Investigation of specific heat in ultrathin two-dimensional superconducting Pb

Superconductivity in two dimensions is nontrivial. One way to achieve global superconductivity is via the Berezinskii-Kosterlitz-Thouless (BKT) transition due to proliferation of vortex-antivortex pairs. This transition is expected to have a clear signature on the specific heat. The singularity at the transition temperature $T_{\mathrm{BKT}}$ is predicted to be immeasurable, and a broad nonuniversal peak is expected at $T>T_{\mathrm{BKT}}$. Up to date, this has not been observed in two-dimensional superconductors; this work is then dedicated to investigate $c_p$ signatures in the limit of ultrathin 2d superconductors. We use a unique highly sensitive technique to measure the specific heat of quench condensed ultrathin Pb films. We find that thick films exhibit a specific heat jump at $T_C$ that is consistent with BCS theory. As the film thickness is reduced below the superconducting coherence length and the systems enter the 2D limit, the specific heat reveals BKT-like behavior in what can appear as to be a continuous BCS-BKT crossover as a function of film thickness. However, a number of problems arise with this interpretation. We discuss the experimental results and the possible significance of various scenarios involving BKT physics.

Figure 1

(a) The quench condensation setup consists of evaporation baskets used for growing sequential continuous Pb layers, the substrate being held at cryogenic temperatures and in UHV conditions. [(b) and (c)] The suspended membrane acting as the thermal cell contains a copper meander, used as a heater, and a niobium nitride strip, used as a thermometer. These are lithographically fabricated close to the two edges of the active thermal sensor. The quench-condensed films are evaporated through a shadow mask which, together with the measurement leads, defines its geometry.

Figure 2

(a) Resistance per square $R_{\mathrm{sq}}$ vs temperature for the 22 sequential quench condensed lead films. Purple is for thin films and red-brown for thick films. This color code is maintained throughout the paper. (b) Heat capacity of the films (same color code) in form of $C_p/T$ vs $T^2$ highlighting the cubic behavior above 7.2 K.

Figure 3

(a) Superconducting electronic heat capacity $C_{\mathrm{es}}$ of the films normalized to temperature as extracted from the data presented in Fig. 2 along with an identical color code. The squares mark the $T_{C_{\mathrm{res}}}$ of each layer extracted from the RT curves of Fig. 2. The curves for the 1.2-nm and 55-nm-thick films are shown in (b) and (c), respectively. In (c), the arrow represents the heat capacity jump normalized to temperature.

Figure 4

(a) Specific heat of electrons in the superconducting state $c_{\mathrm{es}}$ normalized to temperature vs $T$. The squares mark the $T_{C_{\mathrm{res}}}$ of each layer extracted from the RT curves of Fig. 2. The yellow dashed line is a fit to BCS expectation. (b) The amplitude of the specific heat jump at $T_C$, $\mathrm{\Delta }{c}_p$, as a function of thickness. (c) The maximum value of the specific heat, normalized to the specific heat of layer 22, as a function of thickness.

Figure 5

Fitting the electronic specific heat obtained from the last stage with $\alpha$ model.