K. Grove-Rasmussen

Quantum transport in carbon nanotubes

Carbon nanotubes are a versatile material in which many aspects of condensed matter physics come together. Recent discoveries have uncovered new phenomena that completely change our understanding of transport in these devices, especially the role of the spin and valley degrees of freedom. This review describes the modern understanding of transport through nanotube devices. Unlike in conventional semiconductors, electrons in nanotubes have two angular momentum quantum numbers, arising from spin and valley freedom. The interplay between the two is the focus of this review. The energy levels associated with each degree of freedom, and the spin-orbit coupling between them, are explained, together with their consequences for transport measurements through nanotube quantum dots. In double quantum dots, the combination of quantum numbers modifies the selection rules of Pauli blockade. This can be exploited to read out spin and valley qubits and to measure the decay of these states through coupling to nuclear spins and phonons. A second unique property of carbon nanotubes is that the combination of valley freedom and electron-electron interactions in one dimension strongly modifies their transport behavior. Interaction between electrons inside and outside a quantum dot is manifested in SU(4) Kondo behavior and level renormalization. Interaction within a dot leads to Wigner molecules and more complex correlated states. This review takes an experimental perspective informed by recent advances in theory. As well as the well-understood overall picture, open questions for the field are also clearly stated. These advances position nanotubes as a leading system for the study of spin and valley physics in one dimension where electronic disorder and hyperfine interaction can both be reduced to a low level.

Kondo-enhanced andreev tunneling in InAs nanowire quantum dots

We report measurements of the nonlinear conductance of InAs nanowire quantum dots coupled to superconducting leads. We observe a clear alternation between odd and even occupation of the dot, with subgap peaks at |V(sd)| = Delta/e markedly stronger (weaker) than the quasiparticle tunneling peaks at |V(sd)| = 2Delta/e for odd (even) occupation. We attribute the enhanced Delta peak to an interplay between Kondo correlations and Andreev tunneling in dots with an odd number of spins, and we substantiate this interpretation by a poor mans scaling analysis.

Singlet–triplet physics and shell filling in carbon nanotube double quantum dots

Carbon nanotube double quantum dots, whose shell-like electronic structure is reminiscent of that of a simple molecule, provide a useful system to study the interaction of just a few electrons at a time.

Fabry–Perot interference, Kondo effect and Coulomb blockade in carbon nanotubes

Abstract High quality single wall carbon nanotube quantum dots have been made showing both metallic and semiconducting behavior. Some of the devices are identified as small band gap semiconducting nanotubes with relatively high broad conductance oscillations for hole transport through the valence band and low conductance Coulomb blockade oscillations for electron transport through the conduction band. The transition between these regimes illustrates that transport evolves from being wave-like transmission known as Fabry–Perot interference to single particle-like tunneling of electrons or holes. In the intermediate regime four Coulomb blockade peaks appear in each Fabry–Perot resonance, which is interpreted as entering the SU(4) Kondo regime. A bias shift of opposite polarity for the Kondo resonances for one electron and one hole in a shell is in some cases observed.

Superconductivity-enhanced bias spectroscopy in carbon nanotube quantum dots

PACS numbers: 73.21.La, 73.23.Hk, 73.63.Fg, 74.50.r Superconducting electrodes provide a useful means of sharpening the spectroscopic features observed in tunneling experiments. In the superconducting phase, an otherwise nearly constant density of states DOS acquires a gap of width 2 centered at the Fermi level and characteristic sharp coherence peaks at the gap edges . These peaks transform a featureless metallic electrode into a high-resolution tunneling probe. This widely used investigative tool 1 was demonstrated already by Giaever’s seminal work 2 from 1960 and more recently used to obtain a high-resolution bias spectrum of the levels in a metallic Al nanoparticle. 3 Here we report low-temperature transport measurements in which this type of BCS focusing promotes an otherwise featureless elastic EL cotunneling conductance to sharp peaks at bias voltages Vsd= 2 /e, corresponding to the onset of quasiparticle QP cotunneling. In the same way, inelastic INEL cotunneling processes involving transitions between two subbands in the nanotube are revealed as sharp peaks rather than steps or cusps in the nonlinear conductance. This sharpening of cotunneling lines inside the Coulomb diamonds allows us to investigate more closely the tunneling-induced gate voltage dependence of the orbital splitting. 4 Finally, we discuss an unusual subgap structure observed in a particularly well-coupled device signaling the importance of both multiple Andreev reflections MARs and dynamically generated bound states in spinful dots. A number of experiments have already investigated interesting aspects of quantum dots with superconducting electrodes, such as supercurrent, 5–7 MAR, 8 and effects of size and charge quantization in the Fabry-Perot, 9–12 Kondo, 13–18

Magnetic-field dependence of tunnel couplings in carbon nanotube quantum dots.

By means of sequential and cotunneling spectroscopy, we study the tunnel couplings between metallic leads and individual levels in a carbon nanotube quantum dot. The levels are ordered in shells consisting of two doublets with strong- and weak-tunnel couplings, leading to gate-dependent level renormalization. By comparison to a one- and two-shell model, this is shown to be a consequence of disorder-induced valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel renormalization.

Single wall carbon nanotube double quantum dot

The authors report on two top-gate defined, coupled quantum dots in a semiconducting single wall carbon nanotube, constituting a tunable double quantum dot system. The single wall carbon nanotubes are contacted by titanium electrodes and gated by three narrow top-gate electrodes as well as a back gate. The authors show that a bias spectroscopy plot on just one of the two quantum dots can be used to extract the addition energy of both quantum dots. Furthermore, honeycomb charge stability diagrams are analyzed by an electrostatic capacitor model that includes cross capacitances, and they extract the coupling energy of the double quantum dot.

Gate-dependent tunneling-induced level shifts observed in carbon nanotube quantum dots

We have studied electron transport in clean single-walled carbon nanotube quantum dots. Because of the large number of the Coulomb blockade diamonds simultaneously showing both shell structure and the Kondo effect, we are able to perform a detailed analysis of tunneling renormalization effects. Thus, determining the environment induced level shifts of this artificial atom. In shells where only one of the two orbitals is strongly coupled, we observe a marked asymmetric gate dependence of the inelastic cotunneling lines together with a systematic gate dependence of the size (and shape) of the Coulomb diamonds. These effects are all given a simple explanation in terms of second-order perturbation theory in the tunnel coupling.

Critical and excess current through an open quantum dot: Temperature and magnetic-field dependence

We present measurements of temperature and magnetic-field dependence of the critical current and excess current in a carbon nanotube Josephson quantum dot junction. The junction is fabricated in a controlled environment which allows for extraction of the full critical current. The measurements are performed in the open quantum dot regime and fitted to theory with good qualitative agreement. We also show how to extract level spacing, level broadening, and charging energy of an open quantum dot from a bias spectroscopy plot.

Kondo resonance enhanced supercurrent in single wall carbon nanotube Josephson junctions

We have contacted single wall carbon nanotubes (SWCNTs) grown by chemical vapour deposition to superconducting Ti/Al/Ti electrodes. The device we here report on is in the Kondo regime exhibiting a fourfold shell structure, where a clear signature of the superconducting electrodes is observed below the critical temperature. Multiple Andreev reflections are revealed by sub-gap structure and a narrow peak in the differential conductance around zero bias is seen depending on the shell filling. We interpret the peak as a proximity induced supercurrent and examine its interplay with Kondo resonances.