Secondary Electron Interference from Trigonal Warping in Clean Carbon Nanotubes

We investigate Fabry-Perot interference in an ultraclean carbon nanotube resonator. The conductance shows a clear superstructure superimposed onto conventional Fabry-Perot oscillations. A sliding average over the fast oscillations reveals a characteristic slow modulation of the conductance as a function of the gate voltage. We identify the origin of this secondary interference in intervalley and intravalley backscattering processes which involve wave vectors of different magnitude, reflecting the trigonal warping of the Dirac cones. As a consequence, the analysis of the secondary interference pattern allows us to estimate the chiral angle of the carbon nanotube.

Figure 1

(a) Differential conductance $G(V_g,V_b)$ of a clean CNT device in the hole conduction regime, as a function of back gate voltage $V_g$ and bias voltage $V_b$ ($G_0=e^2/h$). (b) Zero bias conductance $G(V_g)$ extracted from (a). (c) Average conductance $\overline{G}(V_g)$ obtained over a sliding 0.4-V-wide Gaussian window. A slow modulation is observed. The peak positions are marked with filled circles. The distance $\mathrm{\Delta }{V}_{g,n}=V_{g,n}-V_{g,n+1}$ between the $n$th and the ($n-1$)th peak decreases with $n$. (d) Fourier transform of a sliding 0.4-V-wide window in the signal in (b) as a function of the gate voltage.

Figure 2

Graphene dispersion relation $ϵ(\mathbf{k})$ (contour plots in the top left panel of each subfigure) in the vicinity of a Dirac point and simplified lowest 1D subbands (line plots, top right) [25]. The solid red line in the contour plot marks the direction of $k_{\parallel }$. The chiral angle $\theta$ is measured with respect to the direction of the zigzag CNT (dashed line). The bottom panels show exemplary transmission patterns obtained by numerical tight-binding calculations. The green line represents the sliding average $\overline{T}$ of the transmission signal. (a) Zigzag: Dispersion relations at the two Dirac points $-K_{\perp }$ (green) and $K_{\perp }$ (red) are identical and symmetric with respect to the $k_{\parallel }=0$ axis. The transmission curve of a (12,0) CNT shows a simple, single-channel FP interference pattern. (b) Zigzaglike: Right- and left-moving branches within each valley exhibit different wave vectors $k_{j,r/l}$ at finite energy. No intervalley scattering is possible in (a) and (b); see the text. A single-channel-like transmission pattern can be observed for the (6,3) CNT (bottom left). (c) Armchair: Parity symmetry forbids scattering between $a$ and $b$ branches. At finite energy, the two Kramers channels $a$ and $b$ have different wave vectors associated with the right- and left-moving states and a beat in the interference pattern is observed in the tight-binding calculation for the (7,7) CNT. (d) Armchairlike: In the armchairlike CNTs, the parity symmetry is broken and interchannel scattering is enabled (see the text). A slow modulation of the transmission pattern can be observed in the average transmission $\overline{T}$ of a (10,4) CNT (bottom right).

Figure 3

Computed phase differences $\mathrm{\Delta }{\varphi }^{\theta }$ between modes as a function of energy measured from the Dirac point for different chiral angles $\theta$. The phase difference is a monotonically increasing function of $\theta$ starting from the zigzag CNT with $\theta =0$ (left inset) to the armchair CNT with $\theta =30°$ (right inset). The filled circles are obtained using the experimental positions $E_n+\mathrm{\Delta }{E}_{\text{gap}}$ of the slow modulation and requiring $\mathrm{\Delta }\varphi /2\pi =n$. The error bars indicate the uncertainty in $\alpha$ and in $\mathrm{\Delta }{E}_{\text{gap}}$ (see the text). Acceptable fits are obtained by chiral angles in the range $22°\le \theta <30°$ [11], as indicated by the gray shaded area.