Probing electron-electron interaction along with superconducting fluctuations in disordered TiN thin films

In two-dimensional (2D) disordered superconductors prior to superconducting (SC) transition, the appearance of a resistance peak in the temperature dependent resistance $R\left(T\right)$ measurements indicates the presence of weak localization (WL) and electron-electron interaction (EEI) in the diffusion channel and SC fluctuations in the Cooper channel. Here, we demonstrate an interplay between SC fluctuations and EEI by low-temperature magnetotransport measurements for a set of 2D disordered TiN thin films. While cooling the sample, a characteristic temperature $T^{*}$ is obtained from the $R\left(T\right)$ at which SC fluctuations start to appear. The upturn in $R\left(T\right)$ above $T^{*}$ corresponds to WL and/or EEI. By the temperature and field dependences of the observed resistance, we show that the upturn in $R\left(T\right)$ originates mainly from EEI with a negligible contribution from WL. Further, we have used the modified Larkin's electron-electron attraction strength $\beta \left(T/T_{\mathrm{c}}\right)$, containing a field-induced pair-breaking parameter, in the Maki-Thompson SC fluctuation term. Here, the temperature dependence of $\beta \left(T/T_{\mathrm{c}}\right)$ obtained from the magnetoresistance (MR) analysis shows a diverging behavior close to $T_{\mathrm{c}}$, and it remains almost constant at higher temperature within the limit of $\ln \left(T/T_{\mathrm{c}}\right)<1$. Interestingly, the variation of $\beta \left(T/T_{\mathrm{c}}\right)$ on the reduced temperature $\left(T/T_{\mathrm{c}}\right)$ offers a common trend which has been closely followed by all the concerned samples presented in this paper. Finally, the temperature dependence of inverse phase scattering time (${\tau }_{\varphi }^{-1}$), as obtained from the MR analysis, clearly shows two different regimes; the first one close to $T_{\mathrm{c}}$ follows the Ginzburg-Landau relaxation rate $\left({\tau }_{\mathrm{GL}}^{-1}\right)$, whereas the second one at high temperature varies almost linearly with temperature, indicating the dominance of inelastic electron-electron scattering for the dephasing mechanism. These two regimes are followed in a generic way by all the samples despite being grown under different growth conditions.

Figure 1

Semilogarithmic presentation of temperature-dependent resistance $R\left(T\right)$ characteristics for TiN thin films having different annealing temperature ($T_{\mathrm{a}}$) and film thickness. Here, resistance $R_s$ is in Ω/sq. (a) Two sets of zero-field $R\left(T\right)$ characteristics measured from room temperature down to 2 K with each set containing three different samples of the same thickness but with different annealing temperature. (b) Transport properties of three samples grown at the same annealing temperature but different film thicknesses.

Figure 2

Investigation of zero-field $R\left(T\right)$ data in more detail by using quantum correction to the conductivity (QCC) theory for individual TiN thin films grown at the same annealing temperature ($T_{\mathrm{a}}=780{}^{\circ }\mathrm{C}$) but with different thicknesses. (a)–(c) Main panels: QCC fitting to $R\left(T\right)$ data for a selective range of temperature in the vicinity of the normal metal-to-superconductor transition. Insets: $R\left(T\right)$ for a wider range of temperature demonstrating an upturn with a maximum in resistance appearing at $T_{max}$ as well as a minimum in resistance at $T_{min}$. $T_{max}$ represents the onset of resistance drop causing the superconducting (SC) transition, and $T_{min}$ represents the deviation from the metallic behavior on further lowering the temperature. The region between $T_{min}$ and $T_{max}$ is fitted with [weak localization (WL)+interaction between the particles with close momenta in the diffusion channel (ID)] model as shown by the black dashed curves. (d) Temperature-conductance phase diagram constructed by the characteristic temperatures and conductance measured at 300 K for the samples shown in (a)–(c). With the help of the characteristic temperatures $T_{min}, T_{max}, T_{\mathrm{c}}$, and $T^{*}$, different regimes from metallic state to SC state are defined and highlighted in the phase diagram. $T_{min}, T_{max}$, and $T^{*}$ are shown in the insets of (a)–(c), where $T_{max}$ and $T_{min}$ correspond to the temperature points with maximum and minimum resistance, respectively, and $T^{*}$ is the temperature point just above the $T_{max}$ at which (WL+ID) fit starts to deviate from the experimental data. Here, $T_{\mathrm{c}}$ values are extracted from the QCC fitting performed on the $R\left(T\right)$ data for individual samples.

Figure 3

Logarithmic temperature dependence of conductance presented in semilogarithmic scale for the samples TN10, TN9, and TN8 in (a)–(c), respectively. The black dashed lines are the linear fit to the experimental data.

Figure 4

Field-dependent $R\left(T\right)$ measurements for the sample TN10A (from the same batch of TN10) with magnetic field applied perpendicular to the sample plane. (a) Temperature dependence of conductance ($G$) in semilogarithmic scale under various applied magnetic field. The red solid lines are the linear fits indicating that the temperature dependence for the conductance remains logarithmic even after applying magnetic field. (b) Corresponding $R\left(T\right)$ measured under the magnetic field as mentioned in (a). The field-dependent $R\left(T\right)$'s are fitted with the contribution from [weak localization (WL)+interaction between the particles with close momenta in the diffusion channel (ID)] theory by using Eq. (2), and the fits are shown by the solid red curves. Inset: Three selective $R\left(T\right)$'s measured under 0, 3, and 5 T from the main panel for a clearer view. For each field, the deviation of the WL+ID fit from the experimental data occurs at $T^{*}$, which is marked by the dashed vertical line. (c) The variation of $T_{max}$ and the corresponding maximum resistance value with the applied magnetic field. Here, the saturation in $T_{max}$ after 3.75 T appears due to the temperature limitation of the measuring instrument which is limited by 2 K as the lowest achievable temperature. (d) $B-T$ phase diagram obtained from the extracted $T^{*}$ and $T_{max}$ from the field-dependent $R\left(T\right)$ data.

Figure 5

Magnetoresistance isotherms for the samples (TN10, TN9, and TN8) annealed at $780{}^{\circ }\mathrm{C}$, and their respective $T_{\mathrm{c}}$'s are obtained through total quantum correction to the conductivity (QCC) fitting.

Figure 6

Magnetoconductance (MC) isotherms for samples (a) and (b) TN10, (c) TN9, and (d) TN8. The red solid curves are the fits obtained by the total quantum corrections to the MC that include mainly superconducting fluctuations [Aslamazov-Larkin (AL) and Maki-Thompson (MT)] and weak localization (WL), as expressed in Eqs. (9), (10), and (12), respectively.

Figure 7

Dependence of modified Larkin's parameter ${\beta }_{\mathrm{LdS}}\left(T/T_{\mathrm{c}},\delta \right)$ and inverse phase scattering time $\left({\tau }_{\varphi }^{-1}\right)$ on reduced temperature $T/T_{\mathrm{c}}$. (a) A semilogarithmic presentation of the variation of ${\beta }_{\mathrm{LdS}}\left(T/T_{\mathrm{c}},\delta \right)$ with reduced temperature $\left(T/T_{\mathrm{c}}\right)$ for all the TiN samples presented in this paper from different batches annealed at 820, 780, and $750{}^{\circ }\mathrm{C}$ and with different film thickness. The inset shows the magnified view of ${\beta }_{\mathrm{LdS}}\left(T/T_{\mathrm{c}},\delta \right)$ variation with the reduced temperature for $T$ close to $T_{\mathrm{c}}$. (b) Logarithmic plot of inverse phase scattering time $\left({\tau }_{\varphi }^{-1}\right)$ with reduced temperature for all the measured TiN samples. The red dashed curves correspond to ${\tau }_{\varphi }^{-1}\propto \left(T-T_{\mathrm{c}}\right)$, the black solid curve represents the Ginzburg-Landau relaxation rate ${\tau }_{\mathrm{GL}}^{-1}=\left(8k_B/\pi \hslash \left)\right(T-T_{\mathrm{c}}\right)$ with $T_{\mathrm{c}}=2.43$ K, and the solid red curves are the fits using Eq. (15) for inverse electron-electron scattering time. The open symbols represent the inverse characteristic time obtained from the characteristic field $B_{\mathrm{SF}}$.