Coulomb Blockade and Bloch Oscillations in Superconducting Ti Nanowires

Quantum fluctuations in quasi-one-dimensional superconducting channels leading to spontaneous changes of the phase of the order parameter by $2\pi$, alternatively called quantum phase slips (QPS), manifest themselves as the finite resistance well below the critical temperature of thin superconducting nanowires and the suppression of persistent currents in tiny superconducting nanorings. Here we report the experimental evidence that in a current-biased superconducting nanowire the same QPS process is responsible for the insulating state—the Coulomb blockade. When exposed to rf radiation, the internal Bloch oscillations can be synchronized with the external rf drive leading to formation of quantized current steps on the $I\mathrm{\text{−}}V$ characteristic. The effects originate from the fundamental quantum duality of a Josephson junction and a superconducting nanowire governed by QPS—the QPS junction.

Figure 1

Scanning electron microscope image of a typical sample with high-Ohmic contacts enabling four-probe transport measurements and introduction of rf radiation.

Figure 2

All-titanium structure: The nanowire length $L=20\mu \mathrm{m}$, ${\sigma }^{1/2}=40\pm 2\mathrm{nm}$, and $R_p\simeq 15\mathrm{k}\Omega$. $dV/dI(I)$ in the presence of external rf radiation with frequency $f_{\mathrm{rf}}=1\mathrm{GHz}$ at $T=70\mathrm{mK}$. Arrows indicate the positions of the expected current singularities $I_n=2e{f}_{\mathrm{rf}}n$. Inset: Positions of the first current singularity $I_1$ (left axis, circles) and the corresponding voltage $V_1$ (right axis, stars) as function of the rf frequency $f_{\mathrm{rf}}$. Note the acceptable linear fit (solid line) for the current singularities, as well as the absence of any rationality for the $V_1(f_{\mathrm{rf}})$ dependency, which one might expect in the case of a conventional Shapiro effect.

Figure 3

Multiterminal titanium nanostructure with three adjacent nanowires each with length $L=20\mu \mathrm{m}$, ${\sigma }^{1/2}=24\pm 2\mathrm{nm}$, and $R_p\sim 10\mathrm{M}\Omega$ bismuth contacts. (a) The $I\mathrm{\text{−}}V$ characteristics demonstrate the Coulomb blockade for all three neighboring parts of the same nanostructure. Arrows indicate the direction of the current sweep. The Coulomb gap $\delta V_{\mathrm{CB}}$ decreases with increase of the temperature and disappears above $\sim 450\mathrm{mK}$ (right inset). Application of the gate voltage $V_{\mathrm{gate}}$ quasiperiodically modulates the Coulomb gap (left inset). (b) Application of external radiation with the frequency $f_{\mathrm{rf}}$ generates nonmonotonic peculiarities at positions $I_n=2e{f}_{\mathrm{rf}}n$. Inset: Schematics of the structure.

Figure 4

Multiterminal nanostructure with each nanowire length $L=20\mu \mathrm{m}$ and the high-Ohmic bismuth contacts. (a) Sample 3 with ${\sigma }^{1/2}=15\pm 3\mathrm{nm}$ and $R_N\approx 1.5\mathrm{M}\Omega$. Zoom of the $I\mathrm{\text{−}}V$ characteristic at the transition point from insulating to current-carrying state at $f_{\mathrm{rf}}=50\mathrm{MHz}$ and two amplitudes $V_{\mathrm{rf}}$ of the rf signal. Arrows indicate the positions of the expected current singularities $I_n=2e{f}_{\mathrm{rf}}n$. Note the characteristic backbended shape of the $I\mathrm{\text{−}}V$ dependence: the nose. Left inset: Larger scale $I\mathrm{\text{−}}V$ characteristics of the three adjacent sections of the same nanostructure. Arrows indicate the directions of the current sweep. Right inset: $dV/dI(I)$ at $f_{\mathrm{rf}}=350\mathrm{MHz}$. (b) Sample 1 with ${\sigma }^{1/2}=12\pm 4\mathrm{nm}$ and $R_N\approx 2.5\mathrm{M}\Omega$. Magnified view of the current singularities at $f_{\mathrm{rf}}=3.8\mathrm{GHz}$. Arrows indicate the positions of the expected current singularities. Inset: Dependence of the step width $\delta V$ on rf magnitude for the first Cooper pair singularity ($2/1$) and the Coulomb gap ($0/1$). Note that for the Coulomb gap the scale is reduced by factor of 10. (c) Positions of the current singularities $I_{n,m}$ as functions of the rf frequency $f_{\mathrm{rf}}$.