Enhancement of penetration field in vortex matter in mesoscopic superconductors due to Andreev bound states

We study the field for the penetration of a first vortex, $H_{\mathrm{P}}$, for vortex matter nucleated in micron-sized samples with edges aligned along the nodal and antinodal directions of the $d$-wave superconducting order parameter of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-\delta }$. Here we present evidence that the $H_{\mathrm{P}}$ for vortex matter nucleated in mesoscopic samples with edges parallel to the nodal direction is larger than for the antinodal case, $\sim 72$% at low temperatures. This finding supports the theoretical proposal that surface Andreev bound states appearing in a sample with edges parallel to the nodal direction would produce an anomalous Meissner current that increases the Bean-Livingston barrier for vortex penetration.

Figure 1

Some of the nodal (N) and antinodal (AN) ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ square thin plates studied located on top of $16\times 16\mu {\mathrm{m}}^2$ Hall sensors. (a) N square plates with sides $L=20$, 30, and $50\mu \mathrm{m}$; (b) AN square plates with $L\sim 50\mu \mathrm{m}$. Vortex structure in mesoscopic samples imaged by magnetic decoration in (c) N (8.5 Gauss) and (d) AN (5 Gauss) square plates (see the orientation of the nodes of the $d$-wave superconducting parameter).

Figure 2

Detection of the penetration field $H_{\mathrm{p}}$ in nodal (N) and antinodal (AN) square plates (with $L=50\mu \mathrm{m})$ from (a) dc and (b) ac magnetic hysteresis loops. The $H_{\mathrm{p}}$ determined from the deviation of the linear Meissner response coincides with the field at which transmittivity becomes non-negligible (see vertical dashed lines). Measurements performed at 54 K; ac measurements with $h_{\mathrm{ac}}$ of 1 Gauss and 7.1 Hz.

Figure 3

(a) Temperature dependence of the penetration field $H_{\mathrm{p}}$ for nodal (N) and antinodal (AN) square plates from dc and ac (open and full symbols) measurements. (b) $H_{\mathrm{p}}^{\mathrm{N}}/H_{\mathrm{p}}^{\mathrm{AN}}$ data (symbols) and temperature dependence predicted theoretically in Ref. [24] (black dotted line) for a superconductor with $\kappa =200$. This line is just the digitalization of the data shown in Fig. 3 of Ref. [24]. Inset: Data of (a) compared with the theoretical expectation for geometrical (orange) and BL (pink) barriers for ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ disks with radius $L$.

Figure 4

Temperature dependence of the penetration field for nodal square plates with different square sides of $L=20$, 30, and $50\mu \mathrm{m}$. Data from ac magnetization $(h_{\mathrm{ac}}$ of 1 Gauss rms and 7.1 Hz) in agreement with dc data within the error.