Spintronics refers commonly to phenomena in which the spin of electrons in a solid state environment plays the determining role. In a more narrow sense spintronics is an emerging research field of electronics: spintronics devices are based on a spin control of electronics, or on an electrical and optical control of spin or magnetism. While metal spintronics has already found its niche in the computer industry-giant magnetoresistance systems are used as hard disk read heads-semiconductor spintronics is yet to demonstrate its full potential. This review presents selected themes of semiconductor spintronics, introducing important concepts in spin transport, spin injection, Silsbee-Johnson spin-charge coupling, and spindependent tunneling, as well as spin relaxation and spin dynamics. The most fundamental spin-dependent interaction in nonmagnetic semiconductors is spin-orbit coupling. Depending on the crystal symmetries of the material, as well as on the structural properties of semiconductor based heterostructures, the spin-orbit coupling takes on different functional forms, giving a nice playground of effective spin-orbit Hamiltonians. The effective Hamiltonians for the most relevant classes of materials and heterostructures are derived here from realistic electronic band structure descriptions. Most semiconductor device systems are still theoretical concepts, waiting for experimental demonstrations. A review of selected proposed, and a few demonstrated devices is presented, with detailed description of two important classes: magnetic resonant tunnel structures and bipolar magnetic diodes and transistors. In view of the importance of ferromagnetic semiconductor materials, a brief discussion of diluted magnetic semiconductors is included. In most cases the presentation is of tutorial style, introducing the essential theoretical formalism at an accessible level, with case-study-like illustrations of actual experimental results, as well as with brief reviews of relevant recent achievements in the field. 72.25.Rb, 75.50.Pp, PACS

We consider quasi-one-dimensional Ruderman-Kittel-Kasuya-Yosida (RKKY) systems in proximity to an s-wave superconductor. We show that a 2k(F) peak in the spin susceptibility of the superconductor in the one-dimensional limit supports helical order of localized magnetic moments via RKKY interaction, where k(F) is the Fermi wave vector. The magnetic helix is equivalent to a uniform magnetic field and very strong spin-orbit interaction (SOI) with an effective SOI length 1/2k(F). We find the conditions to establish such a magnetic state in atomic chains and semiconducting nanowires with magnetic atoms or nuclear spins. Generically, these systems are in a topological phase with Majorana fermions. The inherent self-tuning of the helix to 2k(F) eliminates the need to tune the chemical potential.

We study hybrid superconducting-semiconducting nanowires in the presence of Rashba spin-orbit interaction (SOI) as well as helical magnetic fields. We show that the interplay between them leads to a competition of phases with two topological gaps closing and reopening, resulting in unexpected reentrance behavior. Besides the topological phase with localized Majorana fermions (MFs) we find new phases characterized by fractionally charged fermion (FF) bound states of Jackiw-Rebbi type. The system can be fully gapped by the magnetic fields alone, giving rise to FFs that transmute into MFs upon turning on superconductivity. We find explicit analytical solutions for MF and FF bound states and determine the phase diagram numerically by determining the corresponding Wronskian null space. We show by renormalization group arguments that electron-electron interactions enhance the Zeeman gaps opened by the fields.PACS numbers: 73.63. Nm; 74.45.+c Introduction. Majorana fermions 1 (MF) in condensed matter systems 2 , interesting from a fundamental point of view as well as for potential applications in topological quantum computing, have attracted wide interest, both in theory [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and experiment [17][18][19] . One of the most promising candidate systems for MFs are semiconducting nanowires with Rashba spin-orbit interaction (SOI) brought into proximity with a superconductor 7-9 . In such hybrid systems a topological phase with a MF at each end of the nanowire is predicted to emerge once an applied uniform magnetic field exceeds a critical value 6-9 . As pointed out recently 20 , the Rashba SOI in such wires is equivalent to a helical Zeeman term, and thus the same topological phase with MFs is predicted to occur in hybrid systems in the presence of a helical field but without SOI 21,22 .Here, we go a decisive step further and address the question, what happens when both fields are present, an internal Rashba SOI field as well as a helical-or more generally-a spatially varying magnetic field. Quite remarkably, we discover that due to the interference between the two mechanisms the phase diagram becomes surprisingly rich, with reentrance behavior of MFs and new phases characterized by fractionally charged fermions (FF), analogously to Jackiw-Rebbi fermion bound states 23 . Since the system is fully gapped by the magnetic fields at certain Rashba SOI strengths (in the absence of superconductivity), these FFs act as precursors of MFs into which they transmute by turning on superconductivity.The main part of this work aims at characterizing the mentioned phase diagram. For this we find explicit solutions for the various bound states, which allows us to derive analytical conditions for the boundaries of the topological phases. We also perform an independent numerical search of the phases and present numerical results illustrating them. We show that the phases can be controlled with experimentally accessible parameters, such as the uniform field or the chemical potential. We formulat...

We propose a setup for universal and electrically controlled quantum information processing with hole spins in Ge/Si core/shell nanowire quantum dots (NW QDs). Single-qubit gates can be driven through electric-dipoleinduced spin resonance, with spin-flip times shorter than 100 ps. Long-distance qubit-qubit coupling can be mediated by the cavity electric field of a superconducting transmission line resonator, where we show that operation times below 20 ns seem feasible for the entangling √ iSWAP gate. The absence of Dresselhaus spinorbit interaction (SOI) and the presence of an unusually strong Rashba-type SOI enable precise control over the transverse qubit coupling via an externally applied, perpendicular electric field. The latter serves as an on-off switch for quantum gates and also provides control over the g factor, so single-and two-qubit gates can be operated independently. Remarkably, we find that idle qubits are insensitive to charge noise and phonons, and we discuss strategies for enhancing noise-limited gate fidelities. PACS numbers: 73.21.Hb, 73.21.La, 42.50.Pq, 03.67.Lx In the past decade, the idea of processing quantum information with spins in quantum dots (QDs) [1] was followed by remarkable progress [2]. While the workhorse systems are highly advanced, such as self-assembled (In)GaAs QDs [3-10] and negatively charged, lateral GaAs QDs [11][12][13][14][15][16][17], an emerging theme is the search for systems that allow further optimization. In particular, Ge and Si have attracted attention because they can be grown nuclear-spin-free, which eliminates a major source of decoherence [18][19][20] Prime examples for novel qubits are hole spins in Ge/Si NW QDs [25, 26, 31,42,48], because they combine all the advantages of group-IV materials, VB states, and strong confinement along two axes. The Si shell provides a large VB offset ∼0.5 eV [22], induces strain, and removes dangling bonds from the core. Furthermore, the holes feature an unusually strong Rashba-type spin-orbit interaction (SOI), referred to as direct Rashba SOI (DRSOI), that is not suppressed by the band gap [48]. We show here that these properties are highly useful for implementing spin qubits.In this work, we propose a setup for quantum information processing with holes in Ge/Si core/shell NW QDs. In stark contrast to previous systems [13,[43][44][45][46][47]49], where the EDSR relies on conventional Dresselhaus and Rashba SOI [50], the dynamics in our setup are governed by the DRSOI whose origin fundamentally differs. We find that EDSR allows flipping of hole spins within less than 100 ps. Two-qubit gates can be realized via circuit quantum electrodynamics (CQED), i.e., with an on-chip cavity [51][52][53], where we estimate that operation times below 20 ns are feasible for √ iSWAP. The long-range spin-spin interactions [49,[54][55][56] enable upscaling. Compared to the original proposal for electron spins in InAs [49], which was recently followed by encouraging results [46], we find several new and striking features. First, because o...

Phonon-induced spin relaxation in coupled lateral quantum dots in the presence of spin-orbit coupling is calculated. The calculation for single dots is consistent with experiment. Spin relaxation in double dots at useful interdot couplings is dominated by spin-hot spots that are strongly anisotropic. Spin-hot spots are ineffective for a diagonal crystallographic orientation of the dots with a transverse in-plane field. This geometry is proposed for spin-based quantum information processing. DOI: 10.1103/PhysRevLett.96.186602 PACS numbers: 72.25.Rb, 03.67.Lx, 71.70.Ej, 73.21.La Understanding spin relaxation in coupled quantum dots is important for setting the efficiency of spin-based applications of information processing, such as spin quantum computing [1] or controlled generation of spin entanglement [2]. Phonon-induced spin relaxation has already been studied theoretically in single dots for electrons [3][4][5][6][7][8][9][10][11][12], holes [13], and excitons [14], and in one-dimensional coupled dots [15]. Recently, spin relaxation of electrons in single dots has been measured [16].Here we present a realistic calculation of phononinduced spin relaxation in single and coupled lateral quantum dots formed over a depleted two-dimensional electron gas in GaAs grown along 001 , the most typical growth direction. We show that our calculation is consistent with the single-dot experiment [16]. We predict the following: (i) Spin relaxation in coupled dots is strongly anisotropic with respect to the orientation of both an in-plane magnetic field (due to the interplay of the Bychkov-Rashba and Dresselhaus spin-orbit terms) and the dots's axis. The anisotropy is limited by the in-plane inversion symmetry only. (ii) The spin relaxation rate varies strongly with the interdot coupling, having a giant enhancement at a significant range of useful tunneling amplitudes due to spin-hot spots (anticrossings caused by spin-orbit coupling [9,[17][18][19]). This variation, which is over several (4 to 5) orders of magnitude, should be included in any realistic modeling of spin coherence phenomena in coupled quantum dots with controlled temporal evolution of the coupling. (iii) Fortunately, the effects of (ii) are absent at specific configurations. The most robust (with respect to materials parameters) such a configuration is with dots oriented along 110 (or 1 10 ) with the in-plane magnetic field along 1 10 ( 110 ). We propose to use this configuration for spinbased quantum information experiments.Our single-electron Hamiltonian is H T V H SO H Z . Here T is the operator of the kinetic energy with the magnetic field B introduced by minimum coupling, and V is the double-dot confinement potential,The plane radius vector is r x; y , where x 100 and y 010 are the crystallographic axes, while d defines the distance (as well as tunneling energy at B 0) and the orientation of the dots; the angle between d andx is denoted below as . The conduction electron mass is m and the single-dot (d 0) confining energy is @! 0 . The spin-orbit coupli...

Abstract. For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.

Magnetic skyrmions are highly mobile nanoscale topological spin textures. We show, both analytically and numerically, that a magnetic skyrmion of an even azimuthal winding number placed in proximity to an s-wave superconductor hosts a zero-energy Majorana bound state in its core, when the exchange coupling between the itinerant electrons and the skyrmion is strong. This Majorana bound state is stabilized by the presence of a spin-orbit interaction. We propose the use of a superconducting trijunction to realize non-Abelian statistics of such Majorana bound states.

Spin-orbit effects in single-electron states in laterally coupled quantum dots in the presence of a perpendicular magnetic field are studied by exact numerical diagonalization. Dresselhaus ͑linear and cubic͒ and Bychkov-Rashba spin-orbit couplings are included in a realistic model of confined dots based on GaAs. Group theoretical classification of quantum states with and without spin-orbit coupling is provided. Spin-orbit effects on the g factor are rather weak. It is shown that the frequency of coherent oscillations ͑tunneling amplitude͒ in coupled dots is largely unaffected by spin-orbit effects due to symmetry requirements. The leading contributions to the frequency involves the cubic term of the Dresselhaus coupling. Spin-orbit coupling in the presence of a magnetic field leads to a spin-dependent tunneling amplitude, and thus to the possibility of spin to charge conversion, namely, spatial separation of spin by coherent oscillations in a uniform magnetic field. It is also shown that spin hot spots exist in coupled GaAs dots already at moderate magnetic fields, and that spin hot spots at zero magnetic field are due to the cubic Dresselhaus term only.