0-$\pi$ quantum transition in a carbon nanotube Josephson junction: Universal phase dependence and orbital degeneracy

In a $\pi$-Josephson junction, the supercurrent's sign is reversed due to the dephasing of superconducting pairs upon their traversal of the nonsuperconducting part. 0-$\pi$ quantum transitions are extremely sensitive to electronic and magnetic correlations, providing powerful exploration tools of competing orders. In a quantum dot connected to superconducting reservoirs, the transition is governed by gate voltage. As shown recently, it can also be controlled by the superconducting phase in the case of strong competition between the superconducting proximity effect and Kondo correlations. We investigated here the current-phase relation in a clean carbon nanotube quantum dot, close to orbital degeneracy, in a regime of strong competition between local electronic correlations and superconducting proximity effect. We show that the nature of the transition depends crucially on the occupation and the width of the orbital levels, which determine their respective contribution to transport. When the transport of Cooper pairs takes place through only one of these levels, we find that the phase diagram of the phase-dependent 0-$\pi$ transition is a universal characteristic of a discontinuous level-crossing quantum transition at zero temperature. In the case where the two levels are involved, the nanotube Josephson current exhibits a continuous 0-$\pi$ transition, independent of the superconducting phase, revealing a different physical mechanism of the transition.

Figure 1

(a) Scanning electron microscopy image of the measured asymmetric SQUID, containing two reference JJs in parallel with a CNT-based Josephson junction (see text and [31]). To phase bias the CNT junction, a magnetic flux $\mathrm{\Phi }$ is applied with a magnetic field perpendicular to the SQUID. (b) Schematics of the layers constituting the tunnel junctions and the contacts of the CNT. The first layer of sample S-Al is made of Pd(7 nm)/Al(70). The first one of S-NbAl is made of Pd(7 nm)/Nb(20 nm)/Al(40 nm).

Figure 2

(Color plot) Differential conductance $dI/d{V}_{sd}$ as a function of bias voltage $V_{sd}$ and gate voltage $V_g$ for samples S-Al and S-NbAl in the normal state, applying $B=0.13\mathrm{T}$ for S-Al (diamond A) and $B=1\mathrm{T}$ for S-NbAl (diamonds B to L). In white is written the number of electrons in the last occupied shell. (Middle-row graphs) Horizontal cuts showing $dI/d{V}_{sd}$ at zero bias versus gate voltage $V_g$ in the normal state. (Bottom graphs) Supercurrent $\delta I_s\left(V_g\right)$ for a superconducting phase difference of $\phi =\pi /2$. Orange circles indicate Kondo-induced 0 junction for odd occupancies. Purple squares show $\pi$ junctions with a single-level behavior. The green arrows indicate two-level 0-$\pi$ junctions. The consequences of two-level physics are more spectacular for F and J than for H, where $\delta E$ is larger than in F and J (see Table 1).

Figure 3

Typical stability diagram expected in a clean CNT, of charging energy $U$, where the energy levels are separated by $\mathrm{\Delta }E$ and the orbital degeneracy is lifted by $\delta E$. The occupancy is indicated on the top of each Coulomb diamond. We call $\alpha$ the proportionality factor between the energy level $\epsilon$ and the applied gate voltage $V_g$. The spacing between the inelastic cotunneling peaks in the double occupancy $(N=2$) is called $2\delta E^{\prime }$. $\delta E^{\prime }\ne \delta E$ because of exchange interactions [37]. (Inset) Corresponding electronic configuration for $N=1$.

Figure 4

(a) Modulation of the switching current of the SQUID $\delta I_s$, proportional to the CPR, as a function of the magnetic field B and the gate voltage $V_g$, for diamonds A, B, C, G, and I. Vertical cuts at the 0-$\pi$ transition are represented for ridge I, showing the whole transition. For diamond G, the color plot exhibits a discontinuity, probably due to the trapping of a vortex. (b) CPR, for diamond I, near the transition (green continuous line) versus the superconducting phase $\phi$. The dashed lines are guides to the eyes and represent the contributions of the singlet (0 junction, in blue) and the doublet state ($\pi$ junction, in red).

Figure 5

Phase-dependent transition: definition of the relevant quantities (a) $T_K$ (red line), $\mathrm{\Delta }$ (red dotted line), and $\delta I_S(\phi =\pi /2)$ (blue line) for diamond I as a function of the energy level $\epsilon$, defined as equal to zero at the middle of the diamond and such that, in a diamond, $\epsilon \in [-U/2,U/2]$. One can notice that the sign of $\delta I_s$ changes when $T_K\approx \mathrm{\Delta }$. ${\epsilon }_t$ is the value of $\epsilon$ for which the junction switches from $\pi$ to 0 at $\phi =\pi /2$. (b) Definition of the critical phase ${\phi }_C$ such that the CPR has 0 behavior for $\phi \in [0,{\phi }_C]$ and $\pi$ behavior for $\phi \in [{\phi }_C,\pi ]$. (c) This quantity is plotted as a function of $\epsilon$ for diamond I, yielding a phase diagram of the $\phi$-controlled transition. We call $\delta \epsilon$ the width of the transition.

Figure 6

Universal scaling of the phase-induced transition. Critical phase ${\phi }_C$ plotted as a function of $(\epsilon -{\epsilon }_t)/\delta \epsilon$ for diamonds B, G, I, and J (left and right sides of the diamond) such that the various curves collapse on an arccosine curve (black line). (Left inset) Scaling quantity $\delta \epsilon$, normalized by $U/2$, as a function of the ratio $T_K/\mathrm{\Delta }$. It shows that $2\delta E/U$ varies linearly with $T_K/\mathrm{\Delta }$, the quantity that controls the 0-$\pi$ transition. (Right inset) Same quantity for diamond C, where the 0-$\pi$ transition is incomplete. To obtain an arccosine shape, one has to plot ${\phi }_C$ as a function of ${\epsilon }^2$ instead of $\epsilon$ (see text).

Figure 7

(a) Modulation of the switching current of the SQUID versus the magnetic field, proportional to the CPR, as a function of the gate voltage $V_g$, for diamond J. The supercurrent at the transition being very low, the color scale is saturated. (b) Supercurrent versus the superconducting phase $\phi$ at some particular gate voltage, indicated by the numbers on panel (a). Note the cancellation of CPR (4) and the fact that CPR (2), in the two-level 0 junction, has a stronger anharmonicity than (8), on the degeneracy point. Dashed line on (2): guide for the eye, representing a sinus, showing that the continuous line is not perfectly harmonic.

Figure 8

(a) Critical current $I_c$, defined as the maximum amplitude of the measured switching current, as a function of the energy level $\epsilon$. $I_c$ is defined as positive for a 0 junction and negative for a $\pi$ junction. This quantity is plotted for the two diamonds I ($N=1$, blue dots) and J ($N=3$, orange squares), respectively, in the single-level and two-level regimes. The dashed lines materialize discontinuities of $I_c$, specific to first-order transitions. (Inset) Focus on two 0-$\pi$ transitions, centered around ${\epsilon }_t$ [see Fig. 5 for the definition]: at $\epsilon =-1.1\mathrm{meV}$ in diamond I and at $\epsilon =0.1\mathrm{meV}$ in diamond J. (b) Schematic explanation of the $N=1/3$ symmetry breaking observed in the supercurrent. The lower energy level A is better coupled to the reservoirs than the higher one B (see text).

Figure 9

(a) Differential conductance $dI/d{V}_{sd}$ in the normal state as a function of $V_{sd}$ at half filling of six different diamonds: A (S-Al) and C, E, I, J, and K (S-NbAl). (b) Temperature dependence of the same quantity for ridges A (S-Al) and C (S-NbAl). To destroy the superconductivity in the contacts, a magnetic field is applied perpendicular to the sample: $B=0.13\mathrm{T}$ for diamond A and $B=1\mathrm{T}$ for diamonds C, E, I, J, and K.

Figure 10

Modulation of the switching current at the 0-$\pi$ transition in diamond C with optimization of the environment (in black dots) and without (in red lines). The amplitudes of the optimized curves have been readjusted in order to compare the shapes of the curves.

Figure 11

Differential conductance as a function of the bias voltage $[dI/dV\left(V_{sd}\right)]$ in the superconducting state for two diamonds: (a) I ($\pi$ junction ) and (b) K (0 junction).

Figure 12

Data in the normal and superconducting state for a probable SU(4) Kondo effect zone. (Color plot) Differential conductance as a function of $V_{sd}$ and $V_g$. A magnetic field of 1 T is applied perpendicularly to the sample to destroy the superconductivity in the contacts. On the right are plotted vertical cuts of the color plot, yielding $dIdV\left(V_{sd}\right)$ at the middle of $N=1,N=2$, and $N=3$ diamonds. Red curve, $dIdV(V_g,V_{sd}=0)$ in the normal state; black curve, supercurrent $\delta I_s\left(V_g\right)$ for a superconducting phase difference of $\phi =\pi /2$. On the right are plotted current-phase relations at the center of $N=0$, 1, and 3 diamonds.