Electrostatically-tuned superconductor-metal-insulator quantum transition at the ${\text{LaAlO}}_3/{\text{SrTiO}}_3$ interface

Recently superconductivity at the interface between the insulators ${\text{LaAlO}}_3$ and ${\text{SrTiO}}_3$ has been tuned with the electric-field effect to an unprecedented range of transition temperatures. Here we perform a detailed finite-size scaling analysis to explore the compatibility of the phase-transition line with Berezinskii-Kosterlitz-Thouless (BKT) behavior and a two-dimensional–quantum-phase (2D-QP) transition. In an intermediate regime, limited by a gate voltage dependent limiting length, we uncover remarkable consistency with a BKT-critical line ending at a metallic quantum critical point, separating a weakly localized insulator from the superconducting phase. Our estimates for the critical exponents of the 2D-QP-transition, $z\simeq 1$ and $\overline{\nu }\simeq 2/3$, suggest that it belongs to the three-dimensional-$xy$ universality class.

Figure 1

(Color online) ${\left(d\text{ln}R/dT\right)}^{-2/3}$ vs $T$ for $V_g=40\text{V}$ where $R=3/5R_{◻}$. The solid line is ${\left(d\text{ln}R/dT\right)}^{-2/3}=6.5\left(T-T_c\right)$ yielding the estimates $T_c=0.27\text{K}$ and ${\left(2/b_R\right)}^{2/3}=6.5$; the inset shows $R\left(T\right)\text{exp}\left(b_R{\left|T-T_c\right|}^{-1/2}\right)/R_0$ vs $\left(1/R_0\right)\text{exp}\left(b_R{\left|T-T_c\right|}^{-1/2}\right)$ with $R_0=1.67\text{k}\Omega$. The upper branch corresponds to $T>T_c$ and the lower one to $T<T_c$. The solid line is $R\left(T,L\right)\simeq R\left(T,\infty \right)$ and the dashed one $R\text{exp}\left(b_R{\left|T-T_c\right|}^{-1/2}\right)/R_0=\left(g/R_0\right)\text{exp}\left(b_R{\left|T-T_c\right|}^{-1/2}\right)$ with $g\simeq 501\propto 1/L^2$.

Figure 2

(Color online) $T_c$ vs $R_{◻}\left(T^{\ast }\right)$ (●) and $V_g$ vs $R_{◻}\left(T^{\ast }\right)$ $(\star )$ at $T^{\ast }=0.4\text{K}$. The error bars indicate the uncertainty in the finite-size estimates of $T_c$. The solid line is $T_c=1.17\times {10}^{-4}\Delta R\left(T^{\ast }\right)$ and the dashed one $V_g=V_{gc}+1.39\times {10}^{-3}\Delta R^{3/2}\left(T^{\ast }\right)$ with $\Delta R\left(T^{\ast }\right)=\left[R_{◻c}\left(T^{\ast }\right)-R_{◻}\left(T^{\ast }\right)\right]$, $R_{◻c}\left(T^{\ast }\right)=4.28\text{k}\Omega$ and $V_{gc}=-140\text{V}$.

Figure 3

(Color online) Vortex radius ${\xi }_0\propto {\left(R_{0c}-R_0\right)}^{-1/2}$ (●) and $b$ $(★)$ vs $T_c$ where $R=3/5R_{◻}$. The solid line is ${\xi }_0\propto {\left(R_{0c}-R_0\right)}^{-1/2}=8\times {10}^{-3}/T_c$ with $R_{0c}=2.7\text{k}\Omega$.

Figure 4

(Color online) Gate voltage dependence of the limiting length $L$ in terms of $L\propto g^{-1/2}$ vs. $V_g$. The inset shows $T_c$ vs $V_g$. The solid line is $T_c=8.9\times {10}^{-3}{\left(V_g-V_{gc}\right)}^{2/3}\left(\text{K}\right)$ indicating the leading quantum critical behavior [Eq. (14)] with $z\overline{\nu }\simeq 2/3$.

Figure 5

(Color online) (a) ${\sigma }_{◻}-{\sigma }_{◻0}$ vs $T$ for various gate voltages $V_g$. The solid line is ${\sigma }_{◻}-{\sigma }_{◻0}=d\text{ln}\left(T\right)\left({\Omega }^{-1}\right)$ with $d\simeq 1.23\times {10}^{-5}{\Omega }^{-1}$ and ${\sigma }_{◻0}\left(V_g\right)$ taken from Fig. 5b. The dot at the origin marks the quantum critical point and the dashed line is Eq. (19) indicating the low-temperature behavior of the sheet conductivity at the quantum critical point. (b) ${\sigma }_{◻0}$ vs $V_g$. The solid line is Eq. (18) with $V_{gc}=-140\text{V}$.

Figure 6

(Color online) ${\sigma }_{◻}\left(V_{gc},T\right)$ vs $T$ and in the inset vs $T^2$ at $V_{gc}=-140\text{V}$. The solid line, indicating consistency with $T^2$ is Eq. (19), while the dashed line is ${\sigma }_{◻}\left(V_{gc},T\right)=2.275\times {10}^{-4}+1.54\times {10}^{-5}\text{T}\left({\Omega }^{-1}\right)$. The dash-dot one in the inset is again Eq. (19).

Figure 7

(Color online) $\left[{\sigma }_{◻}\left(T\right)-{\sigma }_{◻0}\right]/\left[d\text{ln}\left(T\right)\right]$ vs $1/\left[d\left|\text{ln}\left(T\right)\right|\right]$ at $V_g=-220$, $-240$ and $-300\text{V}$ with $d=1.23\times {10}^{-5}{\Omega }^{-1}$. The lines correspond to Eq. (21) providing a measure for $L$ in terms of $\left|g_L\right|\propto L$.

Figure 8

(Color online) $-g_L\propto L$ vs $V_g$ derived from finite-size scaling plots as shown in Fig. 7 with Eq. (21).

Figure 9

(Color online) Magnetoconductivity ${\sigma }_{◻}$ vs $H$, applied perpendicular to the interface, at $T=0.03\text{K}$ and $V_g=-300\text{V}$ and $-340\text{V}$. The solid line is ${\sigma }_{◻}=4.51\times {10}^{-2}+1.2\times {10}^{-2}\text{ln}\left(H\right)\text{k}{\Omega }^{-1}$, the dashed one ${\sigma }_{◻}=2.6\times {10}^{-2}+1.1\times {10}^{-2}\text{ln}\left(H\right)\text{k}{\Omega }^{-1}$, the dotted and dash-dot curves are Eq. (22) with the ${\sigma }_{◻0}$, $8\pi L_{TH}^2/{\Phi }_0$ and $c={\alpha }^{\ast }{e}^2/\left(\pi h\right)$ values $0.0284\text{k}{\Omega }^{-1}$, $2.906{\text{T}}^{-1}$, $1.46\times {10}^{-5}{\Omega }^{-1}$ for $V_g=-340\text{V}$ and $0.044\text{k}{\Omega }^{-1}$, $4.068{\text{T}}^{-1}$, $1.46\times {10}^{-5}{\Omega }^{-1}$, at $V_g=-300\text{V}$. Note that $8\pi L_{TH}^2/{\Phi }_0=2.906{\text{T}}^{-1}$ corresponds to $L_{TH}\simeq 1.55\times {10}^{-6}\text{cm}$ whereupon $L_H=L_{TH}$ at $H=1.37\text{T}$.

Figure 10

(Color online) $R_{KT}\left(V_g,T\right)/R_0\left(V_g\right)$ vs $\left(V_g-V_{gc}\right)/T^{3/2}$ for various $V_g$’s. The solid line is Eq. (29) with $\tilde{a}=0.0248$ and $\tilde{b}=0.0081$ in ${\text{K}}^{3/2}{\text{V}}^{-1}$.