Discrete versus continuous wires on quantum networks

Mesoscopic systems and large molecules are often modeled by graphs of one-dimensional wires, connected at vertices. In this paper we discuss the solutions of the Schr\"odinger equation on such graphs, which have been named "quantum networks". Such solutions are needed for finding the energy spectrum of single electrons on such finite systems or for finding the transmission of electrons between leads which connect such systems to reservoirs. Specifically, we compare two common approaches. In the "continuum" approach, one solves the one-dimensional Schr\"odinger equation on each continuous wire, and then uses the Neumann-Kirchoff-de Gennes matching conditions at the vertices. Alternatively, one replaces each wire by a finite number of "atoms", and then uses the tight binding model for the solution. Here we show that these approaches cannot generally give the same results, except for special energies. Even in the limit of vanishing lattice constant, the two approaches coincide only if the tight binding parameters obey very special relations. The different consequences of the two approaches are demonstrated via the example of a T-shaped scatterer.Comment: Special P G de Gennes memorial issue, JP

Phase measurements in Aharonov-Bohm interferometers

In this paper we address measurements of the resonant quantum transmission amplitude $t_{QD}=-i|t_{QD}|e^{i\alpha_{QD}}$ through a quantum dot (QD), as function of the plunger gate voltage $V$. Mesoscopic solid state Aharonov-Bohm interferometers (ABI) have been used to measure the "intrinsic" phase, $\alpha_{QD}$, when the QD is placed on one of the paths. In a "closed" interferometer, connected to two terminals, the electron current is conserved, and Onsager's relations require that the conductance ${\cal G}$ through the ABI is an even function of the magnetic flux $\Phi=\hbar c\phi/e$ threading the ABI ring. Therefore, if one fits ${\cal G}$ to $A+B\cos(\phi+\beta)$ then $\beta$ only "jumps" between 0 and $\pi$, with no relation to $\alpha_{QD}$. Additional terminals open the ABI, break the Onsager relations and yield a non-trivial variation of $\beta$ with $V$. After reviewing these topics, we use theoretical models to derive three results on this problem: (i) For the one-dimensional leads, the relation $|t_{QD}|^2 \propto \sin^2(\alpha_{QD})$ allows a direct measurement of $\alpha_{QD}$. (ii) In many cases, the measured ${\cal G}$ in the closed ABI can be used to extract {\it both} $|t_{QD}|$ and $\alpha_{QD}$. (iii) For open ABI's, $\beta$ depends on the details of the opening. We present quantitative criteria (which can be tested experimentally) for $\beta$ to be equal to the desired $\alpha_{QD}$: the "lossy" channels near the QD should have both a small transmission and a small reflection.Comment: 14 pages, lectures at Summer School on Quantum Computation on the Atomic Scale, Istanbul, June 2003. to appear in the Turkish Journal of Physic

Elastic scattering and absorption of surface acoustic waves by a quantum dot

We study theoretically the piezoelectric interaction of a surface acoustic wave (SAW) with a two-dimensional electron gas confined to an isolated quantum dot. The electron motion in the dot is diffusive. The electron-electron interaction is accounted for by solving the screening problem in real space. Since the screening in GaAs/AlGaAs heterostructures is strong, an approximate inversion of the dielectric function epsilon(r,r') can be utilized, providing a comprehensive qualitative picture of the screened SAW potential and the charge redistribution in the dot. We calculate the absorption and the scattering cross-sections for SAW's as a function of the area of the dot, A, the sound wave vector, q, and the diffusion coefficient D of the electrons. Approximate analytical expressions for the cross-sections are derived for all cases where the quantities A*q^2 and A*omega/D are much larger or smaller than unity; omega is the SAW frequency. Numerical results which include the intermediate regimes and show the sample-specific dependence of the cross-sections on the angles of incidence and scattering of surface phonons are discussed. The weak localization corrections to the cross-sections are found and discussed as a function of a weak magnetic field, the frequency, and the temperature. Due to the absence of current-carrying contacts, the phase coherence of the electron motion, and in turn the quantum corrections, increase as the size of the dot shrinks. This shows that scattering and absorption of sound as noninvasive probes may be advantageous in comparison to transport experiments for the investigation of very small electronic systems.Comment: 35 pages, 6 Postscript figure

Flux-dependent Kondo temperature in an Aharonov-Bohm interferometer with an in-line quantum dot

An Aharonov-Bohm interferometer (ABI) carrying a quantum dot on one of its arms is analyzed. It is found that the Kondo temperature of the device depends strongly on the magnetic flux penetrating the ring. As a result, mesoscopic finite-size effects appear when the Kondo temperature of the dot on the ABI is significantly smaller than the nominal one of the quantum dot (when not on the interferometer), leading to plateaus in the finite-temperature conductance as function of the flux. The possibility to deduce the transmission phase shift of the quantum dot from measurements of the ABI conductance when it is opened (i.e., is connected to more than two leads) is examined, leading to the conclusion that finite-size effects, when significant, may hinder the detection of the Kondo phase shiftComment: 13 pages, 10 figure

Spin selectivity through time-reversal symmetric helical junctions

Time-reversal symmetric charge and spin transport through a molecule comprising two-orbital channels and connected to two leads is analyzed. It is demonstrated that spin-resolved currents are generated when spin-flip processes are accompanied by a flip of the orbital channels. This surprising finding does not contradict Bardarson's theorem [J. H. Bardarson, J. Phys. A: Math. Theor. 41, 405203 (2008)] for two-terminal junctions: the transmission does possess two pairs of doubly-degenerate eigenvalues as required by the theorem. The spin-filtering effect is explicitly demonstrated for a two-terminal chiral molecular junction, modeled by a two-orbital tight-binding chain with intra-atomic spin-orbit interactions (SOI). In the context of transport through organic molecules like DNA, this effect is termed "chirality-induced spin selectivity" (CISS). The model exhibits spin-splitting without breaking time-reversal symmetry: the intra-atomic SOI induces concomitant spin and orbital flips. Examining these transitions from the point of view of the Bloch states in an infinite molecule, it is shown that they cause shifts in the Bloch wave numbers, of the size of the reciprocal single turn, whose directions depend on the left-and right-handedness of the helix. As a result, spin-up and spin-down states propagate in the opposite directions, leading to the CISS effect. To further substantiate our picture, we present an analytically-tractable expression for the 8$\times$8 scattering matrix of such a (single) molecule.Comment: 19 pages, 7 figure

Three-terminal semiconductor junction thermoelectric devices: improving performance

A three-terminal thermoelectric device based on a $p$-$i$-$n$ semiconductor junction is proposed, where the intrinsic region is mounted onto a, typically bosonic, thermal terminal. Remarkably, the figure of merit of the device is governed also by the energy distribution of the {\em bosons} participating in the transport processes, in addition to the electronic one. An enhanced figure of merit can be obtained when the relevant distribution is narrow and the electron-boson coupling is strong (such as for optical phonons). We study the conditions for which the figure of merit of the three-terminal junction can be greater than those of the usual thermoelectric devices made of the same material. A possible setup with a high figure of merit, based on Bi$_2$Te$_3$/Si superlattices, is proposed.Comment: Published in New Journal of Physics: Focus on Thermoelectric Effects in Nanostructures (open access). For published version, see http://dx.doi.org/10.1088/1367-2630/15/7/07502

Thermoelectricity near Anderson localization transitions

The electronic thermoelectric coefficients are analyzed in the vicinity of one and two Anderson localization thresholds in three dimensions. For a single mobility edge, we correct and extend previous studies, and find universal approximants which allow to deduce the critical exponent for the zero-temperature conductivity from thermoelectric measurements. In particular, we find that at non-zero low temperatures the Seebeck coefficient and the thermoelectric efficiency can be very large on the "insulating" side, for chemical potentials below the (zero-temperature) localization threshold. Corrections to the leading power-law singularity in the zero-temperature conductivity are shown to introduce non-universal temperature-dependent corrections to the otherwise universal functions which describe the Seebeck coefficient, the figure of merit and the Wiedemann-Franz ratio. Next, the thermoelectric coefficients are shown to have interesting dependences on the system size. While the Seebeck coefficient decreases with decreasing size, the figure of merit first decreases but then increases, while the Wiedemann-Franz ratio first increases but then decreases as the size decreases. Small (but finite) samples may thus have larger thermoelectric efficiencies. In the last part we study thermoelectricity in systems with a pair of localization edges, the ubiquitous situation in random systems near the centers of electronic energy bands. As the disorder increases, the two thresholds approach each other, and then the Seebeck coefficient and the figure of merit increase significantly, as expected from the general arguments of Mahan and Sofo [J. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. U.S.A. 93, 7436 (1996)] for a narrow energy-range of the zero-temperature metallic behavior.Comment: 16 pages, 11 figures, close to the published versio

Robustness of spin filtering against current leakage in a Rashba-Dresselhaus-Aharonov-Bohm interferometer

In an earlier paper [Phys. Rev. B 84, 035323 (2011)], we proposed a spin filter which was based on a diamond-like interferometer, subject to both an Aharonov-Bohm flux and (Rashba and Dresselhaus) spin-orbit interactions. Here we show that the full polarization of the outgoing electron spins remains the same even when one allows leakage of electrons from the branches of the interferometer. Once the gate voltage on one of the branches is tuned to achieve an effective symmetry between them, this polarization can be controlled by the electric and/or magnetic fields which determine the spin-orbit interaction strength and the Aharonov-Bohm flux.Comment: 9 pages, 4 figure

Spin filtering in a Rashba-Dresselhaus-Aharonov-Bohm double-dot interferometer

We study the spin-dependent transport of spin-1/2 electrons through an interferometer made of two elongated quantum dots or quantum nanowires, which are subject to both an Aharonov-Bohm flux and (Rashba and Dresselhaus) spin-orbit interactions. Similar to the diamond interferometer proposed in our previous papers [Phys. Rev. B {\bf 84}, 035323 (2011); Phys. Rev. B {\bf 87}, 205438 (2013)], we show that the double-dot interferometer can serve as a perfect spin filter due to a spin interference effect. By appropriately tuning the external electric and magnetic fields which determine the Aharonov-Casher and Aharonov-Bohm phases, and with some relations between the various hopping amplitudes and site energies, the interferometer blocks electrons with a specific spin polarization, independent of their energy. The blocked polarization and the polarization of the outgoing electrons is controlled solely by the external electric and magnetic fields and do not depend on the energy of the electrons. Furthermore, the spin filtering conditions become simpler in the linear-response regime, in which the electrons have a fixed energy. Unlike the diamond interferometer, spin filtering in the double-dot interferometer does not require high symmetry between the hopping amplitudes and site energies of the two branches of the interferometer and thus may be more appealing from an experimental point of view.Comment: 15 pages, 3 figure

Comment on: "Spin-orbit interaction and spin selectivity for tunneling electron transfer in DNA"

The observation of chiral-induced spin selectivity (CISS) in biological molecules still awaits a full theoretical explanation. In a recent Rapid Communication, Varela et al. [Phys. Rev. B 101, 241410(R) (2020)] presented a model for electron transport in biological molecules by tunneling in the presence of spin-orbit interactions. They then claimed that their model produces a strong spin asymmetry due to the intrinsic atomic spin-orbit strength. As their Hamiltonian is time-reversal symmetric, this result contradicts a theorem by Bardarson [J. Phys. A: Math. Theor. 41, 405203 (2008)], which states that such a Hamiltonian cannot generate a spin asymmetry for tunneling between two terminals (in which there are only a spin-up and a spin-down channels). Here we solve the model proposed by Varela et al. and show that it does not yield any spin asymmetry, and therefore cannot explain the observed CISS effect