Quantum phase slips and number-phase duality in disordered TiN nanostrips

We have measured the electric transport properties of TiN nanostrips with different widths. At zero magnetic field, the temperature-dependent resistance $R\left(T\right)$ saturates at a finite resistance toward low temperatures, which results from quantum phase slips in the narrower strips. We find that the current-voltage $\left(I\text{−}V\right)$ characteristics of the narrowest strips are equivalent to those of small Josephson junctions. Applying a transverse magnetic field drives the devices into a reentrant insulating phase, with $I\text{−}V$ characteristics dual to those in the superconducting regime. The results provide evidence that our critically disordered superconducting nanostrips behave like small self-organized random Josephson networks.

Figure 1

Sheet resistance, $R_{\square }$, vs temperature, $T$, at zero magnetic field of TiN nanostrips of length 780 nm and with different widths, $w$. Inset: Scanning electron micrograph of a typical sample and sketch of the measurement setup.

Figure 2

(a) Arrhenius plot of the resistance at zero magnetic field for different strip width of chip A. The lines are fits according to Eq. (1). The resistance of the narrowest strips saturates at low temperature. The lines are fits according to Eq. (1). (b) Activation temperatures $T_0$ vs cross section $A=wt$ extracted from the fits in (a). The slope of the dashed line corresponds to an effective Ginzburg-Landau parameter of ${\kappa }_{\mathrm{eff}}\simeq 490$.

Figure 3

Temperature evolution of $I\text{−}V$ characteristics for a 250-nm-wide strip. Inset: Voltage source and parallel connection of the active phase slip element and the environmental impedance $Z\left(\omega \right)$. At zero frequency $\text{Re}\left[Z(\omega =0)\right]=R_N$ corresponds to the slopes of $I\left(V\right)$. The lead capacitance renders the parallel and the serial connection equivalent at high frequencies where $\text{Re}\left[Z(\omega ={\omega }_p)\right]=R_B$, the bias resistor in the IZ theory (see text).

Figure 4

Supercurrent peaks of the four narrowest wires on chip A. The solid lines are fits according to the Ivanchenko-Zilberman model with $V=V_{\text{IZ}}-R_{B}I$ (see text).

Figure 5

(a) Critical current $I_0$ vs $T$ for $w=250,130115$, and 75 nm (from top to bottom) as extracted from the IZ fits (see text). (b) Effective temperature used in the IZ analysis of the supercurrent peak of the four narrowest strips at zero magnetic field. The dashed line corresponds to $T=T_{\text{eff}}$. (c) Corresponding fitparameters $R_N$ and $R_B$ extracted from the IZ analysis. (d) Resistance $R\left(T\right)$ of the same samples. The dashed horizontal lines denote the $R_N$ values from (c).

Figure 6

Critical current density $J_c$ extracted from the maximal supercurrent vs nanostrip width. The dashed line indicates the expectation for a standard superconductor with $J_c$ independent of the strip width. Inset: Schematic of droplet structure of the superconducting order parameter.

Figure 7

$I\text{−}V$ characteristics of nanostrip $\mathsf{A}\left(w=85\mathrm{nm}\right)$ for $B=0$ (purple) and in perpendicular magnetic field $B=2.4\mathrm{T}$ (orange). Two 50 $\mathrm{k}\mathrm{\Omega }$-resistors were placed adjacent to the device to access the back-bending part of the $I\text{−}V$ characteristics. Insets: $R\left(T\right)$ for nanostrips $\mathsf{A}$ and $\mathsf{B}\left(w=75\mathrm{nm}\right)$ with an inititially superconducting and reentrant behavior with increasing magnetic field and for nanostrip $\mathsf{A}$ very high values of the linear resistance.

Figure 8

(a) Evolution of $I\text{−}V$ characteristics of nanostrip $\mathsf{A}\left(w=85\mathrm{nm}\right)$ with $T$ at $B=2\mathrm{T}$ in the insulating regime. Upper inset: Dual measurement circuit with a serial connection of the active phase slip element and the environmental impedance $Z\left(\omega \right)$. Lower inset: Droplet structure of superconducting order parameter with magnetic flux induced weakest spot (red). (b) Fits with the dual IZ theory after subtraction of $I/G_N$ from the measured voltage. (c) Magnetic field dependence of the critical voltages $V_c$ and $V_0$ (see text).

Figure 9

(a) Critical voltage $V_0$ vs $T$ for several magnetic fields as extracted from the IZ fits (see text). (b) Effective temperature used in the dual IZ analysis of the Bloch nose in the reentrant insulating regime. (c) Corresponding fitparameters $G_N$ and $G_B$ extracted from the dual IZ analysis. (d) Conductance $G\left(T\right)$ in the insulating reentrance observed for the 85-nm-wide strip. The dashed horizontal lines denote the $G_N$ values from (c).

Figure 10

Magnetoresistance of the wire with $w=85\mathrm{nm}$ (black diamonds). The resistances $R_{\mathrm{N}}=1/G_{\mathrm{N}}$ extracted from the high voltage part of the $I\text{−}V$ characteristics (blue squares) and $R_{\mathrm{B}}=1/G_{\mathrm{B}}$ from the IZ fits (red squares) are shown. The inset shows the evolution of the fit parameters $R_N$ and ${}_B$ (see last section).

Figure 11

Resistance vs temperature in perpendicular magnetic field for the wider TiN strips.

Figure 12

$I\text{−}V$ characteristics for different nanostrips' widths and temperatures in zero magnetic field.