The chemical potential of a quantum dot parabolically confining one or two electrons is studied in an axial magnetic field. The number of electrons in the dot is given by combination between the chemical potentials of the dots and leads. The equilibrium-charge diagram of the dot is obtained in the phase space determined by the magnetic field and the lead voltage. The influence of the magnetic field on the electron number and the electron transport is discussed. The dependence of the electron number on the magnetic fields and the lead voltage is clear at a glance in the equilibrium-charge diagram. Moreover, the diagram is a compact tool to show the condition of the electron transport.
We study theoretically the electron states in a system of two vertically stacked quantum dots. We investigate the influence of the geometrical symmetry breaking (caused by the displacement as well as the ellipticity of the dots) on the electron states. Our modeling is based on the 8-band kp method. We show that the absence of axial symmetry of the system leads to a coupling of the s state from one dot with the p and d states from the other. Our findings indicate, that this coupling can produce a strong energy splitting at resonance (on the order of several meV) in the case of closely spaced qu…
We present the full three dimensionality of an electrostatically calculated stability diagram for triple quantum dots. The stability diagram maps out the favored charge configuration of the system as a function of potential shifts due to gate voltages. For triple dots only a three dimensional visualization allows for the complete identification of all its components. Those are most notably the so called quadruple points where four electronic configurations are degenerate, and quantum cellular automata (QCA) processes. The exact positions of these features within the stability diagram are now r…
The charge stability diagram for two coupled quantum dots containing up to two electrons is computed in magnetic fields. One- and two-particle Schroedinger equations are solved by exact diagonalization to obtain the chemical potentials and exchange energy in these systems. By analyzing the chemical potentials variation with external biases and magnetic fields, it is possible to distinguish between the weak and strong inter-dot couplings. The variation of the chemical potential curvatures and the double-triple point separations in the stability diagrams confirms the inter-dot coupling decrease …
Quantum dots are considered building blocks for future quantum information circuits. We present here experimental results on a quantum dot circuit consisting of three quantum dots with controlled electron numbers down to one per dot and tunable coupling. We experimentally map out for the first time the stability diagram of the triatomic system and reveal the existence of quadruple points, a signature of the three dots being in resonance. In their vicinity we observe a surprising effect, a 'cloning' of charge transfer transitions related to charge and spin reconfigurations. The experimental res…
We report both photon- and phonon-assisted tunneling transitions in a linear array of three quantum dots, which can only be understood by considering the full three-dimensionality of the charge stability diagram. Such tunneling transitions potentially contribute to leakage of qubits defined in this system. A detailed understanding of these transitions is important as they become more abundant and complex to analyze as quantum dot arrays are scaled up.
A basis set expansion is employed to calculate spectra and eigenstates of charge carriers within a toroidal volume characterized by major radius $R$ and minor radius $a$ immersed in an azimuthally symmetric magnetic field. The angular variables appearing in the Schrodinger equation are eliminated by contour methods, yielding effective potentials that reduce computational time by a large factor. An approximation formula for the single particle spectrum is presented that allows efficient construction of the partition function necessary for rapid calculation of thermodynamic quantities. The heat …
We measure a triple quantum dot in the regime where three addition lines, corresponding to the addition of an electron to each of three dots, pass through each other. In particular, we probe the interplay between transport and the tridimensional nature of the stability diagram. We choose the regime most pertinent for spin qubit applications. We find that at low bias transport through the triple quantum dot circuit is only possible at six quadruple point locations. The results are consistent with an equivalent circuit model.
We study a two-electron quantum dot molecule in a magnetic field by the direct diagonalization of the Hamiltonian matrix. The ground states of the molecule with the total spin S=0 and S=1 provide a possible realization for a qubit of a quantum computer. Switching between the states is best achieved by changing the magnetic field. Based on an analysis of the wave function, we show that the system consists of composite particles formed by an electron and flux quanta attached to it. This picture can also be used to explain the spin phase diagram.
The Fano factor stability diagram of a C$_{3v}$ symmetric triangular quantum dot is analysed for increasing electron fillings $N$. At low filling, conventional Poissonian and sub-Poissonian behavior is found. At larger filling, $N\ge 2$, a breaking of the electron-hole symmetry is manifested in super-Poissonian noise with a peculiar bias voltage dependence of the Fano factor at Coulomb and interference blockade. An analysis of the Fano map unravels a nontrivial electron bunching mechanism arising from the presence of degenerate many-body states combined with orbital interference and Coulomb in…
Neutral excitons in strained axially symmetric In(Ga)As/GaAs quantum dots with ringlike shape are investigated. Similar to experimental self-assembled quantum rings, the analyzed quantum dots have volcano-like shapes. The continuum mechanical model is employed to determine the strain distribution, and the single-band envelope function approach is adopted to compute the electron states. The hole states are determined by the axially symmetric multiband Luttinger-Kohn Hamiltonian, and the exciton states are obtained from an exact diagonalization. We found that the presence of the inner layer cove…
Spin properties of two interacting electrons in a quantum dot (QD) embedded in a nanowire with controlled aspect ratio and longitudinal magnetic fields are investigated by using a configuration interaction (CI) method and exact diagonalization (ED) techniques. The developed CI theory based on a three-dimensional (3D) parabolic model provides explicit formulations of the Coulomb matrix elements and allows for straightforward and efficient numerical implementation. Our studies reveal fruitful features of spin singlet-triplet transitions of two electrons confined in a nanowire quantum dot (NWQD),…
We present a density-functional theory (DFT) approach to the study of the phase diagram of the maximum density droplet (MDD) in two-dimensional quantum dots in a magnetic field. Within the lowest Landau level (LLL) approximation, analytical expressions are derived for the values of the parameters $N$ (number of electrons) and $B$ (magnetic field) at which the transition from the MDD to a ``reconstructed'' phase takes place. The results are then compared with those of full Kohn-Sham calculations, giving thus information about both correlation and Landau level mixing effects. Our results are als…
We propose a quantum dot (QD) implementation of a quantum state transfer channel. The proposed channel consists of N vertically stacked QDs with the nearest neighbor tunnel coupling, placed in an axial electric field. We show that the system supports high-fidelity transfer of the state of a terminal dot both by free evolution and by adiabatic transfer. The protocol is to a large extent insensitive to inhomogeneity of the energy parameters of the dots and requires only a global electric field.
During the last years there has been an increasing excitement in nanomotors and particularly in current-driven nanomotors. Despite the broad variety of stimulating results found, the regime of strong Coulomb interactions has not been fully explored for this application. Here we consider nanoelectromechanical devices composed by a set of coupled quantum dots interacting with mechanical degrees of freedom taken in the adiabatic limit and weakly coupled to electronic reservoirs. We use a real-time diagrammatic approach to derive general expressions for the current-induced forces, friction coeffic…
We explore a strongly correlated quantum dot in the presence of a weak confinement potential and a weak magnetic field. Our exact diagonalization studies show that the groundstate property of such a quantum dot is rather sensitive to the magnetic field and the strength of the confinement potential. We have determined rich phase diagrams of these quantum dots. Some experimental consequences of the obtained phase diagrams are discussed.
Monolithic axial InGaAs quantum dot emitters in GaAs-based nanowires via Sb-mediated facet engineering
Walter Schottky Institute, TUM School of Natural Sciences, Technical University of Munich（慕尼黑工业大学沃尔特·施科特基研究所自然科学学院）；Institute for Physics and Astronomy, Technical University Berlin（柏林工业大学物理与天文学研究所）；Department of Chemistry, Ludwig-Maximilians-University of Munich（慕尼黑大学化学系）；Walter Schottky Institute, TUM School of Computation, Information and Technology, Technical University of Munich（慕尼黑工业大学沃尔特·施科特基研究所计算、信息与技术学院）；Department of Physics, University of California, Berkeley（加州大学伯克利分校物理系）
We present a theoretical analysis of the phonon-assisted relaxation in a system composed of two self-assembled vertically stacked quantum dots. We construct realistic model, which takes into account the geometry and strain distribution in the system. We calculate phonon-assisted relaxation rates between the two lowest states (in one- and two-electron cases). The relaxation rates and energy levels are studied as a function of external (axial) electric field and geometry of the structure (dot sizes). We show that the relaxation times can be as low as 1~ps but efficent relaxation occurs only for …
Within the effective mass approximation an adiabatic description of spheroidal and dumbbell quantum dot models in the regime of strong dimensional quantization is presented using the expansion of the wave function in appropriate sets of single-parameter basis functions. The comparison is given and the peculiarities are considered for spectral and optical characteristics of the models with axially symmetric confining potentials depending on their geometric size making use of the total sets of exact and adiabatic quantum numbers in appropriate analytic approximations.
While the dynamics for three-dimensional axially symmetric two-electron quantum dots with parabolic confinement potentials is in general non-separable we have found an exact separability with three quantum numbers for specific values of the magnetic field. Furthermore, it is shown that the magnetic properties such as the magnetic moment and the susceptibility are sensitive to the presence and strength of a vertical confinement. Using a semiclassical approach the calculation of the eigenvalues reduces to simple quadratures providing a transparent and almost analytical quantization of the quantu…
We present numerical investigations based on the Luttinger-Kohn four-band $k \cdot p$ theory and, accordingly, establish a quantitatively valid model of the excitonic fine structures of droplet epitaxial GaAs/AlGaAs quantum dots under uni-axial stress control. In the formalisms, stressing a photo-excited quantum dot is equivalent creating a pseudo-magnetic field that is directly coupled to the pseudo-spin of the exciton doublet and tunable to tailor the polarized fine structure of exciton. The latter feature is associated with the valence-band-mixing of exciton that is especially sensitive to …
In this letter we calculate the energies corresponding to the different magnetic phases present in a ferromagnetic cylinder by means of analytical calculations. From the comparison of these energies, it is possible to construct magnetic phase diagrams as a function of the uniaxial anisotropy of the sample and the external magnetic field applied. As proof of concept, we analyzed the magnetic phase diagrams for a cobalt dot of 240 nm in diameter and 70 nm in length, with an easy axis parallel to the dot axis, and with a magnetic field applied towards or perpendicular to this axis. From these dia…
We analyze the nonequilibrium transport properties of a quantum dot with a harmonic degree of freedom (Holstein phonon) coupled to metallic leads, and derive its full counting statistics (FCS). Using the Lang-Firsov (polaron) transformation, we construct a diagrammatic scheme to calculate the cumulant generating function. The electron-phonon interaction is taken into account exactly, and the employed approximation represents a summation of a diagram subset with respect to the tunneling amplitude. By comparison to Monte Carlo data the formalism is shown to capture the basic properties of the st…
We investigate the linear ac-conductance for tunneling through an arbitrary interacting quantum dot in the presence of a finite dc-bias. In analogy to the well-known Meir-Wingreen formula for the dc case, we are able to derive a general formula for the ac-conductance. It can be expressed entirely in terms of local correlations on the quantum dot, in the form of a Keldysh block diagram with four external legs. We illustrate the use of this formula as a starting point for diagrammatic calculations by considering the ac-conductance of the noninteracting resonant level model and deriving the resul…
We studied a vertical ``quantum dot molecule'', where one of the dots is occupied with electrons and the other with holes. We find that different phases occur in the ground state, depending on the carrier density and the interdot distance. When the system is dominated by shell structure, orbital degeneracies can be removed either by Hund's rule, or by Jahn-Teller deformation. Both mechanisms can lead to a maximum of the addition energy at mid-shell. At low densities and large interdot distances, bound electron-hole pairs are formed.
Quantum dots in magnetic fields are studied within the current spin density functional formalism avoiding any spatial symmetry restrictions of the solutions. We find that the maximum density droplet reconstructs into states with broken internal symmetry: The Chamon-Wen edge co-exists with a modulation of the charge density along the edge. The phase boundaries between the polarization transition, the maximum density droplet and its reconstruction are in agreement with recent experimental results.
We report low-temperature transport studies of parallel double quantum dots formed in GaSb/InAsSb core-shell nanowires. At negative gate voltages, regular patterns of Coulomb diamonds are observed in the charge stability diagrams, which we ascribe to single-hole tunneling through a quantum dot in the GaSb core. As the gate voltage increases, the measured charge stability diagram indicates the appearance of an additional quantum dot, which we suggest is an electron quantum dot formed in the InAsSb shell. We find that an electron-hole interaction induces shifts of transport resonances in the sou…
We show that Berry's geometrical (topological) phase for circular quantum dots with an odd number of electrons is equal to πand that eigenvalues of the orbital angular momentum run over half-integer values. The non-zero value of the Berry's phase is provided by axial symmetry and two-dimensionality of the system. Its particular value (π) is fixed by the Pauli exclusion principle. Our conclusions agree with the experimental results of T. Schmidt {\it at el}, \PR B {\bf 51}, 5570 (1995), which can be considered as the first experimental evidence for the existence of a new realization of Berry's …
We analyze theoretically and experimentally the electronic structure and charging diagram of three coupled lateral quantum dots filled with electrons. Using the Hubbard model and real-space exact diagonalization techniques we show that the electronic properties of this artificial molecule can be understood using a set of topological Hunds rules. These rules relate the multi-electron energy levels to spin and the inter-dot tunneling $t$, and control charging energies. We map out the charging diagram for up to N=6 electrons and predict a spin-polarized phase for two holes. The theoretical chargi…
We present a real-time diagrammatic theory for transport through interacting quantum dots tunnel coupled to normal and superconducting leads. Our formulation describes both the equilibrium and non-equilibrium superconducting proximity effect in a quantum dot. We study a three-terminal transistor geometry, consisting of a single-level quantum dot tunnel coupled to two phase-biased superconducting leads and one voltage-biased normal lead. We compute both the Josephson current between the two superconductors and the Andreev current in the normal lead, and analyze their switching on and off as wel…
A prototype Hamiltonian for the generic patterned magnetic structures, of dipolar interaction with perpendicular anisotropy, is investigated within the finite-size framework by Landau-Lifshift-Gilbert classical spin dynamics. Modifications on the ground state phase diagram are discussed with an emphasis on the disappearance of continuous degeneracy in the ground state of in-plane phase due to the finite lattice size effect. The symmetry-governed ground state evolution upon the lattice size increase provides a critical insight into the systematic transition to the infinite extreme.
A computation scheme for solving elliptic boundary value problems with axially symmetric confining potentials using different sets of one-parameter basis functions is presented. The efficiency of the proposed symbolic-numerical algorithms implemented in Maple is shown by examples of spheroidal quantum dot models, for which energy spectra and eigenfunctions versus the spheroid aspect ratio were calculated within the conventional effective mass approximation. Critical values of the aspect ratio, at which the discrete spectrum of models with finite-wall potentials is transformed into a continuous…
The superconducting proximity effect is probed experimentally in Josephson junctions fabricated with InAs nanowires contacted by Nb leads. Contact transparencies $t \sim 0.7$ are observed. The electronic phase coherence length at low temperatures exceeds the channel length. However, the elastic scattering length is a few times shorter than the channel length. Electrical measurements reveal two regimes of quantum transport: (i) the Josephson regime, characterized by a dissipationless current up to $\sim 100$ nA, and (ii) the quantum dot regime, characterized by the formation of Andreev Bound St…
We present a theoretical study of the Pauli or spin-valley blockade for double quantum dots in semiconducting carbon nanotubes. In our model we take into account the following characteristic features of carbon nanotubes: (i) fourfold (spin and valley) degeneracy of the quantum dot levels, (ii) the intrinsic spin-orbit interaction which is enhanced by the tube curvature, and (iii) valley-mixing due to short-range disorder, i.e., substitutional atoms, adatoms, etc. We find that the spin-valley blockade can be lifted in the presence of short-range disorder, which induces two independent random (i…
We apply the recently introduced Hubbard model approach to quantitatively describe the experimental charge stability diagram and tunnel coupling of silicon double quantum dot systems. The results calculated from both the generalized Hubbard model and the microscopic theory are compared with existing experimental data, and excellent agreement between theory and experiment is found. The central approximation of our theory is a reduction of the full multi-electron multi-band system to an effective two-electron model, which is numerically tractable. In the microscopic theory we utilize the Hund-Mu…
We study electron transport through double quantum dots in series. The tunnel coupling of the discrete dot levels to external leads causes a shift of their energy. This energy renormalization affects the transport characteristics even in the limit of weak dot-lead coupling, when sequential transport dominates. We propose an experimental setup which reveals the renormalization effects in either the current-voltage characteristics or in the stability diagram.
We consider electron states in n-p double quantum dots defined in a semiconducting carbon nanotube (CNT) by an external potential. We describe formation of extended single-electron orbitals originating from the conduction and valence bands confined in a minimum and a maximum of the external potential, respectively. We solve the problem of a confined electron pair using an exact diagonalization method within the tight-binding approach, which allows for a straightforward treatment of the conduction and valence band states, keeping an exact account for the intervalley scattering mediated by the a…
We found that a downwardly concave entanglement evolution of the ground state of a two-electron axially symmetric quantum dot testifies that a shape transition from a lateral to a vertical localization of two electrons under a perpendicular magnetic field takes place. Although affected, the two-electron probability density does not exhibit any prominent change.
We use the entanglement measure to study the evolution of quantum correlations in two-electron axially-symmetric parabolic quantum dots under a perpendicular magnetic field. We found that the entanglement indicates on the shape transition in the density distribution of two electrons in the lowest state with zero angular momentum projection at the specific value of the applied magnetic field.
Since the realization of high-quality microwave cavities coupled to quantum dots, one can envisage the possibility to investigate the coherent interaction of light and matter in semiconductor quantum devices. Here we study a parallel double quantum dot device operating as single-electron splitter interferometer, with each dot coupled to a local photon cavity. We explore how quantum correlation and entanglement between the two separated cavities are generated by the coherent transport of a single electron passing simultaneously through the two different dots. We calculate the covariance of the …
In this work, thermal and magnetic properties for an electron with cylindrical confinement in presence of external electric and magnetic fields have been investigated. We found that the corresponding time-independent Schrödinger equation can be separated into the product of the radially symmetric and an axial equation. Moreover, we have obtained a quasi-exact expression for the energy spectrum of the system in terms of the exact solutions for the radial equation and an approximation up to first order for the axial equation. We have calculated the well-known thermal and magnetic properties as t…
We analyze the subgap excitations and phase diagram of a quantum dot (QD) coupled to a semiconducting nanowire fully wrapped by a superconducting (S) shell. We take into account how a Little-Parks (LP) pairing fluxoid (a winding in the S phase around the shell) influences the proximity effect on the dot. We find that under axially symmetric QD-S coupling, shell fluxoids cause the induced pairing to vanish, producing instead a level renormalization that pushes subgap levels closer to zero energy and flattens fermionic parity crossings as the coupling strength increases. This fluxoid-induced sta…
Rapid Charge Stability Diagram Generation from Device-level Modeling of Semiconductor Quantum Dots
University of California, Los Angeles（加州大学洛杉矶分校）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Joint Quantum Institute, University of Maryland（马里兰大学联合量子研究所）
We calculate steady-state properties of a strongly correlated quantum dot under voltage bias by means of non-equilibrium Cluster Perturbation Theory and the non-equilibrium Variational Cluster Approach, respectively. Results for the steady-state current are benchmarked against data from accurate Matrix Product State based time evolution. We show that for low to medium interaction strength, non-equilibrium Cluster Perturbation Theory already yields good results, while for higher interaction strength the self-consistent feedback of the non-equilibrium Variational Cluster Approach significantly e…
We investigate gain in microwave photonic cavities coupled to voltage-biased double quantum dot systems with an arbitrary strong dot-lead coupling and with a Holstein-like light-matter interaction, by adapting the diagrammatic Keldysh nonequilibrium Green's function approach. We compute out-of-equilibrium properties of the cavity : its transmission, phase response, mean photon number, power spectrum, and spectral function. We show that by the careful engineering of these hybrid light-matter systems, one can achieve a significant amplification of the optical signal with the voltage-biased elect…
An odd-occupied quantum dot in a Josephson junction can flip transmission phase, creating a π-junction. When the junction couples topological superconductors, no phase flip is expected. We investigate this and related effects in a full-shell hybrid interferometer, using gate voltage to control dot-junction parity and axial magnetic flux to control the transition from trivial to topological superconductivity. Enhanced zero-bias conductance and critical current for odd parity in the topological phase reflects hybridization of the confined spin with zero-energy modes in the leads.
Double quantum dots (DQDs) hold great promise as building blocks for quantum technology as they allow for two electronic states to coherently couple. Defining QDs with materials rather than using electrostatic gating allows for QDs with a hard-wall confinement potential and more robust charge and spin states. An unresolved problem is how to individually address these quantum dots, which is necessary for controlling quantum states. We here report the fabrication of double quantum dot devices defined by the conduction band edge offset at the interface of the wurtzite and zinc blende crystal phas…
We demonstrate fast readout of a double quantum dot (DQD) that is coupled to a superconducting resonator. Utilizing parametric amplification in a nonlinear operational mode, we improve the signal-to-noise ratio (SNR) by a factor of 2000 compared to the situation with the parametric amplifier turned off. With an integration time of 400 ns we achieve a SNR of 76. By studying SNR as a function of the integration time we extract an equivalent charge sensitivity of 8 x 10^{-5} e/root(Hz). The high SNR allows us to acquire a DQD charge stability diagram in just 20 ms. At such a high data rate, it is…
The time evolution of a quantum dot exciton in Coulomb interaction with wetting layer carriers is treated using an approach similar to the independent boson model. The role of the polaronic unitary transform is played by the scattering matrix, for which a diagrammatic, linked cluster expansion is available. Similarities and differences to the independent boson model are discussed. A numerical example is presented.
An ambipolar $n$-$p$ double quantum dot defined by potential variation along a semiconducting carbon-nanotube is considered. We focus on the (1e,1h) charge configuration with a single excess electron in the conduction band state confined in the $n$-type dot and a single missing electron in the valence band state of the $p$-dot for which lifting of the Pauli blockade of the current was observed in the electric-dipole spin resonance [E. A. Laird et al. Nat. Nanotech. 8 , 565 (2013)]. The dynamics of the system driven by periodic electric field is studied with the Floquet theory and the time-depe…
We study the properties of a few-electron system confined in coupled elongated quantum dots (QDs) using a model Gaussian potential and the numerical exact diagonalization technique. In the absence of magnetic fields, as the aspect ratio $r$ between the QD extensions in the direction perpendicular and parallel to the coupling directions increases, the exchange energy exhibits a sharp variation at the specific value $r=3.9$, before (after) which the exchange energy increases (declines). The sharp variation occurs because of a sudden change in the single-particle configuration of the triplet stat…
We report fast, simultaneous charge sensing and transport measurements of gate-defined carbon nanotube quantum dots. Aluminum radio frequency single electron transistors (rf-SETs) capacitively coupled to the nanotube dot provide single-electron charge sensing on microsecond timescales. Simultaneously, rf reflectometry allows fast measurement of transport through the nanotube dot. Charge stability diagrams for the nanotube dot in the Coulomb blockade regime show extended Coulomb diamonds into the high-bias regime, as well as even-odd filling effects, revealed in charge sensing data.
We complete our recently introduced theoretical framework treating the double quantum dot system with a generalized form of Hubbard model. The effects of all quantum parameters involved in our model on the charge stability diagram are discussed in detail. A general formulation of the microscopic theory is presented, and truncating at one orbital per site, we study the implication of different choices of the model confinement potential on the Hubbard parameters as well as the charge stability diagram. We calculate the charge stability diagram keeping three orbitals per site and find that the ef…
We describe the electronic properties of a double dot for which the lateral coupling between the two vertical dots can be controlled in-situ with a center gate voltage (Vc) and the current flows through the two dots in series. When Vc is large and positive, the two dots merge. As Vc is made less positive, two dots are formed whose coupling is reduced. We measure charging diagrams for positive and negative source-drain voltages in the weak coupling regime and observe current rectification due to the Pauli spin blockade when the hyperfine interaction between the electrons and the nuclei is suppr…
A semiconductor quintuple quantum dot with two charge sensors and an additional contact to the center dot from an electron reservoir is fabricated to demonstrate the concept of scalable architecture. This design enables formation of the five dots as confirmed by measurements of the charge states of the three nearest dots to the respective charge sensor. The gate performance of the measured stability diagram is well reproduced by a capacitance model.These results provide an important step towards realizing controllable large scale multiple quantum dot systems.
