Suspended Carbon Nanotube Quantum Dots Research Articles

Quantum capacitance mediated carbon nanotube optomechanics

Cavity optomechanics allows the characterization of a vibration mode, its cooling and quantum manipulation using electromagnetic fields. Regarding nanomechanical as well as electronic properties, single wall carbon nanotubes are a prototypical experimental system. At cryogenic temperatures, as high quality factor vibrational resonators, they display strong interaction between motion and single-electron tunneling. Here, we demonstrate large optomechanical coupling of a suspended carbon nanotube quantum dot and a microwave cavity, amplified by several orders of magnitude via the nonlinearity of Coulomb blockade. From an optomechanically induced transparency (OMIT) experiment, we obtain a single photon coupling of up to g0 = 2π ⋅ 95 Hz. This indicates that normal mode splitting and full optomechanical control of the carbon nanotube vibration in the quantum limit is reachable in the near future. Mechanical manipulation and characterization via the microwave field can be complemented by the manifold physics of quantum-confined single electron devices.

Sensing Kondo correlations in a suspended carbon nanotube mechanical resonator with spin‐orbit coupling

We study electron mechanical coupling in a suspended carbon nanotube (CNT) quantum dot device. Electron spin couples to the flexural vibration mode due to spin-orbit coupling in the electron tunneling processes. In the weak coupling limit, i.e. electron-vibration coupling is much smaller than the electron energy scale, the damping and resonant frequency shift of the CNT resonator can be obtained by calculating the dynamical spin susceptibility. We find that strong spin-flip scattering processes in Kondo regime significantly affect the mechanical motion of the carbon nanotube: Kondo effect induces strong damping and frequency shift of the CNT resonator.

Ground state cooling of nanomechanical resonators by electron transport

We discuss two theoretical proposals for controlling the nonequilibrium steady state of nanomechanical resonators using quantum electronic transport. Specifically?, we analyse two approaches to achieve the ground-state cooling of the mechanical vibration coupled to a quantum dot embedded between (i) spin-polarised contacts or (ii) a normal metal and a superconducting contact. Assuming a suitable coupling between the vibrational modes and the charge or spin of the electrons in the quantum dot, we show that ground-state cooling of the mechanical oscillator is within the state of the art for suspended carbon nanotube quantum dots operating as electromechanical devices.

Electrically tunable quantum interfaces between photons and spin qubits in carbon nanotube quantum dots

We present a new scheme for quantum interfaces (QIs) to accomplish the interconversion of photonic qubits and spin qubits based on optomechanical resonators and the spin–orbit-induced interactions in suspended carbon nanotube quantum dots (CNTQDs). This interface implements quantum spin transducers and further enables electrical manipulation of local electron spin qubits, which lays the foundation for all-electrical control of state transfer protocols between two distant quantum nodes in a quantum network. We numerically evaluate the state transfer processes and proceed to estimate the effect of each coupling strength on the operation fidelities. The simulation suggests that high operation fidelities are obtainable under realistic experimental conditions.

Ground-State Cooling of a Mechanical Oscillator by Interference in Andreev Reflection.

We study the ground-state cooling of a mechanical oscillator linearly coupled to the charge of a quantum dot inserted between a normal metal and a superconducting contact. Such a system can be realized, e.g., by a suspended carbon nanotube quantum dot with a capacitive coupling to a gate contact. Focusing on the subgap transport regime, we analyze the inelastic Andreev reflections which drive the resonator to a nonequilibrium state. For small coupling, we obtain that vibration-assisted reflections can occur through two distinct interference paths. The interference determines the ratio between the rates of absorption and emission of vibrational energy quanta. We show that ground-state cooling of the mechanical oscillator can be achieved for many of the oscillator's modes simultaneously or for single modes selectively, depending on the experimentally tunable coupling to the superconductor.

Control of vibrational states by spin-polarized transport in a carbon nanotube resonator

We study spin-dependent transport in a suspended carbon nanotube quantum dot in contact with two ferromagnetic leads and with the dot's spin coupled to the flexural mechanical modes. The spin-vibration interaction induces spin-flip processes between the two energy levels of the dot. This interaction arises from the spin-orbit coupling or a magnetic field gradient. The inelastic vibration-assisted spin flips give rise to a mechanical damping and, for an applied bias voltage, to a steady nonequilibrium occupation of the harmonic oscillator. We analyze these effects as function of the energy-level separation of the dot and the magnetic polarization of the leads. Depending on the magnetic configuration and the bias-voltage polarity, we can strongly cool a single mode or pump energy into it. In the latter case, we find that within our approximation, the system approaches eventually a regime of mechanical instability. Furthermore, owing to the sensitivity of the electron transport to the spin orientation, we find signatures of the nanomechanical motion in the current-voltage characteristic. Hence, the vibrational state can be read out in transport measurements.

Ground-state cooling of a carbon nanomechanical resonator by spin-polarized current.

We study the nonequilibrium steady state of a mechanical resonator in the quantum regime realized by a suspended carbon nanotube quantum dot in contact with two ferromagnets. Because of the spin-orbit interaction and/or an external magnetic field gradient, the spin on the dot couples directly to the flexural eigenmodes. Accordingly, the nanomechanical motion induces inelastic spin flips of the tunneling electrons. A spin-polarized current at finite bias voltage causes either heating or active cooling of the mechanical modes. We show that maximal cooling is achieved at resonant transport when the energy splitting between two dot levels of opposite spin equals the vibrational frequency. Even for weak electron-resonator coupling and moderate polarizations we can achieve ground-state cooling with a temperature of the leads, for instance, of T = 10 ω.

Few-Hundred GHz Carbon Nanotube Nanoelectromechanical Systems (NEMS)

We study 23-30 nm long suspended single-wall carbon nanotube quantum dots and observe both their stretching and bending vibrational modes. We use low-temperature DC electron transport to excite and measure the tubes' bending mode by making use of a positive feedback mechanism between their vibrations and the tunneling electrons. In these nanoelectromechanical systems (NEMS), we measure fundamental bending frequencies f(bend) ≈ 75-280 GHz and extract quality factors Q ∼ 10(6). The NEMS's frequencies can be tuned by a factor of 2 with tension induced by mechanical breakjunctions actuated by an electrostatic force or tension from bent suspended electrodes.

Spin-Orbit-Induced Strong Coupling of a Single Spin to a Nanomechanical Resonator

We theoretically investigate the deflection-induced coupling of an electron spin to vibrational motion due to spin-orbit coupling in suspended carbon nanotube quantum dots. Our estimates indicate that, with current capabilities, a quantum dot with an odd number of electrons can serve as a realization of the Jaynes-Cummings model of quantum electrodynamics in the strong-coupling regime. A quantized flexural mode of the suspended tube plays the role of the optical mode and we identify two distinct two-level subspaces, at small and large magnetic field, which can be used as qubits in this setup. The strong intrinsic spin-mechanical coupling allows for detection, as well as manipulation of the spin qubit, and may yield enhanced performance of nanotubes in sensing applications.

Interplay of magneto-elastic and polaronic effects in electronic transport through suspended carbon-nanotube quantum dots

We investigate the electronic transport through a suspended carbon-nanotube quantum dot. In the presence of a magnetic field perpendicular to the nanotube and a nearby metallic gate, two forces act on the electrons: the Laplace and the electrostatic force. They both induce coupling between the electrons and the mechanical transverse oscillation modes. We find that the difference between the two mechanisms appears in the cotunneling current.

Spectrum and Franck–Condon factors of interacting suspended single-wall carbon nanotubes

A low-energy theory of suspended carbon nanotube quantum dots in weak tunnelling coupling with metallic leads is presented. The focus is on the dependence of the spectrum and the Franck–Condon factors on the geometry of the junction including several vibronic modes. The relative size and the relative position of the dot and its associated vibrons strongly influence the electromechanical properties of the system. A detailed analysis of the complete parameters space reveals different regimes: in the short vibron regime the tunnelling of an electron into the nanotube generates a plasmon–vibron excitation, whereas in the long vibron regime polaron excitations dominate the scenario. The small, position-dependent Franck–Condon couplings of the small vibron regime convert into uniform, large couplings in the long vibron regime. Selection rules for the excitations of the different plasmon–vibron modes via electronic tunnelling events are also derived.

Phonon-assisted transport through suspended carbon nanotube quantum dots

Motivated by several recent experiments, we investigate phonon-assisted electronic transport through suspended carbon nanotube quantum dots including electron-electron, electron-phonon, and spin-orbit interactions. By effectively decoupling the electron and phonon thermodynamics, we present an explanation for the puzzling thermoinduced vibrational absorption sidebands, as well further predict that these absorption sidebands are obviously diminished in the partially filled Coulomb diamonds due to the destructive superposition of electron and hole states. The spin-orbit coupling is shown to split all phonon-assisted sidebands while the Kramers degeneracy remains. Interestingly, in the strong spin-orbit coupling regime, some split absorption sidebands are suppressed by the Coulomb blockade effect, while all split emission sidebands always survive.

Local Franck–Condon factors in suspended carbon nanotube quantum dots

We investigate the effects of the coupling between quantized vibrational modes (vibrons) and electronic degrees of freedom in suspended carbon nanotube quantum dots. The elastic deformations couple both to the total dot charge and to its spatial density fluctuations. The latter, neglected in the Anderson–Holstein model, have profound consequences if the size of the vibron and of the dot do not coincide, as is generically the case in experimental devices. When the vibron is smaller than the quantum dot, spatially varying local Franck–Condon factors emerge with striking effects on the transport properties of the system. The theoretical results are supported by our experimental observations.

Asymmetric Franck-Condon factors in suspended carbon nanotube quantum dots

Electronic states and vibrons in carbon nanotube quantum dots have, in general, different location and size. As a consequence, the conventional Anderson-Holstein model, coupling vibrons to the dot total charge only, may no longer be appropriated in general. Here we explicitly address the role of the spatial fluctuations of the electronic density, yielding space-dependent Franck-Condon factors. We discuss the consequent marked effects on transport which are compatible with recent measurements. This picture can be relevant for tunneling experiments in generic nanoelectromechanical systems.

Electron-vibron coupling in suspended carbon nanotube quantum dots

Motivated by recent experiments, we investigate the electron-vibron coupling in suspended carbon nanotube quantum dots, starting with the electron-phonon coupling of the underlying graphene layer. We show that the coupling strength depends sensitively on the type of vibron and is strongly sample dependent. The coupling strength becomes particularly strong when inhomogeneity-induced electronic quantum dots are located near regions where the vibronic mode is associated with large strain. Specifically, we find that the longitudinal stretching mode and the radial breathing mode are coupled via the strong deformation potential, while twist modes couple more weakly via a mechanism involving modulation of the electronic hopping amplitudes between carbon sites. A special case are bending modes: for symmetry reasons, their coupling is only quadratic in the vibron coordinate. Our results can explain recent experiments on suspended carbon nanotube quantum dots which exibit vibrational sidebands accompanied by the Franck-Condon blockade with strong electron-vibron coupling.

Pumping of Vibrational Excitations in the Coulomb-Blockade Regime in a Suspended Carbon Nanotube

Low-temperature transport spectroscopy measurements on a suspended few-hole carbon nanotube quantum dot are presented, showing a gate-dependent harmonic excitation spectrum which, strikingly, occurs in the Coulomb-blockade regime. The quantized excitation energy corresponds to the scale expected for longitudinal vibrations of the nanotube. The electronic transport processes are identified as cotunnel-assisted sequential tunneling, resulting from nonequilibrium occupation of the mechanical mode. They appear only above a high-bias threshold at the scale of electronic nanotube excitations. We discuss models for the pumping process that explain the enhancement of the nonequilibrium occupation and show that it is connected to a subtle interplay between electronic and vibrational degrees of freedom.

Franck–Condon blockade in suspended carbon nanotube quantum dots

Understanding the influence of vibrational motion of the atoms on electronic transitions in molecules constitutes a cornerstone of quantum physics, as epitomized by the Franck–Condon principle1,2 of spectroscopy. Recent advances in building molecular-electronics devices3 and nanoelectromechanical systems4 open a new arena for studying the interaction between mechanical and electronic degrees of freedom in transport at the single-molecule level. The tunnelling of electrons through molecules or suspended quantum dots5,6 has been shown to excite vibrational modes, or vibrons6,7,8,9. Beyond this effect, theory predicts that strong electron–vibron coupling strongly suppresses the current flow at low biases, a collective behaviour known as Franck–Condon blockade10,11. Here, we show measurements on quantum dots formed in suspended single-wall carbon nanotubes revealing a remarkably large electron–vibron coupling that, owing to the high quality and unprecedented tunability of our samples, allow a quantitative analysis of vibron-mediated electronic transport in the regime of strong electron–vibron coupling. This enables us to unambiguously demonstrate the Franck–Condon blockade in a suspended nanostructure. The large observed electron–vibron coupling could ultimately be a key ingredient for the detection of quantized mechanical motion12,13. It also emphasizes the unique potential for nanoelectromechanical device applications based on suspended graphene sheets and carbon nanotubes.

Quantum dots in carbon nanotubes

In this overview paper, we present low-temperature electronic transport measurements of carbon nanotube quantum dots with a back gate. In a semiconducting tube, charge carriers could be completely depleted. The addition energy and the excitation spectrum have been studied as a function of the number of charges (electrons or holes), one by one. We observe electron–hole symmetry, which is a direct consequence of the symmetric band structure of the nanotube. The excitation spectrum for metallic nanotubes exhibits four-fold shell filling and is completely described by an extended constant-interaction model. Furthermore, nanotubes with a four-fold shell structure are investigated in a parallel magnetic field. The magnetic field induces a large splitting between the two orbital states of each shell, demonstrating their opposite magnetic moment and determining transitions in the spin and orbital configuration of the quantum dot ground state. Also, a small coupling is found between orbitals with opposite magnetic moments leading to anti-crossing behaviour at zero field. Current–voltage characteristics of suspended carbon nanotube quantum dots show an additional series of steps equally spaced in voltage. The energy scale of this harmonic, low-energy excitation spectrum is consistent with that of the longitudinal low-k phonon mode (stretching mode) in the nanotube. Finally, we report on a fully tunable carbon nanotube double quantum dot. We perform inelastic transport spectroscopy via the excited states in the double quantum dot, a necessary step towards the implementation of new microwave-based experiments for quantum information technology.

Tunneling in Suspended Carbon Nanotubes Assisted by Longitudinal Phonons

Current-voltage characteristics of suspended single-wall carbon nanotube quantum dots show a series of steps equally spaced in voltage. The energy scale of this harmonic, low-energy excitation spectrum is consistent with that of the longitudinal low-k phonon mode (stretching mode) in the nanotube. Agreement is found with a Franck-Condon-based model in which the phonon-assisted tunneling process is modeled as a coupling of electronic levels to underdamped quantum harmonic oscillators. A comparison with this model indicates a rather strong electron-phonon coupling factor of order unity.

Coupling between electronic transport and longitudinal phonons in suspended nanotubes

Current–voltage characteristics of suspended single-wall carbon nanotube (NT) quantum dots show a series of steps equally spaced in voltage. The energy scale of this harmonic, low-energy excitation spectrum is consistent with that of the longitudinal low-k phonon mode in the NT. Agreement is found with a Franck–Condon-based model in which the phonon-assisted tunnelling process is modelled as a coupling of electronic levels to underdamped quantum harmonic oscillators. Comparison with this model indicates a rather strong electron–phonon coupling factor of order unity. We investigate different electron–phonon coupling mechanisms and give estimates of the coupling factor.