Thermal conductivity of Al-doped $\mathrm{Mg}{\mathrm{B}}_2$: Impurity scattering and the validity of the Wiedemann-Franz law

We report data on the thermal conductivity $\kappa (T,H)$ along the basal plane of the hexagonal crystal structure of superconducting ${\mathrm{Mg}}_{1-y}{\mathrm{Al}}_y{\mathrm{B}}_2$ with $y=0.02$ and 0.07 at temperatures between 0.5 and $50\mathrm{K}$ and in external magnetic fields between 0 and $70\mathrm{kOe}$. The substitution of Al for Mg leads to a substantial reduction of the heat transport via electronic quasiparticles. The analysis of the $\kappa (T,H)$ data implies that the Al impurities provoke an enhancement of the intraband scattering rate, almost equal in magnitude for both the $\sigma$ and the $\pi$ bands of electronic excitations. This is in contrast with conclusions drawn from analogous data sets for material in which carbon replaces boron and where mainly the intraband scattering rate of the $\sigma$ band is enhanced. Our complete data set, including additional results of measurements of the low-temperature thermal conductivity of pure $\mathrm{Mg}{\mathrm{B}}_2$, confirms the validity of the Wiedemann-Franz law for both pure and doped $\mathrm{Mg}{\mathrm{B}}_2$.

Figure 1

Thermal conductivity vs temperature in the $ab$ plane of single-crystalline ${\mathrm{Mg}}_{1-y}{\mathrm{Al}}_y{\mathrm{B}}_2$ ($y=0.02$ and 0.07) in zero magnetic field (open symbols) and $H\Vert c=50\mathrm{kOe}$ (solid symbols). The arrows indicate the critical temperatures in zero magnetic field.

Figure 2

Thermal conductivity in the basal plane of ${\mathrm{Mg}}_{1-y}{\mathrm{Al}}_y{\mathrm{B}}_2$ $(y=0.02,0.07)$ vs $H$ at several fixed temperatures. The closed and open symbols correspond to the field directions perpendicular and parallel to the $c$ axis, respectively. The arrow in the upper left panel demonstrates how $H_{c2}$ for a given field orientation and at a given temperature was established.

Figure 3

The upper critical fields $H_{c2}(T∕T_c)$. The closed and open symbols correspond to the field directions perpendicular and parallel to the $c$ axis, respectively. For comparison, our earlier results for pure $\mathrm{Mg}{\mathrm{B}}_2$ (Ref. 15) are shown by the dashed lines. The inset shows the anisotropies of the upper critical fields $\gamma =H_{c2}^a∕H_{c2}^c$.

Figure 4

Reversible thermal conductivity $\kappa$ normalized to its normal-state value ${\kappa }^n$ vs $H$ at $1.0\mathrm{K}$ for pure and Al-doped $\mathrm{Mg}{\mathrm{B}}_2$. The closed and open symbols correspond to the field directions perpendicular and parallel to the $c$ axis, respectively. The inset shows doping dependence of ${\kappa }^n$ at $1.0\mathrm{K}$.

Figure 5

Normalized Lorenz number $L\left(T\right)∕L_0$ for three samples of pure $\mathrm{Mg}{\mathrm{B}}_2$ and for C- and Al-doped $\mathrm{Mg}{\mathrm{B}}_2$. C6 denotes $\mathrm{Mg}{\left({\mathrm{B}}_{0.94}{\mathrm{C}}_{0.06}\right)}_2$ (data from Ref. 6).