Electron-hole crossover in gate-controlled bilayer graphene quantum dots
Luca Banszerus, Alexander Rothstein, Thomas Fabian, Samuel Möller, Eike Icking, Stefan Trellenkamp, Florian Lentz, Daniel Neumaier, Kenji Watanabe, Takashi Tani… 展开作者
Luca Banszerus, Alexander Rothstein, Thomas Fabian, Samuel Möller, Eike Icking, Stefan Trellenkamp, Florian Lentz, Daniel Neumaier, Kenji Watanabe, Takashi Taniguchi, Florian Libisch, Christian Volk, Christoph Stampfer
Electron and hole Bloch states in gapped bilayer graphene exhibit topological orbital magnetic moments with opposite signs near the band edges, which allows for tunable valley-polarization in an out-of-plane magnetic field. This intrinsic property makes electron and hole quantum dots (QDs) in bilayer graphene interesting for valley and spin-valley qubits. Here we show measurements of the electron-hole crossover in a bilayer graphene QD, demonstrating the opposite sign of the orbital magnetic moments associated with the Berry curvature. Using three layers of metallic top gates, we independently control the tunneling barriers of the QD while tuning the occupation from the few-hole regime to the few-electron regime, crossing the displacement-field controlled band gap. The band gap is around 25 meV, while the charging energies of the electron and hole dots are between 3-5 meV. The extracted valley g-factor is around 17 and leads to opposite valley polarization for electron and hole states at moderate B-fields. Our measurements agree well with tight-binding calculations for our device.
Quantum dots with large Thouless number $g$ embody a regime where both disorder and interactions can be treated nonperturbatively using large-N techniques (with $N=g$) and quantum phase transitions can be studied. Here we focus on dots where the noninteracting Hamiltonian is drawn from a crossover ensemble between two symmetry classes, where the crossover parameter introduces a new, tunable energy scale independent of and much smaller than the Thouless energy. We show that the quantum critical regime, dominated by collective critical fluctuations, can be accessed at the new energy scale. The n…
We present theoretical results based on mean-field and exact many-body approaches showing that in bilayer triangular graphene quantum dots with zigzag edges the magnetism can be controlled by an external vertical electric-field. We demonstrate that without electric field the spins of the two layers of the quantum dot interact ferromagnetically. At a critical value of the electric-field, the total spin of the bilayer structure can be turned off or reduced to a single localized spin, a qubit isolated from contacts and free from interaction with nuclear spins.
Graphene and bilayer graphene quantum dots are promising hosts for spin qubits with long coherence times. Although recent technological improvements make it possible to confine single electrons electrostatically in bilayer graphene quantum dots, and their spin and valley texture of the single particle spectrum has been studied in detail, their relaxation dynamics remains still unexplored. Here, we report on transport through a high-frequency gate controlled single-electron bilayer graphene quantum dot. By transient current spectroscopy of single-electron spin states, we extract a lower bound o…
Interacting electrons in quantum dots with large Thouless number $g$ in the three classical random matrix symmetry classes are well-understood. When a specific type of spin-orbit coupling known to be dominant in two dimensional semiconductor quantum dots is introduced, we show that a new interacting quantum critical crossover energy scale emerges and low-energy quasiparticles generically have a decay width proportional to their energy. The low-energy physics of this system is an example of a universal interacting crossover regime.
We demonstrate theoretically that quantum dots in bilayers of graphene can be realized. A position-dependent doping breaks the equivalence between the upper and lower layer and lifts the degeneracy of the positive and negative momentum states of the dot. Numerical results show the simultaneous presence of electron and hole confined states for certain doping profiles and a remarkable angular momentum dependence of the quantum dot spectrum which is in sharp contrast with that for conventional semiconductor quantum dots. We predict that the optical spectrum will consist of a series of non-equidis…
Electronic properties of triangular and hexagonal nano-scale quantum dots (QDs) of Silicene and bilayer graphene are studied. It is shown that the low-energy edge-localized electronic states, existing within the size-quantized gap are easily tunable by electric field. The appearance and field evolution of the electronic gap in these zero energy states (ZES) is shown to be very sensitive to QD geometry that permits to design the field-effect scalable QD devices with electronic properties on-demand.
We study numerically the optical properties of low-buckled silicene and AB-stacked bilayer graphene quantum dots subjected to an external electric field, which is normal to their surface. Within the tight-binding model, the optical absorption is calculated for quantum dots, of triangular and hexagonal shapes, with zigzag and armchair edge terminations. We show that in triangular silicene clusters with zigzag edges a rich and widely tunable infrared absorption peak structure originates from transitions involving zero energy states. The edge of absorption in silicene quantum dots undergoes red s…
This work investigates the electronic properties of the energy spectrum of a hybrid system composed of (i) a circular quantum dot of monolayer graphene surrounded by an infinite sheet of AA-stacked bilayer graphene and (ii) a circular quantum dot of AA-stacked graphene bilayer surrounded by infinite monolayer graphene. We establish analytical findings for the related energy levels and wave functions using the continuum model and the zigzag boundary conditions at the graphene monolayer-bilayer interface. We investigate the effects of perpendicular magnetic, dot radius, and electrical fields on …
We present a highly controllable double quantum dot device based on bilayer graphene. Using a device architecture of interdigitated gate fingers, we can control the interdot tunnel coupling between 1 to 4 GHz and the mutual capacitive coupling between 0.2 and 0.6 meV, independently of the charge occupation of the quantum dots. The charging energy and hence the dot size remains nearly unchanged. The tuning range of the tunnel coupling covers the operating regime of typical silicon and GaAs spin qubit devices.
Dirac fermions interacting with a cylindrically symmetric quantum dot potential created in single and bilayer graphene are not confined but form quasi-bound states. The broadening of these quasi-bound states (i. e. the inverse of their lifetimes) decreases (increases) with the orbital momentum of the electron in the case of graphene (bilayer). Quasi-bound states with energy below (above) the barrier height are dominantly electron(hole)-like. A remarkable decrease of the energy level broadening is predicted for electron energies close to the barrier height, which are a consequence of the total …
A parabolic quantum dot (QD) as realized by biasing nanostructured gates on bilayer graphene is investigated in the presence of electron-electron interaction. The energy spectrum and the phase diagram reveal unexpected transitions as function of a magnetic field. For example, in contrast to semiconductor QDs, we find a novel valley transition rather than only the usual singlet-triplet transition in the ground state of the interacting system. The origin of these new features can be traced to the valley degree of freedom in bilayer graphene. These transitions have important consequences for cycl…
We present gate-controlled single, double, and triple dot operation in electrostatically gapped bilayer graphene. Thanks to the recent advancements in sample fabrication, which include the encapsulation of bilayer graphene in hexagonal boron nitride and the use of graphite gates, it has become possible to electrostatically confine carriers in bilayer graphene and to completely pinch-off current through quantum dot devices. Here, we discuss the operation and characterization of electron-hole double dots. We show a remarkable degree of control of our device, which allows the implementation of tw…
We report tunneling spectroscopy experiments on a bilayer graphene double quantum dot device that can be tuned by all-graphene lateral gates. The diameter of the two quantum dots are around 50 nm and the constrictions acting as tunneling barriers are 30 nm in width. The double quantum dot features addition energies on the order of 20 meV. Charge stability diagrams allow us to study the tunable interdot coupling energy as well as the spectrum of the electronic excited states on a number of individual triple points over a large energy range. The obtained constant level spacing of 1.75 meV over a…
Spin transitions driven by a periodically varying electric potential in dilute fluorinated graphene quantum dots are investigated. Flakes of monolayer graphene are considered as well as electrostatic electron traps induced in bilayer graphene. The stationary states are obtained within the tight-binding approach and are used to the basis of eigenstates to describe the system dynamics. The dilute fluorination of the top layer lifts the valley degeneracy of the confined states and attenuates the orbital magnetic dipole moments due to current circulation within the flake. Moreover, the spin-orbit …
We report on charge detection in electrostatically-defined quantum dot devices in bilayer graphene using an integrated charge detector. The device is fabricated without any etching and features a graphite back gate, leading to high quality quantum dots. The charge detector is based on a second quantum dot separated from the first dot by depletion underneath a 150 nm wide gate. We show that Coulomb resonances in the sensing dot are sensitive to individual charging events on the nearby quantum dot. The potential change due to single electron charging causes a step-like change (up to 77 %) in the…
In bilayer CrI3, experimental and theoretical studies suggest that the magnetic order is closely related to the layer staking configuration. In this work, we study the effect of dynamical lattice distortions, induced by non-linear phonon coupling, in the magnetic order of the bilayer system. We use density functional theory to determine the phonon properties and group theory to obtain the allowed phonon-phonon interactions. We find that the bilayer structure possesses low-frequency Raman modes that can be non-linearly activated upon the coherent photo-excitation of a suitable infrared phonon m…
Electrostatic confinement of charge carriers in bilayer graphene provides a unique platform for carbon-based spin, charge or exchange qubits. By exploiting the possibility to induce a band gap with electrostatic gating, we form a versatile and widely tunable multi-quantum dot system. We demonstrate the formation of single, double and triple quantum dots that are free of any sign of disorder. In bilayer graphene we have the possibility to form tunnel barriers using different mechanisms. We can exploit the ambipolar nature of bilayer graphene where pn-junctions form natural tunnel barriers. Alte…
In bilayer graphene, electrostatic confinement can be realized by a suitable design of top and back gate electrodes. We measure electronic transport through a bilayer graphene quantum dot, which is laterally confined by gapped regions and connected to the leads via p-n junctions. Single electron and hole occupancy is realized and charge carriers $n = 1, 2,\dots 50$ can be filled successively into the quantum system with charging energies exceeding $10 \ \mathrm{meV}$. For the lowest quantum states, we can clearly observe valley and Zeeman splittings with a spin g-factor of $g_{s}\approx 2$. In…
Artificial molecular states of double quantum dots defined in bilayer graphene are studied with the atomistic tight-binding and its low-energy continuum approximation. We indicate that the extended electron wave functions have opposite parities on each of the sublattices at both graphene layers and that the ground-state wave function components change from bonding to antibonding with the interdot distance. In the weak coupling limit -- the most relevant for the quantum dots defined electrostatically -- the signatures of the interdot coupling include -- for the two-electron ground state -- form…
We measure telegraph noise of current fluctuations in an electrostatically defined quantum dot in bilayer graphene by real-time detection of single electron tunneling with a capacitively coupled neighboring quantum dot. Suppression of the second and third cumulant (related to shot noise) in a tunable graphene quantum dot is demonstrated experimentally. With this method we demonstrate the ability to measure very low current and noise levels. Furthermore, we use this method to investigate the first spin excited state, an essential prerequisite to measure spin relaxation.
Understanding how the electron spin is coupled to orbital degrees of freedom, such as a valley degree of freedom in solid-state systems is central to applications in spin-based electronics and quantum computation. Recent developments in the preparation of electrostatically-confined quantum dots in gapped bilayer graphene (BLG) enables to study the low-energy single-electron spectra in BLG quantum dots, which is crucial for potential spin and spin-valley qubit operations. Here, we present the observation of the spin-valley coupling in a bilayer graphene quantum dot in the single-electron regime…
We report on temperature-dependent magnetoresistance measurements in balanced double quantum wells exposed to microwave irradiation for various frequencies. We have found that the resistance oscillations are described by the microwave-induced modification of electron distribution function limited by inelastic scattering (inelastic mechanism), up to a temperature of T*~4 K. With increasing temperature, a strong deviation of the oscillation amplitudes from the behavior predicted by this mechanism is observed, presumably indicating a crossover to another mechanism of microwave photoresistance, wi…
Graphene p-n junctions provide an ideal platform for investigating novel behavior at the boundary between electronics and optics that arise from massless Dirac fermions, such as whispering gallery modes and Veselago lensing. Bilayer graphene also hosts Dirac fermions, but they differ from single-layer graphene charge carriers because they are massive, can be gapped by an applied perpendicular electric field, and have very different pseudospin selection rules across a p-n junction. Novel phenomena predicted for these massive Dirac fermions at p-n junctions include anti-Klein tunneling, oscillat…
Intrinsic and extrinsic valley Hall effects are predicted to emerge in graphene systems with uniform or spatially-varying mass terms. Extrinsic mechanisms, mediated by the valley-dependent scattering of electrons at the Fermi surface, can be directly linked to quantum transport simulations. This is a promising route towards more complete experimental investigation of valleytronic phenomena in graphene, but a major obstacle is the difficulty in applying the sublattice-dependent potentials required. Here we show that strongly valley-dependent scattering also emerges from bilayer graphene quantum…
We report on ground- and excited state transport through an electrostatically defined few-hole quantum dot in bilayer graphene in both parallel and perpendicular applied magnetic fields. A remarkably clear level scheme for the two-particle spectra is found by analyzing finite bias spectroscopy data within a two-particle model including spin and valley degrees of freedom. We identify the two-hole ground-state to be a spin-triplet and valley-singlet state. This spin alignment can be seen as Hund's rule for a valley-degenerate system, which is fundamentally different to quantum dots in carbon nan…
We study shot noise for generic quantum dots coupled to two leads and allow for an arbitrary strength of diffractive impurity scattering inside the dots. The ballistic quantum dots possess a mixed classical phase space, where regular and chaotic regions coexist. In absence of disorder, the noise is systematically suppressed below the universal value of fully chaotic systems, by an amount which varies with the positions of the leads. This suppression is due to the deterministic nature of transport through regular regions and along short chaotic trajectories, and disappears by increasing the sca…
We derive closed expressions for the universal weak localization peak of the average conductance peak heights in Coulomb blockade quantum dots in the crossover from orthogonal to unitary symmetry. The scale for the crossover is independent of the number of channels in each lead, in contrast with the case of open dots. The functional form of the weak localization peak is independent of temperature. We also derive analytically the variance of the conductance peak heights as a function of the crossover parameter.
A gate-defined quantum dot in bilayer graphene is utilized as a sensitive electrometer for probing the charge density of its environment. Under the influence of a perpendicular magnetic field, the charge carrier density of the channel region next to the quantum dot oscillates due to the formation of Landau levels. This is experimentally observed as oscillations in the gate-voltage positions of the Coulomb resonances of the nearby quantum dot. From the frequency of the oscillations, we extract the charge carrier density in the channel and from the amplitude the shift of the quantum dot potentia…
We report on finite bias spectroscopy measurements of the two-electron spectrum in a gate defined bilayer graphene (BLG) quantum dot for varying magnetic fields. The spin and valley degree of freedom in BLG give rise to multiplets of 6 orbital symmetric and 10 orbital anti-symmetric states. We find that orbital symmetric states are lower in energy and separated by $\approx 0.4 - 0.8$ meV from orbital anti-symmetric states. The symmetric multiplet exhibits an additional energy splitting of its 6 states of $\approx 0.15 - 0.5$ meV due to lattice scale interactions. The experimental observations …
We show that in bilayer graphene it is possible to achieve a very restrictive confinement of the massless Dirac fermions zero-modes by using inhomogeneous magnetic fields. Specifically, we show that, using a suitable nonuniform magnetic fields, the wave function may be restricted to a specific region of the space, being forbidden all transmission probability to the contiguous regions. This allows to construct mesoscopic structures in bilayer graphene by magnetic fields configurations.
Second order perturbation theory and a Lipkin-Nogami scheme combined with an exact Monte Carlo projection after variation are applied to compute the ground-state energy of $6\le N\le 210$ electron-hole pairs confined in a parabolic two-dimensional quantum dot. The energy shows nice scaling properties as N or the confinement strength is varied. A crossover from the high-density electron-hole phase to the BCS excitonic phase is found at a density which is roughly four times the close-packing density of excitons.
We present transport measurements through an electrostatically defined bilayer graphene double quantum dot in the single electron regime. With the help of a back gate, two split gates and two finger gates we are able to control the number of charge carriers on two gate-defined quantum dot independently between zero and five. The high tunability of the device meets requirements to make such a device a suitable building block for spin-qubits. In the single electron regime, we determine interdot tunnel rates on the order of 2~GHz. Both, the interdot tunnel coupling, as well as the capacitive inte…
We study a system of two quantum dots connected by a hopping bridge. Both the dots and connecting region are assumed to be in universal crossover regimes between Gaussian Orthogonal and Unitary ensembles. Using a diagrammatic approach appropriate for energy separations much larger than the level spacing we obtain the ensemble-averaged one- and two-particle Green's functions. It turns out that the diffuson and cooperon parts of the two-particle Green's function can be described by separate scaling functions. We then use this information to investigate a model interacting system in which one dot…
We investigate the gap in the single-electron spectrum of twisted bilayer graphene. In a perfect infinite lattice of a twisted bilayer, the gap varies exponentially in response to weak changes of the twist angle. Such a large sensitivity makes theoretical predictions of the gap nearly impossible, since experimentally the twist angle is always known with finite accuracy. To address this issue, we numerically study finite clusters of twisted bilayer graphene. For finite systems, changing the twist angle causes a gradual crossover between gapless and gapped regimes. The crossover occurs when the …
We report the fabrication and characterization of gate-defined hole quantum dots in monolayer and bilayer WSe$_2$. The devices were operated with gates above and below the WSe$_2$ layer to accumulate a hole gas, which for some devices was then selectively depleted to define the dot. Temperature dependence of conductance in the Coulomb blockade regime is consistent with transport through a single level, and excited state transport through the dots was observed at temperatures up to 10 K. For adjacent charge states of a bilayer WSe$_2$ dot, magnetic field dependence of excited state energies was…
We introduce a random interaction matrix model (RIMM) for finite-size strongly interacting fermionic systems whose single-particle dynamics is chaotic. The model is applied to Coulomb blockade quantum dots with irregular shape to describe the crossover of the peak spacing distribution from a Wigner-Dyson to a Gaussian-like distribution. The crossover is universal within the random matrix model and is shown to depend on a single parameter: a scaled fluctuation width of the interaction matrix elements. The crossover observed in the RIMM is compared with the results of an Anderson model with Coul…
Trapping electrons in quantum dots and controlling their collective quantum states is crucial for converting semiconductor structures into bits of quantum information processing. Here, we study single- and two-particle states in quantum dots formed in gapped bilayer graphene (BLG), where the electron's valley states appear in pair with their spin quantum number and we analyse spin- and valley-singlet and triplet states for various BLG and dot parameters, as well as two-particle interaction strength and external magnetic field.
We study quantum dots defined by external potentials within finite flakes of bilayer graphene using the tight-binding approach. We find that in the limit of large flakes containing zigzag edges the dot-localized energy levels appear within the energy continuum formed by extended states. As a consequence no ionization threshold for the carriers contained within the dot exists. For smaller flakes with zigzag boundaries the dot-localized energy levels appear interlaced with the energy levels outside the flake, so in a charging experiment the electrons will be added alternately to the dot area and…
The crossover from weak to strong correlations in parabolic quantum dots at zero magnetic field is studied by numerically exact path-integral Monte Carlo simulations for up to eight electrons. By the use of a multilevel blocking algorithm, the simulations are carried out free of the fermion sign problem. We obtain a universal crossover only governed by the density parameter $r_s$. For $r_s>r_c$, the data are consistent with a Wigner molecule description, while for $r_s<r_c$, Fermi liquid behavior is recovered. The crossover value $r_c \approx 4$ is surprisingly small.
Bilayer graphene is a nanomaterial that allows for well-defined, separated quantum states to be defined by electrostatic gating and, therefore, provides an attractive platform to construct tunable quantum dots. When a magnetic field perpendicular to the graphene layers is applied, the graphene valley degeneracy is lifted, and splitting of the energy levels of the dot is observed. Given the experimental ability to engineer this energy valley splitting, bilayer graphene quantum dots have a great potential for hosting robust qubits. Although bilayer graphene quantum dots have been recently realiz…
The ground states of electrons in two vertically coupled quantum dots in the presence of an external magnetic field have been studied within the density functional theory. A phase diagram of the transition to the quantum Hall state in double quantum dots has been plotted for the Landau level filling factor $ν= 1$. It has been shown that the quantum Hall state can be stable for a zero tunnel splitting. The influence of the impurity potential on the stability condition for the quantum Hall state has been analyzed.
Motivated by recent experiments on electronic transport through a carbon nanotube, we investigate the role of the intra- and inter-orbital Coulomb interactions on the temperature evolution of the conductance. It is shown that small amount (~10%) of asymmetry between these Coulomb repulsions substantially deforms the conductance profile at finite temperature, particularly around half-filling. The nature of such thermal symmetry crossover is elucidated.
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.
Electronic properties of quantum dots (QDs) depend sensitively on their parent materials. Therefore, confined electronic states in graphene QDs (GQDs) of monolayer and Bernal-stacked bilayer graphene are quite different. Twisted bilayer graphene (TBG) is distinct from monolayer and Bernal-stacked bilayer graphene because of the new degree of freedom: twist angle. In the past few years, numerous efforts have been made to realize the GQDs of monolayer and Bernal-stacked bilayer graphene and achieved great success. Thus far, however, strategies for realizing GQDs of TBG have been elusive. Here, w…
The magnetic field dependence of energy levels in gapped single- and bilayer graphene quantum dots (QDs) defined by electrostatic gates is studied analytically in terms of the Dirac equation. Due to the absence of sharp edges in these types of QDs, the valley degree of freedom is a good quantum number. We show that its degeneracy is efficiently and controllably broken by a magnetic field applied perpendicular to the graphene plane. This opens up a feasible route to create well-defined and well controlled spin- and valley-qubits in graphene QDs. We also point out the similarities and difference…
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.
We study the crossover between liquid and solid electron phases in a two-dimensional harmonic trap as the density is progressively diluted. We infer the formation of geometrically ordered phases from charge distributions and pair correlation functions obtained via a large scale configuration interaction calculation.
By applying the infinite-mass boundary condition, we analytically calculate the confined states and the corresponding wave functions of AA-stacked bilayer graphene quantum {dots} in the presence of an uniform magnetic field $B$. It is found that the energy spectrum shows two set of levels, which are the double copies of the energy spectrum for single layer graphene, shifted up-down by $+γ$ and $-γ$, respectively. However, the obtained spectrum exhibits different symmetries between the electron and hole states as well as the intervalley symmetries. It is noticed that, the applied magnetic field…
We discuss and compare two different types of confinement in bilayer graphene by top and bottom gating with symmetrical microelectrodes. Trivial confinement corresponds to the same polarity of all top gates, which is opposed to that of all bottom ones. Topological confinement requires the polarity of part of the top-bottom pairs of gates to be reversed. We show that the main qualitative difference between trivial and topological bound states manifests itself in the magnetic field dependence. We illustrate our finding with an explicit calculation of the energy spectrum for quantum dots and ring…
