Band Filling and Interband Scattering Effects in ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$: Carbon versus Aluminum Doping

We argue, based on band structure calculations and the Eliashberg theory, that the observed decrease of $T_c$ of Al and C doped ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$ samples can be understood mainly in terms of a band filling effect due to the electron doping by Al and C. A simple scaling of the electron-phonon coupling constant $\lambda$ by the variation of the density of states as a function of electron doping is sufficient to capture the experimentally observed behavior. Further, we also explain the long standing open question of the experimental observation of a nearly constant $\pi$ gap as a function of doping by a compensation of the effect of band filling and interband scattering. Both effects together generate a nearly constant $\pi$ gap and shift the merging point of both gaps to higher doping concentrations, resolving the discrepancy between experiment and theoretical predictions based on interband scattering only.

Figure 1

Critical temperature $T_c$ as a function of Al (solid symbols) and C (open symbols) versus doping concentration defined as ${\mathrm{M}\mathrm{g}}_{1-x}{\mathrm{A}\mathrm{l}}_x({\mathrm{B}}_{1-y}{\mathrm{C}}_y{)}_2$. Al doping: (▪) [34], (▲) [28], (●) [26], (◆) [44], (▶) [45]; C doping: (☆) [46], (⊳) [34], (◇) [30], (□) [47], (○) [48], (, ⊠) [29, 49], (△) [50], () [31], (⊲) [51]. The lines present estimates based on Eliashberg theory with different levels of approximation: dotted line with DOS from rigid band model, dashed line with DOS from VCA, and solid line with DOS and phonon renormalization from VCA. The horizontal dotted line is the lower limit for interband scattering only.

Figure 2

(Upper panel) Al doping: Superconducting $\sigma$ gap (upper curve and solid symbols) and $\pi$ gap (lower curve and open symbols) as a function of critical temperature $T_c$ obtained from the solution of the Eliashberg equations with scaled ${\alpha }_{ij}^{2}F$ without interband scattering (solid lines) compared to experimental results (□) [34] and (◇) [28]. (Lower panel) C doping: The solid lines show the solution of the Eliashberg equations with interband scattering rate $2000y{\mathrm{c}\mathrm{m}}^{-1}$ compared to the experimental results (□) [34, 35]. The limiting cases of interband scattering only (dotted lines) or scaling of ${\alpha }_{ij}^{2}F$ (dash-dotted lines) are also shown. The dashed straight line indicates the BCS gap relation for $\Delta /T_{\mathrm{c}}$.