Insulating Josephson Junction Chains as Pinned Luttinger Liquids

Quantum physics in one spatial dimension is remarkably rich, yet even with strong interactions and disorder, surprisingly tractable. This is due to the fact that the low-energy physics of nearly all one-dimensional systems can be cast in terms of the Luttinger liquid, a key concept that parallels that of the Fermi liquid in higher dimensions. Although there have been many theoretical proposals to use linear chains and ladders of Josephson junctions to create novel quantum phases and devices, only modest progress has been made experimentally. One major roadblock has been understanding the role of disorder in such systems. We present experimental results that establish the insulating state of linear chains of submicron Josephson junctions as Luttinger liquids pinned by random offset charges, providing a one-dimensional implementation of the Bose glass, strongly validating the quantum many-body theory of one-dimensional disordered systems. The ubiquity of such an electronic glass in Josephson-junction chains has important implications for their proposed use as a fundamental current standard, which is based on synchronization of coherent tunneling of flux quanta (quantum phase slips).

Figure 1

Scanning electron micrograph showing two array families with high (left panel) and low (right panel) plasma frequencies, and nominal junction areas of $100\times 100$ and $300\times 400\mathrm{nm}$, respectively. The specific capacitance is 95 and $54\mathrm{fF}/\mu {\mathrm{m}}^2$, respectively. The precise junction area within a family is modulated by the exposure dose.

Figure 2

(Main plot) Experimental determination of the critical voltage for a $N=250$, family 1, device CS3 (see SM [26]). The blue data are obtained upon stepping up from zero voltage, and the red when stepping back down from nonzero current. (Lower inset) Closeup of the small voltage region where the critical voltage is extracted. A very small hysteresis region is present in the IVC for this device. The critical voltage is taken to be the average value of the switching voltage, where the latter is defined as the voltage having maximum $dI/dV$ upon stepping up from zero voltage. (Upper inset) Linear dependence of $V_c$ on chain length, $N$, for a family of devices where only length has been varied.

Figure 3

Scaled critical voltage $v=e{V}_c/N\hslash {\omega }_p$, versus scaled Bloch bandwidth $w=W/\hslash {\omega }_p$. Symbols represent different fabrication “families” distinguished by plasma frequency ${\omega }_p$, length, and presence of ground plane (see SM [26]). The solid red line is theory for independent QPS across each junction (additive Coulomb blockade, or “rigid” quasicharge, and no disorder) and has $\text{slope}=1$. Solid lines are the quantum theory of a disordered Luttinger mode, Eq. (4), with fitted values of screening length $\mathrm{\Lambda }=13.1$ (blue), and $\mathrm{\Lambda }=4.0$ (black), respectively. These exhibit a $w$-dependent slope, $4/(3-2K)$, where $K(w)$ is the Luttinger parameter. For small $w$, $K\propto {\mathrm{\Lambda }}^{-1}\ln w$: for the same range of $w$, $K$ is enhanced by the decreased screening length of devices with ground planes, resulting in a stronger departure from the classical result. The dotted lines show the classical depinning result, $\text{slope}=4/3$, for $\mathrm{\Lambda }=13.1$ (dotted blue) and $\mathrm{\Lambda }=4.0$ (dotted black). Also plotted for comparison are the quantum results for $\mathrm{\Lambda }=7.7$ (blue dashed), and $\mathrm{\Lambda }=3.2$ (black dashed), screening lengths inferred from the gate-dependent periodicity of $dI/dV$ at large $w$ and biases $V>V_c$ (see SM [26]).