Vernier Spectrum and Valley Polarization Control in Carbon Nanotube Quantum Dots

Dirac cones can tilt or warp due to spin-orbit or lattice interaction resulting in asymmetric velocities for left- and right-moving electrons. In carbon nanotubes, this effect has manifested in a secondary quantum interference on top of the Fabry-Perot interference [1,2]. In finite-sized systems such as quantum dots, this effect is predicted [3] to result in unequally spaced energy levels for left/right movers or K/K’ valley electrons leading to accidental degeneracies and a Vernier spectrum. In this work, we present the Vernier-like spectrum seen in Coulomb blockade measurements of ultraclean suspended carbon nanotubes. Four-fold shell filling on a slowly oscillating background conductance versus gate voltage is seen to be interrupted periodically to add two additional electrons. We present a theoretical model to describe such behavior in the conductance of finite-sized nanotubes that lifts the four-fold degeneracy and gives rise to the Vernier-like spectrum. The model also shows that valley-polarized energy levels are available at specific gate voltages, and the degree of polarization can be tuned by gate voltage. Our work showcases the continued surprises revealed by quantum transport in ultraclean carbon nanotubes.

[1] Dirnaichner et al, Phys. Rev. Lett. 117, 166804 (2016)

[2] Lotfizadeh et al, Phys. Rev. Lett. 126, 216802 (2021)

[3] Izumida et al, Phys. Rev. B 85, 165430 (2012)