Spin polarized current generation from quantum dots without magnetic fields
Jacob J. Krich, Bertrand I. Halperin
备注：
7 pages, 3 figures. Revised version with reference to Bardarson, Adagideli, and Jacquod, Phys Rev Lett 98, 196601 (2007), who performed an analogous calculation in a different geometry
An unpolarized charge current passing through a chaotic quantum dot with spin-orbit coupling can produce a spin polarized exit current without magnetic fields or ferromagnets. We use random matrix theory to estimate the typical spin polarization as a function of the number of channels in each lead in the limit of large spin-orbit coupling. We find rms spin polarizations up to 45% with one input channel and two output channels. Finite temperature and dephasing both suppress the effect, and we include dephasing effects using a new variation of the third lead model. If there is only one channel in the output lead, no spin polarization can be produced, but we show that dephasing lifts this restriction.
We investigate spin-polarized transport phenomena through double quantum dots coupled to ferromagnetic leads in series. By means of the slave-boson mean-field approximation, we calculate the conductance in the Kondo regime for two different configurations of the leads: spin-polarization of two ferromagnetic leads is parallel or anti-parallel. It is found that transport shows some remarkable properties depending on the tunneling strength between two dots. These properties are explained in terms of the Kondo resonances in the local density of states.
In this paper we show that it is possible to switch the spin polarization of the photocurrent signal obtained from a single self-assembled quantum dot photodiode under the effect of elliptically polarized light by just increasing the light intensity. In the nonlinear mechanism treated here, intense elliptically polarized light creates an effective exchange interaction between the exciton spin states through the biexciton state. This effect can be used as a dynamical switch to invert the spin-polarization of the extracted photocurrent. We further show that the effect persists in realistic ensem…
Using density functional theory, we investigate fluctuations of the ground state energy of spin-polarized, disordered quantum dots in the metallic regime. To compare to experiment, we evaluate the distribution of addition energies and find a convolution of the Wigner-Dyson distribution, expected for noniteracting electrons, with a narrower Gaussian distribution due to interactions. The tird moment of the total distribution is independent of interactions, and so is predicted to decrease by a factor of 0.405 upon application of a magnetic field which transforms from the Gaussian orthogonal to th…
We analyze transport through both a double quantum dot and a triple quantum dot with inhomogeneous Zeeman splittings in the presence of crossed dc and ac magnetic fields. We find that strongly spin-polarized current can be achieved by tuning the relative energies of the Zeeman-split levels of the dots, by means of electric gate voltages: depending on the energy level detuning, the double quantum dot works either as spin-up or spin-down filter. We show that a triple quantum dot in series under crossed dc and ac magnetic fields can act not only as spin-filter but also as spin-inverter.
We report measurements of spin transitions for GaAs quantum dots in the Coulomb blockade regime, and compare ground and excited state transport spectroscopy to direct measurements of the spin polarization of emitted current. Transport spectroscopy reveals both spin-increasing and spin-decreasing transitions as well as higher-spin ground states, and allows g-factors to be measured down to a single electron. The spin of emitted current in the Coulomb blockade regime, measured using spin-sensitive electron focusing, is found to be polarized along the direction of the applied magnetic field regard…
We study the interplay of the Kondo effect and spin-polarized tunneling in a class of systems exhibiting uniaxial magnetic anisotropy, such as magnetic molecules, magnetic adatoms, or quantum dots coupled to a single localized magnetic moment. Using the numerical renormalization group method we calculate the spectral functions and linear conductance in the Kondo regime. We show that the exchange coupling between conducting electrons and localized magnetic core generally leads to suppression of the Kondo effect. We also predict a nontrivial dependence of the tunnel magnetoresistance on the stre…
Spin-dependent electronic transport through a quantum dot side-coupled to two quantum dots and attached to ferromagnetic leads with collinear (parallel and antiparallel) magnetizations is analyzed theoretically. The intra-dot Coulomb correlations are taken into account, whereas the inter-dot ones are neglected. Transport characteristics, i.e. conductance and tunnel magnetoresistance associated with the magnetization rotation from parallel to antiparallel configurations, are calculated by the noneqiulibrium Green function technique. The Green functions are derived by the equation of motion meth…
On a basis of extensive analytical and numerical studies we show that a linear-polarized microwave field creates a stationary magnetization in mesoscopic ballistic quantum dots with two-dimensional electron gas being at a thermal equilibrium. The magnetization is proportional to a number of electrons in a dot and to a microwave power. Microwave fields of moderate strength create in a one dot of few micron size a magnetization which is by few orders of magnitude larger than a magnetization produced by persistent currents. The effect is weakly dependent on temperature and can be observed with ex…
We present a comprehensive investigation of the polarization properties of non-polar a-plane InGaN quantum dots (QDs) and their origin with statistically significant experimental data and rigorous k.p modelling. The unbiased selection and study of 180 individual QDs allow us to compute an average polarization degree of 0.90, with a standard deviation of only 0.08. When coupled with theoretical insights, we show that a-plane InGaN QDs are highly insensitive to size differences, shape anisotropies, and indium content fluctuations. Furthermore, 91% of the studied QDs exhibit a polarization axis a…
We investigate the quench dynamics of a quantum dot strongly coupled to spin-polarized ferromagnetic leads. The real-time evolution is calculated by means of the time-dependent density-matrix numerical renormalization group method implemented within the matrix product states framework. We examine the system's response to a quench in the spin-dependent coupling strength to ferromagnetic leads as well as in the position of the dot's orbital level. The spin dynamics is analyzed by calculating the time-dependent expectation values of the quantum dot's magnetization and occupation. Based on these, …
The coherent spin dependent transport through a set of two capacitively coupled quantum dots placed in a magnetic field is considered within the equation of motion method. The magnetic field breaks the spin degeneracy. For special choices of gate voltages the dot levels are tuned to resonance and the orbital Kondo effect results. For different Zeemann splittings at the dots the Kondo resonance can be formed for only one spin channel. In this case the system operates as an efficient spin filter.
Transport properties in the Kondo regime of a nanosystem displaying uniaxial magnetic anisotropy (such as a magnetic molecule, magnetic adatom or quantum dot coupled to a localized magnetic moment) are analyzed theoretically. In particular, the influence of spin-polarized transport through a local orbital of the system and exchange coupling of conduction electrons to the system's magnetic core on the Kondo effect is discussed. The numerical renormalization group method is applied to calculate the spectral functions and linear conductance in the case of the parallel and antiparallel configurati…
We proposed and demonstrated that the nuclear spins of the host lattice in GaAs double quantum dots can be strongly polarized in either of two opposite directions, parallel or antiparallel to an external magnetic field. The direction is selected simply by adjusting the dc source-drain voltage of the device. This nuclear polarization manifests itself by repeated controlled electron-nuclear spin scattering in the Pauli spin blockade state. Polarized nuclei are also controlled coherently by means of nuclear magnetic resonance (NMR). This work confirmed that the nuclear spins in quantum dots are i…
A recent experiment [E. A. Chekhovich et al., Phys. Rev. Lett. 104, 066804 (2010)] has demonstrated that high nuclear spin polarization can be achieved in self-assembled quantum dots by exploiting an optically forbidden transition between a heavy hole and a trion state. However, a fully polarized state is not achieved as expected from a classical rate equation. Here, we theoretically investigate this problem with the help of a quantum master equation and we demonstrate that a fully polarized state cannot be achieved due to formation of a nuclear dark state. Moreover, we show that the maximal d…
We use an atomistic model to consider the effect of shape symmetry breaking on the optical properties of self-assembled InAs/GaAs quantum dots. In particular, we investigate the energy level structure and optical activity of the lowest energy excitons in these nanostructures. We compare between quantum dots with two-fold rotational and two reflections (C2v) symmetry and quantum dots in which this symmetry was reduced to one reflection only (Cs) by introducing a facet between the quantum dots and the host material. We show that the symmetry reduction mostly affects the optical activity of the d…
We propose a new approximation scheme within equation of motion approach (EOM) to spin polarized transport through a quantum dot coupled to ferromagnetic leads. It has some advantages over a widely used in the literature standard EOM technique, in particular when we are interested in spin polarized quantities. Namely, it gives the values of the dot spin polarization which are closer to the ones obtained within numerical renormalization group (NRG), than the standard EOM approach. While restoring the Kondo effect, the spin polarization vanishes and the transport becomes unpolarized, in agreemen…
Berry phase plays an important role in determining many physical properties of quantum systems. However, a Berry phase altering energy spectrum of a quantum system is comparatively rare. Here, we report an unusual tunable valley polarized energy spectra induced by continuously tunable Berry phase in Bernal-stacked bilayer graphene quantum dots. In our experiment, the Berry phase of electron orbital states is continuously tuned from about pi to 2pi by perpendicular magnetic fields. When the Berry phase equals pi or 2pi, the electron states in the two inequivalent valleys are energetically degen…
We theoretically investigate the optical response of a quantum dot, embedded in a microcavity and incoherently excited by pulsed pumping. The exciton and biexciton transition are off-resonantly coupled with the left- and right-polarized mode of the cavity, while the two-photon resonance condition is fulfilled. Rich behaviours are shown to occur in the time dependence of the second-order correlation functions which refer to counter-polarized photons. The corresponding time-averaged quantities, which are accessible to experiments, confirm that such a dot-cavity system behaves as a good emitter o…
Spatially nonhomogeneously spin polarized nuclei are proposed as a new mechanism to monitor electron states in a nanostructure, or as a means to createn and, if necessary, reshape such nanostructures in the course of the experiment. We found that a polarization of nulear spins may lift the spin polarization of the electron states in a nanostructure and, if sufficiently strong, leads to a polarization of the electron spins. Polarized nuclear spins may form an energy landscape capable of binding electrons with energy up to several meV and the localization radius $ >$ 100Å.
Kinetics of polarized photoluminescence of the negatively charged InP quantum dots in weak magnetic field is studied experimentally. Effect of both the nuclear spin fluctuations and the dynamical nuclear polarization on the electron spin orientation is observed.
Pauli spin blockade (PSB) in double quantum dots (DQDs) has matured into a prime technique for precise measurements of nanoscale system parameters. In this work we demonstrate that systems with site-dependent g-tensors and spin-polarized leads allow for a complete characterization of the g-tensors in the dots by magnetotransport experiments alone. Additionally, we show that special polarization configurations can enhance the often elusive magnetotransport signal, rendering the proposed technique robust against noise in the system, and inducing a giant magnetoresistance effect. Finally, we inco…
We analyze cotunneling transport through two quantum dots in series weakly coupled to external ferromagnetic leads. In the Coulomb blockade regime the electric current flows due to third-order tunneling, while the second-order single-barrier processes have indirect impact on the current by changing the occupation probabilities of the double dot system. We predict a zero-bias maximum in the differential conductance, whose magnitude is conditioned by the value of the inter-dot Coulomb interaction. This maximum is present in both magnetic configurations of the system and results from asymmetry in…
We investigate linear and nonlinear transport across single-walled carbon nanotube quantum dots weakly coupled to spin-polarized leads. We consider tubes of finite length and small diameter, where not only forward scattering contributions of the Coulomb potential, but also short-ranged processes play an important role. In particular, they induce exchange effects leading for electron fillings 4n+2 either to a non-degenerate groundstate of spin S=0 or to a triplet groundstate. In the linear regime we present analytical results for the conductance - for both the S=0 and the triplet groundstate - …
Circularly polarized optical excitation is used to demonstrate the efficient injection of spin-polarized excitons to individual self-assembled CdSe quantum dots in ZnSe barrier. The exciton spin-transfer is studied by means of polarization-resolved single dot spectroscopy performed in magnetic field applied in Faraday configuration. Detailed analysis of the neutral exciton photoluminescence spectra reveals the presence of exciton spin relaxation during its lifetime in a quantum dot. This process is seen for both nonmagnetic dots and those containing single Mn$^{2+}$ ions. Taking this into acco…
The statistical theory of multiple exciton generation in quantum dots which is based on the Fermi approach to the problem of multiple particle in nucleon-nucleon collisions is presented. Our estimates of the multiple exciton generation efficiency in different quantum dots, induced by absorption of a single photon, are in a good agreement with the experimental data.
We present an experimental and theoretical study of the polarized photoluminescence spectrum of single semiconductor quantum dots in various charge states. We compare our high resolution polarization sensitive spectral measurements with a new many-carrier theoretical model, which was developed for this purpose. The model considers both the isotropic and anisotropic exchange interactions between all participating electron-hole pairs. With this addition, we calculate both the energies and polarizations of all optical transitions between collective, quantum dot confined charge carrier states. We …
The nuclear spin fluctuations (NSF) as well as the dynamic nuclear polarization (DNP) and their effects on the electron spins in negatively charged (In,Ga)As/GaAs quantum dots have been studied by polarized pump-probe and photoluminescence spectroscopy techniques. The effective magnetic field of the NSF is about 30 mT at low excitation power. The NSF distribution becomes highly anisotropic at strong optical excitation by circularly polarized light with periodically alternating helicity. This phenomenon is attributed to a decrease of the nuclear spin entropy due to the hyperfine interaction wit…
We study the evolution of a quantum dot controlled by a frequency-swept (chirped), linearly polarized laser pulse in the presence of carrier-phonon coupling. The final occupation of the exciton state is limited both due to phonon-induced transitions between the adiabatic spectral branches and because of phonon-assisted transitions to the biexciton state. When the biexciton shift is large enough, the quantum dot can be modeled as a two-level system, which corresponds to excitation with circularly polarized light. For this case, we compare different methods of simulations: (i) a time convolution…
We develop a theoretical description of transport through quantum dots connected to reservoirs via spin-polarized ballistic contacts. Rate equations account for spin accumulation inside the dot, electron-electron interactions, and stochastic fluctuations. It is shown that both the ac response (admittance) and the frequency-dependent shot noise are governed by spin-flip scattering, which can be used to detect spin polarization and spin-flip processes in the dot.
Higher harmonic generation in plane graphene quantum dots initiated by intense coherent radiation is investigated, using dynamical Hartree-Fock mean-field theory. A microscopic theory describing the extreme nonlinear optical response of plane graphene quantum dots is developed. The closed set of differential equations for the single-particle density matrix at the graphene quantum dots-strong laser field multiphoton interaction is solved numerically. The obtained solutions indicate the significance of the type of edge and lateral size, and bandgap and laser field strength in the high harmonic g…
We discuss the possibility to generate, manipulate, and probe single spins in single-level quantum dots coupled to ferromagnetic leads. The spin-polarized currents flowing between dot and leads lead to a non-equilibrium spin accumulation, i.e., a finite polarization of the dot spin. Both the magnitude and the direction of the dot's spin polarization depends on the magnetic properties of leads and their coupling to the dot. They can be, furthermore, manipulated by either an externally applied magnetic field or an intrinsically present exchange field that arises due to the tunnel coupling of the…
We demonstrate single photon emission from self-assembled m-plane InGaN quantum dots (QDs) embedded on the side-walls of GaN nanowires. A combination of electron microscopy, cathodoluminescence, time-resolved micro-PL and photon autocorrelation experiments give a thorough evaluation of the QDs structural and optical properties. The QD exhibits anti-bunched emission up to 100 K, with a measured autocorrelation function of g^2(0) = 0.28 (0.03) at 5 K. Studies on a statistically significant number of QDs show that these m-plane QDs exhibit very fast radiative lifetimes (260 +/- 55 ps) suggesting …
Spin relaxation of electrons doped in InP quantum dots was studied by means of luminescence pump-probe and Hanle measurements. Optical pumping makes spins of doped electrons to be oriented in parallel to the helicity of the circularly polarized excitation. The luminescence pump-probe showed the spin orientation of the doped electrons decay on a millisecond time-scale. Hanle measurement clarified the spin dephasing relaxation time of doped electrons is 1.7 ns which is explained by the frozen fluctuation of nuclear spins.
The exciton recombination and spin dynamics in (In,Al)As/AlAs quantum dots (QDs) with indirect band gap and type-I band alignment are studied. The negligible (less than $0.2~μ$eV) value of the anisotropic exchange interaction in these QDs prevents a mixing of the excitonic basis states with pure spin and allows for the formation of spin polarized bright excitons for quasi-resonant circularly polarized excitation. In a longitudinal magnetic field, the recombination and spin dynamics of the excitons are controlled by the hyperfine interaction between the electron and nuclear spins. A QD blockade…
We review and summarize recent theoretical and experimental work on electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with surrounding nuclear spins. This topic is of particular interest with respect to several proposals for quantum information processing in solid state systems. Specifically, we investigate the hyperfine interaction of an electron spin confined in a quantum dot in an s-type conduction band with the nuclear spins in the dot. This interaction is proportional to the square modulus of the electron wave function at the location of each n…
Many-particle electron states in semiconductor quantum dots with carrier-mediated ferromagnetism are studied theoretically within the self-consistent Boltzmann equation formalism. Depending on the conditions, a quantum dot may contain there phases: partially spin-polarized ferromagnetic, fully spin-polarized ferromagnetic, and paramagnetic phases. The physical properties of many-body ferromagnetic confined systems come from the competing carrier-mediated ferromagnetic and Coulomb interactions. The magnetic phases in gated quantum dots with holes can be controlled by the voltage or via optical …
Understanding strongly interacting electrons enables the design of materials, nanostructures and devices. Developing this understanding relies on the ability to tune and control electron-electron interactions by, e.g., confining electrons to atomically thin layers of 2D crystals with reduced screening. The interplay of strong interactions on a hexagonal lattice with two nonequivalent valleys, topological moments, and the Ising-like spin-orbit interaction gives rise to a variety of phases of matter corresponding to valley and spin polarized broken symmetry states. In this work we describe a hig…
A detailed theoretical study of the optical absorption in self-assembled quantum dots is presented in this paper. A rigorous atomistic strain model as well as a sophisticated electronic band structure model are used to ensure accurate prediction of the optical transitions in these devices . The optimized models presented in this paper are able to reproduce the experimental results with an error less than 1$\%$. The effects of incident light polarization, alloy mole fraction, quantum dot dimensions, and doping have been investigated. The in-plane polarized light absorption is more significant t…
We study a cluster of quantum dots defined within silicene that host confined electron states with spin and valley degrees of freedom. Atomistic tight-binding and continuum Dirac approximation are applied for few-electron system in quest for spontaneous valley polarization driven by inter-dot tunneling and electron-electron interaction, i.e. a valley counterpart of itinerary Nagaoka ferromagnetic ordering recently identified in GaAs square cluster of quantum dots with three excess electrons [P. Dehollain, {\it et al.}, Nature {\bf 579}, 528 (2020)]. We find that for Hamiltonian without intrins…
We proposed and demonstrated a scheme of surface emitting circularly polarized light source by introducing strong imbalance between left- and right-circularly polarized vacuum fields in an on-waveguide chiral grating structure. We observed circularly polarized spontaneous emission from InAs quantum dots embedded in the wave guide region of a GaAs-based structure. Obtained degree of polarization reaches as large as 25% at room temperature. Numerical calculation visualizes spatial profiles of the modification of vacuum field modes inside the structure with strong circular anisotropy.
Spontaneous nuclear polarization is predicted in double quantum dots in the spin-blocked electron transport regime. The polarization results from an instability of the zero-polarization state when singlet and triplet electron energy levels are brought into resonance by the effective hyperfine field of the nuclei on the electrons. The nuclear spins, once polarized, serve as a cold bath for cooling electrons below the lattice (phonon) temperature. We estimate the relevant time scales and discuss the conditions necessary to observe these phenomena.
We investigated optical spin orientation and dynamic nuclear polarization (DNP) in individual self-assembled InGaAs/GaAs quantum dots (QDs) doped by a single Mn atom, a magnetic impurity providing a neutral acceptor A$^0$ with an effective spin $J=1$. We find that the spin of an electron photo-created in such a quantum dot can be efficiently oriented by a quasi-resonant circularly-polarized excitation. For the electron spin levels which are made quasi-degenerate by a magnetic field compensating the exchange interaction $Δ_e$ with A$^0$, there is however a full depolarization due the anisotropi…
Using the tight-binding approach, we investigate the electronic properties of bilayer phosphorene (BLP) quantum dots (QDs) in the presence of perpendicular electric and magnetic fields. Since BLP consists of two coupled phosphorene layers, it is of interest to examine the layer-dependent electronic properties of BLP QDs, such as the electronic distributions over the two layers and the so-produced layer-polarization features, and to see how these properties are affected by the magnetic field and the bias potential. We find that in the absence of a bias potential only edge states are layer-polar…
Excitons in alloyed nanowire quantum dots have unique spectra as shown here using atomistic calculations. The bright exciton splitting is triggered solely by alloying and despite cylindrical quantum dot shape reaches over $15~μ$eV, contrary to previous theoretical predictions, however, in line with experimental data. This splitting can however be tuned by electric field to go below $1$~$μ$eV threshold. The dark exciton optical activity is also strongly affected by alloying reaching notable $1/3500$ fraction of the bright exciton and having large out-of-plane polarized component.
We consider the electronic transport through a Rashba quantum dot coupled to ferromagnetic leads. We show that the interference of localized electron states with resonant electron states leads to the appearance of the Fano-Rashba effect. This effect occurs due to the interference of bound levels of spin-polarized electrons with the continuum of electronic states with an opposite spin polarization. We investigate this Fano-Rashba effect as a function of the applied magnetic field and Rashba spin-orbit coupling.
Spin entanglement between two spatially separated electrons can be generated in nonequilibrium interacting quantum dots, coherently coupled to a common lead. In this system entangled two-electron states develop which are Werner states with an imbalance between singlet and triplet probabilities. We propose a multi-terminal, multiply-connected setup for the generation and detection of this imbalance. In particular, we identify a regime in which the formation of spin entanglement leads to a cancellation of Aharonov-Bohm oscillations.
Current steps attributed to resonant tunneling through individual InAs quantum dots embedded in a GaAs-AlAs-GaAs tunneling device are investigated experimentally in magnetic fields up to 28 T. The steps evolve into strongly enhanced current peaks in high fields. This can be understood as a field-induced Fermi-edge singularity due to the Coulomb interaction between the tunneling electron on the quantum dot and the partly spin polarized Fermi sea in the Landau quantized three-dimensional emitter.
We investigate theoretically the electron states in HgTe quantum dots (QDs) with inverted band structures. In sharp contrast to conventional semiconductor quantum dots, the quantum states in the gap of HgTe quantum dot with an inverted band structure are fully spin-polarized, and show ring-like density distributions near the boundary of the QD and spin-angular momentum locking. The persistent charge currents and magnetic moments, i.e., the Aharonov-Bohm effect, can be observed in such QD structure and oscillate with increasing magnetic fields. This feature offers us a practical way to detect t…
We report optical orientation experiments in individual, strain free GaAs quantum dots in AlGaAs grown by droplet epitaxy. Circularly polarized optical excitation yields strong circular polarization of the resulting photoluminescence at 4K. Optical injection of spin polarized electrons into a dot gives rise to dynamical nuclear polarization that considerably changes the exciton Zeeman splitting (Overhauser shift). We show that the created nuclear polarization is bistable and present a direct measurement of the build-up time of the nuclear polarization in a single GaAs dot in the order of one s…
We report the enhancement of the optical second harmonic signal in non-centrosymmetric semiconductor CdS quantum dots, when they are placed in close contact with isolated silver nanoparticles. The intensity enhancement is about 1000. We also show that the enhancement increases when the incoming laser frequency $ω$ is tuned toward the spectral position of the silver plasmon at $2ω$, proving that the silver nanoparticle modifies the nonlinear emission.
We investigate coherent transport through open lateral quantum dots using recursive Green's function technique, incorporating exchange-correlation effects within local spin-density approximation (LSDA). At low electron densities the current is spin-polarized and electron density in the dot shows a strong spin polarization. As the electron density increases the spin polarization in the dot gradually diminishes. These findings are consistent with available experimental observations. Results of our LSDA-based modeling indicate that utilization of the simplified approaches that use phenomenologica…
Time-resolved Faraday rotation has recently demonstrated coherent transfer of electron spin between quantum dots coupled by conjugated molecules. Using a transfer Hamiltonian ansatz for the coupled quantum dots, we calculate the Faraday rotation signal as a function of the probe frequency in a pump-probe setup using neutral quantum dots. Additionally, we study the signal of one spin-polarized excess electron in the coupled dots. We show that, in both cases, the Faraday rotation angle is determined by the spin transfer probabilities and the Heisenberg spin exchange energy. By comparison of our …
Spin-polarized transport through a quantum dot strongly coupled to ferromagnetic electrodes with non-collinear magnetic moments is analyzed theoretically in terms of the non-equilibrium Green function formalism. Electrons in the dot are assumed to be coupled to a phonon bath. The influence of electron-phonon coupling on tunnelling current, linear and nonlinear conductance, and on tunnel magnetoresistance is studied in detail. Variation of the main Kondo peaks and phonon satellites with the angle between magnetic moments of the leads is analyzed.
We investigated electron transport through ultra small Si quantum dots. We found that the $B$-dependence of energy levels is dominated by the Zeeman shift, allowing us to measure the spin difference between two successive ground states directly. In some dots the number of electrons N in the dot can be tuned starting from zero, and the total spin of the dot can be mapped as a function of N and B. For one of the dots we deduced that the dot becomes spontaneously polarized at N=6 with a large spin change ΔS=3/2, demonstrating the essential features of spin blockade. Surprisingly, for N>20, the tr…
