Multiple crossing points and possible quantum criticality in the magnetoresistance of thin TiN films

We have measured $R(T,B)$ of a TiN thin film very close to the disorder-driven superconductor-insulator transition but still superconducting at zero field and low temperatures. In a magnetic field we find that three distinct crossing points of the magnetoresistance isotherms occur at magnetic fields $B_{cX}$ in three different temperature regions. Each crossing point in $R(T,B)$ corresponds to a plateau in $R(T,B_{cX})$. We systematically study the evolution of these crossing points near the disorder-induced superconductor/insulator transition, identify the most promising candidate for a quantum phase transition, and provide estimates for the two critical exponents $z$ and $\nu$.

Figure 1

Logarithmic derivative of $R(T,B)$ vs temperature $T$ and magnetic field $B$ (chip A). Blue/green areas indicate an insulating ($\partial R/\partial T<0$) trend; red/orange areas indicate a superconducting ($\partial R/\partial T>0$) trend. The nonmonotonic yellow strip displays the separatrix $B^{*}\left(T\right)$ between superconducting and insulating regimes. The three extrema in $B^{*}\left(T\right)$ are marked with black ellipses labeled LT, MT, and HT. The two panels display states with different $R_{\square }^{300}$, i.e., different levels of disorder (see text).

Figure 2

(a) Temperature dependence of the sheet resistance $R_{\square }\left(T\right)$ for several magnetic fields and $R_{\square }^{300}=4.25\mathrm{k}\mathrm{\Omega }$ (chip A). Three plateaus in the $R\left(T\right)$ curves at the extrema of $B^{*}\left(T\right)$ in Fig. 1 are marked with red ellipses. Inset: schematic of the TiN film with Au contacts. (b)–(d) Crossing points in the $R\left(B\right)$ isotherms at the plateau regions in (a) corresponding to the extrema in the separatix $B^{*}\left(T\right)$ (Fig. 1, left panel). Note that the horizontal scale in (b) and (d) is linear. Insets: scaling analysis for each crossing point.

Figure 3

(a) Critical resistance $R_c$ vs sheet resistance at room temperature $R_{\square }^{300}$ (chips A,B). Green stars correspond to disorder levels, for which the MT and HT regimes are merged. (b) Scaling exponent $z\nu$ vs $R_{\square }^{300}$ resulting from the best data collapse of the $R\left(B\right)$ isotherms in the scaling plots [insets of Figs. 2–2]. The values $z\nu =0.7$ and $z\nu =1.2$ are marked by horizontal dotted lines. (c) Magnetic fields at the crossing points $B_{cL}$, $B_{cM}$, and $B_{cH}$ vs $R_{\square }^{300}$. Dashed lines are guides to the eye. Inset: evolution of $B_{cL}$, $B_{cM}$, and $B_{cH}$ for a wider range of $R_{\square }^{300}$. (d) BKT-transition temperature $T_{\mathrm{BKT}}$ extracted from $R(T,B=0)$ (see text). Inset: $B_{cM}$ vs $T_{\mathrm{BKT}}$. The solid red line corresponds to the exponent $z=1$. The dashed red lines are obtained for $z=4/3$ and $z=2/3$, respectively.

Figure 4

(a) $R\left(B\right)$ isotherms for $R_{\square }^{300}=4.228\mathrm{k}\mathrm{\Omega }$ (chip A). The red lines are fits of Eq. (S5) with the parameters $T_{c0}=0.83\mathrm{K}$ and $B_{c2}\left(0\right)=0.88\mathrm{T}$. (b) $G\left(T\right)$ extracted from the fits in the limit of large $B$. The red line denotes the Altshuler-Aronov contribution to $G$ (see text). (c) Zoom to the experimental and Galitski-Larkin crossing points $B_{cH}=0.73\mathrm{T}$ and $B_{GaL}=0.935\mathrm{T}$.

Figure 5

(a) $I\left(V\right)$ characteristics for $T=37\mathrm{mK}$ ($R_{\square }^{300}=4.25\mathrm{k}\mathrm{\Omega }$ chip A). The red dashed line indicates $B<B_{cL}$. The yellow line corresponds to the critical field $B_{cM}$. The black circle indicates first appearance of a (tiny) jump in $I\left(V\right)$. The slight left tilt of the jumps results from an imperfect voltage bias, leading to a slight reduction of the voltage after the jump. (b) Corresponding differential resistance $dV/dI$ (right) for $I\lesssim 5\mathrm{nA}$.