Introduction
The study of the influence of disorder on electron transport in superconducting films is important for understanding the physical mechanisms that determine the superconducting and electronic properties of ultrathin films. Despite the fundamental importance, these studies also have practical applications in many modern microelectronic devices. For example, it is known that increasing disorder has a positive effect on the quantum efficiency of SSPD [1]. But the degree of disorder affects the electron transport parameters, which also matter for SSPD. For example, the scattering rate contributes to the detector dead time, thus largely determining the SSPD performance.
Thin films of niobium nitride (NbN) has been extensively used for production of modern electronic devices such as SNSPDs[2], HEB (hot electron bolometer) mixers [3], microwave nanoinductors [4], etc. It is a typical material, in which disorder can be tuned from moderate to strong limit [5].
Measurement setup
To study transport properties of NbN, we patterned films into 500-µm-wide and 1000-µm-long Hall bars. The level of disorder was varied by different heating of the film substrate during its deposition or by changing the partial pressure of nitrogen. The thickness of all films was 2.5 nm. The film square resistance $R_S$ in the normal state was measured in a four-probe configuration. The measurements were carried out in a homemade cryogenic insert immersed in a Dewar with liquid helium 4He and performed in a wide temperature range (from 300 K to 1.7 K). At low temperatures, we measured the magnetoresistance $R_S(B)$, the temperature dependences of the second critical magnetic field $B_{c2}(T)$ (to determine the diffusion coefficient), and the Hall constant (to determine the concentration of charge carriers) by applying a perpendicular magnetic field B up to 4 T. Here the critical temperature $T_c$ is determined by us as the temperature at $R_S = R_{max}/2$ (Table 1).
Тable 1. Parameters of NbN films.
| s1 | s2 | s3 | s4 | s5 | s6 | |
| $R_S^{300}, Ohm/sq$ | 437 | 509 | 815 | 912 | 1025 | 1574 |
| $k_Fl$ | 6.3 | 5.5 | 3.5 | 3.2 | 2.8 | 2.1 |
| $T_c, К$ | 11.54 | 10.76 | 8.43 | 7.02 | 6.03 | 3.40 |
| $D, cm^2/s$ | 0.59 | 0.57 | 0.36 | 0.35 | 0.34 | 0.27 |
| $n, x10^{29} m^{-3}$ | 1.9 | 1.8 | 1.7 | 1.6 | 1.6 | 1 |
| $tau_{e-ph}$ | 8.3 | 7.0 | 20.5 | 21.0 | 20.0 | 30.0 |
Results and discussion
The dimensionless change in magnetoconductance at the fixed T was determined from the measured $R_S(B, T)$ using the following expression:
$$ delta G(B, T) =frac{2pi^2hbar}{e^2} [R_s(B,T)^{-1}-R_s(0,T)^{-1}] (Eq. 1)$$
Figure 1 shows typical experimental dependences $delta G(B, T)$ for NbN samples obtained in the experiment. The change in resistance occurs due to the influence of the magnetic field on superconducting fluctuations [6] and weak localization.
The data in Figure 1 was compared with
$$delta G(B, T) = G^{SC}(B, T) – G^{SC}(0, T) (Eq. 2) $$
where $G^{SC}(B, T)$ and $G^{SC}(0, T)$ are a sum of four terms of quantum corrections to conductivity at finite and zero magnetic fields: the weak localization, the Aslamazov-Larkin term, the density of states contribution term, and Maki-Thomson term[7]. The final expression for $delta G(B, T)$ contains only one unknown parameter – the phase breaking time of the electron wave function $tau_phi$ – present in the Maki-Thompson correction and weak localization. We obtain this parameter as a result of data processing.
Figure 1. Experimental dependencies of the normalized magnetoconductance $delta G(B, T)$ for a representative sample (s5). Different colors of the curves correspond to different operating temperatures marked on the $R_s(T)$ curve in the inset. The dashed black curves represent fits by Eq. 2.
The dependence $tau_phi(Т)$ extracted from the magnetoresistance measurements is displayed in fig. 2 (a).
The exact expression for $tau_phi^{-1}$ is represented by sum of scattering mechanisms due to superconducting fluctuations $tau_{SC}^{-1}$, the electron-electron (e-e) scattering rate $tau_{e-e}^{-1}$, the spin-flip scattering rate $tau_s^{-1}$, and the electron-phonon (e-ph) scattering rate $tau_{e-ph}^{-1}$ as follows [8]:
$$ tau_phi^{-1}= tau_{SC}i^{-1}+tau_{e-e}^{-1}+2tau_s^{-1}+tau_{e-ph}^{-1} (Eq. 3)$$
It was found that a passivating Si layer over NbN films prevents the appearance of magnetic moments on the surface of the film. Calculating $tau_{SC}i^{-1}$ and $tau_{e-e}^{-1}$ and using the experimental values of $tau_phi(Т)$, we obtained the dependence of e-ph scattering time on temperature (Fig. 2(b)).
We find that the inelastic scattering rates of electrons and their temperature dependencies are close for NbN films of different microscopic quality and with different levels of disorder. The observed results are not described by existing models of e-ph scattering in disordered metals. The values of $tau_{e-ph}^{-1}$ obtained above 10 K are proportional to $T^3$ expected for electron scattering by three-dimensional acoustic phonons in the pure case. At lower temperatures, changes to $T^2$ (not shown here), which is probably due to a decrease in the dimensionality of the phonons involved in e-ph scattering. Our results call for further theoretical andexperimental studies of the e-ph scattering in thin disordered films. The study is described in more detail in [9].
Figure 2. Temperature dependencies of (a) the electron dephasing rate $tau_phi^{-1}$ extracted from magnetoconductance measurements and (b) the e-ph scattering time $tau_{e-ph}$ extracted from $tau_phi^{-1}$. The data are plotted in symbols on a log-log scale. In (a) the solid curves show the best fits of $tau_phi^{-1}$ by Eq. (3).
Acknowledgments
Authors wishing to acknowledge the RSF project 23-72-00014 for the transport measurements.
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