Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot

Tunneling spectroscopy of a Nb coupled carbon nanotube quantum dot reveals the formation of pairs of Andreev bound states (ABS) within the superconducting gap. A weak replica of the lower ABS is found, which is generated by quasiparticle tunneling from the ABS to the Al tunnel probe. An inversion of the ABS dispersion is observed at elevated temperatures, which signals the thermal occupation of the upper ABS. Our experimental findings are well supported by model calculations based on the superconducting Anderson model.

Figure 1

(a) Atomic force micrograph of a typical device. The niobium fork is strongly, the tunnel probe weakly coupled to the nanotube. (b) Manifestation of ABS at energies $e{V}_{\mathrm{sd}}=\pm ({\varepsilon }_{\mathrm{abs}}+{\Delta }_{\mathrm{Al}})$: a peak in the differential conductance is observed when unoccupied (occupied) states of the tunnel probe are aligned to the energy ($\mp {\varepsilon }_{\mathrm{abs}}$) of the lower (upper) state of an ABS pair.

Figure 2

Overview measurements of the differential conductance $dI/d{V}_{\mathrm{sd}}(V_{\mathrm{g}},V_{\mathrm{sd}})$ as function of back gate voltage $V_{\mathrm{g}}$ and bias voltage $V_{\mathrm{sd}}$, for (a) $B=0\mathrm{T}$ and (b) $B=2\mathrm{T}$. While the superconducting energy gap in the contacts is clearly visible in (a), in (b) superconductivity is suppressed and normal-state Coulomb blockade behavior emerges.

Figure 3

(a), (b) Differential conductance plotted as a function of bias voltage $V_{\mathrm{sd}}$ and back gate voltage $V_{\mathrm{g}}$ for two different gate regimes, (a) $V_{\mathrm{g}}\simeq 4.65\mathrm{V}$ and (b) $V_{\mathrm{g}}\simeq 5.44\mathrm{V}$. The sharp resonances are the signature of the Andreev bound states: a main resonance of high conductance (☆), a weak conductance resonance ($◯$) running parallel to the main resonance peak (☆), and a third additional resonance close to the gap edge ($\diamond$). (c), (d) ABS spectrum calculated using NRG for a two-channel superconducting Anderson model with parameters $E_c\equiv U_1=U_2=8\Delta ,{\Gamma }_1=1.85\Delta ,{\Gamma }_2=4\Delta$ and $E_c=9\Delta ,{\Gamma }_1=1.2\Delta ,{\Gamma }_2=4\Delta$, respectively. Inset: Schematic representation of the theoretical model used to describe the main multilevel features; see text. Two independent transport channels connect the quantum dot with the superconducting reservoir.

Figure 4

(a)–(d) Two-dimensional differential conductance maps of the tunnel probe as function of tunnel probe bias $V_{\mathrm{sd}}$ and back gate voltage $V_{\mathrm{g}}$ taken at indicated temperatures. The two subgap resonance lines are the main ABS (☆) resonance (higher conductance) and a secondary weaker conductance ($◯,\square$) resonance running parallel to the main resonance at $T\lesssim 200$ mK. Panels (e)–(h) are model calculations that include the effect of populating the upper ABS according to the temperatures in (a)–(d).

Figure 5

(a) A replica of the lower ABS ($◯$) at $e{V}_{\mathrm{sd}}=\pm {\varepsilon }_{\mathrm{abs}}$ is observed when the probe DOS is finite at the Fermi energy $E_{F\left(\mathrm{probe}\right)}$ and $E_{F\left(\mathrm{probe}\right)}$ is aligned with the ABS. (b) At higher temperatures also the upper ABS ($\square$) at $+{\varepsilon }_{\mathrm{abs}}$ is thermally populated. In this case a peak at $e{V}_{\mathrm{sd}}=\pm ({\Delta }_{\mathrm{Al}}-{\varepsilon }_{\mathrm{abs}}$) is observed in the differential conductance, where an unoccupied (occupied) DOS of the tunnel probe is aligned to a thermally populated upper (lower) ABS.

Figure 6

Left: Bias positions of the main ABS and its satellite peaks, plotted as a function of gate voltage $V_{\mathrm{g}}$ for (a) $T=30\mathrm{mK}$ and (b) $T=800\mathrm{mK}$. Right: Plots combining the data of the left column graphs to demonstrate that the gate dependence of the peak positions can in each case be reduced to a single dispersion relation ${\varepsilon }_{\mathrm{abs}}\left(V_{\mathrm{g}}\right)$; see the main text for details.

Figure 7

(a) Gate dependence of the differential conductance, extending Fig. 4 and displaying a multiloop pattern. (b) ABS spectrum calculated by NRG assuming a single channel in the leads and using the parameters $U_1=U_2=4.5\Delta ,{\Gamma }_1=0.5\Delta ,{\Gamma }_2=0.2\Delta ,{\varepsilon }_2-{\varepsilon }_1=6.5\Delta$. Inset: Schematic of the NRG model; see text.