Probing the intrinsic state of a one-dimensional quantum well with photon-assisted tunneling

The photon-assisted tunneling (PAT) through a single wall carbon nanotube quantum well (QW) is suggested for probing the Tomonaga-Luttinger liquid (TLL) state. The elementary TLL excitations inside the quantum well are density $\left({\rho }_{\pm }\right)$ and spin $\left({\sigma }_{\pm }\right)$ bosons. The bosons populate the quantized energy levels ${\epsilon }_n^{\rho +}=\Delta n/g$ and ${\epsilon }_n^{\rho -\left(\sigma \pm \right)}=\Delta n$ where $\Delta =h{v}_F/L$ is the interlevel spacing, $n$ is an integer number, $L$ is the tube length, and $g$ is the TLL parameter. Since the external electromagnetic field acts on the ${\rho }_{+}$ bosons only whereas the neutral ${\rho }_{-}$ and ${\sigma }_{\pm }$ bosons remain unaffected, the PAT spectroscopy is able of identifying the ${\rho }_{+}$ levels in the QW setup. The spin ${\epsilon }_n^{\sigma +}$ boson levels in the same QW are recognized from Zeeman splitting when applying a dc magnetic field $H\ne 0$. Basic TLL parameters are readily extracted from the differential conductivity curves.

Figure 1

(a) A quantum well composed of the 1D section $\mathcal{T}$ with attached emitter $\left(\mathcal{E}\right)$ and collector $\left(\mathcal{C}\right)$ electrodes. The potential barriers are shown in black at the $\mathcal{E}/\mathcal{T}$ and $\mathcal{T}/\mathcal{C}$ interfaces. (b) Energy diagram of the PAT process in the QW. (c) The split gate configuration of the QW. The right side inset shows how the electric field is applied to the $\mathcal{T}$ section.

Figure 2

(a) The single-electron density of states inside the $\mathcal{T}$ section of an QW for the Luttinger parameter $g=1$ (curve $A$), 0.4 (curve $B$), and $g=0.2$ (curve $C$). (b) Contour $E\text{−}g$ plot of the single-electron density of states (DOS) inside the $\mathcal{T}$ section.

Figure 3

(a) Splitting of the zero-bias TLL dip in the differential tunneling conductivity $\sigma \left(V_{e,c}\right)$ [in units of $e^2{v}_{F}N\left(0\right)$] in a long carbon nanotube CNT junction due to the photon-assisted tunneling. Spacing between the zero dip and adjacent satellite dips is $\hslash \Omega$. (b) The Coulomb staircase in the PASET current-voltage characteristics $I\left(V_{\text{dc}}\right)$ versus reduced voltage $V_{\text{dc}}$ (see text) of the CNT junction in the TLL state with $g=0.4$ under influence of the ac bias field with the amplitude $e{V}^{\left(1\right)}=3.4$ (in units of $\delta =e^2/C$) and for a symmetric junction $\left(\eta =0.5\right)$. Curve $A$ corresponds to $\Omega =4.7$, curve $B$ to $\Omega =3.7$, and curve $C$ to $\Omega =6.7$.

Figure 4

(a) The steady-state tunneling differential conductivity $\sigma \left(V_{e,c}\right)$ [in units of $e^2{v}_{F}N\left(0\right)$] in the TLL state. The peak at $V_{e,c}=4.3$ (in units of $\Delta /e$) corresponds to the ${\rho }_{+}$ boson. (b) The contour plot $\sigma \left(V_{e,c},V_{e,c}^{\left(1\right)}\right)$ [$V_{e,c}$ and the ac bias amplitude $V_{e,c}^{\left(1\right)}$ being in units of $\Delta /e$]. (c) The Fano factor $F\left(E\right)$ of the QW in conditions of PAT. (d) $\sigma \left(V_{e,c}\right)$ for different $\Omega$.

Figure 5

Splitting of quantized levels due to the photon assisted tunneling and Zeeman effect as pronounced in the single-electron density of states $N\left(\epsilon \right)$ of a short CNT junction. (a) A free-electron quantized level (solid curve) for which the charge ${\rho }_{+}$ and spin ${\sigma }_{+}$ bosons coincide splits by an ac electromagnetic field (with spacing $\propto \hslash \Omega$) and by a dc magnetic field (with spacing $\propto {\mu }_{B}H$) simultaneously. (b) The quantized levels of spin ${\sigma }_{+}$ and charge ${\rho }_{+}$ bosons have different energies ${\epsilon }_n^{\rho +}\ne {\epsilon }_n^{\sigma +}$ in the TLL state. The ac bias splits the charge boson levels (dashed curve on the right) while the dc magnetic field splits the spin boson levels (dotted curve on the left) only. The charge boson localized energy level ${\epsilon }_n^{\rho +}$ splits in the two satellite peaks with spacing $2\hslash \Omega$. Although the ac field has no influence on the neutral spin bosons, the spin level ${\epsilon }_n^{\sigma +}$ splits (Ref. 12) in two sublevels spaced with ${\Delta }_Z\propto {\mu }_{B}H$ (both in units of $\Delta$) due to the Zeeman effect when a dc magnetic field $H\ne 0$ is applied.