Non-equilibrium restoration of duality symmetry in the vicinity of the superconductor-insulator transition

TL;DR

This paper investigates the breakdown and subsequent non-equilibrium restoration of duality symmetry near the superconductor-insulator transition in thin films, combining theoretical analysis with experimental evidence.

Contribution

It demonstrates the restoration of duality symmetry out of equilibrium at very low temperatures where it is broken in equilibrium.

Findings

Duality symmetry is broken at low temperatures in the insulating phase.

Experimental evidence supports duality symmetry across the transition.

Restoration of duality symmetry occurs out of equilibrium at very low temperatures.

Abstract

The magnetic field driven superconductor to insulator transition in thin films was theoretically analyzed via a vortex-charge duality transformation applied to the Hamiltonian. Vortices condensation was conjectured as the underline physical mechanism of the insulating phase. Experimental evidence supported duality symmetry across the magnetic-field driven superconductor to insulator transition in amorphous Indium Oxide films. Counterintuitively, duality symmetry is broken at low temperatures where the insulating phase develops strongly non linear current-voltage characteristics. Here, we follow the breakdown of duality symmetry down to very low temperatures and demonstrate the restoration of duality symmetry out of equilibrium.

Equations

$$R(B)=R_C(\frac{B}{B_C}{)}^{P(T)}$$

$$P(T)=\frac{T_0}{T+\theta }$$

Full text

The superconductor to insulator transition Goldman and Markovic (1998); Gantmakher and Dolgopolov (2010) (SIT) is an experimentally accessible quantum phase transition Sondhi et al. (1997). By varying an externally controlled parameter in the Hamiltonian, a disordered superconducting thin film can be driven between its superconducting and insulating ground states Haviland et al. (1989); Hebard and Paalanen (1990); Shahar and Ovadyahu (1992); Yazdani and Kapitulnik (1995); Baturina et al. (2004); Kevin A. Parendo and Goldman (2005); Bollinger et al. (2011); Allain et al. (2012). Two decades ago Fisher theoretically studied Fisher (1990) a specific case in which an applied magnetic field ($B$) drives the SIT. At low $B$, the induced Abrikosov vortices are localized by the disorder and a superconducting state prevails. Upon increasing $B$, Fisher found that the proliferation of vortices can result in a Bose-Einstein condensation of the vortex state that, in turn, leads to insulating behavior where the Cooper-pairs are now localized Kowal and Ovadyahu (1994); Barber Jr et al. (1994); Gantmakher et al. (1996); Crane et al. (2007); Dubi et al. (2007); Nguyen et al. (2009); Feigel’man et al. (2010); Kopnov et al. (2012); Sherman et al. (2012); Sacépé et al. (2015). The exchange of roles between the Cooper-pairs and vortices across the transition is analyzed via a duality transformation applied to the Hamiltonian Fazio and Schön (1991).

Experimentally, vortex-charge duality will manifest itself via the exchange of roles of current ($I$) and voltage ($V$) between the superconductor and the insulator Shahar et al. (1996); Mehta et al. (2012); Breznay et al. (2016). Duality symmetry implies that, for a given resistance ($R\equiv V/I$) measured at a given $B=B_{SC}$ in the superconductor, there exists a dual $B=B_{Ins}$ in the insulator where the conductivity ($G\equiv I/V$) obeys the condition $G(B_{Ins})=R(B_{SC})$. In previous publications Sambandamurthy et al. (2006); Ovadia et al. (2013) we found that our data follow a phenomenological, power-law, form across the SIT:

[TABLE]

where $P(T)\sim\frac{1}{T}$, $B_{C}$ is the critical $B$ value of the SIT and $R_{C}\simeq R_{Q}=\frac{h}{4e^{2}}$ Fisher et al. (1990). This functional form is duality symmetric: The equality $G(B_{Ins})=R(B_{SC})$ holds whenever the condition $B_{SC}/B_{C}=B_{C}/B_{Ins}$ is fulfilled.

Counterintuitively, duality symmetry breaks down at low temperatures ($T$’s) Ovadia et al. (2013). This is most conveniently illustrated through the deviations from the power-law dependence, graphically shown in Figure 1. Interestingly, these deviations appear only in the insulating side of the SIT. In the superconducting side, the data continue to follow the power-law dependence down to our lowest $T$’s Feigel’man et al. (1990).

Together with the appearance of deviations from duality symmetry, our insulator develops strongly non-linear $I-V$ characteristics ($I-V$’s) Ladieu et al. (1996). At $T\lesssim 0.2$ K, applying a bias $V$ above a well-defined $V=V_{th}$ (which is a function of both $B$ and $T$), results in a discontinuous increase, of several orders of magnitude, in $I$. Upon reducing $V$, a discontinuous decrease in $I$ is observed recovering previous $I$ values (see, for example, the 0.05 K data in Inset (b) of Figure 1 where a $I$ discontinuity is visible at $V\approx 7.5$ mV).

These discontinuities, initially associated with a new and exotic superinsulating phase Vinokur et al. (2008), were later theoretically linked Altshuler et al. (2009) to a bi-stability of the electronic temperature ($T_{el}$). Assuming: 1. Weak electron-phonon coupling, 2. Strong electron-electron interactions enabling self-thermalization to a well-defined $T_{el}$, 3. The Ohmic $R$ demonstrating insulating behavior, and 4. Linearity of the intrinsic $I-V$’s, whereby all deviations from linearity are associated with electron heating, Altshuler et al. Altshuler et al. (2009) numerically solved the heat-balance equation $P\propto T_{el}^{\beta}-T^{\beta}$ and showed that $T_{el}$ can, at low enough $T$, either be near equilibrium, or at a significantly higher $T$ than that of the host lattice resulting in a far from thermodynamic equilibrium, high $I$ state. Several experimental results Ovadia et al. (2009); Doron et al. (2016a); Levinson et al. (2016) support this approach. In what follows we will refer to the low $I$ ($V<V_{th}$), near equilibrium, regime as the high $R$ (HR) state, and to the high $I$ ($V>V_{th}$), out-of-equilibrium, regime as the low $R$ (LR) state.

We begin by following the low-$T$ evolution of the breakdown of duality symmetry in one of our amorphous Indium Oxide (a:InO) films, GTIT1 not (a). In Figure 1 we plot $R/R_{C}$ vs. $B/B_{C}$, measured between 0.05-0.5 K, utilizing a 4-terminal Lock-In configuration (solid lines). We adopted a log-log graph to emphasize the power-law dependence of our data in accordance with Eq. 1. The dashed lines are extensions, to the insulating phase, of power-law fits done in a range limited to the superconducting phase. Deviations from the power-law dependence (indicated by arrows) and, consequently, from duality symmetry are observed only in the insulating phase, leading to higher $R$’s than those expected from duality symmetry. As $T$ is reduced, the starting point of these deviations approach $B_{C}$ (see Inset (a) of Figure 1) and the deviations’ magnitude increases. This trend becomes much more severe if we recall that standard 4-terminal measurements fail in the presence of non-linearities typical of insulators at low-$T$’s. This is apparent as we plot, alongside the 4-terminal data, $R$’s (normalized by $R_{C}$) evaluated from full $I-V$’s (circles). We note that, whenever the $I-V$’s are linear, the two measurement techniques are in agreement (0.2 K data in Figure 1 (b)).

The severe breakdown of duality symmetry accompanies a transition to an insulating state that exhibits an unusual $T$-dependence. The significant upward deviations of our measured $R$’s reveal faster than activated behavior. This is supported by direct $R$ vs. $T$ ($R(T)$) measurements, in the insulating phase, near the SIT Ovadia et al. (2015) where we showed that the $R(T)$’s not only exceeds activation, but seem to approach $R=\infty$ at a finite $T$.

We now show how duality symmetry is restored by driving the system into the LR, out-of-equilibrium, state. This is demonstrated in Figure 2 where we plot $R/R_{C}$ vs. $B/B_{C}$ measured in the superconducting phase (solid line), extended to the insulating phase via fitting to a power-law (dashed blue line). Both $R$, which was measured at zero bias $V$, and the fit are shifted to $V/V_{th}=1$. In the insulating phase we superimpose the discontinuous $V/I$ ($\equiv\overline{R}$ not (b), normalized by $R_{C}$) data measured at constant $B$’s while decreasing $V$ (black circles). Both $R$ and $\overline{R}$ are measured at $T=0.02$ K not (c). In the LR ($V>V_{th}$), out-of-equilibrium, state, $\overline{R}$ gradually increases as we decrease $V$ up to a maximum value measured at $V=V_{th}^{+}$. The values $\overline{R}(V_{th}^{+})$ (measured in the LR state), indicated by yellow diamonds, coincides with the extended power-law fit of the superconducting data up to 3 times $B_{C}$, restoring duality symmetry. Due to the discontinuous nature of the $I-V$’s, any further reduction of the applied $V$ will result in a transition to the HR state and an orders of magnitude increase in $\overline{R}$. We note, that at relatively low $B$’s, $\overline{R}(V_{th}^{+})$ is not consistent with $R$ expected from duality symmetry. We attribute such deviations to the sensitivity of the HR branch near $B_{C}$ discussed in Ref Doron et al. (2016b).

The restoration of duality symmetry can not be accommodated within currently available theoretical models. Duality symmetry, inherent to the superinsulating model Vinokur et al. (2008), is predicted for Ohmic transport at $V<V_{th}$, while we observe the opposite. Adopting the over-heated electrons framework Altshuler et al. (2009) also leads to an apparent contradiction. As we have shown above, duality symmetry is most clearly evident in isotherms of $R(B)$, which follow a power-law (Figure 1). Since the power-law is effectively restored immediately following the jump it seems reasonable that, at $V=V_{th}^{+}$, $T_{el}(V_{th}^{+})=T$. This is contradictory to the over-heated electrons framework where we expect that driving the system beyond $V_{th}$ into the LR state induces a significant increase of $T_{el}$ with respect to $T$ Altshuler et al. (2009), resulting in $T_{el}(V_{th}^{+})>>T$. If we adopt a view where data following the power-law $R(B)$ is necessarily isothermal, we have to conclude that $R(V<V_{th}^{-})$, which is much greater than $R(V>V_{th}^{+})$, is more reasonably seen as being at $T_{el}<T$, as if the electronic system had condensed into a reduced-entropy state. This possibility calls for more experimental and theoretical studies.

Aside from duality symmetry, another analogy can be made between the LR, out-of-equilibrium, state and the superconducting phase: at low $T$’s, $R$ becomes weakly $T$-dependent. In the LR state, this $T$-dependence was previously discussed in Ref. Altshuler et al. (2009). In the superconducting phase, the experimental data deviates from the expected behavior of $P(T)\propto 1/T$ Sambandamurthy et al. (2006) (introduced in Eq. 1), and exhibits a more elaborate $T$-dependence. This observation is in compliance with recent reports of a possible metallic state intervening between the superconducting and insulating phases of disordered superconductors Ephron et al. (1996); Mason and Kapitulnik (1999); Qin et al. (2006); Humbert et al. (2014); Tsen et al. (2015); Couëdo et al. (2016). While at high $T$’s $P(T)\sim\frac{1}{T}$, fitting the full range of our data reveals a different dependence

[TABLE]

where both $\theta$ and $T_{0}$ are sample dependent parameters. In Figure 3 we plot $P^{-1}$ vs $T$, for several different a:InO samples varying in size and disorder, visualizing the non-zero crossing of $P^{-1}$ at $T=0$. The color scale represent $B_{C}$ of each sample from 7.1 T in red to 0.4 T in purple. $P$ is extracted by fitting $R(B)$ data measured at the superconducting phase with a power-law. From the data we obtained $T_{0}$ and $\theta$ which are plotted in the inset of the figure as a function of $B_{C}$. Consequently, one observes a weak $R(T)$ dependence whenever $T\lesssim\theta$. We are not aware of any theoretical prediction of such functional behavior. Interestingly, a similar phenomenon was reported in the past near the quantum Hall-to-insulator transitions Shahar et al. (1998). A broader discussion regarding this behavior will follow in an upcoming publication.

In summary, we showed that the duality symmetry, observed at $T\geq 0.5$ K, does not describe the low $T$ physics of the $B$ driven insulating phase bordering the SIT and that experimental evidence point to the existence of a unique low $T$’s insulating phase. The physical nature behind this state is not yet understood and awaits further research. We also showed that this state is fragile and that duality symmetry, which is related to the continuation of the superconducting phase into the bordering insulating phase, can be restored at low $T$’s by driving the system out of equilibrium. The restoration of duality may point to an intriguing interplay between the insulating behavior at high $T$’s and the LR, out-of-equilibrium, state measured at low $T$’s.

Acknowledgments

We are grateful to B. Altshuler, M. Feigelman, V. Kravtsov and K. Michaeli for fruitful discussions. This research was supported by The Israel Science Foundation (ISF Grant no. 751/13) and The United States-Israel Binational Science Foundation (BSF Grant no. 2012210).