Enhancement of the critical temperature in iron pnictide superconductors by finite-size effects

Recent experiments have shown that in agreement with previous theoretical predictions, superconductivity in nanostructures can be enhanced with respect to the bulk ($L\to \infty$) limit. Motivated by these results, we study finite size effects (FSEs) in iron pnictide superconductors. We employ a five-band mean-field approach that reproduces quantitatively the band structure of these materials around the Fermi energy. For realistic values of the bulk critical temperature $T_c^{\mathrm{bulk}}\sim 20-50$ K, we find that $T_c\left(L\right)$ has a complicated oscillating pattern as a function of the system size $L$. For a simplified two-band model we show analytically that these oscillations are caused by fluctuations of the spectral density around the Fermi energy. We identify a scale $L\sim 10$ nm for which deviations from mean fields, not included in our model, are small but still $T_c\left(L\right)$ is higher than $T_c^{\mathrm{bulk}}$. Similar results are obtained for different boundary conditions and geometries. Finally we show that the differential conductance, an experimental observable, is also very sensitive to FSE.

Figure 1

$\delta T_c\left(L\right)/T_c^{\mathrm{bulk}}$ typical deviation of $T_c\left(L\right)$ from the bulk limit $T_c^{\mathrm{bulk}}=T_c(L\to \infty )$ in a square film of side $L$ for different bulk $T_c$ and PBC. Upper: two-band model [Eq. (1)]. Lower: five-band model.[20] Results of both models are qualitatively similar. Typical enhancement (suppression) of $T_c$ of more than $50\%$ is observed for $L\sim 10$ nm. Inset: Critical temperature $T_c\left(L\right)$ as a function of $L$ for $T_c^{\mathrm{bulk}}\approx 30$ K.

Figure 2

Upper: $\delta T_c\left(L\right)/T_c^{\mathrm{bulk}}$ from Eq. (1) for a square of side $L$ and two rectangular films ($L_x=\alpha L,L_y=L/\alpha$) with $\alpha =1.618034$ (rectangle 1) and $\alpha =1.2334$ (rectangle 2), respectively, and $T_c^{\mathrm{bulk}}\approx 32$ K. The intricate band structure makes the dependence on the shape weaker than in single-band superconductors with a parabolic spectrum. Inset: $T_c\left(L\right)$ for the square and rectangle. Lower: $\delta T_c\left(L\right)/T_c^{\mathrm{bulk}}$ in a square film, $T_c^{\mathrm{bulk}}\approx 32$ K, for Dirichlet [including the effect of the matrix elements $I(k,k^{\prime })$] and PBC. In the latter, relative deviations are stronger due to the additional level degeneracy, but absolute deviations for small sizes can still be stronger for Dirichlet due to the effect of $I(k,k^{\prime })$ (see text). Inset: Comparison between $T_c\left(L\right)$ for Dirichlet and PBC. The solid lines stand for the local $T_c\left(L\right)$ average in the same intervals ($\sim$3 nm) in which $\delta T_c\left(L\right)/T_c^{\mathrm{bulk}}$ is evaluated.

Figure 3

Differential conductance $G\left(V\right)$ [Eq. (3)] as a function of the bias voltage $V$ for $T\approx T_c/3$, $T_c\approx 31$ K, and different sizes. $G\left(V\right)$ is rescaled by its value, $G_0\left(V\right)$, at $T\approx 2T_c$. Left: Two-band model [Eq. (1)]. The peak in $G\left(V\right)$ coincides with $\min \left(\right|{\Delta }_e|,|{\Delta }_h\left|\right)$ in Eq. (1). Right: Five-band model.[20] The peak closer to $V=0$ also corresponds with the value of the energy gap. The additional structure is a consequence of the combined effect of the band structure and level degeneracy. The sensitivity to small changes in $L$ suggests that it is feasible to measure FSE by STM.