Tomonaga-Luttinger liquid correlations and Fabry-Perot interference in conductance and finite-frequency shot noise in a single-walled carbon nanotube

We present a detailed theoretical investigation of transport through a single-walled carbon nanotube (SWNT) in good contact to metal leads where weak backscattering at the interfaces between SWNT and source and drain reservoirs gives rise to electronic Fabry-Perot (FP) oscillations in conductance and shot noise. We include the electron-electron interaction and the finite length of the SWNT within the inhomogeneous Tomonaga-Luttinger liquid (TLL) model and treat the nonequilibrium effects due to an applied bias voltage within the Keldysh approach. In low-frequency transport properties, the TLL effect is apparent mainly via power-law characteristics as a function of bias voltage or temperature at energy scales above the finite level spacing of the SWNT. The FP frequency is dominated by the noninteracting spin-mode velocity due to two degenerate subbands rather than the interacting charge velocity. At higher frequencies, the excess noise is shown to be capable of resolving the splintering of the transported electrons arising from the mismatch of the TLL parameter at the interface between metal reservoirs and SWNT’s. This dynamics leads to a periodic shot-noise suppression as a function of frequency and with a period that is determined solely by the charge velocity. At large bias voltages, these oscillations are dominant over the ordinary FP oscillations caused by two weak backscatterers. This makes shot noise an invaluable tool to distinguish the two mode velocities in the SWNT.

Figure 1

The Fabry-Perot double barrier device: The backscattering with bare amplitude strengths $u_1$ and $u_2$ of electrons primarily takes part at the SWNT–metal-reservoir interfaces where the TLL parameter $g$ changes from $g=1$ in the leads to $g<1$ in the nanotube.

Figure 2

The Keldysh contour: Operators are ordered along the contour with operators evaluated at later times acting on the left of operators evaluated at earlier times. Times on the (+) branch are always earlier than times on the (−) branch.

Figure 3

(Color online) Conductance plots: in (a) we show the conductance as a function of bias voltage $v$ and temperature $\Xi$ both in units of the noninteracting level spacing $\hslash ∕t_{\mathrm{L}}$ for the strongly correlated case $g=0.23$. The backscattering coefficients are taken at zero temperature as $U^{\text{in}}=0.09$, $U^{\mathrm{co}}=0.08$. In (b) we compare the conductance at zero temperature for different interaction strengths: $g=0.23$ (solid line), $g=1$ (dashed line). (c) is devoted to the study of the gate voltage dependence for $g=0.23$: we have chosen $U^{\text{in}}=0.09$, $U^{\mathrm{co}}=0.08$ (solid line), $U^{\text{in}}=0.09$, $U^{\mathrm{co}}=-0.08$ (long-dashed line), $U^{\text{in}}=0.09$, $U^{\mathrm{co}}=0$ (short-dashed line).

Figure 4

Comparison of backscattered current $I_B$ in units of $G_0\hslash ∕t_{\mathrm{L}}e$ at zero temperature (black) with the approximate formula, Eq. (24) (dashed line), showing the power law $I_B\propto V^{1-\gamma ∕2}$. In both cases we use $U^{\text{in}}=U^{\mathrm{co}}=0.1$ and $g=0.23$.

Figure 5

Backscattered current $I_B$ in units of $G_0\hslash ∕t_{\mathrm{L}}e$ at $v\sim 0.137$ as a function of dimensionless temperature $\Xi$. The solid line is obtained from numerical integration of Eq. (14), whereas the dashed line is the approximated incoherent contribution of $I_B$ given in Eq. (23). As expected, they agree for temperatures larger than the noninteracting level spacing—i.e., $\Xi >1$. For both curves we use $U^{\text{in}}=U^{\mathrm{co}}=0.1$ at zero temperature and $g=0.23$.

Figure 6

Leading oscillating contributions to the backscattered current $I_B$ in units of $G_0\hslash ∕t_{\mathrm{L}}e$ at zero temperature, $U^{\text{in}}=U^{\mathrm{co}}=0.1$ and $g=0.23$, as a function of bias voltage. The light gray and dashed curves are the FP oscillations given in Eq. (26) with oscillation frequencies $v_{\mathrm{F}}∕L$ and $v_c∕L$, respectively. The dark gray curve is the leading oscillation contribution of $I_B^{\text{in}}$ with frequency $v_c∕2L$ presented in Eq. (25).

Figure 7

The Fano factor $F$ defined in Eq. (38) for different temperatures and backscattering coefficients. In (a) and (b) we show the strongly correlated case with $g=0.23$ whereas in (c) and (d) we present the noninteracting case $g=1$. In (a) and (c) we use $U^{\text{in}}=0.09$ and $U^{\mathrm{co}}=0.08$, and in (b) and (d) we use $U^{\text{in}}=0.09$ and $U^{\mathrm{co}}=-0.08$ at the lowest temperature. The temperatures in units of the noninteracting level spacing $\hslash v_{\mathrm{F}}∕L$ are $\Xi =0$ (black line), $\Xi =0.3$ (dashed line), and $\Xi =0.7$ (light-gray line).

Figure 8

(Color online) Excess noise $S_e=S(V,\omega )-S(0,\omega )$ in units of $G_0\hslash ∕t_{\mathrm{L}}$ at temperature $\Xi =0.3$ for interaction strength $g=0.23$ using numerical integration of Eq. (32). The backscattering coefficients are $U_1^{\text{in}}=U_2^{\text{in}}=0.06$ and $U^{\mathrm{co}}=0.1$. We have chosen the measurement point to be at one of the barriers. (a) shows the excess noise as a function of bias voltage $eV$ and frequency $\hslash \omega$ (both in units of the noninteracting level spacing $\hslash v_{\mathrm{F}}∕L$). In plot (b) we present the low-frequency noise $\omega =0$ as a function of bias voltage. Clear FP oscillations dominated by the noninteracting frequency $v_{\mathrm{F}}∕L$ are seen as well as the power-law scaling with power $1-\gamma ∕2$. In plot (c) we show the frequency dependence of excess noise at large bias voltage $v\sim 34$. Dominant charge-mode oscillations with frequency $v_{\mathrm{F}}∕2Lg$ are observed.

Figure 9

(Color online) Same parameters as in Fig. 8 but for a noninteracting system with $g=1$. In graph (b) approximately the same periodicity of FP oscillations is found as in the interacting case but the frequency dependence at large bias voltage presented in graph (c) is clearly different. Only much smaller ordinary FP oscillations with frequency $v_{\mathrm{F}}∕L$ are seen. Superimposed are the additional oscillations depending on the measurement point. For the chosen point of measurement $\mid x\mid =L∕2$ this oscillation frequency is $v_{\mathrm{F}}∕2L$.

Figure 10

The excess noise $S_e$ in the same regime as in Figs. 8c, 9c but for different measurement points in the leads $\mid x\mid >L∕2$. In graph (a) we present the strongly interacting case $g=0.23$ and in graph (b) the noninteracting case $g=1$. We show curves for three different measurement points: $d=0.14$ (dark lines), $d=0.3$ (light gray lines), and $d=0.6$ (dashed lines) where $d$ is the distance from $x$ to the nearest barrier in units of SWNT length $L$.