Fate of the Bose insulator in the limit of strong localization and low Cooper-pair density in ultrathin films

A Bose insulator composed of a low density of strongly localized Cooper pairs develops at the two-dimensional superconductor to insulator transition (SIT) in a number of thin film systems. Investigations of ultrathin amorphous PbBi films far from the SIT described here provide evidence that the Bose insulator gives way to a second insulating phase with decreasing film thickness. At a critical film thickness $d_c$ the magnetoresistance changes sign from positive, as expected for boson transport, to negative, as expected for fermion transport, signs of local Cooper-pair phase coherence effects on transport vanish, and the transport activation energy exhibits a kink. Below $d_c$ pairing fluctuation effects remain visible in the high-temperature transport while the activation energy continues to rise. These features show that Cooper pairing persists and suggest that the localized unpaired electron states involved in transport are interspersed among regions of strongly localized Cooper pairs in this strongly localized, low Cooper-pair density phase.

Figure 1

(a) Structure of the NHC films as determined by a combination of atomic force micrographs and transport data [15]. Blue regions are thick enough to support Cooper-pair formation, and pink regions are not. Superconducting islands (dots) are connected by weak links (resistor symbols) and form an organized network. The scale bar is 100 nm. (b) Sheet resistance as a function of temperature for a series of $a$-PbBi NHC films of increasing film thickness. Film thicknesses from top to bottom are 0.350, 0.360, 0.370, 0.385, 0.395, 0.400, and 0.413 nm. Inset: Scanning electron micrograph of AAO substrate. The scale bar is 200 nm. (c) Sheet resistance as a function of perpendicular magnetic field (as in upper inset) for the films in (b) at 500 mK on a $logR$ scale. Film 10 is shown on a linear scale in the lower inset.

Figure 2

(a) Sheet resistance as a function of perpendicular magnetic field for the same data as in Fig. 1 for a limited range to show the low-field region. The definitions of oscillation amplitude $\left(A\right)$ and slope $\left(m\right)$ are illustrated for film 13 (black). (b) Amplitude of the first magneto-oscillation in the resistance as a function of film thickness. (c) Low-field slope of the MR as a function of film thickness. (d) Oscillation amplitude vs slope to show the simultaneous disappearance of the two.

Figure 3

(a) Sheet resistance vs inverse temperature for all films measured, omitting film 5 for lack of data above 1 K. Dashed lines are Arrhenius fits to $R=R_0{e}^{T_0/T}$ showing simply activated behavior. (b) Zero-field activation energies $\left(T_0\right)$ and (d) prefactors $\left(R_0\right)$ obtained from fits to Arrhenius curves shown in (a) vs film thickness. The uncertainty in the fit parameters is comparable to the size of the data points. The vertical dashed lines mark $d_c$ and $d_{\mathrm{SIT}}$ as defined in the text. (d) Differentiated curves in (a), $T_{0,\mathrm{local}}$ [defined by Eq. (3)], vs inverse temperature. Signs of reentrance are clear for each film. The vertical orange bar marks the position of the inflection point in $R\left(T\right)$ for each film. $T_0$ from low-temperature Arrhenius fits are shown as horizontal dashed lines.

Figure 4

Sheet resistance vs temperature for (a) a series of insulating $a$-PbBi NHC films approaching the SIT, and (b) the corresponding superconducting reference films grown simultaneously on a neighboring polished glass substrate. Film thicknesses from top to bottom are approximately 0.42, 0.47, 0.49, 0.51, 0.54, 0.55, and 0.56 nm.