Signatures of Dynes superconductivity in the THz response of ALD-grown NbN thin films

Abstract

The frequency-dependent complex optical conductivity reflects key properties of superconductors, such as the energy gap in the density of states (DOS) and the superfluid density. For disordered superconductors, the optical conductivity often can be described within Bardeen-Cooper-Schrieffer (BCS) theory, while in corresponding tunneling experiments, deviations in the observed DOS typically require modelling by the phenomenological Dynes formula. The implications of such Dynes DOS for optics were rarely discussed so far. Here we probe the terahertz conductivity of superconducting NbN thin films with thicknesses ranging from 4.5 to 20nm, which were grown by atomic layer deposition (ALD). Our frequency range from 0.3 to 2.1 THz covers energies below and above the spectral gap. For 20nm thick NbN, we find in the optical conductivity distinct deviations from the BCS model, including a step-like characteristic in the absorption at half the zero-temperature spectral gap. These observations can be fully captured by Dynes electrodynamics with a small and temperature-independent pair-breaking rate. For the other films, we also observe signs of Dynes electrodynamics, and we discuss the evolution of the energy gap, the superfluid density, and the pair-breaking rate as function of film thickness.

Figures (6)

• Figure 1: The main panel (a) shows the temperature-dependent resistivity $\rho(T)$ for all studied films, while the inset (b) displays a magnified view of the behavior near the superconducting transition temperature $T_\mathrm{c}$, where $T_c$ is defined via the 50% criterion with respect to the maximum of the normal-state resistivity, as indicated by dashed lines.
• Figure 2: Complex optical conductivity $\hat{\sigma} = \sigma_1 + i \sigma_2$ of a $20nm$ NbN thin film with $T_c \approx 13.3K$, obtained with THz time-domain spectroscopy (TDS), where panel (a) shows the real part $\sigma_1$ on linear scales and (b) the imaginary part $\sigma_2$ on logarithmic scales. Solid lines represent fits according to the Dynes model Herman2017 with a finite and constant pair-breaking rate of $\Gamma = 0.036 \, \Delta_0$, while dashed lines are calculated according to the Mattis-Bardeen model Zimmermann1991, corresponding to $\Gamma = 0$. The strongest deviations between the Dynes and Mattis-Bardeen models occur near the temperature-dependent minimum of $\sigma_1(f)$. The normal-state conductivity at 15.4K is sketched as a dashed, horizontal line in (a) and corresponds to the value $\sigma_\textrm{dc} = 4460\ \Omega^{-1}\mathrm{cm}^{-1}$ obtained from transport measurements. The inset in (b) shows the superconducting density of states near the Fermi energy in the limit $T=0$. In the BCS case ($\Gamma =0$), only excitations with $hf > 2 \Delta$ are allowed (dark blue arrow). In the Dynes case (here: $\Gamma = 0.1 \Delta$), also excitations at lower energies are possible, and the transitions indicated by light blue arrows, with $hf = \Delta$, lead to the step near $\Delta$ in $\sigma_1(f)$.
• Figure 3: Fabry-Pérot (FP) resonances, observed for numerous temperatures in THz frequency-domain spectroscopy (FDS) on 20nm NbN for two modes, with FP frequencies $f_{0,n}=\SIlist{0.4;0.49}{\tera\hertz}$ in the metallic state at $T = 15K$. Arrows indicate the positions of the temperature-dependent resonance frequency and its corresponding transmission value. Additionally, dashed arrows are guides to the eye to indicate the shift of the modes as the temperature is lowered.
• Figure 4: Frequency-domain spectroscopy (FDS) studies of the temperature-dependent transmission $\text{tr}(f_0(T))$ at temperature-dependent resonance frequency $f_0(T)$ and fractional resonance frequency shift $\frac{f_0(T) - f_0(0)}{f_0(0)}$ displayed in panel (a) and (b) respectively. Solid lines represent fits according to the Dynes model Herman2017 with a finite pair-breaking rate of $\Gamma = 0.036 \, \Delta_0$, while dashed lines are modeled according to the Mattis-Bardeen model Zimmermann1991, corresponding to $\Gamma = 0$. The vertical line at $T = 13.6K$ indicates $T_c$ obtained from the THz measurements. The inset (c) shows the real part of the optical conductivity $\sigma_1$, obtained from TDS, at $3.1K$ and in its normal state at $15.4K$, with dashed vertical lines indicating the frequency positions of the Fabry-Pérot modes studied by FDS.
• Figure 5: Pair-breaking rates $\Gamma$ for five NbN samples normalized to their zero-temperature energy gap $\Delta_0$, from both TDS (discrete temperature points) and FDS (dashed lines) studies. For comparison, pair-breaking rates obtained by Chockalingam et al. Chockalingam2009 from tunneling spectroscopy measurements on NbN are shown as open diamonds with a dashed line as a guide to the eye. The inset displays $\Gamma/\Delta_0$, as determined from FDS, for the different film thicknesses.