Quasiparticle response of superconducting aluminum to electromagnetic radiation

The response of superconducting aluminum to electromagnetic radiation is investigated in broad frequency (45 MHz–40 GHz) and temperature ranges $\left(T>T_c/2\right)$ by measuring the complex conductivity. While the imaginary part probes the superfluid density (Cooper pairs), the real part monitors the opening of the superconducting energy gap and—most important here—the zero-frequency quasiparticle response. Here we observe the full temperature and frequency dependences of the coherence peak. Varying the mean-free path gives insight into the dynamics, scattering, and coherence effects of the quasiparticles in the superconducting state.

Figure 1

(Color online) Frequency and temperature dependence of the real part of the conductivity ${\sigma }_1/{\sigma }_n$, calculated according to the BCS theory[4, 6] with the ratio of the coherence length to the mean-free path $\pi \xi \left(0\right)/2\ell =10$. The pronounced maximum for low frequencies at a temperature slightly below $T_c$ is the coherence peak.

Figure 2

(Color online) Frequency dependence of the complex conductivity of the aluminum film $C$ $\left(T_c=1.9\text{K}\right)$, normalized to the metallic conductivity, ${\sigma }_n={\sigma }_1\left(T=2.1\text{K}\right)$. (a) Real part of the conductivity; note that ${\sigma }_1$ does not vary monotonously with temperature but goes through a maximum around $T\approx 1.69\text{K}$. (b) The imaginary part is plotted on a double-logarithmic scale in order to demonstrate the ${\sigma }_2\propto 1/f$ behavior (green dots). The inset is a sketch of the Corbino arrangement: $a_1=0.8\text{mm}$ and $a_2=1.75\text{mm}$.

Figure 3

(Color online) Real and imaginary parts of the frequency-dependent impedance $Z$. The dashed lines indicate the uncorrected impedance of a superconducting aluminum film. The solid lines show the impedance after the correction for the oxide layer between aluminum film and gold contacts. This correction is performed, assuming the lumped circuit shown in the inset. $Z_{\text{al}}$ is the impedance of the aluminum film under study, whereas $R_{\text{oxide}}$ and $C_{\text{oxide}}$ represent the resistive and capacitive contributions of the oxide layer.

Figure 4

(Color online) Temperature dependence of the [(a) and (c)] real and [(b) and (d)] imaginary parts of the conductivity of aluminum normalized to the normal-state conductivity ${\sigma }_n$. The frames (a) and (b) show the data measured on sample $C$ at various frequencies as indicated. The solid lines are calculated by the BCS theory using $\ell /\left(\pi \xi \right)=0.03$. The frequencies were chosen in a way to demonstrate the effects most clearly. The results of sample $A$ are displayed in panels (c) and (d). The respective calculations are performed with $\ell /\left(\pi \xi \right)=0.1$.

Figure 5

(Color online) Real part of the conductivity of aluminum film $C$ $\left(T_c=1.9\text{K}\right)$ as a function of frequency and temperature. The coherence peak decreases in height as the frequency increases. For comparison with Fig. 1, it should be noted that the superconducting energy gap $2\Delta \left(0\right)\approx 140\text{GHz}$, which implies that we focus on the very low-frequency part $\left[\hslash \omega /2\Delta \left(0\right)<0.2\right]$.

Figure 6

(Color online) Density of electronic states around the Fermi energy $E_F$. In the normal state $\left(T>T_c\right)$, the density of states is basically constant; the states are occupied according to the Fermi distribution (dark filled area). For $T<T_c$ the gap opens and the density of states exhibits a singularity. At finite temperatures, there are still states occupied above $E_F$ and states that are empty below; these are the ones that contribute to the quasiparticle conductivity. For $T=0$ the gap is complete and no quasiparticle excitations are possible up to $\hslash \omega =2\Delta \left(0\right)$. The gray-shaded area depicts the range that corresponds to the finite energy $\hslash \omega$ of our microwave radiation. It can be considered as a smearing of the border between occupied and empty states similar to the thermal broadening.

Figure 7

(Color online) Temperature dependence of the spectral weight $I^s\left(T\right)$ for sample $C$ (squares), evaluated according to Eq. (2). The dashed red line represents the spectral weight obtained by integrating the conductivity calculated by the BCS model (cf. Fig. 8) up to 40 GHz. If we scale the theoretical curve in temperature and spectral weight to the maximum of the data (shown by the solid blue line), the description of the experimental values is excellent. The increase below $T_c$ is due to the enhanced density of states as the superconducting gap opens. After a maximum is reached around 1.6 K, the spectral weight collapses because fewer quasiparticles can be excited across the gap when $T\to 0$. The low-temperature behavior can be approximated by an exponential law (dotted green line).

Figure 8

(Color online) Frequency-dependent conductivity of aluminum at different temperatures calculated according to the BCS theory,[6] and formulated by Mattis and Bardeen.[4] For the critical temperature, we assumed $T_c=1.9\text{K}$ and the superconducting energy gap $2\Delta \left(0\right)/h=140\text{GHz}$. The inset, with a clear correspondence to our experimental data in Fig. 2a, illustrates the rapid growth of the zero-frequency quasiparticle peak as the temperature drops below $T_c$. If $T<1.6\text{K}$, it becomes narrower and eventually vanishes as $T\to 0$. The frequency range covered by our experiments is indicated by the gray-shaded area.

Figure 9

(Color online) The temperature-dependent scattering rate for the aluminum film $C$. The solid line corresponds to a fit $1/\tau =A+B\cdot T^{\alpha }$ with $A=0$, $B=\left(17.6\pm 3.3\right)\cdot {10}^{10}$, and $\alpha =3.5\pm 0.3$. The inset shows an example of a conductivity spectrum $\left(T=1.116\text{K}\right)$ fitted by the Drude model [Eq. (4)].

Figure 10

(Color online) (a) Width $\left(T_c-T_n\right)/T_c$ and (b) height ${\left({\sigma }_1\right)}_{\text{max}}/{\sigma }_n$ of the conductivity coherence peak for aluminum films with different mean-free path $\ell$, normalized to the effective coherence length $\xi \left(0\right)$. The data are taken at various frequencies from 5 to 25 GHz; the lines correspond to a linear interpolation. The multiple data points indicate independent measurements of different samples.