Temperature dependence of resistivity and Hall coefficient in strongly disordered NbN thin films

We report the temperature dependence of resistivity $\left(\rho \right)$ and Hall coefficient $\left(R_H\right)$ in the normal state of homogeneously disordered epitaxial NbN thin films with $k_{F}l\sim 1.68–10.12$. The superconducting transition temperature $\left(T_c\right)$ of these films varies from 2.7 to 16.8 K. While our least disordered film displays usual metallic behavior, for all the films with $k_{F}l\le 8.13$, both $\frac{d\rho }{dT}$ and $\frac{d{R}_H}{dT}$ are negative up to 285 K. We observe that$R_H\left(T\right)$ varies linearly with $\rho \left(T\right)$ for all the films and $\left[\frac{R_H\left(T\right)-R_H\left(285\text{K}\right)}{R_H\left(285\text{K}\right)}\right]=\gamma \left[\frac{\rho \left(T\right)-\rho \left(285\text{K}\right)}{\rho \left(285\text{K}\right)}\right]$, where $\gamma =0.68\pm 0.11$. Measurements performed on a 2-nm-thick Be film show similar behavior with $\gamma =0.69$. This behavior is inconsistent with existing theories of localization and $e\text{−}e$ interactions in a disordered metal.

Figure 1

(a) Transmission electron micrograph of disordered NbN film with a $k_{F}l\sim 3.3$ and (b) transmission electron micrograph of an NbN film with $k_{F}l\sim 9$. The direction of lattice planes in the film and the substrate is shown by arrows.

Figure 2

(Color online) (a) $\rho$ vs $T$ for the films of different $k_{F}l$ and (inset) an expanded view of the $\rho \left(T\right)$ for the most ordered sample with $k_{F}l\sim 10.12$; (b) $R_H$ vs $T$ for all the films and (inset) $R_H$ at 18 K as a function of $k_{F}l$; the black slanting arrows are in the direction of increasing $k_{F}l$ values: 1.68, 3.27, 3.65, 4.98, 5.50, 8.01, 8.13, 8.82, and 10.12; in both (a) and (b) the scale for the film with $k_{F}l$ 1.68 is on the right-hand axis; [(c) and (d)]: Hall resistivity vs magnetic field at different temperatures for the samples with $k_{F}l\sim 3.27$ and 8.82, respectively.

Figure 3

(Color online) $R_H\left(T\right)$ vs $\rho \left(T\right)$ for the samples with (a) $k_{F}l\sim 1.68$, (b) $k_{F}l\sim 3.27$ and 3.65, (c) $k_{F}l\sim 4.98$ and 5.50, and (d) $k_{F}l\sim 8.01$ and 8.13. The error bars indicate the errors in the absolute values of $\rho$ and $R_H$.

Figure 4

(Color online) (a) $\sigma$ vs $T$ for the disordered films showing the linear behavior at lower temperatures. (b) $\sigma$ vs $T^{1/2}$ along with linear fits. (c) An expanded $\rho \left(T\right)$ for the film with $k_{F}l\sim 8.82$. The upturn in resistivity above 240 K shows the relative magnitude of the contribution from electron-phonon scattering. (d) MR for the film with $k_{F}l\sim 4.98$ at different temperatures.

Figure 5

(Color online) [(a) and (b)] $\frac{\Delta R_H}{R_H}\left[=\frac{R_H\left(T\right)-R_H\left(285\text{K}\right)}{R_H\left(285\text{K}\right)}\right]$ vs $\frac{\Delta \rho }{\rho }\left[=\frac{\rho \left(T\right)-\rho \left(285\text{K}\right)}{\rho \left(285\text{K}\right)}\right]$ (shown as points) for the different NbN samples. The linear fits (shown as lines) have an average value of $0.68\pm 0.11$; the inset in (a) shows $\gamma \left[=d\left(\frac{\Delta R_H}{R_H}\right)/d\left(\frac{\Delta \rho }{\rho }\right)\right]$ as a function of $k_{F}l$ with the average value shown as a dashed line. (c) $\frac{\Delta R_H}{R_H}\left[=\frac{R_H\left(T\right)-R_H\left(290\text{K}\right)}{R_H\left(290\text{K}\right)}\right]$ vs $\frac{\Delta R_{sq}}{R_{sq}}\left[=\frac{R_{sq}\left(T\right)-R_{sq}\left(290\text{K}\right)}{R_{sq}\left(290\text{K}\right)}\right]$ for the ultrathin Be film. The $\gamma$ value is 0.69. The inset shows the temperature variation $R_H\left(T\right)$ and $R_{sq}\left(T\right)$ for the ultrathin Be film.