A d.c. current can be pumped through a quantum dot by periodically varying two independent parameters X1 and X2, like a gate voltage or magnetic field. We present a formula that relates the pumped current to the parametric derivatives of the scattering matrix S(X1, X2) of the system. As an application we compute the statistical distribution of the pumped current in the case of a chaotic quantum dot.PACS numbers: 72.10. Bg, 05.45.+b An electron pump is a device that generates a d.c. current between two electrodes that are kept at the same bias. In recent years, electron pumps consisting of small semiconductor quantum dots have received considerable experimental and theoretical attention.1-11 A quantum dot is a small metal or semiconductor island, confined by gates, and connected to the outside world via point contacts. Several different mechanisms have been proposed to pump charge through such systems, ranging from a low-frequency modulation of gate voltages in combination with the Coulomb blockade 1,2,11 to photon-assisted transport at or near a resonance frequency of the dot. 5-8Their applicability depends on the characteristic size of the system and the operation frequency.Most experimental realizations of electron pumps in semiconductor quantum dots made use of the principle of Coulomb blockade. If the dot is coupled to the outside world via tunneling point contacts, the charge on the dot is quantized, and (apart from degeneracy points) transport is inhibited as a result of the high energy cost of adding an extra electron to the dot. Pothier et al. constructed an electron pump that operates at arbitrarily low frequency and with a reversible pumping direction.2 The pump consists of two weakly coupled quantum dots in the Coulomb blockade regime and operates via a mechanism that closely resembles a peristaltic pump: Charge is pumped through the double dot array from the left to the right and electron-by-electron as the voltage U 1 ∝ sin(ωt) of the left dot reaches its minima and maxima before the voltage U 2 ∝ sin(ωt − φ) of the right one.2 The pumping direction can be reversed by reversing the phase difference φ of the two gate voltages.A similar mechanism was proposed by Spivak, Zhou, and Beal Monod for an electron pump consisting of single quantum dot only.4 In this case a d.c. current is generated by adiabatic variation of two different gate voltages that determine the shape of the nanostructure, or any other pair of parameters X 1 and X 2 , like magnetic field or Fermi energy, that modify the (quantummechanical) properties of the system, see Fig. 1a. The magnitude of the current is proportional to the frequency ω with which X 1 and X 2 are varied and (for small variations) to the product of the amplitudes δX 1 and δX 2 . The direction of the current depends on microscopic (quantum) properties of the system, and need not be known a priori from its macroscopic properties. As in the case of the double-dot Coulomb blockade electron pump of Ref. 2, the direction of the current in the single-dot parametric pum...

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Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of e 2 /h.Introduction.-After the discovery of topological insulators and superconductors and their classification for the ten Altland-Zirnbauer symmetry classes [1][2][3], the concept of nontrivial topological band structures has been extended to materials in which the crystal structure is essential for the protection of topological phases [4]. This includes weak topological insulators [5], which rely on the discrete translation symmetry of the crystal lattice, and topological crystalline insulators [6], for which other crystal symmetries are invoked to protect a topological phase. Whereas the original strong topological insulators always have topologically protected boundary states, weak topological insulators or topological crystalline insulators have protected boundary states for selected surfaces/edges only.In a recent publication, Schindler et al. [7] proposed another extension of the topological insulator (TI) family: a higher-order topological insulator. Being crystalline insulators, these have well-defined faces and well-defined edges or corners at the intersections between the faces. An nth order topological insulator has topologically protected gapless states at the intersection of n crystal faces, but is gapped otherwise [7]. For example, a second-order topological insulator in two dimensions (d = 2) has zeroenergy states at corners, but a gapped bulk and no gapless edge states. Earlier examples of higher-order topological insulators and superconductors avant la lettre appeared in works by (see also [11,12]), who considered insulators and superconductors with protected corner states in d = 2 and d = 3 [13]. Sitte et al. showed that a threedimensional topological insulator in a magnetic field of generic direction also acquires the characteristics of a second-order topological Chern insulator, with chiral states moving along the sample edges [14].Since a second-order TI has a topologically trivial d-dimensional bulk, from a topological point of view its boundaries are essentially stand-alone (d − 1)-dimensional insulators, so that topologically protected states at corners (for d = 2) or edges (for d = 3) arise naturally as "domain walls" at the intersection of two boundaries if these ar...

Electron scattering rates in metallic single-walled carbon nanotubes are studied using an atomic force microscope as an electrical probe. From the scaling of the resistance of the same nanotube with length in the low and high bias regimes, the mean free paths for both regimes are inferred. The observed scattering rates are consistent with calculations for acoustic phonon scattering at low biases and zone boundary/optical phonon scattering at high biases.

A diagrammatic method is presented for averaging over the circular ensemble of random-matrix theory. The method is applied to phase-coherent conduction through a chaotic cavity ͑a ''quantum dot''͒ and through the interface between a normal metal and a superconductor.