantum dots. We consider the limit that the dimensionless conductance of the network is large, so that the quantum corrections are small in comparison to the leading, classical contribution to the conductance. For a quantum dot network the conductance and its quantum corrections can be expressed solely in terms of the conductances and form factors of the contacts and the capacitances of the quantum dots. In particular, we calculate the temperature dependence of the weak localization correction and show that it is described by an effective dephasing rate proportional to temperature.

Weak localization in a system with a barrier: dephasing and weak Coulomb blockade

We non-perturbatively analyze the effect of electron–electron interactions on weak localization (WL) in relatively short metallic conductors with a tunnel barrier. We demonstrate that the main effect of interactions is electron dephasing which persists down to T=0 and yields suppression of WL correction to conductance below its non-interacting value. Our results may account for recent observations of low temperature saturation of the electron decoherence time in quantum dots.

Mesoscopic fluctuations of the supercurrent in diffusive Josephson junctions

We study mesoscopic fluctuations and weak-localization correction to the supercurrent in Josephson junctions with coherent diffusive electron dynamics in the normal part. Two kinds of junctions are considered: a chaotic dot coupled to superconductors by tunnel barriers and a diffusive junction with transparent normal-superconducting interfaces. The amplitude of current fluctuations and the weak-localization correction to the average current are calculated as functions of the ratio between the superconducting gap and the electron dwell energy, temperature, and superconducting phase difference across the junction. Technically, fluctuations on top of the spatially inhomogeneous proximity effect in the normal region are described by the replicated version of the $\ensuremath{\sigma}$ model. For the case of diffusive junctions with transparent interfaces, the magnitude of mesoscopic fluctuations of the critical current appears to be nearly three times larger than the prediction of the previous theory, which did not take the proximity effect into account.

Disorder and temperature dependence of the anomalous Hall effect in thin ferromagnetic films: Microscopic model

We consider the anomalous Hall (AH) effect in thin disordered ferromagnetic films. Using a microscopic model of electrons in a random potential of identical impurities including spin-orbit coupling, we develop a general formulation for strong, finite range impurity scattering. Explicit calculations are done within a short range but strong impurity scattering to obtain AH conductivities for both the skew scattering and side-jump mechanisms. We also evaluate quantum corrections due to interactions and weak localization effects. We show that for arbitrary strength of the impurity scattering, the electron-electron interaction correction to the AH conductivity vanishes exactly due to general symmetry reasons. On the other hand, we find that our explicit evaluation of the weak localization corrections within the strong, short-range impurity scattering model can explain the experimentally observed logarithmic temperature dependences in disordered ferromagnetic Fe films.

Dephasing in the semiclassical limit is system-dependent

Dephasing in open quantum chaotic systems has been investigated in the limit of large system sizes to the Fermi wavelength ratio, L/λF 〉 1. The weak localization correction g wl to the conductance for a quantum dot coupled to (i) an external closed dot and (ii) a dephasing voltage probe is calculated in the semiclassical approximation. In addition to the universal algebraic suppression g wl ∝ (1 + τD/τϕ)−1 with the dwell time τD through the cavity and the dephasing rate τ −1ϕ , we find an exponential suppression of weak localization by a factor of ∝ exp[−\(\tilde \tau \)/τϕ], where \(\tilde \tau \) is the system-dependent parameter. In the dephasing probe model, \(\tilde \tau \) coincides with the Ehrenfest time, \(\tilde \tau \) ∝ ln[L/λF], for both perfectly and partially transparent dot-lead couplings. In contrast, when dephasing occurs due to the coupling to an external dot, \(\tilde \tau \) ∝ ln[L/ξ] depends on the correlation length ξ of the coupling potential instead of λF.

Weak localization in monolayer and bilayer graphene

We demonstrate quantitative experimental evidence for a weak localization correction to the conductivity in monolayer and bilayer graphene systems. We show how inter- and intra-valley elastic scattering control the correction in small magnetic fields in a way which is unique to graphene. A clear difference in the forms of the correction is observed in the two systems, which shows the importance of the interplay between the elastic scattering mechanisms and how they can be distinguished. Our observation of the correction at zero-net carrier concentration in both systems is clear evidence of the inhomogeneity engendered into the graphene layers by disorder.

Semiclassical theory of the Ehrenfest-time dependence of quantum transport

In ballistic conductors, there is a low-time threshold for the appearance of quantum effects in transport coefficients. This low-time threshold is the Ehrenfest time ${\ensuremath{\tau}}_{E}$. Most previous studies of the ${\ensuremath{\tau}}_{E}$ dependence of quantum transport assumed ergodic electron dynamics, so that they could be applied to ballistic quantum dots only. In this paper, we present a theory of the ${\ensuremath{\tau}}_{E}$ dependence of three signatures of quantum transport---the Fano factor for the shot noise power, the weak localization correction to the conductance, and the conductance fluctuations---for arbitrary ballistic conductors.