Band-gap renormalization in carbon nanotubes: Origin of the ideal diode behavior in carbon nanotube $p\text{−}n$ structures

We demonstrate dramatic modifications to the electronic structure in single-walled carbon nanotubes due to band gap renormalization, the many-body induced shrinkage of the fundamental band gap. This is examined within the framework of ideal $p\text{−}n$ diodes formed along individual single-walled carbon nanotubes. A combination of photocurrent spectroscopy with detailed transport measurements provides a complete set of energy levels of the nanotube $p\text{−}n$ structure. These energy levels confirm the large band gap shrinkage, consistent with enhanced many-body correction in one-dimensional confinement, and result in fundamental changes to the nanotubes diode transport properties as compared to their bulk counterparts. We show that the ideal diode behavior is a direct consequence of significant renormalization of the band gap at the doped $p$ and $n$ regions, resulting in formations of heterointerfaces, in stark contrast to a uniform band gap expected of a homogeneous material.

Figure 1

(Color online) The inset shows a schematic diagram of the split gate diode device with a portion of SWNT suspended. Data are $I\text{−}V$ curves of a SWNT diode at three different split gate bias (VG1,2) of $\pm 6$, 8, and $10\mathrm{V}$, and measured at $300\mathrm{K}$. The solid lines are fits to Eq. (1) with $n=1.04$ for all three $I\text{−}V$ curves with $I_0$ of 10.5, 37.0, and $84.0\mathrm{fA}$.

Figure 2

(Color online) (a) The band diagram of our diode showing the doped regions (●-electron, 엯-hole) from the split gates, and the formation of heterointerfaces between the doped and the suspended portion (undoped) $L$ of a SWNT. Each doped region contributes $I_0∕2$ in leakage current due to diffusion of minority carriers $n_p$ and $p_n$. In region $L$, the optical excitation at $E_{11}$ creates the lowest excitons with a binding energy $E_B$. Band bending at S/D interface is due to Schottky contact formation with the metal electrodes. (b) Normalized photocurrent spectra measured at 300C showing the first three peaks to emphasize the onset of continuum (mark by arrows). Each spectra is offset by $\frac{1}{2}$ unit in intensity. The horizontal dotted lines help to highlight the onset of the continuum as shown by the arrows. (c) Comparison between $E_{11}$ and $E_{\mathrm{gap}}$ vs $E_a$ where $E_a$ is the activation energy of the doped regions. (d) Measured $I_0$ vs $E_a$. The solid curve is from model for minority carrier diffusion [Eq. (6)].

Figure 3

(Color online) (a) Effects of BGR on $I_0$ with increased doping from the split gates. The observed increase in $I_0$ with doping is consistent with large BGR as shown in the inset for the $n$ doped subband and predicted by the solid curve, and is contrary to a large decrease expected in the absence of BGR (dotted curve). (b) The diode $I\text{−}V$ curves measured at $T=300$ and $330\mathrm{K}$ for split gate bias of $\pm 5\mathrm{V}$ along with their fits (solid curves) to Eq. (1) using $n=1.05$. The inset shows the measured $E_a$ vs the split gate bias voltage. The decrease in $E_a$ with doping is also consistent with a large BGR with increased doping. The carrier density in (a) corresponds to the gate doping density in (b) for a measured threshold voltage of $4\mathrm{V}$. The calculated band gaps are 0.59, 0.47, and $0.39\mathrm{eV}$ for the three doping densities.