Universal relation between doping content and normal-state resistance in gate voltage tuned ultrathin ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ flakes

Gate voltage tunable ultrathin high-$T_c$ cuprates supply a unique platform to investigate the electronic phase diagram and superconductor-insulator transition. One of the challenges in this field is the precise determination of the doping content in the underdoped nonsuperconducting region. Here we report the discovery of a universal relation between the doping content $p$ and the normal-state resistance at a fixed temperature $R\left(T_f\right)$, $p=\alpha +\beta \ln [1/R\left(T_f\right)]$, in the ultrathin ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ flakes. The in-depth analysis shows that the evolution of carrier scattering probability with doping content and the change of effective mass caused by superconductor-insulator transition are two key factors leading to this logarithmic relation. Based on our finding, the more precise electronic phase diagram can be established. In addition, the superconductor-insulator transition is verified to be a quantum phase transition using a finite size scaling analysis. The scaling exponent $z\nu$ is found to have a close correlation with the disorder levels. The present result provides an important foundation to investigate the fascinating electronic states in the ultra-thin cuprates.

Figure 1

Temperature dependence of the longitudinal resistance of the ultra-thin Bi-2212 samples S-1 (a)–(c) and S-2 (d) and (e). The inset of (a) shows a schematic view of the measurement setup. The arrowed lines indicate the directions in which the positive gate voltage increases. (f) Temperature-dependent resistance (black circles) and its derivative (blue curve) of the sample S-1 (the pristine state, A curve). Different methods for the determination of $T_c$ (including $T_{c,10\%R_n}$, $T_{c,50\%R_n}$, $T_{c,90\%R_n}$, and $T_{c,diff}$), are illustrated.

Figure 2

(a) The SC critical temperatures determined using different methods as a function of $p_R=178/R$ (150 K) for sample S-1. The olive solid line represents the generic relation discovered in the bulk samples [see Eq. (2) in the text]. (b) Doping levels in the SC region calculated using Eq. (2) as a function of $\ln [1/R\left(T_f\right)]$. The data with $T_f$ = 150 K and 200 K are shown together. The data of 200 K are shifted to the left by 0.4 for clarity. The straight dashed lines are the results of linear fitting in the underdoped region $p<0.13$. (c) Doping levels as a function of $\ln [1/R\left(T_f\right)]$ in the non-SC region. The data of 200 K are shifted to the left by 0.3 for clarity. The straight dashed lines are the results of linear fitting. The capital letters correspond to different levels of the gate tuning as shown in Figs. 1 and 1.

Figure 3

(a) The doping levels calculated using Eq. (2) (see text) as a function of $\ln [1/R\left(T_f\right)]$ in a wide temperature range 100 $\mathrm{K}\le T_f\le 250$ K for sample S-1. The temperature interval is 25 K. The data at each fixed temperature has been shifted upwards by 0.02 for clarity. The straight dashed lines are the results of linear fitting in the underdoped region $p<0.13$. (b) Fitting parameters $\alpha$ and $\beta$ as a function of $T$.

Figure 4

Phase diagram for the gate voltage tuned ultra-thin Bi-2212. The contour map represents the magnitude of resistance. The $T_c$ data using the criterion of 50% $R_n$ is also shown. The solid blue lines are the contour lines of selected values of resistance (the unit is $\mathrm{\Omega }$). The white dashed line indicates the position of the SIT.

Figure 5

(a, c) Doping dependence of resistance in the temperature region below 25 K of samples S-1 and S-2 respectively. (b, d) Resistance of samples S-1 and S-2 respectively as a function of scaling variable $|p-p_c|T^{-1/z\nu }$. The blue dashed lines are visual guides.

Figure 6

The evolution of $m^{*}/\tau$ as a function of the doping content $p$ calculated using Eq. (5).