Observation and Spectroscopy of a Two-Electron Wigner Molecule in an Ultra-Clean Carbon Nanotube

Coulomb interactions can have a decisive effect on the ground state of electronic systems. The simplest system in which interactions can play an interesting role is that of two electrons on a string. In the presence of strong interactions the two electrons are predicted to form a Wigner molecule, separating to the ends of the string due to their mutual repulsion. This spatial structure is believed to be clearly imprinted on the energy spectrum, yet to date a direct measurement of such a spectrum in a controllable one-dimensional setting is still missing. Here we use an ultra-clean suspended carbon nanotube to realize this system in a tunable potential. Using tunneling spectroscopy we measure the excitation spectra of two interacting carriers, electrons or holes, and identify seven low-energy states characterized by their spin and isospin quantum numbers. These states fall into two multiplets according to their exchange symmetries. The formation of a strongly-interacting Wigner molecule is evident from the small energy splitting measured between the two multiplets, that is quenched by an order of magnitude compared to the non-interacting value. Our ability to tune the two-electron state in space and to study it for both electrons and holes provides an unambiguous demonstration of the fundamental Wigner molecule state.Comment: SP and FK contributed equally to this wor

Coupling of spin and orbital motion of electrons in carbon nanotubes

Electrons in atoms possess both spin and orbital degrees of freedom. In non-relativistic quantum mechanics, these are independent, resulting in large degeneracies in atomic spectra. However, relativistic effects couple the spin and orbital motion leading to the well-known fine structure in their spectra. The electronic states in defect-free carbon nanotubes (NTs) are widely believed to be four-fold degenerate, due to independent spin and orbital symmetries, and to also possess electron-hole symmetry. Here we report measurements demonstrating that in clean NTs the spin and orbital motion of electrons are coupled, thereby breaking all of these symmetries. This spin-orbit coupling is directly observed as a splitting of the four-fold degeneracy of a single electron in ultra-clean quantum dots. The coupling favours parallel alignment of the orbital and spin magnetic moments for electrons and anti-parallel alignment for holes. Our measurements are consistent with recent theories that predict the existence of spin-orbit coupling in curved graphene and describe it as a spin-dependent topological phase in NTs. Our findings have important implications for spin-based applications in carbon-based systems, entailing new design principles for the realization of qubits in NTs and providing a mechanism for all-electrical control of spins in NTs.Comment: 14 pages, 6 figure

Electronic Transport Spectroscopy of Carbon Nanotubes in a Magnetic Field

We report magnetic field spectroscopy measurements in carbon nanotube quantum dots exhibiting four-fold shell structure in the energy level spectrum. 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. We use inelastic cotunneling spectroscopy to accurately resolve the spin and orbital contributions to the magnetic moment. A small coupling is found between orbitals with opposite magnetic moment leading to anticrossing behavior at zero field.Comment: 7 pages, 4 figure

Dynamic nuclear polarization at the edge of a two-dimensional electron gas

We have used gated GaAs/AlGaAs heterostructures to explore nonlinear transport between spin-resolved Landau level (LL) edge states over a submicron region of two-dimensional electron gas (2DEG). The current I flowing from one edge state to the other as a function of the voltage V between them shows diode-like behavior---a rapid increase in I above a well-defined threshold V_t under forward bias, and a slower increase in I under reverse bias. In these measurements, a pronounced influence of a current-induced nuclear spin polarization on the spin splitting is observed, and supported by a series of NMR experiments. We conclude that the hyperfine interaction plays an important role in determining the electronic properties at the edge of a 2DEG.Comment: 8 pages RevTeX, 7 figures (GIF); submitted to Phys. Rev.

Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes

We have studied the high-bias transport properties of the different shells that constitute a multiwalled carbon nanotube. The current is shown to be reduced as the shell diameter is decreased or the length is increased. We assign this geometrical dependence to the competition between electron-phonon scattering process and Zener tunneling.Comment: 4 pages, 4 figure

Scanned Probe Microscopy of Electronic Transport in Carbon Nanotubes

We use electrostatic force microscopy and scanned gate microscopy to probe the conducting properties of carbon nanotubes at room temperature. Multi-walled carbon nanotubes are shown to be diffusive conductors, while metallic single-walled carbon nanotubes are ballistic conductors over micron lengths. Semiconducting single-walled carbon nanotubes are shown to have a series of large barriers to conduction along their length. These measurements are also used to probe the contact resistance and locate breaks in carbon nanotube circuits.Comment: 4 page

Multi-shell gold nanowires under compression

Deformation properties of multi-wall gold nanowires under compressive loading are studied. Nanowires are simulated using a realistic many-body potential. Simulations start from cylindrical fcc(111) structures at T=0 K. After annealing cycles axial compression is applied on multi-shell nanowires for a number of radii and lengths at T=300 K. Several types of deformation are found, such as large buckling distortions and progressive crushing. Compressed nanowires are found to recover their initial lengths and radii even after severe structural deformations. However, in contrast to carbon nanotubes irreversible local atomic rearrangements occur even under small compressions.Comment: 1 gif figure, 5 ps figure

Conductance oscillations in tunnel-coupled quantum dots in the quantum Hall regime

We present measurements of transport through two tunnel-coupled quantum dots of different sizes connected in series in a strong, variable, perpendicular magnetic field. Double dot conductance was measured both as a function of magnetic field, which was varied across the filling factor nu = 4 quantum Hall plateau, and as a function of charge induced evenly on the two dots. The conductance peaks undergo position shifts and height modulations as the magnetic field is varied. These shifts and modulations form a pattern that repeats over large ranges of magnetic field and with the addition of double dot charge. The robust pattern repetition is consistent with a frequency locking effect.Comment: 12 pages, 4 figure

Correlation effects in a quantum dot at high magnetic fields

We investigate the effects of electron correlations on the ground state energy and the chemical potential of a droplet confined by a parabolic potential at high magnetic fields. We demonstrate the importance of correlations in estimating the transition field at which the first edge reconstruction of the maximum density droplet occurs in the spin polarized regime.Comment: 11 pages (revtex) 3 postscript figures are included at the end of the tex file. To appear in Phys. Rev.

Interference effects in electronic transport through metallic single-wall carbon nanotubes

In a recent paper Liang {\it et al.} [Nature {\bf 411}, 665 (2001)] showed experimentally, that metallic nanotubes, strongly coupled to external electrodes, may act as coherent molecular waveguides for electronic transport. The experimental results were supported by theoretical analysis based on the scattering matrix approach. In this paper we analyze theoretically this problem using a real-space approach, which makes it possible to control quality of interface contacts. Electronic structure of the nanotube is taken into account within the tight-binding model. External electrodes and the central part (sample) are assumed to be made of carbon nanotubes, while the contacts between electrodes and the sample are modeled by appropriate on-site (diagonal) and hopping (off-diagonal) parameters. Conductance is calculated by the Green function technique combined with the Landauer formalism. In the plots displaying conductance {\it vs.} bias and gate voltages, we have found typical diamond structure patterns, similar to those observed experimentally. In certain cases, however, we have found new features in the patterns, like a double-diamond sub-structure.Comment: 15 pages, 4 figures. To apear in Phys. Rev.