Direct observation of the superconducting gap in a thin film of titanium nitride using terahertz spectroscopy

We report on the charge carrier dynamics of superconducting titanium nitride (TiN) in the frequency range 90–510 GHz (3–17 cm${}^{-1}$). The experiments were performed on an 18-nm thick TiN film with a critical temperature of $T_c=3.4$ K. Measurements were carried out from room temperature down to 2 K, and in magnetic fields up to $B=7$ T. We extract the real and imaginary parts of the complex conductivity $\widehat{\sigma }$ as a function of frequency and temperature, directly providing the superconducting energy gap $2\Delta$. Further analysis yields the superconducting London penetration depth ${\lambda }_L$. The findings as well as the normal-state properties strongly suggest conventional BCS superconductivity, underlined by the ratio $2\Delta \left(0\right)/k_B{T}_c=3.44$. Detailed analysis of the charge carrier dynamics of the silicon substrate is also discussed.

Figure 1

(a) Real part of the substrate conductivity versus temperature for different frequencies from 4–23 cm${}^{-1}$. Below 50 K, the conductivity drops to zero and the substrate remains transparent. (b) Sheet resistance of the TiN film versus temperature. We emphasize the good agreement between dc and THz approach (solid line and open circles respectively). The inset shows a closeup of the sheet resistance. The small peak is a signature of quantum contributions.

Figure 2

(a) Transmission $T_f$ and (b) phase shift over frequency ${\phi }_t/\omega$ versus frequency at 2 K and 10 K below and above $T_c=3.4$ K. The solid line is a calculation according to BCS theory. The change in both properties indicate a strong frequency-dependent complex conductivity. The inset shows the temperature evolution of the reduced energy gap together with the BCS prediction leading to $2\Delta \left(0\right)=8.14$ cm${}^{-1}$.

Figure 3

(a) Real and (b) imaginary parts of the complex conductivity versus frequency for four different temperatures below $T_c=3.4$ K and one well above. The solid lines are fits according to BCS theory applied to both ${\sigma }_1$ and ${\sigma }_2$ simultaneously. The kink in ${\sigma }_1$ indicates the energy gap, which shifts towards lower frequencies upon increasing temperature. The insets show the London penetration depth versus frequency and temperature, respectively. The solid line in the upper inset is the two-fluid model prediction yielding ${\lambda }_L\left(0\right)=730$ nm.

Figure 4

Energy gap versus magnetic field (open circles) together with the prediction based on the phenomenological two-fluid model. The fit leads to $B_c=5$ T.

Figure 5

$T_s$ of silicon substrate versus temperature and frequency. The frequency dependence shows the well-developed Fabry-Perot oscillations, which extend over the entire temperature and spectral range except for the low-$T_s$ region at low frequencies around 100 K.

Figure 6

Charge carrier dynamics extracted from $T_s$. (a) Charge carrier concentration (solid line: Fermi-Dirac distribution). (b) Mobility and scattering rate are determined by fitting ${\sigma }_1$ within the Drude model. (c) Real part of conductivity ${\sigma }_1$ versus frequency and temperature. The data is extracted from $T_s$ by single-peak fitting.

Figure 7

Real part ${\sigma }_1$ and imaginary part ${\sigma }_2$ (inset) of the complex conductivity versus frequency for three temperatures. Solid lines are fits according to Eqs. (A1) and (A2).