Anisotropic electron-phonon coupling in the spinel oxide superconductor $\mathrm{LiT}{\mathrm{i}}_2{\mathrm{O}}_4$

Among hundreds of spinel oxides, $\mathrm{LiT}{\mathrm{i}}_2{\mathrm{O}}_4$ (LTO) is the only one that exhibits superconductivity ($T_{\mathrm{c}}\sim 13\mathrm{K}$). Although the general electron-phonon coupling is still the main mechanism for electron pairing in LTO, unconventional behaviors such as the anomalous magnetoresistance, anisotropic orbital/spin susceptibilities, etc. reveal that both the spin and the orbital interactions should also be considered for understanding the superconductivity. Here we investigate tunneling spectra of [111]-, [110]-, and [001]-oriented high quality LTO thin films. Several bosonic modes in tunneling spectra are observed in the [111]- and [110]-oriented films but not in [001]-oriented ones, and these modes still exist at $T=2T_{\mathrm{c}}$ and beyond the upper critical field, which are confirmed as stemming from electron-phonon interaction by DFT calculations. These modes only appear in special surface orientations, indicating that the electron-phonon coupling in LTO system is highly anisotropic and may be enhanced by an orbital-related state. The anisotropic electron-phonon coupling should be taken seriously in understanding the nature of LTO superconductivity.

Figure 1

Temperature dependence of tunneling spectra for spinel superconductor $\mathrm{LiT}{\mathrm{i}}_2{\mathrm{O}}_4$ thin films grown on [001]-, [110]-, and [111]-oriented $\mathrm{MgA}{\mathrm{l}}_2{\mathrm{O}}_4$ substrates. (a), (c), and (e) The crystal planes (001), (110), and (111) of LTO are shown. The red balls stand for O atoms. The green balls stand for Li atoms. The Ti atoms are located at the octahedral sites of oxygen atoms (blue balls). (b), (d), and (f) Differential conductance versus bias voltage for the [001]-, [110]-, and [111]-oriented LTO thin films from superconducting state to normal state with increasing temperature, respectively. $T\in \left[2.5\mathrm{K},10\mathrm{K}\right]$ with $\mathrm{\Delta }T=0.5\mathrm{K};T\in \left[10\mathrm{K},12\mathrm{K}\right]$ with $\mathrm{\Delta }T=0.25\mathrm{K};T\in \left[12\mathrm{K},14\mathrm{K}\right]$ with $\mathrm{\Delta }T=1\mathrm{K};T\in \left[14\mathrm{K},34\mathrm{K}\left(36\mathrm{K}\right)\right]$ with $\mathrm{\Delta }T=2\mathrm{K}$.

Figure 2

The comparison of tunneling spectra between the superconducting state and normal state for both [110]- and [111]-oriented samples.(a)–(d) Differential conductance versus bias voltage for (a) [110]-oriented film, $T=2.5$ and 12 K, $H=0\mathrm{T}$; (b) [110]-oriented film, $T=2.5\mathrm{K},H=0$ and 16 T; (c) [111]-oriented film, $T=2.5$ and 12 K, $H=0\mathrm{T}$; and (d) [111]-oriented film, $T=2.5\mathrm{K},H=0$ and 12 T. The detailed data at negative bias are shown in the inset.

Figure 3

Fitting results of temperature- and field-dependent tunneling spectra for [110]- and [111]-oriented samples by a modified BTK model. (a) and (b) Normalized differential conductance versus bias voltage for the [110] and [111] films from 2.5 to 10 K with $\mathrm{\Delta }T=0.5\mathrm{K}$ and from 10 to 11 K with $\mathrm{\Delta }T=0.25\mathrm{K}$. Experimental data (black circles) are fitted with a modified BTK model with a constant broadening Γ (red lines). The data are shifted vertically above 2.5 K. (c) and (d) Normalized differential conductance versus bias voltage for the (c) [110] and (d) [111] films from 0 to 9 T with $\mathrm{\Delta }H=1\mathrm{T}$. Experimental data (black circles) are fitted with an increasing Γ (red lines). (e) Temperature dependence of normalized energy gap Δ($T$)/Δ(0). The energy gap can be fitted well with BCS theory in all films. $2\mathrm{\Delta }/k_{\mathrm{B}}{T}_{\mathrm{c}}\sim 4$ is obtained, indicating a medium-coupling BCS-like superconductor. (f) Normalized energy gap Δ($H$)/Δ(0) versus normalized field $H/H_{c2}$ in different films, which can be scaled with $1-{\left(H/H_{\mathrm{c}2}\right)}^2$ for different temperatures.

Figure 4

Temperature- and field-dependent $d^{2}I/d{V}^2$ of the [111]-oriented film. (a) The differential conductance versus bias voltage of the [111] film at 2.5 K. Two clear humps can be seen at both negative and positive bias. (b) $d^{2}I/d{V}^2$ calculated from the data presented in (a). The bosonic modes are located at the peaks and dips at both negative bias and positive bias with error bars less than ∼1 meV. (c) and (d) Temperature and field dependence of $d^{2}I/d{V}^2$ for the [111] film.

Figure 5

The origin of anisotropic bosonic modes. Band structures of the LTO films with supposed lattice distortions for (a) the [111]-oriented film with reduced angles (58°) between the lattice vectors of primitive cell and (b) the [001]-oriented film with nonequivalent in-plane ($xy$ plane) and out-of-plane ($z$ axis) lattice constants. (a) and (b) The band structures without and with O atom vibrations according to the $E_g\left(\sim 41.1\mathrm{meV}\right)$ mode in (d) are denoted by solid and dotted/dashed lines, respectively. (c) Band structures of ideal LTO structure without (solid lines) and with (short dashed lines) O atom vibrations. (d) Atomic displacement patterns of the $E_g$ phonon mode. (e) Typical ABF image of LTO lattice. (f) The representative area showing ordered oxygen vacancy structure. Here oxygen sites are marked by red dashed circles and the oxygen vacancy sites are marked by yellow dashed circles. The Ti-O octahedron is marked with the same size of yellow quadrilaterals in the insets of (e) and (f), respectively. The octahedron in the ordered oxygen vacancy area cannot match well with the quadrilateral, which indicates a non-negligible distortion in the LTO system.

Figure 6

The comparison of density of states between Pb and LTO. (a) Gap dependence of density of states in Pb calculated by Eliashberg theory. The hump features disappear gradually with decreasing the gap. The dash gray lines are used for guiding the eyes. It can be clearly seen that the hump features shift to low energy with decreasing the gap. (b) Field dependence of normalized tunneling spectra in the [111] samples.