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, . . . 50 can be filled successively into the quantum system with charging energies exceeding 10 meV. For the lowest quantum states, we can clearly observe valley and Zeeman splittings with a spin g-factor of gs ≈ 2. In the low field-limit, the valley splitting depends linearly on the perpendicular magnetic field and is in qualitative agreement with calculations.
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 nano tubes and GaAs-based quantum dots. The spin-singlet excited states are found to be valley-triplet states by tilting the magnetic field with respect to the sample plane. We quantify the exchange energy to be 0.35 meV and measure a valley and spin g-factor of 36 and 2, respectively.
Excellent fits of the tunneling density of states in disordered superconductors can be often achieved making use of the phenomenological Dynes formula. However, no consistent derivation of this formula has been available so far. The Dynes formula can be interpreted by the simplest causal frequency-dependent gap function ∆(ω) with a vanishing gap at the Fermi level. Here we show, within the coherent potential approximation, that precisely such gap function describes superconductors with a Lorentzian distribution of pair-breaking fields and arbitrary potential disorder. We predict spectral and thermodynamic properties of such superconductors.
Recently a generic model has been proposed for the single-particle properties of gapless superconductors with simultaneously present pair-conserving and pair-breaking impurity scattering (the so-called Dynes superconductors). Here we calculate the optical conductivity of the Dynes superconductors. Our approach is applicable for all disorder strengths from the clean up to the dirty limit and for all relative ratios of the two types of scattering, nevertheless the complexity of our description is equivalent to that of the widely used Mattis-Bardeen theory. We identify two optical fingerprints of the Dynes superconductors: (i) the presence of two absorption edges and (ii) a finite absorption at vanishing frequencies even at the lowest temperatures. We demonstrate that the recent anomalous optical data on thin MoN films can be reasonably fitted by our theory.
The tunneling density of states in dirty superconductors is often well described by the phenomenological Dynes formula. Recently we have shown that this formula can be derived, within the coherent potential approximation, for superconductors with simultaneously present pair-conserving and pair-breaking impurity scattering. Here we demonstrate that the theory of such so-called Dynes superconductors is thermodynamically consistent. We calculate the specific heat and critical field of the Dynes superconductors, and we show that their gap parameter, specific heat, critical field, and penetration depth exhibit power-law scaling with temperature in the low-temperature limit. We also show that, in the vicinity of a coupling constant-controlled superconductor to normal metal transition, the Homes law is replaced by a different, pair-breaking dominated scaling law.
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