Isotope effect on the ${\text{E}}_{2g}$ phonon and mesoscopic phase separation near the electronic topological transition in ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$

We report boron isotope effect on the ${\text{E}}_{2g}$ phonon mode by micro-Raman spectroscopy on the ternary ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ system, synthesized with pure isotopes ${}^{10}\text{B}$ and ${}^{11}\text{B}$. The isotope coefficient on the ${\text{E}}_{2g}$ mode frequency is nearly 0.5 in the wide range of Al, with a tendency to decrease at ${\text{MgB}}_2$ $\left(x=0\right)$. The intraband electron-phonon (e-ph) coupling relative to the sigma band has been extracted from the ${\text{E}}_{2g}$ line-shape parameters. By tuning the Fermi energy near the electronic topological transition (ETT), where the sigma Fermi surface changes from two-dimensional to three-dimensional topology (in range $0<x<0.28$), the ${\text{E}}_{2g}$ mode shows the Kohn anomaly accompanied with a splitting into a hard and a soft component. The results suggest that the intraband hardly plays any role to control the high $T_c$ of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ The common physical features of diborides with the multigap FeAs-based superconductors and cuprates are discussed.

Figure 1

Comparison between Raman spectra on ${\text{Mg}}_{1-x}{\text{Al}}_x{{}^{11}\text{B}}_2$ (filled dots) and ${\text{Mg}}_{1-x}{\text{Al}}_x{{}^{10}\text{B}}_2$ (open symbols) samples for $0<x<0.57$ (panel a). The energy scale of the Raman spectra of ${\text{Mg}}_{1-x}{\text{Al}}_x{{}^{10}\text{B}}_2$ (filled dots) is multiplied by the factor ${\left(10/11\right)}^{0.3}$ (panel b) and by the factor ${\left(10/11\right)}^{0.5}$ (panel c).

Figure 2

Typical fit with three components of the Raman data of ${\text{Mg}}_{1-x}{\text{Al}}_x{{}^{11}\text{B}}_2$ (filled dots) and ${\text{Mg}}_{1-x}{\text{Al}}_x{{}^{10}\text{B}}_2$ (open symbols) samples (the Raman shift of ${\text{Mg}}_{1-x}{\text{Al}}_x{{}^{10}\text{B}}_2$ is multiplied by the factor $\sqrt{10/11}$). The dashed lines represent the ${\text{B}}_{1g}$ contribution, the solid line the soft ${\text{E}}_{2g}$ contribution and the dotted line the hard ${\text{E}}_{2g}$ contribution for aluminum concentration 20%, 10%, and 6% showing the increasing intensity of the hard ${\text{E}}_{2g}$ component with increasing Al concentration.

Figure 3

(Panel a) The frequency of the hard and soft components of the ${\text{E}}_{2g}$ mode for ${}^{10}\text{B}$ (open circles) and for ${}^{11}\text{B}$ (filled dots) is reported. The energy hardening of the ${\text{E}}_{2g}$-mode with increasing Al substitution is shown. The splitting of the hard and soft ${\text{E}}_{2g}$ mode (phase separation) occurs between 0% and 25% of Al content. The weighted average of the ${\text{E}}_{2g}$ frequency weighted by relative intensity of the “hard” and “soft” components are plotted for the ${}^{11}\text{B}$ systems (solid line) and ${}^{10}\text{B}$ systems (dashed line). (Panel b) The isotope coefficient of the phonon mode as a function of $x$. The open symbols correspond to the low energy (soft) mode, while the filled dots to the high energy (hard) mode.

Figure 4

(a) Energy of the ${\text{E}}_{2g}$ mode as a function of the $a$ axis going from the ${\text{AlB}}_2$ to ${\text{MgB}}_2$ samples. The dashed line shows the expected behavior due to lattice expansion for a metallic covalent material. The open circles represent the weighted average energy of the ${\text{E}}_{2g}$ mode for the Al doped system. (b) The linewidth of the ${\text{E}}_{2g}$ phonon mode as a function of the $a$-axis. The open circles represent the weighted average of the ${\text{E}}_{2g}$ linewidth for the Al doped system.

Figure 5

(a) Ratio between the expected frequency due to the variation of the lattice compression and the measured ${\text{E}}_{2g}$ phonon frequency as a function of the $a$ axis in the ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ system. (b) Ratio between the linewidth and the energy of the ${\text{E}}_{2g}$ phonon mode as a function of $a$ axis. The black squares correspond to the soft ${\text{E}}_{2g}$ contribution, while the black disks to the hard ${\text{E}}_{2g}$ contribution. The calculated mean values are reported in open circles.

Figure 6

(a) The measured ${\text{E}}_{2g}$ phonon frequency ${\omega }_{{\text{E}}_{2g}}$ normalized for the frequency shift due to lattice $a$-axis compression ${\Omega }_{{\text{E}}_{2g}}$ in the ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ system (filled dots). The experimental softening of the ${\text{E}}_{2g}$ mode frequency as a function of Al substitution is compared with calculations of Profeta et al. (Ref. 52) (open squares), Zhang et al. (Ref. 53) (open circles), and De la Peña-Seaman et al. (Ref. 59) (filled triangles) (b) The electron-phonon coupling ${\lambda }^{\ast }=\lambda /\left(1+\lambda \right)$ calculated from the softening (Ref. 15) (open circles) and widening (Ref. 11) (open squares) of the ${\text{E}}_{2g}$ mode. The experimental effective coupling $-1/\text{ln}\left(K_B{T}_c/\hslash {\omega }_{{\text{E}}_{2g}}\right)$ (filled circles) obtained from the ratio between the superconducting critical temperature $T_c$ measured by susceptibility measurements (Ref. 55) and the average ${\text{E}}_{2g}$ phonon frequency.