Mott variable range hopping and bad-metal in lightly doped spin-orbit Mott insulator $\mathrm{BaIr}{\mathrm{O}}_3$

The Mott transition and the vicinity of a Mott insulating phase have constantly been a fertile ground for exploring exotic quantum states, most notably the high-$T_c$ cuprates. The layered iridate represents another intriguing example. The nature of the Mott insulator phase and transition mechanism to a metallic state is still under debate. Much of the challenge originates from a series of energy scales involved in the electronics phases. Here, we report synthetization, characterization, and transport measurements on doped and undoped $\mathrm{B}{\mathrm{a}}_{1-x}\mathrm{L}{\mathrm{a}}_x\mathrm{Ir}{\mathrm{O}}_3$ films grown on $\mathrm{SrTi}{\mathrm{O}}_3$. The films are fully strained up to 70 nm thick and have a tetragonal lattice structure. For a doping level of $x=0.1$, a bad-metal state with linear temperature dependence of the resistivity beyond the Mott-Ioffe-Regel limit emerges in a wide temperature range down to a critical temperature $T_c\sim 30\mathrm{K}$, below which the system shows Mott variable range hopping (Mott VRH) conduction behavior. The ground state is confirmed to be insulating by the Mobius criterion. A strong correlation between the bad-metal state and Mott VRH localized state is found as the slope of the linear resistivity is inversely proportional to film thickness and the size of the Mott VRH activation energy $\mathrm{\Delta }$ is linearly proportional to $T_c$. We further show that upon doping the spin-orbit Mott insulator, itinerant metallic regions coexist with localized regions within a nanoscale phase-separated ground state with a small activation energy. Our results shed light on the nature of the metallic state and a crossover to a bad-metal phase for doping the spin-orbit Mott insulator.

Figure 1

Structure characterization of BLIO film on $\mathrm{SrTi}{\mathrm{O}}_3\left(001\right)$ substrate. (a) Atomic force microscopy images of 41 nm BLIO sample. (b) A $\theta -2\theta$ scan where the $\mathrm{B}{\mathrm{a}}_{1-x}\mathrm{L}{\mathrm{a}}_x\mathrm{Ir}{\mathrm{O}}_3$ 002 peaks are clearly labeled. (c),(d) X-ray reciprocal-space map (RSM) near STO ($\overline{1}03$) and (002), respectively.

Figure 2

(a) Electrical resistance of 5-nm-thick $\mathrm{B}{\mathrm{a}}_{1-x}\mathrm{L}{\mathrm{a}}_x\mathrm{Ir}{\mathrm{O}}_3$ ($x=0.1$) film as a function of temperature. The dashed black line is a linear fit to the data between 100 and 300 K. Below a temperature $T^{*}, R\left(T\right)$ deviates from its linear-$T$ behavior and develops a pronounced upturn at low temperature. (b) $ln\left(R\right)$ as a function of $1/T$ showing the Mott VRH activation energy. (c) $ln\left(R\right)$ as a function of $1/T^{1/3}$ indicating the 2D Mott-VRH below $T_c$.

Figure 3

Bad-metal to Mott VRH localized state transition in BLIO films with various thickness. (a) Temperature-thickness phase diagram of BLIO. Filled violet circles represent the MIT critical temperature $T_c$, and filled pink oblique squares represent $T^{*}$ as shown in Fig. 2. (b) Temperature dependence of resistance $R\left(T\right)$ as a function of film thickness $d$. (c) Close-up view of the metal-insulator transition temperature $T_c$ as a function of $d$. The data shows that $d_c\sim 6.5\mathrm{nm}$ is a crossover thickness. (d) Temperature derivative of resistance $k\equiv \partial R/\partial T$ as a function of $d$ at temperatures near the crossover points. The white arrow shows the position of the crossover point at $(d=d_c,T=T_c)$.

Figure 4

Suppression of the activation energy in BLIO as a function of film thickness and the correlation between the Mott VRH localized states and bad-metal states. (a) $1/k$ (left) of the bad-metal states and Mott VRH activation energy $\mathrm{\Delta }$ (right) as a function of thickness $d$. (b) $\mathrm{\Delta }$ as a function of $T_c$ for different thickness showing a linear dependence. Inset: Plot of $\mathrm{\Delta }$ as a function of $|d-d_c|$ showing different scaling behavior for $d<d_c$ and $d>d_c$. The dark solid lines are fitting to the scaling equation $\mathrm{\Delta }={\mathrm{\Delta }}_c-|d-d_c{|}^{\delta }$, where $\delta$ is the scaling parameter.

Figure 5

Mobius criterion $\omega \left(T\right)=dln\sigma \left(T\right)/dlnT$ of the insulating ground states in both (a) undoped ($x=0$) and (b) doped ($x=0.1$) $\mathrm{B}{\mathrm{a}}_{1-x}\mathrm{L}{\mathrm{a}}_x\mathrm{Ir}{\mathrm{O}}_3$ films. The undoped $\mathrm{BaIr}{\mathrm{O}}_3$ ($x=0$) films show either a finite value (for thickness of 24.5 nm) or a diverging value (for thickness of 4.9 nm) of $w\left(T\right)$ as $T\to 0$, indicating that the samples are insulator. The doped $\mathrm{BaIr}{\mathrm{O}}_3$ ($x=0.1$) films show finite values (for all samples) of $w\left(T\right)$ as $T\to 0$, indicating that the samples are also genuine insulators at low temperatures.

Figure 6

Scanning tunneling microscopy/spectroscopy of 5 nm film sample at 300 K. (a) Topography of the sample which has a roughness of $\sim 0.3\mathrm{nm}$. The size of the imaged sample area is around $96\times 96\mathrm{n}{\mathrm{m}}^2$. (b) Density of state (DOS) averaged $dI/dV/(I/V)$ vs $V$ curve showing that the density of state linearly changes with bias voltage with different slopes for positive and negative voltages. (c)–(f) $dI/dV/(I/V)$ mapping with voltage bias at −40, −13, 94, and 147 mV, respectively. The spectra show the coexistence of localized and itinerant regions. (g),(h) $IV$ and $dI/dV$ vs sample bias for two blue (dark) (localized) and red (bright) (itinerant) regions in (c)–(f) indicated by the white cross. Inset in (g) shows $IV$ curve with a different scale, showing a similar gap behavior.