Band-Gap-Dependent Electronic Compressibility of Carbon Nanotubes in the Wigner Crystal Regime

Electronic compressibility, the second derivative of ground-state energy with respect to total electron number, is a measurable quantity that reveals the interaction strength of a system and can be used to characterize the orderly crystalline lattice of electrons known as the Wigner crystal. Here, we measure the electronic compressibility of individual suspended ultraclean carbon nanotubes in the low-density Wigner crystal regime. Using low-temperature quantum transport measurements, we determine the compressibility as a function of carrier number in nanotubes with varying band gaps. We observe two qualitatively different trends in compressibility versus carrier number, both of which can be explained using a theoretical model of a Wigner crystal that accounts for both the band gap and the confining potential experienced by charge carriers. We extract the interaction strength as a function of carrier number for individual nanotubes and show that the compressibility can be used to distinguish between strongly and weakly interacting regimes.

Figure 1

(a) Schematic diagram (left) and SEM image of a device (right). Carbon nanotube is suspended over a $2\mu \mathrm{m}$ wide trench. The vertical spacing between trench and contact electrodes is $d=760\mathrm{nm}$. (b) Color scale plot of differential conductance versus gate voltage $V_g$ and source-drain bias $V_{\mathrm{SD}}$ in CN1. Conductance $G$ versus carrier number in (c) CN1 and (d) CN2. (e),(f) Inverse compressibility as a function of carrier number for the related device.

Figure 2

Effect of magnetic field on ${\kappa }^{-1}$ of CN1 for a range of magnetic fields, from $B=0.4\mathrm{T}$, where the band gap reaches its minimum, to 4 T. (Inset) Comparison of theoretical (solid line) and experimental (dots) results of ${\kappa }^{-1}$ as a function of carrier number for CN1.

Figure 3

Experimental data (a) and theoretical results (b) of ${\kappa }^{-1}$ as a function of carrier number ($N$) for nanotubes with different band gaps. (c) Experimental (left) and theoretical (right) results converted to ${\kappa }^{-1}$ times $L_{\mathrm{eff}}$ as a function of charge density.