Experimental Evidence for Anisotropic Double-Gap Behavior in ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$

The behavior of a type II superconductor in the presence of a magnetic field is governed by two characteristic length scales, the London penetration depth and the coherence length. We present magnetization measurements on ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$ powder showing an anisotropy in the upper critical field and hence the coherence length of 6. Using the technique of small angle neutron scattering we show that this anisotropy is not mirrored in the London penetration depth, which is almost isotropic. This result can be explained by the superconductivity residing in two distinct electronic bands of the material, only one of which is highly anisotropic.

Figure 1

The ring of scattering on the multidetector with the background subtracted. The contours are linear. The sample was field cooled in 0.5 T down to 2.0 K. The applied field and sample have been rotated $0.5°$ about a vertical axis, thus satisfying the Bragg condition on the left side of the ring. The data in Fig. 3 have been extracted from this picture by integrating the intensity as a function of deflected angle $2\theta$ and normalizing by the area of each successive integration ring.

Figure 2

The left shows the scattering on the detector expected from a single grain when ${\gamma }_{\lambda }>1$ and $\psi >0$ where $\psi$ is the angle between the $c$ axis of the crystal and the applied field normal to the page. The right shows a crystal at an arbitrary orientation showing the vortices of supercurrent around each flux line. The angle $\beta$ is the projected angle of the $c$ axis (and major axis of the ellipse) to the horizontal. The model involves summing over the scattering patterns for $\psi =0$ to $\pi /2$, weighting by $\sin \psi$, summing over all $\alpha$ from 0 to $\pi /3$ (or setting $\alpha =0$) and then integrating the result over all azimuthal angles $\beta$. With the eccentricity of each ellipse equal to $1/({cos}^2\psi +{sin}^2\psi /{\gamma }_{\lambda }^2{)}^{1/2}$, the result is an increasing broadening of the intensity as a function of $2\theta$ with increasing ${\gamma }_{\lambda }$. The gray lines show the major and minor axes of the ellipse.

Figure 3

The variation of scattering as a function of deflected angle of the beam integrated $360°$ about the undeflected beam axis. The dashed line and the solid squares are the models with $\alpha =0°–60°$ for ${\gamma }_{\lambda }=6.70$ and 1.32, respectively. The dot-dashed line and the solid circles are the models with $\alpha =0°$ for ${\gamma }_{\lambda }=6.70$ and 1.28, respectively. The solid line is the model for ${\gamma }_{\lambda }=1$ (the width is the instrument resolution) and the open circles are the data. Clearly the value ${\gamma }_{\xi }=6.7$, taken from magnetization measurements does not correspond to the data with either of the possible orientations.

Figure 4

The open squares show ${\gamma }_{\xi }$ from Ref. 14 measured with a sample from the same source. The solid squares are ${\gamma }_{\xi }$ measured on this sample using the same magnetization method. The dotted and solid lines show the theoretical predictions for ${\gamma }_{\xi }$ and ${\gamma }_{\lambda }$ in clean ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$ with a gap ratio of 4 [5, 6]. The solid diamond is the upper limit for ${\gamma }_{\lambda }$ derived from the neutron measurements described here.